Stability and Control Processes: Proceedings of the 4th International Conference Dedicated to the Memory of Professor Vladimir Zubov (Lecture Notes in Control and Information Sciences - Proceedings) 3030879658, 9783030879655

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Lecture Notes in Control and Information Sciences Proceedings

Nikolay Smirnov Anna Golovkina   Editors

Stability and Control Processes Proceedings of the 4th International Conference Dedicated to the Memory of Professor Vladimir Zubov

Lecture Notes in Control and Information Sciences - Proceedings Series Editors Frank Allgöwer, Universität Stuttgart, Institute for Systems Theory and Automatic Control, Stuttgart, Germany Manfred Morari, University of Pennsylvania, Department of Electrical and Systems Engineering, Philadelphia, USA

This distinguished conference proceedings series publishes the latest research developments in all areas of control and information sciences – quickly, informally and at a high level. Typically based on material presented at conferences, workshops and similar scientific meetings, volumes published in this series will constitute comprehensive state-of-the-art references on control and information science and technology studies. Please send your proposals to Oliver Jackson, Editor, Springer. e-mail: [email protected]

More information about this series at https://link.springer.com/bookseries/15828

Nikolay Smirnov · Anna Golovkina Editors

Stability and Control Processes Proceedings of the 4th International Conference Dedicated to the Memory of Professor Vladimir Zubov

Editors Nikolay Smirnov Department of Mathematical Modelling of Economic Systems Saint Petersburg State University Saint Petersburg, Russia

Anna Golovkina Department of Control Systems Theory for Electrophysical Facilities Saint Petersburg State University Saint Petersburg, Russia

ISSN 2522-5383 ISSN 2522-5391 (electronic) Lecture Notes in Control and Information Sciences - Proceedings ISBN 978-3-030-87965-5 ISBN 978-3-030-87966-2 (eBook) https://doi.org/10.1007/978-3-030-87966-2 MATLAB is a registered trademark of The MathWorks, Inc. See https://www.mathworks.com/trademarks for a list of additional trademarks. Mathematics Subject Classification: 49-XX, 91Bxx, 91Cxx, 93-XX, 68Txx, 34Dxx © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to the memory of 90th anniversary of Vladimir Zubov (1930–2000), professor, corresponding member of RAS.

Preface

This volume contains proceedings of the IV International Conference Stability and Control Processes (SCP 2020, October 5–10, 2020) dedicated to the 90th anniversary of Vladimir Zubov (1930–2000), professor and corresponding member of RAS. Vladimir Zubov is a famous Russian scientist who made a great contribution to the development of the qualitative theory of differential equations, the theory of rigid body motion, the optimal control theory, and the theory of electromagnetic fields. He led the development of a wide range of issues related to the theory of control systems, such as questions of motion stability, nonlinear oscillations in control systems, navigation and reliability of control devices, vibration theory, and quantization of orbits. His in-depth studies on the theory of stability of motion, the theory of automatic control, and the theory of optimal processes make it possible to solve important applied problems, in particular, in the field of constructing control automata and stabilizing programmed motions. The methods of Vladimir Zubov are also effective in application to control problems arising in industry, mathematical economics, biology and medicine, and navigation. Vladimir Zubov participated in the solution of the following topical problems: • gyroscopic systems axes deviation in the dependence on nutational vibrations and the kinetic moment of inertia of the gyroscope rotors; • design, creation, and operation of homing systems for cruise missiles; • creation of precision control systems for the position of the Proton system spacecraft; • creation of control systems for the spacecraft rotational motion; • control charged particles beams to transport them in a given physical channel, focusing and acceleration; • information transfer by using the apparatus of recurrent functions, which does not allow for a limited time the influence of counteracting interference on the control process (in engineering practice, the terminology “random number sensors” and control using these sensors are adopted). He has published more than 170 scientific works, including 20 monographs and study guides, three of which have been reprinted abroad in English and French. The works of Vladimir Zubov were highly appreciated by prominent Russian and foreign vii

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Preface

scientists, and his works related to the development of A. M. Lyapunov’s methods for the stability of movement were awarded the University Prize in 1962. The Conference Stability and Control Processes is unique in the sense that it takes place at Saint Petersburg State University every 5 years to honor the memory of this great person and scientist. The venue is chosen at the first faculty of applied mathematics and control processes in the country which had a serious impact on the development of applied mathematics and computer science in the former Soviet Union and nowadays in the Russian Federation. Vladimir Zubov initiated this global transformation becoming the first dean of the faculty. The conference targets a variety of topics where he left the scientific mark and covers almost all the fundamental areas of modern applied mathematics and control processes. At the same time, the scope continues to expand in line with the development of science resulting in new sessions on “hot” topics. The SCP traditionally brings specialists in stability, optimal control, differential games, optimization together with practitioners applying the theory to different problems in the industry, robotics, mechatronics, power and energy systems, economics, biology, and socio-science. This year the global pandemic of coronavirus infection made its corrections with respect to the original plan and forced the organizers to turn the conference to the virtual stage. Nevertheless, we kept the traditions and provoked fruitful discussions around new arising problems to outline important applications even without face-to-face interactions between participants. The volume includes several topical parts corresponding to the conference sessions: • • • • • • • • • • • • •

Stability and Robust Control Dynamic Systems Theory Mechanical Systems Control Control and Optimization of Electro-physical Processes Game Theory and Conflict Systems Control Methods for Analysis and Design of Systems with Time-Delay Optimization Methods Non-linear Mechanics and Solid-State Physics Socio-economic Systems Control Medical and Biological Systems Control Informatics and Control Processes Mathematical Modelling and Image Processing Methods Artificial Intelligence

The corresponding chapters contain fundamental and practical results in control and stability theory, differential games, optimization as well as new and interesting approaches in solving important applied problems using the aforementioned theoretical tools. To address the present-day challenges, new sections dedicated to the intellectual and learning system were introduced. All collected works were presented and discussed during the conference.

Preface

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SCP 2020 featured studies of more than 300 authors from 10 countries (Armenia, Russia, China, Romania, France, Tunisia, Japan, Nigeria, Mexico, and Germany). The next international conference SCP will be organized in 2025. Saint Petersburg, Russia October 2020

Anna Golovkina Nikolay Smirnov

Contents

Part I 1

2

3

Stability and Robust Control

A Dynamical System View on Nonlinear Optimal Control Analysis and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noboru Sakamoto

3

From Control Invariant Sets to an Inverse Optimality Perspective on the Constrained Control Design . . . . . . . . . . . . . . . . . Sorin Olaru

11

On the Stability of Nonlinear Mechanical Systems with Time-Varying Discontinuous Coefficients . . . . . . . . . . . . . . . . . . Alexey Platonov

21

4

Normalizing Random Vector Anisotropy Magnitude . . . . . . . . . . . . . Kirill Chernyshov

5

Minimax Approach in a Multiple Criteria Stabilization of Singularly Perturbed Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stanislav Myshkov, Vladimir Karelin, and Lyudmila Polyakova

39

An Estimation Extension of Domain of Attraction for Second-Order Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Kirill V. Postnov

49

Stability of Weak Solutions of Parabolic Systems with Distributed Parameters in a Network-Like Domain . . . . . . . . . Vyacheslav V. Provotorov and Aleksei P. Zhabko

59

An Algorithm for Solving Local Boundary Value Problems with Perturbations and Delayed Control . . . . . . . . . . . . . . . . . . . . . . . Alexander N. Kvitko, Alexey S. Eremin, and Oksana S. Firyulina

65

On the Stability of Linear Time-Delay Systems with Arbitrary Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irina Alexandrova and Sabine Mondié

73

6

7

8

9

29

xi

xii

10

Contents

Study of the Stability Features of Solutions of Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gennady Ivanov, Gennady Alferov, Vladimir Korolev, and Dzmitry Shymanchuk

11

Economic Evolution with Structural Variations . . . . . . . . . . . . . . . . . Alexander N. Kirillov and Alexander M. Sazonov

12

Mixed Feedback–Feedforward Frequency Control in Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oleg O. Khamisov

13

81

89

97

Diagonal Riccati Stability of a Class of Complex Systems and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Alexander Aleksandrov and Nadezhda Kovaleva

Part II

Dynamic Systems Theory

14

Synchronization in Feedback Cyclic Structures of Oscillators with Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Alexander M. Kamachkin, Dmitriy K. Potapov, and Victoria V. Yevstafyeva

15

Optimal Damping Problem for Diffusion-Wave Equation . . . . . . . . 127 Sergey Postnov

16

Identification of Integral Models of Nonlinear Multi-input Dynamic Systems Using the Product Integration Method . . . . . . . . 137 Svetlana Solodusha

17

Algorithm for Constructing a Cognitive Aggregate-Stream Model of the Automatic Spacecraft Flight Control Process . . . . . . . 149 Vladimir S. Kovtun, Boris V. Sokolov, and Valerii V. Zakharov

18

A Method of Determining of Switching Instants for Discrete-Time Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Sergey Khryashchev

19

Algorithmization of Receiving Orbits of Weierstrass and Orbits of Tangences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Maria A. Shagai, Mikhail D. Iofe, and Alexander V. Flegontov

Part III Methods for Analysis and Design of Systems with Time-Delay 20

Systems with Propagation: A Bunch of Models and a Research Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Vladimir R˘asvan

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21

Fourth-Order Method for Differential Equations with Discrete and Distributed Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Alexey S. Eremin and Aleksandr A. Lobaskin

22

The Prediction Scheme to the Linear Systems with Linearly Increasing and Constant Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Alexey Zhabko, Oleg Tikhomirov, and Olga Chizhova

23

On Asymptotic Quiescent Position in Time-Delay Systems . . . . . . . 215 Svetlana Kuptsova, Sergey Kuptsov, and Uliana Zaranik

24

The Generalized Myshkis Problem for a Linear Time-Delay System with Time-Varying Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Alexey Egorov

Part IV Control and Optimization of Electro-physical Processes 25

Field Emitters Periodic System on Substrate with Dielectric Layer Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Ekaterina M. Vinogradova and Grigoriy G. Doronin

26

An Optimization Approach for Minimization of Charged Particles Orbit Deviation in Synchrotron and Transport Channels Systems Caused by Magnetic Field Tolerances . . . . . . . . . 243 Vladislav Altsybeyev and Vladimir Kozynchenko

27

Factor Values Measurement and Heteroscedasticity by the Example of FEE Signal Identification . . . . . . . . . . . . . . . . . . . . 249 Andrey Yu. Antonov, Nikolay V. Egorov, and Marina I. Varayun’

28

Influence of Dyes on the Electro-optical Properties of Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Tatiana Andreeva and Marina Bedrina

29

Genetic Stochastic Algorithm Application in Beam Dynamics Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Liudmila Vladimirova, Anastasiia Zhdanova, Irina Rubtsova, and Nikolai Edamenko

30

On a New Approach to RFQ Channel Optimization . . . . . . . . . . . . . 273 Oleg I. Drivotin

31

Optimization of a Real-Time Stabilization System for the MIMO Nonlinear MagLev Platform . . . . . . . . . . . . . . . . . . . . . 281 Sergey Zavadskiy, Mikhail Verkhoturov, Anna Golovkina, Dmitri Ovsyannikov, Vladimir Kukhtin, Nicolai Shatil, and Andrei Belov

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Contents

Part V

Informatics and Control Processes

32

Application of Quasidifferential Calculus to Solve Optimal Control Problems with a Nonsmooth Functional . . . . . . . . . . . . . . . . 293 Alexander Fominyh

33

On the State Estimation of Non-linear Discrete-Time Models: Application to Unmanned Aerial Vehicles . . . . . . . . . . . . . . 303 Mohamed Boutayeb and Abir Bouaouda

34

Marine Vehicles’ Automatic Control Based on Optimal Damping Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Evgeny I. Veremey and Margarita V. Sotnikova

35

Multiplication Algorithm for Multivariate Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Levon Babadzanjanz, Irina Pototskaya, Yulia Pupysheva, and Irina Alesova

36

Multipurpose Visual Positioning of the Underactuated Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Ruslan Sevostyanov

37

Adaptive Method for an Actuarial Optimal Control Problem with Dynamic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Alina V. Boiko

38

Spectral Design of H2 Optimal Fault Detection Observer Based on Modal Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Yaroslav V. Knyazkin

39

Kolmogorov Complexity-Based Similarity Measures to Website Classification Problems: Leveraging Normalized Compression Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Andrey A. Pechnikov and Anthony M. Nwohiri

Part VI

Game Theory and Conflict Systems Control

40

DEA Modeling with Cluster Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Vladimir Bure, Elena Parilina, and Kseniya Staroverova

41

Cooperation in Vehicle Routing Game on a Megapolis Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Alexander V. Mugayskikh

42

Minimal Current Payments Algorithm for Sustainable Cooperation in Multicriteria Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Denis Kuzyutin, Yaroslavna Pankratova, and Roman Svetlov

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43

Pursuit of Rigidly Coordinated Evaders in a Linear Problem with Fractional Derivatives, a Simple Matrix, and Phase Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Alena I. Machtakova and Nikolai N. Petrov

44

A Pollution Control Problem for the Aluminum Production in Eastern Siberia: Differential Game Approach . . . . . . . . . . . . . . . . 399 Ekaterina Gromova, Anna Tur, and Polina Barsuk

45

Dynamic Programming Equations for the Game-Theoretical Problem with Random Initial Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Anastasiya Malakhova and Ekaterina Gromova

46

Two Echelon Supply Chain: Market Search Behavior and Dependent Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Suriya Kumacheva and Victor Zakharov

47

Differential and Algebraical Relations in Singular Sets Construction for a One Class of Time-Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Pavel D. Lebedev and Alexander A. Uspenskii

48

Subgame Perfect Pareto Equilibria for Multicriteria Game with Chance Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 Denis Kuzyutin, Nadezhda Smirnova, and Igor Tantlevskij

49

On Nash Equilibrium in Repeated Hierarchical Games . . . . . . . . . . 447 Yaroslavna Pankratova and Leon Petrosyan

50

Dynamic Shapley Value for Two-Stage Cost Sharing Game . . . . . . . 457 Li Yin

Part VII

Mechanical Systems Control

51

Boundary Control of String Vibrations with Given Values of the Deflection Function at Intermediate Moments of Time . . . . . 467 Vanya Barseghyan

52

Resonant Oscillations of a Controlled Reversible Mechanical System in the Vicinity of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Valentin N. Tkhai and Ivan N. Barabanov

53

Representation Forms of the Angular Velocity Vector for an Orthonormal Basis of a Moving Frame . . . . . . . . . . . . . . . . . . . 483 Vladislav S. Ermolin and Tatyana V. Vlasova

54

Attitude Controlled Motion in a Neighborhood of the Collinear Libration Point L 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Alexander Shmyrov, Vasily Shmyrov, and Dzmitry Shymanchuk

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Contents

55

Construction of Connecting Trajectories in the Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Dzmitry Shymanchuk, Alexander Shmyrov, and Vasily Shmyrov

56

An Embedded Explicit Method for Partitioned Systems of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Igor V. Olemskoy and Alexey S. Eremin

57

Constructing a Polynomial Method in the State Space for a Nonlinear Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . 517 Daniel Zlobin and Dzmitry Shymanchuk

Part VIII Medical and Biological Systems Control 58

One-Dimensional Non-Newtonian Models of Arterial Hemodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Gerasim V. Krivovichev

59

Some Problems of Modeling the Human Body Subjected to Vertical Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Vladimir Tregubov and Nadezhda Egorova

60

Detection of the Community-Acquired Pneumonia Factors Leading to Death . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Alexandra A. Arzhanik, Anastasiya B. Goncharova, Daria A. Vinokurova, and Evgeny S. Kulikov

61

HIV Incidence in Russia, Ukraine, and Belarus: SIR Epidemic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 Sergei V. Sokolov and Alexandra L. Sokolova

62

Clusterization of White Blood Cells on the Modified UPGMC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 Andrey V. Orekhov, Victor I. Shishkin, and Nikolay S. Lyudkevich

63

Elasticity’s Influence on Biomechanical Model of Corneoscleral Shell Under Vacuum Compression Ring . . . . . . . . 567 Dmitry V. Franus

Part IX Socio-economic Systems Control 64

On Depth of Immersion in Forecasting Task . . . . . . . . . . . . . . . . . . . . 575 Alexander V. Prasolov and Nikita G. Ivanov

65

Supply Chain Model with Random Demand . . . . . . . . . . . . . . . . . . . . 583 Sergei A. Kalin, Elena A. Lezhnina, and Tatyana V. Vlasova

66

Cyber-Physical System Adaptation in One Control Problem for Supply Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Inna Trofimova, Boris Sokolov, and Dmitry Nazarov

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67

Model of Stakeholders of the Socio-cyber-physical System Life Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 Stanislav V. Mikoni

68

National Healthcare System and Economy’s Competitiveness . . . . 607 Anatoliy V. Sigal, Maria A. Bakumenko, and Elena A. Lukyanova

69

Assessment of the Socio-economic Effectiveness of Innovative Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 Elena A. Lezhnina, Yulia E. Balykina, and Alexander V. Konovalov

70

Structural Analysis of Directed Signed Networks . . . . . . . . . . . . . . . . 621 Elizaveta Evmenova and Dmitry Gromov

71

Dynamic Input–Output Models: Analysis of Possibilities and Trends Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 Nikolay V. Smirnov, Viktor P. Peresada, Kirill V. Postnov, Tatiana E. Smirnova, and Yefim V. Zholobov

72

Queueing Systems with Opposite Queues . . . . . . . . . . . . . . . . . . . . . . . 637 Anastasija Glushakova and Alexander Kovshov

73

Choice Modeling in Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Alexandr V. Sachkov

Part X

Optimization Methods

74

A New Characterization of Cone Proper Efficient Points . . . . . . . . . 653 Vladimir D. Noghin

75

On Degree of Pareto Set Reduction Using Information Quanta . . . 659 Oleg Baskov

76

Particular Structures of the Pareto Set and Its Reduction in Bicriteria Discrete Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 Aleksey Zakharov and Yulia Kovalenko

77

On a Support Function on a Convex Cone . . . . . . . . . . . . . . . . . . . . . . 677 Lyudmila Polyakova, Alexander Fominyh, Vladimir Karelin, and Stanislav Myshkov

78

Methodology of Structural–Functional Synthesis for the Small Spacecraft Onboard System Appearance . . . . . . . . . . 687 Alexander N. Pavlov, Valentin. N. Vorotyagin, Dmitry A. Pavlov, and Valerii V. Zakharov

79

Equivalence of Two Optimality Conditions for Polyhedral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Majid Abbasov

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Contents

A Computational Approach to Estimating Activity Coefficients Using Gibbs Energy Minimization . . . . . . . . . . . . . . . . . . 703 Roman Voronov, Anton Shabaev, and Fedor Vasiliev

Part XI

Mathematical Modelling and Image Processing Methods

81

Computation and Analysis of Two-Phase Filtration Using Averaged Models in Oil Formations with Both Vertically Stratified Heterogeneity and Horizontal Zonal Heterogeneity . . . . 715 Dilbar N. Bikmukhametova, Svetlana R. Enikeeva, Kuan M. Tho, and Sergey P. Plokhotnikov

82

Optimization Method of the Velocity Field Determination for Tomographic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Elena Kotina, Pavel Bazhanov, and Dmitri Ovsyannikov

83

Automatic Recognition of Metal Smelting Quality Using Machine Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 Valery Grishkin, Mikhail Shirobokov, and Artemii Grigorev

84

Mathematical Modeling of Cyclic Chemical Compounds . . . . . . . . . 737 Albina Akhmetyanova, Albina Ismagilova, and Fairuza Ziganshina

85

Parametrization of Orthogonal Hexagonally Symmetric Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 Aleksandr Krivoshein

86

Remote Sensing Data Processing for Plant Production Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Vladimir Bure, Olga Mitrofanova, Evgeny Mitrofanov, and Aleksey Petrushin

Part XII

Artificial Intelligence

87

Characteristics of Lexical Spectra of Texts in the Problem of Establishing Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 Nikolai Moskin, Kirill Kulakov, Alexander Rogov, and Roman Abramov

88

Topic Models with Neural Variational Inference for Discussion Analysis in Social Networks . . . . . . . . . . . . . . . . . . . . . 769 Nikita Tarasov, Ivan Blekanov, and Alexey Maksimov

89

On the Possibility of Using Neural Networks for the Specific Problems of Meteorological Forecasting . . . . . . . . . . . . . . . . . . . . . . . . 777 Elena Stankova and Irina Tokareva

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Multilingual Sentiment Analysis for User Discussions on Social Networks: An Approach Based on a Modified SVM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 Mikhail Kukarkin and Ivan Blekanov

91

Data Crawling Approaches for User Discussion Analysis on Web 2.0 Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 Dmitry Nepiyushchikh and Ivan Blekanov

92

Metric for Comparison of Graph-Theoretic Models of the Same Dimension with Ordered Vertices . . . . . . . . . . . . . . . . . . 801 Nikolai Moskin

Part XIII Non-linear Mechanics and Solid-State Physics 93

Interaction of Finite Amplitude Surface Waves in a Basin with a Floating Elastic Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 Anton A. Bukatov

94

Analytical Solutions for Cylindrical Bending of Multilayered Orthotropic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 Alexander V. Matrosov and Dmitry P. Goloskokov

95

A Rectangular Prism Under Own Weight: Comparison of the Method of Initial Functions and the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829 Guryi N. Shirunov, Alexander V. Matrosov, and Denis A. Sarvilin

96

Surface Dislocation Interaction by the Complete Gurtin– Murdoch Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 Mikhail Grekov and Tatiana Sergeeva

97

On Edge Effect for a Finite Doubly Periodic System of Perpendicular Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 Abdulla Abakarov and Yulia Pronina

98

Large Deformations of a Plane with Elliptical Hole for Model of Semi-linear Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 Venyamin Malkov and Yulia Malkova

99

On Minimization of Metal Costs for a Pipeline Exposed to External Corrosion Under Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 863 Marina Elaeva, Yulia Pronina, and Sergey Kabrits

100 Interaction of Misfit Dislocations with Perturbated Surface in Epitaxial Thin Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 Sergey Kostyrko, Mikhail Grekov, and Takayuki Kitamura

xx

Contents

101 Wave Motions of Liquid with Consideration of the Density Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 Sergey Peregudin, Elena Peregudina, and Svetlana Kholodova 102 Elastic–Plastic Bending of Vertical Supports of Drilling Platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 Galina Pavilaynen, Nataliya Naumova, and Denis Ivanov 103 Sommerfeld Effects in Two-Mass Crusher with Three Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 Serge E. Miheev and Petr D. Morozov 104 On the Account of Transverse Young–Laplace Law Under Stability of a Rectangular Nano-Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 907 Anatolii Bochkarev 105 Stress Analysis of a Spherical Pressure Vessel with Multiple Notches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 Olga S. Sedova and Daria D. Okulova

Part I

Stability and Robust Control

Chapter 1

A Dynamical System View on Nonlinear Optimal Control Analysis and Design Noboru Sakamoto

Abstract This note presents a view from the theory of dynamical system, such as invariant manifolds and λ-lemma, for the analysis and the design of optimal control for nonlinear systems. The optimal control problems considered are optimal stabilization and optimal transfer problems. The theory of stable manifold and its iterative computation plays the central role for the optimal stabilization design and the λ-lemma which describes the flows around the invariant manifolds is used for analysis of the optimal transfer problem including turnpike property in the optimal control system.

1.1 Introduction The problem of optimal control is to find control strategy for a given dynamical system so as for the system to behave in an optimal way. The theory of optimal control is an extension of calculus of variations and played significant roles in the development of modern control theory. In spite of its significance, the design (or construction) of optimal control for nonlinear systems is not satisfactorily developed. For nonlinear systems, the framework that is typically used for linear systems may not be appropriate. For instance, although the stabilization of nonlinear system minimizing a cost functional is a fundamental problem and has been investigated for decades [1, 8, 10], very few applications were reported until recently (see, e.g. [2] for a survey on the results up to the year 2000). This note aims to introduce, based on recent works by the author and his collaborators, useful tools from dynamical system theory that can be effective for the analysis and design of optimal control for nonlinear systems. The tools introduced are invariant manifold and related theories (see, e.g. [11, 23]). They are applied to Hamiltonian systems derived from sufficient conditions for the existence of optimal N. Sakamoto (B) Faculty of Science and Engineering, Nanzan University, Yamazato-cho 18, Showa-ku, Nagoya 466-8673, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_1

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N. Sakamoto

control (same systems are obtained using necessary conditions for optimality). This note tries to systematically present a dynamical system theoretic framework for both optimal stabilization and optimal transfer problems. For the former, the framework includes the works in [15, 18, 20, 21] which are based on symplectic geometry as well as dynamical system theory. For the latter, it turns out to be a suitable machinery to understand turnpike phenomenon in optimal control (see, e.g. [3, 25] for the history and definition of turnpike and see [5, 9, 12, 19, 22] for the recent development of the turnpike theory).

1.2 Optimal Control 1.2.1 Problem Formulation Let us consider nonlinear control systems of the form x˙ = f (x) + g(x)u, x(t0 ) = x0 ,

(1.1)

where f Rn → Rn , g : Rn → Rn×m are C 2 maps, x(t) ∈ Rn is the states and u(t) ∈ Rm is control inputs. The optimal control problems we consider in this note is to find a u in an appropriate function space that minimizes the cost functional  J=

T

L(x(t), u(t)) dt,

(1.2)

t0

where T  +∞ is given. Additionally, we assume that f (0) = 0 and L(x, u) = (x  Qx + |u|2 )/2 in (1.2), where Q  0. In this note, we are concerned with the following two specific problems in optimal control theory. A. Optimal stabilization problem: For (1.1)–(1.2) with T = +∞, find a control law u(x) such that for a neighborhood U of the origin the closed-loop system is asymptotically stable and for each x0 ∈ U the control u(t) = u(x(t)) minimizes (1.2) over all control inputs under which the states converge to the origin as t → +∞. B. Optimal transfer problem: For (1.1)–(1.2) with T < +∞ and given x1 ∈ Rn , find a control u that minimizes (1.2) over all control inputs that take x0 at t = t0 to x1 at t = T .

1 A Dynamical System View on Nonlinear Optimal Control Analysis and Design

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1.2.2 The Topics in Optimal Control Theory Discussed in This Note For the optimal control problems defined above, we will discuss their properties and design approaches in more detail, for which the notions of dynamical system theory play significant roles. (i) (Optimal stabilization and stable manifold) The problem of optimal stabilization is one of the most important control problems and it leads to foundations of modern control theory such as LQR theory, Riccati equation, and H ∞ control theory. The key notion here is to replace finding a solution to a HamiltonJacobi–Bellman (HJB) equation for the sufficiency of the optimal control to finding a stable manifold in an associated Hamiltonian system. This notion turns out to be a natural nonlinear extension of the theory of Riccati equation [15]. (ii) (Turnpike phenomenon and λ-lemma) The mathematical economists first observed the turnpike phenomenon in the contexts of optimal growth [4] and later, in control community, it was independently reported as dichotomy or saddle point property [14, 24]. The turnpike theorem says that the optimal control with time-horizon long enough does not depend on the horizon length but depends only on the system and the cost function except at the beginning and the end of the control horizon [3, 25]. The turnpike is a metaphor for long-distance traveling: “if origin and destination are far enough apart, it will always pay to get on the turnpike to cover distance at the best rate of travel ...” [4, Chap. 12]. The turnpike theory is attracting attentions, for instance, in order to simplify the design process of optimal control [5, 22] and optimal shape design [9, 13]. A typical optimal control problem in which the turnpike appears is the optimal transfer problem where T is large. The turnpike is defined as follows. Let u T be the optimal control for (1.1)–(1.2) and x T be corresponding states. The pair (u T , x T ) is said to satisfy the turnpike property if for any ε > 0, there exists an ηε > 0 such that μ({t  0 | |u T (t)| + |x T (t, x0 )| > ε}) < ηε for all T > 0, where ηε depends only on ε, f , g, x0 , and Q. In [12], a sufficient condition for the turnpike property is proposed which requires u T and x T to satisfy   |u T (t)| + |x T (t, x0 )|  M e−αt + e−α(T −t) for t ∈ [t0 , T ] for any T > 0, where M > 0, α > 0 are constants independent of T .

(1.3)

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1.3 Solutions to the Optimal Control Problems 1.3.1 Sufficient Conditions for the Optimal Control Problems It is known that if there is a C 1 function V (x) such that ∂V 1 ∂V ∂V  1  f (x) − g(x)g(x) + x Qx = 0 ∂x 2 ∂x ∂x 2

(1.4)



V (0) = 0, ∂∂Vx (0) = 0 and x˙ = f (x) − g(x)g(x) ∂∂Vx is locally asymptotically stable at x = 0, then, the optimal stabilization is given by u = −g(x)

∂V  . ∂x

Also, if there is a C 1 function V (x, t) such that ∂V ∂V 1 ∂V ∂V  1  + f (x) − g(x)g(x) + x Qx = 0 ∂t ∂x 2 ∂x ∂x 2 and the solution x(t) of x˙ = f (x) − 21 g(x)g(x) ∂∂Vx optimal transfer control is given by u = −g(x)



(1.5)

satisfies x(T ) = x1 , then, the

∂V  . ∂x

Equations (1.4) and (1.5) agree with the condition originally obtained by Krasovskii [8].

1.3.2 Solutions in Terms of Hamiltonian Systems Equations (1.4), (1.5) are called HJB equations and a typical approach for them is to consider a Hamiltonian system x˙i =

∂H , ∂ pi

p˙ i = −

∂H , i = 1, . . . , n, ∂ xi

(1.6)

where H (x, p) = p  f (x) − 21 p  g(x)g(x) p + 21 x  Qx. As we have discussed in the previous subsection, however, HJB equations in our control problems require peculiar side conditions such as asymptotic stability or terminal constraint on the states and the method of characteristics, which is typically applied for partial differential equations of the first order with boundary conditions, is not appropriate. We

1 A Dynamical System View on Nonlinear Optimal Control Analysis and Design

7

need different framework based on dynamical system theory, the detail of which will be introduced in the subsequent Sect. 1.3.3. A. Optimal stabilization problem: Suppose that the linear part of (1.1)–(1.2) is stabilizable and detectable. Then, there exists a stable manifold S for (1.6) around the origin such that it is locally written as S = {(x, p(x)) | x ∈ W }, where W ⊂ Rn is a neighborhood of x = 0 and p(x) is a continuous function defined on W . Moreover,  there exists a C 1 function V (x) on W such that ∂∂Vx = p(x) and it satisfies (1.4) with the additional stability condition. Furthermore, the optimal control is given by u = − 21 g(x) p(x). B. Optimal transfer problem with turnpike property: Suppose that the linear part of (1.1) is controllable and |x0 | and |x1 | are sufficiently small. Then, for sufficiently large T , there exists a solution (x T (t), pT (t)) for (1.6) such that x T (t0 ) = x0 , x T (T ) = x1 and   |x T (t)| + | pT (t)|  M e−αt + e−α(T −t) for t ∈ [t0 , T ], where M, α are positive numbers independent of T . Moreover, for this T , the optimal transfer problem has a solution u T satisfying (1.3). Remark 1.1 (i) The result for the optimal stabilization problem is originally taken from [20]. The solution to the optimal stabilization problem has recently been implemented in quite a few laboratory experiments. For instance, [6] contains the first experimental result of the optimal stabilization for the inverted pendulum swing up control. It also shows that there exist multiple solutions to an HJE creating totally different controlled behaviors which are confirmed in the experiment. The paper also discusses the geometry of the stable manifold yielding the controls using 3D printer. The paper [7] also shows the application result for the Acrobot swing up control. (ii) The result above for the turnpike analysis is taken from [19]. The turnpike analysis recently attracts attentions from the viewpoints of L 1 optimal control (minimum fuel control) and the maximum hands-off control [17].

1.3.3 Dynamical System Theory for Optimal Control We start with a general nonlinear dynamical system of the form z˙ = X (z), where X : R N → R N is a vector field, for which we assume the following. Assumption 1.1 (i) X (0) = 0. (ii) X is locally C 1 class around z = 0.

(1.7)

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(iii) The equilibrium z = 0 is hyperbolic, namely, (∂ X/∂z)(0) ∈ R N ×N has k eigenvalues in the open left-half plane and N − k eigenvalues in the open right-half plane in the complex plane. Under Assumption 1.1, there exist k, (N − k)-dimensional manifolds, called stable and unstable manifolds of (1.7) at the origin, respectively, which are defined by S := {z ∈ R N | φ(t, z) → 0 as t → ∞}, U := {z ∈ R N | φ(t, z) → 0 as t → −∞}, where φ(t, z) is the solution of (1.7) starting from z at t = 0. It should be noted that S, U are invariant under (1.7). A.Design of optimal stabilization control: Suppose that the linear part of (1.1)– (1.2) is stabilizable and detectable. Then, it can be shown that Hamiltonian system (1.6) satisfies Assumption 1.1 with N = 2n, k = n, where n is the dimension of the system (1.1). Let S be the stable manifold around (x, p) = (0, 0). With a linear transformation, (1.6) is written as ξ˙ = Fξ + n s (ξ, η)

(1.8a)



η˙ = −F η + n u (ξ, η),

(1.8b)

where F ∈ Rn×n is a stable matrix and n s , n u are higher order nonlinearities. Let us define the sequence (ξk (t, ξ0 ), ηk (t, ξ0 ) by 

ξk+1 ηk+1



⎛



⎞

t

e n s (ξk (s), ηk (s)) ds ⎟ ⎜ e ξ0 + ⎟  ∞ t0 (t, ξ0 ) = ⎜ ⎠ ⎝ −F  (t−s) − e n u (ξk (s), ηk (s))) ds F(t−s)

Ft

t

with ξ1 (t) = e At ξ0 , η1 (t) = 0. In [18], it is shown that for sufficiently small ξ0 ∈ Rn , (ξk (t, ξ0 ), ηk (t, ξ0 ) converges to the solution on the stable manifold S in (ξ, η)-space. Using this iterative computation, the optimal stabilization feedback control is constructed. See [16] for more detail on how to construct feedback controllers based on this iterative algorithm. B. Design of optimal transfer control and turnpike analysis: For this problem, the λ-lemma or inclination lemma in dynamical system theory (see [11, 23]) plays an essential role in deriving the turnpike inequality. Lemma 1.1 (The λ-lemma) Under Assumption 1.1, let S and U be k, (N − k)dimensional stable and unstable manifolds of (1.7) at 0, respectively. Take an

1 A Dynamical System View on Nonlinear Optimal Control Analysis and Design Fig. 1.1 A scheme of the proof of Theorem 1.1 and (Reproduced from [19])

D

9

z0

y0

S

B(z0, ρ) ϕ(T /2, D) 0 ϕ(−T /2, E)

ζ U

y1 E

B(z1, ρ) z1

(N − k)-dimensional disc B in U , a point z ∈ S, an (N − k)-dimensional disc D transversal to S at z and ε > 0, all arbitrarily. Then, there exists a T > 0 such that if t > T , φ(t, D) contains an (N − k)-dimensional disc D˜ that is ε-close to B with C 1 -topology. Using the λ-lemma, [19] proves the following, which will be used to find a trajectory of a two-point boundary-value problem. Let B(z, r ) denote the N -dimensional open ball at z with radius r . Theorem 1.1 Let X satisfy Assumption 1.1 and take arbitrary points z 0 ∈ S, z 1 ∈ U ¯ E¯ with D¯ being transversal to S at and arbitrary (N − k), k-dimensional discs D, z 0 and E¯ being transversal to U at z 1 . Then, there exists a T0 > 0 such that for every T > T0 there exist ρ > 0, y0 ∈ B(z 0 , ρ) ∩ D¯ and y1 ∈ B(z 1 , ρ) ∩ E¯ such that φ(T, y0 ) = y1 and   |φ(t, y0 )|  M e−αt + e−α(T −t) for t ∈ [0, T ]. Moreover, the radius ρ can be arbitrarily small as T gets large (Fig. 1.1). Theorem 1.1 is applied to (1.6) to show that optimal transfer problem has a solution with turnpike property in the following manner. First, note that Hamiltonian system associated with (1.5) is hyperbolic from the stabilizability and detectability assumptions on the linear part of the system. So, stable and unstable manifold U , S, respectively, exist around the origin. Under certain topological conditions guaranteeing x0 ∈ π1 (S), x1 ∈ π1 (U ), where π1 is the canonical projection (π1 (x, p) = x), one can find a trajectory connecting (x0 , p0 ) near S and (x1 , p1 ) near U for some suitable p0 , p1 ∈ Rn . This solves a two-point boundary-value problem associated with the original control problem. It is seen that the trajectory thus obtained satisfies the turnpike inequality (1.3).

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References 1. Al’brekht, E.G.: On the optimal stabilization of nonlinear systems. J. Appl. Math. Mech. 25(5), 1254–1266 (1961) 2. Beeler, S.C., Tran, H.T., Banks, H.T.: Feedback control methodologies for nonlinear systems. J. Optim. Theory Appl. 107(1), 1–33 (2000) 3. Carlson, D.A., Haurie, A., Leizarowitz, A.: Infinite Horizon Optimal Control, 2nd edn. Springer, Berlin, Heidelberg (1991) 4. Dorfman, R., Samuelson, P.A., Solow, R.M.: Linear Programming and Economic Analysis. McGraw-Hill, New York (1958) 5. Grüne, L.: Economic receding horizon control without terminal constraints. Automatica 49(3), 725–734 (2013) 6. Horibe, T., Sakamoto, N.: Optimal swing up and stabilization control for inverted pendulum via stable manifold method. IEEE Trans. Control Syst. Technol. 26(2), 708–715 (2017) 7. Horibe, T., Sakamoto, N.: Nonlinear optimal control for swing up and stabilization of the Acrobot via stable manifold approach: theory and experiment. IEEE Trans. Control Syst. Technol. 27(6), 2374–2387 (2019) 8. Krasovskii, N.N.: Stabilization problems of control motions. In: Malkin, I.G. Stability Theory of Motion, pp. 475–514 (1966) 9. Lance, G., Trélat, E., Zuazua, E.: Turnpike in optimal shape design. IFAC-PapersOnLine 52(16), 496–501 (2019). https://doi.org/10.1016/j.ifacol.2019.12.010. In: Proceeding of the 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2019 10. Lukes, D.L.: Optimal regulation of nonlinear dynamical systems. SIAM J. Control. Optim. 7(1), 75–100 (1969) 11. Palis, J., Jr., de Melo, W.: Geometric Theory of Dynamical Systems: An Introduction. Springer, New York (1982) 12. Porretta, A., Zuazua, E.: Long time versus steady state optimal control. SIAM J. Control Optim. 51(6) (2013) 13. Porretta, A., Zuazua, E.: Remarks on long time versus steady state optimal control. Springer INdAM Ser. 15, 67–89 (2016) 14. Rockafellar, R.T.: Saddle points of Hamiltonian systems in convex problems of Lagrange. J. Optim. Theory Appl. 12(4), 367–390 (1973) 15. Sakamoto, N.: Analysis of the Hamilton-Jacobi equation in nonlinear control theory by symplectic geometry. SIAM J. Control. Optim. 40(6), 1924–1937 (2002) 16. Sakamoto, N.: Case studies on the application of the stable manifold approach for nonlinear optimal control design. Automatica 49(2), 568–576 (2013) 17. Sakamoto, N., Nagahara, M.: The turnpike property in the maximum hands-off control. In: Proc. of 59th IEEE Conference on Decision and Control, pp. 2350–2355 (2020) 18. Sakamoto, N., van der Schaft, A.J.: Analytical approximation methods for the stabilizing solution of the Hamilton-Jacobi equation. IEEE Trans. Autom. Control 53(10), 2335–2350 (2008) 19. Sakamoto, N., Pighin, D., Zuazua, E.: The turnpike property in nonlinear optimal control — A geometric approach. Automatica 134 (2021). https://doi.org/10.1016/j.automatica.2021. 109939 20. van der Schaft, A.J.: On a state space approach to nonlinear H∞ control. Syst. Control Lett. 16(1), 1–18 (1991) 21. van der Schaft, A.J.: L 2 -Gain and Passivity Techniques in Nonlinear Control, 3rd edn. Springer International Publishing (2017) 22. Trélat, E., Zuazua, E.: The turnpike property in finite-dimensional nonlinear optimal control. J. Differ. Equ. 258(1), 81–114 (2015) 23. Wen, L.: Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity. American Mathematical Society (2016) 24. Wilde, R.R., Kokotovic, P.V.: A dichotomy in linear control theory. IEEE Trans. Autom. Control 17(3), 382–383 (1972) 25. Zaslavski, A.J.: Turnpike Properties in the Calculus of Variations and Optimal Control. Springer (2006)

Chapter 2

From Control Invariant Sets to an Inverse Optimality Perspective on the Constrained Control Design Sorin Olaru

Abstract The paper aims to review some of the early works on constrained control underlining their influence on the recent development of optimization-based control design. We will trace the path (mainly within a linear time-invariant framework) from vertex controllers, to model-based predictive control and ultimately to interpolation-based control. Interestingly, we will point to the relationship between these techniques and the performance indexes in optimal control design. On their turn, the performance indices can be seen as candidate Lyapunov functions but can also be interpreted in a geometrical perspective as convex liftings over the controllable regions of the state space. This last perspective provides a link to the inverse optimality argument for any stabilizing control law. Finally, we will point to the parameter uncertainty in dynamical models as part of such an inverse-optimal control policy and discuss some open problems.

2.1 Introduction V. Zubov by his pioneering work [23] is recognized to be the first to characterize the region of attraction of an asymptotically stable equilibrium (say, x = 0 ∈ Rn ) for a dynamical system described by an ordinary differential equation: x˙ = f (x). The celebrated Zubov’s method completes Lyapunov’s direct method for stability. In a nutshell, it states the sufficient condition for a region D ∈ Rn to be a domain of attraction in terms of the existence of a function V : D → R satisfying the inequality 0 < V (x) < 1, (x = 0), with lim V (x) = 1 as x approaches the frontier of D and for which  (2.1) V˙ = −ϕ(x)(1 − V (x)) 1 +  f (x)2

S. Olaru (B) Laboratory of Signals and Systems, University Paris Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_2

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for a continuous and positive function ϕ(x) for x = 0 with ϕ(0) = 0. This condition lead to constructive approaches [24] based on the resolution of the Cauchy problem for the Zubov equation:  ∇V (x) · f (x) = −ϕ(x)(1 − V (x)) 1 +  f (x)2

(2.2)

and inspired a large number of works in the control literature. More than 60 years of research on the characterization of domains of attractions and their extensions to constrained control design have been nourished by the rich mathematical notions of set invariance [16], set theoretic methods in control [6], reachability analysis [13] or the mathematical theory of viability [2]. In parallel, a similar interest for the characterization of domain of attractions for discrete dynamical systems followed its path. Moreover, given the technological developments, the implementation of control algorithms had to consider sampled measurements and the interplay between the constraints and dynamical evaluations in discrete time. The present paper considers the discrete-time linear dynamics under constraints and revisits some of the historical concepts and insists on the recent developments proving the actual interest in these subjects nowadays. These recent developments are not intending to provide an exhaustive presentation of the research trends on these topics but are mainly concentrating on the approach followed by the author and his collaborators in the last 10 years.

2.2 From Control Invariant Sets to MPC (Model Predictive Control) Constrained control [10] is a broad subject related to various topics that rescind the implementation of feedback control policies. These constraints were ranging from saturation and state/output limitations in the early studies in automatic control, to modern formulation in terms of restricted information available from the systems measurements or handled by the actuation channels (quantization, delay, loss of communication, etc.). The characterization of domains of attractions or alternatively of the maximal controllable regions in the presence of constraints lead to a series of reach developments from both theoretical and constructive point of view. The discrete-time dynamics allow a particularly attractive framework for the effective constructions (often enhanced by additional structural properties for linear time-invariant systems): • Maximal output admissible sets. • The positive invariance and controlled positive invariance. • Vertex control as a first basic attempt to move from the analysis to the synthesis of constrained control systems. • Model Predictive Control. Let us start by recalling the very basic notion of positive invariance.

2 From Control Invariant Sets to an Inverse Optimality Perspective …

13

Definition 2.1 A set X ⊂ Rn is positively invariant with respect to the system xk+1 = f (xk ) if x ∈ X implies f (x) ∈ X. In the continuous-time case, the positive invariance of a set X ⊂ Rn with respect to a dynamical system x˙ = f (x) which admits a unique solution for any initial condition inside the set X is characterized through Nagumo’s Theorem [15]. This states that the necessary conditions for positive invariance are related to the inclusion f (x) ∈ CX(x) , ∀x ∈ X where CX(x) is the tangent cone to X in x in the Bouligand sense [2]. The tangent cone being non-trivial only on boundary of X, in continuoustime, the invariance conditions are essentially related to the properties of the boundary of X. This points to a fundamental difference with respect to the discrete time, where the positive invariance cannot be resumed to the study of the boundaries (there is no mutandis mutatis correspondence of the Nagumo theorem). Definition 2.1 in itself only characterizes a property of a given set without providing clear conditions for their validation other than f (X) ⊆ X. The question of necessary and sufficient conditions for positive invariance received an elegant answer in [4]. Given a map g : Rn → Rm and w ∈ Rm the aforementioned work resumes the invariance of X = {x ∈ Rn |g(x) w} with respect to xk+1 = f (xk ) by the existence of a monotone map h : Rm → Rm such that g ◦ f ≤ h ◦ g and h(w) w.1 In the particular case of polyhedral set and LTI systems, the invariance of P = {Gx ≤ w} with respect to xk+1 = Axk is equivalent to the existence of a non-negative matrix H such that G A = H G and H w ≤ w which allows a straightforward numerical test. Furthermore, the design of linear stabilizing controller for a LTI system all by enhancing the positive invariance of a pre-imposed polyhedral set can be casted in terms of a linear programming problem [5].2 A step forward in the analysis of constrained dynamical systems with respect to the state constraints is the iterative computation of the collection of states which generate trajectories that satisfy the constraints at least for one step: k+1 = {x ∈ k | f (x) ∈ k } = (k ). The initialization for such an iterative construction is generally done with 0 = X where X ⊂ Rn is the predefined set of state constraints. If this set of constraints contains an equilibrium point for xk+1 = f (xk ) and the mapping f (.) has contractive properties then the set mapping k+1 = (k ) converges to a positive invariant set, the proof being a direct application of Brouwer’s fixed point theorem. For the particular case of bounded polyhedral sets P = {Gx ≤ w} containing the origin in their interior and asymptotically stable LTI systems xk+1 = Axk the set ∞ = lim k is finitely generated within the class of polyhedral sets [9]. This k→∞

construction was extensively used as safe set around an equilibrium point and it was denoted in the literature as Maximal Output Admissible Set (MOAS), pointing to the The symbol is used here for the partial order relation. These developments build on the assumption that the equilibrium point is located in the interior or the set P. Only recently, the design of control laws that enforce the positive invariance of such a set with the regulation on the boundaries has been studied thoroughly in [3].

1 2

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fact that the constraints can be seen as admissible conditions on the output trajectories for a dynamical system. As mentioned earlier, in the general nonlinear case, there is no Nagumo theorem for the set invariance with respect to discrete-time dynamics. The construction of MOAS made a link to the boundary condition which can be stated for the case of LTI systems: the convex set S ⊂ Rn is invariant with respect to a LTI system xk+1 = Axk if and only if for all x ∈ ∂S it holds Ax ∈ S. Is worth to be mentioned that whenever the convex set is a bounded polyhedron P ⊂ Rn , the invariance condition can be verified in a finite number of extreme points, i.e. the countable set of vertices, denoted V er t{P}. With these basic notions, one can move forward to the constrained control framework and introduce the controlled set invariance. Definition 2.2 A set X ⊂ Rn is controlled positively invariant with respect to the system xk+1 = f (xk , u k ) if ∀x ∈ X it exists u ∈ U ⊂ Rm such that f (x, u) ∈ X. ¯ u) ¯ with respect to the state X ⊂ Rn and U ⊂ Rm , such Given a feasible pair3 (x, that x¯ = f (x, ¯ u) ¯ the fundamental question is the characterization of the maximal controllable set, i.e. the subset of state constraints X gathering all the initial conditions for which there exist an admissible control signal U generating a feasible trajectory in the state space. Obviously, the stabilizability is a pre-condition and the study of MOAS based on a local linearization around the feasible equilibrium can offer only a first rough approximation of the maximal controllable set. A generic solution is based on the iterative construction of one-step controllable sets which have the advantage of defining a controlled invariant set at each iteration and ultimately providing an approximation of the maximal controllable set around the equilibrium (x, ¯ u) ¯ ∈ I nt{X × U}: k+1 = {x ∈ X|∃u ∈ U such that f (x, u) ∈ k } with an initialization 0 which enjoys the controlled positive invariance properties. In the case of LTI systems xk+1 = Axk + Bu k , the iterative construction [8] leads to poyhedral computations, one of the classical set iteration (under the assumption of non-singularity of A) taking the form4 :   k+1 = A−1 k ⊕ A−1 B(−U) ∩ X, The convergence toward an ε−approximation of the maximal controllable set ∞ has been proven [7, 12]. The construction remains computationally intensive and has been studied in different perspectives in order to decrease the complexity [14, 22]. The approximation of the maximal controllable set ∞ by means of a polyhedral set is an essential element in the constrained control design for LTI systems. Aside 3

The uniqueness of the pair is greatly simplifying the problem, but even this case is important for the characterization of controllable regions around an equilibrium. 4 The Minkowski addition [21] of two sets is denoted by ⊕.

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15

the universal approximation provided by polyhedra with respect to any convex set, they prove to be a versatile tool in the constrained control design by exploiting the following property: Proposition 2.1 A polyhedral set P ⊂ Rn is controlled invariant if and only if for all v ∈ V er t{P} there exist u ∈ U such that Av + Bu ∈ P. It turns out that each polyhedral set can thus be associated to a control action on each of the vertices V er t{P} and further to a control function defined on the boundary V : ∂P → U. The vertex control law strategy is built on the existence of a controlled invariant set  ⊂ X and extends the existing control action on the boundary over the entire feasible set using the following control law: Kv :  → U, Kv (x) = μ (x)V(μ (x)x),   where μ (x) = inf λ∈R>0 : x ∈ λ is the Minkowski function with respect to the set . The closed-loop system: xk+1 = Axk + BKv (x) is stable with a feasible control low and a domain of attraction  ⊂ X.

2.3 From Model-Based Predictive Control to Interpolation Based Control The weakness of the vertex controller resides in the fact that full control authority (with respect to the input constraints) is provided only on the border of the feasible state set  with progressively smaller control action when the state approaches the origin. In the year ’90 the Model-based Predictive Control (MPC) emerged as a control technique which is able to adjust the performances in the presence of constraints on top of the stability guarantees. In a most popular formulation, which deals with a nominal LTI prediction model and polyhedral constraints on the state and inputs, MPC can be stated as a recursive finite-time optimal control policy5 : min

u k ,u k+1 ,...,u k+N −1

N −1 

xk+i 22,Q + u k+i 22,R + xk+N 22,P

i=0

subject to u k+i ∈ U, xk+i ∈ X, ∀i = 0, . . . , i = N − 1 xk+i+1 = Axk+i + Bu k+i , ∀i = 0, . . . , i = N − 1 xk+N ∈  A cost function built on weighted quadratic norm will be used, i.e. for a given vector x ∈ Rn and a weighting matrix Q ∈ Rn×n one denotes x22,Q = x T Qx.

5

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S. Olaru

Fig. 2.1 Comparison of Vertex control and MPC for a double integrator. State space trajectories

Fig. 2.2 Comparison of Vertex control and MPC for a double integrator. States and input time evolutions

with Q, R the weighting matrices for the state and input terms at each prediction step and P = P∞ the solution of the discrete Riccati equation: P∞ = A T P∞ A + Q − A T P∞ B(R + B T P∞ B)−1 B T P∞ A. The result of the optimization depends on the current state vector xk and corresponds to a feedback function, denoted MPC law u M PC (xk ) = u ∗k which is stable if the terminal set  is positive invariant: (A + B (R + B T P∞ B)−1 B T P∞ A) ⊆ .   K∞

While the performance issue received an appropriate answer (see a typical comparison of MPC with respect to Vertex Control performances in Figs. 2.1 and 2.2), the MPC came with a different problem, the one of the online computational complexity related to the resolution at each sampling instant of a finite-time optimal control problem. Interpolating-Based Control (IBC) was developed in the last years [17, 18] to address this issue by bridging the gap in between the attractive structure of vertex controller and the MPC performances. In short, starting from a local control invariant

2 From Control Invariant Sets to an Inverse Optimality Perspective … 4

4

C 3

C

N 3

2

x

x

1

Ω

−1

−1

v

x

0

x −2

x

o 2

0

x2

2

x

N

2

1

* xv

x

* o

x

o

Ω

v

−2

−3 −4 −10

17

−3

−8

−6

−4

−2

0 x

2

4

6

8

−4 −10

10

−8

−6

−4

−2

1

0 x1

2

4

6

8

10

Fig. 2.3 State decomposition as a convex combination of points in  and , respectively. Left before optimization-free choice of c. Right, interpolation based on the optimization - c∗ Fig. 2.4 Comparison of IBC and MPC performances. State space trajectories

4 3 2

MPC method

x2

1 0 −1

Interpolation based control

−2 −3 −4 −10

−8

−6

−4

−2

0

x1

2

4

6

8

10

set  around the equilibrium point and a control invariant set  ⊃  approximating the maximal controllable set, any point x ∈  can be decomposed as x = cxv + (1 − c)xo with xv ∈ , xo ∈  as illustrated in Fig. 2.3. For a given interpolation factor c the IBC feedback function is defined as K I BC :  → U, K I BC (x) = ck V(μ (xv )) + (1 − ck )K (x0 ) with K (x) = K ∞ x associated to the positive invariant set . The IBC is shown to be recursively feasible for any x ∈  and the closed-loop system xk+1 = Axk + BK I BC (xk ) to be asymptotically stable. In order to bring the best performances for the IBC controller, the choice of the interpolation factor becomes the key factor and its selection needs to be done based on the nonlinear optimization: ⎧ ⎨ xv ∈ , xo ∈  c∗ (x) = min c s.t. cxv + (1 − c)xo = x c,xo ,xv ⎩ 0≤c≤1 With this choice, ck∗ is a positive and non-increasing scalar and can be shown to play the role of a Lyapunov function for the closed-loop system. The IBC performances are closed to the MPC as shown in Fig. 2.4. From a control design perspective the interpolation can be seen as a technique for merging Control Lyapunov Function as those discussed in [11] or “uniting” Lyapunov Functions [1].

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Finally, in order to bring the IBC to a low computational fingertip, a change of variables can be employed rv = cxv and ro = (1 − c)xo thus bringing the overall optimization to a low dimensional LP (with n + 1 optimization arguments). ⎧ ⎨ rv ∈ c, ro ∈ (1 − c) c∗ (x) = min c s.t. rv + ro = x c,rv ,ro ⎩ 0≤c≤1

2.4 From Low Complexity Constrained Control to Inverse Optimality The developments related to the performances of the constrained control policies and their online computational cost brought the discussion to a philosophical level. Given a predefined level of performances, associated to a ideal constrained control and a given domain of attraction, is it possible to obtain (and effectively formulate) a tractable convex optimization problem which embeds its intrinsic performance properties? This problem was shown to be related to an inverse-optimal control argument [20] and solved based on a novel concept denoted convex lifting [19]. This solution covers indirectly all the possible realization of MPC policies which are known to lead to Piecewise Affine (PWA) controllers. In mathematical terms, the problem

can be stated as follows: given a polyhedral partition of the polyhedral set  = i∈I N Xi ⊂ Rn and a continuous piecewise affine function f pwa :  → U, find • A convex cost function: J (x, u, z), • A set of convex constraints describing the feasible domain by the pair of matrices Hx , Hu , Hz , K such that f pwa (x) = Pr ojRm arg min J (x, u, z), [u T z ]T s.t. Hu u + Hx x + Hz z ≤ K . The solution starts from the polyhedral partition  = piecewise affine lifting described by a function:

i∈I N

Xi and builds a

z :  → R x → z(x) = AiT x + ai for x ∈ Xi , where Ai ∈ Rn and ai ∈ R. Furthermore, a piecewise affine lifting is called convex piecewise affine lifting if the following conditions hold true:

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19

• z(x) is continuous over X, • for each i ∈ I N , z(x) > A Tj x + a j for all x ∈ Xi \X j and all j = i, j ∈ I N . Once the convex lifting is available, its epigraph can be used as the feasible set for a multiparametric linear program leading to a pre-imposed continuous PWA function defined over a polyhedral partition. Remarkably, this construction leads to at most one supplementary variable in the argument of the optimization and is thus equivalent to the complexity of a IBC policy. Last but not the least, the continuous explicit solution of a generic linear MPC problem with respect to a linear/quadratic cost function is equivalently obtained through a linear MPC problem with a linear or quadratic cost function and the control horizon at most equal to 2 prediction steps: f pwa (x) = Pr ojRm arg s.t.

min J (xk , u k , u k+1 ), T [u kT u k+1 ]T Hu u k + Hx xk + Hz u k+1 ≤ K .

2.5 Conclusion, Further Research, and Open Problems The constrained control nourished a wide research area with interesting problems for a long period. Obviously, the applications are the ones to impose the pace of the developments, but along the line, various mathematical concepts emerged. We reviewed the ones that remain remarkably modern and important after more than 50 years. Other, more recent concepts which have been characterized recently, need to prove their effectiveness and to broaden their scope. In view of the recent notions of inverse-optimal control, three avenues seem to be of particular importance: • given a MPC controller, obtain an equivalent implicit optimal controller of two prediction steps, with the same performance and same structural properties, by avoiding the explicit characterization; • given a PWA controller, find the particular inverse-optimal solution which takes the form of the constraints over the state trajectories using a LTI model-based prediction (at the price of an increase vector of optimization arguments); • move the inverse optimality argument toward a robust constrained control design. In particular for parametric uncertainty, given a predefined feedback controller, find the representative LTI prediction model, which can lead to the adjustment of the robustness level all by preserving the cost and the parameters of the constrained optimal control problem to be solved at each sampling instant (using a receding horizon formulation).

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References 1. Andrieu, V., Prieur, C.: Uniting two control lyapunov functions for affine systems. In: 2008 47th IEEE Conference on Decision and Control, pp. 622–627. IEEE (2008) 2. Aubin, J.P.: Viability Theory. Springer Science & Business Media (2009) 3. Bitsoris, G., Olaru, S., Vassilaki, M.: The linear constrained control problem for discrete-time systems: Regulation on the boundaries. In: International Conference on Difference Equations and Applications, pp. 215–245. Springer (2017) 4. Bitsoris, G., Truffet, L.: Positive invariance, monotonicity and comparison of nonlinear systems. Syst. Control Lett. 60(12), 960–966 (2011) 5. Bitsoris, G., Vassilaki, M.: Constrained regulation of linear systems. Automatica 31(2), 223– 227 (1995) 6. Blanchini, F.: Set invariance in control. Automatica 35(11), 1747–1767 (1999) 7. Cwikel, M., Gutman, P.O.: Convergence of an algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans. Autom. Control 31(5), 457–459 (1986) 8. Dorea, C.E.T., Hennet, J.: (A, B)-invariant polyhedral sets of linear discrete-time systems. J. Optim. Theory Appl. 103(3), 521–542 (1999) 9. Gilbert, E.G., Tan, K.T.: Linear systems with state and control constraints: The theory and application of maximal output admissible sets. IEEE Trans. Autom. Control 36(9), 1008–1020 (1991) 10. Goodwin, G., Seron, M.M., De Doná, J.A.: Constrained Control and Estimation: An Optimisation Approach. Springer Science & Business Media (2006) 11. Grammatico, S., Blanchini, F., Caiti, A.: Control-sharing and merging control lyapunov functions. IEEE Trans. Autom. Control 59(1), 107–119 (2013) 12. Gutman, P.O., Cwikel, M.: An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans. Autom. Control 32(3), 251–254 (1987) 13. Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. In: International Workshop on Hybrid Systems: Computation and Control, pp. 202–214. Springer (2000) 14. Munir, S., Hovd, M., Olaru, S.: Low complexity constrained control using higher degree Lyapunov functions. Automatica 98, 215–222 (2018) 15. Nagumo, M.: Über die lage der integralkurven gewöhnlicher differential gleichungen. Proc. Phys.-Math. Soc. Jpn. 3rd Ser. 24, 551–559 (1942) 16. Nemytskii, V.V., Stepanov, V.V.: Kachestvennaia teoriia Differentsial’nykh uravnenii. Gosteortekhizdat (1947) 17. Nguyen, H.N., Gutman, P.O., Olaru, S., Hovd, M.: Implicit improved vertex control for uncertain, time-varying linear discrete-time systems with state and control constraints. Automatica 49(9), 2754–2759 (2013) 18. Nguyen, H.N., Gutman, P.O., Olaru, S., Hovd, M.: Control with constraints for linear stationary systems: An interpolation approach. Autom. Remote. Control. 75(1), 57–74 (2014) 19. Nguyen, N.A., Gulan, M., Olaru, S., Rodriguez-Ayerbe, P.: Convex lifting: Theory and control applications. IEEE Trans. Autom. Control 63(5), 1243–1258 (2017) 20. Nguyen, N.A., Olaru, S., Rodriguez-Ayerbe, P., Hovd, M., Necoara, I.: Constructive solution of inverse parametric linear/quadratic programming problems. J. Optim. Theory Appl. 172(2), 623–648 (2017) 21. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 151. Cambridge University Press (2014) 22. Scibilia, F., Olaru, S., Hovd, M.: On feasible sets for mpc and their approximations. Automatica 47(1), 133–139 (2011) 23. Zubov, V.: Problems in the theory of the second method of Lyapunov, construction of the general solution in the domain of asymptotic stability. Prikladnaya Matematika i Mekhanika 19, 179–210 (1955) 24. Zubov, V.I.: Methods of AM Lyapunov and their Application. P. Noordhoff (1964)

Chapter 3

On the Stability of Nonlinear Mechanical Systems with Time-Varying Discontinuous Coefficients Alexey Platonov

Abstract In the paper, one class of mechanical systems with nonlinear force fields is considered. It is assumed that the system is under influence of potential, dissipative, and gyroscopic forces. Moreover, we suppose that there is a time-depending coefficient at potential forces. The case where this coefficient is piecewise continuous and piecewise monotonous on any finite-time interval is investigated. Thus, the system can be considered as a non-stationary switched system. We study the situation where this coefficient can be unbounded. Stability conditions for such class of systems are obtained. The Lyapunov direct method is used. Both multiple Lyapunov functions and single Lyapunov functions are constructed. Some examples are presented to illustrate the results.

3.1 Introduction Mechanical systems have been the object of research in many works, due to their great theoretical and practical importance. In particular, numerous approaches to analyzing the stability of such systems have been developed. It should be noted that the investigation of stability is significantly complicated, if considered system is nonlinear and non-autonomous. In such cases, the direct Lyapunov method is usually used in combination with approaches such as theory of differential inequalities, decomposition method, singular perturbation theory, averaging method, and theory of integral manifolds. One of the important problems is to analyze the situation where the parameters of the system begin to evolve over time. Thus, the coefficients for terms describing the action of different forces in the system can be time varying. For instance, in works [1–7] the stability problem was investigated for nonlinear mechanical systems with non-autonomous coefficient at potential forces. However, in these papers some restrictive assumptions were introduced. Thus, situations where the derivative of the A. Platonov (B) St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_3

21

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A. Platonov

coefficient tends to zero, or where the coefficient has a positive average value, or where it is monotonous were studied. Some attempt to weaken these assumptions was made in [11]. The system with a bounded continuous piecewise monotonous coefficient was investigated in [11]. In the present paper, in contrast to [11], we assume that the coefficient used can be discontinuous and unbounded. Breaks of the coefficient can be caused by sudden changes in external conditions or by a change in the control design. So, the present work develops the results established in [11]. If mechanical system can operate in different modes, then it is named a switched system [8]. For the stability analysis of such systems, various modifications of the direct Lyapunov method were developed (see [13]). Note that discontinuities of the time-varying parameters in the studied system can be considered as some switching in this system.

3.2 Mechanical System Consider the mechanical system ∂T ∂D ∂P d ∂T − =− −  q˙ − u(t) . dt ∂ q˙ ∂q ∂ q˙ ∂q

(3.1)

Here q, q˙ ∈ R n are vectors of generalized coordinates and generalized velocities; ˙ is described by quadratic form T = 21 q˙ T A(q)q˙ with the kinetic energy T = T (q, q) symmetric, positive definite and continuously differentiable for q < η1 (0 < η1 ≤ +∞) matrix A(q), such that the conditions ˙ 2 ≤ T ≤ a2 q ˙ 2, a1 q   ∂T    ˙ 2,  ∂q  ≤ a3 q

  ∂T    ˙  ∂ q˙  ≤ a4 q

are valid for q < η1 , q˙ ∈ R n , where a1 , . . . , a4 are some positive constants. The ˙ is continuously differentiable for q˙ ∈ R n , Rayleigh dissipative function D = D(q) positive definite and homogeneous of degree ν + 1, ν > 1; gyroscopic matrix ˙ is skew-symmetric, continuous and bounded for t ≥ 0, q < η1 ,  = (t, q, q) ˙ < η2 (0 < η2 ≤ +∞); the potential P = P(q) is continuously differentiable for q q ∈ R n , positive definite and homogeneous of degree μ + 1, μ ≥ 1; time-varying coefficient u(t) is defined for t ≥ 0. The presence of a non-stationary coefficient u(t) leads to the evolution of potential forces in the system. At the same time, the structure of these forces does not change. The aim of the paper is to analyze the stability of the solution q = q˙ = 0 of system (3.1). In the present paper, let us suppose that function u(t) is bounded below by a positive constant on the interval [0, +∞), but it can be unbounded above on this interval. Moreover, let function u(t) be piecewise continuous and piecewise monotonous. It

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23

should be noted that the case where function u(t) is positive, continuously differentiable, monotonically increasing, and u(t) → +∞ as t → +∞ was studied in [3]. In [10], it was assumed that function u(t) is piecewise constant. Now, we use more general assumptions.

3.3 Stability Conditions Let {τi }i=1,2,... be time moments where the coefficient u(t) has a break or changes the character of monotony, 0 = τ0 < τ1 < τ2 < . . .. We suppose that the total number of such time moments is infinite, the coefficient u(t) is right-continuous at break points, and it is continuously differentiable at intervals (τi , τi+1 ), i = 0, 1, . . .. Construct a Lyapunov function [3]: ˙ =P+ V1 = V1 (t, q, q)

γ ∂T 1 T + σ qνμ−1 qT . u(t) u (t) ∂ q˙

Here σ = max{1 − 1/ν; (3 − ν)/2}, γ > 0. Also, assume ˙ = u(t)V1 . V2 = V2 (t, q, q) One can find piecewise constant function w = w(t) such that w = 1 for t ∈ [τi , τi+1 ), if the coefficient u(t) increases for t ∈ [τi , τi+1 ), and w(t) = 2, if otherwise; i = 0, 1, . . .. ˙ Next, let us use the discontinuous Lyapunov function Vw = Vw(t) (t, q, q). Choose arbitrary c > 1 and for any i = 1, 2, . . . define the number κi by the rule:  u(τi − 0) , κi = c max 1; u(τi + 0) 

if w = 1 for t ∈ [τi−1 , τi+1 );   u(τi + 0) , κi = c max 1; u(τi − 0) if w = 2 for t ∈ [τi−1 , τi+1 );  κi = c max

 1 1 ; , u(τi + 0) u(τi − 0)

if w = 2 for t ∈ [τi−1 , τi ), and w(t) = 1 for t ∈ [τi , τi+1 ); κi = c max {u(τi + 0); u(τi − 0)} , if w = 1 for t ∈ [τi−1 , τi ), and w(t) = 2 for t ∈ [τi , τi+1 ).

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Introduce the following assumption. Assumption 3.1 Let u(t) ≥ a for t ≥ 0, and |u(t)| ˙ ≤ Mu 3/2 (t) for t ∈ (τi , τi+1 ), i = 0, 1, . . ., where a and M are positive constants. For any t ≥ 0 it is possible to find k such that τk ≤ t < τk+1 . In result, we obtain piecewise constant function k = k(t). Assume  t u l0 (s) ds for t ∈ [0, τ1 ),

χ (t) = g(t)

(3.2)

0

and χ (t) = g(t)

 k−1  

τi+1 τi

i=0

−ρ

u (s) ds (κi+1 . . . κk ) li

 +

t

 u (s) ds lk

τk

for t ≥ τ1 ,

(3.3) where ρ = (νμ − 1)/(μ + 1); g(t) = u −ρ (t) for t ∈ [τi , τi+1 ) and li = 1 − σ , if the coefficient u(t) increases for t ∈ [τi , τi+1 ); g(t) = 1 for t ∈ [τi , τi+1 ) and li = 1 − σ − ρ, if otherwise; i = 0, 1, . . .. Theorem 3.1 Let Assumption 3.1 be fulfilled. If χ (t) → +∞ as t → +∞, then the solution q = q˙ = 0 of (3.1) is asymptotically stable. Proof Find the derivative of Lyapunov functions Vw with respect to Eq. (3.1) for t ∈ (τi , τi+1 ), i = 0, 1, . . .. We have d V1 γ (μ + 1) ν+1 qνμ−1 P − D = − σ −1 dt (3.1) u (t) u(t) γ + σ u (t)



∂T ∂ q˙

T



∂ qνμ−1 q σ γ u(t) ˙ ∂T q˙ − σ +1 qνμ−1 qT ∂q u (t) ∂ q˙

γ + σ qνμ−1 qT u (t)



 ∂T ∂D u(t) ˙ − −  q˙ − 2 T, ∂q ∂ q˙ u (t)

d V2 γ (μ + 1) qνμ−1 P − (ν + 1)D = − σ −2 dt (3.1) u (t) +



γ u σ −1 (t)

+

γ u σ −1 (t) −

∂T ∂ q˙

T



∂ qνμ−1 q q˙ + u(t)P ˙ ∂q

νμ−1 T

q

q



∂D ∂T − −  q˙ ∂q ∂ q˙

(σ − 1)γ u(t) ˙ ∂T qνμ−1 qT . u σ (t) ∂ q˙



3 On the Stability of Nonlinear Mechanical Systems with Time-Varying …

25

Then there exist positive constants γ , δ, β1 , β2 , β3 such that the estimates ˙ 2 ) ≤ V1 ≤ β2 (qμ+1 + u −1 (t)q ˙ 2 ), β1 (qμ+1 + u −1 (t)q ˙ 2 ) ≤ V2 ≤ β2 (u(t)qμ+1 + q ˙ 2 ), β1 (u(t)qμ+1 + q d Vw ≤ −β3 u li (t)Vw1+ρ for t ∈ (τi , τi+1 ), i = 0, 1, . . . , dt (3.1) ˙ ≤ κi Vw(τi −0) (τi − 0, q, q), ˙ Vw(τi +0) (τi + 0, q, q) i = 1, 2, . . . , ˙ < δ. are valid, if t ≥ 0, q < δ, u −1/2 (t)q Next, using the approach suggested in [9], it is easy to establish the required.  Remark 3.1 If η2 = +∞, then assuming g(t) ≡ 1 for t ≥ 0 in (3.2), (3.3) and applying Theorem 3.1, we obtain the conditions of partial asymptotic stability of the solution q = q˙ = 0 of (3.1) with respect to q. Remark 3.2 If the coefficient u(t) is continuous for t ≥ 0, then one can choose c = 1. Example 3.1 Assume that the coefficient u(t) increases on intervals [τ2i , τ2i+1 ), decreases on intervals [τ2i+1 , τ2i+2 ), i = 0, 1, . . ., and (τ2i+1 − τ2i ) → +∞ as i → +∞. Let 0 < a0 (t + 1)α ≤ u(t) ≤ b0 (t + 1)β for t ≥ 0, where 0 < a0 ≤ b0 and 0 ≤ α ≤ β. If η2 = +∞, then the solution q = q˙ = 0 of (3.1) is asymptotically stable with respect to q. If 0 < η2 ≤ +∞ and α(1 − σ ) + 1 − ρβ > 0, then the solution ˙ q = q˙ = 0 of (3.1) is asymptotically stable with respect to q and q. Now, let us investigate the stability problem for system (3.1) using a single continuous Lyapunov function V1 . Assume   u(τi − 0) , κ˜ i = c max 1; u(τi + 0) where c = const > 1, if τi is a break point of the coefficient u(t); and κ˜ i = 1, if the coefficient u(t) is continuous at point τi ; i = 1, 2, . . .. Introduce the following assumption. Assumption 3.2 Let u(t) ≥ a for t ≥ 0 and |u(t)| ˙ ≤ M(t)u 3/2 (t) for t ∈ (τi , τi+1 ), i = 0, 1, . . ., where a is a positive constant and M(t) is a positive bounded function. Define function k = k(t), as before. Assume χ˜ (t) = g(t) ˜ e

−ρ J (t)

 0

for t ∈ [0, τ1 ), and

t

u 1−σ (s) eρ J (s) ds

(3.4)

26

A. Platonov

χ(t) ˜ = g(t) ˜ e

−ρ J (t)

 k−1   i=0

τi+1 τi

u 1−σ (s) eρ J (s) ds (κ˜ i+1 . . . κ˜ k )−ρ  +

t τk

u 1−σ (s) eρ J (s) ds

 (3.5)

t ˜ = u −ρ (t); J (t) = 0 ϕ(s) ds; ϕ(t) = for t ≥ τ1 ; where ρ = (νμ − 1)/(μ + 1); g(t) 0 for t ∈ [τi , τi+1 ), if the coefficient u(t) increases for t ∈ [τi , τi+1 ); ϕ(t) = c M(t) u 1/2 (t) for t ∈ [τi , τi+1 ), if otherwise; i = 0, 1, . . .. Theorem 3.2 Let Assumption 3.2 be fulfilled. If χ˜ (t) → +∞ as t → +∞, then the solution q = q˙ = 0 of (3.1) is asymptotically stable. Proof Find the derivative of Lyapunov functions V1 with respect to equations (3.1). We obtain that there exist positive constants γ , δ, β1 , β2 , β3 , such that if t ≥ 0, ˙ < δ, then the estimates q < δ, u −1/2 (t)q ˙ 2 ) ≤ V1 ≤ β2 (qμ+1 + u −1 (t)q ˙ 2 ), β1 (qμ+1 + u −1 (t)q d V1 1+ρ ≤ ϕ(t)V1 − β3 u 1−σ (t)V1 for t ∈ (τi , τi+1 ), dt (3.1)

(3.6)

i = 0, 1, . . ., and ˙ ≤ κ˜ i V1 (τi − 0, q, q), ˙ i = 1, 2, . . . , V1 (τi + 0, q, q) are valid. By integrating inequality (3.6) along the solutions of system (3.1), it is easy to prove the required [12].  ˜ ≡ 1 for t ≥ 0 in (3.4), (3.5) and Remark 3.3 If η2 = +∞, then assuming g(t) applying Theorem 3.2, we obtain the conditions of partial asymptotic stability of the solution q = q˙ = 0 of (3.1) with respect to q. Corollary 3.1 Let the coefficient u(t) be continuous for t ≥ 0, and Assumption 3.2 be fulfilled. If g(t) ˜ e−ρ J (t)



t

u 1−σ (s) eρ J (s) ds → +∞ as t → +∞,

0

then the solution q = q˙ = 0 of (3.1) is asymptotically stable. Example 3.2 Assume that the coefficient u(t) is continuous for t ≥ 0, |u(t)| ˙ ≤ M0 for t ≥ 0, and 0 < a0 (t + 1) ≤ u(t) ≤ b0 (t + 1) for t ≥ 0. Here M0 , a0 and b0 are positive constants. In this case, we have M(t) = M0 u −3/2 (t) and 0 ≤ ϕ(t) ≤ c M0 u −1 (t) for t ≥ 0. If η2 = +∞, then the solution q = q˙ = 0 of (3.1) is asymptotically stable with respect to q. If 0 < η2 ≤ +∞ and σ + ρ < 2, then the solution ˙ q = q˙ = 0 of (3.1) is asymptotically stable with respect to q and q.

3 On the Stability of Nonlinear Mechanical Systems with Time-Varying …

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3.4 Conclusion In the paper, stability problem for a class of nonlinear non-stationary mechanical systems was considered. It was assumed that the non-stationary coefficient that is present in the system can be discontinuous. The analysis of the system is complicated by the fact that the standard Lyapunov function used for this system also depends on the given coefficient. Note that the discontinuities of the coefficient can be understood as some switching in the system. Therefore, for the study of the system, the approach often used in the study of switched systems was applied. Considered approach can be applied for the analysis of the systems of more general form, as well. In particular, the case with several non-stationary coefficients in the system can be investigated.

References 1. Agafonov, S.A.: The stability and stabilization of the motion of non-conservative mechanical systems. J. Appl. Math. Mech. 74(4), 401–405 (2010) 2. Aleksandrov, AYu.: The stability of the equilibrium positions of non-linear non-autonomous mechanical systems. J. Appl. Math. Mech. 71(3), 324–338 (2007) 3. Aleksandrov, A.Yu., Aleksandrova, E.B., Platonov, A.V.: Stability analysis of equilibrium positions of nonlinear mechanical systems with nonstationary leading parameter at the potential forces. Vestnik of St. Petersburg State University. Ser. 10. Appl. Math. Comput. Sci. Control Process. (1), 107–119 (2015) (in Russian) 4. Aleksandrov, AYu., Kosov, A.A.: Asymptotic stability of equilibrium positions of mechanical systems with a nonstationary leading parameter. J. Comput. Syst. Sci. Int. 47(3), 332–345 (2008) 5. Andreyev, A.S.: The stability of the equilibrium position of a non-autonomous mechanical system. J. Appl. Math. Mech. 60(3), 381–389 (1996) 6. Kosov, A.A.: The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces. J. Appl. Math. Mech. 71(3), 371–384 (2007) 7. Kozlov, V.V.: On the stability of equilibrium positions in non-stationary force fields. J. Appl. Math. Mech. 55(1), 14–19 (1991) 8. Liberzon, D.: Switching in Systems and Control. Birkhauser, Boston, MA (2003) 9. Platonov, A.V.: On the asymptotic stability of nonlinear time-varying switched systems. J. Comput. Syst. Sci. Int. 57(6), 854–863 (2018) 10. Platonov, A.V.: On the problem of nonlinear stabilization of switched systems. Circuits Syst. Signal Process. 38(9), 3996–4013 (2019) 11. Platonov, A.V.: Stability analysis for nonlinear mechanical systems with non-stationary potential forces. 15th Int. Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference) (STAB), Moscow, Russia. Institute of Electrical and Electronics Engineers Inc., pp. 1–4 (2020). https://doi.org/10.1109/STAB49150.2020.9140552 12. Platonov, A.V.: Stability analysis for nonstationary switched systems. Russ. Math. 64(2), 56–65 (2020) 13. Zhai, G., Hu, B., Yasuda, K., Michel, A.N.: Disturbance attention properties of time-controlled switched systems. J. Frankl. Inst. 338, 765–779 (2001)

Chapter 4

Normalizing Random Vector Anisotropy Magnitude Kirill Chernyshov

Abstract In the context of the anisotropic control theory vanishing the system αanisotropic norm corresponds with H2 -theory, while its going to infinity, with H∞ theory. Meanwhile, this definition considerably involves just the magnitude of α, the random vector anisotropy (the mean random vectors sequence anisotropy) that characterizes the uncertainty degree. Thus, proper selection of the random vector anisotropy magnitude to determine the system α-anisotropic norm is of importance. Accordingly, the paper presents an approach to constructing a normalization procedure of the anisotropy magnitude as a mapping of the positive semiaxis in the unit interval.

4.1 Preliminaries The anisotropy-based theory of stochastic robust control and filtering was originated by I.G. Vladimirov in the mid 1990s–early 2000s [1–4]. This theory employs stochastic minimax settings, which seek controllers and observers minimizing the worst-case gain of the closed-loop system output with respect to statistically uncertain random inputs, and numerous control problems were being successfully solved within the theory, e.g. [5–17]. The starting point of the anisotropic control theory [1–4] is the definition of the anisotropy A[z] of a random n-dimensional vector z with probability distribution density p(z). The definition is based on Kullback–Leibler divergence and has the form:    z) G ν ( , (4.1) p( z)ln A[z] = min − ν>0 p( z)d z Rn

K. Chernyshov (B) V.A. Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_4

29

30

K. Chernyshov

where G ν (z) is the probability density function of an n-dimensional Gaussian random vector with a scalar covariance matrix ν · In , where In is the unit n × n-matrix: G ν (z) = 

1 (2π ν)

n

e−

z2 2ν

.

The optimal value ν in (4.1) is expressed as   E z 2 . ν= n

(4.2)

with E (·) standing for the mathematical expectation. Under condition (4.2), definition (4.1) immediately takes on its closed form A [z] =

    n  ln 2π e E z2 /n − S (z) , 2

where S(z) is the differential entropy of the n-dimensional random vector z with z)ln ( p( z)) d z. probability distribution density p(z): S(z) = − R n p( Within the definition, the random vector z is assumed to be absolutely continuously   distributed and having finite second moments, E z2 < +∞, but not necessarily Gaussian. The choice of isotropic Gaussian distributions as reference measures comes from the fact that they are used as a standard nominal model for random noises in Kalman filtering and the LQG control theory. Being always non-negative, the anisotropy A [z] vanishes only when z has an isotropic Gaussian distribution. Also, it is scale and rotation invariant in the sense that A [sU z] = A [z] for any multiplicative constant s = 0 and any orthogonal matrix U of order n [1–4]. The key stone of the anisotropic control theory is the use as a norm (for nonrandom linear operators) of the α-anisotropic norm [1–4]

de f

|||F|||α =

sup P: A[z]≤α

  E F z2   E z2

(4.3)

of a non-random matrix F ∈ R m×n , where the supremum is taken over absolutely continuous probability distribution P of the random vector z with finite second moments subject to the constraint A [z] ≤ α. Here, α is a non-negative parameter quantifying the amount of statistical uncertainty in z in terms of deviation from isotropic Gaussian distributions in R n . The meaning of the α-anisotropic norm |||F|||α is that it is the worst-case root mean-square gain of the matrix F with respect to statistically uncertain input random vectors z (whose probability distributions— not only Gaussian ones—belong tothe α-dependent uncertainty classdescribed     above). Meanwhile, the numerator E F z2 and the denominator E z2 in (4.3) are the norms of the random vectors F z and z (considered as the output and input of the transformation defined by m × n-matrix F, respectively) in the

4 Normalizing Random Vector Anisotropy Magnitude

31

Hilbert space of random vectors with finite second moments. Due to the properties lim α→0 |||F|||α = √1n F F , lim α→+∞ |||F|||α = F∞ , the α-anisotropic norm    bridges the gap between the Frobenius normF F = T r F T F of the matrix F (typical for the mean-square optimality criteria in Kalman filtering and LQG control)    and the operator normF∞ = λmax F T F used in the H∞ -control and filtering theory) as α varies from 0 to +∞. Accordingly, whether |||F|||α is closer to F0 or to F∞ (this question pertains to the comparison of anisotropy-based controllers and observers with their limiting LQG and H∞ -counterparts) depends not only on the anisotropy level α > 0, but also on the matrix F [1–4]. Simultaneously, regarding the above last remark, namely, the anisotropy level α > 0 remains at the researcher’s disposal, and properly selecting its magnitude is a problem just from the point of view of the necessity of properly selecting possible values from zero to infinity. So, the present paper proposes a technique to construct a normalized magnitude (i.e. magnitude that takes its values in unit interval) of the random vector anisotropy and a normalized magnitude of the mean random vectors sequence anisotropy. The technique entity is a justified choice of respective mapping the positive semiaxis into the unit interval. Obviously, selecting the uncertainty degree from the unit interval provides considerably more convenient perception of the uncertainty quantification problem in the comparison to the same problem with regard to the infinite semiaxis within the definition of system α-anisotropic norm (4.3).

4.2 Basic Algorithm Normalization of the random vector anisotropy magnitude may be treated as an answer to a question of interpreting some of its meanings, in the same manner as it occurs in case of probability measures. Such a normalization is easily achieved through any continuous monotone mapping [0; ∞ ) → [ 0; 1]. However, there exist infinitely many mappings of this kind (say, ones based on exponential, fractionally rational, trigonometric functions), and hence a question arises as to the existence of “naturally” sound selection criteria for a needed normalization mapping. The choice of construction algorithm for a normalization mapping needed is based on arguments as follows. (1) The random vector anisotropy is a measure of divergence between the distribution of the vector under study and the corresponding Gaussian vector. (2) In turn, any measure of divergence D ( f  g) becomes a measure of dependence between a pair of random vectors (values), x and y, where one of the density functions, (4.4) f = f x y (x, y),

32

K. Chernyshov

is their joint probability density function, and the other density function, g = gx y (x, y) = gx (x)g y (y),

(4.5)

is the product of their marginal probability density functions. This affirmation means the following. For an n-dimensional random vector T  z = z 1 , . . . z n x , z n x +1 , . . . , z n x +n y , n = n x + n y  with n-variate joint probability distribution density pz z 1 , . . . z n x , z n x +1 , . . . ,  z n x +n y , one can consider two (sub)-vectors  T x = z1, . . . znx ,

T  y = z n x +1 , . . . , z n x +n y ,

  with  corresponding (joint)  probability distribution densities px z 1 , . . . z n x and p y z n x +1 , . . . , z n x +n y . Obviously,   px z 1 , . . . z n x =

 Rn y

  pz z 1 , . . . z n x , z n x +1 , . . . , z n x +n y dz n x +1 . . . dz n x +n y ,

  p y z n x +1 , . . . , z n x +n y =

 Rnx

  pz z 1 , . . . z n x , z n x +1 , . . . , z n x +n y dz 1 . . . dz n x ,

  and can be referred as marginals of pz z 1 , . . . z n x , z n x +1 , . . . , z n x +n y . Consequently, there  are no obstacles to consider  the Kullback–Leibler   divergence between pz z 1 , . . . z n x , z n x +1 , . . . , z n x +n y and px z 1 , . . . z n x · p y   z n x +1 , . . . , z n x +n y . (3) And, finally, for random variables measures of dependence, corresponding set of axioms of A. Rényi [18] (see also [19]) is known, whereby the measure of dependence μ (x, y) between two random variables x and y in case of joint Gaussian probability distribution (according to A. Rényi) shall match the absolute value of the correlation coefficient: μ (x, y) = |corr (x, y)|, where corr (x, y) is the ordinary correlation coefficient between x and y. Therefore, the construction algorithm for positive semiaxis mapping into the unit interval to construct a normalized random vector anisotropy magnitude is as follows: (1) For a divergence measure (upon which anisotropic norm D ( f  g) is based), an appropriate measure of dependence should be constructed,   D ( f  g) = D f x y gx · g y = Mx y ,

(4.6)

between random vectors x and y with (in terms of (4), (5)) joint f x y (x, y) and marginal gx (x) and g y (y) probability density functions. (2) Calculate (6) for a joint bivariate (for x = x, y = y) Gaussian probability density function with correlation coefficient corr (x, y)).

4 Normalizing Random Vector Anisotropy Magnitude

33

(3) Express the obtained expression as function of correlation coefficient module, Mx y =  Mx yy (|corr (x, y))|), and invert this function, i.e. construct mapping   |corr (x, y))| = −1 Mx y M x y

(4.7)

  (4) Obtained expression,−1 Mx y M x y , (as function of original measure of dependence Mx y ) defines the needed mapping [ 0; ∞ ) → [ 0; 1 ] for the anisotropy magnitude that is based on this divergence measure D ( f  g): −1 Mx y (D ( f  g)) .

(4.8)

This algorithm, when applied to anisotropy magnitude [1–4] A [z] of random vector z, provides the result as follows. Since A [z] [1–4] in (4.1) is a Kullback– Leibler divergence measure D K L ( f  g),  D K L ( f  g) = −

Rn

f (z)ln

g(z) dz, f (z)

(4.9)

then a corresponding measure of dependence (4.6) (in terms of (4.4), (4.5)) is the mutual information (relative differential entropy) I S (x, y),   g (x)g y (y) D K L f x y gx g y = I S (x, y) = − R n f x y (x, y)ln xf x y (x,y) d xd y, . dimx + dim y = n. For n = 2, formula (4.7) in step 3 of this algorithm is expressed as |corr (x, y))| =

 1 − e−2·I S (x,y) .

Therefore, based on formula (4.8) in step 4 of this algorithm, the desired normalizing transformation of random vector anisotropy magnitude (4.1) is defined by expression  (4.10) 1 − e−2·D K L ( f g) . Then, with consideration of strict monotonicity of function (4.10), normalized random vector anisotropy magnitude Anor m [z] within the context of definition (4.1) can be written as



 z) G ν ( nor m d z . (4.11) p( z)ln A [z] = min 1 − ex p 2 · ν>0 p( z) Rn Consequently, the solution of optimization problem (4.11) is equivalent to the solution of optimization problem (4.1), and is defined by expression (4.2). So, it can be finally written as

34

K. Chernyshov

Anor m [z] =



1 − e−2·A[z] ,

(4.12)

where A [z] is defined by (4.1). The plot of mapping (4.12) is displayed in Fig. 4.1; and even simple superficial view on it confirms that the interpretation of random vector anisotropy magnitude can be considerably different in the case of non-normalized and normalized ones. Say, anisotropy magnitude A [z] (4.1) of [1–4] of the value about 0.5 (which can be interpreted as a small enough from the point of view of comparison with the whole positive semiaxis) implies corresponding normalized value about 0.8 (which, accordingly, should be considered as large enough from the point of view of the unit interval). Therefore, this illustrates the importance of considering and accounting for normalized random vector anisotropy magnitudes specifically and, accordingly, the importance of constructing a sound normalization algorithm for anisotropy-related magnitudes. To illustrate the diversity in quantifying the uncertainty (when represented by the random vector anisotropy) change, one may consider bivariate Gaussian vector coefficient r. In accordance to defiz 2 (r ) with Laplace marginals and correlation  √ 2 1 − r , while in accordance to expression (4.12) nition (4.1) A [z 2 (r )] = −ln Anor m [z 2 (r )] = |r |. Meanwhile, A [z 2 (r )], as a function in the correlation coefficient r , grows with the loss of the uniform continuity as |r | → 1. Corresponding plots are displayed in Fig. 4.2. Accordingly, by virtue of expression (4.12) definition (4.3) can be directly rewritten in the form involving the normalized random vector anisotropy Anor m [z]:

de f

|||F|||αnor m =

Fig. 4.1 The dependence of the normalized anisotropy magnitude of the non-normalized one as per (4.12)

sup P: Anor m [z]≤α nor m

  E F z2  ,  E z2

4 Normalizing Random Vector Anisotropy Magnitude

35

Fig. 4.2 Plots of the anisotropy magnitude and normalized anisotropy magnitude of the random vector z 2 (r ) as functions in the correlation coefficient r

with α nor m ε [0; 1] and validity of the corresponding inequalities: 1 lim |||F|||αnor m = √ F F , n

α nor m →0

lim |||F|||αnor m = F∞ .

α nor m →1

4.3 Normalized Mean Anisotropy Magnitude of a Random Vectors Sequence The α-anisotropic norm definition represented by expression (4.3) in accordance to [4] should be noted to differ from a corresponding α-anisotropic norm definition used within control application related papers [5–17] (see also [20] and others), which is based on involving the notion of the mean random vector sequence anisotropy. Such a definition is in its entity very close to that of 4.3, but directly oriented to the state-space linear dynamic systems representation and based on determining a “worst-case” linear filter whose output possesses a mean anisotropy not exceeding a given magnitude of α. To characterize and describe quantitatively Gaussian and white-noise (mutual independence) properties of an (infinite) sequence of random vectors, the mean anisotropy [4] is defined on the basis of the anisotropy of a single random vector. Namely, let (4.13) z i , i = 1, 2, . . . be a sequence of n-dimensional random vectors, and T  Z N = z 1T , . . . , z TN .

(4.14)

36

K. Chernyshov

Then the mean anisotropy of random vectors sequence (4.13) is defined as [4] A [z] = lim

N →∞

A [Z N ] , N

(4.15)

where A [Z N ] is understood in the sense of definition (4.1). The normalization algorithm presented in Sect. 4.2 enables one to construct corresponding normalized mean anisotropy magnitude of random vectors sequence (4.13). Namely, for N −1 A [Z N ] in (4.15) one can write in accordance to definition (4.1): A [Z N ] = min ν>0 N

 R n·N

1  ZN) N  pC N (   pC N ( Z N )ln d ZN, G ν ( ZN)

(4.16)

where C N stands for the covariance matrix of Z N in (4.14). Meanwhile, the quantity N −1 A [Z N ] = A(N ) [z] is natural to be referred to as average anisotropy of a finite sequence of random vectors, defined by (4.14). Expression of the form   DβK L pC N G ν =

 Rn

pC N (  Z N )ln



ZN) pC N (  G ν ( ZN)

β

d ZN, β > 0

(4.17)

can be considered as a generalization of Kullback–Leibler divergence (regarding to expression (17), β = N −1 ) between pC N (Z N ) and G ν (Z N ). Then, applying the algorithm in Sect. 4.2, by virtue of expressions (4.4)–(4.8), results, by implementing corresponding in the following expression for the normalized value of derivations,   DβK L pC N G ν :  KL (4.18) 1 − e−2β Dβ ( pC N G ν ) . Hence, expressions (4.17), (4.18) and accounting of all preceding considerations imply the following form of the normalized mean anisotropy magnitude:  Anor m

[z] = lim

N →∞

1−e

−2 A[ Z N ] N

.

4.4 Conclusions In the context of the anisotropic theory for stochastic control systems, this paper presents an approach to normalization (provision of belonging to the unit interval) of anisotropy magnitude values for a random vector. It suggests a universal algorithm based on the interrelation between divergence and measures of dependence, which uses the corresponding A. Rényi axioms.

4 Normalizing Random Vector Anisotropy Magnitude

37

As it was cited in Sect. 4.1, the random vector anisotropy magnitude (4.1) involved in α-anisotropic norm (4.3) reflects the uncertainty level; and namely, the anisotropy level α > 0 remains at the researcher’s disposal, and properly selecting its magnitude is a problem just from the point of view of the necessity of properly selecting possible values from zero to infinity. Obviously, selecting the uncertainty degree from the unit interval provides considerably more convenient perception of the uncertainty quantification problem in the comparison to the same problem with regard to the infinite semiaxis within the definition of system α-anisotropic norm (4.3). The essence of normalizing the anisotropy magnitude is the same as with regard to the probability of an event or the reliability of a system. If the probability or reliability would take their values in the whole positive semiaxis, rather than in the unit interval, it would be absolutely impossible to judge on the suitability of a magnitude of the probability or reliability, while the normalization permits that. Regarding the uncertainty quantification, a natural question could be asked, what is the magnitude of α reflecting, say, 50%-uncertainty within the anisotropic control theory? It cannot be answered remaining within non-normalized definition (4.1), but by use of expression (4.12), the answer is α = −0.5 · ln (0.75) ≈ 0.14384103622589046.

References 1. Vladimirov, I.G., Kurdjukov, A.P., Semyonov, A.V.: Anisotropy of signals and the entropy of linear stationary systems. Dokl. Math. 51, 388–390 (1995) 2. Vladimirov, I.G., Kurdjukov, A.P., Semyonov, A.V.: Asymptotics of the anisotropic norm of linear discrete-time-invariant systems. Autom. Remote. Control. 60, 359–366 (1999) 3. Diamond, P., Vladimirov, I., Kurdjukov, A., Semyonov, A.: Anisotropy-based performance analysis of linear discrete time invariant control systems. Int. J. Control 74, 28–42 (2001) 4. Vladimirov, I.G., Diamond, P., Kloeden, P.: Anisotropy-based robust performance analysis of finite horizon linear discrete time varying systems. Autom. Remote. Control. 67, 1265–1282 (2006) 5. Kurdyukov, A.P., Maksimov, E.A.: Robust stability of linear discrete stationary systems with uncertainty bounded in the anisotropic norm. Autom. Remote. Control. 65, 1977–1990 (2004) 6. Tchaikovsky, M.M., Kurdyukov, A.P.: Normalized problem of anisotropy-based stochastic H∞ optimization for closed-loop system order reduction by balanced truncation. Autom. Remote. Control. 71, 776–789 (2010) 7. Tchaikovsky, M.M.: Anisotropic ε-optimal model reduction for linear discrete time-invariant system. Autom. Remote Control 71, 2573–2594 (2010) 8. Kustov, AYu., Kurdyukov, A.P.: Shaping filter design with a given mean anisotropy of output signals. Autom. Remote. Control. 74, 358–371 (2013) 9. Timin, V.N., Kurdyukov, A.P.: Suboptimal anisotropic filtering in a finite horizon. Autom. Remote. Control. 77, 1–20 (2016) 10. Tchaikovsky, M.M.: Multichannel synthesis problems for anisotropic control. Autom. Remote. Control. 77, 1351–1369 (2016) 11. Tchaikovsky, M.M., Timin, V.N.: Synthesis of anisotropic suboptimal control for linear timevarying systems on finite time horizon. Autom. Remote. Control. 78, 1203–1217 (2017) 12. Tchaikovsky, M.M., Timin, V.N., Kustov, AYu., Kurdyukov, A.P.: Numerical procedures for anisotropic analysis of time-invariant systems and synthesis of suboptimal anisotropic controllers and filters. Autom. Remote. Control. 79, 128–144 (2018)

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13. Tchaikovsky, M.M., Kurdyukov, A.P.: Anisotropic suboptimal control for systems with linearfractional uncertainty. Autom. Remote. Control. 79, 1100–1116 (2018) 14. Belov, A.A., Andrianova, O.G., Kurdyukov, A.P.: Control of Discrete-Time Descriptor Systems. An Anisotropy-Based Approach Springer (2018) 15. Andrianova, O.G., Belov, A.A.: Robust performance analysis of linear discrete-time systems in presence of colored noise. Eur. J. Control. 42, 38–48 (2018) 16. Timin, V.N., Kustov, AYu., Kurdyukov, A.P., et al.: Suboptimal anisotropic filtering for linear discrete nonstationary systems with uncentered external disturbance. Autom. Remote. Control. 80, 1–15 (2019) 17. Tchaikovsky, M.M., Timin, V.N., Kurdyukov, A.P.: Synthesis of anisotropic suboptimal pid controller for linear discrete time-invariant system with scalar control input and measured output. Autom. Remote. Control. 80, 1681–1693 (2019) 18. Rényi, A.: On measures of dependence. Acta Math. Acad. Sci. Hung. 10, 441–451 (1959) 19. Micheas, A.C., Zografos, K.: Measuring stochastic dependence using ϕ-divergence. J. Multivar. Anal. 97, 765–784 (2006) 20. Kurdyukov, A.P., Maximov, E.A.: Solution of the stochastic H∞ -optimization problem for discrete time linear systems under parametric uncertainty. Autom. Remote. Control. 67, 1283– 1310 (2006)

Chapter 5

Minimax Approach in a Multiple Criteria Stabilization of Singularly Perturbed Control Stanislav Myshkov, Vladimir Karelin, and Lyudmila Polyakova

Abstract The linear-quadratic control problem with incomplete information about states is considered in the case when the quality of optimal control is characterized by a set of functionals. A multiple criteria problem arises if the control object has some subsystems which have their own criteria of optimal stabilizing. The dynamical system is supposed as time-invariant and singular- perturbed. The output-feedback contains information only about slow states and it is incomplete. The lack of information doesn’t permit to use main results of classical linear-quadratic control theory. There are some approaches to solving the problem under consideration. The method of averaging the functionals in some initial states domain is used by S. Myshkov. That approach may be applied also in the non-stationary case of the problem. In the time-invariant case which takes place in the paper, we offer the minimax approach. The solution of the initial problem results in discrete minimax problem. Some very effective methods of solving the discrete minimax problem are investigated by V. Demyanov The minimax approach to the solution in the multiple criteria linear-quadratic problem of stabilizing the singular perturbed system is the main difference between this paper and the previous works.

5.1 Introduction The linear-quadratic control problem on [0, ∞) is considered under the following conditions. The dynamics is described by the system of linear time-invariant singular perturbed equations. Only the slow states can be used but the information about them S. Myshkov (B) · V. Karelin · L. Polyakova St. Petersburg State University, Universitetskaya nab, 199034 7/9, St. Petersburg, Russia e-mail: [email protected] V. Karelin e-mail: [email protected] L. Polyakova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_5

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is incomplete. The investigation of singularly perturbed systems (SPCS) began in [14] and many results in the control theory were published in reviews of domestic and foreign authors. In the paper, the quality of optimal control is characterized by a set of quadratic functionals. These restrictions do not allow to use the known results of the classical control theory. Therefore some approaches to modify the problem have been proposed [1, 4, 5, 8]. In the most widely published papers the authors have used the feedback device to estimate the object states (all states or a part of them) [4, 15]. In that case, the system order will be increased but it is not desirable in practice. Other approaches include a modification of the functional [3, 9, 10]. In the paper, the minimax approach is used. The initial linear-quadratic problem is transformed into the discrete minimax problem [2] and all Pareto-optimal solutions are obtained. This modification is possible only for the stationary version of Problem [6, 16].

5.2 Preliminary Notes The problem under consideration is described by the following equations x˙s = A11 (μ)xs + A12 (μ)x f + B1 (μ)u, xs (0) = xs0 ,

(5.1)

μx˙ f = A21 (μ)xs + A22 (μ)x f + B2 (μ)u, x f (0) = x f 0 ,

(5.2)

z(t) = C(μ)x(t) = C1 (μ)xs (t),

(5.3)

u = K z,

(5.4)

wi (x, u) dt, i ∈ 1 : N ,

(5.5)

wi (x, u) = x ∗ Q i (μ)x+u ∗ Ri (μ)u.

(5.6)

∞ Fi (u) = 0

where

Here xs ∈ R n 1 is the vector of slow states, x f ∈ R n 2 is the vector of fast states, x = col(xs , x f ) ∈ R n , n = n 1 + n 2 , is the order of the dynamical system; vector u ∈ R r is a control and a vector z ∈ R m is the output. It is supposed that m ≤ n 1 , i.e. information about states is very incomplete. The parameter μ is small and positive: 0 < μ < μ¯  1. It characterizes the singular perturbations which occur in the control systems. All the matrices are time-invariant and are expanded in power series of μ. For example ∞  μs (s) (5.7) A11 (μ) = Aˆ 11 + A , s! 11 s=1

5 Minimax Approach in a Multiple Criteria Stabilization of Singularly …

where

41

A11 (0) = lim A11 (μ) = Aˆ 11 . μ→0

By assumption, these series are absolutely convergent in domain    = μ ∈ R : |μ| < μ¯ . Moreover, we suppose that • forms x ∗ Q i (μ)x are positive semidefinite but all forms u ∗ Ri (μ)u are positive definite; • all the of A22 are at the left-half plane;   eigenvalues • rank C1 (μ) = m. Furthermore, for more compact form we need to use the following matrices  A=

A11 A12 A21 A22



 ,

B=

B1 B2

.

Let K ⊂ R r ×m be a set of matrices K ∈ R r ×m for which the system (5.1)–(5.2), closed by control (5.4), is asymptotically stable. Control (5.4) with K ∈ K is called an admissible control. For any admissible control the functionals (5.5) are equal to Fi (u) = x0∗ i (K )x0 .

(5.8)

The matrix i (K ) is defined from the following equation    ∗ i A + B K C + A + B K C i + Wi (K ) = 0, where

(5.9)

Wi (K ) = Q i + C ∗ K ∗ Ri K C.

It follows from (5.6)–(5.9) that the optimal control will be a function of the initial state x0 , but this is not desirable. So it is necessary to modify the problem by an appropriate method. There are several approaches to do it. From our point of view, the most successful approach involves introducing a global criterion in the linear form combining all the local criteria: F(u) =

N  i=1

αi Fi (u) ∀αi ≥ 0,

N 

αi = 1.

i=1

ˆ The optimizing F(u) is known gives all Pareto-optimal solutions of the multiple criteria problem. The minimax approach is the effective method to solve it. Further it is supposed that all minimax controls (all the matrices K io ) are distinct and the

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optimal values of all functionals are equal to 1. That is, the multiple criteria problem is normalized. In the minimax approach, a control u o (t) = K o z(t) minimizes the functional ˆ ) = max x0∗ (K ˆ )x0 , F(K x0 =1

(5.10)

ˆ ) is defined by Lyapunov’s equation: where (K     ˆ A + B K C + A + B K C ∗ ˆ + Wˆ (K ) = 0.  Here Wˆ (K ) =

N 

αi Wi (K ).

i=1

Because of

  ˆ . . . , λn () ˆ , ˆ )x0 = max λ1 (), max x0∗ (K

x0 =1

the minimax control is optimal in the following sense   ˆ ) =⇒ min . λmax (K K ∈K

(5.11)

Problem (5.11) is equivalent to Problem (5.10). But the problem (5.11) is a discrete minimax Problem [2] and it may be solved by methods of non-smooth analysis. Many other problems in various science topics have been solved by these methods [11–13]. With μ ∈ (0, μ), ¯ the necessary conditions of optimality in Problem (5.11) are of the form [5, 16]:     ˆ A + B K C + A + B K C ∗ ˆ + Wˆ (K ) = 0, 

(5.12)

  ∗  Vˆ A + B K C + A + B K C Vˆ + yˆ yˆ ∗ = 0,

(5.13)

ˆ yˆ = λˆ yˆ , 

(5.14)

ˆ Vˆ C ∗ = 0. Rˆ K C Vˆ C ∗ + B ∗ 

(5.15)

N Ri , Vˆ is a positive semidefinite n-matrix, λˆ , yˆ are the maximum Here Rˆ = i=1 ˆ Notice that (5.15) is the eigenvalue and the corresponding eigenvector of matrix . equation determining the matrix K of the desired minimax control (5.4). If matrix C Vˆ C ∗ is nonsingular, then the matrix K is uniquely determined:   ˆ Vˆ C ∗ C V C ∗ −1 , K o = − Rˆ −1 B ∗ 

(5.16)

5 Minimax Approach in a Multiple Criteria Stabilization of Singularly …

43

and the minimax control is defined by u o (t) = K o z(t). The minimax approach (5.12)–(5.16) transforms substantially the linear-quadratic problem and from our point of view makes it more suitable for practical use that the optimal control will be a function of the initial state x0 , but this is not desirable. So it is necessary to modify the problem under consideration by an appropriate method.

5.3 Reduced Control Problem For solving Eqs. (5.12)–(5.15), we use the method of asymptotic representations which allows to identify the leading terms of the unknown quantities. The solution of the reduced control problem is well known to play here a pivotal role. In the case under consideration, the reduced problem is derived from (5.1)–(5.5) with μ = 0 and is as follows: x˙e (t) = Ae xe (t) + Be (t)u e (t), xe (t0 ) = xe0 ,

(5.17)

z e (t) = Ce xe (t),

(5.18)

u e (t) = K e z e (t),

(5.19)

Fˆe (u e ) =

∞



 xe∗ Q e xe + u ∗e Re u e dt.

(5.20)

t0

Here xe ∈ R n 1 , u e ∈ R r , z e ∈ R m are the state, the control and the output vectors, xe0 = xs0 , z e (t) = z(t), Ce = Cˆ 1 . Other matrices are uniquely expressed by the blocks of A, B, Q, R [8, 11]. Let  ∗ We (K e ) = Q e + K e Ce Re K e Ce . It is clear that the matrix We (K e ) > 0. Denote Ke =



  K e ∈ R r ×m Re λi Ae + Be K e Ce < 0, i ∈ 1, . . . , n 1 .

The set Ke is nonempty if the system (5.1)–(5.2) with output (5.3) is stabilized by control (5.4). Then there exists the minimax control u eo (t) = Meo z e (t), which is optimal for functional (5.20) and the matrix K eo is defined by the following equations:    ∗ e Ae + Be K e Ce + Ae + Be K e Ce e + We (K e ) = 0,

(5.21)

 ∗   Ve Ae + Be K e Ce + Ae + Be K e Ce L s + ye ∗ ye∗ = 0,

(5.22)

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e ye = λe ye ,

(5.23)

Re K e Ce Ve Ce∗ + Be∗ e Ve Ce∗ = 0.

(5.24)

The matrix e is positive definite and the matrix Ve is positive semidefinite. The number λe is the maximum eigenvalue of the matrix e and ye is the corresponding eigenvector. If the matrix Ce Ve Ce∗ is nonsingular, then the matrix K e can be uniquely determined:  −1 K eo = −Re−1 Be∗ e Ve Ce∗ Ce Ve Ce∗ .

(5.25)

In that case the minimax control u eo (t) is uniquely determined as u eo (t) = Meo z e (t). It is obvious that relations (5.21)–(5.25) have analogous forms as the relations (5.12)–(5.15). However the dimension of reduced problem (5.17)–(5.25) is lower because of n 1 < n and they are more simple because the singularity is absent.

5.4 Main Results Under the above assumptions, the solutions of Eq. (5.12)–(5.15) may be represented in the form [1, 8]: ˆ = 



ˆ 11 μ ˆ 12  ∗ ˆ 12 μ ˆ 22 μ 

V =

Vˆ11 ∗ Vˆ12



Vˆ12 Vˆ22

+

 (n) ∞  μn 11 μ(n) 12 (n) , n! μ(n)∗ 12 μ22 n=1

(5.26)

+

 (n) (n) ∞  μn V11 V12 (n)∗ (n) n! V V22 12 n=1

(5.27)



λ = λˆ +

∞  μn n=1

 y = y(μ) =

yˆ1 μ yˆ2



K (μ) = Kˆ +

n!

λ(n)

 (n) ∞  μn y1 + (n) n! μy 2 n=1 ∞  μn n=1

n!

K (n) .

(5.28)

(5.29)

(5.30)

By substituting these series in Eqs. (5.12)–(5.15) and equating the corresponding coefficients with the same power of parameter μ, we can determine all needed coefˆ i j , Vˆi j , Kˆ , λˆ , yˆi are determined ficients of series (5.26)–(5.30). The leading terms  ˆ 12 ,  ˆ 22 , from (5.26) to (5.30) after equation coefficients with μ0 . If we exclude 

5 Minimax Approach in a Multiple Criteria Stabilization of Singularly …

45

ˆ 11 , Vˆ11 , Kˆ , λˆ , yˆ1 we derive the following Vˆ12 , Vˆ22 from the obtained equations for  equations [8, 10]:     ˆ 11 Ae +Be Kˆ Cˆ 1 + Ae +Be Kˆ Cˆ 1 ∗  ˆ 11 + We ( Kˆ ) = 0,  

(5.31)

  ∗ Ae +Be Kˆ Cˆ 1 Vˆ11 + Vˆ11 Ae +Be Kˆ Cˆ 1 + yˆ1 yˆ1∗ = 0,

(5.32)

ˆ 11 yˆ1 = λˆ yˆ1 , 

(5.33)

ˆ 11 Vˆ11 Cˆ 1∗ = 0. Re Kˆ Cˆ 1 Vˆ11 Cˆ 1∗ + Bˆ e∗ 

(5.34)

ˆ 11 , Vˆ11 , Here matrices Ae , Be , We were determined above. So, the leading terms  Kˆ , λˆ , yˆ1 are defined by Eqs. (5.31)–(5.34). They have the same view as the Eqs. (5.21)– (5.24) for the reduced problem. Therefore ˆ 11 = e , Vˆ11 = Ve , 

Kˆ o = K e , λˆ = λe ,

yˆ1 = ye .

So it has been proved that the control problem under consideration is solvable and the asymptotic representations have the following forms K o = K e + O1 (μ),

F(u o ) = λe + O2 (μ).

Here K e , λe are the solutions of the reduced control problem and for O1 (μ), O2 (μ) there take place the inequalities ¯ O1 (μ) ≤ c1 μ, O2 (μ) ≤ c2 μ ∀μ ∈ (0, μ). From the above we can formulate the following theorem. Theorem 5.1 The singularly perturbed control problem has a unique solution if the reduced control problem has a unique solution. The main parts of the minimax control and the global functional are equal to the analogous values of the reduced problem. So, the problem of optimizing the global functional is always solvable and the main parts of the minimax control and the global criterion are determined by the solution of the corresponded reduced problem which is not singularly perturbed. This means that every Pareto-optimal solution may be calculated and every point of the Pareto set may be found. Moreover, the frontier of the Pareto set may be characterized by solving all the two-criteria problems arising in the problem under consideration.

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5.5 Conclusion The multiple criteria problem of optimizing a singular perturbed control system in the case when only dominant states are measured and information about them is incomplete has been considered. It is known that singular perturbed systems are characterized by certain unusual properties. When the systems are linear and stationary, their modes are divided into slow and fast modes. They have the order of O(1) and O(1/μ), correspondingly. Therefore in practice we must separate feedbacks for fast and slow states. The very important case when in the output-feedback only the slow (dominant) states are used is considered. Furthermore, information about them may be incomplete. The minimax approach to the solution of the multiple criteria linear-quadratic problem has investigated. It has shown that the leader terms of the asymptotic representations for output-feedback gains and functional coincide with the corresponding values for the reduced problem. The minimax approach may be helpful also for the investigation of many various properties of the singular perturbed control systems by mathematical means and by modeling at MATLAB.

References 1. Chow, J.H., Kokotovic, P.V.: A decomposition of near-optimum regulators for systems with slow and fast modes. IEEE Trans. Autom. Contr. 21(5), 701–705 (2015) 2. Demyanov, V.F., Malozemov, V.N.: Introduction to Minimax. Wiley (1974) 3. Kabakova, E.V., Myshkov, S.K.:Stabilization of singularly control perturbed system with incomplete information, St. Petersburg State University, Series 1 (Math., Mech., Astronomy), issue 2, pp. 27–33 (2003) 4. Kwakernaak, H., Sivan, R.: Linear Optimal Control Systems. Wiley (1972) 5. Levine, W.S., Athans, M.: On the determination of the optimal constant output-feedback gains for linear multivariable system. IEEE Trans. Autom. Control AC-15, 44–48 (1970) 6. Myshkov, S.K.: Linear control systems with incomplete information about states. In: Demyanov V.F. (ed.) Nonsmooth Problems of the Theory of Optimization and Control, pp. 248–272. University Press, Leningrad (1982) 7. Myshkov, S.K.: On the multiple linear-quadratic problem with incomplete information. In: Prepr. of IFAC Intern. Workshop Control Applications of Optimum, pp. 103–106, St. Petersburg (2000) 8. Myshkov, S.: Optimizing a singular perturbed control system with incomplete information. In: 20th International Workshop on Beam Dynamics and Optimization (BDO 2014), (2014). https://doi.org/10.1109/BDO.2014.6890056 9. Myshkov, S.: On the minimax approach in a singularly perturbed control problem. In: Constructive Nonsmooth Analysis and Relative Topics (Dedicated to the Memory of V.F. Demyanov (CNSA)), CNSA,: Proceedings (2017). https://doi.org/10.1109/CNSA.2017.7973993 10. Myshkov, S.K., Bure, V.M., Karelin, V.V., Polyakova, L.N.: Minimax control of singularly perturbed system. Appl. Math. Sci. 10(51) (2016). https://doi.org/10.12988/ams.2016.66185 11. Myshkov, S.K., Karelin, V.V.: Minimax control in the singularly perturbed linear-quadratic stabilization problem. In: International Conference on Stability and Control Processes in Memory of V. I. Zubov (SCP 2015). https://doi.org/10.1109/SCP.2015.7342130 12. Polyakova, L.N., Karelin, V.V.: On a continuous method for minimizing of nonsmooth functions. In: International Conference on Stability and Control Processes in Memory of V. I. Zubov (SCP 2015). https://doi.org/10.1109/SCP.2015.7342133

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13. Tamasyan, G.S., Chumakov, A.A.: Finding the distance between ellipsoids. J. Appl. Ind. Math. 8(3), 400–410 (2014) 14. Tikhonov, A.N.: Systems of differential equations containing small parameters at derivatives. Mat. Sbornik, 31(3)(73), 575–586 (1952) 15. Wonham, W.M.: On the separation theorem of stochastic control. SIAM J. Contr. 6(2), 312–326 (1968) 16. Yahagi, T.: Minimax output feedback regulators. J. Dyn. Sys., Meas. Control 98(3), 270–276 (1976)

Chapter 6

An Estimation Extension of Domain of Attraction for Second-Order Dynamic Systems Kirill V. Postnov

Abstract In this paper we offer a method for extension of estimation of domain of attraction (DA) for autonomous nonlinear dynamical system defined on a plane. We consider an estimate of domain of attraction obtained by Zubov’s method and describe the technique of estimation extension using the geometric properties of a vector field. A numerical example illustrates the feasibility of the proposed method.

6.1 Introduction At present, there are different methods for constructing an estimate of domain of attraction (DA). The construction of good estimate of domain of attraction is important for various applications, for example, for multi-program control problems [9, 10]. A large class of methods for estimating domain of attraction is based on Lyapunov function. The major trends of development of Lyapunov-function based methods are presented below: • constructing maximal Lyapunov function [12]; • for a given Lyapunov function constructing optimal estimation of DA [5]; • using convex optimization methods for construction the Lyapunov function that gives the highest estimation of DA [1, 2, 4, 6]. Among the Lyapunov function-based methods for estimating domain of attraction we can separate two groups: methods based on the Zubov Theorem [13] that give necessary and sufficient conditions for the certain set to be the domain of attraction and methods based on LaSalle’s invariance principle [7]. Those methods which do not explicitly employ Lyapunov function stand apart. For example, the trajectory reversing method [3], method of approximating the stability boundary by polytopes [8] and other. K. V. Postnov (B) St. Petersburg State University, Universitetskaya nab, 199034 7/9, St. Petersburg, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_6

49

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K. V. Postnov

Often, the obtained estimates of the stability region are quite conservative, i.e. an estimate of the domain of attraction is only a small part of the real stability region. Among the above methods, only the trajectory reversing method [3] can be used to expand the existing estimate. Therefore, development of methods for estimation extension of DA is of special interest.

6.2 Problem Statement We consider the following autonomous nonlinear dynamical system x˙ = F(x), x ∈ R n .

(6.1)

Assume the function F(x) satisfies the conditions for the existence and uniqueness of the solution at any point of R n . Further we also suppose that system (6.1) has an asymptotically stable zero solution. Definition 6.1 A set A of all initial conditions x0 such that x(t, x0 ) −→ 0 with t −→ +∞ will be called the domain of attraction of a zero solution of system (6.1). Definition 6.2 An estimate of stability region A is an open set E such that E ⊆ A. Let us have the estimation E of domain of attraction A for a zero solution of system (6.1) that has been obtained by some special method. Our challenge is to expand the set E so that it can better approximate domain A. In this paper we offer a technique for estimation extension of DA for autonomous nonlinear dynamical systems defined on a plane.

6.3 Method Description Consider an autonomous nonlinear dynamical system on a plane  x˙ = P(x, y) y˙ = Q(x, y).

(6.2)

Assume an implementation of the conditions for existence and uniqueness of the solution at any point of R n and an asymptotic stability of a zero solution. System (2) defines a vector field V on a plane. Let E be an estimation of DA obtained by any of methods. Choose two points A and B on the boundary so that a part of the  boundary between them marked G d V (x∗ ,y∗ )  = 0. Draw an arc G through these does not contain points (x∗ , y∗ ) with  dt (6.2)

6 An Estimation Extension of Domain of Attraction for Second …

51

Fig. 6.1 Domains E and E ∗

two points outside of the set E. Denote by E ∗ a region formed by the arcs G and G (Fig. 6.1). Proposition 6.1 In order that a domain E ∪ E ∗ be an estimation of DA it is sufficient that • trajectories of system (6.2) intersect the arc G from outside to inside while t −→ +∞. • there are no special points in the domain E ∗ . Proof From the contrary, suppose that proposition conditions are satisfied but the domain E ∪ E ∗ is not an estimation of DA. Then in the domain E ∗ there is at least one point z ∗ such that a semitrajectory L + (z ∗ ) does not tend to the point (0, 0). Let us consider this semitrajectory in more detail. It does not leave the domain E ∗ . Indeed, if L + (z ∗ ) intersected the curve G, the first condition of Proposition 6.1 would be contradicted. And if L + (z ∗ ) intersected the curve G, L + (z ∗ ) would tend to the point (0, 0) and this fact brokes our assumption. Therefore, the set of all ω-limit points of semitrajectory belongs to the domain E ∗ [14]. Note that the semitrajectory L + (z ∗ ) is not a point because it contradicts the second condition of Proposition 6.1. Assume that L + (z ∗ ) is a closed trajectory of the system. Then there is at least one special point inside L + (z ∗ ). The set of all ω-limit points coincides with the set of all points of the trajectory. But we established above that all of the ω-limit points lie in the domain E ∗ . Therefore, the special point belongs to E ∗ . We get a collision with the second condition of Proposition 6.1. We now consider the last possible trajectories type of autonomous dynamical systems for L + (z ∗ ), namely, the non-closed one. As shown above, the set of the ω-limit points of this trajectory locates in the domain E ∗ . The character of the set of the limit points for the semitrajectory entirely lying in a restricted part of a plane is expressed in three cases: • one equilibrium point; • one closed trajectory;

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K. V. Postnov

• some connected set consisting of whole trajectories a part of them are equilibrium points and the remaining are non-closed trajectories tending to these equilibrium points both under t −→ +∞ and t −→ −∞ [12]. Any of these cases presumes existence at least one special point in the domain E ∗ that contradicts the second condition so Proposition 6.1 is proved. To use Proposition 6.1 we describe a way of construction the domain E ∗ satisfying its conditions.   Let E = {(x, y) V (x, y) < μ} with a function V (x, y) defined and continuous in R 2 , V (0, 0) = 0 and V (x, y) > 0 when x 2 + y 2 = 0 be the estimation of DA obtained by the Zubov method [13]. Describe a computational algorithm for its extension. 1. Choose the points A and B on the boundary σ E so that the chord G connecting them does not pass through the point (0, 0) and a part of the boundary between this  = 0. Since the estimation points does not contain points (x∗ , y∗ ) with d V (xdt∗ ,y∗ )  (6.2)

obtained by the Zubov method is a convex set then all inside points of the chord have to belong to E. 2. The chord center is the point  C = (xC , yC ) =

x A + x B y A + yB , 2 2

 .

 ρ = (xC − x A )2 + (yC − y A )2 . We refer to [11] for calculation ρ. Suppose that the condition ρ < ρ is satisfied, otherwise it is necessary to choose other points A and B. Write down a function R(x, y) = (x − xC )2 + (y − yC )2 . 3. Calculate the time-derivative of the function R evaluated along the solutions of system (6.2) d R(x, y)   (6.2) dt 4. Consider the set of points    dR   =0 M = (x, y)  dt (6.2)

(6.3)

5. Determine inf Rρ (x, y) = ρ0 where (x, y) ∈ M and ρ < ρ ≤ ρ. The domain E ∪ E ρ0 is the estimation of DA. If set (6.3) is empty then the estimation of DA is the set E ∪ E ρ .

6 An Estimation Extension of Domain of Attraction for Second …

53

6.4 Numerical Examples In this section we present several numerical examples which illustrate the feasibility of the proposed method. We consider the following autonomous dynamical system  x˙ = −2x(1 − x 2 − y 2 ) (6.4) y˙ = −y(1 − x 2 − y 2 ). In Fig. 6.2 the estimate of domain of attraction V (x, y) = 41 x 2 + 21 y 2 < 41 obtained by the Zubov method is marked in black, and the domain of attraction is marked in red. √ √ 1. Choose the points A = (− 0.5, 0.5) and B = ( 0.5, 0.5) on the boundary of the DA estimate. 2. Connect √ the points A and B by the chord G and mark the point C = (0, 0.5). Here ρ = 0.5, ρ = 0.2. 

1 R(x, y) = x + y − 2 2

2

1 = x 2 + y2 − y + . 4

3. Calculate the time-derivative of the function R evaluated along the solutions of system (6.4)   d R  1 = −2 2x 2 + y 2 − y (1 − x 2 − y 2 ).  dt (6.4) 2 4. Find the set M from the following condition   d R  1 2 2 = −2 2x + y − y (1 − x 2 − y 2 ) = 0.  dt (6.4) 2

Fig. 6.2 Semicircles family for the point C

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K. V. Postnov

The first factor in the product is always positive, therefore 

 M = (x, y) x 2 + y 2 = 1 . 5. On the set M under the constraint 0.2 < ρ ≤

√

0.5, calculate

1 5 1 inf (x 2 + y 2 − y + ) = inf ( − y) = . 4 4 4 So all points of the domain E 0.25 belong to the DA. The extended estimate of DA is shown in Fig. 6.3. Now we consider the more complex example.  x˙ = −(1 + x)(x + y) (6.5) y˙ = 2x(1 + y). As before, in Fig. 6.4 the estimate of DA V (x, y) = 4x 2 + 2x y + 2y 2 < 0.97 obtained by the Zubov method is marked in black, and the DA is marked in red.

Fig. 6.3 An extended estimate of DA for system (6.4)

Fig. 6.4 DA and estimate of DA for system (6.5)

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55

Fig. 6.5 Semicircles family and the points of the set M

1. Choose the points A = (0.393, 0.26687) and B = (−0.443, 0.59775) on the boundary of the DA estimate. 2. Construct the chord G. Then, coordinates of the point C = (−0.025, 0.43231), and ρ = 0.4495491, ρ = 0.23561. R(x, y) = (x + 0.025)2 + (y − 0.43231)2 . 3. Calculate the time-derivative of the function R evaluated along the solutions of system (6.5) d R  = −2((x + 0.025)(1 + x)(x + y) − 2x(1 + y)(y − 0.43231)).  dt (6.5) 4. Find the set M from the condition d R  = −(x + 0.025)(1 + x)(x + y) + 2x(1 + y)(y − 0.43231) = 0.  dt (6.5) In this case, unlike system (6.4), we fail to find a simple analytical expression for the set M. We can solve this problem graphically. We construct the constraints specified by the condition −(x + 0.025)(1 + x)(x + y) + 2x(1 + y)(y − 0.43231) = 0 on a a a plane. In addition, we draw the straight line xx−x = yy−y and the semicircles family b −x a b −ya Rρ = ρ, ρ = 0.23561 < ρ ≤ ρ = 0.4495491. Figure 6.5 shows that the set  is empty under the given restrictions on ρ. Hence, all points of the domain E ρ belong to DA. The extended estimate of DA is shown in Fig. 6.6.

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Fig. 6.6 An extended estimate of DA for system (6.5)

6.5 Conclusion In this study we show that the existing estimate of domain of attraction E can be extended to the domain E ∪ E ∗ if the conditions of Proposition 6.1 are satisfied. The described conditions are sufficient, because there are examples where the first condition is not satisfied, but the domain E ∪ E ∗ is still an estimate of DA. We offer the technique for construction domain E ∗ satisfying the conditions of Proposition 6.1. The proposed method can be applied many times. Two numerical examples illustrate the feasibility of the proposed method.

References 1. Chesi, G.: Estimating the domain of attraction for non-polynomial systems via lmi optimizations. Automatica 45(6), 1536–1541 (2009) 2. Chesi, G., Garulli, A., Tesi, A., Vicino, A.: Lmi-based computation of optimal quadratic lyapunov functions for odd polynomial systems. Int. J. Robust a Nonlinear Control: IFAC-Affil. J. 15(1), 35–49 (2005) 3. Genesio, R., Tartaglia, M., Vicino, A.: On the estimation of asymptotic stability regions: state of the art and new proposals. IEEE Trans. Autom. Control 30(8), 747–755 (1985) 4. Hachicho, O.: A novel lmi-based optimization algorithm for the guaranteed estimation of the domain of attraction using rational lyapunov functions. J. Frankl. Inst. 344(5), 535 – 552 (2007). https://doi.org/10.1016/j.jfranklin.2006.02.032. http://www.sciencedirect.com/ science/article/pii/S0016003206000524. Modeling, Simulation and Applied Optimization Part II 5. Chiang, H-D., Fekih-Ahmed, L.: Quasi-stability regions of nonlinear dynamical systems: theory. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 43(8), 627–635 (1996) 6. Johansen, T.A.: Computation of lyapunov functions for smooth nonlinear systems using convex optimization. Automatica 36(11), 1617–1626 (2000). https://doi.org/10.1016/S00051098(00)00088-1 7. LaSalle, J.: Some extensions of liapunov’s second method. IRE Trans. Circuit Theory 7(4), 520–527 (1960) 8. Psiaki, M.L., Luh, Y.P.: Nonlinear system stability boundary approximation by polytopes in state space. Int. J. Control 57(1), 197–224 (1993)

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9. Smirnov, N.V.: A complete-order hybrid identifier for multiprogrammed stabilization. Autom. Remote. Control. 67(7), 1051–1061 (2006). https://doi.org/10.1134/S0005117906070046 10. Smirnov, N.V., Smirnova, T.Y.: The synthesis of multi-programme controls in bilinear systems. J. Appl. Math. Mech. 64(6), 891–894 (2000). https://doi.org/10.1016/S0021-8928(00)001192 11. Uteshev, A.Y., Goncharova, M.V.: Point-to-ellipse and point-to-ellipsoid distance equation analysis. J. Comput. Appl. Math. 328, 232–251 (2018) 12. Vannelli, A., Vidyasagar, M.: Maximal lyapunov functions and domains of attraction for autonomous nonlinear systems. Automatica 21(1), 69–80 (1985) 13. Zubov, V.I.: The methods of lyapunov and their applications (1957). In Russian 14. Zubov, V.I.: Stability of motion. Vysshaya Shkola, Moscow (1973)

Chapter 7

Stability of Weak Solutions of Parabolic Systems with Distributed Parameters in a Network-Like Domain Vyacheslav V. Provotorov and Aleksei P. Zhabko

Abstract The work devoted to the actual question of application: the stability of differential systems with distributed parameters in the network-like domain (such as the finite set of limited domains, jointed with each other by parts of their boundaries by the type of geometric graph; in applications it’s network carriers of continuous mediums, for example, fluids, gases or their mixtures). When selecting the spaces of admissible solutions, the authors focused on the class of summable functions, which are highly adequate to describe the property of multiphase of transferable continuous mediums. In finding the conditions of stability, approaches are used, having their origins the classic methods of A. M. Lyapunov analysis of the stability of differential systems, developed by the works of V. I. Zubov. The analysis uses the completely continuous and symmetry of the elliptical operator generated by the elliptical part of the differential equation. Namely, its eigenvalues are real positive numbers, forming a sequence with a limit point on infinity, generalized eigenfunctions form a basis in the space of the operator’s task, a weak solution of the differential system is presented in the form of a exponential series of generalized eigenfunctions, the coefficients of which are exponentials with negative indicators. The stability of a weak solution of differential system is a direct consequence of this notation.

7.1 Notation, Concepts, and Basic Statement In the Euclidean space Rn , n ≥ 2, let’s look at a network-like bounded domain I, comprised of N domains Ik (k = 1, N ), pairwise united by means of M nodal N M    place ω j ( j = 1, M, M < N ): I = Iˆ ω, ˆ where Iˆ = I k , ωˆ = ω j , morek=1

j=1

V. V. Provotorov (B) Voronezh State University, 1, Universitetskaya pl., Voronezh 394006, Russia e-mail: [email protected] A. P. Zhabko St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_7

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   over I k I l = Ø (k = l), ω j ω i = Ø ( j = i), I k ω j = Ø [1, 2]. Domains I k in nodal place ω j share common boundaries in the form of adjoining surfaces S j (meas S j > 0). At each nodal place ω j the adjoining surface S j separating to her m j mj  the domains I k0 and I ki (1 ≤ i ≤ m j ≤ N − 1) has a representation S j = Sj i i=1

(meas S j i > 0). In addition S j and S j i are parts of boundary ∂I k0 and ∂I ki of domains I k0 and I ki , respectively; S j i is two-sided surface for each j, i: S −j i is interior surface, S +j i is exterior surface. Thus, the nodal place ω j is determined by the adjoining surface S j , for which S j i are also the adjoining surface I k0 to I ki , i = 1, m j . The boundary ∂I of the domain I is called the union of the boundary ∂Ik of domain Ik (k = 1, N ), which does not include the adjoining surface of all node places: N M   ∂I = ∂Ik \ S j . The domain I has a network-like structure similar to that of k=1

j=1

the geometric graph [3], each domain Ik adjoins to one or two node places and has one or more of the surface adjoining other domains (to compare with the structure of the graph: each edge of the graph has two endpoints, of which one or both are conjugation nodes with the other edges). We use customary Lebesque spaces L q (U ), q = 1, 2, and the Sobolev space W 21 (U ), where U is a bounded domain in Rn . For each fixed k (1 ≤ k ≤ N ) denote 1 (Ik ) the closure in W 21 (Ik ) a set infinitely differentiable on Ik functhrough W2,0 tions equal to zero on ∂Ik ⊂ ∂I. Let a (I) is a set of functions u : I → R1 , u|Ik ∈ 1 W2,0 (Ik ) for each k = 1, 2, ..., N , u satisfies the condition of agreement

 S j ⊂∂Ik0

u|S+ = u|S− , i = 1, m j , j i j i mj   ∂u(x) u(x) a(x) ∂n j d x + a(x) ∂n d x = 0, ji

(7.1)

i=1 S j i ⊂∂Iki

for each node place ω j on surfaces S j =

mj 

S j i , j = 1, M; here vectors n j and n j i

i=1

are outer normals to S j and S j i , respectively, a(x) ∈ L 2 (I) and 0 < a∗ ≤ a(x) ≤ a ∗ < ∞.

(7.2)

Definition 7.1 Closing the set a (I) in norm 1 1 1/2

u

, I = ((u, u)I )   N  n   ∂u(x) ∂v(x) 1 u(x)v(x) + d x, (u, v)I = ∂ xκ ∂ xκ k=1 Ik

(7.3)

κ=1

 01 (a, I). let’s call space W We denote by V2 (IT ) (IT = I × (0, T ), T < ∞) the set of all functions u(x, t) ∈ W21,0 (IT ) (W 1,0 (IT ) as the space of functions u(x, t) ∈ L 2 (IT ) that have the

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generalized first derivative with respect to x belonging to L 2 (IT ), the norm in W 1,0 (IT ) defined by u W 1,0 (IT ) = ( u 2L 2 (IT ) + u x 2L 2 (IT ) )1/2 ) with finite norm

u 2,IT ≡ max u(·, t) L 2 (I) + u x L 2 (IT ) . Let the next a (IT ) is the set of func0≤t≤T

tions u(x, t) ∈ V2 (IT ), whose traces are defined in sections of the domain IT the  01 (a, I). plane t = t0 (t0 ∈ [0, T ]) as a function u(x, t0 ) of class W Definition 7.2 The closing of the set a (IT ) in norm (7.3) relabel V 1,0 (a, IT ); it is clear that V 1,0 (a, IT ) ⊂ W 21,0 (IT ). In space V 1,0 (a, IT ) consider the parabolic equation ∂u(x,t) ∂t

−

∂ ∂ xι



a(x) ∂u(x,t) + b(x)u(x, t) = f (x, t), ∂ xκ

(7.4)

represents a system of differential equations with distributed parameters on I; f (x, t) ∈ L 2,1 (IT ) (L 2,1 (IT ) is the space of summable on IT functions with norm T  1

u L 2,1 (IT ) = ( u 2 (x, t)d x) 2 dt); a(x), b(x) ∈ L 2 (I), a(x) satisfy a ratios (2) and 0 I



n  ∂u(x,t) ∂ ≡ a(x) . |b(x)| ≤ β < ∞, x ∈ I; ∂∂xι a(x) ∂u(x,t) ∂ xκ ∂ xι ∂ xκ ι,κ=1

The state u(x, t) (x, t ∈ IT ) of the system (7.4) in the domain IT is determined by a weak solution u(x, t) of Equation (7.4), satisfying the initial and boundary conditions u |t=0 = ϕ(x), ϕ(x) ∈ L 2 (I). (7.5) Definition 7.3 A weak solution of the initial-boundary value problem (7.4), (7.5) is the function u(x, t) ∈ V 1,0 (a, IT ), that satisfies an integral identity  I

 u(x, t)η(x, t)d x − u(x, t) ∂η(x,t) d xdt + t (u, η) = ∂t I t   = ϕ(x)η(x, 0)d x + f (x, t)η(x, t)d xdt I

It

under any t ∈ [0, T ] and for any function η(x, t) ∈ W 1 (a, IT ) (W 1 (a, IT ) is the closing in norm u = ( u 2L 2 (I) + u t 2L 2 (I) + u x 2L 2 (I) )1/2 the set of differentiable on IT functions u(x, t) satisfy a ratios (1) under any t ∈ (0, T )); here It = I × (0, t),

t (y, η) is bilinear form defined by the ratio of

t (u, η) =

 ∂η(x,t) a(x) ∂u(x,t) + b(x)u(x, t)η(x, t) d xdt, ∂ xκ ∂ xι

It

∂η(x,t) a(x) ∂u(x,t) ∂ xκ ∂ xι

≡

n  ι,κ=1

∂η(x,t) a(x) ∂u(x,t) , t ∈ (0, T ]. ∂ xκ ∂ xι

We will use the following basic statements, the full evidence of which is presented in the works [1, 2].

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 01 (a, I) gener+ b(x)v(x, t) is operator in space W Let  v = − ∂∂xι a(x) ∂v(x) ∂ xκ ated by differential form T (u, η). It’s easy to show that at b(x) ≥ 0, x ∈ I, the spectrum of the operator  consists of a denumerable set {λi } of positive eigenvalues having a limit point on ∞. A set of generalized eigenfunctions {u i (x)} are a basis  01 (a, I) orthonormalized in L 2 (I). in L 2 (I) and W Theorem 7.1 Let a(x) ∈ L 2 (I) satisfy a ratios (7.2) and b(x) ∈ L 2 (I), |b(x)| ≤ β < ∞, x ∈ I. For any f (x) ∈ L 2,1 (IT ), ϕ(x) ∈ L 2 (I) and for any 0 < T < ∞ the initial-boundary value problem (7.4), (7.5) is weak solvability in space V 1,0 (a, IT ). The solution of problem (7.4), (7.5) has the following presentation y(x, t) =

∞ 

  t −λi t −λi (t−τ ) ϕi e + f i (τ )e dτ u i (x),

i=1

where ϕ(x) =

∞ 

ϕi u i (x), ϕi =

i=1

f (x, t) =

∞ 

f i (t)u i (x),

i=1

(7.6)

0

f i (t) =

 I

 I

ϕ(x)u i (x)d x,

f (x, t)u i (x)d x, t ∈ [0, T ].

7.2 Main Result Suppose 0 ≤ b(x) ≤ β for x ∈ I then the eigenvalues λi (i ≥ 1) are positive. Consider the system (7.4) on the set I∞ = I × (0, ∞). Relabel It0 ,t = I × (t0 , t), ∂It0 ,t = ∂I × (t0 , t) (0 < t0 < t < ∞), It0 ,∞ = I × (t0 , ∞), ∂It0 ,∞ = ∂I × (t0 , ∞); it is clear that It0 ,t ⊂ It . Let f (x, t) ∈ L 2,1 (I∞ ). Let the state of system (7.4) describes the function u(x, t) ∈ V 1,0 (a, It0 ,∞ ), what is a weak solution of equation (7.1) in the domain It0 ,∞ with initial and boundary conditions u |t=t0 = ϕ(x), x ∈ I, u |x∈∂It0 ,∞ = 0, (7.7) and function u(x, t) ∈ V 1,0 (a, t0 ,∞ ) is a weak solution of equation (7.4) in the domain It0 ,∞ with initial and boundary conditions u |t=t0 = ϕ(x), x ∈ I, u |x∈∂It0 ,∞ = 0

(7.8)

(initial-boundary value problem (7.4), (7.7) differs from problem (7.4), (7.8) the fact that the function ϕ(x) of first ratio (7.7) replaced by a different function ϕ(x)). The state u(x, t) of system (7.4) call perturbed, and u(x, t) is non-perturbed. From the presentation (7.6) of weak problem solving (7.4), (7.5) implies that states u(x, t), u(x, t) are defined in the domain It0 ,∞ , satisfy the corresponding initial and boundary conditions (7.7), (7.8) and belong to the space V 1,0 (a, It0 ,∞ ) for f (x, t) ∈ L 2,1 (I∞ ).

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Definition 7.4 The non-perturbed system (7.4) is stability, if for any t0 > 0 and  > 0 executed such δ(t0 , ) > 0 that under ϕ − ϕ L 2 (I) < δ(t0 , ) is

u(·, t) − u(·, t) W 1 (a,I) <  under t ≥ t0 , where u(x, t) is the perturbed status of system (7.4). Theorem 7.2 Let a(x) ∈ L 2 (I) satisfy a ratios (7.2) and b(x) ∈ L 2 (I), 0 ≤ b(x) ≤ β, x ∈ I, then the non-perturbed system (7.4) in the domain I∞ is stability. Proof By virtue of the linearity of equation (7.4) function θ (x, t) = u(x, t) − u(x, t) is element of space V 1,0 (a, It0 ,∞ ) and a weak solution of initial-boundary value problem for homogeneous equation (7.4) ( f = 0), satisfies the initial and boundary conditions θ |t=t0 = ϕ(x), x ∈ I, θ |x∈∂It0 ,∞ = 0, (7.9) where ϕ(x) = ϕ(x) − ϕ(x). The initial-boundary value problem (7.4), (7.9) is uniquely weakly solvable, its solution has form (see (7.6)) θ (x, t) =

∞ 

ϕi e−λi t u i (x), ϕi = (ϕ, u i ),

i=1

 and is limit weakly converging sequence θ N N ≥1 of approximations θ N (x, t) =

N 

ϕi e−λi t u i (x),

i=1

moreover

θ N 2,It ≤

N  i=1

ϕi2 e−2λi t , N = 1, 2, ... .

Passage to the limit in the last inequality under N → ∞ we come to the estimate

θ 2,It ≤ C ∗ ϕ L 2 (I) (e−2λi t < 1, i = 1, 2, ...) for any t ∈ [t0 , ∞); C ∗ is constant, not dependent on t. The latter means that (see (7.1))



θ (·, t) W 1 (a,I) ≤ C ∗ ϕ L 2 (I) . Fix  > 0, take δ =

 , C∗

(7.10)

then from (7.10) and

ϕ L 2 (I) = ϕ − ϕ L 2 (I) < δ

it should be inequality θ (·, t) W 1 (a,I) = u(·, t) − u(·, t) W 1 (a,I) <  for any t >  t0 .

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7.3 Conclusion The approach presented by the statement of Theorem 7.2 may be used to obtain conditions of stability (asymptotic stability) of weak solution of initial-boundary value problem (7.4), (7.5). The same approach applies to both tasks, space variable x which has a dimension of large 1 (x ∈ Rn , n ≥ 2) and for many-dimensional functions that describe the state of researched system [2, 3]. The results are fundamental in tasks of optimal control and stabilization of differential systems [4].

References 1. Provotorov, V.V.: Solvability of nonlinear initial boundary value problem with distributed parameters in a netlike domain. In: Proceedings of 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the Memory of V.F. Demyanov) (CNSA), 7974008 (2017) 2. Provotorov, V.V.: Unique weak solvability of a hyperbolic systems with distributed parameters on the graph. In: Proceedings of 14th International Conference “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference), vol. 14, pp. 1–2 (STAB). Institute of Electrical and Electronics Engineers (2018) 3. Baranovskii, E.S.: Mixed initial-boundary value problem for equations of motion of KelvinVoigt fluids. Comput. Math. Math. Phys. 56(7), 1363–1371 (2016). https://doi.org/10.1134/ S0965542516070058 4. Aleksandrov, A.Y., Zhan, J.: Investigation of ultimate boundedness conditions of mechanical systems via decomposition. Vestnik of Saint Petersburg University. Appl. Math. Comput. Sci. Control Process. 15(2), 173–186 (2019). (In Russian). https://doi.org/10.21638/11702/spbu10. 2019.202

Chapter 8

An Algorithm for Solving Local Boundary Value Problems with Perturbations and Delayed Control Alexander N. Kvitko, Alexey S. Eremin, and Oksana S. Firyulina

Abstract In this paper, a class of controllable nonlinear stationary systems of ordinary differential equations with account of external perturbations is studied. The control function has a delay and is norm-bounded. A control transferring the system from a given initial state to an arbitrary neighborhood of the origin is constructed. The algorithm has both numerical and analytical stages and is easy to implement. A Kalman-type constructive sufficient condition under which the transfer is possible is presented. The algorithm efficiency is demonstrated with solving a robot-manipulator control problem.

8.1 Introduction The mathematical theory of control considers, among the other, the problems of construction of controls (usually chosen from a certain class of functions) which provide the transfer of an object from an initial state to a certain final state in finite time. The object behavior is described with a system of ordinary differential equations (ODEs). Those problems lead to boundary value problems (BVPs) for controlled systems of differential equations. Particularly important are problems of stabilization, which can be considered as BVPs for controlled systems over the infinite time interval. Quite a lot of papers are devoted to BVPs for both linear and nonlinear systems with continuous, piecewise continuous, piecewise differentiable, measurable or pulse controls. It should be noticed that when computers and navigation systems are used to form a control signal, certain delays arise, since it takes some time for the information about the object state and the control action to arrive and to be processed. Consequently, A. N. Kvitko (B) · A. S. Eremin · O. S. Firyulina St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. S. Eremin e-mail: [email protected] O. S. Firyulina e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_8

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more complicated BVPs with time delays in the right-hand side of the system of ordinary differential equations describing the behavior of the control object attract great attention of the researchers. Since the late 1960s, a lot of papers devoted to the study of BVPs for linear stationary, nonstationary linear, bilinear, semilinear, and nonlinear systems of ODEs with delays in control or in both state and control were published. Recently the problem of controllability has been considered for various systems with delays (e.g. [2, 3, 5, 6, 11–15]). The study of BVPs over finite time intervals for such types of systems includes finding necessary and sufficient conditions of controllability, studying and estimating the attainability domain (see [16, 17]); and developing exact or approximate methods to construct control functions, under which the solution of an ODE system connects the given points in the phase space (see [4, 17]). Nowadays, BVPs are thoroughly studied for linear and some special types of nonlinear controllable systems. However, the theory of solving BVPs for general nonlinear controllable systems in the class of delayed controls is not yet sufficiently developed and such task meets a lot of difficulties. Moreover, among numerous studies BVPs for controlled systems of delay differential equations, only few are devoted to control functions construction. The aim of this paper is to develop an algorithm for construction of a normbounded control function with a delay transferring a nonlinear stationary system of ODEs into an arbitrary small neighborhood of the origin from an initial point in the presence of external perturbations. We achieve it by reducing the original problem to a problem of stabilization of an auxiliary linear non-stationary system and then solving an initial value problem for an auxiliary system of ODEs. The novelty of the presented result is that an easily verifiable constructive local controllability condition of Kalman type is given for a wider class of nonlinear stationary systems: perturbations and delays in the control are considered. For such systems an easily implementable and numerically stable algorithm is developed for constructing the required control and the corresponding solution in phase coordinates. We present the estimates on the initial state, the magnitude of perturbation and the maximal delay, for which the solution existence is guaranteed under the given control restrictions. The algorithm is easy to implement since its most time-consuming part (stabilization of an auxiliary system, construction of an auxiliary control and return to the original variables) is made analytically with a symbolic algebra software. In the present paper, the method developed and applied in [7–10] to various systems is modified for systems with delayed control.

8.2 Problem Statement Consider a nonlinear stationary controlled system of ordinary differential equations with a delay h > 0 in the control and a constant perturbation F = (F1 , . . . , Fn )T : x(t) ˙ = f (x(t), u(t − h)) + F, t ∈ [0, 1],

(8.1)

8 An Algorithm for Solving Local Boundary Value Problems with Perturbations …

67

where x = (x1 , . . . , xn )T , and the control function u = (u 1 , . . . , u r )T such that r ≤ n. The right-hand side function f = ( f 1 , . . . , f n )T is sufficiently smooth f ∈ C 4n (R n × R r ; R n ),

(8.2)

f (0, 0) = 0.

(8.3)

and satisfies We also consider the matrix S = (B, AB, . . . , An−1 B) with  A=

 ∂f (0, 0) , ∂x

 B=

 ∂f (0, 0) . ∂u

to satisfy rank S = n.

(8.4)

The control is considered to be norm-bounded: u < N ,

N > 0,

N = const.

Problem 8.1 Find a control u(t) ∈ C 1 ([0, 1]) and a function x(t) ∈ C 1 ([0, 1]) satisfying (8.1) and the following conditions: • Initial condition:

x(0) = x, ¯ x¯ = (x¯1 , . . . , x¯n )T ,

(8.5)

• Stabilization condition: for any small ε > 0 there exists some moment t¯ satisfying 1 − t¯ < ε that x(t¯) ≤ ε, • The control starts at the initial time: u(t) ≡ 0, t ∈ [−h, 0].

(8.6)

The main result of the paper is Theorem 8.1 Let the right-hand side of (8.1) satisfy the conditions (8.2)–(8.4). Then for any ε > 0 there exist constants ε01 > 0, ε02 > 0, and ε03 > 0 such that for ¯ < ε01 , any perturbation F ∈ R n : F < ε02 and any any initial point x¯ ∈ R n : x 3 delay h ∈ [0, ε0 ) there exists a solution of the Problem 8.1, which can be obtained by solving a stabilization problem for a linear nonstationary system with exponential coefficients followed by solving an auxiliary initial problem for a system of ordinary differential equations. The dimensions of both systems are n + r .

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8.3 Theorem Proof Overview In [7, 9], it was shown that in case of h = 0, even stronger theorem can be proved. There was given an algorithm of constructing u(t) and x(t) such that x(1) = 0. Here, we first consider the system (8.1) without the delay h = 0, and then estimate for which h values the system can be transferred to ε-neighborhood of 0. We make a change of time t to τ according to t = 1 − e−ατ ,

(8.7)

dy = αe−ατ f (y, w) + αe−ατ F, τ ∈ [0, +∞), dτ

(8.8)

so the system (8.1) takes form

where y(τ ) = (y1 , . . . , yn )T = x(t (τ )) and w(τ ) = u(t (τ )). The function f in (8.8) is then expanded into Taylor series at the point (0, 0), and 4n shifts of phase coordinates are made. The goal of the shifts is to make the norms of those terms in the right-hand side of (8.8), that don’t explicitly contain powers of y and w components, satisfy the estimate O(e−4nατ F) as τ → ∞ and F → 0. For details, see [7, 9]. After the shifts, the system takes form dz = Pz + Qw + R(z, w, τ ), dτ

(8.9)

where z(τ ) is a vector of new phase coordinates, w(τ ) = u(t (τ )) is the control in τ -domain, P is an [n × n]-matrix and Q is an [n × r ]-matrix, which are dependent on τ , α and F, and the initial condition z(0) depends in x¯ and F as well. The vector R contains the terms that can be shown to satisfy R → 0 as τ → ∞, if z and w are bounded (which is the case). Then we introduce an auxiliary control function v(τ ) such that dw = v, v = (v1 , . . . , vr )T , dτ

(8.10)

and we set w(0) = 0. Then we can consider the linear part of (8.9) together with (8.10) to be a controlled system ⎧ dz ⎪ = Pz + Qw, ⎨ dτ ⎪ ⎩ dw = v. dτ

(8.11)

8 An Algorithm for Solving Local Boundary Value Problems with Perturbations …

69

It is shown that under the condition (4), there exists ε2 > 0 such that if F < ε2 there exists a control function v(τ ): v(τ ) = M(τ )

 z , w

(8.12)

guaranteeing the exponential decay of the fundamental matrix of the system (8.11) closed with it, where M(τ ) is a matrix of size [r × n + r ]. After solving the stabilization problem for (8.11), we find the solution of the initial value problem (8.9), (8.10) closed by the auxiliary control function (8.12) at certain interval [0, τ¯ ], where τ¯ depends on ε. This explicitly gives v as a function of τ via (8.12), and expressing τ from (8.7), we get v¯ (t) = v(τ (t)). Finally, the initial problem x˙ = f (x, u(t − h)) + F, u˙ =

1 (1 − t)−1 v¯ (t) α

(8.13)

with initial data (8.5), (8.6) is solved over [0, t¯], t¯ = 1 − e−ατ¯ , so the solution of the Problem 8.1 is constructed. The final part of the Theorem 8.1 proof, showing that for any ε > 0 there exist non-zero delays h providing the transfer of the system to ε-neighborhood of 0 by some t¯ ∈ (1 − ε, 1], hasn’t been presented in previous works. It’s idea is to substitute w(τ ¯ ) = u(t (τ ) − h) instead of w(τ ) = u(t (τ )) into the right-hand side of (8.9) and study the behavior of such system closed with the same control (8.12). However, we don’t include it to the present paper due to the paper size restrictions.

8.4 Test Example As an example of the presented algorithm application, consider the problem of controlling a manipulator when transporting a load to a given point. According to [1], the system of equations describing the motion of a manipulator in presence of perturbations has form

x˙1 = x2 , (8.14) x˙2 = −a2 sin x1 − a1 x2 + u(t − h) + F, where x1 is the manipulator’s deviation angle from the x2 is the deflec vertical axis, −1 M −1 m m , a = gL + , m 1 = m 0 + M3 , tion angle change rate, a1 = γ L −2 m −1 2 0 1 1 2 M is the manipulator’s mass, L is its length, g is the free-fall acceleration, γ is the viscous friction coefficient, m 0 is the load mass, F is the perturbation, x = (x1 , x2 )T , the control u is scalar. We consider the boundary conditions

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x(0) = x, ¯ ¯ x(t ) ≤ ε,

u(t) ≡ 0 for t ∈ [−h, 0], 1 − t¯ ≤ ε.

(8.15)

In this case, the system (8.9), (8.10) takes form dz 1 1 = αe−ατ z 2 − αa1 Fe−4ατ , dτ 6  dz 2 1 1 −2ατ −ατ −3ατ + Fe = − αe a2 sin z 1 + Fa1 e dτ 6 2 1 − αe−ατ a1 z 2 + αe−ατ w + αe−4ατ a12 F, 6 dw = v. dτ

(8.16)

The matrix ⎛ −ατ ⎞T −e a1 a2 α + 3a2 α + 6a2 − α12 e2ατ (8α 3 + 24α 2 + 22α + 6) M(τ ) = ⎝ −a12 α + e−ατ a2 α + 3a1 α + 6a1 − α1 eατ (7α 2 + 18α + 11) ⎠ . αe−ατ a1 − 3α − 6 After the substitution of v = M · (z 1 , z 2 , w)T to (8.16), we get the initial value problem with z 1 (0) = x¯1 − 21 F − 16 Fa1 , z 2 (0) = x¯2 + F − 13 Fa1 , w(0) = 0. Its solution gives the explicit expression of v(τ ). Finally, we solve the initial value problem for the system (8.13) ⎧ x˙1 = x2 , ⎪ ⎪ ⎪ ⎨ x˙ = −a sin x − a x + u(t − h) + F, 2 2 1 1 2  1 ⎪ ln(1 − t) v − ⎪ ⎪ α ⎩ u˙ = α(1 − t) over the interval [0, t¯] with the initial data x1 (0) = x¯1 , x2 (0) = x¯2 , u(t) ≡ 0 for t ∈ [−h, 0]. The found functions x1 (t), x2 (t) and u(t − h) are the solution of the Problem 8.1 for the system (8.14), (8.15). Figure 8.1 shows the solution for x¯1 = −0.5, x¯2 = −0.8, γ = 0.1, L = 10, M = 20, m 0 = 1, F = 0.1. The value ε = 0.05. We choose α = 0.25, and for h = 0.01 s the problem is solved with t¯ = 0.9.

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20 x 1(t )

15

x 2(t ) u( t )

10

5

0

t

-5

-10

0

0.2

0.4

0.6

0.8

1

Fig. 8.1 The solution of the problem (8.14), (8.15)

8.5 Conclusion In the current paper, we consider a class of nonlinear stationary controlled systems. This makes possible to take into account the influence of all acting forces, external perturbations, and the delay of the control signal in the mathematical model of a controlled object. The proposed method of reducing the initial problem to the problem of an auxiliary system stabilization helps to find the solution of the original problem considering the errors in the initial data and the system parameters, and computational errors. The results of numerical simulation show that the proposed algorithm can be used for the construction of control systems for various technical objects described by complex systems of differential equations.

References 1. Afanas’ev, V.N., Kolmanovskii, V., Nosov, V.R.: Mathematical Theory of Control Systems Design, Mathematics and Its Applications, vol. 341. Springer Netherlands (1996). https://doi. org/10.1007/978-94-017-2203-2 2. Arora, U., Sukavanam, N.: Approximate controllability of impulsive semilinear stochastic system with delay in state. Stoch. Anal. Appl. 34(6), 1111–1123 (2016). https://doi.org/10. 1080/07362994.2016.1207547

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3. Balachandran, K., Balasubramaniam, P.: A note on controllability of nonlinear volterra integrodifferential systems. Kybernetika 28(4), 284–291 (1992) 4. Golev, A., Hristova, S., Nenov, S.: Monotone-iterative method for solving antiperiodic nonlinear boundary value problems for generalized delay difference equations with maxima. Abstr. Appl. Anal. (2013). https://doi.org/10.1155/2013/571954.Art.ID571954 5. Karthikeyan, S., Sathya, M., Balachandran, K.: Controllability of semilinear stochastic delay systems with distributed delays in control. Math. Control Signals Syst. 29(4), 17 (2017). https:// doi.org/10.1007/s00498-017-0206-9 6. Klamka, J.: Stochastic controllability of linear systems with state delays. Int. J. Appl. Math. Comp. Sci. 17(1), 5–13 (2007). https://doi.org/10.2478/v10006-007-0001-8 7. Kvitko, A., Yakusheva, D.: On one boundary problem for nonlinear stationary controlled system. Int. J. Control 92(4), 828–839 (2019). https://doi.org/10.1080/00207179.2017.1370727 8. Kvitko, A.N., Firyulina, O.S., Eremin, A.S.: Algorithm of the resolving of a boundary-value problem for a nonlinear controlled system and its numerical modeling. Vestn. St. Petersburg U., Math. 50(4), 372–383 (2017). https://doi.org/10.3103/S1063454117040124 9. Kvitko, A.N., Firyulina, O.S., Eremin, A.S.: Solving boundary value problem for a nonlinear stationary controllable system with synthesizing control. Math. Probl. Eng. (2017). https://doi. org/10.3103/S1063454117040124.Art.ID8529760 10. Kvitko, A.N., Maksina, A.M., Chistyakov, S.V.: On a method for solving a local boundary problem for a nonlinear stationary system with perturbations in the class of piecewise constant controls. Int. J. Robust Nonlin. Control 29, 4515–4536 (2019). https://doi.org/10.1002/rnc. 4644 11. Muni, V.S., Govindaraj, V., George, R.K.: Controllability of fractional order semilinear systems with a delay in control. Indian J. Math. 60(2), 311–335 (2018) 12. Olenchikov, D.M.: Global controllability of sampled-data bilinear time-delay systems. J. Appl. Math. Mech. 68(4), 537–544 (2004). https://doi.org/10.1016/j.jappmathmech.2004.07.006 13. Shukla, A., Sukavanam, N., Pandey, D.N.: Complete controllability of semilinear stochastic systems with delay in both state and control. Math. Rep. 18(2), 247–259 (2016) 14. Shukla, A., Sukavanam, N., Pandey, D.N.: Controllability of semilinear stochastic control system with finite delay. IMA J. Math. Control. Inf. 35(2), 427–449 (2018). https://doi.org/10. 1093/imamci/dnw059 15. Sikora, B., Klamka, J.: Constrained controllability of fractional linear systems with delays in control. Syst. Cont. Lett. 106, 9–15 (2017). https://doi.org/10.1016/j.sysconle.2017.04.013 16. Thuan, M.V., Trinh, H., Huong, D.C.: Reachable sets bounding for switched systems with time-varying delay and bounded disturbances. Int. J. Syst. Sci. 48(3), 494–504 (2017). https:// doi.org/10.1080/00207721.2016.1186248 17. Zubov, V.I.: Lectures in Control Theory [In Russian]. Nauka, Moscow (1975)

Chapter 9

On the Stability of Linear Time-Delay Systems with Arbitrary Delays Irina Alexandrova and Sabine Mondié

Abstract The construction of Lyapunov–Krasovskii functionals with prescribed derivative for linear time-invariant time-delay systems is based on the Lyapunov matrix. This matrix is usually computed with the help of the so-called semianalytic procedure, which is to solve a system of ODE with the boundary conditions. The problem lies in the fact that the dimension of this ODE system may be large in some cases. Moreover, the semianalytic procedure is not applicable in the case when the delays in the system are incommensurate. This is the case addressed in the contribution. We suggest the stability conditions for systems with incommensurate delays which do not require computation of the Lyapunov matrix in this difficult case. The Lyapunov matrix of another system with commensurate delays, for which the semianalytic procedure is applicable, is involved instead. The implementation issues of the method are discussed, and the illustrative example is given.

9.1 Introduction This paper is concerned with the stability problem for linear time-invariant time-delay systems in the framework of the Lyapunov–Krasovskii functionals with prescribed derivative [7]. The main feature of these functionals is that they are adapted to the analysis of a certain system, resulting in the necessary and sufficient stability conditions. The Lyapunov matrix is at the forefront in construction of the functionals with prescribed derivative. Moreover, the possibility to express the necessary and sufficient stability conditions exclusively in terms of the Lyapunov matrix was shown recently [2–4]. The problem lies in the fact that there are no effective techniques for computation of this matrix in the case when the delays are incommensurate, or even if I. Alexandrova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St., Petersburg 199034, Russia e-mail: [email protected] S. Mondié Departamento de Control Automático, Cinvestav, IPN, México D.F., Mexico e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_9

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they are commensurate but the representation of a system in the form where all delays are multiples of the basic one requires large number of delays. The semianalytic procedure [7] which is usually applied fails to be effective or is not available at all in these cases. The different approximating techniques either are not supplied with a constructive estimation of the approximation error (see the piecewise linear approximation scheme in [7] or the polynomial scheme [6]) or require an additional assumption of the exponential stability (see [1]). The aim of this work is to modify the necessary stability conditions presented in [2] so that they were applicable to the analysis of systems with incommensurate delays. To this end, a transformed functional with the Lyapunov matrix of the original system replaced with a different matrix is used [1, 7, 8]. The resulting stability conditions require to verify positive definiteness of the special block matrix as in [2] but involve the Lyapunov matrix of an auxiliary system with commensurate delays instead of the original one. An illustrative example with a detailed discussion is given.

9.2 Preliminaries In this contribution, we consider a system x(t) ˙ =

m 

A j x(t − h j ), t  0,

(9.1)

j=0

where x(t) ∈ Rn , A j ∈ Rn×n , j = 0, m, are constant matrices, and the delays are such that 0 = h 0 < h 1 < . . . < h m = H. Let x(t) be a solution of system (9.1) with x(θ ) = ϕ(θ ), θ ∈ [−H, 0], where an initial function   initial condition ϕ ∈PC [−H, 0], Rn , and the space of piecewise continuous initial functions PC [−H, 0], Rn is supplied with the uniform norm ϕ H = supθ∈[−H,0] ϕ(θ ), where the Euclidean norm is used for vectors. Denote by xt the state of system (9.1): xt (ϕ) :

θ → x(t + θ, ϕ), θ ∈ [−H, 0].

Given a positive definite matrix W, a continuous matrix U (τ ) which satisfies the set of equations U  (τ ) =

m 

U (τ − h j )A j , U (−τ ) = U T (τ ), τ  0,

j=0 m    U (−h j )A j + A Tj U (h j ) = −W, j=0

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is called the Lyapunov matrix of system (9.1) associated with W [7]. Together with system (9.1), consider a system with the same matrices and different delays: y˙ (t) =

m 

A j y(t − hˆ j ), t  0,

(9.2)

j=0

(τ ) be the Lyapunov matrix of system (9.2) where hˆ 0 = 0, hˆ j  0, j = 1, m. Let U associated with W. The following necessary stability conditions which are based exclusively on the Lyapunov matrix were obtained in [2]: if system (9.1) is exponentially stable, then the block matrix r  (9.3) U (τ j − τi ) i, j=1

is positive definite for any r and τ1 , . . . , τr ∈ [0, H ] such that τi = τ j , if i = j. In this work, we assume that the Lyapunov matrix U (τ ) of system (9.1) is unknown (τ ) of system (9.2) is known instead. Our aim is to derive but the Lyapunov matrix U the necessary stability conditions for system (9.1) which are similar to (9.3) but (τ ) instead of U (τ ). An explicit condition on depend on the Lyapunov matrix U |h j − hˆ j |, j = 1, m, will be given. The resulting stability conditions also involve the fundamental matrix of system (9.2) on the segment [0, H ], which is a solution of m (t)  dK (t − hˆ j )A j , t  0, = K dt j=0 (0) = I and K (θ ) = 0 for θ < 0. To guarantee the existence with initial condition K  of the Lyapunov matrix U (τ ), we assume that system (9.2) satisfies the Lyapunov condition, i.e. does not have the eigenvalues located symmetrically with respect to the origin of the complex plane [7].

9.3 Stability Conditions In this section, we present the necessary stability conditions for system (9.1) with incommensurate delays. Following the works [1, 7, 8], we introduce the functional ) = ϕ T (0)U (0)ϕ(0) + 2ϕ T (0) v0 (ϕ, U +

m

 k=1

0

−h k

ϕ T (θ1 )AkT

 m

j=1

m

 j=1

0

−h j

0

−h j

(−θ − h j )A j ϕ(θ )dθ U

(θ1 + h k − θ2 − h j )A j ϕ(θ2 )dθ2 dθ1 . U

(9.4)

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The functional v0 (ϕ, U ) with the original Lyapunov matrix of system (9.1) is the classical functional [5] whose time-derivative along the solutions of system (9.1) is (τ ), equal to −x T (t)W x(t), t  0. In (9.4), the Lyapunov matrix is replaced with U and the time-derivative satisfies ) dv0 (xt , U = −x T (t)W x(t) + l(xt , ), where (9.5) dt m 0

  l(xt , ) = x T (t) (0) + T (0) x(t) + 2x T (t) T (θ + h j )A j x(t + θ)dθ, j=1 −h j

(τ ) =

m 

F j (τ )A j ,

(τ − h j ) − U (τ − hˆ j ), F j (τ ) = U

j=1

see p. 117 in [7]. Estimating the additional part of the derivative, we obtain l(xt , )  ξ0 x(t) + 2

m  j=1

ξj

0

−h j

x(t + θ )2 dθ,

(9.6)

where ξ0 = (1 + α)F,  ξ j = A j F, j =1, m. Here, α = 1 + mj=1 A j h j , F = mj=1 A j  maxτ ∈[0,H ] F j (τ ). Assume that m  ξ j < λmin (W ). (9.7) ξ0 + H j=1

Notice that if the delays in systems (9.1) and (9.2) coincide, i.e. h j = hˆ j , then ξ j = 0, j = 0, m. Hence, inequality (9.7) imposes a restriction on the closeness between h j and hˆ j in fact. Now, we introduce the function m 

ω(θ ) = ξ0 − θ

ξ j , θ ∈ [−H, 0],

j=1

and modify the functional: ) = v0 (ϕ, U ) + v(ϕ, U



0

−H

 ϕ T (θ ) W − ω(θ )I ϕ(θ )dθ.

Due to inequality (9.7) the additional summand here is non-negative, thereby we are able to prove the following useful lemmas. ) satisfies Lemma 9.1 Along the solutions of system (9.1) functional v(ϕ, U  m 

 ) dv(xt , U  −x T (t − H ) W − ξ0 + H ξ j I x(t − H ). dt j=1

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Lemma 9.2 Assume that system (9.2) satisfies the Lyapunov condition and inequality (9.7) holds. If system (9.1) is exponentially stable, then there exists an α > 0 such that 

0   )  α ϕ(0)2 + ϕ(θ )2 dθ , ϕ ∈ PC [−H, 0], Rn . v(ϕ, U −H

) is suitable for the stability analysis of system (9.1). Hence, the functional v(ϕ, U The methodology similar to that used in [2] applied with this functional allows us to prove the following necessary stability condition. The idea of the proof is to substitute initial functions of the special form, which depend on the fundamental ), and then apply Lemma 9.2. matrix of system (9.2), into the functional v(ϕ, U Theorem 9.1 Assume that system (9.2) satisfies the Lyapunov condition and inequality (9.7) holds. If system (9.1) is exponentially stable, then the block matrix r  (τ j − τi ) + (τi , τ j ) U

i, j=1

(9.8)

is positive definite for any r and τ1 , . . . , τr ∈ [0, H ] such that τi = τ j , if i = j. Here, T (τi )

(τi , τ j ) = K T (τi ) +K + + + + −

m 0 

m 0 

k=1 −λk

m  −hˆ k

k=1 −h k

k=1 −h k m 0  k=1 −h k m 0 

(−θ − k )Ak K (τ j + θ )dθ U

T (τi + θ1 )A T K k T (τi + θ1 )A T K k

m 0 

(τ j + θ2 )dθ2 dθ1 Fl (θ1 + h k − θ2 )Al K

l=1 −λl m −hˆ l  l=1 −h l

(θ1 + h k − θ2 − l )Al K (τ j + θ2 )dθ2 dθ1 U

T (τi + θ )A T F T (−θ − τ j )dθ K k k

k=1 −λk m −hˆ k  k=1 −h k

0

(τ j + θ )dθ Fk (−θ )Ak K

T (τi + θ )A T U  K k (θ + τ j + k )dθ

− min{τi ,τ j }

T (τi + θ ) K (τ j + θ )dθ, ω(θ) K

where λk = min{h k , hˆ k }, k = max{h k , hˆ k }, k = 1, m.

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Fig. 9.1 Stability region of equation (9.9) in the space of parameters b and c

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9.4 Example Consider an equation

√  x(t) ˙ = −x(t) + bx(t − 1) + c x t − 5/2 , t  0.

(9.9)

The stability domain for equation (9.9) in the space of parameters b and c was analyzed in [8] with help of the D-subdivision technique and the special numerical approach. Here, we perform the stability analysis using Theorem 9.1. ˆ and set To this end, we first define the delays of equation (9.2) as hˆ 1 = 1, hˆ 2 = h, r and the equidistant points τ1 , . . . , τr ∈ [0, H ] with τ1 = 0 and τr = H. Next, we consider a mesh in the space of parameters b and c, and verify inequality (9.7) for each point (b, c). Finally, if inequality (9.7) holds, we check the positive definiteness of the block matrix (9.8). The stability diagrams for different values hˆ and r are presented on Fig. 9.1. The small points correspond to the parameter values for which (9.7) holds, whereas the large ones correspond to the values for which the matrix (9.8) is positive definite. The points are presented along with the stability boundaries obtained by the D-subdivision technique. If the area contains the points for which (9.7) holds but the matrix (9.8) is not positive definite, then this area corresponds to the instability. It is clear that the only domain containing the point (0, 0) is the domain of the exponential stability of equation (9.9) for b, c ∈ [−14, 14].

9.5 Conclusions The necessary stability conditions for linear time-invariant systems with incommensurate delays are obtained in the paper. The practical experiments show that if inequality (9.7) is satisfied then the set of parameter values for which the obtained condition holds approaches the exact stability domain with the increase of r, i.e. the dimension of the resulting matrix (9.8). This allows us to expect that the obtained conditions may become also the sufficient stability conditions for some values of r under the assumption that (9.7) holds. It is worthy of mention that this is the case for the original necessary stability conditions in [2]. Acknowledgements The work was made during the research stay supported by St. Petersburg State University, project 41128606.

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References 1. Egorov, A.V., Kharitonov, V.L.: Approximation of delay Lyapunov matrices. Int. J. Control 91(11), 2588–2596 (2018) 2. Egorov, A.V., Mondié, S.: Necessary stability conditions for linear delay systems. Automatica 50, 3204–3208 (2014) 3. Egorov, A.V., Cuvas, C., Mondié, S.: Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays. Automatica 80, 218–224 (2017) 4. Gomez, M.A., Egorov, A.V., Mondié, S.: Lyapunov matrix based necessary and sufficient stability condition by finite number of mathematical operations for retarded type systems. Automatica 108, 108475 (2019) 5. Huang, W.: Generalization of Liapunov’s theorem in a linear delay system. J. Math. Anal. Appl. 142, 83–94 (1989) 6. Huesca, E., Mondié, S., Santos, J.: Polynomial approximations of the Lyapunov matrix of a class of time delay systems. In: Proceedings 8th IFAC Workshop on Time Delay Systems, Sinaia, Romania, pp. 261–266 (2009) 7. Kharitonov, V.L.: Time-Delay Systems: Lyapunov Functionals and Matrices. Birkhäuser, Basel (2013) 8. Zhabko, A.P., Medvedeva, I.V.: Stability analysis of linear time-delay systems with two incommensurate delays. In: E. Witrant et al. (ed.) Recent Results on Time-Delay Systems: Advances in Delays and Dynamics, vol. 5, pp. tttytyttrpuioyoii, pp. 69–87. Springer Int. Publ., Switzerland (2016)

Chapter 10

Study of the Stability Features of Solutions of Systems of Differential Equations Gennady Ivanov, Gennady Alferov, Vladimir Korolev, and Dzmitry Shymanchuk Abstract The stability conditions for solutions of systems of ordinary differential equations are considered. The conditions and criteria for the use of partial and external derivatives are proposed. This allows us to investigate the behavior of a function of several variables, without requiring its differentiability, but using only information on partial derivatives. This reduces the restrictions on the degree of smoothness of the studied functions. The use of the apparatus of external derived numbers makes it possible to reduce the restrictions on the degree of smoothness of manifolds when studying the question of the integrability of the field of hyperplanes. Using the apparatus of partial and external derived numbers, it can be shown that the investigation of the stability of solutions of a system of differential equations can be reduced to an investigation of the solvability of a system of equations of a special form.

10.1 Introduction Many sciences are engaged in the creation of mathematical models of various processes. The problems in the study of dynamic processes lead to complex systems of differential equations [1–10]. The concepts of stability of solutions or asymptotic stability are often used in studies of solutions of equations and the ability to control the behavior in the presence of perturbations [11–15]. For their approximation solution or successive approximations to the exact solution necessary to check the conditions and criteria that must be met. The study of control problems and the stability of solutions of systems of differential equations to describe processes that are defined as linear operations makes it possible to divide all tasks into classes and identify important properties inherent in systems of differential equations of the same class. G. Ivanov (B) · G. Alferov · V. Korolev · D. Shymanchuk St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] D. Shymanchuk e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_10

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In the study of the problems of controlling the motion of mechanical systems [1, 4, 17] in the transition from a general formal description to the construction of mathematical models take into account: – – – –

content and properties of functions in the system of equations of dynamics, structure of control functions, restrictions, or boundary conditions, type of functional or quality criterion of solutions, stability conditions for solutions for admissible controls.

The concepts of partial derivatives of numbers and external derivatives of numbers are considered in order to use them to study the stability of solutions of a system of differential equations through the study of the solvability of a system of equations of a special form [18]. The proposed method can be used to obtain necessary or sufficient stability conditions for solutions of systems of differential equations.

10.2 Features of Stability Conditions Let the change of parameters or the object’s behavior be described by a system of ordinary differential equations of the form x˙ = Ax.

(10.1)

From Eq. (10.1) for a linear stationary system follows the validity of the following equation d ∗ x x = x ∗ (A∗ + A)x. (10.2) dt Here, an asterisk means a transpose operation. Let A∗ + A be a nonsingular matrix symmetric with respect to the diagonal. Then applying the Lagrange method to Eq. (10.2) reduction of quadratic forms to the sum of squares [3], it is easy to verify that there is a linear transformation x = L y, reducing Eq. (10.2) to the form d ∗ ∗ y L L y = y ∗ By, dt where B = L ∗ (A∗ + A)L is the diagonal matrix. If the matrix B is negative definite, i.e. all its elements are negative, then system (10.1) is asymptotically stable. In general we can talk about the stability of solutions under additional conditions.

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10.2.1 The Partial Derivatives Numbers Using the apparatus of private and external derivatives of numbers, show that the study of the stability of solutions systems of differential equations can be reduced to the study of the solvability of systems of equations of a special kind. The present studies are based on [1–5]. Let the function f be given in some open region of space R n , and let it go x = (x1 , . . . , xn )T —an arbitrary point of this areas as x = (x1 , . . . , xn )T — arbitrary increment of function arguments f . Let be ωi , i = 1, 2, . . . , n, ψi [ f ](x; x) = n−1 2 xi where ωi =



[ f (x1 + μ1 x1 , . . . , xn + μn xn )−

μ∈νi

− f (x1 + μ1 x1 , . . . , xi−1 + μi−1 xi−1 , xi , xi+1 + +μi+1 xi+1 . . . , xn + μn xn )], and μ = (μ1 , . . . , μn ), and νi , i = 1,…,n, marked a bunch of n-dimensional vectors consisting of zeros and ones and having unit at the ith place. Definition 10.1 The number λ is called the partial derivative of the function f in point x in the variable xi if there is a sequence {x k } such that for any j ∈ (1, . . . , n), x kj → 0 at k → ∞, xik = 0, and lim ψi [ f ](x; x k ) = λ.

k→∞

The fact that λ is a partial derivative functions f at the point x with respect to the variable xi , we will write this: λ = λxi [ f ](x). Perform a study of the stability of solutions of systems of ordinary differential equations.

10.2.2 The External Derivatives Numbers The definition of the external derivative number allows us to find the conditions for the complete integrability of continuous fields of hyperplanes. Let M be a Hausdorff space with a countable base, and let p be an arbitrary point of M. If a point p has a neighborhood U that is homeomorphic to an open subspace of an n-dimensional

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Euclidean space R n , then M is called an n-dimensional topological manifold. Let M n be an n-dimensional topological manifold. Let V be an n-dimensional vector space over a field of real numbers. Every linear mapping f : V ∈ R, i.e. display at which f (av + bw) = a f (v) + b f (w), v, w ∈ V, a, b ∈ R.

Definition 10.2 The form λ[ω]( p) is called the external derivative of the external differential q-form of the class C r , r ≥ 0, on variety M n at the point p ∈ M n , if in R n there is a sequence converging to zero {x k } such that ⎧ ⎨  ∗ −1 κ λ[ω]( p) = lim k→∞ ⎩

=

 j1 0 such that from x0 < δ it follows x(t; t0 , x0 ) < ε for all t ≥ t0 . We introduce the class of functions H , assuming that the function l(r ) belongs to this class (l(r ) ∈ H ), if l(r ) is continuous, strictly increasing for r ∈ [0, H ], H = const > 0, or for r ∈ [0, ∞), the function is l(0) = 0. The function is H, which means that l(r ) is optional this class (l(r ) ∈ H), if l(r )—continuous strictly increasing at r ∈ [0, H ], H = const > 0, or at r ∈ [0, ∞) function, moreover l(0) = 0. Definition 10.3 The function V (t, x), V (t, 0) ≡ 0, t ≥ 0, will call definitely positive if there is a function l(r ) ∈ H, such that in t ≥ 0, x ≤ H inequality holds V (t, x) ≥ l( x ).

(10.4)

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This definition is equivalent to the generally accepted definition of positive definiteness of a function V (t, x). In the future, we will adhere to the following notation: K r (x0 ) = {x : c − x0 ≤ r }, Sr (x0 ) = {x : x − x0 = r },

r = const > 0.

For brevity we put S1 (0) = S.

Theorem 10.1 Suppose that in region (10.4) there exist continuous partial derivatives ∂ Fi , i, j = 1, 2, . . . , n. ∂x j Then, in order for the solution x = 0 of system (10.3) was stable according to Lyapunov, it is necessary and sufficient that in the region t ≥ 0, x ≤ h,

0 < h = const < H,

(10.5)

system a0 (t, x) + a(t, x) · F(t, x) ≤ 0, ω ∧ λ[ω] ≡ 0,

a(t, x) = (a1 (t, x), . . . , an (t, x)),

ω = a0 dt + a1 d x1 + · · · + an d xn ,

(10.6) (10.7)

had a continuous solution (a0 (t, x), a(t, x)), satisfying the following requirements: (1) in the region of (10.8) t ≥ 0, x ∈ K h (0) \ {0} n 

ai2 (t, x) > 0;

i=0

(2) in the region of t ≥ 0, μ ∈ [0, h], x ∈ S 

μ

a(t, μ x) · xdμ ≥ l(μ),

(10.9)

l(μ) ∈ H.

0

The solution x = 0 of system (10.3) will be called uniformly sustainable if for any ε > 0 there is δ(ε) > 0 such that from t0 ≥ 0 and x0 < δ should

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x(t; t0 , x0 ) < ε for all t ≥ t0 . We will say that the solution x = 0 of system (10.3) is evenly attractive if exists 0 = const > 0 such that the condition lim x(t; t0 , x0 ) = 0

t→∞

performed uniformly by (t0 , x0 ) from area t0 ≥ 0, x0 < 0 . If the solution x = 0 of system (10.3) is simultaneously uniformly stable and evenly attractive, then we will call uniformly asymptotically stable.

10.2.4 Stability Conditions Theorem 10.2 Suppose that in region (10.4) there exist continuous partial derivatives. Then, in order for the solution x = 0 of system (10.3) to be Lyapunov stable, it is necessary and sufficient that in the region The system had a continuous solution (a0 (t, x), a(t, x)),satisfying the following requirements: in the region of t ≥ 0. A solution x = 0 of system (10.3) will be called uniformly stable if for any ε > 0 there is δ(ε) > 0 such that t0 ≥ 0 and x0 < δ follows

x(t, t0 , x0 ) < δ for all t ≥ t0 . The proposed method allows one to obtain statements that give necessary or sufficient conditions for uniform stability or asymptotic stability for solutions of systems of differential equations. Theorem 10.3 Suppose that in region (10.4) the functions Fi and their partial derivatives (∂ Fi )/(∂ x j ) are continuous and bounded: |Fi (t, x)| ≤ B, B = const, |

∂ Fi (t, x) | ≤ A, A = const, i, j = 1, 2, . . . , n. ∂x j

Then, for the solution x = 0 of system (10.3) to be uniformly asymptotically stable, it is necessary and sufficient that in region (5), where h is a sufficiently small constant,

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system (10.6)–(10.7) has a continuous solution (a0 (t, x), a(t, x)), satisfying in the area (10.8) or (10.9) the following constraints: (1)

n 

ai2 (t, x) > 0;

i=0



μ

(2) l1 (μ) ≤

a(t, μ x) · xdμ ≤ l2 (μ);

0

max [a0 (t, μx) + a(t, μx) · F(t, μx)] ≤ −l3 (μ), lk (r ) ∈ H.

t≥0, x =1

The proposed method allows to obtain the necessary or sufficient conditions for the stability of solutions of systems of differential equations.

10.3 Conclusion The proposed apparatus of partial and external derived numbers allows us to investigate the behavior of a function of several variables, without requiring its differentiability, but using only information about partial derived numbers. This reduces the limitations imposed on the degree of smoothness of the functions studied. The use of the apparatus of external derived numbers also makes it possible to reduce the restrictions on the degree of smoothness of manifolds when studying the question of the integrability of the hyperplanes field. Acknowledgements This work was carried out the financial support by Russian Foundation for Basic Research, project No. 18-08-00419.

References 1. Zubov, V.I.: Lyapunov’s Methods and Their Application. Leningrad State University Publishing, Leningrad (1957).(in Russian) 2. Demidovich, B.P.: Lectures on the Mathematical Theory of Stability Science. Moscow (1967).(in Russian) 3. Gantmakher, F.R.: Matrix Theory. Fizmatlit, Moscow (2010).(in Russian) 4. Lyapunov, A.M.: The general problem of motion stability. (Classics of Natural Science). Gostekhizdat, Moscow-Leningrad (1950) (in Russian) 5. Arnold, V.I.: Ordinary Differential Equations. Science, Moscow (1975).(in Russian) 6. Ivanov, G., Alferov, G., Gorovenko, P.: Derivatives of numbers of functions of one variable. Vestnik of Perm University. Math. Mech. Comput. Sci. 3(42), 5–19 (2018) 7. Ivanov, G., Alferov, G., Sharlay, A., Efimova, P.: Conditions of Asymptotic Stability for Linear Homogeneous Switched System. Int. Conf. Numer. Anal. Appl. Math. AIP Conf. Proc. 1863(080002) (2017)

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8. Alferov, G., Ivanov, G., Efimova, P., Sharlay, A.: Stability of linear systems with multitask right-hand member. Stochastic Methods for Estimation and Problem-Solving in Engineering, (Book Chapter) pp. 74–112 (2018) 9. Kadry, S., Alferov, G., Ivanov, G., Sharlay, A.: Stabilization of the program motion of control object with elastically connected elements. AIP Conf. Proc. 2040(150014) (2018) 10. Kadry, S., Alferov, G., Ivanov, G., Sharlay, A.: About stability of select or linear differential inclusions. AIP Conf. Proc. 2040(150013) (2018) 11. Kadry, S., Alferov, G., Ivanov, G., Sharlay, A.: Derived numbers of one variable convex functions. IJPAM 41, 649–662 (2019) 12. Kadry, S., Alferov, G., Ivanov, G., Korolev, V., Selitskaya, E.: A New Method to study solutions of ordinary differential equation using functional analysis. Mathematcs 7(8) (2019) 13. Kulakov, F., Alferov, G., Sokolov, B., Gorovenko, P., Sharlay, A.: Dynamic analysis of space robot remote control system. AIP Conf. Proc. 1959(080014) (2018) 14. Korolev, V.: Properties of solutions of nonlinear equations of mechanics control systems. In: International Conference on Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V.F. Demyanov), CNSA – IEEE Conference Proceedings (7973973) (2017) 15. Korolev, V., Pototskaya, I.: Integration of dynamical systems and stability of solution on a part of the variables. Appl. Math. Sci. 9(15), 721–728 (2015) 16. Alferov, G., Ivanov, G., Sharlay, A., Fedorov, V.: Application of derived numbers theory in convec analysis. AIP Conf. Proc. 2116(08003) (2019) 17. Novoselov, V., Korolev, V.: Analytical mechanics of a controlled system. St. Petersburg State University Publ, St. Petersburg (2005).(in Russian) 18. Zubov, V.I.: The Problem of Stability of Control Processes. St. Petersburg State University Publishing, St.Petersburg (2001).(in Russian)

Chapter 11

Economic Evolution with Structural Variations Alexander N. Kirillov and Alexander M. Sazonov

Abstract In this paper, an approach to mathematical modeling of the Schumpeterian theory of the endogenous evolution of the economic systems is presented. In Sect. 11.2 the model of the sector capital distribution dynamics over efficiency levels is proposed. In order to take into account the boundedness of the economic growth which is due to the boundedness of the markets, the resource base, and other factors, the notion of economical niche volume is introduced. The global stability of the presented dynamical system with the Jacobi matrix having at the equilibrium all the eigenvalues equal to zero, except one—negative, is proved. In Sect. 11.3 the approach to the modeling of the structural variations on the base of the system with the variable number of the efficiency levels is presented. The scenarios of the emergence and vanishing of the efficiency levels are proposed. Some results concerning the dynamics of the special discrete system describing the efficiency levels presence are obtained. The proposed models permit to evaluate and predict the dynamics of the efficiency levels of the economic sector firms development.

11.1 Introduction The theory of endogenous economic development proposed by Schumpeter [7] is widely used and actively developed currently (for example, see [1, 2, 5]). The main idea of this theory is that the interacting processes of innovations and imitations account for the economic growth. A. A. Shananin and G. M. Henkin described the mathematical model of the firm capacity dynamics in [3]. Let Mi (t) be the integrated firm capacity at the ith level, λi is the profit per capacity unit at the ith level, ϕi (t) is the share of the investments A. N. Kirillov (B) · A. M. Sazonov Institute of Applied Mathematical Research, Federal Research Center “Karelian Research Center of the Russian Academy of Sciences”, 11 Pushkinskaya str., Petrozavodsk 185910, Russia e-mail: [email protected] A. M. Sazonov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_11

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of the firms at the ith level for the creation of the capacities at the next i + 1th level, 0  ϕi (t)  1, N is the initial number of levels. Here, ϕi = α + β(1 − Fi (t)), α > 0, i 

β > 0 are constants, Fi (t) =

k=0 ∞ 

Mk

. The main result is the existence of the solution Mk

k=0

family Fi∗ (t) such that lim sup |Fi (t) − Fi∗ (t)| = 0 as t → ∞. Here, the function i1

Fi∗ (t) has the form of the logistic curves and varies in the space of the technologies as a traveling wave. However, this model does not take into account the boundedness of the economic growth, which is incorrect from the economic perspective. In [4] it is shown that Mi (t) → ∞ as t → ∞. The authors, developing the described model, in Sect. 11.2 construct the model of the capital distribution dynamics over efficiency levels taking into account the boundedness of the economic growth by introducing the notion of the economic niche volume which is analogous to the ecological niche volume. The economic niche volume is a limit integrated capital value, for which the growth rate is so low that there is no capital growth. Besides, in Sect. 11.3 this model is extended by introducing the structural variations based on the system with variable number of efficiency levels. The scenarios of emerging and vanishing of the efficiency levels are proposed.

11.2 The Model of the Capital Distribution Dynamics over Efficiency Levels Let N be the number of efficiency levels. Assume that the greater i corresponds to the higher level. Consider the following system  C˙ 1 = C˙ i =

1−ϕ1 (C) C1 (V − C1 − · · · − C N ) = f 1 (C), λ1 1−ϕi (C) Ci (V − C1 − · · · − C N ) + ϕi−1 (C)Ci−1 λi

= f i (C), i = 2, . . . , N . (11.1) Here, Ci is the integrated capital of all firms at the ith level (one firm can have the capital at different levels), V is the economic niche volume, ϕi is the share of capital of the firms at the ith level intended to the developing of the production at the next, i + 1th, level, λi is the unit prime cost at the ith level (i.e. the unit goods production cost per unit time), i = 1, . . . , N . Let V > 0, λi > 0 are constants, ϕ1 (C) ∈ (0, 1) are continuously differentiable, i = 1, . . . , N , C = (C1 , . . . , C N ). Denote ai (C) = 1−ϕλii(C) > 0. Denote by C(t, t0 , C 0 ) the solution of (11.1) for which C(t0 , t0 , C 0 ) = C 0 . If t0 = 0 the solution will be denoted as C(t, C 0 ). Denote by r (C 0 ) the positive halftrajectory r (C 0 ) = {C(t, C 0 ) : t  0}. If t0 > 0, then r (t0 , C 0 ) = {C(t, C 0 ) : t  t0 }. Proposition 11.1 The set R+N = {(C1 , . . . , C N ) ∈ R N : Ci  0, i = 1, . . . , N } is invariant.

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Proof From (11.1) we obtain C˙ i  0 if Ci = 0. It means that the positive halftrajectories do not leave R+N through the boundary hyperplanes Ci = 0, therefore, R+N is invariant. (C)  Wi , ∀C ∈ R+N , where Wi > 0 is a preassigned conProposition 11.2 Let ϕai−1 i (C) stant such that Wi  V , for each i = 2, . . . , N . Denote K = {(C1 , . . . , C N ) ∈ R N : 0  C1  V, 0  Ci  max (V, Wi ), i = 2, . . . , N }. Then all positive half-trajectories with initial point C 0 ∈ R+N , except for r˜ (C 0 ) such that C 0 = (0, . . . , 0, R), R > V , enter the parallelepiped K and do not leave it. (C) Remark 11.1 From economic point of view the condition ϕai−1  Wi , ∀C ∈ R+N , i (C) where Wi > 0 means that the contribution of the production at the ith level to the capital growth described by the parameter ai (C) is not arbitrary small in comparison with the contribution of the investments from the i − 1th level with the intensity (C) is unbounded, then the production at the ith level ϕi−1 (C). It is clear, that if ϕai−1 i (C) makes practically no contribution to the economic growth at this level in comparison with the investments from the i − 1th level, which indicates at some problems in economy.

Proof If

ϕi−1 (C) ai (C)

 V , then for Ci = R  Wi we obtain from (11.1)

  C˙ i = ai R V − C1 − · · · − Ci−1 − R − Ci+1 − · · · − C N + ϕi−1 Ci−1    ϕi−1 ai R V − C1 − · · · − Ci−1 − − Ci+1 · · · − C N + ϕi−1 Ci−1  ai   ϕi−1 ϕi−1 V − C1 − · · · − Ci−1 − − Ci+1 · · · − C N + ϕi−1 Ci−1 = ai   ϕi−1 ϕi−1 V − C1 − · · · − Ci−2 − − Ci+1 − · · · − C N  ai ϕi−1 (V − C1 − · · · − Ci−2 − V − Ci+1 − · · · − C N ) < 0. If

ϕi−1 (C) ai (C)

< V , then for Ci = R  V

C˙ i = ai R(V − C1 − · · · − Ci−1 − R − Ci+1 − · · · − C N ) + ϕi−1 Ci−1 = ϕ  i−1 − V + ai R(V − R − C1 − · · · − Ci−2 − Ci+1 − · · · − C N ) < 0. ai Ci−1 ai So, all positive half-trajectories enter any parallelepiped K R = {(C1 , . . . , C N ) ∈ R N : 0  C1  V, 0  Ci  R, i = 2, . . . , N }, where R  max (V, Wi ) and do not leave it. Therefore all positive half-trajectories enter the smallest parallelepiped K . Definition 11.1 The equilibrium C ∗ of the system C˙ = f (C), f (C ∗ ) = 0, is globally stable in Rn+ \ {O} if C(t, C 0 ) → C ∗ , as t → ∞, for every C 0 ∈ Rn+ \ {O}. (C)  Wi , ∀C ∈ R+N , where Wi > 0 is a preassigned conTheorem 11.1 Let ϕai−1 i (C) stant, for each i = 2, . . . , N .

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Then the equilibrium P = (0, . . . , 0, V ) ∈ R+N of the system (11.1) is globally stable in R+N \ {O}. Proof According to Proposition 11.2 all positive half-trajectories, except for r˜ (C 0 ) such that C 0 = (0, . . . , 0, R), R > V , enter the parallelepiped K = {(C1 , . . . , C N ) ∈ R N : 0  C1  V, 0  Ci  max (V, Wi ), i = 2, . . . , N } and do not leave it. Therefore, all positive half-trajectories, except for r˜ (C 0 ), are positively Lagrange stable. Hence, the ω-limit set C = ∅ for any positive half-trajectory r (C 0 ) ∈ R+N [6]. Consider the set J (d) = {(C1 , . . . , C N ) ∈ R+N : 0  C1 + · · · + C N  d, d ∈ [0, V ]}. Denote g(t) = C1 (t) + · · · + C N (t), where Ci (t, C 0 ) are the components ˙ > 0 for C 0 ∈ J (V ) \ {O}. of the solution C(t, C 0 ), C 0 ∈ J (V ) \ {O}. Then g(t) Hence, all positive half-trajectories, except for r˜ (C 0 ), leave the set J (V ). As for r˜ (C 0 ), C 0 = (0, . . . , 0, R), R > V , it is easily to understand that r˜ (t, C 0 ) → P as t → ∞. Denote L = K \ J (V ). Obviously, all positive half-trajectories, except for r˜ (C 0 ), enter L and do not leave it. Therefore, if C 0 ∈ R+N , then C1 (t, C 0 ) → 0 as t → ∞. Hence, C1 = 0 for any point belonging to C , or C = {(C1 , . . . , C N ) ∈ R N : C1 = 0}, i.e. C ⊂ R+N −1 . Let us show that C = P = (0, . . . , 0, V ) ∈ R+N for any positive half-trajectory r (C 0 ) ∈ R+N \ {O}. Assume, on the contrary, that there exists a positive half-trajectory r (C 0 ), C 0 ∈ R+N \ {O} for which there exists an ω-limit point Q ∈ C ⊂ R+N −1 and Q = P. Obviously, Q ∈ L. Since R+N −1 is the invariant set of (11.1), then (11.1) in R+N −1 has the following form ⎧ 1−ϕ2 ⎪ ⎨C˙ 2 = λ2 C2 (V − C2 − · · · − C N ) = f 2 (C), i (11.2) Ci (V − C2 − · · · − C N ) + ϕi−1 Ci−1 = f i (C), C˙ i = 1−ϕ λi ⎪ ⎩ i = 3, . . . , N . Let C − (t, Q) = (C2− (t, Q), . . . , C N− (t, Q)) be the solution of (11.2) such that C − (0, Q) = Q = (C2− (Q), . . . , C N− (Q)). Consider two cases. First, assume C2 (Q) > 0. Then the invariance of the set {(C2 , . . . , C N ) ∈ R+N −1 , C2 = 0} for (11.2) implies that C2− > 0, ∀t  0. Taking into account that C˙ 2− (t, Q) > 0 (see (11.2)), we obtain that the corresponding positive half-trajectory r − (Q) intersects any hyperplane {(C2 , . . . , C N ) : C2 = const > 0}. Thus, r − (Q) leaves the set L ∩ R+N −1 . But the trajectory {C(t, Q), t ∈ R} of (11.2) is the ω-limit set of r (C 0 ) ([6], Theorem 3.01), and we obtain a contradiction, because r (C 0 ) enters and does not leave L. Now, assume C2 (Q) > 0 = 0, i.e. Q ∈ R+N −2 = {C3 , . . . , C N ), Ci  0, i = 3, . . . , N }, and we can repeat the above argument, taking 3 i C3 (V − C3 − · · · − C3 ), C˙ i = 1−ϕ Ci (V − C3 − into account the system C˙ 3 = 1−ϕ λ3 λi · · · − C N ) + ϕi−1 Ci−1 , i = 4, . . . , N . At last, following such way of reasoning, we have to consider the case Q ∈ {(C1 , . . . , C N ) : C j = 0, j = 1, . . . , N − 1, C N > V } ∩ L, i.e. Q belongs to the trajectory belonging to the C N axis for which C N > V . Reversing the direction of time, like in previous argument, we obtain the trajectory leaving L, which cannot be the ω-limit set of r (C 0 ). So, Q is not the ω-limit point of r (C 0 ).

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Thus we have, C = {P = (0, . . . , 0, V )} for any positive half-positive halftrajectory r (C 0 ). Since any positive half-trajectory r (C 0 ) is positively Lagrange stable then, according to Theorem 3.07 [6], ρ(r (C 0 ), P) → 0 as t → ∞, which means that P = (0, . . . , 0, V ) is globally stable in R+N \ {O}.

11.3 The Model with the Variable Number of the Efficiency Levels The global stability of the equilibrium means the stagnation from economic perspective which is extremely unfavorable situation. The further economic development requires a new efficiency level. This is consistent with Schumpeter’s concept of the creative destruction [7]. Let us consider the following scenario of the emergence of the new highest efficiency level. The new level emerges at the moment when the positive half-trajectory intersects the boundary of the neighborhood Uδ (P) = {C ∈ R+N : ||C − P|| < δ} of the equilibrium P = (0, . . . , 0, V ), which certainly happens because of the global stability of P, where δ > 0 is a preassigned constant. Assume, the new level emerged at the time moment tem . The switch from R+N to R+N +1 at the time moment tem arises instantaneously so that the point (C1 (tem ), . . . , C N (tem )) transfers to the point (C1 (tem ), . . . , C N (tem ), C N +1 (tem )), where C N +1 (tem ) = 0. Let us now consider the scenario of the vanishing of the lowest level. As shown above, C1 (t) → 0 for t → ∞ if N  2. Therefore, when the capital at the lowest level C1 is sufficiently small (in comparison with the total capital volume at all levels) it may be neglected and considered as vanished, since the production at this level does not make a substantial contribution to the economy. Thus, since C1 (t) → 0 monotonically, the vanishing of the lowest level arises at the time moment tvan such that C1 (tvan ) = ε, where ε is a preassigned constant, showing how substantial is the capital at the first level in comparison with the total capital volume at all levels. If C1 (t) > ε the production at the first level make a sufficiently substantial contribution to the economy and the capital at the first level cannot be neglected. For example, ε = θ V , where 0 < θ 1 then for any initial values (low0 , N0 ) the system will switch between two states: (low(t) = low0 + N0 − 1 + j, N (t) = 1) (one level) and (low(t) = low0 + N0 − 1 + j, N (t) = 2 (two levels), where j is the number of switchings from the first state to the second one. Proof Since η(ε, δ) > 1, ε > δ. Therefore, for any low(t), N (t) any positive halftrajectory C(t) intersects the hyperplane C1 = ε before intersecting the boundary of the neighborhood Uδ (P), i.e. t ∗ < t ∗∗ , where t ∗ : Clow (t ∗ ) = ε, t ∗∗ : C(t ∗∗ ) ∈ Uδ (P). Thus, the vanishing of the lowest level occurs before emerging of the new level for any low(t), N (t). Therefore, all levels except the highest initial level low0 + N0 − 1 vanish. In the case of one level (N (t) = 1) Clow (t) → V , hence, this positive halftrajectory intersects the boundary of the neighborhood Uδ (P) of the equilibrium P, so the second level low0 + N0 emerges. After that Clow(t)=low0 +N0 −1 → 0, and since ε > δ the positive half-trajectory intersects the hyperplane C1 = ε before intersecting the boundary of the neighborhood Uδ (P), hence, the lowest level low0 + N0 − 1 vanishes and one level low0 + N0 remains. Further, the procedure is similar to the described above one.

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Let us analyze the distribution of the capital Fi (t) =

k=0 ∞ 

Ck

. According to the Ck

k=0

proposed scenarios, next (t) → ∞ as t → ∞. Hence, Ci (t) → 0 as t → ∞ for the fixed i, because, from Theorem 11.1, Ci (t) → 0 for any i < next (t) − 1. So, we 0+···+0 = 0. Besides, we obtain Fi (t) → 0+···+0+V = 1 as i → ∞. have Fi (t) → 0+···+0+V 0+···+0+V These results are in agreement with the results for the capacity dynamics model presented by Shananin and Henkin in [3]. In their framework, Fi (t) → Fi∗ (t), and Fi∗ (t) → 0 as t → ∞, Fi∗ (t) → 1 as i → ∞. Acknowledgements The study was supported by Russian Foundation for Basic Research (RFBR), project No. 18-01-00249.

References 1. Acemoglu, D., Cao, D.: Innovation by entrants and incumbents. J. Econ. Theory (2015). https:// doi.org/10.1016/j.jet.2015.01.001 2. Dosi, G., Napoletano, M., Roventini, A., Treibich, T.: Micro and macro policies in the Keynes+Schumpeter evolutionary models. J. Evol. Econ. (2017). https://doi.org/10.1007/ s00191-016-0466-4 3. Henkin, G.M., Shananin, A.A.: Mathematical modeling of the schumpeterian dynamics of innovation. Math. Mod. 26(8), 3–19 (2014) 4. Kirillov, A.N., Sazonov, A.M.: Global schumpeterian dynamics with structural variations. Bull. SUSU MMCS (2019). https://doi.org/10.14529/mmp190302 5. Klimek, P., Hausmann, R., Thurner, S.: Empirical confirmation of creative destruction from world trade data. CID Working Paper, vol. 238, pp. 1–13 (2012) 6. Nemytskii, V.V., Stepanov, V.V.: The Qualitative Theory of Differential Equations. Princeton University, Princeton (1960) 7. Schumpeter, J.: The Theory of Economic Development (English translation of Theorie der wirtschaftlichen Entwicklung, 1912). Oxford University Press, Oxford (1961)

Chapter 12

Mixed Feedback–Feedforward Frequency Control in Power Systems Oleg O. Khamisov

Abstract The presented work addresses one of the control aspects of power systems: frequency control. In contrast to other transportation networks, here it is not possible to store electrical energy in amounts sufficiently large for reliable system operation. As a result, generation must always be equal to the demand and frequency oscillations are indicator of power imbalance. New proportional-integral controller is derived in order to provide fast response in both feedback and feedforward modes. Linear power system model is analyzed. Second-order turbine governor dynamics is considered in order to ensure high quality of the model. Proof of global asymptotic stability is given. Faster frequency stabilization in comparison with the existing frequency control is shown by numerical simulations.

12.1 Introduction Power systems proved to be effective and secure method of long distance power transportation. However, they are unique due to lack of energy storage capacity sufficient to provide reliable operation. Installation and maintenance cost of the energy storage capacity necessary to cover power consumption even for short period of time are unacceptably high. As a result, generation must always be equal to demand in order to keep power balance. Frequency oscillations are indicator of deviations from power balance and frequency control is designed to keep this balance in the system. If disturbance appears in the system frequency control must adjust power generation in order to restore frequency (deliver it to the nominal value, e.g. 50 Hz). Currently used frequency control is implemented as proportional-integral (PI) controller [10]. However, lag in the turbine governor response to the control actions limits its effectiveness. There exists a large variety of different frequency controls that utilize optimal control approach [1, 2], averaging based methods [3, 4] and primal-dual approaches with gradient descend [5, 6] or interior-point methods [7]. O. O. Khamisov (B) Skolkovo Institute of Science and Technology, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_12

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We suggest a new control for a power system model with the second-order turbine governor dynamics in order to represent realistic impact of the lag on the system stability. It is known [8] that such approach is essential to avoid instability of the control. Our control is a mixed type feedback–feedforward control meaning that it can work as a combination of feedback PI frequency controller with integral power controller or as feedforward integral controller of disturbance. Finally, we support our results with numerical experiments that include a comparison with the frequency control currently implemented in power systems.

12.2 Power System Model Power system is represented by a connected oriented graph  = (N , E), where N is a set of n buses (nodes), E is a set of q lines (edges). Kron reduction is applied [9] in order to exclude load buses, therefore we consider only generator buses. Power system dynamics is represented by system of linear differential equations [10]: M ω˙ = −Dω − C p + p m + r,

(12.1a)

p˙ = BC T ω,

(12.1b)

T m p˙ m = − p m + v,

(12.1c)

T v v˙ = −v + u.

(12.1d)

Here ω(t) ∈ Rn is vector of frequency deviation from the nominal value; p(t) ∈ Rq is vector of line power flows; p m (t) ∈ Rn is vector of mechanical power injections; v(t) ∈ Rn is vector of governor valve positions. System’s parameters: M ∈ Rn×n , M  0 is diagonal matrix of synchronous machines inertias; D ∈ Rn×n , D  0 is diagonal matrix of synchronous machines dampings; r ∈ Rn is vector of unknown disturbances; B ∈ Rq×q , B  0 is diagonal matrix of line parameter reciprocal to reactances; T v ∈ Rn×n , T v  0 is diagonal matrix of time constants characterizing response of governor valves to control; T m ∈ Rn×n , T m  0 is diagonal matrix of time constants characterizing energy carrier dynamics in turbines; C is the incidence matrix of the oriented graph . Control is a continuous vector-function u(t) ∈ Rn . It is known [11] that system (12.1) is stable for any constant u(t) ≡ u ∗ and converges to a vector ω∗ given by ωi∗ = ω∗j =

 ∗ k∈N (r k + u k )  , i, j ∈ N . k∈N Dkk

(12.2)

This result together with linearity of the system ensures stability of the system for any control that converges to a constant value.

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12.3 Problem Statement It can be seen from Formula (12.2) that frequency can be restored if and only if sum of control values is equal to sum of the disturbances. Therefore, any existing frequency control approximates the size of the disturbance explicitly or implicitly. We utilize this fact by separating disturbance approximation and control calculation into two different parts: disturbance approximation and control calculation. Disturbance approximation part uses system state (information about electrical power p e = C p and frequency deviations ω) for feedback operation or disturbance measurements (r ) for feedforward operation. Disturbance approximation part calculates disturbance approximation r˚ and sends it to the control calculation part that returns values of u. As discussed before, we will derive frequency control in PI form. However, not all information about the system is available. Only the following parameters and t measurements are known: frequency deviations and their integral: ω(t), 0 ω(τ )dτ ; t integral of bus electrical powers: 0 p e (τ )dτ ; integral of the disturbances for some t buses 0 r dτ . It is assumed that r i = ri , if disturbance is known for the bus i and r i = 0 otherwise, i ∈ N ; vector of indicators r I , such that riI = 1 if disturbance is known for the bus i and riI = 0 otherwise. Problem 12.1 Find control u with the following properties: (1) control converges to a constant vector: limt→∞ u(t) = u ∗ ∈ Rn ; (2) frequency deviations in the physical system (12.1) converge to zero: limt→∞ ω(t) = 0; (3) among all controls satisfying the items (1) and (2), the derived control must deliver minimum of the function f (u ∗ ) = 21 (u ∗ ) W u ∗ , W = diag(w1 , . . . , wn )  0.

12.4 Control Derivation Let us start control derivation with optimization problem: min

ω, ˆ pˆ m ,ˆv, p, ˆ uˆ

1  uˆ W u, ˆ 2

(12.3a)

0 = ω, ˆ

(12.3b)

0 = −D ωˆ − C pˆ + pˆ m + r,

(12.3c)

ˆ 0 = BC T ω,

(12.3d)

0 = − pˆ m + vˆ ,

(12.3e)

0 = −ˆv + u. ˆ

(12.3f)

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Here symbol ˆ is used to separate functions from vectors. Constraint (12.3b) corresponds to the frequency restoration requirement. Constraints (12.3c)–(12.3f) are introduced by the definition of stationary point for the system (12.1). Since we are interested only in the control values uˆ the problem can be simplified. Vector ωˆ = 0 can be excluded. Matrix C is the incidence matrix. Therefore, sum of  ˆ rows of (12.3c) gives i∈N ( pˆ im + ri ) = 0, from (12.3e) and (12.3f) pˆ m = vˆ = u. Hence, problem (12.3) is reduced to the following convex optimization problem: 1 ˆ min uˆ  W u, uˆ 2

(12.4a)

 (uˆ i + ri ) = 0.

(12.4b)

i∈N

ˆ = 1 uˆ T W uˆ − λ1 ˆ T (uˆ + r ), where 1 is a vector Lagrange function is given by L(u, ˆ λ) 2 of ones. The corresponding stationary point is defined by the system of algebraic equations: (12.5) uˆ = W −1 1λˆ , 1 (W −1 1λˆ + r ) = 0. 

Solving this system gives uˆ ∗ = −W −1 1 11W r−1 1 . Substitution u(t) ≡ uˆ ∗ in (12.1) would make frequency deviations converge to zero; therefore properties (1) and (2) of the problem would be fulfilled. However, disturbance vector r might not be known. Therefore, we approximate it using the following technique. Recall that p e = C p. ˙ p e and p m are Thus, from Eq. (12.1a) we have r = M ω˙ + Dω + p e − p m . Here ω, unknown. Integration of this equation allows us to exclude ω˙ and p e : 



t

r dτ = M(ω(t) − ω(0)) + D

0

0

t



t

ω(τ )dτ +

 p e (τ )dτ −

0

t

p m (τ )dτ.

0

(12.6) Unknown p m is approximated by p˜ m using the following first-order equation: T˜ p˙˜ m = − p˜ m + u, T˜ = diag(T˜11 , . . . , T˜nn )  0.

(12.7)

Choice of the matrix T˜ will be discussed later. Considering the possible availability of the disturbance measurements we introduce disturbance approximation r˚ (t) of r : t ⎧ ⎨ Mii (ωi (t) − ωi (0))  t +mDii 0 ωi (τ )dτ + t e r˚i (τ )dτ = + 0 pi (τ )dτ − 0 p˜ i (τ )dτ, if riI = 0, i ∈ N . ⎩t 0 otherwise, 0 r i dτ, (12.8) As a result, it is possible to approximate not the disturbance itself, but its integral. Based on this information we perform a transition from the algebraic equations of the stationary point (12.5) to the following integral algebraic system: 

t

12 Mixed Feedback–Feedforward Frequency Control in Power Systems

u(t) = W −1 1λ(t), λ(t) = −



t

1 (u(τ ) + r˚ (τ ))dτ + λ0 .

101

(12.9)

0

This system represents control calculation part of the controller. Equations (12.7) and (12.8) define disturbance approximation part. Together Eqs. (12.7), (12.8), and (12.9) form the developed controller. For feedback operation that corresponds to first case (riI = 0) in Eq. (12.8) the developed controller uses a sum of PI controller of ω and integral controller of p e . For feedforward operation that corresponds to second case in Eq. (12.8) the developed controller uses integral controller of the disturbance.

12.5 Control Stability From (12.6) and (12.8) we obtain r˚i (t) = ri + (1 − riI )( pim (t) − p˜ im (t)), i ∈ N . As a result, value of u depends only on p m defined by Eqs. (12.1c) and (12.1d). Thus, in order to prove asymptotic stability of u(t) it is sufficient to prove asymptotic stability of the system (12.9), (12.1c), and (12.1d). This system contains both integral and differential equations. For simplicity they are all reduced to the differential ones: x˙ = Ax − R, (12.10) ⎞ ⎛ ⎞ ⎛  ⎞ −1 W −1 1 −1T 0 1 diag(r I ) λ 1 r ⎜ ⎟ ⎜ 0 ⎟ ⎜ pm ⎟ 0 −(T m )−1 (T m )−1 0 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ , A = ⎝ v −1 −1 x =⎝ ⎠, R = ⎝ 0 ⎠. v ⎠ (T ) W 1 0 −(T v )−1 0 p˜ m 0 0 0 −T˜ −1 T˜ −1 W −1 1 ⎛

Let (λ∗ , p m∗ , v∗ , p˜ m∗ ) be stationary point of (12.10). Then ( p m )∗ = ( p˜ m )∗ and u ∗ = W −1 λ∗ coincides with the solution of the problem (12.4). Therefore, stability proof is sufficient to for the developed control to satisfy all Problem requirements. Theorem 12.1 (Control stability) Let elements of T˜ from (12.7) satisfy inequalities T˜ii ≤ max{Tiim , Tiiv }, i ∈ N . Then system (12.10) is globally asymptotically stable. Proof Let us consider characteristic equation for the matrix A: det(A − I η) = 0. If we take determinant of the lower 3n × 3n block we get P1 (η) =

    1 1 1 − m −η − v −η − −η . Tii Tii T˜ii i∈G

Thus, for P1 (η) = 0 it is necessary η = Re η < 0. If P1 (η) = 0, then according to the Schur complement formula det(A − I η) = P1 (η)P2 (η), P2 (η) = −

  g n   y1i y2i 1 ri y3i , −η+ − wi wi (y1i + η)(y2i + η) y3i + η i=1 i=1

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where y1i = T1m , y2i = T1v , y3i = T˜1 , i ∈ N . Let us show that for Re η = α and ii ii ii I m η = β equations P2 (η) = 0 only has solutions with α < 0. We introduce new functions: z ki (α, β) =



yki (yki + α) + β j

= yki

yki + α (yki + α)2 + β 2

−



β (yki + α) + β 2

j , k ∈ {1, 2, 3}.

 (12.11) n 1  g ri  i i i z − η + (α, β)z (α, β) − z (α, β) . Let us Then P2 (η) = − i=1 1 3 i=1 wi  iwi  2 i i define ϕi (α, β) = Re z 1 (α, β)z 2 (α, β) − z 3 (α, β) . For β = 0:  ϕi (α, 0) =

y1i y2i

From (12.11) α = z 2i (α, 0) =

1 i (y1 + α)(y2i + α)

y1i z 1i (α,0)

 −

y3i = z 1i (α, 0)z 1i (α, 0) − z 3i (α, 0). y3i + α

− y1i . Then

z 1i (α, 0)y2i z 1i (α, 0)y3i i , z (α, 0) = . (12.12) z 1i (α, 0)(y2i − y1i ) + y1i 3 z 1i (α, 0)(y3i − y1i ) + y1i

We introduce auxiliary variables: u ik = z ki (α, 0) = auxiliary variables 0
0 and ψi is monotonously increasing. From ψi (1) = 0 we have ψi (u i1 )  0, u i1 ∈ (0, 1]. Therefore, ϕ(α, 0)  0, α  0. Additionally, u i1 = 1 if and only if α = 0, thus, ϕ(α, 0) < 0 for α > 0. From (12.11) α =

y2i z 2i (α,0)

− y2i .

Same derivation gives ϕ(α, 0) < 0 for α > 0 and y3i − y2i  0. Let us introduce new variables: k1i = y1i y2i (y1i + α)(y2i + α), c1i = (y1i + (β) 2 α) , c2i = (y2i + α)2 , c3i = (y3i + α)2 . Then ϕi (α, β) = (ci +β)(cfii +β)(c . From i +β) 1

2

3

f i (0) = ϕi (α, 0)  0 we have c3i k1i − k2i c1i c2i  0. Furthermore y3i > y1i ⇒ c3i > c1i ⇒ k1i c3i > k1i c1i ⇒ k1i c1i − k2i c1i c2i < k1i c3i − k2i c1i c2i < 0 ⇒ k1i − c3i − k2i c1i − k2i c2i < −c3i − k2i c1i < 0. Thus, all second-order coefficients of f i are negative and f i (0)  0 we have f i (β)  0. Then ϕi (α, β)  0 and n from 1 < 0. Therefore, equation P2 (η) = 0 has solution only if P2 (0) = − i=1 wi

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α < 0, thus, all real parts of eigenvalues of A are negative and system (12.10) is globally asymptotically stable if y3i − y1i  0 or y3i − y2i  0 or in original notations T˜ii ≤ max{Tiim , Tiiv }, i ∈ N . 

12.6 Numerical Experiment New England System is used for the numerical experiments [12]. Turbine and governor constants are taken from [13]. Partial outage of 100 MW appears on the generator 10; thus, each generator should increase its output by 10 MW. Three control types are considered: (1) widely used primary and secondary frequency controls [10]: u I + u I I ; (2) developed control: u; (3) developed control summed with primary frequency control: u + u I . Frequency responses are show in the Fig. 12.1. It can be seen that control u provides fast convergence speed; however, it does not reduce nadir in comparison to the traditional control. Numerical experiments show that the best frequency response can be obtained by adding primary frequency control to the developed one. Such modification does not improve convergence speed; however, reduces nadir. Primary frequency control is represented by the proportional controller and its signal converges to 0, thus, such modification does not change post-transient state of the system. 60.2 60.1

Control type

60

uI+uII u u+uI

59.9

Hz

59.8 59.7 59.6 59.5 59.4 59.3 59.2

0

10

20

30

40

50

t(s)

Fig. 12.1 System frequencies for different control types

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12.7 Conclusion Frequency in power system is restored if and only if sum of the control values is equal to the sum of elements of the disturbance vector; thus, every single frequency control approximates the disturbance size in some way. Here we explicitly separate disturbance approximation into a specific stage. Although the derived approximation depends on the state of the physical system, it converges to a disturbance vector that does not depend on the state. Compared to traditional control scheme, where only frequency deviations are used, such approach provides significantly more reliable input to the control calculation stage. Thus, control can provide faster response without stability loss. As a result, the derived approach provides fast frequency restoration using only frequency and electrical power, disturbance measurements, or combinations of them. Moreover, based on the type of the input it can operate as feedback, feedforward, or mixed type control. Global asymptotic stability of the controller is proven for the power system model with second-order turbine governor dynamics.

References 1. Liu, Q., Ili´c, M.D.: Enhanced automatic generation control (E-AGC) for future electric energy systems. In: IEEE Power and Energy Society General Meeting (2012). https://doi.org/10.1109/ PESGM.2012.6345137 2. Ilic, M.D.: From hierarchical to open access electric power systems. Proc. IEEE (2007). https:// doi.org/10.1109/JPROC.2007.894711 3. Andreasson, M., Dimarogonas, D.V., Johansson, K.H. Sandberg, H.: Distributed vs. centralized power systems frequency control. In: 2013 European Control Conference (2013). https://doi. org/10.23919/ECC.2013.6669721 4. Andreasson, M., Dimarogonas, D.V., Sandberg, H., Johansson, K.H.: Distributed control of networked dynamical systems: static feedback, integral action and consensus. IEEE Trans. Autom. Control (2014). https://doi.org/10.1109/TAC.2014.2309281 5. Li, N., Zhao, C., Chen, L.: Connecting automatic generation control and economic dispatch from an optimization view. IEEE Trans. Control Netw. Syst. (2016). https://doi.org/10.1109/ TCNS.2015.2459451 6. Stegink, T., De Persis, C., van der Schaft, A.: A unifying energy-based approach to stability of power grids with market dynamics. IEEE Trans. Autom. Control (2017). https://doi.org/10. 1109/TAC.2016.2613901 7. Zhang, X., Papachristodoulou, A.: A real-time control framework for smart power networks with star topology. In: American Control Conference (2013). https://doi.org/10.1109/ACC. 2013.6580624 8. Khamisov, O.O., Chernova, T.S., Bialek, J.W., Low, S.H.: Corrective Control: Stability analysis of unified controller combining frequency control and congestion management. In: NEIS 2018. Conference on Sustainable Energy Supply and Energy Storage Systems, pp. 1–6 (2018) 9. Dörfler, F., Bullo, F.: Kron reduction of graphs with application to electrical networks. In: IEEE Transactions on Circuits and Systems I: Regular Papers (2013). https://doi.org/10.1109/TCSI. 2012.2215780 10. Bergen, A.R., Vittal, V.: Power Systems Analysis, 2nd edn. Prentice Hall, Prentice (2000)

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11. Khamisov, O.O.: Direct disturbance based decentralized frequency control for power systems. In: 2017 IEEE 56th Annual Conference on Decision and Control (2017). https://doi.org/10. 1109/CDC.2017.8264139 12. Chow, J., Rogers, G.: Power system toolbox. Cherry Tree Scientific Software (2000) 13. Kundur, P.: Power System Stability and Control. McGraw-Hill, New York (1994)

Chapter 13

Diagonal Riccati Stability of a Class of Complex Systems and Applications Alexander Aleksandrov and Nadezhda Kovaleva

Abstract The problem of diagonal Riccati stability is studied for a class of complex systems describing the interaction of the second-order subsystems. It is assumed that the connection graph has a special structure and there is a constant delay in connections between the subsystems. Conditions under which the problem of diagonal Riccati stability for an original system can be reduced to that one for an auxiliary positive system are derived. If the auxiliary systems are diagonally Riccati stable, then there exist diagonal quadratic Lyapunov–Krasovskii functionals ensuring that zero solutions of the original complex systems are exponentially stable for any nonnegative delay. To demonstrate the effectiveness of the obtained results, some applications of the developed approaches to the stability analysis of mechanical and biological systems are presented.

13.1 Introduction Diagonal Lyapunov functions are effectively used for the stability analysis of wide classes of systems (see, for instance, [1–6] and the bibliography therein). However, methods for constructing such functions are well developed only for differential and difference systems. The problem of the existence of a diagonal Lyapunov–Krasovskii functional for a linear time-invariant system with constant delay was stated in [7]. It is known [7] that this problem can be reduced to finding a pair of diagonal positive definite matrices satisfying a Riccati matrix inequality. Therefore, systems possessing such functionals are called diagonally Riccaty stable (DRS) [8–11]. In [7, 8], constructively verifiable necessary and sufficient conditions of DRS were obtained for positive linear time-delay systems. This motivates the issue of determining classes of systems for which the DRS-problem is equivalent to that

A. Aleksandrov (B) · N. Kovaleva St. Petersburg State University, 7/9, Universitetskaya nab., St., Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_13

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one for auxiliary positive systems. Some approaches to finding such classes were suggested in [8, 10–13]. In this paper, a complex system describing interaction of subsystems of the second order with delay in connections between subsystems is studied. It is assumed that the connection graph has a special structure. Conditions under which the DRS-problem for the considered system can be reduced to that one for a positive system are derived. Furthermore, some applications of the obtained result are presented.

13.2 Notation and Problem Formulation Let R be the field of real numbers, R n and R n×n denote the vector spaces of n-tuples of real numbers and of n × n matrices, respectively,  ·  be the Euclidean norm of a n n refers to the nonnegative cone of the space R n and int R+ is vector. The notation R+ n n×n  the interior of R+ . Consider a matrix P ∈ R . Let P be the transpose of P. The notation P  0 (P ≺ 0) means that P is symmetric and positive (negative) definite. A matrix Q ∈ R n×n is Metzler if its off-diagonal entries are all nonnegative, Q is nonnegative if all of its entries are nonnegative. Let diag{λ1 , . . . , λn } be the diagonal matrix with the elements λ1 , . . . , λn along the main diagonal. Consider the linear difference-differential system x(t) ˙ = Ax(t) + Bx(t − τ ).

(13.1)

Here x(t) ∈ R n , A, B ∈ R n×n are constant matrices, τ is a constant nonnegative delay. We will assume that initial functions for (13.1) belong to the space PC([−τ, 0], R n ) of piecewise continuous functions ϕ(ξ ) : [−τ, 0] → R n with the uniform norm ϕτ = supξ ∈[−τ,0] ϕ(ξ ). Definition 13.1 (see [7, 11]) The system (13.1) is DRS if there exist diagonal matrices  and  such that   0,   0, A  + A +  + B−1 B   ≺ 0.

(13.2)

Remark 13.1 It is worth mentioning that (13.2) is a Riccati matrix inequality. Remark 13.2 It is known (see [7]) that if the system (13.1) is DRS, then it admits a diagonal quadratic Lyapunov–Krasovskii functional guaranteeing delay-independent exponential stability of (13.1). Definition 13.2 (see [14]) The system (13.1) is said to be positive if its state is nonnegative for any nonnegative initial function. Let us note that (13.1) is a positive system iff A is a Metzler matrix and B is a nonnegative matrix. In [7, 8], the following simple and constructive criterion of DRS was obtained for the case where the system (13.1) is positive.

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109

Proposition 13.1 A positive system (13.1) is DRS if and only if A + B is a Hurwitz matrix. Therefore, an interesting and important problem is that of finding classes of systems of the form (13.1) for which conditions of DRS are equivalent to those for auxiliary positive systems. Some approaches to the solution of the problem were developed in [9, 10, 12, 13]. In particular, the following result (see [9]) may be useful. Proposition 13.2 Let D = diag{d1 , . . . , dn }, E = diag{e1 , . . . , en } with di ∈ {−1; +1}, ei ∈ {−1; +1} for i = 1, . . . , n. Then the system (13.1) is DRS if and only if the system x(t) ˙ = D ADx(t) + D B E x(t − τ ) is DRS. The present paper is addressed to the solution of the above problem for the case where n = 2m, m is a positive integer, A = diag{A1 , . . . , Am },  Al =

a2l−1 2l−1 a2l−1 2l a2l 2l−1 a2l 2l

 ,

l = 1, . . . , m,

and B = {bi j }i,n j=1 with b2l−1 j = 0, bi,2l−1 = 0 for l = 1, . . . , m, i, j = 1, . . . , n. Then the system (13.1) takes the form x˙2l−1 (t) = a2l−1 2l−1 x2l−1 (t) + a2l−1 2l x2l (t), x˙2l (t) = a2l 2l−1 x2l−1 (t) + a2l 2l x2l (t) +

m 

b2l 2r x2r (t − τ ), l = 1, . . . , m.

r =1

(13.3) Hence, we will study a complex (multiconnected) system describing interaction of subsystems of the second order with delay in connections between the subsystems. Moreover, a special structure of connection graph is considered: it is assumed that interaction between the subsystems occurs through variables x2 (t), x4 (t), . . . , x2m (t), whereas variables x2l−1 (t) are associated only with x2l (t), l = 1, . . . , m. Our goal is to determine conditions under which the problem of DRS for (13.3) can be reduced to that one for an auxiliary positive system. In addition, we will consider some applications of the obtained results to the stability analysis of mechanical and biological systems.

13.3 DRS Analysis First, let us note that, with the aid of the approaches proposed in [9, 10], it can be shown that DRS-problem for (13.3) is equivalent to that one for the system

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x˙2l−1 (t) = a2l−1 2l−1 x2l−1 (t) + a¯ 2l−1 2l x2l (t), x˙2l (t) = a¯ 2l 2l−1 x2l−1 (t) + a2l 2l x2l (t) +

m 

b2l 2r x2r (t − τ ), l = 1, . . . , m,

r =1

where a¯ 2l−1 2l = |a2l−1 2l |, a¯ 2l 2l−1 = |a2l 2l−1 | for a2l−1 2l a2l 2l−1 ≥ 0; a¯ 2l−1 2l = a¯ 2l 2l−1 = 0 for a2l−1 2l a2l 2l−1 < 0, l = 1, . . . , m. m ˜ Next, construct the matrix  B = {b˜lr }l,r =1 with blr = b2l2r , l, r = 1, . . . , m.   e˜1 , . . . , e˜m } Assumption 13.1 There exist matrices D=diag{ d˜1 , . . . , d˜m }, E=diag{   is nonnegative. BE with d˜l , e˜l ∈ {−1; +1}, l = 1, . . . , m, such that the matrix D Applying Proposition 13.2, we obtain that, under Assumption 13.1, the system (13.3) is DRS iff the positive system x˙2l−1 (t) = a2l−1 2l−1 x2l−1 (t) + a¯ 2l−1 2l x2l (t), x˙2l (t) = a¯ 2l 2l−1 x2l−1 (t) + a2l 2l x2l (t) +

m 

|b2l 2r |x2r (t − τ ), l = 1, . . . , m,

r =1

possesses the same property. Remark 13.3 Conditions ensuring the fulfillment of Assumption 13.1 were found in [12]. However, these conditions are insufficiently constructive. Therefore, we will propose another, more constructive approach to the verification of Assumption 13.1. Theorem 13.1 The matrix  B satisfies Assumption 13.1 if and only if, for any tuple b˜r1 s1 , b˜r1 s2 , b˜r2 s2 , b˜r2 s3 , . . . , b˜rk sk , b˜rk s1

(13.4)

of entries of  B, the inequality b˜r1 s1 b˜r1 s2 b˜r2 s2 b˜r2 s3 . . . b˜rk sk b˜rk s1 ≥ 0

(13.5)

holds.   be nonnegative matrix. Then Proof Necessity. Let D BE b˜r s d˜r e˜s ≥ 0,

s, r = 1, . . . , m.

Hence b˜r1 s1 d˜r1 e˜s1 b˜r1 s2 d˜r1 e˜s2 b˜r2 s2 d˜r2 e˜s2 b˜r2 s3 d˜r2 e˜s3 . . . b˜rk sk d˜rk e˜sk b˜rk s1 d˜rk e˜s1 = b˜r1 s1 b˜r1 s2 b˜r2 s2 b˜r2 s3 . . . b˜rk sk b˜rk s1 d˜r21 e˜s21 d˜r22 e˜s22 . . . d˜r2k e˜s2k

(13.6)

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= b˜r1 s1 b˜r1 s2 b˜r2 s2 b˜r2 s3 . . . b˜rk sk b˜rk s1 ≥ 0. Sufficiency. Without loss of generality, we assume that in each row of  B there are at least two nonzero entries. Let b˜1 s11 , . . . , b˜1 s1 p1 be nonzero elements of the first row. Choose d˜1 = 1 and define e˜s11 , . . . , e˜s1 p1 according to the condition (13.6). Check if b˜kl = 0 exists such that k > 1, l ∈ {s11 , . . . , s1 p1 }. In the case where there are no such an entry, we pass to the next row of  B. Otherwise, denote indices of the corresponding entry by k1 , s21 . Using the condition b˜k1 s21 d˜k1 e˜s21 > 0, choose d˜k1 ∈ {−1; +1}, and after that determine values of e˜l for the remaining nonzero elements b˜k1 s22 , . . . , b˜k1 s2 p2 of the k1 th row. Next, check if b˜kl = 0 exists such that k > 1, k = k1 , l ∈ {s11 , . . . , s1 p1 , s21 , . . . , s2 p2 }. We continue this process until we determine all values of d˜k and e˜l corresponding to nonzero entries of the matrix  B. Let us note that if, at some step of this algorithm, we come across a nonzero component b˜kl for which the values of d˜k and e˜l have already been determined, then such a component should belong to a tuple of the form (13.4) with nonzero components and with given values of d˜r1 , e˜s1 , d˜r2 , e˜s2 , . . . , d˜rk , e˜sk . Then from the  condition (13.5), it follows that b˜kl d˜k e˜l > 0.

13.4 A System with the Closed Feedback Loop In this section, we will apply the obtained results to the DRS analysis of a system with a special structure of connections. Let the system (13.3) be of the form x˙2l−1 (t) = a2l−1 2l−1 x2l−1 (t) + a2l−1 2l x2l (t), x˙2l (t) = a2l 2l−1 x2l−1 (t) + a2l 2l x2l (t)

(13.7)

+b2l 2l−2 x2l−2 (t − τ ) + b2l 2l+2 x2l+2 (t − τ ), l = 1, . . . , m. Here, x0 (t) = x2m (t), x2m+2 (t) = x2 (t), b2m 2m+2 = 0, and the rest notation is the same as for (13.3). The connection graph for (13.7) is presented in Fig. 13.1. In this case the matrix  B has the form ⎛ ⎞ 0 b˜12 0 0 0 · · · 0 b˜1m ⎜ b˜ 0 b˜ 0 0 · · · 0 0 ⎟ ⎜ 21 ⎟ 23 ⎜ ⎟ 0 ⎟ ⎜ 0 b˜32 0 b˜34 0 · · · 0 ⎜  B=⎜ . . . . . . .. .. ⎟ ⎟. ⎜ .. .. .. .. .. . . . . ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 · · · 0 b˜m−1 m ⎠ 0 0 0 0 0 · · · b˜m m−1 0

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Fig. 13.1 Structure of connections for the system (13.7)

Let us verify conditions of Theorem 13.1 for this matrix. If m is an odd number, then any tuple of the form (13.4) contains at least one zero element, and the inequality (13.5) is valid. In the case where m is an even number, the matrix  B can admit only one tuple b˜1m , b˜12 , b˜32 , b˜34 , b˜54 , . . . , b˜m−1 m with all nonzero elements. Thus, we arrive at the following theorem. Theorem 13.2 If one of the conditions (i) m is an odd number, (ii) m is an even number and b˜1m b˜12 b˜32 b˜34 b˜54 . . . b˜m−1 m ≥ 0 is fulfilled, then the DRS-problem for (13.7) is equivalent to that one for the positive system x˙2l−1 (t) = a2l−1 2l−1 x2l−1 (t) + a¯ 2l−1 2l x2l (t), x˙2l (t) = a¯ 2l 2l−1 x2l−1 (t) + a2l 2l x2l (t) +|b2l 2l−2 |x2l−2 (t − τ ) + |b2l 2l+2 |x2l+2 (t − τ ), l = 1, . . . , m.

13.5 Applications Next, consider some applications of the obtained results.

13.5.1 A Mechanical System Let motions of a mechanical system be defined by the equations

(13.8)

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q¨2l−1 (t) + h q˙2l−1 (t) − a2l−1 2l−1 q2l−1 (t) − a2l−1 2l q2l (t) = 0, q¨2l (t) + h q˙2l (t) − a2l 2l−1 q2l−1 (t) − a2l 2l q2l (t)

(13.9)

= b2l 2l−2 q2l−2 (t − τ ) + b2l 2l+2 q2l+2 (t − τ ), l = 1, . . . , m. Here, q(t) = (q1 (t), . . . , q2m (t)) and q(t) ˙ = (q˙1 (t), . . . , q˙2m (t)) are vectors of generalized coordinates and velocities, q0 (t) = q2m (t), q2m+2 (t) = q2 (t); ar s , br s are constant coefficients with b2m 2m+2 = 0; h is a positive parameter (damping coefficient); τ is a constant nonnegative delay. The variables q2l−1 and q2l can be treated as coordinates of bearing bodies and carrying bodies, respectively, l = 1, . . . , m. We assume that bearing bodies interact only with carrying ones, while connections between carrying bodies form a closed feedback loop. With the aid of Theorem 13.2 and results of [15], it can be shown that the following theorem is valid. Theorem 13.3 Let the conditions of Theorem 13.2 be fulfilled and the auxiliary positive system (13.8) be DRS. Then there exists a number hˆ > 0 such that the equilibrium position q = q˙ = 0 of (13.9) is asymptotically stable for any h ≥ hˆ and any τ ≥ 0.

13.5.2 A Model of Population Dynamics Consider a Lotka–Volterra system of the form

x˙2l−1 (t) = x2l−1 (t) c2l−1 + a2l−1 2l−1 x2l−1 (t) + a2l−1 2l x2l (t) , x˙2l (t) = x2l (t) c2l + a2l 2l−1 x2l−1 (t) + a2l 2l x2l (t)

+b2l 2l−2 x2l−2 (t − τ ) + b2l 2l+2 x2l+2 (t − τ ) , l = 1, . . . , m.

(13.10)

Here xi (t) is the population density of the ith species, cr , ar s , br s are constant coefficients, τ is a constant nonnegative delay. We assume that x0 (t) = x2m (t), x2m+2 (t) = x2 (t), b2m 2m+2 = 0. This system describes interaction of species in a biological community [2, 16, 17]. If b2l 2l−2 = b2l 2l+2 = 0 for l = 1, . . . , m, then the system (13.10) is decomposed into isolated second-order subsystems modeling pairwise interactions between species with densities x2l−1 (t) and x2l (t), l = 1, . . . , m. These interactions may be of competition, prey-predator or symbiosis type (see [16]). At the same time, species with densities x2 (t), x4 (t), . . . , x2m (t) form a trophic food chain with a closed feedback loop. Using Theorem 13.2 and results of [8, 10], we arrive at the following theorem.

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n Theorem 13.4 Let the system (13.10) admit an equilibrium position z ∈ int R+ . If the conditions of Theorem 13.2 are fulfilled and the positive system (13.8) is DRS, then n for any τ ≥ 0. the equilibrium position is globally asymptotically stable in int R+

13.6 Conclusion In the present contribution, new DRS conditions are found for a class of multiconnected systems with constant delay. Our approach is based on the reducing DRSproblem for an original system to that one for an auxiliary positive system. Some applications of the obtained results are given. In future work, the authors wish to use the developed approaches in formation control problems. Acknowledgements This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Project No. 075-15-2021-573).

References 1. Zubov, V.I.: Asymptotic stability with respect to a first approximation in the broad sense. Dokl. Math. 53(1), 31–32 (1996) 2. Kaszkurewicz, E., Bhaya, A.: Matrix Diagonal Stability in Systems and Computation. Birkhauser, Boston (2000) 3. Mason, O., Shorten, R.: On the simultaneous diagonal stability of a pair of positive linear systems. Linear Algebra & Appl. 413, 13–23 (2006) 4. Pastravanu, O.C., Matcovschi, M.-H.: Max-type copositive Lyapunov functions for switching positive linear systems. Automatica 50, 3323–3327 (2014) 5. Dirr, G., Ito, H., Rantzer, A., Ruffer, B.S.: Separable Lyapunov functions for monotone systems: constructions and limitations. Discrete & Cont. Dyn. Syst. - Series B 20(8), 2497–2526 (2015) 6. Provotorov, V.V., Ryazhskikh, V.I., Gnilitskaya, Yu.A.: Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain. Vestnik of Saint Petersburg University. Appl. Math. Comput. Sci. Control Proc.13(3), 264–277 (2017) 7. Mason, O.: Diagonal Riccati stability and positive time-delay systems. Syst. Control Lett. 61, 6–10 (2012) 8. Aleksandrov, A., Mason, O.: Diagonal Riccati stability and applications. Linear Algebra & Appl. 492, 38–51 (2016) 9. Aleksandrov, A., Mason, O., Vorob’eva, A.: Diagonal Riccati stability and the Hadamard product. Linear Algebra & Appl. 534, 158–173 (2017) 10. Aleksandrov, A.Yu., Vorob’eva, A.A., Kolpak, E.P.: On the diagonal stability of some classes of complex systems. Vestnik of Saint Petersburg University. Appl. Math. Comput. Sci. Control Proc. 14(2), 72–88 (2018) (in Russian) 11. Aleksandrov, A., Kovaleva, N.: Diagonal Riccati stability of a class of time-delay systems. Cybern. Phys. 7(4), 167–173 (2018) 12. Vorob’eva, A.A.: Diagonal stability conditions for matrices with a special structure. Control Proc. Stabil. 5(1), 59–63 (2018) (in Russian) 13. Shen, J., Lam, J.: On the algebraic Riccati inequality arising in cone-preserving time-delay systems. Automatica 113, 108820 (2020) 14. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000)

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15. Aleksandrov, A.Yu., Zhabko, A.P., Chen, Y.: Stability analysis of gyroscopic systems with delay via decomposition. AIP Conf. Proc. 1959(080002), 1–6 (2018) 16. Britton, N.F.: Essential Mathematical Biology. Springer Press, London (2003) 17. Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998)

Part II

Dynamic Systems Theory

Chapter 14

Synchronization in Feedback Cyclic Structures of Oscillators with Hysteresis Alexander M. Kamachkin, Dmitriy K. Potapov, and Victoria V. Yevstafyeva

Abstract We study synchronous processes in a complex system with cyclic links. The system represents some hysteresis-feedback oscillators coupled in a ring. Besides, each oscillator has an additional feedback with the next oscillator. We establish sufficient conditions for the existence of the periodic or recurrent motions of the system. The periodic motion corresponds to a synchronous process in the system. We obtain conditions for synchronization to arise and also conditions for the stability of synchronous oscillatory processes in two special cases.

14.1 Introduction and Statement of Problem In the paper, we study a complex dynamical system. To investigate this system, we need not only give initial conditions, but also describe the sequence for the system members to respond. The member of the system is an oscillator that contains an integrator and a feedback unit with relay hysteresis. The case when the oscillators are coupled in a ring and such a system has a common half-period is of particular interest for applications. That is because all the members in the ring are synchronized after this system is turned on (see papers [1–3, 8]). Such structures that produce the synchronous processes are often used in industry (for example, in designing delivery devices), in medicine and biology (in designing biological and chemical oscillators). A mathematical model of the system has the form

A. M. Kamachkin (B) · D. K. Potapov · V. V. Yevstafyeva St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg, 199034, Russia e-mail: [email protected] D. K. Potapov e-mail: [email protected] V. V. Yevstafyeva e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_14

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⎧ ⎪ y˙1 (t) = − T21 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y˙2 (t) = − T22 ⎪ ⎪ ⎨ y˙3 (t) = − T23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y˙ (t) = − 2 n Tn

(u 1 (y1 (t)) + γ u n (yn (t)) − γ u 2 (y2 (t))) , (u 2 (y2 (t)) + γ u 1 (y1 (t)) − γ u 3 (y3 (t))) , (u 3 (y3 (t)) + γ u 2 (y2 (t)) − γ u 4 (y4 (t))) ,

(14.1)

...................... (u n (yn (t)) + γ u n−1 (yn−1 (t)) − γ u 1 (y1 (t))) ,

where T1 , T2 , . . . , and Tn are half-periods of oscillators, γ ∈ R\{0}, ⎧ ⎪ −1 ⎪ ⎪ ⎨ u i (yi (t)) = 1 ⎪ ⎪ ⎪ ⎩ 0 ui

if (yi (t) < 1 ∧ u i (t − 0) = −1) ∨ (yi (t) = −1), if (yi (t) > −1 ∧ u i (t − 0) = 1) ∨ (yi (t) = 1),

(14.2)

if yi (τ ) ∈ (−1, 1) for any τ ∈ [t0 , t],

u i0 ∈ {−1, 1} for any i = 1, n. Expression (14.2) describes how a hysteresis-feedback unit responds (switches) in the ith oscillator. On the planes (yi Ou i ), where i = 1, n, the hysteresis loop is bypassed counterclockwise. It is symmetric since relay thresholds are yi = ±1 and outputs are u i = ±1. All the oscillators are connected in a ring. Therefore, the first equation of system (14.1) is determined by the sum of the values u 1 (t) and γ u n (t) at the input of the integrator in the first oscillator. Here and elsewhere, γ is the gain. The second oscillator gets the signal γ u 1 (t) and transmits the signal γ u 2 to the third oscillator, etc. At the same time, each ith member transmits the signal γ u i to the (i + 1)th member of the ring and gets the signal −γ u i+1 . If the oscillators are n . In other words, controllers, one should know the switching sequence {u i (yi (t))}i=1 the order of relay switchings in the feedbacks of oscillators has a significant influence on the solution of (14.1) and (14.2). Note that if n = 2 in system (14.1) and one of the oscillators has constant parameters, then its output is a periodic function that enters the input of the other oscillator. Such systems (the oscillators under periodic external forcing) have been successfully investigated for a long time (for example, see [8]). The latest papers in this direction we would like to mention are [4–7, 9, 10].

14.2 General Dynamical Properties First, we examine the general properties of system (14.1). Let for any switching n no oscillator can switch twice until all the other oscillators switch. sequence {u i }i=1 On the one hand, this assumption narrows the class of systems. On the other hand, it allows us to carry out an analytical research. System (14.1) represents the class of dynamical systems. We define the types of motions that can exist in such systems. Let

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Y = (y1 , y2 , . . . , yn ) ∈ E n , where E n is the n-dimensional Euclidean space. Then Q = {Y ∈ E n : yi ∈ [−1, 1] for any i = 1, n} is a closed cube in E n . We have Theorem 14.1 Let the parameters T1 , T2 , . . . , Tn , and γ of system (14.1) be chosen so that the phase trajectories remain in the set Q for any initial time starting from some finite value t. Then all the solutions Y of system (14.1) with the initial point Y0 ∈ Q are recurrent functions. In addition, there exists at least one periodic solution located in Q. Proof Considering the right-hand sides of (14.1), one may state that for any admisn sible switching sequence {u i }i=1 , the phase trajectories consist of some segments of spatial lines. If, moving along one line segment, the representative point reaches the hyperplane yi = 1 or yi = −1 (i = 1, n), then the representative point goes to another line segment and moves along this segment until it reaches the other hyperplane. This motion is independent of the initial point Y0 = (y10 , y20 , . . . , yn0 ) and the n . The values T1 , T2 , . . . , Tn , and γ are system parameters on initial sequence {u i0 }i=1 which the spatial location of these line segments depends. Due to the assumption regarding the switchings of all the oscillators coupled in the cyclic structure (the ring), we may claim the following: if Y0 ∈ Q, then Y (t, t0 , Y0 ) ∈ Q for any t > t0 . The solution Y (t, t0 , Y0 ) is a superposition of a finite number of mappings for one fixed face of the cube Q into itself. Indeed, let the point Y0 lie on a face of the cube Q that, for example, belongs to the hyperplane yi = 1. n be an initial sequence, where u i0 = −1 or u i0 = 1. Having started a Let {u i0 }i=1 motion at Y0 , the representative point moves along a line segment corresponding to n until it reaches the hyperplane yi = −1 or yi = 1. Then the output u i varies, {u i0 }i=1 i.e., switching of the relay occurs. As a result, the representative point continues its n . Obviously, the motion along the next line segment due to another sequence {u i }i=1 mapping of each face of the cube Q by virtue of one sequence is a continuous function of the initial point. Thus, the mapping Y (t, t0 , Y0 ) of a cube face is a superposition n for a finite number m of continuous mappings. We remind that the sequence {u i }i=1 consists of the values u i = 1 or u i = −1. We see that m ≤ (2n)! . Therefore, we have (n!)2 the continuous mapping Y (t, t0 , Y0 ) on a compact set, which is a face of the cube Q. The function Y (t, t0 , Y0 ) maps any face of Q into itself. Thus, the motion that starts at any point Y0 ∈ Q remains in the set Q. This means that Q is an invariant set of the system. Then there exists at least one periodic motion lying in Q entirely. Next, we consider the other motions that lie in Q. The function Y (t, t0 , Y0 ) with Y0 ∈ Q satisfies all the properties of dynamical systems. The set Q is a non-empty closed invariant set of the dynamical system Y (t, t0 , Y0 ). Moreover, Q is the minimal set since it does not have a subset with the same properties. According to Birkhoff’s theorem, each motion starting in such a set is recurrent. The recurrent motion corresponds to the recurrent function, that is, the solution to system (14.1). Theorem 14.1 is proved.   Thus, in a continuum of the bounded recurrent solutions to (14.1), there exists at least one periodic solution. The periodic motion corresponds to the synchronous

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process in system (14.1) when all the oscillators in the ring have a common halfperiod. Keeping synchronization as t → +∞ means that the periodic motion is stable. Further, we discuss how to identify the periodic solutions. Note that the general theory of dynamical systems provides no constructive methods to do this.

14.3 Sufficient Conditions for Synchronization The phase trajectories of system (14.1) consisting of the spatial line segments fill the closed n-dimensional cube Q. The objective of this section is to identify at least one periodic trajectory from the continuum considered in Sect. 14.2. We give some examples of cyclic structures with three oscillators and investigate the dynamics of these oscillatory systems. Case 1. Let n = 3. We denote the oscillators by O1 , O2 , and O3 . The switching 3 contains the elements u 1 , u 2 , and u 3 such that each of them cannot sequence {u i }i=1 vary its value twice until the two others do. Let tki be the kth switching time of the ith oscillator and tk1 < tk2 < tk3 . In this case, the oscillator O1 lags behind O2 that lags behind O3 . The switching sequence can be represented as {u 1 ↑ u 2 ↓ u 3 ↑ . . . }, where the symbol ↑ indicates the switching from −1 to 1 and the symbol ↓ does i − tki = Ti = 1, where vice versa. To simplify, we put T1 = T2 = T3 = 1. Then tk+1 i = 1, 3. We integrate system (14.1) and consider the periodic solutions to this system. Let t0 = tk1 and y10 = 1. Then we have the function y1 (t) such that y1 (2T1 ) = y1 (2) = 1. Next, we use both the periodicity and identity of all the oscillators to integrate the second and third equations of (14.1) and shift the initial conditions in time. We have y2 (t0 ) = y2 (tk2 ) = −1 then y2 (2T2 ) = y2 (2) = −1 and y3 (t0 ) = y3 (tk3 ) = 1 then y3 (2T3 ) = y3 (2) = 1. Integrating the first equation of system (14.1), we have the 1 ], following subintegral functions: 1 − 2γ on [tk1 , tk2 ], 1 on [tk2 , tk3 ], 1 + 2γ on [tk3 , tk+1 1 2 2 3 3 1 −1 + 2γ on [tk+1 , tk+1 ], −1 on [tk+1 , tk+1 ], and −1 − 2γ on [tk+1 , tk+1 + 1]. Inte1 ], grating the second equation of (14.1), we have −1 + 2γ on [tk2 , tk3 ], −1 on [tk3 , tk+1 1 2 2 3 3 1 −1 − 2γ on [tk+1 , tk+1 ], 1 − 2γ on [tk+1 , tk+1 ], 1 on [tk+1 , tk+1 + 1], and 1 + 2γ on 1 2 + 1, tk+1 + 1]. Finally, integrating the third equation of (14.1), we have 1 − 2γ [tk+1 3 1 1 2 2 3 3 1 , tk+1 ], 1 + 2γ on [tk+1 , tk+1 ], −1 + 2γ on [tk+1 , tk+1 + 1], on [tk , tk+1 ], 1 on [tk+1 1 2 2 3 −1 on [tk+1 + 1, tk+1 + 1], and −1 − 2γ on [tk+1 + 1, tk+1 + 1]. From necessary conditions for the existence of the periodic solution, we obtain ⎧ 1 3 1 ) + 2γ (t 1 3 ⎪ T = 1 = (1 − 2γ )(tk+1 − tk1 ) − 2(1 + γ )(tk+1 − tk+1 ⎪ k+1 − tk ), ⎨ 1 2 3 2 ) − 2γ (t 3 − t 2 ), T2 = 1 = tk+1 − tk2 + 2γ (tk+1 − tk+1 k k ⎪ ⎪ ⎩T = 1 = t 3 − t 3 − 2γ (t 3 − t 1 ) + 2γ (t 3 − t 1 ). 3 k+1 k k+1 k+1 k k

(14.3)

Let tk2 − tk1 = tk3 − tk2 = ξ T , where ξ is the lag coefficient, T is the half-period of the synchronous mode. In this case, we have the same lag ξ T . Then from (14.3), it

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follows that ⎧ ⎪ 1 = (1 − 2γ )T − 2(1 + γ )2ξ T + 2γ (T − 2ξ T ), ⎪ ⎪ ⎨ 1 = T + 2γ ξ T − 2γ ξ T, ⎪ ⎪ ⎪ ⎩ 1 = T − 4γ ξ T + 4γ ξ T.

(14.4)

From the second and third equations of (14.4), we have T = 1. From the first equation when T = 1, we obtain ξ(1 + 2γ ) = 0 for any ξ ∈ (0, 21 ). Thus γ = − 21 . If put different lags, namely, tk2 − tk1 = ξ1 T and tk3 − tk2 = ξ2 T , where ξ1 , ξ2 ∈ (0, 21 ), i.e. ξ1 + ξ2 < 1, we obtain the same result from (14.3). We turn to the question on stability of the synchronous mode. Define the phase variables θki = tki − kT − (i − 1)ξ T , where ξ ∈ (0, 21 ) and i = 1, 2, 3. Then we denote θk by (θk1 , θk2 , θk3 )∗ (∗ means transposition). Let the lags be the same and tki = θki + k + (i − 1)ξ . We substitute it in (14.3). Then we obtain the system describing the iterative process θk+1 = Aθk + B, where ⎛

⎞ 8γ 2 + 2γ + 1 2(1 − 4γ 2 ) 0 ⎜ ⎟ 3 3 ⎜ ⎟ 8γ 2 (2γ + 1) ⎟ ⎜ 8γ 2 (2γ + 1) A=⎜ ⎟, 1 − ⎜ 3(2γ − 1) 3(2γ − 1) ⎟ ⎝ ⎠ 4γ (2γ + 1) 8γ 2 + 4γ − 3 0− 3 3

∗ 4ξ(2γ − 1)(2γ + 1) 16γ 2 ξ(2γ + 1) 8γ ξ(2γ + 1) B= − ,− ,− . 3 3(2γ − 1) 3 ⎛ ⎞ 2/3 0 0 We substitute γ = − 21 in A and B. Then A = ⎝ 0 1 0⎠ and B = (0, 0, 0)∗ . The 0 01 matrix A has the eigenvalues λ1 = 23 and λ2,3 = 1. This means a compressibility as to the component θk1 and no divergence as to the components θk2 , θk3 , i.e., the synchronous mode is stable. Thus, the below theorem holds. Theorem 14.2 Let n = 3 and T1 = T2 = T3 = 1. Let O1 lag with respect to (w.r.t.) O2 and O2 lag w.r.t. O3 . Then system (14.1) can be synchronized with half-period T = 1 for any the same lag ξ T provided ξ ∈ (0, 21 ) and γ = − 21 . Case 2. Consider another switching sequence {u 1 ↑ u 3 ↓ u 2 ↑ . . . }. Here tk1 < tk3 < tk2 . Let T1 = T2 = T3 = 1. Integrating (14.1), for the solution y1 (t), we obtain 1 + 1 1 3 ], −1 − 2γ on [tk+1 , tk+1 ], −1 2γ on [tk1 , tk3 ], 1 on [tk3 , tk2 ], 1 − 2γ on [tk2 , tk+1 3 2 2 1 on [tk+1 , tk+1 ], and −1 + 2γ on [tk+1 , tk+1 + 1]. For y2 (t), we have 1 + 2γ on

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1 1 3 3 2 2 1 [tk2 , tk+1 ], 1 on [tk+1 , tk+1 ], 1 − 2γ on [tk+1 , tk+1 ], −1 − 2γ on [tk+1 , tk+1 + 1], −1 1 3 3 2 on [tk+1 + 1, tk+1 + 1], and −1 + 2γ on [tk+1 + 1, tk+1 + 1]. For y3 (t), we have 1 1 3 3 2 ], −1 + 2γ on [tk+1 , tk+1 ], 1 + 2γ on [tk+1 , tk+1 ], −1 − 2γ on [tk3 , tk2 ], −1 on [tk2 , tk+1 2 1 1 3 1 on [tk+1 , tk+1 + 1], and 1 − 2γ on [tk+1 + 1, tk+1 + 1]. As a result, for the three oscillators, we have

⎧ 1 3 1 ⎪ T1 = 1 = tk+1 − tk1 + 2γ (tk3 − tk1 ) − 2γ (tk+1 − tk+1 ), ⎪ ⎪ ⎨ 2 2 1 T2 = 1 = tk+1 − tk2 − 2γ (tk2 − tk1 ) + 2γ (tk+1 − tk+1 ), ⎪ ⎪ ⎪ ⎩ 3 2 3 T3 = 1 = tk+1 − tk3 − 2γ (tk2 − tk3 ) + 2γ (tk+1 − tk+1 ).

(14.5)

Let T be a new half-period. Then tk3 − tk1 = ξ T and tk2 − tk3 = ξ T , where ξ ∈ (0, 21 ). From (14.5), we obtain T = 1 for γ ∈ R\{0}. From this, it follows that tk1 = θk1 + k, tk2 = θk2 + k + 2ξ , and tk3 = θk3 + k + ξ . We substitute them in (14.5) and obtain the linear system θk+1 = Aθk , where A = E. Hence, we have the constant shift in time along all the components θki (i = 1, 2, 3) and thus the stable synchronous mode. In view of the above, we formulate the theorem as follows: Theorem 14.3 Let n = 3 and T1 = T2 = T3 = 1. Let there be a lag of O1 w.r.t. O3 and a lag of O3 w.r.t. O2 . Then system (14.1) can be synchronized with half-period T = 1 for any γ ∈ R\{0} and either for the same lag ξ T or the different lags ξ1 T and ξ2 T , where ξ , ξ1 , ξ2 ∈ (0, 21 ). Remark 14.1 Comparing Theorems 14.2 and 14.3, we see that the switching sequences change significantly the dynamics of the system at synchrony.

14.4 Conclusion Based on the necessity for the existence of the synchronous mode, we have obtained the conditions on the parameters T , γ , and ξ , which are sufficient conditions for the existence of the stable synchronous process in system (14.1). We have considered the system of the 3th order as an example. The cyclic structures with the same lag or the different lags of switchings have been examined. Generally speaking, the dimension of system (14.1) can be increased, and hence the dimensions of systems (14.3), (14.5) can be increased as well. However, there will be some technical difficulties with the description of the lags, especially if Ti is sufficiently small. In [8], the conditions for the existence of synchronous modes with the other half-periods were obtained.

References 1. Balanov, Z., Krawcewicz, W., Rachinskii, D., Zhezherun, A.: Hopf bifurcation in symmetric networks of coupled oscillators with hysteresis. J. Dyn. Differ. Equ. 24, 713–759 (2012)

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2. Barron, M.A., Sen, M.: Synchronization of four coupled van der Pol oscillators. Nonlin. Dyn. 56, 357–367 (2009) 3. Hooton, E., Balanov, Z., Krawcewicz, W., Rachinskii, D.: Noninvasive stabilization of periodic orbits in O4 -symmetrically coupled systems near a Hopf bifurcation point. Int. J. Bifurcation Chaos 27(1750087), 1–18 (2017) 4. Kamachkin, A.M., Potapov, D.K., Yevstafyeva, V.V.: Existence of periodic solutions to automatic control system with relay nonlinearity and sinusoidal external influence. Int. J. Robust Nonlinear Control 27, 204–211 (2017) 5. Kamachkin, A.M., Potapov, D.K., Yevstafyeva, V.V.: Existence of subharmonic solutions to a hysteresis system with sinusoidal external influence. Electron J. Differ. Equ. 140, 1–10 (2017) 6. Kamachkin, A.M., Potapov, D.K., Yevstafyeva, V.V.: On uniqueness and properties of periodic solution of second-order nonautonomous system with discontinuous nonlinearity. J. Dyn. Control Syst. 23, 825–837 (2017) 7. Kamachkin, A.M., Potapov, D.K., Yevstafyeva, V.V.: Existence of periodic modes in automatic control system with a three-position relay. Int. J. Control 93, 763–770 (2020) 8. Varigonda, S., Georgiou, T.T.: Dynamics of relay relaxation oscillators. IEEE Trans. Automat. Control 46, 65–77 (2001) 9. Yevstafyeva, V.V.: On existence conditions for a two-point oscillating periodic solution in an non-autonomous relay system with a Hurwitz matrix. Automat. Remote Control 76, 977–988 (2015) 10. Yevstafyeva, V.V.: Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of eigenvalue zero. Ukr. Math. J. 70, 1252–1263 (2019)

Chapter 15

Optimal Damping Problem for Diffusion-Wave Equation Sergey Postnov

Abstract In this paper, we consider a model system, which is defined by a onedimensional non-homogeneous diffusion-wave equation. For such system, we investigate an optimal damping problem as an optimal control problem of the following type: we need to transfer the system from the given initial state to the final state with zero time derivative of the system state. The two types of optimality analyzed were: control norm minimization at given control time and time-optimal control search at given restriction on control norm. In the general case, we consider both boundary and distributed controls, which are p-integrable functions (including p = ∞). We use an explicit solution for diffusion-wave equation in order to reduce the optimal control problem to an infinite-dimensional l-problem of moments. We also derived the finite-dimensional l-problem of moments using an approximate solution of the diffusion-wave equation. For this problem, the correctness and solvability are analyzed.

15.1 Introduction Fractional-order models with distributed parameters are widely used now for the description of heat and mass transfer processes, diffusion in non-homogeneous media, diffusion-wave and wave phenomena in complex multi-component systems, oscillations of non-homogeneous bodies, etc. Later, for such models, optimal control problems were discussed using variational methods [1–3] with constraints on quadratic functional including both of system state and control. In paper [4, 5], the moment method was applied for optimal control problem investigation with constraints on control norm in case of systems described by anomalous diffusion equation.

S. Postnov (B) V.A. Trapeznikov Institute of Control Sciences of RAS, 65, Profsoyuznaya str., 117997 Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_15

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In this paper, we learn linear non-homogeneous diffusion-wave equation and investigate an optimal control problem with constraints on control norm. Both distributed and boundary controls were considered, which supposed to be p-integrable functions at the segment. Unlike the aforementioned publications, we consider below the problem of optimal damping, i.e., the optimal control problem, where the final state defined by the condition of first derivative of system state vanishing. This paper is organized as follows. In Sect. 15.2, the formal statement of optimal damping problem is given. In Sect. 15.3, the posed problem was reduced to the generalized infinite-dimensional l-problem of moments using an explicit solution of diffusion-wave equation. In Sect. 15.4, an approximate solution for the equation is used for construction of finite-dimensional l-problem of moments for which correctness and solvability proved. The last section contains the conclusion.

15.2 Problem Statement The model system investigated in this paper is described by the following equation:   ∂ Q(x, t) ∂ w(x) − q(x)Q(x, t) + f (x, t) + u(x, t), ∂x ∂x (15.1) where Q(x, t)—system state, C0 Dtα —Caputo left-side fractional time derivative operator, α ∈ (1, 2), t ≥ 0, x ∈ [0, L], (x, t) ∈  = [0, L] × [0, ∞). For left-side Caputo derivative, we use the following definition [6, Sect. 2.4]: r (x)

C α 0 Dt Q(x, t)

=

 C α 0 Dt Q(x, t)

=0R L

Dtα

Q(x, t) −

[α]  ∂ k Q(x, 0+) t k

∂t k

k=0



k!

,

(15.2)

where 0R L Dtα is the left-side Riemann–Liouville derivative RL α 0 Dt Q(x, t)

∂ [α]+1 1 = (1 − {α}) ∂t [α]+1

t 0

Q(x, τ )dτ . (t − τ ){α}

Function Q(x, t) allowed to be differentiable by time at positive time half-axis and twice differentiable by space variable at segment [0, L]. Functions r (x) > 0, w(x) > 0, and q(x) allowed to be continuous at segment [0, L]. Perturbation f (x, t) allowed to be summable by both of their variables on domain . We will suppose that distributed control u(x, t) belongs either to the space L p1 , p2 (), 1 < p1,2 < ∞ of functions which are p1 -integrable by time and p2 -integrable by space variable or to the space L ∞ () ( p1 = p2 = ∞). Equation (15.1) we will call diffusion-wave equation. Let the initial conditions for Eq. (15.1) is defined as follows:

15 Optimal Damping Problem for Diffusion-Wave Equation

∂ k Q(x, 0+) = ϕ k (x), x ∈ [0, L], k = 0, 1. ∂t k

129

(15.3)

The boundary conditions for Eq. (15.1) will be chosen in the following form: 

∂ Q(x, t) bi + ai Q(x, t) ∂x

 = h i (t) + u i (t), t ≥ 0, i = 1, 2,

(15.4)

x=x i

where ai and bi —constant coefficients, b1 ≤ 0, b2 ≥ 0; h i (t)—some known completely regular functions, x 1 = 0, x 2 = L. Boundary controls u 1,2 (t) allowed to to the space L p [0, T ], 1 < p ≤ ∞ and can be joined in vector U (t) = belong u 1 (t), u 2 (t) . The final condition is formulated in such a way that at some time T > 0 the state of the system must satisfy the following expression: ∂Q (x, T ) = 0. ∂t

(15.5)

Therefore, we will state that the optimal damping problem as the following optimal control problem including two sub-problems [7]. Problem 15.1 To find controls u(x, t) and U (t) such that for the system (15.1) with initial conditions (15.3) and boundary conditions (15.4), the final state (15.5) will be reached and: (A) norm of controls u(x, t) and U (t) will be minimal at given T ; or (B) final state will be reached in minimal time T ∗ at given restriction on control norm u(x, t) ≤ l, U (t) ≤ l, l > 0.

15.3 Generalized l-Problem of Moments Equation (15.1) with initial conditions (15.3) and boundary conditions (15.4) has the explicit solution [8, Exp. (18)]: Q(x, t) = R(x, t) +

∞ 

t X n (x)

n=1 ∞ 

u n (τ ) 0

t

E α,α [−λn (t − τ )α ] dτ + (t − τ )1−α

E α,α [−λn (t − τ )α ] × (15.6) (t − τ )1−α n=1 0

   × v1 wn − (q(x)v1 (x))n u 1 (τ ) + v2 wn − (q(x)v2 (x))n u 2 (τ ) −

+

X n (x)

dτ

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−

∞ 

t X n (x)

n=1

dτ 0

E α,α [−λn (t − τ )α ] × (t − τ )1−α

 × (r (x)v1 (x))n · C0 Dτα u 1 (τ ) + (r (x)v2 (x))n · C0 Dτα u 2 (τ ) ,

where v1 (x) =

a2 (x − L) − b2 ; a 2 b1 − a 1 b2 − a 1 a 2 L

v2 (x) =

b1 − a 1 x ; a 2 b1 − a 1 b2 − a 1 a 2 L

 = v1,2

dv1,2 (x) ; dx

R(x, t) = v1 (x)u 1 (t) + v2 (x)u 2 (t) + V (x, t)+ +

∞ 



E α (−λn t α ) ϕn0 − Vn (0+) − v1n u 1 (0+) − v2n u 2 (0+) X n (x)+

n=1

+t

∞ 

E α,2 (−λn t α )X n (x)×

n=1





× ϕn1 −

+

∞  n=1

      ∂ V (x, 0+) ∂u 1 (0+) ∂u 2 (0+) − v1n − v2n + ∂t ∂t ∂t n n n

t

X n (x)

E α,α [−λn (t − τ )α ] × (t − τ )1−α 0       ∂ V (x, τ ) α V (x, τ ) − (q(x)V (x, τ ))n − r (x)C D × f n (τ ) + wn ; τ 0 n ∂x n dτ

ϕn0,1 , u n (τ ), f n (τ ), Vn (τ ) and v(1,2)n —expansion coefficients of functions ϕ 0,1 (x), u(x, τ ), f (x, τ ), V (x, τ ) and v1,2 (x) by the system of eigenfunctions {X n (x)}, analogously (. . .)n is the expansion coefficient of expression in brackets by the system of eigenfunctions {X n (x)}; V (x, t) = v1 (x)h 1 (t) + v2 (x)h 2 (t); E α,β (t)—twoparametric Mittag–Leffler function; E α (t) = E α,1 (t). Eigenvalues λn and eigenfunctions X n (x) one can find as a result of solution of the following Sturm–Liouville problem [8]: 

∂ ∂x



∂ w(x) ∂x



 − q(x) X (x) + λr (x)X (x) = 0,

  ∂ X (x) bi + ai X (x) = 0, i = 1, 2. ∂x x=x i Further, we will reduce the problem (15.1) to the following l-problem of moments [7]. Problem 15.2 Let the system of functions gn (t) ∈ L p [0, T ], the number l > 0, and the number set cn containing at least one nonzero element are given (n = 1,2,…). It’s needed to find a function W (t) ∈ L p [0, T ] (1/ p + 1/ p  = 1) which satisfy the

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following conditions: T gn (τ )W (τ )dτ = cn , n = 1, 2, . . .

(15.7)

0

W (t) ≤ l.

(15.8)

Let us choose the functions W (t) and gn (t) and numbers cn by the following way:  W (t) = u n (t) + v1 wn − (q(x)v1 (x))n + λn (r (x)v1 (x))n u 1 (t) +  + v2 wn − (q(x)v2 (x))n + λn (r (x)v2 (x))n u 2 (t), gn (t, T ) =

E α,α−1 [−λn (T − t)α ] , (T − t)2−α

(15.9) (15.10)

 ∂ Vn (T ) + λn T α−1 ϕn0 − Vn (0+) E α,α (−λn T α )− ∂t     ∂ V (x, 0+) T E α (−λn T α )− − ϕn1 − ∂t n    T ∂ V (x, τ ) − − gn (τ, T ) f n (τ ) + wn ∂x n 0   −(q(x)V (x, τ ))n − r (x)C0 Dτα V (x, τ ) n dτ + cn (T ) = −

∂u 1 (T ) ∂u 2 (T ) + [(v2 (x)r (x))n − v2n ] + (15.11) + [(v1 (x)r (x))n − v1n ] ∂t ∂t  + λn T α−1 [(v1 (x)r (x))n − v1n ] u 1 (0+)+ + [(v2 (x)r (x))n − v2n ] u 2 (0+) E α,α (−λn T α )−  ∂u 1 (0+) − [(v1 (x)r (x))n − v1n ] + ∂t  ∂u 2 (0+) + [(v2 (x)r (x))n − v2n ] E α (−λn T α ). ∂t Note that dependency from T in expressions (15.10) and (15.11) is parametric. Theorem 15.1 Let the boundary controls u 1,2 (t) ∈ L p [0, T ] defined at the points t = 0+, t = T , T > 0 and have a first derivative in these points. Suppose that function W (t), defined by (15.9), is pointwise discontinuous function and has no more than countable number of discontinuities at [0, T]. Assume also than α ∈ (1, 2), expression in right side of (15.11) is finite real number, nonzero at least for one value of index n. Then the optimal control problem (15.1) at given l and T (for subproblem

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A) is equivalent to the infinite-dimensional l-problem of moments (problem (15.2)) for function (15.9) with moments (15.11) and functions (15.10). Proof Using (15.6), one can calculate the first derivative of system state and substitute it into (15.5). But, firstly, we will transform the expression (15.6) and get rid of control derivatives on its right side. Let us perform eigenfunction expansion for functions Q ∗ (x) and R(x, t) and substitute results in (15.6). Because eigenfunction system X n (x) is complete than expression (15.6) is equivalent to corresponding set of expressions for expansion coefficients. The following formula of integration by parts is valid [9]: T f (t) ·

C α 0 Dt g(t)dt

T =

0

g(t) · tR L DTα f (t)dt +

0

+



[α] 

R L {α}+ j−1 t DT

j=0

[α]− j

f (t) · tR L DT

T  g(t)  , 0

(15.12)

where [α] and {α}—integer and fractional parts of number α correspondingly, Riemann–Liouville derivative of fractional-order σ . If we apply formula (15.12) to the last integral term in (15.6), which contain fractional-order derivatives of boundary controls taking into account α ∈ (1, 2), we can obtain after some calculations the following expression: RL σ t DT —right-side

t 0

E α,α [−λn (t − τ )α ] α 1 C α 2 (r (x)v1 (x))n · C 0 Dτ u (τ ) + (r (x)v2 (x))n · 0 Dτ u (τ ) dτ = (t − τ )1−α = −λn

t

(r (x)v1 (x))n u 1 (τ ) + (r (x)v2 (x))n u 2 (τ ) gn (τ, T )dt − 0



−t E α,2 −λn t α



 (r (x)v1 (x))n

 ∂u 1 (0+) ∂u 2 (0+) + (r (x)v2 (x))n + ∂t ∂t

(15.13)

+(r (x)v1 (x))n u 1 (t) + (r (x)v2 (x))n u 2 (t) −

 − (r (x)v1 (x))n u 1 (0+) + (r (x)v2 (x))n u 2 (0+) E α −λn t α .

In calculations, we used the representation of Mittag–Leffler function as power series, which converge uniformly on real axis [6, Chap. 1]. Further, we substitute (15.13) in (15.6), calculate the first derivative of the obtained expression, and put t = T . Using (15.5) and taking into account expressions (15.9), (15.11), and (15.10), we finally obtain the formula (15.7). So that, we have shown that the problem (15.1) is equivalent to the problem (15.2). Remark 15.1 Expression (15.11) contains the initial and final values of boundary controls and its derivatives. Generally, these values can be defined with the help of

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some additional conditions or assumptions. But in case of r (x) = 1 coefficients at these values zeroize and aforementioned conditions or assumptions are not needed.

15.4 Finite-Dimensional l-Problem of Moments The l-problem of moments (15.2) obtaining for the explicit solution of diffusion-wave equation is infinite-dimensional and generally we can’t prove its solvability and find the explicit solution of such problem. But we can use an approximate solution of Eq. (15.1) which can be obtained by truncation of series in (15.6). Then we can analogously derive a finite-dimensional l-problem of moments which dimension defined by truncation number N . We will call this problem the{l, N }-problem of moments. Definition 15.1 We will call the finite-dimensional {l, N }-problem of moments correct if the norm of functions gn (t) are defined in the space L p [0, T ]. Theorem 15.2 The {l, N }-problem of moments (15.7)–(15.8) taking into account the expressions (15.9)–(15.11) at any fixed N and given l > 0 is correct iff the following condition were evaluated: 1 (15.14) α >1+ . p Proof Let us estimate the norm of functions gn (t) in L p [0, T ]: g˜ n (t, T ) ≤ E α,α−1 [−λn (T − τ )α ] · (T − t)α−2 , n = 1, . . . , N . The first multiplier is bounded [6, p. 42]. The second multiplier can be calculated explicitly. Taking into account that p  ≥ 1 and α ∈ (1, 2), we will obtain 

α−2

(T − t)



(T − t)[ p (α−2)+1]/ p = [ p  (α − 1) + 1]1/ p

T    , t ∈ [0, T ].  0

This expression bounded at any T > 0, p  ≥ 1 iff the condition (15.14) is satisfied. Consequently, the {l, N }-problem of moments is correct iff the condition (15.14) is satisfied. Remark 15.2 The condition (15.14) can be rewritten if we take into account α ∈ (1, 2) 1 {α} > . p Remark 15.3 The finite-dimensional l-problem of moments is solvable iff the functions gn (t) ∈ L p [0, T ] are linear independent [7]. As it can be demonstrated by

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direct calculation, the functions (15.10) satisfy this condition. Consequently, {l, N }problem of moments (15.7)–(15.8) taking into account the expressions (15.9)– (15.11) is solvable. If the finite-dimensional problem of moments is correct and solvable, then it can be solved explicitly [7]. And this solution allows to construct an optimal control problem unique solution. In case of subproblem A of problem (15.1) function W (t) will be defined by the expression W˜ (t) =

p N

 N  p −1  N       ∗ ∗ ξi gi (t) sign ξi gi (t) , t ∈ [0, T ].    i=1

(15.15)

i=1

In case of subproblem B of problem (15.1) function W (t) can be defined analogously [7]. The value  N (T ) is equal to norm of optimal control (15.15) and defined by the following formula: ⎛

 p ⎞1/ p T  N  1   =⎝  ξi∗ gi (t) dt ⎠ .    N (T ) i=1 0

The numbers ξi∗ is a solution of the following conditional minimization problem. Problem 15.3 To find  p ⎞1/ p ⎛T  N  p ⎞1/ p T  N        min ⎝  ξi gi (t) dt ⎠ =⎝  ξi∗ gi (t) dt ⎠ ξ1 ,...,ξ N     ⎛

0

i=1

0

i=1

subject to N  i=1

ξi ci =

N 

ξi∗ ci = 1, i = 1, . . . , N .

i=1

15.5 Conclusions In this paper, we investigated the optimal damping problem for the system modeled by diffusion-wave equation with fractional-order time derivative. We reduced the problem to the infinite-dimensional l-problem of moments and construct an approximate finite-dimensional {l, N }-problem of moments. For the last one, we proved the correctness and solvability.

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References 1. Mophou, G.M.: Optimal control of fractional diffusion equation. Comp. Math. Appl. 61, 68–78 (2011) 2. Tang, Q., Ma, Q.: Variational formulation and optimal control of fractional diffusion equations with caputo derivatives. Adv. Diff. Eq. (2015). https://doi.org/10.1186/s13662-015-0593-5 3. Zhou, Z., Gong, W.: Finite element approximation of optimal control problems governed by time fractional diffusion equation. Comp. Math. Appl. 71, 301–318 (2016) 4. Kubyshkin, V.A., Postnov, S.S.: The optimal control problem for linear systems of non-integer order with lumped and distributed parameters. Discontin. Nonlinearity Complex. 4(4), 429–443 (2015) 5. Kubyshkin, V.A., Postnov, S.S.: Time-optimal boundary control for systems defined by a fractional order diffusion equation. Automn. Remote Control. 79(5), 874–886 (2018) 6. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) 7. Butkovskiy, A.G.: Distributed Control Systems. American, Elsevier (1969) 8. Sandev, T., Tomovski, Z.: The general time fractional wave equation for a vibrating string. J. Phys. A: Math. Theor. 43, 055204 (12 pages) (2010) 9. Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A: Math. Theor. 40, 6287–6303 (2007)

Chapter 16

Identification of Integral Models of Nonlinear Multi-input Dynamic Systems Using the Product Integration Method Svetlana Solodusha

Abstract This paper discusses a method for constructing Volterra polynomials in the case of a vector input signal. The key idea behind the new approach to constructing integral polynomials is based on the identification of integrals from Volterra kernels. We describe in detail the numerical modeling of a multi-input dynamic system using the cubic Volterra polynomial. This approach is based on solving a specific system of linear algebraic equations (SLAE) obtained by approximating multidimensional convolutions by the product integration method ( pi-method). The pi-approximation makes it possible to avoid essential difficulties arising in the solution of multidimensional Volterra equations of the first kind. The numerical efficiency is tested for the problem of modeling the nonlinear dynamics of a heat exchanger element. The results of the computational experiment showed that in most cases the new approach increases the accuracy of modeling.

16.1 Introduction Many technological processes, including those in power engineering, are often investigated using the theory of mathematical modeling of dynamic systems of the “input– output” type. Methods for identifying such systems are widely presented in the scientific literature (see, for example, [1, 2] and the references therein). Due to a variety of applications, the modification of existing and the creation of new approaches are still relevant for solving inverse problems of dynamics [3]. Volterra integral equations of the first kind take a well-deserved place among the mathematical tools for describing dynamic systems. As noted in the monograph [4], this is explained by their priority in situations when it is extremely difficult or technically impossible to use differential equations. To date, a large number of papers and monographs have been devoted to theories and numerical methods for solving Volterra-type integral S. Solodusha (B) Melentiev Energy Systems Institute Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_16

137

138

S. Solodusha

equations (including their generalizations), see, for example, the monograph [5] and the references listed therein. In this paper, we consider a deterministic identification method that satisfies the following set of factors 1. A finite segment of the Volterra integro-power series (polynomial) [6] was chosen as an identification model y(t) =

N 



f λ1 ,...,λm (t), t ∈ [0, T ],

(16.1)

m=1 1≤λ1 ≤...≤λm ≤ p

t f λ1 , ..., λm (t) =

t ...

0

K λ1 , ..., λm (s1 , . . . , sm )

m 

xλ j (t − s j )ds j ,

(16.2)

j=1

0

where t ∈ [0, T ], y(0) = 0, x(t) = (x1 (t), . . . , x p (t))T . 2. The characteristics of a dynamic system are presented in the time domain. 3. The collection of initial data occurs during the execution of an active experiment, which assumes the possibility of influencing the system with test input signals. The analysis of scientific and technical literature has shown that, despite a large number of available methods for constructing Volterra polynomials (16.1), (16.2), the development of such techniques that are focused on practical application and also have a relatively simple implementation remains relevant [7].

16.2 The Problem Statement A new approach to the construction of (16.1), (16.2) for the case N = 2, which is based on the identification of Volterra kernels K λ1 , ..., λm (s1 , . . . , sm ) from (16.2) and takes into account the features of a real-life dynamic system, was developed in [8]. In this paper, we consider a modeling problem based on the identification of integrals of Volterra kernels. As noted in [9], this will allow us to eliminate the problems of ensuring complex conditions for the solvability of multidimensional integral equations for the required kernels [10]. This idea was tested earlier in [11] using the product integration method [12], according to which the pi-approximation of one-dimensional convolutions takes the following form: i h K (s)x(i h − s)ds ≈ 0

i  j=1

j h K (s)ds,

xi− j+ 21

(16.3)

( j−1)h

  xi− j+ 21 = x (i − j + 21 )h , h is a sufficiently small mesh step ti = i h, i = 1, n, nh = T (this method is especially effective if K (t) is a highly oscillating function).

16 Identification of Integral Models of Nonlinear Multi-input …

139

According to [12], the rate of convergence of the to zero is of   pi-method error the first order in approximation of x(t) at integer x(ti ) = x(i h) nodes and of the   second order at approximation of x(t) at “non-integer” x(ti+ 21 ) = x(ti + h2 ) nodes of uniform mesh. Let us extend the technique of identification developed in [11] to the case of vector input disturbances x(t), consisting of p ≥ 2 components. In this case, instead of (16.3), we have i h

i h ...

0

≈

i  i 1 ,...,i m =1

 m

x λk

xλk (i h − sk )dsk ≈

(16.4)

k=1

0



k=1

m 

K λ1 ,...,λm (s1 , . . . , sm )

 1 i − ik + h 2

 i1 h

im h ...

(i 1 −1)h

 K λ1 ,...,λm (s1 , . . . , sm )ds1 . . . dsm ,

(i m −1)h

where kernels K λ1 , ..., λm are not symmetrical, 1 ≤ λ1 ≤ · · · ≤ λm ≤ p, i 1 , . . . , i m = 1, n.

16.3 On the Main Results The identification procedure (16.1) for p = 1 is largely based on the symmetry of the Volterra kernels K λ1 , ..., λm with respect to s1 , . . . , sm , herein λ1 = · · · = λm . The construction algorithm (16.1) for N = 2, p = 1 based on the integration method can be found in [9]. Transition to the vector case ( p > 1) requires additional research of two different situations: first, when λ1 = · · · = λm , and second, so that K λ1 , ..., λm is partly symmetrical. In this paper, we address the second case. Let N = 3, p = 2. To be specific, set λ1 = λ2 = 1, λ3 = 2, so that K 112 is partly symmetrical. We write the difference analog of the cubic Volterra polynomial for the case x(t) = (x1 (t), x2 (t))T with the help of (16.4) ycubi = yscalari + yvectori , yvectori = f 12 (ti ) + f 112 (ti ) + f 122 (ti ) =

i i  

p12μ1 , μ2 x1i−μ1 +1/2 x2i−μ2 +1/2 + (16.5)

μ1 =1 μ2 =1

+

i i i   

q112μ1 , μ2 , μ3 x1i−μ1 +1/2 x1i−μ2 +1/2 x2i−μ3 +1/2 +

μ1 =1 μ2 =1 μ3 =1

+

i i i    μ1 =1 μ2 =1 μ3 =1

q122μ1 , μ2 , μ3 x1i−μ1 +1/2 x2i−μ2 +1/2 x2i−μ3 +1/2 ,

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S. Solodusha

yscalari =

2   i  ν=1

+

m νμ1 xνi−μ1 +1/2 +

μ1 =1

i i  

pνν μ1 , μ2 xνi−μ1 +1/2 xνi−μ2 +1/2 +

μ1 =1 μ2 =1

i i i   

 qννν μ1 , μ2 , μ3 xνi−μ1 +1/2 xνi−μ2 +1/2 xνi−μ3 +1/2 ,

μ1 =1 μ2 =1 μ3 =1

μ1 h m νμ1 =

μ1 h

μ2 h

K ν (s)ds, pννμ1 , μ2 = (μ1 −1)h

K νν (s1 , s2 )ds1 ds2 , (μ1 −1)h (μ2 −1)h

μ1 h

μ2 h

μ3 h

qννν μ1 , μ2 , μ3 =

K ννν (s1 , s2 , s3 ) ds1 ds2 ds3 , (μ1 −1)h (μ2 −1)h (μ3 −1)h

μ1 h

μ2 h

p12μ1 , μ2 =

K 12 (s1 , s2 )ds1 ds2 , (μ1 −1)h (μ2 −1)h

μ1 h

μ2 h

μ3 h

q112μ1 , μ2 , μ3 =

K 112 (s1 , s2 , s3 ) ds1 ds2 ds3 ,

(16.6)

(μ1 −1)h (μ2 −1)h (μ3 −1)h

μ1 h

μ2 h

μ3 h

q122μ1 , μ2 , μ3 =

K 122 (s1 , s2 , s3 ) ds1 ds2 ds3 , (μ1 −1)h (μ2 −1)h (μ3 −1)h

where xν (ti−μ1 +1/2 ) = xνi−μ1 +1/2 (ν = 1, 2), ycub (ti ) = ycubi , yscalar (ti ) = yscalari , yvector (ti ) = yvectori . Without loss of generality, assume that the values of m ν , pνν , qννν for ν = 1, 2, which correspond to the case with scalar inputs, are known. Additionally, we suppose that the problem of decomposing ycubi into f 12 (ti ) = f 12i , f 112 (ti ) = f 112i , f 122 (ti ) = f 122i in the form of pi-approximations (16.2) has been solved. 2 = n (n+1) Then the problem of identification of n + 2n(n − 1) + n(n−1)(n−2) 2 2 unknowns q112 in (16.6) (given symmetry by the first input) can be reduced to solving i i i   

q112μ1 , μ2 , μ3 x1i−μ1 +1/2 x1i−μ2 +1/2 x2i−μ3 +1/2 = f 112 (ti ),

(16.7)

μ1 =1 μ2 =1 μ3 =1

where f 112 (ti ) corresponds to the contribution of the given summand to the response ycub (ti ) of the dynamical system. Due to the symmetry of K 112 with respect to s1 , s2 , the following equality holds: q112μ1 , μ2 , μ3 = q112μ2 , μ1 , μ3 . 3 To identify the values q112 in each of subdomains 3 = 3(r ) , where i=1

16 Identification of Integral Models of Nonlinear Multi-input …

141

 (1) 3 = s1 , s2 , s3 : 0 ≤ s3 ≤ s1 , s2 ≤ T ,

 (2) 3 = s1 , s2 , s3 : 0 ≤ s1 , s2 ≤ s3 ≤ T ,

 (3) 3 = s1 , s2 , s3 : 0 ≤ s1 ≤ s3 ≤ s2 ≤ T ∪ 0 ≤ s 2 ≤ s3 ≤ s1 ≤ T , we choose tests that differ from those introduced earlier in [13] and ensure the minimum possible total duration of input disturbances

x1αj1 (t) = α1 (I (t) − I (t − h) + I (t − j h) − I (t − j h − h)) , x2αj,2 k (t) = α2 (I (t − j h − kh) − I (t − j h − kh − h)) ,

(16.8)

⎧ α1 ⎪ ⎨ x1 j, k (t) = α1 (I (t − j h) − I (t − j h − h)+ +I (t − j h − kh) − I (t − j h − kh − h)) , ⎪ ⎩ α2 x2 (t) = α2 (I (t) − I (t − h)) ,

(16.9)



x1αj,1 k (t) = α1 (I (t) − I (t − h) + I (t − j h − kh) − I (t − j h − kh − h)) , x2αj2 (t) = α2 (I (t − j h) − I (t − j h − h)) , (16.10) where I (t) is Heaviside function, α1 , α2 are amplitudes (height) of test inputs. Substituting of the mesh analogs (16.8)–(16.10) into (16.7), we obtain (1)

q112i, i, i− j−k + q112i− j, i− j, i− j−k + 2q112i, i− j, i− j−k = f 112i, j, k , (2)

q112i− j, i− j, i + q112i− j−k, i− j−k, i + 2q112i− j, i− j−k, i = f 112i, j, k , (3)

q112i, i, i− j + q112i− j−k, i− j−k, i− j + 2q112i, i− j−k, i− j = f 112i, j, k , ( p)

( p)

where f 112 (i h, j h, kh) = f 112i, j, k ( p = 1, 3) denote the corresponding mesh analogs of the system outputs to the first (16.8), second (16.9), and the third (16.10) series of input signals. Thus, we have the formulas for finding a solution for the entire domain 3 1 (1) 1 (2) 1 (3) f f f (i h, 0, 0) = (i h, 0, 0) = (i h, 0, 0), i = 1, n, 4 112 4 112 4 112 1 (1) 1 (2) f f = (i h, 0, j h), q112i− j, i− j, i = (i h, j h, 0), i = 2, n, j = 1, i − 1, 4 112 4 112   (1) 1 f 112 (i h, j h, 0) − q112i, i, i− j − q112i− j, i− j, i− j = q112i, i− j, i− j = 2   1 (3) f 112 (i h, j h, 0) − q112i, i, i− j − q112i− j, i− j, i− j , i = 2, n, j = 1, i − 1, = 2   1 (2) f 112 (i h, 0, j h) − q112i, i, i − q112i− j, i− j, i = q112i− j, i, i = 2 q112i, i, i =

q112i, i, i− j

142

S. Solodusha =

1 2

  (3) f 112 (i h, 0, j h) − q112i, i, i − q112i− j, i− j, i , i = 2, n, j = 1, i − 1,

  1 (1) f 112i, j, k −q112i, i, i− j−k − q112i− j, i− j, i− j−k , 2   1 (2) f 112i, j, k −q112i− j, i− j, i − q112i− j−k, i− j−k, i , = 2   (3) 1 f 112i, j, k −q112i, i, i− j − q112i− j−k, i− j−k, i− j , = 2

q112i− j, i, i− j−k = q112i− j, i− j−k, i q112i, i− j−k, i− j

i = 3, n, 2 ≤ j + k ≤ i − 1. Note that if we apply the series of test signals of the form (16.8)–(16.10) and their inversions with respect to the components of the vector x(t), we will be able to reduce the problem of finding p12 , q112 , q122 from (16.5) to the solution of the closed nonsingular system of linear algebraic equations. In particular, for 5n 2 − 2n unknowns p12μ1 , μ2 , q112μ1 , μ2 , μ3 , q122μ1 , μ2 , μ3 and for μl = i − j, i (l = 1, 3) this SLAE has the form (1) α , α 1 2 i, 0, j ,

(16.11)

β1 α2 p12i, i− j + β12 α2 q112i, i, i− j + β1 α22 q122i, i− j, i− j =

(1) β , α y i,10, j2 ,

(16.12)

α1 β2 p12i, i− j + α12 β2 q112i, i, i− j + α1 β22 q122i, i− j, i− j =

(1) α , β y i,10, j2 ,

(16.13)

α1 α2 p12i, i− j + α12 α2 q112i, i, i− j + α1 α22 q122i, i− j, i− j = y

i = 1, n, j = 0, i − 1, α1 α2 p12i− j, i

+ α12 α2 q112i− j, i− j, i

(2) α , α 1 2 i, j, 0 ,

(16.14)

(2) β , α 1 2 i, j, 0 ,

(16.15)

+ α1 α22 q122i− j, i, i = y

β1 α2 p12i− j, i + β12 α2 q112i− j, i− j, i + β1 α22 q122i− j, i, i = y

(2) α , β 1 2 i, j, 0 ,

(16.16)

α1 α2 p12i, i− j + α1 α2 p12i− j, i− j + α12 α2 q112i, i, i− j + 2α12 α2 q112i, i− j, i− j +

(16.17)

α1 β2 p12i− j, i + α12 β2 q112i− j, i− j, i + α1 β22 q122i− j, i, i = y

(1) α , α (3) α , α 1 2 y i,1j, 02 , i, j, 0 = 2 + α1 α2 q112i, i, i− j + (16.18)

+α12 α2 q112i− j, i− j, i− j + α1 α22 q122i, i− j, i− j + α1 α22 q122i− j, i− j, i− j = y α1 α2 p12i, i− j + α1 α2 p12i, i + α12 α2 q112i, i, i

(2) α , α (3) α , α 1 2 g i,10, j2 , i, 0, j = 2 + α1 α2 q112i− j, i− j, i− j +

(16.19)

(1) (3) + 2α1 α22 q122i− j, i, i− j + α1 α22 q122i− j, i− j, i− j = g i,α1j,, α02 = g i,α1j,, α02 , α1 α2 p12i, i + α1 α2 p12i− j, i + α12 α2 q112i, i, i + 2α12 α2 q112i, i− j, i +

(16.20)

+α1 α22 q122i, i, i + 2α1 α22 q122i, i, i− j + α1 α22 q122i, i− j, i− j = g α1 α2 p12i− j, i + α1 α2 p12i− j, i− j + α12 α2 q112i− j, i− j, i +α1 α22 q122i− j, i, i

+α12 α2 q112i− j, i− j, i + α1 α22 q122i, i, i + α1 α22 q122i− j, i, i =

(2) α , α y i,10, j2

=

(3) α , α y i,10, j2 ,

i = 2, n, j = 1, i − 1, ( p)

y

γ1 , γ 2

(i h, j h, kh) =

( p) γ , γ ( p) y i,1 j, 2k , g γ1 , γ2 (i h,

( p) γ , γ 1 2 i, j, k ,

j h, kh) = g

p = 1, 3.

16 Identification of Integral Models of Nonlinear Multi-input … ( p)

143

γ ,γ

1 2 Here y γ1 , γ2 is the reaction y vector (16.5) to the corresponding series of tests (16.8)– (16.10) with amplitudes γ1 = α1 , β1 (α1 = β1 ) and γ2 = α2 , β2 (α2 = β2 ), whereas

( p)

g α1 , α2 stands for the response of (16.5) to the pth series, whose test signals were subject to inversion with respect to x1 , x2 . The SLAE (16.11)–(16.20) is closed, i.e., the number of equations equals the number of unknowns. Indeed, the number of equations in (16.11)–(16.13) equals to 3

i−1 n  

1=3

i=1 j=0

n 

i=

i=1

3n(n + 1) , 2

whereas in (16.14)–(16.20), we have only 7

n  i−1 

1=7

i=2 j=1

n  7n(n − 1) (i − 1) = 2 i=2

equations. Their sum coincides with the number of unknowns. The matrix A that corresponds to the system (16.11)–(16.20) is such that AQ = Y,  Q = p121, 1 , q1121, 1, 1 , q1221, 1, 1 , . . . , p12n, n , q112n, n, n , q122n, n, n , p122, 1 , q1122, 2, 1 , q1222, 1, 1 , . . . , p12n, n−1 , q112n, n, n−1 , q122n, n−1, n−1 , . . . , . . . , p12n, 1 , q112n, n, 1 , q122n, 1, 1 , p121, 2 , q1121, 1, 2 , q1221, 2, 2 , . . . , . . . , p12n−1, n , q112n−1, n−1, n , q122n−1, n, n , . . . , p121, n , q1121, 1, n , q1221, n, n , q1122, 1, 1 , . . . , q112n, n−1, n−1 , . . . , q112n, 1, 1 , q1222, 2, 1 , . . . ,

 Y =

. . . , q122n, n, n−1 , . . . , q122n, n, 1 , q1221, 1, 2 , . . . , q122n−1, n−1, n , . . . , T . . . , q1221, 1, n , q1121, 2, 2 , . . . , q112n−1, n, n , . . . , q1121, n, n , (1) α , α (1) (1) y 1,1 0, 20 , y β1,1 ,0,α20 , y α1,1 ,0,β20 ,

(1) β , α (1) α , β y 2,1 0, 21 , y 2,1 0, 21 , (2) α , β y 2,1 1, 20 , (2) α , α g 2,1 0, 21 ,

(1) α , α (1) β , α (1) (1) 1 2 y n,1 0, 20 , y αn,1 ,0,β20 , y α2,1 ,0,α21 , n, 0, 0 ,

..., y

(1) α , α (1) β , α (1) (2) (2) 1 2 y n,1 0, 2n−1 , y αn,1 ,0,β2n−1 , y α2,1 ,1,α20 , y β2,1 ,1,α20 , n, 0, n−1 ,

..., y

(2) α , α (2) β , α (2) α , β (1) α , α 2 2 1 2 y n,1 n−1, y n,1 n−1, y 2,1 1, 20 , n, n−1, 0 , 0, 0,

..., y

(2) α , α (1) α , α 1 2 g 2,1 1, 20 , n, 0, n−1 ,

..., g

(1) α , α (2) α , α 1 2 y 2,1 0, 21 , n, n−1, 0 ,

..., g

(1) α , α 1 2 n, n−1, 0 , T (2) . . . , y αn,1 ,0,α2n−1 ,

..., y

and is block-triangular. The form of matrix A is conceptually trivial but rather cumbersome  A=

B O C D



⎛

C1 ⎜... ⎜ , C =⎝ O O

O ... ... ...

... ... O O

... ... C1 C1

O ... O O

C1 ... ... ...

O ... O ...

... ... C1 O

⎞ O   ...⎟ ⎟ , C1 = α1 α2 α 2 α2 α1 α 2 1 2 O⎠ C1

144

S. Solodusha

  withdimensions A—(n(5n − 2) × n(5n − 2)), B— 3n 2 × 3n 2 , C—(2n(n − 1)× 3n 2 , D—(2n(n − 1) × 2n(n − 1)). Here O are matrix blocks with zero elements, D is a diagonal matrix: D = diag{2α12 α2 , . . . , 2α12 α2 , 2α1 α22 , . . . , 2α1 α22 , 2α12 α2 , . . . , 2α12 α2 }, B is a quasi-diagonal matrix with the blocks ⎛

⎞ α1 α2 α12 α2 α1 α22 B1 = ⎝ β1 α2 β12 α2 β1 α22 ⎠ α1 β2 α12 β2 α1 β22 on the diagonal, the remaining elements are zero. The nonsingularity of the SLAE follows from α1 = β1 = 0, α2 = β2 = 0, since  n 2 det A = 22n(n−1) (α1 α2 )n(5n−3) β1 β2 (α1 − β1 ) (α2 − β2 ) = 0.

16.4 Numerical Experiment Let us illustrate the effectiveness of this approach to identification using the example of quadratic Volterra polynomials used in modeling the dynamics of a heat exchanger element with independent heat supply [14]: ⎞ ⎛ t  −λ t D(ς)dς t  −λ D(ς)dς 1 2 Q0 λ1 λ2 ⎠ dη, Q(η) − i(t) = D(η) ⎝e η −e η λ2 − λ1 D0 0

(16.21) where t–time (s), D—fluid flow rate (kg/s), Q—full thermal load (kW), —an increment, for example, D(t) = D0 + D(t), λ1 , λ2 —roots of a characteristic equation for the system of two differential equations for fluid and material of the heatconducting wall. Indices “0” are used to denote the initial parameters: D0 = 0.16 kg/s, Q 0 = 100 kW. Model (16.21) was used to check the accuracy of modeling using integral models that describe changes in enthalpy i(t) ≡ y(t) (i(0) = 0) at disturbances D(t) ≡ x1 (t), Q(t) ≡ x2 (t) of arbitrary form. For different levels of disturbance, the fluid flow rate D¯ = max |D(t)| = α D0 0≤t≤T

and thermal load

Q¯ = max |Q(t)| = α Q 0 , 0≤t≤T

16 Identification of Integral Models of Nonlinear Multi-input …

a)

145

b)

Fig. 16.1 Applicability domains of the quadratic polynomials (16.1) for the set of input signals (16.22). The area where the inequality M AE 1 < M AE 2 holds true was obtained with the accuracies δ = 10−3 (a) and δ = 10−1 (b)

0.1 ≤ α ≤ 0.25 were used to build their quadratic polynomials (16.1) at p = 2. Two approaches to building the responses yr , r = 1, 2, of integral models were implemented in the computational experiment. The first of the approaches is based on the restoration of Volterra kernels using the midpoint rule according to [13] (r = 1). The second one is based on the restoration of integrals from Volterra kernels using the product integration method (r = 2). Let us use the response i(t) ≡ yet of the simulation model (16.21) and compare the accuracy of modeling y1 and y2 for the class of inputs X (B, T ) of the following form Dωβ (t) = D ∗ (I (t) − I (t − ω)), Q βω (t) = Q ∗ (I (t) − I (t − ω)),

(16.22)

¯ Q ∗ ∈ [0, Q], ¯ β ∈ [0, B], B = 25%, t ∈ [0, T ], T = 30 s. The where D ∗ ∈ [0, D], software implementation was in Borland C++ [15]. We take the value of “Mean Absolute Error” (MAE) factor: M AEr =

B 1  β β |y (T ) − yet (T )| B β=1 r

at r = 1, 2, as a criterion for modeling accuracy. The domains where the inequality M AE 1 < M AE 2 for 1 ≤ ω ≤ 29, 10 ≤ α ≤ 25 holds true are highlighted in Fig. 16.1. The arrow is directed inside the domain where a specific inequality holds true (around the domain the inequality M AE 1 < M AE 2 does not hold true). In the lefthand graph, the comparison is performed with the accuracy of δ = 10−3 , and in the right-hand graph—with the accuracy of δ = 10−1 . Gray filling in Fig. 16.1 corresponds to the case where equality M AE 1 = M AE 2 holds true. Thus, in most cases, the use of the product integration method for modeling the nonlinear dynamics of the heat exchanger gives a smaller error at the end of the transition process than the application of the method based on the restoration of Volterra kernels.

146

S. Solodusha

16.5 Conclusions The paper presents a new method designed to construct Volterra polynomials based on the identification of integrals from kernels. Based on the generalization of the product integration method, an algorithm for the numerical restoration of cubic Volterra polynomials in the vector case is proposed. Mesh analogs of Volterra polynomials for a vector input signal are obtained with the aid of a new type of test signals of length h (mesh step) in the form of linear combinations of Heaviside functions with deviating arguments. The results of the computational experiments illustrating the effectiveness of the proposed method for identifying Volterra polynomials are presented. Acknowledgements The research was carried out under State Assignment of the Ministry of Science and Higher Education of the Russian Federation (Project FWEU-2021-0006, theme No. AAAA-A21-121012090034-3).

References 1. Raibman, N.S.: Identification of control objects (a rewiew). Avtomatika i telemekhanica 6, 80–93 (1979) 2. Kleiman, E.G.: Identification of input signals in dynamic systems. Avtomatika i telemekhanica 12, 3–15 (1999) 3. Karelin, A.E., Maistrenko, A.V., Svetlakov, A.A., Kharitonov, S.A.: Synthesis of the method of automatic control of processes that is based on the concept of inverse dynamics problems. Omsk Sci. Bull. 4(154), 83–86 (2017) 4. Verlan’, A.F., Sizikov, V.S.: Integral Equations: Methods, Algorithms, Programs. Naukova Dumka, Kiev (1986) 5. Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press, Cambridge (2017) 6. Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. Dover Publications, New York (1959) 7. Cheng, C.M., Peng, Z.K., Zhang, W.M., Meng, G.: Volterra-series-based nonlinear system modeling and its engineering applications: a state-of-the-art review. Mech. Syst. Signal Process. 87, 340–364 (2017). https://doi.org/10.1016/j.ymssp.2016.10.029 8. Solodusha, S.V.: New classes of Volterra integral equations of the first kind related to the modeling of the wind turbine dynamics. In: 15th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference) (STAB) (2020). https:// doi.org/10.1109/STAB49150.2020.9140662 9. Solodusha, S.V., Spiryaev, V.A., Tairov, E.A.: Numerical modeling of dynamics of thermal power equipment of the power unit at the Nazarovo power station by Volterra polynomial. In: XXI International Conference Complex Systems: Control and Modeling Problems (CSCMP) (2019). https://doi.org/10.1109/CSCMP45713.2019.8976749 10. Solodusha, S.V.: Quadratic and cubic Volterra polynomials: identification and application. Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr. 14, 131–144 (2018). https:// doi.org/10.21638/11702/spbu10.2018.205 11. Apartsyn, A.S., Spiryaev, V.A.: On an approach to the identification of Volterra polynomials. Optimizaciya, upravlenie, intellekt. 2, 109–117 (2005) 12. Linz, P.: Product integration method for Volterra integral equations of the first kind. BIT Numer. Math. 11, 413–421 (1971)

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13. Apartsyn, A.S.: Nonclassical Linear Volterra Equations of the First Kind. VSP, Utrecht (2003) 14. Tairov, E.A.: Nonlinear modeling of heat exchange dynamics in a duct with one-phase heat carrier. Bull. Acad. Sci. USSR. Ser. Energy Transport. 1, 150–156 (1989) 15. Solodusha, S.V.: Software package “Dynamics” for studying dynamic processes by Volterra series. Bull. South Ural State Univ. Ser. Comput. Technol. Autom. Control Radio Electron. 17, 83-92 (2017). https://doi.org/10.14529/ctcr170207

Chapter 17

Algorithm for Constructing a Cognitive Aggregate-Stream Model of the Automatic Spacecraft Flight Control Process Vladimir S. Kovtun, Boris V. Sokolov, and Valerii V. Zakharov Abstract The resource consumption of onboard systems (OS) largely depends on the synergetic phenomena that occur during intersystem interaction in automatic spacecraft (AS). By using these phenomena, it is possible to increase the efficiency of using existing resources, as well as to supplement them with new “synergistic” resources [1]. At the same time, synergetic phenomena can lead to premature development of the OS resource and unforeseen (non-calculated) failures and accidents [2]. For a targeted search for data on synergetic phenomena, special modeling of processes occurring on board is required. The purpose of the research was to move from the system-cybernetic model of the AS as a “black box” to a model that provides “transparency” of the AS as a “white box” for processes occurring on board [3], taking into account synergetic phenomena. The article considers a new algorithm for constructing a cognitive aggregate-flow model of the AS flight control process that meets the stated requirements.

17.1 Introduction The calculation of the estimated flight resource of automatic spacecraft (AS) is based on the primary cause-and-effect topological and functional nominal relationships between the OS and their elements, as well as emergent properties identified as a result of system associations. However, no matter how the decomposition of AS as a complex technical system (CTS) is carried out, at the “primary” level, which is V. S. Kovtun S.P. Korolev Rocket and Space Corporation Energia, 4A Lenin street, Korolev 141070, Moscow, Russia e-mail: [email protected] B. V. Sokolov · V. V. Zakharov (B) SPIIRAS, line 14-ya V. O., 39, 199178 Saint Petersburg, Russia e-mail: [email protected] B. V. Sokolov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_17

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determined by the “system”, due to its openness and non-linearity, there are implicit evolutionary secondary cause-and-effect interactions that the functional and topological schemes of systems do not reflect. Secondary interactions caused by multi-node connections of nominal processes occurring in systems with processes determined by external factors lead to the emergence of synergetic phenomena. Resources due to self-organization and self-development of systems that depend on binary relations in the interaction between factors that determine nominally occurring processes in systems and factors that are determined by external processes in relation to systems will be called “synergetic”. The purpose of this article is to build a formal model of these interactions that take into account synergetic phenomena.

17.2 Static Cognitive Model of the Complex Flight Control Process of an Automatic Spacecraft The synergetic phenomena described above on board AS manifest themselves at the level of physical processes. Therefore, it is necessary to decompose the AS as a material object of research, as well as the corresponding decomposition of the processes of controlling its flight at each specific level of detail of the AS obtained during its decomposition. The process is considered as a physical phenomenon that occurs in a material object and is controlled by a process controller system (spacecraft, onboard equipment, devices, structural elements, etc.) and/or a group of controllers, which leads to a change in the state of processes over time. In this case, the process is called simple if only one material, conceptual, and mathematical model can be developed for it, describing the change of various parameters of the process under study over time through a known mapping of the definition area to the value area. A complex process is considered as a set of simple processes. Using the controller as a control device, the state of the process is monitored, evaluated, and predicted, and control actions are developed to change it. It is proposed to use a stratified description in the form of a cognitive map (creating layers—strata) at 4 levels for modeling AS and its flight control processes: target processes occurring at the AS level (basic processes (BP)) (1st); processes occurring at the system (2nd), element (3rd), and intra-element (4th) levels. The regulators of these processes are: at the stratum of the first level (C1)—AS interacting with the external environment; the second (C2)—all AS systems; third (C3)—elements of all systems, and on the stratum of the fourth level (C4)—intra-element devices of all elements. A set of telemetric parameters measured by sensors, as well as parametersmessages generated by algorithms of onboard computing tools, is assigned to each process that takes place on board the AS [4, 5]. For the transition from the linguistic description of the equilibrium working state of the regulators to their parametric description, fixed at the current time, the assumption is made about the possibility of a static description of the processes taking place. This assumption allows you to initially ignore the time factor, assuming that it does not affect the results of the

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decomposition and stratification of the AS flight control process at any given time. On each stratum, the components are created and supplemented by highlighting, identifying, and describing the observed properties of processes with primary and generalized parameters, the sets of μ(·) simple processes (), for which the notation [6]: 1st stratum, basic processes that characterize the state of the apparatus as a whole through generalized parameters: μ( j ) = { j | j = 1, . . . , J },

(17.1)

where j is the BP designation; 2nd stratum, system processes that characterize the state of individual systems of the apparatus: s,ks s μ(s,k j ) = { j | j = 1, . . . , J ; s = 1, . . . , S; ks = 1, . . . , K },

(17.2)

where s is the designation of the system process number in the ks version of the implementation of structural and functional reserves; 3rd stratum, element processes that characterize the state of individual elements of the BS: s,ks s μ(s,k j p ) = { j p | j = 1, . . . , J ; s = 1, . . . , S; ks = 1, . . . , K ; p = 1, . . . , P}, (17.3) where p is the process designation in the system element; 4th stratum, intra-element processes: s,ks s μ(s,k j p(z) ) = { j p(z) | j = 1, . . . , J ; s = 1, . . . , S; ks = 1, . . . , K ;

p = 1, . . . , P; z = 1, . . . , Z },

(17.4)

where z is the designation of the intra-element process. The investment principle of transition from lower-level processes to higher-level processes is used. In the accepted hierarchy, the fourth (17.4)—level processes are the simplest, i.e., they do not include processes of an even lower level. A hierarchical cognitive map is constructed as a set of oriented functional graphs (united by a dotted line) distributed over strata levels (Fig. 17.1), each of which establishes material, energy, and information connections between processes in regulators. In addition, between the functional graphs of processes that have cause-and-effect relationships that ensure the flow of nominal functional processes, there are connections-factors that have an external synergistic effect on the main processes. The constructed hierarchical cognitive map is a static model of the complex flight control process of AS. Incident matrices are used to describe functional graphs [7]. Each set of processes  on the strata corresponds to a set of vertices of the functional graph, whose weights are determined by the corresponding elements of the set of coefficients

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Fig. 17.1 Hierarchical cognitive map describing the structure of the AS flight control process

(a set of constants) proportional to the available flight times for each controller. Each arc of the graph corresponds to a weight, which is set based on the physical interpretation of the binary relations r1 , . . . , rm , built on the Cartesian product of the sets of control U = {u 1 , . . . , u l } and perturbing effects (PE) ϑ = {ξ1 , . . . , ξq }, forming the set of input effects V = U × ϑ, r1 = U, ϑ, R1 , . . . , rm = U, ϑ, Rm , R = {R1 , . . . , Rm } ⊆ V . To account for the influence of external influence on the processes of AS functioning, an element defining an additional factor of synergetic influence is introduced into the set of disturbing influences.

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17.3 A Cognitive Aggregate-Stream Model of a Complex Flight Control Process for an Automated Spacecraft Simulation of the complex process of flight control AS is performed in the form of a “dynamic cognitive map”, in which the model parameters depend on time and the binary relations of the control set and the PE. The specificity of the onboard systems of the automatic spacecraft (ASA), as the main objects of research in this article, is manifested in that basic space-time, technical and technological limitations that are characteristic of space flight and do not have a place on Earth. These restrictions include the influence of gravitational forces on the spacecraft from such planets as the Moon and the Sun, the negative influence of cosmic radiation and debris, and uneven heating of the body. These constraints are formally described in the form of the following binary or multi-place relations (mappings) of the form (17.5), (17.6), and Tables 17.1, 17.2, 17.3, and 17.4. The structural and functional operational basis of SFO-models is selected [3]. In accordance with the accepted conceptual and terminological definition of SFO-models, functional models of individual complex processes in the form of weighted graphs with arc weights interpreted as throughput belong to the class of flow models [3]. The link weights of weighted graphs quantitatively describe the throughput of its arcs. One of the possible characteristics of the flow through the graph arc is the intensity of the discrete flow, which is determined by the energy indicators of controlling and disturbing influences on the intra-element processes. A secondary characteristic of the flow is its discreteness—the time scale for the formation of the control set. The simulation is performed using incident matrices of weighted graphs on intervals bounded by discrete moments of time t0 , t1 , . . . , ti , forming a set of moments of time T . This takes into account the weights of edges between vertexes associated with both input actions and the processes of their formation in each controller for adjacent controllers. The regularity of AS state change is described by the transition mapping (ϕ) and the output mapping (ψ): ˆ × R × T −→ , ˆ ψ : ˆ × R × T −→  ˜ ϕ:

(17.5)

A special case of a dynamic system—a finite automaton—is used to model the ˆ ⊂ ) and values ( ˜ ⊂ ) process mapping. In this case, the set of definitions ( of the process are divided by time ti , with the mapping function R ˜ = {π˜1 , . . . , π˜i }, ˆ = {πˆ1 , . . . , πˆi },   R = {r1 , . . . , ri }, ri = ri+ ∪ ri− .



˜ πˆ (i−1) , ri+ ) πˆi = ϕ( ˜ πˆi , ri+ , ri− ) π˜i = ψ(

ˆ −→ , ˜ a variant of the For the output function of mapping the process ψ :  “input-output” type finite state machine is set (Moore’s automaton [8]). The modeled processes, denoted by sets (17.1)–(17.4), are considered both simple and complex. To obtain complex processes, processes are aggregated using the

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Fig. 17.2 .

investment principle of switching from lower-level processes to higher-level processes. The procedure is aimed at increasing the dimensionality of processes and taking into account the synergetic components of the PE when moving to each higher level. The resulting model is called an aggregate model [9]. In the accepted hierarchy, the fourth—level processes are the simplest, i.e., by convention, they do not include processes at an even lower level. If necessary, additional strata levels are introduced, with the condition that the lowest-level process describes the controller as a “white box”. It is known how the controller is arranged and functions by the given physical input effect on the controller and the value of the process at its output. The aggregativity of the model allows the researcher to consistently reveal the device of the control object. Disclosure occurs due to the “transparency” of the lower-level processes that make up and determine the composition of the higher-level processes. The algorithm for constructing a cognitive-functional aggregate-flow model of the AS flight control process is developed taking into account the cognitive map and includes the following steps: Step 1. Modeling elementary processes Functional graph of an element process 11 11(z) : p = 1, s = 1, ks = 1, j = 1 (Fig. 17.2) 11 11 G(11 11(z) ) = μ[(11(z) ), K (11(z) ), 11 11 D(11(z)(m z ) ), R(11(z)(m z ) )]; 11 11 μ(11 11(z) ) = {11(1) , . . . , 11(z) , z = 1, 6}; 11 11 μ[K (11 11(z) )] = {k11(1) , . . . , k11(z) , z = 1, 6}; 11 11 11 , m z = 1, 9}; μ[D(11(z)(m z ) )] = {d11(z)(1) , . . . , d11(z)(m z) 11 11 11 μ[R(11(z)(m z ) )] = {r11(z)(1) , . . . , r11(z)(m z ) , m z = 1, 9}; 11 = {u˜ 2 , ξ˜6 (91 ˜ 2 , ξ˜4 , u˜ 2 , ξ˜5 }11 r11(z)(6) 24(3) ), u 11(z)(6) , 11 91 r11(z)(8) = {u˜ 1 , ξ˜6 (24(3) ), u˜ 1 , ξ˜4 , u˜ 1 , ξ˜5 }11 11(z)(8) , 11 11 ˜ ˜ r11(z)(8) = {u˜ 1 , ξ4 , u˜ 1 , ξ5 }11(z)(1) etc.

17 Algorithm for Constructing a Cognitive Aggregate-Stream …   Table 17.1 Mˆ I p(z)i G(11 11(z) )m z , z = 1, 6, dm z = 1, 9

(9×6)i

155

=

Modeling on [t(i−1) , ti ]th tact 11 11 ˆ 11 ˜ 11  11(z) ⊂ 11(z) , 11(z) ⊂ 11(z) , ˆ 11  ˆ z1(i) , . . . , πˆ zz(i) }11 11 , 11(z) = {π 11 ˜ 11  = { π ˜ , . . . , π ˜ } z1(i) zz(i) 11 , 11(z) + − 11 11 11 ˜ πˆ zz(i−1) , rm−z i )11 r11m z i = [rm z i ∪ rm z i ]11 , πˆ zz(i) = ϕ( 11 , + − + − 11 π˜ 11z1(i) = ψ˜ m z [(r2i , r1i ), r2i ∩ r1i = ∅]11 , 11 11 = ψ˜ m z [(r5i+ , r6i+ , r4i− ), r5i+ ∩ r4i− = ∅, π˜ 11z3(i) r6i+ ∩ r4i− = ∅]11 11 etc. (see Fig. 17.2, Table 17.1). The Initial parameters for process modeling are the parameters of the processes on the previous clock cycle 11 11 11 11 , π˜ 11z1(i−1) ; πˆ 11z2(i−1) , π˜ 11z2(i−1) etc. πˆ 11z1(i−1) Aggregate: covering a family of parameter sets of simple intra-element processes in an element  6   11  11 11 A G (11 ) = ( ),  = ∅, ∀z ∈ {1, . . . , 6} .  11 z 11(z) 11(z) 11(z+1) z=1

Image: Euler–Venn diagrams (Fig. 17.3). General description of functional graphs s,ks s of element processes, aggregates, and incident matrices G(s,k j p ) = μ[( j p(z) ), s,ks K ( j p(z) ), s,ks s D(s,k j p(z)(m z ) ), R( j p(z)(m z ) )],   s,ks s ( j p(z) ), A G (s,k jp ) = z∈Z   s,ks s s,k  = ∅, ∀z ∈ {1, . . . , Z } , j p(z) j p(z+1)   s ) , z = 1, Z , d = 1, d . Mˆ I p G(s,k m m M z z j p(z) z m z ×z

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Fig. 17.3 The unit element of the process 11 11(z)

Fig. 17.4 .

Step 2. Modeling system processes Functional graph of the system process 11 1( p) : s = 1, ks = 1, j = 1 (see Fig. 17.4). 11 11 11 G(1 ) = μ[(1( p) ), K (1( p) ), 11 D(11 1( p)(m p ) ), R(1( p)(m p ) )]; 11 11 μ(11 1( p) ) = {1(1) , . . . , 1( p) , p = 1, . . . , 4}; 11 11 11 μ[K (1( p) )] = {k1(1) , . . . , k1( p) , p = 1, . . . , 4}; 11 11 μ[D(11 )] = {d , . . . , d1( 1( p)(m p ) 1( p)(1) p)(m p ) , m p = 1, 7}; 11 11 11 μ[R(1( p)(m p ) )] = {r1( p)(1) , . . . , r1( p)(m p ) , m p = 1, 7}; 11 r1( ˜ 2 , ξ˜6 (91 ˜ 2 , ξ˜4 , u˜ 2 , ξ˜5 }11 23 ), u p)(1) = {u 1( p)(1) ; 11 11 ˜ ˜ r1( p)(2) = {u˜ 1 , ξ4 , u˜ 1 , ξ5 }1( p)(2) etc. Modeling on [t(i−1) , ti ]-th tact 11 ˆ 11 ˆ 11  ˆ p1(i) , . . . , πˆ pp(i) }11 1 , 1( p) ⊂ 1( p) , 1( p) = {π 11 11 11 ˜ ˜ 1( p) ⊂ 1( p) , 1( p) = {π˜ p1(i) , . . . , π˜ pp(i) }11 1 ; + − 11 11 11 r1m = [r ∪ r ] , π ˆ = ϕ( ˜ π ˆ , rm−p i )11 pp(i−1) 1 , m pi m pi 1 1 pp(i) pi  + + − − + − 11 ˜ π˜ 1 p1(i) = m p [(r1i , r3i , r2i , r6i ); r1i r2i = ∅;    r1i+ r6i− = ∅; r3i+ r2i− = ∅; r3i+ r6i− = ∅]11 1 ,  ˜ m p [(r2i+ , r5i+ , r1i− , r4i− ); r2i+ r1i− = ∅; π˜ 111p2(i) = 

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  s 11 Table 17.2 Incidence matrix Mˆ Is,k ( p)i G(1( p) )m p , p = 1, 4, dm p = 1, 7

(7×4)i

=

   r2i+ r4i− = ∅; r5i+ r1i− = ∅; r5i+ r4i− = ∅]11 1 ,  ˜ m p [(r4i+ , r7i+ , r3i− ); r4i+ r3i− = ∅; π˜ 111p3(i) =   r7i+ r3i− = ∅]11 1 and so on (see Fig. 17.4, Table 17.2). The initial parameters for process modeling are the parameters of the processes on the previous clock cycle πˆ 111p1(i−1) , π˜ 111p1(i−1) etc. Aggregate: covering a family of parameter sets of simple element processes in a system 

4  11 11 11 π1( p+1) = ∅, ∀ p ∈ {1, . . . , 4} . (π˜ 1( A G i (π˜ 111 ) p = p) ), π1( p) p=1

Image: Euler–Venn diagrams (Fig. 17.5). General description of functional graphs of system processes, aggregates, and incident matrices: s,ks s,ks s G(s,k j ( p) )m p = μ[( j ( p) ), K ( j ( p) ), s,ks s,ks D( j ( p)(m p ) ), R( j ( p)(m p ) )];

Fig. 17.5 System process aggregate 11 1( p)

i

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Fig. 17.6 .

s A G (s,k j )p =



s (s,k j ( p) ),

s s,k j ( p+1) = ∅, ∀ p ∈ {1, . . . , P} ;    s Mˆ I ( p) G s,k j ( p) m p , p = 1, P, dm p = 1, d M p

s s,k j ( p)



p∈P

m p×p

.

Step 3. Modeling basic processes s) : j = 1 (see Fig. 17.6). Functional graph of the base process (s),(k 1 (s),(ks ) (s),(ks ) G(1 ) = μ[(1 ), K (1 ), (s),(ks ) s) D((s),(k 1(m s,ks ) ), R(1(m s,ks ) )];

(s)1 μ(1 ) = {11 1 , . . . , 1 , s = 1, . . . , 5, ks = 1}; s) μ[K ((s),(k )] = {k111 , . . . , k1(s)1 , s = 1, . . . 5, ks = 1}; 1 (s),(ks ) (s),(ks ) (s),(ks ) μ[D(1(m s,ks ) )] = {d1(1) , . . . , d1(m , m s,ks = 1, 8}; s,ks )

(s),(ks ) (s),(ks ) s) μ[R((s),(k , . . . , r1(m , m s,ks = 1, 8}; 1(m s,ks ) )] = {r 1(1) s,ks ) (s),(ks ) s) 51 r1(7) = {u˜ 2 , ξ6 (2 ), u˜ 2 , ξ˜4 , u˜ 2 , ξ˜5 }(s),(k 1(7) ; (s),(ks ) s) = {u˜ 1 , ξ˜4 , u˜ 1 , ξ˜5 }(s),(k r1(2) 1(2) ; (s),(ks ) s) etc. r1(3) = {< u˜ 2 , ξ˜4 >, < u˜ 2 , ξ˜5 >}(s),(k 1(3) Modeling on [t(i−1) , ti ]-th tact 

s) s) s) ˆ (s),(k ˆ (s),(k  ⊂ (s),(k , = πˆ s1(i) , . . . , πˆ ss(i) 1 , 1 1 1

 s) s) s) ˜ (s),(k ˜ (s),(k  ⊂ (s),(k , = π˜ s1(i) , . . . , π˜ ss(i) , 1 , 1 1 1 r1m s,ks i = [rm+s,ks i ∪ rm−s,ks i ]1 , ˜ πˆ ss(i−1) , rm−s,ks i )1 , πˆ 1 pp(i) = ϕ( ˜ m s,k [(r2i+ , r6i+ , r1i− , r8i− ); r2i+ ∩ r1i− = ∅; π˜ 1s1(i) =  s + − r2i ∩ r8i = ∅; r6i+ ∩ r1i− = ∅; r6i+ ∩ r8i− = ∅]1 , ˜ m s,k [(r7i+ , r5i− , r6i− ); r7i+ ∩ r5i− = ∅; π˜ 1s2(i) =  s ˜ m s,k [(r1i+ , r3i+ , r2i− , r4i− ); r1i+ ∩ r2i− = ∅; r1i+ ∩ r4i− = ∅; r7i+ ∩ r6i− = ∅]1 , π˜ 1s3(i) =  s + − + − r3i ∩ r2i = ∅; r3i ∩ r4i = ∅]1 etc. (see Fig. 17.6, Table 17.3). The initial parameters for process modeling are the parameters of the processes on the previous clock cycle

17 Algorithm for Constructing a Cognitive Aggregate-Stream …   Table 17.3 Incidence matrix Mˆ I ( j)i G((s)(ks )1 )m s,ks , s = 1, 5, ks = 1, dm s,ks = 1, 8

159

(8×5)i

=

Fig. 17.7 Unit basic process (s),(k ) 1 s

πˆ 1s1(i−1) , πˆ 1s2(i−1) ; π˜ 1s1(i−1) , π˜ 1s2(i−1) etc. Aggregate: covering a family of parameter sets of simple system processes in the base process    s) (s),(k , A G ( j )s,ks = j s∈S;ks ∈K s  s) s +1) (s),(k (s+1),(k = ∅, . . . j j    (s),(ks )  j ,... ..., s∈S;ks ∈K s

∀s ∈ {1, . . . , S}, ∀ks ∈ {1, . . . , K s } Image: Euler–Venn diagrams (Fig. 17.7). General description of functional graphs of system processes, incident matrices, and aggregates: s) ), G( j ) = μ[((s),(k j (s),(ks ) s) ), D( K ((s),(k j j (m s,ks ) )],

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Fig. 17.8 Functional graph of flight control process AS  = {1 , . . . ,  j , j = 1, . . . , 5}

 Mˆ I j G((s),(ks ) j )m s,ks , s = 1, S, ks = 1, K s ,  dm s,ks = 1, dm s,ks , m×(s)(ks )   (s+1),(k +1) s) s) s ((s),(k ), (s),(k = ∅, A G ( j )s,ks = j j j s∈S;ks ∈K s

∀s ∈ {1, . . . , S}, ∀ks ∈ {1, . . . , K s } . Step 4. Modeling the AS flight control process Taking into account the weighted vertices and arcs of the functional graph (Fig. 17.8) for the complex process of flight control of AS on the [t(i−1) , ti ]-th cycle, the simulation is performed using an incident matrix of the form shown in Table 17.4. The process mapping at the device level is a function defined on the vertices  ( j) , 

ˆ ( j) ⊂  ( j) ,  ˆ ( j) = πˆ j1(i) , . . . , πˆ j j (i) (where  is the set of process defini  ˜ ˜ tions,  ( j) ⊂  ( j) ,  ( j) = π˜ j1(i) , . . . , π˜ j j (i) is the set of process values) and   arcs rm j i = rm+j i ∪ rm−j i of the functional graph of the base processes

17 Algorithm for Constructing a Cognitive Aggregate-Stream …   Table 17.4 Mˆ I ( )i G ( ( j) )m j ,

j = 1, 5, m j = 1, 12

161

 (12×5)

  G( ) = μ ( ( j) ), K ( ( j) ), D( (m j ) ), R( (m j ) ) 

   = 1 , . . . ,  j , j = 1, . . . , 5 , μ K ( ( j) ) = {k1 , . . . , k j , j = 1, . . . , 5},

μ[D( (m j ) )] = d1 , . . . , dm j , m = 1, . . . , 12 , μ[R( (m j ) )] = {r( j)(1) , . . . , r( j)(m j ) , m j = 1, . . . , 12}, r11 = {u˜ 1 , ξ˜4 , u˜ 1 , ξ˜5 }11 , r27 = {u˜ 2 , ξ˜4 , u˜ 2 , ξ˜5 }27 , r38 = {u˜ 1 , ξ˜4 , u˜ 2 , ξ˜3 }38 etc.

For the complex process of flight control AS on [t(i−1) , ti ]-th modeling tact is performed using the incident matrix having the form shown in Table 17.4. Aggregate: covering a family of parameter sets of simple basic processes ⎧ ⎫ 5 ⎨ ⎬  ˆ ) j = π˜ j+1 = ∅, ∀ j ∈ {1, . . . , 5} . A G i ( (π˜ j ), π˜ j ⎩ ⎭ j=1

i

Image: Euler–Venn  diagrams (Fig. 17.9). General description: ( j ), A G ( ) j = j∈J

  j  j+1 = ∅, ∀ j ∈ {1, . . . , J } ,   Mˆ I G (( j) )m j , j = 1, J , dm j = 1, d M j . mj×j

The initial parameters for process modeling are the parameters of the processes on the previous tact πˆ j1(1−i) , π˜ j1(1−i) etc.

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Fig. 17.9 Unit flight control process AS

In the complex process of flight control, the output functions of mapping basic ˜ πˆ j j (i−1) , rm−j i ) based processes depend on transient mappings formed πˆ j j (i) = ϕ( − on the physical interpretation of binary relations (r m j i ) and the formation of output effects in the controllers themselves for output to adjacent controllers of basic + ): processes based on the physical interpretation of binary relations (r m ji   − ˜ m j (r1i+ , r3i+ , r2i− , r12i π˜ j1(i) = ); r1i+ r2i− = ∅;     − − r12i r2i− = ∅; r3i+ r12i r1i+ = ∅; r3i+ = ∅ ,   + + − − + − ˜ r1i = ∅; π˜ j2(i) = m j (r4i , r7i , r1i , r8i ); r4i     r4i+ r8i− = ∅; r7i+ r1i− = ∅; r7i+ r8i− = ∅



(17.6)

and so on (see Fig. 17.8, Table 17.4). A cognitive-functional aggregate-flow model of a complex flight control process AS is formed by describing the processes at each level and a coordinating mechanism for their integration, which ensures the achievement of the goal in the transition from level to level.

17.4 Conclusion The article presents an algorithm for constructing a formal model of interactions of processes on Board AS with consideration of synergetic phenomena. The practical significance of the algorithm consists of a set-theoretic description of the modeling scheme, which allows initially to identify the most competent binary relations

17 Algorithm for Constructing a Cognitive Aggregate-Stream …

163

between the control and perturbing effects, which should be taken into account when switching to mathematical-functional and/or material-functional models of process controllers. In turn, this gives the researcher a multi-modal view of the processes taking place on Board and the multi-model complexes necessary to accompany the AS flight. Acknowledgements The research described in this paper is partially supported by the Russian Foundation for Basic Research (grants 19-08-00989, 20-08-01046), state research 0073-2019-0004.

References 1. Akhmetov, R.N., Makarov, V.P., Sollogub, A.V.: Bypass as an attribute of survivability of automatic spacecraft in anomalous situations. Bull. Samara State Aerosp. Univ. 14(4), 17–37 (2015). https://doi.org/10.18287/2412-7329-14-4-17-37 2. Ivanov, V.A.: Functional stability of systems. Prospects for improving their reliability. Cosmonaut. Rocket Sci. (19), 181–189 (2000) 3. Mikoni, S.V., Sokolov, B.V., Yusupov, R.M.: Qualimetry of Models and Polymodel Complexes, p. 312. RAS Ed., Moscow (2018) 4. Kravets, V.G., Lyubinsky, V.E.: Fundamentals of Space Flight Control, p. 224. Mashinostroenie, Moscow (1983) 5. Solovyov, V.A., Lysenko, L.N., Lyubinsky, V.E.: Space Flight Control. Part 1, p. 477. Bauman Moscow State Technical University, Moscow (2009) 6. Kovtun, V.S.: Stratification of the complex process of flight control of the spacecraft. Cosmonaut. Rocket Sci. 4, 78–88 (2012) 7. Gorbatov, V.A.: Fundamental Foundations of Discrete Mathematics, p. 544. Fizmatlit, Moscow (1999) 8. Okhtilev, M.Y., Sokolov, B.V., Yusupov, R.M.: Intelligent Technologies for Monitoring the State and Managing the Structural Dynamics of Complex Technical Objects, p. 410. Nauka, Moscow (2006) 9. Manuylov, Y.S., Novikov., E.A., Pavlov, A.N., Kudryashov, A.N., Petroshenko, A.V.: System Analysis and Organization of Automated Control of Space Vehicles: Textbook, p. 266. VKA, St. Petersburg (2010)

Chapter 18

A Method of Determining of Switching Instants for Discrete-Time Control Systems Sergey Khryashchev

Abstract This article discusses dynamical polysystems with piecewise constant controls. Assuming that a dynamical system is controllable in continuous time, the controllability of this system is investigated in discrete time. Discrete-time controls are constructed by using the theory of multidimensional continued fractions. Discrete switching instants were found by using continued fractions approximating continuous switching instants.

18.1 Introduction The paper studies control objects that have a finite set of control actions, i.e. dynamical polysystems. Previously, such control objects were considered in author’s papers [1– 4]. A goal of control is to transfer an object from an initial state to a final state. The control process consists of switching of elements of a polysystem at suitable instants in time. To find the switching instants, it is required to solve some accompanying equation given by evolution operator. This equation is assumed to be solvable. It is true provided that a polysystem has a sufficient number of elements to control an object in continuous time. The following problem is investigated in this work. It is required to define classes of objects such that if an object is controlled in continuous time, then it is also controlled in discrete time. In essence, the problem can be reduced to study when the accompanying equation has Diophantine (integer) solutions with a given accuracy and also to find the solutions. To solve this problem, there are various methods [1, 5]. For some classes of polysystems, we propose a method to solve the problem of finding switching instants in discrete time. This method is based on a recent version of theory of multidimensional continued fractions proposed by the author. This article provides a summary of this version. S. Khryashchev (B) St. Petersburg State Electrotechnical University, St. Petersburg, Russia e-mail: [email protected] Admiral Makarov State University of Maritime and Inland Shipping, St. Petersburg, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_18

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18.2 Basic Definitions and Concepts Let us introduce the following notation. X = {x} is an m-dimensional state space, R = {t} is a one-dimensional time space. For t ∈ R and for i = 1, . . . , l where l ∈ N is fixed, a set {Fit : X → X} of families of maps with the semigroup property Fit+s = Fit ◦ Fis is called a dynamical polysystem. Assume that t j−1 ≤ t j , j = 1, . . . , l, l ≥ m are switching instants and (t j−1 , t j ] is some time interval. For t ∈ (t j−1 , t j ], a dynamical process is given by some Fitj . Assume that x0 = x(t0 ) is a initial state. Then τ

l−1 ◦ · · · ◦ Fiτ11 )(x0 ) := F(x0 , τ ) (Fiτl l ◦ Fil−1

is a state of the polysystem at the last instant of time where τ = (τ1 , . . . , τl ) ∈ Rl . The set Rl is called an l-dimensional time space. The polysystem is ε-controllable from the state x0 to the state x∗ if for any ε > 0 there exists τ = (τ1 , . . . , τl ) depending on ε, i.e. τ = τ (ε), such that |F(x0 , τ ) − x∗ | ≤ ε.

(18.1)

The polysystem is precisely controllable if for some τ the value ε = 0. Assume that some polysystem is precisely controllable in continuous time. We show that this polysystem is also ε-controllable in discrete time for any ε > 0, i.e. the switching instants τ (ε) can be selected as integers. Besides, we give a method of finding of τ (ε) ∈ Nl for some classes of polysystems (see Sect. 18.7).

18.3 Classical Continued Fractions A classical continued fraction can be written in the form a0 + (a1 + · · · + (an−1 + (an + · · · )−1 )−1 · · · )−1 ,

(18.2)

where an ∈ N, n = 0, 1, . . . an , . . . . Short record is (a0 , a1 , . . . , an , . . . ) or a0 + 1 . For a finite continued fraction, the term an + · · · is replaced by the term an . A a1 +··· finite continued fraction can be written as an ordinary fraction in the form rn = qpnn where the numerator and denominator of rn are pn = N(rn ) and qn = D(rn ), respectively. From these numbers, it is possible to generate the sequence (r0 , r1 , . . . , rn , . . . ) which is called a sequence of convergents of a continued fraction. By direct calculations, using the method of mathematical induction, it can be shown that pn and qn are calculated recursively, i.e. they satisfy the equations p−1 = 1, p0 = a0 , pn+1 = an+1 pn + pn−1 , q−1 = 0, q0 = 1, qn+1 = an+1 qn + qn−1 .

(18.3) (18.4)

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167

18.4 Spaces of Scalars We consider some spaces whose elements we represent by generalized continued fractions. Assume that there is a set with two algebraic operations (addition and multiplication). With respect to addition, the set is an abelian group, and with respect to multiplication, it is an arbitrary group. In addition, we assume that this set is also a Euclidean space Rm of dimension m over the field R of reals. This set together with the introduced structures is denoted by T. For example, the sets R, C, H, O (i. e. real numbers, complex numbers, Hamilton numbers (quaternions), Cayley numbers (octanions)) and some other sets can be considered as T. The following notation is used below. Let G be a lattice of integer elements Rm , i. e. G := Zm and T0 := Qm where Q is the field of rational numbers. Let T∗ := T \ {0}, T := T \ T0 .

18.5 Iteration Sequences and Generalized Continued Fractions In this section, we define a continued fraction through an iterative sequence. An iterative sequence {z n } is formed less restrictive than an iterative sequence used for determining of classical continued fractions. Let a map f : T∗ → G be given by a correspondence z → a = f (z), where a is one of vertices of the unit cube containing z. A map f is called a choice function. Using the map f , we associate to each z ∈ T two sequences which are determined inductively. Namely, the sequence {z n } and the sequence {an } are given as follows: First, we define z 0 = z and a0 = f (z 0 ). Next, let z 0 , . . . , z n and a0 , . . . , an be defined for n ≥ 0 where ak = f (z k ), k = 0, 1, . . . , n. If f (z n ) = z n we terminate these sequences and, otherwise, we define the following values z n+1 = (z n − f (z n ))−1 , an+1 = f (z n+1 ).

(18.5)

Notice, the first of condition (18.5) can be rewritten in the form −1 , n ≥ 0. an = z n − z n+1

A sequence {z n }∞ n=0 is called an iteration sequence for a number z if z 0 = z, z 0 − z 1−1 ∈ G, and for n = 1, 2, . . . −1 ∈ G \ {0}. |z n | ≥ 1, z n − z n+1

A sequence {an }∞ n=0 is called a sequence of partial quotients.

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Further, we assume that z ∈ T . An iteration sequence is said to be degenerate if there exists n 0 such that |z n | = 1 for all n ≥ n 0 . It is said to be nondegenerate if it is not degenerate. We assume that the multivalued choice function f can provide that the sequence z n , n = 1, 2, . . . is nondegenerate. For the initial element z and for the algorithm defined by the function f , the sequence (a0 , a1 , . . . , an , . . . ) gives a continued fraction which is defined by the expression in the form (18.2). The sequence (a0 , a1 , . . . , an , . . . ) is the short form of continuous fraction (18.2). If there is a finite sequence (a0 , a1 , . . . , an ) then it corresponds to a finite continued fraction rn where the term (an + · · · ) in Eq. (18.2) is replaced by the term (an ). This continued fraction can be represented as a “rational element” rn , i.e. rn ∈ T0 = Qm . Similar to the classical continued fractions, an element rn can be rewritten in the form of an ordinary fraction qn−1 pn . The fraction rn is called a convergent fraction of an infinity continued fraction. By calculations, it can be shown that the quantities pn and qn satisfy equations of the form (18.3)–(18.4). Note that the commutativity of multiplication is not assumed, i.e. when multiplying, the factors are not rearranged. The following lemma holds which is an analogue of the Proposition 3.3 in [6]. Lemma 18.1 Let z ∈ T , {z n } be an iteration sequence for z and {an } be the corresponding sequence of partial quotients. Let the sequences { pn }, {qn } be given by Eqs. (18.3)–(18.4). Then we have that for all n ≥ 0 qn z − pn = (−1)n (z 1 · · · z n+1 )−1 .

(18.6)

18.6 Convergence Conditions for Convergents of Continued Fractions Next, we study the convergence conditions when |qn z − pn | → 0, n → ∞.

(18.7)

Since qn ≥ 1, condition (18.7) implies as follows: qn−1 pn → z, n → ∞. The converse may not be true. If condition (18.7) is true, then (a0 , a1 , . . . , an , . . . ) is called a continued fraction expansion of z with respect to a choice function f .

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Let Z n := z 1 · · · z n+1 . From condition (18.6), it follows that condition (18.7) is equivalent to the condition Z n → 0, n → ∞. First, we find a necessary condition for convergence, and then we find a sufficient condition for convergence of rn . Next, we assume that the conditions of Lemma 18.1 are satisfied. Theorem 18.1 Let condition (18.7) be satisfied; then the following series diverges, i.e. ∞ 

ln |z n | = +∞.

(18.8)

n=1

Proof From condition (18.6), it follows that 1 1 ≥ = e−(ln |z1 |+···+ln |zn+1 |) . |Z n | |z 1 | · · · |z n+1 |

|qn z − pn | ≥

Therefore, if condition (18.7) is satisfied, then condition (18.8) is satisfied. Theorem 18.1 is proved. Theorem 18.2 Assume the following series diverges, i.e. ∞ 

ln(|z n−1 |−1 ) = +∞,

(18.9)

n=1

then condition (18.7) is satisfied. Proof From condition (18.6), it follows that −1 |qn z − pn | = |(z 1 · · · z n+1 )−1 | ≤ |z n+1 | · · · |z 1−1 | = −1

eln |z1

−1 |+···+ln |z n+1 |

−1 −1

= e−(ln(|z1

−1 −1 | )+···+ln(|z n+1 | ))

.

Therefore, condition (18.9) implies condition (18.7). Theorem 18.2 is proved.

18.7 The Application of Continued Fraction Theory for Control In this section, we find integer solutions τ = (τ1 , . . . , τl ) of inequality (18.1) for given ε. First, for ε = 0, consider condition (18.1) which can be rewritten in the ˜ 0 , x∗ , τ ) = F(x0 , τ ) − x∗ . Assume that the func˜ 0 , x∗ , τ ) = 0, where F(x form F(x ˜ 0 , x∗ , τ ) is asymptotically linear in τ . Then this equation can be represented tion F(x in the form

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ψ(x0 , x∗ ) + (x0 , x∗ )τ + Tx0 ,x∗ (τ ) = 0,

(18.10)

where Tx0 ,x∗ (τ ) → 0 as τ → ∞. Notice, that condition (18.10) is satisfied for some classes of linear in states systems. Then the function Tx0 ,x∗ (τ ) exponentially decreases. Next, we assume that rank (x0 , x∗ ) = m. Let the x0 , x∗ be fixed. We omit the x0 , x∗ in the following formulas. Consider the linear part of equality (18.11), namely, the equation ψ + τ = 0. Let τ = (τ  , τ  ) where τ  := (τ1 , . . . , τm ), τ  := (τm+1 , . . . , τl ) and  = (  ,   ] = 0. Then the equation ψ + τ = 0 can be rewritten as follows  ) where det[ ∂ ∂τ  ψ +   τ  +   τ  = 0 and it is solvable with respect to τ  in the form τ  = ϕ + τ  ,

(18.11)

where ϕ = −(  )−1 ψ,  = −(  )−1   . Approximate the real matrix (ϕ, ) by the rational matrix (r, R), where r = 1 ρ, R = q1 P. The matrix (ρ, P) is integer where ρ ∈ Nm and P = (ρ1 , . . . , ρk ), q ρi ∈ Nm , i = 1, . . . , k. For the matrix (ϕ, ), the matrix (r, R) is convergent of multicomponent continued fraction in the form (18.2) such that |ϕ − r | < δ, | − R| < δ for some δ depending on n ∈ N. Rename τ  =: s where vector s = col(s1 , . . . , sk ) ∈ Nk and consider the rational vector q1 (ρ + Ps) ∈ Nm which approximates the real vector τ  given by Eq. (18.11). Suppose that the vector s can be chosen such that the components of the vector q1 (ρ + Ps) are positive integers. For this, the vector ρ + Ps should be divided component-wise by the integer q. In other words, the positive integer vector s must satisfy the vector equation for the congruence modulo q as follows: ρ + Ps ≡ 0

mod q.

(18.12)

Conditions are known (see for example [7]) when Eq. (18.12) can be solved, i.e. there is constant vector c such that s = (s1 , . . . , sk ) ≡ c = (c1 , . . . , ck )

mod q.

(18.13)

Thus, we get an integer approximation of τ , namely τ = (τ  , τ  ) ≈ (, s) =: τ ∈ Nl where l = k + m, s = (s 1 , . . . , s k ) ∈ Nk is the smallest element of residue class ∈ Nm . (18.13), i. e. the norm |s| is minimal. In this case, the value ρ = ρ+Ps q It is especially simple to find a solution of vector Eq. (18.12) for the case m = k. In the following example, we consider the case m = k = 1. Thus, there is the scalar equation ρ + Ps ≡ 0

mod q.

(18.14)

The ordinary rational fraction Pq can be represented by a continued fraction. Suppose that for some positive integer ν depending on a given accuracy of calculations, there is a finite sequence of convergent fractions (r0 , r1 , . . . , rν−1 , rν ). In this sequence,

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ν−1 rν := qPνν is the latest convergent fraction, rν−1 := qPν−1 is the penultimate convergent fraction, values of the remaining convergents are not significant. Since Pν = P and qν = q then rν = Pq . Since qν Pν−1 − qν−1 Pν = (−1)ν (see [7]), there is the following “chain of congruences” modulo mod q:

qν−1 Pν ≡ (−1)ν−1 ⇒ (−1)ν−1 qν−1 Pν ≡ 1 ⇒ (−1)ν−1 qν−1 Pν s ≡ s (18.14)

⇒ (−1)ν−1 qν−1 (−ρ) ≡ s ⇒ (−1)ν qν−1 ρ ≡ s.

Let q˜ := qν−1 . Note, q˜ depends on q. Thus, the residue class s ≡ (−1)ν ρ q˜ =: c

mod q

(18.15)

is the solution of original congruence (18.14). The smallest positive number s from residue class (18.15) can be easily calculated. Now, we consider the general case k = m for an arbitrary m. It is easy to show that vector Eq. (18.12) is equivalent to the following system of equations δi + si ≡ 0, i = 1, . . . , m.

(18.16)

Here, = det P = 0 and δi = det Pi where P = (ρ1 , . . . , ρ i , . . . ρ k ),

Pi = (ρ1 , . . . , ρ, . . . ρ k ).

Taking into account (18.15), the solution of system (18.16) is as follows: ˜ si ≡ (−1)ν δi

mod q, i = 1, . . . , m,

˜ is the numerator of the penultimate convergent of continued fraction . where q ˜ mod q where i = 1, . . . , m. Thus, in formula (18.13), the values ci = (−1)ν δi From Eq. (18.11), the instants of control switching, which are the components of the vector τ , provide the control error ε = |  |(1 + |s|)δ in discrete time, while for the set of switching instants τ , there is zero control error in continuous time. Example 18.1 Let the two-dimensional state space R2 be represented as the complex plane C. Consider the polysystem defined by the following mappings of the complex plane C → C as follows: z → e(λκ +iβκ )t z, κ = 1, 2, 3, 4, where the i is the imaginary unit, the parameter t is real time. Let us assume that the starting point z 0 should be transferred to the final point z ∗ . Then relation (18.1) for ε = 0 gives the complex equation in the form e(λ1 +iβ1 )τ1 e(λ2 +iβ2 )τ2 e(λ3 +iβ3 )τ3 e(λ4 +iβ4 )τ4 z 0 = z ∗ ,

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which can be rewritten as a system of two equations as follows:    z∗  λ1 τ1 + λ2 τ2 + λ3 τ3 + λ4 τ4 = ln   , z0 z∗ β1 τ1 + β2 τ2 + β3 τ3 + β4 τ4 = arg . z0 Suppose that the matrix of this system is a matrix of rank two. Taking into account formula (18.11) for m = 2 and k = 2, the values τ1 , τ2 can be expressed through the parameters s1 := τ3 , s2 := τ4 as follows: τ1 = ϕ10 + ϕ11 s1 + ϕ12 s2 , τ2 = ϕ20 + ϕ21 s1 + ϕ22 s2 . Approximate the set (ϕ10 , ϕ11 , ϕ12 ; ϕ20 , ϕ21 , ϕ22 ) by a 6-component continued fraction. By formulas given above in Sect. 18.7, the set of switching instants (1 , 2 , s 1 , s 2 ) can be found such that (τ1 , τ2 , τ3 , τ4 ) ≈ (1 , 2 , s 1 , s 2 ) ∈ N4 .

18.8 Conclusion For a class of polysystems operating in continuous time, the problem of control at discrete times is solved. To find the instants of control switching, it is required to find integer solutions of some equation specified by evolution operators of a polysystem. To find integer solutions, a new version of continued fractions proposed by the author is used. Comparing the results of this work with other results, the following can be noted. Let some simulated polysystem have both stochastic properties and algebraicgeometric properties sufficient for its controllability. Then controls using only the stochastic properties usually give a longer control time than a control time in the alternative case. However, in the first case, it is possible to control a wider class of polysystems than in the second case. The results obtained in this work can be used in design of polysystems for which a state space is a certain region of the Euclidean space. In this case, a polysystem is some set of linear systems with given properties sufficient for its controllability. By switching the elements of this polysystem at suitable time instants, it is possible to achieve a given control goal.

References 1. Khryashchev, S.M.: Controllability and Number-theoretic Properties of Dynamical Polysystems. Nonlinear Phenom. Complex Syst. 16(4), 388–396 (2013) 2. Khryashchev, S.M.: On some metric characteristics of polysystems with control switchings in discrete time nonlinear. Phenom. Complex Syst. 21(1), 92–101 (2018)

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3. Khryashchev, S.M.: Statistic methods for control of dynamical polysystems. Nonlinear Phen. Complex Syst. 18(4), 489–501 (2015) 4. Khryashchev, S.M.: On finding switching instants for control of discrete-time dynamical polysystems by using continued fractions. In: Proceedings of 15th International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiys Conference), STAB 2020, 9140678 (2020) 5. Kvitko, A.N.: A method for solving boundary value problems for nonlinear systems in class of discrete controlls. Differ. Equ. 44(11), 1499–1509 (2008) 6. Dani, S.G., Nogueira, A.: Continued fractions for complex numbers and values of binary quadratic forms. Trans. Amer. Math. Soc. 366(7), 3553–3583 (2014) 7. Buchstab, A.A.: Number Theory. Prosveshchenie, Moscow (1966).(in Russian)

Chapter 19

Algorithmization of Receiving Orbits of Weierstrass and Orbits of Tangences Maria A. Shagai, Mikhail D. Iofe, and Alexander V. Flegontov

Abstract This article analyzes families of equations such as the Weierstrass orbit, the tangent orbit solutions of which have a special structure. The relations are derived for these classes, on the basis of these relations (according to a finite set of functions), solutions are constructed for some generalized homogeneous Emden–Fowler equations through a finite set of special functions. This is done for algorithmizing the process of searching for new equations. Such interpretation of equations will be useful to specialists in the field of differential equations.

19.1 Introduction A new method for finding solutions was proposed in V. F. Zaitsev’s Differential puzzles on solutions of nonlinear equations [1], that is, the differential puzzle method, which allows to find subclasses of the studied class of ordinary differential equations, the solutions of which can be expressed in terms of a finite set of elements that can be expressed in terms of given classes of functions (polynomials, Weierstrass functions, etc.). In [4], an algorithm for searching new equations and their solutions which are constructed from some finite set of polynomials was proposed, and moreover, new solutions were also obtained for some generalized Emden–Fowler equations. Thus, there is a question: can the arguments, which allow to create an algorithm for the orbit of polynomials, be applied to other orbits? Let’s take a look at the generalized equations of Emden–Fowler’s class M. A. Shagai · M. D. Iofe · A. V. Flegontov (B) Herzen State Pedagogical University of Russia, 48, Moika Emb., St. Petersburg 191186, Russia e-mail: [email protected] M. A. Shagai e-mail: [email protected] M. D. Iofe e-mail: [email protected] A. V. Flegontov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_19

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Table 19.1 Elements of the puzzle Weierstrass orbit E1 E2 E3 E4 E5 E6 E7 E8 E9

Tangent orbit

=τ = ℘ (τ ) = ℘  (τ ) = τ 2 ℘ (τ ) ∓ 1 = ℘  (τ ) ± 2τ ℘ 2 (τ ) = τ ℘  (τ ) − ℘ (τ ) = τ ℘  (τ ) + 2℘ (τ ) = τ 3 ℘  (τ ) + 3τ 2 ℘ (τ ) ∓ 1 = τ 3 ℘  (τ ) − 4τ 2 ℘ (τ ) ± 6

T1 = cosh(τ + C2 ) cos(τ ) T2 = tanh(τ + C2 ) + tan(τ ) T3 = tanh(τ + C2 ) + tan(τ ) T4 = 3T2 T3 − 4 1 = cos(τ ) − sin(τ + C2 ) 2 = sinh(τ ) + cos(τ + C2 ) 3 = sinh(τ ) − cos(τ + C2 ) 4 = cosh(τ ) + sin(τ + C2 ) 5 = 32 3 − 221

 l y  = Ax n y m y  ,

(19.1)

where x(τ ) = ϕ(τ, C1 , C2 ), y(τ ) = ψ(τ, C1 , C2 ), and the functions ϕ and ψ are elements of a subset of some differential ring.. . We introduce the following function χ =ψ / ϕ . Then (19.1) is converted to .

.

.

χ ∼ ϕ n ψ m (ϕ )1−l (ψ)l .

(19.2)

The triple of functions ϕ, ψ, χ defines the functions x, y, y  . Elements of the puzzle for each of the orbits (Table 19.1). This table shows all the elements that are currently known.

19.2 Identification of Considered Orbits Regularities It was found out that for the Emden–Fowler’s equations type, the solutions of which are polynomials, there are at least three proportions between the polynomials, which are included in all known solutions. These proportions link their first derivatives. In [4], an algorithm was proposed to search for new equations and their solutions. This algorithm, in addition to all the known ones, found new solutions, but based not on the original set of polynomials, but on the other ⎧ √ ⎪ I m(τ + C1 + i 3C1 )w+1 ⎪ ⎪ , √ ⎨ Pw = 3C1 (w + 1) ⎪ ⎪ ⎪ ⎩ P2 = τ 2 + 2C1 τ 2 + 4C12 ,

(19.3)

19 Algorithmization of Receiving Orbits of Weierstrass and Orbits of Tangences

177

Table 19.2 Analysis of orbit elements (E 1 , E 2 , E 3 )

E 1 = 1, E 2 = E 3 , E 3 = ±6E 22 ,

(E 2 , E 3 , E 5 )

(T1 , T2 , T3 )

T1 = T1 T3 , T2 = 2 − T2 T3 , T1 T2 + T1 T2 = 2T1 ,

(1 , 2 , 3 )

E 2 = E 3 , E 3 = ±6E 22 , E 2 E 5 − 2E 2 E 5 = ±2, 1 = 3 , 2 = 1 , 2 3 + 32 = 22 3 ,

where w is a positive integer such that w > 2, and from each triple of polynomials (P2 , Pw−1 , Pw ) 6 solutions can be constructed, one of which corresponds to an , − w−1 , 2w+3 ). equation of class y  = Ax n y m (y  )l given by three parameters (− 2w+1 w w w+2 Functions are of the form: ⎧ −1 ϕ = Pw−1 P2w , ⎪ ⎪ ⎪ ⎪ ⎪ w ⎨ −1 ψ = Pww+1 Pw−1 , (19.4) ⎪ ⎪ ⎪ w+2 ⎪ ⎪ ⎩ χ = Pww+1 . And the remaining 5 groups of equations and their solutions can be obtained from symmetry considerations. Having done a similar research, it was found out that this statement is true for the Weierstrass orbit and for the tangent orbit. While studying the Weierstrass orbit and the tangent orbit, it was found that 4 or 6 solutions can be constructed from some triples of the puzzle elements, one (solution) of them corresponds to an equation of the class y  = Ax n y m (y  )l , and the others 3 or 5 can be obtained due to symmetry. For these triples of elements, three relations that express the relation of the elements must be done. Having analyzed all the equations of the Weierstrass orbit and the tangent orbit from the handbook [3], we managed to obtain three relations for each group of equations (Table 19.2). Let us give an example of the modified algorithm of the propagation operation of equations for some triple of elements, which has got three known relations to elements and one equation (which is also known). The solution of the equation (n, m, l) = (0, 2, 0) is represented in the following form (19.1), where . . ϕ = E 1 , ψ = E 2 , χ =ψ / ϕ = E 3 , and the following relationships are true for the elements:

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E 1 = 1, E 2 = E 3 , E 3 = ±6E 22 . Let us compare the triple (ϕ, ψ, χ ) in some order, with the triple of elements (E 1 , E 2 , E 3 ). Case 1 .

.

.

ϕ = E 1a , ψ = E 3b , χ = ±6E 2c ⇒ ϕ = a E 1a−1 , ψ= ±6bE 22 E 3b−1 , χ = ±6cE 2c−1 E 3 . Let . . us substitute the values (ϕ, ψ, χ ) in χ =ψ / ϕ then

±6E 2c =

±6bE 22 E 3b−1 a E 1a−1

and if we equate the degrees of elements, we get (a, b, c) = (1, 1, 2). Then (ϕ, ψ, χ ) . . . = (E 1 , E 2 , ±6E 32 ). And from χ ∼ ϕ n ψ m (ϕ )1−l (ψ)l , we have the following: E 2 E 3 ∼ E 1n E 2m E 22l

⇒ (n, m, l) = (0, 1, 1/2).

Case 2 ϕ = E 3a , ψ = E 1b , χ = E 2c

⇒ (n, m, l) = (1, 0, 25 )

Case 3 ϕ = E 3a , ψ = E 2b , χ = E 1c where

E 1c =

.

.

.

ϕ = ±6a E 22 E 3a−1 , ψ= bE 2b−1 E 3 , χ = cE 1c−1 ,

⇒

bE 2b−1 E 3 ±6a E 22 E 3a−1

⇒

(a, b, c) = (2, 3, 0)

Case 4 ϕ = E 32 , ψ = E 23 , χ = 1

⇒

.

.

0 ∼ E 22n E 23m E 22−2l E 31−l E 2l E 3l , that is like (n, m, l) it is impossible to select. Case 5 ϕ = E 2a , ψ = E 1b , χ = E 3c ⇒

.

ϕ = ±12E 22 E 3 , ψ= 3E 2 E 3 , χ = 0,

(n, m, l) = (2, 0, 3)

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Case 6 ϕ = E 2a , ψ = E 3b , χ = ±6E 1c ⇒ 0 ∼ E 23n E 12m E 22−2l E 31−l E 1l , that is like (n, m, l) it is impossible to select. Having taken into consideration all possible comparisons, we can make a conclusion that in four of the six cases, they could get the equations, but in the rest two cases, the equations could not be constructed. It is said in [2, 5] that the class of generalized Emden–Fowler equations admits a general group of transformations D3 {g, r}, which can be defined by two generators r : x = u, y = t, (n, m, l) → (m, n, 3 − l) and 1

g:x =u

1 n+1

1m , y =  , (n, m, l) → u



n 2m + 1 1 ,− , . 1−l n+1 m

For any values n, m, l, except for the singular points m = 0, −1; n = 0, −1; l = 1, 2. Due to symmetries which are taken into consideration in [4], we can get the equations (2, 0, 3), (0, 1, 21 ), (1, 0, 25 ) from the equations (0, 2, 0), moreover, two points are singular. This result shows that the algorithm is self-sufficient in terms of the reproduction of solutions and it does not require using a discrete group transformations. By this method, solutions are obtained simply by rearranging the original functions. Singular points are also taken into account when searching for equations in this method.

19.3 Description of the Algorithm for Finding New Equations of the Weierstrass Orbit Let us have a look at the elements of the puzzle of this orbit, their general structure is given below ⎧ E 1 = τ, ⎪ ⎪ ⎪ ⎪ ⎨ E 2 = τ k ℘ (τ ) + f, ⎪ ⎪ ⎪ ⎪ ⎩ E 3 = τ s ℘  (τ ) + cτ r ℘ h (τ ) + d,

(19.5)

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where k, f, s, c, r, h, d—are integers. We will call an ordered set of these numbers the defining set of equations. Representing the functions ϕ(τ ) and ψ(τ ) through the degrees of special functions, it became possible to find some discrete subclass of Eq. (19.1), the solutions of which are special functions, i.e. .

.

.

p

q

a1 E 1 E 2 E 3 + b1 E 1 E 2 E 3 + c1 E 1 E 2 E 3 = E 1 1 E 2 1 E 3r1 .

(19.6)

Let us take the general form of the (19.5) into (19.6) and take out of the brackets τ on the right and left sides in the maximum degree, that is, the degree τ , which is each term. It turns out from equality (19.6) that the maximum degree of τ coincides both on the right and on the left sides.

    a1 +b1 · k + c1 · s τ k+s ℘ (τ ) + c · b1 + c1 · h τ 1+k+r ℘ h (τ )+    + c1 · f · c · h · τ 1+r · ℘ h−1 (τ ) + b1 · d · τ 1+k + f · a1 + c1 · s τ s ℘  (τ )+   + 4 · b1 + 6 · c1 τ 1+k+s ℘ 3 (τ ) + 6 · c1 · f · τ 1+s · ℘ 2 (τ )+

     + c · a1 + b1 · k + c1 · r τ k+r ℘ h (τ ) + d · a1 + b1 · k τ k ℘ (τ )+   + c · f · a1 + c1 · r τ r ℘ h (τ ) − b1 · τ 1+k+s + a1 · f · d =  q1  s  r1 = τ p1 τ k ℘ (τ ) + f τ ℘ (τ ) + cτ r ℘ h (τ ) + d (19.7) Let us have a look at the right side  q1  s  r1 τ ℘ (τ ) + cτ r ℘ h (τ ) + d , τ p1 τ k ℘ (τ ) + f let us take all possible cases of taking out the maximum degree possible of τ , which is contained in each term. The following cases cover all options for a given structure (selection of basic polynomials). 1. f = 0, c = 0, d = 0. The maximum degree τ is p1 + k · q1 + s · r1 . 2. f = 0, c = 0, d = 0. If s < r , so the maximum degree τ is p1 + k · q1 + s · r1 . If s ≥ r , so the maximum degree τ is p1 + k · q1 + r · r1 . 3. f = 0, c = 0, d = 0. The maximum degree τ is p1 + k · q1 . 4. f = 0, c = 0, d = 0. If s < r , so the maximum degree τ is p1 + s · r1 . If s ≥ r , so the maximum degree τ is p1 + r · r1 . 5. f = 0, c = 0, d = 0. The maximum degree τ is p1 . Mind that, on the left side, the maximum degree τ cannot be clearly determined because of the fact that the terms can be reduced depending on the defining set, this

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fact does not allow us to simplify the searching of the algorithm, but the principle itself is convenient for the solution. At the moment, it has been impossible to make a clear algorithm of actions in each of the cases considered above, and an algorithm for searching of the equations of the orbit of the tangents has not been developed yet, but relying on the elements of the puzzle that has already been obtained, as well as on the equations that has been found, we can identify some patterns that help to find a defining set in each case.

19.4 Conclusion The results of our analysis allow us to draw the following conclusions which are important for our study, i.e., Weierstrass orbit and tangent orbit have a structure similar to one of polynomial orbit; Using a modified algorithm for finding equations gives a positive result, but it does not allow to generalize the entire search. For further progress, additional research is needed. The first direction is to develop an algorithm that would expand the studied orbit, find new relationships, elements, and equations. The second direction is the analysis of other orbits. In particular, it is necessary to answer the question: are the arguments that were given for the orbit of polynomials and the orbit of Weierstrass suitable for the orbit of Euler, the orbit of sines, and the orbit of Bessel. The third important direction is the application of this algorithm to other classes of generalized-homogeneous differential equations of the second order.

References 1. Zaitsev, V.F.: Differential puzzles on solutions of nonlinear equations. Some actual problems of modern mathematics and mathematical education. In: Materials of the LXX International Conference Herzen Readings (St. Petersburg, April 10–14, 2017), pp. 58–62 (2017) 2. Zaitsev, V.F., Flegontov, A.V.: Discrete-Group Methods for Integrating Ordinary Differential Equations. LIIAN, Leningrad (1991) 3. Zaitsev, V.F., Polyanin, A.D.: Handbook of Nonlinear Differential Equations. Applications in Mechanics, Exact Solutions. Science, Moskow (1993) 4. Zaitsev, V.F., Iofe, M.D.: New solutions of nonlinear equations representable in the class of polynomials. Some actual problems of modern mathematics and mathematical education. In: Materials of the LXX International Conference Herzen Readings (St. Petersburg, April 10–14, 2017), pp. 63–68 (2017) 5. Zaitsev, V.F., Lynchuk, L.V., Flegontov, A.V.: Differential Equations (Structural Theory). Lan’, Saint-Petersburg (2017)

Part III

Methods for Analysis and Design of Systems with Time-Delay

Chapter 20

Systems with Propagation: A Bunch of Models and a Research Program Vladimir R˘asvan

Abstract There are discussed here several applications of Physics and Engineering which are described by 1D propagation—hyperbolic partial differential equations in two dimensions (time and one space dimension for distributed parameters) having nonlinear and nonstandard boundary conditions. Nonstandard boundary conditions means they contain ordinary differential equations. It is presented the association of a system of functional differential equations and the one to one correspondence between the solutions of the two mathematical objects. In most cases the functional differential equations thus associated are of neutral type having (sometimes) a marginally stable difference operator. A set of open problems for these equations is listed.

20.1 Introduction and Overview We shall consider in this paper the dynamics of physical systems, occurring in Physics but mainly in Engineering, described by (non)linear Partial Differential Equations (PDE) of hyperbolic type “in the plane”, i.e., having the “time” and a “space line”— the so-called 1D systems [2, 22, 32]. The boundary conditions are (non)linear and nonstandard (in the sense that they are coupled through an internal feedback to ordinary differential equations). We give here some examples of such systems which are, generally speaking, quite well known. • Control dynamics of co-generation (combined heat electricity generation) [9, 44, 45]. • Control and dynamics of water hammer and frequency/megawatt for hydroelectric power plants [24, 34, 42]. • Control of open canals [21, 31, 40]. • Control and dynamics in gas networks [14–16] V. R˘asvan (B) Romanian Academy of Engineering Sciences and University of Craiova, A.I. Cuza str. no. 13, 200585 Craiova, Romania e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_20

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All these applications belong to the more comprehensive case of controlling systems of conservation laws. Another class of applications arises from the dynamics and control of the mechanical systems containing distributed parameter beams and rods, such as those presented in [50].  The overhead crane and the marine riser,  The flexible manipulator, and  The oil-well drillstring. To the aforementioned applications, we can add two applications from the field of electrical and electronic engineering. ♦ The distributed parameter waveguides [5]. ♦ The synchronization of local oscillators through one-dimensional distributed media [51] We shall show in the following how these apparently various systems can be approached in a unitary way—from the point of view of stability and stabilization.

20.2 Boundary Value Problems and Functional Differential Equations As pointed out in [41], or, more recently, in [48, 49], the very first known reference on an equation with deviated argument, belonging to Bernoulli [4], has in its rather long title the mention of a weighted stretched vibrating string with distributed masses on it. The equation was as follows y˙ (t) = y(t − 1)

(20.1)

and it is important to mention that this equation arose from a problem for hyperbolic PDEs. More recently, in the papers of A. D. Myshkis and his co-workers, published along half century [1, 36–38], there is presented a methodology of associating certain systems of functional differential and/or integral equations to certain hyperbolic PDEs with dynamic boundary conditions (containing Volterra operators). Being rather general, their results are not easy to follow or apply. Somehow after the first two aforementioned references, Cooke and a co-worker [6, 7] considered a more explicit case. Their principal result, partly proven in [6] and completely proven in [50] will be emphasized in what follows. Consider the following Initial Boundary Value Problem (IBVP) with derivative boundary conditions ∂u + ∂u + + τ + (λ, t) = + (λ, t) ∂t ∂λ ∂u − ∂u − + τ − (λ, t) = − (λ, t), 0 ≤ λ ≤ 1, t ≥ t0 , ∂t ∂λ

(20.2)

20 Systems with Propagation: A Bunch of Models and a Research Program

 m   dk + dk − + − ak (t) k u (0, t) + ak (t) k u (0, t) = f 0 (t) dt dt k=0  m  k  d + dk − + − bk (t) k u (1, t) + bk (t) k u (1, t) = f 1 (t) dt dt k=0

187

(20.3)

+ − u ± (λ, t0 ) = u ± 0 (λ) , 0 ≤ λ ≤ 1 ; τ (λ, t) > 0 , τ (λ, t) < 0

Clearly the equations are written in the Riemann invariants and, moreover, the equations are decoupled. This form of the PDEs cannot be obtained in all cases, as it will appear next. Consider the differential equations of the characteristics 1 dt = ± , τ + (λ, t) > 0 , τ − (λ, t) < 0 dλ τ (λ, t)

(20.4)

and let t ± (σ ; λ, t) the two characteristic curves crossing some point (λ, t) of the strip [0, 1] × [t0 , t1 ). We define the propagation times forward and backward along the characteristics T + (t) := t + (1; 0, t) − t , T − (t) := t − (0; 1, t) − t

(20.5)

Since t + (·; λ, t) is increasing, it can be always extended “to the right” i.e. up to σ = 1 and since t − (·; λ, t) is decreasing it can be always extended “to the left”, i.e., up to σ = 0. Writing down the “progressive (forward) wave” along the increasing characteristic and the “reflected (backward) wave” along the decreasing one, the following representation formulae for u ± (λ, t) are obtained u + (λ, t) = u + (1, t + (1; λ, t)) − u − (λ, t) = u − (0, t − (0; λ, t)) +



1

+ (σ, t + (σ ; λ, t)) dσ τ + (σ, t + (σ ; λ, t))

λ

− (σ, t − (σ ; λ, t)) dσ τ − (σ, t − (σ ; λ, t))

λ

(20.6)

 0

For the cases when t + (·; λ, t) can be extended up to σ = 0, i.e., t + (0; λ, t) > 0 and t − (·; λ, t) up to σ = 1, i.e., t − (1; λ, t) > 0, we obtain from (20.6) and taking into account (20.5) +

+

+



u (0, t) = u (1, t + T (t)) −

1

+ (σ, t + (σ ; 0, t)) dσ τ + (σ, t + (σ ; 0, t))

1

− (σ, t − (σ ; 1, t)) dσ τ − (σ, t − (σ ; 1, t))

0

u − (1, t) = u − (0, t + T − (t)) +

(20.7)

 0

Observe that (20.7) are in fact some functional relations between the boundary values of the forward and backward waves. Denoting

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V. R˘asvan +

+



+

1

y (t) := u (1, t) ,  (t) := 0

y − (t) := u − (0, t) ,  − (t) :=

+ (σ, t + (σ ; 0, t)) dσ τ + (σ, t + (σ ; 0, t)) (20.8)



1 0

−

−

 (σ, t (σ ; 1, t)) dσ τ − (σ, t − (σ ; 1, t))

we find that (y + (t), y − (t)) thus defined satisfy the following system of differential equations with deviated argument  m  m   dk + dk − dk + − + ak (t) k y (t + T (t)) + ak (t) k y (t) = f 0 (t) + ak+ (t) k  + (t) dt dt dt k=0 k=0  m  m   dk dk dk bk+ (t) k y + (t) + bk− (t) k y − (t + T − (t)) = f 1 (t) − bk− (t) k  − (t) dt dt dt k=0 k=0 (20.9) Its solutions can be constructed by steps provided some initial conditions are given. Following [50] we obtain y0+ (t + (1; λ, t0 )) = u + 0 (λ) + y0− (t − (0; λ, t0 )) = u − 0 (λ) −



1

+ (σ, t + (σ ; λ, t0 )) dσ τ + (σ, t + (σ ; λ, t0 ))

λ

− (σ, t − (σ ; λ, t0 )) dσ τ − (σ, t − (σ ; λ, t0 ))

λ

(20.10)

 0

But (20.6) can be also written as +

+

+

u (λ, t) = y (t (1; λ, t)) − u − (λ, t) = y − (t − (0; λ, t)) +



1 λ

 0

+ (σ, t + (σ ; λ, t)) dσ τ + (σ, t + (σ ; λ, t)) (20.11)

λ

−

−

 (σ, t (σ ; λ, t)) dσ τ − (σ, t − (σ ; λ, t))

which appear as representation formulae allowing the statement of the following mathematical result Theorem 20.1 Consider the boundary value problem (20.2)–(20.3). If u ± (λ, t) is a solution satisfying the equations as well as the initial and the boundary conditions, then y ± (t) defined by (20.8) are a solution of (20.9) with the initial conditions defined by (20.10). Conversely, let y ± (t) be a sufficiently smooth solution of (20.9) with a set of initial conditions defined on [t0 , t0 + T ± (t0 )]. Then u ± (λ, t) defined by (20.11) is a solution of (20.2)–(20.3) with the initial conditions u ± 0 (λ) defined also by (20.11) computed at t = t0 .

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Theorem 20.1 ascertains a one to one correspondence between the solutions of two mathematical objects; in this way, all results obtained for one of them are automatically projected back on the other. A long date experience shows that some properties are easier to be obtained for the PDEs, e.g., a Lyapunov functional deduced from the energy identity while others are easier to be obtained for the Functional Differential Equations (FDEs), e.g., the Barbashin Krasovskii LaSalle invariance principle. But the aforementioned approach represents also a rather natural approach of introducing various classes of equations with deviated argument. We define the integers L + = max{k : ak+ (t) = 0} , L − = max{k : bk− (t) = 0} K + = max{k : bk+ (t) = 0} , K − = max{k : ak− (t) = 0}

(20.12)

M = L + + L − − (K + + K − ) According to the sign of M system (20.9) belongs to one of the three classes of systems with deviated argument: if M > 0, it is of delayed type; if M < 0, it is of advanced type; if M = 0, it is of neutral type. This assertion is consistent with the classification of G. A. Kamenskii see [11]. As the experience shows, most systems with deviated arguments associated to the boundary value problems for hyperbolic PDEs are of neutral type. Consequently their solutions are not smoothed in time (unlike those of the systems of retarded type) and this aspect, in particular the propagated discontinuities, reflects the mismatch between the initial and the boundary conditions. For this reason, but for simplicity also, we shall consider in the sequel only (possibly discontinuous) classical solutions.

20.3 Lossless and Distortionless Propagation 20.3.1 The Wave Propagation We shall start here from some considerations in [5], some of them completed in [48, 49]. In the 1D wave equation (20.13) c2 vx x − vtt = 0 the general solution reads v(λ, t) = ϕ(t − λ/c) + ψ(t + λ/c)

(20.14)

with ϕ and ψ—arbitrary functions: ϕ is a wave “traveling to the right” with velocity c (the forward wave), being transmitted distortionless and also lossless but with a time delay τ = λ/c; the same holds for the backward wave “traveling to the right”. The equation of the spherical waves in n dimensions reads

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  n−1 vr = vtt c2 vrr + r

(20.15)

and, if n = 3, the general solution is v(r, t) =

1 (ϕ(t − r/c) + ψ(t + r/c)) r

(20.16)

Now the transmission is also distortionless but lossy, with attenuation r −1 . As mentioned in [8], living in a three-dimensional space is a happy circumstance because signals can be broadcasted without distortion only in one or three dimensions.

20.3.2 The Distortionless Waveguide The waveguide is, generally speaking, an inhomogeneous and dissipative electrical transmission line described by the telegraph equations − vλ = r (λ)i + l(λ)i t , −i λ = g(λ)v + c(λ)vt

(20.17)

The line is called distortionless in voltage if there exists a solution of the form v(λ, t) = f (λ)ϕ(t − τ (λ)) , v(0, t) = ϕ(t)

(20.18)

where f (·) is called attenuation and τ (·) is called propagation delay; they should be independent of ϕ and subject to f (0) = 1, τ (0) = 0. Following [5, 48, 49], we introduce the Riemann invariants through u ± (λ, t) := v(λ, t) ± a(λ)i(λ, t)

(20.19)

to obtain the equations       r (λ) − a  (λ) r (λ) − a  (λ) 1 1 + + g(λ) + u g(λ) − u− −u + = a(λ) u + + t λ 2 a 2 (λ) 2 a 2 (λ)       r (λ) + a  (λ) r (λ) + a  (λ) 1 1 − + g(λ) − u g(λ) + u− = −a(λ) u + + −u − t λ 2 a 2 (λ) 2 a 2 (λ)

(20.20) √ with a(λ) = l(λ)c(λ)—the propagation delay. It is obvious that it is not possible to obtain distortionless propagation for both waves. Choosing this option for the forward wave u + means choosing a(λ) to be subject to the Riccati equation a  (λ) = r (λ) − g(λ)a 2 (λ) , a(0) = 0

(20.21)

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Consequently (20.20) becomes + + −u + λ λ = a(λ)(u t + g(λ)u ) − 2 − 2 + −u − λ = −a(λ)[u t + (r (λ)/a (λ))u + (g(λ) − r (λ)/a (λ))u ]

(20.22)

The distortionless condition would be precisely the Heaviside matching condition   g(λ) = r (λ)/a 2 (λ) ⇔ r (λ) c(λ)/l(λ) = g(λ) l(λ)/c(λ)

(20.23)

More details can be found in [48, 49].

20.3.3 The Drillstring Dynamics We start from the model of Sect. 2.3 of [50] ρ(s)I p (s)θtt + c(s)θt − (G(s)I p (s)θs )s = 0 , 0 < s < L c θ˙m (t) + G(0)I p (0)θs (0, t) = 0 Jm θ¨m (t) + c0 θ˙m (t) = τ (t) − c θ˙ (0, t)

(20.24)

Jb θ¨ (L , t) + G(L)I p (L)θs (L , t) = −T (θ˙ (L , t)) Here, we also can point out a possible distortionless propagation of the drilling vibrations. Introduce the new state variables ω := θt , v := G(s)I p (s)θs

(20.25)

to obtain a system in the Friedrichs form ωt −

1 c(s) vs + ω = 0 , vt − G(s)I p (s)ωs = 0 ρ(s)I p (s) ρ(s)I p (s)

(20.26)

Introducing the Riemann invariants ω± (s, t) = ω(s, t) ∓ we obtain

1 v(s, t) √ I p (s) ρ(s)G(s)

(20.27)

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V. R˘asvan

ωt+

+

ωt− −

G + 1 ω + [(a  + c)ω+ − (a  − c)ω− ] = 0 ρ s 2ρ I p

(20.28)

G − 1 ω + [(a  + c)ω+ − (a  − c)ω− ] = 0 ρ s 2ρ I p

√ with a(s) = I p (s) ρ(s)G(s). By choosing a(s), e.g., from the differential equation a  (s) = c(s)

(20.29)

the forward wave ω+ is propagating distortionless (ωt± ±



G(s)/ρ(s)ωs± ) + (c(s)/(ρ(s)I p (s)))ω+ = 0

(20.30)

Observe that for space constant parameters (uniform drillstring) distortionless would require also losslessness since a  (s) = 0.

20.4 The Functional Differential Equations of the Drillstring We shall consider Eqs. (20.24) in the case of constant parameters and c(s) ≡ 0 ρθtt − Gθss = 0 c θ˙m + G I p θs (0, t) = 0 Jm θ¨m (t) + c0 θ˙m (t) + c θ˙ (0, t) = τ (t)

(20.31)

Jb θ¨ (L , t) + T (θ˙ (L , t)) + G I p θs (L , t) = 0 We define ω := θt , v := θs and obtain the system in the Friedrichs form, introduce the Riemann invariants which, in this case, are angular velocities ω± (s, t) = ω(s, t) ∓



G/ρv(s, t)

(20.32)

Following the methodology of Sect. 20.2, the following new variables are introduced √ η+ (t) := ω+ (0, t) ⇒ ω+ (L , t) = η+ (t − L ρ/G) √ η− (t) := ω− (L , t) ⇒ ω− (0, t) = η− (t − L ρ/G) and the following system of FDEs is associated to (20.31)

(20.33)

20 Systems with Propagation: A Bunch of Models and a Research Program

 Jm ω˙ m + c0 +

c2 √ I p ρG



193

 ωm = −c η− (t − L ρ/G)

   Jb ω˙ b + I p ρGωb + T (ωb ) = I p ρGη+ (t − L ρ/G)  2c η (t) = √ ωm + η− (t − L ρ/G) I p ρG √ η− (t) = 2ωb − η+ (t − L ρ/G)

(20.34)

+

(after eliminating the cyclic variables θm and θ (L , t)). System (20.34) belongs to the general class defined by x˙ = Ax(t) + By(t − τ ) − b0 ϕ(c∗ x(t)) y(t) = C x(t) + Dy(t − τ ) − b1 ϕ(c∗ x(t))

(20.35)

which can be met in several publications of the author; we cite but the monograph [43] (Russian version in 1983) and the surveys [46, 47]. Unlike the applications there, matrix D has its eigenvalues ±ı that is on the unit disk and not inside it. The difference operator of the neutral functional differential system (20.34) is not strongly stable but only marginally stable (in a critical case). Moreover, all applications described in [50], or more recently, described in [51], and arising from Mechanics and/or Mechanical Engineering are such. This implies additional difficulties in tackling qualitative problems (stability, oscillations). We can, however, suggest one way to overcome the difficulty. Assume (based on possibly available numerical data) that the time constant  Tm := Jm c0 +

c2 √ I p ρG

−1 (20.36)

is small. Then, taking Tm ≈ 0, one can obtain  ωm (t) = −c c0 +

c2 √ I p ρG

−1

 η− (t − L ρ/G)

(20.37)

hence 



c 2 η (t) = ρ η (t − L ρ/G) ; ρ := 1 − √ c0 I p ρG  η− (t) = 2ωb − η+ (t − L ρ/G) ; ρ ∈ (0, 1) +

−



c 2 1+ √ c0 I p ρG

−1

(20.38) √ Now matrix D will have its eigenvalues ±ı ρ —inside the unit disk. Consider in the following the usual structure in singularly perturbed systems [19, 54]—the “reduced” system given by

194

V. R˘asvan

   Jb ω˙ b + I p ρGωb + T (ωb ) = I p ρGη+ (t − L ρ/G)  η+ (t) = ρ η− (t − L ρ/G)  η− (t) = 2ωb − η+ (t − L ρ/G) ; ρ ∈ (0, 1)

(20.39)

and the system of the fast variable—the boundary layer system—the equation for ωm , here given by  Jm ω˙ m + c0 +

c2 √ I p ρG



 ωm = −c η− (t − L ρ/G)

(20.40)

If each of them has the equilibrium asymptotically stable, and this is indeed the case, then the overall system is asymptotically stable provided the small parameter Tm is small enough. This aspect gives reasonable hopes that the aforementioned critical case for D can be also solved in a rigorous way.

20.5 A Research Program An experience of 30–35 years of applications of the aforementioned one to one correspondence between the Initial Boundary Value Problems for 1D propagation and the Initial Value Problems for differential equations with deviated arguments led to the statement of certain research problems which might be gathered in a research program. A. The first observation is concerned with the fact that less books and research surveys are concerned with neutral FDEs than with retarded ones. The book of Bellman and Cooke [3] gives a certain extent to the linear neutral equations and their characteristic equations, as the book of Pinney [41]; nothing is mentioned about them in the books of Myshkis [35] (first early version from 1951) or Krasovskii [29]; they are however considered not only in the book of El’sgol’ts and Norkin [11], but also in an earlier book of El’sgol’ts [10], published a couple of years after the early book of Myshkis of 1951 (considered to be the first book dealing with differential equations with deviated argument). In the book of Halanay [18], again, the neutral equations are not even mentioned. After the three versions of the book of Hale (1971, 1977, 1993) [20], where neutral equations enjoy an extended formalized study, the interest for this class of equations increased—see, e.g., the books of V. L. Kharitonov and his co-workers (for the first of them) [13, 23]. The spread of the aforementioned one to one correspondence between 1D propagation and FDEs with deviated arguments via the author’s book [43] (mainly its Russian edition) and surveys [46, 47, 50] also contributed to this interest—see the book of Niculescu [39] and the more recent one of Gil [12]. We have to add here the books of Kolmanovskii and Nosov [27, 28]; there are substantial differences between them—the second is not the English version of the

20 Systems with Propagation: A Bunch of Models and a Research Program

195

first. Other valuable references containing procedures and results on neutral FDEs are the books by Kolmanovskii and Myshkis [25, 26]. Summarizing the aforementioned references, we can give below a list of problems which are still waiting for their solutions. B. We start this list with the problem of the (strong) stability of the difference operator. This property occurred in the pioneering papers on neutral functional differential equations (NFDEs)—a list of them can be “detected” in the reference list of [11]. In these papers, it was shown also that weaker properties would send to some “pathologies”. The assumption of strong stability—for (20.35) this translates as location of the eigenvalues of D inside the unit disk—was promoted by Hale and allowed the construction of the stability theory for NFDEs in close connection and analogy with retarded FDEs. Fortunately, most applications arising from 1D propagation lead to strongly stable difference operators, e.g., Fluid Mechanics, Thermal and Hydro Power Engineering, Electrical and Electronic Engineering. There exist however applications, most of them arising from Mechanical Engineering, some of them listed in [50], which generate marginally stable difference operators (see the previous Section of this paper). Considering the marginally stable difference, operators should be stimulated also by the paper of O. Staffans with its controversial title [53]; a less circulated paper pairing the one of Staffans belongs to Kurbatov [30]. C. Another (more or less) open problem for NFDEs is the so called Bohl Perron result: if a forced linear differential system has bounded solutions with zero initial conditions for all bounded input signals, the free (unforced) system is uniformly asymptotically stable. For NFDEs this result is mentioned in [27] and tackled in [12]. Connected to it is the Persidskii type result: for linear systems uniform asymptotic stability is always exponential (under some boundedness assumptions for the coefficients). For NFDEs this result is not known. To end the discussion on linear systems, we shall mention here the so-called problem of the forced oscillations for the quasi-harmonic systems—due to Malkin [33], pp. 357–361. There are considered forced linear systems with T -periodic coefficients, forced by the input vector signal f (t) =

N 

eıω j t f j (t)

(20.41)

1

where f j (t) are T -periodic and ω j /(2π ) and T are not rationally dependent. It is required to find conditions on the system’s coefficients to ensure existence of a solution of the form N  eıω j t x j (t) (20.42) x(t) = 1

with x j (t) being T -periodic. For retarded FDEs, the problem was solved by Halanay [17], see also [18], Sect. 4.11, for NFDEs it is unsolved.

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D. Another open problem for NFDEs is the validity of the Barbashin Krasovskii LaSalle invariance principle in the absence of the strong stability of the difference operator. A hint on this subject can be found in [52] and it points to an adequate choice of the state space for NFDEs—other than C—as in [20] or PC—as in [23]. This may open the way to the use of “weak” Lyapunov functionals of energy-like class for asymptotic stability.

References 1. Abolinia, V.E., Myshkis, A.D.: Mixed problem for an almost linear hyperbolic system in the plane (russian). Mat. Sbornik 50:92(4), 423–442 (1960) 2. Bastin, G., Coron, J.M.: Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Birkhäuser, Basel (2016) 3. Bellman, R.E., Cooke, K.L.: Differential Difference Equations. Academic, New York (1963) 4. Bernoulli, J.: Meditationes. dechordis vibrantibis... Comm. Acad. Sci. Imp. Petropolitanae 3, 13–28 (1728) 5. Burke, V., Duffin, R.J., Hazony, D.: Distortionless wave propagation in inhomogeneous media and transmission lines. Quart. Appl. Math. XXXIV, 183–194 (1976) 6. Cooke, K.L.: A linear mixed problem with derivative boundary conditions. In: Sweet, D., Yorke, J.A. (eds.) Seminar on Differential Equations and Dynamical Systems (III), Lecture Series, vol. 51, pp. 11–17. University of Maryland, College Park (1970) 7. Cooke, K.L., Krumme, D.W.: Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations. J. Math. Anal. Appl. 24, 372–387 (1968) 8. Courant, R.: Hyperbolic partial differential equations. In: Beckenbach, E.F. (ed.) Modern Mathematics for the Engineer: First Series, pp. 92–109. McGraw Hill, New York (1956) 9. Danciu, D., Popescu, D., R˘asvan, V.: Control of a time delay system arising from linearized conservation laws. IEEE Access 7, 48524–48542 (2019) 10. El’sgol’ts, L.E.: Qualitative Methods in Mathematical Analysis. Fizmatgiz, Moscow, USSR (1955). (in Russian) 11. El’sgol’ts, L.E., Norkin, S.B.: Introduction to the Theory and Applications of Differential Equations with Deviating Arguments. Nauka Publishing House, Moscow, USSR (1971). (in Russian) 12. Gil’, M.I.: Stability of Neutral Functional Differential Equations. Atlantis Press, Amsterdam (2014) 13. Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003) 14. Gugat, M., Dick, M., Leugering, G.: Classical solutions and feedback stabilization for the gas flow in a sequence of pipes. Netw. & Heterogen. Media 5, 691–709 (2010) 15. Gugat, M., Dick, M., Leugering, G.: Gas flow in fan-shaped networks: classical solutions and feedback stabilization. SIAM J. Control. Optim. 49(5), 2101–2117 (2011) 16. Gugat, M., Herty, M.: Existence of classical solutions and feedback stabilization for the flow in gas networks. ESAIM: Control Optim. Calculus Var. 17, 28–51 (2011) 17. Halanay, A.: Almost-periodic solutions of linear systems with time lag (in russian). Rev. Math. Pures Appl. IX, 71–79 (1964) 18. Halanay, A.: Differential Equations. Stability, Oscillations. Time Lags. Academic, New York (1966) 19. Halanay, A., Dr˘agan, V.: Singular perturbations. Asymptotic Expansions (in Romanian). Romanian Academy Publishing House, Bucharest (1983) 20. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)

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21. de Halleux, J., Prieur, C., Coron, J.M., d’Andréa Novel, B., Bastin, G.: Boundary feedback control in networks of open channels. Automatica 39, 1365–1376 (2003) 22. Haraux, A.: Systèmes dynamiques dissipatifs et Applications. No. 17 in Research in Applied Mathematics. Masson, Paris; Wiley, New York (1990) 23. Kharitonov, V.L.: Time-Delay Systems. Lyapunov Functionals and Matrices. Birkhäuser, New York (2013) 24. Kishor, N., Sainia, R.P., Singh, S.P.: A review on hydropower plant models and control. Renew. Sustain. Energy Rev. 11, 776–796 (2007) 25. Kolmanovskii, V.B., D., M.A.: Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht (1992) 26. Kolmanovskii, V.B., D., M.A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht (1999) 27. Kolmanovskii, V.B., Nosov, V.R.: Stability and Periodic Regimes of the Control Systems with Time Delay. Nauka, Moscow, USSR (1981). (in Russian) 28. Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic, New York (1986) 29. Krasovskii, N.N.: Some Problems of the Theory of Stability of Motion. Fizmatgiz, Moscow, USSR (1959). (in Russian) 30. Kurbatov, V.G.: Solvability with respect to the derivative of a stable functional differential equation (in russian). Ukrain. Matem. Jurnal 34, 103–106 (1982) 31. Leugering, G., Schmidt, E.: On the modeling and stabilization of flows in networks of open canals. SIAM J. Contr. Optim. 41, 164–180 (2002) 32. Li, T.: Global classical solutions for quasilinear hyperbolic systems. Wiley, Chichester; Masson; Paris (1994) 33. Malkin, I.G.: Some problems of nonlinear oscillations theory. Fizmatgiz, Moscow, USSR (1956). (in Russian) 34. Munoz-Hernandez, G.A., Mansoor, S.P., Jones, D.I.: Modeling and Control of Hydropower Plants. Advances in Industrial Control. Springer, London (2013) 35. Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow, USSR (1972). (in Russian) 36. Myshkis, A.D., Filimonov, A.M.: Continuous solutions of quasi-linear hyperbolic systems with two independent variables (in russian). Differ. Equ. 17, 488–500 (1981) 37. Myshkis, A.D., Filimonov, A.M.: On the global continuous solvability of the mixed problem for one-dimensional hyperbolic sysems of quasilinear equations (in russian). Differ. Equ. 44, 413–427 (2008) 38. Myshkis, A.D., Shlopak, A.S.: Mixed problem for systems of functional diffeential equations with partial derivatives and volterra operators (russian). Mat. Sbornik 41:83(2), 239–256 (1957) 39. Niculescu, S.I.: Delay Effects on Stability. A Robust Control Approach. Springer, Berlin (2001) 40. Petre, E., R˘asvan, V.: Feedback control of conservation laws systems. part i: Models. Rev. Roum. Sci. Techn. Sér. Electr. Energ. 54, 311–320 (2009) 41. Pinney, E.: Ordinary Difference-Differential Equations. University of California Press, Berkeley (1958) 42. Popescu, M.: Hydroelectric Plants and Pumping Stations (in Romanian). Editura Universitar˘a, Bucharest (2008) 43. R˘asvan, V.: Absolute Stability of Time Lag Control Systems (in Romanian). Editura Academiei, Bucharest (1975) 44. R˘asvan, V.: Stability of bilinear control systems occurring in combined heat electricity generation i: The mathematical models and their properties. Rev. Roumaine Sci. Techn. Série Electrotechn. Energ. 26(3), 455–465 (1981) 45. R˘asvan, V.: Stability of bilinear control systems occurring in combined heat electricity generation ii: Stabilization of the reduced models. Rev. Roumaine Sci. Techn. Série Electrotechn. Energ. 29(4), 423–432 (1984) 46. R˘asvan, V.: Dynamical systems with lossless propagation and neutral functional differential equations. In: Mathem. Theory of Networks and Systems MTNS1998, pp. 527–530. Il Poligrafo, Padova, Italia (1998)

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47. R˘asvan, V.: Functional differential equations of lossless propagation and almost linear behavior. In: IFAC Proceedings Volumes, no. 10 in 39, pp. 138–150. Elsevier (2006) 48. R˘asvan, V.: Functional differential equations and one-dimensional distortionless propagation. Tatra Mount. Math. Publ. 43, 215–228 (2009) 49. R˘asvan, V.: Delays. propagation. conservation laws. In: R. Sipahi, T. Vyhlidal, S.I. Niculescu, P. Pepe (eds.) Time Delay Systems: Methods, Applications and New Trends, no. 423 in Lecture Notes in Control and Information Sciences, pp. 147–159. Springer, Berlin (2012) 50. R˘asvan, V.: Augmented validation and a stabilization approach for systems with propagation. In: Miranda, F. (ed.) Systems Theory: Perspectives, Applications and Developments, pp. 125– 169. Nova Science Publishers, New York (2014) 51. R˘asvan, V.: Huygens synchronization over distributed media – structure versus complex behavior. In: E. Zattoni, A.M. Perdon, G. Conte (eds.) Structural Methods in the Study of Complex Systems, no. 482 in Lecture Notes in Control and Information Sciences, pp. 243–274. Springer (2019) 52. Saperstone, S.H.: Semidynamical Systems in Infinite Dimensional Spaces. No. 37 in Applied Mathematical Sciences. Springer, New York (1981) 53. Staffans, O.J.: A neutral fde with stable d-operator is retarded. J. Differ. Equ. 49, 208–217 (1983) 54. Vasilieva, A.B., Butuzov, V.F.: Asymptotic expansions of singularly perturbed differential equations. Nauka, Moscow (1973). (in Russian)

Chapter 21

Fourth-Order Method for Differential Equations with Discrete and Distributed Delays Alexey S. Eremin and Aleksandr A. Lobaskin

Abstract Differential equations with discrete and distributed delays are considered. Explicit continuous-stage Runge–Kutta methods for state-dependent discrete delays based on functional continuous methods for retarded functional differential equations and Runge–Kutta methods for integro-differential equations based on methods for Volterra equations are combined to get a method suitable for both types of delays converging with order four. A method that requires six right-hand side evaluations and only two of its integral argument evaluations is presented. The questions of the practical implementation for delay differential equations within general nonsmooth solutions are discussed. The numerical solution of test problems confirms the declared fourth order of convergence of the constructed method.

21.1 Introduction Various problems of physics and technology lead to integro-differential equations in order to increase the accuracy of modeling. The theory of such equations has long attracted the attention of both theoretical physicists and mathematicians. The problems with distributed delays which are quite similar to integro-differential equations are widely used in mathematical modeling in biology and medicine, particularly in population dynamics, epidemiology, and immunology. Often models originally using ordinary differential equations (ODEs) are extended with the use of delays (see [1, 2] for examples). It is usually impossible to solve such problems analytically and numerical methods are needed. One-step Runge–Kutta methods are easy to implement and have nice computational characteristics. They are thoroughly studied for various systems of ODEs (e.g., [3, 4], discrete delay differential equations (e.g., [5]), integro-differential equations (e.g., [6]), and theoretically for most general case of functional differential equations [7]. A. S. Eremin (B) · A. A. Lobaskin St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_21

199

200

A. S. Eremin and A. A. Lobaskin

However, we couldn’t find papers devoted to adaptation of Runge–Kutta methods for differential equations with discrete and distributed delays. In the current paper, we marry the Runge–Kutta methods for Volterra integro-differential equations (VIDEs) studied by Ch. Lubich in [6] and a method for state-dependent discrete delay differential equations by L. Tavernini [8] to develop a method convergent with order four suitable for equations with discrete and distributed time- or state-dependent delays. In the following section, we describe the method and discuss the implementation issues. In the last section, numerical tests confirming the convergence order four are presented. We consider a differential equation with discrete and distributed delays ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎛

⎜ y  (t) = f ⎝t, y(t), y α(t, y(t)) ,

t

⎞ ⎟ K (t, s, y(s))ds ⎠ ,

t > t0 ,

σ (t,y(t))

y(t) = φ(t),

(21.1)

t ≤ t0 .

In contrast to VIDEs, the lower integration limit σ is not fixed as t0 , and as well as the discrete delay α can be the function of time and/or the unknown y(t). The only assumption that is made is that the delayed times satisfy α(t, y(t)) ≤ t and σ (t, y(t)) ≤ t for any t and y(t) within the domain of the solution. If this is not true, the problem becomes an advanced differential equation and is out of the scope of studying delay equations. The history function φ(t) is defined over some interval such that all possible values of α and σ either belong to it or are greater than t0 . In general, it can be piecewise-continuous, and the discontinuities in it which give rise to discontinuities in the solution can be treated the same way as for discrete delay differential equations (see [5] for the overview, and [9] for an algorithm of detecting and approximating them). Also, we consider the integral term to be non-vanishing in the current paper, i.e. σ (t, y(t)) < t − σ¯ , σ¯ > 0 to avoid the lower integration limit falling into the current time-step of the numerical method.

21.2 Numerical Method Among the possible approaches to construction of Runge–Kutta method for VIDEs described in [6], we have chosen the so-called Bel’tyukov-type methods. It is not difficult to adapt them to equations with variable lower integration limit. Discrete state-dependent delays, however, demand construction of continuousstage methods satisfying more complicated order conditions than continuous Runge– Kutta methods. A good study of such methods and examples of explicit methods of the kind can be found in [10]. We have chosen the method by Tavernini [8] and slightly tuned it to make it a Bel’tyukov type method for distributed delays as well as

21 Fourth-Order Method for Differential Equations …

201

keeping it suitable for state-dependent discrete delays. The chosen method has only two different new abscissae per step which means that the integral argument of f must be calculated only twice per step. Assume that n steps of the size h have already been taken. The approximation to the solution y(tn + θ h) over the (n + 1)-th step is made according to the explicit formula y(tn + θ h) ≈ ηn+1 (tn + θ h) = yn + h

s 

bi (θ )Pi , θ ∈ [0, 1],

(21.2)

i=1

where tn+1 = tn + h, yn+1 = ηn+1 (tn+1 ) and for i = 1, . . . , s   

 i(n+1) + Z i(n+1) Pi = f Ti(n+1) , Yi(n+1) , η α Ti(n+1) , Yi(n+1) , F

(21.3)

with Ti(n+1) = tn + ci h, Yi(n+1) = yn + h

η(t) =

i−1 

ai j (ci )P j ,

j=1

⎧ φ(t), ⎪ ⎪ ⎪ ⎪ ⎨ ηk (t),

t ≤ t0 , t ∈ [tk−1 , tk ],

i−1  ⎪ ⎪ ⎪ y + h ai j (θ )P j , ⎪ n ⎩

k ≤ n,

t ∈ [tn , tn+1 ], θ =

j=1

Z i(n+1) = h

i−1 

t − tn , h

(21.4)

  ai j (ci )K tn + e j h, T j(n+1) , Y j(n+1) ,

j=1

i(n+1) ≈ Fi(n+1) = F

tn   σ Ti(n+1) ,Yi(n+1)

  K Ti(n+1) , τ, y(τ ) dτ.

In order to provide continuous approximation η(t), the parameters should satisfy the conditions bi (0) = 0, ai j (0) = 0, i = 1, . . . , s, j = 1, . . . , i − 1. As it can be seen, η(t) is updated after every step. The method’s parameters can be nicely collected into the so-called Butcher tableau (see. Table 21.1). Combining the local order conditions of general functional-continuous Runge– Kutta method for retarded functional differential equations [10] reduced to the case of considered method (21.2)–(21.4) and the relation between local order and convergence order of Runge–Kutta methods for delay differential equations [5], we get the following demands on the approximation orders of the method. To get convergence of order four, we must provide:

202 Table 21.1 ei ci e1 0 e2 c2 .. c3 . . es−1 .. cs − bi

A. S. Eremin and A. A. Lobaskin Butcher tableau of an explicit method (21.2)–(21.4) ai j 0 a21 (θ) a31 (θ) a32 (θ) .. .. . . as1 (θ) as2 (θ) b1 (θ) b1 (θ)

..

.

. . . as,s−1 (θ) . . . bs−1 (θ) bs (θ)

• the approximation of y(t) of order four in the time-mesh points tk , k = 1, . . . ; • the approximation of y(t) of order at least three everywhere between the mesh points; • the approximation of Fi(n+1) by quadrature rule of order at least three; • the problem to be smooth enough between the mesh points, i.e., the points where the solution derivatives have discontinuities must be included into the mesh (at least up to the fourth derivative). The last condition is important for delay equations since their solution is in general piecewise smooth. However, detection and calculation of discontinuity points is a different problem almost independent from the method construction and we don’t consider it in the scope of the present paper. As it was mentioned, one of the possible approaches to it can be found in [9]. Due to the discontinuities, approximating Fi(n+1) with Romberg rule (multi-step interpolation combined with trapezoid quadrature) used in [6] for VIDEs becomes impossible for the problem (21.1). We use Simpson rule with step h over the intervals [tk , tk+1 ] and use η((tk + tk+1 )/2) as midpoint values. For those pieces that must be calculated over a part of the step h, we approximate all the values required by the Simpson rule with η(t). i(n+1) , it must be calculated once for each As it follows from the expression for F (n+1) i(n+1) calculations, we choose the method unique value of Ti , so to minimize F with only three different values of ci : 0, 21 , and 1. The method’s parameters providing all the above-stated requirements are presented in Table 21.2. It has six f -evaluations per step and (as it might seem) three  F-evaluations. 1(n+1) is calculated in the same time point tn+1 as However, notice that the value of F (n) 6 . Thus it can be easily obtained from F 6(n) by adding the Simpson term the value F over the previous step [tn−1 , tn ] and if necessary modifying the leftmost Simpson term (which changes since σ changes). This makes it free compared to the full   F-evaluation, and we can say that the method uses six f -evaluations and two Fevaluations per step.

21 Fourth-Order Method for Differential Equations …

203

Table 21.2 Method (21.2)–(21.4) convergent with order four coefficients ei ci ai j 1 0 2 1 1 θ 2 1 1 θ2 θ2 θ− 2 2 2 2 θ2 1 θ2 1 θ− 2 2 2 3 2 4 1 2 1 1 θ − θ 2 + θ 3 0 2θ 2 − θ 3 − θ 2 + θ 3 2 2 3 3 2 3 3 2 4 1 2 − 1 θ − θ2 + θ3 0 θ2 − θ3 − θ2 + θ3 θ2 2 3 3 2 3 3 2 4 1 2 bi θ − θ 2 + θ 3 0 0 0 2θ 2 − θ 3 − θ 2 + θ 3 2 3 3 2 3

21.3 Numerical Tests In order to verify the fourth convergence order of the constructed method, we run two tests with constant time-steps h and check the global error to the time-step ratio. We choose the test problems with known solutions such that the delays are time and/or state-dependent and overlapping occurs for the discrete delay (i.e., α > tn for some steps and ai j (θ ) continuous values are used). We choose the problems without discontinuities to make a “pure” convergence test unaffected by discontinuities detection and approximation algorithm.

21.3.1 Problem 1 Consider the problem with overlapping happening at one or few first steps ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩



t

y (t) = −[y(t)] − t exp(t ) [y (t/2)] 2

2

4

y(s)es−st ds,

t > 0,

t−1

y(t) = e−t ,

(21.5)

t ≤ 0.

We solve it over the interval [0; 5] with N constant steps h = N5 and calculate the global error E as the maximum difference between the exact solution exp(−t) and its continuous approximation η(t). As it can be seen from Table 21.3 and Fig. 21.1, the error behaves as O(h 4 ).

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Table 21.3 Convergence of the method for (21.5) N 10 20 40 80 160 320 640 1280

Global error 1.077239237 · 10−4 6.474600204 · 10−6 4.001772102 · 10−7 2.585829407 · 10−8 1.652691528 · 10−9 1.045746753 · 10−10 6.577947388 · 10−12 4.125901009 · 10−13

Convergence order 4.06 4.02 3.95 3.97 3.98 3.99 4.00

Fig. 21.1 Test results of the example (21.5)

21.3.2 Problem 2 This is a problem with two discrete delays (one of which is state-dependent) and a time-dependent delay σ . ⎧  t+1     2 2 ⎪ 1 ln(y) t −1 1 ⎪  ⎪ ⎪ y − [y(t)]2 C e− 2 , (t) = − y y ⎪ ⎪ t + 1 2 4 ⎪ ⎪ ⎛ ⎞ ⎪ ⎨ t ⎜ ⎟ ⎪ C = ⎝1 + t exp(t 2 ) y(s)es−st ds ⎠ , ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ 2 −1 ⎪ ⎪ ⎩ −t y(t) = e ,

t > 0, (21.6)

t ≤ 0.

Analogously to Problem 1, we solve it over the interval [0; 5] with N constant steps h = N5 and calculate the global error E as the maximum difference between the exact solution exp(−t) and its continuous approximation η(t). Table 21.4 and Fig. 21.2 show convergence of order 4.

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Table 21.4 Convergence of example (21.6) N 10 20 40 80 160 320 640 1280

Global error 4.909334319 · 10−4 1.708048154 · 10−5 9.083847819 · 10−7 5.143638897 · 10−8 3.036482401 · 10−9 1.836685690 · 10−10 1.121132961 · 10−11 6.839346797 · 10−13

Convergence order 4.85 4.23 4.14 4.08 4.05 4.03 4.03

Fig. 21.2 Test results of the example (21.6)

21.4 Conclusion An extension of Runge–Kutta methods for differential equations with discrete and distributed delays meets with the necessity to satisfy the conditions that both methods for discrete delay differential equations and for integro-differential equations must satisfy. It is possible to construct a fourth-order continuous-stage Runge–Kutta method suitable for distributed delays with six stages and just two integral argument evaluations per step. Under such characteristics, it seems to be possible to embed a third-order error estimator to construct a variable time-step method. This is a possible direction to develop the paper results.

References 1. Erneux, T.: Applied Delay Differential Equations. Surveys and Tutorials in the Applied Mathematical Sciences, Springer Science+Business Media, LLC (2009) 2. Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics, Springer Science+Business Media, LLC (2011)

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3. Butcher, J..C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley (2008) 4. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, 3rd edn. Springer, Berlin Heidelberg (2008) 5. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations, 1st edn. Oxford Science Publications, Clarendon Press, Oxford (2013) 6. Lubich, C.: Runge-Kutta theory for Volterra integrodifferential equations. Numer. Math. 40, 119–135 (1982). https://doi.org/10.1007/BF01459081 7. Bellen, A., Guglielmi, N., Maset, S., Zennaro, M.: Recent trends in the numerical solution of retarded functional differential equations. Acta Numerica, pp. 1–110 (2009) 8. Tavernini, L.: One-step methods for the numerical solution of Volterra functional differential equations. SIAM J. Numer. Anal. 8(4), 786–795 (1971). https://doi.org/10.1137/0708072 9. Eremin, A., Humphries, A.R.: Efficient accurate non-iterative breaking point detection and computation for state-dependent delay differential equations. In: Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014) / AIP Conf. Proc. 1648, 150,006 (2015). https://doi.org/10.1063/1.4912436 10. Maset, S., Torelli, L., Vermiglio, R.: Runge-Kutta methods for retarded functional differential equations. Math. Models Meth. Appl. Sci. 15(8), 1203–1251 (2005). https://doi.org/10.1142/ S0218202505000716

Chapter 22

The Prediction Scheme to the Linear Systems with Linearly Increasing and Constant Delays Alexey Zhabko, Oleg Tikhomirov, and Olga Chizhova

Abstract In this paper, we investigate the systems of differential-difference equations of delay type. More specifically, we study the systems with constant input delay and unbounded state delay. These systems have been investigated a lot less than the class of systems with both constant aftereffects. However, the state time-delay is not always constant. Now the mathematical models with time-dependent state delay constitute an important topic for theoretical research and practical applications. In this article, we have researched the stabilization problem of the linear differentialdifference systems with linearly increasing state delay and constant input delay. The theoretical basis of the study is thr prediction scheme for the compensation of the state delay in the construction of stabilizing controller. The possibility of the construction of such a control has been studied. Some sufficient conditions of the asymptotic stability of the close-loop system have been obtained. The proof is constructive; it can be realized as a controlling algorithm. The application of this approach to the stability analysis of the systems with distributed delay is the subject of future studying.

22.1 Introduction The problem of constructing a stabilizing control for systems with both input and state delay is much less studied than the same problem for systems without input delay. Smith [1] first applied the method of compensation with predictor for this problem. Manitius and Olbrot [2] suggested the prediction scheme for the compensation of the input delay in the case of construction of stabilizing control for these A. Zhabko · O. Tikhomirov (B) · O. Chizhova St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. Zhabko e-mail: [email protected] O. Chizhova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_22

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systems. Krstic and Smyshlyaev [3] used the backstepping transformation of the control variable. Besides, Krstic [4] suggested using the quadratic Lyapunov form for delay compensation in nonlinear systems. Kharitonov [5] used a complete type functional and suggested an application of the prediction scheme of Manitius and Olbrot to the linear systems with constant input and state delay; moreover, these results can be generalized to the case of systems with multiple state delays. Veremey [6] applied the compensatory approach for the construction of the stabilizing controls in systems with both input and state delay and suggested new prediction models. Ponomarev [7–9] studied the systems with distributed input delay and problems of robustness analysis. In this paper, we explore a possibility to stabilize the linear difference-differential systems with linear state delay and constant input delay. The linear increasing delay is unbounded, for this reason, some well-known methods of investigation are not applicable for stability analysis of such systems. Zhabko, Chizhova, and Zaranik [10, 11] applied the Razumikhin approach [12] to analyses of the systems. The main goal of the paper is the construction of the stabilizing control based on this approach. It is based on compensation of the state delay in the construction of stabilizing controller. The paper is organized as follows. In Sect. 22.2, we give basic designation used in the paper and formulate the stabilization problem for given systems. In Sect. 22.3, we prove an auxiliary result. As the main result in Sect. 22.4, we obtain the stabilizing control for the system.

22.2 Problem Formulation We consider a linear system with simultaneous input and state delay of the form d x(t) = A0 x(t) + A1 x(t − h(t)) + Bu(t − τ ), dt

τ > 0, t0 ≥ 0.

(22.1)

Here, A0 , A1 are given real n × n matrices, B is given real n × m matrix, n ≥ m, t0 is initial time instant, τ is a positive input delay, h(t) = (1 − α)t is a state delay, 0 < α < 1. ¯ 0 ), t0 ] → R n be an initial function. We ¯ = max{τ, h(t)} and ϕ : [t0 − h(t Let h(t) ¯ 0 ), t0 ], R n ), of piecewise assume that the function belongs to the space, PC([t0 − h(t ¯ continuous functions defined on the segment [t0 − h(t0 ), t0 ]. Let x(t, ϕ) stand for the solution of system (22.1) under the initial conditions x(θ, ϕ) = ϕ(θ ); θ ∈ [t0 − ¯ 0 ), t0 ] and xt (t0 , ϕ) denote the restriction of the solution to the segment [t − h(t ¯h(t), t]. ¯ θ ∈ [−h(t), 0]. xt (t0 , ϕ) : θ → x(t + θ, t0 , ϕ), Recall that the euclidean norm is used for vectors and the induced matrix norm for matrices. We use the uniform norm ϕh¯ = sup ϕ(θ ) for elements of the ¯ 0 ), t0 ], R n ). space PC([t0 − h(t

¯ 0] θ∈[t0 −h,t

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Kharitonov [5] studied the case h(t) ≡ h ≤ τ . He assumed that there are exist matrices F0 and F1 such that the system d x(t) = (A0 + B F0 )x(t) + (A1 + B F1 )x(t − h) dt

(22.2)

is exponentially stable. Then he constructed a control law under which system d x(t) = A0 x(t) + A1 x(t − h) + Bu(t − τ ) dt for t ≥ τ coincides with system (22.2). Let us consider the system (22.1). We assume that there exist matrices F0 and F1 such that the close-loop system d x(t) = (A0 + B F0 )x(t) + (A1 + B F1 )x(αt) dt is asymptotically stable. As you know [13] the system dy(t) = Dy(t) + Gy(αt) dt is asymptotically stable if the following conditions are satisfied: 1. the system dz(t) = Dz(t) is exponentially stable; dt 2. the difference system D z˜ (t) + G z˜ (αt) = 0 is asymptotically stable.

22.3 General Scheme In accordance with [5], we consider an auxiliary system d x(t) = A0 x(t) + Bu 0 (t − τ ), dt

τ > 0,

t0 ≥ 0.

(22.3)

We select a compensating control in the form u 0 (t) = F0 ν(t) ν(t) = e

A0 τ

t x(t) +

e A0 (t−θ) B F0 ν(θ )dθ.

(22.4)

t−τ

It is obvious that the characteristic polynomial of the linear system (22.3)–(22.4) has a form

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⎛ f ( p) = det ⎝

e− pτ B F0

− pE + A0 e A0 τ

−E +

0

−τ

e( pE−A0 )θ dθ B F0

⎞ ⎠ = det( pE − A0 − B F0 ).

The matrix F0 is chosen such that the system (22.3)–(22.4) is exponentially stable. Lemma 22.1 Let the following conditions hold: 1. The matrix (A0 + B F0 ) satisfies Hurwitz criterion; 2. The eigenvalues of the matrix (A0 + B F0 )−1 (A1 + B F1 ) are in the unit circle on the complex plane. Then there exist two matrices C and D such that the system ⎧ ⎨

A0 y(t) + B F0 ν(t) + A1 y(t − σ ) + B F1 ν(t − σ ) = 0, 0 ⎩ e A0 τ y(t) − ν(t) + e−A0 θ dθ B F0 ν(t) + C y(t) + Dν(t − σ ) = 0

(22.5)

−τ

is exponentially stable for σ > 0. Proof We show that all roots of the characteristic polynomial of the system (22.5) are in the unit circle on the complex plane. We assume that matrices A0 and (A1 − A0 − B F0 ) have not zero eigenvalues. Then the characteristic polynomial of the system (22.5) has a form  f (ρ) = det

ρ B F0 + B F1 ρ A0 + A1

e A0 τ − E B F0 + D ρe A0 τ + ρC ρ −E + A−1 0

 .

Now we transform the matrix. We sum the first and n+1 columns, the second and n+2 columns and so on. As a result, we obtain  f (ρ) = det

ρ A0 + A1 0 + B F0 ) + (A1 + B F1 ) ρ(A A0 τ e − E (A0 + B F0 ) + ρC + D ρe A0 τ + ρC ρ A−1 0

 .

Then we introduce two matrices  A0 τ

 ˜ 0 + B F0 ) and D = C˜ + A−1 e − E C = C(A (A1 + B F1 ) , 0

(22.6)

therefore f (ρ) = det (ρ(A0 + B F0 ) + (A1 + B F1 ))   E ρ A0 + A1 det . A0 τ ˜ 0 + B F0 ) C˜ + A−1 ρe A0 τ + ρ C(A − E) 0 (e Let us note that matrices F0 and F1 are chosen such that all roots of the first multiplier are in the unit circle on the complex plane.

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Now we consider the second multiplier. We apply the formula of Frobenius to the determinant of the block structure matrices.   E ρ A0 + A1 det = A0 τ ˜ 0 + B F0 ) C˜ + A−1 ρe A0 τ + ρ C(A − E) 0 (e   ˜ 0 + B F0 ) = det −ρe A0 τ − ρ C(A   A0 τ (−ρ)n det E + A−1 )B F0 . 0 (E − e Then we introduce the matrix C˜ by the equality A0 τ ). C˜ = A−1 0 (E − e

(22.7)

It is obvious that the second multiplier of the characteristic polynomial f (ρ) has zero roots only. Lemma is proven. Now we use the lemma and control (22.4) to solve the stabilizing problem of the system (22.1).

22.4 Main Result We assume that there exist matrices F0 and F1 such that the matrix (A0 + B F0 ) satisfies Hurwitz criterion and eigenvalues of matrix (A0 + B F0 )−1 (A1 + B F1 ) are in the unit circle on the complex plane. Let us consider the system ⎧ ⎪ ⎪ ⎨

= A0 x(t) + A1 x(αt) + Bu(t − τ ) u(t − τ ) = F0 ν(t) + F1 ν(αt) 0 ⎪ ⎪ ⎩ ν(t) = e A0 τ x(t − τ ) + e−A0 θ B F0 ν(t + θ )dθ + C x(t − τ ) + Dν(αt), d x(t) dt

−τ

(22.8) where the matrices C and D are defined by the equalities (22.6) and (22.7). Let the matrices A0 and (A1 − A0 − B F0 ) have not zero eigenvalues. Theorem 22.1 The zero solution of the system (22.8) is asymptotically stable. Proof Without limiting of generality, we can assume that αt0 ≥ τ . Then the solution (x(t), ν(t)) of the system (22.8) has k continuous derivatives for t ≥ α −k t0 . We differentiate the system (22.8) k times and denote x(t) ˜ = x (k) (t), ν˜ (t) = ν (k) (t). In −k accordance with [11], we obtain for t ≥ α t0

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⎧ ⎪ ⎪ ⎨

d x(t) ˜ dt

= A0 x(t) ˜ + α k A1 x(αt) ˜ + B u(t ˜ − τ) u(t ˜ − τ ) = F0 ν˜ (t) + α k F1 ν˜ (αt) 0 ⎪ ⎪ ⎩ ν˜ (t) = e A0 τ x(t ˜ − τ ) + e−A0 θ B F0 ν(t ˜ + θ )dθ + C x(t ˜ − τ ) + α k D ν˜ (αt). −τ

(22.9)

Since the system ⎧ ⎨

d x(t) ˜ dt

= A0 x(t) ˜ + B F0 ν˜ (t) 0 ⎩ ν˜ (t) = e A0 τ x(t ˜ − τ ) + e−A0 θ B F0 ν(t ˜ + θ )dθ −τ

is exponentially stable then there exists number n such that system (22.9) be asymptotically stable for k ≥ n [11]. Now we use two equalities 0 x(t ˜ − τ ) = x(t) ˜ −

˙˜ + θ )dθ and ν˜ (t + θ ) = ν˜ (t) − x(t

−τ

0

ν˙˜ (t + μ)dμ,

−θ

and transform system (22.9) into the form ⎧ A0 x(t) ˜ + B F0 ν˜ (t) + α k A1 x(αt) ˜ + α k B F1 ν˜ (αt) = f˜(t) ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ A τ ⎪ ˜ − ν˜ (t) + e−A0 θ B F0 dθ ν˜ (t) + C x(t) ˜ + α k D ν˜ (αt) = g(t) ˜ ⎨ e 0 x(t) −τ

˜ ⎪ f˜(t) = d x(t) ⎪ dt ⎪ ⎪ 0 0

0 ⎪ ⎪ ˙˜ + θ )dθ + e−A0 θ B F0 ν˙˜ (t + μ)dμdθ, ⎪ ˜ = e A0 τ + C x(t ⎩ g(t) −τ

−τ

(22.10)

−θ

moreover, f˜(t) → 0 and g(t) ˜ → 0 as t → +∞. Let us note that homogeneous part of system (22.10) coincide with system (22.5) for σ = (1 − α)t and the summands with delay have the multiplier α k . So if the eigenvalues ρ j ( j = 1, 2, . . . , 2n) of system (22.5) are in the unit circle on the complex plane, then the eigenvalues of homogeneous part of system (22.10) ρ˜ j = α k ρ j ( j = 1, 2, . . . , 2n) be in the same place for k = 0, 1, . . . , n. All solutions of system (22.10) tend to zero for k = n, so relations x (k) (t) = ˜ → 0 as t → +∞ are true. Then we consider sysx(t) ˜ → 0 and ν (k) (t) = ν(t) tem (22.10) for k = n − 1, n − 2, . . . , 0 and obtain that the relations x(t) → 0 and ν(t) → 0 as t → +∞ for the solutions of system (22.8) are true too. This ends the proof.

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22.5 Conclusion The system of differential-difference equations of delay type with constant input delay and unbounded state delay is investigated. The prediction scheme for compensation of the state delay is applied. The stabilizing control is obtained.

References 1. Smith, O.J.M.: A controller to overcome dead time. ISAJ 6(2), 28–33 (1959) 2. Manitius, A., Olbrot, A.: Finite spectrum assignment problem for systems with delays. IEEE Trans. Autom. Control 24(4), 541–553 (1979) 3. Krstic, M., Smyshlyaev, A.: Backstepping boundary control for first-order hyperbolic PDEs and application to the system with actuator and sensor delays. Syst Control Lett 57(9), 750–758 (2008) 4. Krstic, M.: Delay Compensation for Nonlinear, Adaptive and PDE Systems. Birkhäuser, Boston (2009) 5. Kharitonov, V.L.: An extension of the prediction scheme to the case of systems with both input and state delay. Automatica 50(1), 211–217 (2014) 6. Veremey, E.I.: Prediction-based delay compensation for LTI systems with dynamic feedback. Proceedings: XII All-Russian Meeting on Management Problems, pp. 1333–1342 (2014) (In Russian) 7. Ponomarev, A.A.: Suboptimal control construction for the model predictive controller. Vestn. S.-Peterb. un-ta 10(3), 141–153 (2014) (In Russian) 8. Ponomarev, A.: Reduction-based robustness analysis of linear predictor feedback for distributed input delays. IEEE Trans. Autom. Control 61(2), 468–472 (2016) 9. Ponomarev, A.: Nonlinear predictor feedback for input-affine systems with distributed input delays. IEEE Trans. Autom. Control 61(9), 2591–2596 (2016) 10. Zhabko, A.P., Chizhova, O.N.: Razumikhin approach to analyses of the differential-difference systems with linearly increasing time-delay. In: 20th International Workshop on Beam Dynamics and Optimization (BDO) (2014). https://doi.org/10.1109/BDO.2014.6890103 11. Zhabko, A.P., Chizhova, O.N., Zaranik, U.P.: The stability of difference systems with linear increasing time-delay. In: 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov)(CNSA) (2017). https://doi.org/10.1109/CNSA.2017. 7974034 12. Razumikhin, B.S.: Stability of systems with delay. Appl. Math Mech. 20(4), 500–512 (1956) 13. Zhabko, A.P., Laktionov, A.A., Zubov, V.: I: robust stability of differential-difference systems with linear time-delay. IFAC Proc. Vol. 30(16), 97–101 (1997)

Chapter 23

On Asymptotic Quiescent Position in Time-Delay Systems Svetlana Kuptsova, Sergey Kuptsov, and Uliana Zaranik

Abstract The nonlinear time-delay systems are considered and the limiting behavior of their solutions is investigated. The case in which the solutions have the trivial equilibrium that may not be an invariant set of the system is studied. The Lyapunov– Krasovskii functionals approach is applied to obtain sufficient conditions for the existence of an asymptotic quiescent position in the large. In the case when a general system has a trivial solution, new sufficient conditions for its asymptotic stability are obtained. Examples that illustrate the application of the obtained results are given.

23.1 Introduction The second Lyapunov method is the main tool to analyze the qualitative behavior of the solutions of differential equations. For the differential equations with delay, this method includes two approaches, which are as follows. In accordance with the Krasovskii approach [1], Lyapunov–Krasovskii functionals are constructed as Lyapunov functions for stability analysis. In accordance with the Razumikhin approach [2], the motion equations are studied using the classical Lyapunov function but its derivative along the trajectories of the system is estimated not on the whole set of its integral curves, but on some subset. The junction of these approaches is also successfully used to analyze the stability of time-delay systems [3, 4]. In this paper, we investigate the issue of the existence of an asymptotic quiescent position in nonlinear time-delay systems. The concept of an asymptotic quiescent position for the systems of differential equations was introduced by Zubov in [5] for S. Kuptsova (B) · U. Zaranik St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] U. Zaranik e-mail: [email protected] S. Kuptsov OGS RUSSIA, Saint-Petersburg, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_23

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studying the motions with a limiting behavior for infinitely increasing times in which the limit sets are not the invariant sets of the initial differential equations. A number of papers are devoted to this topic; see, for example, [6–8]. For time-delay systems, the concept of an asymptotic quiescent position was introduced in [9], and in papers [10, 11] some sufficient conditions for its existence have been established. In [10, 11], Razumikhin’s approach was used to study this issue. In this paper, using the method of Lyapunov–Krasovskii functionals, we get new sufficient conditions for the existence of an asymptotic quiescent position. Compared with the result obtained in [9], conditions on the derivative of the functional along the solutions of a system are weakened, but, along with this, the restriction on the choice of the functional itself is strengthened.

23.2 Preliminaries In this part, we present the object of research and give the necessary definitions. Let us consider the system of differential equations with positive delay h: d x(t) = f (t, x(t), x(t − h)), dt

(23.1)

where x(t) ∈ R n . Let the vector function f (t, x, y) be defined, continuous in the variables t ≥ 0, x ∈ R n and y ∈ R n and satisfies the Lipschitz condition with respect to the arguments x and y. Given an initial time instant t0 ≥ 0 and an initial function ϕ, we denote a solution of initial value problem by x(t, t0 , ϕ). So, we will have x(t0 + s, t0 , ϕ) ≡ ϕ(s) as s ∈ [t0 − h, t0 ]. From now on, we assume thatinitial funcn tions belong to the space  vector functions C [−h, 0], R and denote  1 of continuous   n 1  X = C [−h, 0], R , R+ = t ∈ R t ≥ 0 . It is well known from [12], that the above restrictions on the right-hand side of system (23.1) ensure the existence and 1 1 uniqueness of a solution x(t, t0 , ϕ) for any t0 ∈ R+ and ϕ ∈ X . For given t0 ∈ R+ and ϕ ∈ X , the state of the system at time t is defined as: xt (t0 , ϕ) = x(t + s, t0 , ϕ), s ∈ [−h, 0]. From now onward, we use the Euclidian norm for vectors, and for functions ϕ ∈ X , we use the uniform norm: ϕh = sup ϕ(s). s∈[−h,0]

Definition 23.1 The position x = 0 is called an asymptotic quiescent position in the large if all solutions of system (23.1) are defined on the set t ≥ t0 and x(t, t0 , ϕ) −→ 0 as t −→ +∞.

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It is important to note here that, in the general case, the point x = 0 may not be the trajectory of system (23.1). 1 . Let the function λ(t) be defined and continuous for each t ∈ R+ Definition 23.2 A function W (t, x) is called negative definite on the set x ≥ λ(t) if the following condition are satisfied: 1 , x ∈ Rn ; (1) W (t, x) is continuous in the variables on the set t ∈ R+ (2) W (t, x) ≤ −W1 (x) on the set x ≥ λ(t), where a function W1 (x) is continuous and positive definite on the set x ∈ R n . 1 Let a functional V (t, ϕ) be defined on the set X for each t ∈ R+ . This functional 1 1 will be understood as a mapping V : R+ × X → R . 1 Definition 23.3 The functional V (t, ϕ) is said to be continuous on the set R+ ×X 1 if for any ε > 0, t ∈ R+ and ϕ ∈ X there exist a value δ > 0 such that, for any 1 and ψ ∈ X with |t − τ | + ϕ − ψh < δ, the following inequality holds τ ∈ R+ |V (t, ϕ) − V (τ, ψ)| < ε.

If we substitute a solution x(t, t0 , ϕ) into the functional V , we get a function v(t) = V t, xt (t0 , ϕ) . Definition 23.4 The derivative of the functional V (t, xt ) along the solution x(t, t0 , ϕ) is the functional W (t, xt ) which satisfies the following condition:   dv(t) ≡ W t, xt (t0 , ϕ) . dt This identity should hold for all t ≥ t0 for which the right-hand side is defined. Such a functional W (t, xt ), if it exists, will be denoted by V˙ |(1) (t, xt ). And in this case, the functional V (t, xt ) will be called differentiable along the solutions of system (23.1).

23.3 Basic Results In this part, the theorems which give sufficient conditions for the existence of an asymptotic quiescent position and also the asymptotic stability of a trivial solution are formulated. Let us suppose for each H > 0 the function f (t, x, y) is uniformly bounded in t ≥ 0 on the set x ≤ H , y ≤ H , and introduce the set    S = ϕ ∈ X  ϕ(s) < ϕ(0), s ∈ [−h, 0) . 1 Theorem 23.1 Let V (t, xt ) and W (t, xt ) be continuous on the set R+ × X functionals satisfying the conditions:     (1) V1 x(t) ≤ V (t, xt ) ≤ V2 xt h , where the functions V1 (r ) and V2 (r ) are positive definite on the set r ≥ 0, and V1 (r ) → +∞ as r → +∞;

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(2) there exists δ > 0 such that v(t) > v(ξ ) for all ξ ∈ [t − δ, t) and xt ∈ S; (3) V˙ |(1) (t, xt ) = W (t, xt ) ≤ W1 (t, x), where the function W1 (t, x) is negative definite on the set x ≥ λ(t); (4) lim sup W1 (t, x) ≤ 0; t→+∞ x 0 and λ(t) → 0 as t → +∞, then x = 0 is asymptotic quiescent position in the large for trajectories of system (23.1). Further, we abandon the condition uniformly boundedness of the vector function f (t, x, y) with respect to t ≥ 0 on the set x ≤ H , y ≤ H , and assume that system (23.1) has a trivial solution. 1 × X funcTheorem 23.2 Let V (t, xt ) and W (t, xt ) be continuous on the set R+ tionals satisfying the conditions:     (1) V1 x(t) ≤ V (t, xt ) ≤ V2 xt h , where the functions V1 (r ) and V2 (r ) are positive definite on the set r ≥ 0, and 0 ≤ r ≤ H ; (2) there exists δ > 0 such that v(t) > v(ξ ) for all ξ ∈ [t − δ, t) and xt ∈ S; (3) V˙ |(1) (t, xt ) = W (t, xt ) ≤ W1 (t, x), where the function W1 (t, x) is negative definite on the set λ(t) ≤ x ≤ H ; (4) lim sup W1 (t, x) ≤ 0; t→+∞ x 0 and λ(t) → 0 as t → +∞, then the trivial solution of the system (23.1) is asymptotically stable.

Remark 23.1 Note that the restriction on the behavior of the functional W (t, xt ) from the fourth condition of Theorems 23.1 and 23.2 is significantly weaker than the restrictions presented in [9]. This allows us to establish the existence of an asymptotic quiescent position and also the asymptotic stability of a trivial solution for a wider class of systems with delay. Remark 23.2 Note that the verification of the second condition of the above theorems in the general case seems to be very difficult. However, it can be easily verified for a wide class of functionals.

23.4 Examples In this part, the application of the above theorems is illustrated by the examples of scalar nonlinear differential-difference equation. Let us consider the equation d x(t) 1 = −2x 3 (t) + x 3 (t − h) + √ 3 dt 1+t and the functional

(23.2)

23 On Asymptotic Quiescent Position in Time-Delay Systems

 V = x 4 (t) +

t

x 6 (s) ds.

219

(23.3)

t−h

This functional is continuous in the sense  of Definition 23.3  and  satisfies the first condition of the theorem, where V1 x = x4 and V2 xh = x4h + hx6h . The second condition of the theorem is satisfies too, since if x(t) ∈ S, then x 4 (t) > t 6 ξ 6 x 4 (ξ ) and x (s) ds > x (s) ds as ξ ∈ [t − h, t). The functional W (t, xt ) = t−h

ξ −h

(t) √ − x 6 (t − h) is also continuous in the sense of −7x 6 (t) + 4x 3 (t)x 3 (t − h) + 4x 3 1+t Definition 23.3. On an arbitrary solution x(t) = x(t, t0 , ϕ) of Eq. (23.2), the function  v˙ (t) that can be found using the basic rules of differentiation coincides with W t, xt (t0 , ϕ) . Then, by virtue of Definition 23.4, we have 3

V˙ |(23.2) (t, xt ) = W (t, xt ), where 4x 3 (t) − x 6 (t − h) = W (t, xt ) = −7x 6 (t) + 4x 3 (t)x 3 (t − h) + √ 3 1+t  2 4x 3 (t) 4|x(t)|3 ≤ −3x 6 (t) + √ = W1 (t, x(t)). = −3x 6 (t) − 2x 3 (t) − x 3 (t − h) + √ 3 3 1+t 1+t

If we put 2 4|x|3 and λ(t) = √ , W1 (t, x) = −3x 6 + √ 3 9 1+t 1+t then we get 5 2 . W1 (t, x) ≤ − x 6 on the set |x| ≥ √ 9 2 1+t Thus, the function W1 (t, x) is negative definite on the set |x| ≥ λ(t). 32 −→ 0 as t → +∞, sup W1 (t, x) ≤ √ 3 ( 1 + t)2 |x| τ (t0 + 0). If the delay is continuously differentiable, the condition means that the derivative at every point is not greater than 1. Definition 24.2 Let the delay in (24.1) be admissible. System (24.1) is said to be uniformly stable, if for any ε > 0 there exists δ > 0, such that x(t, t0 , ϕ) < ε for any t0  0, any t  t0 , and any initial function ϕ, such that supθ∈[−1,0] ϕ(θ ) < δ. Here x(t, t0 , ϕ), t  t0 − 1, is a solution of system (24.1), satisfying the initial condition x(t, t0 , ϕ) = ϕ(t − t0 ), t ∈ [t0 − 1, t0 ]. The initial instant t0 is assumed to be non-negative, the initial function ϕ is assumed to be piecewise continuous on [−1, 0]. The results in [10] allow to conclude that system (24.1) has a unique solution, which is absolutely continuous on [t0 − 1, ∞), for any initial data and any admissible delay. In this paper, we do not analyze the uniform stability of a fixed system with a fixed delay, we analyze the uniform stability of a class of systems. We call this task the generalized Myshkis problem. Definition 24.3 We say that matrix A solves the generalized Myshkis problem for system (24.1), if system (24.1) with matrix A is uniformly stable for any admissible delay. With this definition we can reformulate the classical 3/2 Myshkis theorem. Theorem 24.1 ([4]) Let n = 1. Number A solves the generalized Myshkis problem for system (24.1), if and only if A ∈ [−3/2, 0].

24.3 The Razumikhin Approach Our analysis is based on a modification of the Razumikhin stability result. The modification is the following. We introduce an artificial delay N 1. How to continue the initial function? On the first segment [−N , −N + 1] we take arbitrary continuous function ϕ, and on [−N + 1, 0] we take the solution of (24.1), corresponding to the initial function ϕ and time instant t0 = −N + 1. Then we can apply the standard Razumikhin theorem. For the linear system (24.1) we can choose the simplest possible Lyapunov function v(x) = x2 . This approach directly leads to the following theorem; the rigorous proof can be found in [11] (see, p. 255). Theorem 24.2 System (24.1) is uniformly stable, if there exists N 1, such that for every t  0 the inequality ϕ T (0)Aϕ(−τ (t))  0 (24.2) holds for any continuous function ϕ, defined on [−N , 0] and satisfying the conditions ϕ(0) = 1,

ϕ(θ ) < 1, θ ∈ [−N , 0),

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and the following system almost everywhere: d ϕ(θ ) = Aϕ(θ − τ (t + θ )), θ ∈ [−N + 1, 0]. dθ

(24.3)

If N = 1, then this is the classical Razumikhin result, and Eq. (24.3) degenerates. The classical result can not be applied to system (24.1), because for any matrix A (except zero matrix) and any delay τ (except the case τ (t) = 0 for any t  0), one can find continuous function ϕ and number t  0, such that ϕ(0) = 1, ϕ(θ ) < 1 for θ ∈ [−1, 0), and ϕ T (0)Aϕ(−τ (t)) > 0. For our task, it is reasonable to consider only integer N (fractional N does not give any advantages). For N = 2 we can obtain an interesting sufficient stability condition, but a much better result can be obtained for the case N = 3. It would be curious to consider N > 3, but the computations become much more complicated, and this does not allow us to obtain better result than for N = 3. Thus, for the moment, we consider the option N = 3. With this, we obtain the following condition. Theorem 24.3 Matrix A solves the generalized Myshkis problem for system (24.1), if matrix A H = A + A T is negative semidefinite, and the set of solutions (γ , μ) of the following system is empty: ⎧ γ T Aμ > 0, ⎪ ⎪ ⎨ γ T γ · μT μ < 1, 1

⎪ ⎪ ⎩ 1 − γ T μ < min A, θ A2  + γ T Aμ dθ.

(24.4)

0

Proof Start from the case A = 0. Obviously, this matrix solves the generalized Myshkis problem, A H is negative semidefinite, and system (24.4) has no solutions. Assume A = 0. If τ (t) = 0 left hand side of (24.2) takes the form ϕ T (0)Aϕ(0) = 1 T ϕ (0)A H ϕ(0). Under the conditions of the theorem, it is a nonpositive number. 2 Discuss the case A = 0, A2 = 0. Such matrices are not diagonalizable and have only zero eigenvalues. Hence, system x(t) ˙ = Ax(t) is not uniformly stable. Prove that matrix A H is not negative semidefinite in this case. We need to find a vector η, such that η T (A + A T )η > 0. As A2 = 0, the trace of matrix A is zero. Hence, there are three possibilities: (1) kth element akk of the diagonal of matrix A is positive; (2) all the diagonal elements are zero, and there is a pair (i, j), i = j, such that ai j + a ji = 0; (3) matrix A is skew-symmetric. Let ek is a vector, where all the elements are zero, except k-th element, which is equal to 1. In the first case, we can take η = ek , in the second case, we can take η = sign(ai j + a ji ) · ei + e j , and the third case is impossible, as 0 = A2 = −A T A implies A = 0. Thus, in what follows, we can assume A2 = 0. As the case τ (t) = 0 has already been considered, assume that τ (t) > 0. Integrate equality (24.3) from −τ (t) to 0:  ϕ(0) − ϕ(−τ (t)) = A

0 −τ (t)

ϕ(θ − τ (t + θ )) dθ.

(24.5)

24 The Generalized Myshkis Problem for a Linear Time-Delay …

227

Construct estimates for the argument of ϕ under the integral. The lower estimate is θ − τ (t + θ ) − τ (t) − τ (t + θ ) − 2, the upper estimate can be obtained via the property 3◦ of Definition 24.1: θ − τ (t + θ )  θ − τ (t) − θ = −τ (t) < 0. Therefore, ϕ(θ − τ (t + θ )) < 1, θ ∈ [−τ (t), 0], (24.6) and we can integrate equality (24.3) from θ − τ (t + θ ) to −τ (t) to obtain that  ϕ(θ − τ (t + θ )) = ϕ(−τ (t)) − A

−τ (t)

θ−τ (t+θ)

ϕ(s − τ (t + s)) ds.

(24.7)

Premultiply now equality (24.5) by ϕ T (0). As ϕ(0) = 1,  1 − ϕ T (0)ϕ(−τ (t)) =

0

−τ (t)

ϕ T (0)Aϕ(θ − τ (t + θ )) dθ.

Our aim is to obtain an upper estimate for the right-hand side expression. By (24.6), ϕ T (0)Aϕ(θ − τ (t + θ ))  ϕ(0) · A · ϕ(θ − τ (t + θ )) < A. By (24.7) we can obtain another estimate: ϕ T (0)Aϕ(θ − τ (t + θ )) = ϕ T (0)Aϕ(−τ (t))  −τ (t) − ϕ T (0)A2 ϕ(s − τ (t + s)) ds < ϕ T (0)Aϕ(−τ (t)) θ−τ (t+θ)

+ A  − τ (t) − (θ − τ (t + θ ))  ϕ T (0)Aϕ(−τ (t)) + A2 1 − (τ (t) + θ ). 2

Combining two obtained estimates, we get the following one:  1 − ϕ (0)ϕ(−τ (t)) < T

τ (t)



min A, A2 1 − θ + ϕ T (0)Aϕ(−τ (t)) dθ.

0

We need to verify conditions of Theorem 24.2. Assume by contradiction that ϕ T (0)Aϕ(−τ (t)) > 0. In this case the expression under the integral is non-negative, hence, we can replace upper limit τ (t) by 1. Thus, we have the following set of equations and inequalities: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

γ T Aμ > 0, γ T γ = 1, μT μ < 1, ⎪ ⎪ 1

⎪ ⎪ ⎩ 1 − γ T μ < min A, θ A2  + γ T Aμ dθ, 0

(24.8)

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where γ = ϕ(0), μ = ϕ(−τ (t)). The connection of γ , μ with function ϕ does not matter anymore, because, the function is excluded from the system. Thus, we conclude that matrix A (such that A H is negative semidefinite) solves the generalized Myshkis problem, if system (24.8) has no solutions. Indeed, in this case we arrive at the contradiction to the assumption that ϕ T (0)Aϕ(−τ (t)) > 0. It remains to show that conditions γ T γ = 1, μT μ < 1 can be replaced by condition γ T γ · μT μ < 1. Obviously, any solution of (24.8) satisfies (24.4). On the other hand, if (γ , μ) is a solution of (24.4), then γ /γ , γ  · μ is a solution of (24.8). Thus, the set of solutions of system (24.4) is empty, if and only if the set of solutions of (24.8) is.  To simplify system (24.4) we need to eliminate “minimum” from the last inequality. As has been shown in the proof of Theorem 24.3, we can assume that A2 = 0. As function θ A2  + γ T Aμ is increasing with respect to θ , we need to consider three mutually exclusive cases: (1) A < θ A2  + γ T Aμ for any θ ∈ [0, 1]; (2) A  θ A2  + γ T Aμ for any θ ∈ [0, 1]; (3) there exists θ0 ∈ [0, 1), such that A = θ0 A2  + γ T Aμ. The first case is impossible, as |γ T Aμ| < A. The second case is equivalent to A  A2  + γ T Aμ, and the third case is equivalent to A < A2  + γ T Aμ. The last two cases generate two systems: ⎧ ⎪ ⎨

0 < γ T Aμ  A − A2 , γ T γ · μT μ < 1, 2 ⎪ ⎩ γ T Aμ + γ T μ > 1 − A  , 2 ⎧

γ T Aμ > max 0, A − A2  , ⎨ γ T γ · μT μ < 1, ⎩ A − γ T Aμ2 < 2A2 A − 1 + γ T μ.

(24.9)

(24.10)

We consider the case A2   A, the case A2  < A can be investigated similarly.

24.4 A Particular Case Assume that A2   A. Obviously, in this case, system (24.9) has no solutions, and it remains to consider system (24.10), where the first inequality simplifies to γ T Aμ > 0. The set (A) of the solutions of this system for a fixed matrix A is not necessarily convex. Therefore, it is not easy to verify numerically, whether the system has a solution or not (in other words, whether set (A) is empty or not). But we can construct a convex set P(A), which contains (A). Thus, emptiness of P(A) would imply emptiness of (A).

24 The Generalized Myshkis Problem for a Linear Time-Delay …

229

Note that any solution (γ , μ) of system (24.10) satisfies the following properties: γ T Aμ ∈ (0, A), γ T μ ∈ (−1, 1). Therefore, we can add these inequalities to the 2 system. Introduce now vector of new variables: x = vec γ μT ∈ Rn . This vector contains all possible cross products of elements of vectors γ and μ. With this new variables, we can rewrite the products γ T μ, γ T Aμ, and γ T γ · μT μ: γ T μ = Tr(γ T μ) = Tr(I μγ T ) = vec I T · x, γ T Aμ = Tr(Aμγ T ) = vec A T · x, γ T γ · μT μ = Tr(γ μT μγ T ) = x T x. Thus, we obtain system with respect to n 2 scalar variables x1 , . . . , xn 2 : ⎧ T x Qx − 2q T x − r < ⎪ ⎪ ⎨ xT x − 1 < 0 < aT x < ⎪ ⎪ ⎩ −1 < e T x
δ or y0 ≤ y ≤ y1 ,

(25.2)

so that τ (y) = lim 2δ y0 ρ(y). δ→0

(25.3)

The solution of the boundary-value problem (25.1)–(25.3) can be written as ⎧ ⎪ ⎪ U0 ⎨

y ε1  + U1 (x, y), 0 ≤ y ≤ y1 , y2 ε2 + y1 ε1 − ε2 U (x, y) = y ε2 + y1 ε1 − ε2 ⎪ ⎪  + U2 (x, y), y1 ≤ y ≤ y2 . ⎩ U0 y2 ε2 + y1 ε1 − ε2

(25.4)

Functions Ui (x, y) (i = 1, 2) are the potential distributions created by a charged line at 0 ≤ y ≤ y1 and y1 ≤ y ≤ y2 accordingly under homogeneous boundary conditions: U1 (x, 0) = 0,

∂U1 (x, y)  ∂U1 (x, y)  = = 0,   x =0 x = x1 ∂x ∂x

U2 (x, y2 ) = 0,

∂U2 (x, y)  ∂U2 (x, y)  = = 0.   x =0 x = x1 ∂x ∂x

Function U1 (x, y) can be represented as U1 (x, y) =

∞

v1,n (y) cos(αn x),

(25.5)

n=0

where αn =

πn . x1

(25.6)

Making use of (25.2), (25.3), (25.5), (25.6) in the Poisson equation (25.1), it turns out that the functions v1,n (y) satisfy the ordinary differential equation

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 v1,n (y) − αn2 v1,n (y) = ϕ1,n (y), 0 ≤ y ≤ y1 , x0 +δ 1 1 ϕ1,0 (y) = − lim ρ(x, y) d x = − τ (y), ε0 x1 δ→0 ε0 x1 y0 x0 −δ x0 +δ

ϕ1,n (y) = −

2 lim ε0 x1 δ→0

ρ(x, y) cos(αn x) d x = −

x0 −δ

2 cos(αn x0 ) τ (y), n = 0. ε0 x1 y0 (25.7)

The solution of Eq. (25.7) at v1,n (0) = 0: 1 v1,0 (y) = − ε0 x1 y0

y τ (η)(y − η) dη + C1,0 y, 0

2 cos(αn x0 ) v1,n (y) = − ε0 x1 y0 αn

y τ (η) sinh(αn (y − η)) dη + C1,n sinh(αn y), n = 0, 0

(25.8) where C1,n —integration constants, n ≥ 0. The potential distribution U2 (x, y) at y1 ≤ y ≤ y2 , as for the function U1 (x, y) (25.5), (25.6), can be written as U2 (x, y) =

∞

v2,n (y) cos(αn x),

(25.9)

n=0

and, accordingly, the functions v2,n (y) are a solution of the ordinary differential equations  v2,n (y) − αn2 v2,n (y) = ϕ2,n (y), y1 ≤ y ≤ y2 , ⎧ x0 +δ ⎪ ⎪ 1 1 ⎨ − lim ρ(x, y) d x = − τ (y), y1 ≤ ϕ2,0 (y) = ε0 x1 δ→0 ε0 x1 y0 ⎪ x0 −δ ⎪ ⎩ 0, y0 ≤ ⎧ x0 +δ ⎪ ⎪ 2 ⎪ ⎪ − lim ρ(x, y) cos(αn x) d x = ⎪ ⎪ ⎨ ε0 x1 δ→0 x0 −δ ϕ2,n (y) = 2 cos(αn x0 ) ⎪ ⎪ ⎪ =− τ (y), y1 ≤ y ⎪ ⎪ ε0 x1 y0 ⎪ ⎩ 0, y0 ≤ y

y ≤ y0 , y ≤ y2 ,

≤ y0 , ≤ y2 , (25.10)

The solution of Eq. (25.10) at v2,n (y2 ) = 0:

25 Field Emitters Periodic System on Substrate with Dielectric …

⎧ ⎪ ⎪ ⎨

1 ε0 x1 y0

239

y0

τ (η)(y − η) dη − C2,0 (y2 − y), y1 ≤ y ≤ y0 , ⎪ y ⎪ ⎩ y0 ≤ y ≤ y2 , ⎧ −C2,0 (y2 − y), y0  (25.11) ⎪ ⎪ 2 cos(αn x0 ) ⎪ ⎪ τ (η) sinh(αn (y − η)) dη+ ⎨ ε0 x1 y0 αn v2,n (y) = y ⎪ ⎪ ⎪ +C2,n sinh(αn (y2 − y)), y1 ≤ y ≤ y0 , ⎪ ⎩ y0 ≤ y ≤ y2 , C2,n sinh(αn (y2 − y)),

v2,0 (y) =

C2,n —integration constants, n ≥ 0. To calculate the unknown coefficients C1,n , C2,n in expansions (25.8), (25.11) the potential continuity (25.4), (25.5), (25.9) can be used and electric displacement vector normal component continuity conditions at interfaces y = y1 : U1 (x, y1 ) = U2 (x, y1 ), ε2

∂U1 (x, y)  ∂U2 (x, y)  = ε1 . (25.12)   y = y1 y = y1 ∂y ∂y

Conditions (25.12) lead to conditions for the functions vi,n (y) (i = 1, 2; n ≥ 0): v1,n (y1 ) = v2,n (y1 ), ε2

∂v1,n (y)  ∂v2,n (y)  = ε1 .   y = y1 y = y1 ∂y ∂y

(25.13)

Formulas (25.8), (25.11), (25.13) allow to calculate the coefficients C1,n , C2,n in an explicit form: C1,0 = y1 ×

1 × ε0 x1 y0 (y2 − (1 − ε1 /ε2 )y1 )

τ (η) (y2 − (1 − ε1 /ε2 )y1 − ε1 /ε2 η) dη + 0

y1

C1,n = −An

 τ (η)

0

y0

τ (η)(ε1 /ε2 )(y2 − η) dη , y1

ε1 sinh(αn (η − y1 )) cosh(αn (y2 − y1 ))− ε2



y0 ε1 − sinh(αn (y2 − y1 )) cosh(αn (η − y1 )) dη − τ (η) sinh(αn (y2 − η))dη , ε2 C2,0 =

1 × ε0 x1 y0 (y2 −(1−ε1 /ε2 )y1 ) y 1  y0

× −

τ (η)η dη + 0

y1

 τ (η) (1 − ε1 /ε2 )y1 − η dη ,

y1

(25.14)

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C2,n

y1  y0 ε1 = An τ (η) sinh(αn η) dη + τ (η) sinh(αn y1 ) cosh(αn (η − y1 ))+ ε2 y 0 1  + sinh(αn (η − y1 )) cosh(αn y1 ) dη , (25.15)

where

2 cos(αn x0 ) × An =  ε0 x1 y0 αn −1 ε1 × sinh(αn y1 ) cosh(αn (y2 − y1 )) + cosh(αn y1 ) sinh(αn (y2 − y1 )) . ε2 (25.16) In accordance with formulas (25.8), (25.11), (25.14)–(25.16), after transformations, the functions v1,n (y), v2,n (y) for n > 0 can be represented as y v1,n (y) = An

 τ (η)

0

y1 + y

+ sinh(αn (y2 − y1 )) cosh(αn (y1 − y)) sinh(αn η)dη+  ε1 τ (η) sinh(αn (y1 − η)) cosh(αn (y2 − y1 ))+ ε2  + sinh(αn (y2 − y1 )) cosh(αn (y1 − η)) sinh(αn y)dη+

y0 + y1

ε1 sinh(αn (y1 − y)) cosh(αn (y2 − y1 ))+ ε2 

ε1 τ (η) sinh(αn (y2 − η)) sinh(αn y)dη , ε2

y1 v2,n (y) = An τ (η) sinh(αn η) sinh(αn (y2 − y)) dη+

(25.17)

0

 y ε1 sinh(αn y1 ) cosh(αn (η − y1 ))+ + τ (η) ε2 y1  + sinh(αn (η − y1 )) cosh(αn y1 ) sinh(αn (y2 − y)) dη+  y0 ε1 + τ (η) sinh(αn y1 ) cosh(αn (y − y1 ))+ ε2 y 

+ sinh(αn (y − y1 )) cosh(αn y1 ) sinh(αn (y2 − η)) dη .

(25.18)

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241

Thus, Formulas (25.4)–(25.6), (25.9), (25.14)–(25.18) are a solution of the boundary-value problem (25.1). The potential distribution U (x, y) is found in the entire area of the unit cell of the field emitters periodic system.

25.3 Conclusion In this article a two-dimensional periodic field emitters array on a flat substrate is modeled. The substrate is covered with a dielectric layer. The multi-tip field cathode is calculated using a diode-type unit cell model. Each individual field emitter is positioned in the center of unit cell. The effect of the field emitter on the electrostatic potential distribution is simulated as the effect of the charged line and emitter surface coincides with zero equipotential. The solution of the boundary-value problem (25.1) is presented in the form of the eigenfunctions series (25.4)–(25.6), (25.9). The coefficients of the potential expansions are calculated in an explicit form (25.8), (25.11), (25.14)–(25.18). The electrostatic potential distribution is found analytically in the entire area of the field emitters periodic system.

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11. Vinogradova, E.M., Egorov, N.V., Starikova, A.V., Varayun’, M.I.: Calculating a multipole cylindrical electrostatic system. Tech. Phys. 62(5), 791–794 (2017) 12. Vinogradova, E.M., Egorov, N.V., Klimakov, A.A.: Mathematical simulation of a diode system with a cylindrical field-emission tip. Tech. Phys. 60(2), 176–179 (2015) 13. Harris, J.R., Jensen, K.L., Shiffler, D.A.: Modelling field emitter arrays using line charge distributions. J. Phys. D: Appl. Phys. 48(38), 385203 (2015) 14. Vinogradova, E.M., Egorov, N.V., Televnyi, D.S.: Calculation of a triode field-emission system with a modulator. Tech. Phys. 59(2), 291–296 (2014)

Chapter 26

An Optimization Approach for Minimization of Charged Particles Orbit Deviation in Synchrotron and Transport Channels Systems Caused by Magnetic Field Tolerances Vladislav Altsybeyev and Vladimir Kozynchenko

Abstract Optimization methods of synchrotrons and transport channels systems are considered in this report. The significant deviation of orbit of charged particles beam in this facilities is caused by magnetic field tolerances. These tolerances may appear because of deviation of real magnet length on theoretically calculated. In this report, we discuss the approach of the fast channel parameter optimization technique by using swarm computations and gradient descend for improving the beam orbit. The optimizations aimed at choosing angles of correctors elements in structure of acceleration system.

26.1 Introduction In the case of presence, the relative magnetic field tolerances deviations of trajectories of charged particles may significantly enlarge and this may cause enlarging beam losses. The typical solution of this problem is using such orbit correction algorithms based on singular value decomposition of response matrix. The purpose of this report is to study the new approach based on possibility to use swarm computations and gradient descend methods to minimize the effect of enlarging deviations of charged particles trajectories. For example, at present, the main purpose of the discussed approach is solving the problems related with booster synchrotron of MegaScience NICA Accelerator Complex.

V. Altsybeyev (B) · V. Kozynchenko St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. Kozynchenko e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_26

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26.2 Orbit Dynamics Simulations and Optimization 26.2.1 Equations and Problem Statement In this work, we use the Design of Accelerators, optImizations, and SImulations (DAISI) code for all calculations [1, 2]. For dynamics modeling of charged particles in synchrotrons and transport channels, we use linear approximation of magnetic fields. ⎞ x  ⎜x ⎟ ⎟ = M X i + b. X = ⎜ ⎝y⎠  y. ⎛

X i+1





Here x, x , y, y are positions and velocities of charged particles; i is number of magnet element; M and b are transport matrix and vector, respectively, which describes influence of corresponding ith magnetic element on the particle trajectory. Below in this report we will use the following magnet elements with corresponded M and b: 1. Drift (empty element): ⎛

1 ⎜0 M =⎜ ⎝0 0 Here L is corrector length. 2. Corrector: ⎛

1 ⎜0 M =⎜ ⎝0 0

L 1 0 0

⎞ ⎛ ⎞ 0 0 ⎜0 ⎟ 0⎟ ⎟, b = ⎜ ⎟. ⎝0 ⎠ L⎠ 1 0

L 1 0 0

0 0 1 0

0 0 1 0

⎞ ⎛ ⎞ 0 0.5Lϕ ⎜ ⎟ 0⎟ ⎟, b = ⎜ ϕ ⎟. ⎠ ⎝ L 0.5Lψ ⎠ 1 ψ

Here ϕ and ψ are corrector angles in horizontal and vertical planes, respectively, L is corrector length. 3. Quadrupole: ⎛ ⎞ 0 ⎜0 ⎟ ⎜ , b=⎝ ⎟ 0⎠ 0 for K > 0 and

√ √ sin(L √ K) cos(L K ) K √ √ ⎜ √ ⎜− K sin(L K ) cos(L K ) M =⎜ ⎜ ⎝ 0 0

⎛

0

0

0

0

⎞

⎟ ⎟ ⎟ ⎟ sinh(L K ) √ ⎠ cosh(L K ) K√ √ √ K sinh(L K ) cosh(L K ) 0

√

0√

26 An Optimization Approach for Minimization of Charged Particles … ⎛ ⎞ 0 ⎜0⎟ ⎜ , b=⎝ ⎟ 0⎠ 0

√ √ sinh(L √ K) cosh(L K ) K √ √ ⎜√ ⎜ K sinh(L K ) cosh(L K ) M =⎜ ⎜ ⎝ 0 0

⎛

0

0

0

245 0

⎞

⎟ 0 0√ ⎟ ⎟ √ ⎟ sin(L √ K) ⎠ cos(L K ) K √ √ √ − K sin(L K ) cos(L K )

for K < 0, where K is quadrupole coefficient. 4. Sector magnet: ⎛

cos(α) R0 sin(α) ⎜− sin(α)/R0 cos(α) ⎜ M =⎝ 0 0 0 0

0 0 1 0

⎛ ⎞ ⎞ R0 σ (1 − cos(α)) 0 ⎜ ⎟ 0 ⎟ σ sin(α) ⎟, b = ⎜ ⎟. ⎝ ⎠ ⎠ α R0 0 1 0

Here R0 and α are rotation radius and angle in horizontal plane, and σ is relative magnetic field tolerance. So, in this system, ϕ and ψ (corrector angles in horizontal and vertical planes, respectively) are control parameters we can choose to change beam dynamics. Let us call trajectory X i obtained using this equations with zero initial conditions X 0 = 0 as orbit. If the value of relative magnetic field error is zero, the deviation of orbit may be zero (X i = 0). In the case of presence, the relative magnetic field tolerances (σ ) X i may significantly enlarge and this may cause enlarging beam losses. To solve the optimization problem, we need to obtain the two sets of channel parameters ϕ = (ϕ1 , . . . , ϕ M ), ψ = (ψ1 , . . . , ψ M ) where M is total number of corrector element and we can formalize problem as minimization of the following function: i=N  xi2 + y 2j . → min F(ϕ, ψ) = i=0

Here N is total number of optic elements.

26.2.2 Optimization Approach We use an approach based on sequential application of particle swarm evolutionary optimization algorithm [3] and gradient descent optimization. At the first stage, we use particle swarm method, it does not guarantee achievement of the local fitness function minimum, but it can provide a good initial approximation of the solution. After the particle swarm method stage work is done, it is possible to significantly improve the obtained solution using the application gradient descend method with initial conditions obtained from particle swarm. We use finite differences for calcula-

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tion of gradient. So we don’t need any analytical calculations. Also the useful “heavy ball” modification of the gradient descent method is used to improve convergence:

xin = xin−1 + σ k

G(xn−1 + xin−1 ) − G(xin−1 ) xin−1

+ β(xn−1 − xn−2 ).

Here xin is the ith component of solution xn−1 on nth algorithm iteration; σ n is the gradient step on the kth iteration; xik−1 is the small argument deviation for finite difference; β is the predefined “weight” parameter; and G is the minimized function.

26.3 Example of Orbit Deviation Minimization in Booster Synchrotron In Fig. 26.1, trajectory of beam orbit in the Booster synchrotron of NICA Accelerator Complex is presented taking into account tolerances of magnetic field cases by deviation of real magnet length on theoretically calculated [4, 5]. The presence of these tolerances results in significant enlargement of orbit amplitude oscillation. We provide a set of calculations using particle swarm (Fig. 26.2) and combination of particle swarm and gradient descent methods with β = 0 (Figs. 26.3, 26.4) to choose angles of correctors elements (its initial values are zero). As you can see from provided figures, oscillation amplitudes of beam orbit trajectories were significantly reduced by using the above-described optimization procedure.

Fig. 26.1 Trajectories of beam orbit before optimization

Fig. 26.2 Trajectories of beam orbit before and after optimization using particle swarm methods

26 An Optimization Approach for Minimization of Charged Particles …

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Fig. 26.3 Trajectories of beam orbit before and after optimization using combination of particle swarm and gradient descent methods

Fig. 26.4 Correctors horizontal angles before and after optimization

26.4 Conclusion Results of modeling given above show that optimization of corrector elements in synchrotrons optic structure by using the combination of stochastic determined optimization methods allows to significantly improve beam orbit deviation in the case of presence of tolerances of magnetic field. Acknowledgements The authors wish to thank O. S. Kozlov, V. A. Mikhaylov, D. A. Ovsyannikov, A. O. Sidorin, A. V. Tuzikov, and G.V. Trubnikov from Joint Institute for Nuclear Research for useful discussions on problems related to NICA Accelerator Complex and attention to the work done. This work was supported by Grants Council of the President of the Russian Federation.

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References 1. Altsybeyev, V.: Configurable code for beam dynamics simulations in electrostatic approximation. In: 2016 Young Researchers in Vacuum Micro/Nano Electronics, VMNE-YR 2016 - Proceedings, p. 7880398 (2017) 2. Altsybeyev, V., Kozynchenko, V.: Development of the distributed information system for the cooperative work under the design and optimization of charged particle accelerators. Cybern. Phys. 8, 195–198 (2019) 3. Kennedy, J.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks IV, pp. 1942–1948 (1995) 4. Altsybeyev, V.V., Butenko, A.V., Emelianenko, V., et al.: Simulation of closed orbit correction for the nuclotron booster. Phys. Part. Nucl. Lett. 15, 854–857 (2018) 5. Altsybeyev, V.V., Kozlov, O., Kozynchenko, V., et al.: Development of software for simulating and analyzing the dynamics of charged-particle beams in synchrotrons and beam lines. Phys. Part. Nucl. Lett. 15, 798–801 (2018)

Chapter 27

Factor Values Measurement and Heteroscedasticity by the Example of FEE Signal Identification Andrey Yu. Antonov, Nikolay V. Egorov, and Marina I. Varayun’

Abstract The problem of parametric identification of a field electron emission (FEE) signal is studied in the paper. The current response dependence on the voltage factor is investigated in terms of mathematical modeling by the least squares method. Attention is paid to the influence of the factor error on the determination of the parameters. It was found that after linearization of the response, the residuals of the regression model should show heteroscedasticity. At the same time, the autocorrelation effect was also checked. The Goldfeld–Quandt and Durbin–Watson tests, respectively, were performed. The significance of the regression model was monitored using Fisher’s statistics. The normality of the residuals was investigated using the Shapiro–Wilk test.

27.1 Introduction Among the methods for investigating a substance, emission ones can be distinguished. If a strong electric field is applied between the cathode and anode, then the FEE phenomenon is observed. The current is strongly dependent on the system configuration and the used materials [1]. This circumstance can be used as a tool for non-destructive testing and research of both electrodes. Based on the research of Sir Ralph Howard Fowler and Lothar Wolfgang Nordheim [2]—FN theory, the dependence of current I on voltage V can be represented as y = f (x; q) = Ax 2 exp [−B/x] ,

(27.1)

A. Yu. Antonov (B) · N. V. Egorov · M. I. Varayun’ St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] N. V. Egorov e-mail: [email protected] M. I. Varayun’ e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_27

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x = V /V0 , y = I /I0 . Coefficients V0 and I0 are used for calibration. Dimensionless parameters q = (A, B) depend on the properties of the particular emission system in question. They can be estimated by regression analysis methods in both nonlinear and linearized formulations of the problem [3, 4]. Indeed, let   X = 1/x, Y = lg y/x 2 ,

(27.2)

what converts (27.1) to the linearized form Y = ϕ(X ; ϑ) = ϑ1 g1 (X ) + ϑ2 g2 (X ),

(27.3)

g1 (X ) = 1, g2 (X ) = X , ϑ1 = lg A, ϑ2 = −B/ ln 10. The transformations (27.2) are known as the FN coordinates. Obviously, measurements of both current and voltage contain errors. This is a concern, since in the regression analysis, the values of factors (X i in this case) are assumed to be known exactly. In addition, the nonlinear transformations (27.2) affect errors—their additivity becomes a question. Can the prerequisites for using a simple linear regression model be satisfied: the normality of residuals, their independence, and heteroscedasticity?

27.2 Signal Modeling and Its Linearization Let us write the current response of the emission system as y˜i = f (xi + εV,i ; q∗ ) + ε I,i , i = 1, N . Here, A∗ and B ∗ values are considered as known, not arbitrary A and B as in (27.1). Let εV,i = xi δV εi , ε I,i = yi δ I εi , where εi and εi are independent realizations of the standard normal random variable, δV and δ I are responsible for the constant signal-to-noise ratio. It is easy to show that  lg

f (x, q) + ε I x2

 = ϕ(X ; ϑ) +

δ I ε + o(δ I ), lg 10

    B δV ε + o (δV ) . f (x + εV ; q) = f (x; q) 1 + 2 + x

(27.4)

(27.5)

Formula (27.4) allows us to rely on the homoscedasticity of the residuals for the linearized regression model (27.3). An attempt to use the exact values of the factor

27 Factor Values Measurement and Heteroscedasticity …

251

p˜(ΔY ) δV = 0% δI = 5%

δV = 5% δI = 0%

15

10

5

0 −0.3

−0.2

−0.1

0.0

0.1

0.2

0.3 ΔY

Fig. 27.1 Response deviations

x (or X respectively) instead of the noisy one is a sadder case due to the presence of the term B/x in (27.5). Thus, there are two limit cases. In the first case, δV = 0, and the signal error is formed only by the fluctuations in the current values. In the second case, δ I = 0, and the signal error is due to voltage variations. Figure 27.1 shows the sample density function p˜ of deviations (not residuals) Y = Y˜ − Y of the noisy signal values from the exact values for these two cases. Here, A∗ = 2.0, B ∗ = 4.0, I0 = 1.0 A, V0 = 1.0 V were used for y˜i , and Y˜i = lg y˜i /xi2 (pay attention to the exact values of the factor). Modeling occurred in the range from xmin = 1.0 to xmax = 10.0. Intermediate values were calculated by formula xi =

xmax xmin (N − 1) , i = 1, N . xmax (N − i) + xmin (i − 1)

Thus, the nodes X i turned out to be equally spaced. In this case, N = 100. At the same time, 100 independent repeated measurements were modeled at each point X i for representativeness of the deviations sample. The results show that at a similar noise intensity δ, the voltage error contribution is more significant. Figure 27.2 shows the process of generating errors introduced by voltage deviavalue tions (δ I = 0). Firstly, the factor X˜ i is replaced with the exact  X i . Secondly,  the  observed value is also interpreted in a different way: lg y˜i /xi2 instead lg y˜i /x˜i2 .

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Y = lg

I V02 V 2 I0

•·

0.0

•·



•·

 

•· •· 

 •· −1.0



•·



•·

•·  •·

−2.0

0.0

0.2

0.4

0.6

0.8

1.0 X = V0 /V

Fig. 27.2 Error formation in case δV = 0

27.3 Estimation and Statistics The least squares method allows to obtain the following estimation of the parameters for the field electron emission signal: ϑˆ = M−1 b. The components of the constant matrix M and the observation based vector b are obtained by the following formulas: m jl =

N  i=1

g j (X i )gl (X i ), bl =

N 

gl (X i )Y˜i ,

j, l = 1, 2.

i=1

There is no ill-posed problem here. Now determine the residuals of the regression model (27.3): ˆ eˆi = Y˜i − Yˆi , Yˆi = f (X i ; ϑ). First of all, we will find out the significance of the regression model. For this, the characteristic F is calculated

27 Factor Values Measurement and Heteroscedasticity …

2  N − 2  ¯ Y − Yˆi , S 2 = eˆi2 . 2 S i=1 i=1 N

F=

253 N

If the condition F > F1−α (1, N − 2) is satisfied at the significance level α, then there is no reason to reject the hypothesis of significance [5]. We always used α = 5%. Since the simulated signal is already based on the desired model, it is practically impossible to obtain an insignificant model. Now we can test the residuals for normality, independence, and homoscedasticity. The normal distribution law for the residuals sample is verified using the Shapiro– Wilk test [6]. For this, the value W is calculated as shown below 1 W = 2 S

 N 



a N +1−i eˆ(N +1−i) − eˆ(i)

2 ,

i=1

N  = N /2, eˆ(i) are the sorted residuals. If the condition W > W (α) is satisfied at the significance level α, then there is no reason to reject the hypothesis of normality. Coefficients ai and critical values W (α) can be found in almost any statistical tables for a small sample size. Previously, residuals were tested for normality for case δV = 0, including other criteria [7]. When modeling the noise, the well-proven Mersenne Twister generator was used [8]. However, due to the small sample size, you can always get unexpected statistical conclusions. We decided to use the Durbin–Watson test to control the level of autocorrelation in the residuals [9, 10]. For this, the value D is calculated as shown below N

2 1  eˆi − eˆi−1 . D= 2 S i=2 If D > DU there is no reason to talk about positive autocorrelation. In case D < D L , vice versa. Similarly, If (4 − D) > DU there is no reason to worry about negative autocorrelation. In case (4 − D) < D L , vice versa. As you can see, the situation may be statistically uncertain. Finally, we consider the problem of homoscedasticity for the case when one value of the factor corresponds to one value of observation. Here the Goldfeld–Quandt test will help us [11]. The test procedure is as follows. The domain of the factor is divided into three . The central part should contain about a quarter of all the points, and it is taken out of consideration. In the remaining subregions, the number of measurements must be the same. So, for example, for case N = 10, two central points are discarded, for N = 20—4, for N = 30—8. Each subdomain has its own optimal regression function, and its own sum of the residuals squares is calculated. If the characteristics s 2 differ significantly, then we are most likely dealing with heteroscedasticity. Figure 27.3 shows the process of the regression model constructing and the Goldfeld–Quandt test application for case δV = 0. Here, Yˆ (1) (x) and Yˆ (3) (x) functions are the result of regression in subintervals 1 and 3. Data called phantom

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Y = lg

I V02 V 2 I0



Y (X) Yˆ (X)

Yˆ (1) (X) Yˆ (3) (X) Y˜ (X)

0.0

−1.0

−2.0

1 0.0

0.2

2 0.4

3 0.6

0.8

1.0 X = V0 /V

Fig. 27.3 Case δV = 0

are pseudo-repeated observations. These points are not taken into account and are intended only to show homoscedasticity. Thus, further we will focus on case δ I = 0, which is presented in Fig. 27.4. Here heteroscedasticity is already obvious, but only if repeated measurements are performed! For single observations shown by blue dots, this is not striking. It is in such cases that we have to rely on the Goldfeld–Quandt test. However, a single data sample may be arbitrarily warring. This fact is referred in Fig. 27.5 (here N = 30 for clarity). And although small deviations at small values of the factor are observed in all data samples, the statistical conclusions turned out to be different. Blue dots correspond to the sample, to which there were no complaints. Orange triangles show autocorrelation revealed by the Durbin–Watson test. Green triangles demonstrate heteroscedasticity, based on the Goldfeld–Quandt test. Finally, brown rhombs are recognized as both autocorrelated and heteroscedastic at the same time.

27 Factor Values Measurement and Heteroscedasticity …  1.0

Y = lg

I V02 V 2 I0

255



Y (X) Yˆ (X)

Yˆ (1) (X) Yˆ (3) (X) Y˜ (X)

0.0

−1.0

−2.0

1 0.0

2

0.2

3 0.6

0.4

0.8

1.0 X = V0 /V

Fig. 27.4 Case δ I = 0





0





















 

 



 

























 













0



0.6





 

           

0



eˆi

 







 

 







0.6 



0.6



•· •· •· •· •· •· •· •· •· •· •· •· •· •· •· •· •· •· •· •· •· •· · · • • •· •· •· •· •· •·

0

0.0

0.2

0.4

0.6

0.8

0.6

1.0 Xi

Fig. 27.5 Residuals for different statistical conclusions

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27.4 Conclusion The paper considers the problem of parametric identification of the FEE signal. Attention is paid to the case when the values of voltage can be measured inaccurately. In the general case, of course, some other circumstances will be superimposed on the total error of required estimations. However, it is the case under consideration that can lead to the presence of heteroskedasticity in the residuals of the regression model. The Durbin–Watson and Goldfeld–Quandt tests are able to indicate a situation where the researcher should caution himself against a simplified approach to the I –V curve study. If there is one of the unfavorable situations for the residuals, then we should refer to the estimates of White [12] or Newey–West [13] when constructing confidence intervals. Repeated measurements still remain the most preferred option, especially since modern automated systems for collecting information in this subject area allow this. Acknowledgements The work has been funded by Russian Foundation for Basic Research (Grant No. 20-07-01086).

References 1. Egorov, N.V., Sheshin, E.P.: On the current state of field-emission electronics. J. Surf. Invest.: X-ray, Synchrotron Neutron Tech. 11(2), 285–294 (2017) 2. Fowler, R.H., Nordheim, L.: Electron emission in intense electric fields. Proc. Roy. Soc. A 119, 173–181 (1928) 3. Antonov, A.Yu., Varayun’, M.I.: I–V curve investigation with regression methods. In: 2016 14th International Baltic Conference on Atomic Layer Deposition (BALD), St. Petersburg, pp. 41–43 (2016) 4. Egorov, N.V., Antonov, A.Yu., Varayun’, M.I.: Analysis of the emission characteristics of field cathodes using regression models. J. Surf. Invest.: X-ray, Synchrotron Neutron Tech. 12(5), 1005–1012 (2018) 5. Draper, N.R., Smith, H.: Applied Regression Analysis. A Wiley-Interscience Publication (1998) 6. Shapiro, S.S., Wilk, M.B.: An analysis of variance test for normality (complete samples). Biometrika 52(3/4), 591–611 (1965) 7. Lifantova, E.E., Varayun’, M.I., Antonov, A.Yu.: A linear regression model for the field emission signal. In: 2016 Young Researchers in Vacuum Micro/Nano Electronics (VMNE-YR), St. Petersburg, pp. 1–3 (2016) 8. Matsumoto, M., Nishimura, T.: Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM T. Model. Comp. S. 8(1), 3–30 (1998) 9. Durbin, J., Watson, G.S.: Testing for serial correlation in least squares regression: I. Biometrika 37(3/4), 409–428 (1950) 10. Durbin, J., Watson, G.S.: Testing for serial correlation in least squares regression: II. Biometrika 38(1/2), 159–177 (1951) 11. Goldfeld, S.M., Quandt, R.E.: Some tests for homoscedasticity. J. Am. Stat. Assoc. 60(310), 539–547 (1965) 12. White, H.: A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48(4), 817–838 (1980) 13. Newey, Wh.K., West, K.D.: A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent. Econometrica 55(3), 703–708 (1987)

Chapter 28

Influence of Dyes on the Electro-optical Properties of Liquid Crystals Tatiana Andreeva and Marina Bedrina

Abstract Different molecules of liquid crystals such as 4-pentyl-4’-cyanobiphenyl (CB5) and 4-(Hexyloxy)phenyl 4-butylbenzoate, and their interaction with dyes such as 1,2-Diamino-4-nitrobenzene, N,N-Dimethyl-4-nitrosoaniline, and dimethylamino-β-nitrostyrene were investigated by the density functional theory (DFT) using hybrid potential method B3LYP/6-31G. The electronic absorption spectra of isolated dye molecules and the resulting complexes with liquid crystals were calculated. It was shown that the shift of the absorption bands in the spectra of the dye depends on the structure of the complex. The shift and splitting of the bands due to minor impurity of dye molecules placed in the liquid crystal characterizes the mesophase structure.

28.1 Introduction Liquid crystal compounds and their colored solutions in the last few years are interesting because such compounds are the main components of industrial liquid crystal display (LCD). The properties of the compositions are determined by orientational order mechanisms upon transition from an isotropic liquid state to a mesophase. In turn, the properties of the mesophase in a cooperative system depend strongly on the ability of its constituent chemical compounds to self-organization. If the mesogen forms intermolecular complexes of various structural types, their effect on the spectral properties of the dye will also be different. It is very important to study the structural features of liquid crystals of IR and UV spectroscopy methods and computer modeling with respect to equations of quantum mechanics.

T. Andreeva (B) · M. Bedrina St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] M. Bedrina e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_28

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28.2 Modeling In DFT methods, the central physical quantity is the electron density ρ, which depends on the coordinates of all the electrons that make the system. For each electron, ρi = (|ϕi (r)|)2 and the electron density created by all the electrons of the molecule is defined as N  |ϕi (r)|2 , (28.1) ρ(r) = i=1

where ϕi is the wave function of the system. In accordance with the theorem of Hohenberg and Kohn, the energy of the ground state of a molecule is described by the electron density functional E, and the energy is minimal if ρ is the exact electron density of the ground state. The total energy of the quantum system in the Born-Oppenheimer approximation can be represented as E total = T + E ne + J + K + E nn ,

(28.2)

where T is the kinetic energy, the energy of electron attraction to the nuclei E ne , the electron repulsion energy J and the exchange energy K are expressed in terms of electron density, and the internuclear repulsion energy E nn is constant. DFT methods differ from each other in the form of the E XC (ρ), functional. As a rule, E XC is divided into the exchange E X and the correlation E C  E XC (ρ) = E X (ρ) + E C (ρ) =

 ρ(r )ε X (ρ(r ))dr +

ρ(r )εC (ρ(r ))dr. (28.3)

There are several well-known forms for the functionals E X and E C . An example of hybrid functionality is the B3LYP correlation exchange functionality B3LY P = (1 − α)E XL S D A + a E XH F + bΔE XB88 + (1 − c)E CV W N + cΔE CLY P , E XC (28.4) where E XL S D A is the exchange energy using the local spin density approximation, E XB88 is the Becke three-parameter functional of the gradient correction method, E XH F is the Hartree-Fock exchange energy, E CLY P is one of the most popular correlation functionals proposed by Lee, Young, and Parr, a, b, and c are constants selected from experimental data for a set of relatively simple chemical compounds. Adding other members to the LSDA functional significantly improves the quality of calculations, including the structural and energy parameters of molecules. The non-stationary theory of the density functional (TD-DFT) is based on the fact that for a given initial wave function, there is a single mapping between the timedependent external potential of the system and the time-dependence of the density, that is, it reduces to the equations of the response theory method

(A − B)1/2 (A + B)(A − B)1/2 Fi = ζ12 Fi ,

(28.5)

28 Influence of Dyes on the Electro-optical Properties of Liquid Crystals

259

where ζ1 is the i-th excitation energy, Fi is the corresponding response function, A and B are the matrices determined by the differences of the Kon-Shem orbital energies and the integrals including the second variational derivatives of the exchangecorrelation functional E XC . The configuration interaction method consists of the Ψ wave function which is written as a linear combination (Slater determinants) that determines a set of different configurations of the location of electrons at the main and virtual levels. The CIS (Configuration Interaction Singles) method includes only single-excited states occ  vir t  Aia Ψia , (28.6) ΨC I S = A0 Ψ0 + a

i

i.e., determinants Ψia included with coefficient Aia in the full wave function are generated by transferring an electron from the occupied orbital i to the virtual orbital a. The configuration interaction with only one excited configuration in it does not affect the energy of the ground state of the system. Since the CIS method does not improve the calculation of correlation energy compared to other Hartree-Fock methods, we used the TD-DFT method when calculating the excited state and such important physical and chemical characteristics of the substance as the electronic absorption spectrum of a molecule. To simplify the calculations without much damage, we can limit the number of configurations taken into account by setting a certain number of occupied and virtual orbitals. The correct description of intermolecular forces requires taking into account electron correlations. Complexes formed by the liquid crystal molecules [1] were investigated with dye molecules using the density functional B3LYP/6-31G method, which takes into account the indirect effect of the electron correlations. The intensity of the absorption band in the electronic spectrum is determined by the strength of the oscillator. f = 4.315 · 10−9

 γ dν,

(28.7)

where γ is the extinction coefficient, v is the frequency expressed in wave numbers. In order to test the possibilities of the TD-DFT and CIS methods in combination with various bases, the electronic absorption spectra of dyes were well studied experimentally. For all compounds, by optimizing the geometry, method equilibrium structures with minimum total energies were determined. The reliability of the found minimum energy on the potential surface was controlled by the absence of imaginary frequencies in the vibrational spectrum.

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Fig. 28.1 Calculated structure of 1,2-Diamino-4-nitrobenzene

Fig. 28.2 Calculated structure of 4-dimethylamino-βnitrostyrene

Fig. 28.3 Calculated structure of N,N-Dimethyl4-nitrosoaniline

Table 28.1 Total energy (E, a.u.) and dipole moments (D, Debays) of calculated dyes 1,2Diamino-44-dimethylamino-βN,N-Dimethyl-4nitrobenzene nitrostyrene nitrosoaniline E D

−547.3004 8.32

−647.9374 11.09

−495.3740 8.46

28.3 Results and Discussion Dye molecules with optimized geometry are shown on the figures (Figs. 28.1, 28.2 28.3). The dipole moment is a very important characteristic in the formation of liquid crystal structures. The Table 28.1 shows the total energies and dipole moments for these molecules in vacuum.

28 Influence of Dyes on the Electro-optical Properties of Liquid Crystals

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Fig. 28.4 Complex of liquid crystal molecule 4-pentyl-4’-cyanobiphenyl with the 1,2-Diamino-4-nitrobenzene dye

Fig. 28.5 Complex of 4-(Hexyloxy)phenyl 4-butylbenzoate molecule with the N,N-Dimethyl-4nitrosoaniline dye

Table 28.2 Total energy of calculated dyes (E, a.e.), intermolecular interaction energies (ΔE, kkal/mol), the minimum distances (Rmin , Å) in studied structures E total ΔE Rmin 1,2-Diamino-4nitrobenzene with CB5 1,2-Diamino-4nitrobenzene with dimer CB5 N,N-Dimethyl-4nitrosoaniline with 4(Hexyloxy)phenyl 4-butylbenzoate

−1299.2505

3.34

2.21

−2051.2012

5.05

2.51

−1615.3039

1.01

2.57

Complexes with the dye molecule which were found by the geometry optimization method are shown on Figs. 28.4 and 28.5. Dye molecule and monomer CB5 lie in the same plane at a distance of 2.21 Å (see Fig. 28.4). In other cases, the dye molecule is perpendicular to the main axis of the liquid crystal 4-(Hexyloxy)phenyl 4-butylbenzoate at a distance of 2.57 Å (see Fig. 28.5). Communication is carried out due to the dipole-dipole and donor-acceptor interaction between two molecules. The main criterion for selecting models for further research was that these structures have the maximum absolute value of total energies, thus their presence in the mesophase is most likely. Table 28.2 shows the total energy, the energy of intermolecular interaction, the distance between the atoms of liquid crystal molecules and dyes.

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Table 28.3 The wavelength (λ, nm) and oscillator strength ( f , p.u.) of electronic transitions λ1 f1 λ2 f2 λ3 f3 1,2-Diamino-4-nitrobenzene 1,2-Diamino-4-nitrobenzene with CB5 N,N-Dimethyl-4-nitrosoaniline N,N-Dimethyl-4-nitrosoaniline with 4-(Hexyloxy)phenyl-4-butylbenzoate

414.4 399.4 335.2 344

0.1 0.13 0.44 0.02

314.1 322.2 249.7 338.4

0.18 0.23 0.08 0.31

250.6 279 222 310.7

0.14 0.86 0.05 0.23

Fig. 28.6 Electronic absorption spectra of 1,2-Diamino-4-nitrobenzene dye

The electron spectra is sensitive to the associate’s structure [2, 3]. The electronic absorption spectra of complexes with dyes and liquid crystals for the optimized structures were investigated by the TD (time dependent) DFT method [4, 5]. Transitions between energy levels of molecular orbital and their frequency values performed are given in Table 28.3. We can see that received changes in the electron spectrum of the dye depending on the liquid crystal (Fig. 28.6). The frequency of the basic high-intensity transition corresponds to a transition from the highest occupied molecular energy levels (HOMO) to the lowest unoccupied molecular level (LUMO) and equal to 414.4 nm (λ1 ) for the 1,2-Diamino4-nitrobenzene in vacuum. The value of the wavelength and intensity for similar transitions changes for the complex CB5 with dye molecule (λ1 = 399.4 nm).

28.4 Conclusion It was shown that the shift of the absorption bands in the dye spectra depends on the structure of the complex. The shift and splitting of the bands of small impurity dye placed in the liquid crystal gives us an indication of the mesophase structure.

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When the mesogen forms an intermolecular complexes of various structural types, their influence on the spectral properties of the dye will also be different.

References 1. Andreeva, T., Bedrina, M., Egorov, N.: Dimerization of 4-cyano-4’-n-pentylbiphenyl in vacuum and under constant electric field. J. Vac. Sci. Tech. B 33, 03D102 (2015) 2. Andreeva, T., Bedrina, M.: Structure and optical properties of the 4-cyano-4’-n-pentylbiphenyl dimers and trimers. Liq. Crys. 47, 125–135 (2020) 3. Maslov, V.: Interpretation of the electronic spectra of phthalocyanines with transition metals from quantum-chemical calculations by the density functional method. Opt. Spectrosc. 101, 853–861 (2006) 4. Koch, W., Holthausen, A.: Chemist’s Guide to Density Functional Theory, 2nd edn. Wiley-VCH Verlag GmbH Press, Weinheim, Germany (2001) 5. Tawada, Y., Tsuneda, T., Yanagisawa, S.: A long-range-corrected time-dependent density functional theory. J. Chem. Phys. 120, 8425–8433 (2004)

Chapter 29

Genetic Stochastic Algorithm Application in Beam Dynamics Optimization Problem Liudmila Vladimirova, Anastasiia Zhdanova, Irina Rubtsova, and Nikolai Edamenko Abstract The article discusses the application of the genetic global search algorithm to the problem of beam dynamics optimization. The algorithm uses normal distribution to form new generations and provides covariance matrix adaptation during random search. The method is easy to use because it does not require calculation of the covariance matrix. The algorithm application is illustrated in the problem of global extremum search for the functional that characterizes beam dynamics quality in a linear accelerator. The extremal problem under study has a large number of variables; the objective function is multi-extreme. Therefore, the use of the stochastic method is the preferred way to achieve the goal. The algorithm quickly converges and can be successfully used in solving multidimensional optimization problems, including its combination with directed methods. The optimization results are presented and discussed.

29.1 Genetic Stochastic Algorithm with Covariance Matrix Evolution Genetic algorithms realize an iterative approach; each iteration deals with a generation of points (individuals). General scheme of stochastic methods of global optimization includes initial generation design and the way of transition to next generation. A goal is to provide the convergence of generation sequence to global extremum point. The various types of genetic algorithms are developed by now. For example, covariance matrix adaptation evolution strategy (CMA-ES) [1, 2] and simulated annealing L. Vladimirova · A. Zhdanova · I. Rubtsova · N. Edamenko (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] L. Vladimirova e-mail: [email protected] I. Rubtsova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_29

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method [3]. Both of these methods provide convergence of generation sequence to global extremum point with probability 1. This paper deals with genetic algorithm [4] belonging to the set of methods allowing covariance matrix evolution in the course of random search. The next generation is modeled using normal distribution of random test points. The special method of new generation modeling permits not to calculate covariance matrix. Genetic Algorithm with Covariance Matrix Evolution Consider the problem of global minimum search of the function F(X ) in the domain D of n-dimensional Euclidean space E n min X ∈D F(X ). Let l be the number of generation, ε be the prescribed accuracy. A. Initial generation: l = 0. 1. Modeling M random points X i , i = 1, M using uniform distribution in the domain D. (0) (0) = mini=1,...,M F(X i ), X min = arg mini=1,...,M F(X i ). 2. Fmin B. Transition to next generation. 1. Selection of m “the best” points Y1 , . . . , Ym among the points X 1 , . . . , X M . 2. Introduction of new points Xj =

 1 m (i)  (l) (l) + X min η j Yi − X min , j = 1, M, i=1 m

where η(i) j , i = 1, m, j = 1, M are independent standard normal random variables. l := l + 1. (l) (l) = mini=1,...,M F(X i ), X min = arg mini=1,...,M F(X i ). 3. Fmin 4. If

(l−1) (l) Fmin −Fmin (l−1) Fmin

< ε, then go to Exit, else goto B.

Exit. The convergence of this algorithm for unimodal function is proved in [4]. Main features of the method are as follows: • Since random variables η(i) j are included in the expressions for vectors X j , the covariance matrix varies from generation to generation and allows one to concen(l) . Thus, trate the sample in the region of the scattering ellipsoid with center X min test points (individuals) more often appear in the vicinity of the best population found in the previous generation. • When finding random normally distributed sampling points, it is not necessary to calculate the covariance matrix and use it in the simulation. This is especially important when the search space dimension is large.

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29.2 Beam Dynamics Model and Optimization Problem Beam dynamics optimization presents a class of specific complex problems, and the approaches to their solving are varied. Directed methods are widely used if mathematical optimization model is smooth [5–7]. These methods may be successfully combined with global search algorithms; a widely known example of stochastic one is Particle Swarm Optimization [8]; some applications are given in [9–11]. If there are several different optimization goals, a multiobjective approach is effective [3, 12, 13]. A number of researchers use genetic algorithms of global search [12, 14, 15]. However, the genetic algorithm described above with adaptation of the covariance matrix was not applied to the problem of beam dynamics optimization prior to our research [16]. Let us investigate the longitudinal dynamics of relativistic beam in linear waveguide accelerator. Beam evolution is considered to be a complex of synchronous particle motion and the motion of particles of a beam [11]. The synchronous phase is supposed to change along the structure. This approach opens up additional opportunities for optimization of beam evolution [5, 7, 10, 11]. Beam dynamics equations without considering Coulomb forces are as follows: (βγ )s d(βγ )s dξs = = −α(ξs , u 1 ) sin(ϕs (ξs , u 2 )), , dτ dτ 1 + (βγ )2s dξ (βγ ) d(βγ ) = = −α(ξ, u 1 ) sin(ϕˆ + ϕs (ξ, u 2 )), , 2 dτ dτ 1+ (βγ )  βγ 1 + (βγ )2s d ϕˆ  = 2π −1 dτ (βγ )s 1 + (βγ )2

(29.1)

with initial conditions ξs (0) = ξs0 , (βγ  )s (0) = (βγ )s0 , ξ(0) = ξ0 , (βγ )(0) = (βγ )0 , 2π ξ0 1 + (βγ )2s0 ϕ(0) ˆ = . (βγ )s0

(29.2)

Here, τ is reduced time; ξ , βγ , β, γ are reduced values of particle coordinate, impulse, velocity, and energy correspondingly; index s marks the characteristics of a synchronous particle; the functions α and ϕs are accordingly the dimensionless amplitude of accelerating wave and synchronous phase; u 1 and u 2 are the vectors of control parameters; ϕˆ is particle phase deviation from synchronous phase. The independent variable is introduced to be time analogue for the convenient account of Coulomb field in future research. Hamilton equations describing longitudinal oscillations of particles near the synchronous one [7] with dynamic variables presenting the differences between asynchronous and synchronous phases (ψ = ϕ − ϕs ) and reduced energies ( pψ = γ − γs ) are as follows:

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∂ H (ξ, ψ, pψ ) pψ =− , dξ ∂ψ

dψ ∂ H (ξ, ψ, pψ ) = , dξ ∂ pψ

where H (ξ, ψ, pψ ) = π(γs2 − 1)−3/2 pψ2 + V (ξ, ψ), V (ξ, ψ) = −α(ξ, u 1 ) (cos (ψ + ϕs (ξ, u 2 )) + ψ sin (ϕs (ξ, u 2 ))) . Under the assumption of adiabatic variation of the functions α(ξ, u 1 ), ϕs (ξ, u 2 ), (γs2 − 1)−3/2 along the structure, we have an equation for the separatrix that restricts the region of particle capture into acceleration mode [7]   pψ = ± (1/π )(γs2 − 1)3/2 V (ξ, −π − 2ϕs (ξ, u 2 )) − V (ξ, ψ).

(29.3)

Consider the problem of beam dynamics optimization by control parameters u = (u 1 , u 2 ) to provide high quality of bunching and accelerating of particle beam. Let us present the optimization objectives and corresponding quality criteria taking into account the experience [5, 7, 10]. 1. The first objective is to provide the synchronous particle output reduced energy in the required interval [γ1 , γ2 ]. The corresponding quality criterion is ⎧ ⎨ (γs (L) − γ1 )2 , γs (L) < γ1 , γs (L) ∈ [γ1 , γ2 ], K 1 (u) = 0, ⎩ (γs (L) − γ2 )2 , γs (L) > γ2 , where L is device exit reduced coordinate. 2. The goal of minimizing beam energy spread at accelerator exit may be achieved by minimizing the criterion K 2 (u) = |Na |−1



2 γn (L) − γs (L) , n∈Na

where Na is a set of numbers of model particles captured in acceleration mode, |Na | is the total number of captured particles, n is the model particle number. 3. For output phase spread minimizing at device exit, the following criterion is introduced: 

2 ϕˆn (L) − ϕ(L) , K 3 (u) = |Na |−1 n∈Na

where ϕ is the average deviation of particle phase from synchronous one. 4. The value of criterion K 4 (u) is a total penalty imposed on particles that have left their bunch or are outside the separatrix (3) at any cross-section of the structure. 5. It is advisable to minimize the defocusing factor influence at the stage of longitudinal motion optimization [5]. To realize this idea, we introduce the functional

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K 5 (u) imposing a penalty when the value of defocusing factor exceeds the specified limit. 6. To provide monotonous bunching, we impose the requirement of negative rate dG(ξ )/dξ of variation along the structure of the mean square spread of particle phases [5]. To satisfy the stated requirement, we minimize the criterion K 6 (u) presenting the accumulated penalty introduced for each point where dG(ξ )/dξ > 0. The resulting beam quality criterion is as follows: K (u) =

6 

b j K j (u),

(29.4)

j=1

where b j , j = 1, 6 are the weight constants. So beam dynamics optimization problem is formulated as a problem of criterion (4) minimization by control u. Note that this is a problem of joint optimization of program motion (of synchronous particle) and the ensemble of beam particle motions [5, 7, 10].

29.3 Numerical Results Numerical simulation and optimization of longitudinal beam dynamics are performed for electron waveguide accelerator with accelerating wavelength of 10 cm, structure length of 80 cm, channel radius of 0.04 m, and average beam current of 0.25 A. Due to the low injection energy (80 keV), initial energy spread influence is not taken into account. The effect of Coulomb repulsion can also be neglected. The functions α(ξ ) and ϕs (ξ ) are modeled by trigonometric polynomials, the components of the vectors u 1 and u 2 are the values of the derivatives of polynomials at grid points and the values α(0), ϕs (0). This allows one to obtain smooth functions α(ξ ) and ϕs (ξ ). The functions used before optimization are presented in Fig. 29.1 (dotted lines). Beam dynamics optimization problem is reduced to criterion (4) minimization by control parameters (the components of u 1 and u 2 ). This extremal problem is treated using the genetic stochastic algorithm with covariance matrix evolution. To apply the algorithm presented above, assume X be the vector of control parameters (dim(X ) = 84), F(X ) = K (u). So, the search is carried out in multidimensional domain D ∈ E 84 . Numerical optimization experience shows that the objective function has many closely spaced extrema. Therefore, to implement the algorithm, a sufficiently large number M of random vectors is necessary. The parameters of the method are chosen as follows: M = 1000, m = 50. The optimization performed provided a significant decrease of objective functional value and appropriate beam dynamics improving. It took only 6 generations to achieve the required accuracy (ε = 0.01) of finding the extremum (see Table 29.1). Regarding beam characteristics at device output, the optimization allowed to reduce the phase spread from 1.63 to 0.98 (radian), to reduce the relative energy

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Table 29.1 Criterion decrease during optimization Generation 1 2 3 K(u)

15.463

6.303

4.940

4

5

6

4.742

4.678

4.674

Fig. 29.1 The functions α(ξ ) and ϕs (ξ ) before optimization (dashed line) and after optimization (solid line)

Fig. 29.2 Phase deviation of particles from synchronous one

spread from 0.36 to 0.31, and to increase the number of particles within the separatrix from 94% to 98%. Capture coefficient remained constant during optimization and is equal to 96%. Synchronous particle energy after optimization is reduced from 11.93 to 11.37. Thus, it belongs to the required interval (11.3; 11.7). So the optimization provided beam quality improvement. The functions α(ξ ) and ϕs (ξ ) obtained after optimization are shown in Fig. 29.1 (solid lines). The plots presenting beam dynamics (for particles in acceleration mode) before optimization (left) and after optimization (right) are given in Figs. 29.2, 29.3, and 29.4. It should be noted that the genetic algorithm used is simple to implement, efficient, and enables high performance. Comparative experiments have shown that the time spent for one iteration processing is two times less than the corresponding time for Particle Swarm Optimization.

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Fig. 29.3 Reduced energy of particles

Fig. 29.4 Separatrix and phase-energy distribution of particles at accelerator exit

29.4 Conclusion The paper presents an approbation of a genetic stochastic algorithm with an adaptation of a covariance matrix on a multi-extreme large-dimensional problem, namely, the problem of beam dynamics optimization in a linear accelerator. The practice of numerical experiments and the results obtained indicate the simplicity, convenience, and effectiveness of this method. The results of successive optimization show beam quality improvement. Acknowledgements The authors are grateful to Professor S.M. Ermakov for his attention to the work and valuable comments.

References 1. Ermakov, S.M., Mitioglova, L.V.: On extreme search method based on the estimation of the covariance matrix. Autom. Comput. Eng. 5, 38–41 (1977). (in Russian) 2. Vladimirova, L.V., Ermakov, S.M.: Random search method with a “Memory” for global extremum of a function. In: Proceedings of 10th International Workshop on Simulation and Statistics. Universitat Salzburg, Salzburg, Workshop booklet, 89, (2019). https://datascience. sbg.ac.at/SimStatSalzburg2019/SimStat2019BoA.pdf

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3. Igel, C., Hansen, N., Roth, S.: Covariance matrix adaptation for multi-objective optimization. Evol. Comput. 15(l), l–28 (2007) 4. Ermakov, S.M., Semenchikov, D.N.: Genetic global optimization algorithms. In: Communications in Statistics, Part B: Simulation and Computation (2019). https://doi.org/10.1080/ 03610918.2019.1672739 5. Ovsyannikov, A.D.: Mathematical models of beam dynamics optimization. VVM, St. Petersburg, p. 181 (2014) (in Russian) 6. Ovsyannikov, D.A., Ovsyannikov, A.D., Vorogushin, M.F., Svistunov, Yu.A., Durkin, A.P.: Beam dynamics optimization: Models, methods and applications. Nucl. Instr. Meth. Phys. Res. Sect. A 558(1), 11–19 (2006) 7. Ovsyannikov, A.D., Shirokolobov, A.Y.: Mathematical model of beam dynamics optimization in traveling wave. In: Proceedings of RuPAC-2012. JACoW, pp. 355–357 (2012). http://www. JACoW.org 8. Kennedy, J.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks IV, pp. 1942–1948 (1995) 9. Altsybeev, V.V., Svistunov, Yu.A., Durkin, A.P., Ovsyannikov, D.A.: Preacceleration of the multicharged ions with the different A/Z ratios in single RFQ channel. Cybern. Phys. 7(2), 49–56 (2018) 10. Rubtsova, I.D., Vladimirova, L.V., Edamenko, N.S., Goncharova, A.B.: Intense beam dynamics study in Alvarez accelerator. Phys. At. Nucl. 82, 1527–1531 (2019). https://link.springer.com/ article/10.1134/S106377881911019X 11. Zhdanova, A.Y., Rubtsova, I.D.: Modeling and optimization of intense beam dynamics in traveling-wave field. In: Proceedings of V International Conference on Laser& Plasma Researches and Technologies (LaPlas-2019), part 2, Moscow, National Research Nuclear University MEPhI, pp. 160–161 (2019) 12. Bartolini, R., Apollonio, M., Martin, I.P.S.: Multiobjective genetic algorithm optimization of the beam dynamics in linac drivers for free electron lasers. Phys. Rev. ST Accel. Beams 15(3), 030701 (2012) 13. Vladimirova, L.V.: Multicriterial approach to beam dynamics optimization problem. J. Phys. Conf. Ser. 747(1), 012070 (2016) 14. Balabanov, M.Yu.: On initial control choice in charged particles beams dynamic optimization problems. Vestnik of Saint Petersburg University. Appl. Math. Comput. Sci. Control Process. 3, 93–99 (2010) 15. Gao, W., Wang, L., Li, W.: Simultaneous optimization of beam emittance and dynamic aperture for electron storage ring using genetic algorithm. Phys. Rev. ST Accel. Beams 14(9), 094001 (2011) 16. Vladimirova, L.V., Zhdanova, A.Y., Rubtsova, I.D.: Application of the genetic global search algorithm in beam dynamics optimization problem. In: Proceedings of VI International Conference on Laser& Plasma researches and technologies (LaPlas-2020), part 1, Moscow, National Research Nuclear University MEPhI, pp. 91–92 (2020)

Chapter 30

On a New Approach to RFQ Channel Optimization Oleg I. Drivotin

Abstract The radio frequency quadrupole (RFQ) channel optimization problem is regarded as an optimal control problem for an ensemble of dynamical systems described by density distribution in the phase space. Previously, for a numerical solution of this problem methods using the first variation of the trajectory were applied. At the present work, a method based on the second variation of the trajectory is proposed. This method allows computing not only the first derivatives, but also the second ones of the quality functional over control parameters.

30.1 Introduction The RFQ accelerating structure was invented by Kapchinsky and Teplyakov [1]. They proposed an algorithm of the channel design based on the adiabatic bunching [2], which is widely used now. The extensive use of the RFQ structure in modern accelerators at an initial stage of acceleration is due to its ability of accelerating, bunching, and focusing of a low-energy beam simultaneously. In 1980, D.A. Ovsyannikov suggested to search for accelerating structures by optimal control theory methods, and developed an indirect first-order method of beam dynamics optimization [3] based on known numerical methods of optimal control theory [4]. This method was successfully applied for numerical optimization of accelerator structures by D.A. Ovsyannikov and his followers [5–7]. The main disadvantage of this method is that it allows to obtain only the gradient of quality functional. Gradient defines a surface separating regions with greater and lesser values of the quality functional, but does not allow to find the direction of descend. Later, it was proposed to use a second order method [8] allowing to find descend direction, which is based on the known expression for the second variation of the quality functional [9–12]. The method consists of the computation of the matrix momenta [12] at each step of descent. A number of matrix momenta is quadratic in O. I. Drivotin (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_30

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number of parameters, and for all of them, a differential equation should be integrated. The number of parameters of an accelerator channel could be as many as several hundred. Therefore, the optimization by the matrix momenta method requires a large amount of computations. Later, a numerical method based on the integral expression for the second variation of the quality functional was proposed for solving the optimal control problems [13, 14]. Here, the number of differential equation to be integrated is quadratic in the phase space dimension. Therefore, the problem of numerical optimization of an accelerator channel can be regarded as relevant for application of this method. As alternative approaches that can be also applied in beam dynamics optimization are methods using the Pontryagin maximum principle [15, 16], and differential algebraic methods [17, 18]. The goal of this article is to formulate the control theory model for optimization of the RFQ channel on the base of the method using the second variation expression.

30.2 Second Variation of Trajectory of a Controlled System Consider a controlled dynamical system [15] dx = f (t, x, u), t ∈ [t0 , T ], x(t) ∈ Ω ⊂ R n , x(t0 ) = x0 ∈ Ω, dt

(30.1)

where u = u(t) is a piecewise continuous control function with values belonging to some set: u(t) ∈ U ⊂ R k . Assume that the set of allowed states of the system Ω and the function f (t, x, u) are such that the solution of the Cauchy problem (30.1) uniquely exists in all cases under consideration. The first variation of trajectory of such system is t δx (t) = i

G ij (t, t  )δu f j (t  )dt  ,

i = 1, n,

(30.2)

t0

where G is the Green function of the linear equation for the trajectory first variation dδx i ∂fi j = δx + δu f i , dt dx j

i = 1, n.

(30.3)

Here and further, we apply the Einstein summation convention according to which the sum is taken over repeating lower and upper indices in tensor component equalities. The second variation of the trajectory can be written by analogous way [13, 14]

30 On a New Approach to RFQ Channel Optimization

t δ x (t) = 2 i

t0

t +

∂f j D ijk (t, t  ) l ∂u



t0

∂2 f j  k (t )δu ∂u k ∂ x l +

t 

G km (t  , t  )

t0

⎡ G ij (t, t  ) ⎣



(t )δu (t ) l

t  t0

275

G lm (t  , t  )

∂ f m  n  (t )δu (t ) dt  dt  + ∂u n

∂ f m  n  (t )δu (t ) dt  + ∂u n

 ∂2 f j  k  l  (t )δu (t )δu (t ) dt  . ∂u k ∂u l

(30.4)

Here D is a tensor of the third rank: twice covariant and one time contravariant. The tensor D has the following integral representation: D ijk (t, t  )

t =

G li (t, t  )

t

∂2 f l (t  )G mj (t  , t  )G nk (t  , t  ) dt  , ∂xm∂xn

and satisfies to the differential equation d D ijk (t, t  )

m 2 m ∂f m  i  ∂f  i  ∂ f (t ) + D (t, t ) (t ) − G (t, t ) (t  ), jm m dt  ∂x j ∂xk ∂x j ∂xk (30.5) and to the condition D(t, t) = 0. i = Dmk (t, t  )

30.3 Optimization Problem Formulation and Solution for a Beam of Low Intensity Let us regard a charged particle beam as an ensemble of a dynamical systems described by the Eq. (30.1). Denote the distribution density by ρ(t, x). Following the works [19–21], we regard ρ(t, x) as a differential form which degree depends on the beam model. The form of the top degree correspond to the nondegenerate case, when the distribution supporter is a region in the phase space. Degenerate distribution corresponds to a case when particles are distributed on some surface whose dimension is less then the dimension of the phase space N . Then the form degree is between 1 and N . Most degenerate distribution can be described by a set of macroparticles. In this case, the dimension of the supporter is 0, and ρ(t, x) is density of the poinlike measure that is the sum of scalar functions whose values are distinct from 0 only at points where particles are located. In all of these cases, evolution of the density satisfies to the Liouville equation [19–21] and to the condition ρ(t + δt, F f, δt x) = F f, δt ρ(t, x), ρ(t0 , x) = ρ(0) (x)

(30.6)

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where F f, δt denote the Lie dragging along the vector field f by a parameter increment δt. The problem is to find such control u that the quality functional  g(x T )ρ(T, x T )

Φ(u) =

(30.7)

Ω

reaches minimal value. Note that in the functional (30.7), density forms of any degree from 0 to the dimension of the phase space are admitted. Further, we shall restrict our consideration by the case of the form of zeroth degree. In this case, integral in (30.7) is regarded as sum over points where ρ(t, x) = 0. In the particle beam theory, such an approach is known as the method of macroparticles. In this case, the Eq. (30.6) ensures that the charge of a macroparticle is conserved along its trajectory. Applying the second-order method described in the previous section in the case under consideration, we get the first and the second variations of the functional (30.7) as a sum of expressions (30.2), (30.4) over all the macroparticles.

30.4 RFQ Channel Optimization The RFQ channel consists of two pairs of modulated vanes [2]. Let us consider it as a sequence of M cells: ζ ∈ [ζ j−1 , ζ j ), j = 1, M, where ζ = z/λ is reduced longitudinal coordinate, λ is a wavelength of the accelerating RF field, ζ j are reduced coordinates of cell limits, which we regard as points where modulation phases are equal to π multiplied by an integer. Let us use ζ as the parameter t. For brevity, consider only the longitudinal dynamics. Let introduce the longitudinal coordinates [5, 6] γ = (1 − β 2 )−1/2 , ϕ = ωt. Here, β is reduced velocity of the macroparticle and ϕ = ωt where t is time when macroparticle passes a point with coordinate ζ, ω = 2π c/λ. Let the initial distribution be the set of macroparticles with the same energy γ = γ0 uniformly distributed over phase ϕ, ϕ ∈ [−2π, 0]. The longitudinal component of the electric field intensity can be approximated by the expression [5, 6] 4kΘ cos η cos ωt. (30.8) E z = U0 π Here, 2U0 is the amplitude of the voltage between opposite vanes. k = π/L , L is the z cell length, η is the vane modulation phase, η(z) = k(z  ) dz  , Θ–is the acceleration z0

efficiency Θ=

π m2 − 1 · 2 , 4 m I0 (ka) + I0 (mka)

30 On a New Approach to RFQ Channel Optimization

277

m is the modulation coefficient, I0 is the modified Bessel function of the zeroth order, a is the aperture of the cell. It is easy to get that the differential equation for phase of the particle has the form 2π γ dϕ = . dζ γ2 − 1

(30.9)

Assume that the vane modulation is such that there exists a particle, at least at the last few sections, so that dϕ/dζ = k, k = λk. Let us call this particle the synchronous particle and denote it by the index s. Call the difference between the phase of that particle and the phase of the vane modulation the synchronous phase and denote it by Φs : Φs = ϕs − η. As for the synchronous particle, the Eq. (30.9) is also satisfied, we get the differential equation for η: dη = 2π γs (γs2 − 1)−1/2 − u 1 . dζ Here u 1 = dΦs /dζ is taken as one of the components of the vector control function u. Take Θ as the second component: u 2 = Θ. Taking into account the expression (30.8), one can write the second equation of the longitudinal motion in the form dγ = C L (2π γs (γs2 − 1)−1/2 − u 1 )u 2 cos η cos ϕ. dζ Here C L = 2eU0 /(π mc2 ). Assume that Θ does not change in the last few sections of the structure. As the value πγ 3 (γ − γs )2 . H (ϕ, γ ) = (ϕ − ϕs )2 − 4C L kΘβs3 sin ϕs is conserved for particles close to the synchronous particle, the function  g(ϕ, γ ) =

0, H (ϕ, γ ) ≤ H0 , (H (ϕ, γ ) − H0 )2 , H (ϕ, γ ) > H0 .

(30.10)

characterizes deviation of a particle from the synchronous particle and can be used in the quality functional (30.7). In this case, the functional (30.7) characterizes the phase and the energy spreads of the beam at the end of the channel. The most commonly used approach in problems of charged particle beam modeling and optimization is the method of macroparticles, where phase density ρ is the sum of delta-functions, each corresponding to a macroparticle. Let N be a number

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of macroparticles. As it was mentioned above, in this case integral in (30.7) reduces to sum, and the functional (30.7) can be written in the form N

Φ=

g(ϕi , γi ).

(30.11)

i=1

Let the control function is piecewise constant: u k (ζ ) = u k j , ζ ∈ [ζ j−1 , ζ j ), j = 1, M. Then the functional (30.11) can be considered as a function of 2M parameters, derivatives over which are found in a common way: ζ j N

∂Φ =− ∂u k j

ψ(ζ, x(i) )

ζ j−1 i=1

∂Φ = ∂u 1 j

∂δu f (ζ, x(i) ) dζ, ∂u k j

 ζ j

N ψγ (i) C L u 2 cos η cos ϕ(i) dζ, ψη + i=1

ζ j−1

∂Φ =− ∂u 2 j

ζ j N

ζ j−1

ψγ (i) C L k cos η cos ϕ(i) dζ.

i=1

j

The equations for conjugate functions (momenta) ψi (ζ ) = −(∂g/∂ x j )G i (T, ζ ) are dψη = ψγ (i) C L ku 2 sin η cos ϕ(i) , dζ i=1 N

+ψϕ(s)

dψγ (s) 2π = ψη 2 + dζ (γs − 1)3/2

N 2π 2π ψγ (i) 2 C L u 2 cos η cos ϕ(i) , (γs2 − 1)3/2 i=1 (γs − 1)3/2

dψϕ(i) = C L ψγ (i) ku 2 cos η sin ϕ(i) , dζ

dψγ (i) 2π = ψϕ(i) 2 , i = s. dζ (γ(i) − 1)3/2

The second derivatives over the parameters are m 2  N m ∂ x(i) ∂Φ ∂ 2 x(i) ∂ 2Φ 1 ∂ 2Φ = (T ) + (T ) , ∂ x m ∂u 2k j 2 ∂(x m )2 ∂u k j ∂u 2k j i=1

(30.12)

30 On a New Approach to RFQ Channel Optimization

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where ∂ x m /∂u k j (T ) and ∂ 2 x m /∂u 2k j (T ) are found from expressions for the first and the second variations, which can be obtained according to the expressions (30.2), (30.4). The expression (30.12) is obtained under the condition ∂ 2 Φ/(∂ x m ∂ x l ) = 0 for m = l, which is realized when Φ is determined by function (30.10).

30.5 Conclusion Here, we presented the new approach for RFQ structure optimization, consisting of the computation of the first and the second derivatives of the quality functional over control parameters. This approach can be regarded as upgrading and refinement of D.A. Ovsyannikov approach [3]. It is expected to be more effective as it allows to obtain direction and step of the descent in the process of optimization. But numerical optimization, according to this approach, requires greater effort in programming than the first-order method, because not only differential equations for momenta ψ should be integrated, but also the Eqs. (30.3) and (30.5) for tensors G and D.

References 1. Kapchinsky, I.M., Teplyakov, V.A.: Linear ion accelerator with space uniform rigid focusing. Prib. Tekh. Eksp. 2, 19–22 (1970) (in Russian) 2. Kapchinsky, I.M.: Theory of Resonance Linear Accelerators. Harwood Academic Publishers, New York (1985) 3. Ovsyannikov, D.A.: Mathematical Methods of Beam Control. Leningrad University Publications, Leningrad (1980).(in Russian) 4. Bryson, A.E., Ho, Y.C.: Applied Optimal Control. Blaisdell Publishing Company, Waltham, Mass (1969) 5. Drivotin, O.I., Ovsyannikov, D.A., Svistunov, Yu.A., Vorogushin, M.F.: Mathematical models for accelerating structures of safe energetical installation. In: Proceedings of the 6-th European Particle Accelerator Conference EPAC’98. Stockholm, June 22–26 (1998) 6. Drivotin, O.I., Ovsyannikov, D.A.: Modeling and optimization of the dynamics of high density beam in RFQ channel. In: Proceedings of the 6th International Workshop “Beam Dynamics and Optimization”. Saratov, Russia, Sept 6–9, 1999 (2000) 7. Drivotin, O.I., Vlasova, K.A.: Numerical optimization of RFQ channel. In: 20th International Workshop On Beam Dynamics and Optimization (BDO 2014). St. Petersburg, Russia (2014) 8. Bublik, B.N., Garashchenko, F.G., Kirichenko, N.F.: Structural-Parametrical Optimization and Stability of Beam Dynamics. Naukova dumka publications, Kiev (1985).(in Russian) 9. Miele, A.: Recent advances in gradient algorithms for optimal control problems. J. Optim. Theory Appl. 17(5/6), 361–430 (1975) 10. Gorbunov, V.K., Lutoshkin, I.V.: Development and experience of application of method of parameterization in degenerate problems of dynamical optimization. Izvestiya Rossiyskoy akademii nauk. Teoriya i sistemy upravleniya 5, 67–84 (2004) (in Russian) 11. Golfetto, W.A., Silva Fernandes, S.: A review of gradient algorithms for numerical computations of optimal trajectories. J. Aerosp. Technol. Manag. 4(2), 131–143 (2012) 12. Gabasov, R., Kirillova, F.M.: Special Optimal Controls. Nauka Publications, Moscow (1973).(in Russian)

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13. Drivotin, O.I.: Second variation of trajectory of a controlled dynamical system. Cybern. Phys. 7(4), 884–913 (2018) 14. Drivotin, O.I.: On numerical solution of the optimal control problem based on a method using the second variation of a trajectory. Vestnik Sankt-Peterburgskogo Universiteta. Prikladnaya Matematika, Informatika, Protsessy Upravleniya 15(2), 283–285 (2019) (in Russian) 15. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience Publishers, New York (1962) 16. Nikol’skii, M.S., Belyaevskikh, E.A.: L.S. Pontryagin maximum principle for some problems of optimal control of trajectories pencils. Vestnik Sankt-Peterburgskogo Universiteta. Prikladnaya Matematika, Informatika, Protsessy Upravleniya 14(1), 59–68 (2018) (in Russian) 17. Makino, K., Berz, M.: Optimal correction and design parameter search by modern methods of rigorous global optimization. Nucl. Instrum. Methods Phys. Res. A 645, 332–337 (2011) 18. Di Lizzia, P., Armellin, R., Bernelli-Zazzera, F., Berz, M.: High order optimal control of space trajectories with uncertain boundary conditions. Acta Astronautica 93, 217–229 (2014) 19. Drivotin, O.I.: Covariant formulation of the Vlasov equation. In: IPAC 2011 - 2nd International Particle Accelerator Conference. San-Sebastian, Spain (2011) 20. Drivotin, O.I.: Degenerate solutions of the Vlasov equation. In: Proceedings of 23 Russian Particle Accelerator Conference (RuPAC 2012). St. Petersburg, Russia (2012) 21. Drivotin, O.I., Ovsyannikov, D.A.: Stationary self-consistent distributions for a charged particle beam in the longitudinal magnetic field. Phys. Part. Nucl. 47(5), 889–913 (2016)

Chapter 31

Optimization of a Real-Time Stabilization System for the MIMO Nonlinear MagLev Platform Sergey Zavadskiy, Mikhail Verkhoturov, Anna Golovkina, Dmitri Ovsyannikov, Vladimir Kukhtin, Nicolai Shatil, and Andrei Belov Abstract The paper discusses optimal stabilization problem for the MagLev transport platform. The platform has a specific levitation system with four combined suspensions which include both electromagnets and permanent magnets. The presence of permanent magnets makes the problem nonlinear and incompletely controllable and causes the main challenge for this paper. In order to address it, we propose to optimize the controller with respect to the bunch of possible trajectories taking into account the mentioned nonlinearities. The controller has a dynamical structure and includes a Kalman filter with feedback on the platform gaps and electromagnet coil currents. The introduced approach allows operating in real-time even when platform parameters and mass change and in the presence of noise and disturbances. Optimization of stabilizing controller improves both energy costs and control accuracy.

31.1 Introduction The paper aims to develop an algorithm synthesizing and optimizing stabilization system of the levitating platform position. The platform is driven into magnetic levitation state by means of combined electromagnetic suspensions. Each of suspension includes a permanent magnet and two electromagnets (upper and lower). Such a combined construction causes instability in the platform motion and stipulates for a reliable digital control. In the same time, commonly adopted approaches used for controller construction may not be enough to cope with the problem. This is caused by the presence of eddy currents, continually sensed noise and two types of magnets. In such a case, the

S. Zavadskiy (B) · M. Verkhoturov · A. Golovkina · D. Ovsyannikov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. Kukhtin · N. Shatil · A. Belov JSC “NIIEFA”, 3, Doroga na Metallostroy, St. Petersburg 196641, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_31

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stabilizing controller requires a condition-specific real-time optimization to achieve a sufficient accuracy, energy consumption, and stabilization quality criteria. Papers [5, 8] describe the mathematical model of the object studied in this paper. The paper [8] considers combinations of different types of magnets for simultaneous application and suggests the solution of the problem of choosing optimal configurations of electromagnets in order to minimize energy costs. The paper [5] presents energy consumption improvements in the case of combined electromagnetic systems and computational models that provide an accurate description of the reactions of the system for the development of effective control algorithms, as well as the optimal parameters of the controller to ensure greater stability. Papers [3, 9] are devoted to numerical modeling of electrodynamic suspension, also they show different variants of design. In the papers [10, 17, 18] the object of the study is the high-speed road in Shanghai. The paper [18] shows an example of subsystem modeling: electromagnet force—air gap. The paper [10] presents a dynamic model of the interaction between the vehicle and the guide. The paper [17] provides a realistic model that includes the dynamics of roll, pitch, and yaw, and also solves the problem of constructing a linear quadratic regulator. Kalman-Bucy filter from works [10, 12–15, 20, 23] found extensive application in control problem [16]. The book [11] considers fundamental works that reveal the principles of solving problems of stabilization and optimization of dynamic objects. In the papers [13, 14, 21], the synthesis of regulators in solving the problems of plasma stabilization in tokamaks for the dynamics of the ensemble of trajectories is carried out by changing the strength of currents in the control coils, which is similar to the presented problem. In [7], the optimization problem of a charged particle beam in an electrostatic field is presented on the basis of variation approaches.

31.2 Maglev Platform Dynamic Equations Combined electromagnet suspensions at the corners of the platform perform the object position change when the current in the coils are changed by control system (see Fig. 31.1). The vertical position of the combined suspension is assumed to be zero when the gap between the rail and the lower electromagnet of this suspension is 5 · 10−3 m. The force of the electromagnet Fem depends on the position of the electromagnet z and the current I supplied to the electromagnet. The following constants and expressions were introduced to simulate the electromagnet force Fem (z, I ). Corresponding functional dependencies were obtained from the results of numerical 3D modeling of an electromagnetic suspension using the program codes KOMPOT [2] and KLONDIKE [2, 7] developed at the NIIEFA for the analysis of EM systems [1], including muggle systems based on the principles of EMS [5] and EDS [3, 4, 6].

31 Optimization of a Real-Time Stabilization System …

283

Fig. 31.1 Maglev platform: 1—platform, 2—ferromagnetic guides (rails), 3—electromagnets, 4— permanent magnets, Fem1 —electromagnetic force of the lower magnet, Fem2 —electromagnetic force of the upper magnet, F pm —magnetic force of the permanent magnet, Fg —gravity, z 1 , z 2 , z 3 , z 4 —vertical positions of combined suspensions, x E , y E , z E —coordinate axes

h = z · 0.001 + 0.005, h 0 = 3, corr = 1 + (h mm − h 0 ) · q, h I , Ik A = , h mm = 1000 1000 f 1 = (a − c) · x + b,   sf 2 · 9.807, Fem = corr

dh = 0.3, a = 15.5, b = 8.5, c = 1.3, d = 5.3, q = 0.025, Ik A · corr x= , h mm + dh f 2 = (a − c) · x − b,    1 sf = c · x + · ( f 12 + d 2 ) − f 22 + d 2 . 2

Let us define Fem (z, I )upper as a force of the upper electromagnet, and Fem (z, I )bottom as a force of the bottom electromagnet. The permanent magnet force [5, 8] is determined by the expression  F pm (z) =

−z + 21.5 84

2

 +

1 7.5

2 −2 · 9.807.

Some disturbances in the suspension system are described by a piecewise constant function, which is constant in the intervals δti and is given by the recurrence equation: (Icoil , t) =

 3   j=1

(ti−1 , Icoil ti−1 ) − j

j wall Icoil



· exp

ti − ti−1 j

τall

 ,

(31.1)

T

T where τall = 0.13 0.025 5 · 10−3 and wall = 0.1 0.35 0.37 are weighting vectors, ti : t ∈ [ti−1 , ti ], Icoil = Icoil (ti ) − Icoil (ti−1 ), Icoil is the current value in the coil.

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Taking into account the decaying nature of the eddy currents, the |(Icoil , t)| → 0 ∗ ∗ , where Icoil is a steadyestimate is valid for (31.1) when t → ∞ and Icoil → Icoil state value of current in the coil. This fact follows from the negative exponent degree in formula (31.1). The length of δti interval is set to 10−3 s. By analogy with [8], the measurement noise ϕ(t) ∈ E 8 is a piecewise constant vector function on the interval ti whose values are uniformly distributed in the range |ϕ(t)| ≤ 5 · 10−6 m for components numbers 1..4 and are equal to zero for components 5.8. The resulting force created by one combined suspension consists of the forces of top bottom , and the permanent the upper electromagnet Fem , the lower electromagnet Fem magnet F pm    i,top i,top i,top −z i , Icoil (I i ) + i,top Icoil (I i ), t + Fi (z i , I i , t) = −Fem    (31.2) i,bottom i i,bottom i i,bottom i +Fem z i , Icoil (I ) + i,bottom Icoil (I ), t + F pm (z i ), where i = 1, 4 is the suspension number, z i is the deviation of the lower magnet of the i-th suspension from the position at which the distance between it and the rail is 5 · 10−3 m, I i is a generalized current, depending on which the currents for the i,top upper and lower electromagnets are calculated using the formulas below, Icoil is the current in the coil supplied to the upper electromagnet of the i-th suspension, determined by the ratio  i,top Icoil (I i )

=

−I i + Imin I i ≤ 0 , Ii > 0 Imin

i,bottom Icoil is a the current in the coil supplied to the lower electromagnet of the i-th suspension, determined by the ratio

 i,bottom i (I ) = Icoil

Ii ≤ 0 Imin , I + Imin I i > 0 i

i,top is the eddy current in the system of the upper electromagnet for the i-th suspension, i,bottom is the eddy current in the system of the lower electromagnet for the i-suspension, Imin = 1 A. Angular velocity components ω = ( p, q, r )T depend on Euler angles ξ , χ , θ . Given that ξ = 0 and r = 0, we conclude p = θ˙ and q = χ˙ cos θ . Then the system characterizing the state of levitation of the platform is given by the expression:

31 Optimization of a Real-Time Stabilization System … 4 

θ˙ = p,

p˙ =

q˙ = 4 

z˙ c = vc ,

v˙ c =

Fi (χ , θ, z c , Ii , t)

i=1

,

2mw 4 

χ˙ = q,

285

Fi (χ , θ, z c , Ii , t)

i=1

2ml

,

(31.3)

Fi (χ , θ, z c , Ii , t)

i=1

I˙i = u i , i = 1..4,

4m

− g,

where u 1 , u 2 , u 3 , u 4 are the components of the control vector, F1 , F2 , F3 , F4 are the forces of the combined suspensions defined by the formulas (31.2).

31.3 Model for Transient Process Optimization 31.3.1 Optimization Model We rewrite (31.3) in a vector form and close loop by the controller. Then the transient optimization model includes: the dynamics equation, stabilizing regulator, quality functional, and has the following form: x˙ = F(x, u, t), y = G(x) + ϕ(t), x(0) = x0 , z˙ = Ac z + Bc y,

(31.4)

u = Cc z, T  N I = (xiT Rxi + u iT Qu i ) dt → min, 0

(31.5)

i=1

where N is a number of trajectories in the bunch. To minimize the functional (31.5), we will vary the elements of the matrices Ac , Bc , Cc by the steepest descent [19, 21]. We represent elements of variable matrices in the form of a vector p with elements p1 , p2 , ..., p210 , here the total number of variable parameters is 210.

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Let us consider the general formulation of parametric optimization problem for the dynamical system. Let a dynamic system be described by a system of the following differential equations:    x˙ = F(t, x , p),  x (0) =  x0 .

(31.6)

where t is time,  x is a vector of phase states, p is a vector of parameters. The function  x, p) is determined and continuous along with in the right part of the equation F(t,   x , p). Let the quality its partial derivatives ∂∂ Fx , ∂∂Fp with respect to all variables (t,  functional be given on the trajectories of the system:

I ( p) =

T  N 0

( g (t,  xi , p) dt → min,

i=1

and problem of its minimization is set. It’s shown that based on representations for the varied functional, the gradient with respect to the parameter pk can be represented as ∂ I ( p) = ∂ pk

 T  N   xi , p) ∂ g (t,  xi , p) T ∂ F(t,  − i dt, + ∂ pk ∂ pk i=1 0

where i , i = 1..N is an auxiliary vector functions satisfying along the entire system trajectories i (31.6) an equation with the condition on the right end   ∂ F(t, xi , p) ∂ g (t,  xi , p) + , ∂ xi ∂ xi iT (T ) = 0,

˙i = −

k = 1, 210 is the parameter number.

31.3.1.1

Optimization Approach

The system (31.4) can be represented in the following, assuming the initial condition is given.     x˙ F(x, Cc z, t), = , z˙ Ac z + Bc (G(x) + ϕ(t)) In order to make a link between the common case and the particular one considered in this paper, we suppose that

31 Optimization of a Real-Time Stabilization System …

 x=

  x , z

  F(t, x , p) =



287

 F(x, Cc z, t) , Ac z + Bc (G(x) + ϕ(t))

 g (t,  x , p) = x T Rx + z T CcT QCc z. Then the following statements are valid for the problem solved in this paper: ⎛ ⎞ ∂ F(t, x, p) ∂ F(t, x, p)  ∂ F(t,  x , p) ⎜ ⎟ ∂x ∂z =⎝ ⎠, ∂G(x) ∂ x Ac Bc ∂x ⎛ ∂ F(t, x, p) ⎞   ∂ F(t, x , p) ⎜ ⎟ ∂ pk = ⎝∂ A ⎠, ∂ Bc c ∂ pk z+ G(x) ∂ pk ∂ pk ∂ g (t,  x , p) = 2R T x T + 2CcT QCc z T , ∂x ∂ g (t,  x , p) ∂CcT ∂Cc = 2x T Q x. ∂ pk ∂ pk ∂ pk Then gradient components takes the form ⎞ ∂ F(t, xi , p) ⎞ T ∂ I ( p) ∂Cc ⎟ ⎜ ⎟ ∂ pk T ⎜ T ∂C c = Q xi ⎠ dt, ⎝− i ⎝ ∂ Ac ⎠ + 2xi ∂ B c ∂ pk ∂ pk ∂ pk z + G(x ) i=1 i 0 ∂ pk ∂ pk (31.7) where auxiliary vector functions i satisfy the equation: T  N

⎛

⎛

⎛ ∂ F(t, x , p) ∂ F(t, x , p) ⎞ i i ⎟ ⎜ T T T T ∂ x ∂z i ˙ i = − ⎝ ⎠ + 2R xi + 2Cc QCc z , ∂G(xi ) Ac Bc ∂ xi (31.8) iT (T ) = 0. Expressions (31.7) and (31.8) allow determining the functional gradient components (31.5) numerically, similar to [22, 23], and program their calculation in the MATLAB system. Further, the steepest descent for each parameter is determined by the standard expression.

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31.4 Simulation Results Next, we consider the results of numerical simulation of system (31.4) with regulators before and after optimization. Initial controller was obtained by well know LQG approach applied to the linearized model. Further, to test the optimization of the transient processes, the worst case in the direction of the most unstable eigenvalue is considered. To check the optimization of an individual transient process, an initial deviation of x0 from the equilibrium position is specified, at which the deviations of the observation vector from the equilibrium position at the initial time are such that y0 = y(0) = (0.0009, −0.0005, −0.0007, 1.45, −0.58, 1.16, −0.87)T . Measurement noise, as shown in [8], is in the range |ϕ(t)| =≤ 5 · 10−6 m. Figures 31.2 and 31.3 show a comparison of trajectories beams for current in coils and shift in the lower gap values from the nominal position with a random

Fig. 31.2 The ensemble of trajectories (|yi | ≤ 1 · 10−3 m) for electromagnets z 1 , z 2 , z 3 , z 4 vertical positions values with a the initial b optimized controller

Fig. 31.3 An ensemble of trajectories (|y j | ≤ 1.5A) for the control values of the currents I1 , I2 , I3 , I4 with a the initial b optimized controller

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initial deviation in the interval from 0 to 0.5 s for the initial and optimized regulators. From the graphs, we can conclude about an increase in the rate of transition to the equilibrium position.

31.5 Conclusions The problem of optimal stabilization of the magnetic levitation position of the transport platform in the nonlinear formulation is solved. The control system is designed to work in real time with MIMO feedback, since it has several inputs and outputs which are processed simultaneously. To operate in the presence of measurement noise and eddy currents, and unexpected platform mass drops, the dynamic controller includes a Kalman filter. The all number of controller components are taken as tunable parameters. By varying these parameters, integral quality criterion is minimized which is given on the trajectories of the original nonlinear equations of closed object. The first variation of the functional includes nonlinear representation. The presented approach effectively extends the existing known methods. It is demonstrated that both control accuracy and reduced energy costs have been achieved. Acknowledgements Anna Golovkina and Sergey Zavadskiy acknowledge Saint Petersburg State University and JTI for organization of rearch visit to Nagoya University in March 2020 (projects ID: 52673807, 52673749). It brought fruitful discussions and promoted further collaborative research with Prof. Noboru Sakamoto on this topic. Anna Golovkina also acknowledges Russian Science Foundation for grant (project No. 19-7100074).

References 1. Amoskov, V.M.: Global computational models for analysis of electromagnetic transients to support ITER tokamak design and optimization. Fusion Eng. Des. 87, 1519–1532 (2012) 2. Amoskov, V.M., et al.: Computation technology based on KOMPOT and KLONDIKE codes for magnetostatic simulations in tokamaks. Plasma Devices Oper. 16(2), 89–103 (2008). https:// doi.org/10.1080/10519990802018023 3. Amoskov, V.M., et al.: Simulation of electrodynamic suspension systems for levitating vehicles. III. continuous track systems. In: Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes, (4), pp. 4–21 (2015) 4. Amoskov, V.M., et al.: Simulation of electrodynamic suspension systems for levitating vehicles. iv. discrete track systems. In: Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes, vol. 3, pp. 4–17 (2016) 5. Amoskov, V.M., et al.: Modeling EMS maglev systems to develop control algorithms. Cybern. Phys. 7(1), 11–17 (2018) 6. Amoskov, V.M., et al.: Simulation of maglev eds performance with detailed numerical models. In: Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes, (4), pp. 286–301 (2018)

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7. Amoskov, V.M., et al.: Magnetic model MMTC-2.2 of ITER tokamak complex. In: Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes, vol. 15(1) (2019). https://doi.org/10.21638/11702/spbu10.2019.101 8. Andreev, E.N., et al.: Combined electromagnetic suspensions with reduced energy consumption for levitation vehicles. Tech. Phys. 64(7), 1060–1065 (2019). https://doi.org/10.1134/ s1063784219070041 9. Long, Z., Wang, Z., Cheng, H., Li, X.: A novel design of electromagnetic levitation system for high-speed maglev train. Transp. Syst. Technol. 4(3 suppl. 1), 212–224 (2018). https://doi.org/ 10.17816/transsyst201843s1212-224 10. Nagurka, M.L., Wang, S.K.: A superconducting maglev vehicle/guideway system with preview control: Part i–vehicle, guideway, and magnet modeling. J. Dyn. Syst. Meas. Control 119(4), 638–643 (1997). https://doi.org/10.1115/1.2802372 11. Nise, N.S.: Control System Engineering, 6th edn. Wiley (2011) 12. Nuchkrua, T., Parnichkun, M.: Identification and optimal control of quadrotor. Sci. Technol. Asia 17(4), 36–53 (2012) 13. Ovsyannikov, D., Zavadskiy, S.: Optimization approach to the synthesis of plasma stabilization system in tokamak ITER. Probl. Atom. Sci. Technol. 116(4), 102–105 (2018) 14. Ovsyannikov, D., Zavadskiy, S.: Pareto-optimal choice of controller dimension for plasma stabilization system. IFAC-PapersOnLine 51(32), 175 – 178 (2018). 17th IFAC Workshop on Control Applications of Optimization CAO 2018 15. Sarhang, A., Mohammadi, A.: A new method for quadrotor attitude estimation based on kalmanbucy filtering. Int. J. Mechatron. Electr. Comput. Technol. 170–186 (2014) 16. Talagaev, Y.: State estimation, robust properties and stabilization of positive linear systems with superstability constraints. Cybern. Phys. 6(1), 32–39 (2017) 17. Wang, S.K., Nagurka, M.L.: A superconducting maglev vehicle/guideway system with preview control: Part II–controller design and system behavior. J. Dyn. Syst. Measur. Control 119(4), 644–649 (1997). https://doi.org/10.1115/1.2802373 18. Wang, Z.L., Xu, Y.L., Li, G.Q., Yang, Y.B., Chen, S.W., Zhang, X.L.: Modelling and validation of coupled high-speed maglev train-and-viaduct systems considering support flexibility. Veh. Syst. Dyn. 57(2), 161–191 (2018). https://doi.org/10.1080/00423114.2018.1450517 19. Zavadskiy, S., Lepikhin, T.: Dynamics characteristics optimization for the UAV ensemble of motions. In: Convergent Cognitive Information Technologies, pp. 175–186. Springer International Publishing (2020). https://doi.org/10.1007/978-3-030-37436-5_16 20. Zavadskiy, S., Sharovatova, D.: Improvement of quadrocopter command performance system. In: 2015 International Conference "Stability and Control Processes" in Memory of V.I. Zubov (SCP). IEEE (2015). https://doi.org/10.1109/scp.2015.7342220 21. Zavadskiy, S.V.: Concurrent optimization of plasma shape and vertical position controllers for ITER tokamak. In: 2014 20th International Workshop on Beam Dynamics and Optimization (BDO). IEEE (2014). https://doi.org/10.1109/bdo.2014.6890102 22. Zavadsky, S., Ovsyannikov, A., Sakamoto, N.: Parametric Optimization for Tokamak Plasma Control System. In: World Scientific Series on Nonlinear Science Series B, pp. 353–358. World Scientific (2010). https://doi.org/10.1142/9789814313155_0054 23. Zavadsky, S.V., Ovsyannikov, D.A., Chung, S.L.: Parametric optimization methods for the Tokamak Plasma control problem. Int. J. Mod. Phys. A 24(05), 1040–1047 (2009). https://doi. org/10.1142/s0217751x09044486

Part V

Informatics and Control Processes

Chapter 32

Application of Quasidifferential Calculus to Solve Optimal Control Problems with a Nonsmooth Functional Alexander Fominyh

Abstract The paper considers the problem of optimal control of an object described by a system with a continuously differentiable right-hand side and a nondifferentiable (but only quasidifferentiable) quality functional. We consider a problem in the form of Mayer with both a free and a fixed right end. Admissible controls are piecewise continuous and bounded vector functions, which belong to a certain polyhedron at each moment of time. Standard discretization of the initial system and control parameterization is performed, and theorems on the convergence of the solution of the obtained discrete system to the desired solution of the continuous problem are presented. Further, for the obtained discrete system, the necessary and, in some cases, sufficient minimum conditions are written in terms of quasidifferential. The quasidifferential descent method is applied to this problem. The developed algorithm is demonstrated by examples.

32.1 Introduction Despite the rich arsenal of methods accumulated over the 60-year history of the development of the theory of optimal control, most of them deal with classical systems, the right-hand sides of which are continuously differentiable functions with respect to their arguments, as well as with the objective functional, which is also assumed to be smooth. This paper is aimed at solving optimal control problems in the form of Mayer with both a free and a fixed right end with a nondifferentiable, but only a quasidifferentiable (see definition below) objective functional. The relevance of considering such functionals is due to the fact that in many cases it is natural to minimize nonsmooth functionals in optimal control problems. To solve the problem posed in this paper, we will use a combination of control parameterization technique (see, e.g. [15]), as well as the gradient method for optimal control problems (see, e.g. [11]), and quasidifferential calculus [3, 4]). A. Fominyh (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_32

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Briefly, note that, many theoretical results are obtained regarding the considered and similar problems (see, e.g. [6, 8, 10, 13, 17]), but application of their rather complicated conditions of a minimum to the specific control problems with a nonsmooth quality functional seems difficult. Also, very briefly listed are some works devoted directly to the construction of numerical methods for the problems considered in the paper and similar problems: [7, 9, 12, 14].

32.2 Statement of the Problem Consider the system of ordinary differential equations x(t) ˙ = f (x(t), u(t), t)

(32.1)

x(0) = x0 .

(32.2)

with the initial condition In formula (32.1) f (x, u, t), t ∈ [0, T ], is the given n-dimensional vector function, T > 0 is the given finite moment of time, x(t) is a n-dimensional continuous vectorfunction of the phase coordinates with a piecewise continuous and bounded on [0, T ] derivative, u(t) is a m-dimensional vector-function of controls which is assumed to be piecewise continuous and bounded on [0, T ]. The vector-function f (x, u, t) and and ∂ f (x,u,t) are continuous in (x, u) at each moment its partial derivatives ∂ f (x,u,t) ∂u ∂x t ∈ [0, T ], piecewise continuous in t ∈ [0, T ] and bounded for each point (x, u). We also impose the standard growth condition on the right-hand side of the system: for each compact subset K ⊂ R m , there exists a number C, such that for all x ∈ R n , u ∈ K , and t ∈ [0, T ], we have || f (x, u, t)|| Rn ≤ C(1 + ||x|| Rn ) (see also Remark 32.3 below). In formula (32.2), x0 ∈ R n is the given vector. In this paper, we will use the following notation. Cn [0, T ] is the space of n-dimensional continuous on [0, T ] vector-functions with derivatives from the space Pn [0, T ]; Pn [0, T ] is the space of piecewise continuous and bounded on [0, T ] ndimensional vector functions. Let a, b denote the scalar product of the vectors a, b from the corresponding space. Also, denote co P the convex hull of the set P. Recall the definition of a quasidifferentiable function and of a quasidifferential. Consider some nonempty set  ⊂ R n . The function ξ :  → R is called quasidifferentiable on the set  if for each ς ∈  there exist convex compact sets, the subdifferential ∂ξ(ς ) ⊂ R n and the superdifferential ∂ξ(ς ) ⊂ R n , such that for each allowable increment ς ∈ R n (i.e., co{ς, ς + ς } ∈ ) the corresponding increment of the function ξ is given by the formula ξ(ς + ς ) = ξ(ς ) + max σ1 , ς  + min σ2 , ς  + o(ς, ς ), σ1 ∈∂ξ(ς)

σ2 ∈∂ξ(ς)

where o(ας, ς )/α → 0, if α → 0.

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The pair Dξ(ς ) = [∂ξ(ς ), ∂ξ(ς )] is called the quasidifferential of the function ξ at the point ς . Introduce the set of admissible controls   U = u ∈ Pm [0, T ] | u i ≤ u i (t) ≤ u i , i = 1, m, t ∈ [0, T ] .

(32.3)

Here, u i , u i ∈ R, i = 1, m, are the given numbers. For simplicity, we consider only this set of admissible controls. Other restrictions on control may also be considered. Denote x(t, u) the solution of Cauchy problem (32.1), (32.2) for some given control u ∈ U (sometimes we will also write simply x(t) instead of x(t, u) for brevity). Under the assumptions made, such a solution exists and is unique [16]. It is required to find such a control u ∗ ∈ U that minimizes the functional ϕ(u) = ψ(x(T, u)). We suppose the function ψ(x) to be quasidifferentiable in the phase variables. Denote x ∗ (t) the corresponding opimal trajectory x(t, u ∗ ) (it must satisfy initial condition (32.2)). We assume that there exists such a control. Remark 32.1 Instead of controls from the space Pm [0, T ] and corresponding trajectories from the space Cn [0, T ] with derivatives from the space Pn [0, T ], one can consider measurable and bounded a. e. on [0, T ] controls and corresponding absolutely continuous on the interval [0, T ] trajectories with measurable and bounded a. e. on [0, T ] derivatives, respectively. Then, under the assumptions made, the considered optimal control problem has a solution [5]. The choice of the control space, as well as corresponding trajectories, in the paper is explained by the possibility of their practical construction.

32.3 Control Parametrization Following work [15], perform the discretization of the original problem. Divide the time interval [0, T ] into 2 p equal parts t (where p is the given natural number) with the points tk , k = 0, 2 p , t0 = 0, . . . , t2 p = T . Consider a control that takes constant values at each of these intervals. It may be written in the form 2  p

u (t) = p

σ p,k χ Ikp (t),

(32.4)

k=1

where σ p,k ∈ R m , k = 1, 2 p , is the vector of constant controls, and χ Ikp (t) is the p p indicator function of the set Ik = [tk−1 , tk ), k = 0, 2 p − 1, I2 p = [t2 p −1 , t2 p ]. Denote p p,k p U the set of all the vectors σ , k = 1, 2 , which belong to the set U .

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Put σ p = (σ p,1 , . . . , σ p,2 ) and then rewrite original system (32.1) with the new control u p (t) (32.5) x(t) ˙ = f (x(t), σ p , t) := f (x(t), u p (t), t) p

and the same initial condition x(0) = x0 .

(32.6)

Denote x(t, σ p ) the solution of Cauchy problem (32.5), (32.6) for some fixed control σ p ∈ U p . Under the assumptions made, such a solution exists and is unique [16]. It is required to find such a control σ p∗ ∈ U p , which minimizes the functional ϕ(σ p ) = ψ(x(T, σ p )). Denote x p∗ (t) the corresponding opimal trajectory x(t, σ p∗ ) (it must satisfy initial condition (32.6)). We assume that there exists such a control. Thus, the initial problem of constructing the piecewise continuous control u ∗ (t) is reduced to the problem (or the sequence of problems with the numincreasing 2p p∗,k p∗ σ χ ber p) of constructing the piecewise constant control u p∗ (t) = Ik (t) k=1 (i.e., definition the parameter vector σ p∗ ). The control parameterization performed will allow us to calculate the quasidifferential of the minimized functional at each point σ p , and thereby to apply the quasidifferential descent method, thus obtaining a constructive algorithm for solving the original problem. One can justify the constructed control parametrization, that is, prove the convergence of the functional values ϕ(u p∗ ) → ϕ(u ∗ ), when p → ∞. Some results on the convergence of the sequence of controls u p∗ and the corresponding trajectories x(t, u p∗ ) are also known. Let us write the required theorems [15] in a form convenient for this paper. For this, we additionally assume that the function ψ(x) is Lipshitz continuous on any bounded subset of R n . (Since only piecewise constant controls are considered in this problem, the corresponding trajectories will be bounded in the uniform metric [15], therefore, it is enough to consider the function ψ(x) only on bounded subsets of R n .). Theorem 32.1 Under the assumptions made, the following relation is true ϕ(u p∗ ) → ϕ(u ∗ ), when p → ∞. Theorem 32.2 Under the assumptions made, the following statement is true: if u p∗ → u a. e. on the interval [0, T ], then u is the optimal control of the initial problem. Then the following relation is also true lim ||x(t, u p∗ ) − x(t, u)||∞ = 0,

p→∞

and x(t, u) is the optimal trajectory (perhaps not the only) of the initial problem by definition.

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Remark 32.2 In fact, in [15], the function ψ(x) is supposed to satisfy a stronger condition of continuous differentiability, but it is clear that the proofs of the required statements in this paper use only the Lipschitz continuity of the function ψ(x) on bounded subsets of R n . Remark 32.3 From the proofs of the required results in [15], it is also clear that they stay true if we replace the noticed growth condition of the function f (x, u, t) with the assumption of uniform (in controls u ∈ U ) boundedness of the solutions of system (32.1), (32.2) (with the assumption that for each admissible control there exists the unique solution of system (32.1), (32.2)).

32.4 The Necessary Minimum Conditions Theorem 32.3 The functional ϕ(σ p ) is quasidifferentiable, and its quasidifferentail at the point σ p is given by the formula   Dϕ(σ p ) = ∂ϕ(σ p ), ∂ϕ(σ p ) , 

∂ϕ(σ p ) = V = [V1 , . . . , V2 p ] Vk = [H k (T )] v, k = 1, 2 p , v ∈ ∂ψ(x(T, σ p )) , 

∂ϕ(σ p ) = W = [W1 , . . . , W2 p ] Wk = [H k (T )] w, k = 1, 2 p , w ∈ ∂ψ(x(T, σ p )) ,

where the matrix H (t) satisfies the matrix differential equation ∂ f (x(t, σ p,l ), σ p,l , t) ∂ f (x(t, σ p,l ), σ p,l , t) k H (t) + δk,l H˙ k (t) = , ∂x ∂σ p t ∈ [tl−1 , tl ), l = k, 2 p − 1, t ∈ [tl−1 , tl ], l = 2 p , with the initial condition H k (t) = 0, t ∈ [0, tk−1 ). We don’t give here the proof of this theorem for brevity. Let us now return to previously introduced control restriction (32.3). We construct the following penalty function in which this restriction is taken into account. ϕ(σ p ) = ϕ(σ p ) + λ

2p  m  k=1 i=1

2  m  p

max{0, σi

p,k

− ui } +

max{0, u i − σi

p,k

} .

k=1 i=1

Let σ p∗∗ be a point of a local minumum of the function ϕ(σ p ). It is known [1] that the constructed function is a local exact penalty function at the point σ p∗∗ . This

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means that there exists a number λ0 < ∞, such that for all λ ≥ λ0 the point σ p∗∗ will p be also a point of a local minumum of the function ϕ(σ p ) on the space R m × R 2 . Calculating the quasidifferential of the penalty term and using the known [4] minimum condition of a quasidifferentiable functional, we obtain the theorem. Theorem 32.4 For the control σ p∗ ∈ U p to minimize ϕ(σ p ) (wherein the corresponding trajectory x(t, σ p∗ ) satisfies the system of differential equations (32.5) with initial condition (32.6)), it is necessary for the inclusion − ∂ϕ(σ p∗ ) ⊂ ∂ϕ(σ p∗ )

(32.7)

to be satisfied. Remark 32.4 The case of a partly fixed right end may be analogously considered via adding to the functional ϕ(σ p ) an exact penalty term, taking in account the restrictions on the right end.

32.5 The Quasidifferential Descent Method Describe the following quasidifferential descent algorithm for finding stationary p p points of the functional ϕ(σ p ). Fix the arbitrary initial point σ(1) ∈ R m × R 2 . Let p p the point σ(k) ∈ R m × R 2 be already constructed. If minimum condition (32.7) is p satisfied, then the point σ(k) is the stationary point of the functional ϕ(σ p ), and the process terminates. Otherwise, put

p p p σ(k+1) = σ(k) − γ(k) G σ(k) ,

p where the vector-function G σ(k) is the direction of the quasidifferential descent p of the functional ϕ(σ p ) at the point σ(k) . Note that in most practical examples the direction of the quasidifferential descent can be obtained if one finds the deviation of a convex polyhedron from another convex polyhedron [4]. This problem can be effectively solved via known methods, e.g., Malozemov-Demyanov-Mitchell method [2]. The value γ(k) is the solution of the following one-dimensional minimization problem  

p 

p  p p min ϕ σ(k) − γ G σ(k) = ϕ σ(k) − γ(k) G σ(k) . γ ≥0

p 

p p Then ϕ σ(k+1) ≤ ϕ σ(k) . If the sequence σ(k) is finite, its last point is the stationary p point of the functional ϕ(σ p ) by construction. If the sequence σ(k) is infinite, this process may not lead to a stationary point of the functional ϕ(σ p ), as the quasidifferential mapping Dϕ(σ p ) is not continuous in the Hausdorff metric [4].

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Remark 32.5 To guarantee the convergence of the algorithm, it is necessary to use the codifferential descent method [4] instead of the quasidifferential descent method. For brevity, we don’t use the concept of codifferential in this paper (in fact, codifferential calculus is very similar to quasidifferential calculus [4]). However, in practice, it is recommended to use the method of the quasidifferential descent, since it is less computationally expensive than the method of the codifferential descent, and in most practical cases leads to a stationary point of the minimized functional.

32.6 Numerical Examples In this section, examples are given, with the results of the numerical implementation of the algorithm. All calculations are written down to the 4-th decimal place. In all the examples examined, the method of the quasidifferential descent led to a solution. Example 32.1 On the interval [0, 1] consider the system x˙1 = u 1 , x˙2 = 2x2 − x1 + u 2 with the initial condition

x(0) = (0, 0) .

It is required to find such a control u (here we don’t impose any restrictions on it), which minimizes the functional ϕ(u) = ψ(x(1, u)) =   = max (|x2 (1)| − |x1 (1)| + 1)(|x2 (1)| + 2|x1 (1)| + 1), 1 . The obvious solution (perhaps not the only one) is the point u ∗ = (0, 0) , ϕ(u ∗ ) = 1. Let p = 0, then u = u 0 = σ 0 by definition (see (32.4)). Let the initial approximation be the point u (1) = (0, 1) . The corresponding value of the functional: ϕ(u (1) ) = 17.5938. At the 3-rd iteration of the algorithm implementation, the point u ∗ = u (3) = (0, 0.001) was constructed, and the value of the minimized functional at this point ϕ(u ∗ ) = 1.005, which differs from the required value of the functional by a magnitude of the order 5 × 10−3 . Example 32.2 On the interval [0, 1] consider the system x˙1 = u, x˙2 = x12 + u 2 with the initial condition

x(0) = (1, 0) .

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Let the control of this example belong to the set U , in which u = −1, u = 1. It is required to find such a control u ∈ U , which minimizes the functional ϕ(u) = ψ(x(1, u)) = 10|x1 (1) − 0.6481| + |x2 (1)|. Book [15] provides the analytical solution to the problem, equivalent to the considered in this example. The optimal value of the cost functional: ϕ(u ∗ ) = 0.7616. Let p = 1, then by definition (see (32.4)) on the interval [0, 0.5) the control is the number σ 1,1 , and on the interval [0.5, 1] the control is the number σ 1,2 . Let 1 = (−0.5, 0.5) . The corresponding value of the functional: the initial point be σ(1) 1 ϕ(σ(1) ) = 1.0208. During the work of the quasidifferential descent method at the 41,1 1,2 th iteration, the point σ 1∗ = (σ(4) , σ(4) ) = (−0.54, −0.168) was constructed, and the corresponding value of the functional ϕ(σ 1∗ ) = 0.776, which shows that the obtained value of the functional differs from the value obtained using the analytical solution by a magnitude of the order 2 × 10−2 . Note that such an accuracy was obtained by dividing the interval [0, T ] into only two intervals (as p = 1), that is, by approximating analytically obtained continuous control (see [15]) with piecewise continuous control consisting of two “pieces”. With an increase in the discretization parameter p, we will obtain more exact solutions that converge to the optimal solution (see Theorems 32.1, 32.2 and Remark 32.3). Acknowledgements This work was supported by Russian Science Foundation (project no. 21-7100021).

References 1. Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optimization 65(2), 1167–1202 (2015) 2. Demyanov, V.F., Malozemov, V.N.: Introduction to Minimax. Nauka, Moscow (1972).(in Russian) 3. Demyanov, V.F., Nikulina, V.N., Shablinskaya, I.R.: Quasidifferentiable functions in optimal control. Math. Program. Study 29, 160–175 (1986) 4. Demyanov, V.F., Rubinov, A.M.: Foundations of Nonsmooth Analysis and Quasidifferential Calculus. Nauka, Moscow (1990).(in Russian) 5. Filippov, A.F.: On certain questions in the theory of optimal control. J. SIAM Control Ser. A 1, 76–84 (1962) 6. Fominyh, A.V.: Open-Loop control of a plant described by a system with nonsmooth right-hand side. Comput. Math. Math. Phys. 59(10), 1639–1648 (2019) 7. Fominykh, A.V.: Methods of subdifferential and hypodifferential descent in the problem of constructing an integrally constrained program control. Autom. Remote Control 78, 608–617 (2017) 8. Frankowska, H.: The first order necessary conditions for nonsmooth variational and control problems. SIAM J. Control Optim. 22(1), 1–12 (1984) 9. Gorelik, V.A., Tarakanov, A.F.: Penalty method and maximum principle for nonsmooth variable-structure control problems. Cybern. Syst. Anal. 28(3), 432–437 (1992) 10. Ioffe, A.D.: Necessary conditions in nonsmooth optimization. Math. Oper. Res. 9(2), 159–189 (1984)

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11. Kelley, H.J.: Method of gradients. Math. Sci. Eng. 5, 205–254 (1962) 12. Mayne, D.Q., Polak, E.: An exact penalty function algorithm for control problems with state and control constraints. In: 24th IEEE Conference on Decision and Control, pp. 1447–1452 (1985) 13. Mordukhovich, B.: Necessary conditions for optimality in nonsmooth control problems with nonfixed time. Diff. Equ. 25(1), 290–299 (1989) 14. Morzhin, O.V.: On approximation of the subdifferential of the nonsmooth penalty functional in the problems of optimal control. Avtomatika i Telemekhanika 5, 24–34 (2009). (in Russian) 15. Teo, K.L., Goh, C.J., Wong, K.H.: A unified computational approach to optimal control problems. In: Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific and Technical, New York (1991) 16. Vasil’ev, F.P.: Optimization Methods. Factorial Press, Moscow (2002).(in Russian) 17. Vinter, R.B., Cheng, H.: Necessary conditions for optimal control problems with state constraints. Transactions of the American Mathematical Society 350(3), 1181–1204 (1998)

Chapter 33

On the State Estimation of Non-linear Discrete-Time Models: Application to Unmanned Aerial Vehicles Mohamed Boutayeb and Abir Bouaouda

Abstract In this work, we are interested in the state estimation of multivariable nonlinear discrete-time systems. The latter are obtained from the Leap-Frog discretization technique that may be applied to a large class of systems composed of nonlinear differential equations. This type of discretization induces a delay in the state space representation. Therefore, a specific state space estimator without change of coordinates was established with convergence under weak conditions. In the second part of this work, we are interested in the estimation of the position, orientation, and speed of an Unmanned Aerial Vehicles (UAV) equipped with few sensors. From the observability condition, to assure convergence of the proposed state estimator, we show which physical variable should be measured.

33.1 Introduction For the past fifty years, we have witnessed a very rich and varied production of research articles dealing with the problem of state estimating of dynamic systems. This effervescence reflects the importance of the subject and the challenges that it creates both on a theoretical and an application level. Indeed, knowing the state of a system is an essential or even essential step to predict its behavior, make a diagnosis or control it. This wealth of papers comes from the fact that there are several classes of dynamic systems with different properties, therefore, it is difficult to devise an universal method for estimating the state of any system. It is useful to note that the state estimation techniques developed to date are mainly continuous-time approaches because most of the dynamic systems that one wishes to observe are described by nonlinear differential equations (ordinary or partial). Far M. Boutayeb (B) CRAN-CNRS-7039 University of Lorraine - France, Nancy, France e-mail: [email protected] A. Bouaouda CRAN-CNRS-7039 IUT-HP-Longwy University of Lorraine - France, Nancy, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_33

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from being exhaustive, the reader could refer to the following papers [5, 8, 10] and to the references inside. With the development of more and more powerful and less and less expensive computers, computational techniques have developed considerably with the aim of making the behavior of systems more reliable and more efficient. In this work, we develop a computational approach to estimate the state of multivariable nonlinear time-discrete systems. The latter, in most cases, are obtained from the numerical discretization of systems described by differential equations whose theoretical solution turns out to be very difficult if not impossible to establish. It is important to note that the estimation techniques in the continuous time case and in the discrete-time case as well as the analysis and synthesis tools are quite different. In the first part of this article, we propose to use the simple and useful Leap-Frog discretization scheme for a class of nonlinear polynomial systems. The two main advantages of this technique lie in the simplicity of the model obtained, compared to a Runge– Kutta type discretization, for example, and good numerical stability (compared to the explicit Euler method for example). However, perhaps, the drawback is that the technique leads to a delay in the state space equation. In the second part, we develop a specific state estimator to this class of systems without change of coordinates with convergence under weak sufficient conditions. In the last part of this work, we are interested in the estimation of the position, orientation and speed of a UAV equipped with few sensors. Indeed, from the observability condition, to assure convergence of the proposed state estimator, we show which physical variable should be measured.

33.2 The State Estimation of Nonlinear Discrete-Time Models Considering a dynamic system described by a couple of equations of the form: X˙ (t) = f (X (t), U (t))

(33.1)

Y (t) = g(X (t), U (t))

(33.2)

X (t) ∈ Rn , U (t) ∈ Rm , Y (t) ∈ R p represent the state of the system, the inputs and outputs (sensors), respectively, with a given initial values : X (0) and U (0) . f (.) and g(.) are nonlinear functions of appropriate dimensions assumed to be Li pschit z. In this study, we focus on polynomial nonlinearities because they model a large class of physical systems [7]: biological, electrical, mechanical... The writing in discrete time that we will adopt is based on the Leap-Frog discretization of the form (33.3) X k+1 − X k = h f (X k , Uk )

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and X k − X k−1 = h f (X k , Uk )

(33.4)

From (33.3) and (33.4), we deduce X k+1 = X k−1 + 2h f (X k , Uk )

(33.5)

X k = X (kt) represents the state at time kh. h is the discretization step with k = 1, 2,... (33.2) and (33.5) then becomes X k+1 = X k−1 + 2h f (X k , Uk )

(33.6)

Yk = g(X k , Uk )

(33.7)

Initializations in the discrete-time case will be : X 0 = X (0), U0 = U (0), and use (33.3) or (33.4) to generate X 1 before using (33.6). On stability of the numerical scheme. Theorem 33.1 The discrete-time model (33.6)–(33.7) of the system (33.1)–(33.2) is stable and consistent if f (.) is assumed to be locally Lipschitz for all bounded inputs U(t) of class C ∞ . Proof Owing to lack of space here, the proof will not be detailed but may be deduced from standard matrix inequalities. The goal of this theorem is to guarantee the convergence of the state estimator which will be developed in the next section toward the real state X (t) of the system (33.1)–(33.2) at time instant kh. State estimator design. The state estimator that we propose is written as follows: Xˆ k+1 = Xˆ k−1 + 2h f ( Xˆ k , Uk ) + K k (Yk − g( Xˆ k , Uk ))

(33.8)

The gain matrix K k should be designed so that: X k − Xˆ k → 0 when k → ∞ for all Uk ∈ Du and X 0 = Xˆ 0 . In this study, an optimization based approach is performed. The construction of the gain matrix K k is given by k )G kT (G k Pk G kT + Rk )−1 K k = (2h Fk Pk + Pk−1

Pk+1 = Pk−1 + k Mk−1

=

k Mk−1

k Pk−1 (2h Fk

+

kT Mk−1

+ Sk + Q k +

K k Rk K kT

− Kk Gk )

Sk = (2h Fk − K k G k )Pk (2h Fk − K k G k )

(33.9a) (33.9b) (33.9c)

T

(k−1)T k = Pk−2 + Pk−1 (2h Fk−1 − K k−1 G k−1 )T Pk−1

(33.9d) (33.9e)

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with Q k and Rk are arbitrary positive definite matrices fixed by the user. Fk and G k are Jacobian matrices with respect to X k (evaluated at Xˆ k ) of f (.) and g(.), respectively. Proof Construction of the gain matrix is based on minimization of the trace of Pk+1 MinT race(Pk+1 ) T = MinT race(X k+1 − Xˆ k+1 )(X k+1 − Xˆ k+1 )  Kk

(33.10)

where Pk+1 is determined from the mean value of the quadratic state estimation error T Pk+1 = (X k+1 − Xˆ k+1 )(X k+1 − Xˆ k+1 ) 

with

T k = (X k−1 − Xˆ k−1 )(X k − Xˆ k )  Pk−1

(33.11) (33.12)

The symbol S represents the mean value of S. The gain matrix is deduced as follows : ∂(trace Pk+1 ) =0 ∂ Kk ⇐⇒

(33.13)

∂(trace Pk+1 ) k = −2Pk−1 G kT + 2K k Rk − 4h Fk Pk G kT + 2K k G k Pk G kT = 0 ∂ Kk (33.14) k )G kT (G k Pk G kT + Rk )−1 (33.15) ⇐⇒ K k = (2h Fk Pk + Pk−1

The main theorem is stated as follows: Theorem 33.2 Under the sufficient conditions: f (.) is assumed to be locally Lipschitz, U(t) is bounded C ∞ and T 0 < λmin I ≤ Oobs M Oobs M ≤ λmax I

(33.16)

then the state estimator (33.8) with (33.9a)–(33.9e) assure: lim (X (kh) − Xˆ k ) = 0 for all Uk ∈ Du and X (0) = Xˆ 0

k→+∞

X(kh) is the state vector of the original system (33.1) at time instant hk. Oobs M represents the observability matrix defined as : T T T T T ¯ T ¯ T ¯ k+2 · · · F¯kT .. F¯k+M−1 ] G¯ k+M Oobs M = [G¯ kT F¯kT G¯ k+1  Fk Fk+1 G   2h Fk I ; with G¯ k = G k 0 ; F¯k = I 0 I is the identity matrix of appropriate dimension. Convergence analysis. The proof of convergence may be deduced, using some mathematical artifacts, from the techniques as used in [2, 3].

33 On the State Estimation of Non-linear Discrete Time Models …

307

Fig. 33.1 To study the movement of the quadrotor, we use two frames: the inertial reference frame E (X, Y, Z) and the body frame B (x, y, z) linked to the body of the quadrotor [1]

33.3 Application to Unmanned Aerial Vehicles The second part of this work concerns the position, attitude and speed estimation of an UAV by using the state estimator developed above. It is useful to emphasize the importance of this step in the control design in order to make these vehicles autonomous [6, 9, 11]. Without being able to go into details, the dynamic model of a Quadrotor UAV is written as follows: Dynamic modeling of quadrotor. A quadrotor is a multirotor Unmanned Aerial Vehicle (UAV) with four rotors, which are varied in speed to change the altitude and/or angular position of the vehicle [4, 11] (Fig. 33.1). Using Newton–Euler formulation as in [9] and Under some assumptions [6], the dynamics of a quadrotor can be written as ⎧ ¯ θ˙ + dU2 } ˙ y − Iz ) − K f ax ϕ˙ 2 − Jr  ϕ¨ = I1x {θ˙ ψ(I ⎪ ⎪ ⎪ ⎪ 1 2 ˙ ¯ ¨ ˙ ⎪ θ = I y {ϕ˙ ψ(Iz − Ix ) − K f ay θ − Jr ϕ˙ + dU3 } ⎪ ⎪ ⎪ ⎨¨ ψ = I1z {θ˙ ϕ(I ˙ x − I y ) − K f az ψ˙ 2 + K d U4 } 1 ⎪ ˙ ⎪x¨ = m {(cos ϕ sin θ cos ψ + sin ϕsinψ)U1 − K f t x x} ⎪ ⎪ ⎪ 1 ⎪ y¨ = {(cos ϕ sin θ sin ψ − sin ϕ sin ψ)U1 − K f t y y˙ } ⎪ m ⎪ ⎩ z¨ = m1 {(cos ϕ cos θ )U1 − K f t z z˙ } − g

(33.17)

x, y, and z: are the coordinates of the quadrotor in the 3D space with respect to the inertial reference frame E. φ, θ , and ψ: are the roll, pitch, and yaw angles correspondent to the vehicle, also in frame E. m: is the total mass of the body. Ix , I y and Iz : moments of inertia with respect to the x, y, and z axis, respectively. K p : thrust coefficient. K f t x , K f t y andK f t z : Drag coefficients. d: moment arm. kd : aerodrag coefficient. K f ax , K f ay , and K f az : coefficients of aerodynamic friction force about the respective reference frame axes.

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Jr and ωi : represent rotor inertia and speed. U1 , U 2, U3 , and U4 : are the input signals of the system which are written in function of rotors speed: ⎞ ⎡ 2⎤ ⎡ ⎤ ⎛ ω1 Kp Kp Kp Kp U1 ⎢U2 ⎥ ⎜−K p 0 K p 0 ⎟ ⎢ω2 ⎥ ⎟ ⎢ 2⎥ ⎢ ⎥=⎜ (33.18) ⎣U3 ⎦ ⎝ 0 −K p 0 K p ⎠ ⎣ω32 ⎦ 2 U4 K d −K d K d −K d ω4 ¯ = (ω1 − ω2 + ω3 − ω4 ) And :  The state space model for the rotational motion of the quadcopter base on model (33.17) could be written: X˙ (t) = f (X (t), U (t)) where X = [x1 ...x12 ]T is the state vector. And: X = [φ φ˙ θ θ˙ ψ ψ˙ x x˙ y y˙ z z˙ ]T ⎧ x˙1 = x2 ⎪ ⎪ ⎪ ⎪ ⎪ x˙2 = a1 x4 x6 + a2 x22 + a3 x4  + b1 U2 ⎪ ⎪ ⎪ ⎪ ⎪ x˙3 = x4 ⎪ ⎪ ⎪ ⎪ x˙4 = a4 x2 x6 + a5 x42 + a6 x2  + b2 U3 ⎪ ⎪ ⎪ ⎪ ⎪ x˙5 = x6 ⎪ ⎪ ⎪ ⎨x˙ = a x x + a x 2 + b U 6 7 2 4 8 6 3 4 ⎪ x˙7 = x8 ⎪ ⎪ ⎪ ⎪ ⎪ x˙8 = a9 x8 + Um1 (cos x1 sin x3 cos x5 + sin x1 sin x5 ) ⎪ ⎪ ⎪ ⎪ ⎪x˙9 = x10 ⎪ ⎪ ⎪ ⎪x˙10 = a10 x10 + U1 (cos x1 sin x3 sin x5 − sin x1 cos x5 ) ⎪ m ⎪ ⎪ ⎪ ⎪ = x x ˙ ⎪ 11 12 ⎪ ⎩ x˙12 = a11 x12 + cos x1mcos x3 U1 − g ⎧ ⎪ a1 ⎪ ⎪ ⎪ ⎨a4 Such as : ⎪ a7 ⎪ ⎪ ⎪ ⎩b 1

= = = =

I y −Iz , Ix Iz −I x , Iy I x −I y , Iz d , b2 Ix

a2 = a5 = a8 = =

d , Iy

−K f ax , Ix −K f ay , Iy −K f az , Iz

b3 =

a3 = a6 = a9 =

−Jr Ix Jr Iy −K f t x m

, a10 =

−K f t y , m

(33.19)

a11 =

−K f t z m

1 Iz

(33.20) In the following section, we try to write the observability condition into an explicit form. First of all, it is worth to notice that physical variables as position, orientation or speed may be measured in a linear way. It means that Y (t) = G.X (t), where G is an appropriate and known matrix, the number of rows represents the number of sensors used. On the other hand, the Jacobian matrix Fk (of dimension 12x12) may be deduced easily from the state space form with a specific structure:

33 On the State Estimation of Non-linear Discrete Time Models …

⎡

0 1 0 0 0 0 ⎢ 0 F22k 0 F24k 0 F26k ⎢ ⎢ 0 0 0 1 0 0 ⎢ ⎢ 0 F42k 0 F44k 0 F46k ⎢ ⎢ 0 0 0 0 0 1 ⎢ ⎢ 0 F62k 0 F64k 0 F66k Fk = ⎢ ⎢ 0 0 0 0 0 0 ⎢ ⎢ F81k 0 F83k 0 F85k 0 ⎢ ⎢ 0 0 0 0 0 0 ⎢ ⎢ F101k 0 F103k 0 F105k 0 ⎢ ⎣ 0 0 0 0 0 0 0 0 F121k 0 F123k 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 a9 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 a10 0 0 0 0

309

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 a11

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(33.21)

The observability matrix becomes then:  ⎤ G  0  ⎢ ⎥   2h Fk I ⎢ ⎥ G0 ⎢ ⎥ I 0 ⎥ =⎢ ⎢ ⎥ ···  ⎢  ⎥  ⎣  2h Fk+M−1 I 2h Fk I ⎦ G0 ··· I 0 I 0 ⎡

Oobs M



(33.22)

Therefore from this specific structure of Oobs M , we may deduce how to determine sensors (i.e., the matrix G) which will be used so that the observability condition in the theorem is satisfied.

33.4 Conclusion In this work, we are interested in the state estimation of a class of nonlinear discretetime systems obtained from the Leap-Frog discretization. This type of discretization concerns a large class of nonlinear continuous time models, however, generating a delay in the state space equation. A specific state estimator was, therefore, established with an application to UAVs. In a future work, we will be interested in the development of a reduced order state estimator which could increase the basin of attraction even guarantee a global convergence (under certain conditions).

References 1. Alderete, T.S.: Simulator Aero Model Implementation. NASA Ames Research Center, Moffett Field, CA (1995)

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2. Boutayeb, M., Aubry, D.: A strong tracking Extended Kalman observer for nonlinear discretetime systems. IEEE Trans. Autom. Control 44(8), 1550–1556 (1999) 3. Boutayeb, M.: Observers design for linear time-delay systems. Syst. Control Lett. 44, 103–109 (2001) 4. Carrillo, L.R.G., López, A.E.D., Lozano, R., Pégard, C.: Modeling the quad-rotor minirotorcraft. In: Quad Rotorcraft Control, pp. 23–34. Springer (2013) 5. Ghattassi, M., Boutayeb, M., Roche, J.R.: Reduced order observer of finite dimensional radiative conductive heat transfer systems. SIAM J. Control Optim. 56(4), 2485–2512 (2018) 6. Habib, M.K., Abdelaal, W.G.A., Saad, M.S. et al.: Dynamic modeling and control of a quadrotor using linear and nonlinear approaches (2014) 7. Khalil, H.K.: Non-Linear Systems, 3rd edn. Prentice HAll (2002) 8. Khalil, H.K., Praly, L.: High Gain observers in nonlinear feedback control. Int. J. Robust Noninear Control 24-6, 993–1015 (2014) 9. Loianno, G., Brunner, C., McGrath, G., Kumar, V.: Estimation, control, and planning for aggressive flight with a small quadrotor with a single camera and IMU. IEEE Robot. Autom. Lett. 2(2), 404–411 (2016) 10. NøRgaard, M., Poulsen, N.K., Ravn, O.: New developments in state estimation for nonlinear systems. Automatica 36–11, 1627–1638 (2000) 11. Thorat, S.R.: Quadcopter flight control using modular spiking neural networks. Ph.D. thesis, Indian Institute of Technology, Bombay, Mumbai (2015)

Chapter 34

Marine Vehicles’ Automatic Control Based on Optimal Damping Concept Evgeny I. Veremey and Margarita V. Sotnikova

Abstract This paper is devoted to the problems of feedback control laws design based on the optimization approach for marine vehicles. To provide desirable stability and performance features of nonlinear and nonautonomous closed-loop connections, it is proposed to construct design procedures using the optimal damping concept firstly developed by V.I. Zubov. Central attention is focused on the questions connected with practical implementation of the optimal damping methods for marine control systems. As an example, tracking problem is considered based on optimal damping control laws.

34.1 Introduction Modern marine vehicles of various classes operate in a range of increasing marine traffic due to the intensive development of world economy. This generates many problems connected with a safety of the motion. These problems seem to be essentially significant for further development of marine transportation services with considerable increasing of theirs efficiency and reliability. To guarantee a safety and efficiency sailing, the highest quality of automatic control systems operation is in urgent demand. As for marine vehicles, many scientific publications are devoted to the control laws design, taking into account specific requirements for theirs operational regimes [1, 2, 5]. The optimization approach can be treated now as the most effective analytical and numerical tool for the mentioned control laws design to provide desirable dynamical features of the closed-loop connections. Correspondent methods are widely presented in numerous works, particularly in [1–3, 5, 10–12]. E. I. Veremey · M. V. Sotnikova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] E. I. Veremey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_34

311

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The paper addresses to the theory of transient processes optimal damping (OD). This theory, firstly proposed and developed by Zubov [10–12], gives effective analytical and numerical methods of control laws synthesis with essentially reduced computational consumptions.

34.2 Control Problems for Marine Vessels’ Motion First, let us introduce a commonly used nonlinear model of the control plant obtained on the base of Lagrange equations of the second kind, which represent marine vessel motion for various regimes of its operation [1, 2, 5] M˙v + C(v)v + D(v)v + g(η) = Gu τ + d, η˙ = J(η)v.

(34.1)

Here, the vector v ∈ E n presents vessel’s velocities defined in a plant-fixed frame; the vector η ∈ E n contains position dynamical parameters (displacements and angles) in an earth-fixed frame. External disturbances and controls are presented by the vectors d ∈ E n and τ ∈ E m , respectively. Let us accept that the inertia matrix is positive definite: M = MT > 0, the matrix of Coriolis-centripetal terms is skew-symmetrical: C(v) = −CT (v), and the damping matrix D(v) > 0 is positive definite but non-symmetrical. The vector g(η) represents gravitational and buoyancy forces and moments. At last, here J(η) is the matrix of rotations. Following [2], let us transform the body-fixed frame representation (34.1) to the Earth-fixed one with respect to the vector η. Taking into account the following notation: τη := J−T (η)Gu τ, dη := J−T (η)d, Mη (η) := J−T (η)MJ−1 (η),   −1 ˙ Cη (v, η) := J−T (η) C(v) − MJ−1 (η)J(η), J (η),

(34.2)

Dη (v, η) := J−T (η)D(v)J−1 (η), gη (η) := J−T (η)g(η), ˙ we obtain Mη (η)η¨ + Cη (v, η)η˙ + Dη (v, η)η˙ + gη (η) = τη + dη , v = J−1 (η)η. Since the matrix Mη (η) is non-singular, the last equation can be presented as follows:   ˙ η¨ = −Mη−1 (η) (Cη (v, η)η˙ + Dη (v, η)η˙ + gη (η) − τη − dη , v = J−1 (η)η. (34.3) Automatic control design procedures are based on certain practical problems posed for marine vehicles presented by equations (34.1) ÷ (34.3) of controlled plants. As results of these problems solution, the correspondent feedback control laws could be accepted for theirs practical implementation.

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One can extract the three most significant practical problems, which take a central place in the area of guidance and control of marine vehicles. 1. Stabilization problem. It is supposed that the plant (34.1) is governed by an action of the feedback controller of the form τ = τ (η, v),

(34.4)

providing zero equilibrium position η = η0 ≡ 0, v = v0 ≡ 0 for the undisturbed (d(t) ≡ 0) closed-loop system (34.1), (34.4). The essence of the stabilization problem is the guaranteeing of desirable stability features for the mentioned system. Most often, it is required that the equilibrium point 0 must be globally asymptotically stable (GAS). Nevertheless, in some cases, this requirement can be weakened until the local asymptotic stability or can be enhanced until the exponential stability (GES). 2. Tracking control problem. This variant guesses the existence of the given presentation η = ηd (t) for the reference vessel’s motion. Herein, the state feedback τ = τ (η, v, ηd (t))

(34.5)

should be implemented to provide the zero equilibrium with respect to the tracking error e(t) := η(t) − ηd (t) for the closed-loop system. Naturally, this equilibrium point must be asymptotically stable to guarantee that e(t) → 0 as t → ∞. Let us especially notice that the mentioned closed-loop system is nonautonomous, if we have no constant reference motion ηd (t). This gives reasons for us to require the uniform asymptotic stability in global (UGAS) or local (UAS) form. 3. Dynamic positioning problem. This is a particular case relative to the previous problem, since the reference motion is determined by the constant vector η = ηd here. In other words, the question is to remove a vessel from any starting point (η0 , v0 ) into desirable reference point (ηd , 0). As before, the final position must be asymptotically stable, guaranteeing that e(t) := η(t) − ηd (t) → 0 as t → ∞, and the closed-loop system should have the same stability properties as for the first case. 4. Path following problem. As for this situation, here we have a given presentation for the reference vessel’s motion not as a vector function of time, but as a curve η = ηd (μ) in the space E n : here μ is some scalar real parameter. It is typical for a problem that the distance ρ = ρ(η, ηd ) is calculated for every moment of time from the current vessel’s position to the desirable curve. The feedback control should be constructed by the way such that ρ(t) → 0 as t → ∞, and the closed-loop system should have the same stability properties as for the first case, providing certain stability features for the equilibrium point ρ0 = 0.

34.3 Optimal Damping Concept To decide the mentioned practical problems based on a stabilizing feedback control, one can use various ideas in the range of the widely accepted optimization approach. Here, we propose to use Zubov’s optimal damping concept as the universal tool

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for effective analytical or numerical design of feedback control systems. To this end, first, let us consider nonlinear and nonautonomous mathematical model for controlled marine vehicle presented in the following normal form: x˙ = f(t, x, u)

(34.6)

with the state space vector x ∈ E n , control vector u ∈ E m , and continuous function f : [t0 , ∞) × E n × E m → E n , f(t, 0, 0) = 0, ∀t ≥ t0 . Together with the controlled plant (34.6), let us introduce a controller u = u(t, x)

(34.7)

to be synthesized as a stabilizing control law for the equilibrium point x = 0 of the closed-loop connection (34.6), (34.7). In addition, controller (34.7) can be oriented to a decision of some optimization problem to provide desirable dynamic features of the motion. In framework of the OD-theory, a central role is played by initially given Lyapunov function candidate V = V (t, x) and an auxiliary functional t f 0 (τ, x, u)dτ,

L = L(t, x, u) = V (t, x) +

(34.8)

t0

to be damped optimally for the given positive definite function f 0 . A rate of the functional L change along the motions of the plant (34.6) can be presented as follows:  ∂ V (t, x) ∂ V (t, x) d L  + f(t, x, u) + f 0 (t, x, u). = dt (6) ∂t ∂x (34.9) The essence of OD-approach consists of the control generation as a function from the current values of variables t, x in the form W = W (t, x, u) :=

u0 = u0 (t, x) := arg min W (t, x, u), u∈U

(34.10)

where U ⊂ E m is the metric compact set. Based on [3, 8], it can be shown that if the function V = V (t, x) is such that α1 ( x ) ≤ V (t, x) ≤ α2 ( x ), W (t, x, u0 (t, x)) ≤ −α3 ( x ),

(34.11)

∀x ∈ E n , ∀t ∈ [t0 , ∞), and for some functions α1 , α2 , α3 ∈ K , then the function V is control Lyapunov function (CLF) for the plant (34.6), and the zero equilibrium for the closed-loop system (34.6), (34.10) is UAS. Let us especially note that the function u0 (t, x) could be obtained not only analytically, but also numerically, using the pointwise minimization of the function W (t, x, u) by the choice of u ∈ U for current values of the variables t, x. This opens a way to use OD-controller (34.10) in real-time regimes of a plant motion.

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34.4 OD-Based Tracking Control Problem As an example of the OD-ideology implementation, let us consider the tracking control problem for marine vessels, which consists of the feedback controller τ = τ (η, v, ηd ),

(34.12)

synthesis for the given vector function η = ηd (t). This controller should be constructed by the way such that under condition d(t) ≡ 0 zero equilibrium for the tracking error e(t) := η(t) − ηd (t) has the required stability features for the closedloop system (34.1), (34.12) under the action of an admissible control τ ∈ Tu . Let us formalize this problem in the range of the OD-concept. To this end, initially suppose that the known vector functions ηd (t), vd (t), and τd (t) satisfy equations (34.1) or (34.3), i.e., the following identities are valid: M˙vd (t) + C(vd (t))vd (t) + D(vd (t))vd (t) + g(ηd (t)) ≡ Gu τd (t), η˙ d (t) ≡ J(ηd (t))vd (t).

(34.13)

Introducing the following additional notations    x˜ 1 v −M−1 [C(v) + D(v)]v − M−1 g(η) x˜ := := , , f(˜x) := x˜ 2 J(η)v η  B :=

M−1 Gu , 0

one can transform Eqs. (34.1) and (34.13) as follows if d ≡ 0: x˙˜ = f(˜x) + Bτ, x˙ d ≡ f(xd ) + Bτ.

(34.14)

Then, we can present equations of the vehicle in the deflections    ev v − vd x1 := , u := τ − τd := x := x˜ − xd := x2 e η − ηd

(34.15)

from the desirable motion. Using notations (34.15) on the base of (34.14), we obtain x˙ = α(t, x) + Bu, α(t, x) := α(t, e, ev ) :=

:=

(34.16)

 −M−1 [C(ev + vd (t)) + D(ev + vd (t))](ev + vd (t)) − M−1 g(e + ηd (t)) − J(e + ηd (t))(ev + vd (t))  −M−1 [C(vd (t)) + D(vd (t))]vd (t) − M−1 g(ηd (t)) − , J(ηd (t))vd (t)

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One can easily check that Eqs. (34.16) have zero equilibrium position, which must be stabilized by the control u = u(t, x). If this controller is known, then initial feedback (34.12) can be presented as v − vd (t) . τ = τ (t, v, η) := τ (η, v, ηd ) = u(t, x) + τd (t), x = η − ηd (t) 

(34.17)

Now let us state the OD-problem to be solved for the synthesis the control u, considering the functional L (34.8) to be damped with the function V = V (t, x), satisfying requirements (34.11). It is necessary to find OD-controller (34.10), using an admissible set U ⊂ E m such that ∀u ∈ U : u(t, x) + τd (t) ∈ Tu , ∀t ≥ t0 . It should be taken into account that, we have   d L  d V  W (t, x, u) := = + f 0 (t, x, u) = (34.18) dt (16) dt (16) ∂ V (t, x) ∂ V (t, x) ∂ V (t, x) + α(t, x) + Bu + f 0 (t, x, u). = ∂t ∂x ∂x Using (34.18), one can design controller (34.10) by two possible ways: the first is based on the numerical calculation of the points u = u0 (t, x) providing the pointwise minimization of the function W by the choice of u for current point (t, x). The second way initially has an analytical nature: if ∀t ≥ t0 , ∀x ∈ Br u0 (t, x) ∈ int U , then with necessity we have  

∂ f 0 (t, x, u) ∂ V (t, x) dW  B + ≡ 0 ⇒ ≡ 0. du u=u0 (t,x) ∂x ∂u u=u0 (t,x)

(34.19)

Using necessary equality (34.19), one can solve the following nonlinear system a(t, x, u) = b(t, x) for any point (t, x) with respect to the vector u, where a(t, x, u) = col[ai (t, x, u)], b(t, x) = col[bi (t, x, u)], i = 1, 2, ..., n,

ai (t, x, u) :=

∂ f 0 (t, x, u) ∂u



, bi (t, x) := −

i

 ∂ V (t, x) B . ∂x i

Based on the reasons presented above, we can make the following statement. Given the affine in control system (34.16) and the function V = V (t, x), satisfying condition (34.11), if there exists a function α3 ∈ K such that ∂ V (t, x) ∂ V (t, x) ∂ V (t, x) + α(t, x) + Bu0 (t, x) ≤ −α3 ( x ), ∀x ∈ Br , ∀t ≥ t0 , ∂t ∂x ∂x

34 Marine Vehicles’ Automatic Control …

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then OD-controller (34.10) locally uniformly asymptotically stabilizes zero equilibrium position for the closed-loop system (34.16), (34.10). The mentioned function is local control Lyapunov function for affine plant (34.16). On this occasion, if we have α1 , α2 , α3 ∈ K ∞ , Br = E 2n , U = E m , then the function V = V (t, x) is global control Lyapunov function, and controller (34.10) is a global uniform asymptotical stabilizer.

34.5 Conclusions The aim of this work is to discuss some vital questions connected with various utilizations of the modern optimization ideology for modeling, analysis and synthesis of automatic marine systems. There are many practical problems to be mathematically formalized in the range of the optimization approach. We extract four of them: stabilization, tracking control, dynamical positioning, and path following. All these problems are widely implemented in practice, and are considered in numerous scientific publications. It is quite suitable to use optimization methods for the solutions of the mentioned variants. This idea is discussed in detail in [6–9], based on multipurpose control concept. Among the various optimization approaches, optimal damping theory deserves particular attention. The main positions of this theory were proposed and developed by V.I. Zubov in the early 60s [10–12]. In modern interpretation, OD-theory is closely connected with control Lyapunov functions ideology [1, 2, 4, 5]. In particular, the essence of this connection is reflected by the various methods, using inverse optimal control principle. Nevertheless, the initial idea was proposed by Zubov essentially more early: he suggested to use Lyapunov constructions for providing stability and performance requirements. In this paper, some efforts are made to apply the optimal damping approach for various marine controllers design. Central attention is paid to the example of such application for the solution of the tracking control problem: both direct numerical and analytical decisions are mentioned. The results of executed investigations presented above can be expanded to take into account robust features of the optimal damping controller, and to take into account transport delays both in input and output of a controlled plant. Acknowledgements This work was supported by the Russian Foundation for Basic Research (RFBR) [research project number 20-07-00531] controlled by the Government of Russian Federation.

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References 1. Do, K., Pan, J.: Control of Ships and Underwater Vehicles. Design for Underactuated and Nonlinear Marine Systems, Springer, London (2009) 2. Fossen, T.I.: Handbook of Marine Craft Hydrodynamics and Motion Control. Wiley, Chichester (2011) 3. Grimble, M.J.: Robust Industrial Control Systems Optimal Design Approach for Polynomial System. Wiley, Chichester (2006) 4. Khalil, H.: Nonlinear Systems. Prentice Hall, Englewood Cliffs (2002) 5. Perez, T.: Ship Motion Control: Course Keeping and Roll Stabilization using Rudder and Fins. Springer, London (2005) 6. Sotnikova, M.V., Veremey, E.I.: Dynamic positioning based on nonlinear MPC. IFAC Papers Online 9(1), 31–36 (2013) 7. Veremey, E.I.: Optimization of filtering correctors for autopilot control laws with special structures. Optim. Control Appl. Met. 37(2), 323–339 (2016) 8. Veremey, E.I.: Separate filtering correction of observer-based marine positioning control laws. Int. J. Control 90(8), 1561–1575 (2017) 9. Veremey, E.I.: Special spectral approach to solutions of SISO LTI H-optimization problems. Int. J. Autom. Comput. 16(1), 112–128 (2019) 10. Zubov, V.I.: Oscillations in Nonlinear and Controlled Systems. Sudpromgiz, Leningrad (1962).(in Russian) 11. Zubov, V.I.: Theory of Optimal Control of Ships and Other Moving Objects. Sudpromgiz, Leningrad (1966).(in Russian) 12. Zubov, V.I.: Theorie de la Commande. Mir, Moscow (1978)

Chapter 35

Multiplication Algorithm for Multivariate Trigonometric Series Levon Babadzanjanz, Irina Pototskaya, Yulia Pupysheva, and Irina Alesova

Abstract Multivariate trigonometric series are used for analytical and numerical solutions of a wide class of applied problems. This mathematical tool is in demand in celestial mechanics, quantum mechanics, electrical engineering, acoustics, optics, and the theory of signal and image processing. Multiplying such series when implementing numerical methods is a complex task that requires a large amount of machine time and memory. The formulas for the coefficients of the product of multivariate series derived in this article, can significantly reduce memory consumption and calculation time. Therefore, the use of these formulas in the software development to solve the mentioned tasks can improve its quality and competitiveness. All the results obtained are presented in the conclusion as final formulas for software implementation.

35.1 Introduction Trigonometric series used in solving applied problems in many areas of knowledge is very effective. This approach is widely utilized in such fields as electrical engineering [1, 10], in the theory of digital signal and image processing [2, 3, 7–9], and in other engineering problems [4]. When constructing analytical theories of planetary motion, it is usually also necessary to decompose functions into multivariate trigonometric series, and calculate the series at some points [5, 6]. In all these cases, it is necessary to add and multiply the series and to collect similar terms. To perform these L. Babadzanjanz · I. Pototskaya (B) · Y. Pupysheva · I. Alesova St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] L. Babadzanjanz e-mail: [email protected] Y. Pupysheva e-mail: [email protected] I. Alesova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_35

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actions, significant machine resources (CPU time and memory) are required. Therefore, the development of analytical tools to reduce these resources is very demanded. In this paper, we propose formulas that allow us to automate such calculations and significantly reduce both their execution time and the necessary RAM.

35.2 Derivation of Formulas for Coefficients of Multivariate Series Product Consider multiplication of the multivariate series and derive formulas for their product coefficients. Let there are two multivariate series of the following type 

f =

i1

g=

i2

 j1

···



 ai1 ,...,ik cos

 k 



j2

i m ϕm

+ bi1 ,...,ik sin

 k 

1

ik

···





 A j1 ,..., j p cos

i m ϕm

,

1



 p

jm ϕm

+ B j1 ,..., j p sin

 p 

1

jp



(35.1)  jm ϕm

,

1

where i m∗ ≤ i m ≤ i m∗ , m = 1, 2, . . . , k, (35.2) jn∗ ≤ jn ≤

jn∗ ,

n = 1, 2, . . . , p,

We will assume that the series are reduced. By the reduced series, one means the series whose coefficients with the first non-zero index are zeros, if this index is negative. In particular, it follows that i 1∗ ≥ 0, j1∗ ≥ 0. Let k ≥ p. Then write down the p-argument series in following form g=

 j1 

···



j2

 

A j1 ,..., jk cos

 k 

 jm ϕm

+ B j1 ,..., jk sin

1

jk



 k 

 jm ϕm

,

1



where A j1 ,..., jk = B j1 ,..., jk = 0 when at least one of the indexes jt (t > p) differs from zero and 

A j1 ,..., j p ,0,...,0 = A j1 ,..., j p , (35.3) 

B j1 ,..., j p ,0,...,0 = B j1 ,..., j p .

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It is advisable to multiply the k-argument series, and after getting the necessary formulas, take into account (35.3). Sometimes vector-indexes i = (i 1 , . . . , i k ), j = ( j1 , . . . , jk ), etc., will be used as well as vector ϕ = (ϕ1 , . . . , ϕk ) and scalar products j · ϕ, i · ϕ. For convenience, we write down the series as f =

+∞ 

+∞ 

i 1 =−∞ i 2 =−∞ +∞ 

+∞ 

g=



[ai cos(i · ϕ) + bi sin(i · ϕ)],

i k =−∞

···

j1 =−∞ j2 =−∞

+∞ 

···

+∞ 





[A j cos( j · ϕ) + B j sin( j · ϕ)],

jk =−∞



where ai = bi = 0, A j = B j = 0 if the components of the vectors i, j do not satisfy the inequalities (35.2). Let us introduce another notation 

+∞ 

=

···

i 1 =−∞

i

+∞ 

,

i k =−∞



=

j

+∞ 

+∞ 

···

j1 =−∞

etc.

jk =−∞

Then fg =

 i





[(ai cos(i · ϕ) + bi sin(i · ϕ))(A j cos( j · ϕ) + B j sin( j · ϕ))] =

j

⎧ 1 ⎨      = [(ai A j + bi B j ) cos((i − j) · ϕ) + (−ai B j + bi A j ) sin((i − j) · ϕ)]+ 2⎩ i

+

j

 i

j

⎫ ⎬ [(ai A j − bi B j ) cos((i + j) · ϕ) + (ai B j + bi A j ) sin((i + j) · ϕ)] . ⎭ 







In the first sum, we will introduce the replacement of the vector-index j ˜ l˜ = (l˜1 , l˜2 , . . . , l˜k ), j = i − l, and in the second sum we will put j = l˜ − i. Then we get

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   1     fg = (ai Ai−l˜ + bi Bi−l˜) + (ai A−i+l˜ − bi B−i+l˜) + cos(l˜ · ϕ) 2 i i l˜

 + sin(l˜ · ϕ)

      (−ai Bi−l˜ + bi Ai−l˜) + (ai B−i+l˜ + bi A−i+l˜) i

(35.4)  .

i

The written down series is not necessarily reduced. Let us introduce the term a reduced vector-index l (35.5) l = (l1 , l2 , . . . , lk ). The components of vector (35.5) are such that the first non-zero component (counting from   the left) is positive. By l , we mean the sum of all possible vectors l. In order to get l instead of  in the expression for product (35.4), it is necessary to collect similar terms. l˜ Taking into account the equality  l˜

=



+

 −l

l(l=0)

expression (35.4) can be rewritten as follows:

   1     fg = (ai Ai−l + bi Bi−l ) + (ai A−i+l − bi B−i+l )+ cos(l · ϕ) 2 l i i        (ai Ai+l + bi Bi+l ) + (ai A−i−l − bi B−i−l ) + +  + sin(l · ϕ)

i

i

      (−ai Bi−l + bi Ai−l ) + (ai B−i+l + bi A−i+l )+ i

+



i 

(ai Bi+l

   − bi Ai+l ) + (−ai B−i−l − bi A−i−l ) 

i

 =

i

=

 (αl cos(l · ϕ) + βl sin(l · ϕ)). l

It can be shown that if the original series are reduced, then the fourth sums in the expressions for αl and βl are equal to zero. Let us write out expressions for αl and βl :

35 Multiplication Algorithm for Multivariate Trigonometric Series

  1      αl = (ai Ai−l + bi Bi−l ) + (ai A−i+l − bi B−i+l )+ 2 i i     (ai Ai+l + bi Bi+l ) , +

323

(35.6)

i

  1      (−ai Bi−l + bi Ai−l ) + (ai B−i+l + bi A−i+l )+ βl = 2 i i     (ai Bi+l − bi Ai+l ) . +

(35.7)

i

Now it is possible to set limits in the sums (35.6), (35.7), (and in (35.7) they will be similar to (35.6)), using (35.2), (35.3). (a) Let us consider the sum    (ai Ai−l + bi Bi−l ). i

The limits in this sum must be such that i m∗ ≤ i m ≤ i m∗ , m = 1, 2, . . . , k; i q − lq = 0,

jn∗ ≤ i n − ln ≤ jn∗ , n = 1, 2, . . . , p; q = p + 1, . . . , k.

Because of this, we can write down max(i n∗ , jn∗ + ln ) ≤ i n ≤ min(i n∗ , jn∗ + ln ), n = 1, 2, . . . , p, (35.8) i q = lq , q = p + 1, . . . , k. (b) Now let us consider    (ai A−i+l − bi B−i+l ). i

Here, it is necessary that i m∗ ≤ i m ≤ i m∗ , m = 1, 2, . . . , k; −i q + lq = 0, Hence, we get

jn∗ ≤ −i n + ln ≤ jn∗ , n = 1, 2, . . . , p; q = p + 1, . . . , k.

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max(i n∗ , − jn∗ + ln ) ≤ i n ≤ min(i n∗ , − jn∗ + ln ), n = 1, 2, . . . , p, (35.9) i q = lq , q = p + 1, . . . , k. (c) It remains to consider the sum    (ai Ai+l + bi Bi+l ). i

Similarly to the previous one, we get i m∗ ≤ i m ≤ i m∗ , m = 1, 2, . . . , k; i q + lq = 0,

jn∗ ≤ i n + ln ≤ jn∗ , n = 1, 2, . . . , p; q = p + 1, . . . , k.

From this follows max(i n∗ , jn∗ − ln ) ≤ i n ≤ min(i n∗ , jn∗ − ln ), n = 1, 2, . . . , p, (35.10) i q = −lq , q = p + 1, . . . , k. Let us introduce the notation (here n = 1, 2, . . . , p) max(i n∗ , jn∗ + ln ) = rn1 , max(i n∗ , − jn∗ + ln ) = rn2 , max(i n∗ , jn∗ − ln ) = rn3 , (35.11) min(i n∗ , jn∗ + ln ) = Rn1 , min(i n∗ , − jn∗ + ln ) = Rn2 , min(i n∗ , jn∗ − ln ) = Rn3 , q

R1  q

r1

q

···

Rp 

R  = , q = 1, 2, 3, q

q

(35.12)

rq

rp

ai1 ,i2 ,...,i p ,l p+1 ,...,lk = ail ,

bi1 ,i2 ,...,i p ,l p+1 ,...,lk = bil .



Given that A j1 ,..., j p ,0,...,0 = A j1 ,..., j p = A j , taking into account (35.8)–(35.13), we get

(35.13)



B j1 ,..., j p ,0,...,0 = B j1 ,..., j p = B j and

⎧ R1 R2  1 ⎨ l l αl = (a Ai−l + bi Bi−l ) + (ail A−i+l − bil B−i+l )+ 2⎩ 1 i 2 r

r

R  3

+δ(l)

r3

⎫ ⎬

(ai−l Ai+l + bi−l Bi+l ) , ⎭

35 Multiplication Algorithm for Multivariate Trigonometric Series



where δ(l) =

325

0, l = 0, 1, l = 0.

The introduction of the notation δ(l) is due to the fact that when l = 0, in the expression for αl , the first sum coincides with the third one. Expression for βl is ⎧ R1 R2  δ(l) ⎨ l l βl = (−ai Bi−l + bi Ai−l ) + (ail B−i+l + bil A−i+l )+ 2 ⎩ 1 r r2 ⎫ R3 ⎬  + (ai−l Bi+l − bi−l Ai+l ) . ⎭ 3 r

35.3 Conclusion The article result is as follows. For two multivariate series f and g given in (35.1) as  k  k        f = ··· i m ϕm + bi1 ,...,ik sin i m ϕm ai1 ,...,ik cos , i1

g=

i2

 j1

j2

···

ik





 A j1 ,..., j p cos

1



1



p

jm ϕm

+ B j1 ,..., j p sin

1

jp

 p 

 jm ϕm

,

1

their product can be represented as ⎧ R1 R2  1  ⎨ l l fg = (a Ai−l + bi Bi−l ) + (ail A−i+l − bil B−i+l )+ 2 l ⎩ 1 i 2 r

r

⎫ R3 ⎬  +δ(l) (ai−l Ai+l + bi−l Bi+l ) cos(l · ϕ) + ⎭ 3 r

⎧ R1 R2 ⎨  1 l l + δ(l) (−ai Bi−l + bi Ai−l ) + (ail B−i+l + bil A−i+l )+ ⎩ 1 2 l r r2 ⎫ 3 R ⎬  + (ai−l Bi+l − bi−l Ai+l ) sin(l · ϕ). ⎭ 3 r

(35.14)

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Here A j1 ,..., j p = A j ,

B j1 ,..., j p = B j , ai1 ,i2 ,...,i p ,l p+1 ,...,lk = ail , bi1 ,i2 ,...,i p ,l p+1 ,...,lk = bil ;

i = (i 1 , . . . , i k ), l = (l1 , . . . , lk ), ϕ = (ϕ1 , . . . , ϕk ), j = ( j1 , . . . , jk ); (l · ϕ) – scalar product; q q  R1 Rp Rq    0, l = 0 = ··· , q = 1, 2, 3, δ(l) = , 1, l = 0 q q q r

r1

rp

rn1 = max(i n∗ , jn∗ + ln ), rn2 = max(i n∗ , − jn∗ + ln ), rn3 = max(i n∗ , jn∗ − ln ), Rn1 = min(i n∗ , jn∗ + ln ), Rn2 = min(i n∗ , − jn∗ + ln ), Rn3 = min(i n∗ , jn∗ − ln ), i m∗ , i m∗ − upper and lower values of the index i m , m = 1, 2, . . . , k; jn∗ , jn∗ − upper and lower values of the index jn , n = 1, 2, . . . , p. The components of the vector-index l are defined as follows: 1

for

R 

: ln = i n − jn , n = [1 : p]; lq = i q , q = [ p + 1 : k]; ⇒ Ai−l = A j , Bi−l = B j ;

r1 2

for

R 

: ln = i n + jn , n = [1 : p]; lq = i q , q = [ p + 1 : k]; ⇒ Al−i = A j , Bl−i = B j ;

r2 3

for

R 

: ln = jn − i n , n = [1 : p]; lq = −i q , q = [ p + 1 : k]; ⇒ Ai+l = A j , Bi+l = B j .

r3

These results allow us to automate the calculation process. Now there is no need to multiply multivariate series by each other every time and to collect the similar terms in order to calculate the coefficients of their product. The product can be obtained immediately by programming formula (35.14). Thus, it speeds up corresponding software products and thereby improves their quality and competitiveness.

References 1. Atabekov, G.I.: Theoretical foundations of electrical engineering. Linear Electrical Circuits: Textbook, 7th edn. LAN Publishing House, Saint Petersburg (2009) 2. Baskakov, S.I.: Radio Engineering Circuits and Signals. LENAND, Moscow (2016)

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3. Bazhenov, A.V.: Digital Methods for Implementing Space-Time Signal Processing in Aviation Radio-Electronic Complexes. SVVAIU, Stavropol (2006) 4. Bracewell, R.: The Fourier Transform and Its Applications. McGraw-Hills, New York (1986) 5. Brower, D., Clemens, J.: Methods of Celestial Mechanics. Academic Press, New York (1961) 6. Duboshin, G.N.: Reference Guide to Celestial Mechanics and Astrodynamics. Nauka, Moscow (1976) 7. Gonorovsky, I.S.: Radio Engineering Circuits and Signals. Sovetskoe Radio, Moscow (1977) 8. Honina, S., Baranov, V., Kotlyar, V.: Spectral method of increasing fragments of digital images. Comput. Opt. 19, 165–173 (1999) 9. Rabiner., L.: Theory and Application of Digital Signal Processing. Prentice Hall, Upper Saddle River (1975) 10. Zhezhelenko, I.V.: Higher Harmonics in Power Supply Systems of Industrial Enterprises. Energoatomizdat, Moscow (1984)

Chapter 36

Multipurpose Visual Positioning of the Underactuated Mobile Robot Ruslan Sevostyanov

Abstract The paper is devoted to the problem of the positioning of the mobile robot in front of some external visual marker which is recognized by the video camera mounted on the robot. The task is complicated by the fact that the robot is underactuated. Algorithms which use the visual information directly in the feedback loop are well known as visual servoing approach, but mostly such algorithms consider fully actuated moving plants. Moreover, the considered algorithm uses the mathematical model of the robot’s dynamics to enforce the control quality and also to use the special multipurpose structure of the feedback which allows meeting the set of requirements, particularly—the presence of the external disturbances. The results of the experiments with the computer model are given.

36.1 Introduction Nowadays, a lot of tasks in automated vehicles involve usage of the visual data. Probably the most important examples are driverless cars and mobile robots used in logistics. There are different ways in that the visual data can be used for automatic control. The approach that uses the data from the camera in the feedback loop is known as image-based visual servoing [2]. This approach calculates the desired velocity of the plant, based on the error between the actual and desired position of the set of image points. In real life, there must be another control loop for providing the calculated velocity. And there is also a wide range of regulators that can solve such task in some way. But besides just achieving the desired values somehow there might be some requirements and restrictions to the quality of the dynamics—for example, the control system should provide the minimum deflection of the stabilized parameters in the case of the constant wind acting on the vehicle. In such tasks, one can use the so-called multipurpose regulators [4] that can meet the individual requirements relatively independently which simplifies the process of synthesizing the regulator. R. Sevostyanov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_36

329

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This paper considers the task of positioning of the underacted mobile robot in front of the visual marker based on the visual servoing approach and using the multipurpose regulator.

36.2 Problem Statement Let us consider the mobile robot of the unicycle type which is controlled by setting the voltage of the two motors. The dynamics of the robot can be described by the equations [1] v˙ = Av + Bτ + τe , η˙ = R(η)v,

(36.1)

where v = (ν, ω) is the velocity vector, τ = (τν , τω ) is the input voltage vector, τe is the external disturbance vector, η = (x, y, ϕ) is the kinematic parameters vector, A and B are the matrices of physical parameters of the robot. The only nonlinearity of the system (36.1) is described by the matrix ⎛

⎞ cos(ϕ) 0 R(η) = ⎝ sin(ϕ) 0 ⎠ . 0 1 Let us assume that there is a video camera rigidly mounted to the robot’s geometrical center perpendicularly to the motion plane and directed forward. Somewhere in the scene, there also exists some visual marker described as a set of image points σ = (xi , yi ), i = 1, N which are in fact the projections of the marker points (X i , Yi , Z i ), i = 1, N in the camera coordinate system. The task is to move the robot towards the marker, i.e. to position the robot in front of the marker. This problem can be formulated in terms of image points dynamics: the goal is to minimize the error e = σ − σd between the actual set of marker points in the image plane and the set of the desired marker points projections σd when the robot is in front of the marker. Another requirement is that the control system must take to account the constant external disturbance and have a specific reaction to it. We can complement the dynamics (36.1) of the robot with the error dynamics in the image plane which can be described by the equations: e˙ = L(σ, Z)v + dc ,

(36.2)

where dc is the external disturbance and L is the interconnection matrix with the components for each point (xi , yi ) as follows:  L(xi , yi , Z i ) =

 xi /Z i −(1 + xi2 ) . yi /Z i −xi yi

36 Multipurpose Visual Positioning of the Underactuated Mobile Robot

331

It is also assumed that we can directly measure vectors η, σ and Z i components which are, in fact, just the distances between the camera and the marker points.

36.3 Multipurpose Regulator In order to provide the minimization of the projection error e, let us introduce the feedback of the special structure which is known as the multipurpose regulator and have the form z˙ v = Azv + Bτ + R T (η)K1 (η − zη ), z˙ η = R(η)zv + K2 (η − zη ), z˙ e = L(e, Z)zv + He (e − ze ), p˙ = αp + βη (η − zη ) + βe (e − ze ), ζ = γ p + μη (η − zη ) + μe (e − ze ), τ = −Ke ze − Kv zv + ζ.

(36.3) (36.4) (36.5)

Equations (36.3) represent the state observers of the system (36.1)–(36.2). The first obvious purpose of those is the state estimation in the cases when not all of the state components are available for the direct measurement. Another reason to use them is that the estimation is the input to the dynamic corrector which is described by the equations (36.4). Its role is to regulate the reaction to the external disturbance— for example, constant wind or periodical curvature of the motion surface. Finally, Eq. (36.5) is the control signal which is passed as the plant’s input. It is worth noting that the dynamic corrector can be switched off in real-time, for example, for those regimes where there are no external disturbances. There are a lot of tunable parameters of the controller. Matrices K1 , K2 , and He affects the convergence of the state estimates and also the reaction to the external disturbance (especially the constant one). These matrices must be positive definite and matrices K1 and K2 must also have the diagonal structure. Further search of these matrices is based on the requirements to the dynamics in the corresponding regimes and can be done, for example, through the optimization procedure. Let us discuss the process of searching the matrices Ke and Kv of the control law (36.5). Consider the basic control law without the dynamic corrector and with e and v as a direct input: (36.6) τ = −Ke e − Kv v, and the quadratic form V =

1 1 T e e + v T v. 2 2

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Its derivative with respect to the system is V˙ = −v T Av + v T (LT e + τ ). Therefore, if we take Ke = LT and positive definite matrix Kv , we ensure the stability of the closed-loop system (36.1),(36.2),(36.6). Further search of the matrix Kv depends on the specific requirements to the dynamics of the system. As a final step, we need to find matrices α, βη , βe , γ , μη , and μe of the dynamic corrector. First obvious requirement is that the matrix α must be Hurwitz. For the further reasoning, let us rewrite the dynamic corrector in the tf-form: ζ = F1 (s)(η − zη ) + F2 (s)(e − ze ), where F1 (s) = γ (Es − α)−1 βη + μη , F2 (s) = γ (Es − α)−1 βe + μe . The form of the transfer matrices F1 (s) and F2 (s) depends on the requirements to the dynamics in the corresponding regimes. Considering our problem statement, we need to ensure the astatic property of the control system. Assuming that in the equilibrium position η = η0 and L = L0 and conducting derivations analogous to [3], we can find that the transfer matrices must satisfy the following conditions: F1 (0) = −R T (η0 )K1 , F2 (0) = −Ke + AT + Kv T, T = −(L0T L0 )−1 L0T H.

36.4 Computer Modeling For the experiments, let us use the following particular dynamic model of the robot: v˙ = −v + τ + τe . The visual marker is a square represented by the four corner points. Initial and desired positions of the marker on the image are demonstrated on Fig. 36.1. The tunable parameters of the multipurpose structure have the following values: K1 = 10E3x3 , K2 = 5E3x3 , H = 0.5E8x8 , α = −E2x2 ,   8 0 . βη = F1 (0), βe = F2 (0), γ = E2x2 , μη = μe = 0, Kv = 0 20

36 Multipurpose Visual Positioning of the Underactuated Mobile Robot Fig. 36.1 Marker projections

333

1.5 initial desired

1

0.5

0

-0.5

-1

-1.5 -1.5

Fig. 36.2 Dynamics without disturbances

-1

-0.5

0

0.5

1

1.5

-1

-0.5

0

0.5

1

1.5

-1

-0.5

0

0.5

1

1.5

1.5

1

0.5

0

-0.5

-1

-1.5 -1.5

Fig. 36.3 Disturbance dynamics

1.5

1

0.5

0

-0.5

-1

-1.5 -1.5

First of all, let us consider the situation when there is no external disturbance (τe = 0) and the dynamic corrector is turned off. Figure 36.2 represents the dynamics of the marker projection. It can be seen that the projection smoothly achieves the desired position which means that the robot is actually positioned in front of the marker and the proposed regulator works well.  T Now let us add the constant external disturbance vector τe = −0.01 −0.03 . The dynamic corrector is still turned off. The corresponding dynamics is shown on the Fig. 36.3. We can observe that the projection fails to converge to the desired position, because the feedback can not overcome the action of the disturbance without the dynamic corrector. Finally, let us turn the dynamic corrector on. Figure 36.4 shows that though the dynamics slightly differs from the case without the disturbance, the projection of the marker achieves the desired position despite the action of the disturbance.

334 Fig. 36.4 Dynamics with the corrector

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36.5 Conclusions In this paper, we consider the task of positioning of the mobile robot in front of the visual marker. The offered control system is based on the visual servoing approach, but our method takes into account the dynamics of the robot and a set of requirements to the control quality through the use of the special multipurpose controller. Experiments with the computer model show that the proposed method is an effective and powerful tool even in the case of the underactuated systems.

References 1. Carona, R., Aguiar, A.P., Gaspar, J.: Control of unicycle type robots: tracking, path following and point stabilization. In: Proceedings of IV Jornadas de Engenharia de Electronica e Telecomunicacoes e de Computadores, pp. 180–185 (2008) 2. Chaumette, F., Hutchinson, S.: Visual servo control: basic approaches. IEEE Robot Autom. Mag. 13(4), 82–90 (2006) 3. Veremey, E.I.: Dynamical correction of positioning control laws. In: Proceedings of the 9th IFAC Conferences on Control Applications in Marine Systems, pp. 31–36 (2013) 4. Veremey, E.I., Sotnikova, M.V.: Visual image based dynamical positioning using control laws with multipurpose structure. IFAC Proc. 48(16), 184–189 (2015)

Chapter 37

Adaptive Method for an Actuarial Optimal Control Problem with Dynamic Constraints Alina V. Boiko

Abstract In this paper, we consider the application of a modern approach to solving nonlinear optimal control problems using as an example a relevant problem applied to the Russian insurance market. The actuarial problem is examined as an optimal control problem with dynamic constraints on the control. The approach for solving optimal control problems is based on R. Gabasov’s adaptive method. The linear problem is reduced to an interval linear programming problem and the linear programming problem is solved by the adaptive method.

37.1 Introduction The insurance market is growing and developing every day. There are many different types of insurance, one of the main types being motor insurance. The Russian market features both compulsory and non-compulsory auto insurance, with the former called CMTPL (compulsory motor third-party liability), and as such is regulated by the government. We will consider the actuarial problem of the Russian insurance market, which directly affects the profit of each motor insurance company. This problem can be considered as an optimal control problem with dynamic control constraints. There are many approaches and methods for solving linear and nonlinear optimal control problems. Some of them are fundamental, such as Pontryagin’s maximum principle [8] and Belman’s optimality principle [3]. Besides, there are also many numerical methods based on the classic theory of optimal control [7]. This article discusses the modern approach to solving nonlinear optimal control problems. While featuring the ability to dynamically construct the model in real time, this approach is based on reducing the optimal control problem to the Interval Linear Programming Problem (ILPP). The optimal plan for ILPP is calculated using the adaptive method of Gabasov [1, 6], which has several advantages over the simplex method. A. V. Boiko (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_37

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37.2 Claim Settlement in Insurance Companies In the Russian insurance market, there is a rule of direct compensation for losses for compulsory motor third-party liability, which called the Belgian system [9]. In the event of an insured accident with two participants, in which the fault of one participant is established, the victim can go to his insurance company (for example, company A) and get a compensation for loss. The insurance company must pay compensation to the non-guilty person (company A client), thus paying for another company’s loss (company B). After that, the company A may demand compensation for loss from the company B. But the compensation for company A is not an equal amount of loss. Within one calendar week, all companies send their requirements with losses to some electronic system, after that the market average is calculated and this average fixed value is returned to the account of each insurance company. This process is repeated every week. The main problem of this system is that for companies that pay less than the market average, this system is beneficial, since they receive compensation more than the actually paid loss, but for the rest of the companies, this process brings losses. To solve this problem, the company may revise its sales and claims settlement policy. But for small companies, this is not a solution of the problem. It is possible to predict the average value of each week, but for a short period of time (not more than 5 weeks in advance). We will consider how to find the optimal policy for a company if the market average is known 5 weeks in advance, i.e. how to send the company’s losses in order to get maximum compensation.

37.2.1 Problem Statement The actuarial problem can be considered as an optimal control problem with control parameter u(t) equal to the sum of losses that the insurance company A reports during the week. Note that, for large companies in the market, this system is not problematic since these companies determine the average. We will consider the policy of small companies that do not affect the market as they have a negligible contribution to the average. Each week the insurance company A decides what losses must be paid and sent in to get compensation. Let the initial capital of the company for this type of loss compensation k(t0 ) = k0 be given. At the end of the period, we want to obtain capital more than k1 . Suppose that we know the average for each time interval c(t). In practice, we can forecast the average value for a few weeks in advance with a small rate of error. Yet such a calculation is not considered in detail as we do not deal with forecasting methods and models of fixed average in this article. The financial result of each week is f (t) = n(t)c(t) − u(t), where n(t) is a number of requirements.

37 Adaptive Method for an Actuarial Optimal Control Problem …

337

Since the capital growth consists of financial result f (t) from claims payment and ˙ = n(t)c(t) − u(t) − δk(t), where δ is norm the amount of capital depreciation k(t) of deprecation. Since at every moment losses cannot be paid in excess of capital, we will apply the restrictions 0 ≤ u(t) ≤ k(t). It is necessary to define control u(t) satisfying all restrictions and providing maximum to the amount of paid losses u(t) for all periods. We finally obtain the following system: t1 u(t)dt → max u(t)

t0

˙ = n(t)c(t) − u(t) − δk(t), k(t0 ) = k0 , k(t)

(37.1)

k(t1 ) ≥ k1 , 0 ≤ u(t) ≤ k(t), u ∈ U.

37.2.2 Adaptive Method To find the optimal policy for the actuarial problem, we use a modern approach based on the R. Gabasov’s adaptive method. The adaptive method is used to solve a wide class of optimal control problems (see [2, 4, 5]). We divide the interval [t0 , t1 ] into N parts, N ∈ N. As admissible controls u(t), 0 we use piecewise constant functions with a period h = t1 −t , i.e. N u(t) = u(t0 + (i − 1)h) = u i , t ∈ [t0 + (i − 1)h, t0 + i h), i = 1, N . (37.2) When substituting the general solution for the phase variable into the dynamic constraints, we get that the current control depends on the values of the previous control values 0 ≤ u i ≤ g(t0 , k0 , ti , u 1 , . . . , u i−1 ), i ∈ 2, N , 0 ≤ u 1 ≤ k0 . To reduce the problem (37.1) to ILPP, we take the vector R ∈ RN , which is close to the upper control constraints (less than real values). For linear system (37.1), we obtain following ILPP: max U, V U ≥ W, U

0 ≤ U ≤ R,

(37.3)

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where vk = e

−δ(t1 −t0 )

t 0 +kh

eδ(τ −t0 ) dτ, k = 1, N , V = (v1 , . . . , v N ),

t0 +(k−1)h

W = −k1 − e

−δ(t1 −t0 )



t1 k0 +

 Y −1 c(τ )n(τ )dτ .

t0

Thus, for system (37.3), the optimal plan u ∗ (t) is the optimal control obtained using the adaptive method.

37.2.3 Numerical Experiment Let us consider a numerical application. Suppose that, at the initial moment of time, the capital is 1 million, in 5 weeks, we want to increase the capital to more than 5 million. At the same time, we know the average fixed value and the number of requirements for each week. We consider just such a small interval of time because in practice we know information about the market only for 5 weeks. We take the following parameters and construct optimal control: t0 = 0, t1 = 5, N = 5, c(1) = 74 932, c(2) = 82 546, c(3) = 53 266, c(4) = 274 959, c(5) = 260 634, k0 = 1 ∗ 106 , k1 = 5 ∗ 106 , δ = 9.6 ∗ 10−4 , n(1) = 68, n(2) = 96, n(3) = 142, n(4) = 50, n(5) = 71. It is necessary for the insurance company to determine such a policy in order to maximize the number of claims for which it is necessary to receive compensation. For the numerical example, we developed several programs in the MATLAB environment. The difference of the objective function for the adaptive method and simplex method is 6.68 ∗ 10−12 . Also the working time of the adaptive method algorithm is 0.434 s and the simplex method working time 2.63 s (Fig. 37.1). In the graph (see Fig. 37.2), the objective function, the adaptive method that started from the left border quickly converges to the required control value.

37.3 Conclusion The resulting control vector is the optimal policy for setting requirements. Thus, using a modern approach based on the adaptive method, it was possible to solve the actuarial problem with real data. We developed a software package in the MATLAB

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environment for solving similar problems. The adaptive method was compared with the simplex method. The resulting control obtained using the adaptive method is located on the right boundary of control restrictions. The advantage of the adaptive method is that, for large-dimensional problems, it does not require the construction of a Hamiltonian, or the solution of conjugate systems. Besides, the adaptive method works faster and more precisely than the simplex method, an essential factor in high-dimension cases. Further research will be aimed at improving this approach for nonlinear systems of optimal control.

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References 1. Balashevich, N.V., Gabasov, R., Kirillova, F.M.: Numerical methods of program and positional optimization of linear control systems. J. Comput. Math. Math. Phys. 40(6), 838–859 (2000). (In Russian) 2. Baranov, O.V., Smirnov, N.V., Smirnova, T.E., Zholobov, Y.V.: Design of a quadrocopter with pid-controlled fail-safe algorithm. J. Wirel. Mob. Netw. Ubiquitous Comput. Dependable Appl. 11(2), 23–33 (2020). https://doi.org/10.22667/JOWUA.2020.06.30.023 3. Bellman, R., Kalaba, R.: Dynamic Programming and Modern Control Theory. Academic Press, New York (1965) 4. Boiko, A.V., Smirnov, N.V.: Approach to optimal control in the economic growth model with a nonlinear production function. In: ACM International Conference Proceeding Series, pp. 85–89 (2018) 5. Boiko, A.V., Smirnov, N.V.: On approaches for solving nonlinear optimal control problems. Stud. Comput. Intell. 183–188 (2020). https://doi.org/10.1007/978-3-030-32258-821 6. Gabasov, R., Kirillova, F.M., Kostina, E.A.: The method of constructing local nash equilibrium points in a linear game problems. J. Comput. Math. Math. Phys. 6(38), 912–917 (1998). (In Russian) 7. Intriligator, M.D.: Mathematical Optimization and Economic Theory. Prentice-Hall Inc, Englewood Cliffs, NJ (1971) 8. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience Publishers Wiley, New York, London (1962) 9. Russian association of motor insurers (2020). https//www.autoins.ru/en/osago

Chapter 38

Spectral Design of H2 Optimal Fault Detection Observer Based on Modal Synthesis Yaroslav V. Knyazkin

Abstract The main focus of this paper is design scheme of slowly varying additive fault detection observer–filter synthesis. It is necessary to suppress effect of the external disturbance, consisting of polyharmonical oscillation with given central frequency and a step function. A specialized spectral approach to the filter design, guaranteeing nonuniqueness of the optimal solution, is implemented to satisfy additional conditions, such as integral action. The improved modal synthesis technique, simplifying the design procedure, is proposed and a case study on marine ship longitude motion is given in the end to illustrate the proposed approaches with numerical simulation in MATLAB package.

38.1 Introduction With enhancing requirements of reliability such areas of control engineering as fault detection (FD), fault estimation (FE), and fault-tolerant control (FTC) have received serious attention. Obviously, it is necessary to correct the control law timely, until the failure causes serious consequences, so design of the fault detection observers, guarantying well-timed detection of malfunctions, is a significant issue. Numerous fault detection techniques, existing in our time, can be split to data-based approaches (e.g. PCA [10]), using statistical criteria and model-based ones [2, 3], implementing given mathematical description of the plant and usually including asymptotic observers. Model-based fault detection has been an important field of automatic control theory and control engineering since 1970s (the monograph [3] contains detailed history of this issue and references to the most significant investigations). However, there are some ways to improve effectiveness of the fault detection process, especially in case of the external disturbance with given spectral features (e.g. marine vehicles control processes). The algorithm, presented in this paper, is based on the special Y. V. Knyazkin (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_38

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spectral approach in frequency domain, such that the optimal solution of the problem is no unique, and the parametrization technique, used as base for the modal synthesis scheme. It has been improved, compared to the previous research [7, 8], and does not include such elements as auxiliary observers, that makes possible to construct the relatively low-order observer and reduces computational effort. Also it can be implemented to the MIMO (Multiple Input Multiple Output) plants, in contrast to such the previous results as [6]. The rest of this paper is organized as follows. In the next section, equations of a controlled plant, external disturbance, structure of the observer–filter and problem statement are posed. Section 38.3 is devoted to description of the used spectral approach and parametrization technique. Derivation of the integral action conditions and modal synthesis procedure are presented in Sect. 38.4. Section 38.5 demonstrates numerical example of fault detection observer synthesis. Finally, in Sect. 38.6, we discuss the overall results of the investigation.

38.2 Problem Statement Consider a linear time invariant(LTI) plant x˙ = Ax + Bu + Ef + hd, y = Cx,

(38.1)

where x ∈ R n is the state space vector, u ∈ R nr is the control, d is the scalar external disturbance (note that it is not difficult to extend the proposed ideas to the case of multidimensional one), f ∈ R n f is the slowly varying fault, and y ∈ R m is the measured signal. All components of the matrices A, B, C, E, h are given constants, the pairs {A, B} and {A, C} are controllable and observable, respectively. The external disturbance d for the system (38.1) can be presented as a composition of polyharmonical oscillation and a step function d(t) =

Nd 

Adi sin(σi t + ϕi ) + 1(t)d0 ,

(38.2)

i=1

where Nd is the number of harmonics, Adi , σi , ϕi are their amplitudes, frequencies and phases, respectively, d0 is a constant value, and 1(t) is Heaviside step function. For simplicity, let us assume that the central frequency of d is ω0 and d ≈ Ad0 sin(ω0 t) + 1(t)d0 (note that the detector can be tuned to the several dominant ones in a similar way). Similar to the well-known research, such as [3], the residual signal, indicating the presence of the fault, is generated by the fault detection observer–filter

38 Spectral Design of H2 Optimal Fault Detection Observer …

x˙ˆ = Aˆx + Bu + ν, ν(s) = L(s)(y − Cˆx), r = cr (x − xˆ ),

343

(38.3)

where ν ∈ R n is the corrective signal, r ∈ R 1 is the residual one, cr can be specified as one of the matrix C rows, and L(s) is the transfer matrix. Denote ex = (ˆx − x), ey = Cex , then the error dynamics is given by e˙ x = Aex − ν + Ef + hd, ey = Cex , ν(s) = L(s)ey , r = cr ex .

(38.4)

Let introduce the indices [3], characterizing influence of the signals f, d to the residual signal: J = J1 /J2 , J1 = min Fr f ( jω)2 , J2 = max |Fr d ( jω)|2 , ω∈1

ω∈2

(38.5)

where 1 , 2 are the frequency ranges of the signals f , d; and Fr d = cr (sI − A + L(s)C)−1 h, Fr f = cr (sI − A + L(s)C)−1 E.

(38.6)

Similar to [3], the problem can be formally defined in the following way: J (L) → max , L∈L

(38.7)

where L is set of L(s) , guarantying stability of the closed-loop system (38.6). The second crucial feature of the filter (38.3) is integral action of r relatively to d: Fr d = 0.

(38.8)

Let consider the conjugated plant (38.4), x˙ 1 = A1 x1 + B1 u1 + h1 d1 , e1 = hT x1 , e2 = ET x1 ,

(38.9)

u1 = LT (s)x1 = W(s)x1 ,

(38.10)

where x1 ∈ R n , u1 ∈ R m , A1 = AT , B1 = −CT , h1 = crT , W(s) = W2−1 (s)W1 (s)x1 , d1 ∈ R 1 , is a harmonic with the frequency ω0 . Taking into account the condition (38.8), minimization of the functional J2 (L) can be reduced to the well-known optimization problem [1, 5]. Let introduce the functional

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1 ∞ T {Fx1 (− jω)RFx1 ( jω)+ J2 = jπ 0 +k 2 FuT1 (− jω)QFu1 ( jω)}Sd1 (ω)dω, where

(38.11)

Fx1 (s) = (sI − A1 + B1 W(s))−1 h1 , Fu1 (s) = W(s)(sI − A1 + B1 W(s))−1 h1 , R = hhT , Sd1 (ω) = δ(ω − ω0 ) is Dirac delta function, Q is symmetrical positive definite matrix (e.g. unit one) and k is a small positive value. It is easy to check that hT Fx1 (s) = Fr d (s) and J2 −→ J2 , so minimization of the value J2 can be considered k→0 as J2 −→ min , W∈W

where W is the set of controllers (38.10), such that the characteristic polynomial (s) = (As )1−m det(W1 B1s − W2 As ),

(38.12)

where B1s = As (sI − A1 )−1 B1 , As = det(A1 ) is a Hurwitz one.

38.3 Suppression of the Harmonic Disturbance In accordance to the approach, presented in [1], introduce the function-parameter

(s) = α(s)Fx1 (s) + β(s)Fu1 (s),

(38.13)

where the parameters α(s), β(s) can be calculated by the formulae [1] α(s) = α0 (s) =

1 −1 T Q B1 S, β(s) = β0 = I, k2

(38.14)

where the matrix S is solution of the matrix Riccati equation SA1 + A1T S −

1 SB1 Q−1 B1T S + R = 0, k2

Let express Fx1 (s), Fu1 (s) as functions of the parameter (s) Fx1 (s) = P−1 (s)h1 + B1s (s) −1 (s)( (s) − α0 P−1 (s)h1 ), Fu1 (s) = As (s) −1 (s)( (s) − α0 P−1 (s)h1 ), where P(s) = (sI − A1 ),

(s) = As β0 + α0 B1s , B1δ = D1s (sI − A1 + B1 α0 )−1 B1 , D1s = det(sI − A1 + B1 α0 ),

(38.15)

38 Spectral Design of H2 Optimal Fault Detection Observer …

345

then substitute (38.15) to (38.11), taking into account, that choice of the parameters α(s), β(s) by the formulae (38.14), guarantees the equalities ∗ 2 ∗ −1

−1 = k 2 Q, ∗ (B1s RB1s + k As QAs ) ∗ 2 −1 2 −1/2 T B1 (P1 + B1 α0 )−1 (k 2 Q)−1/2 [ −1 ∗ B1s R − k Qα0 ]P h1 = (k Q) ∗ Sh1 ,

−1 (s) = (As D1s )−1 (D1s I − α0 B1δ ). The integrand in (38.11) can be transformed to the form F0∗ F0 = (T∗1 + ∗ T∗2 )(T1 + T2 ) + T3 , where   k 2 Q, T1 (s) = ( k 2 Q)−1 B1T (P1 + B1 α0 )−1 ∗ Sh1 , T2 (s) = ∗ −1 T3 (s) = h1T P∗−1 RP−1 h1 − h1T P∗−1 RB1s −1 (k 2 Q)−1 −1 ∗ B1s RP h1 .

In connection with the filtering property of the delta function and independence of T3 (s) on (s), the functional J2 accepts its minimal value if and only if T1 ( jω0 ) + T2 ( jω0 ) ( jω0 ) = 0, or

( jω0 ) = −T−1 2 ( jω0 )T1 ( jω0 ).

(38.16)

38.4 Transfer Matrix of the Optimal Observer The most obvious way of the transfer matrix W(s) construction is a modal synthesis procedure, but dependency of the characteristic polynomial (38.12) from W(s) is nonlinear. The solutions, implemented in [7, 8] are not convenient and need extra computing resources. An alternative way, proposed in this paper, is to design the function-parameter (s) (38.13), maximizing the value J (38.5), providing the equalities (38.8), (38.16), and calculate the corresponding controller W(s). The matrix ( jω0 ), minimizing value of the functional J2 has been received in the previous section. The next step is calculation of the matrix (0), providing the desired dynamics of the observer–filter on the zero frequency. Let introduce the notations M1 = (P−1 h1 − B1s −1 α0 P−1 h1 )|s=0 , M2 = B1s −1 |s=0 . Then the matrix (0) is a solution of the following system of linear equations: Fr d (0) = (hT Fx1 (0))T = M1T h + T (0)M2T h = 0, Fr f (0) = (ET Fx1 (0))T = M1T E + T (0)M2T E = f0 ,

(38.17)

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where f0 ∈ Rn f is a vector with nonzero components, which can be parameterized f0 = f0 (γ1 ), γ1 ∈ R n f .

(38.18)

Then, let calculate the elements of the transfer matrix (s):

i (s) = ξi (s)/ (s), ξi (s) = ξi2 s 2 + ξi1 s + ξi0 , i = 1, m, where (s) is a n order Hurwitz characteristic polynomial of (s) . Choice of the parameters ξik , i = 1, m, k = 0, 2 should provide the desired (0), ( jω0 ) and they can be calculated by the formulae ξi0 = i (0) (0), ξi1 = I m( i ( jω0 ) ( jω0 ))/ω0 , ξi2 = −Re( i ( jω0 ) ( jω0 ) − i (0) (0))/ω02 .

(38.19)

The polynomial (s) with the degree of stability αst can be parameterized [9]: 

˜ ∗ (s, γ ), ∗ (s, γ ) = ˜ ∗ (s, γ ), ∗ (s, γ ) = (s + ad+1 ) k d  ∗ (s, γ ) = (s 2 + ai1 (γ , αst ) + ai0 (γ , αst )), kd = [n /2], ∗ (s) =

ai1 (γ , αst )

i=1

γi12 ,

ai0 (γ , αst )

αst2

= 2αst + = + 2 + αst , ad+1 (γ , αst ) = γd0 γ = {γ11 , γ12 , γ21 , γ2 , . . . , γkd1 , γkd2 , γd0 }.

γi12 αst

+

(38.20)

γi22 ,

Let substitute the designed transfer matrix (s) to the formulae (38.16) and receive the transfer matrices Fx1 , Fu1 . The controller (10) W(s), providing them, can be calculated as solution of the following polynomial equation [5]: W1 (s)F˜ x1 (s) − W2 (s)F˜ u1 = 0,

(38.21)

where polynomial matrices F˜ x1 (s) = D1s Fx1 (s), F˜ u1 (s) = D1s Fu1 (s) are numerators of Fx1 (s), Fu1 . Note that order n W of the controller W(s) should be chosen in such a way that that numerators and denominator of the transfer matrices of the closed-loop system (38.9), (38.10) have no common multiplier, e. g. if elements of F˜ x1 (s), F˜ u1 (s) are coprime polynomials with D1s , then n W = n . The matrix f0 and the polynomial (s) have been parametrized (38.18), (38.20), so we can considered the functional J (38.5) as function of the parameters γ , γ1 J = J (γ , γ1 , k, αst ) and solve the maximization problem (38.7), using any numerical method, e. g. the Nelder–Mead one. Based on foregoing, let formulate the algorithm of the optimal observer–filter (38.3) design.

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Algoritm 1. 1. Set the number n and initial parameters γ = γ 0 , γ1 = γ10 . 2. Pparametrize the set of pairs {Fx1 (s, (s)), Fu1 (s, (s))} by the formulae (38.15). 3. Calculate the complex-value matrix ( jω0 ) (38.16). 4. Compute f0 (γ1 ), and = (s, γ ) using formulae (38.18), (38.20). 5. Receive the matrix (0), solving the system of linear equations (38.17). 6. Design the transfer matrix (s), by the formulae (38.19). 7. Substitute (s) to (38.15), calculate Fx1 (s), Fu1 (s), Evaluate the functional J (38.5). 8. Maximize J , repeating the steps 4–7 with new parameters γ , γ1 , searched with any numerical method, e.g. Nelder-Mead algorithm. Receive the optimal parameters γ = γ ∗ , γ1 = γ1∗ and corresponding (s) = 0 (s) . 9. If J (γ ∗ , γ1∗ , k, αst ) is too small then repeat steps 2–8 with decreased k, αst . 10. Substitute numerators of {Fx1 (s, 0 (s)), Fu1 (s, 0 (s))} to (38.21), calculate W(s) = W0 (s) and construct the optimal L(s) = L0 (s) = W0T (s).

38.5 Example of Synthesis Let us demonstrate the practical implementation of the proposed algorithm by the example of marine ship moving with the constant longitudinal speed. Consider the plant (38.1) and disturbance (38.2) with the parameters [4]: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

−0.0936 0.634 0 0.0196 0.041 100 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ A = 0.0480 −0.717 0 , B = E = 0.016 , h = 0.0076 , c = , 001 0 1 0 0 0

d(t) =  sin(ω  0 t) + 0.1 sin(0.9ω0 t) + 0.1 sin(1.1ω0 t), ω0 = 0.45. Let uschoose cr = 0 0 1 , Q = I, k = 0.05, n = 3, αst = 0.2, γ10 = 1, γ 0 = 1 1 1 . Finally, we receive γ1∗ = 0.222, γ1∗ = 1.50 0.86 1.08 ,

1 0.04s 2 + 0.013s + 0.011

0 (s) = 3 , 0.02s + 0.1475 s + 4s 2 + 4.875s + 1.735 L( s) = ⎛

s3

+

6s 2

1 · + 11s + 6

⎞ 2.15s − 6.73 s 2 + 31.1s + 1.12 −606s 3 − 5517s 2 − 478631s − 2712 ⎠. −0.278 −34.73 ·⎝ 0.06s 3 + 0.2 s 2 + 0.272s + 1.05 4.04s 3 − 13 s 2 − 93.9s − 20.74 3

The frequency responses Ar d , Ar f of the transfer functions Fr d , Fr f and fault detection process are shown on Fig. 38.1. One can see that the external disturbance

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Fig. 38.1 Frequency responses of the transfer functions Fr d , Fr f and the fault detection process

effect is almost completely suppressed, the constant component of the external disturbance increases at 100 s., but the residual signal almost does not respond, and the fault, occurred at 250 s. is detected.

38.6 Conclusion In this paper, the improved modal scheme of the fault detection observer design is presented. Despite the previous research, it does not include cumbersome calculations and allows constructing the enough low-order observers. The proposed approach can be implemented to various plants, affected by polyharmonical oscillations. Notice simplicity of this scheme, which can be very useful for on-board real time implementation for autonomous systems with limited computational resources. Non-uniqueness of the solution is the second significant benefit, providing flexibility of the observer tuning. Working capacity and effectiveness of the formulated algorithm are illustrated by the numerical example. Acknowledgements This work was supported by the Russian Foundation for Basic Research (RFBR) [research project number 20-07-00531] controlled by the Government of Russian Federation.

References 1. Aliev, F.A., Larin, V.B., Naumenko, K.I., Suncev, V.I.: Optimizacija linejnyh invariantnyh vo vremeni sistem upravlenija. Naukova dumka, Kiev (1978) 2. Chen, J., Patton, R.J.: Robust Model-Based Fault Diagnosis for Dynamic Systems. Springer Science and Business Media (2012) 3. Ding, S.X.: Model-Based fault diagnosis techniques: design schemes, algorithms and tools. Springer Sci. Bus. Media (2012). https://doi.org/10.1007/978-1-4471-4799-2 4. Veremey, E.I.: Dynamical correction of control laws for marine ships’ accurate steering. J. Mar. Sci. Appl. 13, 127–133 (2014). https://doi.org/10.1007/s11804-014-1250-1

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5. Veremey, E.I.: H2 -Optimal synthesis problem with nonunique solution. Appl. Math. Sci. 10(38), 1891–1905 (2016). https://doi.org/10.12988/ams.2016.63120 6. Veremey, E.I., Knyazkin, Y.V.: SISO problems of H2 -optimal synthesis with allocation of control actions. WSEAS Trans. Syst. Control 12, 199–200 (2017) 7. Veremey, E.I., Knyazkin, Y.V.: Synthesis of fault estimation observer, based on spectral MIMO H2 optimization. Mod. Inf. Technol. IT-Educ. 14(1), 91–100 (2018). https://doi.org/10.25559/ SITITO.14.201801.091-100 8. Veremey, E.I., Knyazkin, Y.V.: Marine ships’ control fault detection based on discrete H2 optimization. WIT Trans. Built Environ. 187, 73–82 (2019). https://doi.org/10.2495/MT190081 9. Veremey, E.I., Smirnov, M.N., Smirnova, M.A.: Synthesis of stabilizing control laws with uncertain disturbances for marine vessels. In: “Stability and Control Processes” in Memory of VI Zubov (SCP) International Conference, pp. 1–3 (2015). https://doi.org/10.1109/SCP.2015. 7342219 10. Zanoli, S.M., et al.: Application of fault detection and isolation techniques on an unmanned surface vehicle (USV). IFAC Proc. Vol. 45(27), 287–292 (2012). https://doi.org/10.3182/ 20120919-3-IT-2046.00049

Chapter 39

Kolmogorov Complexity-Based Similarity Measures to Website Classification Problems: Leveraging Normalized Compression Distance Andrey A. Pechnikov and Anthony M. Nwohiri

Abstract World Wide Web has become the largest source for all kind of information thanks to its connectivity and scalability. With increasing number of web users and websites, the need for website classification becomes necessary. Due to its enormous size, fetching required information is a challenging task. Owing to the generality of topics, most links direct to websites or domains, instead of single webpages. Therefore, this paper proposes a new Kolmogorov complexity-based approach to website classification that leverages normalized compression distance to examine the similarity of websites. The approach is shown to have some prospects. To fully achieve the potentials of the approach, some questions need to be addressed before the approach could be automated for large-scale studies.

39.1 Introduction A web page is classified by assigning it to one or more predefined category labels. It is usually performed based on the content of the site [13]. Websites are similar to each other, especially the sites of organizations engaging in similar kind of activity. However, some sites are more similar to one another than others. There have been attempts to classify sites under different categories [4, 12, 14].

A. A. Pechnikov (B) Institute of Applied Mathematical Research of the Karelian Research Centre, Russian Academy of Sciences, 11 Pushkinskaya Str., Petrozavodsk 185910, Russia e-mail: [email protected] A. M. Nwohiri Department of Computer Sciences, Faculty of Science, University of Lagos, Akoka Rd, Akoka-Yaba, Lagos 101017, Nigeria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_39

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This paper pursues a more general goal. We do not search for similarities in many specific features that would be relevant to website classification. We search for one generalized characteristic reflecting the interactions of various influences as described in the general theory of similarity. This is based on transition from ordinary physical quantities that affect the simulated system to generalized quantities of a complex type that depend on the specific nature of the process under study [5, 6]. Such a generalized characteristic for websites could be the normalized compression distance (NCD) [1, 2, 10]. However, it needs to be carefully studied to see how feasible this would be. This idea, in turn, is based on the theoretical concept of Kolmogorov complexity [7]. Studies have shown that this universal similarity characteristic is suitable for specific cases, such as automatic construction of phylogenetic trees [8] and classification of musical genres [3]. We conducted a series of experiments based on the approach proposed by [11] for a selected set of websites. Many of the actions were performed “manually”, without developing and using appropriate software. This was mainly to identify the potential of NCD in website research and outline the main questions that need to be answered and the tasks that need to be solved for a large-scale study.

39.2 Kolmogorov Complexity and Normalized Compression Distance By a description method, or a decompressor, we call an arbitrary computable partial self-mapping D from set of binary words . The computability of map D means that there is an algorithm applicable to words from the domain of map D and only to those words; the result of applying the algorithm to word x is D(x). If D(y) = x, then y is a description of x by description method D. For each description method D, we determine the relevant complexity, assuming it is equal to the length of the shortest description K S D (x) = min{l(y)|D(y) = x}. Moreover, the minimum of empty set is considered equal to. It is said that description method D1 is no worse than description method D2, if there exists a constant c such that K D D1 (x) ≤ K S D2 (x) + c, for all words x. A description method is called optimal if it is not worse than any other description methods. Now we determine some (not necessarily optimal) description method. The complexity of word x relative to this description method is denoted by K (x). For simplicity, we will further assume it to be equal to the number of bits in the compressed

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version of x. Moreover, as already mentioned, we will from now on use standard archiver programs as map D. Let y be another binary word. Denote by K (x|y) the minimum number of bits needed to restore x from y. For any pair of strings x and y, we can define the normalized compression distance as N C D(x, y) =

max{K (x|y), K (y|x)} . max{K (x), K (y)}

Broadly speaking, two objects are considered similar if we can significantly “squeeze” one of them using the information contained in the other. The idea here is that if two objects are very similar, then we can describe one of them more briefly using the other. A very sophisticated theorem on the symmetry of algorithmic information was proved in [10], resulting in approximate equality K (y) ≈ K (yx) − K (y), where yx denotes concatenation of binary strings y and x. Then, just as in [2], taking into account the fact that, again, in practice, K (x y) ≈ K (yx), for subsequent approximate calculations, N C D can be represented as follows: N C D(x, y) =

K (x y) − min{K (x), K (y)} . max{K (x), K (y)}

(39.1)

It is clear that distance N C D(x, y) is symmetric, and in [9], it was shown that this is really a metric, since the identity and triangle axioms also hold. Moreover, it also shows that the N C D metric is universal in the sense that for each normalized metric f and each x and y, we have d(x, y) ≤ f (x, y) + O( log(k) ), where k = k max{K (x), K (y)}. This means that if x and y are close by any "well-behaved" metric f , then x and y are also close by universal metric N C D. We will be using the N C D(x, y) metric subsequently in this paper.

39.3 Simple Experiment We present the results of one of a series of very simple experiments we conducted. The results are well interpreted in terms of content. Here, the set consists of ten websites known to us. The details are summarized in Table 39.1. The websites were compared based on their home pages. These web pages were saved as .txt files. This is the simplest and most intuitive option, since text file contains a sequence of characters combined into strings, and in this case is closest to a binary file. Therefore, it can be directly interpreted as a binary word. These files were compressed using the RAR 5.50 archiver [16]. The volume of the RAR archive of the i-th site is taken as K (i), where i = 1, ..10. Files were

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Table 39.1 Experimental websites S/N URL 1 2 3

4 5

6 7 8 9 10

Organization’s name

https://efccnigeria.org

The Economic and Financial Crimes Commission https://www.nuc.edu.ng The National Universities Commission https://www.cpn.gov.ng Computer Professionals (Registration Council of Nigeria) http://www.transportation.gov. Federal Ministry of ng Transportation https://www.nafdac.gov.ng National Agency for Food and Drug Administration and Control https://unilag.edu.ng University of Lagos https://www.unn.edu.ng University of Nigeria https://www.unilorin.edu.ng University of Ilorin https://thenationonlineng.net The Nation Newspaper https://www.ekohotels.com Eko Hotels & Suites

concatenated for sites i and j by simply combining the contents of the files using a text editor. The RAR archive of the combined file weighs K (i j). N C D(i, j), which is the distance between sites i and j, is calculated based on formula (39.1). For example, for site 1 (https://efccnigeria.org), the size of the text file of the home page is 55592 bytes, its archive is 10657 bytes, and for site 2 (https://www.nuc.edu. ng), it is 41130 and 7263 bytes, respectively. For site 1 and site 2, K (12) = 15031 bytes. Then, according to formula (1), N C D(1, 2) = N C D(2, 1) = 0.728910575. The numerical values of the elements of the matrix {N C D(i, j)}, i, j = 1, ..10 are shown in Table 39.2. The matrix is symmetric with respect to the main diagonal. Dendrograms constructed by the method as applied to this distance matrix for the nearest neighbor and by Ward’s method [15] are shown in Figs. 39.1 and 39.2. In both cases, sites numbers 1, 2, and 4 (federal government agencies) were clustered together. Three university sites (sites 6, 7, and 8) were clustered together but (for unknown reasons) with hotel site (site 10). However, in the first case, site number 5 (https://www.nafdac.gov.ng) falls into a cluster with the sites of other federal institutions, while in the second case, the site is very far from it. Curiously, some contextual explanation can be partially given for both the first and second cases. The fact is that sites 1, 2, and 4 are the sites of federal government institutions. However, it’s not yet clear why site 3 was clustered with them, since it was a legal entity charged with the control and supervision of the computing profession in Nigeria. Site 5 is the website of the National Agency for Food and Drug Administration and Control—a federal agency under the Nigerian Federal Ministry of Health. Therefore, it is assumed that

39 Kolmogorov Complexity-Based Similarity Measures to Website … Table 39.2 Distance matrix 1 2 3 1 2 3 4 5 6 7 8 9 10

0.976 0 0.976 0.923 0.896 0.94 0.988 0.407 0.317 0.98 0.923

4

5

6

7

8

9

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0.9

0.899

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0.836

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0 0.935 0.931 0.916 0.922 0.977 0.976 0.918 0.931

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0 0.81 0.856 0.957 0.921 0.929 0.948 0.828

0 0.863 0.963 0.896 0.899 0.952 0.836

0 0.951 0.94 0.944 0.941 0.856

0 0.988 0.988 0.941 0.964

0 0.4 0.98 0.92

0 0.982 0.929

0.953 0 0.953

0

Fig. 39.1 Dendrogram for the single-linkage clustering method for N C D(i, j), i, j = 1, ..10

the site might have been developed by a contractor that also developed sites 1, 2, and 4. The first dendrogram (Fig. 39.1) characterizes the affiliation of sites to one type of activity, and the second (Fig. 39.2) identifies the site developers. In both cases, both dendrograms show site similarity quite close to an “intuitive” presentation. The

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Fig. 39.2 Dendrogram for the Ward method for N C D(i, j), i, j = 1, ...eps

described approach is therefore promising and requires additional research on a much larger target set of sites. The dataset could include all the official websites of all Federal government agencies. In this case, manual research would be difficult to conduct since the set would contain over 300 sites. Therefore, it is necessary to automate the steps involved in collecting site data, compressing this data, building a distance matrix, and performing cluster analysis. However, there is a number of questions arising in the simple” experiments. They require research and answers before one could automate the process.

39.4 Questions for Further Research As said in the previous section, the experiments conducted have generated some vital questions that need to be looked into and addressed. First question. Which page of the site should be the one (first) to be downloaded? Or in a more general sense, which information from a site should be retrieved for similarity comparison? Standard browsers and specialized download programs (offline browsers) offer various options. For example, apart from text representation, you can save a web page as a .htm file and its associated folder containing images from the page and the page styles. But then comes the second question—how do you interpret as a binary word a folder containing a .htm file and the associated folder containing images and styles? The third question follows from the first two—how many pages should be downloaded in order for the study to be adequate? This is not about one page

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and probably not about the entire site (since this can be too expensive). The following approach seems promising here. Let K (s(M)) be the size of the archive of site s from which M pages were downloaded and let K (s(M + 1)) be the size of the archive of the same site from which M + 1 pages were downloaded. N C D(s(M), S(M + 1))—the distance between two site versions that differ by 1 page—can therefore be calculated. It is necessary to study the behavior of series N C D(s(1), S(2)), N C D(s(2), S(3)), ..., N C D(s(M), S(M + 1)), ... with regards to the tendency of its terms to zero as M increases and to compute some optimal point where data download should stop. Results of experiments manually performed for the first pages of the site www.nuc.edu.ng (0.139, 0.032, 0.042, 0.03, ...) look encouraging. Fourth question: is it possible to choose the best archiver, and what properties should such archiver possess? With regards to having the “best” archiver, the answer is far from being obvious. Regarding the properties of such an archiver, it is necessary to at least study idempotency, monotonicity, symmetry, and distributivity properties. Fifth question: it is obvious that hierarchical clustering methods are applicable for a relatively small number of objects. The methods should be analyzed both in terms of applicability on large sets of objects and in terms of interpretation of results obtained.

39.5 Conclusion We have described a technique to the study of website similarity based on Kolmogorov complexity and normalized compression distance. Such an approach has some potentials. However, to fully realize them, a number of tasks need to be investigated before automating the approach for large-scale studies. These tasks are formulated as the main directions for further research.

References 1. Borbely, R.S.: On normalized compression distance and large malware. J. Comput. Virol. Hacking Tech. 12, 235–242 (2016). https://doi.org/10.1007/s11416-015-0260-0 2. Cilibrasi, R., Vitanyi, P.: Clustering by compression. IEEE Trans. Inf. Theory 51(4), 1523–1545 (2005) 3. Cilibrasi, R., Vitanyi, P., de Wolf, R.: Algorithmic clustering of music based on string compression. Comput. Music J. 28(4), 49–67 (2004) 4. Gali, N., Mariescu-Istodor, R., Fränti, P.: Functional Classification of Websites. In: Proceedings of the 8th International Symposium on Information and Communication Technology, pp. 34–41 (2017). https://doi.org/10.1145/3155133.3155178 5. Gukhman, A.A.: Introduction to the Theory of Similarity, 2nd edn., 296 p. Higher School, Moscow (1973) 6. Hampton, J.A.: Similarity-based categorization: the development of prototype theory. Psychologica Belgica 35(2), 103–125 (1995)

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7. Kolmogorov, A.N.: Three approaches to the definition of the concept “quantity of information”. Probl. Peredachi Inf. 1(1), 3–11 (1965) 8. Li, M., Badger, J.H., Chen, X., Kwong, S., Kearney, P., Zhang, H.: An information-based sequence distance and its application to whole mitochondrial genome phylogeny. Bioinformatics 17(2), 149–154 (2001) 9. Li, M., Chen, X., Li, X., Ma, B., Vitanyi, P.: The similarity metric. IEEE Trans. Inf. Theory 50(12), 3250–3264 (2004) 10. Li, M., Vitanyi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn., 809 p. Springer, New York (2008) 11. Pechnikov, A.: About the similarity of the websites and kolmogorov complexity. Nor. J. Dev. Int. Sci. 1(14), 25–29 (2018) 12. Popescu, D.A., Danauta, C.M.: Similarity measurement of web sites using sink web pages. In: 34th International Conference on Telecommunications and Signal Processing, pp. 24–26 (2011) 13. Rajalakshmi, R., Aravindan, C.: Naive Bayes approach for website classification. In: International Conference on Advances in Information Technology and Mobile Communication. Communications in Computer and Information Science, vol. 147, pp. 323–326 (2011) 14. Sivakumar, B.:Chaos in Hydrology: Bridging Determinism and Stochasticity, 394 p. Springer Science (2017) 15. Ward, J.H., Jr.: Hierarchical grouping to optimize an objective function. J. Am. Stat. Assoc. 58, 236–244 (1963) 16. WinRAR, information. https://www.rarlab.com. Accessed 26 Mar. 2020

Part VI

Game Theory and Conflict Systems Control

Chapter 40

DEA Modeling with Cluster Analysis Vladimir Bure, Elena Parilina, and Kseniya Staroverova

Abstract We present a model of clustering the homogeneous firms using the numerical characteristics representing efficiency of their activities during a given time interval. The firms’ efficiency is found by the DEA (Data envelopment analysis) methodology, which is based on solving several optimization problems. The DEA modeling gives an opportunity to compare firms’ efficiencies taking into account several factors of their activities. The second step of the analysis we make is to define the partition of the time series showing the firms’ efficiencies. We find the clusters of firms close to each other in some sense. We also propose the method of finding a stable partition of firms according to their efficiencies.

40.1 Introduction Recently, DEA methodology (Data Envelopment Analysis) of modeling the efficiencies proposed in [5] has been widely used. This method allows us to find estimators of the efficiencies of the objects taking into account the set of indicators of their activities. The method is based on solving specially formulated optimization probThe work of the first author was supported by the Russian Foundation for Basic Research under grant no. 19-29-05184. The first author’s contribution is a DEA modeling with two factors and theoretical results presented in Sect. 40.2. The work of the second author was supported by the Russian Science Foundation under grant no. 17-11-01079. The contribution of the second author is in theoretical part of using stability of coalition structures in the problem of clusterization. The third author’s contribution is in cluster analysis of time series and numerical modeling. V. Bure · E. Parilina (B) · K. Staroverova St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. Bure e-mail: [email protected] K. Staroverova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_40

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lems. A modification of the method is proposed in [1]. The aim of DEA modeling is to evaluate the technical performance of a firm among a set of homogeneous firms and subsequent ordering the firms in terms of their productivities or efficiencies. Therefore, we first evaluate the efficiency of the firm’s management. The firm may be called the DMU (decision-making unit), and we evaluate the effectiveness of DMUs, which manages the firm’s capital and labor resources. Currently, DEA method is used everywhere along with the standard statistical procedures when analyzing the effectiveness of objects that can be enterprises, regions, countries. For example, you can use DEA methodology for ranking grant applicants, given their performance in the previous time period. An example of constructing DEA model for resource allocation in a production network is proposed in [17]. The problem of choosing an effective portfolio, discussed in [9], also uses DEA modeling approach. We propose a method for finding stable partitions of the firms into the subsets of similar firms in two stages. At the first stage, DEA modeling of firms’ activities in a certain time interval is made. As a result of the first stage, we obtain a set of time series of firms’ efficiencies. At the second stage, we propose to use the time series clustering procedure to find a partition of a set of firms into disjoint subsets. The procedure of clustering time series is described in [18]. To solve the problem of dividing the set of firms into clusters, we propose to apply a game-theoretic approach of finding a stable coalition structures described in [14, 16]. This approach is based on the Nash equilibrium concept and allows us to find the partition of players from which there is no incentive to deviate individually. The method of finding stable coalition structures is adapted to find a stable set of clusters in [4]. This method is also useful to find stable partition of firms competing in the market of homogeneous products [15] (see also [11] for examining the models of dynamic Cournout competition). The results of the paper are based on theoretical propositions proved in [3], where the idea of using DEA modeling results for further analysis of stable partition of firms, e.g. selling the homogeneous product. In the paper, we also introduce an example in which we find the clusters of firms close in effectiveness if their work is considered in dynamics. The stability of the partition of the set of firms into clusters means the absence of possibilities to improve the objective function, which depends on the intercluster and intracluster distances in a special way, by transferring an object from one cluster to the other or by separating it from the cluster. Recently, game-theoretic models are used in clustering data, e.g. for partitioning scientific communities [8], partitioning in a network of a given configuration [2]. Elements of game theory, including the theory of fuzzy cooperative games, are also used in DEA modeling [12, 13]. In these works, “efficiencies” of the cities in Iran and the effectiveness of medical institution work are investigated. The article has the following structure. Section 40.2 presents the theoretical foundations of DEA modeling. The clustering method of DEA time series analysis based on the game-theoretic approach is presented in Sect. 40.3. The example in Sect. 40.4 illustrates the proposed methodology. We briefly conclude in Sect. 40.5.

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40.2 DEA Modeling with Two Factors Following the Cobb and Douglas model [7], we consider two important factors of production that are labor and capital. The Cobb–Douglas two-factor production functions are often used in economics. The production function is assumed in the following form: (40.1) Y = AK α L β , where Y is a total production, K is a capital input, L is a labor input (in monetary terms, it can characterize the size of payments to all participants in the production process), α (β) is a constant representing the output elasticity of capital K (labor L), and A is a constant representing the total factor productivity. Suppose that there are n firms producing a homogeneous product or provide similar services (e.g. bakeries). The activity of any firm for a certain time period (e.g. a year) is characterized by a set of parameters Yi , K i , L i (i = 1, . . . , n) having the same meanings as in the Cobb–Douglas function (40.1). To estimate the efficiency of the firms and their ordering, we apply an approach based on DEA modeling. We characterize a firm’s efficiency in a given time period by the value δ0 Yi , i = 1, . . . , n. (40.2) Ti (δ0 , δ1 , δ2 ) = δ1 K i + δ2 L i Define the set D of admissible values of coefficients (δ0 , δ1 , δ2 ):   D = (δ0 , δ1 , δ2 ) | δ0  0, δ1  0, δ2  0, Ti (δ0 , δ1 , δ2 )  1, i = 1, . . . , n (40.3) We formulate n optimization problems: max

(δ0 ,δ1 ,δ2 )∈D

Ti (δ0 , δ1 , δ2 ), i = 1, . . . , n.

(40.4)

Remark 40.1 In case δ1 and δ2 obtained as a part of solution of the optimization problem are equal to zero, the efficiency defined by equation (2) is not defined. We need to exclude the solutions with these zero values. ∗ ∗ We assume that the set of coefficients (δ0i , δ1i∗ , δ2i ) is the solution of problem (40.4), i.e. ∗ ∗ , δ1i∗ , δ2i ) = max Ti (δ0 , δ1 , δ2 ). (40.5) Ti∗ = Ti (δ0i (δ0 ,δ1 ,δ2 )∈D

We call the value Ti∗ as the efficiency coefficient of firm i = 1, . . . , n. It is obvious ∗ ∗ , δ1i∗ , δ2i )  1, i = 1, . . . , n. Efficiency coefficients can be easily organized that Ti (δ0i in an ordered vector1 of efficiency coefficients of n firms: 1

Order the efficiency coefficients Ti∗ in a non-increasing order in a vector hereinafter.

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(40.6)

Series (40.6) represents the ranking of the set of homogeneous firms for a certain time period. Suppose that the first k coefficients of the series are equal to one, i.e. ∗ ∗ ∗ = T(2) = . . . = T(k) = 1, T(1)

then following the methodology of DEA modeling, the firms corresponding to these coefficients can be considered as the firms with maximal efficiency (or optimally operating) in the set of firms in a certain time period. Transform the expressions of coefficients Ti (δ0 , δ1 , δ2 ), i = 1, . . . , n in the following way: Ti (δ0 , δ1 , δ2 ) =

δ0 Yi 1 1 = = , δ1 K i + δ2 L i θ1 x1i + θ2 x2i Si (θ1 , θ2 )

where δ1 δ2 , θ2 = , δ0 δ0 Ki Li x1i = , x2i = , Yi Yi Si (θ1 , θ2 ) = θ1 x1i + θ2 x2i .

θ1 =

We define the set D˜ as   D˜ = (θ1 , θ2 ) | θ1  0, θ2  0, Si (θ1 , θ2 )  1, i = 1, . . . , n and consider n optimization problems min Si (θ1 , θ2 ), i = 1, . . . , n.

(θ1 ,θ2 )∈ D˜

(40.7)

Let (θ1i∗ , θ2i∗ ) be the solution of the ith optimization problem (40.7): Si∗ = Si (θ1i∗ , θ2i∗ ) = min Si (θ1 , θ2 ), i = 1, . . . , n. (θ1 ,θ2 )∈ D˜

(40.8)

Having solutions Si∗ , i = 1, . . . , n, we can construct the ordered series ∗ ∗ ∗ ∗ ∗  S(2)  . . .  S(k)  S(k+1)  . . .  S(n) , S(1)

(40.9)

which is equivalent to the ranking (40.6). In the Cartesian system of coordinates (θ1 , θ2 ), we consider n lines in the first quadrant (θ1  0, θ2  0), defined by equations

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Si (θ1 , θ2 ) = 1,

i = 1, . . . , n.

(40.10)

Taking into account that x2i > 0, equations (40.10) can be solved with respect to θ2 , and we obtain the linear functions: θ2 = −bi θ1 + ai , where bi =

x1i , x2i

i = 1, . . . , n,

ai =

(40.11)

1 . x2i

Consider the piecewise linear envelope of these lines: θˆ2 (θ1 ) = max (−bi θ1 + ai ), i=1,...,n

(40.12)

   where θ1 ∈ 0, max x11i . i=1,...,n

Theorem 40.1 The following statements are true:    1. The envelope θˆ2 (θ1 ) on the set 0, max x11i is a concave function of θ1 . i=1,...,n   2. The set D˜ = (θ1 , θ2 ) | θ1  0, θ2  0, Si (θ1 , θ2 )  1, i = 1, . . . , n coincides with the set      1 (θ1 , θ2 ) | θ1 ∈ 0, max , θ2  θˆ2 (θ1 ) . i=1,...,n x 1i 3. For optimization problems (40.4) and (40.7), it is true that Ti∗ =

1 , i = 1, . . . , n. Si∗

4. The solutions of the problems (40.4) and (40.7) exist. ∗ ∗ = S(1) = 1. 5. It is hold that T(1) Proof The proof can be found in [3].



We consider efficiencies on a certain time period, but it will be more informative to examine the dynamics of firms’ rankings, in periods m = 1, . . . , M. Thus, for each ∗ firm i = 1, . . . , n, a sequence of coefficients Tim , m = 1, . . . , M characterizing firm ∗ , i among the set of firms is constructed. Instead of considering the coefficients Tim ∗ we can study the equivalent coefficients Sim , m = 1, . . . , M. Therefore, each firm i generates time series of coefficients characterizing its efficiency in time. The specific property of the time series is that there is an ideal point, that is, a series consisting of ones, since the maximum (best) value of the coefficient is equal to one. We provide the procedure of clusterization of firms and find the stable partition based on clustering the time series of efficiency coefficients.

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40.3 Clusterization as a Game-Theoretical Problem Let N = {1, 2, . . . , n} be the set of objects such that for any two of them the distance d(i, j) : N × N → [0, 1] and similarity f (i, j) : N × N → [0, 1] are defined. In clusterization problem, we recall a coalition S ⊂ N by a cluster. Let the equities d(i, j) = d( j, i) and f (i, j) = f ( j, i) hold for any i, j ∈ N and d(i, j)  d(k, l) ⇐⇒ f (i, j)  f (k, l) for any i, j, k, l ∈ N . By cluster S we mean a non-empty subset of elements from N which are joined. Now we explain how we solve clustarization problem using the theory of cooperative games with coalition structures. In this theory, the game is defined by (N , v, π ), where N is given set of firms, v is a characteristic function determining the value for any subset S ⊂ N , and π is a partition of firms. We may consider any possible partitions which represents the unions of clusters. The main point is to define characteristic function v in game (N , v, π ). Taking into account the structure of the data we analyze, we suggest to define this function in the following form: v(S) =



f (i, j) + 2

i, j∈S i = j



d(i, j).

(40.13)

i∈S j∈N \S

The value v(S) is large for cluster S which contains the objects with the large value of similarity f (i, j) for the objects belonging to the same cluster i, j ∈ S and, at the same time, the large distance d(i, k) for the objects outside the cluster, k ∈ N \ S. The next step is to define cooperative solution based on this function. The cooperative solution ψ π = (ψkπ : k ∈ N ) can be calculated for any structure π . In the work, we use the Aumann–Dreze and equal surplus values which redistribute v(S) of cluster S among firms. We can consider a coalition structure or partition π which can be interpreted as a possible solution of a clusterization problem. The next problem is to find a stable coalition structure which then can be considered as a solution of clusterization problem. We base our procedure of finding stable partitions of the objects on a game-theoretic approach proposed in [14] and then applied in [4] for clustering purposes. By a stable coalition structure π with respect to some solution concept, if for any player k ∈ N and any cluster B(k) ∈ π and coalition structure π

the inequality

ψkπ  ψkπ , holds, where π = {B(k) \ {k}, B ∪ {k}, π−B(k)∪B }, and B ∈ π−B(k) ∪ ∅, π−Bk such that π−Bk = π \ Bk . We should notice that the stable coalition structure always exists with respect to the Aumann–Dreze value (see [10]) but may be non-unique with respect to this solution. Therefore, there may be several results of clusterization using the method we propose.

40 DEA Modeling with Cluster Analysis

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Fig. 40.1 Solutions of optimization problems (40.8)

40.4 Example We demonstrate the results of the method clustering the firms taking into account their productivity in the recent years. Let for each firm i, characteristics Yi , K i , L i be given for the last 20 years.2 Solving optimization problems (40.8) for each year, we obtain the set of time series corresponding to each firm and containing the values inverse to the efficiencies (see Fig. 40.1). In order to find firms with similar efficiency, we solve the problem of clustering ∗ , i = {0, 1, . . . , 9}, t = {1998, 1999, . . . , 2017}. As a function of time series S(i,t) distance d, we choose coefficient based on time correlation and Euclidean distance (CORT) [6]: n−1 



x(t + 1) − x(t) y(t + 1) − y(t) t=1 , c(x, y) =   n−1

2 n−1

2   x(t + 1) − x(t) y(t + 1) − y(t) t=1

d(x, y) =

t=1

2 ||x − y||2 , 1 + exp (c(x, y))

because it takes into account the dynamics of time series. And let function f be defined as f (x, y) = 1 − d(x, y). The game-theoretic clustering procedure gives the following result. There is a stable partition with respect to the E S -value and the Aumann–Dreze value that is {{1}, {2, 4, 8}, {0, 3, 5, 6, 7, 9}}. The results of DEA modeling for firm 1 confirm the initial hypothesis about its inefficiency. If you look at the initial characteristics of firm 1 in Table 40.1, the total 2

We do not provide all initially given data because of limited number of pages. The data can be provided under request.

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Table 40.1 Characteristics of firm 1 (odd years), thousand of rubles t 1999 2001 2003 2005 2007 2009 2011 K1 L1 Y1

757 437 517

672 479 409

638 543 395

666 521 449

889 712 515

680 634 414

756 658 361

2013

2015

2017

726 819 312

891 893 331

1283 906 340

Fig. 40.2 The result of clusterization into three clusters

production Y1 decreases, while the values of L 1 and K 1 increases. It differs firm 1 from other firms and therefore, it is natural that it forms its own cluster (Fig. 40.2).

40.5 Conclusions We have considered the problem of partitioning the homogeneous firms operating in a certain time period. The parameters of the firms are used in DEA modeling to estimate their efficiencies. Based on these efficiencies we construct the time series describing the firm activities in dynamics. Then we find the stable partitions of firms using time series clusterization procedure using the idea of stable partitions of the objects borrowed from coalition game theory. The work of the method is represented on a numerical example.

References 1. Andersen, P., Petersen, N.C.: A procedure for ranking efficient units in data envelopment analysis. Manag. Sci. 39(10), 1261–1264 (1993) 2. Avrachenkov, K.E., Kondratev, A.Y., Mazalov, V.V., Rubanov, D.G.: Network partitioning algorithms as cooperative games. Comput. Soc. Netw. 5(11), 1–28 (2018) 3. Bure, V.M., Parilina, E., Staroverova, KYu.: Two-factor dea modeling and clusterization of homogeneous firms. Math. Game Theory Appl. 11(4), 24–43 (2019) 4. Bure, V.M., Staroverova, KYu.: Applying cooperative game theory with coalitional structure for data clustering. Autom. Remote Control 80(8), 1541–1551 (2019)

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5. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring efficiency of decision making units. Eur. J. Oper Res. 2(6), 429–444 (1978) 6. Chouakria, A.D., Nagabhushan, P.N.: Adaptive dissimilarity index for measuring time series proximity. Adv. Data Anal. Class. 1(1), 5–21 (2007) 7. Cobb, C.W., Douglas, P.H.: A theory of production. Am. Econ. Rev. 18(1), 139–165 (1928) 8. Ermolin, N.A., Mazalov, V.V., Pechnikov, A.A.: Game-theoretical methods of finding communities in academic web. Trudi SPIIRAN 55, 237–254 (2017) 9. Fiala, F.: Project portfolio designing using data envelopment analysis and De Novo optimisation. Central Eur. J. Oper. Res. 26, 847–859 (2018) 10. Gusev, V.V., Mazalov, V.V.: Potential functions for finding stable coalition structures. Oper. Res. Lett. 47, 478–482 (2019) 11. Haurie, A., Krawczyk, J.B., Zaccour, G.: Games and Dynamic Games. Singapore, Scientific World (2012) 12. Omrani, H., Fahimi, P., Mahmood, A.: A data envelopment analysis game theory approach for constructing composite indicator: an application to find out development degree of cities in West Azarbaijan province of Iran. Socio-Econ. Plan. Sci. 69, art.no. 100675 (2020) 13. Omrani, H., Shafaat, K., Emrouznejad, A.: An integrated fuzzy clustering cooperative game data envelopment analysis model with application in hospital efficiency. Expert Syst. Appl. 114, 615–628 (2018) 14. Parilina, E., Sedakov, A.: Stable Bank Cooperation for cost reduction problem. Czech Econ. Rev. 8(1), 7–25 (2014) 15. Parilina, E., Sedakov, A.: Stable coalition structures in dynamic competitive environment. In: Pineau, P.-O., Sigue, S., Taboubi, S. (eds.) Games in management science: essays in Honor of Georges Zaccour. Int. Ser. Oper. Res. Manag. Sci. 280, 381–396 (2020) 16. Sedakov, A., Parilina, E., Volobuev, Yu., Klimuk, D.: Existence of stable coalition structures in three-person games. Contrib. Game Theory Manag. 6, 407–422 (2013) 17. Shao, Y., Bi, G., Yang, F., Xia, Q.: Resource allocation for branch network system with considering heterogeneity based on DEA method. Central Eur. J. Oper. Res. 26, 1005–1025 (2018) 18. Staroverova, K.Yu., Bure, V.M.: Characteristics based dissimilarity measure for time series. Vestnik of Saint Petersburg University. Applied mathematics. Computer science. Control processes, vol. 13(1), pp. 51–60 (2017). https://doi.org/10.21638/11701/spbu10.2017.105

Chapter 41

Cooperation in Vehicle Routing Game on a Megapolis Network Alexander V. Mugayskikh

Abstract In this work, cooperative multiple depot open vehicle routing problem (MDOVRP) is considered. The underlying model is a time-dependent variant of classic VRP which presents congested traffic in a megapolis more correctly compare to its nontemporal flavor. With the aim to reduce operational expenses on transportation costs or rent for the vehicles, carrier companies can share the customers with each other by forming coalitions. We introduce Direct Coalition Induction Algorithm (DCIA) for constructing the characteristic function of TD-MDOPVR game that satisfies subadditive property. Shapley values calculated for the problem instance of 150 customers and 3 companies are compared with costs before cooperation. All numerical runs are performed on a graph of real road network of Saint-Petersburg which includes 255 nodes and 1251 arcs. Time-dependent travel times are obtained by solving traffic assignment problem in case of Wardrop’s user equilibrium.

41.1 Vehicle Routing Problem The open vehicle routing problem (OVRP) is the flavor of the basic vehicle routing problem (VRP) [1]. The main difference between the OVRP and classic VRP is that vehicles do not return to the depot after completing the routing plan, so the routing is open. Most often, this model is used when the company does not own its fleet of vehicles, but rents them daily for serving customers. As in any other routing problem, the overall travel cost of all vehicles used is minimized under several constraints: each customer should be visited only once and the vehicle capacity should not be exceeded by the amount of goods to deliver.

A. V. Mugayskikh (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_41

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41.1.1 Mathematical Model of TD-MDOVRP In this work, we consider the time-dependent version of the routing problem. This variant of the classic MDOVRP is inspired by the fact that in urban environment variable traffic conditions make significant adjustments into route planning and can not be ignored in order to perform a realistic optimization [5, 9]. Implementation of the time-dependent model of MDVRP decreased the route’s total duration up to the 27% in the congested road network [10]. The TD-MDOVRP can be defined as follows. Let G = (V, E), where V = {1, . . . , N + M} is the nodes array and E = {(i, j) | i, j ∈ V } is the arc set. Vertex set Vcust represents the customers to be served, and set Vdepot represents the uncapacitated depots, and V = Vcust ∪ Vdepot , where Vcust = {1, . . . , N }, Vdepot = {1, . . . , M}. Each arc (i, j) ∈ E shows the travel time between node i and j on the network and it depends on the period of daytime. For evaluating elements of matrix E, we use the notation from [4]:  ti j (bi ) =

, z h−1 + i j,h−1 ≤ bi ≤ z h − i j h

cihj cihj

+

h (cih+1 j −ci j )(bi −z h +i j h ) 2i j h

, z h − i j h < bi < z h + i j h ,

(41.1)

where h = 1, . . . , H ; i j0 = i j H = 0, bi , i ∈ Vcust —the departure time from node i, z h —boundaries of travel times slots in travel time function, cihj , i j H – appropriate values and parameters to smooth travel time function. More details provided in [4]. Each depot d ∈ Vdepot stores and supplies products and has an limited set K of homogeneous vehicles with the same capacity of Q. Each customer i ∈ Vcust has a demand qi , 0 < qi ≤ Q that should be met. We introduce binary variable {xi j }i, j∈V that equals one, if arc (i, j) exists in the routing plan, and zero otherwise. Let {yik }i∈V,k∈K be a binary variable, that equals one, if the route k contains the customer i, and zero otherwise. To make TD-MDVRP open we make the durations from the customer nodes to each depot to equal zero. Thus, The TD-MDOVRP consists of finding the set of routes of minimal traveling cost (41.2) satisfying the constraints (41.3)–(41.6):  ti j (bi )xi j → min, (41.2) i∈V j∈V

subject to



xi j =

i∈V

 i∈V



x ji

∀j ∈ V;

(41.3)

i∈V

xi j = 1

∀ j ∈ Vcust ;

(41.4)

41 Cooperation in Vehicle Routing Game on a Megapolis Network



qi yik ≤ Q k

∀k ∈ K ;

373

(41.5)

i∈Vcust

bj ≥



xi j (bi + ti j (bi )) + x1 j

i∈Vcust



y jv (b0k + ti j (b0k )), ∀ j ∈ Vcust .

(41.6)

k∈K

Target function (41.2) summarizes the travel times of all vehicles in the routing plan. Routing plan stands for the set of routes, constructed to serve all customers in the problem instance and the route is the sequence of customers, which vehicle should follow during the trip. Constraints (41.3) represent the continuity property of the solutions (routing plans). Constraints (41.4) guarantee that each customer is visited exactly one time. Constraints (41.5) ensure capacity restrictions to be held. The last group of constraints (41.6) evaluates low boundaries for the moments of departure from each customer, and b0k is the departure time from depot.

41.1.2 Traffic Assignment Problem The main obstacle in applying time-dependent models to solve VRP is lack of traffic information. To construct routes, one should have several travel-time matrices, each of which will correspond to one period of the day. For this purpose, online monitoring services are commonly used: they collect information on the number of drivers using GPS signals. Control system Cartesio in Italy [2], the ITIS Floating Vehicle Data in the UK [3] and traffic information system LISB in Germany [4] are worth mentioning. Other methods form correspondence matrices (or OD-matrices) and then solve the traffic assignment problem by finding the Wardrop’s user equilibrium assignment [12]. Elements of correspondence matrix show the number of cars following from node i to node j at a given period of the day. Related aspects of the Nash equilibrium and the Wardrop’s user equilibrium assignments are discussed in [7, 8]. In our work, we implemented the second approach. Traffic assignment problem was successfully solved and traffic volumes on edges of graph G were found by Frank–Wolfe algorithm. Using the knowledge on amount of traffic on each edge, we calculated the journey time along the edge using the BRP-delay function (41.7):  di ( f i ) =

ti0

1 + αi



fi pi

βi 

,

(41.7)

where ti0 —driven time along edge i in free-flow contexts, f i —volume of traffic on a link i, and pi —capacity of link i per unit of time. Form (41.7) of BPR-function is convenient to be used in practice because speed equals half the free-flow speed when traffic volume reaches the capacity. Since the initial graph of the network is not fully connected, we are to solve the shortest path for each pair (i, j) of graph G. Applying Dijkstra’s algorithm, we formed travel-time matrices E d f for each period

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of the day. Thus, we carried out some preliminary operations before handling with routing problem on a megapolis network.

41.2 Cooperation in Time-Dependent Vehicle Routing Game Horizontal cooperation between freight-forwarding companies, aimed to delivery any kind of goods for customers, can significantly decrease operational costs for all companies involved. The main idea of cooperation in routing problems is that companies can form coalitions and redistribute customers. A detailed review on cost allocation methods in collaborative transportation is presented in [13]. Due to the heuristic nature of methods applied for solving routing problems, the main property of cooperative game theory, subadditivity of characteristic function, can be violated. This leads to unstable cooperation and infeasible cost allocation, in some cases. The major part of existing studies on cooperative logistics is devoted to possible strategies of transport collaboration and cost allocation. But articles do not refer routing problems in them. In work [14], authors claim that transportation companies are obliged to adopt a collaborative focus to survive under the everincreasing competitive and global pressures to operate more efficiently. Two types of cooperation are discussed: order sharing and capacity sharing. No details in solving routing problems are provided.

41.2.1 Mathematical Model of Cooperative TD-MDOVRP Let U be a set of freight forwarding companies aimed to delivery some amount of goods on a megapolis network. Considered companies have opportunity to form coalitions in order to reduce transportation costs. We denote S ⊆ U —possible coalition of players in a vehicle routing game. Costs of coalition S are composed of two components: fixed costs for rent of one vehicle and transportation costs depending on distance traveled. The latter costs component is calculated according to traveltime matrix on delay-flow network E d f , introduced. We consider costs function of coalition S as follows: cost (S, p S ) = α S · N V (S, p S ) + β S · T T D(S, p S ), where • • • •

PS —set of feasible routing plans for coalition S; p S ∈ PS —feasible routing plan for coalition S; α S —cost for usage of one vehicle for coalition S; N V (S, p S )—number of vehicles used by coalition S in routing plan p S ;

(41.8)

41 Cooperation in Vehicle Routing Game on a Megapolis Network

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• β S —costs for one unit of time in a route by coalition S; • T T D(S, p S )—total travel duration of all vehicles of coalition S used in routing plan p S . We assume that freight forwarding companies have an opportunity to redistribute costs among them via defined payment procedure. The most challenging problem of research in cooperative game theory is the construction of characteristic function of a game, which defines the guaranteed total cost of players involved in cooperation. Definition 41.1 Characteristic function is a function c(s) : 2 M → R, where 2 M states for the set of all possible coalitions of players that satisfies c(Ø) = 0. The constructed characteristic function should possess subadditivity property [15]: c(S ∪ T ) ≤ c(S) + c(T ), S ⊆ U, T ⊆ U, S ∩ T = Ø.

(41.9)

According to the results in [16, 17] which are true for free-flow network, they will be also correct for delay-flow variant of vehicle routing game if all calculations will be done on congested network. Using the direct coalition induction algorithm from [17] for vehicle routing game, we define the value of characteristic function c(S) as follows: ⎧ , |S| = 1 ⎨ f h (S) 

 (41.10) c(S) = ⎩min f h (S), min c(S \ L) + c(L) , |S| > 1, L⊂S

where f h (S)—the solution of min cost (S, p S ) by heuristic algorithm on delay-flow ps ∈Ps

matrices E d f . Hence, to construct characteristic function mentioned in (41.10), we calculate the values of characteristic function c(S) for all singleton coalition, then—for coalitions of two players and so on, increasing the number of players in coalition until the total coalition of U is reached. The characteristic function constructed by such approach satisfies the subadditivity condition.

41.2.2 Numerical Experiments Numerical experiments were held on a road network of Saint-Petersburg which includes 255 nodes and 1251 arcs. According to results from Sect. 41.1, traffic assignment problem was solved for Saint-Petersburg road network for 5 periods of a day. Five time-travel matrices E d f were formed which represent duration between nodes on a network during the daytime. We consider the cooperation of three companies S1, S2, S3, aimed to deliver goods to 40, 40, and 70 customers, respectively. Assume that depots of companies are placed

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(a) Free flow route

(b) Delay flow route

Fig. 41.1 Comparison of delay-flow and free-flow routes in VRP

in different districts of the city, they are marked with numbers 1, 2, 3 in Fig. 41.2. Each company in a depot has a homogeneous fleet of 10 cars with capacity of 500 units. The demand of each customer equals 10 units. Randomly generated TD-MDOVRP problem instances were solved by genetic heuristics. Number of generations to obtain one solution equaled 20 and there were one hundred individuals in each population. The algorithm is described in details in works [6, 11]. Table 41.1 presents characteristic function values for all possible coalitions in this game in two different cases: on free-flow and delay-flow network. Characteristic function values were found by the direct coalition induction algorithm for vehicle routing game with parameters α = 100, β = 1. One can notice that solutions generated for a free-flow network are at average more expensive than delay-flow ones. The percentage of improvement in the instance with 150 clients and cooperation of 3 players was about 17%. It can be possible because route generation on delay-flow network avoids well-predictable traffic congestion in offline vehicle routing. As an example, we assume the route of one of the vehicles starting from depot number 1, in Fig. 41.1. The time horizon for routing starts at 8 AM. The TD-solution in Fig. 41.1b construct the route initially through the center of the city, and it went to the outskirts. It turned out to be more profitable than firstly to serve customers from distant areas and end the route in the center as in Fig. 41.1a. According to Table 41.2, the total minimum expenses of all companies without cooperation equals 2270.05. But if the companies decide to work in cooperation, they save approximately 31% of expenses having total costs in amount of only 1562, 63. Cost distribution for each company was calculated using Shapley value definition, you can see it in Table 41.2. Each player has cost reduction from 24% up to 37%. Figure 41.2 presents the way of redistribution of customers in the problem instance with 60 customers and 3 depots. In Fig. 41.2a, clients of first, second, and third players are marked as triangles, squares, and circles, respectively. In Fig. 41.2a, players work in cooperation and have depots and customers which belong to total coalition. In this example, players managed to save about 6% in cooperation.

Number of customers

40 40 70 80 110 110 150

Coalition

[S1] [S2] [S3] [S1, S2] [S1, S3] [S2, S3] [S1, S2, S3]

590,58 553,07 726,40 847,65 1033,72 1042,40 1162,63

1 1 2 2 3 3 4

690,58 653,07 926,40 1047,65 1333,72 1342,40 1562,63

598,75 579,18 792,02 902,32 1031,33 1139,33 1355,42

1 1 2 3 3 3 4

Num. of veh.

Duration

Characteristic function value

Duration

Num. of veh.

Free flow solution

Delay flow solution

Table 41.1 Characteristic function values in vehicle routing game

698,75 679,18 992,02 1202,32 1331,33 1439,33 1755,42

Characteristic function value 1,38% 4,72% 9,03% 18,25% –0,23% 9,30% 16,58%

Percent of improvemnt

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Table 41.2 Shapley values in vehicle routing game Type of TD-MDOVRP (delay-flow network) network Coalition Costs Costs based Cost before on Shapley reduction cooperation value coefficient S1 S2 S3

690,58 653,07 926,40

437,26 422,84 702,54

0,37 0,35 0,24

(a) Without cooperation

MDVRP (free-flow network) Costs Costs based Cost before on Shapley reduction cooperation value coefficient 698,75 679,18 992,02

482,02 526,24 747,16

0,31 0,23 0,25

(b) With cooperation

Fig. 41.2 VRP instance with 60 customers and 3 depots

41.3 Conclusion This work showed that cooperation in routing problems can significantly reduce the costs of involved companies. Players in cooperative routing game can save from 24 to 37% as in the problem instance described in the article. The percentage of cost reduction depends on the topology of the test problem, and specifically on location of customers on the network. In order to obtain a realistic optimization on a megapolis network, researches are to choose the most suitable model for it. For this reason, we explored time-dependent model with time-varying elements in cost matrix. Acknowledgements The work was jointly supported by a grant from the Russian Science Foundation (No. 19-71-10012 Multi-agent systems development for automatic remote control of traffic flows in congested urban road networks).

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References 1. Dantzig, G.B., Ramser, J.H.: The truck dispatching problem. Manag. Sci. 6, 80–91 (1959) 2. Donati, A.V., Montemanni, R., Casagrande, N., Rizzoli, A.E., Gambardella, L.M.: Time dependent vehicle routing problem with a multi ant colony system. Eur. J. Oper. Res. 185(3), 1174– 1191 (2008) 3. Eglese, R., Maden, W., Slater, A.: A road timetableTM to aid vehicle routing and scheduling. Comput. Oper. Res. 33(12), 3508–3519 (2006) 4. Fleischmann, B., Gietz, M., Gnutzmann, S.: Time-varying travel times in vehicle routing. Transp. Sci. 38(2), 160–173 (2004) 5. Gendreau, M., Ghiani, G., Guerriero, E.: Time-dependent routing problems: a review. Comput. Oper. Res. 64, 189–197 (2015) 6. Liu, R., Jiang, Z., Geng, N.: A hybrid genetic algorithm for the multi-depot open vehicle routing problem. OR Spectr. 36, 401–421 (2014). https://doi.org/10.1007/s00291-012-0289-0 7. Krylatov, A.Y., Zakharov, V.V., Malygin, I.G.: Competitive traffic assignment in road networks. Transp. Telecommun. 17(3), 212–221 (2016) 8. Krylatov, A.Y., Zakharov, V.V.: Competitive traffic assignment in a green transit network. Int. Game Theory Rev. 18(2), 1–14 (2016) 9. Malandraki, C., Daskin, M.S.: Time dependent vehicle routing problems: formulations, properties and heuristic algorithms. Transp. Sci. 26, 185–200 (1992) 10. Mugayskikh, A.V., Tuovinen, T., Zakharov, V.V.: Time-dependent multiple depot vehicle routing problem on megapolis network under wardrop’s traffic flow assignment. In: Proceedings of the 22nd Conference of Open Innovations Association FRUCT pp. 173–178 (2018) 11. Surekha, P., Sumathi, S.: Solution to multi-depot vehicle routing problem using genetic algorithms. World Appl. Program. 1(3), 118–131 (2011) 12. Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civ. Eng. 2, 325–378 (1952) 13. Guajardo, M., Ronnqvist, M.: A review on cost allocation methods in collaborative transportation. Int. Trans. Oper. Res. 23(3), 371–392 (2016) 14. Verdonck, L., et al.: Collaborative logistics from the perspective of road transportation companies. Transp. Rev. 33(6), 700–719 (2013) 15. Osborne, M.J., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge, MA, USA (1994) 16. Zakharov, V., Dementieva, M.: Multistage cooperative games and problem of time-consistency. Int. Game Theory Rev. 6(1), 1–14 (2004) 17. Zakharov, V.V., Shchegryaev, A.N.: Stable cooperation in dynamic vehicle routing problems. Autom. Remote Control 76(5), 935–943 (2015)

Chapter 42

Minimal Current Payments Algorithm for Sustainable Cooperation in Multicriteria Game Denis Kuzyutin, Yaroslavna Pankratova, and Roman Svetlov

Abstract To ensure sustainable cooperation in a dynamic game, we adopt the socalled imputation distribution procedure (IDP) or payment schedule-based method. A novel IDP which satisfies time consistency, irrational-behavior-proof condition and the reward immediately after the move property is designed for extensive-form multicriteria game with perfect information. This payment schedule implies the minimal current positive payments to the players when the game unfolds along the cooperative history.

42.1 Introduction The sustainability of a cooperative agreement is an important issue in a dynamic game. We consider extensive-form games with vector payoffs or multicriteria games (see, e.g. [2, 3, 17]) with perfect information and use an IDP-based approach, i.e. an appropriate current payments scheme (see, e.g. [8, 13, 15, 17, 18, 20]) to enforce the long-term cooperation. This approach was extended to multicriteria games in [8, 9], several payment schedules and their properties were explored in [6, 10, 12]. In particular, the Reward Immediately after the Move (RIM) property introduced in [10] implies that a player which has to move at node x should get some reward, D. Kuzyutin · Y. Pankratova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] D. Kuzyutin e-mail: [email protected] D. Kuzyutin National Research University Higher School of Economics (HSE), Soyuza Pechatnikov ul. 16, St. Petersburg 190008, Russia R. Svetlov The Herzen State Pedagogical University of Russia, Moika Emb. 48, St. Petersburg 191186, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_42

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i.e. a positive payment for the “correct” move according to the cooperative agreement immediately after her move, while the other players (which are inactive at x) receive zero current payments. Another good property of the payment schedule—the irrational-behavior-proof (IBP) condition [8, 9, 22]—means that each player has an incentive to play cooperatively even if she anticipates that the cooperative agreement can be violated at some intermediate position as a result of the other players’ “irrational behavior”. In this paper, we propose a novel imputation distribution procedure which meets RIM property, time consistency (see, e.g. [2, 5, 8, 10, 11, 13–18, 20]), IBP condition, non-negativity, and efficiency property. The proof of the Proposition 1 provides an algorithm how to calculate the current payments to every player along the optimal cooperative history. Moreover, this payment schedule implies the minimal current positive payments to the players among all possible imputation distribution procedures meeting the properties mentioned above. In Sect. 42.2, we remind the main notations for the class of extensive-form multicriteria games and discuss the methods how to construct a unique cooperative history and a characteristic function. The main properties which an IDP should satisfy are formalized in Sect. 42.3. In Sect. 42.4, we introduce the minimal current payments IDP and explore its properties. The conclusions are presented in Sect. 42.5.

42.2 A Cooperative Solution for Multistage Multicriteria Game Let us remind the following notations for a class of multicriteria games in extensive form with perfect information [6, 10, 12, 17] which will be used throughout the paper: • N = {1, . . . , n} denotes the players’ set; • K is the finite game tree with the set of all nodes (positions) P and the game tree root x0 ; • S(x) denotes the set of all direct successors of the position x, while the unique parent of y = x0 is denoted by S −1 (y); • Pi is the set of player i’s decision nodes, Pi ∩ P j = ∅ for i = j, while Pn+1 = {y j }mj=1 denotes the finite ordered set of terminal nodes, i.e. S(y j ) = ∅ ∀y j ∈ n+1 Pi = P; Pn+1 , ∪i=1 • ω = (x0 , . . . , xt−1 , xt , . . . , x T ) is the history (or trajectory) in the game tree, xt−1 = S −1 (xt ), 1  t  T ; x T = y j ∈ Pn+1 , where t in xt equals to the ordinal number of this node within the history ω and could be considered as the “time counter”; • h i (x) = (h i/1 (x), . . . , h i/r (x)) denotes the (vector) payoff of the i-th player at current position x ∈ P\{x0 }, where r is a number of criteria.

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We suppose in the paper that ∀i ∈ N ; k = 1, . . . , r the payoffs are positive, i.e. h i/k (x) > 0 for all nodes x ∈ P\{x0 }. Denote by M G P (n, r ) the class of n-person r -criteria extensive-form games with perfect information and positive stage payoffs. Let Ui denote the set of all pure strategies of the i-th player, while U = i∈N Ui denote a finite set of all possible pure strategy profiles. Each strategy profile u = (u 1 , . . . , u n ) ∈ U determines a unique history ω(u) = (x0 , . . . , xt , xt+1 , . . . , x T ) = (x0 , x1 (u), . . . , xt (u), xt+1 (u), . . . , x T (u)), where xt+1 = u j (xt ) ∈ S(xt ) if xt ∈ P j , 0 ≤ t ≤ T − 1, x T ∈ Pn+1 , and, hence, a set of all the players’ payoffs.  For any strategy profile u, let Hi (u) = (Hi/1 (u), . . . , Hi/r (u)) = h˜ i (ω(u)) = τT=1 h i (xτ (u)) denote the value of the (vector) payoff function of player i ∈ N . For each intermediate node xt ∈ P\Pn+1 denote by u ixt the restriction of the player i’s pure strategy u i (·) in  x0 on the subgame  xt with the subgame root xt . The strategy profile u xt = (u 1xt , . . . , u nxt ) generates the history ω xt (u xt ) = (xt , xt+1 , . . . , x T ) = set of the player’s (vector) (xt , xt+1 (u xt ), . . . , x T (u xt )) in the subgame and, hence, a payoffs in  xt . Finally, let Hixt (u xt ) = h˜ ixt (ω xt (u xt )) = τT=t+1 h i (xτ (u xt )) denote the value of the i-th player’s payoff function in the subgame  xt ,while Uixt is the set  xt xt of all i-th player’s pure strategies in  , and U = i∈N Uixt . Given a, b ∈ R m vector inequality a ≥ b means that ak  bk , ∀k = 1, . . . , m, while at least one inequality of m is strict. When all the players have agreed to cooperate in multictriteria game, they seek n Hi (u). Denote by P O( x0 ) to maximize the grand coalition N vector payoff i=1 the nonempty set (see, e.g. [19]) of all Pareto efficient strategy profiles, i.e.: u ∈ P O( x0 ) i f  v ∈ U :



Hi (v) ≥

i∈N



Hi (u).

i∈N

However, P O( x0 ) may contain multiple strategy profiles, hence, in general the players need to employ a rule how to choose a unique Pareto efficient strategy profile ¯ u) ¯ = u¯ ∈ P O( x0 ) and corresponding unique optimal cooperative history ω¯ = ω( (x¯0 , x¯1 , . . . , x¯ T ). It is worth noting that such a rule has to satisfy time consistency requirement [2, 14, 17], i.e. a fragment of the cooperative history in the subgame ¯ u) ¯ should remain an optimal cooperative history in this subgame. Two  xt , xt ∈ ω( examples of such rules—the refined leximin algorithm and the minimal sum of relative deviations approach were suggested in [6, 12]. We will assume henceforth that the players employ some rule μ in order to specify the unique cooperative history ω¯ = ω(u) ¯ = (x¯0 , . . . , x¯ T ) that was generated by the specific Pareto efficient cooperative strategy profile u¯ ∈ P O( x0 ). Denote by μ

max u∈U

 i∈N

Hi (u) =



Hi (u) ¯

i∈N

the maximal (according to the rule μ) total vector payoff of the grand coalition in  x0 . ¯ the players choose the subgame Moreover, assume that at each subgame  x¯t , x¯t ∈ ω, cooperative strategy profile u x¯t ∈ U x¯t and the optimal cooperative history ω¯ x¯t in  x¯t using the same approach μ as in the original game  x0 .

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The next step to specify a cooperative behavior in  x0 is to derive a vector-valued worth function for coalitions—the so-called Characteristic Function (CF). To this aim, the players can use, for instance, a classical α-characteristic function combined with the rule μ (see [8, 12] for details). Denote by  x0 (N , V x0 ) a multicriteria extensive-form game  x0 ∈ M G P (n, r ) with CF V x0 . Further, suppose that the players have agreed on a single-valued cooperative solution ϕ x0 for the game  x0 (N , V x0 ) (one can adopt, for instance, the vector analogue of the nucleolus or the Shapley value [8, 21]) that meets the efficiency (collective rationality) constraint n 

ϕix0 = V x0 (N ) =

T  n 

h i (x¯τ ),

(42.1)

τ =1 i=1

i=1

and the individual rationality property ϕix0  V x0 ({i}), i ∈ N .

(42.2)

¯ u), ¯ t = 0, . . . , T − 1 a cooperative subgame Denote by  x¯t (N , V x¯t ), x¯t ∈ ω( along the optimal cooperative history with CF V x¯t that can be derived in the subgameusing the same approach as in the original game. Note that V x¯t (N ) =  T i∈N h i ( x¯ τ ). τ =t+1 Again, we assume that a cooperative solution ϕ x¯t at every subgame  x¯t (N , V x¯t ), t = 0, . . . , T − 1 meets the properties (42.1) and (42.2).

42.3 The Properties of a Payment Schedule The IDP-based method (see, e.g. [2, 15, 17]) implies that the players manage the current payments stream to enforce the cooperative agreement. Let βi/k (x¯τ ) corresponds to the actual current payment which the i-th player gets at x¯τ w.r.t. criterion k (instead of originally prescribed payoff h i/k (x¯τ )) if the players employ the payment schedule β. Below, we remind and briefly discuss a number of useful properties that the IDP β may satisfy (see [8, 10, 12] for details). RIM property (Reward Immediately after the Move) [10]. If x¯t ∈ Pi , t = 0, . . . , T − 1, then β j (x¯t+1 ) = 0 for all j ∈ N \{i}, i.e. the only player who can get positive current payment at node x¯t+1 along the cooperative history is the player i who makes a decision (or moves) at the previous position x¯t = S −1 (x¯t+1 ). A positive vector payment βi (x¯t+1 ) can be considered as a reward to player i for his correct move at xt according to the cooperative scenario. Moreover, if node x is the last decision node of the player i along the cooperative history this player is expected to receive the rest of his optimal payoff right after his last and correct move. Note that RIM property corresponds to the so-called “A-subgame” approach (see [4, 10, 17] for details) that takes into account only active players at every subgame, i.e.

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the players who have at least one decision node in this subgame. Similar approach was used in [1]. Denote by (y1i , y2i , . . . , yTi (i) ) the ordered set of the i-th player’s nodes Pi ∩ ω¯ along the cooperative history ω, ¯ where positions {yτi } are numbered in order of their occurrence in ω, ¯ i.e.: y1i = x¯t i (1) , y2i = x¯t i (2) , . . . , yTi (i) = x¯t i (T (i)) . To simplify the notations, henceforth, we will omit superscript i in t i (λ), λ = 1, . . . , T (i), i.e. we will write βi (x¯t (λ)+1 ) instead of βi (x¯t i (λ)+1 ), etc. Definition 42.1 [10]. The IDP β = {βi/k (x t )} meets the efficiency (in the whole game  x0 (N , V x0 )) property if T 

βi (x¯t ) =

T (i) 

βi (x¯t (λ)+1 ) = ϕix¯0 , i = 1, . . . , n.

(42.3)

λ=1

t=1

Definition 42.2 The IDP β = {βi/k (x¯t )} meets the time consistency (TC) inequality if for every player i ∈ N , |T (i)|  2, and for each τ = 1, . . . , T (i) − 1 the following inequality holds: τ  x¯ βi (x¯t (λ)+1 ) + ϕi t (τ )+1  ϕix¯0 . (42.4) λ=1

The vector inequality (42.4) means that the amount of payments to the i-th player is sufficient to ensure his incentive to continue cooperation at every subgame along the cooperative history. Definition 42.3 The IDP β = {βi/k (x¯t )} in  x0 (N , V x0 ) satisfies the strong IBP property, if for every player i ∈ N , |T (i)|  2, and for each τ = 1, . . . , T (i) − 1 the following inequality holds: τ 

βi (x¯t (λ)+1 ) + V x¯t (τ )+1  V x0 ({i}).

(42.5)

λ=1

This property implies that each player has an incentive to play cooperatively even if he anticipates that the cooperative agreement can be violated at some intermediate position as a result of the other players’ “irrational behavior” (see [6, 8, 22] for details). Definition 42.4 The IDP β meets the non-negativity condition if βi/k (x¯t )  0, i = 1, . . . , n; k = 1, . . . , r ; t = 1, . . . , T.

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42.4 The Minimal Current Payments IDP Let us introduce a payment schedule meeting all the properties mentioned above, i.e. (42.3), (42.4), non-negativity constraint and RIM property, which implies the minimal positive current payments (MCP) to the players along the cooperative history. Proposition 42.1 For any extensive-form multicriteria game there exists a MCP imputation distribution procedure that satisfies RIM property, efficiency (42.3), TC inequality (42.4), strong IBP property (42.5), and non-negativity condition. Proof The current payments βi/k (x¯t ), t = 1, . . . , T are specified separately for every player i ∈ N . If T (i) = 1, i.e. Pi ∩ ω¯ = {x¯t = x¯t (1) } then βi (x¯t+1 ) = ϕix0 while β j (x¯t+1 ) = 0 for all j ∈ N \{i}. Note that the TC as well as strong IBP condition (42.4) and (42.5) are not defined for player i in this case, while the RIM property and non-negativity condition are obviously satisfied. Otherwise, i.e. if T (i) ≥ 2, to specify current payments βi/k (x¯t (λ)+1 ), λ = 1, . . . , T (i), we provide the so-called MCP algorithm. Let β j/k (x¯t (λ)+1 ) = 0 for all j ∈ N \{i}, k = 1, . . . , r . Step 1 (λ = 1, calculate  βi/k (x¯t (1)+1 ):  x¯

x¯

x0 t (1)+1 − ϕi/k , Vkx0 ({i}) − Vk t (1)+1 ({i}), 0 . βi/k (x¯t (1)+1 )) = max ϕi/k Step 2 λ (λ = 2, . . . , T (i) − 1; calculate βi/k (x¯t (λ)+1 )): λ−1  x0  x¯t (λ)+1 βi/k (x¯t (λ)+1 ) = max ϕi/k − ϕi/k − βi/k (x¯t (τ )+1 ), Vkx0 ({i}) − τ =1

x¯

Vk t (λ)+1 ({i})− λ−1   − βi/k (x¯t (τ )+1 ), 0 . τ =1

Step 3 T (i) (final non-negative payments at node x¯t (T (i))+1 )): T (i)−1  x0 − βi/k (x¯t (λ)+1 ). βi/k (x¯t (T (i))+1 ) = ϕi/k λ=1

The designed imputation distribution procedure β by its construction meets RIM property, (42.3), (42.4) and (42.5). Moreover, IDP β obviously meets non-negativity condition at positions x¯t (λ)+1 for all λ = 1, . . . , T (i) − 1. What has to be done is to check the non-negativity condition at node x¯t (T (i))+1 . Note that at the original game  x0 as well as at every subgame  x¯t (λ)+1 the optimal imputation satisfies the following inequalities: x¯

x¯

x0 t (λ)+1 ϕi/k  Vkx0 ({i})  0, ϕi/k  Vk t (λ)+1 ({i})  0, λ = 1, . . . , T (i).

(42.6)

We prove by induction that for all λ = 1, . . . , T (i) − 1 IDP β satisfies λ  τ =1

x0 βi/k (x¯t (τ )+1 )  ϕi/k .

(42.7)

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When we apply the MCP algorithm for λ = 1 (step 1), we have three following options: x0 x0 • [βi/k (x¯t (1)+1 ) = ϕi/k − ϕ x¯t (1)+1 ]. Then βi/k (x¯t (1)+1 )  ϕi/k due to (42.6). x¯

• [βi/k (x¯t (1)+1 ) = Vkx0 ({i}) − Vk t (1)+1 ({i})]. Then, taking (42.6) into account we get x¯ x0 x0 βi/k (x¯t (1)+1 )  ϕi/k − Vk t (1)+1 ({i})  ϕi/k . • [βi/k (x¯t (1)+1 ) = 0]. Obviously, inequality (42.7) holds for λ = 1. Now assume that condition (42.7) is valid for (λ − 1), λ ≥ 2, i.e.: λ−1 

x0 βi/k (x¯t (τ )+1 )  ϕi/k .

(42.8)

τ =1

Then, applying the MCP algorithm for λ = 2, . . . , T (i) − 1, we have three following options:  λ t (λ)+1 x0 • [βi/k (x¯t (λ)+1 ) = ϕi/k − ϕi/k − λ−1 Then τ =1 βi/k ( x¯ t (τ )+1 )]. τ =1 βi/k x¯

x0 x0 t (λ)+1 (x¯t (τ )+1 ) = ϕi/k − ϕi/k  ϕi/k due to (42.6).  x¯t (λ)+1 x0 • [βi/k (x¯t (λ)+1 ) = Vk ({i}) − Vk ({i}) − λ−1 Then, again, τ =1 βi/k ( x¯ t (τ )+1 )]. using inequality (42.6) twice, we get λ x¯t (λ)+1 x0 x0 {i}  ϕi/k . τ =1 βi/k ( x¯ t (τ )+1 )  ϕi/k − Vk • [βi/k (x¯t (λ)+1 ) = 0]. Then, according to the inductive assumption (42.8) we obtain λ−1 λ x0 τ =1 βi/k ( x¯ t (τ )+1 ) = τ =1 βi/k ( x¯ t (τ )+1 )  ϕi/k .

Hence, we proved (42.7) for all λ = 1, . . . , T (i) − 1. When substituting λ = (i)−1 x0 T (i) − 1 in (42.7), we obtain that ϕi/k − τT=1 βi/k (x¯t (τ )+1 ) = βi/k (x¯ T (i)+1 )  0. ¯  Therefore, the IDP β meets non-negativity condition at each node x¯k ∈ ω. Remark 42.1 The IDP β constructed above prescribes the minimal positive current payments to the players among all the IDPs meeting RIM property, conditions (42.3), (42.4), (42.5), and non-negativity condition. Finally, let us briefly compare the properties of three payment schedules which meet the RIM assumption for multicriteria extensive-form games, namely, the Aincremental IDP [10], refined A-incremental IDP [7] and the suggested MCP imputation distribution procedure. All three payment schedules satisfy efficiency in the whole game (42.3), and time consistency inequality (42.4). The second IDP in addition meets non-negativity condition, while the last one satisfies both the nonnegativity condition and strong irrational-behavior-proof property (42.5).

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42.5 Conclusion An extensive-form n-person game could be interpreted as the “dynamic game with changing conditions” because the set of active players may vary (shrink) while the game unfolds along the cooperative history. The RIM property as well as the minimal current payments schedule are designed to reveal this interesting property of the game in extensive form. Acknowledgements The research of D. Kuzyutin and Ya. Pankratova was funded by RFBR under the research project 18-00-00727 (18-00-00725). The research of R. Svetlov was funded by RFBR under the research project 18-00-00727 (18-00-00628).

References 1. Chander, P., Wooders, M.: Subgame-perfect cooperation in an extensive game. J. Econ. Theory 187, 105017 (2020). https://doi.org/10.1016/j.jet.2020.105017 2. Haurie, A., Krawczyk, J.B., Zaccour, G.: Games and Dynamic Games. Scientific World, Singapore (2012) 3. Kuhn, H.: Extensive games and the problem of information. Ann. Math. Stud. 28, 193–216 (1953) 4. Kuzyutin, D.V., Romanenko I.A.: On the equilibrium solutions properties for n-person games in extensive form. Vestnik St.Petersburg Univ. Math. 3(15), 17–27 (1998) (in Russian) 5. Kuzyutin, D.: On the consistency of weak equilibria in multicriteria extensive games. In: Petosyan, L.A., Zenkevich, N.A. (eds.) Contributions to Game Theory and Management, vol. V, pp. 168–177 (2012) 6. Kuzyutin, D., Gromova, E., Pankratova, Y.: Sustainable cooperation in multicriteria multistage games. Oper. Res. Lett. 46(6), 557–562 (2018). https://doi.org/10.1016/j.orl.2018.09.004 7. Kuzyutin, D., Lipko I., Pankratova, Y., Tantlevskij I.: Cooperation enforcing in multistage multicriteria game: new algorithm and its implementation. In: Petrosyan, L., Mazalov, V., Zenkevich, N. (eds.) Frontiers of Dynamic Games. Static and Dynamic Game Theory: Foundations and Applications. Birkhäuser, Cham (2020) 8. Kuzyutin, D., Nikitina, M.: Time consistent cooperative solutions for multistage games with vector payoffs. Oper. Res. Lett. 45(3), 269–274 (2017). https://doi.org/10.1016/j.orl.2017.04. 004 9. Kuzyutin, D., Nikitina, M.: An irrational behavior proof condition for multistage multicriteria games. In: Dedicated to the memory of Demyanov, V.F. (ed.) Consrtuctive Nonsmooth Analysis and Related Topics, CNSA 2017, Proceedings. IEEE, pp. 178–181 (2017) 10. Kuzyutin, D., Pankratova, Y., Svetlov, R.: A-Subgame concept and the solutions properties for multistage games with vector payoffs. In: Petrosyan, L., Mazalov, V., Zenkevich, N. (eds.) Frontiers of Dynamic Games. Static and Dynamic Game Theory: Foundations and Applications. Birkhauser, Cham (2019) 11. Kuzyutin, D., Smirnova, N.: Subgame consistent cooperative behavior in an extensive form game with chance moves. Mathematics 8, 1061 (2020). https://doi.org/10.3390/math8071061 12. Kuzyutin, D., Smirnova, N., Gromova, E.: Long-term implementation of the cooperative solution in multistage multicriteria game. Oper. Res. Perspect. 6, 100–107 (2019). https://doi.org/ 10.1016/j.orp.2019.100107 13. Parilina, E., Zaccour, G.: Node-consistent core for games played over event trees. Automatica 55, 304–311 (2015)

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Chapter 43

Pursuit of Rigidly Coordinated Evaders in a Linear Problem with Fractional Derivatives, a Simple Matrix, and Phase Restrictions Alena I. Machtakova and Nikolai N. Petrov Abstract The problem of conflict interaction between a group of pursuers and a group of evaders in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players is described by a system of differential equations with fractional derivatives and a simple matrix. The target sets are the origin. The aim of the group of pursuers is to capture at least one evader. Counterstrategies are acceptable strategies for pursuers. For such a conflictcontrolled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evaders. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.

43.1 Introduction Significant results are obtained in differential pursuit-evasion games involving a group of pursuers and one evader. A generalization of this class of problems is the problem of conflict interaction between a group of pursuers and a group of evaders, in which the goal of the group of pursuers is to capture a given number of evaders, and for the solution of which requires new solution methods [1, 2, 5, 10]. Sufficient conditions were obtained in [10] for the capture of at least one evader in a differential game with many pursuers and evaders under the condition that the evaders use the same control. The problem of pursuit of a group of coordinated evaders without phase restrictions was addressed in [6, 11], and the one with phase restrictions, in [12]. A. I. Machtakova (B) · N. N. Petrov Udmurt State University, 1, Universitetskaya St., bld. 1, 426034 Izhevsk, Russia e-mail: [email protected] N. N. Petrov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_43

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This paper addresses the problem of catching at least one evader under the condition that the motion of all participants is described by a linear system with fractional derivatives and a simple matrix and that all evaders use the same control and do not move out of a convex cone. The terminal sets are the origins of coordinates. Sufficient conditions for capture are obtained in terms of the initial positions and parameters of the game. It should be noted that in [5], two-person game with fractional derivatives was considered, which results are illustrated by the example of the gaming problem for the oscillation process with a second order fractional damping and of the dynamic game for a first-order system (with a fractional component), which describes the relaxation during the glass formation of supercooled liquids.

43.2 Formulation of the Problem Definition 43.1 Let f : [0, ∞) → Rk be an absolutely continuous function, α ∈ (0, 1). A function D (α) f of the form 

 D (α) f (t) =

1 (1 − α)



t 0

f  (s) ds , where (β) = (t − s)α



∞

e−s s β−1 ds ,

0

is the Caputo derivative of order α of the function f . In the space R k (k ≥ 2), we consider the differential game G(n, m) of n + m persons: n pursuers P1 , . . . , Pn , and m evaders E 1 , . . . , E m with the laws of motion D (α) xi = axi + u i , xi (0) = xi0 , u i ∈ V , D (α) y j = ay j + v , y j (0) = y 0j , v ∈ V .

(43.1) (43.2)

Here, xi , y j , u i , v ∈ R k , i ∈ I = {1, . . . , n}, j ∈ J = {1, . . . , m}, V = {v | v ≤ 1}, a ∈ R, α ∈ (0, 1). In addition, xi0 = y 0j for all i, j. It is also assumed that each evader E j does not move out of the convex cone  = {y ∈ R k : ( p j , y) ≤ 0, j = 1, . . . , r } with a nonempty interior, where p1 , . . . , pr are unit vectors in Rk . If  = Rk , then we assume that r = 0. Introducing new variables z i j = xi − y j , we obtain the system D (α) z i j = az i j + u i − v , z i j (0) = z i0j = xi0 − y 0j , u i , v ∈ V .

(43.3)

The measurable function v : [0, ∞) → Rk is called admissible if v(t) ∈ V , y j (t) ∈  for all t ≥ 0, j ∈ J.

43 Pursuit of Rigidly Coordinated Evaders in a Linear Problem …

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Definition 43.2 A capture occurs in the game G(n, m) if there exists a T > 0 such that for any admissible control v(t), t ∈ [0, +∞), of evaders E j , j ∈ J , there are admissible controls u i (t, z i0j , i ∈ I , j ∈ J , v(t), t ∈ [0, +∞)), of pursuers Pi , i ∈ I , time τ ∈ [0, T ], and numbers l ∈ I, s ∈ J such that zls (τ ) = 0.

43.3 Auxiliary Results Let Int X and coX denote the interior and the convex hull of the set X ⊂ Rk , respectively. Let us introduce the following notation: λ(h, v) = sup{λ ≥ 0 | − λh ∈ V − v} ,

E ρ (z, μ) =

∞  l=0

zl (ρ > 0, μ ≥ 0) (lρ −1 + μ)

is the generalized Mittag–Leffler function.   g(t, τ ) = (t − τ )α−1 E 1/α a(t − τ )α , α , f (t) =

t

g(t, τ )dτ = E 1/α (at α , α + 1) .

0

Definition 43.3 [7] The vectors a1 , a2 , . . . , as form a positive basis in Rk if for any x ∈ Rk there exist positive real numbers α1 , α2 , . . . , αs such that x = α1 a1 + α2 a2 + . . . + αs as . Lemma 43.1 Let the vectors b1 , . . . , bn , p1 form a positive basis in Rk , a < 0, α ∈ (0, 1). Then there exists a T > 0 such that, for any admissible function v(·), there is a number q for which E 1/α (aT α , 1) −

T g(T, τ )λ(bq , v(τ )) dτ ≤ 0 . 0

Proof It follows from the condition of the lemma that there exists a δ > 0 such that   min max max λ(bi , v), ( p1 , v) ≥ δ > 0 . v∈V

i

Let v(·) be an admissible function. Define functions h i (t, v(·)) and sets T1 (t), T2 (t) t h i (t, v(·)) = g(t, τ )λ(bi , v(τ )) dτ , 0

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T1 (t) = {τ ∈ [0, t] : ( p1 , v(τ )) ≥ δ} , T2 (t) = {τ ∈ [0, t] : ( p1 , v(τ )) < δ} . Then the inequality max λ(bi v(τ )) ≥ δ holds for all τ ∈ T2 (t). Since v(·) is an admisi

sible function, it follows that ( p1 , y1 (t)) ≤ 0 for all t ≥ 0. From [3] and the system (43.2), we have y1 (t) = E 1/α (at

α

t , 1)y10

+

g(t, τ )v(τ ) dτ . 0

Therefore, t

g(t, τ )( p1 , v(τ )) dτ ≤ μ0 (t) = −E 1/α (at α , 1)( p1 , y10 ) .

0

Consequently,  μ0 (t) ≥ δ





g(t, τ ) dτ −

T1 (t)

g(t, τ ) dτ ,



f (t) =

T2 (t)

g(t, τ ) dτ + T1 (t)

g(t, τ ) dτ . T2 (t)

It follows from the last two relations that  δ f (t) − μ0 (t) . g(t, τ ) dτ ≥ 1+δ T2 (t)

Next, we have max h i (t, v(·)) ≥ i

1 n

t g(t, τ )

δ n

λ(bi , v(τ )) dτ ≥

i

0

≥





g(t, τ ) dτ ≥ T2 (t)

1 n

t g(t, τ ) max λ(bi , v(τ )) dτ ≥ i

0

δ(δ f (t) − μ0 (t)) . n(1 + δ)

Consider the function H0 (t) = E 1/α (at α , 1) −

δ(δ f (t) − μ0 (t)) . n(1 + δ)

Since a < 0, the following asymptotic estimates [5] hold for t → +∞: 1 E 1/α (at , 1) = − α +O at (1 − α) α



1 t 2α

,

43 Pursuit of Rigidly Coordinated Evaders in a Linear Problem …

E 1/α (at α , α + 1) = − Therefore,

1 +O at α



c δ2 H0 (t) = − α + +O t an(1 + δ)

395



1 t 2α



1

.

t 2α

and hence lim H0 (t) < 0. t→∞

Thus, there exists a time T > 0 for which H0 (T ) < 0. Let the number q ∈ I for which max h i (T, v(·)) = h q (T, v(·)). Then i

E 1/α (aT α , 1) − h q (T, v(·)) ≤ H0 (T ) < 0. 

This proves the lemma.

43.4 Sufficient Conditions for Capture Theorem 43.1 Let n ≥ k, a < 0, 0 ∈ Intco{xi0 − y 0j , i ∈ I, j ∈ J, p1 , . . . , pr } .

(43.4)

Then a capture occurs in the game G(n, m). Proof The case r = 0 is examined in [6]. Let r = 1. Condition (43.4) implies that the set of vectors {xi0 − y 0j , p1 } forms a positive basis. Consider a minimal positive basis. If p1 is not contained in the minimal positive basis, then the statement follows from [6]. Let p1 be contained in the minimal positive basis. Denote J (i) = { j ∈ J : xi0 − y 0j is contained in the minimal positive basis} ,

J0 =



J (i) .

i

If |J 0 | = 1, then the statement of the theorem follows from [8]. Let |J 0 | ≥ 2. Denote I 0 = {i : J (i) = ∅}. Assume that I 0 = {1, . . . , q}, J 0 = {1, . . . , l}. β 0 0 = z α1 + c1 for all α, β. Consequently, the set Next, let cαβ = yα0 − yβ0 . Then z αβ 0 {z i1 , i ∈ I 0 , c1α , α ∈ J 0 , α = 1, p1 } forms a positive basis. It may be assumed that this basis is minimal. Since n ≥ k and |J 0 | ≥ 2, it follows that q < n. Therefore, q + α − 1 ∈ {q + 1, . . . , n} for all α ∈ J 0 , α = 1. It follows from [11] that there 0 0 exists a μ > 1 such that the set {z i1 , i ∈ I 0 , z q+α−1,1 + μc1α , α ∈ J 0 , α = 1, p1 } forms a positive basis. We define the controls of pursuers Pi , assuming that 0 , v(t))z 0 , i ∈ I 0 , u i (t) = v(t) − λ(z i1 i1 0 0 u q+α−1 (t) = v(t) − λ(z q+α−1,1 + μc1α , v(t))(z q+α−1,1 + μc1α ) , α ∈ J 0 , α = 1 .

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From the system (43.3), we have 0 h (t) , i ∈ I 0 , z i1 = z i1 i γ

0 0 z q+γ −1,1 (t) = z q+γ −1,1 h q+γ −1,1 (t) − μc1 h q+γ −1 (t) , α ∈ J , α = 1 , (43.5)

where h i (t) = E 1/α (at α , 1) −



t 0

 0 h q+γ −1 (t) =

t 0

0 g(t, τ )λ(z i1 , v(τ )) dτ ,

γ

0 g(t, τ )λ(z q+γ −1,1 + μc1 , v(τ )) dτ ,

0 h q+γ −1 (t) = E 1/α (at α , 1) − h q+γ −1 (t) .

By the lemma 1, there exists a time T and a number s such that h s (T ) = 0. If s ∈ I 0 , then z s1 (T ) = 0 and a capture of evader E s will occur in the game G(n, m). If h q+γ0 −1 (T ) = 0 at some γ0 ∈ J , γ0 = 1, then γ

0 (T ) . z q+γ0 −1,1 (T ) = −μc10 h q+γ 0 −1

We show that 0 ∈ Intco{xi (T ) − y j (T ), i ∈ I, j ∈ J, j = 1, p1 } .

(43.6)

0 = z i1 (T )/ h i (T ). Also, From (43.5), we have z i1 0 0 − z i1 ). z iγ (t) − z i1 (t) = E 1/α (at α , 1)(z iγ

Therefore, the following equation holds for all γ ∈ J 0 , γ = 1: z iγ (T ) − z i1 (T ) z i1 (T ) z iγ (T ) − z i1 (T ) = + = E 1/α (aT α , 1) h i (T ) E 1/α (aT α , 1) z iγ (T ) E 1/α (aT α , 1) − h i (T ) = z i1 (T ) + . h i (T )E 1/α (aT α , 1) E 1/α (aT α , 1)

0 0 = z i1 + z iγ

 By the condition, the system z i0j , i ∈ I 0 , j ∈ J 0 , p1 forms a positive basis. Consequently, the positive basis forms the set

z iγ (T ) z i1 (T ) E 1/α (aT α , 1) − h i (T ) , z i1 (T ) + , α h i (T ) h i (T )E 1/α (aT , 1) E 1/α (aT α , 1)  i ∈ I 0 , γ ∈ J 0 , γ = 1, p1 .

(43.7)

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α Since h i (T ) > 0, E 1/α (aT α , 1) > 0 and E 1/α  (aT , 1) > h i (T ) with a < 0, the 0 0 system of vectors z i j (T ), i ∈ I , j ∈ J , p1 forms a positive basis. γ 0 (T ) and condition (43.7) is satisfied, (43.6) Since z q+γ0 −1,1 (T ) = −μc10 h q+γ 0 −1 follows from [11]. Assuming that T is the initial time, we obtain a game in which m − 1 evaders are involved. Continuing this process further, we find that there exists a time T0 such that

0 ∈ Intco{xi (T0 ) − ys (T0 ), i ∈ I, p1 } at some s ∈ J. Now the capture follows from [8]. Now let r > 1. Since {xi0 − y 0j , i ∈ I, j ∈ J, p1 , . . . , pr } form a positive basis, there exist positive numbers αi j , βs such that 0=

 i, j

αi j (xi0 − y 0j ) +



βs ps .

s

Consider the vector p0 = β1 p1 + . . . αs ps and the set 0 = {y ∈ R k : ( p0 , y) ≤ 0}. Then  ⊂ 0 and the set {xi0 − y 0j , i ∈ I, j ∈ J, p1 } forms a positive basis. Consequently, a capture with phase restrictions 0 occurs in the game G(n, m). This proves the theorem.  Theorem 43.2 Let a = 0, n ≥ k and let condition (43.4) be satisfied. Then a capture occurs in the game G(n, m). Proof The proof of this theorem is similar to that of Theorem 43.1.



Acknowledgements This research was funded by the Ministry of Science and Higher Education of the Russian Federation in the framework of state assignment No. 075-00232-20-01, project FEWS-2020-0010 and under grant 20-01-00293 from the Russian Foundation for Basic Research.

References 1. Blagodatskikh, A.I., Petrov, N.N.: Konfliktnoe vzaimodeistvie grupp upravlyaemykh ob”ektov (Conflict interaction of groups of controlled objects). Udmurt State University, Izhevsk (2009) 2. Chikrii, A.: Conflict-Controlled Processes. Springer, Dordrecht (1997) 3. Chikrii, A.A., Machikhin, I.I.: On an analogue of the Cauchy formula for linear systems of any fractional order. Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky. 1, 50–55 (2007) 4. Chikrii, A.A., Machikhin, I.I.: On linear conflict-controlled processes with fractional derivatives. Trudy Inst. Mat. Mekh. UrO RAN 17(2), 256–270 (2011) 5. Grigorenko, N.L.: Matematicheskie metody upravleniya neskol’kimi dinamicheskimi protsessami (Mathematical methods of control over multiple processes). Moscow State University, Moscow (1990)

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6. Machtakova, A.I.: Persecution of rigidly coordinated evaders in a linear problem with fractional derivatives and a simple matrix. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. (2019). https://doi.org/10.20537/2226-3594-2019-54-04 7. Petrov, N.N.: On the controllability of autonomous systems. Differ. Uravn. 4(4), 606–617 (1968) 8. Petrov, N.N.: One problem of group pursuit with fractional derivatives and phase constraints. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki (2017). https://doi.org/10.20537/vm170105 9. Popov, AYu., Sedletskii, A.M.: Distribution of roots of Mittag-Leffler functions. J. Math. Sci. (2011). https://doi.org/10.1007/s10958-013-1255-3 10. Satimov, N., Mamatov, MSh.: On problems of pursuit and evasion away from meeting and differential games between groups of pursuers and evaders. Dokl. Akad. Nauk UzSSR. 4, 3–6 (1983) 11. Vagin, D.A., Petrov, N.N.: A problem of the pursuit of a group of rigidly connected evaders. J. Comput. Syst. Sci. Int. 40(5), 749–753 (2001) 12. Vagin, D.A., Petrov, N.N.: A problem of group pursuit with phase constraints. J. Appl. Math. Mech. 66(2), 225–232 (2002)

Chapter 44

A Pollution Control Problem for the Aluminum Production in Eastern Siberia: Differential Game Approach Ekaterina Gromova, Anna Tur, and Polina Barsuk

Abstract In this paper, we apply a dynamic game-theoretic model and analyze the problem of pollution control in Eastern Siberia region of Russia. When carrying out the analysis, we use real numerical values of parameters. It is shown that cooperation between the major pollutants can be beneficial not only for the nature but also for the respective companies.

44.1 Introduction Air pollution is a major environmental problem that affects everyone in the civilized world. Emissions from large industrial enterprises have a great adverse impact on the environment and the people’s quality of life. A detailed review of the scientific literature published in 1990–2015 on the topic of climate and environmental changes can be found in [4]. Air pollution is closely linked to climate change. Therefore, one of the most important issues in ecologic management concerns the reduction of the pollutant emission into the atmosphere. Game theory offers a powerful tool for modeling and analyzing situations where multiple players pursue different but not necessarily opposite goals. In particular, it is well suited for analyzing the ecological management problems in which players (countries, plants) produce some goods while bearing costs due to the emitted pollution [5, 6, 12–14, 16]. It should be noted that most results on pollution control turn out to be of more theoretical nature because it is difficult to obtain realistic numerical values of the model parameters. E. Gromova (B) · A. Tur · P. Barsuk St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. Tur e-mail: [email protected] P. Barsuk e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_44

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In contrast to the mentioned approach, we consider local situations that can be modeled with more precision. We hope that the obtained results can be of use when planning local policies aimed at decreasing pollution load in particular regions. Recently, a detailed analysis of the pollution control problem of the city Bratsk from the Irkutsk region of the Russian Federation based on data for 2011 was reported in [17]. Our contribution extends the model presented in that paper. The ecological situation is considered for the largest aluminum enterprises of Eastern Siberia located in Krasnoyarsk, Bratsk, and Shelekhov, which are the largest aluminum plants producing about 70% of aluminum in Russia. In the model, we include absorption which is considered for different weather conditions. It is known that the ecological situation aggravates in the wintertime because of frequent temperature changes, weak winds, and fog [1–3]. The problem of pollution control is formulated with a differential game framework and is considered based on data for 2016 [23, 24]. The paper is structured as follows. In Sect. 44.2, the description of the differential game model is presented. In Sect. 44.2.1, the Nash equilibrium is found. Section 44.2.2 deals with cooperative differential game. The numerical example of pollution control for the aluminum production in Eastern Siberia is presented in the Sect. 44.3.

44.2 A Game-Theoretic Model Consider a game-theoretic model of pollution control based on the models [5, 8]. It is assumed that on the territory of a given region there are n stationary sources of air pollution involved in the game. Each player has an industrial production site. Let the production of each unit is proportional to its pollution u i . Thus, the strategy of a player is to choose the amount of pollution emitted to the atmosphere. We assume that the n sources “contribute” to the same stock of pollution. Denote the stock of accumulated net emissions by x(t). The dynamics of the stock is given by the following equation with initial condition: x(t) ˙ =

n 

u i (t) − δx(t), t ∈ [t0 , T ], x(t0 ) = x0 ,

(44.1)

i=1

where δ > 0 denotes the environment’s self-cleaning capacity. Each player i controls its emission u i ∈ [0, bi ], bi > 0, i = 1, n. The solution will be considered in the class of open-loop strategies u i (t). The net revenue of player i at  time instant t is given by quadratic functional form: Ri (u i (t)) = u i (t) bi − 21 u i (t) , t ∈ [t0 , T ], where bi > 0. Each player i bears pollution costs defined as di x(t), where di ≥ 0 is a fine for environmental pollution.

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401

The revenue of player i at time instant t Ri (u i (t)) − di x(t). The objective of player i is to maximize its payoff 

T

K i (x0 , T − t0 , u 1 , u 2 , . . . , u n ) =



 Ri (u i ) − di x(s) ds.

(44.2)

t0

44.2.1 Nash Equilibrium We choose the Nash equilibrium as the principle of optimality in non-cooperaive game. To find the optimal emissions u 1N E , . . . , u nN E for players 1, . . . , n, we apply Pontrygin’s maximum principle. The Hamiltonians for this problem are as follows:  n    1 Hi (x0 , T − t0 , u, ψ) = u i (t) bi − u i (t) − di x(t) + ψi u i (t) − δx(t) . 2 i=1 (44.3) From the first-order optimality condition, we get the following formulas for optimal controls: u iN E = bi + ψi , i = 1, . . . , n. Adjoint variables ψi (t) can be found from differential equations ∂ Hi (x0 ,T∂ x−t0 ,u,ψ) = − dψdti (t) , ψi (T ) = 0, i = 1, . . . , n. Then u iN E (t) = bi −

di di + eδ(t−T ) , i = 1, . . . , n. δ δ

(44.4)

Here, we assume that for the environment’s self-cleaning capacity δ the following inequalities hold: δ ≥ dbii , i = 1, . . . , n. This condition ensures that u i ∈ [0, bi ], i = 1, n. n n



Let b N = bi , d N = di . Then the optimal trajectory is i=1

i=1

x N E (t) = C1 e−δt +

dN d N δ(t−T ) b N − 2, e + 2 2δ δ δ

(44.5)

d N δ(t0 −T ) ). where C1 = eδt0 (x0 − bδN + dδN2 − 2δ 2e But in the case when for some i: δ < dbii , it may happen that optimal control u iN E  for player i leaves the compact [0; bi ]. Let t i = T + 1δ ln 1 − bdiiδ . If δ < dbii and

t i > t0 , then optimal control for player i has a following form:

u iN E (t) =

0, bi −

di δ

+

di δ(t−T ) e , δ

for t0 ≤ t ≤ t i ; for t i ≤ t ≤ T.

(44.6)

It can be noted that T + 1δ ln(1 − bdiiδ ) ≥ T − bdii for all δ > 0. It means if T ≤ t0 + then optimal controls have no switching points.

bi di

,

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44.2.2 Cooperative Solution Consider now the cooperative case of the game. Assume the players agreed to cooperate and their goal is to achieve the joint optimum. The joint payoff is n 

 K i (x0 , T − t0 , u 1 , u 2 , . . . , u n ) = t0

i=1

T

 n 

Ri (u i ) − d N x(s) ds.

(44.7)

i=1

Similar to the non-cooperative case, we use Pontrygin’s maximum principle to get: u i∗ (t) = bi −

d N δ(t−T ) dN + e , i = 1, . . . , n. δ δ

(44.8)

Here, we assume that for the environment’s self-cleaning capacity δ the following inequalities hold: δ ≥ dbNi , i = 1, . . . , n. This condition ensures that u i∗ ∈ [0, bi ], i = 1, n. Then the optimal cooperative trajectory is x ∗ (t) = C2 e−δt +

nd N nd N δ(t−T ) b N − 2 , e + 2δ 2 δ δ

(44.9)

N δ(t0 −T ) e ). where C2 = eδt0 (x0 − bδN + ndδ2N − nd 2δ 2 Notice that the open-loop Nash equilibrium yields more pollution than the optimal n n



u i∗ (T ) = u iN E (T ), and for all t ∈ [t0 ; T ): strategies in the cooperative game:

i=1

n 

u i∗ (t) −

i=1

n 

i=1

u iN E (t) =

i=1

d N (n − 1) δ(t−T ) (e − 1) < 0. δ

Also consider the situation, when for player i the inequality δ < dbNi holds. In this t, where  ti = T + 1δ ln(1 − bdiNδ ). So, if δ < dbNi case u i∗ (t) becomes negative when t <  and  ti > t0 , then optimal control for player i has a following form: u i∗ (t) =

0, bi −

dN δ

+

d N δ(t−T ) e , δ

ti ; for t0 ≤ t ≤  for  ti ≤ t ≤ T.

(44.10)

It can be noted that T + 1δ ln(1 − bdiNδ ) ≥ T − dbNi for all δ > 0. It means if T ≤ t0 + bi , then optimal cooperative control of player i has no switching points. dN

44 A Pollution Control Problem for the Aluminum Production in Eastern …

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44.3 A Pollution Control Problem in Eastern Siberia Non-ferrous metallurgy is one of the most developed industries in Eastern Siberia. Large aluminum smelters such as Krasnoyarsk, Bratsk, and Irkutsk Aluminum Plants are located in this region. All mentioned factories belong to RUSAL, which is one of the world’s major producers of aluminum. In the considered model, the problem of reducing emissions from smelters during adverse weather conditions is solved by changing the parameter δ, which describes the environment’s self-cleaning capacity. We consider the three-player differential game, where players are the specified companies. To calculate the required model parameters bi , di , we use the data about the sources of air pollution for year 2016. We compute the coefficient bi ≥ 0 as the ratio of operating profit of company (Pi ) to its amount of air emissions (Vi ) and di ≥ 0 as the amount of fine for air pollution depending on the total pollution. To determine the fines, we used the data about companies payments for air pollution in the year 2016. Let L i be the payment for air pollution of the company i, then: bi =

Pi Li , di = . Vi V1 + V2 + V3

(44.11)

Table 44.1 includes the data on the operating profit of each company for the year 2016, their air pollution and the corresponding fines. The operating profit of Krasnoyarsk Aluminum Smelter was taken from [19, 20] gives us the joint operating profit of Bratsk and Irkutsk Aluminum Smelters, which is equal to 4210.43 mln.rub. We estimated the profit of each company in proportion to the volume of aluminum produced by these companies in 2016. According to [20], Bratsk Aluminum Smelter produced 1005500 tons of aluminum and Irkutsk Aluminum Smelter—415400 tons in 2016. So, the operating profit of the two plants accounts for 2979.51 mln.rub. and 1230.92 mln.rub., respectively. The payment for air pollution of Krasnoyarsk Aluminum Smelter amounted to L 1 = 87723.95 ths.rub. [21] in 2016. According to [22], the payment for air pollution of the company Irkutsk Aluminum Smelter accounted for L 2 = 18830 ths.rub. in the same year. Environmental impact fee including waste disposal fee of Bratsk Aluminum Smelter is equals to 65278 in 2016 [22]. According to [22], the payment for air pollution of Bratsk Aluminum Smelter is approximately 90we estimated its payment for air pollution at L 2 = 0.9 · 65278 = 58780.2 ths.rub. Using formulas (44.11), we get the respective coefficients of the model bi , di (Table 44.1). Table 44.2 represents the non-cooperative solutions obtained for some numerical parameters (t0 = 0, T = 0, 4). We consider two cases of meteorological conditions, more precisely, value δ = 0, 02 corresponds to adverse weather conditions, for instance, in winter months and δ = 0, 2 to normal weather conditions. The inequalities T ≤ t0 + bdii , T ≤ t0 + dbNi are satisfied for the chosen parameter values, thus, optimal cooperative controls of players have no switching points (we use (44.4), (44.8) to compute the optimal strategies). Table 44.3 contains the optimal cooperative strategies.

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Table 44.1 The operating profits, air pollution and payments for air pollution of the companies in 2016. The coefficients of the model Company Pi (mln. Vi (tons) L i (ths. bi di rubles) rubles) Krasnoyarsk Aluminum 3412.23 Smelter Bratsk Aluminum Smelter 2979.51 Irkutsk Aluminum Smelter 1230.92

57800

87723.95

59035.12

525.06

83578.707 25694.1

58780.2 18830

35649.15 47906.72

351.64 112,71

Table 44.2 Nash equilibrium strategies. Payoffs of companies in Nash equilibrium Company u iN E , δ = 0.02 u iN E , δ = 0.2 32782.12 + 26253e0.02t−0.008 18067.15 + 17582e0.02t−0.008 42271.22 + 5635.5e0.02t−0.008   K i x0 , T − t0 , u N E , δ = 0.02 691063605.8 − 209.19x0 250177865.6 − 140.09x0 457730701.9 − 44.9x0

KrAS BrAS IrAS KrAS BrAS IrAS

56409.82 + 2625.3e0.2t−0.08 33890.95 + 1758.2e0.2t−0.08 47343.17 + 563.55e0.2t−0.08   K i x0 , T − t0 , u N E , δ = 0.2 691203820 − 201.84x0 250271735 − 135.18x0 457760774.5 − 43.33x0

Table 44.3 Optimal cooperative strategies Company u i∗ , δ = 0, 02

u i∗ , δ = 0, 2

9564.62 + 49470.5e0.02t−0.008 −13821.35 + 49470.5e0.02t−0.008 −1563.78 + 49470.5e0.02t−0.008

KrAS BrAS IrAS

54088.07 + 4947.05e0.2t−0.08 30702.1 + 4947.05e0.2t−0.08 42959.67 + 4947.0e0.2t−0.08

Table 44.4 Differences between total air pollution in cooperative and non-cooperative case and differences between the accumulated emissions. Joint cooperative payoff δ

n n



u iN E (t)− u i∗ (t)

x N E (T )−x ∗ (T )

i=1

n

i=1

  K i x0 , T − t0 , u ∗

n

i=1

 n  

Ki u∗ − Ki u N E

δ = 0.02

i=1 98941(1 − e0.02t−0.008 )

157.044

1398986922 − 394.18x0

14748.7

δ = 0.2

9894.1(1 − e0.2t−0.08 )

146.2124

1399250309 − 380.35x0

13979.5

i=1

If we compare cooperative and non-cooperative emissions of players from Tables 44.3 and 44.3 it is easy to see that the optimal cooperative emissions are smaller. Table 44.3 shows differences between total air pollution in cooperative and non-cooperative cases and between the accumulated emissions. The joint cooperative payoffs and its differences with sum of payoffs in Nash equilibrium are also given in the Table 44.3. See Table 44.4) for differences in air pollution for both cases.

44 A Pollution Control Problem for the Aluminum Production in Eastern … Table 44.5 Characteristic function δ = 0.02 V η ({1}.x

0 .T − t0 ) V η ({2}.x0 .T − t0 ) V η ({3}.x0 .T − t0 ) V η ({1.2}.x0 .T − t0 ) V η ({1.3}.x0 .T − t0 ) V η ({2.3}.x0 .T − t0 )

405

δ = 0.2

691061320.2 − 209.19x0 250173553 − 140.09x0 457722552.2 − 44.9x0 941245436.6 − 349.28x0 1148794743 − 254.09x0 707904167 − 184.99x0

691201653.2 − 201.84x0 250267647.2 − 135.18x0 457753050 − 43.33x0 941479313.4 − 337.02x0 1148965007 − 245.17x0 708028338 − 178.5x0

Table 44.6 Difference between the Shapley value and the Nash equilibrium     Company Sh i (x0 , T − t0 ) Sh i (x0 , T − t0 ) Sh i − K i u N E Sh i − K i u N E δ = 0.02 δ = 0.2 δ = 0.02 δ = 0.2 KrAS BrAS IrAS

691072037.4 − 209.19x0 250182865.8 − 140.09x0 457732018.6 − 44.9x0

691211811.9 − 201.84x0 250276474.5 − 135.18x0 457762022.7 − 43.33x0

8431.6

7991.9

5000.2

4739.5

1316.7

1248.2

Consider the Shapley value as an cooperative solution. To calculate it, we use a non-standard method of construction a characteristic function proposed in [9, 10]. According to [9], players from coalition S use (obtained earlier) strategies u ∗S from the optimal profile u ∗ and the players from N \ S use (obtained earlier) strategies E from the Nash equilibrium strategies: u NN \S  η

V (S, ·) =

S = {∅},

0, E K i (·, u ∗S , u NN \S ), S ⊆ N .

(44.12)

i∈S

Table 44.5 contains the characteristic function for our example. The Shapley values are presented in Table 44.6. It is also interesting to see how much each firm benefits from cooperation as compared to a non-cooperative case. Table 44.6 shows this difference. We can observe that it is profitable to the companies to stick to the cooperative agreement, however to different extent. The results show that cooperation is beneficial for all smelters. It should be noted that the higher value of the fine di , the more profitable the company is to cooperate. In our example, Krasnoyarsk Aluminum Smelter is most motivated for cooperation.

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44.4 Conclusion In this paper, we analyzed the problem of pollution control in Eastern Siberia. In doing so, we considered the real data for 2016–2018 years obtained from statistical and accounting reports. It can be noted that for small values of the absorbtion parameter δ, when unfavorable weather conditions occur, players are more motivated for cooperation. We also observe a greater decrease of accumulated emissions under cooperation during adverse weather conditions. This shows that the model, which takes into account a self-cleaning ability of the atmosphere, allows more effective influence on companies to reduce emissions during adverse weather conditions. Acknowledgements This study was funded by RFBR under the research grant N 18-00-00727 (18-00-00725)

References 1. Akhtimankina, A.V.: Investigation of dynamics of concentration of harmful substances in atmosphere of Shelekhov city. The bulletin of Irkutsk State University. Series “Earth Sciences”, vol. 13, pp. 42–57 (2015) (in Russian) 2. Arguchintseva, A.V., Kochugova, E.A.: Atmospheric self-purification potential. The bulletin of Irkutsk State University. Series “Earth Sciences”, vol. 27 (2019) (in Russian) 3. Avdeeva, E.V., Chernikova, K.V.: The features of formation of the environment of a large industrial city (on the example of Krasnoyarsk). Theor. Appl. Res. J.-Conifers Boreal Area 29(3–4), 183–188 (2011) (in Russian) 4. Edvardsson Björnberg, K., Karlsson, M., Gilek, M., Hansson, S.O.: Climate and environmental science denial: a review of the scientific literature published in 1990–2015. J. Clean. Prod. 167, 229–241 (2017) 5. Breton, M., Zaccour, G., Zahaf, M.: A differential game of joint implementation of environmental projects. Automatica 41(10), 1737–1749 (2005) 6. Dockner, E.J., Van Long, N.: International pollution control: cooperative versus noncooperative strategies. J. Environ. Econ. Manag. 25(1), 13–29 (1993) 7. Dockner, E.J., Jorgensen, S., Van Long, N., Sorger, G.: Differential games in economics and management science. Cambridge University Press (2000) 8. Gromova, E.: The shapley value as a sustainable cooperative solution in differential games of three players. In: Petrosyan, L.A., Mazalov, V.V. (eds.) Recent Advances in Game Theory and Applications, Static and Dynamic Game Theory: Foundations and Applications, pp. 67–91. Springer (2016) 9. Gromova, E.V., Marova, E.V.: Coalition and anti-coalition interaction in cooperative differential games. IFAC PapersOnLine 51(32), 479–483 (2018). https://doi.org/10.1016/j.ifacol.2018.11. 466 10. Gromova, E., Marova, E., Gromov, D.: A substitute for the classical Neumann-Morgenstern characteristic function in cooperative differential games. J. Dyn. Games 7(2), 105–122 (2020). https://doi.org/10.3934/jdg.2020007 11. Jorgensen, S., Quincampoix, M., Vincent, T.L. (eds.): Advances in Dynamic Game Theory: Numerical Methods, Algorithms, and Applications to Ecology and Economics. Springer (2007) 12. Van Long, N.: Dynamic games in the economics of natural resources: a survey. Dyn. Games Appl. 1(1), 115–148 (2011) 13. Van Long, N.: Pollution control: a differential game approach. Ann. Oper. Res. 37, 283–296 (1992)

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14. Mäler, K.G., De Zeeuw, A.: The acid rain differential game. Environ. Resour. Econ. 12(2), 167–184 (1998) 15. Petrosjan, L., Zaccour, G.: Time-consistent Shapley value allocation of pollution cost reduction. J. Econ. Dyn. Control 27(3), 381–398 (2003) 16. Van der Ploeg, F., de Zeeuw, A.: A differential game of international pollution control. Syst. Control Lett. 17(6), 409–414 (1991) 17. Tur, A.V., Gromova, E.V.: On optimal control of pollution emissions: an example of the largest industrial enterprises of Irkutsk oblast. Autom. Remote Control 81, 548–565 (2020) 18. Vikulova, A.A.: Game-theoretic approach to the problem of regulating the volume of harmful emissions of industrial enterprises in the Irkutsk region. Master thesis. SPbSU (2019) 19. Accounting report of JSC “RUSAL Krasoyarsky aluminum smelter” (2016). https://e-ecolog. ru/buh/2016/2465000141 (in Russian) 20. Yearly report of JSC “RUSAL Bratsk” (2016). https://braz-rusal.ru/ (in Russian) 21. http://gkeco-nn.ru/images/docs/DRPN_Presentation.pdf 22. State reports “About the state and protection of the environment in Irkutsk oblast”. In: 2011, 2012, 2013, 2014, 2015, 2016. https://irkobl.ru/sites/ecology/picture/ (in Russian) 23. State report “About the state and protection of the environment in Russian Federation in 2016”. http://www.mnr.gov.ru/docs/gosudarstvennye_doklady/ (in Russian) 24. State reports “About the state and protection of the environment in Russian Federation”. In: 2017, 2018. http://www.mnr.gov.ru/docs/gosudarstvennye_doklady/ (in Russian)

Chapter 45

Dynamic Programming Equations for the Game-Theoretical Problem with Random Initial Time Anastasiya Malakhova and Ekaterina Gromova

Abstract A class of differential games with random initial time is considered. It is shown that payoff function in such class of games can be simplified to the form suitable for using of standard dynamic programming approach for both, the competitive and cooperative schemes. In order to find the Nash Equilibrium, a form of the Hamilton–Jacobi–Bellman equation is obtained.

45.1 Introduction Commonly, in differential game theory, process is considered on the bounded and finite time interval. Another widespread approach is that time interval is assumed to be infinite. These models are popular in economic theory and are frequently used in practice. The payoff function in such models often has a discount component which allows to consider game on infinite time period. See examples in [3, 4, 14]. In addition, there are a lot of works on differential games with random time horizon [2, 7, 8]. In such type of games, the moment of the end of the game is assumed to follow some probability distribution [2, 10, 13, 15]. As these games allow to model natural processes in a more accurate way, they also became very popular in many applications in economy, ecology, finance, etc. [3, 5, 9, 11, 16, 18]. In this paper, we consider a new kind of differential games: differential games with random initial time [6]. In such games, we suppose that game starts in random time instance which responds to some cumulative distribution function on considerable interval. A. Malakhova · E. Gromova (B) St. Petersburg State University, 35, Universitetskii pr., St. Petersburg198504, Russia e-mail: [email protected] A. Malakhova e-mail: [email protected] E. Gromova Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, S. Kovalevskaja st., 16, Yekaterinburg 620990, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_45

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This new class of games has a big practical importance. There are a lot of conflict processes which can be modeled by differential game theory and which initial time is not deterministic, but has a random nature. There can be also the case when only several participates of a game enter the game in random time. This work shows the derivation of Hamilton–Jacobi–Bellman equation (HJB) [1, 17] for non-cooperative scenarios under the class of close-loop strategies used by players. The paper is organized as follows. Section 45.2 is devoted to introduce the game model giving the set of assumptions to the game formulation. The form of payoff function for the game is obtained also in Sect. 45.2. Section 45.3 covers dynamic programming approach to the game: the HJB equation is written for non-cooperative case.

45.2 Game Formulation Let us consider the game of n players  T (x0 , t0 , T f ). Denote the set of players as N = {1, 2, . . . , n}, |N | = n. The game starts at the moment Tstar t from state x0 , the initial time belongs to the interval Tstar t ∈ [t0 , T f ]. The dynamics of the game x˙ = g(x, u 1 (x, t), . . . , u n (x, t)), x(Tstar t ) = x0 .

(45.1)

We make the following Assumptions: • the function g(x, u 1 (x, t), . . . , u n (x, t)) in (45.1) is a differentiable function on [t0 ; T f ]; • the state variable belongs to some open subset of finite-dimensional real Euclidean space: x(t) ∈ X ⊂ Rr for all t ∈ [t0 , T f ]; • the state variable takes constant initial value before the game starts x(t) = x0 , ∀t ∈ [t0 ; Tstar t ]; • the controls u i (x, t) are closed-loop strategies; • the controls u i (x, t) are taken from the sets of admissible controls Ui , which consist of all measurable functions on the interval [t0 , T f ], taking values in the set of admissible control values Ui , which are convex compact subsets of a Rk . For convenience, n we use the nfollowing short notation: u(x, t) = (u 1 (x, t), . . . , Ui , U = i=1 Ui . Thus, u ∈ U, u(x, t) ∈ U, t ∈ [t0 ; T f ]. u n (x, t)), U = i=1 In this class of games, the moment of the beginning of the game is not known in advance, we assume that the game starts at the moment Tstar t that is a realization of some random variable T . For T , there is a cumulative distribution function F(t) on the interval [t0 , T f ]:

45 Dynamic Programming Equations for the Game-Theoretical …



Tf

411

d F(t) = 1.

t0

We assume the existence of the density function of a random variable T : f (t) = F  (t). Let h i (x(τ ), u(x, τ )) denote the instantaneous payoff function for player i at the moment τ ∈ [t0 , T f ]. We assume that h i (x(τ ), u(x, τ )) is continuous in all its arguments. To simplify the notation below, we write h i (x(τ ), u(x, τ )) as h i (τ ). Thus, the player’s i payoff, taking into account the random component of the model, can be written as a mathematical expectation of the payoff: ⎡ ⎢ K i (x0 , t0 , T f , u(x, t)) = E ⎣

T f

⎡ ⎤ T f T f ⎢ ⎥ ⎥ h i (τ )dτ ⎦ = ⎣ h i (τ )dτ ⎦ d F(t). ⎤

t0

Tstar t

t

Theorem 45.1 Under above Assumptions, the payoff of the i-th player in the game  T (x0 , t0 , T f ) is  Tf K i (x0 , t0 , T f , u(x, t)) = h i (t)F(t)dt. t0

Proof Denote

 Hi (t) =

Tf

h i (τ )dτ.

t

As Hi (t0 )—finite, F(t0 ) = 0, F(T f ) = 1, Hi (T f ) = 0, the functional K i (·) can be simplified by integrating by parts as follows:

K i (x0 , t0 , T f , u(x, t)) =

T [Hi (t)F(t)]t0 f



Tf

−

F(t)d H (t)

t0



Tf

= H (T f )F(T f ) − H (t0 )F(t0 ) −

F(t)d H (t)

t0



Tf

= 0 · 1 − H (t0 ) · 0 −



Tf

F(t)d H (t) = −

t0

F(t)d H (t).

t0

By the Leibnitz formula, we have 

Tf

K i (x0 , t0 , T f , u(x, t)) = −

F(t)(0 + h i (T f ) · 0 − h i (t) · 1)dt

t0



Tf

− t0

 F(t)(−h i (t) · 1)dt = t0

Tf

h i (t)F(t)dt.

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Fig. 45.1 An illustration for the definition of conditional probability Fθ (t)

Consider a subgame  T (x(θ ), θ, T f ) starting in the moment θ ∈ [t0 ; T f ] from some point x(θ ) of trajectory of the game  T (x0 , t0 , T f ). It is clear that the CDF (the cumulative distribution function) Fθ (t) in the subgame  T (x(θ ), θ, T f ) can be interpreted as the conditional probability of the game starting before the moment t under condition of the game not starting before θ . By the law of total probability and using some algebraic transformations, we get (Fig. 45.1): Fθ (t) =

F(t) − F(θ ) . 1 − F(θ )

Denote, x ∗ (θ ) as the optimal trajectory’s state in the time instant θ . Now let’s obtain the payoff in the subgame as mathematical expectation ⎡ ⎤ T f T f ⎢ ⎥ F(t) − F(θ ) . K i (x ∗ (θ ), θ, T f , u(x, t)) = ⎣ h i (τ )dτ ⎦ d 1 − F(θ ) θ

t

 Hi (t) =

Tf

h i (τ )dτ.

t

As Hi (θ ) - finite, F(t0 ) = 0, F(T f ) = 1, Hi (T f ) = 0, the functional K i (·) can be simplified by integrating by parts as follows: 

 Tf F(t) − F(θ ) F(t) − F(θ ) T f d Hi (t) K i (x ∗ (θ ), θ, T f , u(x, t)) = Hi (t) − 1 − F(θ ) θ 1 − F(θ ) θ  Tf F(T f ) − F(θ ) F(t) − F(θ ) F(θ ) − F(θ ) − Hi (θ ) − d Hi (t) = Hi (T f ) 1 − F(θ ) 1 − F(θ ) 1 − F(θ ) θ  Tf  Tf  Tf F(t) − F(θ ) F(t) − F(θ ) (−h i (t))dt = h i (t)dt. h i (t)dt · 0 − =0·1− 1 − F(θ ) 1 − F(θ ) θ θ θ

This finally yields K i (x ∗ (θ ), θ, T f , u(x, t)) =



Tf θ

F(t) − F(θ ) h i (t)dt. 1 − F(θ )

45 Dynamic Programming Equations for the Game-Theoretical …

413

45.3 Dynamic Programming Equations Let us consider a non-cooperative case of the game. Assume that all players are not interested in cooperation and aim at only maximizing of their own payoffs. K i (x0 , t0 , T f , u(x, t)) → max, ∀i ∈ N . ui

(45.2)

Richard Bellman in [1] proposed a dynamic programming method to find a solution of maximization problem. This method is based upon the so-called “optimality principle”. The dynamic programming method breaks this decision problem into smaller subproblems. Principle of Optimality: An optimal strategy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal strategy with regard to the state resulting from the first decision. That is, the solution (corresponding trajectory) is optimal in any given subgame. There is a generalization of this method to continuous problems, called the HJB equation. Definition 45.1 If a profile of strategies (u 1N E (x, t), . . . , u nN E (x, t)) responds to the following condition ∀u j ∈ U j , ∀ j ∈ N K j (t0 , x0 , u 1N E (x, t), . . . , u Nj E (x, t), . . . , u nN E (x, t)) ≥ E E (x, t), u j (x, t), u Nj+1 (x, t), . . . , u nN E (x, t)), ≥ K j (t0 , x0 , u 1N E (x, t), . . . , u Nj−1

then this set is called Nash Equilibrium. In the non-cooperative case, in the differential game, the solution of the HJB equation will give the profile of the strategies that is the Nash equilibrium. As we know, from the above results, the conditional probability for the game initial time is F(t) − F(θ ) . Fθ = 1 − F(θ ) Thus, we can write the expression also for the moment θ +  Fθ+ =

F(t) − F(θ + ) Fθ (t) − Fθ (θ + ) = . 1 − Fθ (θ + ) 1 − F(θ + )

For the density function, f θ (t) =

f (t) , 1 − F(θ )

f θ+ (t) =

f (t) , 1 − F(θ + )

f θ (t) =

Suppose the value function for player i is the following:

1 − F(θ + ) f θ+ (t). 1 − F(θ )

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Wi (x(θ ), θ ) = max u∈U

⎧ T ⎪ ⎨ f ⎪ ⎩

T f f θ (t)

θ

h i (x(τ ), u(τ ))dτ dt t

⎫ ⎪ ⎬ ⎪ ⎭

.

(45.3)

For the moment θ + , Wi (x(θ + ), θ + ) = max u∈U

⎧ T ⎪ ⎨f ⎪ ⎩

T f f θ+ (t)

θ+

h i (x(τ ), u(τ ))dτ dt t

⎫ ⎪ ⎬ ⎪ ⎭

.

We can rewrite (45.3)

Wi (x(θ), θ) = max u∈U

T f +

⎧ ⎪ ⎨ θ+ ⎪ ⎩

T f f θ (t)

θ

h i (x(τ ), u(τ ))dτ dt t

T f f θ (t)

θ +

h i (x(τ ), u(τ ))dτ dt t

1 − F(θ + ) + 1 − F(θ)

= max u∈U

⎧ ⎪ ⎨ θ+ ⎪ ⎩

θ

T f

⎫ ⎪ ⎬ ⎪ ⎭

T f f θ (t) t

u∈U

⎪ ⎩

 u∈U

θ

h i (x(τ ), u(τ ))dτ dt t

h i (x(τ ), u(τ ))dτ dt t

⎫ ⎪ ⎬ ⎪ ⎭

⎫ ⎪ ⎬ 1 − F(θ + ) Wi (x(θ + ), θ + ) . h i (x(τ ), u(τ ))dτ dt + ⎪ 1 − F(θ) ⎭

We divide by  and take into account that 0 = max

T f f θ (t)

T f f θ + (t)

θ +

= max

⎧ ⎪ ⎨ θ+

1−F(θ+) 1−F(θ)

=1+

F(θ)−F(θ+) , 1−F(θ)

1 (Wi (x(θ + ), θ + ) − Wi (x(θ), θ)) 

1 F(θ) − F(θ + ) 1 Wi (x(θ + ), θ + ) +  1 − F(θ) 

θ+ 

T f f θ (t)

θ

 h i (x(τ ), u(τ ))dτ dt .

t

Aiming  to 0, we obtain 

∂ Wi (t, x) ∂ Wi (t, x) f (t) + g(x(t), u(x, t)) − Wi (x(t), t) ∂t ∂x 1 − F(t) [t,T f ] ⎫ ⎪ T f ⎬ f (t) h i (x(τ ), u(τ ))dτ . + ⎪ 1 − F(t) ⎭

0 = max  u∈U 

t

45 Dynamic Programming Equations for the Game-Theoretical …

415

Finally, we get the following HJB equation: 

∂ Wi (t, x) g(x(t), u(x, t)) ∂x [t,T f ] ⎫ ⎪ T f ⎬ f (t) + h i (x(τ ), u(τ ))dτ . ⎪ 1 − F(t) ⎭

∂ Wi (t, x) f (t) Wi (x(t), t) = + max  1 − F(t) ∂t u∈U 

t

Wi (x(t0 ), t0 ) = max u∈U

⎧ T ⎪ ⎨ f ⎪ ⎩

h i (x(τ ), u(τ ))F(τ )dτ

t0

⎫ ⎪ ⎬ ⎪ ⎭

.

Thus, the following theorem is true: Theorem 45.2 Assume there exists such a continuous differentiable by all its argument function Wi (x(t), t) which satisfies ⎫ ⎧ T ⎪ ⎪ ⎬ ⎨ f Wi (x(t0 ), t0 ) = max h i (x(τ ), u(τ ))F(τ )dτ , ⎪ u∈U ⎪ ⎭ ⎩ t0



f (t) ∂ Wi (t, x) Wi (x(t), t) = + max  1 − F(t) ∂t u∈U 

∂ Wi (t, x) g(x(t), u(x, t)) ∂x [t,T f ] ⎫ ⎪ T f ⎬ f (t) + h i (x(τ ), u(τ ))dτ , ⎪ 1 − F(t) ⎭ t

under the condition of limτ →T f W (x(τ ), τ ) = 0 and there exists such an admissible T control u ∗ which obtains maximum for t0 f h i (x(τ ), u(τ ))dτ and for f (t) ∂ Wi (t, x) g(x(t), u(t)) + ∂x 1 − F(t) then u ∗ is an optimal control in (45.2).

T f h i (x(τ ), u(τ ))dτ, t

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45.4 Conclusion In this paper, the class of differential games with random initial time is considered. The form of payoff function for this class of games is obtained. A way to find an optimal solution was proposed using the HJB equation. In the future, authors plan to consider the games with several players each of all has its own random variable of the moment of entering the game and develop obtained results. Acknowledgements This work was supported by the grant 17-11-01093 of Russian Science Foundation.

References 1. Bellman, R.: Dynamic programming and a new formalism in the calculus of variations. Proc. Natl. Acad. Sci 40(4), 231–235 (1954) 2. Boukas, E.K., Haurie, A., Michel, P.: An optimal control problem with a random stopping time. J. Optim. Theory Appl. 64(3), 471–480 (1990) 3. Breton, M., Zaccour, G., Zahaf, M.: A differential game of joint implementation of environmental projects. Automatica 41(10), 1737–1749 (2005) 4. Chang, F.R.: Stochastic Optimization in Continuous Time. Cambridge University, Press (2004) 5. Dockner, E., Jorgensen, S., Long, N.V., Sorger, G.: Differential Games in Economics and Management Science. Cambridge University Press, Cambridge (2000) 6. Gromova, E., Lopez-Barrientos, J.D.: A differential game model for the extraction of nonrenewable resources with random initial times the cooperative and competitive cases. Int. Game Theory Rev. 18(2), 1640004 (2016) 7. Gromov, D., Gromova, E.V.: Differential games with random duration: a hybrid systems formulation. Contrib. Game Theory Manag. 7, 104–119 (2014) 8. Gromova, E.V., Malakhova, A., Palestini, A.: Payoff distribution in a multi-company extraction game with uncertain duration. Mathematics 6 (2018). https://doi.org/10.3390/math6090165.s 9. Gromova, E.V., Tur, A.V., Balandina, L.I.: A game-theoretic model of pollution control with asymmetric time horizons. Contrib. Game Theory Manag. 9, 170–179 (2016) 10. Kordonis, I., Papavassilopoulos, G.P.: LQ nash games with random entrance: an infinite horizon major player and minor players of finite horizons. IEEE Trans. Autom. Control 60, 1486–1500 (2015) 11. Kostyunin, S., Palestini, A., Shevkoplyas, E.V.: On a nonrenewable resource extraction game played by asymmetric firms. JOTA 163, 660–673 (2014) 12. Kostyunin, S., Shevkoplyas, E.: On simplification of integral payoff in differential games with random duration. Vestnik S. Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr. 4, 47–56 (2011) 13. Marin-Solano, J., Shevkoplyas, E.V.: Non-constant discounting and differential games with random time horizon. Automatica 47(12), 26262638 (2011) 14. Petrosjan, L.A., Murzov, N.V.: Game-theoretic problems of mechanics. Litovsk. Mat. Sb. 6, 423–433 (1966). (in Russian) 15. Petrosjan, L.A., Shevkoplyas, E.V.: Cooperative solutions for games with random duration. Game Theory Appl. Nova Science Publishers IX, 125–13 (2003) 16. Pliska, S.R., Ye, J.: Optimal life insurance purchase and consumption/investment under uncertain lifetime. J. Bank. Finance 31(5), 1307–1319 (2007)

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17. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishenko, E.: Matematicheskaya teoriya optimal’nykh protsessov. Nauka (1983) 18. Yaari, M.E.: Uncertain lifetime, life insurance, and the theory of the consumer. Rev. Econ. Stud. 32(2), 137–150 (1965)

Chapter 46

Two Echelon Supply Chain: Market Search Behavior and Dependent Demands Suriya Kumacheva and Victor Zakharov

Abstract Nowadays, supply chain management (SCM) is one of the most popular and intensively developed areas of applied mathematics. The necessity of studying and modelling the processes of interaction between the manufacturer, suppliers, and final purchasers of products requires such mathematical tools as game theory and applied statistics. Like many earlier works, the presented paper considers a model of the supply chain with one manufacturer and two retailers. The game-theoretical approach is applied to design and study two echelon supply chain model with market search behavior and dependent demands of customers. At the same time, retailers play Cournot game. Retailers’ demands supposed to be mutually dependent random variables which joint distribution is assumed to be known. Constructive method to find Nash Equilibrium in pure strategies for two-echelon supply chain model with market search behavior of retailers and dependent demands of customers is proposed.

46.1 Introduction The distribution of some specific product between final buyers is one of the most important problem of modern business, trade, and economics. Therefore supply chain management (SCM) focuses on the optimization of the total process from its original state to the customer. In order to increase economic efficiency [7] of each supply chain, it is necessary to take into account such factors as cost reduction, satisfaction of demand for final products, that is synchronizing supply with demand. The practical value and relevance of problems related to supply chain management is reflected in such works as [1, 8, 9]. Game-theoretical approach to the modeling of various dynamic processes is one of the most popular attitudes in modern analysis in economy, sociology, and manageS. Kumacheva (B) · V. Zakharov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. Zakharov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_46

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ment. Last decades SCM developed very intensively and game theory has become an important tool in the analysis of supply chains with multiple agents, often with conflicting interests [3, 12]. A lot of models investigate a simple system with one supplier and two retailers to illustrate game-theoretic approach to supply chain analysis. So, for example, model presented in [4] considers a restriction in capacities of the manufacturer and possibility of a stockout and re-distribution of the customers between retailers. The idea to analyze one manufacturer and two retailers supply chain as a basic model was further developed in the works [6, 7, 15]. The problem of competition between two retailers [15] makes it necessary to coordinate actions and distribute revenue between retailers. In this case, retailers are considered as two opponents in the competition model, and the manufacturer as a leader playing the Stackelberg game with both of them. For the models with such structure in [4], it was shown that the equilibrium between retailers exist and it is the unique Nash Equilibrium. In [6], the study of the coordination mechanism for a supply chain of the same structure is continued. The demand, which can be different for every retailer, is one of the factors of influence on this coordination. Starting with the earliest game theoretic models [10], it is familiar to consider demand as a random variable. In [7], the retailers were represented by their random demand and a supplier by a random lead time. In turn, this led to the need to apply queuing theory along with game-theoretical attitude. In the current paper, we study the issue of optimizing the supply chain of one specific product. To illustrate this problem, we consider an example of a twoechelon chain, which includes one product manufacturer and two retailers. A similar model was investigated in the formulation of [16]. In the mentioned paper, both retailers’ demands were assumed to be random and independent of each other. But we suppose, that in general case such an assumption is incorrect and may lead to false conclusions. On the contrary, we suppose that real practice may design various types of dependence between these variables. Thus, we consider their joint distribution as a given function and try to apply our hypothetical knowledge about it to the gametheoretical model formulated in [16]. We assume that this generalization which as an assumption about the dependence of retailers’ demand will further allow us to consider various cases of their joint distribution, also associated with the analysis of real practical possibilities. This fits well with the concept that the applications of mathematical statistics, along with mathematical game theory, are important tools in supply chain management [12, 13]. The paper has the following structure. Section 46.1 represents the overview of the existing problems of SCM, basically focusing on the models of one manufacturer and two retailers with random demand. Section 46.2 formulates the studied model in comparison with the previous model observed in Sect. 46.1. Section 46.3 is devoted to obtaining the conditions of Nash Equilibrium in the designed game. Sect. 46.4 concludes the results of the research.

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46.2 Model Formulation Consider two-echelon supply chain model in which participators are one manufacturer and two retailers. There is one product made by manufacturer and then sold to two retailers. The unit producing cost of this products is c and the wholesale price is w. The market price of products is p which is fixed common for both retailers. Let δi be a random demand for retailer i products, i = 1, 2. Similarly, δ j is a retailer j random demand, j = 3 − i. In contrast to [16], in which retailers’ demands are considered as mutually independent values, the current study suggests that δi and δ j are continuous variables which have joint cumulative distribution function Fδi ,δ j (x, y) (which is equal to zero in the case when at least one demand δi , i = 1, 2 is negative) and joint probability density function f δi ,δ j (x, y). In the case when the local demand of retailer i exceeds its order quantity qi , then the unsatisfied demand (δi − qi ) goes to the retailer j with a probability βi and the retailer i looses sales which unit cost is bi . If the retailer j also does not have enough order quantity to satisfy demand switched from the retailer i, then the manufacturer looses bm which is the cost of brand loyalty. The studied model can be considered as a hierarchical game with rational players. For each player, the aim is to choose positive value of order quantity to maximize his/her own expected profit. Being the high-level player of this hierarchy, the manufacturer chooses the strategy which is to determine the wholesale price w of the production. The retailers choose the order quantities under the price given by the manufacturer. So the manufacturer can be considered as a leader of the game, the retailers’ behavior is formed as the best response on the leader’s strategy. Therefore, the interaction between manufacturer and retailers is the Stackelberg game [11, 14]. In our paper, we suppose wholesales price w to be fixed and only focus on calculation Nash Equilibrium in the Cournot game between retailers. In general, the strategy profile can be considered as the system of variables (w, qi , q j ). But in the framework of the current study, the manufacturer’s control tool is the wholesale price w and the retailers’ control tools are qi and q j . Therefore, for the further analysis, we consider the players’ profits only as functions of their own strategies. Thus, the manufacturer’s total expected profit is m (w) = (w − c)

2  i=1

⎡ + ⎤ 2  qi + bm E ⎣ (δi − qi ) ⎦ .

(46.1)

i=1

The first summand in the Eq. (46.1) represents the profit that manufacturer receive from the sales. The second summand is related to expected loss of brand loyalty cost. In turn, the relationship of retailers can be designed by the Cournot duopoly [11, 14]. The total demand of retailer i, i = 1, 2, consists of the local demand δi and the

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demand switched from the other retailer j, j = 3 − i, when there is no enough stock. Thus, the retailer i expected profit can be expressed as follows:

i (qi ) = pE min{δi + β j (δ j − q j )+ , qi } − bi E (δi − qi )+ − wqi .

(46.2)

46.3 The Equilibrium Conditions Consider the Eq. (46.2) in detail. The first summand is the market price p multiplied by expected value of the expression min{δi + β j (δ j − q j )+ , qi }. Let’s first represent this expression in another form. Consider two cases: 1. If δi + β j (δ j − q j )+ < qi then min{δi + β j (δ j − q j )+ , qi } = δi + β j (δ j − q j )+ and therefore the mentioned expected value is E(δi ) + β j E(δ j − q j )+ (even in the case when δi and δ j are dependent).

2. If δi + β j (δ j − q j )+ ≥ qi then E min{δi + β j (δ j − q j )+ , qi } = qi as qi is determined real constant. Finally, the expected value of the first summand of the retailer i expected profit (46.2) can be computed according to the formula:

E min{δi + β j (δ j − q j )+ , qi } =

= Pr {δi + β j (δ j − q j )+ < qi } E(δi ) + β j E(δ j − q j )+ + +Pr {δi + β j (δ j − q j )+ ≥ qi }qi =

= Pr {δi + β j (δ j − q j )+ < qi } E(δi ) + β j E(δ j − q j )+ − qi + qi . (46.3) Now let’s analyze the value of Pr {δi + β j (δ j − q j )+ < qi }, or, that is equivalent in the conditions of the problem formulated above, the value of Pr {0 ≤ δi + β j (δ j − q j )+ < qi }. Due to the non-negativity of both summands under probability, this expression can be written in the form Fξ (qi ), where ξ = δi + β j (δ j − q j )+ and Fξ (x) is a cumulative distribution function of the variable ξ for its fixed value x = qi . This form represents a convolution of probability distributions in the case the variables δi and (δ j − q j )+ are independent. But we consider another case assuming the joint probability of δi and δ j . To obtain the main result, let’s first prove the auxiliary lemma. Lemma 46.1 The probability Pr {0 ≤ δi + β j (δ j − q j )+ < qi } depends only on the joint distribution of demands δi and δ j , i = j, and the values of the order quantities qi and q j in the following form:

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Pr {0 ≤ δi + β j (δ j − q j )+ < qi } = =

+∞  α1 qi −∞ 0

f δi ,δ j (x, y)d xd y +

1−α1 βj

+q j



+∞  −∞

qj

f δi ,δ j (x, y)d xd y −

1 + q j ) − Fδi ,δ j (0, q j ) + −Fδi ,δ j (α1 qi , 1−α βj 1 +Fδi ,δ j (α1 qi , q j ) + Fδi ,δ j (0, 1−α + q j ), βj

(46.4)

where α1 is a real constant, α1 ∈ (0, 1). Proof First, we denote the right side of the equality (46.4) as . Then, let’s consider two double inequalities: (46.5) 0 ≤ δi < α1 qi and qi ≤ δ j
q j }.

Thus, if we know the joint distribution of the retailers’ random demands δi and δ j , we can obtain the first summand (46.3) of the retailer i expected profit (46.2). The similar way of reasoning can be applied to analysis of the second summand in Eq. (46.2): bi E (δi − qi )+ = bi  = bi

+∞  +∞  qi −∞

+∞  +∞  −∞ −∞

(x − qi )+ f δi ,δ j (x, y)d xd y = 

x f δi ,δ j (x, y)d xd y − qi Pr {δi > qi }

(46.7)

Now we can use the Eqs. (46.4) and (46.7) to compute the first derivative of the retailer i expected profit (46.2) and obtain the extremum conditions: ∂i = p(1 − ) + bi Pr {δi > qi } − w = 0, ∂qi

(46.8)

where  is defined as a right side of the Eq. (46.4). The similar extension can be obtained for the expected profit of retailer j: ∂ j = p(1 − ) + b j Pr {δ j > q j } − w = 0, ∂q j where can be defined as

(46.9)

46 Two Echelon Supply Chain: Market Search Behavior and Dependent Demands

=

α2 q j +∞  0 −∞

f δi ,δ j (x, y)d xd y +

+∞  −∞

1−α2 βi



425

+qi

f δi ,δ j (x, y)d xd y −

qi

2 −Fδi ,δ j ( 1−α + qi , α2 q j ) − Fδi ,δ j (qi , 0) + βi 2 + qi , 0), +Fδi ,δ j (qi , α2 q j ) + Fδi ,δ j ( 1−α βi

(46.10)

where α2 is a real constant α2 ∈ (0, 1). Following proof of the proposition in [16] for the game with independent demands, it is easy to demonstrate that non-negative solution (q1 , q2 ) of the system (46.8)– (46.9) forms Nash Equilibrium in the considered game with dependent demands. Thus, the following theorem holds. Theorem 46.1 Non-negative solution of the system (46.8)–(46.9) forms Nash Equilibrium in two-echelon supply chain model with market search behavior and dependent demands of customers. Remark 46.1 The manufacturer’s equilibrium expected profit m (w) can be defined from (46.1) for any solution of the system (46.8) and (46.9) which can be obtained in general terms if the joint distribution Fδi ,δ j (x, y) is a known function.

46.4 Conclusion In the paper, a new model of two-echelon supply chain with market search behavior and dependent demands of customers has been investigated. The specific feature of the presented model is the dependence of retailers’ demands. For the case when demands joint distribution is supposed to be known, the conditions of the equilibrium are defined. To obtain the final result, we have proved the lemma about representation of probability Pr {0 ≤ δi + β j (δ j − q j )+ < qi }, which depends on a joint distribution of random demands. Further prospects for development of the model involve clarifying the form of the joint distribution function and obtaining the results for the problem in each case.

References 1. Bonci, A., Pirani, M., Longhi, S.: An embedded database technology perspective in cyberphysical production systems. Procedia Manuf. 11, 830–837 (2017) 2. Borovkov, A.A.: Probability Theory. Editorial URSS, Moscow (2009).(in Russian) 3. Cachon, G., Netessine, S.: Game theory in supply chain analysis. In: SimchiLevi, D., Wu, S., Shen, Z. (eds.) Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era. Kluwer, Boston (2004) 4. Dai, Y., Chao, X., Fang, S., Nuttle, H.: Game theoretic analysis of a distribution system with customer market search. Ann. Oper. Res. 135, 223–238 (2005)

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5. Dekking, F.M., Kraaikamp, C., Lopuhaa, H.P., Meester, L.E.: A Modern Introduction to Probability and Statistics: Understanding Why and How. Springer, London Limited (2005) 6. Ghiami, Y., Williams, T.: A two-echelon production-inventory model for deteriorating items with multiple buyers. Int. J. Prod. Econ. 159, 233–240 (2015) 7. Hennetx, J.C., Ardax, Y.: Supply chain coordination; a game theory approach. Eng. Appl. Artif. Intell. 21(3), 399–405 (2008) 8. Kherbach, O., Mocan, M.L.: The importance of logistics and supply chain management in the enhancement of romanian SMEs. Procedia—Soc. Behav. Sci. 221, 405–413 (2016) 9. Kumara, V., Chibuzob, E.N., Garza-Reyesc, J.A., Kumaria, A., Rocha-Lonad, L., LopezTorrese, G.C.: The impact of supply chain integration on performance: evidence from the UK food sector. Procedia Manuf. 11, 814–821 (2017) 10. Parlar, M.: Game theoretic analysis of the substitutable product inventory problem with random demands. Nav. Res. Logist. 35, 397–409 (1988) 11. Pechersky, S.L., Belyaeva, A.A.: Game Theory for Economists. Europian University, St. Petersburg (2001).(in Russian) 12. Sharma, A., Dwivedi, G., Singh, A.: Game-theoretic analysis of a two-echelon supply chain with option contract under fairness concerns. Comput. Ind. Eng. 137, 106096 (2019). https:// doi.org/10.1016/j.cie.2019.106096 13. Sukati, I., Hamid, A.B., Rohaizat, B., Yusoff, RMd.: The study of supply chain management strategy and practices on supply chain performance. Procedia—Soc. Behav. Sci. 40, 225–233 (2012) 14. Tirole, J.: The Theory of Industrial Organization. MIT Press, Cambridge, MA (1988) 15. Yao, Z., Leung, S.C.H., Lai, K.K.: Manufacturer’s revenue-sharing contract and retail competition. Eur. J. Oper. Res. 186, 637–651 (2008) 16. Yuqing, Q., Weihong, Ni., Kuiran, Shi.: Game theoretic analysis of one manufacturer two retailer supply chain with customer market search. Int. J. Prod. Econ. 164, 57–64 (2015)

Chapter 47

Differential and Algebraical Relations in Singular Sets Construction for a One Class of Time-Optimal Control Problems Pavel D. Lebedev and Alexander A. Uspenskii

Abstract We consider the Dirichlet boundary value problem for the Hamilton– Jacobi equation, the minimax solution of which coincides with the optimal result function for one class of time-optimal control problems. The approach to constructing a solution to a boundary value problem applied in this article is based on identifying the conditions for the appearance of a singularity of a solution depending on the geometry of the boundary set and the differential properties of its boundary. The effectiveness of the developed theoretical methods and numerical procedures is illustrated with an example.

47.1 Introduction The problem of constructing a minimax [1] solution of the plane Dirichlet problem for one class of the Hamilton–Jacobi equations is investigated. The boundary value problem is studied in the context of solving the corresponding time-optimal control problem, for which the minimal time function coincides with the minimax solution of the mentioned above Dirichlet problem. The connection between minimax solution of the Hamilton–Jacobi equations and solutions of optimal control problems was studied by many authors (see, for example, [2]). In the present paper, we consider time-optimal control problems for a dynamical system [3] with circular restrictions on velocities and a non-convex target set, which boundary has points, where the curvature is not smooth is broken. The time-minimal function in the problem is studied by the use of the selection of the bisector [4, 5] of the target set. In the plane case, the bisector of the target coincides with the union of dispersing curves. We study properties of pseudo-vertices, which are the characteristic points of the target P. D. Lebedev · A. A. Uspenskii (B) Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya str., 16, Yekaterinburg 620990, Russia e-mail: [email protected] P. D. Lebedev e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_47

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set. The dispersing curve is constructed as an integral curve. Moreover, the initial conditions of the corresponding Cauchy problem are determined by the properties of the pseudo-vertex. One of the numerical characteristics of the pseudo-vertex, the pseudo-vertex marker, determines the initial velocity of a particle which describes a smooth portion of a dispersion curve. The value of the marker depends on the differential properties of the boundary of the target set. Earlier, a number of particular cases were investigated in detail [5, 6]. There was considered a case where the curvature of the target boundary, in the pseudo-vertex is a smooth function [7]. Also, we have investigated the case in which the classical curvature in the pseudo-vertex is not determined, but at the same time, there exist its one-sided non-equal limit values [8]. It is of natural interest to identify the structure of the singularity of the timeoptimal function (its non-smoothness) in a situation, where the target set boundary curvature is defined in the neighborhood of the pseudo-vertex, but is not smooth at this point. In this paper, we find limit, differential and algebraic relations for the coordinates of points generating dispersing curves in the neighborhood of the pseudovertex with the described above curvature properties. It should be emphasized that the case considered in the paper is quite specific, in particular, because of the revealed connection between the dynamic problem and the problem of polynomial algebra. It is proved that the pseudo-vertex marker is the non-positive root of some cubic polynomial whose coefficients are determined by the one-sided derivatives of the curvature of the pseudo-vertex of the target set boundary. Section 47.2 includes the definition of the control problem and connected boundary value problems for the first-order PDE. Sect. 47.3 includes the description of the dispersing curves, pseudo-vertex, and their role in a time-optimal problem. Sect. 47.4 consist of two theorems about the structure of pseudo-vertex in a case of brake of the derivative of the curvature of the boarder of  the target set A. Sect. 47.5 consists of an example of the solution of the time-optimal problem when the curve  is parametrically defined.

47.2 Statement of the Problem and Basic Definitions We consider the time-optimal control problem on the Euclidean plane R2 with the target set A ⊂ R2 . It is required in this problem from the system’s trajectory to reach the set A from the current coordinates x = (x, y) within the shortest possible time. The motion of point x˙ = v (47.1) is given by the control v = (v1 , v2 ), which is restricted by the unit circle centered at the origin, i.e. v ∈ O(0, 1), where O(c, r ) = {x ∈ R2 : x − c ≤ r }, 0 = (0, 0). The control problem is considered as solved, if we construct the optimal result function. For the class of problems which is under consideration, this function is nonsmooth in the case of a nonconvex target set, its level lines contain points of

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derivatives discontinuity, the pointwise union of which forms a set of dispersing curves. The time-optimal problem of dynamics (47.1) is of independent interest. It is worth noting that the development of methods for construction its solution enriches the procedures for constructing nonsmooth solutions of problems of allied branches of mathematics with new approaches. Let us consider the Dirichlet boundary value problem for the Hamilton–Jacobi Eq. [1]: min ∇u(x), v + 1 = 0

v∈O(0,1)

(47.2)

with the boundary condition u|∂ A = 0.

(47.3)

Here, ∂ A is the boundary of the set A, ∇u(x) means the gradient of function u(x) at point x; ·, · is the scalar product of vectors. The minimax solution of the Dirichlet problem (47.2), (47.3) (introduced by A.I. Subbotin [1]) coincides with the optimal result function u(x, y) of the time-optimal problem on the set G = R2 \ A (see [4, Theorem 1]). When x ∈ / A in the time-optimal problem with dynamics (47.1), the optimal control v is the unit norm vector, codirectional with the vector, pointing from point x to the nearest point y on the boundary of the set A in the Euclidean metric. The optimal result function u(x) = u(x, y) (equal to the minimal time for which the point, moving with a given control, can reach A) coincides with the Euclidean distance ρ(x, A) = min{x − y : y ∈ A} from point x = (x, y) ∈ R2 to the set A. Let us consider the fundamental equation of geometrical optics, the eikonal equation  ∂u 2  ∂u 2 + = 1, (47.4) ∂x ∂y which describes the propagation of radiation (in this case in an isotropic medium). In the Dirichlet problem for Eq. (47.4) with the boundary condition (47.3), S.N. Kruzhkov introduced the so-called fundamental (generalized) solution, which differs   from u(x, y) in the sign: u k (x, y) = −ρ (x, y), A (for details, see [9]). Its meaning is the value of the optical path between the point x and the set A. In this paper, we will consider the case of the set A such that its boundary  of which is a planar curve given by the parametric equation  = {y ∈ R2 : y = y(t), t ∈ }.

(47.5)

Here,  ⊆ R is a closed simply connected set, and the mapping y :  → R2 is continuous on , twice differentiable at all interior points of  and thrice differentiable at all interior points of  except possibly for a finite number of points.

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47.3 Dispersing Curves in a Time-Optimal Problem If the set A is convex, the function u(x) = ρ(x, A) is convex on the whole plane R2 and differentiable on sets G = R2 \ A and int A, the only domain of nonsmoothness is the curve  (see [10, Chap. II, Sect. 8]). If the set A is not convex, then u(x) has singular sets on which u(x) loses smoothness. Definition 47.1 We denote the union of all points y ∈ A closest to x in the Euclidean metric as the set  A (x) of projections of point x onto the set A. Definition 47.2 We denote the set of all points for which the set  A (x) consists of two or more elements as the bisector L(A) of the closed set A ⊂ R2 :   L(A) = x ∈ R2 : ∃y1 ∈  A (x), y2 ∈  A (x)(y1 = y2 ) .

(47.6)

According to R. Isaacs classification [11, Example 6.10.1], the bisector is the union of dispersing curves in the time-optimal problem. More than one optimal trajectory originates from each point of this set. At  the sametime, (47.6) can be interpreted as the locus of the centers of circles O x, ρ(x, A) , such that the circumferences bounding them touch the boundary ∂ A of the set A at two or more points. A bisector is a special case of the so-called symmetry sets [12, 13]. We will further denote t1 and t2 as parameters defining the projections of a single point x ∈ L(A) of the bisector of the set A, and x = x(t1 , t2 ) as this point. Let us also introduce the set () of all pairs (t1 , t2 ) of parameters that define the projections of the bisector points. We further assume that we know (we have identified) the connection between the parameters t1 and t2 , and that connection is expressed by a smooth scalar mapping t2 (t1 ). Analytical and computational approaches to the construction of such functions are given in the paper [6]. The characteristic target points (pseudo-vertices) are the case of the initiation of dispersing curves in the time-optimal problem. Definition 47.3 Let us call a pseudo-vertex of the set A the point y0 = y(t0 ), and call the corresponding extreme point of the bisector x0 , if there exists the sequence ∞ {(yn ,  yn )}∞ n=1 ⊂ A of pairs of the set A points and the sequence {xn }n=1 ⊂ L(A) of the bisector points for which the following conditions are met. yn ) = (y0 , y0 ), lim (yn ,

n→∞

lim xn = x0 ,

n→∞

∀n ∈ N (yn , yn ) ⊂  A (xn ). Various types of pseudo-vertices were previously classified in paper [5]. But in the scope of this paper, we restrict ourselves to considering the target set boundary sections that have the second order of smoothness.

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The key characteristic of a plane curve (47.5) is its curvature [14, Chap. III, Sect. 26] calculated by the formula k(t) =

y (t) ∧ y (t) , y (t)3/2

where (a1 , a2 ) ∧ (b1 , b2 ) = a1 b2 − a2 b1 means the pseudoscalar product of vectors. When solving the time-optimal problem, it is necessary to know the behavior pattern of the coordinates of projections of the bisector points in the neighborhood of the pseudo-vertex. This allows us to construct the fragments of dispersing curves in the form of arcs of accumulation curves for the differential equation dt2 = g(t1 , t2 ), dt1

(47.7)

that relates the parameters t1 and t2 , and find the points x(t1 , t2 ) ∈ L(A) based on these parameters [6]. Here, g(t1 , t2 ) = −

k(t1 ) − r −1 (t1 , t2 ) , k(t2 ) − r −1 (t1 , t2 )

  r (t1 , t2 ) = ρ x(t1 , t2 ), A =

y(t2 ) − y(t1 )   2 sin ∠(y (t2 ), y (t1 ))/2

is a function determined on the set () and is equal to the distance from the bisector point x(t1 , t2 ) to the set A, ∠(a, b) is the angle between the vectors a and b. Knowing the coordinates of the pseudo-vertex makes, it possible to find the initial conditions to solve this equation. However, since the pseudo-vertex itself in the general case is not a projection of a point from L(A), it is required to define the value of the 1 ) − t0 function g(t2 , t1 ) for t1 = t2 = t0 as the limit lim t2 (t t − t . Knowing it allows t1 →t0 −0

1

0

us to correctly formulate the Cauchy problem for the differential Eq. (47.7).

47.4 Structure of Optimal Trajectories in the Neighborhood of the Pseudo-vertex Consider the complicated case of the pseudo-vertex y0 = y(t0 ) at which the curvature k(t) of the curve  is determined, but its one-sided derivatives on the left and on the right do not coincide. This means that at the point t0 there is a discontinuity of the third-order derivative y (t) of the vector function y(t). Theorem 47.1 Let y0 = y(t0 ) be the pseudo-vertex of the set A. Let y  (t0 ) = 0 and let one-sided limits k  (t0 − 0) and k  (t0 + 0) of the curvature derivative are determined, and k  (t0 + 0) = 0. Then, if the following limit relation is satisfied

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t2 (t1 ) − t0 =c t1 →t0 −0 t1 − t0 lim

(47.8)

for the parameters t1 < t0 < t2 , where t2 = t2 (t1 ), which specify the coordinates of the projections of the bisector points, the value (47.8) (the so-called pseudo-vertex marker c) is equal to the negative or zero root of the equation k  (t0 + 0)λ3 − 3k  (t0 + 0)λ2 + 3k  (t0 − 0)λ − k  (t0 − 0) = 0.

(47.9)

Theorem 47.2 Let y0 = y(t0 ) be the pseudo-vertex of the set A at which the conditions of Theorem 47.1 are satisfied. Then, among the solutions of Eq. (47.9) there is exactly one real negative root. Theorem 47.2 guarantees that the value of function (47.7) is uniquely determined for t1 = t2 = t0 . Therefore, the solution of the Cauchy problem

dt2 =  g (t1 , t2 ), dt1 t2 (t0 ) = t0 ,



where  g (t1 , t2 ) =

(47.10)

g(t1 , t2 ), t1 < t0 < t2 ; c, t1 = t2 = t0 ;

in which the initial condition has the form t2 (t0 ) = t0 , is also unique (in some neighborhood of the point t0 ).

47.5 Example of Solving a Time-Optimal Problems The software package [17] developed by the authors was improved to be able to solve time-optimal problems with the boundary of the target set having pseudovertices at which the curvature is determined, but its derivative has a discontinuity. The dispersing curve is constructed by solving the Cauchy problem (47.10). We consider the time-optimal problem with dynamics (47.1), in which the tar2 get  set A is  bounded upside from above a curve  = {y(t) ∈ R : t ∈ R}, y(t) = x(t), y(t) , x(t) =



t, t ≤ 0, et − 1,

t > 0,

y(t) =

t 3 /3 + t 2 /2 − t, t ≤ 0, −t, t > 0.

(47.11)

It is required to construct the optimal result function in the form of level lines of the optimal result function u(x, y).

47 Differential and Algebraical Relations in Singular Sets Construction … y

433

5

4

3

2

1

Γ L Φ

0

−1

−2

x 0

2

4

Fig. 47.1 Curve , level lines of the optimal result function u(x, y) and the singular set L(A).

Note that the function y(t) is continuously differentiable. The analysis of the boundary  of the target set shows that it contains one pseudo-vertex (when t0 = 0) y0 = y(t0 ) = (0, 0)—the point of the curvature’s local maximum. The corresponding extreme point of the √ bisector has coordinates (2, 2). The curvature calculated at point y0 is k(0) = 2/4. The one-sided √ derivatives of the √curvature calculated at the pseudo-vertex are equal k  (−0) = 7 2/8, k  (+0) = − 2/8. The difference in the one-sided derivatives of curvature at the point y(t0 ) is due to the discontinuity of the highest derivatives of the functions x(t) and y(t). Therefore, conditions of Theorem 47.1 hold. A polynomial (47.9), defining the roots, among which the value c lies, is √ √ √ √ 7 2 3 2 2 7 2 2 λ −3· λ −3· λ+ = 0. 8 8 8 8

(47.12)

The array of the approximate roots of Eq. (47.12) is C = {3.1226, −0.2838, 0.1612}. According to Theorem 47.2, C contains exactly one real negative number c ≈ −0.2838. The curve , given by Eq. (47.11), the dispersing curve L(A), and the level lines

of the optimal result function are shown in Fig. 47.1.

47.6 Conclusion The time-optimal problem with a circular velocity restrictions and features of the curvature of the boundary  of the target set is investigated in the paper. The features are that the curvature k(t) is determined, but its one-sided derivatives on the left

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k  (t0 − 0) and on the right k  (t0 + 0) do not coincide at the pseudo-vertices y0 = y(t0 ). This situation is, in a sense, intermediate between the case of a pseudo-vertex at which the curvature is determined and is differentiable, and the case when the one-sided limits of curvature on the left k(t0 − 0) and on the right k(t0 + 0) are determined and they differ (for more detailed information on the latter case see [8, Theorem 3]). A relation of the limit relation (marker) c is established for the coordinates of projections of points of the bisector L(A) in the neighborhood of y0 with the solution of the cubic equation, whose coefficients depend on one-sided derivatives k  (t0 − 0) and k  (t0 + 0) of the curvature. Acknowledgements Pavel D. Lebedev’s research is supported by the Russian Science Foundation (project no. 19–11–00105).

References 1. Subbotin, A.I.: Generalized Solutions of First Order PDEs: The Dynamical Optimization Perspective, vol. 314. Boston, Birkhäuser, XII (1995). https://doi.org/10.1007/978-1-46120847-1 2. Subbotina, N.N., Kolpakova, E.A., Tokmantsev, T.B., Shagalova, L.G.: Metod kharakteristik dlya uravneniya Gamiltona-Yakobi-Bellmana. Method of characteristics for HamiltonJacobiBelman’s equation. Yekaterinburg: Ural Branch of RAS, 244 (2013) 3. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems, vol. 517. Springer, New York (1988). ISBN 978-1-4612-8318-8 4. Lebedev, P.D., Uspenskii, A.A.: Analytical and numerical construction of the optimal outcome function for a class of time-optimal problems. Comput. Math. Model. 19(4), 375–386 (2008). https://doi.org/10.1007/s10598-008-9007-9 5. Lebedev, P.D., Uspenskii, A.A.: Geometry and asymptotics of wavefronts. Russ. Math. 52(3), 24–33 (2008). https://doi.org/10.3103/S1066369X08030031 6. Lebedev, P.D., Uspenskii, A.A.: Construction of the optimal result function and dispersing lines in time-optimal problems with a nonconvex target set. Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 22(2), 188–198 (2016). (in Russian). https://doi.org/10.21538/0134-48892016-22-2-188-198 7. Uspenskii, A.A., Lebedev, P.D.: Construction of the optimal outcome function for a timeoptimal problem on the basis of a symmetry set. Autom. Remote. Control 70(7), 1132–1139 (2009). https://doi.org/10.1134/S0005117909070054 8. Lebedev, P.D., Uspenskii, A.A.: Construction of a nonsmooth solution in a time-optimal problem with a low order of smoothness of the boundary of the target set. Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 25(1), 108–119 (2019). (in Russian). https://doi.org/10.21538/01344889-2019-25-1-108-119 9. Kruzhkov, S.N.: Generalized solutions of the Hamilton–Jacobi equations of eikonal type. I. Formulation of the problems; existence, uniqueness and stability theorems; some properties of the solutions. Math. USSR– Sb. 27(3), 406–446 (1975). https://doi.org/10.1070/ SM1975v027n03ABEH002522 10. Dem’yanov, V.F., Vasil’ev, L.V.: Nedifferentsiruemaya optimizatsiya. In: Non-differentiable Optimization, vol. 384. Moscow, Nauka (1981) 11. Isaacs, R.: Differential Games. Wiley, N.Y. (1965) 12. Arnold, V.I.: Singularities of caustics and wave fronts. Springer, Dordrecht (1990). https://doi. org/10.1007/978-94-011-3330-2

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13. Sedykh, V.D.: On the topology of wave fronts in spaces of low dimension. Izv.: Math. 76(2), 375–418 (2012). https://doi.org/10.1070/IM2012v076n02ABEH002588 14. Rashevskii, P.K.: A Course in Differential Geometry. URSS, Moscow (2003) 15. Ushakov, V.N., Uspenskii, A.A., Lebedev, P.D.: Geometry of singular curves for one class of velocity. Vestnik Sankt-Peterburgskogo Universiteta. Seriya 10. Prikladnaya Matematika. Informatika. Protsessy Upravleniya (3), 157–167 (2013). (in Russian) 16. Vinberg, E.B.: Algebra Mnogochlenov. Algebra of polynomials, Prosveshchenie, Moscow (1980) 17. Lebedev, P.D., Uspenskii, A.A.: Program for constructing wave fronts and functions of the Euclidean distance to a compact nonconvex set. Certificate of state registration of the computer program no. 2017662074 (2017)

Chapter 48

Subgame Perfect Pareto Equilibria for Multicriteria Game with Chance Moves Denis Kuzyutin, Nadezhda Smirnova, and Igor Tantlevskij

Abstract For n-person multicriteria game with chance moves in extensive form, we prove the existence of pure strategy subgame perfect Pareto equilibrium (SPPE). Then we provide and demonstrate an algorithm which allows to reasonably select and construct a unique SPPE.

48.1 Introduction We deal with the n-person multicriteria extensive-form games with chance moves (MEGCM) and perfect information (see, e.g. [2, 5–7, 9, 19]). The main challenge here is that due to the chance moves occurrence any pure strategy profile does not create a unique history in the game tree but rather the whole set or bunch of histories. In the paper, we use and expand the extensive-form game specification from [11] and the Attitude SPE algorithm from [12] which was originally published open access under a CC BY 4.0 license at https://doi.org/10.3390/math8071061. However, we differ from [12] in at least two respects. First, we consider a multicriteria game, whereas in [12], all the results are obtained for single-criterion games. Second, we focus on the (subgame perfect) Pareto equilibria existence and its refinement, whereas paper [12] deals with construction of a unique and subgame perfect Nash equilibrium (as well as with subgame consistent cooperative behavior). D. Kuzyutin · N. Smirnova (B) · I. Tantlevskij St. Petersburg State University, 7/9, Universitetskaya nab., Saint Petersburg 199034, Russia e-mail: [email protected] D. Kuzyutin e-mail: [email protected] I. Tantlevskij e-mail: [email protected] D. Kuzyutin · N. Smirnova National Research University Higher School of Economics (HSE), Soyuza Pechatnikov ul. 16, 190008 Saint Petersburg, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_48

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We also adopt and develop the minimal sum of relative deviation (MSRD) approach from [10] which was originally published open access under a CC BY 4.0 license at https://doi.org/10.1016/j.orp.2019.100107. However, we differ from [10] since we explore the non-cooperative behavior for the games with chance moves, whereas [10] focuses on consistent cooperative behavior in the games without chance moves (which implies different methods and solutions). While we consider the same class of games as in [11], we differ from this related research since in [11] noncooperative behavior, Pareto equilibria and its refinement are not at all on the menu, whereas here, they are the main focus. The contribution of this paper is twofold: – We prove that any finite n-person multicriteria game with chance moves and perfect information possesses a (subgame perfect) Pareto equilibrium in pure strategies. – We provide an SPPE refinement in the form of an algorithm that allows to reasonably select a unique pure strategy (subgame perfect) Pareto equilibriium. It is worth noting that a Pareto equilibria refinement, called μ-SPPE, is constructed as a result of the MSRD approach [10] extension and the Attitude SPE algorithm [12] application to multicriteria games with chance moves. Section 48.2 recalls the main concepts and notations for the class of MEGCM. In Sect. 48.3, we prove the existence of pure strategy subgame perfect Pareto equilibrium for multicriteria games with chance moves and specify a refinement of SPPE as well as an algorithm how to construct this non-cooperative solution. In Sect. 48.4, we demonstrate the algorithm implementation using an example of bicriteria game in extensive form. The conclusions are presented in Sect. 48.5.

48.2 Pareto Equilibria in Multicriteria Game Let us briefly remind the following notations for a class of MEGCM with perfect information [9–12, 17] which will be used throughout the paper: • N = {1, . . . , n} denotes the players’ set; • K is the finite game tree with the set of all nodes (positions) P and the game tree root x0 ; • S(x) denotes the set of all direct successors of the position x, while the unique parent of y = x0 is denoted by S −1 (y); • P0 is the set of all nodes at which a chance moves, where π(y|x) > 0 denotes the probability of transition from node x ∈ P0 to node y ∈ S(x); • Pi is the set of player i’s decision nodes, Pi ∩ P j = ∅ for i = j, while Pn+1 = {y j }mj=1 denotes the (ordered) finite set of terminal nodes, i.e. S(y j ) = ∅ ∀y j ∈ n+1 Pi = P; Pn+1 ; ∪i=0 • ω = (x0 , . . . , xt−1 , xt , . . . , x T ) is the history (or trajectory) in the game tree, xt−1 = S −1 (xt ), 1  t  T ; x T = y j ∈ Pn+1 , where t in xt equals to the ordi-

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nal number of this node within the history ω and could be considered as the “time index”; • h i (x) = (h i/1 (x), . . . , h i/r (x)) denotes the (vector) payoff of the i-th player at current position x ∈ P, where r is a number of criteria. We suppose that ∀i ∈ N ; k = 1, ..., r the payoffs are positive, i.e. h i/k (x) > 0 for all nodes x ∈ P. Denote by M E GC M(n, r ) the class of multicriteria extensiveform games with chance moves, perfect information [4, 13, 17] and positive stage payoffs, while  x0 denotes a game with  root x0 . Let Ui denote the set of all pure strategies of the i-th player, and U = i∈N Ui denote a (finite) set of all possible pure strategy profiles. Because of the chance moves occurrence, in general each pure strategy profile creates the finite set (or the bunch) of histories (u) = {ωk (u)| p(ωk , u) > 0}, where p(ωk , u) denotes the probability of realization of the history ωk = (x0 , . . . , xτ , xτ +1 , . . . , x T (k) ), x T (k) ∈ Pn+1 , xτ +1 ∈ S(xτ ), τ = 0, . . . , T (k) − 1, when the players use the strategies u i from the strategy profile u = (u 1 , . . . , u n ) (see [11, 12] for details). T  h i (xτ ) denote the i-th player’s vector payoff corresponding to Let h˜ i (ω) = τ =0

the history ω = (x0 , . . . , xt , xt+1 , . . . , x T ). Then we can calculate the (expected) value of the vector payoff function of each player i ∈ N given strategy profile u = (u 1 , . . . , u n ) 

Hi (u) =

p(ωk , u) · h˜ i (ωk ) =

ωk ∈(u)



p(ωk , u) ·

T (k) 

h i (xτ ).

(48.1)

τ =0

ωk ∈(u)

For each intermediate node xt ∈ P\Pn+1 denote by u ixt the restriction of the player i’s pure strategy u i (·) in  x0 on the subgame  xt with the subgame root xt . The strategy profile u xt = (u 1xt , . . . , u nxt ) generates the bunch of the subgame histories xt (u xt ) = {ωkxt (u xt )| p(ωkxt , u xt ) > 0}. Similarly to (48.1), let

Hixt (u xt )



= p(ωkxt , u xt ) x ωk t ∈xt (u xt )

·

T (k)  τ =t

h i (xτ ) =



p(ωkxt , u xt ) · h˜ i (ωkxt )

x

ωk t ∈xt (u xt )

denote the value of the i-th player’s payoff function in the subgame  xt , while Uixt  xt xt is the set of all i-th player’s pure strategies in  , and U = i∈N Uixt . Given a, b ∈ R m vector inequality a ≥ b means that ak  bk , ∀k = 1, . . . , m, while at least one inequality of m is strict. The concept of Nash equilibria has been extended to multicriteria games in [21]. Definition 48.1 The strategy profile u = (u¯ 1 , . . . , u¯ n ) is called Pareto equilibria (or strong Pareto equilibria) [6, 21–23] in multicriteria game  x0 ∈ M E GC M(n, r ) with chance moves, if ∀i ∈ N u i ∈ Ui :

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Hi (u i , u¯ −i ) ≥ Hi (u¯ i , u¯ −i ).

(48.2)

Note that for single-criterion game  x0 ∈ M E GC M(n, 1) Pareto equilibria coincides with Nash equilibria. Denote by P E( x0 ) the set of all Pareto equilibria (in pure strategies) for the game  x0 . Definition 48.2 The strategy profile u = (u¯ 1 , . . . , u¯ n ) is called subgame perfect Pareto equilibria (SPPE) [6, 20] in game  x0 ∈ M E GC M(n, r ) if u x = (u¯ 1x , . . . , u¯ nx ) ∈ P E( x ) at each subgame  x , x ∈ P \ P n+1 . Let S P P E( x0 ) denote the set of all subgame perfect Pareto equilibriums (in pure strategies) for the game  x0 ∈ M E GC M(n, r ).

48.3 Choosing a Unique Subgame Perfect Pareto Equilibrium Any multistage multicriteria game with perfect information (and without chance moves) was proved to possess a subgame perfect Pareto equilibria in pure strategies (see [6]). To extend this result to the class of MEGCM, we first need to consider single-criterion multistage game  x0 ∈ M E GC M(n, 1) with chance moves. The backward induction procedure ([4, 13, 17]) is an efficient algorithm to construct subgame perfect equilibrium (SPE) in a game with perfect information, however, it may generate multiple subgame perfect equilibriums with different players’ payoffs (see, e.g. [3, 13, 17]). The “lack of refinement procedures that can select a unique SPE” (even for extensive-form games without chance moves) was recently noted in [1]. At the same time, to make a correct prediction on the players’ non-cooperative behavior in a game  x0 ∈ M G cm (n, 1), one need to use some rule or refinement procedure for selecting a unique SPE that creates a unique optimal bunch of histories. To this aim one may adopt, in particular, the so-called Attitude SPE algorithm, which takes into account a system of the players’ attitudes to each other, formalized via the players’ attitude vectors (see [12, 17] for details and rigorous algorithm specification). Proposition 48.1 [12]. The Attitude SPE algorithm allows to construct a unique SPE u = (u¯ 1 , . . . , u¯ n ) in pure strategies for any game  x0 ∈ M E GC M(n, 1) as well as a unique bunch of histories (u). ¯ Further, we employ the weighted game approach, suggested in [21], Prop.1 and the so-called minimal sum of relative deviations (MSRD) approach [10] to design a unique SPPE in a multicriteria game. Proposition 48.2 A finite multicriteria extensive-form game  x0 ∈ M E GC M(n, r ) possesses a Pareto equilibrium in pure strategies.

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Proof For each multicriteria game  x0 ∈ M G cm (n, r ), one can consider the corresponding single-criterion weighted game λx0 ∈ M G cm (n, 1), where λ = (λ1 (i), . . . , λr (i)), λk (i) > 0, k = 1, . . . , r, i ∈ N , (see [6, 21] for details) that can be obtained from  x0 , if we suppose that Hiλ (u) =

r 

λk (i) · Hi/k (u), i ∈ N .

(48.3)

k=1

The game λx0 ∈ M E GC M(n, 1) with payoff function (48.3) is a finite singlecriterion extensive-form game that possesses a subgame perfect equilibria u = (u¯ 1 , . . . , u¯ n ) due to Prop.1. / P E( x0 ), i.e. u does not satisfy (48.2). Then, for some player Suppose that u ∈ i ∈ N and u j ∈ Ui , we have that Hi/k (u i , u¯ −i )  Hi/k (u¯ i , u¯ −i ), k = 1, . . . , r,

(48.4)

and at least one inequality in (48.4) is strict. If we multiply each inequality in (48.4) by λk (i) and then sum up all the inequalities, we get Hiλ (u i , u¯ −i ) > Hiλ (u¯ i , u¯ −i ). The last inequality contradicts the fact that u ∈ S P E(λx0 ). Hence, the pure strategy  profile u is a Pareto equilibrium in multicriteria game  x0 . Remark 48.1 It follows from Prop.1 and the Attitude SPE algorithm specification that each finite multicriteria game  x0 ∈ M E GC M(n, r ) has a subgame perfect Pareto equilibrium u = (u¯ 1 , . . . , u¯ n ) in pure strategies. However, one can construct different equilibria using different weighted games λx0 and the Attitude SPE algorithm. Hence, to make a precise prediction on the players non-cooperative behavior in multicriteria game  x0 ∈ M E GC M(n, r ), we need to specify the vector of coefficients λ(i) = (λ1 (i), . . . , λr (i)) that the i-th player is expected to use to take into account all the criteria of her vector payoffs. ∗ = max Hi/k (u) denote the maximal possible payoff of the player i in Let Hi/k u∈U

 x0 with respect to criterion k = 1, . . . , r , while H i/k = min Hi/k (u). The vector u∈U   ∗ ∗ can be considered as the ideal (absolutely perfect) payoffs , . . . , Hi/r Hi∗ = Hi/1 ∗ vector for player i ∈ N , and the difference Hi/k − H i/k is the range of the i-th ∗ > H i/k } = ∅ for player’s criterion k values. Assume that θi = {k = 1, . . . , r : Hi/k each i ∈ N . The MSRD rule was introduced in [10] to reasonably select a unique cooperative history in a multicriteria game (without chance moves). Now we suggest to employ similar approach to specify the vector of coefficients which the i-th player should use to properly account for different criteria. Namely, suppose that for each player i ∈ N

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μk (i) =

∗ Hi/k

1 , k ∈ θi , − H i/k

(48.5)

/ θi . Let  u = ( u1, . . . ,  u n ) denote the SPE in weighted game while μk (i) = 1 for k ∈ μx0 ∈ M E GC M(n, 1) that is constructed applying the Attitude SPE algorithm. According to Remark 48.1,  u is a subgame perfect Pareto equilibria in multicriteria game  x0 ∈ M E GC M(n, r ). Remark 48.2 The strategy profile  u ∈ S P P E( x0 ) defined above generates a u )| p(ωk ,  u ) > 0} and uniquely determines unique bunch of histories ( u ) = {ωk ( u ). the players’ behavior at all nodes x ∈ ωk ( We will refer to the Pareto equilibria refinement introduced above as the μ-SPPE. Note that  u obviously satisfies time consistency [3, 5, 14–17].

48.4 The Algorithm Implementation Let us provide an example of bicriteria two-person game to demonstrate how one can construct μ-SPPE using MSRD approach, Prop.2 and the Attitude SPE algorithm for weighted game. Example 48.1 μ-SPPE construction for two-person bicriteria game with chance moves. Let P0 = {x1 , x3 }, P1 = {x0 , x22 , x42 }, P2 = {x21 , x43 }, Pn+1 = {z 1 , . . . , z 9 }. The players’ stage payoffs (the payoffs at each node) and probabilities of transition π(y|x), x ∈ P0 are written near the corresponding node in the game tree (see Fig. 48.1). Note that stage payoffs are given by matrices, where the columns correspond to the players, while the rows correspond to the criteria. First, we calculate ∗ , H i/k and the coefficients μk (i) according to the ideal and the worst payoffs Hi/k (48.5).

Fig. 48.1 Bicriteria two-person extensive-form game

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Fig. 48.2 The weighted game μx0 : construction of μ-SPPE

1 ∗ = 4, H 1/2 = 2, μ2 (1) = ; H1/2 10 1 ∗ = 1, μ1 (2) = ; H2/2 = 10, H 2/2 = 0, μ2 (2) = 4

∗ H1/1 = 10, H 1/1 = 0, μ1 (1) = ∗ = 5, H 2/1 H2/1

1 ; 2 1 . 10

Then, we can calculate the players’ payoffs for the weighted game μx0 (see Fig. 48.2). Finally, we construct μ-SPPE using the backward induction procedure. Note that there exists unique subgame perfect equilibrium  u for weighted game μx0 in Ex.1, hence, the bunch of optimal trajectories ( u ) (marked in bold in Fig. 48.2) does not depend on the players’ attitude vectors. In general (when there exist multiple subgame perfect equilibriums for the weighted game), one needs to employ the Attitude SPE algorithm to select one of them.

48.5 Conclusion The main contribution of this paper is that we proved the existence of Pareto equilibrium (in pure strategies) for MEGCM and introduced a Pareto equilibria refinement— μ-SPPE—which allows to reasonably select a unique SPPE. One interesting feature of the game with chance moves in extensive form is that some conditions of the game (the set of active players, the expected optimal players’ payoffs, the set of optimal trajectories) may change while the game unfolds along a history. Hence, the MEGCM can be considered as an example of “the dynamic games with changing conditions”.

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Acknowledgements The research of D. Kuzyutin and N. Smirnova was funded by RFBR under the research project 18-00-00727 (18-00-00725). The research of I. Tantlevskij was funded by RFBR under the research project 18-00-00727 (18-00-00628).

References 1. Chander, P., Wooders, M.: Subgame-perfect cooperation in an extensive game. J. Econ. Theory 187, 105017 (2020). https://doi.org/10.1016/j.jet.2020.105017 2. Climaco, J., Romero, C., Ruiz, F.: Preface to the special issue on multiple criteria decision making: current challenges and future trends. Intl. Trans. in Op. Res. 25, 759–761 (2018). https://doi.org/10.1111/itor.12515 3. Haurie, A., Krawczyk, J.B., Zaccour, G.: Games and Dynamic Games. Scientific World, Singapore (2012) 4. Kuhn, H.: Extensive games and the problem of information. Ann. Math. Stud. 28, 193–216 (1953) 5. Kuzyutin, D.: On the consistency of weak equilibria in multicriteria extensive games. In: Petosyan, L.A., Zenkevich, N.A. (eds.) Contributions to Game Theory and Management, vol. V, pp. 168–177 (2012) 6. Kuzyutin, D., Nikitina, M., Pankratova, Y.: O Svoystvakh Ravnovesiy v Mnogokriterialnikh Pozicionnikh Igrakh N Lic. Matematicheskaya Teoriya Igr i eyo Prilojeniya 6(1), 19–40 (2014). (in Russian) 7. Kuzyutin, D., Nikitina, M., Razgulyaeva, L.: On the A-equilibria properties in multicriteria extensive games. Appl. Math. Sci. 9(92), 4565–4573 (2015) 8. Kuzyutin, D., Nikitina, M.: An irrational behavior proof condition for multistage multicriteria games. In: Consrtuctive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov), CNSA 2017, Proceedings. IEEE, pp. 178–181 (2017) 9. Kuzyutin, D., Gromova, E., Pankratova, Y.: Sustainable cooperation in multicriteria multistage games. Oper. Res. Lett. 46(6), 557–562 (2018). https://doi.org/10.1016/j.orl.2018.09.004 10. Kuzyutin, D., Smirnova, N., Gromova, E.: Long-term implementation of the cooperative solution in multistage multicriteria game. Oper. Res. Perspect. 6, 100107 (2019). https://doi.org/ 10.1016/j.orp.2019.100107 11. Kuzyutin, D., Gromova, E., Smirnova N.: On the cooperative behavior in multistage multicriteria game with chance moves. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds.) Mathematical Optimization Theory and Operations Research, MOTOR 2020. Lecture Notes in Computer Science, vol. 12095. Springer, Cham (2020). https://doi.org/10.1007/978-3-03049988-4_13 12. Kuzyutin, D., Smirnova, N.: Subgame consistent cooperative behavior in an extensive form game with chance moves. Mathematics 8, 1061 (2020). https://doi.org/10.3390/math8071061 13. Myerson, R.: Game Theory. Harvard University Press, Analysis of Conflict (1997) 14. Parilina, E., Zaccour, G.: Node-consistent core for games played over event trees. Automatica 55, 304–311 (2015) 15. Petrosyan, L.: Stable solutions of differential games with many participants. Vestn. Leningr. Univ. 19, 46–52 (1977). (in Russian) 16. Petrosyan, L.A., Kuzyutin, D.V.: On the stability of E-equilibrium in the class of mixed strategies. Vestnik St. Petersburg Univ. Math. 3(15), 54–58 (1995). (in Russian) 17. Petrosyan, L., Kuzyutin, D.: Games in extensive form: optimality and stability. Saint Petersburg University Press (2000). (in Russian) 18. Podinovskii, V., Nogin, V.: Pareto-optimal solutions of multicriteria problems. Nauka (1982). (in Russian) 19. Puerto, J., Perea, F.: On minimax and Pareto optimal security payoffs in multicriteria games. J. Math. Anal. Appl. 457(2), 1634–1648 (2018). https://doi.org/10.1016/j.jmaa.2017.01.002

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20. Selten, R.: Reexamination of the perfectness concept for equilibrium points in extensive games. Int. J. Game Theory 4, 25–55 (1975) 21. Shapley, L.: Equilibrium points in games with vector payoffs. Nav. Res. Logist. Q. 6, 57–61 (1959) 22. Voorneveld, M., Vermeulen, D., Borm, P.: Axiomatizations of Pareto equilibria in multicriteria games. Games Econom. Behav. 28, 146–154 (1999) 23. Wang, Y.: Existence of a Pareto equilibrium. J. Optim. Theory Appl. 79, 373–384 (1993)

Chapter 49

On Nash Equilibrium in Repeated Hierarchical Games Yaroslavna Pankratova and Leon Petrosyan

Abstract In the paper, non-cooperative and cooperative versions of repeated rhomboidal games with hierarchical structure are investigated. In non-cooperative case as solution concept the Nash Equilibrium is considered. Moreover, a special subclass of Nash equilibrium, based on threat and punishment strategies, is derived. Additionally, we compute the Price of Anarchy (PoA) and the Price of Stability (PoS).

49.1 Introduction Repeated games with hierarchical structure can be used for modeling complicated management systems and multilevel decision problems. We can see applications of hierarchical games in different areas such that the electricity market [14], communication market, or other market models [1, 12], etc. Here, we analyze three-level repeated hierarchical games, namely, the rhomboidal hierarchical games. Following the principal approaches used in the Folk Theorem’s [2, 3, 5, 10, 13] variety of Nash equilibrium are constructed with the use of threat and punishment strategies. Such kind of Nash equilibrium often appears in reality. The cooperative model of behavior is also investigated. A similar but much more simple problem in the case of hierarchical games with tree-like structure is considered in [10]. To define the Price of Anarchy (PoA) and the Price of Stability (PoS), the worst and the best Nash equilibrium are constructed. Under the PoA is understood the ratio between the maximal sum of players payoffs and the sum of players payoffs in the worst Nash equilibrium (Nash equilibrium in which the sum of players payoffs takes its minimal value among other Nash equilibrium outcomes). Under the PoS, we understand the ratio between the sum of payoffs in the best Nash equilibrium (Nash Y. Pankratova (B) · L. Petrosyan St. Petersburg State University, 7/9, Universitetskaya nab., Saint Petersburg 199034, Russia e-mail: [email protected] L. Petrosyan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_49

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equilibrium in which the sum of players payoffs takes its maximal value among other Nash equilibrium outcomes) and the maximal sum of players payoffs. This paper is organized as follows. In Sect. 49.2, we describe the rhomboidal game with m + 2 players A0 , B1 , . . . , Bm , C and present the solution for such game as a Nash equilibrium [7]. In Sect. 49.3, we construct the repeated hierarchical game and find Nash equilibrium in punishment strategies and Nash equilibrium in threat strategies. In Sect. 49.4, we construct a cooperative version of the infinitely repeated game and calculate the PoA and PoS [6, 11]. We briefly conclude in Sect. 49.5.

49.2 Rhomboidal Game Consider hierarchical systems with m + 2 players A0 , B1 , . . . , Bm , C. The set of controls of player C depends on controls of B1 , . . . , Bm . The set of controls of player Bi depends on control chosen by player A0 One can imagine the situation in which one center A0 can represent the interests of society, and others (B1 , . . . , Bm ) can represent regional interests, including the issues of environment protection and C is manufacturing enterprise. The further interpretation of the proposed hierarchical system (see Fig. 49.1) can be found in [9]. According to the hierarchical structure under consideration player A0 makes a first move and selects a behavior (strategy) u = (u 1 , . . . , u m ) from a given set U , where U is interpreted as strategy set of Player A0 . Each chosen behavior restricts the abilities of players B1 , . . . , Bm to make the selection of their decisions at the next stage. This means that the set of possible decisions of player Bi depends upon parameter u i (denote this set Bi (u i ), i = 1, . . . , m), and by ωi ∈ Bi (u i ), i = 1, . . . , m denote the elements this decision set. Decisions ωi , i = 1, . . . , m made by players Bi impose restrictions on the decision set of player C at the third stage. And finally, we get that the set C depends upon decisions ωi made by Bi , i = 1, . . . , m. To underline the dependence on ωi , we shall denote it by C(ω1 , . . . , ωm ). The elements of this set we denote by v. In what follows we consider a special case when the payoffs of A0 , B1 , . . . , Bm , C depend only on v—the choice made by C on the last level of hierarchy. Denote them by f 0 (v), f 1 (v), . . . , f m (v), f (v) where f 0 (v) ≥ 0 (the payoff of A0 ), f i (v) ≥ 0 (the payoff of Bi ), i = 1, . . . , m and f (v) ≥ 0 (the payoff of C).

Fig. 49.1 Rhomboidal game

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Suppose that the following conditions hold. Condition 1. For any ω=(ω1 , . . . , ωm ), ωi ∈ Bi (u), u ∈ U in the set C(ω1 , . . . , ωm ), there exist v0 ∈ C(ω1 , . . . , ωm ) such that f i (v0 ) = 0, i ∈ 0, 1, . . . , m, f (v0 ) = 0. Condition 2. For any u ∈ U , there exist such ωi0 ∈ Bi (u), that the set C(ω10 , . . . , ωm0 ) will contain one element v0 (see Condition 1). Condition 3. There exist such u 0 ∈ U that the sets Bi (u 0 ) will contain only one element ωi0 . These conditions allow higher level players to paralyze the actions of lower level players completely. On the other hand, the lowest level players can paralyze the actions of higher level players. This hierarchical game can be represented as a noncooperative n-person game in normal form if the strategies for player A0 are taken to be the elements u = (u 1 , . . . , u m ) ∈ U , while the strategies for players B1 , . . . , Bm and C are taken to be the functions ω1 (u 1 ), . . . , ωm (u m ) and v(ω1 , . . . , ωm ) with values in the sets B1 (u 1 ), . . . , Bm (u m ), C(ω1 , . . . , ωm ), respectively, (the sets of such functions will ¯ Setting ¯ 1 , . . . , B¯ m , C). be denoted by B K 0 (u, ω1 (·), . . . , ωm (·), v(·)) = f 0 (v(ω1 (u 1 ), . . . , ωm (u m ))) K i (u, ω1 (·), . . . , ωm (·), v(·)) = f i (v(ω1 (u 1 ), . . . , ωm (u m ))), i = 1, m K (u, ω1 (·), . . . , ωm (·), v(·)) = f (v(ω1 (u 1 ), . . . , ωm (u m ))), we define the payoff functions and obtain the normal form of the game ¯ K 0 , {K i }i=1,m , K ).  = (U, {B¯ i }i=1,m , C, Nash equilibrium. m We shall consider ∗a Nash equilibrium in the game . For Bi , we denote by v (ω1 , . . . , ωm ) a solution to the parametric (ω1 , . . . , ωm ) ∈ i=1 optimization problem max

v∈C(ω1 ,...ωm )

f (v) = f (v ∗ (ω1 , . . . , ωm )).

(49.1)

The maximum in (49.1) is supposed to be achieved. The solution v ∗ (·) = v ∗ (ω1 , . . . , ωm ) of problem (49.1) is function of the parameters ω1 , . . . , ωm and v ∗ (·) ∈ C. Consider an auxiliary parametric (with parameters u 1 , . . . , u m ) m-person (B1 , . . . , Bm ) game   (u 1 , . . . , u m ) = {B1 (u 1 ), . . . , Bm (u m ), f 1 , . . . , f m }, where f i = f i (v ∗ (ω1 , . . . , ωm )), i = 1, . . . , m. The elements ωi ∈ Bi (u i ) are strategies of player Bi in   (u 1 , . . . , u m ) and f i (v ∗ (ω1 , . . . , ωm )) is a payoff function of player Bi , i = 1, . . . , m. Suppose the game   (u 1 , . . . , u m ) has Nash equilibrium in pure strategies denoted as (ω1∗ (u 1 ), . . . , ωm∗ (u m )). Note that ωi∗ (·) is function of parameter u i and ωi∗ (·) ∈ Bi , i = 1, . . . , m. Further, let u ∗ = (u ∗1 , . . . , u ∗m ) be a solution to the optimization problem maxu∈U f 0 (v ∗ (ω1∗ (u 1 ), . . . , ωm∗ (u m ))).

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The game  is generalization of rhomboidal 4 person ( A0 , B1 , B2 , C) game considered in [9] and the following Lemma is similar to one proved in [9]. Lemma 49.1 The strategy profile (u ∗ , ω1∗ (·), . . . , ωm∗ (·), v ∗ (·)) is a Nash equilibrium in the game .

49.3 Repeated Game In this section, we consider the multistage game G which represents the infinite repetition of game  on each stage [2]. The payoffs in game G are defined as follows. If on stage l (1 ≤ l < ∞), the m + 2 tuple of strategies l (u lm ))) (u l , ω1l (u l1 ), . . . , ωnl (u ln ), vl (ω1l (u l1 ), . . . , ωm

∞ l−1 l l l l is used then the payoff of A0 is equal to H0∞ = l=1 0 (v (ω1 (u 1 ), . . . , ωm δ∞ fl−1 l l ∞ l l (u m ))), the payoff of Bi (i = 1, . . . , m) equals to Hi = l=1 δ f i (v (ω1 (u 1 ), . . . , ∞ l (u lm ))) and the payoff of player C equals to H ∞ = l=1 δl−1 f (vl (ω1l (u l1 ), . . . , ωm l l ωm (u m ))). Denote by u¯ = (u 1 , . . . , u l , . . .) the strategies of player A0 , where u l = (u l1 , . . . , l l 1 1 l ¯ ¯ ω( ¯ u)) ¯ = v( ¯ ω¯ 1 (u), ¯ . . . , ω¯ m (u)) ¯ = u m ), by ω¯ i (u)=(ω i (u ), . . . , ωi (u ), . . .) and by v( 1 1 1 1 1 l (u lm )), . . .) the strategies of play(v (ω1 (u 1 ), . . . , ωm (u m )), . . . , vl (ω1l (u l1 ), . . . , ωm ers Bi , i = 1, . . . , m and C in game G. Then we get H0∞ = H0∞ (u, ¯ ω¯ 1 (u), ¯ . . . , ω¯ m (u), ¯ v( ¯ ω¯ 1 (u), ¯ . . . , ω¯ m (u))), ¯ ¯ ω¯ 1 (u), ¯ . . . , ω¯ m (u), ¯ v( ¯ ω¯ 1 (u), ¯ . . . , ω¯ m (u))) ¯ Hi∞ = Hi∞ (u, ¯ ω¯ 1 (u), ¯ . . . , ω¯ m (u), ¯ v( ¯ ω¯ 1 (u), ¯ . . . , ω¯ m (u))) ¯ H ∞ = H ∞ (u,

49.3.1 Nash Equilibrium in Repeated Game In the infinitely repeated game G, there is a rich variety of Nash equilibrium [2, 7]. The trivial one is the repetition of equilibrium (u ∗ , ω1∗ (·), . . . , ωm∗ (·), v ∗ (·)) in each stage game defined in Sect. 49.2. Denote this equilibrium by E1. Also, there is a large variety of Nash equilibrium, which use the so-called threat or punishment strategies. Consider now two special types of equilibrium.

49.3.1.1

Nash Equilibrium in Punishment Strategies

Consider in stage game l of the repeated game G the following strategy profile: l (u˜ lm ))). (u˜ l , ω˜ 1l (u˜ l1 ), . . . , ω˜ nl (u˜ ln ), v˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m

(49.2)

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for players A0 , B1 , . . . , Bm , C. Is it possible that this arbitrary strategy profile generates a Nash Equilibrium? It is obvious that after getting on some stage l − 1 the l player C can instead of following the information about the choices ω˜ 1l , . . . , ω˜ m l instructions made by A0 , B1 , . . . , Bm (choose v˜ l (ω˜ 1l , . . . , ω˜ m )) improve his payoff l l l ˜ by choosing v˜ (ω˜ 1 , . . . , ω˜ m ) such that Hˆ =

l H (u˜ l , ω˜ 1l (u˜ l1 ), . . . , ω˜ nl (u˜ ln ), vl (ω˜ 1l (u˜ l1 ), . . . , ω˜ m (u˜ lm ))) max l ) vl ∈C(ω˜ 1l ,...,ω˜ m l (u˜ lm ))) ≥ H (u˜ l , ω˜ 1l (u˜ l1 ), . . . , ω˜ nl (u˜ ln ), v˜˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m

=

l H (u˜ l , ω˜ 1l (u˜ l1 ), . . . , ω˜ nl (u˜ ln ), v˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m (u˜ lm ))) = H

But in all remaining stages B1 , . . . , Bm have the possibility to punish C selecting ωi0 , i = 1, . . . , m, from Condition 2. ¯ δ¯ ∈ (0, 1) such that for δ ∈ (δ, ¯ 1) player C (or A, We prove the existence of δ, l l l l (u˜ lm )) will B1 ,. . . , Bm ) deviating from the prescribed behavior v˜ (ω˜ 1 (u˜ 1 ), . . . , ω˜ m lose in the infinitely repeated game. For justification of this, we shall apply the standard method usually used in Folk theorems. If C deviates from v˜ l (ω˜ 1l (u˜ l1 ), . . . , l ω˜ m (u˜ lm )) on stage l he can get at most Hˆ , but in all other stages, he will be punished by B1 , . . . , Bm by restriction strategies ωi0 . l (u˜ lm )), his payoff will be If C follows the strategy v˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m H∞ =

∞ 

l δl−1 f (v˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m (u˜ lm )))

l=1

If C changes this strategy on stage l, he can not get more than l−1 

l δ t−1 f (v˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m (u lm ))) + δl Hˆ = Hˆ ∞ .

t=1

We have H

∞

=

l−1 

δ

t−1

f (v˜

l

l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m (u˜ lm )))

+

∞ 

t=1

l δ t−1 f (v˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m (u˜ lm )))

t=l

which is the payoff of C under strategy profile (49.2). To justify the inequality H ∞ ≥ Hˆ ∞ , it is necessary to prove δl Hˆ ≤ δl−1

∞ 

l δ t−1 f (v˜ l (ω˜ 1l (u˜ l1 ), . . . , ω˜ m (u˜ lm ))) = δl−1 H ∞

t=1

δ Hˆ ≤ H ∞

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The last inequality is true for some δ, δ ∈ (0, 1). The deviation of players A0 , B1 , . . . , Bm from (49.2) gives also the similar result, since Bi can punish A0 (if he deviates from u) ˜ by choosing ω0 from Condition 2 and as result C will chose v0 , and C also can always punish the deviator on the next stage selecting v = v0 . We get the following theorem. ¯ 1), the Theorem 49.1 In the game G, there exists δ¯ ∈ (0, 1) such that for δ ∈ (δ, ˜ can be achieved payoffs under arbitrary strategy profile (u(·), ˜ ω˜ 1 (·), . . . , ω˜ m (·), v(·)) in some Nash equilibrium (Denote it by E2).

49.3.1.2

Nash Equilibrium in Threat Strategies

Now we show how the threat strategies can be used to construct Nash equilibrium. Consider the following strategy profile for the coalition of players B1 , . . . , Bm in ¯¯ stage game  ω(u) = (ω¯¯ 1 (u), . . . , ω¯¯ i (u), . . . , ω¯¯ m (u)), and define the strategy of player C as  ∗ ¯¯ ¯¯ ω(u) = ω(u) v (ω(u)), v(ω) ¯¯ = ¯ v0 , ω = ω(u) ¯ Here, v ∗ is solution of (49.1). Player C can declare in the beginning of stage game, ¯¯ that he will use strategy v(ω). Then feeling the threat B1 , . . . , Bm will be forced to ¯ ¯ use ω(u) ¯ = (ω¯ 1 (u), . . . , ω¯¯ i (u), . . . , ω¯¯ m (u)) in either case they can get zero payoff. ¯¯ 1 (u), . . . , ωm (u))), Theorem 49.2 The strategy profile (u ∗ , ω¯¯ 1 (u), . . . , ω¯¯ m (u), v(ω ¯¯ 1 (u), . . . , ωm (u))), where u ∗ is the best reply of A0 against (ω¯¯ 1 (u), . . . , ω¯¯ m (u), v(ω is Nash equilibrium in  (Denote it by E3), and in infinitely repeated game G. ¯¯ The proof follows directly from the definition of threat strategy v(ω).

49.4 Cooperation in Infinitely Repeated Game Suppose now that the payoffs in the game G are transferable. Then according to the classical cooperative game theory approach, players decide to maximize the sum of their payoffs in G. This is equivalent to the maximization of joint payoff in each stage game. Denote the corresponding strategy profile by l ˜l (u˜ m ))). (u˜˜ l , ω˜˜ 1l (u˜˜ l1 ), . . . , ω˜˜ nl (u˜˜ ln ), v˜˜ l (ω˜˜ 1l (u˜˜ l1 ), . . . , ω˜˜ m

49 On Nash Equilibrium in Repeated Hierarchical Games

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We have [ f 0 (v(ω1 (u), . . . , ωm (u)))+ m max u∈U,ω(u)∈ i=1 Bi (u), v(ω(u))∈C(ω(u)) m 

f i (v(ω1 (u), . . . , ωm (u))) + f (v(ω1 (u), . . . , ωm (u)))] =

i=1

˜˜ . . . , ω˜˜ m (u))) ˜˜ + = f 0 (v(ω˜˜ 1 (u),

m 

˜˜ ω˜˜ 1 (u), ˜˜ . . . , ω˜˜ m (u)))+ ˜˜ f i (v(

i=1

˜˜ . . . , ω˜˜ m (u))) ˜˜ ˜˜ ω˜˜ 1 (u), + f (v( The payoff under cooperation in G will be ∞  l=1

˜˜ . . . , ω˜˜ m (u))) ˜˜ + δ m−1 [ f 0 (v(ω˜˜ 1 (u),

m 

˜˜ ω˜˜ 1 (u), ˜˜ . . . , ω˜˜ m (u)))+ ˜˜ f i (v(

i=1

˜˜ . . . , ω˜˜ m (u)))] ˜˜ ˜˜ ω˜˜ 1 (u), = + f (v(  1 ˜˜ . . . , ω˜˜ m (u))) ˜˜ + ˜˜ ω˜˜ 1 (u), . . . , ω˜˜ m (u)))+ ˜˜ [ f 0 (v(ω˜˜ 1 (u), f i (v( 1−δ i=1 m

=

˜˜ . . . , ω˜˜ m (u)))] ˜˜ ˜˜ ω˜˜ 1 (u), = V (A0 , B1 , . . . , Bm , C) + f (v( As we have seen in Sect. 49.3.1, an arbitrary strategies profile (49.2) can generate an outcome in some Nash equilibrium of type E2 (using so called “punishment strategies”). If we have u˜˜ = u, ˜ ω˜˜ = ω, ˜ v˜˜ = v, ˜ we get that the cooperative behavior can be obtained in some Nash equilibrium (this will be the best Nash equilibrium). Denote by N¯ = {A0 , B1 , . . . , Bm , C} the set of players in game G. V ( N¯ ) Definition 49.1 The PoA A = W , where V ( N¯ ) is maximal joint payoff in the ( N¯ ) game, and W ( N¯ ) is the joint payoff of players in the worst Nash equilibrium of type E3.

The payoffs in worst Nash equilibrium are equal to 0 since, according to Theorem 49.2, we can define  ∗ v (ω0 ), ω(u) = ω0 (u) v(ω) ¯¯ = v0 , ω(u) = ω0 (u), which will give us zero payoffs for all players (see Condition 2). In the infinitely repeated game, the price of Anarchy is the same as in stage games. And we get that A = ∞ in case when V ( N¯ ) is a finite number.

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¯ ( N¯ ) Definition 49.2 The PoS is S = W , where V ( N¯ ) is maximal joint payoff of playV ( N¯ ) ers in the game, and W¯ ( N¯ ) is the joint payoff of players in the best Nash equilibrium.

The PoS S in G (infinitely repeated hierarchical game) is equal to 1. Since according to theorem 1, there exists the best Nash equilibrium with maximal joint payoff V ( N¯ ) of players. In the paper [10], the tree-like hierarchical game is considered, and the different types of Nash Equilibrium in this game were proposed. Additionally, in the paper [10], the question about Subgame Perfectness of constructed equilibrium was investigated. Using the results of the paper [10], we can calculate the PoA and PoS. For this repeated hierarchical game, the worst Nash equilibrium is (0, 0, . . . , 0), so it is clear that the PoA A also equals ∞. The PoS S is equal to 1 since there exists a Nash equilibrium giving the maximal joint payoff (see [10]) of all players.

49.5 Conclusion In the paper, the three-level rhomboidal-type hierarchical game is considered. The corresponded infinitely repeated games are constructed, and different types of Nash equilibrium are found. Also, we investigate a cooperative version of the game and calculate the prices of Anarchy and Stability. We get an interesting result that the PoA equals to ∞. The same result we have for the tree-like hierarchical game from [10]. The important conclusion can be made: “for this type of game, players’ individual behavior can give infinitely worse results than cooperation”. Acknowledgements The research was funded by Russian Science Foundation grant “Optimal Behavior in Conflict-Controlled Systems” (N 17-11-01079).

References 1. Ageev, P., Pankratova, Y. and Tarashnina, S.: On Competition in the telecommunications market. Contrib. Game Theory Manag. 10, 7–21 (2018) (Petrosyan, L.A., Zenkevich, N.A.) 2. Aumann, R.J., Maschler, M.: Repeated Games with Incomplete Information. MIT Press, Cambridge (1995) 3. Fudenberg, D., Maskin, E.: The folk theorem in repeated games with discounting or with incomplete information. Econometrica 54(3), 533–554 (1986) 4. Germeyer, Y.B.: Non-Zero Sum Games. Nauka, Moskva (1976).(in Russian) 5. Maschler, M., Solan, E. and Zamir, S.: Game Theory. Cambridge University Press (2013) 6. Mazalov, V.V.: Mathematical Game Theory and Applications. Wiley (2013) 7. Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951) 8. Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior, Princeton (1947) 9. Petrosyan, L., Zenkevich, N.: Game Theory. World Scientific Publishing Co. Pte. Ltd. Singapore-London (1996) 10. Petrosjan L.A., Pankratova, Y.B.: Equilibrium and cooperation in repeated hierarchical games. In: Lecture Notes in Computer Science. Springer, pp. 685–696 (2019). https://doi.org/10.1007. 2F978-3-030-22629-9

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11. Christodoulou G., Koutsoupias E.: On the price of anarchy and stability of correlated equilibria of linear congestion games. In: Brodal G.S., Leonardi S. (eds.) Algorithms ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol. 3669. Springer, Berlin, Heidelberg 12. Teng, Y., Song, M., Zhang, Y., Xu, Y. and Song J.: Hierarchical game theory analysis in subscribers cooperative relaying network with OFDMA orthogonal channels. In: 2009 IEEE International Conference on Communications Technology and Applications, Beijing, pp. 664– 670 (2009). https://doi.org/10.1109/ICCOMTA.2009.5349116 13. Vasin A.A.: Sil’nye situatsii ravnovesiya v nekotorykh sverkhigrakh. Vestnik Moskovskogo Universiteta, ser. Matem. I mekhanika, Vyp. 1, pp. 30–39 (1978) (in Russian) 14. Weibelzahl, M., Märtz, A.: Optimal storage and transmission investments in a bilevel electricity market model. Ann. Oper. Res. 287(2), 911–940 (2020)

Chapter 50

Dynamic Shapley Value for Two-Stage Cost Sharing Game Li Yin

Abstract The problem of constructing the dynamic Shapley values in a two stage game is studied. During the dynamic game, each stage game can be considered as a minimum cost spanning tree game. From the first stage, the players’ strategy profiles construct the graph in stage games, and the minimum cost spanning tree of the graph is defined by Prim (1957). At the second stage, the graph built by the players will be changed in some possible ways, with several specified probabilities. These probabilities are determined by the strategy profiles of players in the first stage. The meaning of the change is to break several edges on the graph. Then the players’ cooperative behavior is defined. Along the cooperative trajectory, characteristic functions are defined for all coalitions. The IDP (Imputation Distribution Procedure) was used to construct dynamic Shapley Values.

50.1 Introduction In recent years, many problems in practice can be modeled by dynamic games. How to cooperate is an important task to be solved in dynamic games. According to classical game theory, there are two critical problems to consider when dealing with dynamic cooperative games. The first problem is how to define a principle for players to allocate payoff in the games. The second problem is to consider the stability of the solution of this dynamic game. After determining the optimal principle for the distribution of payoff, players should receive at the end of the game payoffs that is equal to what they agreed to receive at the beginning of the game. In [1], the time consistency of cooperative solutions in the dynamic game is studied for the first time. In [2], the imputation distribution procedure is proposed. In stochastic games, the property of time consistency is not available because the cooperative trajectory is indeterminate. In order to make a replacement for cooperative trajectories, we define a bunch of cooperative trajectories in stochastic games. To ensure that players’ behavior L. Yin (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_50

457

458

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is optimal, we propose an extension of the time-consistent cooperative solution rules for the game, i.e. the time-consistent cooperative solution rules. In a minimum cost spanning tree game, the researchers are concerned with two questions, how to form a minimum cost spanning tree, and how to allocate costs between players. In [3–5], several algorithms for the minimum cost spanning tree have been proposed. And the solution to allocate costs among players in a minimum cost spanning tree game was first proposed in paper [6]. The present work can be considered as an extension of the classical literature on dynamic games with spanning tree. In [7], a SHOCK case in a two-stage dynamic stochastic game with spanning tree is considered. However, in dynamic cooperative game with spanning tree, the game model in [7, 8] is simple. Because in these games, there are two such strong conditions. The first condition is that the game model only takes into account the disappearance of a particular player. But in the practical scenario, the players’ strategy profiles in the first stage would cause some complex changes to the game in the second stage. The second condition is that the probability of the particular player leaving the game in the game mentioned above is strongly related to the subtree in the first stage. The purpose of this paper is to improve these conditions mentioned above. That is the arbitrary edge could break out in the second stage, and that break depends on the strategy profiles of players in the first stage. So in the following, we define a matrix called an α-matrix. Different from before, in the second stage, with the help of the α-matrix and strategy profiles of players, the cost matrix is constructed. This process can be seen as the reduction of the edges on the graph. And the deleted edges are considered as broken edges in the second stage. And the probability of different α-matrices appearing is determined by the strategy profiles of players in the first stage. Thus, in the game, each edge on the graph might break out. And the probability of break is related to the specific setting of the game in the first stage. Similar to the approach in [7], we define the characteristic function for coalitions in dynamic stochastic game, and construct the dynamic Shapley value by [2]. At last verify its time consistency.

50.2 The Model In the paper, a two-stage game with spanning tree is considered. The finite game tree is denoted by H = (Z , F) with the initial vertex z 1 is defined, where Z is set of vertices in the game tree. And F(z 1 ) is a point-to-set mapping, which is defined as following: F(z 1 ) ⊂ Z . In the second stage, the set of vertical on the tree-like graph is the set F(z 1 ). i.e. F(z 1 ) = Z \ {z 1 }. The number of elements in the set F(z 1 ) is denoted by m. The game starting from initial vertex z 1 is denoted by (z 1 ). Similarly, the subgame starting from the vertex z k ∈ F(z 1 ) is denoted by (z k ). Define the set of players as N = {1, . . . , n}. N 0 = N ∪ {0}, where {0} is the source. We define G(N 0 , E) is a graph over N 0 , where E = {(i, i  ) : ∀i, i  ∈ N 0 }—

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the set of all edges on the graph. An edge in a graph G is denoted by (i, i  ), if (i, i  ) ∈ E, ∀i, i  ∈ N . In a graph G(N 0 , E), i and i  are called to be connected, if ∃(i 1 , i 2 ),(i 2 , i 3 ), . . . , (i n−1 , i n ) such that (i k , i k+1 ) ∈ G(N 0 , E), 1 ≤ k ≤ n − 1, and i 1 = i, i n = i  , i, i  ∈ N 0 . If i, i  are connected in a graph G(N 0 , E), ∀i, i  ∈ N 0 , then the graph G(N 0 , E) is a connected graph over the set N 0 . We denote the set of all connected graphs over the set N 0 is G N 0 . We define the cost matrix as below. In the cost matrix, each element is the cost of connections between two players in the graph. C = (cii  )(n+1)×(n+1) ,

(50.1)

where cii  = ci  i > 0 it is the cost of connecting from the player i to the player i  , i = i  ∈ N 0 . In this paper, ci0 = c0i is always equal to a nonnegative constant. At each stage, player i selects a vector u i = (u i,1 , . . . , u i,i−1 , u i,i+1 , . . . , u i,n ). Each element is a strategy of the player i against another player i  ∈ N \ {i}. For instance, u i,i  ∈ Ui,i  is strategy of player i against player i  ∈ N \ {i}. Now define the cost of an edge between the player i and the player i  : cii  = f cost (u i,i  , u i  ,i ) = ci  i , ci0 = c0i > 0, ∀i, i  ∈ N .

(50.2)

where f cost is a mapping from the set of strategies of players i, i  to the set of the cost of edge (i, i  ). Furthermore, a cost matrix C = (cii  )(n+1)×(n+1) could be completely decided by a strategy profile u = (u 1 , . . . , u n ), where cii  = ci  i , ∀i, i  ∈ N . The minimum cost spanning tree (MCST) Tu (N 0 , C) over N 0 is defined as below Tu (N 0 , C) = arg min

G∈G N 0



cii  ,

(i,i  )∈G(N 0 ,E)

where C = (cii  )(n+1)×(n+1) is defined in (50.1), and the graph G(N 0 , E) is build by u = (u 1 , . . . , u n ). The total cost of edges in Tu (N 0 , C) is defined as below Ct [Tu (N 0 , C)] =



cii  .

(i,i  )∈Tu (N 0 ,C)

At the first stage, all players construct a graph together by selecting a strategy individually and simultaneously. 1 1 1 1 1 1   u 1 = (u 11 , . . . , u 1n ), u i1 = (u i,1 , . . . , u i,i−1 , u i,i+1 , . . . , u i,n ), u i,i  ∈ Ui,i  , i, i ∈ N , ∀i = i

According to the definitions in (50.1) and (50.2), this means that both the graph and the cost matrix will be determined at the same time.

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Stage 2: After the first stage, the game proceeds to the second stage with probability, which depends on strategies of players in the previous stage. The probability is described as following: p(z 1 , z 2 , u 11 , . . . , u 1n ) = p(z 1 , z 2 , u 1 ) ≥ 0,  p(z 1 , z k , u 1 ) = 1

(50.3)

z k ∈F(z 1 )

where p(z 1 , z 2 , u 1 ) is the probability that the game moves from initial vertex z 1 to the vertex z 2 . z 1 and z 2 are vertexes in the tree-like graph H = (Z , F). Assume that each vertex z k ∈ F(z 1 ) associate with a matrix called is α-matrix. The dimension of α-matrix is (n + 1) × (n + 1), and every element of the α-matrix is equal to 0 or 1. But all elements on the main diagonal of the α-matrix are +∞. Definition 50.1 The α-matrix of the stage game on the vertex z k is described as follows: ⎞ ⎛ ∞ 1 1 ... 1 1 ⎜ 1 ∞ α1,2 . . . α1,n−1 α1,n ⎟ ⎟ ⎜ ⎜ 1 α2,1 . . . . . . . . . ... ⎟ 1 ⎟ , z k ∈ F(z 1 ) ⎜ α[ p(z 1 , z k , u )] = ⎜ ... ⎟ ⎟ ⎜. . . . . . . . . . . . . . . ⎝ 1 αn−1,1 . . . . . . ∞ αn−1,n ⎠ 1 αn,1 . . . . . . αn,n−1 ∞ where u 1 is the strategies of players in the first stage game (z 1 ), and αi,i  = αi  ,i = 1 or 0, k ∈ {k : z k ∈ F(z 1 )}, for each i = i  ∈ {1, . . . , n}. Definition 50.2 For two matrices A and B with the same dimension (m × n), the Hadamard product A ◦ B is a matrix which elements are given by ⎛

⎞ ⎛ ⎞ ⎛ ⎞ a11 . . . a1n b11 . . . b1n a11 × b11 . . . a1n × b1n ⎠ ...⎠ ◦ ⎝... ...⎠ = ⎝ ... ... A ◦ B = ⎝... am1 . . . amn bm1 . . . bmn am1 × bm1 . . . amn × bmn (50.4) where the matrices are A = (aii  )m×n and B = (bii  )m×n . The definition of the Hadamard product is defined in [9]. When the game continues to the stage game (z 2 ), the cost matrix on the stage game (z 2 ) is equal to the Hadamard product between α-matrix and the cost matrix which established by all players as they did in the first stage. So in the stage game (z 2 ), the cost matrix of the stage game is defined as follows: C2 = α[ p(z 1 , z 2 , u 1 )] ◦ {cii2  }(n+1)×(n+1) , 2 2  cii2  = ci2 i = f cost (u i,i  , u i  ,i ), ci0 = c0i > 0, ∀i, i ∈ N .

(50.5)

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50.3 Characteristic Function Characteristic Function of the Game (z 1 ): According to the method in [10], let us consider defining characteristic functions for all players N 0 . It is assumed that path z 1 , z 2 is realized in the game. We therefore consider the mathematical expectation of the total cost to the player in two stage games. Using the Bellman equations, we calculate the cooperative solution 

W 1 (N 0 ) = min{Ct [Tu 1 (N 0 , C1 )] + x(·)

p(z 1 , z k , u 1 )Ct [Tu 2 (N 0 , C2 )]}

k∈{k:z k ∈F(z 1 )}

= Ct [Tu¯ 1 (N , C1 )] + 0



p(z 1 , z k , u¯ 1 )Ct [Tu¯ 2 (N 0 , C2 )]

k∈{k:z k ∈F(z 1 )}

(50.6) where p(z 1 , z k , u¯ 1 ), k ∈ {k : z k ∈ F(z 1 )} are defined by (50.3). u¯ i (·), i ∈ N are called cooperative strategies. Thus, in the game, cooperative strategy profile u(·) ¯ = (u¯ 1 , . . . , u¯ n ) is the behavior of all players. In the same way, the definition of the characteristic function is given for the coalition S ⊂ N . It is assumed that players in N , but not in S, are not involved at all. In other words, none of players N \ S are connected to the source {0}. 

0

W 1 (S 0 ) = min{Ct [Tu 1 (S 0 , C1S )] + x(·)

0

p(z 1 , z k , u 1 )Ct [Tu 2 (S 0 , C2S )]}

k∈{k:z k ∈F(z 1 )}

= Ct [Tu¯ 1 (S

0

0 , C1S )]

+



0

p(z 1 , z k , u¯ 1 )Ct [Tu¯ 2 (S 0 , C2S )]

k∈{k:z k ∈F(z 1 )}

(50.7) where S ⊂ N , S 0 = S ∪ {0}, and p(z 1 , z k , u¯ 1 ), k ∈ {k : z k ∈ F(z 1 )} are defined by 0 0 (50.3). C1S and C2S are the cost matrices restricted to S 0 and is determined by (50.5). u¯ i (·), i ∈ S are called cooperative strategies. Thus, in the game, cooperative strategy profile u¯ i (·) = (u¯ i , i ∈ S) is the behavior of all players. Characteristic Function of Subgame (z 2 ): The definition of characteristic function for N 0 at the second stage is described as below. Suppose that a subgame (z 2 ) happened in the vertex z 2 on the tree-like graph H = (Z , F). According to the definition (50.3), p(z 1 , z 2 , u 1 ) is the probability of the game proceeds from initial vertex z 1 to the vertex z 2 . Thus, the characteristics function for the set N 0 is defined as follows: W 2 (N 0 ) = min Ct [Tu 2 (N 0 , C2 )] = Ct [Tu¯ 2 (N 0 , C2 )] u 2 (·)

where u¯ i2 (·), i ∈ N are called cooperative strategies. Thus, in the game, cooperative strategy profile u¯ 2 (·) = (u¯ 21 , . . . , u¯ 2n ) is the behavior of all players. In the vertex z 2 with the probability p(z 1 , z 2 , u¯ 1 ), z 2 ∈ F(z 1 ), the characteristic function for the coalition S ⊂ N is defined as follows:

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0

W 2 (S 0 ) = min Ct [Tu 2 (S 0 , C2S )] = Ct [Tu¯ 2 (S 0 , C2S )] u 2 (·)

0

where C2S is the cost matrix restricted to S 0 and is determined by (50.5). u¯ i (·), i ∈ S are called cooperative strategies. Thus in the game, cooperative strategy profile u¯ i (·) = (u¯ i , i ∈ S) is the behavior of all players.

50.4 The Dynamic Shapley Value In two-stage MCST game the Shapley value is defined as Sh i1 (N 0 ) =

1  1 0 0 [W (Sπ(i) ∪ {i}) − W 1 (Sπ(i) )], ∀i ∈ N , S 0 = S ∪ {0}, S ⊆ N n! π∈

where  is the set of all permutations over the set N , and Sπ(i  ) = {i|π(i) < π(i  )}. At the second stage, if the game proceeds to the stage game on the vertex z k ∈ F(z 1 ) with probability p(z 1 , z k , u 1 ), in one-stage MCST game the Shapley value is defined as follows:

Sh i2 (N 0 ) =

1  2 0 0 [W (Sπ(i) ∪ {i}) − W 2 (Sπ(i) )], ∀i ∈ N , S 0 = S ∪ {0}, S ⊆ N n! π∈

where  is the set of all permutations over the set N , and Sπ(i  ) = {i|π(i) < π(i  )}. The first time that the concept of time consistency of cooperative solution was discussed was in [11]. with the help of definition of IDP, the Dynamic Shapley Value is constructed. IDP of the Shapley value in two-stage MCST game is a scheme β = (β 1 , β 2 ) s.t. β 1 = Sh 1 (N 0 ) −



p(z 1 , z k , u 1 )Sh 2 (N 0 )

k∈{k:z k ∈F(z 1 )}

β = 2



p(z 1 , z k , u 1 )Sh 2 (N 0 )

(50.8)

k∈{k:z k ∈F(z 1 )}

If there exists a nonnegative IDP, i.e. (∃βi1 ≥ 0, and βi1 ≥ 0, ∀i ∈ N ) in the game, such that the following condition holds: Sh 1 (N 0 ) = β 1 +



p(z 1 , z k , u 1 )Sh 2 (N 0 )

k∈{k:z k ∈F(z 1 )}

 k∈{k:z k ∈F(z 1 )}

p(z 1 , z k , u 1 )Sh 2 (N 0 ) = β 2

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the Dynamic Shapley value Sh 1 is called time consistent. Proposition 50.1 With IMP β defined by (50.8), the Dynamic Shapley value Sh 1 = (Sh 11 , . . . , Sh 1n ) is time inconsistent.

50.5 Example A two-person game as an illustration is considered. The set of players N = {1, 2}, and the source is {0}, N 0 = N ∪ {0}. The sets of strategy, which of player 1 against 1 2 = {3, 4}, U1,2 = {6, 7}, and the sets of strategy, which of player 2 player 2, are U1,2 1 2 = {8, 3}. Assume that there are two vertexes against player 1, are U2,1 = {6, 2}, U2,1 z 2 , z 3 after the initial vertex z 1 in the game . The tree-like graph as shown in Fig. 50.1 on the left side. As shown in Fig. 50.1, in each stage, there is a graph over N 0 . Assume that edges (0, 1), (0, 2) are fixed and the cost of edges are c01 = c10 = 80, c02 = c20 = 10. According to the definition (50.2), the function f cost is defined as f cost = u 1,2 × u 2,1 , u 1,2 ∈ U1,2 , u 2,1 ∈ U2,1 . The α-matrices of the stage game on the vertexes z 2 and z 3 are described as follows: ⎛ ⎞ ⎛ ⎞ ∞ 1 1 ∞ 1 1 α[ p(z 1 , z 2 , u 1 )] = ⎝ 1 ∞ 1 ⎠ , α[ p(z 1 , z 3 , u 1 )] = ⎝ 1 ∞ 0 ⎠ 1 1 ∞ 1 0 ∞ In the case of different strategy profiles of players, the game’s probabilities proceeds from z 1 to z 2 or z 3 are shown in Table 50.1. According to above-mentioned analysis, we have

Fig. 50.1 The figure on the left side is the tree-like graph of the game. The figure on the right side is the graph at each stage game Table 50.1 The game’s probabilities from the first stage to the second stage under different strategy profiles u 1 = (3, 6) u 1 = (3, 2) u 1 = (4, 6) u 1 = (4, 2) p(z 1 , z 2 , u 1 ) p(z 1 , z 3 , u 1 )

0.5 0.5

0.7 0.3

0.9 0.1

0.15 0.85

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W 1 (N 0 ) = Ct [Tu¯ 1 (N 0 , C1 )] +

k={2,3}

p(z 1 , z k , u¯ 1 )Ct [Tu¯ 2 (N 0 , C2 )] = 62.6

where cooperative strategies of the game are u¯ 1 = (3, 2) on vertex z 1 , u¯ 2 = (6, 3) on vertex z 2 , and u¯ 2 = (6, 3) on vertex z 3 . p(z 1 , z 2 , u¯ 1 ) = 0.7, p(z 1 , z 2 , u¯ 1 ) = 0.3. The results are as follows: Sh 11 (N 0 ) = 101.3, Sh 12 (N 0 ) = −38.7. The subgame on z 2 with probability p(z 1 , z 2 , u¯ 1 ) = 0.7: Sh 21 (N 0 ) = 49, Sh 22 (N 0 ) = −21. The subgame on z 3 with probability p(z 1 , z 3 , u¯ 1 ) = 0.3: Sh 21 (N 0 ) = 80, Sh 22 (N 0 ) = 10. Then, the IDP β11 = 43, β21 = −27, β12 = 58.3, β22 = 11.7. The Dynamic Shapley Value is time inconsistent in the example.

50.6 Conclusion In the paper, we consider time-consistency in two-stage game with spanning tree. Unfortunately, using the IDP, the Dynamic Shapley Value was defined; however, it is time inconsistency. In the future, the construction of the cost matrix in the game could apply the definition of an irreducible network proposed by Bird, CG in 1976. The new solution may satisfy more properties.

References 1. Petrosyan, L.: Time-consistency of solutions in multi-player differential games. Vestn. Leningr. State Univ. 4, 46–52 (1977) 2. Petrosyan, L., Danilov, N.: Stability of solutions in non-zero sum differential games with transferable payoffs. Vestn. Leningr. Univ. 1, 52–59 (1979) 3. Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7(1), 48–50 (1956) 4. Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36(6), 1389–1401 (1957) 5. Dijkstra, E.W., et al.: A note on two problems in connexion with graphs. Numer. Math. 1(1), 269–271 (1959) 6. Bird, C.G.: On cost allocation for a spanning tree: a game theoretic approach. Networks 6(4), 335–350 (1976) 7. Yin, L.: The dynamic shapley value in the game with spanning tree. In: 2016 International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference), pp. 1–4. IEEE (2016) 8. Yin, L.: Dynamic shapley value in the game with spanning forest. In: 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of VF Demyanov) (CNSA), pp. 1–4. IEEE (2017) 9. Hadamard, J.: Resolution d’une question relative aux determinants. Bull. des Sci. Math. 2, 240–246 (1893) 10. Parilina, E.M.: Stable cooperation in stochastic games. Autom. Remote Control 76(6), 1111– 1122 (2015) 11. Petrosjan, L.A.: Cooperative stochastic games. In: Advances in Dynamic Games, pp. 139–145. Springer (2006)

Part VII

Mechanical Systems Control

Chapter 51

Boundary Control of String Vibrations with Given Values of the Deflection Function at Intermediate Moments of Time Vanya Barseghyan Abstract A boundary control problem for the string vibration equation with given initial and terminal conditions and with given values of the deflection function of the string points at intermediate moments of time is considered. The problem is reduced to a problem with zero boundary conditions and further using the variable separation method, the problem is reduced to a control problem with a countable number of ordinary differential equations with given initial, terminal, and multipoint intermediate conditions. In practice, the modal method is widely used on the basis of which the problem is solved for arbitrary numbers of the first harmonics. In this paper, employing methods of control theory for finite-dimensional systems, a control action for arbitrary numbers of the first harmonics is constructed. The approach proposed for the string vibration equation allows usage for other (not one-dimensional) vibrational systems.

51.1 Introduction A large class of physical processes associated with vibrating systems is modeled by the wave equation [1–3]. Moreover, in practice, problems of boundary control and optimal control often arise when it is necessary to generate the desired waveform satisfying the multipoint intermediate conditions. Multipoint boundary problems of control and optimal control, where along with the classical boundary (initial and terminal) conditions, multipoint intermediate conditions are also given, are studied in [4–10]. Elastic vibration control problems described by the one-dimensional wave equation are studied in [1–3, 9–12], using boundary controls for various types of boundary conditions. Problems of control of elastic vibrations described by an onedimensional wave equation are considered and approaches for constructing boundary controls are presented in [3]. The works [12, 13] (and other works of these authors) are devoted to the problem of boundary control (optimal control) of free-flow processes V. Barseghyan (B) Institute of Mechanics of NAS of RA, Yerevan State University, Yerevan, Armenia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_51

467

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in the class of generalized solutions and boundary controls are obtained. A problem of optimal control of string vibrations with given intermediate states using external forces acting along the string was considered in [6]. The problem of optimal boundary control of string vibrations with given constraints at intermediate moments of time is studied in [8]. In [9, 10], boundary value problem for the string vibration equation with a given velocity of string vibration at some moments of time is considered and a solution to the problem is constructed. In this work, a boundary control problem for the string vibration equation with given values of the deflection function at intermediate moments of time is considered. The problem is reduced to the problem of control of distributed actions with zero boundary conditions. Using the method of separation of variables and methods of the control theory of finite-dimensional systems, a control action for arbitrary numbers of the first harmonics is constructed.

51.2 The Problem Setting Let the state of the distributed vibrating system (small transverse vibrations of the stretched string), i.e. the deviations from the equilibrium state, are described by a function Q(x, t), 0 ≤ x ≤ l, 0 ≤ t ≤ T , that obeys for 0 < x < l and t > 0 the wave equation 2 ∂2 Q 2∂ Q = a (51.1) ∂t 2 ∂x2 with the initial conditions Q(x, 0) = ϕ0 (x),

 ∂ Q  = ψ0 (x), 0 ≤ x ≤ l ∂t t=0

(51.2)

Q(l, t) = ν(t), 0 ≤ t ≤ T,

(51.3)

and boundary conditions Q(x, 0) = μ(t),

where functions μ(t) and ν(t) are boundary controls. In Eq. (51.1), a 2 = Tρ0 , where T0 is the string tension and ρ specifies the density of the homogeneous string. The set of functions Q(x, t) satisfying Eqs. (51.1) is twice continuously differentiable up to the boundary of the region 0 < x < l, 0 < t < T . Let at some intermediate moments of time 0 = t0 < t1 < ... < tm < tm+1 = T the values of the state and the velocities of the points of the string are given Q(x, ti ) = ϕi (x), 0 ≤ x ≤ l, i = 1, m.

(51.4)

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The boundary control problem of string vibrations is posed as follows: among the possible controls μ(t) and ν(t), 0 ≤ t ≤ T it is required to find a control that transfers the system from a given initial state (51.2) through intermediate states (51.4) to the final state  ∂ Q  = ψT (x) = ψm+1 (x), 0 ≤ x ≤ l. Q(x, T ) = ϕT (x) = ϕm+1 (x), ∂t t=T (51.5) We assume that the functions ϕi (x), i = 0, m + 1, belong to the space C (2) [0, l], and functions ψT (x) belong to the space C (1) [0, l].

51.3 Reduction of the Problem to a Problem with Zero Boundary Conditions Since the boundary conditions (51.3) are non-homogeneous, the solution of the stated problem is reduced to a problem with zero boundary conditions [14]. We seek for the solution of the Eq. (51.1) in the form of a sum Q(x, t) = V (x, t) + W (x, t),

(51.6)

where V (x, t) is an unknown function, satisfying Eq. (51.1) with the homogeneous boundary conditions V (0, t) = V (l, t) = 0, (51.7) and W (x, t)—solution of the Eq. (51.1) with the non-homogeneous boundary conditions W (0, t) = μ(t), W (l, t) = ν(t). (51.8) The function W (x, t) has the form W (x, t) = (ν(t) − μ(t))

x + μ(t). l

(51.9)

Substituting (51.6) in (51.1) and taking into account (51.9), we find

where

2 ∂2 V 2∂ V = a + F(x, t), ∂t 2 ∂x2

(51.10)

 x − μ (t). F(x, t) = μ (t) − ν  (t) l

(51.11)

By virtue of the initial, intermediate, and boundary conditions, respectively (51.2), (51.4), and (51.5), the function V (x, t) must satisfy the initial conditions

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x V (x, 0) = ϕ0 (x) − (ν(0) − μ(0)) − μ(0), l     ∂ V  x = ψ0 (x) − ν (0) − μ (0) − μ (0), ∂t t=0 l

(51.12)

intermediate conditions V (x, ti ) = ϕi (x) − (ν(ti ) − μ(ti ))

x − μ(ti ), l

(51.13)

and final conditions x V (x, T ) = ϕT (x) − (ν(T ) − μ(T )) − μ(T ), l     ∂ V  x − μ (T ). = ψT (x) − ν (T ) − μ (T ) ∂t t=T l

(51.14)

From the condition (51.7), it follows that   ∂ V (0, t)  ∂ V (l, t)  V (0, ti ) = V (l, ti ) = 0, = = 0, i = 0, m + 1. ∂t t=ti ∂t t=ti (51.15) From the conditions (51.12), (51.13) and (51.14) taking into account (51.15), we obtain the following compatibility conditions: μ(0) = ϕ0 (0), μ (0) = ψ0 (0), ν(0) = ϕ0 (l), ν  (0) = ψ0 (l), (51.16) μ(ti ) = ϕi (0), ν(ti ) = ϕi (l), i = 1, m, (51.17) μ(T ) = ϕT (0), μ (T ) = ψT (0), ν(T ) = ϕT (l), ν  (T ) = ψT (l). (51.18) Thus, taking into account conditions (51.16)–(51.18), the conditions (51.12)– (51.14) are written as follows, respectively: x V (x, 0) = ϕ0 (x) − (ϕ0 (l) − ϕ0 (0)) − ϕ0 (0), l  ∂ V  x = ψ0 (x) − (ψ0 (l) − ψ0 (0)) − ψ0 (0) ∂t t=0 l x V (x, ti ) = ϕi (x) − (ϕi (l) − ϕi (0)) − ϕi (0), i = 1, m, l x V (x, T ) = ϕT (x) − (ϕT (l) − ϕT (0)) − ϕT (0), l  ∂ V  x = ψT (x) − (ψT (l) − ψT (0)) − ψT (0).  ∂t t=T l

(51.19)

(51.20)

(51.21)

Thus, the boundary control problem of string vibrations is reduced to control problem (51.10), (51.11) with boundary conditions (51.7), which is formulated as follows: it is required to find the boundary controls μ(t) and ν(t) for 0 ≤ t ≤ T and transferring the motion (51.10) with boundary conditions (51.7) from the given initial state (51.19) through the intermediate states (51.20) to the final state (51.21).

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51.4 The Problem Solution The solution of the Eq. (51.10) is searched for in the following form: V (x, t) =

∞ 

Vk (t) sin

k=1

πk x. l

(51.22)

The functions F(x, t), ϕi (x) (i = 0, m + 1), ψ0 (x) and ψT (x) are represented in the form of Fourier series and substituting their values together with V (x, t) into the Eqs. (51.10), (51.11) and into the conditions (51.19)–(51.21), it is obtained V¨k (t) + λ2k Vk (t) = Fk (t) , λ2k =



aπ k l

2 ,

2a  ϕ0 (0) − ϕ0 (l)(−1)k , λk l 2a  ψ0 (0) − ψ0 (l)(−1)k , V˙k(0) = ψk(0) − λk l 2a  (i) Vk (ti ) = ϕk − ϕi (0) − ϕi (l)(−1)k , i = 1, m, λk l 2a  Vk (T ) = ϕk(T ) − ϕT (0) − ϕT (l)(−1)k , λk l  2a ψT (0) − ψT (l)(−1)k , V˙k (T ) = ψk(T ) − λk l Vk(0) = ϕk(0) −

where Fk (t) =

2a   ν (t)(−1)k − μ (t) , λk l

(51.23) (51.24)

(51.25) (51.26)

(51.27)

ϕk(i) (i = 0, m + 1), ψk(0) and ψk(T ) denote the Fourier coefficients, corresponding to the functions ϕi (x) (i = 0, m + 1), ψ0 (x) and ψT (x). The general solution of the Eq. (51.23) with initial conditions (51.24) and its derivative with respect to time have the form 1 1 Vk (t) = Vk (0) cos λk t + V˙k (0) sin λk t + λk λk V˙k (t) = −λk Vk (0) sin λk t + V˙k (0) cos λk t +

t Fk (τ ) sin λk (t − τ ) dτ, 0

t Fk (τ ) cos λk (t − τ ) dτ. (51.28) 0

Now, taking into account the intermediate (51.25) and final (51.26) conditions, using the approaches given in [4, 8], from (51.28), it is obtained that the functions

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Fk (t) for each k must satisfy the relation:

T

H¯k (τ )dτ = Ck (t1 , ..., tm , T ), k = 1, 2, ... ,

(51.29)

0

where ⎞ ⎛ ⎞ C1k (T ) sin λk (T − τ ) −(−1)k sin λk (T − τ ) k ⎜ C (T ) ⎟ ⎜ cos λk (T − τ ) −(−1) cos λk (T − τ ) ⎟ ⎟ ⎜ 1k ⎟ ⎜ ⎜ (1) (1) k ⎟ , Ck (t1 , ..., tm , T ) = ⎜ C1k (t1 ) ⎟ H¯ k (τ ) = ⎜ −(−1) h k (τ ) h k (τ ) ⎟, ⎟ ⎜ ⎟ ⎜ .. ⎠ ⎝ ··· ⎠ ⎝ . h (m) −(−1)k h (m) C1k (tm ) k (τ ) k (τ )    sin λk (t j − τ ), 0 ≤ τ ≤ t j , μ(τ ) ( j) j = 1, m, u(τ ) = , h 1k (τ ) = 0, t j < τ ≤ T, ν(τ ) ⎛

 λk l ˜ C1k (T ) + X 1k − (−1)k Y1k , 2a   1 λk l ˜ C2k (T ) + X 2k − (−1)k Y2k , C2k (T ) = 2 λk 2a   1 λk l ˜ ( j) ( j) C1k (t j ) = 2 C1k (t j ) + X 1k − (−1)k Y1k , j = 1, m, λk 2a C1k (T ) =

1 λ2k



C˜ 1k (T ) = λk Vk (T ) − λk Vk (0) cos λk T − V˙k (0) sin λk T, C˜ 2k (T ) = Vk (T ) + λk Vk (0) sin λk T − V˙k (0) cos λk T, C˜ 1k (t j ) = λk Vk (t j ) − λk Vk (0) cos λk t j − V˙k (0) sin λk t j X 1k = λk ϕT (0) − ψ0 (0) sin λk T − λk ϕ0 (0) cos λk T, Y1k = λk ϕT (l) − ψ0 (l) sin λk T − λk ϕ0 (l) cos λk T, X 2k = ψT (0) − ψ0 (0) cos λk T + λk ϕ0 (0) sin λk T, Y2k = ψT (l) − ψ0 (l) cos λk T + λk ϕ0 (l) sin λk T, ( j) X 1k = λk ϕ j (0) − ψ0 (0) sin λk t j − λk ϕ0 (0) cos λk t j , ( j) Y1k = λk ϕ j (l) − ψ0 (l) sin λk t j − λk ϕ0 (l) cos λk t j .

Thus, to find the functions u(τ ), τ ∈ [0, T ], the infinite integral relations (51.29) are obtained. In practice, based on the modal method, some first n harmonics of elastic vibrations are selected and then a control synthesis problem is solved using methods from the theory of control of finite-dimensional systems. Following the modal method, for the first few n harmonics, the following notations for block matrices are introduced

51 Boundary Control of String Vibrations with Given Values …

⎞ ⎞ ⎛ H¯ 1 (τ ) C1 (t1 , ..., tm , T ) ⎜ H¯ 2 (τ ) ⎟ ⎜ C2 (t1 , ..., tm , T ) ⎟ ⎟ ⎟ ⎜ ⎜ Hn (τ ) = ⎜ . ⎟ , ηn = ⎜ ⎟ .. ⎠ ⎝ .. ⎠ ⎝ . ¯ Cn (t1 , ..., tm , T ) Hn (τ )

473

⎛

(51.30)

with dimensions Hn (τ )—(2n(m + 1) × 2), η—(2n(m + 1) × 1). For the first few n harmonics (here and in what follows, the presence of “n” in subscript denotes the first few n harmonics), taking into account (51.30) from (51.29) we have

T Hn (τ )u n (τ )dτ = ηn . (51.31) 0

For an arbitrary number of the first harmonics the boundary control action u n (t), satisfying the integral relation (51.31) is constructed in the form [4, 15] u n (t) = HnT (t)Sn−1 ηn + f n (t), where HnT (t) is a transposed matrix, f n (t) is some vector-function, such that

T

T Hn (t) f n (t)dt = 0, Sn =

0

Hn (t)HnT (t)dt. 0

Here Sn is a known matrix of dimension 2n(m + 1) × 2n(m + 1) and it is assumed that det Sn = 0. Further, having obtained u n (t) from (51.27) and (51.28), Vk (t) is found on the time interval t ∈ [0, T ]. Hence, from the formula (51.22), the function V (x, t) is obtained for the first n harmonics.

51.5 Conclusions The boundary control problem for the equation of string vibrations with given states at intermediate moments of time is reduced to a problem with zero boundary conditions and by the method of separation of variables for arbitrary numbers of the first harmonics, the control action is constructed. The approach proposed for the wave equation (using the Fourier method instead of the D’Alembert’s method) allows usage for other (not one-dimensional) vibrational systems.

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References 1. Butkovskii, A.G.: The Theory of Optimal Control of Systems with Distributed Parameters. Nauka, Moscow (1965) 2. Butkovskii, A.G.: Control Methods for Systems with Distributed Parameters. Nauka, Moscow (1975) 3. Znamenskaya, L.N.: Control of Elastic Vibrations. Fizmatlit, Moscow (2004) 4. Barseghyan, V.R.: Control of Composite Dynamic Systems and Systems with Multipoint Intermediate Conditions. Nauka, Moscow (2016) 5. Barseghyan, V.R.: Optimal control of string vibrations with nonseparate state function conditions at given intermediate instants. Autom. Remote Control 81, 226–235 (2020). https://doi. org/10.1134/S0005117920020034 6. Barseghyan, V.R., Saakyan, M.A.: The optimal control of wire vibration in the states of the given intermediate periods of time. Proc. NAS RA. Mech. 61, 52–60 (2008) 7. Barseghyan, V.R.: A control problem for string vibrations with nonseparated multipoint conditions at intermediate times. Proc. RAS. Mech. Solids. 6, 108–120 (2019). https://doi.org/10. 1134/S0572329919060047 8. Barseghyan, V.R.: On one problem of optimal boundary control of string vibrations with restrictions in the intermediate moment of time. In: Analytical Mechanics, Stability and Control, vol. 3, pp. 119–125 (2017) 9. Korzyuk, V.I., Kozlovskaya, I.S.: Two-point boundary problem for string oscillation equation with given velocity in arbitrary point of time. I. In: Tr. Inst. Mat. NAS of Belarus. 18, 22–35 (2010) 10. Korzyuk, V.I., Kozlovskaya, I.S.: Two-point boundary problem for string oscillation equation with given velocity in arbitrary point of time. II. In: Tr. Inst. Mat. NAS of Belarus. 19, 62–70 (2011) 11. Andreev, A.A., Leksina, S.V.: The boundary control problem for the system of wave equations. J. Samara State Tech. Univ. Ser. Phys. Math. Sci. 1(6), 5–10 (2008). https://doi.org/10.14498/ vsgtu565 12. Il’in, V.A., Moiseev, E.I.: Optimization of boundary controls of string vibrations. Russ. Math. Surv. 60, 1093–1120 (2005). https://doi.org/10.1070/RM2005v060n06ABEH004283 13. Moiseev, E.I., Kholomeeva, A.A.: Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end. Differ. Equat. 49, 1317–1322 (2013). https://doi.org/10.1134/S0012266113100133 14. Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Nauka, Moscow (1977) 15. Zubov, V.I.: Lectures on Control Theory. Nauka, Moscow (1975)

Chapter 52

Resonant Oscillations of a Controlled Reversible Mechanical System in the Vicinity of Equilibrium Valentin N. Tkhai and Ivan N. Barabanov

Abstract Forced resonant oscillations of the reversible mechanical system are studied in the case when the frequencies of the linearized system coincide with the frequencies of the exerting control. The regulator’s gain k is supposed to be small. Given the nondegenerate case, all isolated unsymmetrical oscillations, as well as families of symmetrical oscillations, are found. The resonant effect is manifested by O(k 1/3 ) increase in amplitude.

52.1 Introduction For a system that admits the first integral, the Lyapunov center theorem is known (see [4]). In a perturbed Lyapunov system, resonant oscillations are generated out of equilibrium under the action of a periodic k-perturbation [4]. This occurs when the natural frequency coincides with the perturbation frequency. In this case, the resonance reveals itself in oscillations of the amplitude O(k 1/3 ). The result applies to systems of a general type: in [1] systems without internal resonances up to the fourth order were studied. In this paper, reversible mechanical systems are studied. The analog of Lyapunov’s center theorem holds for such systems. We study a system that allows n oscillations in linearization. We establish the existence of n unsymmetrical oscillations and n families of symmetrical oscillations in the controlled system. The possibility of implementing resonant oscillations with special properties is of particular interest in control problems.

V. N. Tkhai · I. N. Barabanov (B) V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya str., Moscow, Russia e-mail: [email protected] V. N. Tkhai e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_52

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52.2 Reversible Mechanical System A reversible dynamical system with phase vector x and a non-degenerate map G has the property of space–time symmetry in the sense of invariance with respect to the transformation (x, t) → (Gx, −t) [3]. When    Il 0   G=  0 −In  , l ≥ n, where I j is ( j × j) identity matrix, we deal with a reversible mechanical system. Phase space of this system is given by the vectors u and v so that dim u = l, dim v = n, and the symmetry transformation has the form (u, v, t) → (u, −v, −t). The set M = {u, v : v = 0} is called the fixed set of reversible mechanical systems. Reversible mechanical systems describe the dynamics of most models of classical and celestial mechanics [6]. In this case, the vectors u and v are usually taken as the vectors of generalized coordinates (quasicoordinates) and generalized velocities (quasi-velocities), respectively. In the neighborhood of equilibrium, the reversible mechanical system is written as u˙ = Av + U (u, v), v˙ = Bu + V (u, v), u ∈ Rl , v ∈ Rn , l ≥ n, U (u, −v) = −U (u, v), V (u, −v) = V (u, v), U (0, 0) = 0, V (0, 0) = 0.

(52.1)

Here, A and B are costant matrices, U (u, v) and V (u, v) are sufficiently smooth nonlinear functions. Motions of (52.1) are symmetrical with respect to fixed set M = {u, v : v = 0} of the reversible mechanical system. The structure of the linear approximation of (52.1) implies that the characteristic equation admits (l − n) simple zero roots, whereas the remaining roots form pairs ±λ that can be calculated from the equation det B A − I λ2  = 0,

(52.2)

where I is the identity matrix. We consider the case when Eq. (52.2) admits purely imaginary roots. Periodic motions of (52.1) are given by the Lyapunov–Bruno–Devaney theorem, which is the analog of the Lyapunov center theorem for reversible systems. In the case of l = n and in the presence of the first integral, the Lyapunov theorem is obtained. However, the Lyapunov–Bruno–Devaney theorem also holds in the absence of the first integral, the requirement of the first integral being replaced by the requirement of symmetry of the phase space of the system. Moreover, it holds for arbitrary l ≥ n. Proof of the theorem can be found in [7, 9]. The linear approximation of (52.1) can be reduced to the form in which it splits into independent subsystems, while subsystems remain coupled in the nonlinear formulation of the problem. The most interesting case for research occurs when the linear approximation is stable. Here, in a non-resonant situation (in the sense of

52 Resonant Oscillations of a Controlled Reversible Mechanical System …

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internal resonance) in a normalized (up to an arbitrarily finite order) system, there are no nonlinear couplings between subsystems [5]. However, the complete system still remains coupled.

52.3 Resonant Oscillations In local analysis, it is convenient to use the normal form [2] up to members of finite order. Let the variable χ ∈ Rl−n correspond to the zero roots in the normalized system. Then the equation for χ does not contain quadratic terms that are free of χ . In the case of n pairs of purely imaginary roots λs = ±iωs , s = 1, . . . , n for 2n variables in normal form, it is convenient to use complex-conjugate variables z s , z¯ s , s = 1, . . . , n. Then the quadratic terms in the right-hand sides of the equations for z, z¯ turn to zeros along with the variable χ . Now, by adding periodic control, we get a controlled system in real variables χ , x, y (z = x + i y) χ˙ = (χ , x, y) + k R(t), x˙ = ωy + f (χ , x, y) + X (χ , x, y, ) + k F(t), y˙ = −ωx + g(χ , x, y) + Y (χ , x, y) + kG(t), x = (x1 , . . . , yn ),

(52.3)

y = (x1 , , . . . , yn ), ω = (ω1 , . . . , ωn ).

Consider the frequency ωs . Then the resonance takes place if the system is under 2π/ωs -periodic control. We take into account the structure of nonlinear terms [2] in the normal form. The system restricted to the terms of the third order appears to admit a unique 2π/ωs -periodic oscillation such that x˙s = ωs ys + f s∗ (xs , ys ) + k Fs (t), χ = 0, x j = y j = 0,

y˙s = −ωs xs + gs∗ (xs , ys ) + kG s (t),

j = 1, . . . , n,

j = s.

(52.4)

The functions f s∗ and gs∗ in normal form are given by the terms of the identical resonance and have the form f s∗ = f s (χ , x1 , y1 , . . . , xn , yn )∗ = −css ys (xs2 + ys2 ), g1∗ = gs (χ , x1 , y1 , . . . , xn , yn )∗ = css xs (xs2 + ys2 ). Here, the subscript ∗ denotes that the functions are taken at χ = 0, x j = y j = 0, j = 1, . . . , n, j = s; css is a constant value. The very solution of the restricted system is written as

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ξ=

k 1/3 xs0

+k

t 

 f ss∗ (xs0 , ys0 ) + F˜s (t) dt,

0

η = k 1/3 ys0 + k

t 

 ∗ gss (xs0 , ys0 ) + G˜ s (t) dt,

(52.5)

0

χ = 0, x j = y j = 0,

j = 1, . . . , n;

j = s, Fs + iG s = ( F˜s + iG˜ s )exp(iωs t).

(xs + iys ) = (ξ + iη)exp(iωs t),

The initial point of the oscillation is in the plane (xs , ys ) whereas xs0 , ys0 are as follows: xs0 = −  3

I ys 3 (I 2 + I 2 )/ω 2π css s xs ys

where ωs Ixs (t) = 2π

t

,

ys0 = −  3

F˜s (τ )dτ,

0

Ixs = Ixs (2π/ωs ),

Ixs 3 (I 2 + I 2 )/ω 2π css s xs ys

ωs I ys (t) = 2π

t 0

G˜ s (τ )dτ,

,

(52.6)

(52.7)

I ys = I ys (2π/ωs ).

It turns out that when there is no zero roots (i.e. l = n) the solution of the restricted system yields the resonant oscillation of the coupled controlled system in the vicinity of equilibrium. The following theorem holds. Theorem 52.1 Let the characteristic equation of the reversible mechanical system near equilibrium have a pair of purely imaginary roots ±iωs . Let the remaining roots not be multiple to the above ones and not be equal to zero. Then the system under 2π/ωs -periodic controls admits a unique 2π/ωs -periodic resonant oscillation. For the system in the form (52.3), the solution is in the O(k)-vicinity of the set {x, y : x j = y j = 0, j = 1, . . . , n; j = s}, where the variables xs and ys are given by (52.5)–(52.7) to within o(k). Proof of Theorem 52.1 is based on choosing appropriate measurement scale for each variable, then constructing the generating system, and applying the perturbation theory. This theorem has been proved this way previously in [1] for the particular case of the system without internal resonances. The generalization of proof in [1], which is appropriate to prove Theorem 52.1, is omitted here, since it is of technical nature. Resuming these reasonings, we can state that n resonant oscillations can be realized in the controlled reversible mechanical system in the general case under appropriate periodic controls.

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52.4 A Family of Symmetrical Resonant Oscillations For a reversible mechanical system with l = n, there exists a control other than found in Theorem 52.1 such that a symmetrical resonance oscillation is realized. The necessary and sufficient conditions for a symmetrical periodic motion to exist are obtained by using the condition that the solution intersects the set M at time instants multiple to 2π/ω. For a 2π/ω-periodic system, those conditions are written as [8] (52.8) vs (u 01 , . . . , u l0 , 2π/ω) = 0, s = 1, . . . , n, where (u 01 , . . . , u l0 ) is the initial point on M. The above solutions form an (l − n)family. Consider system (52.3). Since the restricted system admits a solution such that xs and ys oscillate with O(k 1/3 ) amplitude, while the remaining variables oscillate with O(k) amplitude, we apply the following scaling: (xs , ys ) → k 1/3 (xs , ys ), (x j , y j ) → k 2/3 (x j , y j ). The scaling allows us to identify the most significant terms for the existence of the resonant oscillations in the complete system. These terms are defined by the terms in the lowest degree of k in each equation. In the restricted system, the necessary conditions for the existence of the manifold with the oscillation described by the variables xs and ys , is the identical vanishing of the remaining equations. Therefore, the result of the scaling is written as χ˙ = O(k 2/3 ), ξ˙ = k 2/3 [ f s∗ (ξ, η) + F˜s (t)] + O(k), η˙ = k 2/3 [gs∗ (ξ, η) + G˜ s (t)] + O(k), x˙ j = ω j y j + k 1/3 [ f j∗ +  j (t)] + O(k 2/3 ),

(52.9)

y˙ j = −ω j x j + k 1/3 [g ∗j + H2 (t)] + O(k 2/3 ), xs + iys = (ξ + iη) exp (iωs t). System (52.9) inherits the reversibility property from (52.4), in addition, it contains the fixed set {χ , ξ, η, x j , y j ( j = 1, . . . , n; j = s) : η = 0, y2 = . . . = yn = 0}. System (52.9) admits a 2π -periodic solution such that χ = 0, ξ = xs0 = const, η = ys0 = const, x j = 0, y j = 0, j = 1, . . . , n; j = s. Constant values xs0 and ys0 are found from the system of amplitude equations

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f s∗ (ξ, η) + Ixs (2π/ωs ) = 0, gs∗ (ξ, η) + I ys (2π/ωs ) = 0. This system has a unique simple solution (52.6). If Ixs (2π/ωs ) = 0, then ys0 = 0, and the oscillation is symmetrical. Introduce the parameter μ = k 1/3 . Since (52.9) depends on μ, system (52.8) depends on μ as well. In addition, the possibility to extend a solution by the small parameter μ is determined by the implicit function theorem applied to (52.8) at μ = 0. Rearrange (52.8) as follows: ⎡ ⎢ η(μ, χ 0 , ξ 0 , xˆ 0 , 2π/ωs ) ≡ μ2 ⎣

2π/ω  s





⎤

⎥ gs∗ (ξ 0 , 0) + G˜ s (t) dt + O(μ)⎦ = 0,

0 0 0 0 0 y j (μ, χ , ξ , xˆ , 2π/ωs ) ≡ x j sin 2π(ω j /ωs ) + O(μ) = 0, xˆ 0 = (x10 , . . . , x 0j , . . . , xn0 : j  = s).

j = 1, . . . , n; j  = s,

Since there is no frequencies multiple to ωs , we obtain x 0j = O(μ), j = 1, . . . , n; j = s. By putting it to the equations for η and taking into account that the root ξ 0 = xs0 of Eq. (52.6) is simple, we conclude that (52.9) admits a χ 0 -family of solutions. This means the existence of symmetrical 2π -periodic oscillations in (52.3). Theorem 52.2 Let a reversible mechanical system be described by (52.1) in the vicinity of equilibrium. Let the appropriate characteristic equation have l − n zero roots and n pairs of purely imaginary roots ±iω j , j = 1, . . . , n. Let (52.1) undergo 2π/ωs -periodic controls. Then a unique (l − n)-family of 2π/ωs -periodic resonant oscillations is realized. Those oscillations are symmetrical with respect to the set M ∗ = {u, v, t : v = 0, sin(ωs t = 0)} if the frequencies ±ω j , j = 1, . . . , n, j = s are not multiple to ±ωs . For a system written in the form of (52.3), the solution is in an O(k)-vicinity of the set {χ , x, y : x j = y j = 0, j = 1, . . . , n; j = n}, where the variables xs and ys are determined to within o(k) by (52.4)–(52.7) with ys0 = 0, Ixs = 0. Theorem 52.2 remains valid if the frequency ωs of the uncontrolled system is O(k)-close to a natural number. By summarizing the above reasoning, we conclude the following. Families of symmetric resonant oscillations are realized in a reversible mechanical system under the appropriate periodic controls. The number of families is l − n, the frequencies of the oscillations are equal to those of the linearized system.

52.5 Example Consider the pendulum equation x¨ + sin x = 0, which is invariant with respect to two transformations: (1) (x, x, ˙ t) → (x, −x, ˙ −t), (2) (x, x, ˙ t) → (−x, x, ˙ −t). Hence,

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this system belongs to the class of reversible mechanical system with two fixed sets: (1) M1 = {x, x˙ : x˙ = 0}, (2) M2 = {x, x˙ : x = 0}. This fact can be readily seen on the phase portrait of the system, which is symmetrical with respect to both the abscissa and the ordinate axes. Note that l = n = 1. The pendulum performs local oscillations with the frequency ω = 1. Hence, by applying a 2π -periodic control with a small regulator gain k we obtain, according to Theorems 52.1, 52.2, 2π -periodic oscillations of the O(k 1/3 )-amplitude. Certain controls are chosen according to the control purposes. In view of this, the controlled system can be written two ways: either (1) u˙ = v + k F(t), v˙ = − sin u + kG(t), u = x, v = x˙ or (2) u˙ = v + k F(t), v˙ = − sin u + kG(t), u = x, ˙ v = x, F(t + 2π ) = F(t), G(t + 2π ) = G(t). Then, by applying Theorem 52.2 to the first case, we obtain resonant oscillations sym˙ t : x˙ = 0, sin t = 0}. An even function metrical with respect to the set M1∗ = {x, x, F(t) with nonzero average value, as well as an odd function G(t) will be sufficient for the existence of those oscillations. According to (52.6), the pendulum must have a nonzero initial deviation, while the initial angular velocity must be zero. ˙ t: Resonant oscillations that are symmetric with respect to the set M2∗ = {x, x, x˙ = 0, sin t = 0} are obtained by applying Theorem 52.2 to the second case. In this case, the initial angular velocity is nonzero, while the initial deviation is zero. Finally, choose F(t) and G(t) with nonzero average values in order to obtain an unsymmetrical resonant oscillation. Then, by applying Theorem 52.1 to either the first or the second case, we obtain the existence of the resonant oscillation where both initial deviation and initial angular velocity are nonzero.

52.6 Conclusion In the vicinity of equilibrium, resonant oscillations occur in a controlled reversible mechanical system. The frequency of periodic control with a small gain k coincides with the frequency of the system, and the oscillation amplitude is equal to O(k 1/3 ). Resonant oscillations form an α-family of symmetrical oscillations, where α is the exceeding of the system’s dimension over twice the dimension of the fixed set. Thus, depending on the control purpose, it is always possible to implement a symmetrical or an unsymmetrical resonant oscillation. Acknowledgements This paper is financially supported in part by the Russian foundation for basic researches (Project No. 19-01-00146)

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References 1. Barabanov, I.N., Tkhai, V.N.: Oscillations of a coupled controlled system near equilibrium. Automat. Remote Control 80(12), 2126–2134 (2019) 2. Bruno, A.D.: Local Method of Analysis of Differential Equations. Nauka, Moscow, Russia (1979). (In Russian) 3. Lamb, J.S.W., Roberts, J.A.G.: Time-reversal symmetry in dynamical systems: a survey. Physica D 112(1–2), 1–39 (1998) 4. Malkin, I.G.: Some Problems in the Theory of Nonlinear Oscillations. Gostekhizdat, Moscow, Russia (1956). (In Russian) 5. Tkhai, V.N.: On stability of mechanical systems under the action of position forces. J. Appl. Math. Mech. 44(1), 24–29 (1980) 6. Tkhai, V.N.: The reversibility of mechanical systems. J. Appl. Math. Mech. 55(4), 461–468 (1991) 7. Tkhai, V.N.: Continuation of periodic motions of a reversible system in non-structurally stable cases. Application to the N -planet problem. J. Appl. Math. Mech. 62(1), 51–56 (1998) 8. Tkhai, V.N.: The rotational motions of mechanical systems. J. Appl. Math. Mech. 6(2), 173–188 (1999) 9. Tkhai, V.N.: Lyapunov families of periodic motions in a reversible system. J. Appl. Math. Mech. 64(1), 41–52 (2000)

Chapter 53

Representation Forms of the Angular Velocity Vector for an Orthonormal Basis of a Moving Frame Vladislav S. Ermolin and Tatyana V. Vlasova

Abstract In this paper, we consider a Cartesian moving reference frame. Its angular velocity vector is introduced as a solution to a system of kinematic equations of basis vectors. These equations connect the position of the basis vectors with their velocity. The construction of a formula for the angular velocity vector of an orthonormal basis is described. It is shown that the angular velocity vector in the found form is a solution to the system of the equations. Using transformations of the constructed solution, four more representation forms of the angular velocity vector are derived. It is shown that all the obtained forms define the same angular velocity vector of the moving space, though they contain different elements. All of the forms are also solutions of the system of kinematic equations. Presented results can be applied both to a solid body and to any rigid system.

53.1 Introduction The concept of the angular velocity for a moving frame and for a rigid body is one of the fundamentals and difficult to explain in kinematics. There are various ways of introducing this concept [1, 2, 6–15]. In this paper, we propose the approach based on the construction of a general solution for the system of kinematic equations of a moving basis in vector form. Ideas for this approach are outlined in [3–5]. The Cartesian coordinate system rigidly connected with a moving space is considered as a moving reference frame. A kinematic equation is determined for each of its basis vectors. It is shown that each kinematic equation of the basis vector has not a unique solution. Sets of solutions of these equations are characterized. Further, vector ω(t) is considered as a solution to the system of constructed equations. It is called the angular velocity vector of the moving space. The algorithm for V. S. Ermolin · T. V. Vlasova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. S. Ermolin e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_53

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constructing the angular velocity vector formula is described and the verification is carried out that ω(t) is a solution to the system of equations. The structure of this solution is used to find another representation forms. As a result, five different representation forms of the angular velocity vector for the orthonormal basis of a moving frame are obtained.

53.2 Representation Forms for the Angular Velocity Vector Problem Statement In absolute space with a coordinate system Oa x yz, another space moves (we will call it a moving space) with its own reference frame O x  y  z  . Let e1 , e2 , e3 be the basis of this frame. We will call this the moving basis. The orientation and motion of this basis in absolute space is given by equations e1 = e1 (t), e2 = e2 (t), e3 = e3 (t). Here, ei (t) (i = 1, 2, 3) are vector functions twice continuously differentiable for t ∈ J . For each t, the moving basis e1 , e2 , e3 forms the right orthonormal triple of vectors. Hence, ei (t) (i = 1, 2, 3) are related as follows: e3 (t) ≡ e1 (t) × e2 (t), e1 (t) ≡ e2 (t) × e3 (t), e2 (t) ≡ e3 (t) × e1 (t) (53.1) (ei (t), ei (t)) ≡ 1, i = 1, 2, 3, (53.2) (ei (t), e j (t)) ≡ 0,

i, j = 1, 2, 3,

i = j.

(53.3)

Differentiating (53.2), (53.3) w.r.t. (with respect to) t, we obviously get additional restrictions on the velocities of the basis vectors (˙ei (t), ei (t)) ≡ 0, i = 1, 2, 3;     e˙ i (t), e j (t) ≡ − ei (t), e˙ j (t) , i, j = 1, 2, 3, i = j.

(53.4) (53.5)

The purpose of this paper is to prove the existence of five possible representation forms for the instantaneous angular velocity vector of a moving frame and to construct them explicitly. Constructing the First Representation Form of the Angular Velocity Vector Let us consider identities (53.4). It follows that the vectors ei (t) and e˙ i (t) (i = 1, 2, 3) are orthogonal to each other for any t. Moreover, ei (t) = 0 for any t. It is known from analytical geometry, that under such conditions, for every t, there exist vectors ωi (t) connecting ei (t) and e˙ i (t) by equalities e˙ i (t) = ωi (t) × ei (t), i = 1, 2, 3, t ∈ J.

(53.6)

The vector function ωi (t) continuously differentiable for all t ∈ J , that turns the i-th relation (53.6) into an identity w.r.t. t is called the solution of this equation. Based on [5], it is easy to show that each set i of vector functions ωi (t) (i = 1, 2, 3)

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  i = ωi (t)|ωi (t) = ei (t) × e˙ i (t) + αi (t)ei (t), ∀αi (t) ∈ C 1 (t ∈ J ) contains all solutions of the ith Eq. (53.6). We now consider the following system of kinematic equations e˙ i (t) = ω(t) × ei (t), i = 1, 2, 3, t ∈ J.

(53.7)

Here, the vector ω(t) is the same to all of basis vectors e1 , e2 , e3 . In (53.7), either a non-unique solution may exist, or it may not exist at all. This follows from the fact that (53.7) is redundant. It consists of three vector equations under connections (53.1)–(53.5). Only one vector ω(t) (its three components) is unknown in (53.7). System (53.7) is called the system of kinematic equations of the moving basis. The set of solutions of this 3 system is determined by the intersection of the sets i . i . Thus, any solution to the system (53.7) is a solution Denote this set  = i=1 of each of three equations in (53.7). Elements of the set  we denote ω(t) and call the angular velocity vector of the moving basis. According to [3], a necessary and sufficient condition for the existence of a solution to system (53.7) is the fulfillment of relations (53.4), (53.5). 1. We will now describe the method of constructing the solution to system (53.7). We assume that the vector ω(t) is a solution of (53.7). Substituting ω(t) in the righthand side of (53.7) turns the system into identities. Let us multiply vectorially from the left both sides of the i-th identity in (53.7) by the corresponding vector ei (t) (i = 1, 2, 3). Then we transform the right-hand sides using the triple vector product formula (hereafter, to simplify the notation, we omit the argument t). Using (53.2), we get ei × e˙ i = ei × (ω × ei ) = ω(ei , ei ) − ei (ω, ei ) = ω − ei (ω, ei ) (i = 1, 2, 3). Summing separately the left and the right sides of the obtained identities for all i = 1, 2, 3, we have 3 

ei × e˙ i = 3ω − [e1 (ω, e1 ) + e2 (ω, e2 ) + e3 (ω, e3 )] = 3ω − ω = 2ω.

i=1

So, we get a representation of the vector ω for any t ∈ J in the following form ωI =

1 [e1 × e˙ 1 + e2 × e˙ 2 + e3 × e˙ 3 ] . 2

(53.8)

2. Let us check that vector function ω I is a solution to system (53.7). We show that the equality ω I × e1 = e˙ 1 holds. For this, we multiply vectorially from the right both sides of (53.8) by the vector e1 , then multiply out the triple vector product of each term. Using (53.2) – (53.5), we have

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(e1 × e˙ 1 ) × e1 = e˙ 1 (e1 , e1 ) − e1 (˙e1 , e1 ) = e˙ 1 + e1 (˙e1 , e1 ) ; (e2 × e˙ 2 ) × e1 = e˙ 2 (e1 , e2 ) − e2 (˙e2 , e1 ) = e2 (˙e1 , e2 ) ; (e3 × e˙ 3 ) × e1 = e˙ 3 (e1 , e3 ) − e3 (˙e3 , e1 ) = e3 (˙e1 , e3 ) .

(53.9)

In the first relation of (53.9), we add the term 2e1 (˙e1 , e1 ) ≡ 0. Finally, we obtain 1 1 1 1 e˙ 1 + ei (˙e1 , ei ) ≡ e˙ 1 + e˙ 1 ≡ e˙ 1 . 2 2 i=1 2 2 3

ω I × e1 =

This means that the vector ω I is a solution of the first equation of system (53.7). The second and third equations of system (53.7) are verified in a similar way. Then, we have ω I × e2 = e˙ 2 , ω I × e3 = e˙ 3 . Thus, the vector ω(t) = ω I , determined by formula (53.8), is a solution to system (53.7). This means that ω(t) = ω I is the angular velocity vector of the moving basis. Its value at a fixed time is usually called the instantaneous angular velocity. 3. Let us show that solution (53.8) of system (53.7) is unique. Suppose there exist two solutions of system (53.7) that do not coincide at least at one time t¯ : ω 1 (t¯) = ω 2 (t¯). Substitute them in (53.7). We get two systems of identities. Subtracting one system from another, we get the identities: 0 = (ω 2 (t¯) − ω 1 (t¯)) × e i (t¯) (i = 1, 2, 3). Hence, we have: ω 2 (t¯) − ω 1 (t¯) = 0. We get a contradiction with the initial assumption. This means that the set  contains only one element, and the basis of the moving space has a unique instantaneous angular velocity vector. 4. Let us show now that the moving space has a unique angular velocity vector, which can be determined by the motion of any chosen reference frame in the space. In addition to the reference frame O x  y  z  with the basis e1 (t), e2 (t), e3 (t), we consider a reference frame with an orthonormal basis e1 , e2 , e3 . The motion of the basis e1 (t), e2 (t), e3 (t) is associated with the motion of the basis of the first system as follows 3  ei (t) ≡ αi j e j (t), i = 1, 2, 3. (53.10) j=1

Here, αi j = (ei (t), e j (t)) = const., (i, j = 1, 2, 3). The kinematic equation of the basis e1 (t), e2 (t), e3 (t) has the form e˙ i (t) = ω(t) × ei (t), i = 1, 2, 3

(53.11)

where ω(t) is the angular velocity vector for the basis of the second frame. Differentiating w.r.t. t system (53.10), we get

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e˙ i (t) ≡

3 

487

αi j e˙ j (t), i = 1, 2, 3.

(53.12)

j=1

In (53.7), replace ω(t) by ω I , and then substitute the right-hand side of (53.7) into the right-hand side of (53.12). We have e˙ i (t) ≡

3 

αi j [ω I × e j (t)] ≡ ω I ×

j=1

3 

αi j e j (t) ≡ ω I × ei (t), i = 1, 2, 3.

j=1

Therefore, the vector ω I is the solution to system (53.11), i.e. ω(t) ≡ ω I . Due to the arbitrariness of the second reference frame, we can state that ω I is the angular velocity of the whole moving space, despite the choice of the moving frame. Transformation of Formula (53.8) for Vector ω I We will consider equality (53.8) as an analytical representation of the vector function ω I that gives an explicit dependence of ω I on ei and e˙ i . By transforming formula (53.8), we construct four more representation forms for the instantaneous angular velocity vector of the moving space (vectors ω II , ω III , ω IV , ω V ). Let us simplify formula (53.8). For this, we introduce the notation ω 01 = e1 × e˙ 1 ,

ω 02 = e2 × e˙ 2 ,

ω 03 = e3 × e˙ 3 .

(53.13)

Then (53.8) has the form ωI =

1 (ω 01 + ω 02 + ω 03 ). 2

(53.14)

Denote γ 1 = −ω 01 + ω 02 + ω 03 , γ 2 = ω 01 − ω 02 + ω 03 ,

γ 3 = ω 01 + ω 02 − ω 03 . (53.15)

It follows easily that (53.14) can be written in three ways: 1 1 ω I = ω 01 + γ 1 , ω I = ω 02 + γ 2 , 2 2

1 ω I = ω 03 + γ 3 . 2

(53.16)

Now let us prove that the following equalities hold: e i × γ i = 0,

i = 1, 2, 3.

(53.17)

To do this, we first verify that equality (53.17) holds for i = 1. We substitute in its left part the vector γ 1 from (53.15) and vectors ω 0i (i = 1, 2, 3) from (53.13), and multiply out the triple vector products. We obtain

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e 1 × γ 1 = e 1 × (−ω 01 + ω 02 + ω 03 ) = [−e 1 (e 1 , e˙ 1 ) + e˙ 1 (e 1 , e 1 )] + [e 2 (e 1 , e˙ 2 ) − e˙ 2 (e 1 , e 2 )] + [e 3 (e 1 , e˙ 3 ) − e˙ 3 (e 1 , e 3 )] . In the first square brackets: (1) the first term equals zero, but we will save it (this will not change the result); (2) in the second term (e 1 , e 1 ) = 1, so it is saved. In the second square brackets: (1) the first term is replaced by −e 2 (˙e1 , e 2 ); (2) the second term equals zero since (e 1 , e 2 ) = 0. In the third square brackets: (1) the first term is replaced by −e 3 (˙e1 , e 3 ); (2) the second term equals zero since (e 1 , e 3 ) = 0. Therefore, e 1 × γ 1 = e˙ 1 − e 1 (˙e1 , e 1 ) − e 2 (˙e1 , e 2 ) − e 3 (˙e1 , e 3 ) = e˙ 1 − e˙ 1 = 0. The validity of (53.17) for i = 1 is proved. Continuing this line of reasoning for i = 2, 3, we prove the validity of all equalities of (53.17). From (53.17), we conclude that vectors γ i and e i are collinear (i = 1, 2, 3). This means that there exist real numbers λi such that the following equalities hold: γ i = λi e i , i = 1, 2, 3.

(53.18)

An explicit dependence of γ i on e i and e˙ 1 is easily constructed using (53.18) and (53.13). Indeed, scalarly multiplying the i-th equality (53.18) by e i , we obtain λi = (e i , γ i ), i = 1, 2, 3.

(53.19)

Substituting (53.15) in (53.19), we get λ1 = (e 1 , −ω 01 + ω 02 + ω 03 ), λ3 = (e 3 , ω 01 + ω 02 − ω 03 ). λ2 = (e 2 , ω 01 − ω 02 + ω 03 ), Using (53.13) and multiplying out scalar products, we obtain λ1 = − (e1 , e1 , e˙ 1 ) + (e1 , e2 , e˙ 2 ) + (e1 , e3 , e˙ 3 ) . Taking into account (53.1) and (53.5), we have λ1 = (e3 , e˙ 2 ) + (−e2 , e˙ 3 ) = (e3 , e˙ 2 ) + (e3 , e˙ 2 ) = 2 (e3 , e˙ 2 ) . Arguing as above for i = 2, 3, we see that λ2 = 2 (e1 , e˙ 3 ), λ3 = 2 (e2 , e˙ 1 ). Substituting λi (i = 1, 2, 3) in (53.18), we get γ 1 = 2 (e3 , e˙ 2 ) e1 , γ 2 = 2 (e1 , e˙ 3 ) e2 , γ 3 = 2 (e2 , e˙ 1 ) e3 .

(53.20)

In (53.16), replace γ i by (53.20) and ω 0i (i = 1, 2, 3) by (53.13). The obtained forms denote

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ω II = e1 × e˙ 1 + (e3 , e˙ 2 ) e1 ; ω III = e2 × e˙ 2 + (e1 , e˙ 3 ) e2 ; ω IV = e3 × e˙ 3 + (e2 , e˙ 1 ) e3 .

(53.21) (53.22) (53.23)

Given these notations, we get three more variants of the representation form for the vector ω I , i.e. we can write ω I = ω II ,

ω I = ω III ,

ω I = ω IV ,

(53.24)

where ω II , ω III , ω IV are determined by (53.21)–(53.23). The Fifth Representation Form for Vector ω I One more time, we consider formula (53.8). Transforming it, we get the fifth representation form of the vector ω I : ω I = ω V,

(53.25)

ω V = (e3 , e˙ 2 ) e1 + (e1 , e˙ 3 ) e2 + (e2 , e˙ 1 ) e3 .

(53.26)

where

To do this, consider the elements (53.13) included in formula (53.8). Replace the first multipliers e1 , e2 , and e3 by (53.1). Then each of the terms in (53.8) will take the form of a triple vector product. Using triple vector product identity and (53.5), we get ω 01 = e1 × e˙ 1 = (e2 × e3 ) × e˙ 1 = (e2 , e˙ 1 ) e3 + (e1 , e˙ 3 ) e2 ; ω 02 = e2 × e˙ 2 = (e3 × e1 ) × e˙ 2 = (e3 , e˙ 2 ) e1 + (e2 , e˙ 1 ) e3 ; ω 03 = e3 × e˙ 3 = (e1 × e2 ) × e˙ 3 = (e1 , e˙ 3 ) e2 + (e3 , e˙ 2 ) e1 .

(53.27) (53.28) (53.29)

Summing separately the left and right sides of equalities (53.27)–(53.29) and equating the results to each other, we obtain 3  i=1

ω 0i =

3 

ei × e˙ i = 2 [(e3 , e˙ 2 ) e1 + (e1 , e˙ 3 ) e2 + (e2 , e˙ 1 ) e3 ] .

(53.30)

i=1

Comparing (53.30) with (53.8) and (53.26), we get equality (53.25). It is easy to show that equality (53.25) for ω I can be constructed via any of the three equalities (53.24). For this, we replace the first terms on the right-hand sides of equalities (53.21)–(53.23) by their values (53.27)–(53.29). We have ω II = ω V ,

ω III = ω V ,

ω IV = ω V .

Substitution (53.31) in equalities (53.24) leads them to the form ω I = ω II = ω V ,

ω I = ω III = ω V ,

ω I = ω IV = ω V .

(53.31)

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Fig. 53.1 The angular velocity vector ω = ω II ; −→ ω II = ω 01 + O A1 ; −→ −→ ω 01 = e1 × e˙ 1 = O A2 + O A3

Fig. 53.2 The angular velocity vector ω = ω III ; −→ ω III = ω 02 + O A2 ; −→ −→ ω 02 = e2 × e˙ 2 = O A1 + O A3

These results can be summarized as follows. All vectors ω I , ω II , ω III , ω IV , ω V define the same angular velocity vector of the moving frame in absolute space. Figures 53.1, 53.2, 53.3, and 53.4 illustrate the representation of the vector ω (t) −→ in the forms of (53.21)–(53.23) and (53.26). In all the figures O A1 = (e3 , e˙ 2 )e1 ; −→ −→ O A2 = (e1 , e˙ 3 )e2 ; O A3 = (e2 , e˙ 1 )e3 . Remark 53.1 Each of the constructed vectors ω I , ω II , ω III , ω IV , ω V is a solution to each equation of system (53.7) separately. The validity of this statement for (53.21) – (53.23) and (53.26) was established in the same way as for the vector ω I , but the volume of the paper does not allow us to present these proofs. Remark 53.2 All the obtained results are easily transferred to a solid body. For this, a moving space is conditionally considered as an absolutely solid body, and a reference frame as a Cartesian coordinate system that is rigidly attached to the solid body.

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Fig. 53.3 The angular velocity vector ω = ω IV ; −→ ω IV = ω 03 + O A3 ; −→ −→ ω 03 = e3 × e˙ 3 = O A1 + O A2

Fig. 53.4 The angular velocity vector ω = ω V ; −→ −→ −→ ω V = O A1 + O A2 + O A3

53.3 Conclusion In this paper, five different representation forms ω I , ω II , ω III , ω IV , ω V for the angular velocity vector of a moving space are obtained. In our opinion, the form ω I is the most convenient for using in theoretical research. The other forms may be useful when considering practical problems. In addition, the presented results provide students with a better understanding for the relevant material of the course of theoretical mechanics. These representation forms of the angular velocity vector may be used in any variant depending on objectives of a researcher.

References 1. Buchholz, N.N.: The Basic Course of Theoretical Mechanics. Part I. Lan’, St. Petersburg (2009).(in Russian) 2. Crampin, M.: On the concept of angular velocity. Eur. J. Phys. 7, 287–293 (1986) 3. Ermolin, V.S.: The canonical representation of the instantaneous angular velocity. In: Smirnov, N.V. et al. (eds.) Control Processes and Stability, Proceedings of XXXIV Conference on Control Processes and Stability, pp. 147–153. SPbU, St. Petersburg (2003) (In Russian)

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4. Ermolin, V.S., Korolev, V.S., Pototskaya, I.Y.: Theoretical Mechanics. Part 1. Kinematics. SPbU, St. Petersburg (2012) (In Russian) 5. Ermolin, V.S., Vlasova, T.V.: The generalized formula for angular velocity vector of the moving coordinate system. AIP Conf. Proc. 1959, 030008 (2018) 6. Hentschke, R.: Classical Mechanics. Springer, Berlin (2017) 7. Kilchevsky, N.A.: The Course of Theoretical Mechanics, vol. I. Nauka, Fizmatlit, Moscow (1972).(In Russian) 8. Lur’ye, A.I.: Analytical Mechanics. Nauka, Fizmatlit, Moscow (1961).(In Russian) 9. Markeev, A.P.: Theoretical Mechanics. CheRo, Moscow (1999).(In Russian) 10. Nikitin, N.N.: The Course of Theoretical Mechanics. Lan’, St. Petersburg (2010).(In Russian) 11. Polyakhov, N.N., Zegzhda, S.A., Yushkov, M.P.: Theoretical Mechanics. Yurait, Moscow (2012).(In Russian) 12. Suslov, G.K.: Theoretical Mechanics. Gostekhizdat, Moscow (1946).(In Russian) 13. Wilke, V.G.: Theoretical Mechanics. Lan’, St. Petersburg (2003).(in Russian) 14. Wittenburg, J.: Dynamics of Systems of Rigid Bodies. Springer, Berlin (2013) 15. Zhukovsky, N.E.: The Mechanics of the System. Editorial URSS, Moscow (2011).(In Russian)

Chapter 54

Attitude Controlled Motion in a Neighborhood of the Collinear Libration Point L 1 Alexander Shmyrov, Vasily Shmyrov, and Dzmitry Shymanchuk

Abstract This paper considers an attitude controlled motion of a celestial body in a neighborhood of the collinear libration point L 1 of the Sun–Earth system. The equation of the attitude controlled motion within the restricted three-body problem are investigated. This attitude controlled motion is described by the Euler’s dynamic equations and the quaternion kinematic equation. We solve the stability problem of the celestial body attitude motion in relative equilibrium positions and programmed attitude motions with proposed control laws.

54.1 Introduction An attitude motion in outer space depends essentially on the position of a celestial body center of mass and an orientation of a celestial body can affect orbital motion. We research the attitude motion of a celestial body of mass m in the gravitational field of two massive centers, the Earth of mass M1 and the Sun of mass M2 (m  M1 < M2 ). In this case, the celestial body motion can be described by a mathematical model of the circular restricted three body problem [4, 5]. We note that in [6] represented some statements of the orientation problems of a celestial body and their solutions. In [1] represented the uncontrolled attitude motion of a symmetrical celestial body at libration point L 2 . In celestial mechanics, there are five solutions which are called libration points or Lagrange points. So-called collinear libration points L 1 , L 2 , L 3 are unstable equilibrium positions. So-called triangular libration points L 4 , L 5 are stable. These stable A. Shmyrov · V. Shmyrov (B) · D. Shymanchuk St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. Shmyrov e-mail: [email protected] D. Shymanchuk e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_54

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Fig. 54.1 Related frames

equilibrium positions take place in systems for witch a certain mass ratio of attracting massive centers [2]. To research the celestial body attitude motion, we introduce the frames (Fig. 54.1). O X Y Z is the barycentric frame with origin at the center of mass of the Sun–Earth system, O X Y is the plane of the Earth and the Sun orbits. M1X Y Z is a rotating frame. C X Y Z is the orbital frame with the origin in the celestial body center of mass. The axes of the C X Y Z parallel to the axes of the rotating frame, and ii , i = 1, 2, 3, are its basis. C x yz is the rigidly connected frame with the celestial body. The origin of the C x yz in the celestial body center of mass, and ei , i = 1, 2, 3, are its basis vectors. The axes of the C x yz are directed along the principal central axes of inertia of the celestial body.

54.2 Equations of Attitude Controlled Motion The orientation of the celestial body in the frame C X Y Z will be set by using the Rodriguez–Hamiltonian parameters λ0 , λ1 , λ2 , λ3 : λ20 + λ21 + λ22 + λ23 = 1.

(54.1)

It is known, that by the Rodriguez–Hamiltonian parameters can be uniquely determine the transition matrix from the frame C x yz to the frame C X Y Z : e1 e2 e3 ⎛ ⎞ i1 2(λ20 + λ21 ) − 1 2(λ1 λ2 − λ0 λ3 ) 2(λ1 λ3 + λ0 λ2 ) i2 ⎝ 2(λ0 λ3 + λ1 λ2 ) 2(λ20 + λ22 ) − 1 2(λ2 λ3 − λ0 λ1 ) ⎠. i3 2(λ1 λ3 − λ0 λ2 ) 2(λ2 λ3 + λ0 λ1 ) 2(λ20 + λ23 ) − 1

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In the Hill problem of the circular restricted three-body problem the libration point L 1 in the rotating frame is stationary, and have coordinates [3, 5]: R1∗ = 1; 0; 0 ,

(54.2)

where · is a transpose operation. We get that the units were chosen so that the unit of distance is approximately equal to 1.5 × 106 km and the unit of time is 58.0916 days (year divided by 2π ). To describe the attitude controlled motion of the celestial body with respect to its center of mass when R1 = R1∗ (see (54.2)). In this way the celestial body center of mass will be located at the L 1 . Then the gravitational torque are expressed in the terms of the Rodrigues–Hamiltonian parameters, and the dynamic equations are [5]: p˙ + N x qr = 2ηN x (λ1 λ2 − λ0 λ3 )(λ1 λ3 + λ0 λ2 ) + u p = ε p + u p , q˙ + N y r p = ηN y (λ1 λ3 + λ0 λ2 )(2(λ20 + λ21 ) − 1) + u q = εq + u q , r˙ + Nz pq = ηNz (2(λ20 + λ21 ) − 1)(λ1 λ2 − λ0 λ3 ) + u r = εr + u r ,

(54.3)

where p, q, r are coordinates of the angular rate in the frame C x yz ; N x =

Iz −I y , Ix

Ny =

I x −Iz , Iy

Nz =

I y −I x Iz

; Ix , I y , Iz are the moments of inertia relative to the principal   1 central axes of inertia; η = 6 3 + 0.99 3 ; ε p , εq , εr are coordinates of the gravitational angular acceleration in the frame C x yz ; ua = u p ; u q ; u r  is the vector of the angular acceleration control. To describe of the rotation around the C X Y Z , we consider the quaternionic kinematic equation [5]: ⎛

⎛ ⎞ ⎞⎛ ⎞ λ˙ 0 0 −p −q −r + 1 λ0 ⎜ λ˙ 1 ⎟ 1 ⎜ p ⎟ ⎜ λ1 ⎟ 0 r + 1 −q ⎜ ⎟= ⎜ ⎟⎜ ⎟. ⎝ λ˙ 2 ⎠ 2 ⎝ q −r − 1 0 p ⎠ ⎝ λ2 ⎠ λ3 r −1 q −p 0 λ˙ 3

(54.4)

Equations (54.3), (54.4) are the system of differential equations of the celestial body attitude controlled motion at the libration point L 1 of the Sun–Earth system.

54.3 Linearized Equations of the Attitude Motion We assuming that the celestial body center of mass is located at the libration point L 1 , then there is an relative equilibrium position of the celestial body in the C X Y Z : λ0 = 1, λ1 = λ2 = λ3 = 0, and

(54.5)

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p = q = 0, r = 1. In the framework of the our research problem of the celestial body attitude dynamics in the gravitational field of two centers when Ix = I y , there is an unlimited set of relative equilibrium positions in the frame C X Y Z : λ20 + λ23 = 1, λ1 = λ2 = 0,

(54.6)

then C z coincides with the C Z . It follows from (54.6), there can be a stationary rotation the celestial body (the Euler’s case): p = q = 0, r = rr e f = const. To estimate the stability of the relative equilibrium positions (54.5), (54.6) at the libration point L 1 , we linearize the uncontrolled system of attitude motion (54.3), (54.4): ζ˙ = Aζ, where A is the matrix of dimension 7 × 7, ζ =  p; q; r − rr e f ; λ0 − cos α; λ1 ; λ2 ; λ3 − sin α , |rr e f |  ∞, α ∈ [0, π ). In the case of the equilibrium position (54.5) (rr e f = 1, α = 0), we get the following eigenvalues of the matrix A: 1 = 0,



4+N y (−4η+8(1+η)N x +(η+2N x )2 N y ) , 2 √ −2+(η+2N x )N y − 4+N y (−4η+8(1+η)N x +(η+2N x )2 N y ) ±

, 2 ηNz ±i 2 .

2,3 = ±

4,5 = 6,7 =

√

−2+(η+2N x )N y +

Note that, it is necessary to satisfy Ix ≤ I y + Iz , I y ≤ Ix + Iz , Iz ≤ Ix + I y , than |N x | ≤ 1, |N y | ≤ 1, |Nz | ≤ 1. Thus, for the existence of small oscillations at the equilibrium position (54.5), it is necessary that 1. N x > 0, N y < 0, Nz > 0 or Ix < I y < Iz . 2. N x < 0, N y > 0, Nz > 0 or Iz < Ix < I y .

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In the case of the equilibrium position (54.6) (Nz = 0, N y = −N x ), we get the following eigenvalues of the matrix A: 1 = 0, 2,3 = ±



√

−1−2r ∗ −2ηN x −r ∗2 (1+4N x )+

−16r ∗ (1+r ∗ )(η+r ∗ +r ∗2 )N x2 +((1+r ∗ )2 +2(η+2r ∗2 )N x )2 √ , 2 2 √ ∗ ∗2 ∗ ∗ ∗ ∗2 2 ∗ 2 ∗2 2 −1−2r −2ηN x −r (1+4N x )− −16r (1+r )(η+r +r )N x +((1+r ) +2(η+2r )N x ) √ , 2 2 √ ∗ 2 (r −1) . 2



4,5 = ± 6,7 = ±i

From the form of the eigenvalues, it follows that small oscillations are possible outside the interval −1 < r ∗ < 0 in the regions: 1. N x > 0 or Ix = I y < Iz ; 2. N x < 0 or Ix = I y > Iz at a sufficiently large value of the angular rate r ∗ .

54.4 Construction of the Control Laws Let the attitude controlled motion of the celestial body describes by (54.3), (54.4). Theorem 54.1 The control ua u p = −ε p − lI1x p − kIx1 λ0 λ1 , u q = −εq − lI2y q − kI 2y λ0 λ2 , u r = −εr − lI3z (r − 1) − kI3z λ0 λ3 ,

(54.7)

when ki > 0, li > 0, i = 1, 2, 3, ensures the Lyapunov stability of the equilibrium position (54.5) (C x yz and C X Y Z coincide): ∗ = 1; 0; 0; 0 . Proof Let we define the Lyapunov function in the form V = T + W,   where T = 21 Ix p 2 + I y q 2 + Iz r 2 is the celestial body kinetic energy,      W = k 1 − λ20 = 2k 1 − λ20 + λ21 + λ22 + λ23 is the “kinematic function” where k > 0. V is positive definite and equals zero at the equilibrium (54.5). To the function V the total time derivative, by virtue of the equations of motion (54.3), (54.4), is equal to

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V˙ = Ix p(ε p + u p ) + I y q(εq + u q ) + Iz r (εr + u r ) + k ( pλ0 λ1 + qλ0 λ2 + r λ0 λ3 ) . (54.8) Let us substitute the control (54.7) into the expression (54.8), than we get V˙ = −l1 p 2 − l2 q 2 − l3 r (r − 1) − pλ0 λ1 (k1 − k) − qλ0 λ2 (k2 − k) − r λ0 λ3 (k3 − k). (54.9)

Thus, we can put in the expression (54.9) that ki = k, i = 1, 2, 3, then for li > 0, i = 1, 2, 3, the V˙ is a negative definite for r > 1 and is positive definite for r < 1.  Remark 54.1 The values λ0 λi , i = 1, 2, 3, are the measures of the small angular deviation of the C x yz from the C X Y Z : ψ (r oll) ≈ 2λ0 λ1 , θ ( pitch) ≈ 2λ0 λ2 , ϕ (yaw) ≈ 2λ0 λ3 . Remark 54.2 Control (54.7) ensures the stabilization in the entire region of the celestial body positions with the exception λ0 := 0. Theorem 54.2 The control ua u p = −ε p − lI1x p − kIx1 (λ2 λ3 + λ0 λ1 ), u q = −εq − lI2x q − kIx2 (λ0 λ2 − λ1 λ3 ), u r = − lI3z (r − rr e f ).

(54.10)

when ki > 0, i = 1, 2, li > 0, i = 1, 2, 3, ensures the Lyapunov stability to the stationary rotation λ20 + λ23 := 1 when p = q = 0, r = rr e f . Proof The proof is similar to the proof of the Theorem 54.1. Let we define the Lyapunov function as V = T + W,   where T = 21 Ix p 2 + I y q 2 + Iz r 2 ,        2 W = k 1 − λ0 + λ23 = k2 1 − λ20 + λ23 + λ21 + λ22 and k > 0. V is positive definite and equals zero at the positions of relative equilibrium (54.6). To the function V the total time derivative, by virtue of the equations of motion (54.3), (54.4), taking into account (54.10) will be V˙ = −l1 p 2 − l2 q 2 − l3 r (r − rr e f ) − p(λ2 λ3 + λ0 λ1 )(k1 − k) − q(λ0 λ2 + λ1 λ3 )(k2 − k).

(54.11)

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Thus, we can put in the expression (54.11) that ki = k, i = 1, 2, then for li > 0, i = 1, 2, 3, the V˙ is a negative definite for |r | > |rr e f | and is positive definite for  |r | < |rr e f |. Remark 54.3 To stabilize the rotational motion relative to stationary rotation, it is necessary to take into account the condition according to the study of the equilibrium position (54.6) so that there are no roots with a positive real part among the eigenvalues. Theorem 54.3 The control ua u p = −ε p − lI1x ( p − 2(λ∗1 λ∗3 − λ∗0 λ∗2 )) − kIx1 (λ∗0 λ1 − λ∗1 λ0 + λ∗3 λ2 − λ∗2 λ3 ), u q = −εq − lI2y (q − 2(λ∗2 λ∗3 + λ∗0 λ∗1 )) − kI 2y (λ∗0 λ2 − λ∗2 λ0 + λ∗1 λ3 − λ∗3 λ1 ), k3 ∗2 ∗ ∗ ∗ ∗ u r = −εr − lI3z (r − (2(λ∗2 0 + λ3 ) − 1)) − Iz (λ0 λ3 − λ3 λ0 + λ2 λ1 − λ1 λ2 ), when ki > 0, li > 0, i = 1, 2, 3, ensures the Lyapunov stability to the programmed position   ∗ = λ∗0 ; λ∗1 ; λ∗2 ; λ∗3 . Theorem 54.4 The control ua u p = −ε p − lI1x ( p − (2(λ∗1 λ∗3 − λ∗0 λ∗2 ) + p  ))− − kIx1 (λ∗0 λ1 − λ∗1 λ0 + λ∗3 λ2 − λ∗2 λ3 ), u q = −εq − lI2y (q − (2(λ∗2 λ∗3 + λ∗0 λ∗1 ) + q  ))− − kI 2y (λ∗0 λ2 − λ∗2 λ0 + λ∗1 λ3 − λ∗3 λ1 ), ∗2  u r = −εr − lI3z (r − ((2(λ∗2 0 + λ3 ) − 1) + r ))− k3 ∗ ∗ ∗ ∗ − Iz (λ0 λ3 − λ3 λ0 + λ2 λ1 − λ1 λ2 ), when ki > 0, li > 0, i = 1, 2, 3, ensures the Lyapunov stability to the programmed motion   ∗ (t) = λ∗0 (t); λ∗1 (t); λ∗2 (t); λ∗3 (t) and where p  , q  , r  are the projections of a programmed motion angular rate on the axes of the C x yz , which represented by ∗ (t). Remark 54.4 To the proof of the Theorems 54.3 and 54.4, we can define the Lyapunov function as V = T + W,   where T = 21 Ix p 2 + I y q 2 + Iz r 2 , W = 2k(1 − λ∗0 λ0 − λ∗1 λ1 − λ∗2 λ2 − λ∗3 λ3 ) and k > 0, where in the case of the Theorem 54.4 proof ∗ is the function of time.

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54.5 Conclusion We should note that this research is based on the study [4] and continues it with a significant difference. This difference is due to the description of the attitude control motion in the rotating frame which entails the change of the equations of motion, the analysis of these equations, and the construction of the control laws. We can say that similarly to [4], but in the rotating frame, conditions for the existence of small oscillations of the celestial body were obtained and control laws according to the formulations of the Theorems 54.1 and 54.2 were constructed. The results of constructing the control laws according to the formulations of the Theorems 54.3 and 54.4 have a generalizing character for the attitude controlled motion and are new. The study results extend the scope of controlled orbital motion, not only in the neighborhood of the collinear libration points when the control torque more than the gravitational torque. They can be useful at the initial stage of the orientation problems of the celestial bodies in outer space. Due to the small values of the gravitational forces [3] and the gravitational torque [4] in the neighborhood of the collinear libration point, the use of small control actions here is effective, for example, in case when controlling huge celestial bodies.

References 1. Kane, T.R., Marsh, E.L.: Celest. Mech. 4, 78–90 (1971) 2. Markeev, A.P.: Libration Points in Celestial Mechanics and Cosmodynamics. Nauka, Moscow, p. 312 (1978). (in Russian) 3. Shmyrov, A.S., Shmyrov, V.A.: Numerical algebra. Control Optim. 7, 185–189 (2017) 4. Shmyrov, A.S., Shmyrov, V.A., Shymanchuk, D.V.: AIP Conf. Proc. (2017). https://doi.org/10. 1063/1.5007413 5. Shymanchuk, D.V.: Vestnik of Saint Petersburg University. Series 10. Appl. Math. Comput. Sci. Control Process. 13(2), 147–167 (2017). https://doi.org/10.21638/11701/spbu10.2017.203 6. Zubov, V.I.: Lectures on Control Theory. Nauka, Moscow, p. 496 (1975). (in Russian)

Chapter 55

Construction of Connecting Trajectories in the Circular Restricted Three-Body Problem Dzmitry Shymanchuk, Alexander Shmyrov, and Vasily Shmyrov

Abstract In this paper, we consider the problem of construction of the connecting trajectories between neighborhoods of collinear libration points of the Sun–Earth system. The motion of spacecraft in these regions of space is described using the equations of the circular restricted three-body problem and with the help of Hill’s model. At the first stage, such trajectories are constructed using the symmetry properties that Hill’s model possesses. Further, the obtained trajectories are modeled in a circular three-body problem.

55.1 Introduction Neighborhoods of collinear libration points participate in many space projects. Many papers devoted to orbital maneuvering. The problem of constructing connecting trajectories is very interesting, that allows switching between these areas of space with relatively small energy and time costs. In this case, the choice of a dynamic model of motion is very important. This model should be adequate enough. On the other hand, it should have special properties that ensure the search for connecting trajectories. In this paper, we consider and compare two dynamic models: the equations of the circular restricted three-body problem and the equations of Hill’s approximation. It will be shown that the second model has additional symmetry properties, that greatly facilitate the search for connecting trajectories.

D. Shymanchuk (B) · A. Shmyrov · V. Shmyrov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. Shmyrov e-mail: [email protected] V. Shmyrov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_55

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55.2 Circular Restricted Three-Body Problem. Symmetry Properties The equations of motion of a spacecraft in geocentric coordinates in circular threebody problem have the form γ m 1 x˜ + γ m2 x¨˜ = − ||x|| ˜ 3



 Rl(t) − x˜ 1 − l(t) , ||Rl(t) − x|| ˜ 3 R2

(55.1)

where x˜ = (x˜1 , x˜2 , x˜3 ) are the geocentric coordinates of the spacecraft; γ — gravitational constant; m 1 , m 2 —mass of the Earth and the Sun; R—distance between the Earth and the Sun (1 AU); l(t)—unit vector directed along the Earth–Sun line; || · || is the Euclidean norm. If we put y˜ =  x˙ , Eqs. (55.1) are representable in the Hamiltonian form with the Hamiltonian   γ m2 1 1 γ m1 l(t)x˜ 2   − − ˜ y˜ , t) = ||y|| − H1 (x, . (55.2) l(t) − x˜  2 ||x|| ˜ R R R per unit of time, and 10−2 AU per unit of distance [Vestnik2005]. We choose year 2π The Hamiltonian (55.2) of the equations of motion in the rotating coordinate system takes the form ⎛ ⎞ x 1 1 3 1  − ⎠ + x2 y1 − x1 y2 . − R 2 ⎝  H2 (x, ˜ y˜ , t) = ||y||2 −  ||x||2  2 ||x|| R 1 − 2x1 +  R

R2

(55.3) where x = (x1 , x2 , x3 ) is the position of a spacecraft in the rotating coordinate system, the axis O X 1 is directed to the Sun, y = (y1 , y2 , y3 )—impulses. The equations of motion corresponding to the Hamiltonian (55.3) have the form 3x1 x˙1 = y1 + x2 , y˙1 = − ||x|| 3 + 3x2 x˙2 = y2 − x1 , y˙2 = − ||x|| 3 −

x˙3 = y3 ,

y˙3 =

3x3 − ||x|| 3

−

R−x1   −  2x1 ||x||2 3 1− R + R 2  x2   −  2x1 ||x||2 3 1− R + R 2  x3   .  2x1 ||x||2 3 1− R + R 2 

R + y2 ; y1 ;

(55.4)

The system (55.4) has stationary solutions close to the Earth—collinear libration points L 1 and L 2 . Note that the energy constants at these points are different [2]. The location of the libration points L 1 and L 2 is not symmetrical with respect to the center of the Earth. So L 1 is located at a distance of 0.99499 units of distance in the direction to the Sun from the center of the Earth, and L 2 is at a distance of 1.004995 units in the direction from the Sun.

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The system (55.4) has symmetry in the following sense. Together with some solution (x1 (t), x2 (t), y1 (t), y2 (t)), there is a solution (x1 (−t), −x2 (−t), −y1 (−t), y2 (−t)). Thus, with some trajectory (x1 (t), x2 (t)), there is a symmetric one [3] with respect to the axis O X 1 .

55.3 Hill’s Model. an Additional Property of Symmetry With the help of Hill’s method for expanding the solar potential in the Hamiltonian (55.2), the equations of motion are obtained in form [1, 4] 3x1 x˙1 = y1 + x2 , y˙1 = − ||x|| 3 + 2x 1 + y2 + u; 3x2 x˙2 = y2 − x1 , y˙2 = − ||x|| 3 − x 2 − y1 ; 3x3 x˙3 = y3 , y˙3 = − ||x||3 − x3 ,

(55.5)

where x = (x1 , x2 , x3 ) is the position of a spacecraft in the rotating coordinate system, the axis O X 1 is directed to the Sun, y = (y1 , y2 , y3 )—impulses. In the system (55.5), the libration points L 1 and L 2 are fixed and have coordinates L 1 : x ∗ = (1, 0, 0), y ∗ = (0, 1, 0); L 2 : x ∗∗ = (−1, 0, 0), y ∗∗ = (0, −1, 0). The system (55.5) has a Hamiltonian form with a Hamiltonian H3 (x, y) =

3 1 3 ||x||2 ||y||2 − − x12 + + x2 y1 − x1 y2 . 2 ||x|| 2 2

The values of the energy constants (Hamiltonians) at the libration points L 1 and L 2 are the same, which allows us to pose the transition problem between the neighborhoods of these points. The system of equations (55.5) preserves the symmetry property as in the circular problem, but the system (55.5) has an additional symmetry property. Namely, with the decision (x1 (t), x2 (t), y1 (t), y2 (t)), there is a solution in the ecliptic plane (−x1 (−t), x2 (−t), y1 (−t), −y2 (−t)).

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Thus, with the trajectory (x1 (t), x2 (t)), there is the trajectory, that symmetric about the axis O X 2 .

55.4 Construction of Connecting Trajectories The symmetry properties of Hill’s model provide an opportunity for the direct construction of connecting trajectories. We construct the point (x10 , x20 , y10 , y20 ) on the connecting trajectory, based on the following conditions: x10 = 0, x˙20 = 0. In this case, the energy constant should not differ too much from the value of the Hamiltonian at the libration points L 1 and L 2 , that equal to −4.5. We construct a connecting trajectory with initial data in the ecliptic plane, as a result of direct search. Figure 55.1 shows the achievement of a neighborhood of L 2 for 2.35 units of time. Figure 55.2 shows the flight from a neighborhood of L 1 to a neighborhood of L 2 for 5 units of time. This trajectory can be used as an initial approximation for the construction of connecting trajectories in the circular three-body problem. Figure 55.3 shows the

Fig. 55.1 Reaching the neighborhood of L 2 from the starting point

Fig. 55.2 Connecting trajectory in Hill’s model

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Fig. 55.3 The connecting trajectory in circular restricted three-body problem

Fig. 55.4 The flight time along the presented trajectory is 18.818 time units

2

0.4 0.2 1.0

0.5

0.5

1.0

1

0.2 0.4

connecting trajectory between neighborhoods of libration points L 2 and L 1 . This simulation was implemented for equations of the circular three-body problem (55.4). The initial data is the same as for Fig. 55.2, the flight time is 5 time units. Perhaps this simulation can be used as an initial approximation in constructing connecting trajectories in the elliptic restricted three-body problem. Figures 55.4 shows connecting trajectories with other different initial data. The simulation is implemented in Hill’s model.

55.5 Conclusion This research showed that Hill’s method has significant advantages in the study of connecting trajectories in the circular restricted three-body problem. This is due to the fact that the equations of motion in Hill’s model have additional symmetry. This makes it possible to significantly simplify the algorithm of connecting trajectories constructing. The resulting trajectories serve as an initial approximation for research in a more general model—the three-body problem.

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References 1. Aminov, R., Shmyrov, A., Shmyrov, V.: Impulse control flight to the invariant manifold near collinear libration point. Cybern. Phys. 8(2), 51–57 (2019). https://doi.org/10.35470/22264116-2019-8-2-51-57 2. Markeev, A.P.: Libration Points in Celestial Mechanics and Astrodynamics. Nauka, Moscow (1978) 3. Poincare, H.: New Methods of Celestial Mechanics. Springer, New York (1992) 4. Shmyrov, V.A.: Stabilization of controlled space vehicle orbital motion in the neighborhood of collinear libration point L1. Vestnik of Saint Petersburg University. Series 10. Appl. Math. Comput. Sci. Control Process. 2, 193–199 (2005)

Chapter 56

An Embedded Explicit Method for Partitioned Systems of Ordinary Differential Equations Igor V. Olemskoy and Alexey S. Eremin

Abstract In the paper, an explicit Runge–Kutta-type method for partitioned systems of ordinary differential equations is considered. The partitioning is made on base of structural properties of the system right-hand side functions. With use of such properties, an explicit method, that requires fewer stages than classic Runge-Kutta methods applied to systems of ordinary differential equations, is constructed. A system of order conditions for an embedded pair of methods of orders six and four is presented. Coefficients of such pair of methods are chosen to provide convenient local error estimation in order to organize a variable time-step solver. Numerical testing of the constructed method and its comparison to widely used Dormand–Prince pair of orders six and five is held. The test results confirm better performance of the constructed method in sense of global error to amount of right-hand side evaluations ratio.

56.1 Introduction As it is shown in [1], any system of ordinary differential equations (ODEs) ηk = ϕk (x, η1 , . . . , ηg ), k = 1, . . . , g, with reordering of unknowns can be transformed into the form with three groups y0 = f 0 (x, y0 , . . . , yn ), yi = f i (x, y0 , . . . , yi−1 , yl+1 , . . . , yn ),

i = 1, . . . , l,

(56.1) (56.2)

y j = f j (x, y0 , . . . , y j−1 ),

j = l + 1, . . . , n,

(56.3)

I. V. Olemskoy · A. S. Eremin (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] I. V. Olemskoy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_56

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where ys and f s can be vectors of sizes rs , s = 0, . . . , n. The two groups of Equs. (56.2) and (56.3), which we call structural, have similar structure. Each equation within them has a fixed place such that its right-hand side does not depend on the unknowns which behavior is described by this and the following equations of the same group. The group (56.1) contains all the equations without the mentioned structural properties and is called general. The general group or the group (56.2) can be missing. Systems that can be rewritten in the form (56.1)–(56.3) by variables reordering appear, for instance, in the problems of celestial mechanics, high energy physics, Hamiltonian problems, etc. In the papers [2–4], an extension of explicit Runge–Kutta methods for partitioned systems (56.1)–(56.3) is presented. The structural properties of the structural groups are algorithmically used in the considered methods. Their effectiveness in comparison with classic Runge–Kutta methods follows from the fact that, for the same method’s order, they require fewer right-hand side evaluations (stages) for the equations belonging to the structural groups. If the general group is absent this effect is even more noticeable: in this case, fifth-order explicit method can be constructed with four stages instead of six [5] and sixth-order method with six stages instead of seven [6–10]. In the current paper for the full system (56.1)–(56.3), an embedded pair of methods of orders six and four is constructed with eight stages for the general group and seven stages for the structural groups. The last stages are reused at the next step, so the overall amount of right-hand side evaluations is lower than for the classic Runge– Kutta method of order six.

56.2 Numerical Method In the following to simplify the notations without loss of generality, we assume rs = 1, s = 0, 1, . . . , n. An embedded pair of explicit Runge–Kutta-type methods for a structurally partitioned system (56.1)–(56.3) with initial conditions ys (x0 ) = y¯s , s = 0, 1, . . . . , n provides two approximations Z s and z s to the solutions ys (x0 + h), s = 0, 1, . . . . , n according to the formulae Z 0 = y¯0 + h Z i = y¯i + h Z j = y¯ j + h

m 0

b0,ν K 0,ν , mν=1 1 ν=1 b1,ν K i,ν , m 2

ν=1

b2,ν K j,ν ,

z 0 = y¯0 + h z i = y¯i + h z j = y¯ j + h

m 0

mν=1 1 ν=1

m 2

ν=1

d0,ν K 0,ν , d1,ν K i,ν ,

(56.4) i = 1, . . . , l, (56.5)

d2,ν K j,ν ,

j = l + 1, . . . , n, (56.6)

where the values K s,w ≡ K s,w (h) are computed one by one in the strict order K 0,1 , K 1,1 , . . . , K n,1 , K 0,2 , K 1,2 , . . . , K n,2 , K 0,3 , K 1,3 , . . .

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as (56.7) K 0,ν = f 0 (x + c0,ν h, Y00,0,ν , Y01,1,ν , . . . , Y01,l,ν , Y02,l+1,ν , . . . , Y02,n,ν ), K i,ν = f i (x + c1,ν h, Y10,0,ν , Y11,1,ν , . . . , Y11,i−1,ν , Y12,l+1,ν , . . . , Y12,n,ν ), (56.8) K j,ν = f j (x + c2,ν h, Y20,0,ν , Y21,1,ν , . . . , Y21,l,ν , Y22,l+1,ν , . . . , Y22, j−1,ν ), (56.9) i = 1, . . . , l, j = l + 1, . . . , n (56.10) with Y00,0,ν = y¯0 + h Y02, j,ν = y¯ j + Y10,0,ν = y¯0 + Y12, j,ν = y¯ j + Y20,0,ν = y¯0 + Y22, j,ν = y¯ j +

ν−1

μ=1 a00,ν,μ K 0,μ , ν−1 h μ=1 a02,ν,μ K j,μ ,  h νμ=1 a10,ν,μ K 0,μ ,  hh ν−1 μ=1 a12,ν,μ K j,μ ν h μ=1 a20,ν,μ K 0,μ ,  h νμ=1 a22,ν,μ K j,μ ,

Y01,i,ν = y¯i + h

ν−1

μ=1

a01,ν,μ K i,μ , (56.11)

i = 1, . . . , l, Y11,i,ν

j = 1, . . . , n, (56.12) ν = y¯i + h μ=1 a11,ν,μ K i,μ , (56.13)

i = 1, . . . , l − 1, j = 1, . . . , n, (56.14)  Y21,i,ν = y¯i + h νμ=1 a21,ν,μ K i,μ , (56.15) i = 1, . . . , l,

j = 1, . . . , n − 1.

(56.16)

The groupwise numbers of stages m 0 , m 1 and m 2 , and the parameters bu,ν , du,ν , cu,ν and auv,ν,μ are chosen to provide the desired order of the methods. We call explicit methods designed for the system (56.1)–(56.3) “structural” methods. In the current paper, we choose m 0 = 8 and m 1 = m 2 = 7 to construct a pair of methods with orders six (main method giving Z -values) and four (local error estimator providing z-values, see [11] for details), i.e. for any system (56.1)–(56.3) with sufficiently smooth right-hand side functions, |ys (x0 + h) − Z s | ≈ O(h 7 ) and |ys (x0 + h) − z s | ≈ O(h 5 ), s = 0, 1, . . . , n. Consider vectors of abscissae {Cu }, weights {Bu } and {Du }, and matrices of stage weights {Auv } (u = 0, 1, 2 is a number of a group, v = 0, 1, 2 is a number of a block within the group): ⎛

⎞

⎛

⎞

⎛

⎞



cu,1 bu,1 du,1 auv,1,1 ⎜ cu,2 ⎟ ⎜ bu,2 ⎟ ⎜ du,2 ⎟ ... ... Cu = ⎝ . . . ⎠, Bu = ⎝ . . . ⎠, Du = ⎝ . . . ⎠, Auv = . auv,m u ,1 . . . auv,m u ,m v cu,m u bu,m u du,m u Thus, three blocks of stage weights Auv correspond to each group of equations. Butcher’s tableau of the method (56.4)–(56.11) (see Table 56.1) demonstrates structural properties of the system and their algorithmic use for the method’s construction and implementation. The matrices A00 , A01 , A02 , A12 are strictly lower triangular, while A10 , A20 , A11 , A21 , and A22 are lower triangular with nonzero diagonals.

510

I. V. Olemskoy and A. S. Eremin

In the following section, we will present order conditions and the parameters of the method denoted as SRK6(4)[8,7,7]F, where SRK stands for “Structural Runge– Kutta”, 6(4) shows the orders of the main method and the error estimator, [8,7,7] are groupwise numbers of stages and the last F means that the “First Same as Last” (or reuse) idea is implemented (the first stage of the current step is the same and the last stage of the previous step), so only 7, 6, and 6 groupwise right-hand side evaluations are required per step [12]. If in the system (56.1)–(56.3) any group w ∈ {0, 1, 2} is missing, then the corresponding parameters Cw , Bw , Dw , Aww , Auw , Awv , u, v ∈ {0, 1, 2}\{w} can be omitted. If both structural groups (56.2) and (56.3) cannot be formed the method (56.4)–(56.11) reduces to an explicit m 0 -stage Runge–Kutta pair with parameters A00 , B0 , D0 and C0 .

56.3 Order Conditions In order to provide the sixth order of the method (56.4)–(56.11), its parameters must satisfy a system of nonlinear equations called order conditions which are derived by comparison of Taylor series in respect of h of the exact solution and the numerical approximation. The number of order conditions depends on the amount of groups formed in the system (56.1)–(56.3). If we consider the general group only, we get the most simple case, when the order conditions for a sixth-order Runge–Kutta method with seven stages form a system of 37 equations with 35 parameters [11]. For a system (56.2)–(56.3), without the general group, it is possible to construct methods with six stages for each group [7–10]. If l = 1, n = 2, there are 74 order conditions for 61 parameters C1 , C2 , B1 , B2 , A12 , A21 . Otherwise, if l > 1 and n − l > 1, the number of order conditions rises to 292 and the number of parameters together with A11 and A22 becomes 103. We will use the standard conditions (extended for three groups) η 

auw,ν,ξ = cu,ν , u, w ∈ {0, 1, 2}, ν = 1, . . . , m u ,

(56.17)

ξ =1

where η = ν − 1 if (u, v) ∈ {(0, 0), (0, 1), (0, 2), (1, 2)}, and η = ν otherwise. With their use the order conditions for SRK6(4)[8,7,7]F embedded pair form a system of 1290 nonlinear algebraic equations with 320 unknowns:

56 An Embedded Explicit Method for Partitioned Systems …



v bq,ν cq,ν =

ν



λ bq,ν cq,ν



ν



aq p,ν,μ c p,μ =

μ

λ bq,ν cq,ν

ν



1 , v = 0, 1, . . . , 5, v+1



aq p,ν,μ c2p,μ =

μ





511



q, p, r, e, t ∈ 0, 1, 2 ;

1 , λ = 0, 1, 2, 3; 2 · (3 + λ) 1 , λ = 0, 1, 2; 3(4 + λ)

1 , λ = 0, 1, 2; 6(4 + λ) ν μ ξ



   1 θ , θ = 0, 1; bq,ν cq,ν aq p,ν,μ c p,μ · aq p,ν,μ c p,μ = 4(5 + θ ) ν μ μ   1 θ , θ = 0, 1; bq,ν cq,ν aq p,ν,μ c3p,μ = 4(5 + θ ) ν μ    1 u , bq,ν aq p,ν,μ cλp,μ a pr,μ,ξ cr,ξ = 10(λ + u)(1 + u) ν μ ξ 

λ bq,ν cq,ν

aq p,ν,μ

a pr,μ,ξ cr,ξ =

(λ, u) ∈ {(0, 2), (0, 3), (1, 1), (1, 2), (2, 1)};    ρ u bq,ν aq p,ν,μ cλp,μ a pr,μ,ξ cr,ξ ar s,ξ,ψ cs,ψ =

ν

μ

ξ

ψ

1 , 60(λ + 2u + 2ρ)

(λ, u, ρ) ∈ {(0, 0, 1), (1, 0, 1), (0, 1, 1)};



   1 2 , bq,ν aq p,ν,μ c p,μ · aq p,ν,μ c p,μ = 36 ν μ μ     1 1 , bq,ν aq p,ν,μ c4p,μ = du,ν aup,ν,μ c p,μ = , 30 6 ν μ ν μ ⎛ ⎞

    1 , bq,ν ⎝ aq p,ν,μ a pr,μ,ξ cr,ξ ⎠ · aq p,ν,μ c p,μ = 72 ν μ μ ξ







1 , λ = 1, 2; 24(1 + λ) ν μ ξ ⎛ ⎞ ⎛ ⎞     1 , bq,ν aq p,ν,μ ⎝ a pr,μ,ξ cr,ξ ⎠ · ⎝ a pr,μ,ξ cr,ξ ⎠ = 120 ν μ ξ ξ



bq,ν cq,ν

bq,ν cq,ν

ν

 ν

bq,ν

 μ



aq p,ν,μ

aq p,ν,μ

μ

aq p,ν,μ

 ξ



λ a pr,μ,ξ cr,ξ =

a pr,μ,ξ

ξ

a pr,μ,ξ

 ψ



ar e,ξ,ψ ce,ψ =

ψ 2 ar e,ξ,ψ ce,ψ =

1 , 144

1 , 360

(56.18)

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I. V. Olemskoy and A. S. Eremin

 ν



bq,ν



aq p,ν,μ

μ



a pr,μ,ξ



ξ

ψ

ar e,ξ,ψ

 ϕ

aet,ψ,ϕ ct,ϕ =

1 , 720



1 v du,ν cu,ν = , v = 0, 1, . . . , 3; u, p, r ∈ 0, 1, 2 , v+1 ν      1 1 2 , . du,ν aup,ν,μ cu,μ = du,ν aup,ν,μ a pr,μ,ξ cr,ξ = 12 24 ν μ ν μ ξ

In addition to these order conditions, there are equations providing First Same as Last approach: auv,m v ,w = bv,w , cv,m v = 1, bv,m v = 0, u, v ∈ {0, 1, 2}, w = 1, . . . , m v , (56.19) and the explicitness condition c0,1 = c1,1 = c2,1 = 0. The parameters of SRK6(4)[8,7,7]F method are presented in the Table 56.1.

56.4 Numerical Test We consider a test system of ODEs   y0 = x y3 y1 y2−1 + 7 y0 y1 = 10x exp(5(y4 − 1))y3 1

= f 0 (x, y0 , y1 , y2 , y3 ), = f 1 (x, y3 , y4 ),

y2 = 2x y15 y3 + 41 ln y0 − y4 + 1 = f 2 (x, y0 , y1 , y3 , y4 ), = f 3 (x, y0 , y2 ), y3 = − 25 x ln (y0 y2 ) = f 4 (x, y0 y1 y2 , y3 ), y4 = 2x y0 y1−1 y2 y3

(56.20)

with the exact solution y0 = exp(4 sin x 2 ), y1 = exp(5 sin x 2 ), y2 = exp(sin x 2 ), y3 = cos x 2 , y4 = sin x 2 + 1 satisfying the initial conditions ys (0) = 1, s = 0, 1, . . . , 4. The test problem contains all the three groups of the system (56.1)–(56.3). The general group consists of the first equation of (56.20), two other groups have two equations each. The method SRK6(4)[8,7,7]F calculates f 1 , f 2 , f 3 , and f 4 one time fewer than f 0 . All nine blocks of parameters Auv are used and their correctness can be checked. If we don’t use the structural properties in the algorithm (all five equations for the general group), the method reduces to a classic Runge–Kutta method application to ODE systems with eight stages and FSAL. Let’s denote this variant SRK6(4)[8]F. Two popular explicit schemes of the same class by Dormand and Prince were used as opponents for SRK6(4)[8,7,7]F and SRK6(4)[8]F: DOPRI5(4)7F (a pair of orders five and four with seven stages and FSAL [12]) and DOPRI6(5)8 (a pair of orders six and five with eight stages and no FSAL [13]).

1

1

1 5 1 3 1 4 8 11

c2,p 0

1

1

1 5 1 5 3 10 8 11

c1,p 0

1

1

2 15 1 5 1 3 2 3 7 9

c0,p 0

0

3 20 −5 36 −5 18 385 972 −105 292

10 27 −35 54 260 243 −5830 6643 3125 17472

3 1 20 20 11 −5 108 36 15 17 256 256 30 1214 14641 1331 181 −15 2492 356 31 0 420

0

10 27 35 256 1070 14641 1810 8099 3125 17472

3 1 20 20 3 1 0 20 20 69 −9 9 800 160 32 30 6118 7250 73205 1331 102487 119 250 −15 1660 332 1079 31 3125 0 17472 420

0

2 15 1 20 11 108 23 54 −119 324 1067 2044 31 420

0

−3 256 5226 14641 111 356 81 320

a20,p,ν

−9 800 26274 73205 51 166 81 320

a10,p,ν

7 6 −182 243 108 73 81 320

a00,p,ν

4374 6643 6561 73 29120 960

0

2808 14641 −108 623 27 140

0

0

0

0

0

1 10

0

1 10 −5 54 −5 27 385 1458 −35 146

1 10 1 4 −1340 3993 20365 14442 125 1392

10 27 −35 27 20930 21141 140 2117 125 1392

a21,p,ν

−1 60 9280 11979 −32320 35109 1000 2961

a11,p,ν

4095 29986 658845 970456 161051 392544

3 22 307461 452516 161051 392544

0

2 15 1 10 1 18 16 27 −308 729 29 73 13 160

0

0

1 10 5 18 −110 27 5845 1458 −395 146 56 81 −648 949 81 520

0

0

a12,p,ν

112 27 −8320 2187 5248 1533 256 945

a02,p,ν

1331 4374 22627 0 39858 161051 89 393120 1080

b1,p

0

3125 17472 81 320 27 140 6561 29120 73 960

d1,p

1 12

0

0

125 168 −3 8 33 56

0

−1 24

31 420

0

d0,p

b0,p

0 83 1008

1 10 1 18 1 12 14093 87846 −407 2136 13 160

0

0

1 10 5 18 5 24 −27980 43923 1045 534

4536 14641 −324 1157 81 520

0

0 −1 24 33280 43923 −2176 1869 256 945

a22,p,ν

3 22 43923 0 64792 161051 89 393120 1080

0

81 520 256 945 161051 393120 89 1080

1 12

0

9 52 16 63 1331 3276

0

1 12

13 160

0

d2,p

b2,p

23 288

1 12 1 0 0 5 1 1 125 25 10 1392 10 348 39 1000 50 −9 27 160 2961 400 800 141 −47852 195970 516954 −1395712 161051 6655 73205 14641 73205 73205 392544 16356 1601 −11255 −36555 83 40448 14641 0 0 415 166 1079 415 1008 10790 83 256 161051 89 13 1 81 0 0 1008 160 520 945 393120 1080 12

140 81 −58240 605605 102789 1987254 113135 4160 0 10293 198998 83 1000 161051 2961 392544 1008

a01,p,ν

1 1 10 10 1 −5 10 18 54 27 49 5 −5 55 128 576 384 288 329 20 −40240 39520 2178 1331 115797 51183 19683 −1067 −5 11060 −34360 32396 6408 178 7743 37647 23 125 6561 73 1000 0 29120 960 288 1392 2961

0

1 10 1 10 1 15 3637 98136 23958 512435 −505 −72 6561 10790 415 2988 23 27 6561 73 140 29120 960 288

104 243 −216 511 27 140

2 15 1 10 1 18 34 81 −469 2187 44 219 23 288

0

56 An Embedded Explicit Method for Partitioned Systems … 513

Table 56.1 The method SRK6(4)[8,7,7]F

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I. V. Olemskoy and A. S. Eremin

Fig. 56.1 Accuracy to cost ratio

Fig. 56.2 Accuracy to number of steps ratio

log10 N step

We solve (56.20) with each method over the interval [0, 5] with the absolute local error tolerances 10−k , k integer varying from 6 to 20. We use 40-digit mantissa and calculate the global error Err glob in the final point. The total number of steps Nstep includes accepted as well as rejected steps, and the total value of right-hand side functions evaluations N f equals Nstep multiplied by total amount of f s evaluations per step. Figure 56.1 shows Err glob  to N f ratio, while Figure 56.2 shows Err glob  to Nstep ratio in double logarithmic scale.

56 An Embedded Explicit Method for Partitioned Systems …

515

56.5 Conclusion The numerical test shows that the methods SRK6(4)[8,7,7]F and SRK6(4)[8]F demonstrate their correctness and effectiveness in comparison with DOPRI5(4)7F and DOPRI6(5)8. It confirms that economical explicit methods within the approach described in the paper can be constructed with a reliable step-size control procedure.

References 1. Olemskoy, I.V.: Modifikatsiya algoritma vydeleniya strukturnykh osobennostei [Modification of structural properties detection algorithm]. Vestnik St-Petersburg Uni. 2, 55–64 (2006). [In Russian] 2. Olemskoy, I.V.: Structural approach to the design of explicit one-stage methods. Comput Math. and Math. Phys. 43(7), 918–931 (2003) 3. Olemskoy, I.V.: A fifth-order five-stage embedded method of the Dormand-Prince type. Comput Math. and Math. Phys. 45(7), 1140–1150 (2005) 4. Olemskoy, I.V., Eremin, A.S.: An embedded method for integrating systems of structurally separated ordinary differential equations. Comput Math. and Math. Phys. 50(3), 414–427 (2010) 5. Olemskoy, I.V.: Fifth-order four-stage method for numerical integration of special systems. Comput Math. and Math. Phys. 42(8), 1135–1145 (2002) 6. Eremin, A.S., Kovrizhnykh, N.A., Olemskoy, I.V.: An explicit one-step multischeme sixth order method for systems of special structure. Appl. Math. Comp. 347, 853–864 (2019) 7. Olemskoy, I.V., Eremin, A.S.: An embedded pair of method of orders 6(4) with 6 stages for special systems of ordinary differential equations. AIP Conf. Proc. 1738, 160010 (2016) 8. Olemskoy, I.V., Eremin, A.S., Ivanov, A.P.: Sixth order method with six stages for integrating special systems of ordinary differential equations. In: 2015 International Conference "Stability and Control Processes" in Memory of V.I. Zubov (SCP), pp. 110–113 (2015) 9. Olemskoy, I.V., Eremin, A.S., Kovrizhnykh, N.A.: Embedded methods of order six for special systems of ordinary differential equations. Appl. Math. Sci. 11(1), 31–38 (2017) 10. Olemskoy, I.V., Kovrizhnykh, N.A.: A family of sixth-order methods with six stages. Appl. Math. Comput. Sci. Control Process. 14(3), 215–229 (2018) 11. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2008) 12. Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comp. Appl. Math. 6(1), 19–26 (1980) 13. Prince, P.J., Dormand, J.R.: A family of embedded Runge-Kutta formulae. J. Comp. Appl. Math. 7(1), 67–75 (1981)

Chapter 57

Constructing a Polynomial Method in the State Space for a Nonlinear Optimal Control Problem Daniel Zlobin and Dzmitry Shymanchuk

Abstract The paper presents a method for taking into account phase constraints and control constraints in the direct construction of a quasi-optimal polynomial trajectory in the state space of a system of a special form, the left-hand side of which preserves the polynomiality of the input arguments, and the right-hand side contains an orthogonal linear control transformation parameterized by the trajectory, which allows one to exclude control from tasks and build a trajectory directly in the state space. To take into account the phase constraints, we use an asymptotically exact monotone estimate of the range of values of the polynomial based on the expansion in Bernstein polynomials. Boundary conditions are taken into account using the Hermite polynomial. The presented method can also be applied to complex systems provided that they are approximated. The presented method is illustrated by the example of the task of terminal control of an aircraft.

57.1 Introduction The task of controlling a given system of differential equations is ubiquitous in practice. Difficulties in obtaining an analytical solution determine the high demand for numerical methods. To construct a numerical method, a system of the form with a given quality functional and constraints on the control action usually reduces to a finite-dimensional nonlinear programming problem by discretizing, generally speaking, the state space, control, and Lagrange multipliers of the corresponding constraints [1] (collocation method).1 Another option for discretization may be to

1 If

there are no constraints, then effective sequential approximation methods have been developed, for example [2, 3].

D. Zlobin · D. Shymanchuk (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_57

517

518

D. Zlobin and D. Shymanchuk

reduce the controlled system to differential inclusion and approximate the attainability set of this inclusion [4]. It should be noted that both methods are successfully used in practice. However, for the method of reducing to differential inclusion it is difficult to obtain higher order approximations for attainability sets, and the collocation method has enormous computational complexity and, moreover, as shown [5], in poorly conditioned cases does not have strong advantages in terms of convergence rate. In this paper, we propose a method for reducing a problem of a fairly general form to a finite-dimensional problem on the coefficients of a polynomial of a certain type. Increased attention is paid to guaranteed admissibility of the trajectory of the system.

57.2 Controlled System Formulation of the Problem Consider a controlled system of differential equations of the following form:     p Dn χ | T = ϕ(u | Dn χ , T ), u ∈ PWC [0, 1] → Rm .

(57.1)

  Hereinafter, we use the notation Dn χ = ∂ n χ , ∂ n−1 χ , . . . , ∂ 0 χ . We will assume that for a fixed T ∈ R the function p is a polynomial   p ( · |T ) ∈ P Rm(n+1) → Rm and using T ≥ 0, the coefficients of the polynomial are parameterized. The function ϕ (· | Dn χ , T ) is a linear transformation contained in the orthogonal group of some given nonsingular quadratic form Q:     ϕ · | Dn χ , T ∈ Om (Q) ⇐⇒ Q ϕ(u | Dn χ , T ) = Q (u) .

(57.2)

We set the trajectory and control limits as follows: Q − ≤ Q(u) ≤ Q + ,

(57.3)

D χ ∈ MT , j Dn χ ( j) ∈ GT , j = 0, 1.

(57.4) (57.5)

n−1

complex; in this paper, we restrict The specific form of the set MT can be very p  ourselves to the assumption that MT = MT McT . McT is an arbitrary convex set, p and MT is a set with an algebraic boundary in the following sense:   p MT = ξ ∈ Rmn | q(ξ | T ) ≤ 0, w(ξ | T ) = 0 .

(57.6)

57 Constructing a Polynomial Method in the State Space …

519

    where q and w are polynomial q( · | T ) ∈ P Rmn → Rl1 , w( · | T )∈P Rmn → Rl2 . Given all of the above, we consider the problem of minimizing a given functional J J [χ (·), u(·), T ] −→ inf .

(57.7)

Exclusion of Control We restrict ourselves to considering only polynomial trajectories. For this, we add the following condition:   (57.8) χ (·) ∈ P R → Rm . Problem (57.1)–(57.7) together with condition (57.8) will be called a quasiapproximation of the original problem. The condition (57.8) does not always entail continuity of control, so if the right side of (57.1) contains switches, then the control will have corresponding points of discontinuous. Using condition (57.2) and constraint (57.3), we exclude control from the problem   Q ( p (Dn χ | T )) = Q (ϕ (u | Dn χ , T )) = Q (u) ∈ Q − , Q + ,   J [χ (·), u(·), T ] = J χ (·), ϕ −1 (· | Dn χ , T ) ◦ p (Dn χ | T ) , T = J1 [χ (·), T ] . (57.9) The circle is a composition operator. Thus, the quasi-approximation of the original problem can be formulated as follows: inf

(57.8), T ≥0

J1 [χ (·), T ] .

(57.10)

Moreover, constraints (57.4), (57.5), (57.9) are fulfilled.

57.3 Boundary Conditions Constraints (57.5) are boundary for polynomial (57.8). It is known from the theory of interpolation that a polynomial of minimal degree satisfying boundary conditions is a Hermite polynomial [6, 7], which in this case can be written as 1 n   χ H s | ∂ i χ ( j) = ∂ i χ ( j) μi j (s), j=0 i=0

μi j (s) =

(57.11)

 n−i (−1)(q+n+1) j n + q (s − j)i+q (1 − j − s)n+1 . q i! q=0

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Substituting explicitly the boundary conditions in (57.11), we obtain χ H (s | v) =

n 1

v i j μi j (s),

  j v j = v 0 j , . . . , v n j ∈ GT

j=0 i=0

Now, any polynomial (57.8) satisfying the boundary conditions (57.5) is uniquely expressed in terms of the Hermite polynomial:   χ (s | v) = χ H (s | v) + s n+1 (s − 1)n+1 r (s), r (·) ∈ P R → Rm .

(57.12)

57.4 Method of Estimation for State-Space Constraints Estimating the Range of Values of the Polynomial Consider an arbitrary polynomial q(·) ∈ P [R → R], denote the vector of its coefficients as   ϒ [q(s)] = ϒ a0 + a1 s 1 + . . . + ad s d = [a0 , a1 , . . . , ad ] As shown in [8, 9], the range of the polynomial on the segment s ∈ [0, 1] can be estimated by representing the polynomial q(·) in the expansion according to Bernstein polynomials. To do this, we define an arbitrary k ≥ d and write the Bernstein form q(s) =

k

bik

i=0

 k i s (1 − s)k−i . i

(57.13)

The coefficients of this form are recalculated based on the power expansion q(·) as follows [8]:

  min(i,d) i k k . (57.14) bi = al l l l=0

Now, using the Bernstein form (57.13), we write the estimate of the range of values. min bik ≤ q(s) ≤ max bik

0≤i≤k

0≤i≤k

k  k i=0

i

s i (1 − s)k−i = max bik . 



0≤i≤k

(57.15)

(s+1−s)k =1

The most remarkable thing about value range estimates (57.15), is that due to the parameter k, as shown in [9], they are asymptotically accurate:

57 Constructing a Polynomial Method in the State Space …

521

lim min bik = min q(s), s∈[0,1]

k→∞ 0≤i≤k

lim max

k→∞ 0≤i≤k

bik

= max q(s). s∈[0,1]

We use these estimates to construct asymptotically exact constraints on the coefficients of the polynomial q(·) so that the values of the polynomial are guaranteed to lie in some admissible region when the argument changes within the segment s ∈ [0, 1]. Let us want to provide q − ≤ q(s) ≤ q + , these restrictions will be guaranteed to be fulfilled if q − ≤ min bik ≤ max bik ≤ q + , 0≤i≤k

which is equivalent to

0≤i≤k

∀i = 0, k q − ≤ bik ≤ q + .

(57.16)

We denote

  (k − l)!l! (i − l + 1) · . . . · i i! (i)l i k · = = = . λilk = l l l!(i − l)! k! (k − l + 1) · . . . · k (k)l Now we write (57.16) in matrix form using (57.14): ⎛ k⎞ ⎛ k λ00 b0 ⎜b1k ⎟ ⎜λk10 ⎜ ⎟ ⎜ ⎜ .. ⎟ ⎜ .. ⎜.⎟ ⎜ . ⎜ k⎟ = ⎜ k ⎜b ⎟ ⎜λ ⎜ d ⎟ ⎜ d0 ⎜.⎟ ⎜ . ⎝ .. ⎠ ⎝ .. bkk λk

k0

0 λk11 .. .

··· ··· .. .

0 0 .. .

⎞

⎟⎛ ⎞ ⎟ a0 ⎟⎜ ⎟ ⎟ ⎜ a1 ⎟ k,d ⎟ ϒ [q] , k k ⎟ ⎜ .. ⎟ =  λd1 · · · λdd ⎟ ⎝ . ⎠ .. . ⎟ . · · · .. ⎠ ad λkk1 · · · λkkd   k,d

q − ≤ k,d ϒ [q] ≤ q + .

(57.17)

Comparison of a vector and a number is understood in the coordinate sense. Thus, if a polynomial satisfies the relation (57.17), then its range of values is included in the segment [q − , q + ]. On the other hand, for k → ∞, the inequality (57.17) is satisfied by all polynomials whose range of values is included in [q − , q + ], which guaranteed by the asymptotic accuracy of the estimates used. Thus, we obtain (in the sense of monotonic convergence) the necessary and sufficient conditions for the region of values of the polynomial for s ∈ [0, 1] to be included in [q − , q + ]. All of the foregoing directly applies to q(·) ∈ P [R → Rm ]. To do this, we define ϒ [q(s)] as the matrix (d + 1) × m, in the rows of which are the vector coefficients of the  corresponding  power expansion q(·). For simplicity, we will use the linear map (k+1) j corresponding to k,d and matrix multiplication k : i,∞j=1 Ri j → ∞ j=1 R for a given d.

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Conditions for a Trajectory to Lie in a Convex Region Consider the set McT . Since this set is convex by definition, condition Dn−1 χ ([0, 1]) ⊂ McT is equivalent to conv Dn−1 χ ([0, 1]) ⊂ McT , where conv denotes a convex hull. It is completely analogous to the case of complex coefficients presented in [9], for Dn−1 χ (·) ∈ P [R → Rm ] it can be shown that if b0k , . . . , bkk are Bernstein form coefficients of a given polynomial trajectory, then conv χ ([0, 1]) =

∞ 

j

j

conv{b0 , . . . , b j } ⊂ conv{b0k , . . . , bkk }.

j=d

Then, setting conv{b0k , . . . , bkk } ⊂ McT , we obtain asymptotically exact conditions on the coefficients that ensure that the trajectory is admissible. These conditions can be equivalently expressed as ∀i = 0, k bik ∈ McT . And if the set McT can be represented as   McT = ξ ∈ Rmn | f (ξ | T ) ≤ 0, g(ξ | T ) = 0 ,

(57.18)

then these conditions can be written in the form ∀i = 0, k

f (bik | T ) ≤ 0, g(bik | T ) = 0.

(57.19)

57.5 Reduction to a Finite-Dimensional Problem We start from (57.12), (57.17) and (57.19). Suppose we want to find a quasioptimal trajectory of degree 2n + 1 + d, then we consider vector coefficients a = (a 0 , . . . , a d−1 ), a i ∈ Rm . According to (57.12): j

χ (s | v, a) = χ H (s | v) + s n+1 (s − 1)n+1 (a 0 + . . . + a d−1 s d−1 ), v j ∈ GT . (57.20) The expression (57.20) represents the parameterization of all polynomials of degree at most 2n + 1 + d among those satisfying the boundary conditions. We use this parameterization to reduce (57.4), (57.9), (57.10) to a finite-dimensional form. Suppose that in addition to (57.6) and (57.18) the following representations are also true:    j GT = ξ ∈ Rm(n+1)  h j (ξ | T ) ≤ 0, r j (ξ | T ) = 0 .

57 Constructing a Polynomial Method in the State Space …

523

Then the original problem, in accordance with the above, reduces to the following form: inf a, v, T J1 [χ ( · |v, a), T ] , −

(57.21) +

Q ≤  ϒ [Q ◦ p (· | T ) ◦ D χ (s | v, a)] ≤ Q , k

n

  k ϒ q (· | T ) ◦ Dn−1 χ (s | v, a) ≤ 0,     f λik ϒ Dn−1 χ (s | v, a)  T ≤ 0,

  ϒ w (· | T ) ◦ Dn−1 χ (s | v, a) = 0, (57.23)  k  n−1    g λi ϒ D χ (s | v, a) T = 0, (57.24)

h j (v j | T ) ≤ 0,

r j (v j | T ) = 0, (57.25)

k is fixed ≥ 2n + 1 + d, i = 0, k, λik is the i-th row of k , a ∈ R , v = (v , v ) ∈ R md

0

1

(57.22)

2(n+1)m

j = 0, 1,

, T ≥ 0.

As you can see, problem (57.21)–(57.25) is a nonlinear programming (NLP) problem. The number of parameters to be minimized is md + 2m(n + 1) + 1. As the experience of implementing this method shows, for the numerical solution of this NLP problem, it is reasonable to use SQP method [10], which converges much better than, for example, the interior-point method [11]. The interior-point method is characterized by a certain jamming of the algorithm in the vicinity of some trajectories. It should be noted that under conditions (57.23), one can carry out transformations and represent the left-hand sides of the expressions as polynomials in the coefficients of the original polynomials, and the multiplication will be defined as the operation of convolution on the sequence of coefficients of the original polynomials. Such a study of the structure of constraints (57.23) would be useful, however, this is beyond the scope of this study.

57.6 Non-Polynomial Case Without going into details, we note that if the right-hand side of the differential equation (57.1) is not polynomial, then we can try to choose the appropriate change of variables in order to reduce the problem to the required form. Another option would be to approximate the left side of the equation with Bernstein polynomial [12]. In the important case of a affine controlled system, one can use polar decomposition, if, for example, ∂χ = f (∂χ , T ) + A(∂χ , T )u, then expanding the matrix A = S P, assuming that det A = 0, we get system S −1 (∂χ − f ) = Pu, P ∈ Om (·2 ), the left-hand side of which can be approximated.

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57.7 Example Let us give an example of the application of the presented method for constructing a quasi-optimal polynomial flight path of an aircraft in terms of time costs. Previously, this problem was considered in [13], where a minimal polynomial trajectory (Hermite polynomial) is constructed, however, the phase constraints are checked at a finite number of test points. ⎧ h˙ = v sin(α), v˙ = g (n 1 − sin(α)) , ⎪ ⎪ ⎪ ⎨ n 2 cos(γ ) − cos(α) , y˙ = v cos(α) cos(β), α˙ = g v ⎪ ⎪ n sin(γ ) ⎪ ⎩z˙ = −v cos(α) sin(β), β˙ = −g 2 . v cos(α)

(57.26)

Here, h is the flight altitude, y is the range, z is the lateral displacement, v is the velocity, α is the angle of inclination of the trajectory, β is the track angle, γ is the angle of heel, n 1 is the longitudinal overload, n 2 is the transverse overload, g is the gravity acceleration. By changing variables t = T s, s ∈ [0, 1] and u 1 = n 2 sin(γ ),

ξ1 = v sin(α),

χ1 = h,

u 2 = n 2 cos(γ ), u3 = n1,

ξ2 = v cos(α) cos(β), ξ3 = v cos(α) sin(β),

χ2 = y, χ3 = z,

the system (57.26) is reduced to 1 g



∂ 2χ − fc T2

 = B (∂χ ) u, ∂χ22 + ∂χ32 = 0.

(57.27)

  Moreover, ∂ = dsd , f c = −g, 0, 0 and B(∂χ )u = ϕ(u | ∂χ ) ∈ O3 (·2 ). Thus, system (57.27) directly has the form (57.1). If the system has the following constraints,   ⎧  ⎨C − ≤   ∂χ  ≤ C + , C − ≤ u ≤ C + , v v u u T  ⎩ − + C h ≤ χ1 ≤ C h , ∂ i χ ( j) = T i i j (T ) , i = 0, 2, j = 0, 1. Then the problem is reduced to form (57.1) and the method presented can be used to find a quasi-optimal landing trajectory, for example, on a ship moving in a circle. The Fig. 57.1 presents a visualization of the process of solving this problem using the constructed method and the SQP algorithm.

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Fig. 57.1 Used parameters and visualization of the solution process. View from above. The solution found is marked by a bold black line

57.8 Conclusion Thus, the problem of finding a quasi-optimal polynomial trajectory is approximated by the finite-dimensional nonlinear programming problem on the coefficients of a special type polynomial. An obvious addition of the method obtained is that there is no need to worry about the behavior of the trajectory between test points, because optimization is carried out according to coefficients, restrictions on which guarantee the admissibility of the trajectory. Also, parameterization using the Hermite polynomial ensures the exact fulfillment of boundary conditions on the trajectory. Also, the presented method is suitable for cases with non-convex constraints and allows us to separate the convex part.

References 1. Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., Rao, A.V.: Direct trajectory optimization and costate estimation via an orthogonal collocation method. J. Guid. Control Dyn. 6, 1435– 1440 (2006) 2. Mehne, H.H., Borzabadi, A.H.: A numerical method for solving optimal control problems using state parametrization. Numer. Algorithm. 2, 165–169 (2006) 3. Kafash, B., Delavarkhalafi, A., Karbassi, S.M.: Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems. Sci. Iran. 3, 795–805 (2012) 4. Seywald, H.: Trajectory optimization based on differential inclusion. J. Guid. Control Dyn. 3, 480–487 (1994) 5. Conway, B.A., Larson, K.M.: Collocation versus differential inclusion in direct optimization. J. Guid. Control Dyn. 5, 780–785 (1998)

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6. Hermite, M.Ch., Borchardt, M.: Sur la formule d’interpolation de Lagrange. J. Reine Angew. Math. 84, 70–79 (1878) 7. Uteshev, A.Yu., Tamasyan, G.Sh.: On the problem of polynomial interpolation with multiple nodes (In Russian). Vestnik of Saint Petersburg University. Ser. 10(3), 76–85 (2010) 8. Lane, J.M., Riesenfeld, R.F.: Bounds on a Polynomial. BIT Numer. Math. 1, 112–117 (1981) 9. Rivlin, T.J.: Bounds on a polynomial. J. Res. Natl. Bur. Stand. Sec. B: Math. 1, 47–54 (1970) 10. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research. Springer, New York (2006) 11. Waltz, R.A., Morales, J.L., Nocedal, J., Orban, D.: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math. Program. 107, 391–408 (2006) 12. Lorentz, G.G.: Bernstein Polynomials. University of Toronto Press, Toronto (1953) 13. Kanatnikov, A.N., Shmagina, E.A.: The problem of terminal control of aircraft movement (In Russian). In: Emelyanov, S.V., Korovina, S.K. (eds.) Nonlinear Dynamics and Control, vol. 7, pp. 79–94. Fizmatlit, Moscow (2010)

Part VIII

Medical and Biological Systems Control

Chapter 58

One-Dimensional Non-Newtonian Models of Arterial Hemodynamics Gerasim V. Krivovichev

Abstract The one-dimensional models of non-Newtonian hemodynamics are considered. The models are constructed by the averaging of 3D incompressible Navier–Stokes system on the vessel cross-section. The Power Law, Carreau-Yasuda, and Cross non-Newtonian models are compared with the inviscid and Newtonian models. In dimensionless variables, it is demonstrated that the small parameter exists in the system, and the perturbation method can be applied for the solution of nonlinear problems. For the smooth initial condition, it is demonstrated that in comparison with the Newtonian model, the strongest damping of the solutions takes place for the non-Newtonian models.

58.1 Introduction The modeling of blood flow in a human arterial system plays an essential role in the prediction of the results of cardiovascular surgeries and the influence of the pathologies [5]. Blood as fluid can be considered as a non-Newtonian media with the shear-dependent viscosity [2, 4, 9]. In most of the works, the blood is modeled as a non-Newtonian fluid for the cases of 2D and 3D problems [1, 8, 10, 11]. Only in [7], a time-dependent non-Newtonian Oldroyd-B model for 1D blood flow is considered. In the presented paper, we try to construct the time-independent non-Newtonian models for 1D case, based on the well-known rheological models of blood, previously used for 2D and 3D problems. The paper has the following structure. In Sect. 58.2, the 1D models are constructed by the averaging of 3D models on the vessel cross-section. In Sect. 58.3, the perturbation method, used for the comparison of the models, is described. Section 58.4 is devoted to the numerical results. Some concluding remarks are made in Sect. 58.5.

G. V. Krivovichev (B) St. Petersburg State University, 7/9, Universitetskaya nab., St., Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_58

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58.2 One-Dimensional Models of Blood Flow The main mechanical properties of blood are its incompressibility and viscous nature. So the general equations, describing blood behavior, are written as ∇ · V = 0, ρ

dV = ρf + ∇ · S, dt

S = − pI + T, T = F(D),

(58.1) (58.2)

where V is the velocity, ρ is a constant density, ρf is the body force (in the presented paper the case of f = 0 is considered), S is the stress tensor, p is the pressure, I is a unity tensor, T is a tangential stress tensor, and D is a strain rate tensor. In the presented paper, the so-called generalized Newtonian fluids are considered, and tensor function F is represented as F = 2μ(I2 )D, where I2 is the second invariant of tensor D. Equations (58.1) represent the incompressibility condition and motion equation, respectively. Equations (58.2) are the rheological relations, which define the model of blood as a fluid. The case of F = 0 corresponds to the model of inviscid (ideal) fluid. In the presented paper, we consider the rheological models, widely used for the description of blood. They are presented in Table 58.1. According to the averaging procedure on the vessel cross section [6], in the case of cylindrical coordinates, Eqs. (58.1) can be reduced to the following system: ∂A ∂Q + = 0, ∂t ∂z

∂Q ∂ +α ∂t ∂z



Q2 A

 +

A ∂P + f (A, Q) = 0, ρ ∂z

(58.3)

where A = A(t, z) is a cross-sectional area, Q = Q(t, z) is a flow rate, P = P(t, z) is the averaged pressure, corresponding to the cross section at point z, α is a Boussinesq  coefficient, defined as: α = s 2 dσ/A, where S is vessel cross section and s = s(y), S

Table 58.1 Rheological models of blood Model Expression for μ(I2 ) Newtonian

μ = const,

Power Law model Carreau–Yasuda model

μ = k I2 2 μ = μ∞ + (μ0 −  a  n−1 a μ∞ ) 1 + λa I22

Cross model

μ = μ∞ + (μ0 −   1 m −1 μ∞ ) 1 + λI22

n−1

Parameters μ = 0.0035 Pa·s [1] n = 0.9, k = 0.0035 Pa·sn [11] λ = 1.902 s, n = 0.22, a = 1.25, μ0 = 0.056 Pa·s, μ∞ = 0.00345 Pa·s [4] μ0 = 0.056 Pa·s, μ∞ = 0.00345 Pa·s, λ = 1.007 s, m = 1.028 [4]

58 One-Dimensional Non-Newtonian Models of Arterial Hemodynamics Table 58.2 Expressions for f (A, Q) Model Expression for f (A, Q)

Parameters 

Q A

K = − 2π μsρ (1)

Q|Q|n−1

K = − 2ks (1)|s (1)| ρ

Newtonian

K

Power Law model

K

Carreau–Yasuda model

K 1 QA +   n−1 a a K 2 1 + K 3 |Q|3a



3n−1 A 2

A

K 1 QA 

Cross model

531

2

+

K 2 1 + K 3 |Q|3m

m

A

−1

2



n−1 π

n+1 2



∞ K 1 = − 2π s (1)μ , ρ 

Q A

0 −μ∞ ) , K 2 = − 2π s (1)(μ ρ a 2

K 3 = λa π |s  (1)|a  ∞ K 1 = − 2π s (1)μ , ρ 

Q A

0 −μ∞ ) , K 2 = − 2π s (1)(μ ρ m 2

K 3 = λm π |s  (1)|m

where y is a dimensionless radius, is a dimensionless velocity profile. Nonlinear term f is obtained as the result of the averaging of ∇ · T (case of f = 0 corresponds to the inviscid fluid). The expressions for f , corresponding to the models, considered at Table 58.1, are presented at Table 58.2. The system (58.3) is closed by the equation-of-state P = P(A). For the arteries, the following dependence could be used [13]: P − Pext = Pmin +

 β √ 4√ ( A − Amin ), β = π Eh, Amin 3

where Pext is the external pressure, Amin and Pmin are the diastolic cross-sectional area and pressure, E is the Young modulus, and h is the vessel wall thickness. It must be noted that the dimensionless profile s(y) analytically could be obtained only for the cases of Newtonian and Power Law models, the corresponding expressions are written as s(y) = 2(1 − y 2 ), s(y) =

 3n + 1  1 1 − y 1+ n . n+1

For the other models, s(y) should be modeled by some smooth function, satisfying the following condition: 1 1 (58.4) ys(y)dy = , 2 0

which can be considered as a consequence of the representation for the mean velocity  U (t, z): U = Vz dσ/A, where Vz is a z-component of vector V. S

In the presented paper, the following model representation of s(y), satisfying to (58.4) is used:

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G. V. Krivovichev

  2 (1 − y d ). s(y) = 1 + d For this representation, the values of α and s  (1) are obtained as α=

d +2 , s  (1) = −(d + 2). d +1

The case of d = 2 corresponds to the fully parabolic profile.

58.3 Perturbation Method Let the dimensionless variables are introduced: z˜ =

t z , t˜ = , LM TM

A A˜ = , AM

P˜ =

P , 2 ρU M

Q˜ =

Q , A M UM

where L M is a typical length, TM is a typical time, U M = L M /TM is the typical velocity, and A M is the typical cross-sectional area. In all formulas presented below in the text, the tilde sign will be ignored. In new variables, Eqs. (58.3) are rewritten as ∂A ∂Q + = 0, ∂t ∂z

∂Q ∂ +α ∂t ∂z



Q2 A

 +χ

√ ∂A + f (A, Q) = 0, A ∂z

(58.5)

√ 2 where χ = γ A M /U M , γ = β/(2 Amin ρ). The dimensionless expressions for f (A, Q) are presented in Table 58.3. Table 58.3 Dimensionless expressions for f (A, Q) Model

Expression for f (A, Q)

Parameters

Newtonian

εQ A

ε= AM M

Power Law model

ε Q|Q| 3n−1

KT

n−1

ε=

A 2

Carreau–Yasuda model

1+n

A M2

 n−1  a Q |Q|a εQ + ξ 1 + ζ 3a A A A 2

Cross model

n−1 T K UM M

−1  |Q|m Q εQ A + ξ 1 + ζ 3m A A 2

T K

K T

1 2 M ε= M AM , ξ = AM ,

ζ =

a K3UM a 2 AM

T K

K T

1 2 M ε= M AM , ξ = AM ,

ζ =

m K3UM m

A M2

58 One-Dimensional Non-Newtonian Models of Arterial Hemodynamics

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Dimensionless parameters can be estimated from the physiological data, presented in [3]. The value of parameter χ , associated with the equation-of-state, for large arteries is estimated as χ ∼ 10 − −30. The estimation for all of the parameters are realized at d = 2 for Newtonian, Carreau-Yasuda, and Cross models, and for d = 1 + 1/n (theoretical value) for Power Law model. The values of ε for all models are estimated as: ε ∼ 0.01–0.06 for the ascending aorta, ε = 0.08–0.13 for the descending thoracic aorta, and ε ∼ 0.2–0.4 for the femoral artery. The value of ξ for Carreau–Yasuda and Cross models can be estimated as ξ = Cε, where C ≈ 15. So, in the considered problems, the small parameter exists, and the perturbation method can be applied for the solution. Let the infinite vessel is considered, and the following initial conditions are stated: A(0, z) = A0 (z),

Q(0, z) = Q 0 (z), z ∈ (−∞, +∞),

(58.6)

where |A0 (z)|, |Q 0 (z)| < ∞ at z → ±∞. Let Q ≥ 0 and initial functions are presented as A0 (z) = A0 + εϕ1 (z) + ε2 ϕ2 (z) + · · · ,

Q 0 (z) = Q 0 + εψ1 (z) + ε2 ψ2 (z) + · · · , (58.7)

where A0 and Q 0 are the constants. According to the perturbation method, solution of the Cauchy problem (58.5), (58.6) is obtained in the following form: A(t, z) = A0 + ε A1 (t, z) + · · · ,

Q(t, z) = Q 0 + ε Q 1 (t, z) + · · · .

(58.8)

The problems for the i-th order terms Ai (t, z),Q i (t, z) are obtained by the substitution of (58.7) and (58.8) in (58.5), (58.6). The details of its solution are presented in [12].

58.4 Numerical Results For the comparison of models, the example with following parameter values is considered: A0 = π, Q 0 = 1, ε = 0.01, χ = 18. The following functions in initial conditions are used: ϕ1 (z) = ϕ2 (z) = 0, ψ1 (z) = arctan(κz), ψ2 (z) = 0. The value κ = 8 is used. The values of parameter ζ are computed for the case of the ascending aorta. The computations are realized at space interval z ∈ [−10, 10] and at time interval t ∈ [0, 5]. As can be seen, the described choice of initial conditions is based on the functions, which are bounded at z → ±∞ and are smooth enough. In the presented paper, the expressions for first and second perturbations are not presented, but they can be easily obtained in computer algebra systems. At Fig. 58.1 the plots of flow rate Q(t, z) at fixed value of t(t = 1) as a function of z are presented. In Fig. 58.2, the same plots are presented at a fixed value of z

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Q(1,z)

1.01

1 2 3 4 5

1.005 1 0.995 0.99 0.985 0.98 −10

10

5

0

−5

z

Fig. 58.1 Plots of the flow rate Q(t, z) at fixed value of t for several models: 1—inviscid model, 2—Newtonian model, 3—Power Law model, 4—Carreau–Yasuda model, 5—Cross model 1.025 1 2 3 4 5

1.02 1.015 1.01

Q(t,5)

1.005 1 0.995 0.99 0.985 0.98 0.975 0

1

2

3

4

5

t

Fig. 58.2 Plots of the flow rate Q(t, z) at fixed value of z for several models: 1—inviscid model, 2—Newtonian model, 3—Power Law model, 4—Carreau–Yasuda model, 5—Cross model

(z = 5) as plots of time functions. As can be seen, the plots corresponding to the viscous models demonstrate the damping of the solution at t → +∞. The strongest damping takes place for the Carreau–Yasuda and Power Law models. The weakest damping is realized for the Newtonian model.

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58.5 Conclusion In the paper, 1D models of non-Newtonian hemodynamics are considered. The models are constructed from the general hydrodynamical 3D model by the averaging procedure, presented in [6]. By considering the system in dimensionless variables, it is demonstrated that the small parameter exists in the model, so the perturbation method can be applied for the solution of initial and initial-boundary problems. On the numerical example, it is demonstrated, that the solutions, corresponding to nonNewtonian models, lead to the strongest damping of the solution at t → +∞ in comparison with the Newtonian model. The obtained solutions can be used for the comparison with other non-Newtonian models and for the testing of the programs, which realize the algorithms of the numerical methods.

References 1. Abbasian, M., Shams, M., Valizadeh, Z., Moshfegh, A., Javadzadegan, A., Cheng, S.: Effects of different non-Newtonian models on unsteady blood flow hemodynamics in patient-specific arterial models with in-vivo validation. Comp. Meth. Prog. Biomed. 186, 105185 (2020) 2. Ameenuddin, M., Anand, M., Massoudi, M.: Effects of shear-dependent viscosity and hematocrit on blood flow. Appl. Math. Comput. 356, 299–311 (2019) 3. Caro, C.G., Pedley, T.J., Schroter, R.C., Seed, W.A.: The Mechanics of Circulation. Cambridge University Press, Cambridge (2012) 4. Cho, Y.I., Kensey, K.R.: Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part 1: Steady flows. Biorheology. 28, 241–262 (1991) 5. Fasano, A., Sequeira, A.: Hemomath: The Mathematics of Blood. Springer, Berlin (2017) 6. Formaggia, L., Lamponi, D., Quarteroni, A.: One-dimensional models for blood flow in arteries. J. Eng. Math. 47, 251–276 (2003) 7. Ghigo, A.R., Lagree, P.-Y., Fullana, J.-M.: A time-dependent non-Newtonian extension of a 1D blood flow model. J. Non-Newtonian Fluid Mech. 253, 36–49 (2018) 8. Karimi, S., Dabagh, M., Vasava, P., Dadvar, M., Dabir, B., Jalali, P.: Effect of rheological models on the hemodynamics within human aorta: CFD study on CT image-based geometry. J. Non-Newtonian Fluid Mech. 207, 42–52 (2014) 9. Marcinkowska-Gapinska, A., Gapinski, J., Elikowski, W., Jaroszyk, F., Kubisz, L.: Comparison of three rheological models of shear flow behavior studied on blood samples from postinfarction patients. Med. Biol. Eng. Comput. 45, 837–844 (2007) 10. Razavi, A., Shirani, E., Sadeghi, M.R.: Numerical simulation of blood pulsatile flow in a stenosed carotid artery using different rheological models. J. Biomech. 44, 2021–2030 (2011) 11. Razavi, A., Shirani, E.: Development of a general method for designing microvascular networks using distribution of wall shear stress. J. Biomech. 46, 2303–2309 (2013) 12. Tkachenko, P.S., Krivovichev, G.V.: Analytical solutions of the problems for equations of one-dimensional hemodynamics. J. Phys. Conf. Ser. 1400(4), 044031 (2019) 13. Toro, E.F.: Brain venous haemodynamics, neurological diseases and mathematical modelling. A review. Appl. Math. Comput. 272, 542–579 (2016)

Chapter 59

Some Problems of Modeling the Human Body Subjected to Vertical Vibration Vladimir Tregubov and Nadezhda Egorova

Abstract The problem of determining the structure and parameters of a mechanical model of a human body subjected to vibration is analyzed. This is extremely important for using this model to construct a vibration protection system. Based on the example of a two-mass mechanical model of a human body it was found that to determine the model structure and its parameters, it is necessary to know the total model mass and the amplitude–frequency response (AFR) for both solids in the model structure. An alternative is to have an input mechanical impedance (IMI) and to know the mass of one of the solids in the model structure. In addition, using mechanical models with an arbitrary number of degrees of freedom, the influence of multi-articular muscles on the frequency characteristics of a human body was found out. The possibility of additional antiresonance frequencies and their presence in the upper mass is shown.

59.1 Introduction The main purpose of mechanical models of a person subjected to vibration is the development (creation) of human protection systems against vibration. The development of such models is preceded by an experimental study of the frequency properties of a human body, which are determined by the transfer function, AFR, IMI, or apparent mass. A large number of papers have been devoted to the experimental determination of these characteristics. The first researcher to measure the impedance of the human body was von Bekesi [11]. He used this frequency response to determine the limit of a person’s sensitivity to vibration. The advent of more modern sensors for measuring force and acceleration has generated a stream of experimental studies of the human body frequency characteristics. Thus, in [2] the results of measuring the IMI of a sitting person in tense and relaxed positions with prolonged V. Tregubov · N. Egorova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. Tregubov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_59

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vibration in the frequency range from 0 to 20 Hz were presented. The graph clearly shows two resonant peaks in the region of 5–6 Hz and 11–12 Hz for all postures. It was also noted that the behavior of the sitting human body impedance is similar to that of a mechanical system consisting of two rigid bodies connected in series by springs and dampers. Later papers repeatedly drew attention to the strong influence of the subject’s posture on the transmission of vibration from the seat to the head [4, 9], as well as to significant variability in the measurement results for the different volunteers [3, 8]. The first mechanical models of a human body exposed to vertical vibration appeared half a century ago [10]. These models were based on experimental studies for three different poses of subjects. These models had a structure consisting of sequentially arranged solid bodies connected by springs and dampers but differed from each other in the number of degrees of freedom. The model parameters were selected so that the frequency response of the model was as close as possible to the experimental frequency response. In subsequent papers the structure of mechanical models became more complex. In [6] a lumped-parameter model of a human body in the sitting position is formulated. It includes the head, vertebral column, upper torso, abdomen-thorax viscera, pelvis, and legs. The deformability of some of them is modeled by springs and dampers. The same authors proposed a mechanical model in which the number of degrees of freedom was reduced to three [7], and according to the authors, the reproducibility of experimental results deteriorated only slightly. A similar model with three degrees of freedom was presented in [12]. These examples show that the problem of determining the structure was not solved and the reasons for the resonance of the human body have not been fully studied [5]. The uncertainty of the structure of models is also a characteristic of recent works. Thus, in [13] a group of models with four degrees of freedom was presented, in which solid bodies were connected by springs and dampers in different combinations not only in series but also in parallel. While doing so, the model elements were not associated with specific parts of the human body. Therefore, the proposed models are not models of the human body, but only models of experimental AFR or IMI. This is also confirmed by models from [1, 15]. In this regard, the number one problem in modeling the human body is to determine the structure of the mechanical model. The second problem is the problem of determining a unique set of parameter values for a mechanical model of a human body. This is extremely important because having two or more values for at least one parameter will make it impossible to be used for building a vibration protection system. Unfortunately, this problem has neither been resolved yet nor has it been formulated. Another problem is that the previously proposed mechanical models did not take into account and did not study the influence of multi-joint muscles on the frequency properties of a sitting human body. However, it is well known that they provide mobility in the cervical, thoracic, and lumbar spine and maintain the equilibrium position. These problems are the subject of the research in this article.

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59.2 Defining the Model Structure and the Unique Set of Values for Its Parameters The problem of determining the model structure and determining the unique set of its parameters’ values are interrelated. As stated in the introduction, in order to determine the structure of a model, all its elements modeled by solid bodies must correspond to certain parts of a human body. For all elements, the mass must be determined based on the average anatomical data. The sum of these masses must be equal to the mass of the human body per seat, which is determined by weighing. In addition, it is necessary to experimentally determine the transfer function from the seat to this simulated area of the human body for each solid body in the model. Such measurements are absent in the vast majority of previous works. A rare exception is the model presented in [14]. The disadvantage of this article is that vibration measurements were made not on the vertebrae themselves but on the skin area adjacent to the vertebra. As for searching for a single set of model parameters’ values, for clarity, we will consider a simple mechanical model of a chain structure with two degrees of freedom that is shown in Fig. 59.1. In it m 1 and m 2 are the upper and lower body mass, c1 and c2 are the spring stiffness, b1 and b2 are the damping coefficients, respectively, and y(t) is displacement of the model vibrating base. The motion equations of the given model in the absolute coordinate system have the following form: m 1 x¨1 + b1 (x˙1 − x˙2 ) + c1 (x1 − x2 ) = 0 , m 2 x¨2 − b1 (x˙1 − x˙2 ) − c1 (x1 − x2 ) + b2 (x˙2 − y˙ ) + c2 (x2 − y) = 0 . (59.1) Applying the Laplace transformation to the system of motion equations of (59.1), we can obtain the transfer functions for masses m 1 and m 2 : H1 ( p) =

α2 p 2 + α1 p + α0 X 1 ( p) = , 4 Y ( p) δ4 p + δ3 p 3 + δ2 p 2 + δ1 p + δ0

(59.2)

H2 ( p) =

β3 p 3 + β2 p 2 + β1 p + β0 X 2 ( p) = . Y ( p) δ4 p 4 + δ3 p 3 + δ2 p 2 + δ1 p + δ0

(59.3)

Fig. 59.1 Two-mass mechanical model of a human body subjected to vibration

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The coefficients in these transfer functions are related to the model parameters in the following way: δ0 = α0 = β0 = c1 c2 ,

δ1 = α1 = β1 = c1 b2 + c2 b1 ,

δ3 = (m 1 + m 1 )b1 + m 1 b2 , (59.4) δ2 = (m 1 + m 1 )c1 + m 1 c2 + b1 b2 , δ4 = m 1 m 2 , α2 = b1 b2 , β2 = m 1 c2 + b1 b2 , β3 = m 1 b2 . The AFR corresponding to the transfer functions (59.2) and (59.3) has the form  |H1 (ω)| =

(59.5)

(β0 − β2 ω2 )2 + (α1 ω − δ3 ω3 )2 . (δ4 ω4 − δ2 ω2 + δ0 )2 + (β1 ω − δ3 ω3 )2

(59.6)

(δ4 

|H2 (ω)| =

(α0 − α2 ω2 )2 + (α1 ω)2 , − δ2 ω2 + δ0 )2 + (δ1 ω − δ3 ω3 )2

ω4

The values of the coefficients δi , αi , and βi are determined in a way to bring the theoretical frequency responses (59.5) and (59.6) as close as possible to the experimental ones. Having thus their values determined, in accordance with the relations (59.5) we will have eight equations, from which we can determine the values of six unknown parameters of the model m 1 , m 2 , c1 , c2 , b1 , b2 provided that the total mass of the model is known M = m 1 + m 2 : c1 =

δ2 − β2 δ3 − β3 Mδ0 Mα0 Mβ0 = = , , b1 = , c2 = M M δ2 − β2 δ2 − β2 δ2 − β2 b2 =

Mα2 β3 (δ3 − β3 ) , m1 = , δ3 − β3 Mα2

m2 =

Mα2 δ4 . β3 δ3 − β3

As you can see in this case, you do not need to pre-set the masses m 1 and m 2 based on the average anatomical data as it was required above to determine the structure of the model. However, in most of the studies, only the experimental frequency response for mass m 1 is available to determine the values of the model parameters. In this case, only six equations out of the eight (59.5) have remained. As a result, we will not be able to uniquely determine the values of all the model parameters, since for parameters m 1 , c1 , and b1 we will get square equations, namely m 1 2 − Mm 1 + δ4 = 0 , b2 c1 2 − α1 c1 + α0 b1 = 0 ,

Mb1 2 − δ3 b1 + m 1 α2 = 0 .

As we can see, prior knowledge of the masses m 1 and m 2 does not save the situation.

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For a mechanical model of a chain structure with three degrees of freedom, similar results are obtained, but their proof is more cumbersome. Additional research is necessary for mechanical models of a chain structure with an arbitrary number of degrees of freedom. In a number of works the IMI is used as an experimentally determined frequency characteristic of a human body. The modulus of IMI for a mechanical model (Fig. 59.1) has the form  |Z (ω)| =

( f 3 ω 4 − f 1 ω 2 )2 + ( f 0 ω − f 2 ω 3 )2 , (δ4 ω4 − δ2 ω2 + δ0 )2 + (δ1 ω − δ3 ω3 )2

where f 0 = Mδ0 ,

f 1 = Mδ1 ,

f 2 = Mb1 b2 + m 1 m 2 c2 ,

f 3 = m 1 m 2 b2 ,

δi are the same as in (59.5). As for the values of f i and δi , as indicated above, they are determined based on the condition of the maximum approximation of the theoretical module of the IMI to the experimental one. Having these values determined, we can proceed to define the model parameters. In doing so, the unique set of their values we can get only if we know the value of one of the masses, for example m 1 . Then we get

c1 =

δ1 m 1 − δ2 b1 + b1 2 b2 δ0 δ3 − b2 m 1 f3 , b2 = , c2 = , b1 = , m2 = M − m1. m 1 b2 + Mb1 c2 M δ4

By the way, you can also determine the total mass of the model without weighing because M = f 0 /δ0 . Thus, to uniquely determine the parameters of a two-mass model of a human body using IMI, you need to know the value of one of the masses.

59.3 Research on the Role of Multi-articular Muscles Another issue that is up for discussion is the effect of multi-joint skeletal muscles on the dynamics of a sitting human body. It is known that multi-joint muscles largely determine the mechanical properties of a spine, and in their absence, the vertebral column is not able to maintain its configuration. However, the influence of multi-joint skeletal muscles on the mechanical characteristics of a human body under vibration has not been studied, including when modeling the human body exposed by vibration. Since these muscles are in a constant tension under vibration conditions, they can be modeled by multi-link viscoelastic joints as shown in Fig. 59.2 for the simplest human body model with two degrees of freedom. The transfer functions for masses m 1 and m 2 are written as follows:

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Fig. 59.2 Mechanical chain structure system with two degrees of freedom in the presence of two-link connections

α3 p 3 + α2 p 2 + α1 p + α0 X 1 ( p) = , Y ( p) δ4 p 4 + δ3 p 3 + δ2 p 2 + δ1 p + δ0 β3 p 3 + β2 p 2 + β1 p + β0 X 2 ( p) = H2 ( p) = , Y ( p) δ4 p 4 + δ3 p 3 + δ2 p 2 + δ1 p + δ0

H1 ( p) =

where δ0 = α0 = β0 = c1 c2 + c0 (c1 + c2 ) , δ1 = α1 = β1 = c1 b2 + c2 b1 + b0 (c1 + c2 ) + c0 (b1 + b2 ) , δ2 = c1 (m 1 + m 1 ) + c2 m 1 + b1 b2 + b0 (b1 + b2 ) + c0 m 2 , δ3 = b1 (m 1 + m 1 ) + b2 m 1 + b0 m 2 , δ4 = m 1 m 2 , α2 = b1 b2 + c0 m 2 , α3 = b0 m 2 , β2 = m 1 c2 + b1 b2 + b0 (b1 + b2 ) , β3 = m 1 b2 . The frequency responses for masses m 1 and m 2 are as follows:  |H1 (ω)| =  |H2 (ω)| =

(α0 − α2 ω2 )2 + (α1 ω)2 − α3 ω)3 , (δ4 ω4 − δ2 ω2 + δ0 )2 + (δ1 ω − δ3 ω3 )2 (β0 − β2 ω2 )2 + (α1 ω − δ3 ω3 )2 . (δ4 ω4 − δ2 ω2 + δ0 )2 + (β1 ω − δ3 ω3 )2

It should be noted that the numerator of frequency response |H1 (ω)| for b j = 0 is equal to α0 − α2 ω2 and vanish when  ω=

α0 = α2



c1 c2 + c0 (c1 + c2 ) . c0 m 2

This means that when two-link connections are applied, an anti-resonance frequency appears on the upper mass, but this is not possible in the absence of two-link connections. It should be noted that multi-link joints can bind different masses of the system, depending on which multi-joint muscles are included in the developed model. Consider, for example, the case of the existence of antiresonance frequencies on the upper mass of a model with 3 degrees of freedom, depending on how the

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Fig. 59.3 Three variants for applying multi-link connections to a model with three degrees of freedom

Fig. 59.4 AFR for the models with 8 degrees of freedom. Curve 1—the usual chain structure, curve 2—the chain structure with multi-link connections

additional multi-link connection is imposed. In this case, there are 3 variants for applying an additional connection (Fig. 59.3). As an illustration, we have given numerical calculations of the AFR for the human body mechanical model with 9 degrees of freedom (Fig. 59.4) in the presence (curve 2) and absence (curve 1) of multi-link connections. In this model m 1 is the mass of the head, m 0 is the mass of the trunk, and m 2 , . . . , m 8 are the masses of seven cervical vertebrae. Fig. 59.4 shows that the presence of multi-link connections (curve 2), which model multi-joint neck muscles, can lead to a decrease in the maximum values of the frequency response and the appearance of a frequency at which the AFR is close to zero.

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59.4 Conclusion To sum up, the requirements for defining the structure of a human body mechanical model intended for building a vibration protection system were formulated. Based on the example of a two-mass human body mechanical model, it was shown that unambiguous determination of its parameters is possible only if there is an experimental frequency response for both masses and the value of the total mass, or if there is an experimental IMI and the value of one of the masses. In addition, the influence of multi-articular muscles on the frequency characteristics of the human body was found using mechanical models. In particular, the possibility of antiresonance frequencies, including at the upper mass, was shown.

References 1. Abbas, W., Abouelatta, O.B., El-Azab, M., Elsaidy, M., Megahed, A.A.: Optimization of biodynamic seated human models using genetic algorithms. Engineering 2, 710–719 (2010) 2. Coermann, R.R.: The mechanical impedance of the human body in sitting and standing position at low frequencies. Hum. Factors 4, 227–253 (1962) 3. Fairley, T.E., Griffin, M.J.: The apparent mass of the seated human body: vertical vibration. J. Biomech. 22(2), 81–94 (1989) 4. Griffin, M.J.: Vertical vibration of seated subjects: effects of posture, vibration level, and frequency. Aviat. Space Environ. Med. 46, 269–276 (1975) 5. Matsumoto, Y., Griffin, M.J.: Modelling the dynamic mechanisms associated with the principal resonance of the seated human body. Clin. Biomech. 16(1), S31–S44 (2001) 6. Muksian, R., Nash, C.D.: A model for response of seated humans to sinusoidal displacements of the seat. J. Biomech. 17, 207–215 (1974) 7. Muksian, R., Nash, C.D.: On frequency-depended damping coefficient in lumped-parameter models of human being. J. Biomech. 9, 339–342 (1976) 8. Paddan, G.S., Griffin, M.J.: The transmission of translational seat vibration to the head - I. Vertical seat vibration. J. Biomech. 21(3), 191–197 (1988) 9. Paddan, G.S., Griffin, M.J.: A review of the transmission of translational seat vibration to the head. J. Sound Vibr. 215, 863–882 (1998) 10. Potemkin, B.A., Frolov, K.V.: Constructing the dynamic model of the human body for manoperator exposed to broadband random vibration. In the collection of machine vibro-isolation and vibroprotection. Moscow, Nauka, pp. 17–30 (1973) (In Russian) 11. Von Bekesy, G.: Uber die Empfindlichkeit des stehenden und sitzenedn Menschen gegen sinusfoermige Erschuetterungen. Akustische Zeitschrift 4, 360–369 (1939) 12. Wan, Y., Schimmels, J.M.: A Simple model that captures the essential dynamics of a seated human exposed to whole body vibration. Adv. Bioeng., ASME-publication-BED (Bioeng. Div.) 31, 333–334 (1995) 13. Bai, X.X., Xu, S.X., Cheng, W., Qian, L.J.: On 4-degree-of-freedom biodynamic models of seated occupants: Lumped-parameter modeling. J. Sound Vibr. 402(18), 122–141 (2017) 14. Yoshimura, T., Nakai, K., Tamaoki, G.: Multi-body dynamics modelling of seated human body under exposure to whole-body vibration. Ind. Health 43, 441–447 (2005) 15. Zhang, E., Xu, L.A., Liu, Z.H., Li, X.L.: Dynamic modeling and vibration characteristics of multi-DOF upper part system of seated human body. Chine J. Eng. Des. 15, 244–249 (2008)

Chapter 60

Detection of the Community-Acquired Pneumonia Factors Leading to Death Alexandra A. Arzhanik, Anastasiya B. Goncharova, Daria A. Vinokurova, and Evgeny S. Kulikov

Abstract The aim of the paper was to detect the factors leading to the death of patients classified as mild on the CURB-65 scale. We performed a retrospective analysis of 1412 case histories of hospitalized community-acquired pneumonia patients in all hospitals in the Tomsk Region in 2017. All patients were categorized into different groups based on lethality and CURB-65 score. We analyzed age, gender, and laboratory indicators. Significant indicators of an adverse outcome were identified. It is necessary to modify the scale and take into account a larger number of indicators.

60.1 Introduction Community-acquired pneumonia is one of the leading causes of death worldwide. According to the World Health Organization, lower respiratory tract infections occupy the fourth highest mortality rate in the world [1]. The Ministry of Health of Russian Federation updates every three years the clinical guidelines for the diagnosis and treatment of community-acquired pneumonia [2]. Prediction of the risk of death in community-acquired pneumonia is becoming increasingly important, as this determines the choice of treatment site, the amount of diagnostic, and treatment procedures. In this regard, the problem of predicting the outcome of the disease based on the available patient data is relevant. Pneumonia is a rapidly progressing disease that can lead to a life-threatening condition in a short time. Therefore, when a patient is admitted to the hospital, it is necessary to correctly assess their condition and determine the necessity for hospitalization. A. A. Arzhanik · A. B. Goncharova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. A. Arzhanik e-mail: [email protected] D. A. Vinokurova · E. S. Kulikov Siberian State Medical University, 2 Moskovsky trakt, Tomsk 634050, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_60

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There are scales for patients with community-acquired pneumonia that help to categorize patients into different risk groups based on mortality. Different scales have their advantages and disadvantages as they were developed in different socioeconomic and geographical conditions, thus, none of them are generally accepted in the world [3]. Assessment of the prognosis of community-acquired pneumonia according to the clinical guidelines of the Ministry of Health of Russian Federation for communityacquired pneumonia for all patients is recommended to be performed by using the CURB-65 scale [2]. The CURB-65 scale has been developed by Lim W.S. et al. The scale includes simple clinical and biochemical measurements [4]. There are some researches that confirm the effectiveness of the CURB-65 scale. The effectiveness of the scale has been shown on 223 patients in the research done by Falcone M. et al. [5] and on 883 patients in the research by Aronsky D. and Dean N. [6]. However, some authors show that the CURB-65 scale should be used cautiously because there are patients who were classified as mild but later they died [7, 8]. For example, in the research done by Ilg A. et al. [9] in the group of patients classified as mild, the mortality rate was 0.6%.

60.2 Materials and Research Methods Mortality of community-acquired pneumonia was studied based on the data of all patients hospitalized with community-acquired pneumonia in all hospitals in the Tomsk Region in 2017. The database contained information about 1412 patients. For each patient, 250 indicators were requested: age, gender, social, clinical, laboratory, and radiological characteristics, and the patient’s death was considered as a predicted variable. The CURB-65 scale includes an analysis of 5 signs: • • • • •

impaired consciousness due to pneumonia; an increase in the level of urea nitrogen > 7 mmol/l; respiration rate ≥ 30 breaths/min; a decrease in systolic blood pressure < 90 mm Hg or diastolic < 60 mm Hg; patient age ≥ 65 years.

The presence of each symptom is valued at 1 point. The total amount of points can vary from 0 to 5, and the risk of death increases with an increasing amount of points [4]. For each patient, a CURB-65 scale factor was calculated. Of all patients, the following groups were identified based on lethality and CURB-65 score: Group 1: Light onset—Dead (L–D)—the total score is 0 or 1, patients died, and there were 20 patients (1.4%); Group 2: Heavy onset—Dead (H–D)—the sum of points is 3, 4, or 5, patients died, and there were 16 patients (1.1%);

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Group 3: Light onset—Recovered (L–R)—the total score is 0 or 1, patients were discharged from the hospital in a satisfactory condition, and there were 970 patients (68.7%); Group 4: Heavy onset—Recovered (H–R)—the total score is 3 and 4, patients were discharged from the hospital in a satisfactory condition, and there were 112 patients (7.9%). There were no severe patients with a total score of 5 who recovered. The Kruskal-Wallis test was used to find the differences among groups in general. The Kruskal-Wallis H-test is used to evaluate the differences between more than two unrelated samples simultaneously according to the level of a particular attribute [10]. The method of multiple comparisons of mean ranks for all groups was used to assess the differences between group 1 (L–D) and group 3 (L–R).

60.3 Results and Discussion After dividing into groups, a calculation was performed for each group according to the following indicators: duration of hospitalization, age (years), body mass index (BMI), systolic and diastolic blood pressure (mmHg), heart rate (beats/min.), respiration rate (breaths/min.), oxygen saturation blood level (%), temperature (◦ C), hemoglobin level (g/L), erythrocyte level (1012 /L), platelet count (109 /l), white blood cell count (109 /l), segmented neutrophils (%), stab and young neutrophils (%), hematocrit level (%), bilirubin level (mcmol/L), total protein (g/L), glucose level (mmol/L), Alanine Aminotransferase (ALT) (Unit/L), Aspartate aminotransferase (AST) (U/L), serum urea level (mmol/L), serum creatinine level (mmol/L), serum sodium level (mmol/L), serum potassium level (mmol/L), the level of C-reactive protein (Mg/L), and the level of albumin (G/L). Figure 60.1 shows the indicators included in the CURB-65 scale. The results are presented as the median, upper and lower quartiles, maximum and minimum values. For all indicators, the p-level is less than 0.05, therefore, there are statistically significant differences. The Kruskal-Wallis test showed differences in all groups in general. However, group 1 (L-D) is of interest because these patients were classified as mild but later they died. It is necessary to determine which factors lead to death using the method of multiple comparisons of mean ranks. Table 60.1 presents the indicators for groups 1 and 3 for which there are statistically significant differences (significance level p < 0.05). The data are presented as Me (Q 1 ; Q 3 ), where Me is the median, Q 1 and Q 3 are the first and third quartiles, respectively. According to Table 60.1, it can be noted that a decrease in systolic pressure, an increase in respiratory rate and heart rate, a decrease in saturation and hematocrit, and an increased level of bilirubin and segmented neutrophils indicate an increased probability of death of a patient. Systolic pressure and respiratory rate are already included in the CURB-65 scale; however, other indicators are not involved in it.

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Fig. 60.1 Group differences of the median of a age; b respiration rate; c systolic blood pressure; d diastolic blood pressure; e serum urea level

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Table 60.1 The results of the comparison of the indicators in groups 1 and 3 using multiple comparisons of mean ranks Indicator Group 1 (L-D) Group 3 (L-R) p-level Systolic blood pressure, mmHg Respiration rate, breaths/min. Heart rate, beats/min. Saturation blood level, % Segmented neutrophils, % Hematocrit level, % Bilirubin level, mcmol/L

110(100; 120) 20(19; 22) 100(94.5; 103.5) 93(85; 95) 78(67.8; 87) 31.7(27.9; 41) 30.55(12.85; 38.15)

120(115; 130) 18(17; 20) 87(78; 97) 97(95; 98) 67(58; 75) 38, 4(35; 42) 14(10.8; 20)

0.0024 0.0142 0.0013 0.0004 0.0053 0.0457 0.0495

60.4 Conclusion In the presented study, we have performed a retrospective analysis of 1412 case histories of hospitalized community-acquired pneumonia patients in all hospitals in the Tomsk Region in 2017. All patients were categorized into different groups based on lethality and CURB-65 score. According to the Kruskal-Wallis test, the groups have statistically significant differences on the indicators included on the CURB-65 scale. However, there was a group of patients who were classified as mild on the CURB-65 scale, who later died. Two groups of patients (recovered and dead patients classified as mild) were compared with each other to determine the factors that increase the probability of occurrence of death. Significant indicators of an adverse outcome were identified. Thus, it is necessary to modify the scale and take into account a larger number of indicators for a more accurate classification of patients when they are admitted to the hospital.

References 1. 10 leading causes of death in the world. World health Organization (2018) Available via DIALOG. https://www.who.int/en/news-room/fact-sheets/detail/the-top-10-causesof-death. Cited 12 Oct 2019 2. Ministry of Health of the Russian Federation Clinical recommendations. Community-acquired pneumonia (ICD 10: J13-J18) (2018) Available via DIALOG. http://spulmo.ru. Cited 10 May 2020 3. Fesenko, O., Sinopalnikov, A.: Severe community-acquired pneumonia and prognostic scores. Pract. Pulmonol. 2, 20–26 (2014) 4. Lim, W.S., van der Eerden, M.M., Laing, R., et al.: Defining community acquired pneumonia severity on presentation to hospital: an international derivation and validation study. Thorax. (2003). https://doi.org/10.1136/thorax.58.5.377 5. Falcone, M., Corrao, S., Venditti, M., Serra, P., Licata, G.: Performance of PSI, CURB-65, and SCAP scores in predicting the outcome of patients with community-acquired and healthcareassociated pneumonia. Intern. Emerg. Med. (2013). https://doi.org/10.1007/s11739-011-0521y

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6. Aronsky, D., Dean, N.: Admission decision for pneumonia: a validation study of the CURB-65 criteria. CHEST J. (2004). https://doi.org/10.1378/chest.126.4_MeetingAbstracts.738S-a 7. Ananda-Rajah, M.R., Charles, P.G., Melvani, S., Burrell, L.L., Johnson, P.D., Grayson, M.L.: Comparing the pneumonia severity index with CURB-65 in patients admitted with community acquired pneumonia. Scand. J. Infect. Dis. (2008). https://doi.org/10.1080/ 00365540701663381 8. Leshchenko, I.V., Trifanova, N.M.: Community-acquired pneumonia: risk factors for an adverse outcome and the results of the implementation of the territorial standard in the Sverdlovsk region. Pulmonology. Doctor.ru. 56, 57–63 (2009) 9. Ilg, A., Moskowitz, A., Konanki, V., et al.: Performance of the CURB-65 score in predicting critical care interventions in patients admitted with community-acquired pneumonia. Ann. Emerg. Med. (2019). https://doi.org/10.1016/j.annemergmed.2018.06.017 10. Kruskal, W.H., Wallis, W.A.: Use of ranks in one-criterion variance analysis. J. Am. Stat. Assoc. 260, 583–621 (1952)

Chapter 61

HIV Incidence in Russia, Ukraine, and Belarus: SIR Epidemic Analysis Sergei V. Sokolov and Alexandra L. Sokolova

Abstract The problem of predicting the incidence rate of the human immunodeficiency virus (HIV) in Russia, Ukraine, and Belarus is considered. The official morbidity levels as initial data for numerical modeling are taken. The search for the coefficients of the model is examined in detail using gradient descent with an auxiliary system applied. The R0 value is calculated. The incidence of HIV in Russia, Ukraine, and Belarus is compared and the difference is discussed.

61.1 Introduction The human immunodeficiency virus (HIV) infection is a disease that affects and alters the human immune system. It causes a gradual decrease in the general immunity of a person resulting in an increase of an individual’s susceptibility to a variety of infections as the organism loses its ability to resist pathogenic bacteria. Without treatment HIV infection develops to the advanced stage, where the destruction of vital systems of the body occurs, acquired immunodeficiency syndrome (AIDS). However, modern advances in medicine enable people to live a regular life in case of receiving proper treatment and efficient support. Being currently relatively small in Russia, Ukraine, and Belarus the scale of the HIV epidemic keeps on increasing [1–3]. In addition, the annual incidence is getting higher and higher. Having considered the number of new infections in the population, one can see that Russia holds a leading position in the world.

S. V. Sokolov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. L. Sokolova St. Petersburg State Electrotechnical University ‘LETI’, 5, ul. Professora Popova, Saint Petersburg 197376, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_61

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No effective inexpensive treatment of HIV is available now. Cases of complete recovery for HIV infection after stem cell transplantation are detected [4], but they are single, prohibitively expensive, and thus can be neglected, assuming that HIV infection is almost incurable in Russia. However, effective suppression of the virus is possible. Moreover, without suppressive therapy, the patient develops AIDS and dies within a 5–10-year period on average. It is important to mention that the spread of HIV causes serious economic losses and the therapy, which improves the quality of life, prolongs life expectancy, improves the ability to work for an HIV-infected individual, and is expensive. Thus, one of the key tasks is to control the spread of HIV infection. Investments allocated for the study of epidemic mechanisms and methods of counteraction, the development of a vaccine and drugs are rising all around the world. Another issue is to reduce the number of new cases, i.e. infection prevention, and to support individuals already infected with HIV, prolonging and improving their quality of life. The life expectancy of HIV-infected patients depends on a number of factors: age, lifestyle, emotional state, region of residence, history of chronic diseases, etc. The average life expectancy (without treatment) is about 9–11 years from the moment of infection till the death. If the patient does not receive any therapy or refuses to follow the treatment protocol, life expectancy is reduced to 2–5 years. On the other hand, there are cases when people have lived with the virus relatively healthy and long life—more than 40 years from the moment of infection. The paper analyzes the key features of HIV infection and disease statistics in Russia, Ukraine, and Belarus for the further use of this information for the epidemic course prediction. Much attention is paid to the basic reproductive number R0 .

61.2 Literature Review One of the most famous works on mathematical modeling of the spread of diseases [5] describes the dynamics of the epidemiological process using a system of differential equations, the solutions of which characterize the dynamics of changes in the number of subgroups in the population in question. Kermak and McKendrick introduced one of the simplest models of the dynamics of the epidemic—the SIR model, which considers three groups of individuals: susceptible to the disease S(t) (Susceptible), infected I (t) (Infected), and R(t) (Removed) dropped out from the group due to recovery or death. The analytical solution of the SIR model, as well as its modification taking into account fertility and mortality, is considered in [6]. There are many models examining the division of a population into a number of different groups of individuals, depending on the stage of the disease, the presence of immunity, etc. In addition to the three groups of SIR models, the article [7] deals with the MSEIR disease spread model, which describes a group of individuals with passive immunity

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from birth (M) and a group of individuals in the latent stage of infection (E) when the virus has already entered the body, but the disease has not developed yet, and the individual is not able to spread it. In [8] various modifications of the SIR model of the dynamics of the epidemic: taking into account fertility and mortality, the virulence of the pathogen, the latent phase of the disease are considered. The types of anti-epidemic measures are also described with examples of their application and optimization. Among other things, measures are being taken to control the HIV epidemic, such as chemoprophylaxis and the isolation of patients. A lot of existing HIV models consider heterogeneous models, dividing populations into different risk groups. In this case, basically, two stages are considered— susceptibility to the disease and infection. One of such models was considered in [9]. In this model, there are three risk groups: the core of the infection (people most at risk of infection due to risky behavior), the bridge group (people in contact with both the nuclear group and the rest of the population), and the main population (non-core people or bridge group). The main problem of such models is a large number of uncertain parameters, the errors in the determination of which significantly affect the simulation result. In [10] a SIR model taking into account the birth rate, mortality, as well as chemoprophylaxis and isolation of a group of infected but not epidemically dangerous patients, is considered. The actual article uses the same model but focuses on the difference between countries’ epidemic behavior.

61.3 The Problem Formulation We consider a mathematical model with as small amount of uncertain coefficients as possible and focus on the most accurate determination of parameters. For analysis, we use statistics of the HIV infection incidence in Russia, Ukraine, and Belarus for 2009–2019. Earlier data are less accurate due to the low degree of awareness of HIV infection in that period and do not affect the accuracy of the resulting model. HIV statistics in Russia, Ukraine, and Belarus for 2009–2019 are presented in Table 61.1 The main goal of this work is to find the most probable coefficients of the equation, derive R0 , and compare all countries with result achieved.

Table 61.1 HIV statistics in Russia, Ukraine, and Belarus according to [1–3] Year 2009 2010 2011 2012 2013 2014 2015 2016 2017 Russia, % Ukraine, % Belarus, %

2018

2019

0,334 0,366 0,378 0,412 0,451 0,504 0,540 0,595 0,643 0,651 0,686 0,308 0,347 0,383 0,425 0,466 0,546 0,561 0,596 0,633 0,671 0,704 0,087 0,094 0,101 0,110 0,117 0,128 0,143 0,162 0,182 0,205 0,229

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61.4 SIR Model with Fertility and Mortality Rate, Chemoprophylaxis, and Isolation At first, the simplest SIR model is considered ˙ = −β I (t)S(t), S(t)

I˙(t) = β I (t)S(t) − γ I (t), ˙ = γ I (t), R(t)

(61.1) (61.2) (61.3)

with S(t) + I (t) + R(t) = N > 0, S(t0 ) = S0 > 0, I (t0 ) = I0 > 0, R(t0 ) = R0 ≥ 0, β > 0, γ > 0. In order to consider the shares of the total number of individuals, dividing all the equations by N , we can take N = 1. Thus, each of the values will correspond to its share of the total number. More accurate prediction of the epidemiological situation requires taking into account additional factors. One of these factors is the demographic situation in the country: there is a generational change that affects the size of a particular group of individuals in mathematical models. Thus, fertility and mortality rates should be taken into account. For HIV infection, unlike many diseases, a vertical transmission mechanism is characteristic: infected mothers are more likely to have already infected children. Thus, the influx of individuals into the group of susceptible individuals will occur both at the expense of the susceptible and at the expense of a certain proportion of the infected. The inflow and outflow in the group of retired individuals for HIV infection do not occur, i.e. for this disease, this group characterizes the number of individuals who died from the disease. The outflow from the other two groups added to consideration should take into account the death of people from all the other factors. In addition to β and γ , the following coefficients are also present in the new model: α > 0, μ > 0—fertility and mortality rate in the country (average value of the number of newborns per year per individual), we take the average value of these coefficients over the past years ϕ1 > 0—the probability to give birth to a healthy child for an infected mother. According to statistics, the average probability is ϕ1 ≈ 0.7. Chemoprophylaxis in the general case is the prevention of the development of the disease in the early stages of infection. In the case of HIV infection, there are two areas of chemoprophylaxis. 1. Chemoprophylaxis of parenteral and sexual infection. Such methods have been developed primarily for prophylaxis in health workers who have been exposed to infected blood on the mucous membrane or who have been injured by an HIVcontaminated tool. The effectiveness of the use has been proven. As a result, the risk of infection is significantly reduced. The onset of chemoprophylaxis is considered inconsistent if more than 72 h have passed since the infection. It can also be used for sexually transmitted infections or transfusion of blood infected with HIV. 2. Decrease in the probability of giving birth to an infected child in an HIV-infected mother. This method of prevention on average reduces the risk of infection with a vertical transmission mechanism to 8%. Thus, under chemoprophylaxis ϕ2 = 0.92.

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For HIV infection, the following interpretation of isolation is relevant. A part of infected individuals after the diagnosis of the disease changes their lifestyle, limiting themselves in actions that can cause infection of others. Thus, they move into a group of isolated individuals Iis who are not able to spread the infection. In this group the coefficient characterizing the probability of having a healthy child will be higher due to the use of chemoprophylaxis. Consider the coefficient ω, which characterizes the probability of transition from an infected group to a group of isolated ones. By increasing this probability, the course of the epidemic can be positively affected. The main factors for this value to depend on are timely diagnosis of the disease and the conscious behavior of infected individuals, in which they limit themselves to actions dangerous for the environment. Launched in 2014, the UNAIDS Fast-Track strategy outlined plans to step up the HIV response in low- and middle-income countries to meet the SDG 3 target to end AIDS by 2030 [11]. The strategy acknowledges that, without rapid scale-up, the HIV epidemic will continue to outrun the response. To prevent this it underlines the need to reduce new HIV infections and AIDS-related deaths by 90% by 2030, compared to 2010 levels. To achieve this, the Fast-Track strategy sets out targets for prevention and treatment, known as the 90-90-90 targets (90% people who are aware of their status, 90% among them are on HIV treatment, and 90% among them are virally suppressed; the latter we will refer to as isolated, the product still gives only 73% of all infected population). All the countries considered declare similar targets, but it is evident that it fails to date since less than a half of infected are on HIV treatment and less than 25% are annually tested. This gives actual ω not exceeding 0.15. The transition diagram for this model is presented in Fig. 61.1.

Fig. 61.1 Transition diagram for a model with isolation, chemoprophylaxis, and birth and death rates

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The model with isolation of patients, chemoprophylaxis, and fertility and mortality rates will be the following: ˙ = −β I (t)S(t) + α(S(t) + ϕ2 Iis (t) + ϕ1 I (t)) − μS(t) , S(t) S(t) + I (t) + Iis (t) α(1 − ϕ1 )I (t) − μI (t) I˙(t) = β I (t)S(t) − γ I (t) + − ωI (t), S(t) + I (t) + Iis (t) α(1 − ϕ2 )Iis (t) − μIis (t) , I˙is (t) = ωI (t) − γ Iis (t) + S(t) + I (t) + Iis (t) ˙ = γ (I (t) + Iis (t)). R(t)

(61.4) (61.5) (61.6) (61.7)

Since the number of people who died from AIDS at 2009–2019 and the total number of infected are known, one can calculate R(2019) − R(2009) . γ =  2019 2009 (I (t) + Iis (t))dt The typical time until recovery (or death for HIV/AIDS model) is Tr ≈ γ −1 . Simple calculation gives the following numbers for countries studied (Table 61.2). The typical time for Russia and Belarus is very close and the mortality data for Ukraine seems to be at least halved. The following simple explanation can be given to this fact: in Ukraine, only those patients who are registered in the mortality statistics are taken into account. For further calculations we assume γ = 0.04 for Ukraine. The average life expectancy is relatively high, and this value reflects the fact that approximately half of the patients receive therapy and this does not lead to a decrease in their life expectancy. For a qualitative assessment of the epidemiological situation, it is necessary to have sufficiently accurately estimated parameters of the model under consideration. To find them, statistical data on the disease are used. In the SIR model, we are interested in the coefficient β. To obtain the model coefficients for a number of measurements, we solve the following problem. Let k be the number of available measurements of the number of people infected with HIV in the country by year. Thus, we have k values I1 , I2 , . . . , Ii , . . . , Ik . For different β, the system will have different solutions. It is necessary to select such values of these parameters so that the error in the solution obtained, when comparing with real statistics is minimal so that

Table 61.2 Typical time until recovery Country γ Russia, % Ukraine, % Belarus, %

0,037 0,021 0,040

Tr 26,7 48,7 24,8

61 HIV Incidence in Russia, Ukraine, and Belarus: SIR Epidemic Analysis Table 61.3 Basic reproductive number Country ω = 0, 05 Russia Ukraine Belarus

ω = 0, 1

557

ω = 0, 15

β = 0, 150, R0 = 4, 0 β = 0, 182, R0 = 4, 8 β = 0, 217, R0 = 5, 8 β = 0, 155, R0 = 3, 9 β = 0, 188, R0 = 4, 7 β = 0, 223, R0 = 5, 5 β = 0, 180, R0 = 4, 5 β = 0, 230, R0 = 5, 7 β = 0, 264, R0 = 6, 5

k  [I (ti ) − Ii ]2 → 0, i=i

where I (ti ) is the solution of the system for I (t) for fixed β. Using a grid of the values of β solving the corresponding systems numerically, one can obtain an approximate value for available data (Table 61.3). More accurate approximation can be achieved using local minimum search methods [12] that is based on minimization of the function 1 (I (ti , β) − Ii )2 . 2 i=1 k

(β) = Gradient equations have the form

k  ∂(β) dβ = =− (I (ti , β) − Ii )y11 (ti ) dτ ∂β i=1

(61.8)

with auxiliary equation for yi j as it is shown in [10]. For different initial conditions, the solution can converge to different local minima. Therefore, a preliminary search for the approximate global minimum is necessary. As a result, we obtain the most accurate value of the coefficient β and we can calculate R0 for all three countries. Since ω is unknown we vary it from 0,05 (pessimistic) to 0,15 (optimistic). Bigger ω gives higher R0 because of retrospective analysis. Even with ω = 0 we get the estimate 3, 2 < R0 < 3, 8. Finally, the basic reproductive number for all countries R0 = 3.9 − 6.5 that is close to Africa [13].

61.5 Conclusion Based on the calculations made, the following conclusions can be proposed: • All three countries have a full-blown HIV epidemic with Belarus having a lower relative number of infected, but higher growth rates.

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• The official mortality rate in Ukraine is at least half or real even with official data on infected number. The real infected number and mortality rate are higher in all countries. • The basic reproductive number R0 is likely between 4.8 and 5.7.

References 1. Gosudarstvennyy doklad O sostoyanii sanitarno-epidemiologicheskogo blagopoluchiya naseleniya v Rossiyskoy Federatsii v 2018 godu [State report On the state of the sanitaryepidemiological well-being of the population in the Russian Federation in 2018] Russian Federal Service for Surveillance on Consumer Rights Protection and Human Wellbeing (Rospotrebnadzor), pp. 116–120 (2018) (in Russian) 2. Statistichna informaciya pro VIL/SNID [HIV/AIDS statistic data] https://www.phc. org.ua/kontrol-zakhvoryuvan/vilsnid/statistika-z-vilsnidu/statistichni-dovidki-pro-vilsnid (in Ukrainian) 3. Epidsituaciya po VICH/SPID v Belarusi [HIV/AIDS surveillance in Belarus] https://www. belaids.net/epidsituaciya-po-vichspid-v-belarusi/ 4. Gupta, R.K., Abdul-Jawad, S., McCoy, L.E., et al.: HIV-1 remission following CCR5 32/ 32 haematopoietic stem-cell transplantation. Nature 568, 244–248 (2019) 5. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. 700–721 (1927) 6. Harko, T., Lobo, F.S.N., Mak, M.K.: Exact analytical solutions of the susceptible-infectedrecovered (SIR) epidemic model and of the SIR model with equal death and birth rates. Appl. Math. Comput. 236, 184–194 (2014) 7. Hethcote, H.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000) 8. Kolesin, I.D., Zhitkova, E.M.: Matematicheskiye modeli epidemiy. [Mathematical models of epidemics]. Solo, Saint Petersburg (2004) (in Russian) 9. Nosova, E.A.: Models of control and spread of HIV-infection. Math. Biol. Bioinform. (7.2), 632–675 (2012). https://doi.org/10.17537/2012.7.632 10. Sokolov, S.V., Sokolova, A.L.: HIV incidence in Russia: SIR epidemic modelbased analysis. Vestnik of Saint Petersburg University. Appl. Math. Comput. Sci. Control Process. 15(4), 616– 623 (2019). https://doi.org/10.21638/11702/spbu10.2019.416 11. World Health Organization. Available at: https://www.who.int/news-room/fact-sheets/detail/ hiv-aids 12. Babadzanjanz, L.K., Boyle, J.A., Sarkissian, D.R., Zhu, J.: Parameter identification for oscillating chemical reactions modelled by systems of ordinary differential equations. J. Comput. Methods Sci. Eng. 3(2), 223–232 (2003) 13. Nsubuga, R.N., White, R.G., Mayanja, B.N., Shafer, L.A.: Estimation of the HIV basic reproduction number in rural South West Uganda. PLoS ONE 9(1), 1991–2008, e83778 (2014). https://doi.org/10.1371/journal.pone.0083778 14. UNAIDS. Ambitious treatment targets: writing the final chapter of the AIDS epidemic. Geneva, UNAIDS (2014). https://www.unaids.org/sites/default/files/media$_$asset/ JC2670$_$UNAIDS$_$Treatment$_$Targets$_$en.pdf 15. Rasporyazheniye Pravitel’stva RF ot 20 oktyabrya 2016 g. N 2203-r O Gosudarstvennoy strategii protivodeystviya rasprostraneniyu VICH-infektsii v RF na period do 2020 g. i dal’neyshuyu perspektivu [Decree of the Government of the Russian Federation of October 20, 2016 No. 2203-r On the State Strategy for Counteracting the Spread of HIV Infection in the Russian Federation for the Period Until 2020 and the Future] http://static.government.ru/media/files/ cbS7AH8vWirXO6xv7C2mySn1JeqDIvKA.pdf (in Russian)

Chapter 62

Clusterization of White Blood Cells on the Modified UPGMC Method Andrey V. Orekhov, Victor I. Shishkin, and Nikolay S. Lyudkevich

Abstract Modern methods of flow cytometry make it possible to characterize cell populations with unprecedented detail, but the traditional analysis of data using the “manual gating” method under these conditions is ineffective and unreliable. A large number of research papers have been published in recent years that describe specialized clustering algorithms for detecting and determining populations of white blood cells. However, problems related to the presence of noise and different data density remain relevant. The internal problem of cluster analysis associated with determining the preferred number of clusters, and the moment the process itself stops remains unresolved. It is proposed to use the modified UPGMC (Unweighted Pair Group Method with Centroid average) with the Markov moment of stopping the clustering process from eliminating noise and overcoming problems associated with different data density.

62.1 Introduction A flow cytometer allows measuring the optical properties of single biological cells in dispersed media. The forward scatter (FS) detector determines their size, and the side scatter (SS) detector allows judging the complexity of the internal structure of the cells [1]. In a cytometry study of blood, three subpopulations of leukocytes are distinguished: lymphocytes, monocytes, and granulocytes (neutrophils, eosinophils, basophils) [1, 2]. The differentiation of leukocytes into three groups due to the technical specifics of flow cytofluorometry and the morphological features of white blood cells. It is A. V. Orekhov (B) · V. I. Shishkin · N. S. Lyudkevich St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. I. Shishkin e-mail: [email protected] N. S. Lyudkevich e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_62

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Fig. 62.1 Lymphocytes, monocytes, and granulocytes standard distribution of the FS and SS axes

possible to perform an initial analysis of leukocyte populations on the Euclidean plane (in terms of size and complexity of the internal structure of cells) using the FS and SS scales. Figure 62.1 shows the standard distribution of different leukocyte populations in the FS and SS axes. However, at present, the isolation of leukocyte subpopulations is done manually using gating. Specialized software allows the operator to graphically select groups of cells on the console screen (select a rectangular region, select a polygonal region, select an elliptical region) [1]. Recent technological developments in flow cytometry allows to characterize the cell populations with unprecedented detail. Traditional data analysis using the “manual gating” method under these conditions is inefficient and unreliable [3, 4]. It led to the development of automated methods based on cluster analysis algorithms. In recent years, a large number of research papers have been published that describe specialized clustering algorithms for detecting and determining populations of white blood cells. Some of them have found practical application, for example, FlowSOM, X-shift, PhenoGraph, Rclusterpp, and flowMeans [5, 6]. However, problems related to the presence of noise and different data density during the clustering of leukocytes by flow cytometry remain relevant. The internal problem of cluster analysis associated with determining the preferred number of clusters, and the moment the process itself stops remains unresolved. In Lepsky’s paper [7] presented a comparative analysis of various methods of clustering leukocytes inside the FS and SS axes. The most popular clustering methods were investigated in numerical experiments on real dataset. The worst result was shown by the K-means method. The division of granulocytes into two clusters is erroneous in principle. Results with the EM algorithm are not better. Moreover, both methods assume a priori knowledge of the number of clusters. In general, the hierarchical single linkage algorithm allows satisfactory results, but no more. It is not possible to reliably separate lymphocytes from fine debris. The problem of

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completing the clustering process and determining the preferred number of clusters remains open. The best results were obtained using the DBSCAN method. However, in the presence of a large volume of noise or a significant difference in the distribution density of leukocytes, selecting the parameters necessary for the implementation of this algorithm causes great difficulties. In 1973, Sneath and Sokal first described centroid linkage clustering or UPGMC (Unweighted Pair Group Method with Centroid average) [8]. It is proposed to use a modified UPGMC method with the Markov moment of stopping the clustering process for the automatic gating of leukocyte.

62.2 Modified UPGMC Method When clustering numeric data, the standard Euclidean metric is usually used    n ρ(X , Y ) =  (x j − y j )2 , j=1

where X (x1 , x2 , . . . , xn ) and Y (y1 , y2 , . . . , yn ) vectors from n-dimensional Euclidean space En . An arbitrary clustering algorithm A of a certain set of vectors X ⊂ En is a map that assigns to any X i ∈ X a unique a natural number k, which is called its label and is the number of the corresponding cluster [9, 10]. Algorithm A divides the set of vectors X into clusters that are disjoint subsets of X . Consequently, the clustering process defines an equivalence relation on X , for which the clusters are equivalence classes. Usually, in this case, so-called “centroids” are chosen as independent representatives of equivalence classes. The centroids’ coordinates in Euclidean space are equal to the arithmetic mean of the corresponding coordinates of the vectors belonging to the cluster. If we identify vectors with points of unit mass, then centroids are the centers of mass clusters [9, 11]. The formally modified UPGMC method can be described as follow. The formula calculates the “distance” between centroids in the modification of the UPGMC method: l ) = ρ(  l ) − w · n k · n l , Xk, X ρ (  Xk, X

(62.1)

l are the centroids of the clusters k and l, ρ(  l ) is the Euclidean Xk, X where  X k and X distance between the centroids of these clusters, n k and n l are the number of elements in the corresponding clusters, w is the parameter called the coefficient attraction. This parameter used to fine-tune the modified UPGMC algorithm interactively. The “distance” between centroids in the modification of the UPGMC method given by the formula (62.1) is not a metric.

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If the set X contains m elements (vectors), then it is assumed that X is divided into m equivalence classes each containing one element: X 1 = x 1 , X 2 = x 2 , . . . , X m = x m . The clusters consisting of a single element coincide their centroids: X h =  Xh for ∀ h | 1 ≤ h ≤ m. Without loss of generality, we assume that at the beginning of the j-th iteration of the A j agglomerative algorithm UPGMC A the X is divided into j clusters. An arbitrary iteration A j of the algorithm A consists of the following operations. First, the diagonal matrix of distances between cluster centroids is calculated. Then the minimum element of this matrix is determined, which is denoted as F j . X h and cluster X l with centroid  Xl Two clusters, for example, X h with centroid  for which the distance turned out to be minimal, unite in one cluster X j . The centroid of  X j has coordinates equal to the arithmetic mean of the corresponding coordinates of vectors from the clusters X h and X l . Thus, after the completion of A j , the sample X is divided into j − 1 elements [9, 11].

62.3 Markov Moment of Stopping for Modified UPGMC Method If there is no rule to complete the clustering process, the sample set X will be combined into one cluster after the m − 1 iteration of the modified UPGMC algorithm. This is an absurd result. We will use the set of minimum distances: {F1 , F2 , . . . , Fm−1 } to derive the formal rule for completing the modified UPGMC clustering process in n-dimensional Euclidean space En . It is formed during the execution of the algorithm A. If F j < F j−1 then F j is assigned the value F j−1 . Therefore, the values of the elements of the set of minimum distances monotonically increase F1 ≤ F2 ≤ · · · ≤ Fm−1 . When the clusters are merged, there is a sharp jump in the numerical value of the minimum distance Fi . This jump coincides with the end of the clustering process [9, 11]. It is approximated by an incomplete quadratic parabola (no linear term) rather than a straight line. The moment of transition of the monotonic growth of a numerical sequence from linear to nonlinear (parabolic) can be determined using the quadratic form δ 2 (40 ), “approximation estimation test” [12, 13], δ 2 (40 ) =

1 (19y12 − 11y22 + 41y32 + 12y1 y2 − 64y1 y3 − 46y2 y3 ). 245

(62.2)

The “approximation-estimating test” is a formalization of the well-known heuristic of “elbow” method [14]. Can say that near the element yk the character of increasing the numerical sequence yn has changed from linear to parabolic if the following conditions are fulfilled. For nodes y0 , y1 , . . . , yk−1 the linear approximation is no worse than incomplete parabolic, that is, δ 2 ≤ 0 is fulfilled. And for a set of points, y1 , y2 , . . . , yk , shifted by one step of discreteness, the incomplete parabolic approxi-

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mation became more accurate than linear, that is, the inequality δ 2 > 0 is fulfilled [9, 11]. In addition to the set of minimum distances {F1 , F2 , . . . , Fm−1 }, the elements of which were defined in the formal description of the UPGMC algorithm, we consider the set of “primitive minimum distances”: {D1 , D2 , . . . , Dm/2 }. Elements of this set are values of a random variable, which are calculated as follows. The matrix of Euclidean distances of the elements of the sample X is constructed, the minimum element of this matrix is determined and its value is assigned D1 , and the corresponding elements are deleted from X . The procedure is repeated, the new minimum value of the distance matrix is assigned D2 , and corresponding elements are deleted from X , and so on until all elements from X are deleted. If X has an odd number of elements, then D m2 = D m2 −1 . If we consider the clustering of a finite set X from the Euclidean space En as a discrete random process ξ = ξ(t, ω), then the random event ω extract the set X from En . We do not care whether white blood cells are clustered or, for instance, texts. From the point of view of mathematical formalization, the same logical reasoning is valid [14, 15]. Let τ be the moment of the occurrence of some event in the random process ξ . If for ∀ t0 ∈ T we can definitely say whether the event τ occurred or not, provided that the values of ξt are known only in the past (to the left of t0 ), then τ is the Markov moment with respect to the random process ξ = {ξt , t ∈ T }. And if the onset of τ at a finite moment in time is a reliable event, then τ is the Markov moment of stopping the random process ξ = {ξt , t ∈ T }. In the Euclidean space En , one of the main characteristics of the modified UPGMC clustering process is the set of minimum distances. It is natural to consider its values as a random variable ξt , assuming that t is the iteration number of the clustering algorithm A. For any fixed random event ω0 , the corresponding trajectory ξt (ω0 ) = Ft is a monotonically increasing random sequence. Let us construct a statistical criterion for the completion of the clustering process as the Markov stopping time τ [16, 17]. We consider the binary problem of testing the statistical hypotheses H0 and H1 , where the null hypothesis H0 is that the random sequence ξt (ω0 ) grows linearly, and the alternative hypothesis H1 is that the random sequence ξt (ω0 ) increases nonlinearly (parabolic). It is necessary to construct the criterion as a strict mathematical rule that allows it to be accepted or rejected, as such a rule, we use the “approximation-estimating test” to test the statistical hypothesis [14, 15]. Then, by definition, the Markov moment of stopping the clustering process will be statistics: τ = min{t ∈ T | δt2 > 0}, where δt2 is given by the formula (62.2). That is, the Markov moment of stopping the clustering process is the minimum value t, at which the null hypothesis—H0 is rejected (the sequence of minimum distances increases linearly) and the alternative hypothesis is accepted—H1 (sequence minimum distances increases parabolic) [14, 15]. Consider the “sensitivity problem” of the “approximation-estimating test” δ 2 . Let us introduce the linear transformation yi = Fi + q · i, and we get a set {y1 , y2 , . . . , yk }, which we call the “trend set”, where q—“trend coefficient”. The

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clustering result changes qualitatively when the criterion is applied to the set {y1 , y2 , . . . , ym−1 }. The stopping criterion has the maximum sensitivity at q = 0, and in this case, clustering gives the largest number of clusters. The higher the value of q, the lower the criterion’s sensitivity for stopping the clustering process. With the maximum value of q, the process continued until all m vectors are combined into one cluster. The main task is to the interactive mode select the value of q to obtain an acceptable number of clusters [15]. In our case, four clusters of white blood cells—lymphocytes, monocytes, granulocytes, and fine debris.

62.4 Results of Computational Experiments Let us now consider some results of computational experiments on the automatic gating of leukocytes. For numerical experiments and the implementation of the modified UPGMC algorithm, we used program code written in the programming language Python 3.7, using the libraries NumPy, SciPy and using the PyCharm shell developed by JetBrains based on IntelliJ IDEA, and the free access environment Colaboratory for Jupiter Notebook from Google. First, flow cytometry data is cleared of noise. For the random variable D j , the standard deviation d is calculated as the square root of its variance. Then a lot of “conditionally isolated points” are formed. A point x ∈ X is assumed to be conditionally isolated if its neighborhood of radius r contains less than l other points from the set X . In computational experiments, r = d/n. Data is cleaned from noise by removing all “conditionally isolated points.” Figure 62.2 shows the data before clearing the noise. Figure 62.3 shows the result of the job of the modified UPGMC method after cleaning from noise. After cleaning the data from noise, white blood cells are clustered in two stages. At the first stage, a very dense cluster of lymphocytes is allocated, after which it is removed from X . At the second stage, the remaining elements are clustered again. Then lymphocytes return to the X sample, and all small clusters are removed from X . Clustering was done interactively. The following parameters were used as control parameters: q is a trend coefficient, w is an attraction coefficient, n is a coefficient determining the size of the radius of a neighborhood to highlight conditionally isolated points, and l is a number of points in a neighborhood. It was found empirically that n = 0.5; l = 20 are the optimal parameters for noise removal in most experimental datasets. At the first stage of clustering w = 0.0000001, q = 2. At the second stage of clustering, w = 0.0000005, q = 18. The difference with the results obtained with “manual gating” does not exceed 2%.

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Fig. 62.2 White blood cells with noise

Fig. 62.3 Result of the job of the modified UPGMC method after cleaning from noise

62.5 Conclusion The main problem of the proposed approach is the fact that the following parameters are chosen heuristically. The radius of the neighborhood and the number of points used to clean the data from noise. The attraction coefficient w, which is using in metric, and the trend coefficient of q that affects the sensitivity of the approximationestimated test. However, some regularities were discovered between these parameters during numerical experiments. It is planned to automate their choice in the future.

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References 1. Zurochka, A.V., Khaydukov, S.V., Kudryavtsev, I.V., Chereshnev, V.A.: Protochnaya tsitometriya v meditsine i biologii. [Flow cytometry in medicine and biology.] Ural Branch of the Russian Academy of Sciences Publ., Yekaterinburg (2014) (in Russian) 2. Gulati, G., Caro, J.: Blood Cells: Morphology and Clinical Relevance, 2nd edn. ASCP Press, Chicago (2019) 3. Daneau, G., et al.: CD4 results with a bias larger than hundred cells per microliter can have a significant impact on the clinical decision during treatment initiation of HIV patients. Cytom. Part B: Clin. Cytom. 92(6), 476–484 (2017) 4. Omana-Zapata, I., et al.: Accurate and reproducible enumeration of T-, B-, and NK lymphocytes using the BD FACSLyric 10-color system: A multisite clinical evaluation. PLoS One 14(1), e021120 (2019) 5. Weber, L.M., Robinson, M.D.: Comparison of clustering methods for high-dimensional singlecell flow and mass cytometry data. Cytom. Part A 89A(12), 1084–1096 (2016) 6. Zhang, C., Xiao, X., Li, X., Chen, Y.-J., Zhen, W., Chang, J., Zheng, C., Liu, Z.: White blood cell segmentation by color-space-based K-means clustering. Sensors 14(9), 16128–16147 (2014) 7. Lepsky, A.I.: Comparative analysis of leukocyte clustering algorithms according to FS and SS parameters in a cytofluorimetric blood test. Inf. Technol. 26(1), 56–61 (2020). (In Russian) 8. Sneath, P.H.A., Sokal, R.R.: Numerical Taxonomy: The Principles and Practices of Numerical Classification. Freeman, San-Francisco (1973) 9. Orekhov, A.V., Kharlamov, A.A., Bodrunova, S.S.: Network presentation of texts and clustering of messages. In: El Yacoubi, S., Bagnoli, F., Pacini, G. (eds.) Internet Science. INSCI 2019. Lecture Notes in Computer Science, vol. 11938. Springer, Cham (2019). https://doi.org/10. 1007/978-3-030-34770-3_18 10. Orekhov, A.V.: Markov moment for the agglomerative method of clustering in Euclidean space. Vestnik Sankt-Peterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya 15(1), 76–92 (2019) (in Russian). https://doi.org/10.21638/11702/spbu10.2019. 106 11. Orekhov, A.V.: Agglomerative method for texts clustering. In: Bodrunova, S., et al. (eds.) Internet Science. INSCI 2018. Lecture Notes in Computer Science, vol. 11551, pp. 19–32. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17705-8_2 12. Orekhov, A.V.: Criterion for estimation of stress-deformed state of SD-materials. In: AIP Conference Proceedings, vol. 1959, p. 070028 (2018). https://doi.org/10.1063/1.5034703 13. Orekhov, A.V.: Approximation-evaluation tests for a stress-strain state of deformable solids. Vestnik Sankt-Peterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya. 14(3), 230–242 (2018) (in Russian). https://doi.org/10.21638/11702/spbu10. 2018.304 14. Bodrunova, S.S., Orekhov, A.V., Blekanov, I.S., Lyudkevich, N.S., Tarasov, N.A.: Topic detection based on sentence embeddings and agglomerative clustering with Markov moment. Futur. Internet 12(9), 144 (2020). https://doi.org/10.3390/fi12090144 15. Kharlamov, A.A., Orekhov, A.V., Bodrunova, S.S., Lyudkevich, N.S.: Social network sentiment analysis and message clustering. In: El Yacoubi, S., Bagnoli, F., Pacini, G. (eds.) Internet Science. INSCI 2019. Lecture Notes in Computer Science, vol. 11938. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34770-3_2 16. Wald, A.: Sequential Analysis. Wiley, New York (1947) 17. Sirjaev, A.N.: Statistical Sequential Analysis: Optimal Stopping Rules. American Mathematical Society (1973)

Chapter 63

Elasticity’s Influence on Biomechanical Model of Corneoscleral Shell Under Vacuum Compression Ring Dmitry V. Franus

Abstract The research deals with the biomechanical model for the stress–strain state of the corneoscleral shell of the human eye under loading by a vacuum compression ring. A three-dimensional finite-element model of the contact problem of loading of the multilayer isotropic corneoscleral shell of variable thickness is presented. The influence of such parameters as sclera elasticity, corneal stroma elasticity, vacuum level, corneal thickness, and length of the longitudinal axis of the eyeball on the value of IOP is studied.

63.1 Introduction The reason for the elasticity of sclera to change is the age or various eye diseases or anomalies [1]. Therefore, a better understanding of its nature is important for better results, for example, in refractive surgery. This research deals with the first stage of operation for vision correction (LASIK type) when a vacuum compression ring is applied to create a corneal flap. The main reason for a person to have a vision correction is refraction anomalies. Most often, vision correction operations are performed to correct myopia (shortsightedness). Another reason for surgery is the diagnosis of hypermetropia (farsightedness). This is a feature of the refractive properties of the eye, in which the image of distant objects is focused behind the retina. The most common occasion for these two anomalies is the deformed ellipsoidal shape of the eyeball (instead of spherical). The third main reason for vision correction is astigmatism—non-spherical shape of the cornea. Intraocular pressure (IOP) is the principal source of mechanical stress in ocular tissues. As a result, the level of IOP is a fundamental indicator in the diagnosis of various eye diseases and anomalies. LASIK (Laser-Assisted In Situ Keratomileusis) is today one of the most effective methods of vision correction. The essence of the operation is to first cut out a corneal D. V. Franus (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_63

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flap sufficient for the operation of vision correction. Then the laser ablation is performed by narrowly directed pulses on the main layer of the cornea; this process is controlled by a computer in accordance with the required profile of the cornea [1]. A vacuum compression ring (vacuum ring) is used during the refractive surgery (LASIK) for vision correction for the purpose of better formation of the corneal flap. The vacuum ring allows increasing the intraocular pressure to values above 65 mmHg, by applying vacuum from 520 to 550 mmHg [2]. The increase of IOP to these values makes flap creation easier and more accurate layer-by-layer corneal incision during refractive surgery and allows improving the accuracy of vision correction. Therefore, the purpose of this study is to analyze the influence of corneoscleral elasticity moduli, thickness of corneal stroma, and length of the longitudinal axis of the eyeball on IOP level and stress–strain state under the loading of a vacuum compression ring.

63.2 Mathematical Model A finite element (FE) model for estimating changes in the stress–strain state of multilayer corneoscleral shell under the pressure of vacuum compression ring is performed. The FE model is geometrically nonlinear with linear isotropic materials. The corneoscleral shell is modeled with two main parts: 1. Spherical segment cornea of variable thickness is divided into four layers: a. Epithelium—thin anterior layer. Young’s modulus—E = 0.06 MPa. b. Bowman’s membrane—thin layer separating the epithelium from stroma (E = 0.6 MPa). c. Stroma of the cornea—the main layer of the cornea is modeled with variable central thickness from 0.4 to 0.5 mm; Young’s modulus varies from 0.29 MPa to 0.31 MPa. d. Descemet membrane—monocell posterior layer (E = 0.9 MPa). 2. The sclera is modeled by one spherical segment, which is in contact with the corneal shell with a radius of 11.6 mm at α + β = 62◦ and with the second ellipsoidal segment. The outer radius of the sclera is Rsc = 12 mm, and the thickness varies from h sc_r = 1 mm at the base of the cornea to h sc_inter = 0.8 mm (α = 50◦ ), and further to h sc_cent = 0.6 mm in the equatorial zone of the scleral shell and then increases back to h sc_b = 1 mm [3, 4]. Individual eyes vary in size and volume. In general, myopic eyes are larger than emmetropic eyes, and emmetropic eyes are larger than hyperopic ones. In the case of myopia ellipsoidal segment is set so that the length of the front–back axis varies from 24 up to 28 mm, and in the case of hyperopia from 24 mm down to 19 mm; in the model it is done with parameter asc . A vacuum compression ring has three main parameters, r1 (inner radius), r2 (outer radius), and h (height), which are shown in Fig. 63.1 (left). It is assumed that the

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Fig. 63.1 Parameters of vacuum compression ring (left); boundary load of vacuum compression ring (upper right); initial curvilinear coordinate system (bottom right)

corneal tissue and scleral shell are close to incompressible, so ν = 0.48. The vacuum compression ring is modeled as a rigid domain. The corneoscleral shell is initially loaded with normal IOP (15 mmHg), which is set in the direction perpendicular to the inner surface. Boundary load of vacuum compression ring is shown in Fig. 63.1 (upper right). As boundary condition, the displacements and rotations are zero in all directions for vacuum compression ring (rigid domain). Contact pair of vacuum compression ring and sclera is modeled as a roller. The roller node adds a roller constraint as the boundary condition; that is, the displacement is zero in the direction perpendicular (normal) to the boundary, but the boundary is free to move in the tangential direction. The curvilinear coordinate system is built to create a base vector system of normalized vector field on each step of solution, whereby corneal and scleral moduli of elasticity is always normal to a deformed corneoscleral shell. The initial curvilinear coordinate system is shown in Fig. 63.1 (bottom right).

63.3 Results Figure 63.2 shows von-Mises stress distribution on scleral shell under a load of vacuum compression ring in case of normal IOP (15 mmHg) for scleral elasticity

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Fig. 63.2 von-Mises stress on the posterior surface of corneoscleral shell for scleral elasticity modulus from 3.5 to 8 MPa while loading with a vacuum pressure of 500 mmHg

Fig. 63.3 Increase of IOP level above normal for scleral elasticity modulus from 3.5 to 8 MPa while loading with vacuum pressure (normal IOP—15 mmHg)

modulus—8 MPa, corneal stroma elasticity modulus—0.3 MPa, the length of the front–back axis—28 mm, and central cornea thickness—0.5 mm. Figure 63.3 shows growing change in IOP level (above normal IOP—15 mmHg) in the center of the cornea on the posterior surface while increasing vacuum pressure up to 500 mmHg for various scleral elasticity modulus from 3.5 to 8 MPa; corneal stroma elasticity modulus—0.3 MPa; the length of the front–back axis—24 mm; and central cornea thickness—0.5 mm. Almost 10% deviation of IOP level in the center of the cornea on posterior surface for range of corneal stroma elasticity modulus from 0.29 MPa to 0.31 MPa (scleral elasticity modulus from 3.5 to 8 MPa; the length of the front–back axis—24 mm, and central cornea thickness—0.5 mm) is presented in Fig. 63.4. Figure 63.5 shows von-Mises stress distribution on the posterior surface of corneoscleral shell under the load of vacuum compression ring in the case of normal IOP (15 mmHg) for scleral elasticity modulus—from 3.5 to 8 MPa, corneal stroma elasticity modulus—0.3 MPa, the length of the front–back axis—24 mm, and cen-

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Fig. 63.4 Deviation of IOP level for corneal stroma elasticity modulus E st from 0.29 to 0.31 MPa for a variety of scleral elasticity modulus E sc from 3.5 to 8 MPa while loading with vacuum pressure; the length of the front–back axis—24 mm; central cornea thickness—0.5 mm

Fig. 63.5 von-Mises stress on the posterior surface of corneoscleral shell for scleral elasticity modulus from 3.5 to 8 MPa while loading with vacuum pressure of 500 mmHg; the length of the front–back axis—24 mm; central cornea thickness—0.5 mm

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tral cornea thickness—0.5 mm. Arc length is measured beginning from point in the center of cornea on the posterior surface (r = 0; z = 12.5; see Fig. 63.1 (left)) until equator of eyeball (z = 0; r = 11.4; see Fig. 63.1 (left)). The first peak around 8 mm (of arc length) on the posterior surface corresponds to a point of contact with radii r1 of vacuum ring; the level of this peak shows reverse dependency from scleral elasticity modulus. The second peak around 11 mm corresponds to a point of contact with radii r2 of a vacuum ring. A significant collapse of stress level in the middle of these points on the posterior surface is opposite to maximum level of stress on the anterior surface of sclera under vacuum ring.

63.4 Conclusion • The higher the scleral elasticity modulus, the higher the IOP level increases. • Stress level on the posterior surface of corneoscleral shell doesn’t depend on elasticity modulus of corneal layers and central corneal thickness. • 3% deviation of corneal stroma elasticity modulus results in almost 10% deviation of IOP level while loading with vacuum pressure up to 500 mmHg. • In terms of total stress on the posterior surface of corneoscleral shell, the scleral elasticity modulus mainly affects only the area corresponding to the contact of vacuum ring with sclera for the radius r1 ; for all other areas the total stress level almost doesn’t depend on scleral elasticity modulus. Acknowledgements This work was supported in part by Russian Foundation for Basic Research (RFFI), project No. 18-01-00832-a.

References 1. Iomdina, E., Bauer, S., Kotliar, K.: Eye Biomechanics: Theoretical Aspects and Clinical Applications. Real Time, Moscow (2015) 2. Chen, C., Reed, J.F., Rice, D.C., Gee, W.D., Updike, P., Salathe, E.P.: Biomechanics of ocular pneumoplethysmography. J. Biomech. Eng. 115(3), 231–238 (1993) 3. Vit, V.: The Structure of the Human Visual System. Astroprint, Odessa (2003) 4. Zakharov, V.: Vitreoretinal Surgery. Moskva, Moscow (2003)

Part IX

Socio-economic Systems Control

Chapter 64

On Depth of Immersion in Forecasting Task Alexander V. Prasolov and Nikita G. Ivanov

Abstract The forecast of dynamic processes is based on mathematical models which usually describe determinate and stochastic characteristics of processes adequately. Parameter values of considered models are estimated by observation data over the processes in the past. It often occurs that there is data excess, which means that an additional number of data does not contribute to the precision of forecasting but makes it more expensive. This work is devoted to the estimation of forecasting depth at the fixed horizon. It was assumed that time series, of which forecast is considered, possess some informative features that allow establishing the balance between the horizon and the depth of the forecast. The algorithm of depth estimation has been offered and real non-stationary time series has been analyzed. This analysis demonstrated that there exists quasi-optimal depth for a fixed horizon of forecasting.

64.1 Introduction The important question emerging at building the model of time series is connected to the length of interval chosen to identify the best model parameters for the given class of functions. In the theory of forecasting with the help of time series models, we call in such a way the immersion depth into past data when the forecast is being done for the given horizon on the future. The optimization of depth here is reached by means of decreasing the scatter of predicted values. It definitely concerns only those tasks that have plenty of data. For example, the macroeconomic parameters of Great Britain can be obtained for hundreds of years: we mean parameters such as GDP (gross domestic product), money stock in the country, index of life and inflation, percentage on credits, etc. with the step of one year (or sometimes A. V. Prasolov (B) · N. G. Ivanov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] N. G. Ivanov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_64

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one quarter). When forecasting the weather, it is possible to collect information for decades of years with the step of several hours. However, for the working algorithm that is actually used in practice, a researcher is not likely to be willing to create and use such an inconvenient apparatus, when every time at forecasting it is necessary addressing to the whole database. Moreover, a large database requires constant attention and service, which leads to an increase in forecast cost. Thus, the aspiration to optimize the volume of the database on expenses in its exploitation is natural at creating the working algorithms of forecasting. Let us discuss this question in more detail. In this work we as usual understand time series as realization of stochastic process x(t) on the time interval t ∈ (0, N ). As a model we consider the additive relation x(t) = u(t) + v(t),

(64.1)

where u(t) and v(t) correspond to systematic and random components, at this E(v(t)) = 0; E(v(t), v(t1 )) = 0; i.e. random components have zero expected value and they are inter-independent. Time t below can be continuous as well as discrete with constant discretization step. Besides, a researcher is limited by a certain class of functions {u(t, B)} where t ∈ (0, N ) and B are vector parameters, giving specific representative from the class {u(t, B)}. The task of building the model reduces to the choice of such parameter B ∗ so that the corresponding function u(t, B ∗ ) was the best approximation of the model x(t) at the interval t ∈ (0, N ) in the sense of mean-square deviation. If we wanted to use all the data, then the definition B ∗ would be implemented by minimizing functional,  N

S(B) =

(x(t) − u(t, B))2 dt

0

for continuous t ∈ (0, N ), or S(B) =

N 

(x(t) − u(t, B))2 ,

t=1

when variable t is a discrete one. This is a typical task of regressive analysis [1–3]. This work is dedicated to building the algorithm to reduce the depth of forecasting, and in some sense the algorithm is optimal (which will be clear below in this work). However, in this case a required parameter B ∗ would depend on the whole volume of available information. We modify the task (in this statement it appeared in [4]). We consider the last part of information massive (the latest information) of length M. As a criterion of proximity, we select the mean from square deviations: in the continuous case  N 1 S(B, M) = (x(t) − u(t, B))2 dt. M N −M

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Or in the discrete case S(B, M) =

1 M

N 

(x(t) − u(t, B))2 .

(64.2)

t=N −M+1

If M = N , then to forecast we use the entire time series. If M = 1, then the forecast is built on the last point of information. Now by changing the M value, we can select the immersion depth in the past. Statement 64.1 For the given time series {x(t)}, t = 1, N , and the given class of approximation {u(t, B)}, the optimal solution of the task from the functional S(B, M) point of view is carried out by the solution of the equation system ⎧ N ∂u(t,B) ⎪ ⎨ N −M (x(t) − u(t, B)) ∂ B dt = 0, (64.3)

⎪ ⎩1 N M

N −M

(x(t) − u(t, B)) dt = [x(N − M) − u(N − M, B)] . 2

2

Indeed, the derivative of S(B, M) with respect to B gives the left part of the first equation. We formally express from it B = B(M) and substitute it in S(B, M) and then from the received expression we take derivative with respect to M: 1 ∂S =− 2 ∂B M



N N −M

1 [x(N − M) − u(N − M, B(M))]2 + M

 N ∂u ∂ B 1 dt. (x(t) − u(t, B(M))) −2 M N −M ∂B ∂M

[x(t) − u(t, B)]2 dt +

The last summand is equal to zero owing to the first equation from (64.3) and independency B from t. Setting the obtained derivative to zero and multiplying by M, we confirm the second equation in (64.3).  Note that the proven statement gives us the necessary condition of optimality. Besides, the dependence S(B, M) is substantially nonlinear with respect to M and so we should not expect the uniqueness of solution (64.3). The difference in analogue of system (64.3) is reached by simple discretization of corresponding function and by substitution of integrals by the sums: ⎧ N −1 ∂u(t,B) ⎪ ⎨ t=N −M [x(t) − u(t, B)] ∂ B = 0, ⎪ ⎩ 1 N M

t=N −M+1

[x(t) − u(t, B)]2 = [x(N − M) − u(N − M, B)]2 .

So, for example, for the function x(t) = t 2 á on the time interval t ∈ (0, 1) and the class of approximating functions u(t, B) = bt, we obtain after no complicated calculations on a computer that the optimal interval of identification according to the formulae (64.3) is equal to 0, 775, and the optimal value of parameter is equal to 0, 757. To the point, in the given example the optimal value appears to be the only one.

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In fact, we would like to obtain the best forecast, i.e. the approximation of the future rather than the analysis of the past. We transform the task so that the optimal parameter is selected from the past (otherwise we cannot use a model in real conditions), but the optimal interval would provide us with the best forecast in the future.

64.2 Task Formulation As mentioned previously, let N be the interval length of data about a dynamic process, h is the given horizon of the forecast, M is the required length of the optimal interval of model identification (or to be more exact, the length of a known set of data which are used to build the optimal vector of parameters B ∗ = B(M)). For each value of M the algorithm has been quite well studied. If the parameters of a class of functions enter u(t, B) linearly, then it is the task of linear regression and it can be solved by the least-square method. More complicated task settings require special analysis, which we will see below later. So far we will consider that there is quite an adequate algorithm of calculation B ∗ = B(M). Then the optimization on M means finding the best forecast in the horizon h, i.e. M ∗ = argmin

N +h 



2 x(t) − u(t, B ∗ ) .

(64.4)

t=N +1

Note that as the horizon h is constant, so we do not need to search for the minimum of mean square error; it is enough to follow the summary errors. The calculation M ∗ obligatory has the result because the set of possible M is finite. The serious difficulty is brought by the fact of randomness x(t), and as a result, the fact of randomness of obtained values B ∗ = B(M) and M ∗ . So the procedure of averaging all suspicious M ∗ that are delivered by realization (64.4) is inserted into the algorithm. The examples below will deliberately underline this selection.

64.2.1 The Choice of Functions Class This is the most disputable and subjective part of the task: different researchers can give preference to their familiar methods in specific cases of analysis. We point out the most typical peculiarities while selecting the class of functions. It is completely impossible for this class to include polynomials of high degrees on t, as forecasting with the help of polynomials is unstable in connection to observation errors, and outside the identification interval, a polynomial grows quickly on module [3]. However, a linear function of time reflects the tendency of monotonicity well, so its inclusion is usually justified. It is possible to approximate the rest of the data by means of the combination of harmonic oscillations. The exceptions are

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“wavelets”, “jumps”, “outbursts”, but for such tasks special methods are carried out. These methods are aimed at forecasting the appearance of such changes in data as something extraordinary. Our consideration concerns standard forecasts observed in economics, biology, and other fields of knowledge on comparatively small horizons. Remark 64.1 The method of expansion into Fourier’s series has a wide application. Actually, this method uses the set of trigonometric polynomials for modeling the observation series. This is the most universal and precise mathematical tool [5]. Its important characteristic is the requirement of periodicity outside the interval of known data. This leads to the fact that the point at which the forecast begins is not the last point of known observations but the first point according to the periodicity. Since in this work we offer the algorithm to estimate the optimal interval of identification on the formula (64.4), the above given arguments allow using the simplest system of functions which is easily programmed and analyzed: u(t, B) = b0 + b1 t + b2 sin(ω1 t) + b3 cos(ω1 t) + b4 sin(ω2 t) + b5 cos(ω2 t). (64.5) This is the combination of linear time function and two harmonic oscillations with unknown frequencies. All coefficients come into the formula linearly, so they are easily defined by the least-square method. Difficulties appear only at the optimal selection of frequencies ω1 and ω2 . Let us divide the selection into two stages: rough estimation of frequency and precise adjustment on squares of approximation errors. There exists a set of different ways of approximate assessment of frequencies: from sense analysis to computation procedures. If we know the periods of seasonality, then it is already simple to assess one of the frequencies. It is possible to carry out smoothing until the moment when periods are defined visually, and so on. Researchers do so when using Holt-Winters’ method. The below examples of real-time series will demonstrate how this works.

64.3 An Example of Intellectual Property Dynamics in the USA We have chosen time series of annual data about products of intellectual property in the USA from 1925 to 2018 [6]. The choice is based on the fact that these data were not often analyzed and it is difficult to distinguish seasonal or some other regular harmonics. Exteriorly a time series of volumes is like an exponent, so we turn into indexes, i.e. in annual relative changes of volumes in percent. Figure 64.1 shows data in percent on the analyzed parameter. Numerical values are not important for an algorithm illustration. Let us give the horizon of forecasting for 5 years and keep the last five indexes to search the optimal immersion depth. The class of functions is represented in the form (64.5). We select the frequencies of harmonics in two stages: by appearance and then more exact by fitting. As a result, we obtain ω1 = 0, 615, ω2 = 0, 02. Coefficients bk come into formula (64.5) linearly and are easily calculated by using MS Excel for each value M of immersion depth.

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Fig. 64.1 Indexes of intellectual property volumes in the USA from 1925 to 2018

Fig. 64.2 Forecast precision (sum of errors squares) on five control data in dependence from the depth of forecast M

Below there is a graph (Fig. 64.2) showing dependency of modeling preciseness from M, notably the sum of squares of deviations on five control data. The numerical analysis has shown that at M, having values in the interval [41, 50], the sum of deviations squares is approximately equal to 6, 6 and a mean deviation is about 1. It is not possible to get precise values of M ∗ , as measurements contain a random component, and so we think that the optimal value is equal to M ∗ = 45. Let us note that the absence of clearly expressed least value is connected not only with a random character of time series but also with difficulties of frequency selection for each value of M. Figure 64.3 shows the result of calculations. We did not set the goal to analyze concrete data, so those who are willing to get the forecast on the given horizon with a precision assessment can use the algorithm. We

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Fig. 64.3 The comparison of real data on indexes in percent and on modeling results on the segment of 45 years. The last five years show the comparison in control points

Fig. 64.4 We have shown standard errors of forecasting as a function of depth M

have carried out computational operations at other horizons of forecasting: h = 10, h = 15 (Fig. 64.4). At this, for h = 10 the optimal depth is 42 years and the average error is 3,5%, and for h = 15 the optimal depth is 41 years with the average error 4,3%. All the results have a random character and, nevertheless, each time series has its own relation between the horizon and the depth. For some unknown reasons

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the given series identifies the “optimal” depth in the interval [41, 50] for different horizons (5, 10, and 15 years). We have conducted a check with two depths: 42 and 60 years. The check showed a substantial difference in the average errors: in the first case for the horizon h = 5, an average error is 3%, while in the second case it is 5%.

64.4 Conclusion The offered algorithm to estimate the identification interval of time series on the fixed horizon can be used for arbitrary series with known frequencies of harmonics as well as with selected ones in an empirical way. It is obvious that the precision of such selection substantially depends on the selection of class of functions. Different technique of observations and the nature of stochastic processes can detect further recommendations in the use of this method. However, the availability (possibly on the infinity) of a stationary point, about which it is said in the statement, encourages optimism. Acknowledgements The reported study was funded by RFBR, project number 20-31-90063.

References 1. Eliseeva, I.: Econometrika: Textbook/Under the Editorship of I. Eliseeva. Prospect, Moscow (2009) 2. Magnus, Ya.R., Katyshev, P.K., Peresetskii, A.A.: Econometrika: Nachal’nyi kurs. Textbook, 8th edn. Delo, Moscow (2007) 3. Prasolov, A.V.: Mathematical Methods of Economic Dynamics: Textbook, 2nd edn. Lan publisher (2015) 4. Prasolov, A.V., Wei, K.C.: On forecast of exchange rate of a foreign currency. In: Proceedings of the 2000 IEEE International Conference on Control Applications, pp. 14–19. Anchorage, USA (2000) 5. Ivanov, N.G., Prasolov, A.V.: Analysis of various methods of the trend approximation of the time series. In: Smirnov, N.V. (ed.) The XLVI Annual International Conference on Control Processes and Stability (CPS’15), pp. 623–628. Publishing House Fedorova G.V., St. Petersburg (2015) 6. Bureau of Economic Analysis. Table 1.1. Current-Cost Net Stock of Fixed Assets and Consumer Durable Goods. Intellectual property products. http://www.bea.gov

Chapter 65

Supply Chain Model with Random Demand Sergei A. Kalin, Elena A. Lezhnina, and Tatyana V. Vlasova

Abstract The ability to timely and efficiently satisfy customer requests is one of the most important competitive advantages of any organization. A valuable and effective tool to achieve this is the competent modeling and optimization of supply chains. Modeling is complicated by the often encountered combination of external and internal uncertainties, which significantly affects the planning of the optimal allocation of resources. In this paper, we solve the problem of constructing an optimization model of a supply chain with random demand, close to real life. A supply chain model with external and internal uncertainty was built and checked for correctness. In the paper, we model all stages of production taking into account the randomness of demand and possible disruptions at every stage.

65.1 Introduction The lack of adequate control over the reliability of the supply chain leads to huge losses of both monetary and material resources. The authors of many scientific publications, recognizing the importance of the considered problems, offer various approaches to solve them. In [8], the modern concept of logistics, its tasks, and functions are presented. Researchers divide logistic models into several categories. Classifications of logistic models are presented in [11]. Numerous works related to logistics are dedicated to the modeling and optimization of supply chains [2]. For example, in [9], the supply chain is considered from the perspective of an object-oriented system approach. The article developed the classification of supply chains, presented a variant of the network model in the supply chain model system, and performed its comparative analysis with the traditional network model. In [10], the influence of external and internal uncertainty on the functioning of the supply S. A. Kalin · E. A. Lezhnina (B) · T. V. Vlasova St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] T. V. Vlasova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_65

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chain is investigated. The development of modeling a multi-level supply chain is presented in [13, 16]. The authors use a combination of the Wagner–Whitein inventory management model and the classic transportation problem. Modeling the strategic development of supply chains, based on the methods of ontological engineering, integrated system-dynamic and simulation modeling, is considered in [12]. In [18], problems of ensuring the sustainability of supply chains are considered. The reasons for their instability in the market environment and the specifics of ensuring reliability are studied. A comparative analysis of modeling methods is carried out. In [20], the maintenance of the reliability of supply chains based on a probabilistic approach is investigated. In [7], a set of indicators and methods is proposed for assessing the reliability of the supply chain. This component is one of the key elements in the development of circuit models, therefore, this problem is given special attention in science. Article [14] is devoted to the criteria for the correct choice of a method for forecasting market demand. The article discusses the principles on which methods should be based and conducts a comparative analysis of some of them. The concept of forecasting market demand in a competitive environment is presented in [15]. Methods for forecasting market demand are described in detail in [5, 6]. A systematic presentation of the methods for forecasting sales volumes that are most often used in practice is presented in [6]. In [5], the mathematical-statistical prediction apparatus is described. The authors explain in detail the methods of forecasting and simulation. The main inventory management models are presented in [19]. In the work, the author raises the issue of inventory structuring. Models based on the use of classical models of inventory control theory are considered in monographs [3, 4]. The problem of optimizing an inventory management system in the format of a decision-making model under uncertainty is considered in [1]. It is believed that the unit price and annual consumption are unknown, and only random loss of profit is taken into account. In the modern world, one of the most important competitive advantages of any organization is the ability to timely and efficiently satisfy customer requests. There are several ways to maintain timely customer satisfaction and increase profits, one of which is to reduce and optimize costs in production and transportation. A valuable and effective tool to achieve this is the competent modeling and optimization of supply chains.

65.2 Problem Statement We will consider a company that has a manufacturer with a warehouse and sailing points of products. Suppliers Let I be the set of companies producing components that are supplied to the manufacturer. The i-th supplier company produces L i types of components from Ri types of resources. Let xli be the number of components of the l-th type, l = 1, . . . , L i planned for release by the i-th supplier; xli be the number of components of the l-th type, which must be produced at the company i in order to fulfill the production plan.

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A component unit of the l-th type at enterprise i is manufactured in time tli . Total Li  time for the production of all components at the enterprise i equals Ti = tli xli . l=1

Delivery of Components Delivery can be carried out either directly or through J intermediate transshipment points P1 , P2 , . . . , PJ , which have their own warehouses with limited capacity. The storage capacities of the transshipment points are limited. j Let xli be the number of components of the l-th type of company i, which is delivered through an intermediate point P j ; xliS be the number of components of the l-th type, planned for delivery directly to the destination. The minimum time for the manufacture of all components and their transportation to the manufacturer’s company, provided that the order arrived at all manufacturers at the same time is T1 = maxi (Ti + TiCh ), where TiCh is the time for which transportation is carried out selected by the company method. Production The company manufactures K -name products. In order to produce a unit of the final product of the k-th type, it is necessary alik (l = 1, L i , k = 1, K ) components of the l-type. Let yk be the planned quantity of finished products of the k-th type, and yk be the quantity of products of the k-th type that must be produced at the manufacturer. Then the number of components of the l-th type produced at the i-th manufacturer of components and delivered for the production of goods is equal K yk alik . The unit of production of the k-th type is produced in the time to xli = k=1 K tk . the Total time for the production of all products is equal to T  = k=1 tk yk . Storage of Components Before use, components are stored in the manufacturer’s warehouse. In a situation where it is not possible to store all components at the manufacturer’s warehouse, third-party premises are rented. Costs for storing components in third-party warehouses are equal to Rxp1 =

G  I  L 

clig xlg ,

g=1 i=1 l=1

where clig is the storage cost of a product unit, xlg is the number of storage units. Product Delivery The manufactured products are delivered to M points of sale D1 , D2 , . . . , D M . Delivery can be carried out either directly or through N transn be the shipment points P1 , P2 , . . . , PN , which have their own warehouses. Let ykm quantity of finished products of the k-th type, which is planned for transshipment at Pn during transportation to the final destination Dm . The quantity of products of the S-th type for delivery directly from to the desS . The time of transportation of products from the tination Dm is denoted by ykm manufacturer directly to the destination Dm is indicated by TmS . The transportation time of the product to destination m through the point n, including the simple one at the point is equal to Tmn = tn1 + tn + tnm , where tn1 is the transportation time to the n-th transshipment point, tn is the downtime at the transshipment point, tnm is the time of transportation from the transshipment point to destination Dm . Product Storage Let ckmh be the cost of product storage of the k-th kind in the warehouse h. Then the cost of storing ykmh units of product in third-party warehouses

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is equal to Rxp2 =

K H  

ckmh ykmh .

h=1 k=1 (1) Transportation Cost Let ci1 be the cost of delivery of a consignment of components from the i-th supplier to the manufacturer; ci(2) j1 —the cost of transporting a consignment of components from the i-th supplier company, including its transshipment and storage at the intermediate point P j . The size of one batch of the order from enterprise i will be denoted by qcomp,i , whereas transport costs for the transportation of components to the manufacturer will be as follows:

Rtp1 =

I 

(1) ci1

i=1

L i l=1

xliS

qcomp,i

I 

+

ci(2) j1

L i

i=1

l=1

j

xli

qcomp,i

.

(4) Let c1m be the cost of transportation of products of the k-th type from the warehouse (5) be the cost of transporting a consignment of of the company to destination Dm ; c1nm  products of the k-th type, including its transshipment to Pn . The size of one batch of the order we denote by qprod,m . Then the transportation costs for the transportation of products to points Dm will take the form

Rtp2 =

M 

(4) c1m

m=1

K k=1

S ykm

qprod,m

+

M  m=1

(5) c1nm

K k=1

n ykm

qcomp,i

.

General Expenses Let cli(0) be the costs of purchasing a unit of a component of the l-th type from enterprise i; ck(3) be the costs of producing a unit of finished goods of the k-th type. Then the total costs of the company associated with the production, storage of components and finished products, their transshipment and transportation from the manufacturer to the destination can be represented as follows: Rc =

Li I  

cli(0)

i=1 li =1

K  k=1

yk alik Rtp1

+

K 

ck(3) yk + Rtp2 + Rxp1 + Rxp2 .

k=1

Demand at Points of Sale We assume that the demand for the k-th type of product at the point of consumption with the number m in the considered time period t is a random variable dkm with a normal distribution law with a mathematical expectation of M Okm and a standard deviation of σkm , all random variables dkm are continuous and independent of each other. Denote by  vkm =

yk1m , if delivery is direct; yk1nm , if delivery is via transshipment point.

65 Supply Chain Model with Random Demand

Then R 0 =

K  M 

587

pkm max(0, dkm − vkm ) shows the total losses of the enterprise

k=1 m=1

due to loss of potential profit. Here, pkm is the selling price per unit of output k-th species. For vkm > dkm , there is a need to store excess production and then the loss is M K   R1 = sk max(0, vkm − dkm ), where sk is the cost for storage or disposal of a k=1

m=1

unit of production. The Problem Our goal is to increase the company’s profits by managing the supply chain, that is, to maximize the function: Pr =

K  M 

pkm vkm − R(qcomp, i , qprod, m ),

k=1 m=1

where R(qcomp, i , qprod, m ) is the total costs of the company, calculated as follows: R(qcomp, i , qprod, m ) = Rc + R 0 + R 1 , while one of the elements R 0 or R 1 equals zero. Thus, it is necessary to find a solution to the problem: min R(qcomp, i , qprod, m ). In other words, it is necessary to find the optimal transportation plan, size, and frequency of orders for components and manufactured products of each type in order to minimize costs in the chain.

65.3 The Optimal Order Size The solution of this problem is to determine the optimal order size for components from each factory and to determine the optimal order size for transportation from the manufacturer of the final product to the points of sale and the time of their order. We assume that the projected demand obeys the normal distribution law and M Okm is the mathematical expectation of goods that consumers will buy at each point of sale for a certain time period. We divide the supply chain into two stages: (1) an order, transportation of components to the manufacturer of the final product and storage; (2) manufacturing of the final product at the factory, transportation of products to sale points and storage. Since a continuous production process is being modeled, it is necessary to take into account the probability of a disruption in the supply chain. For this, we use the Snyder model [17]. We shall say that the interval in which the supplier number i is functioning normally, is a wet interval, and the interval in which its functioning is interrupted is a dry interval. The durations of wet and dry periods are distributed exponentially with parameters λi a (disruption rate) and μi a (recovery rate). The duration of the wet interval of a manufacturing company is determined by an exponential distribution

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with the parameter γ , and the dry interval with the parameter η. In [17], Snyder obtained the optimal order size for the manufacturer:  (β D f )2 + 2 f μ(K Dμ + D 2 pβ) − β D f , Q= fμ where β = γ γ+η ; D is the demand for the product; f is the storage cost per unit of goods; p is price per a unit of goods. Then the optimal order size for components of type l from enterprise i will be found as follows:  ⎛    2  (1) S (αi xliS clig )2 + 2clig μi ⎝ci1 xli μi + xliS ∗ qcomp, l, i =

clig μi

M 

⎞ pkm

m=1

Malik

αi ⎠ −

αi xliS clig . clig μi

If the components are delivered through transshipment points, then the order size will be calculated as follows:  ⎛ ⎞  M   2  pkm  j (2) j m=1 (αi xlij clig )2 + 2clig μi ⎝ci1 xli μi + xli αi ⎠ Malik j αi xli clig ∗ qcomp, = − , l, i clig μi clig μi where αi =

λi λi +μi

.

Analysis of Parameters for Sensitivity The main result of this paper is not only the construction of an adequate model but also the study of the model for sensitivity to changes in parameters. The Study for the Sensitivity of the Model on the Parameter λi (Disruption Rate) for the Active Period of the Manufacturer I of the Final Product Figure 65.1 shows that even small changes of the parameter λi in the interval [0.01; 0.3] lead to significant changes, namely, to increase the optimal order size of the final product. At the same time, for λi > 0.5, changes in the parameter of the active period slightly affect the order size. Influence on the Optimal Order Size of the Parameter η (Recovery Rate) Figure 65.2 shows that even small changes of the parameter η in the interval [0.01; 0.3] lead to significant changes, namely, a decrease in the optimal order size. At the same time, with an increase in the inactive period parameter, its influence on the optimal order size decreases. Sensitivity of the Optimal Order Size to Changes in the Cost of Storage of Products in a Third-Party Warehouse Figure 65.3 shows that even small changes of the storage cost in the interval [10; 20] lead to significant changes, namely, a decrease in the optimal order size. At the same time, with a further increase in the cost of storage of products, the effect on the optimal order size decreases. This is due to the fact

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Fig. 65.1 The sensitivity of the model on the parameter λi (disruption rate)

Fig. 65.2 Influence on the optimal order size of the parameter η (recovery rate)

Fig. 65.3 The dependence of the optimal order size on the cost of storage

that the batch size reaches a point that ensures the satisfaction of consumer demand in a short period of time and allows one to spend a minimum number of resources on storing goods due to the small amount of products that remains in storage at a third-party warehouse.

65.4 Conclusion In this paper, a mathematical model of the supply chain has been developed. This model can be both the basis for research and support for the optimization of the company’s logistics processes. Companies that understand the benefits of the optimization process should consider the uncertainty associated with random demand or supplier termination. As a result, organizations can correctly calculate size orders and avoid significant additional storage costs or losses due to insufficient quantities of goods.

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References 1. Bochkarev, A.A.: Planirovaniye i Modelirovaniye Tsepi Postavok. Publ. House Alfa-Press, Moscow (2008) (in Russian) 2. Bondarenko, L.A., Zubov, A.V., Orlov, V.B., Ugegov, N.S., Kolyada, L.G.: Use of ERP system to manage the economy of agricultural complex. Int. J. Innov. Technol. Explor. Eng. 8(12), 3585–3590 (2019) 3. Bramel, J.: The Logic of Logistics: Theory, Algorithms, and Applications for Logistics Management. Springer, Berlin (1997) 4. Brandimarte, P.: Introduction to Distribution Logistics. Wiley, New York (2007) 5. Bure, V.M., Parilina, E.M., Svirkin, M.V.: Matematicheskaya Statistika. Solo, St.Petersburg (2007) (in Russian) 6. Bushueva, L.I.: Metody prognozirovaniya ob’yema prodazh. Marketing v Rossii i za Rubezhom 1(27), 15–30 (2002) (in Russian) 7. Churilov, R.L.: Metody Otsenki i Povysheniya Nadezhnosti Tsepey Postavok. Dissertation (2012) (in Russian) 8. Gadzhinsky, A.M.: Logistics, 15th edn. Publi. and Trad. Corp. Dashkov and K0, Moscow (2008) (in Russian) 9. Kovalev, M.N.: Modelirovaniye tsepey postavok v promyshlennosti. Vestnik Gomel’skogo Gos. Tekhn. Univ. im. P. O. Sukhogo 4(56), 117–124 (2014) (in Russian) 10. Kuruji, Yu.V.: Razrabotka modeli optimizatsii plana vypuska i dostavki produktsii s uchetom faktorov neopredelennosti. Vostochno-Yevropeyskiy Zhurnal Peredovykh Tekhnologiy 4(3), 12–15 (2015) (in Russian) 11. Lukinsky, B.S.: Modeli i Metody Teorii Logistiki, 2nd edn. Piter, SPb (2007) (in Russian) 12. Lychkina, N.N.: Strategicheskoye razvitiye i dinamicheskiye modeli tsepey postavok: poisk effektivnykh model’nykh konstruktsiy. In: Sergeev, V.I. (eds.) Innovatsionnyye Tekhnologii v Logistike i Upravlenii Tsepyami Postavok, pp. 133–144. SCM Consulting, Moscow (2015) (in Russian) 13. Morozova, I.V.: Dynamic optimization model for planning of integrated logistical system functioning. In: Morozova, I.V., Postan, M.Ya., Dashkovskiy, S.N. (eds.) Proceedings of 3d International Conference Dynamics in Logistics, LDIC2012, pp. 291–300. Springer, Berlin (2013) 14. Pilinkiene, V.: Selection of market demand forecast methods: criteria and application. Eng. Econ. 3(58), 19–25 (2008) 15. Pilinkiene, V.: Market demand forecasting models and their elements in the context of competitive market. Eng. Econ. 3(60), 24–31 (2008) 16. Postan, M.Ya.: Dynamic model for optimization of production and finished products delivery plans in supply chain. Logistyka 4, 2345–2352 (2014) 17. Snyder, L.: A tight approximation for a continuous review inventory model with supplier disruptions. Working Paper, P.C. Rossin College of Engineering and Applied Sciences, Lehigh University, Bethlehem (2011) 18. Tkach, V.V.: Problemy modelirovaniya tsepey postavok. Vestnik Yuzhno-Ural’skogo Gosudarstvennogo Universiteta. Ser.: Ekonomika i Menedzhment 39(215), 106–110 (2010) (in Russian) 19. Tyukhtina, A.A.: Modeli Upravleniya Zapasamie. Nizhegorodskiy Gosuniversitet, Nizhniy Novgorod (2017) (in Russian) 20. Vokhmyanina, A.V.: Metodologicheskiye aspekty obespecheniya nadezhnosti logisticheskikh tsepey postavok. Transport Rossiyskoy Federatsii 5(48), 55–59 (2013) (in Russian)

Chapter 66

Cyber-Physical System Adaptation in One Control Problem for Supply Chain Inna Trofimova, Boris Sokolov, and Dmitry Nazarov

Abstract This paper discusses the supply chain optimization problem in the event of happening of unexpected circumstances and the relevant data about the actual system state (delivery status, preserving temperature, actual stock, availability of transport facilities, etc.) could be timely obtained. To optimize the receiving process of current information, the scheduling problem of retrieving and handling the information is considered. To describe supply chains operation and optimize it, we use the dynamic systems with control and design optimal program and position controls. The advance of this paper is that cyber-physical system has been applied in the SC optimization problem when it is essential to do real-time reaction when unexpected circumstances happen for improving the timeliness of the relevant data about the actual supply chain state. We combine the considered models and apply the classical control theory methods to solve this supply chain management problem.

66.1 Introduction Actual supply chains (SC) are massive systems with geographically distributed companies and a wide variety of products. Responsiveness SC is highly significant for the companies because they tend to seek the competitive advantage, to rise the high delivery level, to keep of delivery dates, to provide land-wise work, to satisfy I. Trofimova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] B. Sokolov · D. Nazarov St. Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences, 14-th Linia VI, 39, St. Petersburg 1999178, Russia e-mail: [email protected] B. Sokolov Petersburg State University of Aerospace Instrumentation, Bolshaya Morskaya Str., 67A, St. Petersburg 190000, Russia Petersburg Marine State University, Lotsmanskaya,3, St. Petersburg 190121, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_66

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international standards, and to have an opportunity to do real-time reaction when unexpected circumstances happen. Thus the optimization problems of operating SC are relevant. Recently, a great number of mathematical models were introduced and different applications of control theory methods were used for solving these problems [7, 8, 15, 16]. In this paper to describe SC dynamic behavior and to optimize it, we consider it as the dynamic control systems. Following works [5, 9] the scheduling SC model is considered and will be complemented in this paper. For quality improvement of the relevant data about the actual SC state and timeliness of the data entry, we offer to consider the dynamic model of retrieving and handling information with the model [5, 19] together. Here we define the cyber-physical system (CPS) after the manner in which [10, 20, 22] the automatic control system is used to combine various objects through multiple channel system and measuring the tools with embedded applications with a variety of multilayered structure types. In broad terms, it is the cooperation of computers and physical systems [11]. Thus different kinds of CPS could be employed in digital and computer-integrated production and the digital economy. Lots of examples of CPS applications are in practice, including management information systems, manufacturing, computer-assisted manufacturing, medical engineering, intelligent house, and traffic engineering. Using these applications the self-management traffic technology could be introduced, the machine usage at a manufacturing site could be scheduled, the supply chain dynamic behavior could be optimized, etc. It is provided to discuss the SC optimization problem in the event of unexpected circumstances happen [4, 18] and the relevant data about the actual system state (delivery status, preserving temperature, actual stock, availability of transport facilities, etc.) could be timely obtained. In this problem, we suggest applying CPS for improving the timeliness of the data entry. Thus we propose to consider this problem with the scheduling problem of retrieving and handling information together. The advance of this paper is that CPS has been applied in the SC optimization problem when it is essential to do real-time reaction when unexpected circumstances happen for improving the timeliness of the relevant data about the actual SC state. The classical control theory methods have been transferred and modified and applied to design the optimal program and position control in this SC management problem.

66.2 Model of Supply Chain Behavior We consider the model of SC to describe the dynamic behavior of SC as it was done in [5, 9]. The SC is described using differential equations and is represented as a complex equipment site in the parlance of a dynamic explication of the job realization. The job realization is marked by results value (score of supply, delivery time, etc.) and resources consumptions. In line with [1] we propose to use the scheduling procedure, which comprised of two parts: to put jobs to deliverer and to coordinate the allocated orders subject to the limitation on the production and transport costs. It comprised

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two models: the dynamic model of job control and the dynamic model of flow control [5] and could be introduced as follows:   S = s˙ = f (s, v, t), g1 (s(t0 )) ≤ 0, g2 (s(T )) ≤ 0, l (1) (s, v) ≤ 0, l (2) (s, v) = 0. (66.1) Here the vector s characterizes the SC state; it includes the job state variables that indicate the relations to jobs (orders), the total period of operation of the j-deliverer ( j = 1, n), and the stream state variables that indicates the relation of the activity progress to flows. The corresponding vector of controls is v, g1 , g2 describe start and final states. Restrictions on control actions and SC states we denote with l (1) , l (2) (control actions are constrained by maximal potential intensities and capacities of the resources, maximal potential channels intensities and capacities of the channel delivering the product flows to the customers, job progression is constrained by precedence relationships (only a single order can be processed at any time by the manufacturer). Estimation of the quality of the control processes could be carried out using a number of objective functions [9, 14]. Let us introduce a vector indicator I S to characterize the accuracy of the end conditions’ accomplishment, i.e. the service level, to refer to the estimation of a job’s execution time with regard to the planned supply terms and reflects the delivery reliability, i.e. accomplishing the delivery to the fixed due dates, and to estimate the equal resource utilization [5]. Indicator I S = I (s, v, t, α1 , . . . , αr ), where α1 , . . . , αr are parameters of the model S.

66.3 A Dynamic Model of Retrieving and Handling Information Together with the model S let us consider a dynamic model of retrieving and handling information Sg . It is used to optimize the process of receiving the relevant data about the actual SC state [13, 19, 21]. (g)

s˙i

(g)

= Hi (t)si ,

Z˙ i = −Z i Hi − HiT Z i −

m   j=1 γ ∈i

(g)

T y (i) j = d j (t)si

(e) viγ j

d j d Tj σ j2

+ ξ (e) j ,

(e) (e) (o) , i = j 0 ≤ viγ j ≤ cγ j viγ j .

Here, the vector s (g) describes the state of the object and the elements of the matrix H specify the dynamics of variables change describing the object state. Vector ξ (e) describes uncorrelated measurement errors of object parameters. Assume measurement errors follow the normal law of errors with zero mathematical mean value and variance σ 2 , v (e) is the control, which characterizes the intensity of remote measurement parameters y, Z is the inverse matrix of the correlation matrix K (t) of

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measurement of the object parameters errors, d marks the technological characteristics of the tool, which takes the measurements, i is the set of interaction operations with the object. The indicator for the model Sg denotes I Sg : m  m  

T

I Sg =

(e) viγ j (τ )dτ , j  = i.

i=1 j=1 γ ∈i t 0

We could combine indicators in one vector: It (s, v) = (I S , I Sg ), or one scalar value Its using convolution methods of performance indicators system [12] (for example, in part 1 of this book).

66.4 Problem Statement At first, let us design a schedule for considered SC. To accomplish this we find the optimal program control for the model S as it was done in [2, 5, 9]. The optimal program control (OPC) problem in this situation could be posed thuswise: to construct the admissible control v(t) at the time interval t ∈ (t0 , T ], which should meet the restrictions on control actions and SC states l (1) , l (2) and follows the dynamic system in (66.1) from the start condition to the targeted terminal state. As there is a need to select one control from the number of admissible controls, it could be done using maximization (or minimization) Its . For the model S and SC, the program control and the schedule are the same. To receive the relevant data about the actual SC state, we use CPS. To optimize this process the scheduling problem of retrieving and handling information is considered and solved. The optimal moments tσ for receiving the processing data are obtained from the OPC problem for the model Sg . In solving the considered problem for S and Sg with Its , we design control vector v pr (t), and respective s pr (t). Later we are concerned with model S and Sg under disturbances and allow for external actions in the right-hand side of equation in (66.1): s˙ = f (s, v, t, ζ ).

(66.2)

Assume perturbation ζ (t) is sectionally continuous and a restricted function; it describes external actions, but in an explicit form ζ (t) is unknown. Assume the relevant data about the actual SC state sσ are obtained at the optimal moments of time tσ from the scheduling problem of retrieving and handling information. We intend that the company is desirous of minimizing incipient deviations of system behavior under external actions attempting to reduce its additional expenses. SC is able to realize the performance of production and transportation execution and reject

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the disturbances. In this context, during the system operation the SC optimization problem is to construct the admissible control  v(t) (on the basis of v pr (t)), at the time interval [t0 , T ) in the process of the system activity (66.2) on it, satisfying constraints and boundary conditions in (66.1), and minimizing performance parameter T  s(t) − s pr (t)  dt → min. I =  t0

66.5 Algorithm For the two-stage procedure of real-time control construction for the SC optimization problem, we formulate the algorithm. On the first base the SC scheduling problem, and the scheduling problem of retrieving and handling information is solved by control theory methods [5, 9]. The OPC v pr (t), corresponding vector s pr (t), and optimal moments of time tσ ∈ [t0 , T ) were obtained: 1. Using the original data, the technical performance, resources, and capacity of deliverers and customers, the possible interaction time is obtained. The estimation of attainable set is found [6]. 2. The OPC problem for (66.1) could be tackled if the terminal conditions are specified; in other case to increase the resource stock is required. 3. If the OPC problem for (66.1) and the scheduling problem of retrieving and handling information is solved, v pr (t), s pr (t), and optimal moments of time tσ ∈ [t0 , T ) are obtained. v(t) = v pr (t) and  s(t) = s pr (t), this is admissible control 4. If τ = t0 we assume  and it is accorded with constraints of the (66.1), (66.2). The position optimization method is appropriate for the considered SC optimization problem in the following step. This method is found in the reducing OPC problem procedure to linear programming subproblems. To deal with solving these subproblems, the linear programming (adaptive) method [3] could be applied. The position optimization method was developed just as for linear, so for nonlinear systems, and there are a lot of its applications for various kinds of systems [3, 7, 17, 21]. Let us set Tτ = {τ = tσ | tσ ∈ [t0 , T )}. In the class of step functions, we’ll define v(t) =  v(tσ ) at t ∈ [tσ , tσ +1 ), for all positional control v(t), t ∈ [t0 , T ) as function  tσ ∈ Tτ , tσ +1 ∈ Tτ . 1. The moments tσ were obtained as the solution of the scheduling problem of retrieving and handling information on the first base. At these moments tσ we obtain the relevant data about the actual system state sσ . The number of OPC subprobp , and lems for (66.1), (66.2) we consider consistently with admissible  v(t) ∈ V conditions at initial time τ = tσ : s(τ ) = sσ , and at the moment T : h 1 ( s(T )) ≤ 0 and  I . This collection of subproblems is related on τ ∈ Tτ and vector sσ . 2. Upon delivery of the relevant data we formulate the subproblem of optimal program control with new initial conditions (tσ , sσ ). Denominate the optimal program

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control  v(τ | τ, sσ ) for the position (τ, sσ ) (positional solution). Here S(τ ) is a s(τ ), for which the optimal program control could number of all initial states sσ =  s(τ ), s(τ ) ∈ S(τ ), t ∈ Tτ . be designed for subproblem at the time τ with sσ =  3. We realize the control constructing procedure in the class of step functions for the period Tτ , τ = tσ , and reduce the OPC subproblem to the linear programming problem. 4. We employ the adaptive method to the linear programming problem, and we get the control vector. 5. The elements of the positional control vector for the period [tσ , tσ +1 ), awaiting the relevant data about the actual system state (tσ +1 , sσ +1 ), are used. Using the relevant data about the actual system state sσ in the initial state τ = tσ we consider a series of OPC subproblems. Then in any one of them, we design the control in the class of step functions using adaptive method [3] and then it is applied to the model under external actions for the period of time till the next data entry.

66.6 Conclusion In this paper the SC optimization problem is discussed in the event of happening of unexpected circumstances and the relevant data about the actual system state (delivery status, preserving temperature, actual stock, availability of transport facilities, etc.) could be timely obtained. In this problem, we suggest applying CPS for improving the timeliness of the relevant data about the actual SC state. The advance of this approach is that CPS have been applied to this problem when it is essential to do real-time reaction for improving the timeliness of the relevant data about the actual SC state. Acknowledgements This study was carried out with the partial financial support of the RFBR (grants 18-08-01505, 19-08-00989), state research theme 0073-2019-0004.

References 1. Chen, Z.L., Pundoor, G.: Order assignment and scheduling in a supply chain. Oper. Res. 54(3), 555–572 (2006). https://doi.org/10.1287/opre.1060.0280 2. Chernousko, F.: Order Assignment and Scheduling in a Supply Chain. SRC Press, Boca Raton, FL (1994) 3. Gabasov, R., Dmitruk, N., Kirillova, F.: Numerical optimization of time-dependent multidimensional systems under polyhedral constraints. Comput. Math. Math. Phys. 45(4), 593–612 (2005) 4. Garcia, C., Ibeas, A., Herrera, J., Vilanova, R.: Inventory control for the supplychain: an adaptive control approach based on the identification of the lead-time. Omega 40, 314–327 (2012). https://doi.org/10.1016/j.omega.2011.07.003 5. Ivanov, D., Sokolov, B.: Adaptive Supply Chain Management (2010). https://doi.org/10.1007/ 978-1-84882-952-7

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6. Ivanov, D., Sokolov, B., Dolgui, A., Solovyeva, I.: Application of control theoretic tools to supply chain disruption management, pp. 1926–1931 (2013). https://doi.org/10.3182/201306193-RU-3018.00294 7. Ivanov, D., Sokolov, B., Solovyeva, I.: Integrated planning and scheduling with dynamic analysis and control of service level and costs. Oper. Res./Comput. Sci. Interfaces Ser. 60, 263–283 (2016). https://doi.org/10.1007/978-3-319-23350-5_12 8. Ivanov, D., Sokolov, B., Solovyeva, I., Dolgui, A., Jie, F.: Dynamic recovery policies for timecritical supply chains under conditions of ripple effect. Int. J. Prod. Res. 54(23), 7245–7258 (2016). https://doi.org/10.1080/00207543.2016.1161253 9. Kalinin, V., Sokolov, B.: Optimal planning of the process of interaction of moving operating objects. Int. J. Differ. Equ. 21(5), 502–506 (1985) 10. Kupriyanovsky, V., Namiot, D., Sinyagov, S.: Cyber-physical systems as a base for digital economy. Int. J. Open Inf. Technol. 4(2), 18–24 (2016) 11. Lee, E.: Present and future of cyber-physical systems. Sensors 15, 4837–4869 (2015). https:// doi.org/10.3390/s150304837 12. Lotov, A.: Introduction to Economical-Mathematical Modelling (1984) 13. Nazarov, D.: Models and program complex for solving planning problems of measuring and computing operations in cyber-physical systems. J. Instrum. Eng. 61, 947–955 (2018). https:// doi.org/10.17586/0021-3454-2018-61-11-947-955 14. Okhtilev, M., Sokolov, B., Yusupov, R.: Intelligent monitoring and control technology of the structural dynamics of complex technical objects (2006) 15. Ortega, M., Lin, L.: Control theory applications to the production-inventory problem: a review. Int. J. Prod. Res. 42(11), 2303–2322 (2004). https://doi.org/10.1080/00207540410001666260 16. Perea, E., Grossman, I., Ydstie, E., Tahmassebi, T.: Dynamic modeling and classical control theory for supply chain management. Comput. Chem. Eng. 24, 1143–1149 (2000). https://doi. org/10.1016/S0098-1354(00)00495-6 17. Popkov, A., Baranov, O., Smirnov, N.: Application of adaptive method of linear programming for technical objects control, pp. 141–142 (2014). https://doi.org/10.1109/ICCTPEA.2014. 6893326 18. Schwartz, J., Wang, W., Rivera, D.: Simulation-based optimization of process control policies for inventory management in supply chains. Automatica 42(8), 1311–1320 (2006). https://doi. org/10.1016/j.automatica.2006.03.019 19. Sokolov, B., Trofimova, I., Nazarov, D., Zakharov, V.: Modification of multiple-model description and planning and update control algorithms of supply chain, pp. 1972–1977 (2019). https:// doi.org/10.1016/j.ifacol.2019.11.492 20. Sokolov, B., Zelentsov, V., Mustafin, N., Kovalev, A., Kalinin, V.: Methods and algorithms of ship-building manufactory operation and resources scheduling, pp. 81–86 (2017) 21. Trofimova, I., Sokolov, B., Nazarov, D., Potryasaev, S., Musaev, A., Kalinin, V.: Application of cyber-physical system and real-time control construction algorithm in supply chain management problem. Stud. Comput. Intell. 868, 394–403 (2020). https://doi.org/10.1007/978-3030-32258-8_46 22. Wolf, W.: Cyber-phisical systems. Computer 3, 88–89 (2009). https://doi.org/10.1109/MC. 2009.81

Chapter 67

Model of Stakeholders of the Socio-cyber-physical System Life Cycle Stanislav V. Mikoni

Abstract It was proposed to take the cyber-physical system as the core of the sociocyber-physical system, and the society for the external environment. The external environment is divided into representatives of society involved in the various stages of the life cycle of the cyber-physical system. The interests of the participants of the life cycle to the various properties of the cyber-physical system are defined. Taking into account these interests allows minimizing design errors at an early stage of shaping the appearance of the system. On the example of a truck, the heuristic structure of indicators and the structure representing the interests of the participants in its life cycle are compared.

67.1 Introduction Modern complex objects distributed in space and containing, in addition to an effector, ramified system of controls and communications belong to the class of cyberphysical systems (CPS) [7]. Due to its complexity, the cyber-physical system is characterized by many dozens of indicators, representing both its own properties and any interactions with the external environment during the entire life cycle (LC). The active part of the external environment is represented by people interacting with the CPS during its life cycle. In conjunction with the CPS, the society belonging to it forms a socio-cyber-physical system (SCPS) [2]. As an active entity, each of the participants (or group of participants) of the life cycle of the CPS has its own interest in its properties. It manifests itself in a different attitude to the indicators of the system, which characterize different stages of the life cycle, from which follows the contradictory interests of the life cycle participants of the CPS. These people are called stakeholders.

S. V. Mikoni (B) St. Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_67

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Since the effectiveness of the functioning of the CPS depends both on the properties of the CPS and on the interest of the personnel interacting with it, it is necessary to take this factor into account at all stages of the life cycle of the CPS. This means that the interests of various groups of society should be taken into account, starting with the formation of the image of the system and the development of technical specifications for the development of CPS. The interest of the participants in the life cycle of the CPS at this stage is manifested, both in the formation of a set of properties of the designed system and in setting the desired values to the corresponding indicators. The set values are used to compare various variants of the CPS project. The problem of constructing a hierarchy of indicators of a complex system does not have a single solution and is usually solved on the basis of accumulated experience [6]. The structure of the hierarchical model affects the values of the weighting coefficients of the generalizing function [4]. Thus, the quality of evaluation of project options depends on the quality of the hierarchical model. The solution to this problem is proposed to look for in the way of structuring a system of indicators of a complex system, taking into account the interests of the participants in its life cycle.

67.2 Model of Society Since, according to the above, a socio-cyber-physical system, in addition to the CPS, includes people involved in it, it is logical for the CPS to take it as the subsystem of the SCPS. In Fig. 67.1 it is presented in the form of SCPS core. Society is represented by the external environment, divided into three parts: the active and passive environment and the external system. The active external environment is subject to impact on the cyber-physical system. It includes the following representatives of the society (in Fig. 67.1 from the bottom-up): the customer and

Fig. 67.1 Model of socio-cyber-physical system

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the designer of the CPS, the manufacturer, the seller, the buyer, the service staff, and the operator. Either the user or another system can act as an object interacting with the CPS. The passive environment includes the resources consumed by the CPS, and the nature polluted by waste from the activities of the CPS. Here, waste means any kind of impact on nature. If resources belong to the transformed part of nature, then nature itself is understood as its “untouched” part. In the economic system, each of these components has its owner in the person of the state or private companies. The owner of the resources for a certain fee supplies energy and other resources, such as the renewable properties of the cyber-physical system. The organization responsible for the preservation of the environment charges a fee for its pollution. A cyber-physical system, open to interaction with other CPS, may be with them in a relationship of cooperation and confrontation. In the first case, the systems agree on the separation of powers and resources, and in the second case, they try to worsen the properties of the opposer, up to his destruction. Regardless of the degree of their self-organization, external systems realize the goals of specific representatives of society. Thus, all representatives of the external environment are personified by the subjects of society and play an active role at various stages of the life cycle of the CPS. The interactions between the components of the SCPS are shown by arrows. A double solid arrow indicates an effect on the properties of a component, and a double dotted arrow indicates the consumption of properties of a component. The dashed single arrow represents the economic interest of the system component. A solid fat single arrow means control of the cyber-physical system, and a dotted fat single arrow means feedback. The designer forms the properties of the future system, the manufacturer materializes it, and the service staff restores the properties of the CPS. The finished system is alienated from its manufacturer and through the seller is available to the buyer/customer. From this point on, the buyer/customer becomes the owner of the CPS. If the buyer is an individual, then the owner, the operator (with independent maintenance), and the operator are combined in his person. In the organization, the role of the service staff is performed by a special unit that maintains the CPS in working condition. The maintenance of the cyber-physical system by the service staff continues throughout the active stage of its life and ends with utilization at the end of the life cycle (LC) due to physical or moral aging. Economic relations CPS with a passive external environment are realized through their owners in the face of the state or private companies. The owner of the resources for a certain fee supplies energy and other resources, such as the renewable properties of the cyber-physical system. The organization responsible for the preservation of the environment charges a fee for its pollution. The direct control of the cyber-physical system is carried out by the operator in the person of the individual or team. Under his control, the cyber-physical system carries out a material, energy, or informational influence on some external object. If, for example, the CPS implements the functions of a social network, then its users are the objects of impact. Consider the interests of each member’s life cycle CPS.

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67.3 Interests of Stakeholders of the Cyber-Physical System The designer creates all the properties of CPS at the model level. He is interested, first of all, in ensuring a high technical level of CPS for reasons of competitiveness of the future product [6]. It is important to note that the improvement of the integrated indicators of CPS is achieved at the level of the technical characteristics of its parts. Hence the need to assess the technical level of the CPS at the level of individual components (units). Each unit of CPS is characterized by its own set of individual indicators. Single indicators and requirements for them are set in the technical specifications for the design of CPS. The manufacturer of the CPS forms the properties of the system at the physical and program level. He is interested in design-technological, patent-legal, and economic indicators characterizing the technological process of production [3]. From the economic indicators of the manufacturer, first of all, they are interested in the cost of production and profit from its implementation. These indicators are assigned from outside (trader, exchange, and auction). The seller and the buyer consider CPS as a commodity, realizing commodity– money relations. The seller of the CPS can be both the manufacturer of the CPS and the trading company. The seller is interested in maximizing profits from the sale of CPS. In addition to the cost price, the wholesale and retail price of CPS depends on market conditions, as well as on sales organization (advertising, etc.). By themselves, the technical and operational characteristics of the CPS are interesting to the seller only in terms of advertising for the purpose of selling goods at a profit. In the purchase of CPS interested buyer is represented by either a private person or an organization. The interest of the buyer of CPS is to minimize the price of CPS. In addition to the price he is interested in operating costs, the prices for which are assigned by the relevant organizations [3]. The operator is primarily interested in the indicators characterizing the purpose of CPS. The effectiveness of its work depends on the use of these properties in the maximum amount. In the second place, the operator is interested in minimizing the cost of servicing the CPS, but, unlike the buyer, not in terms of price, but in terms of availability ratio of the cyber-physical system, i.e. time factor. The service staff in the face of a repair service considers the CPS as a source of their income. In this regard, he is interested in minimizing the costs of restoring the properties of CPS and reducing the cost of repairs. Due to the specialization of repairmen, their interest is directed to specific units of the CPS. In contrast to the buyer, the service staff is interested in increasing the value of his work, i.e. the cost of repair and utilization of CPS. The interests of suppliers of energy and other resources are represented by their owners. They are interested in increasing the amount of resources supplied and minimizing damage to their infrastructure. Nature, as a passive participant in the life cycle of the CPS, has an unintentional effect on the CPS. It manifests itself in the form of adverse weather conditions, such as poor visibility, ice, and precipitation while driving a vehicle. Environmental organizations show an interest in preserving nature from the effects of CPS, making demands to minimize the harmful effects on nature.

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The interest of the external system in relation to CPS depends on the type of their interaction. In the case of cooperation, the external system is interested in maximizing the use of common resources to achieve a common goal. In this sense, the interests of the collaborating systems are contradictory and can be satisfied with an agreed allocation of resources. The interest of the external system in conflict with the CPS is manifested in the maximum opposition to the goals of the CPS. The degree of resistance varies from neutralizing the opposer to his destruction. We will express the interests of the stakeholders of CPS through generalized indicators.

67.4 Indicators of the Upper Levels of the Cyber-Physical System Estimation Model The division of participants in the life cycle of the CPS into groups according to the type of interaction with the CPS provides information for the formation of indicators of the upper level of the assessment model. These include indicators characterizing all stages of the life cycle of the CPS1 : design (customer, designer), production (manufacturer), buy/sell (buyer/seller), storage (storekeeper), performance of work (operator), and maintenance and utilization (operator). The listed groups of indicators characterizing the CPS as a whole are complemented by indicators characterizing the individual units (modules) of the CPS. They are in demand mainly at the stages of design, production, and maintenance of CPS. Each stage of life cycle correlated with its performers (in brackets). Since all products of any complexity go through all the mentioned life cycle stages, characterizing their indicators are common for objects of any type, regardless of their specificity and complexity. The general indicators of the second level for each stage of the life cycle of the CPS are the complexity and cost of its implementation. The target generalized indicator among the indicators of the first level of the hierarchy is the indicator of performance of work. In varying degrees, it is of interest to all participants in the life cycle of the CPS. He is more interested in the operator and the user of the CPS, since the effectiveness of the CPS is manifested in the process of its use for its intended purpose. It is divided into the following generalized indicators of the second level of the hierarchy: 1. 2. 3. 4.

an indicator of purpose, an indicator of the quality of functioning, operational control indicator, an indicator of interaction with the external environment.

The purpose indicators are determined by the role of the CPS. For example, for a vehicle, these include speed, mass, and dimensions indicators.

1

For the indicators systematization, here definitions given at [8] are used.

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Indicator of the quality of functioning is detailed on the following generalized indicators of the third level of the hierarchy: steadiness, safeness, aesthetics, and convenience. Steadiness in a broad sense, we define how the property of an object perform its functions in a changing internal and external environment. The object is stable as long as its ability to achieve the intended goal is maintained. In relation to various impacts, steadiness can be divided into the following types [5]: physical resistance, fault tolerance as resistance to the specified types of failures, noise immunity as resistance to specified types of interference, reliability as steadiness of operation for a period of time, for example, a warranty period, and vitality as steadiness of functioning in the conditions of aggressive environmental impacts. Steadiness can be divided into passive and active ones in ways to ensure it. Under the passive steadiness it will be understood as the natural resistance of the object to internal and external impacts. By active steadiness it is meant a change in the state of an object in order to counter the effects of internal and external factors. Under the conditions of predictable (calculated) changes in the internal and external environment, steadiness is specified in the homeostasis of the system. This property is implemented by means of automatic adjustment, ensuring the control of the constancy of the parameters of CPS. Under the conditions of unpredictable changes in the internal and external environment, steadiness is specified in the property of adaptation. Adaptation of the system is carried out, first of all, due to the accumulation of relevant experience. On its basis, it is decided to change the parameters or structure of the object, adequate to the ongoing changes in the environment. With a long period of operation, the self-organizing system summarizes the experience gained in knowledge of a higher level. This allows it to anticipate future changes and the consequences of choosing a development path. Resilience provided on the basis of foresight characterizes a higher level of self-organization of the system. This property, called pro-activity, allows pre-empting by their actions the possible damage that possible changes in the internal and external environment can bring. The highest level of steadiness is provided by choosing the optimal path for the development of the system at the branch point. The system is stable if it can independently choose the most preferable way of development in the context of the action of various attractors on it. And this is feasible if the system has sufficient knowledge to analyze the usefulness of the outcomes of choice and the resources for its implementation. The fundamental difference between the stability and safety of the system is that the first property is realized by restoring equilibrium when exposed to disturbing factors, and safety is ensured by preventing them. Controls are also applied to ensure safety. The convenience of the CPS in relation to the operator controlling it is considered in terms of ergonomics. The convenience of a CPS user is to provide him with material, energy, and information support in the required and possible volume.

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The condition for the steadiness of CPS under the influence of poorly predictable factors is an advanced control system. In [5], each type of active steadiness is assigned a method of controlling an object. According to the degree of automation, control methods are divided into manual, automated, and automatic control. Due to its complexity, the cyber-physical system has a multi-level control system both at the level of the CPS as a whole and at the level of its components. Each level is characterized by its own way of control. The goal of control is to ensure the active steadiness of the object. Depending on the type of control functions used, the corresponding level of system steadiness with respect to internal and external impacts is ensured. Thus, ordering the systems with control according to the levels of active steadiness, we thereby order also the corresponding controls. It is obvious that every next control tool embedded in an object frees the external control entity from performing the corresponding control function. From this, it follows that the ordering of systems with control by levels of active steadiness corresponds to their ordering by independence from external control. The more controls built into the object, the higher the independence of its operation and the greater the independence from external control, which is consistent with the principle of external addition proposed by S. Beer [1]. Here, the role of external addition to the system with control is played by external control. By increasing the level of active steadiness, the properties of systems with control are ordered as follows: equilibrium, adaptability, proactivity, self-organization, and autonomy. Each subsequent property extends the application of the previous properties. Ensuring the equilibrium state (homeostasis) of an object is the primary goal of control. It is provided by means of automatic control, designed to eliminate deviations from the calculated values of the parameters. To ensure the stable functioning of an object under conditions of a wide range of effects of the internal and external environment, the means of adaptation to these effects are provided. Adaptation to the new conditions of operation is carried out by changing the parameters or structure of the object. The proactivity of an object is an alternative to its reactivity. Responding to the next environmental change is contrasted with the prediction of its future state with the aim of anticipating adverse consequences. Self-organization in accordance with its definition involves the improvement of the functioning of the object, taking into account past experience. Improvement is the result of development, and development is associated with the choice of the most favorable option for the functioning of the object. Thus, an essential feature of a self-organizing system is its ability to choose a path of development. It is important to note here that the goal of the functioning of an object is set from the outside, and its independence is manifested only in the ways of achieving the goal. In this sense, autonomy of an object represents the greatest freedom from external control. It is characterized by the ability of an object to formulate its own goal for the most favorable functioning in a changing environment. This property is inherent in organizational and social systems. A technical object with this property is dangerous to humans, since their goals may be incompatible.

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Environmental impact is studied in the framework of ecology. The environmental hazard of CPS is determined by the amount of waste that pollutes the land, water, and air, as well as noise and radiation exposure. Indicators of the first two levels of the hierarchy are applicable with respect to CPSs of any type, which indicates a high degree of their unification. Of the indicators of the third level of the hierarchy, the indicators belonging to the groups of assignment and operational control most depend on the type of the CPS.

67.5 Conclusion Taking into account the interests of all participants in the CPS life cycle made it possible to unify the upper levels of the hierarchy of indicators. The role of built-in control in ensuring active stability of CPS is shown. The proposed approach allows to simplify the construction of hierarchical models for evaluating CPS and to improve their quality and increase the reliability. Comparison with the empirical system of truck indicators has demonstrated a greater validity of the proposed hierarchy of indicators. The estimates of the quality and technical level of the system obtained on its basis are more reliable. Acknowledgements The studies were carried out with the financial support of the RFBR grants No. 19-08-00989-a, 20-08-01046 within the framework of the budget theme No. 0073-2019-0004.

References 1. Beer, S.: Cybernetics and Management. English University Press (1959) 2. Hansen, T.M., Snauryarayanan, S., Roche, R.: Cyber-Physical-Social Systems and Constructs in Electric Power Engineering. Institution of Engineering and Technology (2016) 3. Horoshev, A.N.: Management of the solution of design tasks at the enterprise. Modern Scientific Research and Innovations, No 7 (in Russian) (2011) 4. Mikoni, S.V.: Theory of Administrative Decision Making. Lan’ Pupl, St. Petersburg (in Russian) (2015) 5. Mikoni, S.V.: Formation of generalized indicators of the transport system from the positions of stakeholders. Ontol. Des. 8(2), 296–304 (2018). (in Russian) 6. Semenov, S.S.: Assessment of the quality and technical level of complex systems. The Practice of Expert Assessments Applying. LENAND Publ, Moscow (in Russian) (2015) 7. Seshia, S.A., Lee, E.A.: Introduction to Embedded Systems: A Cyber-physical Systems Approach, 2nd edn. MIT Press (2017) 8. Volkova, V.N., Emelyanov, A.A.: Systems Theory and Systems Analysis in the Management of Organizations: Handbook: Textbook Manual. Finance and Statistics. INFRA-M, Moscow (in Russian) (2009)

Chapter 68

National Healthcare System and Economy’s Competitiveness Anatoliy V. Sigal, Maria A. Bakumenko, and Elena A. Lukyanova

Abstract In the paper, the authors test the hypothesis about the correlation of ranks between a country’s healthcare system and the national competitiveness. The research rests upon data that are given in reports by the World Health Organization, the World Economic Forum, and the consulting firm Bloomberg. The authors measure the statistical relationship between the examined couples of order variables with the help of Kendall’s coefficient of concordance and assess the statistical significance of a sample value of the concordance coefficient using Pearson’s Chi-square test. The found sample values of the concordance coefficient are rather close to 1, so there is a strong correlation of ranks between the examined couples of order variables. The conclusion can be made that the level of a country’s healthcare system has a strong impact on the national competitiveness.

68.1 Introduction The issues of public health should be a priority of any state policy irrespective of economic development level. “The notion that health is wealth serves as a precondition for both personal productivity and national development” [9]. It is commonly known that public health is determined by many factors, one of which is the healthcare system itself [8]. If a country aims to have healthy citizens, it should make significant investments in public health, at the same time creating a favorable environment for individual investments [9]. A. V. Sigal (B) · M. A. Bakumenko, Institute of Economics and Management, V.I. Vernadsky Crimean Federal University, Sevastopolskaya Str. 21/4, Simferopol 295015, Russia M. A. Bakumenko, e-mail: [email protected] E. A. Lukyanova Taurida Academy, V.I. Vernadsky Crimean Federal University, Vernadsky avenue 4, Simferopol 295007, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_68

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The sustainable development of a national economy is impossible without the sustainable development of the healthcare system. The sustainability of a healthcare system depends on the following factors: efficient preventive healthcare and healthy lifestyle promotion; enhancing the healthcare system by using advanced technologies; funding models that provide the required change in behavior; encouraging the use of innovation and creativity; effective development and necessary support for people employed in the healthcare system; and defining and introducing health determinants into the healthcare system [3]. The importance of building an efficient healthcare system has attracted the attention of many researchers. Many research papers focus on studying a variety of aspects of the country’s healthcare system. Thus, for instance, Boulware L. E. et al. [2] explore the problem of trust and race discrimination in medicine. The paper of Eckelman M. J. and Sherman J. [4] analyzes the impact of the healthcare system on the environment. The paper of Tormusa D. O. and Idom A. M. [9] studies the influence of corruption on national healthcare system efficiency. Coman A. and Grigore A.-M. [3] analyze how innovations affect the sustainable development of the national healthcare system. The challenges of building sustainable healthcare systems in developing countries are dealt with in the paper of Karamat J. et al. [6]. Verulava T. and Maglakelidze T. [10] treat the issues of healthcare system financing. Emanuel E. J. [5] examines the healthcare system in terms of excessive costs and ways to reduce them. Despite a large number of studies on the given topic, there are still many unsolved challenges, in particular the quantitative assessment of the impact of a national healthcare system on the competitiveness of the national economy. Certain steps in this direction have been taken in the research paper by Sigal A.V. and Bakumenko M.A. [8]. Using econometrics (regression equations), the article [8] numerically evaluates the impact of a healthcare system on the economy’s competitiveness. The given paper continues the above research and calculates one more important index, concordance coefficient, in order to measure the statistical relationship between several order variables. The authors aim to test the hypothesis about the correlation of ranks between the level of a country’s healthcare system and the economy’s competitiveness, which are neither more nor less than two order variables. With this object in mind, the authors have to draw samples, calculate respective statistical values to confirm (reject) the hypothesis, verify the statistical significance of the obtained results, and make certain conclusions.

68.2 Describing Calculation Base and Research Methods The research rests upon data that are given in The Global Competitiveness Reports of 2000 [12] and 2018 [13], provided by the World Economic Forum, The World Health Report 2000 [11] prepared by the World Health Organization, and Bloomberg’s 2018 Health Care Efficiency [7].

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From the above official documents, we have extracted the following indices: Health System Performance, Current Competitiveness Index, Global Competitiveness Index 4.0, Health, Bloomberg Health Care Efficiency. See Tables from 68.1 to 68.2. Designed by the World Health Organization, the Health System Performance 2000 was the first index in the world to rank 191 economies by the progress of their medical systems. Table 68.1 gives the values of the index for 56 economies only, with the economies being ordered from 1 (the best level) to 56 (the worst level). Table 68.1 Ranking 56 economies by Health System Performance and the Current Competitiveness Index, 2000a Economy Health System Current Com- Economy Health System Current ComPerformance petitiveness Performance petitiveness (rank) Index (rank) (rank) Index (rank) Argentina Australia Austria Belgium Bolivia Brazil Bulgaria Canada Chile China Colombia Costa Rica Czech Rep. Denmark Ecuador Egypt El Salvador Finland France Germany Greece Hungary Iceland India Indonesia Ireland Israel Italy a Formed

40 22 5 15 50 49 45 20 23 53 16 25 29 24 46 37 48 21 1 18 9 38 10 47 44 13 19 2

43 10 13 12 56 29 53 11 24 42 46 41 32 6 55 37 49 1 15 3 31 30 16 35 45 20 17 22

by the authors based on [11, 12]

Japan Jordan Korea, Rep. Malaysia Mauritius Mexico Netherlands New Zealand Norway Peru Philippines Poland Portugal Russia Singapore Slovak Rep. South Africa Spain Sweden Switzerland Thailand Turkey UK Ukraine USA Venezuela Viet Nam Zimbabwe

6 42 33 30 43 35 11 27 7 51 34 31 8 52 3 36 56 4 17 14 28 39 12 41 26 32 55 54

14 33 25 28 36 40 4 18 19 47 44 39 26 50 9 34 23 21 7 5 38 27 8 54 2 52 51 48

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The ranking of the national economies in terms of Health System Performance 2000 was based on the following criteria: health, responsiveness, and financial contribution [11]. Preceding the Global Competitiveness Index, the Current Competitiveness Index 2000 was designed in 2000 by the World Economic Forum for 58 economies. The Current Competitiveness Index gives data and a conceptual framework for the comparative analysis of national economies and it considers the microeconomic bases of GDP per capita of a national economy. Table 68.1 presents the given index only for 56 economies, which are ranked by their competitiveness from 1 (the best level) to 56 (the worst level). The number of economies in Table 68.1 is determined by the fact that data about them are contained in both official reports: The Global Competitiveness Report 2000 and The World Health Report 2000. While being the revised version of the Global Competitiveness Index, the Global Competitiveness Index 4.0 (GCI 4.0) was first calculated by the World Economic Forum in 2018 for 140 economies. GCI 4.0 is made up of 98 indices, 44 of which are based upon Executive Opinion Survey by the World Economic Forum and 54 of which are based upon official statistics. The Global Competitiveness Index 4.0 consists of 12 enlarged components called pillars. All pillars are of equal importance for GCI 4.0. The pillars of GCI 4.0 form four groups: Markets, Human Capital, Enabling Environment, and Innovation Ecosystem. The report [13] lists several essential factors for economies to achieve long-term competitiveness, one of which is human capital. Health is an essential component of human capital. Health represents the fifth pillar of the Global Competitiveness Index 4.0 and rests upon healthy life expectancy only [13]. The Global Competitiveness Report 2018 includes the ranking of 140 economies by Health and GCI 4.0 in 2018. Please note that in 2018 four economies (Singapore, Japan, Hong Kong, and Spain) ranked first by Health. Proper allowance must be made for this; while calculating the concordance coefficient, we should apply a special equation. It should be noted that Table 68.2 also contains this repetition of the values of order variables (see Health). Bloomberg’s ranking of Health Care Efficiency has been built on data published by the United Nations Organization, the World Health Organization, and the World Bank, and is based on the following major indices: life expectancy, per capita total health expenditure, and total health expenditure as a percentage of GDP [7]. Table 68.2 gives the level of the criterion for 54 economies only. To define the statistical relationship between several orderings of the same finite set of analyzed objects, we should turn to rank correlation analysis. We measure the statistical relationship between several orders of variables with the help of Kendall’s coefficient of concordance and we assess the statistical significance of a sample value of the concordance coefficient using Pearson’s Chi-square test [1]. The above mentioned were used in our research to prove the hypothesis about the correlation of ranks between the level of a country’s healthcare system and the national competitiveness.

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68.3 Calculation Results The economies in Tables 68.1 and 68.2 are listed in descending order in terms of desirable characteristics, which means that the strongest economy has the ordinal number 1, while the poorest economy is characterized by the largest ordinal number in the list. In Tables 68.1 and 68.2, the environment is characterized by expert information that can be introduced as the following input data matrices: R = Rn×k = (ri j ), where k is the number of indices and n is the number of economies. We shall define the statistical relationship between Health System Performance and Current Competitiveness Index 2000 using a sample value of the concordance

Table 68.2 Ranking 54 economies by Health and Bloomberg Health Care Efficiency, 2018b Economy Health Bloomberg Economy Health Bloomberg Health Care Health Care Efficiency Efficiency Algeria Australia Austria Azerbaijan Belgium Brazil Bulgaria Canada Chile China Colombia Costa Rica Czech Republic Denmark Dominican Rep. Ecuador Finland France Germany Greece Hong Kong Hungary Iran Ireland Israel Italy Japan b Formed

43 8 13 52 23 48 46 11 25 33 28 12 31 26 41 30 18 7 21 17 1 45 51 20 10 6 1

32 7 31 52 37 49 54 15 30 19 46 24 29 40 48 42 18 15 44 13 1 41 38 12 5 4 6

by the authors based on [7, 13]

Jordan Kazakhstan Lebanon Malaysia Mexico Netherlands New Zealand Norway Peru Poland Portugal Romania Russia Saudi Arabia Serbia Singapore Slovak Rep. Spain Sweden Switzerland Taiwan Thailand Turkey U.A.E. UK USA Venezuela

49 53 29 40 37 16 15 9 27 36 19 47 54 42 44 1 38 1 14 5 22 32 35 50 24 34 39

47 43 22 28 19 27 14 10 35 23 17 35 51 45 50 2 33 3 21 11 8 26 25 9 34 52 38

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Table 68.3 The calculation results of the sample values of concordance coefficientc Wˆ (k) Time period Indices n k Conclusion Health System Performance and the Current 56 Competitiveness Index Health and Global Competitiveness Index 4.0 140 Health and Bloomberg Health Care Efficiency 54

2000

2018 2018 c Calculated

Strong correlation 2

0.8526

2

0.9249

2

0.8870

Strong correlation Strong correlation

by the authors

coefficient. In Table 68.1, there are no united ranks, so we can define a sample value of the concordance coefficient [1] by the formula (68.1):  k 2 n   k · (n + 1) 12 ri j − . Wˆ (k) = 2 · k · (n 3 − n) j=1 i=1 2

(68.1)

The value that the concordance coefficient takes on ranges between [0; 1]. The closer a sample value of the concordance coefficient is to 1, the stronger rank correlation is characterized between order variables under study. In the environment under study (2000), n = 56 (the number of economies) and k = 2 (the number of indices). The sample value of the concordance coefficient for the year of 2000 constitutes 0.8526; it has been calculated by the formula (68.1) using the data of Table 68.1. See Table 68.3. We shall define the statistical relationship between Health and Global Competitiveness Index 4.0 in 2018 using a sample value of the concordance coefficient. Among the values of Health, there are united ranks, so we can define a sample value of the concordance coefficient [1] by the formula (68.2): Wˆ ∗ (k) =

12

·   k k 2 · n 3 − n − k · i=1 T (m i )

 k n   j=1

k · (n + 1) ri j − 2 i=1

2 . (68.2)

The correction factor T (m i ) (which corresponds to the variable ri j ) is calculated by the formula (68.3): T (m i ) =

 k(m )  1 i (m i ) 3 nt · − n t(m i ) , 2 t=1

(68.3)

where k(m i ) is the number of groups of indiscernible ranks of the variable ri j ; n t(m i ) is the number of elements (ranks) that fall under t group of indiscernible ranks.

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Table 68.4 The test results of the statistical significance of the concordance coefficient sample valuesd Wˆ (k) Time Indices n χ 2f act. χ 2f act. (α, n − 1) Conclusion period

2000

2018

2018

Health System Performance and the Current Competitiveness Index Health and Global Competitiveness Index 4.0 Health and Bloomberg Health Care Efficiency

d Calculated

Statistically significant 0.8526

56

93.78

73.31

Statistically significant 0.9249

140

257.12

167.51 Statistically significant

0.8870

54

94.02

70.99

by the authors

In the environment under study (2018), n = 140 (the number of economies) and k = 2 (the number of indices). The sample value of the concordance coefficient for the year 2018 constitutes 0.9249; it has been calculated by the formula (68.2) using the data of The Global Competitiveness Report 2018. See Table 68.3. We shall define the statistical relationship between Health and Bloomberg Health Care Efficiency in 2018 using a sample value of the concordance coefficient. In Table 68.2, there are united ranks, so we can define a sample value of the concordance coefficient [1] by the formula (68.2). In the environment under study (2018), n = 54 (the number of economies) and k = 2 (the number of indices). The sample value of the concordance coefficient for the year 2018 constitutes 0.8870; it has been calculated by the formula (68.2) using the data of Table 68.2. See Table 68.3. We shall assess the statistical significance of the found sample values of the concordance coefficient using Pearson’s Chi-square test. The test results are summarized in Table 68.4. When analyzing Table 68.4, we see that the found concordance coefficients are to 2 for the two periods under study, and there is a strong be trusted, as χ 2f act. > χtabl. correlation dependence between the three examined couples of order variables.

68.4 Conclusions The found sample values of the concordance coefficient are rather close to 1, so there is a strong correlation of ranks between the three examined couples of order variables: (1) Health System Performance and the Current Competitiveness Index (2000); (2) Health and Global Competitiveness Index 4.0 (2018); and (3) Health

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and Bloomberg Health Care Efficiency (2018). The test results of the found sample values of the concordance coefficient for significance using Pearson’s Chi-square test allow us to regard the relationship between the order variables under study as statistically significant. The conclusion can be made that the level of a country’s healthcare system has a strong impact on the national competitiveness. According to the authors, the obtained results are important for all levels of management of a national economy.

References 1. Aivazian, S.A., Mkhitarian, V.S.: Applied Statistics and Essentials of Econometrics. UNITY, Moscow (1998) 2. Boulware, L.E., Cooper, L.A., Ratner, L.E., LaVeist, T.A., Powe, N.R.: Race and trust in the health care system. Public Health Rep. 118, 358–365 (2003) 3. Coman, A., Grigore, A.-M.: Innovation as a driver of the sustainable healthcare systems: the case of Romania. J. Innov. Bus. Best Pract. 706791 (2017). https://doi.org/10.5171/2017.706791 4. Eckelman, M.J., Sherman, J.: Environmental impacts of the U.S. health care system and effects on public health. PLOS ONE, June 9 (2016). https://doi.org/10.1371/journal.pone.0157014 5. Emanuel, E.J.: The real cost of the US health care system. JAMA 319(10), 983–985 (2018) 6. Karamat, J., Shurong, T., Ahmad, N., Afridi, S., Khan, S., Khan, N.: Developing sustainable healthcare systems in developing countries: examining the role of barriers, enablers and drivers on knowledge management adoption. Sustainability 11, 954 (2019). https://doi.org/10.3390/ su11040954 7. Miller, L.J., Lu, W.: These are the economies with the most (and least) efficient health care (2018). https://www.bloomberg.com/news/articles/2018-09-19/u-s-near-bottomof-health-indexhong-kong-and-singapore-at-top 8. Sigal, A.V., Bakumenko, M.A.: The level of the country’s health care as an indicator of its economic development. Adv. Econ. Bus. Manag. Res. 128, 1098–1103 (2020) 9. Tormusa, D.O., Idom, A.M.: The impediments of corruption on the efficiency of healthcare service delivery in Nigeria. Online J. Health Ethics 12(1) (2016). https://doi.org/10.18785/ ojhe.1201.03 10. Verulava, T., Maglakelidze, T.: Health financing policy in the South Caucasus: Georgia, Armenia, Azerbaijan. Bull. Georgian Natl. Acad. Sci. 11(2), 143–150 (2017) 11. WHO: The World Health Report 2000. WHO, Geneva (2000) 12. WEF: The Global Competitiveness Report 2000. Oxford University Press, New York (2000) 13. WEF: The Global Competitiveness Report 2018. WEF, Geneva (2000)

Chapter 69

Assessment of the Socio-economic Effectiveness of Innovative Projects Elena A. Lezhnina, Yulia E. Balykina, and Alexander V. Konovalov

Abstract In the decision-making process on the construction of large-scale projects, it is necessary to take into account not only the economic benefits for investors but also the possible long-term socio-economic impact on the region. Moreover, this influence is both positive and negative. To carry out such an assessment, it is necessary to determine the financial indicators of the project, as well as the factors of possible impact on the social life of the region, and environmental impact. This article addresses the issue of using migrant labor. Its positive and negative socio-economic impact on the region is explained.

69.1 Introduction Large-scale regional investment projects are the projects where implementation has a significant long-term impact on the social, economic, and environmental situation in the region. Moreover, the consequences of the implementation of such projects for the region can be either positive or negative. Besides an impact on various areas of the economic system of the region, large-scale investment projects affect the development of industrial and transport infrastructure, strengthen the investment activity in the region, and change the regional business climate. The implementation of these projects requires, as a rule, the attraction of significant labor resources, which cannot but affect the living standards of the population of the considered region. On the other hand, there may be a violation of the usual way of life, destruction of E. A. Lezhnina (B) · Y. E. Balykina St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] Y. E. Balykina e-mail: [email protected] A. V. Konovalov Higher School of Economics, Saint Petersburg, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_69

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the indigenous population territories, and an increase in the human impact on the environment. Not every innovative project is good. The development of a new technology may be associated with a high level of environmental pollution, which in turn can affect the level of morbidity of the population, which negatively affects the economy as a whole. At the same time, there are innovative projects with the purpose not to obtain monetary profits, but, for example, to increase the population standard of living, well-being or public health, etc. Quite often, such projects remain in the shadows, and preference is given to projects that look more attractive in the short term, but at the same time are destructive in the long term. Each innovation project is fraught with significant risk because it is difficult to predict in advance how the market will perceive one or another innovation. Therefore, an important stage in the implementation of an innovative project is its evaluation, which allows predicting its perception by the market, industry, and the state, depending on the evaluation perspective. For such an assessment, it is necessary to determine the economic and financial indicators, which are primarily aimed at assessing the economic efficiency of the project. At the same time, innovative projects in addition to economic efficiency often have non-zero social efficiency, which can be expressed in terms of the increase in various indicators of regional activity. For example, almost all projects require a certain workforce that will carry out these projects. Thus, each project creates a certain number of jobs, thereby increasing employment. New jobs are important for the regions because they make it possible to lower unemployment. One of the goals of innovative projects is to make a profit, which means that the level of wages in projects is usually higher than the average level of wages in the region. This, in turn, increases the performance indicators of the regions. Currently, four main approaches have been adopted to assess the socio-economic effectiveness of projects: 1. 2. 3. 4.

cost–effectiveness analysis (CEA); cost–utility analysis (CUA); weighted cost–effectiveness analysis (WCEA); cost–benefit analysis (CBA).

Each of these approaches has specific features and practical uses. However, the CBA method [4] is the most common approach to evaluating projects, especially projects involving government authorities. The CBA method is used when public benefits are of monetary value. To determine the net current benefit for the duration of the project, the difference between the discounted cash flows of benefits and costs is calculated. The CEA method is a set of analytical techniques that allow one to determine the consumption of resources to achieve a particular goal and choose the optimal solution from the given perspective [6]. This method is not limited only to performance as such, but also includes consideration of productivity and profitability. However, the CEA method does not imply a comparison of results that are heterogeneous in nature. Typically, the CEA method is used by decision-makers regarding resource optimization tasks in two cases: (1) focus on

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obtaining maximum results in a fixed budget and the availability of several alternative projects and (2) focus on minimizing the costs in the context of the need to achieve a certain level of efficiency. The CUA method is based on an analysis of costs and utility, which is expressed in a comparison of costs in terms of money and benefits for the population, measured in so-called units of utility [1, 3]. The result is conditional coefficients, which one can take into account to describe the specifics of project implementation. In the presented work, another approach is proposed to assess the socio-economic impact of the project, taking into account the active migration processes that affect the economy around the world.

69.2 Modeling The socio-economic costs of the project may be the following: • destruction of natural complexes and depletion of natural resource potential; • growth of social inequality as a result of rising incomes among workers employed in the project; • destruction of historically developed territorial communities of the population and the change in the usual living conditions of the population living in the project region; • aggravation of social problems due to the involvement of a significant number of shift workers and temporary workers for the project; • increased load on transport infrastructure; • environmental degradation in the project area; • the constructed facility can create competition for small and medium enterprises engaged in the same industry, which will lead to their collapse. This, in turn, will increase unemployment rates, as well as increase unemployment benefits. In order to assess the socio-economic impact of the facility on the surrounding areas, we construct an estimation impact function. Let us introduce the following notation: The first group of indicators: Rs —number of jobs created; Z —average salary in the company; t1 —the amount of income tax rate paid by employees to the budget; this money goes to the federal or local budget, depending on tax laws; P1 —planned annual profit; P2 —the average profit of small and medium enterprises in the area competing with the object, expected in the next time period; t2 —corporate income tax rate; similar to the personal income tax, this money goes either to the federal or local budget, depending on tax legislation; t3 —the number of small and medium enterprises that may close due to competition of the newly created enterprise; Rl —the number of jobs in small and medium enterprises that may close due to competition. Since we can consider their collapse as a random variable, then we take the expected value M(Rl ); Z l —average salary in the region; Z p —unemployment benefits;

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The second group of indicators: E is the combined coefficient of the impact of increased transport load and environmental degradation. The increased transport load is expressed in terms of the deterioration of the environmental situation in the region. It can also be assessed as an increase in the time spent on the movement of vehicles. In addition to transport, the project itself can also influence the environmental situation, especially if it is related to production. For its evaluation, direct environmental assessment studies are used. It is more difficult to assess the impact of environmental degradation on residents of the region. There are various indicators for this kind of evaluation. For example, there are studies proving a direct correlation between the amount of emissions of harmful substances into the atmosphere and the consequences for the economy in the form of a decrease in gross regional product [2]. Com is the change in the quality of life coefficient. Com < 0, if the quality of life of the residents of the region is deteriorating, and Com > 0 if it is improving. S—advantages of building an object over the use of land in agriculture or as an landscaping area. An object is evaluated according to the first group of indicators by calculating cash flows. Assessment according to the second group of indicators can be performed with the help of experts. A rating scale is created in which points for positively influencing factors have a value greater than zero, and points for negatively influencing factors have a value less than zero. As a result, we get a combined indicator of socio-economic impact Q 2 . For Q 2 > 0 we can talk about the positive impact of the project on the region, while Q 2 < 0 tells us about the negative impact on the region. The combined indicator for the first group of indicators looks as follows: Q 1 = t1 Rs Z + t2 P1 − t1 M(Rl )Z l − M(P2 )t2 t3 , i f

Rs − Rl > 0

(that is, the number of jobs created is enough to provide jobs to everyone unemployed; no increase in unemployment benefits) Q 1 = t1 Rs Z + t2 P1 − (M(Rl ) − Rs )Z p − t1 M(Rl )Z l − M(P2 )t2 t3 , i f

Rs − Rl < 0

(i.e. the number of jobs created is not enough to provide jobs to all who lose their jobs; an increase in unemployment benefits is needed). Here, t1 Rs Z —tax payments of workers; t2 P1 —tax payments of a new enterprise; M(Rl )Z l —unemployment benefits; M(P2 )t2 t3 —tax payments of small and medium enterprises. If Q 1 > 0, then the project has a positive economic impact on the region. One of the important influencing factors, which we can control, is the number of migrants involved in the construction and operation of the facility. Without attracting migrant labor, big projects create a large number of jobs. This can compensate for jobs lost due to competition. However, jobs created are often occupied by migrants. The generally accepted view that migrant labor is cheap is a myth. E. Tyuryukanova showed [5] that migrant labor is paid on a par with “local” workers. Unscrupulous

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employers are attracted by the possibility of “shadow employment” when they do not hire workers officially, which eliminates the need to pay part of the taxes. With strict control, the benefits of “external” labor disappear. The Russian government issued a decree determining the share of migrants in “certain types of economic activity”. We are talking about permissible quotas for hiring foreign workers. In some countries, the rules for hiring foreign citizens have restrictions that are more stringent: an employer cannot hire migrants if there are unemployed people in a given specialty in the region. Let us consider a situation where a region can influence recruitment quotas. Let coefficient α(0 < α < 1) be the share of migrant workers. Then the Q 1 coefficient takes the following form: Q 1 = t1 Rs Z + t2 P1 − (Rl − (1 − α)Rs ) Z p − t1 M(Rl )Z l − M(P2 )t2 t3 , if (1 − α)Rs − Rl < 0 (i.e. the number of jobs created is not enough to provide jobs to all who lose their jobs; an increase in unemployment benefits is needed). Q 1 = t1 Rs Z + t2 P1 − t1 M(Rl )Z l − M(P2 )t2 t3 , i f (1 − α)Rs − Rl > 0 (i.e. the number of jobs created is enough to provide jobs to everyone unemployed; no increase in unemployment benefits). In order to decide on the feasibility of locating the object under consideration in the region, it is necessary to consider Table 69.1. To determine the effective quota for hiring, we conducted a numerical simulation with the following parameter values (data in rubles for the St. Petersburg region) (see Fig. 69.1): Rs = 500; Z = 25000 − 35000; t1 = 0, 13; t2 = 0, 2; t3 = 10; P1 = 55800000, P2 = 16250000; Rl = 300; Z l = 25000 − 35000; Z p = 5000. Figure 69.1 shows the amount of money received by the regional budget, depending on the size of job quotas for migrants. In the absence of migrants, unemployment does not grow, so there is no need to pay unemployment benefits. If part of the jobs is occupied by migrants, they pay taxes, but unemployment benefits are paid from the budget. The modeling showed that the effective quota for migrants depends on the average level of wages in the region and the number of jobs created. Table 69.1 Decision table Classes No Migrants To build

Not to build

Q 1 = t1 Rs Z + t2 P1 − t1 M(Rl )Z l − M(P2 )t2 t3 , if Rs − Rl > 0; Q 1 = t1 Rs Z + t2 P1 − (Rl − (1 − α)Rs )Z p − t1 M(Rl )Z l − M(P2 )t2 t3 , if (1 − α)Rs − Rl < 0, Q 1 = t1 Rs Z + t2 P1

Migrants Q 1 = t1 Rs Z + t2 P1 − (Rl − Rs )Z p − t1 M(Rl )Z l − M(P2 )t2 t3 , if Rs − Rl < 0 Q 1 = t1 Rs Z + t2 P1 − t1 M(Rl )Z l − M(P2 )t2 t3 , if (1 − α)Rs − Rl > 0 —

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Fig. 69.1 Results of modeling depending on the average salary and share of migrants

69.3 Conclusion The article considers the influence of using migrant labor on the assessment of socioeconomic impact of a large-scale project on the region. Moreover, such an influence has both positive impact and negative impact. Such an assessment should take into account not only economic benefits for investors but also possible changes in the life of the region. The decision-making process on the appropriateness of this project is proposed to be modeled as a matrix game. The decision-making process on the appropriateness of the project under consideration is discussed and proposed to be modeled as a decision table.

References 1. French, J., Blair-Stevens, C., McVey, D., Merritt, R.: Social Marketing and Public Health: Theory and Practice. Oxford University Press (2017) 2. Ilin, S.N., Koshel, I.S.: Evaluation of socio-economic benefits of innovative projects. Izvestiya MGTU “MAMI” 1(4), 111–117 (2013) 3. Jakubiak-Lasocka, J., Jakubczyk, M.: Cost-effectiveness versus cost-utility analyses: what are the motives behind using each and how do their results differ? Value Health Reg. Issues 4, 66–74 (2014) 4. Sartori, D., et al.: Guide to Cost-Benefit Analysis of Investment Projects. Economic appraisal tool for Cohesion Policy 2014–2020. https://ec.europa.eu/regional_policy/sources/docgener/ studies/pdf/cba_guide.pdf. Cited 5 March 2020 5. Tyuryukanova, E.: Labor migration to Russia. Demoskop weekly, vol. 315–316, January 2008. http://www.demoscope.ru/weekly/2008/0315/tema01.php. Cited 20 February 2020 6. World Health Organization, Baltussen, R., Taghreed, A., Tan-Torres Edejer, T., Hutubessy, R., et al.: Making choices in health: WHO guide to cost-effectiveness analysis. World Health Organization. https://apps.who.int/iris/handle/10665/42699. Cited 20 February 2020

Chapter 70

Structural Analysis of Directed Signed Networks Elizaveta Evmenova and Dmitry Gromov

Abstract Signed networks represent a particularly interesting class of complex networks with many applications in sociology, recommender, and voting systems. There are a couple of theories developed for the description of the structure of such networks, including the structural balance, status, and sentiment analysis. Most results devoted to signed networks are based on the assumption that the network under study is undirected. In this contribution we present initial results aimed at quantifying the amount to which real directed networks differ from undirected ones. We carry out our analysis for the network describing the Wikipedia adminship elections.

70.1 Introduction In the last decades the theory of complex networks has become an area of active research. There has been published dozens of monographs, see, e.g., [1, 2, 9] and a wealth of papers devoted to this topic. We note that the scope of complex networks theory is mostly restricted to the study of statistical properties of undirected and unweighted graphs. However, there are a number of applications that cannot be considered within this framework and require an extension of the baseline graph structure. In particular, there has been a growing interest in studying signed networks, i.e. the networks where two nodes are connected by a signed (i.e. weighted) relation. Signed graphs had been widely used in sociological applications, where one associates friendship with a positive relation and antagonism with a negative one. This line of research resulted in the theory of structural balance, see, e.g., [11, Ch. 9], [5], and references therein. In brief, the structural balance theory says that an unoriented signed triangle is stable if the product of all signs of edges is positive. Furthermore, E. Evmenova · D. Gromov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] E. Evmenova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_70

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the graph is said to be balanced if it is possible to decompose the set of vertices in a number of subsets such that within each subset all vertices are connected by positive relations while the edges connecting the vertices from different subsets are all negative. However, this result cannot be easily generalized to the case of oriented signed graphs despite some possible extensions were proposed, see, e.g., [4]. In contrast to that, the case of graphs with more than two weights has not been addressed within the context to the best of authors’ knowledge. In the last two decades a large amount of data related to different signed interaction networks have appeared, which largely stimulated a new wave of interest towards this subject. We mention highly influential papers [7, 8], where the authors analyze and systematize different approaches to the interpretation of the structure and evolution of such networks, including the status and sentiment analysis (see also [10]). However, all mentioned theories have the drawback that they mostly apply to undirected graphs while in practice the amity/enmity relations are not necessarily reciprocal. Transformation of a directed graph into an undirected one is accompanied by some information loss as we need to either introduce new edges (when transforming a single directed edge into an undirected one) or modify the weight of an edge (when transforming two reciprocal edges with different weights into a single undirected edge). In this contribution we present initial results aimed at developing a systematic approach to determining the extent by which a directed signed network differs from an undirected one. We apply our methods to the signed network describing the relationships between Wikipedia users that participated in administrator elections. This paper is organized as follows. In Sect. 70.2 we present the basic facts from graph theory that will be used later on; in Sect. 70.3 a number of graph-theoretic characteristics are introduced; an application of the developed methods is presented in Sect. 70.4. The paper ends with a conclusion.

70.2 Graph Theory Background and Terminology A directed weighted graph (weighted digraph) G is defined as a tuple G = (V, E, W ), where V is the set of vertices, E ⊆ V × V is the set of edges, and W : E → {w1 , . . . , wn } is a weight function that assigns a unique weight from a finite set to each edge e ∈ E. For a directed graph, (v1 , v2 ) ∈ E does not imply that (v2 , v1 ) ∈ E. Furthermore, if both edges ei j = (vi , v j ) and e ji = (v j , vi ) exist, we say that the edge e ji is reciprocal of ei j . We let N V denote the number of vertices and N E the number of edges in the graph, i.e. N V = |V | and N E = |E|. We will occasionally use the notation vert(G) and ed(G) to denote the number of vertices and edges of a graph G, respectively. / E for all vi ∈ V . The graph We consider graphs without self-loops, i.e. (vi , vi ) ∈ is said to be fully connected if any two distinct vertices are connected by an edge. In this case, N E = N V (N V − 1), which is the maximal possible number of edges, denoted by N¯ E .

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Let S ⊂ V be a subset of vertices V . We say that G S = (S, E S , W S ) is a subgraph of G generated by S if E S = E ∩ (S × S) and W S = W | E S . To avoid multiple counting of the same subgraph, we assume throughout the text  the vertices in the set  that S are ordered. With this convention, there are at most NkV subgraphs generated by sets of vertices consisting of exactly k ≤ N V elements. Finally, we let k (G) denote the set of all subgraphs of G generated by ordered subsets S ⊂ V containing exactly k elements. From now on, we will use small Greek letters, say, γ when specifically referring to subgraphs of the graph G. In practice, when considering different subgraphs in a graph we are interested in identifying the structures that are invariant to the way we enumerate the vertices. Such structures are closely related to the notion of graph isomorphism. We say that two graphs G  = (V  , E  , W  ) and G  = (V  , E  , W  ) are isomorphic, denoted by G  ∼ G  , if there exists a bijection  : V  → V  such that for all v1 , v2 ∈ V  , (v1 , v2 ) ∈ E  ⇒ ((v1 ), (v2 )) ∈ E  and for all (v1 , v2 ) ∈ E  , W  (v1 , v2 ) = W  ((v1 ), (v2 )). Two isomorphic graphs have the same number of vertices and edges and share a number of properties. Plainly speaking, two graphs are isomorphic if one graph can be transformed into another one without breaking or reweighting edges. In Fig. 70.1, a complete set of signed isomorphic digraphs is shown for illustration. The isomorphism relation is obviously an equivalence relation. Hence, for each γ ∈ k (G), it defines an equivalence class [γ ] = {γ  ∈ k (G)|γ  ∼ γ }. The set of all equivalence classes defines a partitioning of the set k (G). An important question consists in determining the cardinality of a given equivalence class γ , denoted by #[γ ], i.e. the number of graphs isomorphic to γ . To start with, we note that in a graph with k vertices, the vertices can be permuted in k! ways, so there are k! possible bijections of the set of vertices to itself. It remains to determine which of these permutations generate distinct graphs. To do so we use the notion of symmetry of a graph. Let σ (γ ) denote the number of transformations that map a graph to itself. Obviously, any graph has at least one symmetry that coincides with the identity transformation,

Fig. 70.1 A set of isomorphic graphs with 3 vertices

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Fig. 70.2 A digraph with 3 vertices, 3 edges, and 3 symmetries: identity, rotation by 2π/3, and rotation by 4π/3

thus σ (γ ) ≥ 1. Figure 70.2 shows a digraph γ such that σ (γ ) = 3. While for simple graphs like that shown in Fig. 70.2 the number of symmetries can be determined by a mere observation, for more complex graphs this can pose a serious computational challenge. We refer the interested reader to [3, Ch. 14] for a thorough discussion of graph symmetries and related problems within a group-theoretic context. Finally, the cardinality of the equivalence class [γ ] is computed as #[γ ] =

vert(γ )! . σ (γ )

We can easily observe that the graph shown in Fig. 70.2 has only two unique isomorphic forms, whereas the second form corresponds to the reflection of the graph shown in Fig. 70.2. Specific notation. In this paper, we use some notation adjusted to the needs of the considered problem. For a given graph γ , we let ed(γ , wi ) denote the number of edges with weight wi .

70.3 Statistical Analysis of Signed Graphs We will consider a slightly more general class of signed graphs, namely the graphs in which the set of weights consists of more than two elements. We recall that in a typical signed graph the set of weights is {−1, +1}. We allow for more weights to account for the cases where there can be an additional relation between the agents, say, when two agents are acquainted but do not experience any positive/negative feelings towards each other. We consider a number of statistical characteristics that will be used to analyze the structure of the graph. In doing so we will mostly concentrate on the properties related to reciprocal edges, i.e. two edges that connect two vertices in opposite directions, and the properties related to the triangles, i.e. the groups of 3 edges where any two edges are connected by at least one edge. To start with, we consider a number of rather general parameters. Density coefficient characterizes the ratio of edges in the graph to the number of possible edges: δ = NV (NN EV −1) . We will also consider a modification of the density

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  coefficient, namely the triangle density coefficient: δ = N / N3V , where N is the total number of triangles in the graph whereas each triangle is counted only once (i.e. we consider only the triangles with ordered sets of nodes). A closely related characteristic is the average degree < k >= NNVE . We recall that the vertex degree is the number of outgoing vertices. The following characteristics are introduced to describe the amount to which an edge is likely to be accompanied by a reciprocal edge of the same or different type. Edge mutuality, denoted as βwi , is the ratio of the number of edges with weight wi ∈ W accompanied by a reciprocal one to the total number of edges with the same weight. If the edge mutuality coefficient is sufficiently high, then a relation of a particular type is likely to be accompanied by a mutual relation (not necessarily of the same type). Such mutual relations can be characterized using the following coefficient. Reciprocity coefficient ρ(wi , w j ) ∈ [0, 1] measures the probability that given an edge with weight w j , there will be a reciprocal edge with weight wi . The reciprocity coefficient is computed as conditional probability  ρ(wi , w j ) = P(wi |w j ) =

G S ,|S|=2



ed(G S , wi ) · ed(G S , w j ) ed(G, w j )

,

where the sum is taken over all subgraphs G S generated by two-element ordered subsets S ⊂ V . Note that the reciprocity coefficient is not symmetric, i.e. ρ(wi , w j ) = ρ(w j , wi ). The relation between mutuality coefficient and reciprocity coefficient can be expressed as  ρ(wi , w j ). (70.1) βj = wi ∈W

To illustrate the introduced coefficients we consider the graph shown in Fig. 70.3. The set of weights consists of two elements, denoted by + and −. For this graph, the reciprocity coefficients are ρ(+, +) = 21 , ρ(+, −) = 21 , ρ(−, +) = 14 , and ρ(−, −) = 0. The mutuality coefficients are β+ = 43 and β− = 21 . One can check that the relation (70.1) holds.

Fig. 70.3 A signed digraph with W = {+, −}

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Finally, since we wish to consider different types of triangles (i.e. balanced or unbalanced ones), the total number of triangles of different types has to be determined. In doing so one should not make a distinction between isomorphic triangles as these represent the same structure up to the vertices relabeling. It turns out that already for the triangles with weights of three types there can be quite a lot of isomorphic variants as shown in Table 70.4 (third row). We note that—to the best of authors’ knowledge—there is no analytical expression for the number of all isomorphism classes of a given directed weighted graph. Therefore, an algorithm for enumerating all isomorphism classes was developed and implemented.

70.4 Results of the Wikipedia Election Graph Analysis In his section we present data obtained for the graph representing the results of the Wikipedia administrator election. Wikipedia is a free encyclopedia, whose operation is supervised and controlled by a relatively small number of administrators, who are users with additional rights and access to technical features that aid in maintenance. If a regular user wants to become an administrator, she/he fills a Request for Adminship (RfA). Subsequently, the Wikipedia community publicly votes on whom to promote to adminship. A person can cast one of three types of votes: for (positive), against (negative), and neutral. Note that a vote can be cast both by existing admins and by ordinary Wikipedia users. In [6], a complete dump of Wikipedia page edit history from January 3, 2008 was used to extract the information about 2,800 elections. Out of these 1,200 elections resulted in a successful promotion, while about 1,500 elections did not result in the promotion. This information about the elections is represented as a graph. The vertices of the graph are the users (both admins and regular users) and the edges correspond to the votes cast. Table 70.1 presents some basic numbers about the graph. We immediately compute the density coefficient δ = 0.0021 and the triangle density δ = 8.6214 · 10−6 , which implies that the network is pretty sparse. However, the relative degree is < k >= 15.3026, which indicates sufficiently large voting activity. Table 70.2 shows the values of the edge mutuality coefficient for three weights. Note that we use symbols “+”, “−”, and “∗” to denote the positive, negative, and Table 70.1 Basic statistics Number of Total number vertices, N V of edges, N E 7194

111247

Number of Number of Number of positive edges negative edges neutral edges

Number of triangles, N

81862

534758

22497

Table 70.2 Values of the edge mutuality coefficient β+ β− 0.1331

0.0344

6888

β∗ 0.0553

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Table 70.3 Values of the reciprocity coefficient ρ(+, +) ρ(+, −) ρ(+, ∗) ρ(−, +) ρ(−, −) ρ(−, ∗) 0.0629

0.0156

0.03615

0.0043

0.0083

Table 70.4 Statistics on triangles Number of edges 3 4 Total number of triangles Fraction of triangles Number of isomorphism classes Number of isomorphism classes in the network

0.0067

ρ(∗, +)

ρ(∗, −)

ρ(∗, ∗)

0.003

0.0021

0.0061

5

6

407677

106297

18203

2581

0.7624

0.1988

0.034

0.0048

38

162

243

130

38

159

188

61

neutral votes. One can see that while the fraction of mutual votes is relatively small for negative votes, it becomes more tangible for the case of positive votes. To analyze the structure of mutual votes we compute the reciprocity coefficient ρ(·, ·) as shown in Table 70.3. One can observe that while there is a relatively high probability that a positive edge will be accompanied by a reciprocal edge of the same type, the probabilities of having a reciprocal edge of different kind are not negligible. For instance, the probability that taken a negative edge, its reciprocal will be positive is twice as big as the probability that there will be a double negative edge. This emphasizes the need for more elaborate analysis taking into account the directed and inhomogeneous character of inter-agent relations. Finally, Table 70.4 contains the data describing the triangle structure of the Wikipedia election graph. The average number of edges in a triangle can be computed to be < e >= 3.2812. While the majority of triangles comprised 3 edges, about a quarter of all triangles have 4–6 edges. This analysis shows that there is a substantial number of triangles with reciprocal edges that support the claim that an additional analysis of the structure of such connections has to be analyzed in more detail. These results will be reported in a subsequent work.

70.5 Conclusions In this paper we introduced a number of characteristics to describe a weighted directed graph and applied them to the analysis of the Wikipedia election graph. We showed

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that there is a nonnegligible number of triangles that cannot be adequately considered within the undirected graph framework. Acknowledgements The authors are grateful to the anonymous reviewers for their helpful comments.

References 1. Chkhartishvili, A.G., Gubanov, D.A., Novikov, D.A.: Social Networks: Models of Information Influence, Control and Confrontation, Studies in Systems, Decision and Control, vol. 189. Springer (2018) 2. Easley, D., Kleinberg, J.: Networks, Crowds, and Markets. Cambridge University Press (2010) 3. Harary, F.: Graph Theory. Addison-Wesley (1969) 4. Higuchi, Y., Sato, I.: A balanced signed digraph. Graphs Comb. 31, 2215–2230 (2015). https:// doi.org/10.1007/s00373-014-1496-z 5. Kunegis, J.: Applications of structural balance in signed social networks. arXiv: 1402.6865 (2014). https://arxiv.org/abs/1402.6865 6. Leskovec, J.: Wikipedia adminship election data. Stanford Network Analysis Project. https:// snap.stanford.edu/data/wiki-Elec.html 7. Leskovec, J., Huttenlocher, D., Kleinberg, J.: Predicting positive and negative links in online social networks. In: Proceedings of the 19th International Conference on World Wide Web, pp. 641–650 (2010). https://doi.org/10.1145/1772690.1772756 8. Leskovec, J., Huttenlocher, D., Kleinberg, J.: Signed networks in social media. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, pp. 1361–1370 (2010). https://doi.org/10.1145/1753326.1753532 9. Newman, M.: Networks, 2nd edn. Oxford University Press (2018) 10. Pang, B., Lee, L.: Opinion mining and sentiment analysis. Found. Trends® Inf. Retr. 2(1–2), 1–135 (2008) 11. Roberts, F.S.: Graph theory and its applications to problems of society. Regional Conference Series in Applied Mathematics, vol. 29. SIAM (1978)

Chapter 71

Dynamic Input–Output Models: Analysis of Possibilities and Trends Control Nikolay V. Smirnov, Viktor P. Peresada, Kirill V. Postnov, Tatiana E. Smirnova, and Yefim V. Zholobov

Abstract The input–output (IO) models proposed by W.W. Leontief are an effective tool for scientific modeling of various economic processes. At the same time, dynamic IO models are of particular importance. They are used to analyze macroeconomic trends. The authors of this work are confident that the theoretical and applied results of modern mathematical control theory can be effectively used in dynamic IO models. It is shown that the process of implementing investment programs is equivalent to the problem of constructing program controls, and their corrections in the presence of some disturbances can be modeled as problems of synthesis of stabilizing feedbacks. Moreover, the notions of an investment scenario and a group of acceptable scenarios are introduced. In the framework of the proposed model, the problem of choosing the structure of the control system is discussed. The results of numerical experiments are presented. In conclusion, the problem of multi-program control is formulated.

71.1 Introduction It is well known that the economy of any state is a set of interconnected industries. The composition of sectors of the economy and the nature of their interconnections are constantly changing under the influence of continuously developing and deepening N. V. Smirnov (B) · K. V. Postnov · T. E. Smirnova · Y. V. Zholobov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] K. V. Postnov e-mail: [email protected] T. E. Smirnova e-mail: [email protected] Y. V. Zholobov e-mail: [email protected] V. P. Peresada ZAO “STO Inforesurs”, 9, ul. Akademika Baykova, St. Petersburg 199427, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_71

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processes of division and cooperation of social labor. Each industry consumes products of other industries, and, in turn, their products are consumed by households. In world practice, intersectoral balances are widely used to identify intersectoral connections, analyze and form the structure of the economy for the forecast period. W.W. Leontief (winner of the 1973 Nobel Prize in Economics) proposed the input– output (IO) model, which describes the interaction of industries at the macro level with mathematical strictness [9, 10]. This model continues to evolve and attract the attention of a wide range of professionals around the world. These days the IO model is used for the scientific analysis of the state of regional socio-economic systems, as well as macroeconomic trends in these systems (see, e.g., [4, 6, 16]). The International Input-Output Association [8] has existed for more than 30 years and is currently active. The European Union has created and is developing the World Input-Output Database (WIOD). The WIOD website [23] contains national input–output tables for the economies of some 40 of the world’s most developed countries. These countries together account for more than 90% of world production. The WIOD database provides a unique opportunity for any analyst to try his hand at modeling and forecasting macroeconomic trends. Here are some examples of applications of IO models. The monograph Raa [15] gives a general overview of the possibilities of such models for static analysis. Most notably, it demonstrates how different problems are interconnected. However, Chap. 13 is the only one devoted to dynamic models in this book. The publications [7, 16, 24, 25] provide examples of forecasting, planning, and management based on the dynamic IO model. The author of [11] uses elements of the optimal control theory for a linear discrete IO model. Here, the final consumption acts as the vector of control parameters. In our previous works, a fairly general approach to the construction of dynamic IO models was developed [5, 13, 19]. The said works give algorithms for and examples of the identification of the system’s coefficients. We will employ these results. In this paper, we consider the problem of program investment control and simultaneous correction of the investment process in real conditions of economic development.

71.2 Types of Input–Output Models and Their Capabilities 71.2.1 Static Input–Output Model The basic, now classic, example of the IO table was developed by Leontief for the US economy in 1958 [10]. Since then, this table has not changed much. There are four main quadrants in it. The first one is (n × n)-matrix of production with elements pi j . The elements sum of each column of this matrix is the internal consumption P p j in j-industry. The second quadrant is n-dimensional column of the final consumption Y with elements Yi . The third quadrant is (n + 1)th value-added row V . Added value V j created in every industry is the difference between the annual output I j = P j I n j and the internal consumption P p j , i.e. V j = I j − P p j . Added value V j consists of

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three parts: the compensation of employees W j , taxes T x j , which are set by the government, and the net operating surplus Pr h j . Finally, the fourth element of the table is the state budget Vb . Under the gross domestic product (GDP) we mean the sum of added values and budget. In a balanced economy, the total value-added equals total consumption V = Y . All these are just general notations that will be needed below. The possibilities of the static IO model are widely presented in [13, 15].

71.2.2 Dynamic Input–Output Model There are several approaches to the construction of dynamic input–output models [4, 6]. Here we will use an approach presented in [13] and consider the following system of differential equations: X˙ = P X,

P = M R,

(71.1)

where ⎞ Y r1 ⎜ Y r2 ⎟ ⎟ ⎜ ⎜ .. ⎟ , R=⎜ . ⎟ ⎟ ⎜ ⎝ rn1 rn1 · · · rnn Y rn ⎠ 1 − r p1 1 − r p2 · · · 1 − r pn rg ⎛

r11 r21 .. .

r12 r22 .. .

··· ... .. .

r1n r2n .. .

⎛ ⎜ ⎜ M =⎜ ⎝

m1 Fe1

.. . 0 0

··· .. . ··· ···

0 .. .

0 .. . 0

0

m n+1 Feb

mn Fen

⎞ ⎟ ⎟ ⎟, ⎠

X = (X 1 , . . . , X n , X n+1 )T is a vector of annual outputs, X n+1 is the GDP, ri j = pi j / X j are relative prices, pi j are elements of Leontief’s IO table, Y ri = Yi / X n+1 are relative final consumptions, Fe j are capital intensities of each industry, including Feb for the consumption sphere, values r p1 , . . . , r pn , rg, m 1 , . . . , m n+1 are macroeconomic parameters [13]. Controlled dynamic model. Using investments in the economic system as control parameters, we get a controlled IO system X˙ = P X + Qu, 0 ≤ u j ≤ Lu j ,

(71.2)

where u = (u 1 , . . . , u n+1 )T is an investments vector (control), Lu j are restrictions constants, and Q is diagonal matrix describing the investments structure.

71.2.3 Macroeconomic Trends Control The main goal of this section is to show how mathematical control theory could be used for planning and correcting macroeconomic trends.

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Formulation of the problem. First of all, we note that one of the macroeconomic development scenarios is the planned growth of the regional economy. This problem can be interpreted as a program control problem [1, 17, 21, 27] or an optimal control problem [2, 3, 14, 22]. Suppose that the initial level of production at t = 0 is X 0 , and, by the moment t = T , it must be increased to the level X 1 . Taking into account the solution representation of the system (71.1), we obtain the integral equation ⎛ X 1 = S(T ) ⎝ X 0 +

T

⎞ S −1 (τ )Qu(τ )dτ ⎠ ,

(71.3)

0

where S(t) is the fundamental matrix of the system (71.1). Construction of a program control. To find a program control function from Eq. (71.3), we must perform the following steps. First of all, we should construct an auxiliary matrix B(t) = S −1 (t)Q, and compute T D=

B(τ )B T(τ )dτ, d = S −1 (T )X 1 − X 0 ,

0

then find a vector z as a solution of the system of linear equations Dz = d. Finally, we obtain the solution of integral equation (71.3) as follows: u(t) = B T (t)z.

(71.4)

Vector function (71.4) is the investment plan for dynamic IO model (71.2), which provides the given trend of macroeconomic growth. Correction of investment programs. When implementing the investment plan in real time, one must plan for random disruptions which are possible due to unaccounted factors. In this case, a plan correction or adjustment is necessary. In control theory, this is equivalent to the implementation of stabilizing control for system (71.2). Let the investment program u p (t) be implemented. Then, under this control function, system (71.2) has a particular solution X p (t) as a planned output, and X p (T ) = X 1 . Correction of the planned mode u p (t), X p (t) of the economy functioning can be provided by the linear feedback u s = C(X (t) − X p (t)), where X (t) is a real process. Finally we obtain the resulting control function u(t, X (t), X p (t)) = u p (t) + u s = u p (t) + C(X (t) − X p (t)).

(71.5)

Function (71.5) describes the investment program and its correction for each industry.

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71.2.4 Numerical Experiment Identification of the IO model. Input data for the input–output model can be acquired from the World Input-Output Database (WIOD) [23] and the Organization for Economic Co-operation and Development (OECD) [12]. These resources contain IO tables for some of the most developed economies in the world, including the countries of the European Union, the USA, and Russia. The IO table for each country describes the interaction of 56 industries. This is quite a lot of data, so we applied the standard industry aggregation algorithm proposed by Leontief [9, 10], and obtained a model of (4 × 4)-dimensions. The following matrix P is constructed for Russia’s economy in 2013 ⎛

0.0594 ⎜ 0.0297 P=⎜ ⎝ 0.0005 0.0605

0.0195 0.0418 0.0015 0.0819

0.0135 0.0305 0.0039 0.0946

⎞ 0.0408 0.0762 ⎟ ⎟, 0.0264 ⎠ 0.0088

⎛

0 ⎜0 Q=⎜ ⎝0 1

0 1 0 0

0 0 1 0

⎞ 1 0⎟ ⎟. 0⎠ 0

(71.6)

Industries selected as the basic ones were industry as a whole, infrastructure, livelihoods, and GDP. Choice of the structure of the matrix Q. While solving the problem of program (planned) growth, we were faced with the problem of selecting the structure of the matrix Q. It determines the investment directions for each industry. From the theoretical point of view, system (71.2) is completely controllable even when Q =

T 1 0 0 0 , i.e. investments in the first sector lead to structural changes in other sectors, which finally bring the system to the given final state X 1 . However, numerical experiments have shown that in this case, programmed (calculated) regimes cannot be applied to an actual economy as unacceptable due to major fluctuations in industry outputs. It became clear that the structure of the matrix of control coefficients should in some sense correspond to the intrinsic dynamics of the system (71.1), i.e. dynamics when no control is applied. As a result, we selected the matrix Q to have the structure reflected in (71.6). Results of a numerical experiment. The vector of initial data X 0 was obtained using information from resources [12, 23]. It has the form

X 0 = 1336370.1 2041165 488413.7 2297595.6 . All values are in millions of USD. To select the planned level X 1 , we used the following algorithm. First, a solution of system (71.1) with initial condition X (0) = X 0 was constructed. Then all of its components were increased by 2% (planned growth). Thus, the vector X 1 was obtained for the program control problem. Afterwards, a program control (71.4) for vectors X 0 , X 1 was constructed. Figures 71.1 and 71.2 show the integral curves of system (71.2) closed by the program control found.

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Fig. 71.1 Investment program (71.5) and its correction for one random disturbance

Fig. 71.2 Investment program (71.5) and its correction for two random disturbances

At the next stage, during the implementation of program controls, two situations of random disturbances in the first phase variable were simulated. In the first case, a perturbation arose in the middle of the control interval, at t = 0.5, and in the second, also at the initial moment of time. To compensate for deviations from the program mode, a stabilizing control (71.5) was constructed (see Figs. 71.1 and 71.2).

71.3 Conclusions and Continued Research The discussion of the possibilities of the IO models and the mathematical control theory can be continued by analyzing nonlinear systems. Consider variants for continuing research in this direction. So in [5, 13] it is shown that exogenous macroeconomic parameters, such as the profit tax rate or rates of wages, can be considered as controls. In this case, the dynamic IO model becomes nonlinear. In [5], one particular problem of the optimal control of the profit tax rate in a regional economy was solved.

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Consider another important problem. It is known that for effective management it is important to have several possible scenarios for the development of the regional economy. Each scenario can be simulated by the control function u p (t) (71.4) and the corresponding program mode X p (t), which is a solution of the closed system (71.2) and (71.4). There may be several such scenarios u pj (t), X pj (t), j = 1, . . . , N . Next, it is necessary to construct one common control, which ensures the sustainable implementation of each scenario, depending on the current situation in the economy. Obviously, the current situation is connected with the real initial conditions, therefore, the control system must implement the scenario that corresponds to the initial data. Such a control can be constructed as follows [18, 20, 26]: u(X, t) =

N 

u pj + C(X − X pj ) − 2u pj

j=1

p j (X, t) =

N  i=1,i= j

(X − X ) p j (X, t), pj (X pj − X pi )2 i=1,i = j N 

(X pj − X pi )

(X − X pi )2 , (X pj − X pi )2

(71.7)

j = 1, N .

Control function (71.7) is the interpolation polynomial and has the property u(X pj (t), t) ≡ u pj (t). System (71.2) closed by control function (71.7) has a family of particular solutions (program modes) X p1 (t), . . . , X pN (t), that are asymptotically stable. Controls of the form (71.7) are called multi-program controls. The reliability of the implementation of each scenario depends on the asymptotic stability domain of each particular solution X pj (t). Therefore, algorithms for evaluating these domains are important for applications.

References 1. Andreev, Y.N.: Control of Finite Linear Objects. Nauka, Moscow (1976). (In Russian) 2. Baranov, O.V., Smirnov, N.V., Smirnova, T.E., Zholobov, Y.V.: Design of a quadrocopter with pid-controlled fail-safe algorithm. J. Wirel. Mobile Netw. Ubiquitous Comput. Dependable Appl. 11(2), 23–33 (2020). https://doi.org/10.22667/JOWUA.2020.06.30.023 3. Boiko, A.V., Smirnov, N.V.: Approach to optimal control in the economic growth model with a nonlinear production function. ACM International Conference Proceeding Series, pp. 85–89 (2018). https://doi.org/10.1145/3274856.3274874 4. Fedoseev, V.V., Garmash, A.N., Daiitbegov, D.M., et al.: Economic-Mathematical Methods and Applied Models. UNITI, Moscow (1999). (In Russian) 5. Girdyuk, D.V., Smirnov, N.V., Smirnova, T.E.: Optimal control of the profit tax rate based on the nonlinear dynamic input-output model. ACM International Conference Proceeding Series, pp. 80–84 (2018). https://doi.org/10.1145/3274856.3274873 6. Granberg, A.G.: Dynamical Models of the Economy. Economics, Moscow (1985). (In Russian)

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7. Hoekstra, R., Janssen, M.A.: Environmental responsibility and policy in a two-country dynamic input-output model. Econ. Syst. Res. 18(1), 61–84 (2006). https://doi.org/10.1080/ 09535310500440894 8. International Input-Output Association (2020). http://www.iioa.org/ 9. Leontief, W.W.: Input-Output Economics. Oxford University Press, New York (1986) 10. Leontief, W.W.: Essays in Economics: Theories, Theorizing, Facts, and Policies. Politizdat, Moscow (1990). (In Russian) 11. Livesey, D.A.: Control theory and input-output analysis. Int. J. Syst. Sci. 2(3), 307–318 (1971). https://doi.org/10.1080/00207727108920197 12. Organisation for Economic Co-operation and Development (2020). https://data.worldbank.org 13. Peresada, V.P., Smirnov, N.V., Smirnova, T.E.: Static and Dynamic Models of Multi-commodity Economy: Textbook. Publishing house Fedorova G.V, St. Petersburg, Russia (2017). (In Russian) 14. Popkov, A.S., Smirnov, N.V., Smirnova, T.E.: On modification of the positional optimization method for a class of nonlinear systems. ACM International Conference Proceeding Series, pp. 46–51 (2018). https://doi.org/10.1145/3274856.3274866 15. ten Raa, T. (ed.): Handbook of Input-Output Analysis. Edward Elgar Publishing, Cheltenham, UK; Northampton, MA, USA (2017) 16. Ryaboshlyk, V.: A dynamic input-output model with explicit new and old technologies: an application to the UK. Econ. Syst. Res. 18(2), 183–203 (2006). https://doi.org/10.1080/ 09535310600653040 17. Smirnov, N.V., Smirnov, A.N., Smirnov, M.N., Smirnova, M.A.: Combined control synthesis algorithm. In: 2017 Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V.F. Demyanov), no. 7974014 in CNSA 2017. IEEE Inc. (2017). https://doi.org/ 10.1109/CNSA.2017.7974014 18. Smirnov, N.V., Smirnova, T.E.: The stabilization of a family of programmed motions of the bilinear non-stationary system. Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya (2), 70–75 (1998). (in Russian) 19. Smirnov, N.V., Smirnova, T.E., Volik, K.M., Peresada, V.P.: Modelling of investment programs based on the impulse program controls. In: Petrosyan, L.A., Zhabko, A.P. (eds.) 2015 International Conference on “Stability and Control Processes” in Memory of V. I. Zubov, SCP 2015, pp. 494–497. IEEE Inc. (2015). https://doi.org/10.1109/SCP.2015.7342182 20. Smirnov, N.V., Smirnova, T.Y.: The synthesis of multi-programme controls in bilinear systems. J. Appl. Math. Mech. 64(6), 891–894 (2000). https://doi.org/10.1016/S0021-8928(00)00119-2 21. Smirnova, M.A., Smirnov, M.N., Smirnova, T.E., Smirnov, N.V.: Astaticism in tracking control systems. In: Lecture Notes in Engineering and Computer Science, International Multiconference of Engineers and Computer Scientists 2016, IMECS 2016, vol. 1, pp. 200–208. Newswood Limited (2016) 22. Smirnova, M.A., Smirnov, M.N., Smirnova, T.E., Smirnov, N.V.: Optimization of the size of minimal invariant ellipsoid with providing the desired modal properties. In: Lecture Notes in Engineering and Computer Science, International Multiconference of Engineers and Computer Scientists 2016, IMECS 2016, vol. 1, pp. 238–241. Newswood Limited (2016) 23. World Input-Output Database (2020). http://www.wiod.org/ 24. Xu, W., Wang, Z., Hong, L., He, L., Chen, X.: The uncertainty recovery analysis for interdependent infrastructure systems using the dynamic inoperability input-output model. Int. J. Syst. Sci. 46(7), 1299–1306 (2015). https://doi.org/10.1080/00207721.2013.822121 25. Zhang, J.S.: A multi-sector nonlinear dynamic input-output model with human capital. Econ. Syst. Res. 20(2), 223–237 (2008). https://doi.org/10.1080/09535310802075463 26. Zubov, V.I.: Synthesis of multiprogram stable controls. Rep. Acad. Sci. USSR 318(2), 274–277 (1991). (In Russian) 27. Zubov, V.I.: Lectures on Control Theory. Lan’, Moscow (2009). (In Russian)

Chapter 72

Queueing Systems with Opposite Queues Anastasija Glushakova and Alexander Kovshov

Abstract The queueing system with one server and two arrival streams is under consideration. Arrival streams are independent and Poisson with different rates. Thus, the queueing system accepts two types of jobs. If the server is busy, jobs are sent to the queue. Each job type has its own queue and waiting room. The server can serve two jobs simultaneously. In this pair of jobs, one job must be of first type, and the other must be of second type. The server cannot serve only one job, and it cannot serve two jobs of the same type. The service time is distributed exponentially. In this research, formulas were obtained that calculate the probability of all states of the system for two particular cases. Two computer programs were created that calculate the probabilities of all states of the system in the general case.

72.1 Introduction Let us consider the queueing system. The system has one server which serves two types of jobs A and B. Jobs of each type arrive independently according to a Poisson process. Poisson process for type A is of rate λ and the Poisson process for type B is of rate μ. The server accepts two jobs for servicing at the same time. It serves each job in this pair for an equal amount of time. Both jobs leave the queueing system immediately after the service is completed. The jobs in the servicing pair should be of different types. One of the jobs is of A type, and the second is of B type. The service time is distributed exponentially with the probability density function ηe−ηt . This means that the server can serve η job pairs per unit of time if it runs continuously. The mathematical expectation of the service time is 1/η. If the server is busy or it cannot accept an arriving job to serve immediately due to the absence of the other type of job, the arriving job is sent to the queue. There are two queues for each type A. Glushakova · A. Kovshov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. Glushakova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_72

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of job. Let it be a queue A and a queue B. Let the queue A has a waiting room with length n − 1 and the queue B has a waiting room with length m − 1. If there are no jobs in the system and a job of the type A or B arrives in the system, it will be placed on the server and will wait for the arrival of a different type of task. So the maximal number of A type jobs in the queueing system is equal to n and the maximal number of B type jobs in the queueing system is equal to m. If a job of type A arrives in the queueing system in a time when n jobs of type A are already in the queueing system, so the waiting room of queue A is full, this job will leave the queueing system without service. Similarly, a job of type B will not be served, if in its arrival time there are already m jobs of type B in the queueing system. Let us say that the system is in state (i, j) when it contains i jobs of type A and j jobs of type B. This means that i − 1 jobs are in the queue A and j − 1 jobs in the queue B. Two other jobs, a job of type A and a job of type B, are serving on the server. When the server finishes serving a pair of jobs and both queues are not empty, the server immediately starts serving the next pair of jobs. If at least one of the queues is empty, the server will wait for a job of the missing type to arrive. The diagram below shows possible state changes. λ

λ

λ

λ

(0, m) −→ (1, m) −→ (2, m) −→ · · · −→ (n, m)  ⏐

μ⏐

η 

 ⏐

μ⏐

η 

 ⏐

μ⏐

η η  ··· 

 ⏐

μ⏐

............................................................     μ⏐ η μ⏐ η μ⏐ η η μ⏐ ···  ⏐  ⏐  ⏐  ⏐ λ

λ

λ

λ

(0, 2) −→ (1, 2) −→ (2, 2) −→ · · · −→ (n, 2)  ⏐

μ⏐

η 

 ⏐

μ⏐

λ

η 

 ⏐

μ⏐

λ

η η  ···  λ

 ⏐

μ⏐

λ

(0, 1) −→ (1, 1) −→ (2, 1) −→ · · · −→ (n, 1)  ⏐

μ⏐

η  λ

 ⏐

μ⏐

η  λ

 ⏐

μ⏐

η η  ···  λ

 ⏐

μ⏐

λ

(0, 0) −→ (1, 0) −→ (2, 0) −→ · · · −→ (n, 0)

This state transitions diagram looks like one for a tandem queueing system with two stations described by Zijm [1], but there is a difference. Here the zero string is the lowest line, but in the tandem system the zero string is the highest. Also, in the tandem system there are unlimited waiting rooms. This makes it impossible to use the results obtained for tandem queues in the case of opposite queues.

72 Queueing Systems with Opposite Queues

639

Let the queueing system is in state (i, j), where 0 < i < n and 0 < j < m. There are three possible events that change this state of the queueing system. 1. If the server finishes servicing a pair of jobs, they leave the queueing system and the state changes to (i − 1, j − 1). Such events occur with a frequency of η. 2. If the job of type A arrives, it will be added to queue A. The state of the queueing system changes to (i + 1, j). Such events occur with a frequency of λ. 3. If a type B job arrives, it will be added to queue B. The state changes to (i, j + 1). Such events occur with a frequency of μ. Denote by p(i, j) the probability of finding the queueing system in the state (i, j). Then the probability pˆ of denial of service for the job of type A and B will be calculated by the following formulas: p(A) ˆ =

m 

p(n, j) ,

p(B) ˆ =

j=0

n 

p(i,m) .

i=0

So we can obtain the number of jobs that are entered into the queueing system per unit of time ⎛ λ˜ = λ · ⎝1 −

m 

⎞



p(n, j) ⎠ ,

μ˜ = μ · 1 −

j=0

n 

p(i,m) .

i=0

The average number of jobs in the queueing system will be calculated, so n¯ =

n  m 

i · p(i, j) ,

m¯ =

i=0 j=0

n  m 

j · p(i, j) .

i=0 j=0

According to Little’s law, we can get formulas for the average time that a job spends in a queueing system m n  

n¯ t¯(A) = = λ˜

i · p(i, j)

i=0 j=0

⎛

λ · ⎝1 −

m 

⎞, p(n, j) ⎠

j=0 n  m 

m¯ = t¯(B) = μ˜

j · p(i, j)

i=0 j=0



μ· 1−

n  i=0

. p(i,m)

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Queueing systems with a few queues have been studied before. For example in [2] systems with parallel queues were considered. Here we consider a queueing system with two queues which contain requests that cannot be served separately. So these queues seem to be moving towards each other. An example of two opposite queues was described by Foreest [3] in the taxi problem near a large hotel. In that example, the service time is zero, so the state space is one-dimensional, and the queueing system is reduced to the case of M/M/1.

72.2 Queueing System Without Waiting Rooms Consider a queueing system such that n = 1, m = 1. The state change diagram will look like the following: λ

(0, 1) −→ (1, 1)  ⏐

μ⏐

η 

 ⏐

μ⏐

λ

(0, 0) −→ (1, 0) Since the intervals between the three types of events described on Sect. 72.1 are independent and exponentially distributed, and the queueing system is steady-state, we can write the system of equations for state probabilities: ⎧ (λ + μ) p(0,0) ⎪ ⎪ ⎨ μp(1,0) λp(0,1) ⎪ ⎪ ⎩ ηp(1,1)

= = = =

ηp(1,1) λp(0,0) μp(0,0) λp(0,1) + μp(1,0) .

The rank of this system is 3, so we cannot solve it, but we can express all probabilities in p(0,0) . λ+μ p(0,0) , η λ = p(0,0) , μ μ = p(0,0) . λ

p(1,1) = p(1,0) p(0,1)

Since the sum of all state probabilities is 1, we can find expressions for them p(0,0) +

λ+μ λ μ p(0,0) + p(0,0) + p(0,0) = 1, η μ λ

(72.1)

72 Queueing Systems with Opposite Queues

hence p(0,0) =

641

1 . λ μ λ+μ + + 1+ η μ λ

(72.2)

Substituting (72.2) in Eqs. (72.1) we obtain formulas for all state probabilities. λ+μ ,  λ μ λ+μ + + η 1+ η μ λ λ , =  λ μ λ+μ + + μ 1+ η μ λ μ . =  λ μ λ+μ + + λ 1+ η μ λ

p(1,1) =

p(1,0)

p(0,1)

72.3 Queueing System with Unit Waiting Rooms Consider a queueing system with n = 2, m = 2. Here is its state transitions diagram: λ

λ

(0, 2) −→ (1, 2) −→ (2, 2)  ⏐

μ⏐

 ⏐

μ⏐

η  λ

η 

 ⏐

μ⏐

λ

(0, 1) −→ (1, 1) −→ (2, 1)  ⏐

μ⏐

 ⏐

μ⏐

η  λ

η 

 ⏐

μ⏐

λ

(0, 0) −→ (1, 0) −→ (2, 0). We can write the following equations for the system state probabilities: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(λ + μ) p(0,0) (λ + μ) p(1,0) μp(2,0) (λ + μ) p(0,1) (λ + μ + η) p(1,1) ⎪ ⎪ (μ + η) p(2,1) ⎪ ⎪ ⎪ ⎪ λp(0,2) ⎪ ⎪ ⎪ ⎪ (λ + η) p(1,2) ⎪ ⎪ ⎩ ηp(2,2)

= = = = = = = = =

ηp(1,1) , λp(0,0) + ηp(2,1) , λp(1,0) , μp(0,0) + ηp(1,2) , λp(0,1) + μp(1,0) + ηp(2,2) , λp(1,1) + μp(2,0) , μp(0,1) , λp(0,2) + μp(1,1) , λp(1,2) + μp(2,1) .

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Expressions for all probabilities in p(0,0) are derived from these equations: λ(λ + 2μ + η) · p(0,0) , μ(λ + μ + η) λ2 (λ + 2μ + η) · p(0,0) , = 2 μ (λ + μ + η) μ(2λ + μ + η) · p(0,0) , = λ(λ + μ + η) λ+μ = · p(0,0) , η   λ λ λ λ(λ + 2μ + η) λ + + · − · p(0,0) , = η μ η μ μ(λ + μ + η)   μ μ μ μ μ(2λ + μ + η) + + · − · p(0,0) , = λ η λ η λ(λ + μ + η) 

 λ + μ 2 λ + μ λ(λ + 2μ + η) + μ(2λ + μ + η) − = + · p(0,0) , η η η(λ + μ + η)

p(1,0) = p(2,0) p(0,1) p(1,1) p(2,1) p(2,1) p(2,2)

Substituting these equations into n  m 

p(i, j) = 1,

i=0 j=0

we can solve it with respect to p(0,0) and obtain expressions for all state probabilities. ρ22 + ρ3 ρ4 ρ24 + ρ5 , p(0,2) = , p(2,0) = , ρ6 + ρ7 + ρ8 ρ6 + ρ7 + ρ8 ρ6 + ρ7 + ρ8 ρ2 + ρ4 ρ1 ρ2 ρ3 (1 + ρ2 ) ρ22 + ρ5 , p(1,1) = , p(1,0) = , p(0,1) = ρ6 + ρ7 + ρ8 ρ6 + ρ7 + ρ8 ρ6 + ρ7 + ρ8   ρ 2 ρ3 (ρ1 + ρ2 + ρ1 ρ2 ) − ρ2 ρ5 ρ3 (ρ1 + ρ2 + ρ1 ρ2 ) − ρ2 ρ4 , p(1,2) = , p(2,1) = 2 ρ6 + ρ7 + ρ8 ρ6 + ρ7 + ρ8 ρ1 ρ2 ρ3 (1 + 2ρ1 + ρ2 + ρ1 ρ2 ) − ρ1 ρ2 (ρ2 ρ5 + ρ4 ) + ρ12 ρ3 , p(2,2) = ρ6 + ρ7 + ρ8 p(0,0) =

where ρ1 =

λ λ , ρ2 = , ρ3 = λ + μ + η, ρ4 = 2λ + μ + η, ρ5 = λ + 2μ + η, η μ

ρ6 = ρ1 ρ2 ρ3 (3 + 2ρ1 + 3ρ2 + ρ1 ρ2 + ρ22 ),

ρ7 = −ρ1 ρ2 (ρ2 ρ5 + ρ4 )

ρ8 = ρ1 ρ3 (1 + ρ1 ) + ρ2 ρ3 (1 + ρ2 + ρ22 ) + ρ24 ρ5 + ρ4 .

72 Queueing Systems with Opposite Queues

643

72.4 General Case For arbitrary n and m, the state probability equations can be constructed according to the state transition diagram on Sect. 72.1. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(λ + μ) p(0,0) (λ + μ) p(i,0) μp(n,0) (λ + μ) p(0, j) (λ + μ + η) p(i, j) ⎪ ⎪ (μ + η) p(n, j) ⎪ ⎪ ⎪ ⎪ λp(0,m) ⎪ ⎪ ⎪ ⎪ (λ + η) p(i,m) ⎪ ⎪ ⎩ ηp(n,m)

= = = = = = = = =

ηp(1,1) , λp(i−1,0) + ηp(i+1,1) , λp(n−1,0) , μp(0, j−1) + ηp(1, j+1) , λp(i−1, j) + μp(i, j−1) + ηp(i+1, j+1) , λp(n−1, j) + μp(n, j−1) + ηp(n+1, j+1) , μp(0,m−1) , λp(i−1,m) + μp(i,m−1) , λp(n−1,m) + μp(n,m−1) ,

here i = 1, n − 1, j = 1, m − 1, We cannot find the common formula for queueing system states probabilities. This equation system can be solved numerically. To get a numerical result for any particular case, a computer program in C++ was created. This program used the Gauss method to solve a linear system of equations and obtain numerical values of the probabilities of the system states. However, the use of such a computer program is limited by high memory consumption. To solve a system of equations it is necessary to operate a matrix with (nm)2 elements. For example, if the system can contain 100 jobs of type A and 100 jobs of type B, the matrix will be of 108 elements. For large numbers, we have developed a computer simulation program in Java that is free of this restriction and shows the result with good accuracy. Tables 72.1 and 72.2 show the result of computer simulation of the system with λ = μ = 15, n = m = 10. In the first case η = 20, ν = 1, in the second—η = 5, ν = 4, where ν is the number of servers. In spite of the same throughput νη = 20 for both cases, the productivity in the first case is higher. These data are shown in Table 72.3. Table 72.1 One-server queueing system state probabilities (×0.0001) 0 1 2 3 4 5 6 7 10 9 8 7 6 5 4 3 2 1 0

159 159 159 158 157 154 149 141 127 106 069

119 120 120 120 119 118 115 112 107 103 106

090 091 091 092 092 093 094 095 099 107 127

068 069 070 072 074 077 080 086 095 112 141

052 054 056 058 061 066 071 080 094 115 149

041 043 046 049 053 058 066 077 093 118 154

033 035 039 043 048 053 061 074 092 119 157

027 031 035 039 043 049 058 072 092 120 158

8

9

10

024 029 034 035 039 046 056 070 091 120 159

024 031 029 031 035 043 054 069 091 120 159

036 024 024 027 033 041 052 068 090 119 159

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Table 72.2 Four-server queueing system state probabilities (×0.0001) 0 1 2 3 4 5 6 7 10 9 8 7 6 5 4 3 2 1 0

029 029 028 028 027 026 024 021 017 012 006

087 087 087 086 084 080 074 066 053 035 012

131 132 132 132 130 126 119 107 087 053 017

133 134 136 137 137 137 133 126 107 066 021

101 103 106 109 113 117 122 133 119 074 024

078 082 086 091 097 105 117 137 126 080 026

062 067 073 080 088 097 113 137 130 084 027

052 058 066 073 080 091 109 137 132 086 028

8

9

10

045 054 063 066 073 086 106 136 132 087 028

045 060 054 058 067 082 103 134 132 087 029

067 045 045 052 062 078 101 133 131 087 029

Table 72.3 The results of simulation for λ = μ = 15, n = m = 10, νη = 20 Number Throughput of Throughput Percentage of lost Average number of servers (ν) of system (νη) jobs (%) of jobs in system one server (η) 1 4

20 5

20 20

6.7 8.3

4.17 × 2 5.16 × 2

72.5 Conclusion This study shows that the model with opposite queues differs significantly from models with tandem queues and parallel queues. Further research may concern more complex conditions. For example, there may be three or more types of tasks.

References 1. Zijm, W.: Manufacturing and Logistic Systems Analysis, Planning and Control (1992) 2. Yao, H., Knessl, C.: Some first passage time problems for the shortest queue model. Queueing Syst. 58, 105–119 (2008). https://doi.org/10.1007/s11134-008-9062-0 3. Foreest, N.: Analysis of Queueing Systems with Sample Paths and Simulation (2020). https:// github.com/ndvanforeest/queueing_book/blob/master/queueing_book.pdf

Chapter 73

Choice Modeling in Insurance Alexandr V. Sachkov

Abstract Choice problems arising in modern insurance are considered in this article. After introducing all the necessary concepts, two distinct approaches to modeling choice are discussed: actuarial science and its instruments, along with axiomatic choice theory. Afterwards, one relevant example is analyzed and both approaches are shown in action. Lastly, some conclusions are drawn and potential further research directions are given.

73.1 Introduction Choice problems are some of the oldest encountered by human beings in all of recorded history. This is only natural, as our every action is tied one way or the other to some sort of a choice—should we study for an exam or spend the time relaxing instead? There is a problem with choice in a purely mathematical sense; however, it is not formalized enough to be considered precise. There clearly had to have been some criteria involved as a means to measure different choices. But all of them are very diverse—we measure the price in dollars or rubles, weight in kilograms or pounds, size in meters or feet. How does one decide which is more important? Most often the so-called weighted sum of criteria was used—we assign weights to be based on our preferences and add them together. The option which scored the largest is proclaimed the “optimal” one. This seemed very intuitive and, therefore, was never really proven to be the correct way of doing things (for an in-depth analysis, see [5]). There is, however, another approach—axiomatic one, very much in the spirit of mathematical precision. It was (and is still being) developed by Prof. Noghin V.D. and his apprentice Baskov O.V. Based on a few simple axioms, this theory builds A. V. Sachkov (B) Leonhard Euler International Mathematical Institute (SPbU Department), 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_73

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up a convenient way to evaluate all options based on decision-maker’s (who will be properly introduced later) preferences. In this article we will take a look at how this approach could be used in insurance and actuarial mathematics, comparing a more “classical” approach with this one, while also discussing potential directions of further research in the field.

73.2 Preliminaries: Actuarial Mathematics We assume the most common definitions from probability theory as known and will instead give those specific to actuarial science and relevant to our topic of discussion. For a brief overview refer to [3], and for a comprehensive one refer to [2]. For the remainder of this section X denotes the age-at-death of an individual, which is, of course, a random variable (RV). We will make no difference between an individual and his age-at-death RV. Its probability distribution function is given by FX (x). Definition 73.1 The distribution function of the form s(x) = 1 − FX (x) = P(X ≤ x) is called the survival function (SF). Definition 73.2 The RV of the form T (x) = X − x is called the time-until-death RV. To make probability statements about an individual X, we introduce for our convenience the following notation: t qx = P(T (x) ≤ t)—the probability of X to die earlier than at x + t, and its complement, t px = 1 − t qx —the probability of X to survive for at least t years from the current year. Consider m individuals (called the insured) X 1 , . . . , X m who are x years old at the moment. Each of them wants to buy a policy which would oblige the company (called the insurer), if the insured lives at least n more years, to pay him 1 unit (called the claim; e.g. 1 unit equals 100.000 rubles). The insured, in return, has to make a singular payment (called the premium). Our task is to calculate that payment. The above is an example of a problem typically solved by actuaries. It is very much simplified, of course, and serves only expository purposes. Later in our discussion we will use an even more simplified model.

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73.3 Preliminaries: Axiomatic Choice Theory In this section we will try to establish what exactly the axiomatic approach to decisionmaking is. We will only cover the basics; for a more in-depth approach refer to [4–6]. First of all, there has to be a set of feasible alternatives X , which is of an arbitrary nature (it could contain cars, houses, etc.). As the name suggests, our task is to choose some of the alternatives contained in X . To compare alternatives to one another we have to introduce some criteria which will measure how good (or bad) each alternative is. Strictly speaking, the vector criterion is a vector function of the form f : X → Rn , where n is the number of criteria, all of which are represented by a real value. Finally, we define a preference relation  X on X , which describes the preferences of a decision-maker (DM), that is, if the DM prefers x1 ∈ X to x2 ∈ X , then x1  X x2 . From that we can construct what is called a multicriteria choice problem: given X , f , and  X , find C(X )—the set of selectable alternatives. C(X ) represents the solution to the problem—the alternatives which actually end up being chosen. There is a problem with the above formulation, however—elements of X are abstract and, therefore, harder to work with numerically. We solve this by inducing the set of feasible vectors Y = f (X )—criteria values of all feasible alternatives, and Y —the preference relation on Y defined as follows: y1 = f (x1 ) Y y2 = f (x2 ) ⇔ x1  X x2 . Now any multicriteria choice problem takes Y and Y and returns C(Y ), which can be easily matched to corresponding alternatives. In order to actually solve such problems we have to make sure that our model is sound and makes sense. For that purpose, we introduce 4 axioms which bind our preference relation Y (it did remain rather enigmatic up to this point). They are as follows: / C(Y ); 1. y1 Y y2 ⇔ y2 ∈ 2. Y is transitive; 3. Y is compatible with each of the criteria f 1 , . . . , f n ; it means that ∀i if y1 and y2 differ only at their i-th element and y1i > y2i , then y1 Y y2 ; 4. Y is invariant with respect to a positive linear transformation. If the above axioms hold, we can present the most important (to our discussion) definition: elementary information quantum. We will employ here its simplified form (for a full derivation see [6]). Definition 73.3 Let i, j ∈ 1 : n. We say that there is an elementary information quantum with parameters wi , w j , if y ∈ Rn of the form y i = wi , y j = −w j , y k = 0, k = i, j satisfies y Y (0, . . . , 0) ∈ Rn .

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While the formal definition says little, its “practical” interpretation is simple—DM is ready to sacrifice w j of criteria f j to get wi of criteria f i . One last thing to discuss is how to use the quantum should we get one. That is done as follows: from the old criteria f we calculate the new one f 1 , where j f 1 = f i × w j + f j × wi , f 1k = f k , k = j. f 1 is then employed to recalculate all y ∈ Y.

73.4 One Choice Problem in Insurance—Which Policy to Get? Consider an insured L. He has to choose one of the 3 policies which pay out if L survived for at least n years since the current one. Recall that [7] the indifference between a larger but random risk loss and a smaller but a certain premium loss can be expressed as follows: E(ω − L) = ω − p, (73.1) where ω is an insured’s capital, L is the risk RV, p is premium. If we were to find out when a small loss of p is preferable, we modify the equation as follows: E(ω − L) ≤ ω − p.

(73.2)

Let us come back to the above problem. There L is the benefit an individual is to receive should he survive (and to lose if they do not buy any policy). Now consider 3 policies: 1. p = 100, L = 1000 2. p = 100, L = 500 3. p = 100, L = 700. Set ω = 1000. Then clearly, policy 1 is the best one, since premium p is the same for all policies, while L is the largest for the first policy. If we consider the way people choose in real life, however [6], it becomes clear that the above approach is way too narrow, as it only considers one criteria—monetary value. If we add some other ones and put into account DM’s preferences, the situation might change drastically.

73.5 One Choice Problem in Insurance—Which Policy to Get? Axiomatic Approach We will now examine the same problem using the axiomatic approach. To do that, we have to outline all relevant criteria. The first one is the value, and it was already

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discussed above. Set criteria two as the insurer’s reputation—naturally, people are more inclined to contract a better known one with established reputation. Let us modify the policies accordingly: 1. L = 1000, R = −10 2. L = 500, R = −1 3. L = 700, R = −5, where R is the current reputation aggregate rating placement for each insurer. R is negative here since the lower it is, the better off we are; premium p is eliminated altogether since it is the same for all 3 policies. Assume that the DM (the insured) made a choice from policies 1 and 2: 2 Y 1. From this, we extract the quantum: w1 = 500, w2 = 9. Recalculating criteria, we get: f 1 = (9 × f 1 + 500 × f 2 , f 2 ). Therefore, our policy choices transform into the following: 1. L = 4000, R = −10 2. L = 4000, R = −1 3. L = 3800, R = −5. It is obvious now that policy 2 is the best one. Therefore, we conclude that introduction of just one more criteria was crucial and our initial approach was, indeed, too narrow.

73.6 Conclusion. Future Research Directions In this article we have acquainted ourselves with axiomatic choice theory and seen how it could be applied to real-world problems—in our case, choosing the best policy available based on DM’s preferences. The axiomatic approach—while rigorously strict at its core—is very much usable in actual calculations and can make decisionmaking process easier and more accurate. It is worth noting that the example used is intentionally kept as simple as possible so as not to take the focus away from the main idea—how one can employ the axiomatic approach in practice. As for future research directions, we can suggest at least two: 1. Consider more practical problems employing higher number of criteria; 2. Try to optimize the quanta collection process; 3. Develop some sort of an application (probably mobile) to automate the entire process. The implementation of the first one requires us to introduce the so-called general information quantum, which compares the relative importance of not two, but an arbitrary number of criteria. Formulas for criteria recalculation do become a bit more cumbersome [1], but it does offer a lot more flexibility in terms of which practical

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problems one can tackle using the axiomatic approach. The second one is seen by this author as especially interesting, as the implementation of required algorithms does not appear too tedious, yet will make things much easier for both clients and companies (e.g. comparing smartphones based on their parameters—screen size, cameras, etc.). Acknowledgements This work was supported by the Ministry of Science and Higher Education of the Russian Federation (research project number No 075-15-2019-1619 dated 08.11.2019, Pure SPbU ID 51510229) controlled by the Government of Russian Federation.

References 1. Baskov, O.V.: Algorithm for sequential accounting of information on the relative criteria importance in the multicriteria choice problem. In: Control Processes and Stability, Proceedings of the XLI International Conference, pp. 553–558. (In Russian) 2. Bowers, N.L., Gerber, H.U., Hickman, L.C., Jones, D.A., Nesbitt, C.J.: Actuarial Mathematics, 2nd edn. The Society of Actuaries, Schaumburg, Illinois (1997) 3. Dorofeev, B.V., Zamurayev, K.A., Smirnov, N.V.: Actuarial Calculations in Pension Insurance (Lecture Notes). Fedorova G.V. Publishing House, St. Petersburg (2018). (In Russian) 4. Noghin, V.D.: Estimation of the set of nondominated solutions. Numer. Funct. Anal. Optim. 5,6(12), 507–515 (1991). (In Russian) 5. Noghin, V.D.: Pareto set reduction: different solution approaches. AI Decis. Mak. 1(1), 98–112 (2008). (In Russian) 6. Noghin, V.D.: Reduction of the Pareto Set: An Axiomatic Approach. Springer Inc. (2018) 7. Sachkov, A.V.: Actuarial calculations and random interest rates. Control Process. Stab. 6(1), 495–498 (2019). (In Russian)

Part X

Optimization Methods

Chapter 74

A New Characterization of Cone Proper Efficient Points Vladimir D. Noghin

Abstract The paper deals with a vector optimization problem in which the outcome space is partially ordered by some cone relation. A cone proper efficiency which was introduced by M.I. Henig is considered as the main optimality notion. Instead of Henig’s characterization of vector optimization problem using the weighted sum of criteria, we present a new characterization of cone proper efficient points in terms of goal programming, i.e. by minimizing the distance between the outcome set and some totally dominating point.

74.1 Introduction Efficient (Pareto-optimal) points play an important role in multicriteria optimization and its numerous applications. The set of efficient points may be considered as the set of maximal elements of partially ordered set by a cone relation with respect to the nonnegative orthant. P.L. Yu [6] introduced cone efficient points with respect to an arbitrary convex cone. A.M. Geoffrion [3] observed that some efficient points may be “improper” in the sense that such points cannot be satisfactorily characterized by a weighted sum of criteria, even if the outcome set is convex. He introduced the so-called proper efficient points with respect to the nonnegative orthant. Later, this notion was generalized by H. Benson [1] and J. Borwein [2] with respect to domination sets which are closed convex cones. M.I. Henig [4] proposed his own notion of cone proper efficiency and established that each cone proper efficient point may be characterized as the maximum point of some weighted sum of criteria if some extended outcome set is convex. In this paper we propose a new characterization of cone proper efficient point as the nearest point from the outcome set to some totally dominating vector. The proposed characterization generalizes the corresponding result by the author [5] which was received for the nonnegative orthant as a cone. V. D. Noghin (B) Saint-Petersburg State University, Universitetsky pr, 35, Saint-Petersburg, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_74

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74.2 Preliminaries Recall some definitions from convex analysis and vector (multicriteria) optimization. Let R m be a real m-dimensional vector space and  ⊂ R m be a convex set.  is a cone if αy ∈  for all α > 0 and y ∈ . A convex cone  is said to be acute if  ∩ − ⊂ {0m }. Given two sets A, B ⊂ R m , their addition is defined by {y ∈ (A + B)| y = a + b ∃a ∈ A, ∃b ∈ B}. The acute convex cone  induces a strict partial order  on R m due to the equivalence y  y  ⇔ y ∈ {y  } + ( \ {0m }). For a convex cone , a dual cone o is defined by o = {y ∈ R m | y, z ≥ 0 ∀z ∈ } and a strict dual cone is defined by ∗ = {y ∈ R m | y, z > 0 ∀z ∈  \ {0m }}, m yi z i . where y, z = i=1 Let f = ( f 1 , f 2 , . . . , f m ) be a vector-valued function defined on X ⊂ R n and Y be the outcome set Y = f [X ] = {y ∈ R m | ∃x ∈ X such that y = f (x)}. Due to P.L. Yu [7] we have the following. Definition 74.1 A point x ∗ ∈ X (vector y ∗ = f (x ∗ ) ∈ Y ) is said to be a cone efficient (or maximal) with respect to acute convex cone , if there is no x ∈ X such that f (x)  f (x ∗ ) or, equivalently, Y ∩ ( + {y ∗ }) ⊂ {y ∗ }. If  is the nonnegative orthant, i.e., m = {y ∈ R m | yi ≥ 0, i = 1, 2, . . . , m, },  = R+

then a cone efficient point coincides with an efficient (Pareto-optimal) point. An important notion of proper efficiency was introduced by A.M. Geoffrion [3]. Definition 74.2 A point x ∗ ∈ X is said to be a proper efficient if there exists a scalar M > 0 such that, for each i, we have f i (x ∗ ) − f i (x) f j (x ∗ ) whenever x ∈ X and f i (x ∗ ) > f i (x). M.I. Henig [4] proposed the following generalization.

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Definition 74.3 A point x ∗ ∈ X (vector y ∗ = f (x ∗ ) ∈ Y ) is said to be a cone proper efficient with respect to an acute closed convex cone  on X (respectively Y ), if ˆ ⊃  \ {0} and x ∗ is a cone ˆ ⊂ R m such that int  there is an acute convex cone  ˆ efficient point with respect to . ˆ may not be closed. It must be noted that according to Definition 74.3 the cone  The author of the paper proved [6] that Henig’s cone proper efficiency with a m ˆ and  = R+ is equivalent to Geoffrion’s proper efficiency. Hence, polyhedral cone  m includes some polyhedral since any acute closed convex cone  such that int ⊃ R+   m cone  , int ⊃ R+ , Henig’s cone proper efficiency is equivalent to Geoffrion’s m . proper efficiency if  = R+

74.3 Characterization of Cone Proper Efficient Points Given an acute closed convex cone  ⊂ R m , let us introduce a set of totally dominating points (vectors) U = {u ∈ R m | u − y ∈ ∗ \ {0m } ∀y ∈ Y }. Usually, Y ∩ U = ∅. Each point of the set U can be interpreted as some unattainable goal that it would be desirable to achieve as a result of solving the multicriteria optimization problem. But due to equality Y ∩ U = ∅ this goal can never be achieved. So among the acceptable points of the outcome set Y, it is proposed to choose the one that is closest to this goal. This is the main idea of the so-called goal programming. It remains only to choose one or another metric according to which the distance between points should be measured. Further, we will use the Euclidean metric. m , we have For  = R+ U = {u ∈ R m | u i > supx∈X f i (x) ∀x ∈ X, i = 1, 2, . . . . , m}.

(74.1)

A new characterization of cone proper efficient points in terms of minimizing the distance between the outcome set and some totally dominating points is given in the following statement. Theorem 74.1 Suppose that  is an acute closed convex cone, U = ∅, and Y −  is a convex set. A point x ∗ ∈ X is a cone proper efficient with respect to  on X if and only if there exists a vector u ∈ U such that the equality ||u − f (x ∗ )|| = min x∈X ||u − f (x)|| holds.1 1

The Euclidean distance ||a − b|| =



m i=1 (ai

− bi )2 is used for points a, b ∈ R m .

(74.2)

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Fig. 74.1 Geometric illustration to Theorem 74.1 (m = 2)

Proof A vector y ∗ ∈ Y is a cone proper efficient on Y if and only if this vector is a cone proper efficient on Y −  since  is an acute cone (Fig. 74.1). Necessity. Part 1. Let a vector y ∗ = f (x ∗ ) ∈ Y be a cone proper efficient with respect to  on Y . Then y ∗ is a cone efficient vector with respect to  on Y − , i.e. (Y − ) ∩ ( + {y ∗ }) = {y ∗ }. According to the hyperplane separation theorem two convex sets Y −  and  + {y ∗ } can be separated by a hyperplane L = {y ∈ R m | c, y = c, y ∗ } for some c ∈ R m \ {0m }, such that c, y ≤ c, y ∗ ∀y ∈ (Y − ), c, y ≥ c, y ∗ ∀y ∈ ( + {y ∗ }). (74.3) From the right inequality (74.3) we have c ∈ o . Furthermore, since y ∗ is a cone proper efficient vector, c ∈ ∗ . Then there exists a positive number α such that u = y ∗ + αc ∈ U . Part 2. Consider a closed ball B centered at the point u with the radius ||u − y ∗ || > 0. This ball has only one common point y ∗ with the hyperplane L as well as the convex set Y \  (and the set Y also). This means that the equality (74.2) holds. Sufficiency. Denote by y ∗ = f (x ∗ ) ∈ Y a point realizing the minimum distance from Y to some vector u ∈ U and by L = {y ∈ R m | u − y ∗ , y = u − y ∗ , y ∗ } a supporting hyperplane2 for the closed ball B centered at u and the radius ||u − y ∗ || passing through the point y ∗ . Since B is a strictly convex set and (Y − ) is convex, B ∩ (Y − ) = {y ∗ }. Thus, L separates two convex sets Y −  and B. 2

Here a supporting hyperplane is the boundary of a half-space which includes the ball B.

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ˆ = {y ∈ R m | u − y ∗ , y > 0}. This is a convex cone Consider the open half-space  ˆ + {y ∗ }) = ˆ = ˆ ⊃  \ {0m } since u − y ∗ ∈ ∗ . Moreover, (Y − ) ∩ ( and int  ∗ ∗ ˆ + {y }) = ∅ ⊂ {y }. According to Definition 74.3, y ∗ is a ∅ and, hence, Y ∩ ( proper efficient vector with respect to . P.L. Yu proved [7] that a set Y −  is convex if f 1 , f 2 , . . . , f m are concave on a m . Hence, Theorem 74.1 implies convex set X and  ⊃ R+ Corollary 74.1 Let U = ∅. Suppose that all components of a vector function f are m . A point x ∗ ∈ X is a cone proper efficient concave on a convex set X and  ⊃ R+ with respect to  on X if and only if there exists a vector u ∈ U such that the equality (74.2) is valid. Corollary 74.2 Suppose that all components of a vector function f are concave and bounded above on a convex set X . A point x ∗ ∈ X is Geoffrion proper efficient if and only if there exists a vector u ∈ U such that the equality (74.2) is true. Corollary 74.2 was established by the author in the paper [5]. The next statement belongs to M.I. Henig [4]. It is a characterization of cone proper efficient points as the maximum points of some weighted sum of criteria. Theorem 74.2 Assume that  is an acute closed convex cone and Y −  is convex. A point x ∗ ∈ X is a cone proper efficient with respect to  on X if and only if there exists a vector c ∈ ∗ such that c, f (x ∗ ) = maxx∈X c, f (x) .

(74.4)

Proof Necessity takes place due to Part 1 of the proof of Theorem 74.1. Sufficiency. Let (74.4) be true for some c ∈ ∗ . Then a linear function c, y

attains its maximum over the set Y −  at y ∗ , i.e. the left inequality (74.3) holds. In other words, for the convex set Y −  there exists a supporting hyperplane L = {y ∈ R m | c, y = c, y ∗ } passing through the point y ∗ = f (x ∗ ). Evidently, L separates two convex sets Y −  and o + {y ∗ } since c ∈ ∗ . Consider the open half-space ˆ = {y ∈ R m | c, y > 0}. We have int  ˆ = ˆ ⊃  \ {0m } and, moreover, (Y −  ∗ ∗ ˆ ˆ ) ∩ ( + {y }) = ∅. Thus, Y ∩ ( + {y }) = ∅ ⊂ {y ∗ } and by Definition 74.3 y ∗ is a cone proper efficient vector.

74.4 Conclusion Vector optimization problems are those where we ask for a certain “optimal” elements of a nonempty subset of a partially ordered outcome space. We suppose that the partial ordering is generated by some cone relation. In this case, there are several concepts of optimality and one of the most important of them is the cone proper efficiency introduced by M.I. Henig. If the ordering cone coincides with the nonnegative orthant, the concept of cone proper efficiency becomes the concept of proper

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efficiency introduced by A.M. Geoffrion. In order to characterize the cone efficient points M.I. Henig used a weighted sum of criteria just like A.M. Geoffrion did. In this paper, a new characterization (see Theorem 74.1) of cone proper efficient points has been obtained, according to which each cone proper efficient point is the closest among the outcome set to some totally dominating vector. And vice versa, for any totally dominating vector the nearest point of an outcome set is a cone proper efficient. The obtained characterization justifies the use of goal programming with the Euclidean metric for solving multicriteria problems. Moreover, due to Theorem 74.1 a new short proof of Henig’s characterization can be obtained (see Theorem 74.2). Acknowledgements The reported study was funded by RFBR (project number 20-07-00298).

References 1. Benson, B.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Mat. Anal. Appl. 71, 232–241 (1979) 2. Borwein, J.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optimiz. 15, 57–63 (1977) 3. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Mat. Anal. Appl. 22, 613–630 (1968) 4. Henig, M.I.: Proper efficiency with respect to cones. J. Optimiz. Theory Appl. 36, 387–407 (1982) 5. Noghin, V.D.: Pareto set reduction based on an axiomatic approach with application of some metrics. Comput. Mat. Mat. Phys. 57, 645–652 (2017) 6. Podinovsky, V.V., Noghin, V.D.: Pareto-Optimal Decisions in Multicriteria Problems, 2nd edn. Fizmatlit, Moscow (2007). (in Russian) 7. Yu, P.L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multi-objectives. J. Optimiz. Theory Appl. 14, 319–377 (1974)

Chapter 75

On Degree of Pareto Set Reduction Using Information Quanta Oleg Baskov

Abstract The axiomatic approach to Pareto set reduction is considered. The possibility of reducing the set of possible choices to a single optimal choice is investigated. A necessary and sufficient condition of existence of a set of information quanta that allow reducing the Pareto set to a single solution is obtained.

75.1 Introduction Decision-making problems are often encountered in various fields. In these problems, a decision-maker is responsible for choosing one or several “best” alternatives from a given set of possible choices. What constitutes being the “best” is often a subjective question. Usually, the possible choices are evaluated with the use of several numeric criteria, and the decision-maker tries to choose alternatives at which these criteria attain optimal values. The criteria represent different characteristics of choices, and often alternatives that are optimal with respect to one criterion are far from optimal with respect to the other criteria. Thus, decision-making with multiple criteria is a problem of finding a compromise. In this class of problems the Edgeworth–Pareto principle is widely adopted [1]. It postulates that optimal choices cannot be improved with respect to any criterion without losses by some other criteria. The set of choices that fulfill this condition is called a Pareto set. It is generally agreed that the “best” choices are to be selected from the Pareto set. In practice, the Pareto set is often quite large, so it is not possible to immediately select the desired Pareto-optimal solutions. Hence, a method of the Pareto set reduction is needed. We consider an axiomatic approach to the Pareto set reduction that relies on the so-called axioms of reasonable choice [2]. If they are accepted, then by interrogating the decision-maker it is possible to gather information about his preferences and perform the reduction of the Pareto set. After a sufficient number of O. Baskov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_75

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steps, the Pareto set is expected to become narrow enough for the decision-maker to select the final choices directly. The aim of this paper is to study the degree of reduction that can be achieved using this method. In [4] bicriteria discrete problems were thoroughly investigated. Our paper tackles the general case. Ideally, there should be an algorithm that produces questions for the decision-maker to answer in order to reduce the Pareto set to a single optimal choice. But we will show that, unfortunately, in most cases the optimal choice cannot be pinpointed by this approach, so the final choice is to be made by the decision-maker.

75.2 The Axiomatic Approach to Pareto Set Reduction Let X be a set of possible choices. The task of the decision-maker is to select one or several “best” choices from this set X . The set of selected choices will be denoted by C(X ). Suppose that every possible choice x ∈ X can be evaluated by several numeric criteria f 1 , . . . , f m . In other words, m functions f i : X → R, i = 1, . . . , m, are given. We will assume w.l.o.g. that the decision-maker aims at maximizing all the criteria. If some criterion g needs to be minimized, it can be replaced with the opposite criterion −g which is to be maximized. We will also define the vector criterion f : X → R m that combines all estimates of a given choice into a possible vector f (x) = ( f 1 (x), . . . , f m (x)). The set of all such vectors Y = f (X ) is called a set of possible vectors. In order to describe individual preferences of the decision-maker, we introduce a preference relation  X [3]. We will say that x   X x  if from  these two choices the decision-maker chooses x  and does not choose x  , i.e. C {x  , x  } = {x  }. This relation naturally induces a binary relation Y on the set of possible vectors Y : f (x  ) Y f (x  ) ⇔ x   X x  . We will assume that the following axioms hold [2]. / C(X ). Axiom 1. If x   X x  , then x  ∈ This is the axiom of rejecting dominated choices. If the decision-maker, comparing x  and x  , prefers the former choice over the latter, then x  should not be selected from the entire set of possible choices X , as there is a better alternative. Axiom 2. There exists an irreflexive transitive binary relation  on R m such that its reduction on Y is Y . This axiom postulates transitivity of preferences. Extending the preference relation Y to the set of all vectors R m makes it possible to reason about not only possible choices but also hypothetical ones. Axiom 3. ei  0 for all unit basis vectors ei ∈ R m . This axiom reflects the assumption that the decision-maker is interested in maximization of all the criteria. Axiom 4. If y   y  , then αy  + c  αy  + c for any c ∈ R m , α > 0.

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This axiom of invariance imposes by far the strongest restriction on the decisionmaker’s preference relation. It mandates that the preference of one choice over the other depends only on the difference of the value of the criteria, and does not depend on their actual values. Thus, it becomes possible to transfer the results of comparison of a pair of choices to other pairs that have similar difference in the criteria values. All these axioms represent quite rational and expected qualities of preferences, so they are called axioms of reasonable choice. They are sufficient to justify the Edgeworth–Pareto principle, which states that the selected choices must be Paretooptimal. Recall that a choice x is Pareto-optimal if there is no other choice x  such that f (x  ) ≥ f (x), where the Pareto relation ≥ means that f i (x  ) ≥ f i (x) for all i = 1, . . . , m, and for at least one index i the inequality is strict. If we denote  the Pareto set by P f (X ) = x ∈ X : ∃x  ∈ X : f (x  ) ≥ f (x) , then the Edgeworth– Pareto principle may be written as C(X ) ⊆ P f (X ). Thus, the Pareto set P f (X ) may be viewed as an upper bound on the set of selected choices C(X ). If some additional information about the decision-maker’s preferences is available, then this bound may be improved. Note that any two Pareto-optimal choices x  and x  are incomparable by the Pareto relation: f (x  ) ≥ f (x  ) and f (x  ) ≥ f (x  ). This means that f i (x  ) > f i (x  ) and f j (x  ) < f j (x  ) for some i, j. Suppose, however, that the decision-maker prefers the first choice over the second: x   X x  . Then f (x  )  f (x  ), and by the axiom of invariance f (x  ) − f (x  )  0. The vector f (x  ) − f (x  ) in this case will have at least one positive and at least one negative component. This inspires the following definition. Definition 75.1 A vector u with at least one positive and at least one negative component such that u  0 is called an information quantum. Using the information quanta, one may construct a stricter upper bound on the set of selected choices, with the aid of the following theorem [2]. Theorem 75.1 Suppose that the axioms 1–4 hold. Let u be an information quantum. Then for any set of selected choices C(X ) C(X ) ⊆ Pg (X ) ⊆ P f (X ), where Pg (X ) is the Pareto set with respect to a new vector criterion g that has the following components: gi = f i , ∀i : u i ≥ 0, gi j = u i f j − u j f i , ∀i, j : u i > 0, u j < 0. Consider an example. Let X = {x1 , x2 , x3 , x4 }, and f (x1 ) = (1; 5), f (x2 ) = (3; 4), f (x3 ) = (4; 3), f (x4 ) = (5; 1) (Fig. 75.1). It is easy to see that all presented choices are Pareto-optimal, i.e. P f (X ) = X . Suppose that the decision-maker prefers x2 over x3 : x2  X x3 . Then f (x2 )  f (x3 ), and by the axiom of invariance f (x2 ) − f (x3 )  0. Thus, this information can be expressed as a quantum

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Fig. 75.1 The set of possible vectors. Observe that the vector from f (x2 ) to f (x3 ) is collinear to the vector from f (x1 ) to f (x4 ), so the preference x2  X x3 by axiom 4 implies x1  X x4

u = f (x2 ) − f (x3 ) = (−1; 1)  0. By the Theorem 75.1, we can construct a new vector criterion g with components g1 = f 2 , g2 = u 2 f 1 − u 1 f 2 = f 1 + f 2 . Then we compute g(x1 ) = (5; 6), g(x2 ) = (4; 7), g(x3 ) = (3; 7), g(x4 ) = (1; 6), and conclude that Pg (X ) = {x1 , x2 }. Thus, we were able to rule out not only x3 but also the choice x4 . This example illustrates the methodology of applying the axiomatic approach to Pareto set reduction. By interrogating the decision-maker, we obtain information quanta and use them to modify the criteria so that the Pareto set becomes smaller. Consider now the case when we have several information quanta u 1 , . . . , u k . Let M be a conical hull of these quanta and all unit basis vectors e1 , . . . , em of R m : M=

⎧ k ⎨ ⎩

i=1

αi u i +

m j=1

⎫ ⎬

β j e j : (α1 , . . . , αk , β1 , . . . , βm ) ≥ 0 . ⎭

(75.1)

Let K = {y : y  0}. It can be shown that under the axioms of reasonable choice m m the set K is a sharp convex cone. Moreover, R+ ⊆ M ⊆ K , where R+ = {y ∈ m ∗ R : y ≥ 0}. As the cone M is finitely generated, so is its dual cone M . Denote i  i  the generators of M ∗ by h 1 , . . . , h p . Then,  if h · f (x ) ≥ h · f (x ∗∗) for all i = i   1, . . . , p, we will have h · f (x ) − f (x ) ≥ 0, f (x ) − f (x ) ∈ M = M ⊆ K , / C(X ). This leads to the so f (x  ) − f (x  )  0, i.e. f (x  )  f (x  ), x   X x  , x  ∈ following result [2]. Theorem 75.2 Suppose that the axioms 1–4 hold. Let u 1 , . . . , u k be information quanta. Let M be a conical hull of the vectors u 1 , . . . , u k , e1 , . . . , em , as in (75.1). Then for any set of selected choices C(X ) C(X ) ⊆ Ph (X ) ⊆ P f (X ), where h is a new vector criterion with components h i · f , and h i are the generators of the dual cone M ∗ .

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Fig. 75.2 The set of possible vectors. Observe that for any line of the form γ1 f 1 + γ2 f 2 = const with nonnegative coefficients γ1 , γ2 that passes through f (x2 ) there always exists a possible vector in the upper right semiplane

We will say that the vector criterion h is induced by the information quanta u1, . . . , uk .

75.3 Reduction to a Single Optimal Choice Using information quanta, we may construct upper bounds on the set of selected choices C(X ). By the Edgeworth–Pareto principle, the selected choices must be Pareto-optimal, so the initial upper bound is given by the Pareto set P f (X ). In this section we study whether it is possible to reduce the Pareto set to a single choice and thus solve the decision-making problem. We will start with an example. Consider the set of possible choices X = {x1 , x2 , x3 } with f (x1 ) = (4; 1), f (x2 ) = (2; 2), f (x3 ) = (1; 4) (Fig. 75.2). Suppose that there is a vector criterion h such that Ph (X ) = {x2 }. That would imply x2  X x1 and x2  X x3 , whence f (x2 )  f (x1 ) and f (x2 )  f (x3 ). By the axiom of invariance then f (x2 ) − f (x1 ) = (−2; 1)  0, and f (x2 ) − f (x3 ) = (1; −2)  0, and (−2; 1) + (1; −2) = (−1; −1)  0. But the last statement contradicts the axiom 3. Thus, using information quanta, it is not possible to reduce the Pareto set to the single choice x2 . On the other hand, taking a quantum (1; −1)  0, we construct a new vector criterion g = ( f 1 ; f 1 + f 2 ), compute g(x1 ) = (4; 5), g(x2 ) = (2; 4), g(x3 ) = (1; 5), and find that Pg (X ) = {x1 }. This example shows that not every choice can be obtained as a result of the Pareto set reduction with the use of information quanta. Our main result establishes a necessary and sufficient condition of the possibility of an ultimate reduction of a Pareto set to a given choice. Theorem 75.3 Let x ∗ ∈ X be a possible choice. Information quanta u 1 , . . . , u k that induce a vector criterion h such that Ph (X ) = {x ∗ } exist if and only if there exist m linearly independent vectors γ 1 , . . . , γ m ∈ R m with nonnegative components such

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that functions γ1i f 1 (x) + · · · + γmi f m (x), i = 1, . . . , m, attain their maximum value on X at the single point x ∗ . Proof Observe that the components of the vector criterion h are linear combinations of the criteria f with nonnegative coefficients: h i = h i · f , i = 1, . . . , p. Consider p p

some linear combination αi h i = αi h i · f of the criteria h with nonnegative i=1

i=1

coefficients αi ≥ 0. Suppose that it attains its maximum on X at some point x = / Ph (X ), so there exists a choice x  such that h(x  ) ≥ h(x). But then x ∗ . Then x ∈ p p

αi h i (x  ) ≥ αi h i (x), a contradiction. Therefore, any linear combination of the i=1

i=1

criteria h attains its maximum on X at x ∗ . Recall the cone M defined by (75.1). The vectors h i are the generators of its dual cone M ∗ . Any of their linear combination with nonnegative coefficients is a vector from M ∗ , and vice versa. As M is sharp, we can find m linearly independent vectors γ 1 , . . . , γ m in M ∗ . Linear combinations of f with coefficients from each of these vectors γ i will have a single maximum on X at x ∗ , as desired. Now suppose that there are vectors γ 1 , . . . , γ m ∈ R m such that functions γ i · f (x) attain maximum on X at x ∗ . Consider the set M =  y ∈ R m \ {0} : γ i · y ≥ 0, i = 1, . . . , m . It is a finitely generated sharp convex m . Denote its generators by u 1 , . . . , u k . By the theorem 75.2 we cone containing R+ will have C(X ) ⊆ Ph (X ), where the components of the new vector criterion h have the form h i · f , h i being the generators of M ∗ . But by construction, the generators of M ∗ are exactly the vectors γ 1 , . . . , γ m . Since functions γ i · f (x) attain their maximum on X at x ∗ , for any other choice x ∈ X \ {x ∗ } we have γ i · f (x ∗ ) ≥ γ i · f (x), i.e. h(x ∗ ) ≥ h(x), so Ph (X ) = {x ∗ }.

It is worth noting one case when the conditions of this theorem are fulfilled. Suppose that the possible vectors are vertices of some convex polyhedron. Take any Pareto-optimal choice x ∈ P f (X ), and compute u i = f (x) − f (xi ) for all xi ∈ X \ {x}. The conical hull of these vectors u i will be a sharp convex cone due to f (xi ) being the vertices of a convex polyhedron. Any set of m linearly independent vectors from the interior of this cone may be taken as the vectors γ1 , . . . , γm from the Theorem 75.3. For instance, when the set of possible choices X is a subset of some Euclidean space R n given by linear inequalities, and the criteria f are also linear, the set of possible vectors f (X ) is polyhedral. Then every its vertex can be found as a result of the reduction of the Pareto set using suitable information quanta.

75.4 Discussion Theorem 75.3 presents a necessary and sufficient condition of the possibility of reduction of the Pareto set to a single choice. Recall the example from Fig. 75.2. Note that there does not exist a linear combination γ1 f 1 + γ2 f 2 of the criteria f

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with nonnegative coefficients that attains maximum on X at x2 . In this respect the requirement of the Theorem 75.3—the existence of m linear combinations that attain their maximum at x ∗ —seems to be very strict. This makes us believe that in most practical cases one should not expect that the axiomatic approach to the Pareto set reduction will allow to reduce the Pareto set to a single choice and thus solve the decision-making problem. The task of determining the selected choices C(X ) is still the duty of the decision-maker, and the axiomatic approach can only simplify it by removing the choices that are guaranteed to not be optimal.

75.5 Conclusions The axiomatic approach to Pareto set reduction has been considered. We have developed a necessary and sufficient condition under which it is possible to reduce a Pareto set to a single optimal choice using solely the axiomatic approach. One class of problems that satisfy this condition is linear multicriteria optimization problems, where the set of possible choices is given by linear inequalities, and the criteria are also linear. In most cases there are Pareto-optimal choices that cannot be found by reducing the Pareto set to a single choice. Thus, our study justifies the usage of the axiomatic approach in combination with other methods of multicriteria decision-making. Acknowledgements This paper is supported by Russian Foundation for Basic Research, project No. 20-07-00298a.

References 1. Noghin, V.D.: A logical justification of the Edgeworth — Pareto principle. Comput. Math. Math. Phys. 42(7), 915–920 (2002) 2. Noghin, V.D.: Reduction of the Pareto Set: An Axiomatic Approach. Springer (2018) 3. Noghin, V.D.: Relative importance of criteria: a quantitative approach. J. Multi-Criteria Decis. Anal. 6, 355–363 (1997) 4. Zakharov, A.O., Kovalenko, Y.V.: Structures of the Pareto set and their reduction in bicriteria discrete problems. J. Phys. Conf. Ser. 1260, 1–8 (2007)

Chapter 76

Particular Structures of the Pareto Set and Its Reduction in Bicriteria Discrete Problems Aleksey Zakharov and Yulia Kovalenko

Abstract Bicriteria discrete problems are analyzed in the context of the axiomatic approach of the Pareto set reduction proposed by V. Noghin. We investigate the question of reduction for instances with special structures of the Pareto set. A practical application of the results is presented for the bicriteria set covering problem.

76.1 Preliminaries We consider the following choice problem < X, f, > with two criteria according to [4]. Here X is the finite set of feasible solutions (alternatives), among which the best solution should be chosen, goals are represented by a vector criterion f : X → R2 , and could be expressed by an asymmetric binary preference relation of the decisionmaker (DM) , specified on the set of points Y = f (X ). Also, we recall that the Pareto set is the set of nondominated points such that P(Y ) = {y ∈ Y | y ∗ ∈ Y : y1∗  y1 , y2∗  y2 , y ∗ = y }. Further, we state the elements of the Pareto set reduction approach as formulated in [4]. If the DM compares two feasible solutions x ∗ and x, and chooses the first one, then we write it as f (x ∗ )  f (x). Assume (hypothetically) that the DM expresses all its preferences, then we call the set of feasible outcomes that satisfy all these preferences as the set of selectable outcomes C(Y ). Suppose that the relation  satisfies the axioms: (1) irreflexivity: y  y does not takes place for any y ∈ Y ; (2) transitivity: if y   y and y   y  , then y   y for any y, y  , y  ∈ Y ; (3) y   y then αy  + c  αy + c for any y, y  ∈ Y , α > 0, c ∈ R2 ; (4) if f 1 (x ∗ ) > f 1 (x), f 2 (x ∗ ) = f 2 (x) or f 1 (x ∗ ) = f 1 (x), f 2 (x ∗ ) > f 2 (x) for some feasible solutions x ∗ , x ∈ X , A. Zakharov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia Y. Kovalenko Sobolev Institute of Mathematics, 13, Pevtsov str., Omsk 644043, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_76

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then f (x ∗ )  f (x); (5) if f (x ∗ )  f (x) is valid then solution x does not belong to the set of selectable outcomes C(Y ). In [4] the author proved that if conditions (1)–(5) hold, then the “optimal” choice of the DM should not be beyond the Pareto set P(Y ), i.e. C(Y ) ⊆ P(Y ) for any C(Y ). This inclusion is called the Edgeworth–Pareto principle. The main idea of the Pareto set reduction axiomatic approach is to use the information about DM’s preferences in order to get a narrower upper bound than in inclusion C(Y ) ⊆ P(Y ). We suppose that vector y  ∈ R2 with one positive component and one negative component is proposed to the DM. If the DM prefers this vector to the zero one 02 = (0, 0), then we can claim that it gives some additional information, an elementary information quantum [4], as vectors y  and 02 are Pareto-optimal to each other. And component having positive value is more important, while the other component is less important. More formal it can be expressed as follows, if for vector y  ∈ R2 , yi = wi > 0 and y j = −w j < 0, the relation y   02 is valid, then criterion i is more important than criterion j. Here i, j ∈ {1, 2}, i = j. The relative characteristics of elementary information quantum is coefficient of compromise θ = w j /(wi + w j ), θ ∈ (0, 1). We note, in the aforementioned definition the vector y  ∈ R2 with components yi = 1 − θ and y j = −θ could be taken instead of vector y  . Further, we take an elementary information quantum only with coefficient θ and denote it as f i → f j : θ . The following two theorems show how to take into account one and two elementary information quanta in a bicriteria problem. We should construct the bicriteria problem, where a new vector criterion based on linear combination of the initial criteria is used. The Pareto set with respect to the new vector criterion will compose the reduction of the Pareto set. We note that the set of feasible solutions remains the ˆ ). same. Further, we denote the reduced Pareto set by notation P(Y ˆ ) ⊆ P(Y ) Theorem 76.1 [4]. Let f i → f j : θ , then the inclusions C(Y ) ⊆ P(Y ˆ ) = f (P ˆ (X )), and P ˆ (X ) is the set of pareto-optimal are valid ∀ C(Y ). Here P(Y f f solutions with respect to vector criterion fˆ = ( fˆ1 , fˆ2 ), where fˆj = θ f i + (1 − θ ) f j , fˆi = f i ; i, j ∈ {1, 2}, i = j. Theorem 76.2 [4] Let f i → f j : θi j and f j → f i : θ ji (θi j + θ ji < 1). Then the ˆ ) ⊆ P(Y ) are valid ∀ C(Y ). But here vector criterion fˆ = inclusions C(Y ) ⊆ P(Y ˆ ˆ ( f 1 , f 2 ) has components fˆi = (1 − θ ji ) f i + θ ji f j , fˆj = θi j f i + (1 − θi j ) f j .

76.1.1 Our Results As we can see from the previous results for discrete problems (e.g. [5, 6]), when all points of the Pareto set belongs to one line, the reduction of the Pareto set either does not hold or consists of one element. We may easily generalize this result to the case of an arbitrary set of parallel lines.

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For any bicriteria problem instance there exists minimum number of parallel lines with a negative slope such that all elements of the Pareto set belong to them. Indeed, for the Pareto set P(Y ) the minimum number of such parallel lines can be found in O(|P(Y )|4 ) time. We draw a line ( pl ) through every pair of points, )|−1) . Then for the line ( pl ) we construct a parallel line through l = 1, . . . , |P(Y )|(|P(Y 2 each point and count the number of different lines. p Theorem 76.3 Let P(Y ) = i=1 {(y1 , y2 ) : y2 = ai − ky1 , y1 ∈ Y˜1 }, where ai , i = 1, . . . , p, and k > 0 are arbitrary constants, Y˜1 is the discrete set of numbers. If f 1 → ˆ )|  p. f 2 : θ  , and θ   k/(k + 1), or f 2 → f 1 : θ  , and θ   1/(k + 1), then | P(Y In this paper we present new structures of the Pareto set, and identify the values of the compromise coefficient that guarantee the reduction.

76.2 Generalization of Cascade and Stairs Structures We proposed cascade and stairs structures of the Pareto set in [7] as follows. We say that some set of points y = (y1 , y2 ) has stairs structure (i, j)

St (k, n, m, z¯ , zˆ , a, b) = {(y1 (1, j)

a (1) < . . . < a (n) , y2

(2, j)

> y2

(i, j)

, y2

(n, j)

> . . . > y2

(i, j)

) : y2

(i, j)

+ ky1

− a (i) = 0,

, y1(n,s) < y1(1,s+1) , y1(1,1) = z¯ ,

y1(n,m) = zˆ , a (1) = a, a (n) = b, i = 1, . . . , n, j = 1, . . . , m, s = 1, . . . , m − 1}, where k > 0, n, m ∈ N, z¯ , zˆ , a, b ∈ R, z¯ < zˆ , a < b (see, e.g., Fig. 76.1). We say that some set of points y = (y1 , y2 ) has cascade structure (i, j)

Cs(k, n, m, d, z¯ , zˆ , a (1) , . . . , a (n) ) = {(y1 y1(i,1)

< ...
. . . > a (n) , z¯ , zˆ ∈ R2 , z¯ 1 < zˆ 1 , z¯ 2 > zˆ 2 (see, e.g., Fig. 76.2). Sets of these structures lay on n parallel lines in different ways. In [7] we consider the Pareto sets having stairs and cascade structures and study the influence of the values of compromise coefficient on the reduction of the Pareto set in the case of an elementary information quantum. In further sections we study more complex structures, obtained by superposition of stairs and cascade.

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Fig. 76.1 Example of stairs structure of the Pareto set (n = 4, m = 3)

y2

y1 y2

Fig. 76.2 Example of cascade structure of the Pareto set (n = 3, m = 12)

y1

76.2.1 “Cascade of Stairs” Structure We introduce a “cascade of stairs” set as a collection of sets Cs O f St (k) = {St (k, n 1 , m 1 , z¯ (1) , zˆ (1) , a1 , b1 ), St (k, n 2 , m 2 , z¯ (2) , zˆ (2) , a2 , b2 ), . . . , St (k, n t , m t , z¯ (t) , zˆ (t) , at , bt )}, where zˆ (i) < z¯ (i+1) , ai > bi+1 , i = 1, . . . , t − 1. This complex structure consists of isolated stairs (blocks) combined into cascade (see, e.g., Fig. 76.3). Here t is the number of blocks, n i is the number of lines in block i, and m i is the number of points on each line in block i, i = 1, . . . , t. Let P(Y ) = Cs O f St (k).

Fig. 76.3 Example of “cascade of stairs” structure

y

k

a b y

y

ky

b

ky

a

a b

k y

y

ky

ky

b

a

y

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k ˆ )|  Theorem 76.4 When f 1 → f 2 : θ1 we have: (1) if θ1 ∈ (0, k+1 ), then | P(Y t k ˆ i=1 m i ; (2) if θ1 ∈ [ k+1 , 1), then | P(Y )|  t. 1 ˆ )|  n 1 + m 1 − 1; When f 2 → f 1 : θ2 we obtain: (1) if θ2 ∈ (0, k+1 ), then | P(Y 1 ˆ )|  n 1 . (2) if θ2 ∈ [ k+1 , 1), then | P(Y When f 1 → f 2 : θ12 and f 2 → f 1 : θ21 simultaneously we have: (1) if θ12 ∈ k 1 ˆ )|  m 1 ; (2) if θ12 ∈ [ k , 1) and θ21 ∈ ) and θ21 ∈ (0, k+1 ), then | P(Y (0, k+1 k+1 1 ˆ )|  t; (3) if θ12 ∈ (0, k ) and θ21 ∈ [ 1 , 1), then | P(Y ˆ )|  n 1 . (0, k+1 ), then | P(Y k+1 k+1

Proof The proof is based on the composition of proofs of Theorems 76.2 and ˆ ) 76.3 from [7]. It uses the condition that the reduction of the Pareto set P(Y is the set of nondominated elements of the set Y with respect to cone M: (1) if f 1 → f 2 : θ1 , M = cone{(0, 1), (1 − θ1 , −θ1 )} \ {02 }; (2) if f 2 → f 1 : θ2 , M = cone{(1, 0), (−θ2 , 1 − θ2 )} \ {02 }; (3) if f 1 → f 2 : θ12 and f 2 → f 1 : θ21 , M =  cone{(1 − θ12 , −θ12 ), (−θ21 , 1 − θ21 )} \ {02 } (θ12 + θ21 < 1 should be valid).

76.2.2 “Stairs of Cascades” Structure We consider a set of points y = (y1 , y2 ) that is organized as cascades combined into stairs structure. St O f Cs(k) = {Cs(k, n i , m i j , di j , a (i,1) , . . . , a (i,ni ) , z¯ (i j) , zˆ (i j) ), (1 j) (2 j) (2 j) (3 j) (t−1, j) (t j) > z¯ 2 , zˆ 1(t1) < z¯ 1(12) , zˆ 1(t2) < z¯ 1(13) , . . . , zˆ 2 > z¯ 2 , zˆ 2 > z¯ 2 , . . . zˆ 2 zˆ 1(t,τ −1) < z¯ 1(1τ ) , i = 1, . . . , t, j = 1, . . . , τ } (see, e.g., Fig. 76.4). Here t is the number of blocks, τ is the number of parts in each block (isolated cascades), n i is the number of lines in block i, m i j is the number of points in part j of block i, and di j is the number of points on the line with constant term a (i,1) in part j of block i, i = 1, . . . , t, j = 1, . . . , τ . Let P(Y ) = St O f Cs(k). k ˆ )|  ), then | P(Y Theorem 76.5 When f 1 → f 2 : θ1 we have: (1) if θ1 ∈ (0, k+1 τ k ˆ j=1 m t j ; (2) if θ1 ∈ [ k+1 , 1), then | P(Y )|  n t . t−1 1 ˆ )|  i=1 ), then | P(Y di1 + When f 2 → f 1 : θ2 we obtain: (1) if θ2 ∈ (0, k+1 τ 1 ˆ d ; (2) if θ ∈ [ , 1), then | P(Y )|  t. 2 j=1 t j k+1 When f 1 → f 2 : θ12 and f 2 → f 1 : θ21simultaneously we have: (1) if θ12 ∈ k 1 ˆ )|  τj=1 dt j ; (2) if θ12 ∈ [ k , 1) and θ21 ∈ ) and θ21 ∈ (0, k+1 ), then | P(Y (0, k+1 k+1 1 ˆ )|  n t ; (3) if θ12 ∈ (0, k ) and θ21 ∈ [ 1 , 1), then | P(Y ˆ )|  t. (0, k+1 ), then | P(Y k+1 k+1

Proof Arguments are identical to Theorem 76.4.



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y y

ky ky

y y

ky

y

ky y

ky

y Fig. 76.4 Example of “stairs of cascades” structure

76.3 Bicriteria Set Covering Problem Now we investigate the bicriteria set covering problem (bi-SCP), and construct families of instances with “stairs of cascades” and “cascade of stairs” structures. Consider a set of objects Nr , |Nr | = n r , and a collection Nc of subsets of these objects, |Nc | = n c . Let c = (ci ), u = (u i ) be n c -dimensional vectors, representing weights of subsets. Note that Nc can include identical subsets with different weights. A feasible subcollection S ⊆ Nc is such that every element j ∈ Nr is “covered” by (namely, belongs to) at  least one of the subsets in S. The total sums of weights f 1 (S) =  u and f (S) = 2 i∈S i i∈S (−ci ) are maximized over all feasible subcollections S such that |S| ≤ β, where 1 ≤ β ≤ n c is the given constant. The set covering problem and its generalizations have various practical applications (see, e.g., [1–3]). For example, supply or production management problem, where objects represent tasks and companies specify subsets of the performed tasks. And the problem of centers location, where objects correspond to zones of some region and subsets indicate zones “covered” by available places for centers. Let n r = 2v , β = 2 and n c = ( p + q)v, where p, q, v ∈ N, p > 1. We consider v groups of ( p + q) subsets. In the j-th group, the first p subsets consist of elements 2 j z + 1, . . . , 2 j z + 2 j , z = 0, 2, . . . , 2v− j − 2, and the next q subsets consist of elements 2 j z + 1, . . . , 2 j z + 2 j , z = 1, 3, . . . , 2v− j − 1, where j = 0, 1, . . . , v − 1. So, the set of feasible solutions is decomposed into v groups, in which any solution includes one of the first p subsets and one of the next q subsets (so the number of

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feasible solutions is vpq). This situation corresponds to the practical case, where there are alternative companies performing the same tasks with different costs and levels of performance, or alternative centers in approximately the same place with different coefficients of efficiency and values of cost.

76.3.1 Cascade of Stairs We define the weights as u i+ j ( p+q) = i + j pq, ci+ j ( p+q) = i + (2 p − 1)q j, i = 1, . . . , p, j = 0, 1, 2, . . . , v − 1; u l+ p+ j ( p+q) = T + (l − 1) p, cl+ p+ j ( p+q) = T + (l − 1)(2 p − 1) + pj, l = 1, . . . , q, j = 0, 1, 2, . . . , v − 1.

Here and later, T is an arbitrary positive constant. Images of all feasible solutions compose the Pareto set P(Y ) = {(T + (l − 1) p + i + j pq; −T − (l − 1)(2 p − 1) − i − (2 p − 1)q j − pj) , i = 1, . . . , p, l = 1, . . . , q, j = 0, 1, 2, . . . , v − 1}

This Pareto set illustrates “cascade of stairs” structure, where k = 2 p−1 , and we p have vp parallel lines by q points on each line (t = v blocks of p parallel lines with pq points). From Theorem 76.4 we indicate the following results. When f 1 → f 2 : θ1 we p−1 ˆ )|  qv; (2) if θ1 ∈ [ 2 p−1 , 1), then | P(Y ˆ )|  v. ), then | P(Y have: (1) if θ1 ∈ (0, 23 p−1 3 p−1 p ˆ )|  p + q − 1; When f 2 → f 1 : θ2 we have: (1) in case θ2 ∈ (0, ), then | P(Y p ˆ )|  p. (2) in case θ2 ∈ [ 3 p−1 , 1), then | P(Y

3 p−1

p−1 ) and θ21 ∈ When f 1 → f 2 : θ12 and f 2 → f 1 : θ21 we have: (1) if θ12 ∈ (0, 23 p−1 p 2 p−1 p ˆ ˆ )|  (0, ), then | P(Y )|  q; (2) if θ12 ∈ [ , 1) and θ21 ∈ (0, ), then | P(Y 3 p−1

3 p−1

p−1 p ˆ )|  p. v; (3) if θ12 ∈ (0, 23 p−1 ) and θ21 ∈ [ 3 p−1 , 1), then | P(Y

3 p−1

76.3.2 Stairs of Cascades We define the weights as u i+ j ( p+q) = ci+ j ( p+q) = i + ( pq + (2q − 1)( p − 1)) j, i = 1, . . . , p, j = 0, . . . , v − 1; u l+ p+ j ( p+q) = T + (l − 1) p, l = 1, . . . , q, j = 0, 1, 2, . . . , v − 1; cl+ p+ j ( p+q) = T + (l − 1) p + (l − 1)( p − 1), l = 1, . . . , q, j = 0, 2, 4, . . . , v − 2; cl+ p+ j ( p+q) = T + (l − 1)(2 p − 1) − ( p − 1)q, l = 1, . . . , q, j = 1, 3, 5, . . . , v − 1.

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Images of all feasible solutions compose the Pareto set P(Y ) = {(T + (l − 1) p + i + ( pq + (2q − 1)( p − 1)) j; −T − (l − 1)(2 p − 1) − i − ( pq + (2q − 1)( p − 1)) j) , i = 1, . . . , p, l = 1, . . . , q, j = 0, 2, 4, . . . , v − 2} ∪ {(T + (l − 1) p + i + ( pq + (2q − 1)( p − 1)) j; −T − (l − 1)(2 p − 1) − i − ( pq + (2q − 1)( p − 1)) j + ( p − 1)q) , i = 1, . . . , p, l = 1, . . . , q, j = 1, 3, 5, . . . , v − 1} . This Pareto set illustrates “stairs of cascades” structure, where k = 1, and we have points on each line (t = 2 blocks of q parallel lines each and 2q parallel lines by vp 2 τ = v/2 identical parts of pq points each). From Theorem 76.5 we obtain the following results. When f 1 → f 2 : θ1 we have: ˆ )|  vpq ; (2) if θ1 ∈ [ 1 , 1), then | P(Y ˆ )|  q. (1) if θ1 ∈ (0, 21 ), then | P(Y 2 2 1 ˆ )|  vp + p; (2) in When f 2 → f 1 : θ2 we have: (1) in case θ2 ∈ (0, 2 ), then | P(Y 2 ˆ )|  2. case θ2 ∈ [ 21 , 1), then | P(Y When f 1 → f 2 : θ12 and f 2 → f 1 : θ21 we have: (1) if θ12 ∈ (0, 21 ) and θ21 ∈ ˆ )|  vp ; (2) if θ12 ∈ [ 1 , 1) and θ21 ∈ (0, 1 ), then | P(Y ˆ )|  q; (3) if (0, 21 ), then | P(Y 2 2 2 1 1 ˆ )|  2. θ12 ∈ (0, 2 ) and θ21 ∈ [ 2 , 1), then | P(Y

76.4 Conclusion The Pareto set reduction was theoretically analyzed for superposition of stairs and cascade structures in the case of bicriteria discrete problems. We applied the obtained results to the well-known set covering problem. Acknowledgements The reported study in Sects. 76.2.1, 76.3.1 was funded by RFBR, project number 20-07-00298 (A. Zakharov). The reported study in Sect. 76.2.2 was funded by RFBR, project number 19-47-540005, and the study presented in Sect. 76.3.2 was funded in accordance with the state task of the IM SB RAS (Yu. Kovalenko).

References 1. Eremeev, A.V., Zaozerskaya, L.A., Kolokolov, A.A.: A set covering problem: complexity, algorithms, experimental research. Diskretn. Anal. Issled. Oper. 7(2), 22–46 (2000) 2. Kolokolov, A.A., Zaozerskaya, L.A.: Solving a bicriteria problem of optimal service centers location. J. Math. Model. Algor. 12, 105–116 (2013) 3. Nechepurenko, M.I., Popkov, V.K., Majnagashev, S.M., Kaul’, S.B., Proskuryakov, V.A., Kohov, V.A., Gryzunov, A.B.: Algoritmy i programmy resheniya zadach na grafah i setyah (Algorithms and Programs for Solving on Graphs and Networks). Nauka, Novosibirsk (1990)

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4. Noghin, V.D.: Reduction of the Pareto Set: An Axiomatic Approach. Springer International Publishing AG, Cham (2018) 5. Zakharov, A., Kovalenko, Yu.: Reduction of the Pareto set in bicriteria asymmetric traveling salesman problem. In: Eremeev, A., Khachay, M. (eds.) OPTA 2018. CCIS, vol. 871, pp. 93– 105. Springer, Cham (2018) 6. Zakharov, A., Kovalenko, Yu.: Construction and reduction of the Pareto set in asymmetric travelling salesman problem with two criteria. Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr. 14(4), 378–392 (2018) 7. Zakharov, A., Kovalenko, Yu.: Structures of the Pareto set and their reduction in bicriteria discrete problems. J. Phys. Conf. Ser. 1260(8), 1–8 (2019)

Chapter 77

On a Support Function on a Convex Cone Lyudmila Polyakova, Alexander Fominyh, Vladimir Karelin, and Stanislav Myshkov

Abstract The support function is a very powerful instrument in convex analysis. In this paper, we consider some properties of support functions on a closed convex cone. We also introduce and study a forming set with respect to some closed convex cone. The forming set is important, as it allows to calculate the value of the support function on this cone without information about all the elements of the considered set. The paper continues the research of [7]. Some new examples are constructed, which illustrate the concept of the forming set.

77.1 Introduction Support functions in convex analysis are widely used and play a fundamental role in recent developments of optimization and variational analysis. The concept of the support function can be defined for any set, but it is especially informative for convex sets. For example, if a function is convex, then its directional derivative is the support function of the subdifferential [9]. If the function is quasidifferentiable, then its directional derivative can be represented as the difference of the support functions of some convex compact sets [5]. In nondifferential optimization, in many problems the directional derivatives are used while proving necessary and sufficient conditions of the extremum both on the whole space and under constraints. In this paper, some properties of a support function defined on a closed convex cone are considered. Known facts from the theory of convex sets and convex functions L. Polyakova · A. Fominyh (B) · V. Karelin · S. Myshkov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] L. Polyakova e-mail: [email protected] V. Karelin e-mail: [email protected] S. Myshkov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_77

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used in this article can be found in books on convex analysis, for example, in [6, 8, 9]. Some properties of support functions were considered in [1–4]. The paper continues the research of [7]. Some new examples are constructed, which illustrate the concept of the forming set. The results obtained may be useful while formulating minimum conditions in nondifferentiable optimization problems in terms of some objects of nonsmooth analysis, such as quasidifferentials, exhausters, etc. The paper is organized as follows. In section II, we introduce the definition of a support function of a set X and give some known properties of this function. In section III, we consider support functions on a convex closed cone K , give the definition of the set X (K ) for a convex compact X and prove that X (K ) = X − K ∗ . As an important corollary of this fact, we note that support functions on a cone of the convex compact sets X and Y are equal iff X − K ∗ = Y − K ∗ . In section IV, we consider a forming set of a convex compact set relative to a convex cone. The forming set is important, as it allows to calculate the value of the support function on this cone without information about all the elements of the set X . We give some examples illustrating the definition of a forming set. In the end of this section, we also consider some auxiliary properties of the forming set and of support functions on a cone.

77.2 Some Properties of Support Functions Let a set X ⊂ R n be nonempty. Definition 77.1 A function s(g, X ) = supx, g, g ∈ R n , x∈X

is called the support function of the set X . Here, the symbol ∗, ∗ denotes the scalar product of two vectors. Formulate some well-known properties of support functions: 1. A support function is a convex closed function. 2. If a set X is convex and compact, then for each g ∈ R n , ||g|| = 0, the inequality x, g ≤ s(g, X ) defines a closed half-space bounded by the hyperplane  H (g, s(g, X )) = {x ∈ R n  x, g = s(g, X )}, which is support for the set X with the external normal g. 3. A closed convex set X ⊂ R n is completely defined by a support function, since it can be defined as a set of solutions of the system of inequalities X=





g∈R , ||g|| = 0 n

  x ∈ R n  x, g ≤ s(g, X ) .

(77.1)

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n 4. Let X i ⊂  mR , i ∈ I = 1, . . . , m, be nonempty compact convex sets. If  X = co X i , then s(g, X ) = max s(g, X i ) ∀g ∈ R n . i∈I

i=1

If X =

m

λi X i , λi ≥ 0, ∀i ∈ I, then s(g, X ) =

i=1

m

λi s(g, X i ) ∀g ∈ R n .

i=1

Here, the convex hull of the set X is denoted by co X . 5. The support function of a nonempty convex compact set is positively homogeneous and subadditive, i.e. s(λg, X ) = λs(g, X ), λ ≥ 0, ∀g ∈ R n , s(g1 + g2 , X ) ≤ s(g1 , X ) + s(g2 , X ) ∀g1 , g2 ∈ R n . 6. Let X ⊂ R n be a convex compact set. Since the function s(g, X ) is positively homogeneous and convex, then for its subdifferential at the point g ∈ R n the inclusion  ∂s(g, X ) = {x ∈ X  x, g = s(g, X ) } is true. If g = 0n , then the subdifferential of the support function coincides with X . At other points g ∈ R n , the next inclusion ∂s(g, X ) ⊂ bd X holds. Here, bd X denotes the boundary of the set X .

77.3 A Support Function on a Convex Closed Cone Let us formulate one of the remarkable properties of the support functions. Lemma 77.1 Let X, Y ⊂ R n be two convex compact sets. Then the equality X = Y holds iff s(g, X ) = s(g, Y ) ∀g ∈ R n . If we consider the support functions of these sets on a convex closed cone other than R n , then the last equality may not hold. Lemma 77.2 Let K ⊂ R n be a closed convex cone, then the next statement  0, g ∈ K , sup v, g = (77.2) +∞, g∈ / K, v∈K ∗ holds. Here, K ∗ denotes the cone conjugate to the cone K . Proof Take an arbitrary v ∈ K ∗ . Then v, g ≤ 0 ∀g ∈ K . Therefore, sup v, g ≤ 0 ∀g ∈ K .

v∈−K ∗

Since K ∗ is a closed cone, then it contains the zero point, so

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sup v, g = 0 ∀g ∈ K .

v∈−K ∗

 If g ∈ / K , then there exists a point vˆ ∈ −K ∗ such that v, ˆ g > 0. And since the set −K ∗ is a cone, then

 ˆ g = +∞. sup λv, λ>0

Let K ⊂ R n be a closed convex cone and X ⊂ R n be a nonempty compact convex set. Define the set     X (K ) = x ∈ R n  x, g ≤ s(g, X ) . g∈K ||g|| = 0 Since the set X is compact, the set X (K ) is closed and convex. Theorem 77.1 The next equality X (K ) = X − K ∗

(77.3)

holds. Proof By virtue of the property (77.1), the set X ⊂ X (K ). Let us first prove the inclusion X (K ) ⊂ X − K ∗ . Suppose this inclusion is false. Take any arbitrary point z ∈ X (K ). Let z ∈ / X − K ∗ . Then, by using separation theorems, there are such n vectors g¯ ∈ R , ||g|| = 0, and ε > 0, for which the inequality x, g ¯ + v, g ¯ ≤ z, g ¯ − ε ∀x ∈ X, ∀v ∈ −K ∗ , holds. From here, it follows s(g, ¯ X ) + v, g ¯ ≤ z, g ¯ − ε ∀v ∈ −K ∗ . If the vector g¯ ∈ / K , then from (77.2), we have z, g ¯ − ε ≥ +∞. A contradiction is obtained. If the vector g¯ ∈ K , then from (77.2), we have s(g, ¯ X ) ≤ z, g ¯ − ε. This inequality contradicts our assumption that z ∈ X (K ).

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Prove the inclusion X − K ∗ ⊂ X (K ). Choose an arbitrary point z ∈ X − K ∗ . Then it can be represented as z = x + v, x ∈ X, v ∈ −K ∗ . For any arbitrary g ∈ K , ||g|| = 0, z, g = x, g + v, g, x ∈ X, v ∈ −K ∗ . From this equality, it follows that z, g ≤ x, g and z, g ≤ s(g, X ). Therefore, for any g ∈ K , ||g|| = 0, z, g = x, g ∀g ∈ K , ||g|| = 0. Thus z ∈ X (K ). And we have equality (77.3).



Corollary 77.1 The next equality sup x, g = maxx, g ∀g ∈ K , x∈X

x∈X (K )

(77.4)

holds. Proof Since condition (77.3) holds, then sup x, g =

x∈X (K )

sup x, g = maxx, g + sup x, g ∀g ∈ K . x∈X

x∈X −K ∗

x∈−K ∗



By using (77.2), we have (77.4).

Remark 77.1 The set X (K ) is unbounded, but nevertheless for any g ∈ K the support function s(g, X (K )) is finite. Corollary 77.2 Let X ⊂ R n , Y ⊂ R n be convex compacts, K ⊂ R n be a closed convex cone. In order for the equality maxx, g = maxy, g ∀g ∈ K x∈X

y∈Y

to be true, it is necessary and sufficient that the equality X − K∗ = Y − K∗ holds.

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77.4 A Forming Set of a Convex Compact Relative to a Convex Cone Let X ⊂ R n be a convex compact, K ⊂ R n be a closed convex cone. To find a value of the support function s(g, X ) on this cone, you don’t have to know all elements of the set X . Calculate the subdifferential of the support function s(g, X ) at the point g ∈ R n , ||g|| = 0. A linear function other than a constant reaches its maximum value at a boundary point of a convex closed set. Put ||x(g)|| =

min

x∈∂s(g,X )

||x||,

√ where || · || denotes the Euclidean norm (||x|| = x, x). The vector x(g) is unique and x(g) ∈ bdX . From the necessary and sufficient minimum conditions of the differentiable function x, x on the set ∂ p(g, X ), the inclusion x(g) ∈ Γ ∗ (x(g), ∂s(g, X )) holds, where Γ (x(g), ∂s(g, X )) denotes the cone of possible directions at the point x(g) to the set ∂s(g, X ). From here we have x, x(g) ≥ x(g), x(g) ∀x ∈ ∂s(g, X ). Definition 77.2 A set X (K ) =



x(g)

g∈K ||g|| = 0 is called a forming set of a convex compact X relative to a convex cone K . It is obvious that s(g, X ) = s(g, X (K )) ∀g ∈ K , ||g|| = 0. The set X (K ) for each cone K is defined uniquely. It is bounded because the set X is bounded. In addition, due to the construction of this set, the next equality maxx, g = sup x, g = x∈X

x∈X (K )

sup

x, g ∀g ∈ K ,

x∈co{X (K )}

is true. From Corollary 77.2 and the properties of a support function we have

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X − K ∗ = X (K ) − K ∗ = co {X (K )} − K ∗ . Consider some examples that illustrate this definition. Example 77.1 Let X = co{(0, 0), (1, 0), (1, 1), (2, 1)} ⊂ R 2 . (a) If K = R 2 , then X (K ) = X = co{(0, 0), (1, 0), (1, 1), (2, 1)}. (b) If K = {(x, y) ∈ R 2 | x ≥ 0, y ≥ 0}, then X (K ) = co{(1, 0), (1, 1), (2, 1)}. (c) If K = co{λ1 (1, 1), λ2 (1, 0)}, λ1 , λ2 ≥ 0, then X (K ) = co{(1, 0), (1, 1), (2, 1)}. (d) If K = {(x, y) ∈ R 2 | y = 0.5x, x ≥ 0}, then X (K ) = {(2, 1)}. Example 77.2 Let X = X 1 + X 2 ⊂ R 2 , where X 1 = co{(1, 1), (−1, 1), (1, −1), (−1, −1)}, X 2 = {(x, y) ∈ R 2 | x 2 + y 2 ≤ 1}. (a) If K = R 2 , then X (K ) = {(x, y) ∈ R 2 | 1 < x < 2, 1 < y < 2, (x − 1)2 + (y − 1)2 = 1}∪ ∪{(x, y) ∈ R 2 | 1 < x < 2, −2 < y < −1, (x − 1)2 + (y + 1)2 = 1}∪ ∪{(x, y) ∈ R 2 | − 2 < x < −1, 1 < y < 2, (x + 1)2 + (y − 1)2 = 1}∪ ∪{(x, y) ∈ R 2 | − 2 < x < −1, −2 < y < −1, (x + 1)2 + (y + 1)2 = 1}∪ ∪(2, 0) ∪ (0, 2) ∪ (−2, 0) ∪ (0, −2). Here the forming set X (K ) consists of four circumference arcs and of four points. See Fig. 77.1. (b) If K = {(x, y) ∈ R 2 | x ≥ 0, y ≥ 0}, then X (K ) = {(x, y) ∈ R 2 | 1 < x < 2, 1 < y < 2, (x − 1)2 + (y − 1)2 = 1} ∪ (2, 0) ∪ (0, 2).

Here the forming set X (K ) consists of a circumference arc and of two points. Theorem 77.2 Let K ⊂ R n be a closed convex cone and a closed convex set Y ⊂ R n contains the zero point, then the forming set [Y ∩ (−K ∗ )](K ) relative to the cone K consists only of the zero point. Proof Since 0n ∈ [Y ∩ (K ∗ )] and v, g ≤ 0 ∀v ∈ Y ∩ (−K ∗ ), ∀g ∈ K , then sup

v, g = 0 ∀g ∈ K .

v∈Y ∩(−K ∗ )

This supremum is achieved at the set containing the zero point. Therefore,  [Y ∩ (−K ∗ )](K ) = 0n .

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Fig. 77.1 Example 77.2, (a): The sets X (with the interior) and X (K )

Corollary 77.3 Let K ⊂ R n be a closed convex cone and X ⊂ R n be a nonempty compact convex set. Let a closed convex set Y ⊂ R n contain a zero point, then the next equality holds. maxx, g = x∈X

max

x∈X +[Y

x, g ∀g ∈ K .

(−K ∗ )]

Corollary 77.4 1. The support function of a set X ⊂ R n relative to a convex cone K ⊂ R n will not change if we add to this set another convex set containing the zero point and lying in the cone −K ∗ . 2. If for some convex set Y ⊂ R n , y(g), g = 0 ∀g ∈ K , then max x, g = maxx, g ∀g ∈ K .

x∈X +Y

x∈X

Thus, any set Y contained in the linear space orthogonal to the linear hull of the cone K can be added to the set X , and the value of the support functions of the sum of these sets is equal to the value of the support function of the set X for each g ∈ K .

77.5 Conclusion Thus, the paper considers the properties of the forming set of a convex compact relative to a convex cone. The forming set allows to calculate the value of the support function on this cone without information about all the elements of the considered set. Conditions under which support functions of the two sets on the cone are equal are derived. Some examples illustrating these properties are given.

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References 1. Bishop, E., Phelps, P.: The support functionals of a convex set. Proc. Symp. Pure Math., 27–35 (1962) 2. Caprari, E., Demyanov, V.F.: Conically equivalent sets and their minimality. In: Giorgi, G., Rossi, F. (eds.) Generalized Convexity and Optimization for Economic and Financial Decisions, pp. 81–95. Pitagora Editrice Bologna, Verona (1998) 3. Demyanov, V.F., Aban’kin, A.E.: Conically equivalent pairs of convex sets. In: Gritzmann, P. et al. (eds.) Recent Advances in Global Optimization, L.N.E.M.S., vol. 452, pp. 19–33. Springer, Berlin (1997) 4. Demyanov, V.F., Kaprari, E.: The minimality of conically equivalent convex sets. Vestn. of St.-Peterbg. Univ. Ser 1. 2, 32–37 (1998) 5. Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Peter Lang Verlag, Frankfurt (1995) 6. Leichtweiss, K.: Von Konvexe Mengen. Springer, Berlin (1980) 7. Polyakova, L.N.: Some properties of the support function of a convex set on a convex cone. Vestn. of St.-Peterbg. Univ. Ser 10. 3, 70–78 (2012) 8. Pschenichny, B.N.: Convex Analysis and Extremal Problems. Nauka, Moscow (1980) 9. Rockafellar, R.: Convex Analysis. Princeton University Press, Prinston (1970)

Chapter 78

Methodology of Structural–Functional Synthesis for the Small Spacecraft Onboard System Appearance Alexander N. Pavlov, Valentin. N. Vorotyagin, Dmitry A. Pavlov, and Valerii V. Zakharov

Abstract To increase the degree of validity of design decisions when creating a small spacecraft, a methodological approach is proposed for the structural–functional synthesis of the appearance of its onboard systems. The proposed approach allows to perform multi-criteria selection of effective configuration variants for onboard system of a small spacecraft, taking into account various types of structural reservation of onboard equipment, a wide assortment of element base. The conceptual and mathematical formulations of the problem of structural and functional synthesis of the appearance of the onboard system of a small spacecraft are presented. An algorithm has been developed for multi-criteria synthesis of the appearance of the onboard system of a small spacecraft, which is based on the interval lexicographic method for removing criterial uncertainty. An illustrative example of finding a given number of rational variants for implementing a motion control system for a small spacecraft uniformly located in the Pareto region is given. This technique allows even at the stage of the spacecraft designing to reduce significantly the number of design errors by the decision-maker, as well as to improve the quality and efficiency of the application of the created space systems.

A. N. Pavlov (B) · Valentin. N. Vorotyagin · D. A. Pavlov Mozhaisky Military AeroSpace Academy, Zhdanovskaya st., 197082 St. Petersburg, Russia e-mail: [email protected] Valentin. N. Vorotyagin e-mail: [email protected] D. A. Pavlov e-mail: [email protected] V. V. Zakharov St. Petersburg Federal Research Center of the Russian Academy of Sciences, 39, 14th Line V.O., St. Petersburg 199178, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_78

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78.1 Introduction Nowadays, in place of traditional approaches to create a new appearance for the spacecraft’s board, smart systems of supporting design solutions are used more often within the framework of model-oriented design [1], which allow to form the appearance of space systems with the best tactical and technical characteristics within shorter time. Model-oriented design is based on constructing a system model, in which various components and subsystems are combined. The analysis of existing methods of modeling and calculating the reliability, survivability, and safety indicators for structurally complex systems [2–5] to create the above-mentioned system model has showed that modeling methods are usually based on structural diagrams of systems properties under study, in which elements (vertices) are the states of individual functional elements (FE) and the edges represent the logical conditions of the fact, whether the elements fulfill their functions in the system or not. As a rule, a structural analysis of SSC BS functioning starts from constructing an object’s functional integrity scheme (FIS) [4, 6]. The most common way to improve reliability of the systems is to introduce structural redundancy (FE SSC redundancy). There are two main types of redundancy: multiple FE redundancies (duplication, triplication , etc.); combinatorial redundancy (work on the principle of “two of three”, “two of four”, “three of four”, etc.). It should be mentioned that when constructing the FIS for implementation of the operating modes of the SSC BS, as a rule, combinatorial redundancy was taken into account [4]. Therefore, with a narrow assortment of the element base, it is required to set the maximum possible multiplicity of the FE SSC redundancy for each FE SSC. In this case, the mass-size and energy characteristics of the FE will increase according to the redundancy rate. In case of wide assortment of elemental base designing the SSC BS, it is required to take into account various reliability indicators, mass-size, and energy characteristics when selecting the composition of the SSC BS elements from the whole item group. The substantive formulation of the problem of the SSC BS appearance structuralfunctional synthesis leads to the following fact that it is required to determine the configuration options for the SSC BS used by the SSC FE taking into account the redundancy rate and the wide range of element base, which provide the maximum reliability of implementation of the SSC BS operating modes, the minimum power consumption of the SSC BS and its minimum mass.

78.2 Mathematical Formulation of Structural and Functional Synthesis of BS SSC Appearence Taking into account the conceptual model for generalized formalization of the considered problem mentioned above, we use the set theory and define the following sets of its elements and characteristics as the main ones:

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A = {Ai , i ∈ N }, N = {1, 2, . . . , n} – a set of FE BS SSC; B = {B ij , j ∈ D i , D i = {1, 2, . . . , m}, i ∈ N } – range of the SSC element base (B ij – j-type Ai FE SSC BS); PF ({ pi , qi = 1 − pi }, i ∈ N ) – probability function (PF) of the SSC BS uptime for implementation of the operating modes specified by logical criterion F; ci j , i ∈ N , j ∈ {1, 2, . . . , m i } – FE SSC power consumption; ϑi j , i ∈ N , j ∈ {1, 2, . . . , m i } – FE SSC mass; pi j , i ∈ N , j ∈ {1, 2, . . . , m i } – FE SSC reliability. Let’s designate the alternative design for SSC BS to be developed as X = xi j , where xi j = 1, if element of B ij type is used as FE Ai , xi j = 0, otherwise. mi n   As the objective functions we use C(X ) = ci j xi j —the SSC BS power consumption, V (X ) = P(X ) = PF (

m1  j=1

mi n  

i=1 j=1

ϑi j · xi j – the SSC BS mass,

i=1 j=1 m2 

mn 

j=1

j=1

p1 j x 1 j ,

p2 j x 2 j , . . . ,

pn j xn j )—the reliability of implementa-

tion of operating modes for the SSC BS. For the task with a wide range of constituent elements, restrictions for the selection of options for implementing the SSC BS of the following type are introduced: mi  xi j ≤ 1, ∀i ∈ N in the SSC BS to be designed, some element of B ij type from j=1

its range is used as a FE Ai . For the task with narrow range of constituent elements, introduced restriction mi  xi j ≤ 1, ∀i ∈ N has the following interpretation—the maximum redundancy rate j=1

FE Ai im SSC BS to be designed is not more m i , and xi j = 1 when FE Ai is duplicated by j rate. Moreover, pi1 = pi , pi2 = 1 − (1 − pi )2 , . . . , pim i = 1 − (1 − pi )m i ,∀i ∈ N , ϑi1 = ϑi , ϑi2 = 2ϑi , . . . , ϑim i = m i ϑi ,∀i ∈ N , and ci1 = ci ,ci2 = 2ci , . . . , cim i = m i ci ,∀i ∈ N. These restrictions specify  area for acceptable options for the SSC BS implementation. Thus, the task of synthesizing the appearance of a designed SSC BS leads to the multi-criteria problem of selecting the valid alternatives of the following form in a discrete set: min C(X ), min V (X ), max P(X ) X ∈

X ∈

X ∈

In other words, it is necessary to find effective (Pareto) variants of the SSC BS project nd ⊆ , which ensure minimum power consumption of the SSC BS, its minimum mass, and maximum reliability of implementation of the SSC BS operating modes.

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78.3 Overcoming Criterial Uncertainty During Designing of SSC BS It is known [7–12] that the basis of multi-criteria selection methods is regularizationadditional determination (clarification) of the problem by appealing to additional qualitative and quantitative information about the properties of criteria functions, alternatives and principles of optimality. The main source of additional information in the process of searching for the best alternatives is the experts, who are well aware of a given subject area and a decision-maker (DM) pursuing a specific aim (s), for which the considered task is being solved. Nowadays, a wide variety of methods for solving the tasks of multi-criteria selection have been developed [7–12]. When classifying these methods, various principles and features can be proposed. For example, in [12] it is proposed to distinguish between classes: a priori, posteriori and adaptive methods, and multi-criteria optimization models. Among a priori methods, methods for constructing convolution of indicators (scalarization) are distinguished. In this case, heuristic and axiomatic convolutions are distinguished [7]. Another group of a priori methods for solving the multi-criteria problems is based on componentwise construction of the resulting preference relationships. At the same time, Pareto, lexicographic and major resultant preference relations are distinguished, the first two of which are subdivided into classic, interval, and threshold ones and the latter is subdivided into non-interval and interval resultant preference relations. When forming the resulting preference relation for the considered we will take into account the following aspects that are most essential for the designed SSC BS. Firstly, the efficiency of the SSC BS functioning is associated with an implementation reliability of its functioning modes and secondly, the requirement to reduce its energy consumption and mass characteristics is essential for the SSC. So, the first aspect is mn m1 m2    p1 j x 1 j , p2 j x 2 j , . . . , pn j xn j ), and reflected by the indicator P(X ) = PF ( j=1

the second − by indicators C(X ) =

mi n  

j=1

ci j xi j and V (X ) =

i=1 j=1

j=1

mi n  

ϑi j xi j .

i=1 j=1

Overcoming the criteria uncertainty of this problem is firstly associated with a compromise between the group of indicators C(X ) and V (X ) reflecting the massenergy parameters of the SSC BS and indicator P(X ) reflecting the reliability characteristics of implementation of the SSC BS area operating modes. Moreover, DM compromise area is being narrowed significantly and decision rules can be considered rational as per which the relation P  C ≈ V is ensured, i.e. indicator P(X ) is more important than indicators C(X ) and V (X ). That is why, lexicographic methods can be used to construct the resulting preference relationship [11, 12]. The lexicographic problem of multi-criteria selection results in solving the following sequence of optimization problems: 1 = arg max P(X ) ⇒ 2 = arg min C(X ) ⇒ X ∗ = arg min V (X ). X ∈

X ∈1

X ∈2

78 Methodology of Structural–Functional Synthesis ...

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One of the lexicographic method features lies in the fact that the set 1 obtained by optimizing the criterion function P(X ) on initial set  can contain a single alternative X ∗ = arg max P(X ). In this situation the opportunity to optimize by other criteria X ∈

functions is lost. To expand the possibilities of applying the lexicographic methods for other criteria functions we will use the interval lexicographic order—the method of consecutive concessions [11, 12]. In this case, the method and corresponding algorithm for successive narrowing the set of alternatives X ∗ ∈ 2 ⊆ 1 ⊆  result in following: 1 = {X ∈  | P(X ) ≥ Pmax − ε1 } ⇒ 2 = {X ∈ 1 | C(X ) ≥ Cmin + ε2 } ⇒ X ∗ = arg min V (X ), where Pmax = max P(X ), Cmin = arg min C(X ). X ∈2

X ∈

X ∈1

The following interpretation can be given to the above ratios: at first, optimization is carried out according to the first objective function P(X ) (SSC reliability in implementation of operating modes), the maximum value of this function Pmax = max P(X ) is determined, and the maximum allowable decrease in this indiX ∈

cator (concession) ε1 is introduced. Secondly, optimization is carried out according to C(X ) (the SSC BS power consumption), the minimum value of this function Cmin = arg min C(X ) is determined, and the maximum allowable indicator increase X ∈1

in this indicator (concession) ε2 is introduced. Finally, optimization by V (X ) indicator (the SSC BS mass) is carried out. The fewer are concessions on the previous indicators, the fewer is a possibility to improve the subsequent indicators. At the same time, it is obvious that there is no sense to assign such concessions that would lower the values of indicators below the minimum (maximum) values accepted by these functions in the Pareto set (non-dominated alternatives). By varying with concessions, we obtain various solutions from the Pareto region (the region of compromises).

78.4 Example of Multi-Criteria Structural–Functional Synthesis of Appearance of SSC BS Motion Control System In this subsection, we illustrate the proposed approach using an example of the appearance synthesis for the motion control system (MCS) Aist-2d SSC [13]. MCS is designed to measure the current angular position of SSC in the given coordinate systems (CS) and its angular velocity with inertial CS, the orientation, and stabilization of the SSC in the specified CS. MCS includes the following functional elements (FE): separate angular velocity meter (SAVM)—4 pcs; flywheel control engine (FCE)—4 pcs; device of orientation on the Earth (DOE)—2 pcs; satellite navigation system (SNS)—1pcs; optical star sensor (OSS)—2 pcs; magnetometer (MA)—2 pcs; electromagnets (EM)—3 pcs.

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Fig. 78.1 Orientation mode of SSC MCS Table 78.1 Initial data on reliability of elements FE 1 2 3 4 SAVM FCE SNS OSS DOE MA EM

0.75 0.60 0.74 0.90 0.84 0.50 0.75

0.94 0.84 0.93 0.99 0.97 0.75 0.94

0.80 0.75 0.80 0.85 0.90 0.89 0.98

0.96 0.94 0.96 0.98 0.91 0.94 0.98

5

6

7

0.85 0.80 0.85 0.80 0.99 0.80 0.99

0.90 0.96 0.98 0.96 0.92 0.96 0.92

0.95 0.99 0.96 0.70 0.94 0.99 0.94

6

7

7 7.8 44 24 6.5 4 28

7 12 22 8 6.5 6 10

Table 78.2 Initial data on energy consumption of elements FE 1 2 3 4 5 SAVM FCE SNS OSS DOE MA EM

6 1.7 15 11 3.5 1.5 4

12 3.4 30 22 7 3 8

7 3 15 9 6.5 4.5 12

14 6 30 18 6.5 6 16

7 3.9 22 12 13 2 14

Structural and functional diagram of interactions of mentioned FE BS MCS SSC when performing the orientation mode is shown in Figure 78.1. The initial data of FE range, out of which options for the MCS SSC appearance can be composed, are given in Tables 78.1, 78.2 and 78.3. As a result of applying a multi-criteria synthesis algorithm for the appearance of the MCS SSC, which are based on the method of sequential concessions (interval lexicographic ordering), 21 options were obtained. Further narrowing of the set of non-dominated alternatives nd (Pareto set) is carried out in interactive mode with the active involvement of decision-makers DM. Such a cutoff is based on a mathematical study of the Pareto set properties and obtaining additional information from the decision-maker DM (estimation of the set power, the range of variation of the indicators values, their inconsistency). So, for example, the developer puts forward requirements to the SSC MCS reliability: relia-

78 Methodology of Structural–Functional Synthesis ... Table 78.3 Initial data on mass of elements FE 1 2 3 SAVM FCE SNS OSS DOE MA EM

0.2 0.18 3.4 4.5 3 0.1 0.03

0.4 0.36 6.8 9 6 0.2 0.06

0.35 0.3 6.4 4 1.6 0.3 0.09

693

4

5

6

7

0.7 0.6 12.8 8 1.82 0.4 0.12

0.9 0.56 4.8 1.7 3.64 0.8 0.1

1.7 1.12 9.6 3.4 2.4 1.6 0.2

2.5 2.7 6.5 1.3 6.2 2.4 0.21

Table 78.4 Effective (Pareto) options for implementing the SSC MCS. Part 1 Version SAVM 1

SAVM 2

SAVM 3

SAVM 4

FCE 1

FCE 2

FCE 3

FCE 4

SNS

OSS 1

OSS 2

16

4

7

7

2

4

4

4

4

1

-

-

18

4

4

4

4

7

7

7

7

1

5

7

20

4

4

4

4

7

7

7

7

1

5

5

Table 78.5 Effective (Pareto) options for implementing the SSC MCS. Part 2 Version

DOE 1

DOE 2

MA 1

MA 2

EM 1

EM 2

EM 3

Mass

Energy consumption

Reliability

16

3

4

4

4

5

5

5

16.42

146.0

0.9559

18

4

4

7

7

5

5

5

28.74

206.0

0.9584

20

3

5

7

4

5

5

5

28.74

216.5

0.9600

bility of the product to be designed should be at least 0.95. The energy consumption indicator for the product is not critical, but the mass indicator should not exceed 30.0 conventional units. In this case, to select the final implementation option for the product, the developer will be offered the following 3 options from Table 78.4 for further analysis: option 16; option No18; option No20 ({x16 , x18 , x20 } ⊆ nd ) (Table 78.5). Let’s comment on one of the effective embodiments of the product under number 18. In the whole, this variant of orientation mode implementation for SSC MCS is characterized by the following values of reliability, mass, and power consumption indicators: the product reliability is 0.9584, and its mass is equal to 28.74, and the power consumption is equal to 206.

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78.5 Conclusion This approach of multi-criteria synthesis of the SSC BC appearance can, even at the design stage, significantly reduce the number of decision-making errors and, therefore, improve the quality and efficiency of use for the space systems to be designed. Acknowledgements The research described in this paper is partially supported by the Russian Foundation for Basic Research (grants 17-29-07073, 18-07-01272, 18-08-01505, 19-08-00989, 20-08-01046), state research 0073-2019-0004.

References 1. Balukhto, A.N., Romanov, A.A.: Artificial intelligence in space technology: State, development prospects. Rocket-space device engineering and information systems 6, 65–75 (2019) 2. The software package Risk Spectrum probabilistic analysis of the reliability and security of systems. Designed by the Swedish company Relcon AB. The form of the initial structural diagram of the system is a fault tree. http://www.riskspectrum.com 3. Kurenkov, V.I., Kapitonov, V.A.: Methods of calculation and reliability of space rocket complexes. Publishing House Samar State Aerospace University, Samara (2007) (In Russian) 4. Polenin, V.I., Riabinin, I.A., Svirin, S.K., Gladkova, I.A.: The use of a common logical - probabilistic method for the analysis of technical, military organizational and functional systems and armed confrontation. The Russian Academy of Natural Sciences, St.Petersburg (2011) (In Russian) 5. Viktorova, V.S., Kuntcher, H., Petrukhin, B.P., Stepanyants, A.S.: Relex: Program for analysis of reliability, safety, risks. Nadezhnost 4(7), 42–64 (2003) 6. Aleshin, E.N., Zinovev, S.V., Kopkin, E.V., Osipenko, S.A., Pavlov, A.N., Sokolov, B.V.: System analysis of organizational and technical systems for space applications. VKA imeni A.F.Mozhajskogo, St.Petersburg (2018) (In Russian) 7. Kini, R.L., Rayfa, K.H.: Decision making under many criteria: preferences and substitutions. Radio i svyaz’, Moscow (1981) (In Russian) 8. Larichev, O.I.: Verbal analysis of decisions. Nauka, Moscow (2006) (In Russian) 9. Nogin, V.D.: Decision-making in a multi-criteria environment: a quantitative approach. FIZMATLIT, Moscow (2005) (In Russian) 10. Nogin, V.D.: Problems of narrowing the pareto set: approaches to solution. Iskusstvennyy intellekt i prinyatiye resheniy 1, 98–112 (2008) 11. Podinovskiy, V.V., Nogin, V.D.: Pareto—Optimal Solutions of Multicriteria Problems. Nauka, Moscow (1982) (In Russian) 12. Sokolov, B.V., Moskvin, B.V., Pavlov, A.N., et al.: Military systems engineering and systems analysis. Models and decision-making methods in complex organizational and technical complexes in the face of uncertainty and multicriteria. VIKKU imeni A. F. Mozhayskogo, St. Petersburg (1999) (In Russian) 13. Kirilin, A.N., Akhmetov, R.N., Shakhmatov, Y.V., Tkachenko, S.I., et al.: Experimental technological small satellite. “AIST-2D” Samarskiy nauchnyy tsentr Rossiyskoy akademii nauk, Samara (2017) (In Russian)

Chapter 79

Equivalence of Two Optimality Conditions for Polyhedral Functions Majid Abbasov

Abstract We study two different optimality conditions for polyhedral functions in the paper. The first condition follows straightforward from the optimality conditions formulated in terms of coexhausters. Coexhausters are families of convex compact sets, via which one can approximate the increment of the considered function at a studied point as a minmax or maxmin of affine functions. The second condition was derived in a recent publication of Dolgopolik. It is a core result for building of a new optimization algorithm that allows one to find a minimum of a piecewise affine function. Piecewise affine function is a significant notion that is used in different areas of mathematics and its applications. Coexhausters are important tools of nondifferentiable optimization that is used for the study of a wide class of nonsmooth functions. Therefore the problem of proving the equivalence of these two types of conditions and connection between them are of high importance. The study of these problems is the main aim of the current work.

79.1 Introduction Piecewise affine functions [1] arise in different branches of mathematics and applications [2–5]. Therefore it is not surprising that this object bring attention of many researchers. In the paper we deal with results recently obtained in [6]. Authors propose iterative algorithm for minimizing a piecewise affine function. This algorithm is based on new optimality conditions for a polyhedral function. It is interesting that optimality conditions for the same problem can be also stated in terms of coexhausters. Coexhausters are a new notion of nonsmooth analysis introduced by V. F. Demyanov in [7, 8]. These are families of convex compact sets, via which one can approximate the increment of the considered function at a studied point as a minM. Abbasov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, 61, Bolshoj pr. V.O., St. Petersburg 199178, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_79

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max or maxmin of affine functions. There is well-developed theory of coexhausters [9–11]. In the work we compare this two optimality conditions and prove their equivalence. The paper is organized as follows. In Sect. 79.1 we provide necessary facts from the theory of codifferentials and coexhausters. The definition of polyhedral function is also given in this section. Optimality conditions for the polyhedral functions as well as the proof of their equivalence and an illustrative numerical result are presented in Sects. 79.2 and 79.3, respectively.

79.2 Codifferentiable Functions and Coexhausters Before going to the definition of coexhausters, we need some additional preliminaries.

79.2.1 Codifferentiable Functions Let an open set X ⊂ Rn be given. It is said that a function f : X → R is codifferentiable at a point x if and only if there exist convex compact sets d f (x) ⊂ Rn+1 and d f (x) ⊂ Rn+1 such that f (x + Δ) = f (x) +

max [a + v, Δ] +

[a,v]∈d f (x)

min

[b,w]∈d f (x)

[b + w, Δ] + ox (Δ), (79.1)

where lim α↓0

ox (αΔ) = 0, ∀Δ ∈ Rn . α

(79.2)

Here a, b ∈ R; v, w ∈ Rn . The pair D f (x) = [d f (x), d f (x)] is called a codifferential of the function f at the point x. Codifferential is a pair of sets from Rn+1 . Equation (79.1) implies f (x + Δ) = f (x) +

max

min

[a + b + v + w, Δ] + ox (Δ) =

[a,v]∈d f (x) [b,w]∈d f (x)]

= f (x) + max

min [b + w, Δ] + ox (Δ),

C∈E(x) [b,w]∈C

where E(x) = {C ⊂ Rn+1 |C = [a, v] + d f (x)}, [a, v] ∈ d f (x)}. Similarly we get

(79.3)

79 Equivalence of Two Optimality Conditions for Polyhedral Functions

f (x + Δ) = f (x) +

697

max [a + b + v + w, Δ] + ox (Δ) =

min

[b,w]∈d f (x) [a,v]∈d f (x)

= f (x) + min max [a + v, Δ] + ox (Δ), C∈E(x) [a,v]∈C

(79.4)

where E(x) = {C ⊂ Rn+1 |C = [b, w] + d f (x)}, [b, w] ∈ d f (x)}. The functions h 1 (x, Δ) = max

min [b + w, Δ] and h 2 (x, Δ) = min max [a + v, Δ]

C∈E(x) [b,w]∈C

C∈E(x) [a,v]∈C

represent approximations of the increment of the function f in the neighborhood of x. The codifferential notion was proposed in [12]. Necessary optimality conditions were stated in the same works. Via the expansions (79.3) and (79.4) we can get the following generalization of the codifferential notion.

79.2.2 Upper and Lower Coexhausters Let a function f be continuous at a point x ∈ X . It is said that at the point x the function f has an upper coexhauster iff the following expansion is true: f (x + Δ) = f (x) + min max [a + v, Δ] + ox (Δ), C∈E(x) [a,v]∈C

(79.5)

where E(x) is a family of convex compact sets in Rn+1 , and ox (Δ) satisfies (79.2). The set E(x) is called an upper coexhauster of the function f at the point x. We say that at the point x the function f has a lower coexhauster iff the following expansion is valid: f (x + Δ) = f (x) + max

min [b + w, Δ] + ox (Δ),

C∈E(x) [b,w]∈C

(79.6)

where E(x) is a family of convex compact sets in Rn+1 , and ox (Δ) satisfies (79.2). The set E(x) is called a lower coexhauster of the function f at the point x. Since the function f is continuous, it follows from (79.5) and (79.6) (for Δ = 0) that (79.7) min max a = max min b = 0. C∈E(x) [a,v]∈C

C∈E(x) [b,w]∈C

The coexhauster notion was proposed in [7, 8].

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Polyhedral Functions

Let us give the definition (see [13]) of the function for which optimality conditions are stated. Definition 79.1 The function of the form f (x) = max[ai + vi , x], where I = i∈I

1, . . . , m and ai ∈ R, vi ∈ Rn for all i ∈ I , is called polyhedral. It is obvious that a polyhedral function has an upper coexhauster at any point x∗ ∈ Rn . We can chose E(x∗ ) = {C}, where C = conv{[ai − f (x∗ ), vi ] | i ∈ I }. Without loss of generality, assume in what follows that the function equals zero at the studied point, i.e. f (x∗ ) = 0. Then E(x∗ ) = {C}, C = conv{[ai , vi ] | i ∈ I } and x∗ is a global minimizer of f iff max [a + v, x] ≥ 0 ∀x ∈ Rn .

[a,v]∈C

79.3 Equivalence of the Two Optimality Conditions for Polyhedral Function Theorem 79.1 (see [10]) For the condition max [a + v, x] ≥ 0

[a,v]∈C

to hold for any x ∈ Rn it is necessary and sufficient that C



L+ 0  = ∅,

  n+1 where L + |a≥0 . 0 = [a, 0n ] ∈ R Theorem 79.2 (see [6]) For the condition max [a + v, x] ≥ 0

[a,v]∈C

to hold for any x ∈ Rn it is necessary and sufficient that one of the following two conditions are true: • 0n ∈ C, • the function f is bounded below and a ∗ > 0, where [a ∗ , v ∗ ] = arg min [a, v] 2 , [a,v]∈C

79 Equivalence of Two Optimality Conditions for Polyhedral Functions

699

and . is a euclidian norm. Theorem 79.3 (main result) Conditions in Theorems 79.1 and 79.2 are equivalent, i.e. the condition  C L+ 0 = ∅ holds iff one of the following two conditions are true: • 0n ∈ C, • the function f is bounded below and a ∗ > 0.  Proof Let the condition C L + 0  = ∅ holds. Consequently, there exists (a, 0n ) ∈ C, such that a ≥ 0. If a = 0 then 0n ∈ C. Therefore, consider the case a > 0. For every x ∈ Rn we have f (x) = max [a + v, x] ≥ a ≥ 0. [a,v]∈C

Hence f is bounded below. Now let us show that a ∗ > 0. Assume the contrary, i.e. a ∗ ≤ 0. Consider the expression [a(α), v(α)] = α[a, 0n ] + (1 − α)[a ∗ , v ∗ ] for any α such that 0 ≤ α ≤ 1. Omitting trivial calculations we get

[a(α), v(α)] 2 = [a ∗ , v ∗ ] 2 + (α 2 − 2α) v ∗ 2 + α 2 (a ∗ − a)2 + 2α(aa ∗ − a ∗ 2 ). Whence [a(α), v(α)] 2 < [a ∗ , v ∗ ] 2 for sufficiently small α, what contradicts the definition of a ∗ .  Now show that the conditions of Theorem 79.2 imply C L + 0  = ∅. If 0n ∈ C then this condition holds. Assume that 0 ∈ / C and f is bonded below and a ∗ > 0. There exists M ∈ R such that for all x ∈ Rn the inequality f (x) ≥ M is true, i.e. max [a − M + v, x] ≥ 0.

[a,v]∈C

 Hence C M L = ∅, where C M = {[a − M, v)] | [a, v] ∈ C}. Therefore there exists a − M ≥ 0. Note that since 0 ∈ / C is true, we have  a = 0. [ a , 0n ] ∈ C such that  Assuming that  a < 0 and considering the point [a(α), v(α)] = α[ a , 0n ] + (1 − α)[a ∗ , v ∗ ] we can show (the same as it was done above) that for sufficiently small α,

[a ∗ , v ∗ ] 2 holds. This contradicts the 0 ≤ α ≤ 1 the inequality [a(α), v(α)] 2 0 and C L +  definition of a . Consequently  0  = ∅. Example 79.1 Let the function f 1 : R → R is given by f 1 (x) = max {−2 + x, −x} . The family E = {C}, where C = conv

 

−2 0 , , 1 −1

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a)

b)

Fig. 79.1 The upper coexhauster and the graph of the function f 1

is an upper coexhauster of the function f 1 . It is  obvious from the Fig. 79.1a) that neither the condition of Theorem 79.1 (since C L + 0 = ∅) nor the conditions of / C, a ∗ < 0) are fulfilled here. Thus, the origin is not a Theorem 79.2 (since 02 ∈ global minimizer of the function f 1 (see Fig. 79.1b).

79.4 Conclusion We proved the equivalence of the two recently proposed optimality conditions for the class of polyhedral functions. The obtained results are important for the development of the coexhausters theory (since one of these conditions is formulated in terms of coexhausters) and the theory of piecewise affine functions. Acknowledgements The results in Section 79.3 were obtained in the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences with the support of Russian Science Foundation (RSF), project No. 20-71-10032.

References 1. Scholtes, S.: Piecewise affine functions. In: Introduction to Piecewise Differentiable Equations. Springer Briefs in Optimization, pp. 13–63. Springer, New York (2012) 2. Coppersmith, D., Lee., J.: Parsimonious binary-encoding in integer programming. Discret. Optim. 2, 190–200 (2005) 3. Croxton, K.L., et al.: Variable disaggregation in network flow problems with piecewise linear costs. Oper. Res. 55(1), 146–157 (2007) 4. Silva, T.L., Camponogara, E.A.: computational analysis of multidimensional piecewise-linear models with applications to oil production optimization. Eur. J. Oper. Res. 232(3), 630–642 (2014) 5. de Farias Jr., I. R., Kozyreffv, E., et al.: Branch-and-cut for separable piecewise linear optimization and intersection with semi-continuous constraints. Math. Program. Comput. 5(1), 75–112 (2013)

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6. Dolgopolik, M.V.: The method of codifferential descent for convex and global piecewise affine optimization. Optim. Methods Softw. (2019). https://doi.org/10.1080/10556788.2019. 1571590 7. Demyanov, V.F.: Exhausters and convexificators – new tools in nonsmooth analysis. Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol. 43, pp. 85–137. Kluwer Academic Publishers, Dordrecht (2000) 8. Demyanov, V.F.: Exhausters of a positively homogeneous function. Optimization 45(1–4), 13–29 (1999) 9. Abbasov, M.E., Demyanov, V.F.: Adjoint coexhausters in nonsmooth analysis and extremality conditions. J. Optim. Theory Appl. 156, 535–553 (2013) 10. Demyanov, V.F.: Proper exhausters and coexhausters in nonsmooth analysis. Optimization 61, 1347–1368 (2012) 11. Abbasov, M.E.: Constrained optimality conditions in terms of proper and adjoint coexhausters (in Russian), Vestnik of St Petersburg University. Appl. Math. Comput. Sci. Control. Process. 15(2), 160–172 (2019) 12. Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Approximation and Optimization, vol. 7. Peter Lang, Frankfurt am Main, iv, 416 (1995) 13. Polyakova, L.N.: The hypodifferential and the ε-subdifferential of polyhedral function. Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 3, 64–71 (2011)

Chapter 80

A Computational Approach to Estimating Activity Coefficients Using Gibbs Energy Minimization Roman Voronov, Anton Shabaev, and Fedor Vasiliev

Abstract Systems that are of interest in thermodynamic modeling for industrial applications contain solution phases, which are not ideal mixtures. To account for deviations from ideal behavior in a mixture of chemical substances, activity coefficients are used. They can be determined experimentally by making measurements of non-ideal mixtures: theoretical laws provide a value for an ideal mixture, against which the experimental value is compared to obtain the activity coefficient. The research novelty is the computational procedure for estimating missing activity coefficients assuming that the system is in equilibrium state, and some activity coefficients are known, which in a sense is the inverse problem to Gibbs energy minimization. The proposed computational procedure allows the mathematical evaluation of some missing activity coefficients instead of running an experiment, which constitutes its main practical contribution.

80.1 Introduction Computational thermodynamics plays a major role in process simulation with its focus on applications in which complex multi-phase equilibria are involved. Among other things, it provides a means to calculate global chemical equilibria. In quite a

R. Voronov (B) · A. Shabaev Petrozavodsk State University, Lenin Str., 33, 185910, Petrozavodsk, Russia e-mail: [email protected] A. Shabaev e-mail: [email protected] F. Vasiliev Metso Outotec Finland Oy, Kuparitie 10, FI-28101, Pori, Finland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_80

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few industrial processes the thermodynamic equilibrium acts as the driving force and determines operating conditions. Computational methods in chemical equilibrium thermodynamics have found numerous application areas in diverse fields such as metallurgy, petrochemistry, the pulp and paper industry, the study of advanced inorganic materials, environmental science, and biochemistry [11]. Computational thermodynamics has been used to model the decarburization of steel in a basic oxygen furnace [9], the char and biomass conversions during gasification, pyrolysis, and torrefaction [7], and pulp suspension in papermaking [6] to mention just a few. Typically, systems that are of interest in thermodynamic modeling for industrial applications contain solution phases, which are not ideal mixtures. For this reason, the proper selection of an activity model to account for non-ideality is often the central problem in thermodynamic modeling. An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture of chemical substances. Modifying mole fractions or concentrations using activity coefficients gives the effective activities of the components, and hence allows expressions such as Raoult’s law and equilibrium constants to be applied to both ideal and non-ideal mixtures. Activity coefficients can be determined experimentally by making measurements of non-ideal mixtures. Raoult’s law or Henry’s law provides a value for an ideal mixture, against which the experimental value is compared to obtain the activity coefficient. Other colligative properties, such as osmotic pressure in aqueous solutions, may also be used. Activity coefficient models for concentrated aqueous solutions were developed based on the Debye–Hückel theory of electrolytes of which the Pitzer model is the most advanced [2, 13]. Several general activity coefficient model formulations, namely, Wilson, NRTL, UNIQUAC, and Quasi-chemical were developed and tested in applications for liquid and solid solutions [1, 12]. The models mentioned above involve complex equations derived from the principles of thermodynamics with certain assumptions and limitations. They feature a number of parameters for each component in a system, which are regressed from data collected in equilibrium experiments. In this paper, we present a computational procedure for estimating activity coefficients. In a sense, the approach constitutes the solution of an inverse problem, where activity coefficients are calculated from a known equilibrium state using Gibbs energy minimization. The rest of the paper is structured as follows. In Sect. 80.2 we provide basics of Gibbs free energy minimization and introduce the required notation. In Sect. 80.3 we provide the computational procedure for the calculation of activity coefficients. Section 80.4 illustrates the procedure with a practical example of the dissolution in water of selected mineral salts and acids. Section 80.5 concludes the paper.

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80.2 Gibbs Free Energy Minimization Models predicting the thermodynamic equilibrium state are often based on minimizing the Gibbs free energy of the system, following the thermodynamic principle that a closed system with a constant temperature and pressure is at an equilibrium state when its Gibbs free energy has a minimum value. The accuracy of such models, thanks to their ability to reproduce rich multi-phase chemistry and thermodynamic state properties, is often very good, even when applied to complex industrial processes [8]. An in-depth analysis of the topic and overview of numerical methods can be found in [3, 4, 16]. Denote N – set ofspecies, P – set of phases, N k – subset of species in the k-th N k , M – set of elements. Let k(i) = k for k ∈ P at which phase, k ∈ P, N = k∈P

i ∈ Nk. A typical Gibbs free energy minimizer finds the minimum of function G: G(n) =



μi n i −→ min,

(80.1)

k∈P i∈N k

An = b,

n ≥ 0,

(80.2)

where G is the total Gibbs free energy of a multicomponent multi-phase system, entry ai j of matrix A is the stoichiometric coefficient between element j and species i, n i —the molar amount of species i, component b j of vector b—the molar amount of element j, μi —the dimensionless chemical potentials of species i in the phase k(i) (i ∈ N , j ∈ M). We note that (80.3) μi = gi0 + ln ai = gi0 + ln(γi xi ), xi =

ni , n k(i)

nk =



ni ,

i ∈ N,

k ∈ P,

(80.4)

i∈N k

where gi0 is the dimensionless chemical potential of a component in a standard state, ai are effective activities of components accounting for effective concentrations of components in non-ideal solutions. The second term on the right-hand side of formula (80.3) is the dimensionless entropy of mixing, where γi is the activity coefficient that is introduced to account for the non-ideal behavior of a component in the system. We put forward two assumptions: n i = 0 if and only if n k(i) = 0; if n i = 0 then xi = 1. In general, the parameters in formula (80.3) are functions of temperature and pressure, but we will consider them at constant values of temperature and pressure. Function (80.1), with Conditions (80.3) satisfied, is convex and its local minimum is also the global minimum [17]. Solving methods of Problem (80.1)–(80.4) are described in [10, 15, 17, 18].

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In ideal solutions the activity coefficients γi are assigned to be unity. For example, the precipitates (condensed phases), dilute aqueous solutions, and gas phases can often be regarded as ideal. However, many solution phases in industrial applications are not ideal. The value of activity coefficients is always positive, as seen from equation (80.3), and may vary within a wide range. The challenge of estimating the values of activity coefficients γi in real operating conditions in order to solve Problem (80.1)–(80.4) is approached by a combination of theoretical methods, which provide the general dependence function, with multiple regression to define specific values of parameters [14]. However, as we will show in the next section, assuming that the system is in equilibrium state, and some activity coefficients are known (e.g. from reference tables, or from theory), the missing activity coefficients may be estimated using a computational procedure, which in a sense solves the inverse problem to Gibbs energy minimization.

80.3 Calculation of Activity Coefficients Our main problem is formulated as follows: for a given n∗ find the values of γ , for which n∗ is the local minimum of Problem (80.1)–(80.4): min G(n) = G(n∗ ). n

We denote n, x, y, z, u, γ , β, λ – vectors consisting, respectively, of elements n i , xi , yi , z i , u i , γi , βi , λi . Corresponding to the Problem (80.1)–(80.4) the Lagrangian function can be formed:    n i gi0 + ln(γi xi ) + L(n, y, z, x, u) = − k∈P i∈N k

+

   j∈M

 ai j n i − b j

yj +

i∈N



ni zi +

i∈N



 n i − xi

k∈P i∈N k



 nt ui .

t∈N k

The Karush–Kuhn–Tucker conditions are L ni = −gi0 − ln(γi xi ) +



 ai j y j + z i + 1 −

j∈M



 xt u i = 0,

i ∈ N,

t∈N k(i)

(80.5) L xi = −

 ni − ui n t = 0, xi k t∈N

i ∈ N,

(80.6)

80 A Computational Approach to Estimating Activity Coefficients …



ai j n i = b j ,

707

j ∈ M,

(80.7)

i∈N

n i z i = 0,

z i ≥ 0,

n i ≥ 0,

i ∈ N.

(80.8)

Considering (80.4) and (80.6) we obtain u i = −1 for all i ∈ N . Note that n∗ satisfies the constraints (80.7)–(80.8). Denote di = gi0 + ln xi , βi = ln γi and rewrite the condition (80.5) as 

ai j y j + z i = βi + di + 1 −

j∈M



xt ,

i ∈ N.

t∈N k(i)

Let us assume that the values of some γi (and βi ) are known. Denote N1 ⊂ N containing indices i with known γi , and N2 = N \ N1 . Let di = βi + di for i ∈ N1 . Further, let us consider two cases separately: a special case n∗ > 0 and a general case n∗ ≥ 0.  If n∗ > 0, then z i = 0 and t∈N k(i) xt = 1, which results in a system of linear equations with unknowns y and β: 

ai j y j = di ,

i ∈ N1 ,

(80.9)

j∈M



ai j y j = βi + di ,

i ∈ N2 .

(80.10)

j∈M

If |M| = |N1 | and the matrix of system (80.9) is invertible, then system (80.9)– (80.10) has a unique solution. Otherwise, we find values of βi closest to zero using the method of least squares: 

βi2 =

i∈N2

 i∈N2



⎛ ⎝



⎞2 ai j y j − di ⎠ −→ min,

j∈M

ai j y j = di ,

i ∈ N1 .

j∈M

Applying the method of Lagrange multipliers with the Lagrange function

L(n, λ) =

 i∈N2

⎛ ⎝

 j∈M

⎞2 ai j y j − di ⎠ +

 i∈N1

⎛ λi ⎝

 j∈M

we obtain a linear equation system with unknowns: y, λ:

⎞ ai j y j − di ⎠ ,

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2

   q∈M

 ai j aiq

yq +

i∈N2



ai j λi = 2

i∈N1





di ai j ,

j ∈ M,

(80.11)

i∈N2

ai j y j = di ,

i ∈ N1 .

(80.12)

j∈M

Next, we find ⎛ γi = eβi = exp ⎝



⎞ ai j y j − di ⎠ ,

i ∈ N2 .

j∈M

Now we consider the general case with n∗ ≥ 0. Let N11 = {i | i ∈ N1 , n i > 0},

N12 = {i | i ∈ N1 , n i = 0},

N21 = {i | i ∈ N2 , n i > 0},

N22 = {i | i ∈ N2 , n i = 0}.

Note that xi = 1 for i ∈ N12 ∪ N22 . We obtain a system of linear equations with unknowns y and β:  ai j y j = di , i ∈ N11 , (80.13) j∈M



ai j y j ≤ di + 1 − |N k(i) |,

i ∈ N12 ,

(80.14)

j∈M



ai j y j = βi + di ,

i ∈ N21 ,

(80.15)

j∈M



ai j y j ≤ βi + di + 1 − |N k(i) |,

i ∈ N22 .

(80.16)

j∈M

We suggest calculating the minimum: t = max |βi | −→ min .

(80.17)

i∈N2

From (80.15) it follows that





|βi | = ai j y j − di

≤ t,

j∈M

i ∈ N21 .

Obviously, there is an optimal solution in which βi ≥ 0 for all i ∈ N22 . Then

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t ≥ |βi | = βi ≥



ai j − di − 1 + |N k(i) |,

709

i ∈ N22 .

j∈M

Thus, we obtain the following problem equivalent to problem (80.13)–(80.17) with unknowns y and t: t −→ min, 

ai j y j = di ,

(80.18) i ∈ N11 ,

(80.19)

j∈M



ai j y j ≤ di + 1 − |N k(i) |,

i ∈ N12 ,

(80.20)

j∈M

−t ≤



ai j y j − di ≤ t,

i ∈ N21 ,

(80.21)

j∈M



ai j y j − t ≤ di + 1 − |N k(i) |,

i ∈ N22 .

(80.22)

j∈M

Problem (80.18)–(80.22) can be solved using a standard algorithm, e.g. the simplex method. Next, we find ⎛ ⎞  γi = eβi = exp ⎝ ai j y j − di ⎠ , i ∈ N21 , j∈M

⎛ γi = e

βi+

= exp ⎝



⎞+ ai j y j − di − |N k(i) | + 1⎠ ,

i ∈ N22 ,

j∈M



where βi+

=

βi , if βi ≥ 0, 0, if βi < 0.

Thus the algorithm for solving the inverse problem is as follows. If |N12 | = 0 and |N22 | = 0, then solve Problem (80.11)–(80.12). Otherwise solve Problem (80.18)– (80.22).

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Table 80.1 Dissolution of selected mineral salts and acids in water at 70◦ C and 1 bar. The estimated activity coefficients marked in bold were fixed prior to the estimation of the other activity coefficients Initial Equilibrium Activity Activity Relative Species composition, composition, coefficients coefficients error, mol mol calculated estimated % H2 O H(+a) OH(–a)

55.51 0.0 0.0

Cu(+2a) SO4 (–2a) Na(+a) NaSO4 (–a) HSO4 (–a) HMoO4 (–a) MoO2 (+2a)

0.0 0.0 0.0 0.0 0.0 0.0 0.0

MoO4 (–2a)

0.0

CuSO4 Na2 SO4 H2 SO4 H2 MoO4

0.1 1.0 0.001 0.01

Phase 1: Aqueous solution 55.51 1.013 1.000 0.001002667 0.291 0.289 1.01201 · 0.567 0.563 10−9 0.1 0.039 0.040 0.593555483 0.045 0.045 1.497492615 0.526 0.530 0.502507385 0.370 0.376 0.004937131 0.550 0.551 0.003318444 0.370 0.372 2.53881 · 0.016 0.016 10−10 0.000310677 0.016 0.016 Phase 2: Precipitates 0.0 1.000 1.000 0.0 1.000 1.000 0.0 1.000 1.000 0.006370879 1.000 1.000

1.314 0.659 0.659 −2.708 −0.878 −0.791 −1.676 −0.214 −0.663 −1.331 −1.331 0.000 0.000 0.000 0.000

80.4 Numerical Example A practical application of the computational approach presented in this paper is demonstrated using an example that simulates the dissolution of selected mineral salts and acids in water at 70◦ C and 1 bar (Table 80.1). Many mineral salts and acids dissolve in water completely or partially depending on their composition, total concentration, temperature, and pressure. As a result of the dissolution, the salts and acids dissociate, and the aqueous solution contains ions. In this system, precipitates are regarded as the pure phase. Thus, the activity coefficients of the species in the precipitate phase are set to unity. The species in phase 1 are regarded as being in an aqueous solution and so the values of their activity coefficients are positive numbers that may vary within a wide range. The example consists of two parts: First, the system equilibrium composition is calculated using the Gibbs free energy minimization approach where the initial composition, temperature, and pressure are given. As a step in the Gibbs free energy minimization calculations, activity coefficients are calculated using the Pitzer activity model. Then, for the species in the aqueous solution, activity coefficients are

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estimated using the approach proposed in this paper, i.e. with a given equilibrium composition, temperature, pressure, and with some activity coefficients fixed at a cerR tain value. The Gem module and thermodynamic database of HSC Chemistry 10 R  were used in these calculations [5]. HSC Chemistry is a software suite with rich databases which allow thermodynamic and mineral processing calculations to be made. Its modeling and simulation platform is a valuable tool for research and development in chemical engineering. One limitation of the proposed approach is the possibility of multiple sets of activity coefficients. To decrease the uncertainty, some coefficients have to be fixed at user defined values, which in practice means that some other method has to be used to provide a rough estimation of their values. For example, in dilute aqueous solutions, the activity coefficient of water is usually around unity, whereas for simple ions like Cu(+2a), SO4 (–2a) and Na(+a) the activity coefficients can be estimated using the Debye–Hückel equation. In general, the number of activity coefficients fixed at a certain value equals the number of elements in the system. As can be seen, the largest error is introduced by the error in the fixed activity coefficients; the estimated activity coefficients are accurate to within 3%.

80.5 Conclusion The computational procedure that is proposed in the paper constitutes an original approach for estimating missing activity coefficients. According to the literature review, so far no unified, comprehensive theory has been developed that allows the unambiguous calculation of the activity coefficients of hydrated or solvated ions and neutral molecules (nonelectrolytes). Typically, a number of adjustment parameters are used to accommodate for changes in the concentrations of the same solute and the nature of the solvent. This approach has a practical significance, since it enables the rough estimation of activity coefficients for complex systems, which often suffices for simulating certain complex industrial processes in cases where developing a rigorous thermodynamically sound activity model is not possible. However, the proposed approach has certain limitations, for example, in certain conditions it may result in multiple sets of solutions. Unambiguous determination of activity coefficients for species present at equilibrium necessitates that the number of known activity coefficients is no less than the number of elements. The authors expect to research this issue further. Another direction for further research is a wider comparison of the computed values of activity coefficients with values obtained from reference tables and by using other methods for solving the problem.

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References 1. Ansara, I.: Thermodynamic modelling of solution phases and phase diagram calculations. Pure Appl. Chem. 62(1), 71–78 (1990) 2. Casas, J., Papangelakis, V., Liu, H.: Performance of Three Chemical Models on the HighTemperature Aqueous Al2(SO4)3-MgSO4-H2SO4-H2O System. Ind. Eng. Chem. Res. 44(9), 2931–2941 (2005) 3. Davies, R.H., et al.: MTDATA-thermodynamic and phase equilibrium software from the national physical laboratory. Calphad 26(2), 229–271 (2002) 4. Gorban, A.N., et al.: Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibrium. Springer Science & Business Media, Berlin (2006) 5. HSC Chemistr® : Outotec, Pori 2020. http://www.outotec.com/HSC (2020) 6. Kalliola, A., Pajarre, R., Koukkari, P., Hakala, J., Kukkamaki, E.: Multi-phase thermodynamic modelling of pulp suspensions: Application to a papermaking process. Nord. Pulp Pap. Res. J. 27(3), 613–620 (2012) 7. Kangas, P., Koukkari, P., Hupa, M.: Modeling biomass conversion during char gasification, pyrolysis, and torrefaction by applying constrained local thermodynamic equilibrium. Energy Fuels 28(10), 6361–6370 (2014) 8. Koukkari, P.: Introduction to Constrained Gibbs Energy Methods in Process and Materials Research. VTT, Espoo (2014) 9. Kruskopf, A., Visuri, V.: A Gibbs energy minimization approach for modeling of chemical reactions in a basic oxygen furnace. Metall. Mater. Trans. B 48(6), 3281–3300 (2017) 10. Krylatov, AYu.: Reduction of a minimization problem for a convex separable function with linear constraints to a fixed point problem. J. Appl. Ind. Math. 12(1), 98–111 (2018) 11. Pajarre, R., Koukkari, P., Kangas, P.: Constrained and extended free energy minimisation for modelling of processes and materials. Chem. Eng. Sci. 146, 244–258 (2016) 12. Pelton, A.D.: Phase Diagrams and Thermodynamic Modeling of Solutions. Elsevier, Amsterdam (2018). https://doi.org/10.1016/C2013-0-19504-9 13. Pitzer, K.: Activity Coefficients in Electrolyte Solutions. CRC Press, Boca Raton, FL (1991) 14. Tanganov, B.B.: Vzaimodejstviya v rastvorah elektrolitov: modelirovanie sol’vatacionnyh processov, ravnovesij v rastvorah polielektrolitov i matematicheskoe prognozirovanie svojstv himicheskih sistem: monografiya. Akad. Estestvoznaniya, Moskva (2009) 15. Voronin, A.V., Kuznetsov, V.A., Shabaev, A.I., Spirichev, M.V., Vilaev, D.G.: Calculation algorithm for chemical equilibrium based on Gibbs energy minimization. Scientific Journal Proceedings of Petrozavodsk State University 6(127), 106–109 (2012) 16. Weber, C.F.: Convergence of the equilibrium code SOLGASMIX. J. Comput. Phys. 145(2), 655–670 (1998) 17. White, W.B., Johnson, S.M., Dantzig, G.B.: Chemical equilibrium in complex mixtures. J. Chem. Phys. 28(5), 751–755 (1958) 18. Zaika, Yu.V.: Algorithm of Gibbs energy minimization: computation of chemical equilibrium. Computational Technologies 16(2), 45–54 (2011)

Part XI

Mathematical Modelling and Image Processing Methods

Chapter 81

Computation and Analysis of Two-Phase Filtration Using Averaged Models in Oil Formations with Both Vertically Stratified Heterogeneity and Horizontal Zonal Heterogeneity Dilbar N. Bikmukhametova, Svetlana R. Enikeeva, Kuan M. Tho, and Sergey P. Plokhotnikov

Abstract The paper considers averaged models of two-phase filtration in oil formations with both vertically stratified heterogeneity and horizontal zonal heterogeneity. Comparative analysis of numerical solutions of averaged two-dimensional models, with modified relative phase permeability and laboratory phase permeability, and solutions of a three-dimensional model was carried out. As a numerical solution, the following task was performed. A ten-layered strip-like oil formation is developed in three oil wells in a waterflood zone. The central well pumps water into the reservoir, while the other two, being symmetrically to the right and to the left of it, produce oil under preset differential pressure. Oil reservoir boundaries are non-permeable. This results in oil displacement, completely symmetrical about the center, with two-phase filtration. The calculations used mathematical models of two-phase filtration within the framework of the Buckley–Leverett model. Particular computation was carried out for uniform probability distribution using a known numerical algorithm and a certified hydrosimulator Tempest. Verification of each of these two-dimensional models relative to the initial three-dimensional numerical solution of the problem was carried out after conducting the computational experiment. The positive result confirmed the possibility of applying averaged models to solve tasks in heterogeneous oil reservoirs.

D. N. Bikmukhametova · S. R. Enikeeva (B) · K. M. Tho · S. P. Plokhotnikov Kazan National Research Technological University, Kazan, Russia e-mail: [email protected] D. N. Bikmukhametova e-mail: [email protected] S. P. Plokhotnikov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_81

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81.1 Introduction Mathematical problem statements in aeromechanics, hydromechanics, filter theory, and others are based on general theorems of continuum dynamics [1]. They are determined by sets of equations of the second order in partial derivatives of elliptic and parabolic types with boundary conditions of the first and the second kind. In some cases, hyperbolic equations are used. In numerical implementation of practical problems in oil and gas field development, one often deals with averaging physical parameters of equations of a three-dimensional system and with transition to averaged mathematical models of lower dimension. Lower dimensional models are also applied for solving the problems of wave propagation through random media, particularly in atmosphere and ocean. Medium heterogeneity can be taken into consideration using methods of averaging. In the process where average medium characteristics depend only on vertical coordinate, the task is significantly simplified.

81.2 Purpose of Research The paper verifies two proposed averaged models in comparison with the threedimensional initial solution for non-isothermal filtration in reservoirs with complex absolute permeability.

81.3 Method of Research For oil and gas fields with layered heterogeneity by absolute rock permeability, this computation is of practical importance. It is especially notable when heterogeneity is set both for thickness and strike, which is determined by physical structure of some multilayer formations [2–5]. Absolute permeability function (APF) of porous medium, in general terms for such formations, can be represented in a multiplicative way by the formula K (x, z) = a(z) · k(x). (81.1) Absolute permeability of each vertical layer in such oil formations is set by the following formula: K i (x, z) = a i · k(x) (i = 1, n), (81.2) where k(x) is an average value of oil formation thickness of a zonal heterogeneity function of absolute permeability, which is presented in analytical and sectionally continuous function. This function can also be defined from the two arguments— x, y. Dimensionless parameter a(z), taking account of heterogeneity in layer thickness (number of layers equals n), is represented by particular probability series in the table of values a i , Pi , (i = 1, n). This series can be subject to general probability

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distribution. Then, all computation formulas for modified relative permeabilities of averaged models C, B, suggested in the works [3–6], are mathematically valid, and for this case, absolute permeability function is set as in (81.1). Particular numerical computation was conducted for uniform probability distribution law for the parameter a(z) in a ten-layered oil formation, where n = 10. Zonal heterogeneity by absolute permeability (an average value of formation thickness) is determined according to the formulas k(x) = (1 + 9x)/5.5,

or

k(x) = (10 − 9x)/5.5.

(81.3)

For each of them zonal heterogeneity k(x) changes through the length of oil formation by a factor of 10. Absolute permeability of each of the ten layers along the upright section is defined by the formula (81.2). In the calculations, three mathematical models of two-phase filtration within the framework of the Buckley–Leverett model were used. The simplest averaged model was model C (in the center), it was used for initial laboratory K B (s), K H (S), and average absolute permeability by thickness k(x) in each upright section of a multilayer oil field. Together with the model, we used the model B based on correction factors and jet stream displacement scheme. This model worked with modified permeabilities K BM (s), K HM (S), obtained on the basis of correction of laboratory permeabilities by particular correction factors. Their formulas were unified for general nonlinear case for laboratory relative permeabilities in the works by Plokhotnikov S. P., Bogomolov V. A. [3–6]. We solve the problem of two-phase filtration of a ten-layered strip-like oil reservoir developed in three oil wells in a waterflood zone. The central oil well pumps water into the reservoir, the other two, situated symmetrically to the right and to the left of it, produce oil under preset differential pressure. As a result, we get oil displacement, completely symmetrical about the center, with two-phase filtration. A case of uniform law of setting the parameter a(z) for taking account of stratified vertical heterogeneity of reservoir was considered, (for variation permeability coefficient V = 0.55). In calculations, we applied standard functions of laboratory relative permeabilities of water and oil [8] of this type K B (S) = K B0 · [S (s)]α , K H (S) = K H0 · [1 − S (s)]β , S − S∗ , α = β = 1, 2, 3. where S (s) = ∗ S − S∗ As for modified permeabilities for uniform law, they can be expressed in terms of the following formulas [3–6]: √ K BM (S) = K B (S) · [1 + V 3 · (1 − S (S))] K HM (S) = K H (S) · [1 − V ·

√ 3 · S (S)].

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Fig. 81.1 Graphs are given for uniform law at quadratic and cubic (on the right) of laboratory and their modified relative permeabilities

Fig. 81.2 Graphs of oil produced, uniform law, square relative permeability functions, k(x) = (1 + 9x)/5.5

Their graphs are presented in Fig. 81.1. Mathematical setting of three-dimensional (x, y, z)—problem of two-phase filtration with known boundary conditions is presented in the works [4–8]. The cited works also contain all physical parameters of porous medium and liquids that are used in calculations for this paper. Numerical computation was carried out with hydrodynamic simulator Tempest by Roxar [7], at Tatar Oil Research and Design Institute (TatNIPIneft), Bugulma (Fig. 81.2). Calculations were made for the case of setting absolute permeability values for calculated blocks of the finite-difference mesh with dimension 34 × 5 × 10 (X × Y × Z), where axis OX is throughout overall height of the table (34 points), and axis OZ is along the length of each table (10 points). The data were taken for models C, B, A3 . Table 81.1 shows values of absolute permeability function for 10 layers using different models (mdarcy) for even points along the axis OX. The other seven three-dimensional Ai solutions use the values of the same tables, but at different vertical positions of layers as it was suggested in the works [3–6]. K is the average

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Table 81.1 Values of absolute permeability function for 10 layers using different models (mdarcy) Uniform law. Distribution in the order of decreasing (from up to down) k(x) = (((10 − 9 ∗ X/33)5.5) ∗ 0.5)) Models A3 Models K = K = K = K = K = K = K = K = K = K = C, B 50 150 250 350 450 550 650 750 850 950 k0 k2 k4 k6 k8 k10 k12 k14 k16 k18 k20 k22 k24 k26 k28 k30 k32 k33

91 86 81 76 71 66 61 56 51 46 41 36 31 26 21 17 12 9

273 258 243 228 213 198 183 169 154 139 124 109 94 79 64 50 35 27

455 430 405 380 355 331 306 281 256 231 207 182 157 132 107 83 58 45

636 602 567 532 498 463 428 393 359 324 289 255 220 185 150 116 81 64

818 774 729 684 640 595 550 506 461 417 372 327 283 238 193 149 104 82

1000 945 891 836 782 727 673 618 564 509 455 400 345 291 236 182 127 100

1182 1117 1053 988 924 860 795 731 666 602 537 473 408 344 279 215 150 118

1364 1289 1215 1140 1066 992 917 843 769 694 620 545 471 397 322 248 174 136

1545 1461 1377 1293 1208 1124 1040 955 871 787 702 618 534 450 365 281 197 155

1727 1633 1539 1445 1350 1256 1162 1068 974 879 785 691 597 502 408 314 220 173

909 860 810 760 711 661 612 562 512 463 413 364 314 264 215 165 116 91

Fig. 81.3 Graphs of oil produced, uniform law, square relative permeability functions, k(x) = (10 − 9x)/5.5

OX value of the function (81.1) for each layer. In this case, the average of absolute permeability function in the oil formation is 0.5 darsi (Fig. 81.3). The graphs, depicting the produced oil, reflect the following regularity: averaged solutions of B and C represent lower and upper bounds for multitude of model solutions A8 − A7 . Calculations were also carried out for cubic functions of relative

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permeability for this law. The results obtained were similar to those shown in the figures with quadratic functions.

81.4 Conclusions As a result of the work, we came to the following conclusions: 1. The character of carrying out two-phase isothermal filtration and indicators of oil field performance in a multilayer formation depend on interlayer spacing by thickness, presence of hydrodynamic association of layers, and analytical type of laboratory functions K B (S), K H (S). 2. Numerical calculation graphs for the amount of oil produced, oil recovery efficiency, development time, and other indicators of oil field performance for the simplest averaged model C show overstated results compared with range of dispersion of graphs of three-dimensional solutions of the problem—model solutions. 3. Calculation graphs of the averaged model B at linear and nonlinear functions K B (S), K H (S) show understated results compared with range of dispersion of model solution graphs. 4. Model solutions form a set of curves in relation to interlayer spacing in a multilayer oil formation, the range of dispersion of model solutions is in the range of two approximate values of models B and C; the averaged models in total can be recommended for each of the stated oil field performance indicators in approximate hydrodynamic computation. 5. The formulas suggested for absolute permeability function (APF) (81.1), (81.2) and the conducted computational experiment lead to verification of the two averaged models B and C that can be applied for two-phase filtration calculations in multilayer oil formations with both vertically stratified heterogeneity and horizontal zonal heterogeneity by absolute rock permeability. 6. The results of the work significantly expanded the possible application of the method of modified permeabilities using correction factors, on the assumption of jet stream in multilayer oil formations, as well as in composite multilayer oil formations, in order to solve the problems of mathematical simulation of oil recovery in subsurface hydromechanics and oil field development theory. In conclusion, it should be noted that computational experiment for setting absolute permeability function (APF) by Maxwell’s law and the exponential law was conducted. The results obtained were favorable as well.

References 1. Lagrange, J.L.: Analytical Mechanics. Gostechizdat, Moscow (1950) (In Russian)

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2. Ahmad, R.K., Plokhotnikov, S.P., Nikiforova, S.V.: Al Jabry Adel Yahya Aly: the solution of the problem of averaging coefficients of elliptic and parabolic equations set. Vestnik KGTU im. A.N. Tupolev. 1, 30–36 (2019) 3. Ahmad, R.K., Plokhotnikov, S.P., Galimyanov, Al Jabri, A.Y., Aly, F.A.: Verification of averaged models of two-phase filtration. Vestnik KGTU im. A.N. Tupolev. 1, 97–99 (2019) 4. Instruction notes on making up permanent geological and technological models of oil and gas fields (Part 2. Filtration models). M.: VNIIOENG, 2003, p. 228 5. Plokhotnikov, S.P., Bogomolov, V.A., Nizaev, RKh., Bogomolova, O.I., Malov, P.V.: Mathematical averaging of coefficients of system of elliptic and parabolic equations in continuum mechanics. Lobachevskii J. Math. 5, 553–561 (2019) 6. Eliseenkov, V.V., Plokhotnikov, S.P.: Hydrodynamic calculations of layered seams on the basis of modified relative permeabilities. J. Appl. Mech. Tech. Phys. 5, 833–838 (2001) 7. Plokhotnikov, S.P., Bogomolov, V.A., Salimyanov, I.T., Al Jabri, A.Y., Kamal, A.R.: Verification of two averaged models of three-phase filtration in inhomogeneous layers of oil reservoirs, obeying uniform distribution. J. Phys.: Conf. Ser. 1328 (2019). https://doi.org/10.1088/17426596/1328/1/012062 8. Tempest-MORE. User’s Guide, version 6., Roxar, p. 373 (2006)

Chapter 82

Optimization Method of the Velocity Field Determination for Tomographic Images Elena Kotina, Pavel Bazhanov, and Dmitri Ovsyannikov

Abstract The digital image processing problem basing on velocity field determination is considered. An optimization method for constructing the velocity field basing on the study of the integral functional on the ensemble of trajectories is developed. The analytical form of functional variation allows using directed optimization methods to find the desired parameters. This method is considered for 3D tomographic image processing, in particular, for PET images for the movement correction in dynamic studies.

82.1 Introduction Currently, processing of images is becoming one of the most important areas of fundamental and applied scientific research. This is due to the fact that the methods of analysis and digital image processing are used in a wide variety of information and technical systems. These methods are also widely used in modern medical diagnostics, which uses a variety of digital imaging methods: radiography, computed tomography (CT), magnetic resonance imaging (MRI), single-photon emission computed tomography (SPECT), and positron emission tomography (PET). A large number of works are devoted to various methods and algorithms for image processing, such as methods for pattern recognition, image analysis [18], reconstruction methods [5, 7], classification methods, contour detection, image segmentation methods [16], etc. All these processing techniques are important for medical image processing. There is a trend towards complete automation of processing and interpretation of medical studies. It should be noted that the automated processing of tomographic images is a particularly challenging problem. E. Kotina (B) · P. Bazhanov · D. Ovsyannikov Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] D. Ovsyannikov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_82

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Essential is the task of sequential image processing to obtain information about changes that occur in sequence from one image to another. In this case, the problem of identifying the direction and velocity of movement of the object in the images is most often considered. With this information for the sequence of images, it is possible to automatically correct the position of the regions of interest in the images. Detection and correction of the patient motion are one of the most important steps of image processing. Even small displacement of the patient or the target organ during the data acquisition may affect the accuracy of diagnostic results [12, 15, 17]. For example, while dynamic studies in PET are carried out, information is recorded and displayed on a whole series of 3D frames. Using the resulting sequence of threedimensional arrays, the functional evaluation of the internal organs of the patient is performed. For this purpose, in particular, various mathematical methods are widely used, the accuracy of which can be affected by various involuntary displacements of the patient or his internal organs. Image processing based on the velocity field determining was considered in many papers [4, 8–11, 14]. The most famous problem statement using the notion of optical flow assumes the constancy of the distribution density function (brightness) along the trajectories of the system under consideration [6]. In this statement, functionals of quality are constructed, which also include additional requirements of spatial smoothness for the required velocity field [3, 6, 14]. Minimization of the constructed functionals is reduced to solving the Euler– Lagrange equations by numerical methods. These equations can be reduced to sparse linear systems, which are solved by block iterative methods. This work develops the optimization approach proposed by the authors in the articles [1, 2]. This approach is based on the variation of the integral functional in the problems of trajectory ensemble control. This paper discusses the optimization method for constructing the velocity field for 3D PET images, which can be used for automated movement correction in dynamic PET studies, for example, in brain studies.

82.2 Mathematical Model Let us consider a system of differential equations x˙ = f (t, x, u),

(82.1)

where t—time, x ∈ R 3 —spatial coordinate vector, u—parameter vector, u ∈ R r , and f —sufficiently smooth vector function. We assume that ρ = ρ(t, x) is a distribution density function that is a quantitative characteristic of the image brightness, depending on spatial coordinates and time, or the intensity of the distribution of radiopharmaceutical for dynamic PET studies.

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The equation that for a given vector function f (t, x, u) determines the change in the function ρ = ρ(t, x) in space over time has the following form [1]: ∂ρ(t, x) ∂ρ(t, x) + f (t, x, u) = 0, ∂t ∂x

(82.2)

under the assumption that divx f (t, x, u) =

3  ∂ f i (t, x, u) i=1

∂ xi

= 0.

(82.3)

Initial condition for equation (82.2) is ρ(0, x) = ρ0 (x),

(82.4)

where ρ0 (x)—given function. It follows from equation (82.2) that the brightness (density) along the trajectories of the system (82.1) remains constant, i.e. we have the case of the so-called optical flow [6]. dρ |(82.1) = 0. (82.5) dt In this paper, we will consider this case. Let ρ0 (x)—a known density (brightness) at the moment t = 0, which defines a certain image. In our case, this is a three-dimensional array of data obtained by the PET method, we will call it a tomographic or three-dimensional image. Let us further assume that we know the density (brightness) ρ(x), ˆ characterizing changed image in time t, here t is small enough. Let us fix time T = t. The problem is to find the vector of parameters u that defines the function f (t, x, u) so that at the time T the density distribution calculated by the equation (82.2) with condition (82.4) would coincide with the density distribution ρ(x): ˆ ρ(T, x) = ρ(x). ˆ

(82.6)

By defining f (t, x, u) we determine the velocity field given by formula (82.1).

82.3 Optimization Method Let M0 ∈ R 3 —the set of initial values for the system (82.1). We assume that the set M0 is a closed set and has a nonzero Lebesgue measure. We will use the following notation: x(t) = x(t, x0 , u), x0 ∈ M0 —the solutions of the system (82.1). The set of these solutions is called trajectory beam (or ensemble of trajectories), coming from the set M0 for a given parameter vector u. We denote by Mt,u = {x(t) = x(t, x0 , u), x0 ∈ M0 } cross section of the trajectory beam at time t for the fixed vector u.

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Let us consider the problem of minimization on the solutions of the system (82.1)– (82.2) of the following functional:  g(x, ρ(T, x))d x,

J (u) =

(82.7)

MT,u

where u ∈ U ⊂ R r , U a compact convex set, MT,u —a cross section of trajectory beam for a moment t = T , g(x, ρ)—nonnegative continuously differentiable on x and ρ function. As a function g(x, ρ(T, x)) we can take the function 2  ˆ , g(x, ρ) = q(x) ρ(T, x(T, x0 , u)) − ρ(x)

(82.8)

where ρ(x)—given ˆ density in R 3 and q(x)—a weight function that allows us to select significant areas of the image. As an example of such a weight function, we ¯ 2 that allows us to select the center of the can consider the function q(x) = e−(x−x) image, here x—coordinates ¯ of the center, or any smooth function with compact support. Let us note that the moment T is fixed here, but it also could be varied. By minimizing the functional (82.7), we determine the parameters of the vector u, thereby solving the problem of restoring the function f (t, x, u), thus we define the velocity field set by the formula (82.1). Taking into account that divx f = 0, we can write a variation of the functional (82.7) as [2]   T

δJ = − 0

ψ ∗ (t, xt )u f (t, xt , u)d xt dt.

(82.9)

Mt,u

Here the superscript ∗ denotes the transposition of the vector, u f (t, xt , u) = f (t, xt , u + u) − f (t, xt , u),

(82.10)

and ψ(t, x)—an auxiliary function that satisfies along the trajectories of the system (82.1) the following equation:   ∂ f (t, x(t), u) ∗ dψ =− ψ, dt ∂x

(82.11)

with final condition ψ ∗ (T, x(T )) = −

∂g(x(T ), ρ(T, x(T ))) . ∂x

(82.12)

The output of variation (82.9) is based on the use of the transformation of trajectory beam on cross sections using the variational equations for (82.1), (82.2) [13]. Let the function f be differentiable with respect to u. Then, taking into account convexity of set U and using (82.9), we can get an expression for the gradient of the

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functional (82.7)

∂J =− ∂u



T 0



ψ∗ Mt,u

∂f d xt dt. ∂u

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(82.13)

Based on this functional gradient expression, various methods of directed search for the parameter vector u can be constructed. In image processing problems, the form of the function f (t, x, u) is unknown. Therefore, the function can be considered as a function represented by a segment of some series, such as the Taylor series. In particular, we can consider the function as a linear vector function, i.e. x˙ = Ax + C. (82.14) We investigate the case of three-dimensional (tomographic) images, so we will 3 . We previassume that A—a square matrix: A = {ai j }i,3 j=1 , and vector C = {ci j }i=1 ously assumed that divx f = 0. It follows that a33 = −a11 − a22 . The parameter vector u consists of matrix A components and vector C, i.e. u = (a11 , a12 , a13 , a21 , a22 , a23 , a31 , a32 , c1 , c2 , c3 )∗ . As before, we assume that u belongs to some compact. Finding the parameter vector u defines the system (82.14) and this gives us the desired velocity field. The gradient of the functional (82.13) has the following form:  T ∂J =− (ψi xi − ψ3 x3 ) d xt dt, i = 1, 2, ∂aii 0 Mt,u  T ∂J =− ψi x j d xt dt, i = j, i, j = 1, 2, 3, ∂ai j 0 Mt,u  T ∂J =− ψi d xt dt, i = 1, 2, 3. ∂ci 0 Mt,u

(82.15) (82.16) (82.17)

Formulas (82.15)–(82.17) can be used to implement an optimization method for ˆ are given and T constructing the velocity field. Let us assume that M0 , ρ0 (x), ρ(x) (small enough) is fixed. Then we define the vector of initial values for the unknown parameter vector u = u 0 , k—iteration number, k = 0. After that we calculate the value of the integral functional J (u k ) using the formula (82.7), and find the value of the auxiliary function ψ, integrating equations (82.11) in reverse order from T to 0 under condition (82.12). And then we calculate the gradient ∂∂uJ |u=u k of the integral functional (82.7) by formulas (82.15)–(82.17) and obtain the optimized value of the parameter vector using the formula u k+1 = u k − μ

∂J |u=u k , ∂u

(82.18)

where μ—parameter of gradient descent algorithm. Then the procedure can be repeated (k = k + 1), ending the calculation in accordance with the stop conditions: achieving a given accuracy or a given number of iterations.

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82.4 Application for PET Images Let us consider the implementation of this method for dynamic PET studies of the brain. This acquisition mode allows us to observe the spatial and temporal distribution of the radiopharmaceutical in the patient’s body. Dynamic PET study can last more than an hour, for example, dynamic brain study [17]. In this case, there may be involuntary movements of the patient’s head resulting in the displacement of the received images in relation to each other, which negatively affects the quality of analysis of these images. Thus motion correction is an important part of image processing. We consider the radiopharmaceutical intensity distribution as a function of the distribution density ρ(t, x), t ∈ [0, T ], x ∈ M ∈ R 3 . Taking into account the discrete nature of the data obtained, we have a sequence of three-dimensional matrices (tomographic images) that display the distribution of the radiopharmaceutical in the region of interest ρ1 (i, j, k), ρ2 (i, j, k), . . . , ρ N (i, j, k), i, j, k = 0, 1, 2, . . . , n.

(82.19)

We consider pairs of neighboring 3D images of this series and denote them as ˆ j, k). Using the method proposed above, we determine the velocρ0 (i, j, k) and ρ(i, ity field at the nodes of a rectangular grid with a step equal to one pixel along any axis. The partial derivatives and integrals considered above are also determined by numerical methods. Then we correct image according to the found velocity field.

82.5 Conclusion In this paper, an optimization method for constructing the velocity field for 3D images is proposed. This method can be applied to correct the movement for dynamic tomographic studies. The development of motion correction methods for PET studies becomes especially important with the development of PET hardware, with the improvement of the spatial resolution of new PET cameras when motion artifacts become a limiting factor. This method can also be used to correct movement in the different studies of organs and systems of the human body.

References 1. Bazhanov, P., Kotina, E.: On optimisation approach to velocity field determination in image processing problems. Bull. Irkutsk State Univ.-Ser. Math. 24, 3–11 (2018) 2. Bazhanov, P., Kotina, E., Ovsyannikov, D., Ploskikh, V.: Optimization algorithm of the velocity field determining in image processing. Cybern. Phys. 7(4), 174–181 (2018) 3. Black, M., Anandan, P.: The robust estimation of multiple motions: Parametric and piecewisesmooth flow fields. Comput. Vis. Image Underst. 63(1), 75–104 (1996)

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4. Fleet, D., Weiss, J.: Optical flow estimation. In: Mathematical Models in Computer Vision: The Handbook, pp. 239–258. Springer, Berlin (2005) 5. Gaitanis, A., Kontaxakis, G., Spyrou, G., Panayiotakis, G., Tzanakos, G.: PET image reconstruction: A stopping rule for the MLEM algorithm based on properties of the updating coefficients. Comput. Med. Imaging Graph. 34–2, 131–141 (2010) 6. Horn, B., Schunck, B.: Determining optical flow. Artif. Intell. 17(11), 185–203 (1981) 7. Kotina, E., Latypov, V., Ploskikh, V.: Universal system for tomographic reconstruction on GPUs. Probl. Atom. Sci. Technol. 88(6), 175–178 (2013) 8. Kotina, E., Ovsyannikov, D.: Velocity field based method for data processing in radionuclide studies. Probl. Atom. Sci. Technol. 115(3), 128–131 (2018) 9. Kotina, E., Pasechnaya, G.: Optical flow-based approach for the contour detection in radionuclide images processing. Cybern. Phys. 3(2), 62–65 (2014) 10. Kotina E., Pasechnaya G.: 3D velocity field for heart tomography. In: Proceedings of 2015 international Conference on “Stability and Control Processes” in memory of V. I. Zubov (SCP), pp. 646–647 (2015). https://doi.org/10.1109/SCP.2015.7342231 11. Kotina E., Ploskikh V.: Data processing and quantitation in nuclear medicine. In: RuPAC 2012 Contributions to the Proceedings—23rd Russian Particle Accelerator Conference, pp. 526–528 (2012). https://accelconf.web.cern.ch/accelconf/rupac2012/papers/weppc039.pdf 12. Ovsyannikov, D., Kotina, E., Shirokolobov, A.: Mathematical methods of motion correction in radionuclide studies. Probl. Atom. Sci. Technol. 88(6), 137–140 (2013) 13. Ovsyannikov, D.A.: Mathematical Methods of Beam Control. Leningrad University Publishing, Leningrad (1980) 14. Papenberg, N., Bruhn, A., Brox, T., et al.: Highly accurate optic flow computation with theoretically justified warping. Int. J. Comput. Vision 67(2), 141–158 (2006) 15. Wardak, M., Wong, K., et al.: Movement correction method for human brain PET images: Application to quantitative analysis of dynamic [18F]-FDDNP scans. J. Nucl. Med. 51(2), 210–218 (2010) 16. Xin-Yi Gong, HuSu., De, Xu., Zhang, Zheng-Tao., Shen, Fei, Yang, Hua-Bin.: An overview of contour detection approaches. Int. J. Autom. Comput. 15(6), 656–672 (2018) 17. Ye, H., Wong, K., Wardak, M., Dahlbom, M., Kepe, V., Barrio, J., Nelson, L., Small, G., Huang, S.: Automated movement correction for dynamic PET/CT images: Evaluation with phantom and patient data. PLoS ONE (2014). https://doi.org/10.1371/journal.pone.0103745 18. Zhuravlev, Y.I., Laptin, Yu.P., Vinogradov, A.P., Zhurbenko, N.G., Lykhovyd, O.P., Berezovskyi, O.A.: Linear classifiers and selection of informative features. Pattern Recognit Image Anal. 27(3), 426–432 (2017)

Chapter 83

Automatic Recognition of Metal Smelting Quality Using Machine Vision Valery Grishkin, Mikhail Shirobokov, and Artemii Grigorev

Abstract The paper proposes a prototype of a computer vision-based algorithm, which allows automatic quality control of one of the intermediate stages of metal smelting at a steelworks. During the transfusion of molten metal, the algorithm works with the data obtained from IP camera and makes it possible to determine deviations from the norm of the quality parameters of smelting from a real-time video stream. The algorithm consists of two stages. At the first stage, every frame is checked for containing the process of transfusion of molten metal. If the process is registered, at the second stage the frame is processed in order to calculate metrics that allow to estimate the amount of molten metal or bright fire in it, as well as the presence and amount of dim flame and smoke. If the metric values exceed thresholds, the operator receives a warning about possible problems and takes appropriate action.

83.1 Introduction In the recent decades, production quality control at enterprises has been increasingly changing from manual to automatic [4, 6, 7], which reduces labor costs, the cost of the product, and, in many cases, the chance of flaws in finished products [2]. One of the stages of production at a steelworks is the transportation of molten metal and transferring it to a trolley. At this stage, an operator, using visual inspection, should determine how well the melt was made. Quality assessment is made subjectively, by the presence of a dim flame, dark smoke of various shades, and soaring during the metal transfusion. This fact indicates that the raw materials obtained in this operation are substandard. In the case of detection of substandard condition of the raw material, the operator must take actions to prevent it from reaching the next stage of production. Currently, a human factor plays a significant role in this process. V. Grishkin · M. Shirobokov (B) · A. Grigorev St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. Grishkin e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_83

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Fig. 83.1 An example of a video frame that shows the process of pouring metal into a trolley. Although there are some flames and smoke, this case is described by factory workers as near to the ideal

In this paper, we propose an algorithm that allows to identify possible cases of insufficient quality of raw materials using the video sequence from the camera located next to the place of metal transfusion into the trolley. Figure 83.1 shows an example of a frame from the video stream of the molten metal transfusion process. In the case of under-melting of metal in a trolley, the combustion process of part of the material is observed, and therefore the main criterion that an operator is guided in assessing the quality of the melt is the quantity and color shade of the flame, as well as the amount of smoke in the working part of the mechanism. The same criteria underlie the work of the proposed algorithm. Therefore, in the proposed algorithm, the video stream from the camera is processed and the beginning of the transfusion process is determined first, and then the video image is analyzed according to the specified criteria.

83.2 Detecting the Process of Metal Transfusion in a Video Stream To facilitate the searching for fire and smoke in the video stream and reduce the likelihood of false-positive results, before assessing the amount of fire and smoke for each frame, a simpler task is solved: it is determined if the frame contains the process of metal transfusion. This section describes the process of obtaining the criteria by which this can be established. These criteria are obtained by preprocessing a real video stream, representing the entire unloading process—before the metal is transfused, the transfusion itself, and also when the metal is poured. At the same time, a classifier is built that separates the frames of the video sequence into two classes, with only one of them containing the metal transfusion process. To form a class of frames containing the metal transfusion process, a reference video footage of metal transfusion is used. The frames of the footage significantly differ from each other during the metal transfusion process and outside it in color and brightness of the image; therefore, for easier separation, the image format of each frame is converted from standard RGB to HSV. The informative part of the frame is isolated (this part is known in advance since the position of the trolley in the frame is known beforehand), and hue, saturation, and value for each pixel are treated as samples of the values of some random variables. Mean, variance, skew, and kurtosis of the indicated parameters are used as features; thus, each frame is described with a 12-dimensional vector, to which the k-means clustering algorithm is applied. Since

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Fig. 83.2 The distribution of footage between two clusters. The points corresponding to the frames related to the metal transfusion process are highlighted in red. Values on axis are linear combinations of features obtained with PCA

k-means tends to produce equal-sized clusters, part of the video sequence was cut so that the metal pouring process took about half the frames. The resulting clusters adequately reflect reality: frames at the beginning of the video, before the start of the process, and frames relating to the time when the metal has already been poured turn out to be in one cluster, and frames belonging to the central part of the video in which the metal was poured into the trolley— in another. Figure 83.2 shows a visualization of the distribution of frames between clusters obtained using PCA.

83.3 Recognition of Fire and Smoke During the Metal Transfusion Process The quality of the smelting is estimated after determining the beginning of the metal transfusion process. The main criterion by which the quality of the melt should be estimated is the amount of flame and smoke in the frame. The areas containing fire and smoke can differ greatly from each other during the transfusion of molten metal, so there are two different metrics used to detect them. These metrics are based on the calculation of areas in the frame occupied by the objects “bright fire” and “smoke and dim fire”, which ratio to the overall “working area” of the frame determines the decision on the quality of the melting. The acceptable values for the metrics are determined experimentally during the tuning of the algorithm on the available sample video recordings.

83.3.1 Bright Fire Recognition Since the main criterion for recognizing an object “bright fire” is its brightness, the frame is primarily transferred from the HSV color space to grayscale. In order to reduce the amount of noise, Gaussian blur is applied to the resulting image, and then it is converted to black and white with a fixed threshold. The dynamic threshold is

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Fig. 83.3 Sample image with a selected work area and recognized molten metal and bright flame

not used since the goal is to isolate the brightest areas in the image in absolute terms, and not in relation to neighboring areas. With the purpose of a further riddance of the noise, the image goes through the erosion stage, which removes small bright spots, and dilation, which returns large objects approximately to the original size. To highlight the areas of “bright fire”, the method of searching for connected components [3] is applied to the resulting binary image. The result is a grayscale image, where the brightness of each pixel corresponds to the number of the area it belongs to. Moreover, the background area has a number equal to zero. In addition, this segmentation creates a list of these areas. Each associated area is described by a set of parameters. This set includes the coordinates of the center of the area region, the coordinates and sizes of the bounding box rectangles. The resulting list of related areas is then filtered by area. After sifting out the background and areas that occupy less than 1 500 of the total size of the studied area, the remaining areas are displayed on the original  image, which is showed to the user to observe the process. The ratio Rb f = Sb f S0 is used as a metric to estimate the amount of bright flame and molten metal in the image, where Sb f is the total area of the connected components and S0 is the area of the working area (which is in a blue rectangle in Fig. 83.3).

83.3.2 Soft Fire and Smoke Recognition The soft fire and smoke in the image in the video footage have a characteristic red tint and medium brightness, which allows to select them using a simple filter by HSV parameters. The image sections that fall into the specified filter are drawn separately, and the ratio of their area to the area of the working region is takenas a metric to estimate the amount of soft fire and smoke in the image: Rs f = Ss f S0 .

83.4 Experimental Results The proposed algorithm is implemented in Python using the OpenCV [1] image processing library and the Scikit-learn [8] machine learning library. The Matplotlib [5] library was used for graph visualization. An application that uses the algorithm processes the video stream. For each frame received from the camera, it is first checked if it belongs to a cluster with a metal

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Fig. 83.4 Example of selection of soft fire and smoke from a source image

transfusion process using a classifier trained on clusters. The clusters themselves are obtained as a result of preprocessing of the technological process video recording. The procedure for obtaining clusters is described in Sect. 83.2. If the frame belongs to the class of the active process, the metrics Rb f and Rs f , which are described in Sect. 83.3, are calculated for it. If the amount of fire or smoke exceeds a predetermined value, an assumption is made that there are malfunctions in the process and the operator receives a warning. The results of the algorithm for determining the flame regions of various types, fumes, as well as the corresponding metrics are shown in Figs. 83.3 and 83.4.

83.5 Conclusion The proposed algorithm has shown its fundamental applicability to the available source data. With its help, the beginning and end of the metal unloading process are reliably determined. When analyzing a video of direct metal discharge, areas of bright and soft fire are clearly detected, as well as areas of smoke, which are used to construct metrics for assessing the quality of the melt. The application of the proposed metrics related to the distribution of brightness and color characteristics in the frame allows to estimate the quality of the melt. The main disadvantage of the proposed algorithm is noise sensitivity, which is partially handled using the filtering process described in Sect. 83.3.1. However, parts of the trolley that reflect light from a molten metal are sometimes mistaken for areas of fire or smoke. One way to deal with this phenomenon is the “skeptical” behavior of the evaluation algorithm, which reports a possible malfunction only when atypical behavior persists for a certain period of time. At the moment, a complete assessment of the method’s operability is not possible due to the absence of any additional data, except for the sample footage. In particular, due to the lack of examples in which the algorithm should work and report a problem, it is impossible to evaluate its precision and recall. Nevertheless, in the images obtained synthetically (initial frames with added areas of fire and smoke), the algorithm showed a positive result.

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83.6 Further Work First of all, it is necessary to verify the algorithm on an extended series of real data, which will improve the quality of the obtained prototype. In particular, at the stage of deciding on the quality of metal smelting, it is assumed that instead of a simple threshold filtering by metric values, a classifier trained in this extended series should be used. This classifier can be constructed, for example, using the support vector machine or neural network. The use of such a classifier will reduce the number of false-positive results arising from the presence of a bright sky and reflecting metal parts in the field of view of the video camera. New video data will allow to automatically determine the alert thresholds, and also more accurately determine the criteria for selecting parts of the image that have fire or smoke. Processing an extended series will allow getting rid of the incorrect definition of bright or red areas as fire or smoke, automate the determination of binarization thresholds, and the maximum permissible area of fire and smoke.

References 1. Bradski, G.: The OpenCV Library. Dr. Dobb’s Journal of Software Tools (2000) 2. Columbis, L.: 10 Ways Machine Learning Is Revolutionizing Manufacturing in 2018 (2018) https://www.forbes.com/sites/louiscolumbus/2018/03/11/10-ways-machine-learningis-revolutionizing-manufacturing-in-2018 (Cited 5 Apr 2020) 3. Dillencourt, M., Samet, H., Tamminen, M.: A general approach to connected-component labeling for arbitrary image representations. J. ACM 39(2), 253–280 (1992). https://doi.org/10.1145/ 128749.128750 4. Ghorai, S., Mukherjee, A., Gangadaran, M., Dutta, P.K.: Automatic defect detection on hot-rolled flat steel products. IEEE Trans. Instrum. Meas. 62(3), 612–621 (2012) 5. Hunter, J.D.: Matplotlib: A 2D Graphics Environment (2007). https://doi.org/10.1109/MCSE. 2007.55 6. Lenty, B., Kwiek, P., Sioma, A.: Quality control automation of electric cables using machine vision (2018). https://doi.org/10.1117/12.2501562 7. Luiz, A.M., Flávio, L.P., Paulo, E.A.: Automatic detection of surface defects on rolled steel using computer vision and artificial neural networks. In: IECON 2010-36th Annual Conference on IEEE Industrial Electronics Society, pp. 1081–1086. IEEE (2010) 8. Pedregosa, F. Gaël, Gramfort, A., Michel, V., Thirion, B., Grisel, O. et al.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12, 2825–2830 (2011)

Chapter 84

Mathematical Modeling of Cyclic Chemical Compounds Albina Akhmetyanova, Albina Ismagilova, and Fairuza Ziganshina

Abstract This work describes the mathematical apparatus for constructing homodesmic reactions using the example of cyclic compounds cis-1,2-dimethylcyclobutane and trans-1,2-dimethylcyclobutane. The created software generates a set of independent homodesmic reactions for the selected compound, which in turn increases the reliability of the theoretical determination of the standard enthalpy of formation. The main stages of building a program that implements the algorithm for determining the HDR basis for cyclic compounds are described in this work. For clarity of the investigated method, an example of constructing the basis of homodesmic reactions, a graph, and an adjacency matrix for the above molecules is given.

84.1 Introduction In the modern world, information technologies have penetrated almost all spheres of human life. In industrial production and in scientific research, new computer programs are increasingly used. The process of informatization has also affected the area of research related to organic chemistry. As we know, the analysis of the mechanisms of chemical reactions requires the thermodynamic parameters of chemical compounds. There are various methods of theoretical calculation of thermodynamic quantities, which appeared due to a lack of experimental data and unreliable data [2]. In the studies of the Russian scientist Khursan S.L., it was shown that “a comparative calculation method based on the use A. Akhmetyanova (B) · A. Ismagilova Bashkir State University, Ufa, Russia e-mail: [email protected] A. Ismagilova e-mail: [email protected] F. Ziganshina Ufa State Oil Technical University, Ufa, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_84

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of homodesmic reactions (HDR) is the most reliable method for theoretical prediction of the enthalpy of formation of a chemical compound” [13].

84.2 Goals and Methodology The key goal of this study is to develop a mathematical apparatus for assessing the thermodynamic parameters of cyclic compounds [2]. When developing the software, the theory of graphs was used, which makes it possible to interpret chemical compounds [4, 6], as well as the foundations of chemical thermodynamics and methods for calculating the main thermodynamic quantities described in the works of scientists [9, 19, 22]. The HDR method was applied in the works of Russian scientists Khursan S.L., Ismagilova A.S. et al. in the study of acyclic non-aromatic compounds [2, 3, 10–12, 15, 25]. It was found that the calculation results are not interrelated with the level of complexity of the quantum-chemical method. The HDR method was used to develop a special computer program for complicated molecular systems, including cyclic compounds. For identification by the program of a chemical compound, it is presented as a combination of thermochemical groups [4]. When finding the basis of homodesmic reactions, the decomposition method is used for a chemical compound. The decomposition method is based on the analysis of the valence bond matrix, group bond matrix, group matrix, and atomic matrix.

84.3 Literature Review Well-known foreign scientist W.J. Henre, who defined isodesmic reactions [7, 8]. Isodesmic reactions are also taken into account in our work. The problem of ambiguity when choosing a reference reaction was investigated by I. Fishtik [5]. He, in his works, uses a limited set of reference compounds and constructs isodesmic reactions in which only selected compounds participate. The results of measurements of the calorific value and formation of cyclopropane are given in the scientific work of J.W. Knowlton and F.D. Rossini [17]. In turn, the work of J. R. Lacher et al. describes a constant-flow isothermal calorimeter and presents the results of the heat of chlorination of some simple fluoroolefins, as well as a modified version of the calorimeter adapted for the hydrobromination reaction [18]. American scientists Steven E. W., Steven M.B., and others in their studies proved that the two definitions of “homodesmotic” reactions are not equivalent. They constructed a consistent hierarchy of reaction classes in such a way that each class sequentially retained larger molecular fragments [20, 21].

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Bulgarian scientists D. Todor et al. proposed a new approach that uses an initio calculations to evaluate the deformation of ring molecules, and to assess the relative contribution of various deformation sources to the ring deformation energy [23]. P.A Vijaya et al. described an algorithm for representing a limited set of dependent elementary reaction steps for a reaction mechanism by using the graph of the reaction route of the mechanism [24]. In our opinion, one of the main and reliable methods for determining the independent basis of homodesmic reactions is the graph theory method. In our works [4, 25], algorithms for constructing the basis of homodesmic reactions are described, which are implemented in the Delphi programming language.

84.4 Results of Research The implemented software constructs the basis of the GDR and calculates the enthalpies of formation of the studied chemical compound. All available experimental data were obtained from NIST Standard Reference Database Number 69 [1]. A specialized recursion-based brute force method has been applied. The intuitive interface of the program allows any user to enter data on a new chemical compound or select from a list of existing ones. All information about the structure and composition of a chemical compound, as well as its energy characteristics, is stored in a relational database, in which some of the tables play the role of reference books. To interact with the database application SQLite DBMS is used. The architecture of this DBMS allows you to store data on the user’s computer. The algorithm for constructing the basis of the GDR for a cyclic connection includes the following main steps: 1. Introduce all sorts of options for a “deployed” molecule, which is obtained from the original “break” of the cyclic bond. Identify all possible combinations of the internal groups that make up the chemical compound. 2. For each term, select end groups from those that are present in the original chemical compound. Form the right parts of the GDR. If there are no end compounds in the compound or no suitable ones are available, construct a “new” one by adding valence-bound atoms. 3. To determine the reagent(s) of the initial chemical compound, take into account the group composition of the products on the right side of the GDR. Reagents are made up of end groups of products. 4. Put down the stoichiometric coefficients in the GDR according to the group balance—maintaining the number of groups (internal and end) of each type. Here is an example of the algorithm working on cis-1,2-dimethylbutane (C6H12) and trans-1,2-dimethylbutane (C6H12) molecules (Fig. 84.1). The structure of cis1,2-dimethylbutane (C6H12) and trans-1,2-dimethylbutane (C6H12) is represented as a combination of two types of internal thermochemical groups B01 and B02 and one type of terminal group K01. There are two types of chemical bonds in this molecule: C–C and C–H.

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Fig. 84.1 a 1,2-Dimethylcyclobutane molecule; b Structural formula of molecules cis-1,2dimethylcyclobutane and trans-1,2-dimethylcyclobutane, group composition; c Internal groups B01 and B02, end group K01

All kinds of independent variants of the cis-1,2-dimethylcyclobutane molecule, which are obtained from the initial “cleavage” of the bond, are presented in the form of following formulas: C6 H12 + C2 H6 → C8 H18 (3, 4 − Dimethylhexane); C6 H12 + C2 H6 → C8 H18 (2, 3 − Dimethylhexane); C6 H12 + C2 H6 → C8 H18 (2, 5 − Dimethylhexane); C6 H12 + 2C2 H6 → C3 H8 (Pr opane) + C7 H16 (2, 3 − Dimethylpentane); C6 H12 + 2C2 H6 → C4 H10 (I sobutane) + C6 H14 (2 − Methylpentane); C6 H12 + 2C2 H6 → 2C5 H12 (2 − Methylbutane); C6 H12 + 2C2 H6 → C4 H10 (Butane) + C6 H14 (2, 3 − Dimethylbutane); C6 H12 + 3C2 H6 → C3 H8 (Pr opane) + C4 H10 (I sobutane) + C5 H12 (2 − Methyl − butane); 9. C6 H12 + 3C2 H6 → 2C3 H8 (Pr opane) + C6 H14 (2, 3 − Dimethylbutane); 10. C6 H12 + 3C2 H6 → C4 H10 (Butane) + 2C4 H10 (I sobutane); and 11. C6 H12 + 3C2 H6 → 2C3 H8 (Pr opane) + 2C4 H10 (I sobutane); 1. 2. 3. 4. 5. 6. 7. 8.

As can be seen, the construction of the cis-1,2-dimethylcyclobutane molecule resulted in 11 basic GDRs. For cis-1,2-dimethylcyclobutane and trans-1,2-dimethylcyclobutane molecules, the GDR basis will look the same, only the calculated data will be different. Below are the experimental and calculated data for each compound of the basic GDR (Fig. 84.2). Thus, in the present work, a graph-theoretic interpretation of the structure of cis-1,2-dimethylcyclobutane and trans-1,2-dimethylcyclobutane molecules is given. Based on the algorithm, a basis for homodesmic reactions for the studied compounds is constructed. At the moment, we are faced with the task of expanding the database and testing the resulting software on various organic compounds in order to identify shortcomings and eliminate them in the software.

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Fig. 84.2 Experimental and calculated data of the molecules cis-1,2-dimethylcyclobutane and trans-1,2-dimethylcyclobutane for calculating the enthalpy of formation program

Acknowledgements This work was supported by the Russian Foundation for Basic Research and the Government of the Republic of Bashkortostan as part of scientific project no. 18-07-00584.

References 1. Afeefy, H.Y., Liebman, J.F., Stein, S.E.: NIST Standard Reference Database Number 69. National Institute of Standards and Technology, Gaithersburg Md. 20899 (2018). https://doi. org/10.18434/T4D303 2. Akhmetyanova, A.I., Ziganshina, F.T., Ismagilova, A.S.: Mathematical apparatus for the construction of homodesmic reactions. In: 2019 1st International Conference on Control Systems, Mathematical Modelling, Automation and Energy Efficiency (SUMMA) (2019). https://doi. org/10.1109/SUMMA48161.2019.8947496

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3. Antonovsky, V.L., Khursan, S.L.: Thermochemistry of organic peroxides in solution. Russ. Chem. Rev. 72(11), 939 (2003) 4. Benson, S.: Thermochemical Kinetics. In: Benson, S. Wiley, New York (1968) 5. Fishtik, I.: Unique stoichiometric representation for computational thermochemistry. J. Phys. Chem. A 116, 1854–1863 (2012) 6. Harari, F.: Graph Theory. Translation from English. Mir, Moscow (1973) 7. Henre, W.J., Ditchfield, R., Radom, L., Pople, J.A.: Isodesmic reactions are defined. Am. Chem. (92), 4796 (1970) 8. Henre, W.J., Radom, L., Schleyer. P.R., Pople, J.A.: Ab Initio Molecular Orbital Theory. Wiley, New York (1986) 9. Jensen, F.: Introduction in Computational Chemistry. 3rd edn., p. 429. Wiley, New York (2017) 10. Khursan, S.L.: Accounting for intramolecular non-valent interactions in the additivity method of thermochemical increments. Comparative analysis of methods for calculating the formation enthalpies. Bashkir Chem. J. 1(4), 37–41 (1997) 11. Khursan, S.L.: The standard enthalpies of formation of fluorinated alkanes: nonempirical quantum-chemical calculations Russ. J. Phys. Chem. 78(l), S34–S42 (2004) 12. Khursan, S.L.: Quantum-chemical ab initio calculations of the enthalpies of formation and hydrogenation of imines. Russ. J. Phys. Chem. 76(3), 405–411 (2002) 13. Khursan, S.L.: Comparative analysis of theoretical methods for determining the thermochemical characteristics of organic compounds. Bull. Bashkir Univ. 2(19), 395–401 (2014) 14. Khursan, S.L., Ismagilova, A.S., Akhmerov, A.A., Spivak, S.I.: Constructing homodesmic reactions for calculating the enthalpies of formation of organic compounds. Russ. J. Phys. Chem. 90(4), 796–802 (2016) 15. Khursan, S.L., Ismagilova, A.S., Spivak, S.I.: A graph theory method for determining the basis of homodesmic reactions for acyclic chemical compounds. Dokl. Phys. Chem. 474, 99–102 (2017) 16. Khursan, S.L., Ismagilova, A.S., Akhmetyanova, A.I.: Determining the basis of homodesmotic reactions of cyclic chemical compounds. Doklady Phys. Chem. 7(92), 1312–1320 (2018) 17. Knowlton, J.W., Rossini, F.D.: Heats of combustion and formation of cyclopropane. J. Res. NBS 43, 113–115 (1949) 18. Lacher, J.R., Walden, C.H., Lea, K.R., Park, J.D.: Vapor phase heats of hydrobromination of cyclopropane and propylene. J. Am. Chem. Soc. 72, 331–333 (1950) 19. Pedley, J.Â., Naylor, R.D., Kirby, S.P.: Thermochemical Data of Organic Compounds. Chapman & Hall, London (1986) 20. Steven, M.B.: The group equivalent reaction: an improved method for determining ring strain energy. J. Chem. Educ. 67(11), 907 (1990) 21. Steven, E.W., Kendall, N.H., Paul, R.S., Wesley, D.A.: A hierarchy of homodesmotic reactions for thermochemistry. J. Am. Chem. Soc. 131(7), 2547–2560 (2009) 22. Stull, D.R., Westrum, E.F., Sinke, G.C.: The Chemical Thermodynamics of Organic Compounds. Wiley, New York (1969) 23. Todor, D., Carmay, L.: Ring strain energies from ab initio calculations. J. Am. Chem. Soc. 120(18), 4450–4458 (1998) 24. Vijaya, P., Tadea, M.O., Fishtik, I., Dattab, R.: A graph theoretical approach to the elucidation of reaction mechanisms: analysis of the chlorine electrode reaction. Comput. Chem. Eng. 49, 85–94 (2013) 25. Ziganshina, F.T., Ismagilova, A.S., Akhmetyanova, A.I., Akhmetshina, E.S., Akhmerov, A.A.: Computer simulation of the problem of determining the basis of homodesmic calls. Control Syst. Inf. Technol. 4(78), 10–15 (2019)

Chapter 85

Parametrization of Orthogonal Hexagonally Symmetric Low-Pass Filters Aleksandr Krivoshein

Abstract A complete parametrization of hexagonally symmetric orthogonal lowpass filters for the case of dyadic matrix dilation and small supports is given. Methods for the construction of high-pass filters are also discussed.

85.1 Introduction and Preliminaries The construction of low-pass filters (also known as refinable masks) with appropriate properties is the first step in the algorithms for designing wavelet systems. Although the general scheme for such algorithms is known (for instance, various extension principles), concrete realizations are not an easy task in the multivariate setting, especially when a variety of additional requirements should be satisfied (see, e.g. [8]). In this article, we concentrate on the construction of low-pass filters in two-dimensional case with hexagonal symmetry and orthogonality. The orthogonality can lead to the construction of an orthogonal wavelet basis (or at least tight wavelet frames). Hexagonal symmetry may be useful in applications of the obtained filters in hexagonal image processing (which has some advantages over the traditional image processing, see, e.g. [10]). The aim of the article is to suggest complete parametrization in the case of dyadic dilation for low-pass filters with indicated above properties for some filter supports. An example of such filters can be found in [1] (see the next section for details). Also, related constructions of hexagonally symmetric biorthogonal wavelets can be found in [2, 3, 5] and references therein. We use the standard multi-index notations. There is a lot of notions related to wavelet theory, which are briefly introduced below. Most of them in more details can be found, for instance, in [8, 11]. The space of finite sequences will be denoted by 0 (Z2 ). A sequence a ∈ 0 (Z2 ), a = {a(k)}k∈Z2 is called a filter. We will deal only with the dyadic dilation matrix M = 2I2 , a set D = {s0 , s1 , s2 , s3 } := {(0, 0), (0, 1), (1, 0), (−1, −1)} is called the set of digits of M. The polyphase A. Krivoshein (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_85

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component of a filter a ∈ 0 (Z2 ) corresponding to the digit sn ∈ D is a filter an ∈ 0 (Z2 ) defined by an (k) := | det M| a(M k + sn ), k ∈ Z2 , n = 0, 1, 2, 3. A filter a ∈ 0 (Z2 ) is called orthogonal, if 3 

| an (ξ )|2 = | det M| = 4,

n=0

 an (ξ ) := k∈Z2 an (k) where  an is the discrete-time Fourier transform of filter an ,  e2πi(k,ξ ) . The support of a ∈ 0 (Z2 ) is defined to be supp(a) := {k ∈ Z2 : a(k) = 0}. We say that filter a is low pass if  a (0) = 1, here and below 0 = (0, 0). We will work with specific symmetric properties. These properties can be defined via the notion of a symmetry group. Our main attention is devoted to the hexagonal symmetry group, defined by  H =

     0 −1 −1 1 10 , , . 1 −1 −1 0 01

Then filter a ∈ 0 (Z2 ) is called H -symmetric , if a(k) = a(Ek), ∀k ∈ Z2 , ∀E ∈ H . An important property, which is connected with approximation properties of a wavelet system generated by a low-pass filter, is the order of sum rule. For instance, filter a obeys sum rule of the first order, if  an (0) = | det M| k∈Z2 a(Mk + sn ) = 1 for n = 0, 1, 2, 3. In fact, if low-pass filter a is orthogonal, then sum rule of order 1 for this low-pass filter a is always valid. The benefit of this condition is that during the construction of a low-pass filter, sum rule always gives linear equations (instead of the orthogonality condition, see the next section). Of course, a low-pass filter is only the first step during the construction of wavelet systems, but, in some sense, this is the most important step. An orthogonal low-pass filter uniquely determines a refinable function φ ∈L 2 (R2 ), and this function is a solution of a refinement equation φ(x) = | det M| k∈Z2 a(k)φ(M x + k). Also, it is known that the orthogonality of a low-pass filter a is only a necessary condition for the orthogonality of the corresponding refinable function (orthogonality of φ means the orthogonality of integer shifts {φ(· + k)}k∈Z2 ). Sufficiency needs an additional condition. For instance, we will use an algorithm based on [9, Prop. 4.2]. The quality of constructed low-pass filter will be estimated using the Sobolev smoothness for the corresponding refinable functions, defined by ν2 (φ) =  exponent (ξ )|2 (1 + |ξ |2 )ν dξ < ∞ . This can be done using an algorithm in [4, sup ν : R2 |φ Th. 7.1]. If we have already obtained some good orthogonal low-pass filter, then tight wavelet frames always can be constructed. The case of orthogonal basis is more involved. However, in some specific cases, such as interpolatory case, the construction of orthogonal wavelet basis can be done explicitly (for more details see the last subsection and, e.g. [8] or [7]).

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85.2 Main Results We present parametrization of hexagonally symmetric orthogonal low-pass filters for two cases. The first case deals with filter supports inside [−2, 2]2 ∩ Z2 . The second one deals with the special case of filter support inside [−3, 3]2 ∩ Z2 .

85.2.1 General Case for Small Filter Supports Let us begin with a simple example. Let filter a ∈ 0 (Z2 ) be such that supp(a) ⊂ [−1, 1]2 ∩ Z2 . If filter a is hexagonally symmetric, then its coefficients should be as in the following template: ⎞ ⎛ 0 B 0 a : ⎝ 0 A B⎠. B 0 0 The boxed element corresponds to a(0). The polyphase components (in terms of trigonometric polynomials) are a0 (ξ ) = 4 A, a1 (ξ ) =  a2 (ξ ) =  a3 (ξ ) = 4B. The sum rule of order 1 yields that 4 A = 4B = 1. The orthogonality condition implies 16A2 + 48B 2 = 4. If (and only if) A = B = 14 , then all conditions are satisfied. The Sobolev smoothness exponent of the corresponding refinable function φ is ν2 (φ) ≥ 0.2075. By [9, Prop. 4.2] it can be established that φ is indeed orthogonal. Next, consider a filter a ∈ 0 (Z2 ) such that supp(a) ⊂ [−2, 2]2 ∩ Z2 . If filter a is hexagonally symmetric, then its coefficients should be as in the following template: ⎛

0 ⎜0 ⎜ a:⎜ ⎜D ⎝C B

0 E G F E

B F A G D

C G F C 0

⎞ D E⎟ ⎟ B⎟ ⎟. 0⎠ 0

The polyphase components a0 , a1 , a2 , a3 are ⎛ ⎞ ⎞     0 4C 0 4B 4D 4E 4G 0 4F 4E ⎝ 4D 4 A 4B ⎠ , , , ⎝ 4G 4F ⎠ , 4C 4G 0 4F 4C 4E 0 4B 4D 0 ⎛

respectively. The sum rule of the first order gives us 4(A + 3B + 3D) = 1,

(85.1)

4 C + 4E + 4 F + 4G = 1.

(85.2)

The orthogonality condition yields that

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B D = 0,

(85.3)

B + D + EC = 0, 2B D + A(B + D) + (C + E)F + (C + E + F)G = 0, 16 A2 + 48(B 2 + C 2 + D 2 + E 2 + F 2 + G 2 ) = 4.

(85.4) (85.5) (85.6)

2

2

Note that coefficients B and D are interchangeable and at least one should be equal to zero, so let D = 0. Also note that the last condition (85.6) is redundant. In fact, it can be obtained as some combination of other equations: (85.1)2 + 3· (85.2)2 − 96· (85.3) −96· (85.4) −96· (85.5) = (85.6). Overall, it remains to solve the following system of equations: 4 A + 12B = 1, B + EC = 0, 2

4C + 4E + 4F + 4G = 1,

AB + (C + E)F + (C + E + F)G = 0.

2 We suggest let √ α = EC, α ≤ 0. Then B = −α or √ the following1 parametrization: 1 B = ± −α. Then A = 4 − 3B = 4 − 3(± −α). Next, let β = E + C. These α and β will be our parameters. Unknowns E and C can be explicitly expressed via α, β   1 1 C = (β ∓ −4α + β 2 ), E = (β ± −4α + β 2 ). 2 2

It remains to get F, G via α and β. Note that F + G =

1 4

− β and

    1√ 1 F G = −AB − (C + E)(F + G) = − ± −α + 3α − β −β = 4 4 β 1√ −α − 3α − + β 2 . ∓ 4 4 Note that F and G are also interchangeable, then    √ 1 2 1 − 4β + 1 ± 16 −α + 192α + 8β − 48β , F= 8    √ 1 2 1 − 4β − 1 ± 16 −α + 192α + 8β − 48β . G= 8 √ Condition 1 ± 16 −α + 192α + 8β − 48β 2 ≥ 0 gives the restriction on the set of possible choices for α and β. For a fixed α ≤ 0, parameter β should be       √ √ 1 1 1 − 2 1 ± 12 −α + 144α ≤ β ≤ 1 + 2 1 ± 12 −α + 144α . 12 12 √ √ 1 For “+” in ± such β exists, if 1 + 12 −α + 144α ≥ 0 or 288 (−3 − 5) ≤ α ≤ 0. √ √ 1 For “−” such β exists, if 1 − 12 −α + 144α ≥ 0 or 288 (−3 + 5) ≤ α ≤ 0.

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Investigating the Sobolev smoothnesses of refinable functions that correspond to the obtained low-pass filters we get that the maximum is achieved for parameters α = 0 and β = −1/12. In this case ν2 (φ) ≥ 0.5185 and the coefficients of that filter a are ⎛ ⎛ ⎞ ⎞ 1 0 0 0 0 0 0 0 0 − 12 0 ⎜0 − 1 1 1 − 1 ⎟ ⎜ 0 0 1 1 0⎟ 12 6 6 12 ⎟ 6 6 ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ 1 1 1 1 1 1 0 0 ⎜ 0 6 4 6 0 ⎟ or ⎜ ⎟ 6 4 6 ⎜ ⎜ ⎟ ⎟ ⎝0 1 1 0 0 ⎠ ⎝− 1 1 1 − 1 0⎠ 6 6 12 6 6 12 1 0 0 0 0 − 12 0 0 0 0 0 depending on the choice of ± in the expressions for coefficients E and C. By [9, Prop. 4.2] it can be established that the corresponding refinable functions are indeed orthogonal.

85.2.2 Special Case for 7 by 7 Support In this section, we consider filters with bigger support: supp(a) ⊂ [−3, 3]2 ∩ Z2 . In general, the number of parameters in this case is too big to deal with them. That is why we restrict ourselves with so-called “snowflake-like” support ⎛

⎞ D ⎜ E F⎟ ⎜ ⎟ ⎜ F ⎟ B C ⎜ ⎟ ⎟. a:⎜ E C A B ⎜ ⎟ ⎜D B C ⎟ ⎜ ⎟ ⎝ ⎠ E D F An example of the low-pass filter with such support was mentioned in [1]. For better readability 0’s are replaced by empty spaces. The Sobolev smoothness of the refinable function that was presented in [1] is ν2 (φ) ≥ 0.8968. However, it is possible to find a filter with better properties based on the complete parametrization for this case of support. The polyphase components a0 , a1 , a2 , a3 are ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 4D 0 0 4C 0 0 4F 0 0 4E ⎝ 4E 4 A 0 ⎠ , ⎝ 4F 4B 0 ⎠ , ⎝ 4D 4B 0 ⎠ , ⎝ 4C 4B 0 ⎠ , 0 4E 0 0 4C 0 0 4F 0 0 4D 0 ⎛

respectively. Sum rule yields that 4 A + 12E = 1, 4(D + F + B + C) = 1. The orthogonality condition yields that E 2 + FC + D(F + C) = 0,

(85.7)

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AE + B(D + F + C) = 0,

(85.8)

16(A + 3(D + B + C + E + F )) = 4. 2

2

2

2

2

2

And again the latter condition is redundant. Using replacement A = 41 − 3E and D +     F + C = 41 − B in (85.8), we get E 41 − 3E + B 41 − B = 0. This is a secondorder curve (an ellipse), which can be parametrized as   1 1 cos α + , E= 12 2

 √  1 3 B = √ sin α + , 2 4 3

where α ∈ [0, 2π ]. Now consider the following equations: 1 F + C = − B − D, 4

 FC = −E − D 2

 1 −B−D . 4

Next, we assume that D is another parameter, then F and C can be expressed explicitly via α and D F=

  √ √ 1  1  3 − 24 · D − 2 3 sin α ± S , C = 3 − 24 · D − 2 3 sin α ∓ S , 48 48

where  √ S = 16 cos α + 2 cos(2α) − 12 3(8 · D + 1) sin α + 144 · D(1 − 12 · D) + 27. Note that parameters α and D should be such that S ∈ R. Thus, the expression under the root should be positive. It is a parabola with respect to D and appropriate D should be chosen from the interval    √ √ √ 1 3 − 2 3 sin α − 4 3 − 3 sin α + cos α + 2 ≤ D ≤ 72    √ √ √ 1 3 − 2 3 sin α + 4 3 − 3 sin α + cos α + 2 72 and α ∈ [0, 2π ]. It appears that for a special choice of parameters it is possible to achieve sum rule of order 2. Namely, let √   √ 1 2 3 sin α , α = arccos (±3 13 − 5) . D=− 72 16   √ 1 Taking α = arccos 16 (−3 13 − 5) , the corresponding refinable function will have the best estimation of Sobolev smoothness exponent among others, i.e. ν2 (φ) ≥ 0.9425. By [9, Prop. 4.2], φ is indeed orthogonal. In this case, D = F

85 Parametrization of Orthogonal Hexagonally Symmetric Low-Pass Filters

F=

749

√ √ √ 1 √ 1 1 1 ( 13 − 5), E = (1 − 13), B = (13 − 13), C = (19 + 13). 192 64 64 192

7 A low-pass filter from [1] can be obtained, if α = arccos(− 337 ) and D = − 1014 . 338

85.2.3 Wavelet Construction In this section, we give some ideas for the construction of wavelets from orthogonal low-pass filters. Basic scheme is related to the problem of extension of one row of trigonometric polynomials up to a unitary matrix of trigonometric polynomials. In more details, let a be an orthogonal low-pass filter and ai , i = 0, 1, 2, 3, are its polyphase components. The task is to find matrix ⎞  a0  a1  a2  a3 ⎜ b0(1)  b1(1)  b2(1)  b3(1) ⎟ ⎟ , r ≥ 3, M := ⎜ ⎠ ⎝ ··· ··· (r ) (r ) (r ) (r )  b1  b2  b3 b0  ⎛

whose elements are trigonometric polynomials such that M ∗ M = 4I4 . These bi(ν) are the polyphase components of b(ν) , which are called high-pass filters. Set (ν) (ν) ting ψ (x) = | det M| k∈Z2 b (k)φ(M x + k), we obtain wavelet functions. By j/2 (ν) default, wavelet system {ψ (ν) ψ (M j x + k) : j ∈ Z, k ∈ Z2 , ν = jk (x) = | det M| 1 . . . r }, generated by these wavelet functions ψ (ν) , is a tight wavelet frame in that ∀ f ∈ L 2 (R2 ) and for some constant C > 0 we have L 2 (R2 ), which  means 2 | f, ψ (ν) | . Providing φ to be orthogonal and r = 3, the wavelet sysC f 2 = jk ν, j,k

tem generates an orthogonal basis of L 2 (R2 ) (see, e.g. [11]). The problem of extension of the matrix M in the multivariate setting is hard to solve in general (see, e.g. [8]) and there is no common algorithm in contrast to univariate setting. But in some specific cases, matrix M can be extended explicitly. This is so-called interpolatory case of a low-pass filter, which means that  a0 ≡ 1. In a0 ≡ 1. Let P detail, let a ∈ 0 (Z2 ) be an interpolatory orthogonal low-pass filter,  a2 , a3 ). Note that by orthogonality PP ∗ = 3. Then be a row defined by P := ( a1 , matrix extension is   1 P . (85.9) M = P ∗ 2I3 − P ∗ P It can be checked directly that M M ∗ = | det M|I4 . If a is hexagonally symmetric, then the high-pass filters obtained by (85.9) are connected to each other with the help of the matrices in the  symmetry denote matrices from hexagonal  group.Let us  0 −1 −1 1 , E2 = . symmetry group by E 1 = 1 −1 −1 0

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Theorem 85.1 Let a be an orthogonal and hexagonally symmetric low-pass filter. Then high-pass filters b(i) , i = 1, 2, 3 obtained by (85.9) are mutually symmetric, b(1) (E 1∗ ξ ) and  b(3) (ξ ) =  b(1) (E 2∗ ξ ). i.e.  b(2) (ξ ) =  This statement can be justified repeating the proof of [6, Th. 15]. A low-pass filter constructed in Sect. 85.2.1 for parameters α = 0 and β = −1/12 is interpolatory since B = D = 0. Then one high-pass filter constructed by (85.9) is ⎛

b(1)

0 0 0 ⎜ 0 0 −1 9 ⎜ ⎜ 0 −1 1 ⎜ 1 91 16 :⎜ − ⎜ 18 9 4 ⎜ ⎝ 0 − 19 16 1 18

− 19

− 19

1 18 − 19

1 0 − 36

⎞

0 0

1 18 1 − 12 1 18

− 19

0

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ 0 ⎠

1 18

0

0

1 18 1 18 1 − 36

Other high-pass filters are related to b(1) via the relations stated in Theorem 85.1.

85.3 Conclusion For the case of dyadic matrix dilation, a complete parametrization of hexagonally symmetric orthogonal low-pass filters is given for several choices of the coefficient support. The technique for the construction of high-pass filters, which leads to tight wavelet frames in L 2 (R2 ), is also presented. Acknowledgements The author is supported by the Russian Science Foundation under grant no. 18-11-00055.

References 1. Allen, J.D.: Perfect reconstruction filter banks for the hexagon grid. In: 2005 5th International Conference on Information Communications and Signal Processing, pp. 73–76. IEEE (2005) 2. Cohen, A., Schlenker, J.-M.: Compactly supported bidimensional wavelet bases with hexagonal symmetry. Const. Appr. 9, 209–236 (1993) 3. Fujinoki, K., Ishimitsu, S.: Triangular biorthogonal wavelets with extended lifting. Int. J. Wav. Multires. Inf. Proc. 11, 1360002 (2013) 4. Han, B.: Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix Anal. Appl. 24, 693–714 (2002) 5. Jiang, Q., Pounds, D.K.: Highly symmetric 3-refinement Bi-frames for surface multiresolution processing. Appl. Num. Math. 118, 1–18 (2017) 6. Krivoshein, A.: Symmetric interpolatory dual wavelet frames. St. Petersburg Math. J. 28, 323– 343 (2017) √ 7. Krivoshein, A.: Hexagonally symmetric orthogonal filters with 3 refinement. Multidim. Syst. Sign. Process. (2020). https://doi.org/10.1007/s11045-020-00735-y

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8. Krivoshein, A., Protasov, V., Skopina, M.: Multivariate Wavelet Frames. Springer, Singapore (2016) 9. Lawton, W., Lee, S.L., Shen, Z.: Stability and orthonormality of multivariate refinable functions. SIAM J. Math. Anal. 28, 999–1014 (1997) 10. Middleton, L., Sivaswamy, J.: Hexagonal Image Processing, a Practical Approach. Springer, London (2005) 11. Novikov, I., Protasov, V., Skopina, M.: Wavelet Theory. American Mathematical Society, Providence (2011)

Chapter 86

Remote Sensing Data Processing for Plant Production Control Vladimir Bure , Olga Mitrofanova , Evgeny Mitrofanov , and Aleksey Petrushin

Abstract Sustainability of agricultural production largely depends on the management of soil heterogeneity and field topography (site-specific management). Precision agriculture provides automation of such management using information technology. Remote sensing data obtained and processed to control the plant production process using the example of aerial photography is considered. Presents the main stages of obtaining and processing aerial photographs in precision agriculture: flight plan, image processing, orthophoto generation, and others. Usage of the remote sensing seems an effective approach to solve a wide set of problems in agriculture related to monitor the agrolandscapes state, assessing, and managing their productivity.

86.1 Introduction Currently, the use of remote sensing (RS) data is becoming more perspective and actual for the effective solution of problems in information support of crop production, precision agriculture, and ecology [1, 4, 8]. Both satellite images and aerial photographs can serve as carriers of such information. Methods of processing and analysis of RS allow rapid detection of emergencies in agricultural fields [9, 12], assess the state of crops and reclamation systems [3], manage crop yields [5, 6, 16], V. Bure (B) · O. Mitrofanova · E. Mitrofanov · A. Petrushin Agrophysical Research Institute, 14, Grazhdanskiy pr., St. Petersburg 195220, Russia e-mail: [email protected] O. Mitrofanova e-mail: [email protected] E. Mitrofanov e-mail: [email protected] A. Petrushin e-mail: [email protected] V. Bure · O. Mitrofanova · E. Mitrofanov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_86

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and contribute to solve other environmental problems. The main RS advantage is the ability to quickly collect data with high information content of their values. Remote sensing is particularly suitable for monitoring large areas used in agriculture. In recent decades, the concept of “sustainability” of agricultural production, which is customarily considered in conjunction with a complex of economic, environmental, and sociological factors, has been actively discussed [13]. Sustainability of agricultural production largely depends on the management of soil heterogeneity and field topography (site-specific management). Precision agriculture provides automation of such management using information technology. In 2000, Hatfield identified three main classes of variation that exist on the agricultural field: natural (soil and topography), random (e.g. weather), and controlled application of agrochemicals [7]. Taking into account all three components, agrotechnological decisions are taken together.

86.2 Methods There are many open sources with processed satellite images, so it is advisable to consider remote sensing data obtaining and processing to control the plant production process using the example of aerial photography. A set of tasks related to the management of agroecosystems and requiring the receipt and preliminary processing of RS data is considered on the basis of experimental agricultural fields located on the territory of the branch of the Agrophysical Research Institute (ARI) in the village of Menkovo, Leningrad Region. Among such tasks, it suffices to single out the main ones: monitoring the crops status [2]; differential application of fertilizers and agrochemicals [14]; yield forecast [11, 15, 17]; calculation of vegetation indices [18]; monitoring of meliorative systems: non-working line detection for drainage systems; determination of overgrowing dimension, of water stagnation, of a drainage system scheme; soil erosion detection; planning of repair work [10]; and others. It is necessary to obtain geo-referenced orthophotomaps in the visible and nearinfrared ranges for information support of the listed problems. In addition, aerophotos must be with high resolution because some tasks have scientific research nature, and there is also a need for a digital terrain model for some tasks. In recent years, Russia has rapidly developed the production of unmanned aerial vehicles (UAV). The market is represented with great varieties for research work. Since 2015, the ARI has been using the Geoscan-401 helicopter-type UAV, which allows for automatic flight and shooting, as well as high-quality images. That is greatly simplifies the task solve. In addition to the copter, this system includes an onboard complex, payload, and ground control station (Fig. 86.1).

86 Remote Sensing Data Processing for Plant Production Control

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Geoscan - 401

UAV

-integrated navigation system (GPS- receiver) -autopilot(automatic flight along a given route, take-off and landing, maintaining a given height and speed, stabilization, automatic shutter release of the camera)

Airborne complex

-сameras (visible and infrared)

Payloads

Ground control station

-rugged laptop with mission planning software installed -digital control and telemetry communication channel -modem for communication UAV

Fig. 86.1 Unmanned aerial system Geoskan-401

86.3 Results and Its Discussion The general scheme for remote sensing data obtaining and processing on an ARI experimental fields can be represented by four main steps. Step 1. Collection of the requests from ARI employees to aerial photographs planning. At this stage, experimental fields for aerial photography are determined; the optimal shooting date is selected (based on the request date, weather forecast, and satellite imagery dates, if necessary). In addition, a knowledge base for each declared field is formed: field number, crop characteristics, experience description, and available archived aerial photographs for this field (digital terrain model, map of heterogeneities, etc.). A flight timetable is being compiled. Step 2. Implementation of flights on schedule. At this stage, the flight team departs to the experimental fields for aerial photography on appointment dates. At the same time, important stage sections are the equipment preparation, the flight plan development, the flight implementation and support using a ground control station, as well as the storage of obtained images. Step 3. Preprocessing aerial photographs. This stage is implemented automatically using special software that comes bundled with an unmanned aerial system. Step 4. Downloading the received and processed data, as well as the knowledge base, into a single interactive information system where all available experimental research information will be displayed for further scientific work. The advantages of such system are not only in the exchanging data between the ARI departments (that ensures multidisciplinary research), but also in its economic potential (licenses for the other institutions and organizations in the future: to store their data or to use the downloaded experimental data). The flight plan is formed by the ground control station using the program Geoscan Planner 2.3. The flight group in the program sets the boundaries of the survey site, the area of which does not exceed the maximum allowable, as well as other aerial

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Fig. 86.2 An example of a developed flight plan for the Geoscan-401 UAV using Geoscan Planner 2.3 program

photography parameters (flight altitude, number of points on the route, etc.). Figure 86.2 shows an example of the developed flight plan for the UAV Geoscan-401. Two digital cameras are used as unmanned aerial complex payload to obtain high-quality aerophotos in the visible and near-infrared ranges: Sony RX1 and Sony A6000. The PhotoScan program automates the calculation of calibration parameters at the photoalignment stage (parameters of the internal and external camera orientations, including nonlinear radial distortions). It is possible to calibrate the camera lens using a calibration image with a chessboard if the alignment results are unstable. Data pre-processing is carried out automatically in the GeoScan Planner and PhotoScan programs. Data for georeferencing is transmitted during the flight by the radio modem, and recorded by the GeoScan Planner program. After that, georeference files are created for further processing in PhotoScan. Further data processing is carried out in PhotoScan and includes the following actions: upload aerophotos to PhotoScan; check photos, delete unwanted images; image alignment; dense point clouds building; three-dimensional polygonal model building; texturing an object; tile model building; digital terrain model building; digital orthophoto generation; and saving results. An orthophoto is constructed on the basis of the obtained image mosaic at the last stage of aerophoto pre-processing. The orthophoto appears as a digitally transformed object image created from overlapping source images. The program automatically detects the so-called singular points (sparse image points, contours, etc.) on the aerophotos. Then the images are stitched together. In this regard, it is advisable to provide the maximum possible overlap of aerial photographs. Figure 86.3 shows an example of a generated orthophoto for the agricultural field in the visible and near-infrared ranges. The use of high-precision orthophotos has shown its significant efficiency in solving a number of plant production control problems (Sect. 86.2) in comparison with traditional agricultural methods. For example, we can build a map of the NDVI (normalized difference vegetation index) distribution (Fig. 86.4) to monitor the plants’ status and the plants’ nitrogen content, as well as to generate a technological map that allows for differentiated fertilization. It should be noted that the ARI studies

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Fig. 86.3 An example of the generated orthophoto for the ARI experimental field in the visible (a) and near-infrared (b) ranges

Fig. 86.4 An example of the generated NDVI map

have shown a significant economic efficiency of the differentiation in comparison with traditional fertilization.

86.4 Conclusion The use of remote sensing seems quite an effective approach to solve a wide range of problems in the interests of agriculture related to monitoring the state of agrolandscapes, assessing, and managing their productivity. At the same time, aerial photography helps to increase the mobility of data acquisition, reduce financial costs and labor costs, speed up the conduct, and increase the level of experimental research, while ensuring high reliability and completeness of the data.

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References 1. Blondlot, A., Gate, P., Poilve, H.: Providing operational nitrogen recommendations to farmers using satellite imagery. In: Stafford, J.V. (ed.) Precision Agriculture ’05, pp. 345–351. Wageningen University Publ., Wageningen, The Netherlands (2005) 2. Bure, V.M., Mitrofanova, O.A.: Analysis of aerial photographs to predict the spatial distribution of ecological data. Contemp. Eng. Sci. 10, 157–163 (2017) 3. Caffey, R.H., Kazmierczak, R.F., Avault, J.W.: Incorporating Multiple Stakeholder Goals into the Development and Use of a Sustainable Index: Consensus Indicators of Aquaculture Sustainability. Department of AgEcon and Agribusiness of Louisiana State University, USA, Staff Paper (2001) 4. Emadi, M., Baghernejad, M., Pakparvar, M., Kowsar, S.A.: An approach for land suitability evaluation using geostatistics, remote sensing, and geographic information system in arid and semiarid ecosystems. Environ. Monit. Assess. 164, 501–511 (2010) 5. Flowers, M., Weisz, R., Heiniger, R.: Remote sensing of winter wheat filler density for early nitrogen application decisions. Agron. J. 93, 783–789 (2001) 6. Graeff, S., Pfenning, J., Claupein, W., Liebig, H.P.: Evaluation of image analysis to determine the N-fertilizer demand of broccoli plants. Adv. Opt. Technol. 1–8 (2008) 7. Hatfield, J.: Precision Agriculture and Environmental Quality: Challenges for Research and Education. National Soil Tilth Laboratory, Ames, Iowa, USA (2000) 8. Liang, S.: Advances in Land Remote Sensing System: Modeling, Inversion and Application. Springer Science (2008) 9. Pan, Z., Huang, J., Wei, Ch., Zhang, H.: Remote sensing of agricultural disasters monitoring: recent advances. Int. J. Remote Sens. Gisciences 6, 4–19 (2017) 10. Petrushin, A.F., Mitrofanov, E.P., Mitrofanova, O.A.: Digital terrain model for monitoring meliorative facilities. Agroecosystems in natural and regulated conditions: from a theoretical model to the practice of precision control, St. Petersburg, pp. 447–451 (2016) 11. Poluektov, R.A., Smolyar, E.I., Terleev, V.V., Topazh, A.G.: Models of the Production Process of Crops. St. Petersburg University, SPb (2006) 12. Sankaran, S., Mishra, A., Ehsani, R., Davis, C.: A review of advanced techniques for detecting plant diseases. Comput. Electron. Agric. 72, 1–13 (2010) 13. Tilman, D., Cassman, K.G., Matson, P.A., Naylor, R., Polasky, S.: Agricultural sustainability and intensive production practices. Nature 418, 671–677 (2002) 14. Voropaev, V., Shpanev, A., Lekomtsev, P., Voropaeva, E., Ilyinskaya, Ya.: Remote means and methods of definition of homogeneous technological eras for precision management of mineral nutrition and phytosanitary condition of agrocenosis. In: 19th International Multidisciplinary Scientific GeoConference SGEM 2019 Conference Proceedings, pp. 609–616. Sofia (2019) 15. Wallach, D., Makowski, D., Jones, J.W.: Working with Dynamic Crop Models: Evaluation, Analysis, Parameterization, and Applications. Elsevier, Amsterdam, Netherland (2006) 16. Yakushev, V.P., Yakushev, V.V.: Prospects for “Smart agriculture” in Russia. Herald Russ. Acad. Sci. 88, 330–340 (2018) 17. Yakushev, V.P., Bure, V.M., Yakushev, V.V.: Stochastic modeling in agriculture. Agrophysics 1, 5–13 (2011) 18. Yakushev, V.P., Kanash, E.V., Rusakov, D.V., Blokhina, S.Yu.: Specific and non-specific changes in optical characteristics of spring wheat leaves under nitrogen and water deficiency. Adv. Anim. Biosci. 8, 229–232 (2017)

Part XII

Artificial Intelligence

Chapter 87

Characteristics of Lexical Spectra of Texts in the Problem of Establishing Authorship Nikolai Moskin, Kirill Kulakov, Alexander Rogov, and Roman Abramov

Abstract The lexical spectrum is a significant characteristic for solving the problem of determining the authorship of texts (for example, when solving the problem of attribution of texts that may belong to F. M. Dostoevsky). However, the application of some data mining methods (for example, decision trees) requires representing the spectrum as a single number that would adequately reflect its structure. We consider the approximation of lexical spectra (at the dictionary level and at the text level) by hyperbolic and exponential curves. Based on the articles from the pre-revolutionary journal “Time” (1861–1863), it is shown that the coefficients of hyperbolic regression approximate the data much better than the coefficients of the exponential curve. In this case, the distance χ 2 was used to determine the difference between the spectra. The research was carried out using the SMALT information system, where automated marking of works was implemented with manual control of specialists-philologists.

87.1 Introduction When solving pattern recognition problems, a number of features can have a rather complicated structure. For example, a certain attribute may be represented by a graphical model, a histogram, and so on. However, when solving the problems of clustering and classification (for example, in task of text attribution [2, 5, 11, 14]), the features should be uniform. This is especially important when using decision trees and random forests [3, 4]. Therefore, it is required to replace a certain characteristic with a single value. For example, how it was done in solving the problem of distinguishing folklore texts from texts stylized as folklore, when the graph-theoretic models N. Moskin (B) · K. Kulakov · A. Rogov · R. Abramov Petrozavodsk State University, Lenin Str., 33, Petrozavodsk, Russia e-mail: [email protected] K. Kulakov e-mail: [email protected] A. Rogov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_87

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Table 87.1 Source texts for analysis Code Name Author 3 4 5 7 22 23

Bezcvetnyya yavleniya Dvoryanin Zhurnalnye interesy Zapiski Talejrana Vopros ob universitetah Zhukovskij i romantizm

33 44 77

Devyatnadcatyj numer Zhurnalnaya zametka Ryad statej o russkoj literature. G. -bov i vopros ob iskusstve Dva lagerya teoretikov. (Po povodu...) Gavanskie chinovniki v ih domashnem bytu... Golos za peterburgskogo Don-Kihota (Po povodu statej G. Teatrina) Durnye priznaki

94 95 96

116 117

Eshche o Peterburgskoj literature

Journal Year

Number of journal

Number of fragments

Dubia Dubia Dubia Dubia Dubia F. M. Dostoevsky Dubia Dubia F. M. Dostoevsky

Time Time Time Time Time Time

1861 1861 1861 1862 1861 1862

8 12 9 2 11 1

10 4 8 8 19 15

Time Time Time

1862 1863 1861

2 1 2

3 9 28

Dubia

Time

1862

2

14

Dubia

Time

1861

2

2

F. M. Time Dostoevsky

1862

10

2

N. N. Strakhov N. N. Strakhov

Time

1862

11

12

Time

1861

6

11

obtained as a result of semantic analysis were presented in the form of a histogram, which was further approximated by a hyperbolic curve y = ax + b (i.e. represented by two coefficients a and b) [8]. We will show the possibility of using a similar technique in the analysis of lexical spectra of texts [12, 15]. Table 87.1 presents a list of 14 analyzed texts. These are articles from the magazines “Time”, which were edited by F. M. Dostoevsky (1861–1863). Nine of them as an author have the “Dubia” mark (lat. “doubtful”). It means that these are works supposedly attributed to one or another author. Regarding the rest of the works, it can be said with certainty that the author is either F. M. Dostoevsky or N. N. Strakhov. The works were divided into fragments of 500 words (if a text fragment has less than 500 words, then it is excluded). The total number of fragments is 149.

87.2 The Lexical Spectra of the Texts In [7], G. Kjetsaa notes that it is possible to research the stylistic handwriting of the writer also from the point of view of the lexical spectrum of a text, i.e. in terms

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of the distribution of word frequencies in the text. In particular, he applied two characteristics: 1. Lexical spectrum at the dictionary level ( f ); 2. Lexical spectrum at the text level (m f ). The lexical spectrum is set based on the frequency dictionary of the text. It is necessary for comparison to extract data from samples of the same length (500 word forms). All word forms were divided into separate groups based on the frequency of occurrence m (1, 2, 3, . . . , 10 or more times). Then the program determined their number in each group f and text coverage m f (here the number of word forms in each group is multiplied by the frequency of occurrence of words from this group). In the first case, the frequency distribution was obtained at the dictionary level, and in the second case at the text level. These characteristics can be further analyzed using the Kolmogorov–Smirnov criterion to verify the uniformity of the text [10]. As an example, Table 87.2 shows the characteristics of f and m f for two texts: “Ryad statej o russkoj literature. G.-bov i vopros ob iskusstve” (F. M. Dostoevsky, “Time”, 1861, No.2) and “Durnye priznaki” (N. N. Strakhov, “Time”, 1862, No.11). Figure 87.1 shows a comparative diagram of the lexical spectrum at the dictionary level ( f ) for texts 77 and 116 (first fragment, 500 words). It is clearly noticeable that as the parameter m increases, the value of the attribute f decreases. In this case, we can conduct an analogy with the famous Zipf’s law [16] used in philological sciences: if all words of a sufficiently large text are ordered by decreasing frequency of their use, then the frequency of the n word in such a list will be approximately inversely proportional to its ordinal number n (the rank of the word). Note that the law is mathematically described by the Pareto distribution. The construction of lexical spectra requires preliminary processing of texts. To correctly compare lexical spectra, texts should have the same length. This is achieved by dividing texts into blocks of the same length and building a lexical spectrum for

Table 87.2 A fragment of the table with the values of the lexical spectra m Text 77 ( f ) Text 116 ( f ) Text 77 (m f ) 1 2 3 4 5 6 7 8 9 10+ Sum

167 48 16 4 7 4 3 3 2 3 257

144 56 12 10 6 7 3 0 0 5 243

249 58 10 0 4 4 2 0 1 2 330

Text 116 (m f ) 202 69 11 10 3 2 2 0 1 2 302

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Fig. 87.1 The lexical spectrum at the dictionary level ( f ) for texts 77 and 116 (first fragment, 500 words)

each block. In addition, the construction of the lexical spectrum at the dictionary level is based on the initial form of the word. There are algorithms for automatically determining the initial form of a word based on the structure of the word or using the database of words; however, under the conditions of using text fragments of pre-revolutionary journals, this approach is not applicable. Firstly, pre-revolutionary graphics are used in the spelling of words, and, secondly, a word can be written in different variations (with the addition, modification, or deletion of letters and signs). Thus, a preliminary analysis of text fragments by philologists is required. Within the framework of the SMALT information system (“Statistical Methods of the Analysis of Literary Texts”) that we are developing [9], we are analyzing both textual works by philologists and analyzing textual works. Text works go through three steps: import, verification, and analysis. During the import, the text product is divided into paragraphs, sentences, and words. Each word is compared with one of the existing morphological analyzes. If it is impossible to select a morphological analysis for a word, then it is marked. In the second step, the text is checked by a philologist. During the test, a specialist can divide a word into two independent words or combine two words into one; change the selected morphological analysis (including the initial form of the word) or create a new one. The results are stored in a database and used in subsequent analyses. Analysis of text works is performed by a group or individually. In a group analysis, the researcher selects one or more textual works and the block size. As a result of group analysis, an Excel table is formed containing the constructed lexical spectra for each whole block. Short texts (the number of words in the text is less than the block size) and incomplete blocks are ignored. When analyzing a single text product, the researcher can choose the block size and the indent from the beginning of the text to build lexical spectra of the block.

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87.3 Approximation of the Spectra by Means of Regression Curves However, the use of multidimensional methods (for example, decision trees) to solve the problem of text attribution requires a diagram to be represented by a single characteristic. This dependence can be approximated: 1. Hyperbolic curve y = ax + b (how it is done in [8]); 2. Exponential curve y = c · e−λ·x . Parameters a, b, λ, and c can be found using the least squares method, having previously normalized the original array. Table 87.3 shows examples of calculating these characteristics for the lexical spectrum at the dictionary level for two texts. We will analyze which of these characteristics can be used to represent lexical spectra in the text attribution problem. To test the hypothesis, we found the Pearson correlation coefficient between the modulus of the difference between the 2 regression coefficients di(1) j = |ai − a j | and the distance χ between the diagrams (lexical spectrum at the dictionary level). Denote z i j as the value of the spectrum with the number j for text fragment i (i = 1, 2, . . . , 149), if j ≤ 9, and the sum of the remaining values, if j = 10. The distance was calculated using the following formula: 10 2  z 2jk z ik di(2) = n ( ( + ) − 1), (87.1) ij j z (z + z jk ) z j (z ik + z jk ) k=1 i ik where n i j =

10 

(z ik + z jk ), z i =

k=1

10 

z ik (i, j = 1, . . . , 149) [1].

k=1

As the calculations showed (see Table 87.4), the Pearson correlation coefficient was r ≈ 0, 681808. According to the scale of Chedoke such a relationship Table 87.3 Examples of calculation of the characteristics (first fragment, 500 words) Text code a b λ c Text 77 Text 116

70,55209 65,79613

−10,6645 −9,27148

0,413414 1,790008

29,27691 1474,175

Table 87.4 Pearson correlation coefficients (500 words in text fragments) Coefficients Lexical spectrum at the dictionary Lexical spectrum at the text level ( f ) level (m f ) a b λ c

0,681808 0,681808 0,206101 0,18129

0,757219 0,757219 0,168497 0,082559

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believes notable. If we similarly consider the second coefficient b, then the corre(3) lation between di(2) j (found by the formula (1)) and di j = |bi − b j | also is equal to 0,681808. Another type of relationship is shown by the exponential curve. Correlations for both coefficients λ and c will be weak (0,206101 and 0,18129, respectively). It means that these characteristics do not reflect the structure of lexical spectra. If we consider another linguistic-statistical parameter m f (lexical spectrum at the text level), the contrast between the coefficients of the two types of regression is even more noticeable. Using the Student’s table, the hypothesis of the significance of the correlation coefficients was tested. It turned out that all the values in Table 87.4 showed statistical significance at the level of 0.05.

87.4 Analysis of Hyperbolic Regression Coefficients Let us consider in more detail the coefficients a and b. Table 87.5 shows the Pearson correlation coefficients for text fragments that consist of 750 and 1000 words. There are 72 and 95 such fragments in total, respectively. The values for a and b showed statistical significance at the level of 0.05. According to the scale of Chedoke, the correlation for text fragments consisting of 750 words is high (for text fragments consisting of 1000 words is very high). It is also clearly seen that the correlation for m f in all three cases (500, 750, 1000 words) is slightly higher for f . This suggests that characteristics a and b better approximate the lexical spectrum at the text level than lexical spectrum at the dictionary level. Using the analysis of 133 text fragments (73 belong to F.M. Dostoevsky and 60 to other authors), the characteristics a and b were calculated. Based on them a decision tree was constructed for classifying texts of F. M. Dostoevsky. It is shown in Fig. 87.2. The accuracy of classification at level 5 was quite low, equal to 0.68. It can be concluded that this feature is not very significant for solving this classification problem. However, it can be used to solve the classification problem by the random forest method along with other features. For example, it may be the frequency of occurrence of certain parts of speech, n-grams, average sentence length, and so on. Among them there are also no strongly significant features.

Table 87.5 Pearson correlation coefficients (750 and 1000 words in text fragments) Characteristics a b Lexical spectrum at the dictionary level f (750 words) Lexical spectrum at the text level m f (750 words) Lexical spectrum at the dictionary level f (1000 words) Lexical spectrum at the text level m f (1000 words)

0,773316 0,84108 0,908395 0,940894

0,773316 0,84108 0,908395 0,940894

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Fig. 87.2 Decision tree

87.5 Conclusion Using the automated procedure implemented in the SMALT information system, lexical spectra were constructed for text fragments from the pre-revolutionary magazine “Time” (1861–1863), articles of which still do not have been established by the author. The research has shown that the coefficients of hyperbolic regression are much better approximate of the data obtained on the basis of lexical spectra (at the dictionary level and at the text level) than the coefficients of the exponential curve. The coefficients a and b of y = ax + b curve can be used in the text attribution problem along with other features (the frequency of occurrence of certain parts of speech, n-grams, average sentence length, and so on). For example, one of these

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problems is determining the authorship of an article “Poems by A. S. Khomyakov” (A. Grigoriev or F. M. Dostoevsky) [6]. Also this approach can be used to solve the problem of distinguishing folklore texts from texts stylized as folklore [13]. The obtained results were presented for further consideration to the specialists of the Department of Russian Language and the Department of Classic Philology, Russian Literature and Journalism (Petrozavodsk State University). Acknowledgements This work was supported by the Russian Foundation for Basic Research, project no. 18-012-90026.

References 1. Ayvazyan, S.A., Enyukov, I.S., Meshalkin, L.D.: Applied statistics: fundamentals of modeling and primary data processing. Reference book. Moscow, Finance and statistics (1983) 2. Batura, T.V.: Formal methods for determining the authorship of texts. Novosibirsk State University Bulletin. Series “Information Technology”. Novosibirsk 10(4), 81–94 (2012) 3. Breiman, L.: Random forests. Mach. Learn. 45(1), 5–32 (2001) 4. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth, Belmont (1984) 5. Calle-Martin, J., Miranda-Garcia, A.: Stylometry and authorship attribution: introduction to the special issue. Engl. Stud. 93(3), 251–258 (2012) 6. Gurova, E.I.: Methods of authorship attribution in contemporary national philology. New Philol. Bull. 3(38), 29–44 (2016) 7. Kjetsaa, G.: Attributed to Dostoevsky: the problem of attributing to Dostoevsky anonymous articles in time and epoch. Solum Forlag A. S, Oslo (1986) 8. Moskin, N.D., Rogov, A.A.: Characteristics of distribution of vertex degrees of graph-theoretic models of texts. In: Extended Abstracts of the X International Petrozavodsk Conference “Probabilistic Methods in Discrete Mathematics”, pp. 101–103. Petrozavodsk: Karelian Research Centre, Russian Academy of Sciences (2019) 9. Rogov, A.A., Kulakov, K.A., Moskin, N.D.: Software support in solving the problem of text attribution. Softw. Eng. 10(5), 234–240 (2019) 10. Rogov, A.A., Sedov, A.V., Sidorov, Y.V., Surovtsova, T.G.: Mathematical Methods for Text Attribution. PetrSU Publ, Petrozavodsk (2014) 11. Romanov, A.S.: Methodology and software complex for identifying the author of an unknown text. Tomsk (2010) 12. Sidorov, Y.V.: Mathematical and Information Support of Literary Text Processing Methods Based on Formal Grammatical Parameters. PetrSU Publ, Petrozavodsk (2002) 13. Shchegoleva, L.V., Lebedev, A.A., Moskin, N.D.: Methods of data mining in the task of distinguishing between folklore and author’s texts. Voprosy Jazykoznanija 2, 61–74 (2020) 14. Stamatatos, E.: A survey of modern authorship attribution methods. J. Am. Soc. Inf. Sci. Technol. 60(3), 538–556 (2009) 15. Surovtsova, T.G.: Multivariate Quantitative Analysis and Classification of Texts Based on Linguistic and Statistical Characteristics. PetrSU Publ, Petrozavodsk (2008) 16. Zipf, G.K.: Human Behavior and the Principle of Least Effort. Addison-Wesley Press (1949)

Chapter 88

Topic Models with Neural Variational Inference for Discussion Analysis in Social Networks Nikita Tarasov, Ivan Blekanov, and Alexey Maksimov

Abstract Topic models and their extensions are widely used to represent documents using generative probabilistic models. Many models were developed and applied to analyze the content of news articles, scientific papers, and other sources of textual data. However, these classic algorithms have severe problems when dealing with short texts, processing multilingual, and noisy data. These problems are most prevalent in the task of analyzing corpora collected from social networks. Authors consider different topic models for users’ discussion analysis in social networks. Authors evaluate topic modeling results of these approaches using a modified NPMI measure. The standard news20 dataset was used as a test collection as well as two user discussions on Twitter in both Russian and English. The experiment showed that a combination of ETM and pre-trained embeddings based on ELMO vastly outperforms traditional topic models such as BTM while analyzing short and unbalanced texts such as Twitter messages.

88.1 Introduction Topic models and their extensions have been applied to many fields, such as marketing, sociology, political science, and the digital humanities [1]. They analyze documents to learn meaningful patterns of words which can be useful in a wide array of natural language processing tasks such as information retrieval, hot words extraction, statistical inference, multilingual modeling, and linguistic understanding [2]. One of the most interesting applications of topic models is the task of identifying and analyzing user discussions in social networks. In this application, a topic model, given a corpus of user messages, can output a set of distributions (topics over documents and words over topics) [3]. This can be helpful while identifying key points of discussions, performing summarization, and clusterization both as independent models and as parts of a larger system in conjunction with other models such as N. Tarasov · I. Blekanov (B) · A. Maksimov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_88

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sentiment analysis [4], text generation models for summarization, and others. However, traditional topic models have some downsides while analyzing user discussions. The issues include poor short text processing, ineffective noisy data analysis, and inability to work with multilingual data. Traditional models can be used to solve these issues using specific auxiliary datasets such as graphs of co-occurrence networks [5] and different translation techniques [6]. However, these solutions introduce new variables that cause the solutions to require additional computing power and reduce the algorithm precision due to the translation process. Taking these complications into account, we are interested in comparing these traditional models with emerging solutions using neural networks for variational inference.

88.2 Models There are multiple ways to address common issues of traditional topic models ranging from simple pre-processing techniques to fully developed methods designed specifically to be used for short and unbalanced texts.

88.2.1 Biterm Topic Model and Word Network Topic Model Biterm topic model is an algorithm which applies the inference process very similar to that of LDA. The key difference of these algorithms is the direct usage of word co-occurrence patterns in the BTM which makes the original data much more diverse and produces a much better result when applied to short texts commonly found in social networks [7]. Previous work with this model has shown its effectiveness compared to the baseline LDA model and another modification of its inference process—Word Network Topic Model. WNTM employs another way of altering the original data by creating a fully connected word network moving away from document-word space to wordword [8]. Previous work [9] has shown that results obtained using BTM are more interpretable in terms of NPMI score and therefore it was chosen as a baseline model for analysis in the current paper.

88.2.2 Neural Variational Document Model The standard approach of LDA model and its various improvements (such as BTM) is to model the underlying set of topics using mean-field variational inference to compute the variational parameters describing the underlying marginal likelihood for a document in collection [10]. The result of this algorithm is a set of

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distributions: words over topics and topics over the documents in the original dataset. On the other hand, the NVDM model introduces a neural network to parametrize the multinomial topic distribution. This results in a model which has the following generative process: θd ∼ G(μ0 , σ02 ), for d ∈ D z n ∼ Multi(θd ), for n ∈ [1, Nd ]

(88.1) (88.2)

wn ∼ Multi(βzn ), for n ∈ [1, Nd ].

(88.3)

It can be seen that the main point of difference of this architecture from a classic LDA approach is the choice of σd (distribution of topics in document d). LDA uses Dirichlet distribution with a symmetric sparse parameter alpha to model this distribution, while NVDM model implements a function G(μ0 , σ02 ) which is composed of a neural network σ = G(x) conditioned on a draw from an isotropic Gaussian x ∼ N (μ0 , σ02 ). There are three proposed variants on the neural network architecture: Gaussian Softmax, Gaussian Stick Breaking, and Recurrent Stick Breaking. The latter is applied in this case to effectively model textual data. The Recurrent Stick Breaking Construction [11] uses the advantages of RNNs to effectively model sequences of textual data. This architecture has advantages over the traditional types of neural networks such as the ability to process inputs of any length (although in this case the maximum length is still bounded by the parametric model’s capacity), the ability to use information from many steps back, and robustness in terms of the memory consumption based on an input length.

88.2.3 Embedded Topic Model ETM differs from the classic approaches like LDA and its newer more robust implementations and modifications such as BTM and NVDM in that it uses the embeddings to represent both words and topics. This allows the model to learn topic and word similarities without using any external data by directly modeling their representations. Prior to the discovery of such models, external graph data on word similarities was used to obtain this information. ETM uses CBOW a type of language model which given the list of surrounding words can output the missing word in a sequence based on their likelihood [12]. The resulting model has the following architecture:

wdn

σd ∼ L N (0, I )

(88.4)

z dn ∼ Cat (σd ) ∼ so f tmax(ρ τ αzdn ),

(88.5) (88.6)

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where L N denotes the logistic-normal distribution which can be obtained by applying the Softmax filter to the Gaussian distribution and thus transforming the variable into a simplex. There are two main use cases of this model: it can either learn the embeddings as part of its inference process simultaneously obtaining the embeddings of topics, or use an external, previously fitted embeddings, to obtain the result given an already existing embedding space. The latter allows the usage of words not present in the training data and can act in the same way as the graphs of word similarities. It improves the model’s lexical capacity by modeling synonyms and words closely related by meaning which is especially important in the case of social networks because of specificity and sparsity of the language used. Another advantage of this particular approach is the robustness in terms of stop words in the vocabulary. Other methods such as LDA require a thorough pre-processing and in the process introduce unwanted variables such as the lower bound cutoff for vocabulary. Meanwhile, embeddings bypass this issue by recognizing the place of the stopwords in embedding spaces and putting all of them in a separate topic.

88.2.4 FastText and ELMO The original ETM paper provides another model based on topic embeddings— a model utilizing pre-trained word embeddings. There are a variety of ways to learn these vector representations from classic context-independent methods such as word2vec, Glove, and FastText to context-aware state-of-the-art models such as XLnet, BERT, and ELMO. FastText is a model developed by Facebook which learns word embeddings by utilizing subword information. The model not only increases the performance compared to the baseline approaches such as word2vec, but also accommodates for the words not present in the training data. Model’s authors provide a variety of pre-trained models for different languages which helps to alleviate the data sparsity problem common for texts obtained from social networks [13]. ELMo model is a three-layer neural network. The first layer is a character-level CNN. It is context independent, so each word always gets the same embedding, regardless of its context. The next part is a two-layer biLMs with character convolutions. A biLM layer consists of two concatenated LSTMs, the first of which predicts the word based on the sequence of previous words and the second LSTM predicts the word using the information on the next words in that sequence [14]. In this case, it’s impossible to use context-aware models such as ELMO to their full potential due to the structural limitations of ETM in that it requires uncontextual word embeddings. However, due to the sparsity of the data used in this case averaging over word embeddings in different context can provide a meaningful improvement over standard models such as word2vec, FastText, and other context-independent methods.

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88.3 Experiment Described models are tested on two distinct datasets using different coherence measures. The results are plotted to find the best performing and most consistent topic model. Prevalent topics for this model are shown as a combination of words with highest probabilities in the topic-word distribution vector.

88.3.1 Corpora Two different datasets are used to compare the results: two directly tied to the main task of this paper (analysis of user discussions in social networks) and another is the standard news20 dataset. The first dataset consists of user massages from Twitter social network. Data was collected based on the hashtags associated with the events of Ferguson unrest (2014). The corpus consists of 193,812 user messages with additional metainformation such as user names and timestamps. Data was collected during a following period of time: 22.08.2014–31.08.2014. The second dataset was collected from Twitter social network (time period: 01.10.2013–31.10.2013) and consists of 10 215 user messages by the riots Byrulevo in Russia. News20 was chosen as a benchmark dataset. It is a standard test collection used in a vast array of different NLP tasks. News20 is a collection of newsgroup posts which consists of 19 848 documents and contains 16 532 unique tokens (after preprocessing). Standard pre-processing procedures are employed to clean up the initial data such as removal of stop words and words with more than 80% document frequency or less than 10 total frequency.

88.3.2 Metrics Topic diversity is a measure of the unique word counts in the learned topics that is calculated by taking the percentage of unique words in the top 25 words of all topics. Diversity close to 0 indicates redundant topics, while diversity close to 1 indicates more varied topics. Normalized pointwise mutual information (NPMI) is another metric utilizing word co-occurrence patterns and can obtained using the following formula: NPMI (wi , w j ) =

N −1 log  j

P(wi ,w j ) P(wi )P(w j )

− log P(wi , w j )

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Coherence score and diversity were calculated as their combination not only shows general quality of topics but also displays variety between words. Their normalized product plotted for all of the present corpora and varying number of topics. The results are present in Figs. 88.1 and 88.2. Tables 88.1 and 88.2 contain the most popular topics for a model with the best coherence score and consistent diversity (above 0.6 in all cases, 0.9 if the number of topics is low). It can be seen that the difference between models is especially big for the Twitter data where ETM in combination with different embedding techniques greatly outperforms other models mainly due to the usage of pre-trained multilingual models.

Fig. 88.1 Resulting model quality for News20 dataset

Fig. 88.2 Score for Twitter discussions data Table 88.1 Popular topics obtained from ETM + ELMO model for News20 dataset Topic 1 Topic 2 Topic 3 Topic 4 Topic 5 Game Team Play Win Hockey

Article Writes Posting News Reply

People Government State Gun Control

Windows Drive System Software Dos

God People Jesus Christian World

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Table 88.2 Popular topics obtained from ETM + ELMO model for Twitter discussions data Topic 1 Topic 2 Topic 3 Topic 4 Topic 5 People Riot Mayhem Justice Movement

News Broadcast Support Community Media

Police Support Authorities Arrest Death

Movement Support Speak Join Change

Black White Racism Jail Justice

88.4 Conclusion The experiment shows that a combination of embedded topic model and pre-trained embeddings obtained using ELMO vastly outperforms traditional topic models, for example, BTM, in the task of analyzing short and unbalanced texts such as user messages on Twitter. Even if used without any customization it learns the most diverse and coherent topics. It is worth mentioning that word embeddings can be reused to optimize a software for user discussions analysis in social networks. In such software topic, modeling is just one part of the whole system. Future work mainly involves implementation of the model that can take full advantage of context-aware embeddings from models such as BERT and sentence-level encoders such as Universal Sentence Encoder. Acknowledgements This work was supported by Russian Science Foundation [grant number 1618-10125-P].

References 1. Boyd-Graber, J., Hu, Y., Mimno, D.: Applications of topic models. Found. Trends Inf. Retr. 143–296 (2017) 2. Zhai, C.: Probabilistic topic models for text data retrieval and analysis. In: Proceedings of the 40th International ACM SIGIR Conference on Research and Development in Information Retrieval, pp. 1399–1401 (2017) 3. Hofmann, T.: Learning the similarity of documents: an information-geometric approach to document retrieval and categorization. In: Advances in Neural Information Processing Systems, pp. 914–920 (2000) 4. Blekanov, I., Kukarkin, M., Maksimov, A., Bodrunova, S.: Sentiment analysis for ad hoc discussions using multilingual knowledge-based approach. In: Proceedings of the 3rd International Conference on Applications in Information Technology, pp. 117–121 (2018) 5. Petterson, J., Buntine, W., Narayanamurthy, S.M., Caetano, T.S., Smola, A.J.: Word features for latent dirichlet allocation. In: Advances in Neural Information Processing Systems, pp. 1921–1929 (2010) 6. Vuli, I., De Smet, W., Tang, J., Moens, M.F.: Probabilistic topic modeling in multilingual settings: an overview of its methodology and applications. Inf. Process. Manag. 111–147 (2015)

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7. Yan, X., Guo, J., Lan, Y., Cheng, X.: A biterm topic model for short texts. In: Proceedings of the 22nd International Conference on World Wide Web, pp. 1445–1456 (2013) 8. Zuo, Y., Zhao, J., Xu, K.: Word network topic model: a simple but general solution for short and imbalanced texts. Knowl. Inf. Syst. 379–398 (2016) 9. Blekanov, I., Tarasov, N., Maksimov, A.: Topic modeling of conflict ad hoc discussions in social networks. In: Proceedings of the 3rd International Conference on Applications in Information Technology, pp. 122–126 (2018) 10. Miao, Y., Grefenstette, E., Blunsom, P.: Discovering discrete latent topics with neural variational inference. In: Proceedings of the 34th International Conference on Machine Learning, vol. 70, pp. 2410–2419 (2017) 11. Sethuraman, J.: A constructive definition of Dirichlet priors. Stat. Sin. 639–650 (1994) 12. Dieng, A.B., Ruiz, F.J., Blei, D.M.: Topic modeling in embedding spaces (2019). arXiv:1907.04907 13. Bojanowski, P., Grave, E., Joulin, A., Mikolov, T.: Enriching word vectors with subword information. Trans. Assoc. Comput. Linguist. 5, 135–146 (2017) 14. Peters, M.E., Neumann, M., Iyyer, M., Gardner, M., Clark, C., Lee, K., Zettlemoyer, L.: Deep contextualized word representations (2018). arXiv:1802.05365

Chapter 89

On the Possibility of Using Neural Networks for the Specific Problems of Meteorological Forecasting Elena Stankova and Irina Tokareva

Abstract The paper explores the possibility of forecasting such dangerous meteorological phenomena as a thunderstorm by applying a neural network to the output data of a hydrodynamic model that simulates dynamic and microphysical processes in convective clouds. The results show that the use of the proposed approach allows us to achieve a forecast accuracy of 91.6%. This accuracy was obtained using one of the perceptron complexes.

89.1 Introduction Throughout human history, meteorological processes prediction has always been a complex task mainly because the Earth’s atmosphere system is very complex and dynamic. Weather forecasts are calculated based on meteorological data collected by a network of weather stations, radiosondes, radars, and satellites around the world. Data is sent to meteorological centers, where it is entered into forecast models for atmospheric conditions calculation. Such models are based on physical laws and work according to extremely complex algorithms. Precipitation is determined by the physical processes occurring in the cloud, namely, the physics of the interaction of water droplets, ice particles, and water vapor. Convective clouds are very variable due to the large vertical speeds within the cloud and its environs. It is also difficult to conduct control experiments involving them. All this leads to the fact that the development of the cloud is usually analyzed using computer simulation, which allows us to do this without resorting to expensive field experiments. E. Stankova (B) · I. Tokareva St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected]; [email protected] I. Tokareva e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_89

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As a result of computer simulation of the cloud, we get a dataset that can be further used for forecasting various dangerous convective phenomena such as thunderstorm, hail, and heavy rain. The use of machine learning methods allows us to automate the forecasting process, which greatly facilitates data analysis. These methods conduct a series of computational experiments with the aim of analyzing, interpreting, and comparing the simulation results with the given behavior of the object under study and, if necessary, subsequently refining their input parameters. The paper [4] uses a generative adversarial network to predict typhoon trajectories. The neural network generates an image showing the future location of the typhoon center and cloud structure using satellite images as an input. In [1], a multilayer perceptron is used to predict changes in tropical cyclone intensity in the northwestern Pacific Ocean. In this paper, we continue the studies described in [5, 6] and analyze the possibility of the use of neural networks for dangerous convective phenomena forecasting by the example of a thunderstorm forecasting.

89.2 Initial Data Research using machine learning methods is based on data; therefore, in order to obtain the best results, it is necessary to use reliable sources of information to obtain data and form their correct structure. In this work, the data were obtained using the following algorithm: 1. We receive data on the date and place of meteorological phenomena occurrence. 2. We select the data corresponding to the presence of a thunderstorm or the absence of any meteorological phenomena. 3. We obtain data from atmosphere radio sounding for the certain date and place. 4. We convert the radio sounding data to the model input data format. 5. Using the hydrodynamic model, we obtain the integral and spectral characteristics of the cloud. 6. We determine the height and time corresponding to the maximum development and maximum water content of the cloud. The cloud parameters corresponding to these height and time will be used for the thunderstorm forecasting. Formed dataset contains 416 records, where 220 samples correspond to the presence of a thunderstorm, and 196 samples to its absence. This data was divided into training and test datasets. The training one contains 333 samples and the test one contains 83. Due to the small amount of data we decided to use test dataset for validation. We also created labels for each sample in the dataset. Since there are only two cases, the presence and absence of phenomenon, we could have created one label per sample. But we decided to use two labels per sample, one for each case, mainly because we will need to divide the output variables of the neural network at some point.

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89.3 Data Preprocessing Neural networks, like all machine learning algorithms, depend significantly on the quality of the source data. Therefore, before proceeding to the construction of a neural network, we will need to prepare the data. First, we normalize the data using the Standard Scaler method from the Scikitlearn library, which converts the data to the standard normal distribution. Then we select the most significant features. To do this, we use the recursive feature elimination method from the Scikit-learn library with Random Forest algorithm as an estimator. The method is as follows. The estimator is firstly trained on the initial set of features, then the least important feature is pruned and the procedure is recursively repeated with smaller and smaller set. Figure 89.1 shows the resulting graph of the prediction accuracy versus the number of features used. As can be seen from the figure, maximum accuracy is achieved when using eight features. Their names and their importance are shown in Fig. 89.2. Thus, we will use the following features:

Fig. 89.1 Graph of prediction accuracy versus the number of features involved

Fig. 89.2 Selected features and their importance

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vapor, aerosol, relative humidity, density, delta temperature, pressure, velocity, and temperature.

89.4 Multilayer Perceptron Since the multilayer perceptron is a classical structure for solving problems similar to that considered in this work, we will use it for its solution. The network structure that gives the highest accuracy value on the test dataset was found experimentally and is shown in Fig. 89.3. The test accuracy of the trained network is 89.1%. The article [2] mentions that the ratio of the volume of the training dataset and the number of trainable network parameters is one of the factors that affects the modeling ability of the perceptron. If this ratio is close to 1, the perceptron will simply remember the training set, and if it is too large, the network will average the data without taking the details into account. In this regard, in most cases, it is recommended that this ratio falls in the range from 2 to 5. In our case, this ratio is 333 = 2, 7, 123 which falls into this range. However, our training dataset is small and the use of algorithms based on neural networks may be ineffective with small amounts of

Fig. 89.3 Multilayer perceptron structure

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experimental data [3]. So we decided to use one of the methods that can help to increase the efficiency of our neural network.

89.5 Perceptron Complex The method is described in [2]. It consists in dividing the set of input and output variables into several perceptrons with a simpler structure and then combining them into a single perceptron complex. Figure 89.4 shows the general structure of such a complex. The perceptron complex training algorithm is as follows [3]: 1. For each first-level perceptron: a. Given the input and output variables of the current perceptron, we construct the training and test datasets for it based on the initial data. b. Perceptron training is executed. c. For all samples of training and test datasets, the values of the perceptron outputs are calculated and stored.

Fig. 89.4 General structure of the perceptron complex

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2. For the resulting perceptron: a. Given the input and output variables of the perceptron, we construct the training and test datasets for it based on the initial data and the calculated output values of the first layer perceptrons. b. Perceptron training is executed. In this paper, we considered two variants of the perceptron complexes, the structures of which are shown in Figs. 89.5 and 89.6. The input and output parameters of single perceptrons were chosen experimentally. The test accuracy for the first perceptron complex is 90.0%, for the second one— 91.6%.

Fig. 89.5 First structure of perceptron complex that is used in this paper

Fig. 89.6 Second structure of perceptron complex that is used in this paper

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89.6 Conclusion The work analyzed the possibility of using neural networks to build forecasts of dangerous convective phenomena by the example of a thunderstorm. The initial dataset was obtained using numerical modeling of a convective cloud. Using machine learning methods at the stage of data analysis and processing of features, the most significant features were identified. The forecast was made using three different neural networks, the structures of which are shown in Figs. 89.3, 89.5, and 89.6. The best result was obtained using the second perceptron complex. Forecast accuracy in this case was 91.6%.

References 1. Baik, J.J., Paek, J.S.: A neural network model for predicting typhoon intensity. J. Meteor. Soc. Japan. (2000). https://doi.org/10.2151/jmsj1965.78.6_857 2. Dudarov, S.P., Diev, A.N.: Neural network modeling based on perceptron complexes with small training data sets. Math. Methods Eng. Technol. 26, 114–116 (2013). (in Russian) 3. Dudarov, S.P., Diev, A.N., Fedosova, N.A., Koltsova, E.A.: Simulation of properties of composite materials reinforced by carbonnanotubes using perceptron complexes. Comput. Res. Model. 7(2), 253–262 (2017) (in Russian) 4. Ruttgers, M., Lee, S., Jeon, S., Jeon, S., You, D.: Prediction of a typhoon track using a generative adversarial network and satellite images. Sci. Rep. 9 (2019). https://doi.org/10.1038/s41598019-42339-y 5. Stankova, E.N., Grechko, I.A., Kachalkina, Y.N., Khvatkov, E.V.: Hybrid approach combining model-based method with the technology of machine learning for forecasting of dangerous weather phenomena. In: Gervasi, O., et al. (eds.) ICCSA 2017, Part V, LNCS, vol. 10408, pp. 495–504 (2017) 6. Stankova, E.N., Balakshiy, A.V., Petrov, D.A., Korkhov, V.V.: OLAP technology and machine learning as the tools for validation of the numerical models of convective clouds. Int. J. Bus. Intell. Data Min. 14(1/2), 254–266 (2019)

Chapter 90

Multilingual Sentiment Analysis for User Discussions on Social Networks: An Approach Based on a Modified SVM Algorithm Mikhail Kukarkin and Ivan Blekanov Abstract Sentiment analysis of users’ messages on social networks is a compelling task in terms of both academic research and application of the research results to real-world business analytics. One of the key problems of sentiment detection in realworld discussions on social media is their mixed languages and, thus, the demand for multilingual instruments. In this paper, we, first, shortly describe the existing research on multilingual sentiment analysis. We suggest our own approach of data processing for classification and an enhanced sentiment detection method based on the SVM algorithm. To evaluate these methods, we apply them to marked-up noisy data from ad hoc discussions on social networks (in English, Russian, French, and German language) and clean data from Twitter US Airline Sentiment collection and assess the quality of sentiment detection by standard metrics. We show that our method reaches highest values of F1-score (0.78 and 0.94) for noisy and clean datasets.

90.1 Introduction The amount of textual information on the Internet is growing every day (according to Netcraft since 1995 [1], an exponential increase has been observed) and its analysis becomes more difficult. Text is the prevailing data format on the Web, as it is easy to generate and publish. With the development of Internet technologies, it became necessary to extract useful information from the text both for academic purposes and in order to extract material benefits [2, 3] However, extracting information from a huge amount of data in the proper form is very difficult due to the fact that the text can have a completely different structure. It is extremely difficult to collect data in a single generalized format [4, 5]. Manual data collection immediately seems impractical since millions of messages per second are being published in social networks nowadays. Therefore, in the first place, the problem of studying the automatic organization and categorization of information is solved [4]. M. Kukarkin · I. Blekanov (B) Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_90

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There are different ways to use the collected text data. For example, studying the data of social networks (Twitter, Facebook, YouTube, etc.) or various information sites (digg.com, medium.com, etc.), you can find out the relationship between users [6], their attitudes to events or products [7], their political disposition [8], and so on, which in the end can be used to optimize the work of specific users with sites and services. However, over the past few years, due to the high activity of users on social networks (for example, the number of active users per month on Facebook is more than 2 billion, on Twitter—more than 300 million) [9] and due to the generation of large volumes of different types of content the most interesting task for researchers of various fields of knowledge is analysis of user discussions in social networks [10]. Usually, the task of analyzing user discussions comes down to solving a whole chain of separate technical and scientific problems. It primarily includes data collection, which is an interesting task for researchers in terms of creating and optimizing crawling algorithms [11, 12]. Recently, it has become more and more difficult to create a universal collection algorithm, as companies tighten restrictions on possible attempts to automatically collect user data. The next task after collection may be the structuring and storage of the received raw data. It is necessary to be able to distinguish information by its structure for subsequent correct and simpler processing. Along with structuring, a separate place in the analysis of user discussions is occupied by the thematic classification of information, the so-called topic modeling [13]. This task is the task of classification or clustering from the field of data analysis. One of the stages along with topic modeling is the analysis of the tonality of the message or the author (his views and opinions). Sentiment analysis belongs to a separate class of tasks in the field of processing natural languages and is of particular interest to business. In this paper, the main emphasis is on the creation of combined approaches, including heuristics, knowledge base, and machine learning. Quality assessment is carried out using recall, precision, f1 and accuracy metrics, showing how the developed method works in conditions of noisy data in multiple languages and well-labeled data from standard test collections in comparison with the previously developed approach [14].

90.2 Current Research in Sentiment Analysis There are different approaches to the analysis of textual information [15]. First of all, you can collect simple statistics—the number of reviews (comments), their length, the number of likes and dislikes, the product page traffic, and the number of relevant queries in popular search engines. Often a product has more than 100 comments, so processing them becomes more difficult and longer. Therefore, it is necessary to determine the number of reviews reviewed and how to choose them. Sometimes this sorting can be done manually. In the analysis of statements, the so-called “grounded theory” [16] can be used. This approach implies collecting qualitative data, extracting repeated elements, and

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marked with codes, which in turn form the concepts and categories that make up the theory. The analysis of textual information can be combined with the analysis of visual, since comments and reviews can contain photos that carry a large amount of useful data. In such a study, it is necessary to work with computer vision, as well as the correct combination of the results of text and graphic analysis. Today, there are three main approaches to solve the problem of sentiment analysis: • an approach based on dictionaries of positive, neutral, and negative words [17]; • an approach using technologies and algorithms of classical machine supervised learning [18]; and • an approach using unsupervised learning algorithms [19]. Moreover, modern approaches tend to use deep learning algorithms and technologies, artificial neural networks. The reason for this is the development of affordable computing technologies on GPUs. Let us first turn to language models based on recurrent artificial neural networks. An important advantage of recurrent architecture over statistical methods is the ability to represent more complex patterns in serial data [20]. Another language model that uses the artificial neural network learning approach is the word2vec model [21]. Word2vec is a group of related models that are used to create so-called embeddings of the words. These models are two-layer neural networks that learn to reconstruct linguistic contexts of words. Word2vec takes a large body of text as input and creates a vector space, with each unique word in the body corresponding to a vector in space. Next very important step in sentiment analysis research as well as in natural language processing is applying transfer learning for such tasks [22]. The authors of the latter paper propose a “universal language model fine-tuning method” (ULMFiT), which can be applied to any NLP task. Authors also suggest methods that are key to correctly configure the language model. The obtained approach significantly surpasses previous approaches in six problems of text classification, reducing the error by 18–24% for most datasets. In this paper, we will pay more attention to working with the data before direct classification, to show how different approaches can affect the results and what the choice of text pre-processing algorithms should depend on.

90.3 Proposed Sentiment Analysis Method Classic machine learning SVM, one of the most common algorithms, is used as a classifier. The main difficulties arising while constructing the discussion classifier on Twitter social network are the string noise of data and possible intersection of classes. Initially, the data was partitioned into four classes: negative, neutral, positive, and mixed. Classification of mixed class is challenging, since when a message of a given class is displayed in the vector space of the entire building, it can appear in any part

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of the remaining three classes. To avoid this, an attempt was made to transfer the dimension of the source space to a space of dimension n + 1, which did not produce a tangible result on the accuracy of the classifier. In this regard, the mixed message class was excluded from the training and test sample. Another significant problem of the classification is the high noise level of the source data. For example, if you look at some instances of the sample, you can see that they can only consist of the mention of a user and a link to an Internet resource. Such messages do not contain the user opinion in relation to the issue under discussion. In addition, many of the instances contain a large number of hashtags that are common between all discussion posts. This feature leads to a possible retraining of the model on such repeating data. This behavior can be observed in three of the four discussions. An exception is the discussion devoted to Ferguson (in English), the number of statements containing unique opinions of people there is much larger than in the other discussions in French, Russian, and German. To further improve the accuracy score of multilingual sentiment analysis for user discussion in social networks, the following steps were taken: • First, experiments showed that the exclusion of “stop words” from user messages leads to a decrease in the accuracy of the classifier. Therefore, all stop words are returned to the original dataset. This change made it possible to raise the classification accuracy by 4–5%. • Secondly, the following message characteristics were introduced into the training set: the number of mentions of other users and the number of links used in messages. These counters made it possible to more accurately determine neutral messages, which often contain mentions and links. • Third, synonyms began to be used to combat noisy data. For each word from the user message, synonyms were selected from the lexical database wordnet. These synonyms, if possible, were selected taking into account the context of the message. The resulting synonyms were added to the text of the original message, and subsequently the classifier was trained on such extended messages. The most noticeable effect of this improvement was seen on the case of the Ferguson discussion, which once again suggests that this was initially the least noisy discussion. In order to recognize synonyms, a database compiled from the pages of the Russian-language Wikipedia was used. These three steps describe the method we are using in this work.

90.4 Experiment The experiment assessed the quality of the proposed methods of sentiment analysis on noisy data in Twitter user discussions (Charlie, Biryulyovo, Cologne, Ferguson) and standard test collection (Twitter US Airline Sentiment [23]) using metrics: recall, precision, f1, and accuracy [24].

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90.4.1 Datasets The results of different approaches are evaluated using f1-score and accuracy measure. Four famous events are analyzed as test collections: • the riots in Biryulyovo (Russia) in October 2013 (3,602 labeled user messages in Russian language with 1208 negative, 1329 neutral, and 1065 positive messages); • the riots in Ferguson (Ferguson unrest) (USA) in August 2014 (2,999 labeled user messages in English with 907 negative, 165 neutral, and 1927 positive messages); • shooting at the offices of the French satirical weekly newspaper Charlie Hebdo (4,200 labeled user messages in French with 1429 negative, 865 neutral, and 1906 positive messages); and • mass public harassment of females on New Year’s Eve in Cologne, Germany, January 2016 (5,770 labeled user messages in German with 2188 negative, 277 neutral, and 3305 positive messages). All the information about these datasets and how they were obtained are in “Please Follow Us” paper [25]. To compare the operation of the algorithms, the public test collection (Twitter US Airline Sentiment) was also used. It is currently used to test algorithms in this research area. This dataset contains user references to six US airlines: Virgin America, Inc; American Airlines, Inc; Delta Air Lines, Inc; and Southwest Airlines United Airlines and US Airways, Inc. The data presented in the body covers the period of February 2015 and contains the following information: • the tweet itself and labeling information; • the name of the user who posted the tweet; • the date the tweet was created. In the paper, we are only interested in the tweet itself and its labeling information. The existing dataset contains 11.541 labeled tweets with 7271 negative, 2424 neutral, and 1846 positive messages. It is worth mentioning that all datasets were splitted in test/train collections with ratio 0.2.

90.4.2 Results During the experiment, the proposed algorithms described in Sect. 90.3 were tested on real ad hoc discussions about four important events and on an open dataset (Twitter US Airline Sentiment). Despite the complex and unpredictable structure of these discussions, it was possible to obtain very good results for the machine learning approach with the teacher. In both cases, the constructed classifiers showed results much better than “random fortune telling” for the three classes—“positive”, “neutral”, and “negative”.

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Table 90.1 The best results of classification Ferguson Birulyovo Accuracy Precision Recall F1

0.78 0.78 0.78 0.78

0.75 0.69 0.73 0.71

Charlie

Cologne

Airlines

0.63 0.59 0.61 0.60

0.73 0.69 0.71 0.70

0.96 0.94 0.94 0.94

Table 90.1 shows best results for the datasets. According to this table, the complex structure of social ad hoc discussions noticeably complicates the work of classifiers, unlike the Airlines dataset that has a similar message structure across itself. Considering classic machine learning algorithms and simple language models, it is necessary to adapt these models to a specific dataset each time.

90.5 Conclusion In this article, a combined approach to sentiment analysis of Twitter user discussion was presented. It includes the use of heuristics, knowledge base, and machine learning approaches. The experiment showed that when studying large and complex ad hoc discussions with unpredictable structure, it is necessary to adjust the algorithms to the data as shown in Sect. 90.4.2. The main goal of the work was to consider possible options for preliminary data processing and to find out how they affect the results. In the future, we plan to compare these approaches using modern artificial neural networks on the same datasets. Acknowledgements This work was supported by Russian Science Foundation [grant number 1618-10125-P].

References 1. www.netcraft.com 2. Gitto, S., Mancuso, P.: Improving airport services using sentiment analysis of the websites. Tour. Manag. Perspect. 22, 132–136 (2017) 3. Arvidsson, A.: General sentiment: how value and affect converge in the information economy. Sociol. Rev. 59, 39–59 (2011) 4. Anandan, B., et al.: t-plausibility: generalizing words to desensitize text. Trans. Data Priv. 5(3), 505–534 (2012) 5. Jonassen, D.H.: The technology of text: principles for structuring, designing, and displaying text. Educ. Technol. 2 (1982) 6. Xiang, R., Neville, J., Rogati, M.: Modeling relationship strength in online social networks. In: Proceedings of the 19th International Conference on World Wide Web (2010)

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7. Niu, T., et al.: Sentiment analysis on multi-view social data. International Conference on Multimedia Modeling. Springer, Cham (2016) 8. Barberà, P., Rivero, G.: Understanding the political representativeness of Twitter users. Soc. Sci. Comput. Rev. 33(6), 712–729 (2015) 9. https://www.statista.com/ 10. Bodrunova, S.S., et al.: Negative a/effect: sentiment of French-speaking users and its impact upon affective hashtags on Charlie Hebdo. International Conference on Internet Science. Springer, Cham (2018) 11. Pant, G., Srinivasan, P., Menczer, F.: Crawling the Web, chapter Web Dynamics (2004) 12. Bošnjak, M., et al.: Twitterecho: a distributed focused crawler to support open research with twitter data. In: Proceedings of the 21st International Conference on World Wide Web (2012) 13. Blekanov, I., Tarasov N., Maksimov A.: Topic modeling of conflict ad hoc discussions in social networks. In: Proceedings of the 3rd International Conference on Applications in Information Technology (2018) 14. Blekanov, I., et al.: Sentiment analysis for ad hoc discussions using multilingual knowledgebased approach. In: Proceedings of the 3rd International Conference on Applications in Information Technology (2018) 15. Raghuvanshi, N., Patil, J.M.: A brief review on sentiment analysis. In: 2016 International Conference on Electrical, Electronics, and Optimization Techniques (ICEEOT). IEEE (2016) 16. Charmaz, K., Belgrave, L.: Grounded theory. The Blackwell encyclopedia of sociology (2007) 17. Suchdev, R., et al.: Twitter sentiment analysis using machine learning and knowledge-based approach. Int. J. Comput. Appl. 103(4) (2014) 18. Pang, B., Lillian, L., Shivakumar, V.: Thumbs up? Sentiment classification using machine learning techniques. arXiv:cs/0205070 (2002) 19. Hu, X., et al.: Unsupervised sentiment analysis with emotional signals. In: Proceedings of the 22nd International Conference on World Wide Web (2013) 20. Mikolov, T.: Statistical language models based on neural networks. Presentation at Google, Mountain View, 2nd April, vol. 80(26) (2012) 21. Mikolov, T., et al.: Efficient estimation of word representations in vector space. arXiv:1301.3781 (2013) 22. Jeremy, H., Ruder, S.: Universal language model fine-tuning for text classification. arXiv:1801.06146 (2018) 23. Data For Everyone https://www.figure-eight.com/data-for-everyone/ 24. Powers, D.M.: Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation (2011) 25. Bodrunova, S.S., Litvinenko, A.A., Blekanov, I.S.: Please follow us: media roles in Twitter discussions in the United States, Germany, France, and Russia. J. Pract. 12(2), 177–203 (2018)

Chapter 91

Data Crawling Approaches for User Discussion Analysis on Web 2.0 Platforms Dmitry Nepiyushchikh and Ivan Blekanov

Abstract This article discusses approaches to collecting data from Web 2.0 platforms such as social networks and messengers. The authors propose the implementation of a flexible architecture for a focused web crawler to collect data of users’ discussion in the social network Facebook and the Telegram messenger. The proposed crawler is based on interaction with a platform’s API, get/post requests, and simulating actions in a browser. The authors set up an experiment comparing implementation of proposed data crawling approaches. The data on the COVID-19 virus was collected from Facebook social network and Telegram messenger using RUM Extractor for Facebook and large number open-source Telegram crawler. Developed focused crawler reached the speed of 15 participants per second and 12 posts per second without blocking account when processing a user discussion in Facebook. Telegram crawler showed the speed of 200 participants per second and 300 posts per second without blocking.

91.1 Introduction Currently, the number of network users is more than 4 billion users. The audience of social networks is more than 3 billion users. About 5,135 billion users use mobile devices, and 2,958 use social resources from mobile devices. Every day last year about one million users started using social networks—about 11 new users per second [1]. The large number of users can be explained by the fact that the Web 2.0 [2] class platforms currently prevail on the Web, which include social networks, instant messengers, and open repositories. A distinctive feature of such platforms is providing users with the opportunity (through the platform’s built-in tools) to create content without limits, exchange information, and interact with each other. Due to the involvement of users, their high activity in content generation, and technological features of web 2.0 platforms, we can observe a high rate of dissemination D. Nepiyushchikh · I. Blekanov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_91

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of information about any events in social networks, which significantly exceeds the rate of information distribution in web 1.0 networks (for example, in the media). The scientific area, which is engaged in the study of user activity in web 2.0 networks’ analysis of user content and user relationship, is called Social Network Analysis (SNA) [3]. Actual tasks of SNA include following tasks: • Data crawling—collecting data from web pages. • Sentiment analysis—section of computer linguistics which deals with determining the sentiment of a text [4]. For example, some authors [5] combine this task with image analysis. In particular, the authors selected the “Zooniverse” service, which is dedicated to image analysis in order to identify animals in photographs from Tanzania Serengeti National Park. The authors used sentiment analysis of posts on the forum and on Twitter to assess the quality of images and the service as a whole. • Topic modeling—defining text message topics in users’ discussions in social network [6]. • Bot detection—identifying social bots (better known as spambots) and robots which exists in social network, automatically generate content, and communicate with another users trying to simulate their behavior [7] or generate spam noise in user discussions in social networks [8]. In this work, authors developed and implemented the architecture of focused web crawlers for collecting data about user discussions from Facebook social network and Telegram messenger.

91.2 Current Research in Data Crawling on Web 2.0 There are different approaches to solving the problem of data collection, taking into account the features of platforms web 2.0. • • • •

Using ready-made data banks [9]. Using ready-made software for collecting data [10]. Using API (application programming interface) [11]. Developing own software.

91.2.1 Using Ready-Made Data Banks The usage of the approach as a source of data for analyzing user activity on web 2.0 platforms has several significant drawbacks: • Data completeness. • Data authenticity.

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• Data integrity. • Data relevance. It is important to note that most of the data bank resources are not free to use.

91.2.2 Using Ready-Made Software It is impossible to evaluate the completeness and integrity of the information received. The tools for data crawling in the Web 2.0 platform are not free to use either.

91.2.3 Data Crawling Based on API Like previous solutions, API-based crawlers have several disadvantages: a limit on the number of calls per unit of time, a periodically changing mark-up data (interface), access to important data is paid [12]. Using this approach, it is not always possible to guarantee the authenticity and integrity of data [12]. But it is worth noting that this approach is mostly used when collecting data from the services that have an API. Relevant platforms can provide various types of API access: 1. Fully accessible API—enough to create an account on a platform. After that, an access key is issued—a token, which can be either constant or issued for a certain period of time. An example of such a service is Telegram API.1 2. Partially accessible API. In this case, it is usually enough to just have an account, but it is required to register an application for data collection. Most of the time the token and application identifier are issued for further use. Services may also have a limit on the number of requests per unit of time: minute, hour, day, etc. To expand access, it is necessary to provide proper information to the service: the goal of service usage, the financial information, and the requested access rights. Yandex maps API is a good example.2 3. Fully paid API. To access the necessary data, you need to pay for the ability to view the data. Sometimes demo access may be provided for a certain period of time (day, week, month, several months, and so on). HeadHunter (job search service) is an example of service with this type of API.3 When working with any kind of API, it is necessary to be aware of the restrictions that the corresponding services can impose on the web crawler. These restrictions may include the following: 1

https://core.tlgr.org/api. https://tech.yandex.ru/maps. 3 https://hh.ru/. 2

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• Query frequency. The frequency is determined empirically, but there may be cases when services in the API documentation explicitly indicate the number of requests per unit time. • Query uniqueness. Obviously, identical bulk requests lead to blocking, as this falls out of typical user behavior. For example, it is difficult to imagine a situation where a user of a service will massively add new friends or view profile photos. • Various identifiers of devices. These identifiers can contain application identifiers that are registered in the services from which data is collected (for example, Application Id and Application Hash in Telegram API). • The privacy of the data collected. The more closed the data is the more caution should be taken while collecting data. For example, phone number and user ID in the Telegram messenger are the most private information in the service.

91.2.4 Own Web Crawling Approaches This approach allows you to control both the integrity and authenticity of the data. The software can be divided into two large classes: 1. Solution based on simulating user activity in browser or application [13]. 2. Solution based on requests to the platform server [14].

91.2.4.1

Solution Based on Simulating User Activity in Browser or Application

Using this approach, it is important to understand that the speed of data crawling is comparable to the speed of the user activity. When receiving content, you need to download the html page, all media, and CSS. Another important issue may be memory overflow during long-term work with browser simulating. The right way to use the approach is to interact with the DOM model, which may also become an issue. When simulating actions in a mobile application, the process of obtaining data becomes even more complicated and longer due to the exchange of commands at the device level and, possibly, the instability of the connection.

91.2.4.2

Solution Based on Requests to the Platform Server

This approach is most interesting for data collecting. When building such a crawler, it can be based on the web version of the resource from which the collection takes place, or also on a mobile application. To build such a bot, it is necessary to differentiate platform traffic and fully emulate all service requests.

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91.2.5 Proposed Focused Crawler for Facebook and Telegram To solve the problem of collecting data from user discussions, a focused web-crawler architecture was developed and implemented. It allows us to work with Telegram chats/channels and communities on Facebook social network. Crawler for collecting data from Facebook was implemented using python3 (Selenium library4 ) to control a browser “Google Chrome”. The media data is being downloaded from a separate thread, while the links are being saved in the main thread. The implemented method requires a set of credentials for authorization in a social network. This method can be described step by step as follows: 1. Authorization in the social network. 2. Verification of account subscription on the community under investigation. If the account is not signed, then the subscription should be done. 3. With an empirically obtained interval of 2 s, the software should scroll down the page, but every 25–30 scrolls (randomly selected for each new iteration) wait 3 s. 4. Collection of the content from the page. 5. Saving the result of the iteration to the database. It is important to note that for RAM optimization, the browser is working in stealth mode, so that it is not visualized on the screen. But the study showed that with a large number of participants and posts in communities, the collection rate of the method simulating activity in a browser becomes low. In this regard, another crawler was developed based on get/post requests. This crawler also allows you to collect community members and posts. Every new request from the Facebook server adds only a new piece of data, which does not contain the previous ones. This greatly simplifies the processing of the received data. Each request contains 15 users and 12 posts. Next, a random time delay of about 2 to 4 s is needed. After that there is a request for the next piece of data. Crawler for collecting data from Telegram was also developed in Python3 based on the Telegram API using the Pyrogram5 library. To fully automate the collection of data from the Telegram service, automatic creation of a Telegram account using the SMS activation service is provided in case of blocking the account during data collection. If the current account is blocked, a new account is registered and the collection process continues from the message on which the lock occurred. For one request, it is possible to obtain data on 300 posts. If the post contains media, then the links are saved, and a separate process is responsible for downloading it. All data is stored in a database under the control of the PostgreSQL DBMS. This focused web crawler implements a system for recognizing crawler locks, both complete when the account is deactivated on the platform, and temporary when empty answers come to the platform server. The crawler also recognizes which 4 5

https://www.selenium.dev. https://docs.pyrogram.org.

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service the community link belongs to and applies the necessary collection method on its own. The developed crawler can be expanded with any other social networks and messengers. It is enough to implement the selection of the necessary content in any convenient way. For faster data collection, multithreading was implemented for each crawler.

91.3 Experiment Using the implemented crawler, it was necessary to measure and compare the speed of receiving data in various ways: using the API, using a browser-oriented solution and using get/post requests. The following indicators were used for evaluation: data acquisition speed and the amount of data received.

91.3.1 Crawling Datasets Currently, the urgent problem is the involvement of a large number of people in the situation associated with the spread of the COVID-19 virus (01.03.2020–31.03.2020). To analyze user activity in social resources in connection with the spread of the virus and the introduction of various protective measures by many countries of the world, communities were identified on Facebook as well as chats and channels in the Telegram messenger. If possible, posts and participants in these communities were collected, since it is not possible to get channel members in the Telegram messenger.

91.3.2 Results As a result, data was obtained from the considered social resources on Web 2.0 platform. The volume of data obtained is presented in Table 91.1. As can be seen from Table 91.2, the total amount of information received from the platforms differs by an order of magnitude. This is due to the fact that for one request each platform gives a different number of posts, as well as you need to search them in the response to each request. During the experiment, it was revealed that on Facebook, the displayed data in the web browser on the number of participants in the community or posts in this community does not always relate to the research community. The actual number of users in the community is less than about 10%, and some posts are related to other communities of similar subjects.

91 Data Crawling Approaches for User Discussion Analysis on Web 2.0 Platforms Table 91.1 Total number of data collected from social service Service Total number users Total number posts collected collected Telegram Facebook based on a browser Facebook by get/post requests

Time spent collecting

97569 63587

28649711 1581049

about 26 h about 44 h

63587

1581049

about 19 h

Table 91.2 Time intervals between requests Service Time interval Extra interval

799

Amount of Amount of Amount of information in new information the request information in per day the request

Telegram

3s

7 s per 1800 messages

100 messages

100 messages

Facebook based on a browser Facebook by get/post requests

2s

3 s per 25–35 scrolls

Content of a 8 posts html web page

2–4 s

Empty

12 posts

12 posts

About 2,4 million messages About 170 thousand posts About 296 thousand posts

91.4 Conclusion Each of the proposed approaches considered has both its pros and cons. When working with the API, not all methods proposed by the service can allow you to collect the necessary data, but the availability of good documentation is a good advantage. When working with emulation of actions in an application or browser, a large amount of resources is required for collecting. Critical errors may occur when disconnecting from the automation services, but, at the same time, you can always visually evaluate at what stage data collection is. When approaching get/post requests, the disadvantage is the complexity of the implementation. It is not always obvious how some important tokens are formed, but the speed and ease of preprocessing of the received data is an obvious advantage. As can be seen from the experiment, the amount of information obtained using a browser solution is significantly less than the amount of information obtained using the API or based on the requests. This is due to the large amount of data available in the browser, and the constant increase of content. Also, when studying communities on Facebook, features, associated with unreliable data displayed to the user, were identified.

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References 1. Digital 2019: GLOBAL INTERNET USE ACCELERATES. https://wearesocial.com/blog/ 2019/01/digital-2019-global-internet-use-accelerates 2. Zajicek, M.: W4A’07: Proceedings of the 2007 International Cross-Disciplinary Conference on Web Accessibility, pp. 35–39 (2007) 3. Marin, A., Wellman, B.: Social network analysis: an introduction (2009). https://pdfs. semanticscholar.org/aa4d/5f9ae3fbe6ae16a1de02f7b6daddf615a238.pdf 4. Ding, J., Sun, H., Wang, Xu., Liu, Xu.: IEEE/ACM 3rd International Workshop on Emotion Awareness in Software Engineering (SEmotion). Entity-level sentiment analysis of issue comments (2018) 5. Woldemariam, Y.: IEEE International Conference on Big Data Analysis (ICBDA). Sentiment analysis in a cross-media analysis framework (2016) 6. Boyd-Graber, J., Blei, D.: SemEval’07: Proceedings of the 4th International Workshop on Semantic Evaluations, pp. 277–281 (2007) 7. Ping, H., Qin, S.: IEEE 18th International Conference on Communication Technology (ICCT). A social bots detection model based on deep learning algorithm (2018) 8. Cresci, S., Di Pietro, R., Petrocchi, M., Spognardi, A., Tesconi, M.: WWW’17 Companion: Proceedings of the 26th International Conference on World Wide Web Companion. The ParadigmShift of Social Spambots: Evidence, Theories, and Tools for the Arms Race, pp. 963–972 (2017) 9. Service for search content in some platforms. https://smmbox.com 10. Service for collecting data from some platforms. https://pepper.ninja 11. API VK.COM. https://vk.com/dev 12. Peleshchyshyn, A., Mastykash, O.: 12th International Scientific and Technical Conference on Computer Sciences and Information Technologies (CSIT). Analysis of the methods of data collection on social networks (2017) 13. Gorro, K.D., Sabellano, M.J., Gorro, K., Maderazo, C., Capao, K.: 3rd International Conference on Computer and Communication Systems (ICCCS). Classification of Cyberbullying in Facebook Using Selenium and SVM (2018) 14. Yuvarani, M., Iyengar, N.ch.s.n., Kannan, A.: IEEE/WIC/ACM International Conference on Web Intelligence (WI 2006 Main Conference Proceedings). LSCrawler: A Framework for an Enhanced Focused Web Crawler Based on Link Semantics (2006)

Chapter 92

Metric for Comparison of Graph-Theoretic Models of the Same Dimension with Ordered Vertices Nikolai Moskin

Abstract The work is dedicated to methods of comparison and classification of graph-theoretic models which are known within the direction of graph matching. It contains an overview of metrics for comparing graphs based on a maximum common subgraph. A modification of the measure based on a maximum common subgraph is proposed, which takes into account the ordering of vertices (each vertex is associated with its unique serial number). If the graphs have the same dimension (i.e. same number of vertices) it is shown that this measure satisfies all the properties of the metric (nonnegativity, identity, symmetry, triangle inequality). It is assumed that this metric can be used to solve the problem of text attribution.

92.1 Introduction Currently, graph-theoretic models are used in various fields of science and can be constructed on different principles. In this work, we mean by such a model (hereinafter the graph) a four G = (V, E, α, β) in which: • • • •

V —finite nonempty set of vertices; E ⊆ V × V —the set of edges connecting vertices; α : V → L V —function that labels the vertices of the graph; and β : E → L E —function that labels the edges of the graph,

where L V and L E are the sets of labels and attributes of objects and relations, respectively, defined in a certain subject area. Denote by m the number of vertices and n the number of edges of G. One of the tasks that arises during the construction of such models is the task of comparison and classification. This direction is known as graph matching [4]. These methods have found application in image and video processing [8, 17], molecular biology [16, 18], fingerprinting [11], recognition of handwriting [6], research of N. Moskin (B) Petrozavodsk State University, Lenin Str., 33, Petrozavodsk, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_92

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social networks [10], analysis of documents [19], etc. On the set of graphs a distance is established that allows to evaluate how the structures are similar to each other. As a rule this measure expresses the degree of inaccuracies that occur when finding an isomorphism of graphs or subgraphs. One of the applications of such metrics is comparison and classification of texts. Graph-theoretic models of texts can be constructed on various principles (at the level of lexis, syntax, or semantics) [9]. A peculiarity of such models is the ordering of their elements. For example, when building dependency trees, the vertices correspond to the words and can be numbered as they occur in the text. The vertices of graphs can be not associated only with individual words, but also with paragraphs of text, sentences, certain parts of sentences, etc.

92.2 Metrics for Comparing Graphs Based on the Maximum Common Subgraph One way to compare graphs is using the maximum common subgraph. The maximum common subgraph of graphs G 1 and G 2 is graph mcs(G 1 , G 2 ) that is isomorphic G 1 ∈ G 1 , G 2 ∈ G 2 and contains the maximum number of vertices. The distance between nonempty graphs G 1 and G 2 can be calculated as follows [3]: d1 (G 1 , G 2 ) = 1 −

|mcs(G 1 , G 2 )| . max(|G 1 |, |G 2 |)

(92.1)

Here |G 1 |, |G 2 |, and |mcs(G 1 , G 2 )| are the number of vertices of the graphs G 1 , G 2 , and mcs(G 1 , G 2 ), respectively. The distance d1 (G 1 , G 2 ) takes values from 0 to 1. If the graphs are isomorphic, then d1 (G 1 , G 2 ) = 0. We can also note that based on the definition the maximum common subgraph is not necessarily unique. It may be a set of graphs. The second measure based on the maximum common subgraph was proposed in [20]: |mcs(G 1 , G 2 )| . (92.2) d2 (G 1 , G 2 ) = 1 − |G 1 | + |G 2 | − |mcs(G 1 , G 2 )| The values of d2 (G 1 , G 2 ) are also from 0 to 1. Another similar distance is suggested in [1], but it is not normalized for [0, 1]: d3 (G 1 , G 2 ) = |G 1 | + |G 2 | − 2 · |mcs(G 1 , G 2 )|.

(92.3)

It is shown in [5] that a maximum common subgraph can be obtained from a minimum common supergraph and backward. This article also suggests a distance that combines these two concepts: d4 (G 1 , G 2 ) = |MC S(G 1 , G 2 )| − |mcs(G 1 , G 2 )|.

(92.4)

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Here MC S(G 1 , G 2 ) is a minimum common supergraph of G 1 and G 2 , i.e. such smallest graph that G 1 ∈ MC S(G 1 , G 2 ) and G 2 ∈ MC S(G 1 , G 2 ). When setting a measure on a set of graphs, it is desirable that d(G i , G j ) satisfies the following properties of the metric (for arbitrary i, j, k): 1. 2. 3. 4.

d(G i , G j ) ≥ 0 (nonnegativity); d(G i , G j ) = 0 ⇔ G i = G j (identity); d(G i , G j ) = d(G j , G i ) (symmetry); and d(G i , G j ) ≤ d(G i , G k ) + d(G k , G j ) (the triangle inequality).

For example, it was shown in [3] that a measure (92.1) satisfies all the properties of the metric. Note that in addition to metrics based on common subgraphs (92.1)– (92.4) there are also other metrics for comparing graphs (for example, graph edit distance [14]).

92.3 Metric for Comparison of Graph-Theoretic Models of the Same Dimension with Ordered Vertices When comparing graph-theoretic models of texts, it is important to consider vertex numbering. Figure 92.1 shows three isomorphic graphs with different vertex numbers G, F, and H (excluding vertex numbering, the distance between any pair of these graphs, which is found using metrics based on the largest common subgraph, is zero). However, taking into account the numbering of the vertices, it is obvious that the graphs G and F are more similar to each other than G and H (or F and H ).

Fig. 92.1 Same graphs with different vertex numbers

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Fig. 92.2 Chain of generating subgraphs for graph G

Therefore, it is necessary to modify the considered metrics so that they take into account the order of vertices. Let’s represent such graph as a chain of subgraphs g1 , g2 , g3 , . . . , gm generating it, as shown in Fig. 92.2. Here the graph gi is a subgraph of G that contains vertices with numbers from 1 to i and all edges of G that are incident to these vertices. Accordingly, the subgraph gm is equal to G. The distance between graphs G and F with the same number of vertices (|G| = |F| = m) can be found as follows: d(G, F) = 1 − min ( i=1,...,m

|mcs(gi , f i )| ). i

(92.5)

Theorem 92.1 The distance d satisfies all the properties of the metric. Proof (a) Prove the first property. Since the number of vertices in the graphs gi and f i does not exceed i for ∀i = 1, . . . , m, then |mcs(gi , f i )| ≤ i. Then the inequality |mcs(gi , f i )| ≤ 1 is fulfilled. Therefore mini=1,...,m ( |mcs(gi i , fi )| ) ≤ 1. Thus d(G, F) = i 1 − mini=1,...,m ( |mcs(gi i , fi )| ) ≥ 0. (b) Prove the second property.

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⇒ Let d(G, F) = 0. Then according to (92.5) mini=1,...,m ( |mcs(gi i , fi )| ) = 1, so |mcs(gi i , fi )| ≥ 1 for ∀i = 1, . . . , m, i.e. |mcs(gi , f i )| ≥ i. On the other hand, |mcs(gi , f i )| ≤ i. Thus, |mcs(gi , f i )| = i, i.e. for ∀i = 1, . . . , m graphs gi and f i are isomorphic and consequently G = F. ⇐ Let G = F. Then gi = f i for ∀i = 1, . . . , m and consequently maximum common subgraph gi and f i will have a dimension i, i.e. |mcs(gi , f i )| = i for ∀i = 1, . . . , m. Hereof mini=1,...,m ( |mcs(gi i , fi )| ) = 1, so d(G, F) = 0 according to (92.5). (c) The third property is true, since for ∀i = 1, . . . , m |mcs(gi , f i )| = |mcs( f i , gi )|. (d) Prove the fourth property. Let H be a random graph with m vertices. It must be shown that d(G, F) ≤ d(G, H ) + d(H, F), i.e. according to (92.5): 1 − min ( i=1,...,m

|mcs(gi , f i )| |mcs(gi , h i )| ) ≤ 1 − min ( )+ i=1,...,m i i |mcs(h i , f i )| 1 − min ( ). i=1,...,m i

Since graphs gi , f i , h i contain i vertices, i.e. |gi | = | f i | = |h i | = i, then make the following change: 1 − min ( i=1,...,m

|mcs(gi , f i )| |mcs(gi , h i )| ) ≤ 1 − min ( )+ i=1,...,m max(|gi |, |h i |) max(|gi |, | f i |) |mcs(h i , f i )| ). 1 − min ( i=1,...,m max(|h i |, | f i |

Moving from the search for the minimum to the maximum, we get

max (1 − (

i=1,...,m

|mcs(gi , f i )| |mcs(gi , h i )| ) ≤ max (1 − )+ i=1,...,m max(|gi |, | f i |) max(|gi |, |h i |) |mcs(h i , f i )| ). max (1 − i=1,...,m max(|h i |, | f i |

According to (92.1), we rewrite the inequality max d1 (gi , f i ) ≤ max d1 (gi , h i ) + max d1 (h i , f i ).

i=1,...,m

i=1,...,m

i=1,...,m

If the maximum d1 (gi , f i ) for i = 1, . . . , m is reached for index i ∗ , then max d1 (gi , f i ) = d1 (gi ∗ , f i ∗ ) ≤ d1 (gi ∗ , h i ∗ ) + d1 (h i ∗ , f i ∗ ) ≤

i=1,...,m

max d1 (gi , h i ) + max d1 (h i , f i ).

i=1,...,m

i=1,...,m

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The inequality is true, since the distance satisfies all the properties of the metric. Therefore, the theorem is proved.  Let’s calculate the distance between the graphs shown in Fig. 92.1. The distance d(G, F) is equal to 0, while d(G, H ) = d(H, F) = 41 . A measure based on the maximum common subgraph (92.1), which does not take into account the order of vertices, does not allow you to determine the differences in graphs.

92.4 Application of the Metrics to the Task of Text Attribution Graph-theoretic models have been used for a long time in solving the text attribution problem. For example, such a problem arises in the study of the corpus of texts from the journals “Time” and “Epoch” (1861–1865). It is known that F. M. Dostoevsky (together with his brother M. M. Dostoevsky) edited and headed these journals, so research has long been conducted on the subject of his writing of these works (also N. N. Strakhov, A. A. Golovachev, I. N. Shill, A. Grigoriev, A. U. Poretsky, Ya. P. Polonsky, and others were published in these journals). More details about this task are written in [7, 13]. In some works, the method of graph of strong links was used to solve this problem [15]. Let’s consider an algorithm for constructing such graphs. In this case, V is a set of grammatical forms which are found in the texts. The number of grammatical forms is determined by a specialist in philology, depending on the type of the study. E is a set of strong links of grammatical forms. Two vertices vi and v j of a graph are connected by an edge if the frequency of occurrence of this pair of grammatical classes is equal or exceeds the assigned threshold. Note that for comparing of such graphs the Rogers–Tanimoto coefficient was used, which is calculated by the following formula: d5 (G, F) =

m(G, F) . |G| + |F| − m(G, F)

(92.6)

Here in (92.6) m(G, F) is the number of common vertices in G and F, i.e. the number of grammatical forms which occur in two comparable texts. In our opinion, a model in the form of a graph with strong links can be complicated and supplemented. The grammatical forms and their corresponding vertices can be sorted in descending order according to the degree of their occurrence in the certain texts (or certain group of texts). Also if a vertex does not have incident edges as a result of dropping “weak” links, it is considered as an isolated vertex. Further, using a measure (92.5), a matrix of distances between the graphs of a certain texts can be constructed. This matrix can be explored, for example, using the multidimensional scaling method for identifying hidden factors. To carry out this study, you can also use metrics (92.1)–(92.3) (without taking into account the

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ordering of vertices), however, as shown by pilot experiments, (92.5) is better at this task.

92.5 Conclusion In this paper, we proposed the distance between graphs with ordered vertices, which is based on the maximum common subgraph. It is shown that this measure satisfies all the properties of the metric (nonnegativity, identity, symmetry, triangle inequality). Note that the problem of finding the maximum common subgraph is NP-hard [2, 12]. In the future, this metric can be used to solve the problem of text attribution. In particular, it applies to the corpus of texts from the magazines “Time” and “Epoch” (1861–1865). It is known that F. M. Dostoevsky and M. M. Dostoevsky edited and headed these journals, so research has long been conducted on the subject of his writing of these works. Since the texts are presented in pre-revolutionary orthography, a lot of philological work is necessary to parse texts. Storing texts and corresponding graph-theoretic models, as well as implementing a metric calculation algorithm supposed to be done in the SMALT information system (“Statistical Methods of the Analysis of Literary Texts”), that is, developing at Petrozavodsk State University [13]. Note that this metric can be also used in other problems of pattern recognition (for example, comparing images). Acknowledgements This work was supported by the Russian Foundation for Basic Research, project no. 18-012-90026.

References 1. Bunke, H.: On a relation between graph edit distance and maximum common subgraph. Pattern Recognit. Lett. 18, 689–694 (1997) 2. Bunke, H., Foggia, P., Guidobaldi, C., Sansone, C., Vento, M.: A comparison of algorithms for maximum common subgraph on randomly connected graphs. In: Proceedings of the Joint IAPR International Workshop on Structural, Syntactic, and Statistical Pattern Recognition. Lecture Notes in Computer Science, vol. 2396, pp. 123–132. Springer (2002) 3. Bunke, H., Shearer, K.: A graph distance metric based on the maximal common subgraph. Pattern Recognit. Lett. 19(3–4), 255–259 (1998) 4. Conte, D., Foggia, P., Sansone, C., Vento, M.: Thirty years of graph matching in pattern recognition. Int. J. Pattern Recognit. Artif. Intell. 18(3), 265–298 (2004) 5. Fernandez, M.-L., Valiente, G.: A graph distance metric combining maximum common subgraph and minimum common supergraph. Pattern Recognit. Lett. 22(6–7), 753–758 (2001) 6. Fischer, A., Suen, C.Y., Frinken, V., Riesen, K., Bunke, H.: A fast matching algorithm for graphbased handwriting recognition. In: Proceedings of the 8th International Workshop on GraphBased Representations in Pattern Recognition (GbRPR 2013). Lecture Notes in Computer Science, vol. 7877, pp. 194–203. Springer, Berlin, Heidelberg (2013)

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7. Lebedev, A.A., Rogov, A.A., Kulakov, K.A., Moskin, N.D.: To the problem of creating of marked corpuses of texts in the graphics of the XIX century. In: Proceedings of the International Conference “Corpus Linguistics-2019” St. Petersburg, pp. 296–302 (2019) 8. Morrison, P., Zou, J.J.: Inexact graph matching using a hierarchy of matching processes. Comput. Vis. Media 1(4), 291–307 (2015) 9. Moskin, N.D.: Graph-Theoretic Models of Folklore Texts and Methods for Their Analysis. PetrSU Publ, Petrozavodsk (2013) 10. Ogaard, K., Roy, H., Kase, S., Nagi, R., Sambhoos, K., Sudit, M.: Discovering patterns in social networks with graph matching algorithms. In: 6th International Conference on Social Computing, Behavioral-Cultural Modeling and Prediction (SBP 2013). Lecture Notes in Computer Science, vol. 7812, pp. 341–349. Springer, Berlin, Heidelberg (2013) 11. Pawar, V., Zaveri, M.: K-means graph database clustering and matching for fingerprint recognition. Intell. Inf. Manag. 7(4), 242–251 (2015) 12. Quer, S., Marcelli, A., Squillero, G.: The maximum common subgraph problem: a parallel and multi-engine approach. Computation 8(2), 48 (2020) 13. Rogov, A.A., Kulakov, K.A., Moskin, N.D.: Software support in solving the problem of text attribution. Softw. Eng. 10(5), 234–240 (2019) 14. Riesen, K., Ferrer, M., Bunke, H.: Approximate graph edit distance in quadratic time. IEEE/ACM Trans. Comput. Biol. Bioinform. 17(2), 483–494 (2020) 15. Rogov, A.A., Sedov, A.V., Sidorov, Y.V., Surovceva, T.G.: Mathematical Methods for Text Attribution. PetrSU Publ, Petrozavodsk (2014) 16. Schirmer, S., Ponty, Y., Giegerich, R.: Introduction to RNA secondary structure comparison. RNA Sequence, Structure, and Function: Computational and Bioinformatic Methods. Methods in Molecular Biology, vol. 1097, pp. 247–273. Humana Press, Totowa (2014) 17. Sharma, H., Pawar, A., Chourasia, C., Khatri, S.: Implementation of face recognition system based on elastic bunch graph matching. Int. J. Eng. Sci. Res. Technol. (IJESRT) 5(3), 888–895 (2016) 18. Shen, R., Guda, C.: Applied graph-mining algorithms to study biomolecular interaction networks. BioMed Res. Int. 2014(article 439476), 11 (2014) 19. Stauffer, M., Fischer, A., Riesen, K.: Speeding-up graph-based keyword spotting in historical handwritten documents. Graph-Based Representations in Pattern Recognition. GbRPR 2017. Lecture Notes in Computer Science, vol. 10310, pp. 83–93. Springer, Cham (2017) 20. Wallis, W., Shoubridge, P., Kraetz, M., Ray, D.: Graph distances using graph union. Pattern Recognit. Lett. 22, 701–704 (2001)

Part XIII

Non-linear Mechanics and Solid-State Physics

Chapter 93

Interaction of Finite Amplitude Surface Waves in a Basin with a Floating Elastic Plate Anton A. Bukatov

Abstract On the basis of multiple scales method, a solution of the problem about progressive surface waves nonlinear interaction in a finite depth basin with a floating elastic plate is constructed. Asymptotic expansions are obtained up to the third order of smallness for the velocity potential of liquid particles movement and elevation of the plate–fluid surface. The analysis of the dependence of the dispersion properties of the formed disturbance on the elastic and mass characteristics of the plate, its longitudinal compression, is carried out. The influence of nonlinearity of vertical displacements acceleration of the elastic plate on the fluctuations amplitude– phase characteristics is studied. It is shown that the presence of compressive force is expressed in the lag of the fluctuation phase from the phase obtained in the absence of compression.

93.1 Introduction The study of wave perturbations in a fluid with a floating elastic plate in a linear formulation without taking into account the compressive force was performed in [1, 9, 11, 13, 15], and if it was present in [3, 8, 14]. Finite amplitude waves in a homogeneous fluid with an elastic plate without taking into account the nonlinearity of the acceleration of its vertical displacements were considered in [10]. The effect of floating absolutely flexible plate on the amplitude–phase characteristics of nonlinear surface waves in the case of nonlinear interaction of two harmonics was considered in [7]. The effect of nonlinearity of the acceleration of vertical displacements of an elastic plate on the propagation of periodic surface waves in the absence of longitudinal compression was studied in [6], and if it was present in [5]. In this paper, the effect of an elastic plate on wave fluctuations formed during the nonlinear interaction of two periodic wave harmonics has been considered. The

A. A. Bukatov (B) Marine Hydrophysical Institute of RAS, Sevastopol, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_93

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nonlinearity of acceleration of plate vertical displacements is taken into account, both in the presence and absence of ice compression.

93.2 Formulation of the Problem Suppose that a thin longitudinally compressed elastic plate float on the surface of homogeneous ideal incompressible fluid which is filling basin of constant depth H . In the horizontal directions, the plates and fluid are unlimited. The fluid motion is potential, and the plate vibrations occur √ without separation. Then, in dimensionless variables x = kx1 , z = kz 1 , t = kgt1 , where k is the wave number, the problem is to solve the Laplace equation ϕ = 0, −∞ < x < ∞, −H ≤ z ≤ ζ,

(93.1)

for the velocity potential ϕ(x, z, t) with boundary conditions on the plate–fluid surface (z = ζ ) D1 k 4 ∂∂ xζ4 + Q 1 k 2 ∂∂ xζ2 + κk ∂∂z    2  2 + 21 ∂∂ϕx + ∂ϕ =0 ∂z 4

2

  1 2

∂ϕ ∂x

2

−

∂ϕ ∂t

 +ζ −

∂ϕ ∂t

+ (93.2)

and at the basin bottom (z = −H ) ∂ϕ = 0. ∂z

(93.3)

At the initial time (t = 0) ζ = f (x),

∂ζ = 0. ∂t

(93.4)

The velocity potential ϕ(x, z, t) and the elevation of the plate–fluid surface ζ (x, t) are related by the kinematic condition ∂ζ ∂ζ ∂ϕ ∂ϕ − + = 0. ∂t ∂x ∂x ∂z Here D1 =

D Q Eh 3 ρ1 , D= , Q1 = , κ=h , ρg 12(1 − ν 2 ) ρg ρ

(93.5)

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E, h, ρ1 , ν—modulus of normal elasticity, thickness, density, Poisson’s ratio of the plate, Q is the longitudinal compressive force per unit width of the plate, ρ is the fluid density, and g is the gravity acceleration. In the dynamic condition (93.2), the expression with the coefficient κ represents the inertia of the plate vertical displacements. Moreover, the first term in the brackets is due to the nonlinearity of the acceleration of its vertical displacements during bending.

93.3 Equations for Definition of Nonlinear Approximations We find a solution of problem (93.1)–(93.5) by the multiple scales method [12], which allows one to obtain uniformly converging expansions for ζ and ϕ. As a result, equations for determining nonlinear approximations are obtained ϕn = 0, −∞ < x < ∞, −H ≤ z ≤ 0, D1 k 4

2 ∂ 4 ζn ∂ 2 ϕn ∂ϕn 2 ∂ ζn + Q k − κk − + ζn = Fn∗ , 1 ∂x4 ∂x2 ∂z∂ T0 ∂ T0

ζn = f n (x) ,

(93.6) z = 0,

(93.7)

∂ζn ∂ϕn + = L n , z = 0, ∂ T0 ∂z

(93.8)

∂ϕn = 0, z = −H, ∂z

(93.9)

∂ζn = Gn, ∂ T0

t =0

(n = 1, 2, 3) .

(93.10)

Here Fn∗ = Fn + Fn0 , F1 = F10 = L 1 = G 1 = 0. All variables are known, but due to cumbersomeness they are not represented [2]. Note that the terms F20 , F30 are caused by account of the acceleration nonlinearity of the plate vertical displacements.

93.4 Expressions for the Fluid Velocity Potential and Plate–Fluid Surface Elevation Equations (93.6)–(93.10) are obtained in the case of unsteady perturbations of finite amplitude. Assume the first approximation in the form ζ1 = cos θ + a1 cos 2θ . Then the solution up to the values of the third order of smallness in dimensional quantities is determined as follows:

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  2 3 2 ζ =  a2 cos θ + 3aa21 cos 2θ + a ka3232 + a k a33 cos2 3θ2 + + a ka24 + a k a34 cos 4θ + a k a35 cos 5θ + a k a36 cos 6θ,

(93.11)

  1 2  ϕ = a gk / shτ H ch (z + H ) sin θ + b12 ch2 (z + H ) sin 2θ + √ kg (b23 ch3 (z + H ) + +b24 ch4 (z + H ) sin 4θ + b20 t) + + a2 √ + a 3 k kg (b33 ch3 (z + H ) sin 3θ + b34 ch4 (z + H ) sin 4θ + + b35 ch5 (z + H ) sin 5θ + b36 ch6 (z + H ) + b30 t) ,

(93.12)

θ = kx + σ t, σ =

   kg τ + akσ1 + a 2 k 2 σ2 ,

(93.13)

here a is the amplitude of the initial harmonic. The expression found for the amplitude of the second interacting harmonic has the following form:

μ2 r1   a1 = ± 2 2 4r2 2τ cth2H + 4τ κk + μ2 (1 + 2κk th2H )

1/ 2

,

(93.14)

here     2  1 5 τ (cthH + κk) + μ1 , cthH + 3κk − r1 = 2 cthH + th2H cthH 2 2  r2 = τ

2

1 + cth2H cthH − κk 2

  1 5 cth2H − cthH + μ1 cthH + cth2H 2 2

  τ 2 = 1 − Q 1 k 2 + D1 k 4 (1 + κk thH )−1 thH,   τ 2 = 1 + D1 k 4 (1 + κk thH )−1 thH, μi = 1 − i 2 Q 1 k 2 + i 4 D1 k 4 , i = 1 . . . 6. All other quantities for solution (93.11)–(93.12) are known [2], but due to cumbersomeness they are not represented.

93.5 Analysis of the Results Solution (93.11) is valid outside small neighborhoods of the resonance values of the wave number ki (i = 1…4) and k1 > k2 > k3 > k4 . To quantitatively assess the plate characteristics effect and nonlinearity of acceleration of its vertical displacements on the amplitude–phase perturbation characteristics and resonance values of ki , the calculations were performed at elastic modulus E = 0.5 × 109 ∼ 3 × 109 N/m2 , Poisson’s ratio ν = 0.34, and density ρ1 /ρ = 0.87

93 Interaction of Finite Amplitude Surface Waves in a Basin …

815

corresponding to the ice plate √ [11]. The longitudinal compressive force was chosen under the condition Q 1 < 2 D1 necessary for the ice plate stability [3]. For the case when Q 1 = 0, the dependence of the critical wave numbers of solution (93.11) on the fluid depth at E = 3 × 109 N/m2 and several thickness of the plate (0.5 and 1 m) is presented in [4]. Analysis showed that for a fixed value of the plate elastic modulus an increase in its thickness leads to a decrease in the values of ki . Moreover, an increase in the basin depth leads to an increase in the values of ki , although the growth rate ki decreases with increasing depth and at depths exceeding 70 m, the values of ki remain almost constant. For Q 1 = 0, increase in the compressive force leads to an increase in ki values at fixed h and E. The distribution of the vertical displacement height of the plate along the wavelength of its bending deformation depends not only on the thickness and Young’s modulus of the plate material, but also on the slope ε = ak of the initial fundamental harmonic and on the amplitude of the second interacting harmonic, determined by formula (93.14). The dependence of a1 on the wave number for a plate thickness of 0.5 m (lines 1, 2) and 1 m (line 3, 4) is shown in Fig. 93.1 at H = 100 m and Q 1 = 0. Here the lines with numbers 1, 3 were obtained at E = 3 × 109 N/m2 , and with numbers 2, 4 at E = 109 N/m2 . Line number 5 corresponds to the case of nonlinear interaction in open water in the absence of a plate. The dashed line is taking into account the nonlinearity of acceleration of vertical displacement of the plate, without the solid line. √ For k = 0, the amplitude a1 takes the value 1/ 2. It can be seen that an increase in the plate thickness manifests in an increase in a1 . The function a1 (k) has a local minimum, the value of which increases with increasing both the plate thickness and Young’s modulus. Moreover, the position of the minimum value shifts towards lower values of the wave number. The influence of nonlinearity of acceleration of vertical plate displacements manifests itself in decrease of a1 with wave number increasing. For a fixed plate thickness, the difference in the effect of its elasticity disappears with a decrease in the wavelength. Accordingly, in the short-wavelength range, there is only difference in value a1 with taking into account and without the nonlinearity of the acceleration of the plate’s vertical displacement [4]. A quantitative estimate of the change in the fluctuation frequency σ (93.13) depending on the wave number at Q 1 = 0 and H = 100 m allows us to obtain Fig. 93.2. Here, lines 1, 3 are shown for a1 > 0 and lines 2, 4 for a1 < 0. Lines 1, 2 are the plate thickness 1 m, and lines 3, 4 are 0.5 m. Dashed lines are the frequency distribution when taking into account the acceleration nonlinearity of vertical displacements of the ice plate, and solid—without taking into account. It can be seen that taking into account the acceleration nonlinearity leads to an increase in the fluctuation frequency. It rises with increase of the elastic plate thickness too. A change in the sign of the amplitude a1 from plus to minus reduces the value of the fluctuation frequency at a fixed wave number. For Q 1 = 0, the dependence of a1 on k at the plate thickness of 0.5 m (lines 1, 2) and 1 m (line 3, 4√) at H = 100 m is shown in Fig. 93.3. Here E = 3 × 109 N/m2 , lines 1, 3—Q 1 = 1.5 D, lines 2, 4—Q 1 = 0. The solid lines characterize the dependence of a1 on k without taking into account the acceleration nonlinearity of the vertical

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Fig. 93.1 The dependence of the distribution of the second interacting harmonic amplitude by the wave number at Q 1 = 0

Fig. 93.2 Distribution of the fluctuation frequency formed under nonlinear interaction of wave harmonics by the wave number at Q 1 = 0

Fig. 93.3 The dependence of the distribution of the second interacting harmonic amplitude by the wave number in cases when Q 1 = 0 and Q 1 = 0

displacements of the plate, and the dashed lines—with taking into account. It is seen that the compressive force decreases the local minimum of the function a1 (k) in comparison with the value obtained at Q 1 = 0. However, it increases with increasing plate thickness. At large values of the wave number, the influence of acceleration nonlinearity of plate vertical displacements is enhanced (dashed lines) [2]. Dependence of the fluctuation frequency σ of the wave perturbation (93.14) formed during the nonlinear interaction of wave harmonics on the wave num9 2 ber at a = √ 1 m, H = 100 m, h = 1 m, E = 3 × 10 N/m ; Q 1 = 0 (lines 1, 3), Q 1 = 1.5 D (lines 2, 4) for the cases a1 > 0 (lines 1, 2) and a1 < 0 (lines 3,

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Fig. 93.4 Distribution of the fluctuation frequency formed under nonlinear interaction of wave harmonics by the wave number in cases when Q 1 = 0 and Q 1 = 0

4) are shown in Fig. 93.4. The solid lines in the figure are plotted without taking into account the acceleration nonlinearity of the plate vertical displacements, and the dashed lines—with taking into account (F20 = 0, F30 =0). The behavior of the presented lines for wave numbers from the range k > k1 shows that under given initial conditions, the effect of taking into account the acceleration nonlinearity of vertical displacements of a longitudinally compressed elastic plate is noticeably manifested in the case of deep water (k H >> 1). For a1 > 0 (lines 1, 2), it manifests itself in an increase in the value of the fluctuation frequency (dashed line), and for a1 < 0 (lines 3, 4) in its decrease. With an increase in the wavelength of the initial harmonic, the influence of the acceleration nonlinearity of the plate vertical displacement decreases. The presence of compressive force (Q 1 = 0) is manifested with a decrease in the fluctuation frequency of the wave perturbation in comparison with the frequency obtained at Q 1 = 0, both for a1 > 0 and a1 < 0.

93.6 Conclusion On the basis of the multiple scales method, asymptotic expansions are obtained up to third-order smallness for elevation of the plate–fluid surface and the potential velocity of liquid particles formed during the nonlinear interaction of two harmonics of progressive surface waves. The analysis of the dependence of the amplitude–phase characteristics of wave perturbations and resonance values of the wave number on the elastic and mass characteristics of the ice plate, wavelength, and slope of the initial fundamental harmonic wave and fluid depth is done. The expression of the second interacting harmonic amplitude is obtained. It is shown that, in the interaction of wave harmonics, the contribution of taking into account the acceleration nonlinearity of the elastic plate vertical displacements into the fluctuation phase depends on the sign of amplitude of the second interacting harmonic. In this case, the presence of compressive force is expressed in the lag of the fluctuation phase from the phase obtained in the absence of compression, and in a slight decrease in the amplitude.

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Acknowledgements The present study is carried out within the framework of the State Order No. 0827-2019-0003.

References 1. Batyaev, E.A., Khabakhpasheva, T.I.: Hydroelastic waves in a channel covered with a free ice sheet. Fluid Mech. (2015). https://doi.org/10.1134/S0015462815060071 2. Bukatov, A.A.: Nonlinear vibrations of a floating longitudinally compressed elastic plate in the interaction of wave harmonics of finite amplitude. Fluid Dyn. (2019). https://doi.org/10. 1134/S0015462819020046 3. Bukatov, A.E.: Effect of longitudinal compression on transient vibrations of an elastic plate floating on the surface of a liquid. Int. Appl. Mech. (1981). https://doi.org/10.1007/ BF00885650 4. Bukatov, A.E., Bukatov, A.A.: Vibrations of a floating elastic plate upon nonlinear interaction of flexural-gravity waves. J. Appl. Mech. Tech. Phys. (2018). https://doi.org/10.1134/ S0021894418040120 5. Bukatov, A.E., Bukatov, A.A.: Nonlinear oscillations of a floating elastic plate. Int. Appl. Mech. (2011). https://doi.org/10.1007/s10778-011-0406-9 6. Bukatov, A.E., Bukatov, A.A.: Finite-amplitude waves in a homogeneous fluid with a floating elastic plate. J. Appl. Mech. Tech. Phys. (2009). https://doi.org/10.1007/s10808-009-0107-x 7. Bukatov, A.E., Bukatov, A.A.: Interaction of surface waves in a basin with floating broken ice. Phys. Ocean. (2003). https://doi.org/10.1023/B:POCE.0000013230.35798.a4 8. Bukatov, A.E., Zharkov, V.V.: Formation of the ice cover’s flexural oscillations by action of surface and internal ship waves. Part 1. Surface waves. Int. J. Offshore Polar Eng. 7(1), 1–12 (1997) 9. Bukatov, A.E., Cherkesov, L.V.: Transient vibrations of an elastic plate floating on a liquid surface. Int. Appl. Mech. (1970). https://doi.org/10.1007/BF00889434 10. Gladun, O.M., Fedosenko, V.S.: Nonlinear steady-state vibrations of an elastic plate floating on the surface of a fluid of finite depth. Fluid Dyn. (1989). https://doi.org/10.1007/BF01051589 11. Kheisin, D.E.: Nonstationary problem of infinite plate fluctuations floating on the ideal fluid surface. Izv. AN SSSR, OTN. Mekhanika i Mashinostroyenie 1, 163–167 (1962) (in Russian) 12. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973) 13. Pogorelova, A.V., Kozin, V.M.: Flexural-gravity waves due to unsteady motion of point source under a floating plate in fluid of finiate depth. J. Hydrodyn. (2010). https://doi.org/10.1016/ S1001-6058(09)60172-4 14. Schulkes, R.M.S.M., Hosking, R.J., Sneyd, A.D.: Waves due to a steadily moving source on a floating ice plate. Part 2. J. Fluid Mech. (1987). https://doi.org/10.1017/S0022112087001812 15. Sturova, I.V.: Wave generation by an oscillating submerged cylinder in the presence of a floating semi-infinite elastic plate. Fluid Mech. (2014). https://doi.org/10.1134/S0015462814040103

Chapter 94

Analytical Solutions for Cylindrical Bending of Multilayered Orthotropic Plates Alexander V. Matrosov

and Dmitry P. Goloskokov

Abstract This work is devoted to the study of the application of two analytical solutions of the cylindrical bending of orthotropic plates for the analysis of layered structures. The first one is the famous Pagano’s solution and the second one is a solution obtaned by a method of initial functions (MIF). It is shown that when applying the Pagano solution, the resolving system of linear algebraic equations depends on the number of layers in the structure. When using the MIF solution, the resolving linear system always has a second order. It is noted that this fact is associated with arbitrary constants in the two solutions obtained: in the Pagano solution, they have no mechanical meaning, and in the MIF solution they represent displacements and stresses on the initial line. The results of calculations of multilayer composites and the time of their execution for two investigated solutions are presented.

94.1 Introduction Multilayered and laminated plates are elements currently widely used in building and engineering structures. Simplified models based on kinematic or force hypotheses are used to analyze the behavior of multilayer plates [9]. Constructions of a model for multilayered plate using kinematic hypotheses are based on the introduction of an a priori behavior of horizontal displacement of the plate. The most popular is the zig-zag model, in which the change in the horizontal displacement of the plate along the thickness of each layer changes in a straight line with the continuity in the interfaces between layers. Models based on force hypotheses determine the nature of the change in tangential stresses in each layer by its thickness and most often in accordance with some parabola [6, 7]. All these approaches are approximate. A. V. Matrosov (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] D. P. Goloskokov The Bonch-Bruevich Saint-Petersburg State University of Telecommunications, 61 Moika, St. Petersburg 191186, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_94

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Fig. 94.1 A design scheme of a multilayered plate

However, the solutions obtained on the basis of the equations of elasticity without involving additional hypotheses about the behavior of any component of the stress– strain state of the orthotropic layer of the plate are of interest. We are aware of two such solutions: Pagano’s solution [8] and the method of initial functions (MIF) solution [2]. Both of these solutions are exact analytical solutions of the differential equations of the orthotropic elastic rectangle. The main goal of this work is a comparative analysis of the application of these solutions to the problems of bending of multilayered plates.

94.2 Formulation of the Problem Let’s consider a multilayered elastic plate in a state of plain strain under a normal p p σx = q0 sin( py) and tangential τx y = t0 cos( py), p = mπ/a (m = 1, 2, . . . ) loads applied on the upper face x = 0 (Fig. 94.1). The bottom face x = h is free from loads: σx (h, y) = 0, τx y (h, y) = 0. On the sides y = 0, l simply supported boundary conditions are realized: σ y (x, 0) = 0, u(x, 0) = 0 and σ y (x, a) = 0, u(x, a) = 0 (u(x, y) and v(x, y) are displacements along the axis O x and O y, respectively). The lamina consists of n orthotropic layers. Each ith layer (i = 1, . . . , n) has a i . In each layer, a thickness h i and technical elastic constants E 1i , E 2i , G i12 , and ν12 i i local coordinate system O x y is introduced, the origin of which is located in the upper left corner of the layer rectangle, and the directions of the axes coincide with the directions of the axes of the global coordinate system shown in Fig. 94.1. On the interface of two layers, the continuity conditions of the following components of the stress–strain state are satisfied: u i (h i , y) = u i+1 (0, y), v i (h i , y) = v i+1 (0, y), σxi (h i , y) = σxi+1 (0, y), τxi y (h i , y) = τxi+1 y (0, y). In the next two sections, a brief derivation of two analytical solutions for the analysis of the bending of a multilayered plate will be presented.

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94.3 Pagano’s Solution A solution for an orthotropic rectangle was received by N.J. Pagano in his work [8]. For the problem of cylindrical bending of an orthotropic simply supported plate, he found a solution to the equilibrium equations of the plane strain problem of the theory of elasticity σx,x + τx y,y = 0,

σ y,y + τx y,x = 0

(94.1)

in the form σx = − p 2 f (x) sin py,  σ y = f (x) sin py,

(94.2)



τx y = − p f (x) cos py. This solution satisfies identically the equilibrium Eq. (94.1) with an arbitrary function f (x). The comma in the index of the stress components in (94.1) means the partial derivative with respect to the variable following it. And the stroke and two strokes of an arbitrary function f (x) in the solution (94.2) mean the first and second derivatives with respect to the variable x. An arbitrary function f (x) is found from the equation 



R22 f (x) − (R66 + 2R12 ) p 2 f (x) + R11 p 4 f (x) = 0. Si3 S j3 (i, j = 1, 2, 6) and Si j are the compliance coefficients with S33 respect to the axis of material symmetry. The general solution of this differential equation can be expressed as

Here Ri j = Si j −

f (x) =

4 

C j exp(m j x).

(94.3)

j=1

Here C j are arbitrary constants and 

a+b m1, m2 = ± p c where a = R66 + 2R12 ,

1/2



,

a−b m3, m4 = ± p c

1/2  b = a 2 − 4R11 R22 ,

1/2 ,

c = 2R22 .

It should be noted that the solution obtained is valid if the coordinate system for the rectangle is such that the origin is located in the middle of its left side, and the

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axis O x is directed up. To go to the coordinate system used in our work, we should replace the variable x by −(x + h/2), where h is the height of the rectangle. Pagano’s solution for ith layer can be written down as σxi

= − p sin pyi 2

4 

C j,i exp(−m j,i (xi + h i /2)),

j=1

σ yi = sin pyi

4 

C j,i m 2j,i exp(−m j,i (xi + h i /2)),

j=1

τxi y = − cos pyi u i = sin pyi

4 

 C j,i

j=1

vi =

4 

C j,i m j,i exp(−m j,i (xi + h i /2)),

j=1

R12 m j,i −

R22 2 p m j,i

(94.4)

 exp(−m j,i (xi + h i /2)),

4   sin pyi  C j,i R12 p 2 − R11 m 2j,i exp(−m j,i (xi + h i /2)). p j=1

For a multilayered plate with n layers 4n arbitrary constants C j,i , j = 1, . . . , 4, i = 1, . . . , n must be determined. These constants are determined from the solution of a linear system of algebraic equations obtained by satisfying the boundary conditions on the upper and lower faces of the plate and the conditions of interlayer interaction (see Sect. 94.2).

94.4 MIF Solution MIF is an analytical method for receiving a solution for an orthotropic rectangle, as Pagano’s solution, but arbitrary constants in this solution are the coefficients of the four initial functions defined in the form of trigonometric functions on an initial line x = 0. The equilibrium equations in displacements for an orthotropic body in the coordinate system O x y (see Fig. 94.1) in the operator form can be written as    A1,1 ∂x2 + A6,6 ∂ y2 u (x, y) + A1,2 + A6,6 ∂x ∂ y v (x, y) = 0,     A1,2 + A6,6 ∂x ∂ y u (x, y) + A6,6 ∂x2 + A2,2 ∂ y2 v (x, y) = 0.



(94.5)

Here Ai, j are elastic constants of the material and symbols ∂x and ∂ y are used for differential operators with respect to the variables x and y, respectively. Let’s consider Eq. (94.5) as the system of ordinary differential equations with respect to the variable x assuming the differential operator ∂ y as a symbolic parameter.

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The solution of this system is searched in the form of linear combinations of the stress–strain state components defined on the initial line x = 0 u(x, y) = L 1,1 u 0 (y) + L 1,2 v 0 (y) + L 1,3 σ 0 (y) + L 1,4 τx0y (y) , v(x, y) = L 2,1 u 0 (y) + L 2,2 v 0 (y) + L 2,3 σ 0 (y) + L 2,4 τx0y (y) .

(94.6)

Here L i, j (i = 1, 2, j = 1, . . . , 4) are unknown operator functions that depend on the symbolic parameter ∂ y and variable x and act on functions u 0 (y), v 0 (y), σx0 (y), τx0y (y) defined on the initial line x = 0. These are so-called initial functions. The MIF operator functions L i, j (i = 1, 2, j = 1, . . . , 4) are obtained as [2]     −A1,1 A2,2 + A21,2 + A1,2 A6,6 + A1,1 A6,6 α12 cos α1 ∂ y x +     + A1,1 A2,2 − A21,2 − A1,2 A6,6 − A1,1 A6,6 α22 cos α2 ∂ y x d,      2 L 1,2 = A6,6 α2 −A2,2 − A1,2 α1 sin α1 ∂ y x +     d, + α1 A2,2 + A1,2 α22 sin α2 ∂ y x      2 L 1,3 = α2 −A2,2 + A6,6 α1 sin α1 ∂ y x +       ∂ y α1 α2 d , + α1 A2,2 − A6,6 α22 sin α2 ∂ y x         L 1,4 = A1,2 + A6,6 cos α1 ∂ y x − cos α2 ∂ y x ∂y d ,      2 L 2,1 = A6,6 α2 −A1,2 − A1,1 α1 sin α1 ∂ y x + (94.7)     + α1 A1,2 + A1,1 α22 sin α2 ∂ y x (α1 α2 d),     2 L 2,2 = A6,6 A1,2 + A1,1 α1 cos α1 ∂ y x −     d, − A1,2 + A1,1 α22 cos α2 ∂ y x         L 2,3 = A1,2 + A6,6 cos α1 ∂ y x − cos α2 ∂ y x ∂y d ,      L 2,4 = α2 −A6,6 + A1,1 α12 sin α1 ∂ y x +       ∂ y α1 α2 d . + α1 A6,6 − A1,1 α22 sin α2 ∂ y x

L 1,1 =



 2d4 2d4 , α2 = , d = d22 − 4d0 d4 , d0 = A2,2 A6,6 , d2 = d2 − d d2 + d A1,1 A2,2 − 2 A1,2 A6,6 − A1,2 2 , d4 = A1,1 A6,6 . Now to get a solution (94.6) it is necessary to choose the type of initial functions and to perform an impact on them by the MIF operators (94.7). Let the initial functions be chosen as trigonometric functions u 0 = u 0 sin ( py), 0 v = v 0 cos ( py), σx0 = σ 0x sin ( py), τx0y = τ 0x y cos ( py), p = mπ/A and the overlined coefficients are real constants. In this case, the MIF solution can be written in the matrix form as 0 (94.8) U = TLU . Here α1 =

Here U = {u, v, σx , σ y , τx y } is the vector of the stress–strain state components, 0 U = {u 0 , v 0 , σ 0x , τ 0x y } is the vector of the coefficients of the initial functions,

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T = sin( py), cos( py), sin( py), sin( py), cos( py) is a diagonal matrix of dimensions (5 × 5) and L is a matrix of dimensions (5 × 4) with following elements:  L 1,1 = L 3,3 = A1,1 A6,6 α1 2 − A1,1 A2,2 + A1,2 2 + A1,2 A6,6 cosh (α1 px) +

  + −A1,1 A6,6 α2 2 + A1,1 A2,2 − A1,2 2 − A1,2 A6,6 cosh (α2 px) /d,  L 1,2 = −L 4,3 = −A6,6 α2 −A1,2 α1 2 − A2,2 sinh (α1 px) +

  + α1 A1,2 α2 2 + A2,2 sinh (α2 px) / (α1 α2 d) ,  L 1,3 = α2 A6,6 α1 2 − A2,2 sinh (α1 px) +

  + α1 −A6,6 α2 2 + A2,2 sinh (α2 px) / (α1 α2 d) ,    L 1,4 = −L 2,3 = A1,2 + A6,6 (cosh (α1 px) − cosh (α2 px)) / ( pd) ,  L 2,1 = −L 3,4 = A6,6 α2 −A1,1 α1 2 − A1,2 sinh (α1 px) +

  + α1 A1,1 α2 2 + A1,2 sinh (α2 px) / (α1 α2 d) ,  L 2,2 = L 4,4 = A6,6 A1,1 α1 2 + A1,2 cosh (α1 px) +

  + −A1,1 α2 2 − A1,2 cosh (α2 px) /d,  L 2,4 = α2 A1,1 α1 2 − A6,6 sinh (α1 px) + 

 + α1 −A1,1 α2 2 + A6,6 sinh (α2 px) / (α1 α2 d) .

(94.9)

To analyze a multilayer plate, we see that two initial functions are the specified p p load on the upper plane of the first layer: σx0 = σx , τx0y = τx y . The two remaining initial functions u 0 = u 0 sin ( py), v 0 = v 0 cos ( py) are unknown. We find them from satisfying the boundary conditions on the lower face of the last layer. To get a system for determining unknown coefficients u 0 and v 0 , it should calculate four components on the bottom face x = h 1 of the first layer using (94.8). They are initial functions for the second layer. The four components on the bottom face x = h 2 of the second layer using (94.8) are evaluated. They are initial functions for the third layer and so on. Finally, the stresses σx and τx y on the bottom face of the nth layer are calculated. They depend linearly from the unknown coefficients u 0 and v 0 . And so the linear algebraic system of two equations is obtained regardless the number of layers in the lamina.

94.5 Numerical Results and Discussion A 43-layer plate with layers of the same thickness h i = h/43 and aspect ratio h/a = 1/2 has been analyzed. All odd layers are orthotropic with following technical constants: E x = 1 × 106 psi, E y = 25 × 106 psi, E z = 1 × 106 psi, ν yx = 0.25,

94 Analytical Solutions for Cylindrical Bending of Multilayered Orthotropic Plates

a

825

b

Fig. 94.2 Normal stress σ y /q0 in the center section y = a/2 (a) and horizontal displacement v E 1 /q0 h on the left side y = 0 (b) of the rectangle

ν yz = 0.25, νzx = 0.25, G yx = 5 × 105 psi, G yz = 5 × 105 psi, G zx = 2 × 105 psi. And a material of even layers are the odd layers material rotated 90 degrees: E x = 25 × 106 psi, E y = 1 × 106 psi, E z = 1 × 106 psi, ν yx = 0.25, ν yz = 0.25, νzx = 0.25, G yx = 5 × 105 psi, G yz = 5 × 105 psi, G zx = 2 × 105 psi. p Loads on the upper face of the lamina x = 0 are as follows: σx = q0 sin(π x/a), p τx y = 0. In Fig. 94.2, normal stress σ y /q0 in the center section y = a/2 and horizontal displacement v E 1 /q0 h on the left side y = 0 are presented. The even layers work for bending taking most of the load applied because their elastic modulus E y is greater than the modulus E y of the odd layers. So their bending stresses σ y /q0 are greater than the same stresses in the odd layers. The graph of the horizontal displacement v E 1 /q0 h shows that its change confirms the well-known hypothesis of a broken line (zig-zag model). This hypothesis is used often to build approximate theories of bending of layered plates. This lamina has been analyzed by two solutions and the results were identical. But the calculation times were significantly different: it took 0.0001 s for the initial function method in the Maple system while 0.156 s for Pagano’s solution, 103 times more. This fact is explained that in the MIF solution the order of the resolving linear algebraic system doesn’t depend on the number of layers in a lamina and is always equal to three. Whereas in the Pagano method the order of the resolving algebraic system depends on the number of layers in the construction and is equal to N = 4n (n is a number of layers). In the example considered n = 43 and the order of the system was equal to 172. Table 94.1 contains the average system solution time for finding unknown coefficients using Pagano’s solution depending on the number of layers of the lamina in the Maple system using the specialized LinearSolve function from the LinearAlgebra

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Table 94.1 Average time to solve a system for Pagano’s solution by two Maple solvers Number of layers LinearSolve solve 11 23 43 83 103 203 253

0.016 0.063 0.156 0.656 0.968 2.620 5.070

0.577 1.124 4.820 31.278 62.151 191.414 393.809

package and the universal solve function. This shows that the specialized function, rather than the universal solver function, should be used to solve systems of linear algebraic equations of large order.

94.6 Conclusion The two solutions studied in this paper are essentially the same solution, but presented in different forms. Their difference lies in the arbitrary constants involved in both solutions. And this difference leads to two different algorithms for analyzing layered plates based on these solutions. The Pagano solution-based algorithm has lower computational efficiency compared to the MIF-based algorithm. For a layered structure of 10 layers, its effectiveness is ten times higher than the first algorithm effectiveness. And for a structure of 200 layers, its advantage is already expressed tens of thousands of times. MIF was popular in the 70–90 of the last century, but with its limited ability to satisfy all the boundary conditions for both three- and two-dimensional plate bending problems, it was forgotten by the wide scientific community. However, with the advent of analytical computing systems (Maple, Mathematica and others), its further development continued on a new computational base. It should be noted that a number of works devoted to the construction of approximate analytical solutions to the plane problems of analyzing complex structures satisfying all boundary conditions using solutions based on the MIF [1, 3]. A number of works are devoted to the construction of an exact analytical solution for rectangular regions (finite and semi-infinite) [4, 5].

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References 1. Goloskokov, D.P., Matrosov, A.V.: Comparison of two analytical approaches to the analysis of grillages. In: 2015 International Conference on “Stability and Control Processes” in Memory of V.I. Zubov – Proceedings 7342169, pp. 382–385 (2015) 2. Goloskokov, D.P., Matrosov, A.V.: Approximate analytical approach in analyzing an orthotropic rectangular plate with a crack. Mater. Phys. Mech. 36, 137–141 (2018) 3. Goloskokov, D.P., Matrosov, A.V.: Approximate analytical solutions in the analysis of thin elastic plates. In: 2018 AIP Conference Proceedings 1959.070012 (2018) 4. Kovalenko, M.D.: The Lagrange expansions and nontrivial null-representations in terms of homogeneous solutions. Dokl. Phys. 42, 90–92 (1997) 5. Kovalenko, M.D., Menshova, I.V., Kerzhaev, A.P.: On the exact solutions of the biharmonic problem of the theory of elasticity in a half-strip. Z. Angew. Math. Phys. 69, 121 (2018) 6. Librescu, L., Hause, T.: Recent developments in the modeling and behavior of advanced sandwich constructions: a survey. Compos. Struct. 48, 1–17 (2000) 7. Noor, A.K., Burton, W.S., Bert, C.W.: Computational models for sandwich panels and shells. Appl. Mech. Rev. 49, 155–199 (1996) 8. Pagano, N.J.: Exact solutions for composite laminates in cylindrical bending. J. Compos. Mater. 3, 398–411 (1969) 9. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press LLC, Florida (2004)

Chapter 95

A Rectangular Prism Under Own Weight: Comparison of the Method of Initial Functions and the Finite Element Method Guryi N. Shirunov, Alexander V. Matrosov, and Denis A. Sarvilin Abstract In this paper, we study the bending of a massive elastic linearly deformable body under its own weight. Two approaches are analyzed: analytical one based on the method of initial functions (MIF) and numerical one based on finite element modeling (FEM). An algorithm for constructing a general solution for a linearly elastic parallelepiped using the superposition method based on three MIF solutions is described. The results of analyzing a massive bridge using two approaches are presented. The advantages and disadvantages of the analytical approach are proposed and numerical modeling is analyzed. Inaccurate satisfaction of the boundary conditions on the horizontal load-free faces when using the FEM is noted. The inability of the FEM to track some of the nuances in the behavior of shear stresses on clamped faces is also noted.

95.1 Introduction In practical analysis, the own weight of plates is taken into account in the form of an equivalent surface load applied to the horizontal plane of the plate. This plane can be upper, lower, or middle one. Such approaches are justified if the plate is thin. However, for thick plates and even for medium-thickness plates, such a technique for taking into account its own weight leads to serious discrepancies with actual distribution of stresses over the plate thickness under their own weight. Nowadays, the most popular numerical model is a finite element method (FEM). It takes into account that own weight of the body is not a surface load but a load G. N. Shirunov (B) Admiral Makarov State University of Maritime and Inland Shipping, 5/7, Dvinskaya str, St. Petersburg 198035, Russia A. V. Matrosov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] D. A. Sarvilin Tekton Ltd., 13 Mozhayskaya, St. Petersburg 190013, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_95

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Fig. 95.1 A design scheme of the rectangular prism under its own weight

distributed over its volume at the grid nodes in accordance with its ideology of approximating an elastic body by finite elements. However, some analytical approaches are developed to solve problems with structures taking into account dead weight. We note several works: optimal design of columns and plates [1, 2], buckling of vertical plates [10], investigation of effects of dead loads on static analysis of plates and prestretched plate [9, 11], and large deformation analysis of Euler–Bernoulli beamshell [8]. An exact analytical solution to the problem of bending a rectangular prism under its own weight does not exist. However, using the superposition method [3], an approximate analytical solution in the form of trigonometric polynomials can be constructed. In this paper, we analyze the results of an approximate analytical solution constructed by the superposition of three solutions obtained by the MIF and the FEM numerical simulation.

95.2 Formulation of the Problem Let’s consider an isotropic rectangular prism (E is a Young’s modulus, ν is a Poisson’s ratio, ρ is a prism density) in the Cartesian rectangular coordinate system O x yz with dimensions L x × L y × h subjected by its own weight (see Fig. 95.1). Two vertical faces x = 0, L x are rigidly fixed (clamped ones). The displacements on them are equal to zero: u = 0, v = 0, and w = 0. The remaining four faces are free of loads. On the faces y = 0, L y , the following stresses are equal to zero: σ y = 0, τx y = 0, and τ yz = 0. The faces z = 0, h have following boundary conditions: σz = 0, τzy = 0, and τx z = 0. We assume that the force of gravity is directed along the Oz-axis.

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95.3 Solution with the Superposition Method Under assumptions about the direction of the self-weight force, the differential equations of equilibrium of an isotropic body in displacements in operator form can be written as follows:    2(ν − 1)∂x2 + (2ν − 1) ∂ y2 + ∂z2 u − ∂x ∂ y v − ∂x ∂z w = 0,    −∂x ∂ y u + 2(ν − 1)∂ y2 + (2ν − 1) ∂x2 + ∂z2 v − ∂ y ∂z w = 0,     ρg (2ν − 1) . −∂x ∂z u − ∂ y ∂z v + (2ν − 1) ∂x2 + ∂ y2 + 2(ν − 1)∂z2 w = − G

(95.1)

In (95.1) G = E/2/(1 + ν), g is the gravitational acceleration, ∂x, ∂ y, and ∂z are symbols for the operators of differentiation with respect to the variables x, y, and z, respectively, and u, v, and w are displacements along the axes O x, O y, and Oz. A general solution of the inhomogeneous system (95.1) is a sum of a general solution of the homogeneous system and a particular solution of the inhomogeneous system: inhom (95.2) W = Whom gen + W par . Here W = {u, v, w} is a vector of displacements, Whom gen is a vector of general solution of the homogeneous system, and Winhom is a partial solution. par The general solution of the homogeneous system can be found by the superposition method [6]. In accordance with it, the general solution is obtained as the sum of three solutions, each of which allows satisfying arbitrary boundary conditions on two opposite faces of a rectangular prism. As such ones, we have taken three MIF solutions hom + Whom + Whom . Whom gen = W x y z

The MIF solution expresses all components of the stress–strain state (SSS) in the solid through the components determined on one of the plane x = 0, y = 0, or z = 0 called with the initial plane x = 0 allows to satisfy the initial plane. The solution Whom x arbitrary boundary conditions on the faces x = 0 and x = L x of the parallelepiped. hom with the initial planes y = 0 and z = 0, respectively, The two others Whom y , Wz allow to meet arbitrary boundary conditions on the faces y = 0, y = L y and z = 0, z = h, respectively. We present an algorithm for obtaining the MIF solution in the case of the initial plane z = 0. The other two solutions are obtained similarly taking into account the corresponding initial plane. The main idea of the MIF is to express the solution of homogeneous system through displacements and stresses determined on an initial plane z = 0: = Lz U0z . (95.3) Whom z

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0,z Here U0z = {u 0,z , v 0,z , w 0,z , σz0,z , τ yz , τ 0,z } is a vector of initial functions deter  xz  ij mined on the plane z = 0, Lz = L z ∂x , ∂ y , E, ν, z , i = 1, . . . , 3, j = 1, . . . 6, is a matrix of the MIF operators. Substituting (95.3) in the homogeneous system (95.1) and taking into account independentness and arbitrariness of the initial functions, we receive the six system of three partial differential equations of the second order to find the MIF operators (each system for three ones). Let’s consider these equations as the ordinary differential equations in respect of the variable z with symbolic parameters ∂x and ∂ y . To find the MIF operators, it should solve a series of Cauchy problems with following initial values: Lz |z=0 = E (an identical matrix of three order) and the values of the operator derivatives when z = 0 can be found using Hooke’s law and the Cauchy relations. Finally, the MIF operators have been received in a closed-form [4]





sin ∂x2 + ∂ y2 z  ij  L z = ai j ∂x2 + ∂ y2 sin ∂x2 + ∂ y2 z + bi j + ∂x2 + ∂ y2  

⎞ ⎛ 

z cos sin ∂x2 + ∂ y2 z ∂x2 + ∂ y2 z ⎜ ⎟ ⎟.   +ci j cos +  ∂x2 + ∂ y2 z + di j ⎜ ⎝− ⎠ 3  ∂x2 + ∂ y2 ∂x2 + ∂ y2 2

(95.4)

The constants ai j , bi j , ci j , and di j (some of them can be equal to zero but not simultaneously) for every operator are products of powers of z from zero up to the first, powers of operators ∂x and ∂ y from zero up to the second, and a factor depending on E and ν. Assume that the initial functions can be represented in trigonometric series (sm = sin (αm x), cm = cos (αm x), sn = sin (βn y), cn = cos (βn y), αm = mπ/L x , βn = nπ/L y , and m and n are integers) ∞ 

u 0,z =

∞ 

u 0,mn cm sn , v 0,z = z

m,n=0 ∞ 

σz0,z =

m,n=0 ∞ 

wz0,mn sm sn ,

m,n=0 ∞ 

0,z qz0,mn sm sn , τ yz =

m,n=0

∞ 

vz0,mn sm cn , w 0,z = pz0,mn sm cn , τx0,z z =

m,n=0

tz0,mn cm sn ,

(95.5)

m,n=0

then the vector Wzhom will also be obtained using (95.3) as trigonometric series Whom z

=

∞ 

mn 0,mn Tmn , z Lz U z

(95.6)

m,n=0

= {u 0,mn , vz0,mn , wz0,mn , qz0,mn , pz0,mn , tz0,mn } are vectors where U0,mn z z  mnof unknown  mn mn coefficients, Tz = cm sn , sm cn , sm sn  is a diagonal matrix, Lz = L z,i j (z, E, ν) , i = 1, . . . , 3, j = 1, . . . 6 are matrices in which elements are results of impacting

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(values) of the MIF operators on the initial functions in the form of the product of 2 trigonometric functions (95.5). To get these values, it should replace ∂x2 with −αm 2 2 and ∂ y with −βn in the MIF operator expressions (95.4). At this case, the four main terms in these expressions become [5] 

2 + β 2 sinh αm n



 

sinh α2 + β 2 z 

m n 2 + β2 z , 2 + β2 z ,  αm , cosh α n m n α2 + βn2     m 2 + β2 x 2 + β2 x sinh αm α cosh n m n −  3/2 . 2 2 2 2 αm + βn αm + βn

Similarly, we have obtained two other MIF solutions taking into account the initial functions on the planes x = 0 and y = 0. These solutions can be written in uniform form as [5] = Whom x

∞  m,n=0

mn 0,mn Tmn , Whom = x Lx U x y

∞ 

mn 0,mn Tmn . y Ly Uy

(95.7)

m,n=0

mn Here the matrices Tmn x and T y are diagonal ones of the form

Tmn x = sin βm y sin γn z, cos βm y sin γn z, sin βm y cos γn z, Tmn y = sin γm z cos αn x, sin γm z sin αn x, cos γm z sin αn x, in which αn = nπ/L x , βm = mπ/L y , γm = mπ/ h, and γn = nπ/ h. Lmn x and Lmn are the results of the MIF operators on the trigonometriy cal functions, U0,mn = {u 0,mn , vx0,mn , wx0,mn , qx0,mn , px0,mn , tx0,mn } and U0,mn = x x y 0,mn 0,mn 0,mn 0,mn 0,mn {u 0,mn , v , w , q , p , t } are vectors of unknown coefficients. y y y y y y The partial solution of the inhomogeneous system has been found in the following form [7]: ρg(1 + ν)(1 − 2ν) 2 z . u par = v par = 0, w par = − (95.8) 2E(1 − ν)   So the Winhom = u par , v par , w par . par Using Hooke’s law and the Cauchy relations, the stresses σx , σ y , σz , τx y , τx z , and τ yz can be expressed using the general solution (95.2). To satisfy the boundary conditions, the corresponding components of the stress– strain state (see Fig. 95.1) are calculated on all six faces of the prism using the general solution (95.2) and the formulae (95.6)–(95.8). The obtained expressions of the components and the given values of the boundary conditions on every face are expanded in the corresponding double trigonometric Fourier series. After equating the coefficients with the corresponding harmonics, an infinite system of linear algebraic equations with respect to unknown coefficients in solutions (95.6), (95.7) is

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(a)

(b)

Fig. 95.2 Dimensionless displacement y = L y /2

(a) Fig. 95.3 Dimensionless normal y = L y /2

wE σz (a) and stress (b) in the section x = L x /2 ρgh 2 ρgh

(b ) σx τx z (a) and tangential (b) stresses in the section x = 0 ρgh ρgh

obtained. This infinite system is solved by the reduction method, leaving it in terms with indices not exceeding a given integer value M. The error in satisfying the boundary conditions on the faces of the prism is a criterion for the accuracy of the solution obtained since it identically satisfies the equilibrium, Eq. (95.1), for any values of the unknown coefficients. Retention of a larger number of terms in the solution leads to a decrease in the error of satisfying the boundary conditions making it possible to obtain an approximate analytical solution with an arbitrary degree of accuracy.

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95.4 Numerical Results and Discussion As an example, a massive isotropic (E = 2 · 1010 Pa, ν = 0.2, ρ = 2500 kg m−3 , g = 9.81 m s−2 ) bridge with dimensions L x = 2 m and L y = h = 1 m was considered. Two solutions were analyzed: MIF and FEM. In the first one, harmonics of up to 23 orders were held in double trigonometric series for each coordinate. Increasing of the number of harmonic did not lead to a significant improvement in the numerical values of the stress–strain state components at the solid boundary and therefore the calculations were performed with the specified number of harmonics in the trigonometric series. The second solution was received by a FEM program Ing+ using hybrid finite elements. The bridge was divided into rectangular elements: 30 ones along the axes O x, 10 ones along the axes O y, and 25 ones along the axes Oz. Thus, the partition grid consisted of 7500 finite elements with dimensions 6.67 × 10 × 4 cm. An attempt to increase the number of finite elements resulted in the values of some stress–strain state components at the solid boundary improved while others deteriorated dramatically such as tangent stresses. On Fig. 95.2, the vertical displacement wE/ρgh 2 and the normal stress σz /ρgh in the center cross section x = L x /2, y = L y /2 are presented. On the figures of this paper, the MIF solutions are shown as a solid (red) line, and the FEM solutions are shown as a dashed (green) line. It’s seen that the vertical displacement (Fig. 95.2a) is greater in FEM model. This fact can be explained by the fact that the finite element model did not allow us to accurately satisfy the boundary conditions on the horizontal faces of the plate. The normal stress σz graph (Fig. 95.2b) shows that this stress is not equal to zero on these surfaces. Thus, the plate is affected by an additional surface loads on the upper and bottom faces. As for the MIF solution, it exactly satisfies the boundary conditions on these faces with respect to the normal stress σz . Figure 95.3 presents the bending stress σx /ρgh and the tangential stress σx z /ρgh in the cross section x = 0, y = L y /2. The discrepancy in the bending stresses σx (Fig. 95.3a) is explained by a small number of terms held in the trigonometric series of the MIF solution. As the number of members held increases, the results will converge. The graph of the tangential stress τx z (Fig. 95.3b) is of interest. On the upper and lower faces of the plate, this stress should be equal to zero. If the MIF solution has a tendency to go to a neighborhood of zero at these points, then the FEM solution is far from zero value. It should also be noted about two small local maxima in tangential stresses in the neighborhood of horizontal planes, marked on the plots of the MIF solution. They are not presented so clearly, as the graph smoothed by Lanczos’s factors is given. These maxima must necessarily be present in the shear stresses in the clamped face. They are absent on the graph of the FEM solution.

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95.5 Conclusion The study of the approximate analytical solution by the superposition method using three MIF solutions and the numerical FEM solution of the bending of the massive bridge under the influence of its own weight has showed a slight advantage of the analytical approach over the numerical one. To obtain more accurate results at the points on the border of the parallelepiped, it is necessary to keep a larger number of terms in the trigonometric series in the approximate analytical solution, whereas for internal points 23 terms for each coordinate were sufficient. Unfortunately, the FEM did not allow exactly (or even with a given accuracy) to satisfy the boundary conditions on the free horizontal surfaces of the bridge. This additional surface load affects the stress–strain state of the entire structure, increasing the values of all its components (displacements and stresses). Moreover, the FEM solution gives incorrect results when calculating the tangential stress in the clamped faces of the bridge: it does not tend to zero on their upper and lower edges.

References 1. Atanackovic, T.M.: Optimal shape of column with own weight: Bi and single modal optimization. Meccanica 41, 173–196 (2006) 2. Erbatur, F., Mengi, Y.: Optimal design of plates under the influence of dead weight and surface loading. J. Struct. Mech. 5, 98–105 (1977) 3. Matrosov, A.V.: A superposition method in analysis of plane construction. In: 2015 International Conference on “Stability and Control Processes” in Memory of V.I. Zubov – Proceedings 7342156, pp. 414–416 (2015) 4. Matrosov, A.V., Shirunov, G.N.: Algorithms for obtaining closed forms of operators of the method of initial functions for 3D problems of elasticity theory. Bull. Civ. Eng. 1, 136–144 (2014) (in Russian) 5. Matrosov, A.V., Shirunov, G.N.: A superposition method for solving a problem of an elastic isotropic parallelepiped Bull. Saint-Petersburg State Univ. Ser. 10. 2, 79–93 (2015) (in Russian) 6. Matrosov, A.V., Shirunov, G.N.: Numerical-analytical computer modelling of a clamped isotropic thick plate. In: 2014 International Conference on Compute Technology in Physics and Engineering Application – Proceedings 6893300, p. 96 (2014) 7. Matrosov, A.V., Shirunov, G.N.: Analyzing thick layered plates under their own weight by the method of initial functions. Mater. Phys. Mech. 31, 36–39 (2017) 8. Sohouli, A.R., Kimiaeifar, A., Mohsenzadeh, A., Mohebpour, S.R.: Large deformation analysis of Euler-Bernoulli beamshell under own weight based on HAM. Cent. Eur. J. Eng. 2, 146–153 (2012) 9. Takabatake, H.: Effects of dead loads on the static analysis of plates. Struct. Eng. Mech. 42, 761–781 (2012) 10. Wang, C.M.: Buckling of standing vertical plates under body forces. Int. J. Struct. Stab. Dyn. 2, 151–161 (2002) 11. Yesil, U.B.: The effect of own weight on the static analysis of a pre stretched plate-strip with a circular hole in bending. Mech. Compos. Mater. 53, 243–252 (2017)

Chapter 96

Surface Dislocation Interaction by the Complete Gurtin–Murdoch Model Mikhail Grekov and Tatiana Sergeeva

Abstract The interaction of an edge dislocation array with a free surface of an elastic solid is considered within the framework of the complete Gurtin–Murdoch surface elasticity model. The generalized Young–Laplace boundary equation of the plane elasticity problem is derived in terms of complex variables for the general case of a curvilinear cylindrical surface. Using this boundary equation in the case of a line edge dislocation array placed near the planar surface, the solution of the corresponding boundary-value problem is reduced to the integro-differential equation in the unknown complex displacement. Based on the analytical solution of this equation in terms of complex Fourier series, we present the numerical results for the stress field at the surface and discuss the difference between them from those obtained with the simplified Gurtin–Murdoch models.

96.1 Introduction Surface energy the concept of which in solids was first introduced by Gibbs in the 70th years of the nineteenth century on the basis of the thermodynamics of solid surfaces is of primary importance in analyzing properties of nanometer-sized subsurface regions, materials with nano-inhomogeneities, nanosized objects and structures, and so on. Theoretical investigations of various mechanical problems at the nanoscale within framework of continuum mechanics have been made possible since Gurtin and Murdoch (GM) [12, 13] developed in 1975 the theory of surface elasticity incorporating surface stress and surface tension (residual surface stress). Original linearized GM model and its different simplifications became a widely used instrument to study the surface stress effects at the nanoscale. In the GM model, a surface is represented as a membrane-like layer adhering to the bulk material without slipping. According to Gurtin and Murdoch [12], the complete (original) linearized constitutive equation M. Grekov (B) · T. Sergeeva Department of Computational Methods in Continuum Mechanics, St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_96

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of a surface corresponding to the small deformations is written in the general case as  s = σ0 A + (λs + σ0 )A trEs + 2(μs − σ0 )Es + σ0 ∇s u, Es =

(96.1)

1 (∇s u) · A + A · (∇s u)T 2

where  s is the Piola–Kirchhoff surface stress tensor of the first kind, which is used in the Lagrangian description [6], Es is the surface strain tensor, u is the displacement vector, A = I − n ⊗ n is the unit tangent tensor, I is the three-dimensional unit tensor, n is the unit vector normal to the surface that points away from the bulk material, ∇s = ∇ − n ∂/∂n is the two-dimensional Hamilton operator acting in the reference configuration of the surface (before deformation caused by the external load and perturbation source), σ0 is the residual surface stress (surface tension), and λs , μs are the surface elastic constants similar to the Lamé constants λ, μ of the bulk material. In most papers in the literature, Eq. (96.1) has been simplified by neglecting either the surface tension σ0 (e.g. [2, 6–10]), or a whole (e.g. [14, 23–25]) or a part (e.g. [1, 11, 15–17]) of the last term or three last terms [4, 21, 26–28] by keeping the influence of only the surface tension on the stress field in the bulk of nanostructures. In addition to the approaches based on these simplified versions, many problems at the nanoscale have been solved with the original GM model corresponding to Eq. (96.1) (see, for example, [5, 9, 18, 19]). It should be noted that all authors who used simplified versions of GM model didn’t adduce any reasons for those simplifications. The only effect that has been estimated is the effect of neglecting the surface tension [3, 5, 18]. This effect is negligible for the case of a circular cavity in an infinite plane under remote loading [5, 18] and can be substantial for an elliptic cavity [5], multiple circular cavities [18], and the edge dislocation near the interface [3]. The objective of the paper is to analyze and compare the results obtained with the original and all simplified versions of the GM model [12, 13] in the problem on the surface dislocation interaction at the nanoscale.

96.2 Governing Equations for Two-Dimensional Material Surface In this section, based on the original Gurtin and Murdoch surface elasticity theory, we will derive the boundary conditions in complex variables for the 2D problems assuming that the surface of the bulk material is cylindrical with generatrix parallel to the x3 -axis of the Cartesian coordinates x j ( j = 1, 2, 3). According to the GM model [12, 13], the adherence of a surface with a bulk material without slipping means continuity of the displacements:

96 Surface Dislocation Interaction by the Complete Gurtin–Murdoch Model

us = ub = u, (x1 , x2 ) ∈ ,

839

(96.2)

where the superscript s(b) describes the surface (bulk material), the curve  is the intersection of the surface and the plane x3 = const. The boundary conditions at the surface follow the generalized Young–Laplace law [12, 13, 22]: (96.3) n ·  = −∇s ·  s + q = qs + q, where q is the traction created by an external load. Introducing the local Cartesian coordinates n, t, x3 with basis vectors n, t, k, one can represent the displacement vector as follows: u = u n n + u t t + u 3 k.

(96.4)

Applying the operator ∇s to the displacement vector (96.4) in the case of the 2D problems yields the following expression for the surface gradient tensor ∇s u: s k ⊗ k, ∇s u = εtts t ⊗ t + υ s t ⊗ n + ε33

(96.5)

where n and t are the normal and tangential vectors to the surface, respectively, and (see, for example, [18, 20]) εtts =

un ∂u n ∂u t ut ∂u 3 s + , υs = − + , ε33 = . ∂l R R ∂l ∂ x3

(96.6)

In Eq. (96.6), l is the arc length of the undistorted surface and R its local curvature radius. Taking into account Eqs. (96.1) and 96.5, represent surface stress tensor  s in terms of its components in the coordinate system n, t, x3 as s k ⊗ k.  s = σtts t ⊗ t + σtns t ⊗ n + σ33

(96.7)

According to the inseparability condition (96.2), εisj = εi j at the surface and so the components of the surface stress tensor can be written as s = σ0 + (λs + σ0 )εtt , σtts = σ0 + Ms εtt , σtns = σ0 υ s , σ33

(96.8)

where Ms = λs + 2μs , εi j is the strain tensor component of the bulk material (i, j = n, t, 3 or i, j = 1, 2, 3). Using representation (96.7) and taking into account that ∂σ33 /∂ x3 = 0 for the 2D problems, one can arrive at the following equality:  ∇s ·  s =

σs ∂σtts + tn ∂l R



 t−

σtts ∂τ s − R ∂l

 n.

(96.9)

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Components of the vector ∇s ·  s can be expressed through the complex displacement u = u 1 + iu 2 , where u 1 and u 2 are the displacements along x1 and x2 axes, respectively. The relation between the complex displacements u and u n + iu t at the point ζ ∈  is (96.10) u = −i(u n + iu t )eiα0 , where α0 is the angle between the tangent to  and the x1 -axis at the point ζ . Using relations dζ = dleiα0 = Rdα0 eiα0 , one can obtain from equality (96.10) the following expression: ∂u = −i ∂ζ



∂u n ∂u t +i ∂l ∂l

 +

u n + iu t , R

(96.11)

and so taking into account representation (96.11), two first equalities of Eq. (96.6) can be rewritten as follows: εtts = Re

∂u ∂u , υ s = −Im . ∂ζ ∂ζ

(96.12)

Using Eqs. (96.8), (96.9), (96.12), one can rewrite the boundary condition (96.3) in the form:    2 Ms ∂u σ0 ∂ u iα0 − + Re + σ0 Im e σnn + iσnt = R R ∂ζ ∂ζ 2     2 σ0 ∂u ∂ u iα0 − Im + q(ζ ) = q s (ζ ) + q(ζ ). −i Ms Re e (96.13) ∂ζ 2 R ∂ζ

96.3 Problem Formulation The surface is considered as a free boundary of a half-space which has elastic properties different from the same properties of the surface. The plane strain conditions are assumed to be satisfied with the presence of an array of the line edge dislocations and the surface stresses. So, we come to the 2D boundary-value problem for the elastic half-plane = {z : Im z < 0, Re z ∈ (−∞, +∞)} of the complex variable z = x1 + i x2 (i is the imaginary unit) with the rectilinear boundary  (Fig. 96.1). Dislocations with Burgers vectors b = (b1 , b2 ) are located in the points z k = ak − i h (a > 0, h > 0, k = 0, ±1, ±2, . . . ). Due to the fact that the surface is flat and the external load is absent, the boundary condition (96.13) is transformed to the following (see [9]): σ22 (x1 ) − iσ12 (x1 ) = −

i p ∂ 2u im ∂ 2 u − ≡ −q s (x1 ), x2 = 0, 2 ∂ x12 2 ∂ x12

(96.14)

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Fig. 96.1 Half-plane under the array of periodic edge dislocations with Burgers vectors b and traction qs induced by the surface s and σ s stresses σ11 12

where p = Ms + σ0 , m = Ms − σ0 , a bar over a symbol denotes complex conjugation. It should be noted that the simplified version neglecting surface tension σ0 or the part of the surface gradient tensor ∂u 2 /∂ x1 leads to the equality p = m = Ms , and neglecting the surface gradient tensor ∇s u leads to p = m = Ms − σ0 . In our problem with the flat surface, if we simplify Eq. (96.1) by omitting three last terms, we will arrive at the classical solution as q s ≡ 0.

96.4 Solution of the Problem Based on the superposition technique, Goursat–Kolosov’s complex potentials, and Muskhelishvili’s representations as in [10], we arrive at the following formulas for the stresses in any point z ∈ out of the boundary , similar to those derived in [10]: 



σ22 − iσ12 = 2Re [ (z) − (z)] + z (z) + (z) − z (z) − (z) −       (z − z) (z) − z (z) −  (z) + T (z) − T (z) + (z − z)T (z),    σ11 + σ22 = 4Re (z) − (z) − z (z) − (z) + 4Re T (z),

(96.15)

where a prime denotes the derivative with respect to the argument and π(z − z 0 ) , a  π(z − z 0 ) π(z − z 0 ) π H − (z + 2i h) cosec2 , (96.16) (z) = H − H ctg a a a

(z) = −H ctg

λ + 3μ iμ(b1 + ib2 ) , æ= . a(æ + 1) λ+μ The unknown function T is defined by the Cauchy-type integral:

H=

1 T (z) = 2πi

+∞ −∞

q(t) dt. t−z

(96.17)

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To find the function T , pass to the limit in the first Eq. (96.15) when x2 → −0. We come to the following integral equation in the unknown complex displacement u(x1 ):  8μu  (x1 ) − i(1 − æ) pu  (x1 ) + mu  (x1 ) −

+∞  pu (t) + mu  (t) 1+æ dt = 4Q(x1 )), 2π t − x1

(96.18)

−∞

where  Q(x1 ) = (æ + 1) H ctg (ξ − ζ 0 ) + H (ζ0 − ζ 0 )cosec2 (ξ − ζ 0 + (1 − æ)H ctg (ξ − ζ0 ) + 2i H, ξ = π x1 /a. Allowing for the periodicity of the function Q(x1 ), we seek the unknown function u  (x1 ) in terms of the following series: u  (x1 ) = a0 +

∞

 ilk x1 ak e + bk e−ilk x1 , lk = 2π k/a.

(96.19)

k=1

After substituting the series (96.19) into Eq. (96.18) and expanding the function Q into the complex Fourier series, coefficients ak and bk can be expressed in terms of the expansion coefficients of the function Q by the formulas derived in [9]. Once the function u  has been obtained from Eq. (96.19) through Fourier coefficients of the function Q, the function T (z) can be evaluated analytically from Eq. (96.17) using definition of the function q s in Eq. (96.14) and properties of Cauchytype integrals: ∞  1 ( pak + mbk )eilk z , Im z > 0, T (z) = − 2 k=1 (ma k + pbk )e−ilk z , Im z < 0.

(96.20)

Equations (96.20) and (96.15), (96.16) provide a way of evaluating the stress field everywhere in the bimaterial, arising due to the presence of the dislocation array.

96.5 Numerical Results and Discussion In the numerical investigation, we take the data related to the surface elastic properties of Al[111]: λs = 6.8511 N/m, μs = −0.3755 N/m that was determined by the embedded atom method in [17] and σ0 = 1 N/m [25]. The elastic constants of the bulk material (aluminum) are λ = 58.17 GPa, μ = 26.13 GPa.

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Fig. 96.2 The stress distribution at the surface within a half period when Burgers vector b = (b1 , 0)

Fig. 96.3 The stress distribution at the surface within a half period when Burgers vector b = (0, b2 )

The distribution of the dimensionless stress components σikj = σi j Ms /2μ2 bk × 10 (k = 1, 2) at the surface within a half period for classical (dotted lines) and nonclassical (solid lines) solutions is depicted in Figs. 96.2, 96.3 for two different Burgers vector orientations b = (b1 , 0) and b = (0, b2 ), respectively. These dependencies have been obtained using the original GM constitutive Eq. (96.1) in the case a = 10 nm and h = {1, 2, 5, 10} nm (curves 1, 2, 3, 4, respectively). Moreover, the stress field at the surface has been studied for two simplified versions of the GM model: (1) σ0 = 0 and (2) ∇s u = 0. The simplified version related to the case ∂u 2 /∂ x1 = 0 coincides with the case σ0 = 0 which is considered in [10] by solving the same problem of the surface dislocation interaction. Comparison of the numerical results shows that the maximum discrepancy between σ11 , σ12 calculated with the origin and simplified GM models is in the region from 2 to 13% in the case h = 1 nm. But the main difference between the origin and simplified GM models is that the last ones lead to the absence of the normal stress σ22 at the surface. 2

Acknowledgements This research was supported by the Russian Foundation for Basic Research under grant 18-01-00468.

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References 1. Bochkarev, A.O., Grekov, M.A.: Influence of surface stresses on the nanoplate stiffness and stability in the Kirsch problem. Phys. Mesomech. 22, 209–223 (2019) 2. Dai, M., Schiavone, P.: Analytic solution for a line edge dislocation in a bimaterial system incorporating interface elasticity. J. Elast. 132, 295–306 (2018) 3. Dai, M., Schiavone, P.: Edge dislocation interacting with a Steigmann-Ogden interface incorporating residual tension. Int. J. Eng. Sci. 139, 62–69 (2019) 4. Dai, M., Schiavone, P., Gao, C.F.: Surface tension-induced stress concentration around an elliptical hole in an anisotropic half-plane. Mech. Res. Commun. 73, 58–62 (2016) 5. Dai, M., Yang, H.B., Schiavone, P.: Stress concentration around an elliptical hole with surface tension based on the original Gurtin-Murdoch model. Mech. Mater. 135, 144–148 (2019) 6. Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2009) 7. Fang, Q.H., Liu, Y.W.: Size-dependent interaction between an edge dislocation and a nanoscale inhomogeneity with interface effects. Acta Mater. 54, 4213–4220 (2006) 8. Gorbushin, N., Eremeyev, V.A., Mishuris, G.: On the stress singularity near the tip of a crack with surface stresses. Int. J. Eng. Sci. 146, 103183 (2020) 9. Grekov, M.A., Sergeeva, T.S.: Interaction of edge dislocation array with bimaterial interface incorporating interface elasticity. Int. J. Eng. Sci. 149, 103233 (2020) 10. Grekov, M.A., Sergeeva, T.S., Pronina, Y.G., Sedova, O.S.: A periodic set of edge dislocations in an elastic solid with a planar boundary incorporating surface effects. Eng. Fract. Mech. 186, 423–435 (2017) 11. Grekov, M.A., Yazovskaya, A.A.: Effect of surface elasticity and residual surface stress in an elastic body weakened by an elliptic hole of a nanometer size. J. Appl. Math. Mech. 78, 172–180 (2014) 12. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291–323 (1975) 13. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978) 14. Kim, C.I., Schiavone, P., Ru, C.Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc. A. 467, 3530–3549 (2011) 15. Kostyrko, S.A., Grekov, M.A.: Elastic field at a rugous interface of a bimaterial with surface effects. Eng. Fract. Mech. 216, 106507 (2019) 16. Kostyrko, S., Grekov, M., Altenbach, H.: Stress concentration analysis of nanosized thin-film coating with rough interface. Contin. Mech. Thermodyn. 31, 1863–1871 (2019) 17. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000) 18. Mogilevskaya, S.G., Crouch, S.I., Stolarski, H.K.: Multiple interacting circular nanoinhomogeneities with surface/interface effects. J. Mech. Phys. Solids 56, 2298–2327 (2008) 19. Mogilevskaya, S.G., Pyatigorets, A.V., Crouch. S.I.: Green function for the problem of a plane containing a circular hole with surface effects. J. Appl. Mech. Trans. ASME 78, 021008 (2011) 20. Novozhilov, V.V.: Theory of Elasticity. Pergamon Press, Oxford (1961) 21. Ou, Z.Y., Wang, G.F., Wang, T.J.: Effect of residual surface tension on the stress concentration around a nanosized spheroidal cavity. Int. J. Eng. Sci. 46, 475–485 (2008) 22. Povstenko, Yu.Z.: Theoretical investigation of phenomena caused by heterogeneous surface tension in solid. J. Mech. Phys. Solids 41, 436–439 (1993) 23. Sharma, P., Ganti, S.: Sized-dependence Eshelby’s tensor for embedded nanoinclusions incorporating surface/interface energies. J. Appl. Mech. 71, 663–671 (2004) 24. Shodja, H.M., Ahmadzadeh-Bakhshayesh, H., Gutkin, MYu.: Size-dependent interaction of an edge dislocation with an elliptical nano-inhomogeneity incorporating interface effects. Int. J. Solids Struct. 49, 759–770 (2012) 25. Tian, L., Rajapakse, R.K.N.D.: Elastic field of an isotropic matrix with nanoscale elliptical inhomogeneity. Int. J. Solids Struct. 44, 7988–8005 (2007)

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26. Wang, G.F., Wang, T.J.: Deformation around a nanosized elliptical hole with surface effect. Appl. Phys. Lett. 89, 161901 (2006) 27. Wang, S., Dai, M., Ru, C.Q., Gao, C.F.: Stress field around an arbitrarily shaped nanosized hole with surface tension. Acta Mech. 225, 3453–3462 (2014) 28. Wang, Z.Q., Zhao, Y.P., Huang, Z.P.: The effects of surface tension on the elastic properties of nano structures. Int. J. Eng. Sci. 48, 140–150 (2010)

Chapter 97

On Edge Effect for a Finite Doubly Periodic System of Perpendicular Cracks Abdulla Abakarov and Yulia Pronina

Abstract The aim of the work was to assess the minimum size of a finite doubly periodic square array of cracks in an infinite plane so that the fracture characteristics in the center of this array would hardly change with its further increase. For this purpose, we investigated the influence of the size of the crack array on the energy release rate near the central (and side) cracks for various types of loading: uniaxial and biaxial tension and in-plane shear. We also observed the effect of the crack density on the relative change in the energy release rate with the increase in the number of cracks.

97.1 Introduction The fracture problem of doubly periodic (DP) arrays of cracks in an infinite solid has been addressed for more than half a century. Such an idealized model can be used to analyze the fracture behavior and effective elastic properties of materials [12, 18]. The problems of DP array of cracks in an infinite isotropic linearly elastic medium under conditions of generalized plane stress or plane strain have been mostly investigated by using the method of complex potentials of Muskhelishvili [2, 3, 7, 8, 12, 14, 16]. Such problems are often reduced to a system of singular or hypersingular integral equations, which have to be solved numerically [3, 7, 14, 18]. The basic functions are expressed in terms of either “pseudotractions” for certain superimposed single crack problems, displacement discontinuities, or dislocations distributed along the crack line. Using the Green functions from [4–6] allows one to investigate the behavior of periodic cracks near an interface. Other approaches are based on the eigenfunction expansions of unknown functions for properly chosen unit regions in terms of resultant forces and displacements and some homogenization techniques A. Abakarov · Y. Pronina (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] A. Abakarov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_97

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[8]. Based on the Jacobi elliptic functions, Chang obtained general solutions for the stress intensity factor (SIF) of parallel DP cracks [2]. To avoid numerical solution of integral equations in the problems of finite multi-crack arrays, Kachanov introduced a simple traction-based influence function method [9], which works well for arbitrary oriented straight and not too close cracks. A procedure of estimating the SIFs for closely located cracks is suggested in [15]. Finite element method is also popular for analysis of DP problems [11, 13]. To study some effects in multi-crack arrays, it is useful to determine the minimum size of a finite DP crack array so that the fracture characteristics in the center of this array would hardly change with its further increase. It is also interesting to compare the energy release rate (ERR) for the central and the side cracks. The present paper deals with these issues for DP square array of perpendicular cracks in an infinite plane under three types of loading by using the Kachanov approach [9], since it is suitable for our further research.

97.2 Formulation of the Problem Consider an infinite isotropic linear elastic solid under conditions of generalized plane stress or plane strain which contains a finite DP array of rectilinear tractionfree cracks of the same length 2l. This array is arranged into a square grid with the equal periods a and the cracks lying along the sides of each cell in their centers (Fig. 97.1). The whole grid forms a square with the number of cracks along its side being N . Thus, the total number of cracks in the array is 2N (N + 1). The solid is subjected at infinity to external load, described by a stress tensor σ 0 . In this work, three types of the load are examined: uniaxial and biaxial tension, and shear loading. The question of the interest is to evaluate the edge (or size) effect, which is expressed in terms of the dependency of the ERR for the central cracks on the size of the array. The latter is controlled by varying the parameter N . Another issue of the interest is to compare the ERR for the central and side cracks (calculated at the tips closest to the center, like the tips flagged in Fig. 97.1) for various array sizes.

97.3 Research Method To succinctly describe the method [9], consider an arbitrary system of M rectilinear traction-free cracks in an infinite elastic solid subjected to remote load σ 0 . The main features of the method include the following: • Representation of this problem by a superposition of M subproblems containing a single crack loaded by unknown normal pi (ξ ) and shear qi (ξ ) tractions (i =

97 On Edge Effect for a Finite Doubly Periodic System of Perpendicular Cracks

849

Fig. 97.1 Overview of a crack grid for N = 4. The circles depict crack tips of the interest

1..M), which consist of the external load contribution and tractions due to crack interactions. Here, ξ runs over the ith crack. • Accepting the key simplifying assumption, which states that the crack interactions are well described only by the uniform average tractions ( pi , qi ), meaning that the nonuniform parts of them, pi (ξ ) −  pi  and qi (ξ ) − qi , may be neglected. j

j

If σ n and σ t denote the known stress tensors generated by the jth crack standing alone and loaded by uniform tractions of unit intensity, normal, and shear correspondingly  field created at a point ξ of the ith crack by the jth  [9], then the stress j j crack is  p j σ n + q j σ t . ξ

Thus, the total tractions on the ith crack are

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⎧   j 0 j ⎪ p  p (ξ ) = p + n · σ + q σ · ni ⎪ i i i j j t n ⎪ ⎪ ξ ⎪ j=i ⎪ ⎨   j j 0  p q (ξ ) = q + n · σ + q σ · ti i i i j j t n ⎪ ⎪ ξ ⎪ j=i ⎪ ⎪ ⎪ ⎩ i = 1..M ,

(97.1)

where pi 0 = ni · σ 0 · ni and qi 0 = ni · σ 0 · ti are the external load contributions, ni and ti are the normal and tangent vectors of the ith crack line. By averaging each of Eq. (97.1) along the ith crack line, the problem is eventually reduced to the system of 2M linear algebraic equations (SLAE) for the average normal  pi  and shear qi  tractions. After the average tractions  pi  and qi  are found from the SLAE, the variable tractions pi (ξ ) and qi (ξ ) can be retrieved by using (97.1). The mode I (tensile) and mode II (in-plane shear) SIFs at both tips of the ith crack can then be obtained by general formulas





l l ± ξ pi (ξ ) 1 K I (±l) =√ dξ , K I I (±l) l ∓ ξ qi (ξ ) πl −l

where the integration is performed along the crack line, and ξ = 0 corresponds to the crack center. The ERR can then be found as G = (K I2 + K I2I )/E  , where E  = E in the case of plane stress and E  = E/(1 − ν 2 ) for plane strain; E is Young’s modulus and ν is the Poisson ratio. The method applicability is limited by a spacing between cracks. In worst cases of irregular crack interposition, the results remain accurate at spacings of about onethird of the crack lengths. We consider spacings between collinear cracks of 0.2 and 1 of the cracks length, which are acceptable for such regular and periodic geometry we study in this work. The described algorithm requires a lot of numerical integrations, which may cause an overhead during the work. Firstly, 2M(2M − 2) elements of the matrix of the basic SLAE must be determined by integrating the terms on the right-hand sides of (97.1). Secondly, calculation of the SIFs for all cracks takes 4M integrations (two modes for two tips). However, we limit ourselves to the four cracks of interest. Some of those integrals could have been taken analytically, however the gain of that would be small while the complexity of the algorithm implementation would increase. The best way we have found in dealing with the computational overhead is parallelization, which is admissible because of the way how the method is constructed. It allows drastic improvement of the performance, which in that case is mostly (but not completely) limited by available computational resources. The program used here was created in MATLAB® software ver. R2016b. The computations were carried out on a computer with a CPU Intel(R) Core(TM) i7-

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4700HQ 2.40 GHz, which contained 4 physical cores and 8 logical processors, with 8 GB RAM. The calculations for N = 24 and a particular crack density took 1 h 40 min (mainly for assembling a 2400-by-2400 matrix), while for the case N = 28 it took 3 h (for assembling a 3248-by-3248 matrix). Without the parallelization even a configuration corresponding to a 624-by-624 matrix (N = 12) has not been evaluated in a reasonable time on this computer.

97.4 Calculation Results The calculations have been conducted for two periods: a = 2.4l and a = 4l, which correspond to the crack densities ρ = 0.347(2) and ρ = 0.125, respectively. The concept of crack density in 2D and 3D cases is addressed in [10, 12]. Three types of the loads are examined: (a) uniaxial tension, (b) biaxial tension, and (c) shear loading. Figure 97.2 demonstrates the dependencies of the normalized ERR at the tips of the central and side cracks on the size of the DP array of cracks. The ERR, G, is normalized by the ERR G 0 for a single crack of the same length and under the same load. Notice that the scales on these plots are different. Table 97.1 provides the relative change (in percents) in the ERR for the step-by-step expansions of the crack array. Note that in the cases of biaxial tension and pure shear, the ERR at the tips of the vertical and horizontal cracks located at the same distances from the center of the array are the same. In the case of uniaxial tension, Fig. 97.2 shows the ERR for the cracks perpendicular to the tension direction. For the cracks collinear to it, the ERR does not exceed 15% of the ERR for perpendicular cracks, so graphs for them were omitted. However, the relative changes in the ERR for those cracks are given in Table 97.1. As one can see from Fig. 97.2, at a relatively large N , all the curves for the ERR near the central cracks (marked with asterisks) practically approach their horizontal asymptotes. This means that the fracture behavior in the center of such arrays is similar to that in an infinite DP array. The relative change in the ERR with the increase in N is higher for the larger crack density (dashed curves). As can be seen from Table 97.1, at ρ = 0.347(2), this change for the central cracks is less than 1% when N increases from 16 to 20, while at ρ = 0.125, this is observed for the increase in N from 12 to 16. It is interesting that under uniaxial and biaxial tension, the ERR for the central cracks increases with the growth of N , while under in-plane shear, this ERR decreases (a little). This means that the ERR for the central cracks in a finite DP array of cracks under both types of tension provides an underestimated value of the ERR in an infinite DP array, while for shear loading the former ERR gives an overestimated value of the latter one. The curves for the side cracks (marked with circles) also approach their horizontal asymptotes at large N . In contrast to the case of the central cracks, the ERR for the

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1.35 1.25 1.15 1.05 0.95 0.85 4

8

12

16

20

24

28

20

24

28

20

24

28

(a)

0.65 0.6 0.55 0.5 4

8

12

16 (b)

1.05 1 0.95 0.9 0.85 0.8 0.75 4

8

12

16 (c)

Fig. 97.2 Dependency of the normalized ERR on the number N for the central crack tips (marked with asterisks) and the side crack tips (marked with circles) for a = 2.4l (dashed lines) and a = 4l (solid lines). a Uniaxial load. b Biaxial load. c Shear load

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Table 97.1 Relative change (in percents) in the ERR for the corresponding increment of N , for a = 2.4l (case 1) and a = 4l (case 2) under uniaxial (a) and biaxial (b) tension, and shear loading (c). For uniaxial tension, the results are presented for the cracks perpendicular (P.) and collinear (C.) to the tension direction Increment of N

1(a) P.

1(a) C.

1(b)

1(c)

2(a) P.

2(a) C.

2(b)

2(c)

4→8

8.85

9.03

8.77

−0.07

3.08

3.35

3.03

−0.40

8 → 12

2.92

3.02

2.87

−0.68

1.01

1.16

0.98

−0.12

12 → 16

1.45

1.48

1.44

−0.32

0.50

0.58

0.49

−0.05

16 → 20

0.87

0.88

0.87

−0.17

0.30

0.35

0.29

−0.02

20 → 24

0.58

0.58

0.58

−0.10

0.20

0.23

0.19

−0.01

24 → 28

0.41

0.41

−0.07

−0.10

0.14

0.16

0.14

−0.01

4→8

−6.33

−20.0

0.13

4.96

−2.20

−11.2

−0.22

3.23

8 → 12

−3.51

−12.5

0.18

3.81

−1.12

−5.75

−0.17

1.42

12 → 16

−2.07

−7.32

−0.08

2.12

−0.64

−3.24

−0.13

0.76

16 → 20

−1.35

−4.72

−0.13

1.30

−0.41

−2.05

−0.10

0.47

20 → 24

−0.95

−3.28

−0.13

0.88

−0.29

−1.41

−0.07

0.31

24 → 28

−0.70

−2.41

−0.11

0.63

−0.21

−1.02

−0.06

0.23

Central cracks

Side cracks

side cracks decreases with the increase in N under uniaxial and biaxial tension (Fig. 97.2a, b), and it increases for shear loading (Fig. 97.2c). Note that the difference in the stress fields around the central and side cracks also imply that stress-assisted processes, such as stress-assisted growth of defects [1, 17], in the center and at the edge of the array will proceed at different rates.

97.5 Conclusion Thus, we investigated the influence of the size of a finite DP square array of perpendicular cracks in an infinite plane under various in-plane loads on the ERR near the central and side cracks. The relative change in the ERR with an increase in the size of the crack array is given in Table 97.1. It was observed that the relative change in the ERR with the increase in the array size is higher for the higher crack density. Numerical analysis also revealed that the ERR for the central cracks in a finite DP array of cracks under uniaxial and biaxial tension provides an underestimated value of the ERR in an infinite DP array, while for shear loading, the ERR for the central cracks in a finite DP array gives an overestimated value of the ERR in an infinite DP system of cracks.

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In contrast to the behavior of the ERR for the central cracks, the ERR for the side cracks decreases with the increase in the size of the crack array under uniaxial and biaxial tension, and it increases for shear loading.

References 1. Butusova, Y.N., Mishakin, V.V., Kachanov, M.: On monitoring the incubation stage of stress corrosion cracking in steel by the eddy current method. Int. J. Eng. Sci. 148, 103212 (2020) 2. Chang, S.S.: The general solutions of the doubly periodic cracks. Eng. Fract. Mech. 18(4), 887–893 (1983) 3. Fil’shtinskii, L.A.: Interaction of a doubly-periodic system of rectilinear cracks in an isotropic medium. PMM 38(5), 906–914 (1974) 4. Grekov, M.A.: Two types of interface defects. J. Appl. Math. Mech. 75(4), 476–488 (2011) 5. Grekov, M.A., Sergeeva, T.S.: Interaction of edge dislocation array with bimaterial interface incorporating interface elasticity. Int. J. Eng. Sci. 149, 103233 (2020) 6. Grekov, M.A., Sergeeva, T.S., Pronina, Y.G., Sedova, O.S.: A periodic set of edge dislocations in an elastic semi-infinite solid with a planar boundary incorporating surface effects. Eng. Fract. Mech. 186, 423–435 (2017) 7. Ioakimidis, N.I., Theocaris, P.S.: Doubly-periodic array of cracks in an infinite isotropic medium. J. Elast. 8(2), 157–169 (1978) 8. Isida, M., Igawa, H.: Doubly-periodic array and zig-zag array of cracks in solids under uniaxial tension. Int. J. Fract. 53(3), 249–260 (1992) 9. Kachanov, M.: Elastic solids with many cracks and related problems. Adv. Appl. Mech. 30(C), 259–445 (1993) 10. Kachanov, M., Mishakin, V.V.: On crack density, crack porosity, and the possibility to interrelate them. Int. J. Eng. Sci. 142, 185–189 (2019) 11. Lapin, R.L., Kuzkin, V.A., Kachanov, M.: Rough contacting surfaces with elevated contact areas. Int. J. Eng. Sci. 145, 103171 (2019) 12. Lapin, R.L., Kuzkin, V.A., Kachanov, M.: On the anisotropy of cracked solids. Int. J. Eng. Sci. 124, 16–23 (2018) 13. Levandovskiy, A.N., Melnikov, B.E., Shamkin, A.A.: Modeling of porous material fracture. Mag. Civ. Eng. 69(1), 3–22 (2017) 14. Linkov, A.M., Koshelev, V.F.: Complex variables BIE and BEM for a plane doubly periodic system of flaws. J. Chin. Inst. Eng. 22(6), 709–720 (1999) 15. Martynyuk, M., Kachanov, M.: Elastic compliances and stress intensity factors of multi-link zig-zag cracks. Int. J. Eng. Sci. 148, 103225 (2020) 16. Panasiuk, V.V., Savruk, M.P., Datsishin, A.P.: Stress distribution in plates and shells. Naukova Dumka, Kiev (1976) (in Russian) 17. Pronina, Y.: An analytical solution for the mechanochemical growth of an elliptical hole in an elastic plane under a uniform remote load. Eur. J. Mech. A/Solids (2017). https://doi.org/10. 1016/j.euromechsol.2016.10.009 18. Shi, P.: Singular integral equation method for 2D fracture analysis of orthotropic solids containing doubly periodic strip-like cracks on rectangular lattice arrays under longitudinal shear loading. Appl. Math. Model. 77, 1460–1473 (2020)

Chapter 98

Large Deformations of a Plane with Elliptical Hole for Model of Semi-linear Material Venyamin Malkov and Yulia Malkova

Abstract The exact analytical solution of the nonlinear problem of elasticity for a plane with an elliptical hole is obtained for semi-linear material. External loads are constant nominal (Piola) stresses at infinity and the boundary of a hole is free. The complex variable formulation and conformal mapping techniques are used. The hoop stress distribution along the boundary of a hole is analyzed for the case, when the material is deformed by uniaxial tension/compression. A comparison is made with the stresses of a similar linear problem.

98.1 Introduction The needs of various fields of modern technology are increasingly forcing to turn to nonlinear problems, since classical linear models of an elasticity have a limited area of applicability. For example, rubber-like materials are capable of experiencing large elastic deformations (order of hundreds of percent), where the linear elasticity is not applicable. Therefore, the study of elastomeric materials and structures, used in many branches of technology, must be carried out using the equations of nonlinear elasticity. Solutions to a number of linear problems for a plane with an elliptical hole are presented in many works, including well-known monographs [1–3] and others. Nonlinear problems for materials with holes (or cavities), in particular elliptical ones, were considered in works by [4, 5] and others. Exact solutions [4] are constructed for John’s harmonic material which are compared with experiments. The agreement is remarkably good. In [6–8], analytical solutions to problems for an elastic half-plane with a circular tunnel or cavity are presented, which are having initial deformations and are loaded at the boundary of the cavity. Interest in these types of problems arises in the mining industry mainly due to displacements of the soil surface that can cause V. Malkov · Y. Malkova (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] V. Malkov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_98

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significant damage. Calculations by analytical and numerical methods confirm that the presence of holes really causes an increase in displacements. In [9], the method of complex functions and superposition was used to study the effect of an elliptical hole located near the interface boundary of two half-planes on the magnitude of stresses. The problem for a composite plane with an elliptical hole was also considered in [10]. The effect of the stress of a crack or an elliptical hole near a rectangular cutout in a plate was studied in [11]. This issue is important for passenger aircraft when a crack is located near the door. The applicability of the model of harmonic material for large deformations is confirmed by experiments on rubber plates with circular or elliptical holes [4].

98.2 General Relations of the Boundary Problem The nonlinear problem of elasticity for a plane with an elliptical hole is considered. At infinity, constant nominal stresses are set and the hole boundary is free. To solve the problem, we use a system of equations consisting of equilibrium equations for the tensor of nominal stresses and compatibility equations for the deformation gradient [12]: div S = 0, rot GT = 0,

(98.1)

where S = sαβ eα eβ —tensor of nominal stress, G = gαβ eα eβ —deformation gradient tensor, and ei is a orthonormal vector basis of Cartesian coordinates of undeformed configuration. Tensor S is defined through the Cauchy stress tensor S = G−1 · J T, where J = det G is volume change rate. In Eq. (98.1), we pass to the tensor components and write in complex form for the plane problem (98.2) (s11 + is12 )1 + i(s22 − is21 )2 = 0, (g22 − ig12 )1 + i(g11 + ig21 )2 = 0,

(98.3)

prime and index at brackets denote partial derivative on Cartesian coordinates (x1 , x2 ). Equations (98.2), (98.3) are exact for plane strain and plane stress, since the stresses s31 , s32 , which are not in equilibrium Eq. (98.2), for the model of semi-linear material are equal to zero. We introduce complex variables of the initial and current configurations z = x1 + i x2 , ζ = ξ1 + iξ2 and complex function of the nominal stresses σ = f 1 + i f 2 . Stresses and deformations are represented by the following expressions: s11 + is12 =

∂σ ∂σ ∂σ ∂σ − , s22 − is21 = + , ∂z ∂z ∂z ∂z

(98.4)

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g11 + ig21 =

∂ζ ∂ζ ∂ζ ∂ζ + 1 , g22 − ig12 = − 1. ∂z ∂z ∂z ∂z

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(98.5)

Equations (98.2), (98.3) are identically satisfied on substituting stresses and strains (98.4), (98.5) into them. The complex functions ζ (z, 1z ) and σ (z, 1z ) are found from the constitutive equations (the law of elasticity) and the boundary conditions of the problem.

98.2.1 Semi-linear Material The elastic potential of a semi-linear material has the form [1, 13]  = 0, 5λtr 2 ( − I) + μtr( − I)2 .

(98.6)

From (98.6), we obtain the law of elasticity for the tensor of nominal stresses S = 2μGT + κQT , κ = λ(tr − 3) − 2μ.

(98.7)

In formulas (98.6), (98.7) are indicated: λ, μ—Lame’s parameters, —tensor of elongations, and Q—orthogonal tensor. These tensors are involved in the polar decomposition of the deformation gradient G = Q · and 2 = GT · G. We write the law of elasticity (98.7) for a plane problem in the components of tensors in complex form s11 + is12 = (λ + 2μ)(g11 + ig21 ) + λ(g22 − ig12 ) + k eiω , s22 − is21 = (λ + 2μ)(g22 − ig12 ) + λ(g11 + ig21 ) + k eiω .

(98.8)

An angle ω represents the rotation of a neighborhood of body point, it is the harmonic function; k = λ(λ3 − 3) − 2μ. Substituting expressions (98.4), (98.5) into relations (98.8), we obtain equations for the functions σ (z, 1z ) and ζ (z, 1z ) ∂ζ ∂σ − 2μc = −2μc eiω , ∂z ∂z

∂σ ∂ζ + 2μ 1 = 0. 1 ∂z ∂z

(98.9)

The constant c is different for plane strain and plane stress states. For these problems, we have, respectively, c = 1/(1 − 2ν), c = (1 + ν)/(1 − ν), ν—Poisson’s ratio. The solution of Eq. (98.9) has the form

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 1 [ϕ (z) + ψ (z) + 2μc eiω dz], 1+c  1 [cϕ (z) − ψ (z) − 2μc eiω dz], σ = 1+c

2μζ =

(98.10)

where ϕ (z), ψ (z)—analytic functions of complex variable z.

98.2.2 Plane with Elliptical Hole The hole boundary is free, then σ =

1 1+c

   cϕ (z) − ψ (z) − 2μc eiω dz = 0.

(98.11)

At infinity of the plane, conditions are given si j → si∞j , |z| → ∞. We map the complex plane z with an elliptical hole to the exterior of the unit  circle of the complex plane ξ : z = η (ξ ) = R ξ + m ξ −1 ; R > 0, 0 ≤ m < 1. Let us set ξ = r eiθ , where the parameters (r, θ ) are the polar coordinates of the plane ξ and the curvilinear coordinates of the plane z, r is the dimensionless radius. We pass in formulas (98.10) to the variable ξ  1  ϕ (ξ ) + ψ (ξ ) + 2μc (ξ, ξ ) , 1+c  1  cϕ (ξ ) − ψ (ξ ) − 2μc (ξ, ξ ) , σ = 1+c 2μζ =

where  (ξ, ξ ) =

ϕ  (ξ )  |η (ξ )| dξ = |ϕ  (ξ )|



η (ξ ) ϕ  (ξ )



ϕ  (ξ )η (ξ ) dξ.

(98.12)

The boundary condition (98.11) on the contour of the ellipse becomes the condition on the unit circle of the plane ξ cϕ (t) − ψ (t) − 2μc (t) = 0, t = eiθ ,

(98.13)

ϕ(t) is the boundary value of function (98.12) on the hole contour. In analytic functions, we separate the holomorphic and non-holomorphic parts in an infinite domain ϕ(z) = Az + ϕ0 (z), ψ(z) = Bz + ψ0 (z), ϕ(ξ ) = A Rξ + ϕ0 (ξ ), ψ(ξ ) = B Rξ + ψ0 (ξ ),

(98.14)

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ϕ0 (z), ψ0 (z), ϕ0 (ξ ), and ψ0 (ξ ) are holomorphic functions. The complex constants A and B are obtained from conditions at infinity A=

s 1+c ∞ ∞ ∞ ∞ s ± 2μ , s = s11 + s22 + i (s12 − s21 ), 2c |s| 1 ∞ ∞ ∞ ∞ − s22 + i (s12 + s21 )], B = (1 + c)[s11 2

plus is taken if |A| > 2μ, minus if |A| < 2μ. Expansions (98.14) are used in Eq. (98.13) 1 cϕ0 (t) − ψ0 (t) − 2μc (t) = −c A Rt + B R . t

(98.15)

We find a solution to the boundary-value problem (98.15) in the domain |ξ | > 1 cϕ0 (ξ ) + 2μc

1 2πi



(t) dt 1 1 = B R , ψ0 (ξ ) = 2μc t −ξ ξ 2πi



(t) dt 1 − cAR . t −ξ ξ

(98.16)

The first equation in (98.16) is a functional equation. Let’s put further 1 ϕ(ξ ) = A Rξ + ϕ0 (ξ ) = A Rξ + K , ξ

(98.17)

where K —unknown complex constant. We calculate the Cauchy-type integral in formulas (98.16) using function (98.17), 1 2πi



(t) dt 1 = t −ξ 2πi





R(1 − mt 2 ) AR − K t2



 R(1 − m/t 2 )(A R − K /t 2 ) dt

dt = t −ξ

 K 1 1 A m+ R . =− 2 |A| AR ξ From the functional equation, we find the value of the constant K K =

|A|B + mμc A R. c(|A| − μ)

98.3 Stress Calculation We write stresses and strains in polar coordinates (r, θ ) 1 srr + isr θ =  η (ξ )

 

∂σ ∂σ −2iθ ∂σ ∂σ −2iθ 1 , sθθ − isθr =  , − + e e ∂ξ η (ξ ) ∂ξ ∂ξ ∂ξ

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Fig. 98.1 Plane stretching along the x1 and x2 axes

Fig. 98.2 Plane compression along the x1 and x2 axes

1 grr + igθr =  η (ξ )

 

∂ζ ∂ζ 1 ∂ζ −2iθ ∂ζ −2iθ e e , gθθ − igr θ =  . + − ∂ξ η (ξ ) ∂ξ ∂ξ ∂ξ

On the free contour of the hole, the stress srr + isr θ = 0 and the stress sθθ − isθr are found by the formula sθθ − isθr

  

 η (t)  2c ϕ  (t) 1 − 2μ    . =  1 + c η (t) ϕ (t)

(98.18)

In [2] for the case of linear problem of a free hole, formulas are given for hoop stresses under tension of a plate by stresses p at an angle α to the coordinate axis x1 . In particular, for α = 0 and α = π/2 we obtain, respectively, σθθ =

1 − m 2 + 2m − 2 cos 2θ 1 − m 2 − 2m + 2 cos 2θ p, σ p. (98.19) = θθ 1 + m 2 − 2m cos 2θ 1 + m 2 − 2m cos 2θ

The calculations of stresses on the hole boundary are performed using nonlinear and linear theories. Plane strain is considered. The material and elliptical hole

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parameters are: μ = 1 MPa, ν = 0.4902, semi-axes of the ellipse a = 3 cm, b = 1 cm, m = 0.5. The uniaxial tension and compression of the plane along the x1 and x2 ∞ ∞ axes are taken as the external load: s11 = ±5 MPa, s22 = ±5 MPa. In Figs. 98.1 and 98.2, the dashed line shows the hoop stresses of the nonlinear theory, the solid line indicates the stresses of the linear theory depending on the angle 0 ≤ θ ≤ 2π . To calculate the stresses, formulas (98.18) and (98.19) were used, respectively. In Fig. 98.1, the stresses of the plane under stretching to the left—along the x1 axis—and to the right—along the x2 axis—are shown. Similarly, Fig. 98.2 corresponds to the compression of a plane along the indicated axes. It can be seen from the figures that the stresses obtained by the nonlinear theory are less than the stresses of the linear theory. In addition, there are significant differences in the nature of the distribution of stresses along the contour of the hole.

98.4 Conclusion An exact analytical solution of the nonlinear problem of elasticity for a plane with an elliptical hole is obtained. Many materials and structures used in various fields of technology work under conditions of large deformations, in particular, rubbers and rubber products. The use of equations of the nonlinear theory of elasticity is necessary when solving boundary-value problems for such materials. The results obtained in the work are of theoretical and practical values. It is of interest for practice that the stresses on the hole contour obtained by the equations of nonlinear elasticity are much less than the stresses of the linear problem. The difference in the nature of the distribution of stresses along the contour of the hole is also significant.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Lurie, A.I.: Nonlinear Theory of Elasticity. North Holland, Amsterdam (1990) Novojilov, V.V.: Theory of Elasticity. Pergamon Press, New York (1961) Savin, G.N.: Stress Concentration Around Holes. Pergamon Press, New York (1961) Varley, E., Cumberbatch, E.: Finite deformation of elastic materials surrounding cylindrical holes. J. Elast. 10, 341–405 (1980) Wheeler, L.T.: Finite deformation of a harmonic elastic medium containing an ellipsoidal cavity. Int. J. Solids Struct. 21, 799–804 (1985) Kooi, C.B., Verruijt, A.: Interaction of circular holes in an infinite elastic medium. Tunn. Undergr. Space Technol. 16, 59–62 (2001) Verruijt, A.: Deformations of an elastic half plane with a circular cavity. Int. J. Solids Struct. 35, 2795–2804 (1988) Verruijt, A.: A complex variable solution for a deforming circular tunnel in an elastic half-plane. Int. J. Numer. Anal. Methods Geomech. 21, 77–89 (1997) Malkov, V.M., Malkova, Yu.V., Petrukhin, R.R.: Interaction of an elliptic hole with an interface of two bonded half-planes. Vestn. St. Petersburg State Univ. Ser. 10. Appl. Math. Inform. Control Process. 3, 73–87 (2016)

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10. Mikiya, O., Norio, H., Takuji, N.: Bimaterial plane with elliptic hole under uniform tension normal to the interface. Int. J. Fract. 71, 293–310 (1995) 11. Ukadgaonker, V.G., Awasarem, P.J.: Interaction effect of rectangular hole and arbitrarily oriented elliptical hole or crack in infinite plate subjected to uniform tensile loading at infinity. Indian J. Eng. Mater. Sci. 6, 125–134 (1999) 12. Malkov, V.M.: Introduction to Non-linear Elasticity. St. Petersburg State University, St. Petersburg (2010) 13. John, F.: Plane strain problems for a perfectly elastic material of harmonic type. Commun. Pure Appl. Math. (13), 239–296 (1960)

Chapter 99

On Minimization of Metal Costs for a Pipeline Exposed to External Corrosion Under Pressure Marina Elaeva, Yulia Pronina, and Sergey Kabrits

Abstract The paper concerns the problem of minimization of metal consumption for a pipeline with unlimited service life and the possibility to replace its parts, without reuse of the material of the old parts replaced. The pipeline is exposed to external mechanochemical corrosion under pressure. The presence of a protective film and corrosion inhibition is taken into account. The problem is reduced to finding the minimum of the corresponding objective function. For an uncoated pipe, this function has only one minimum at a relatively large initial thickness, which cannot always be set in practice for technological reasons. In such cases, it is more advantageous to set the initial thickness of the pipe as large as possible for a specific manufacturing technology. For pipes with coatings, the objective function may have two points of the local minima: at the minimal allowable thickness and at a relatively large one.

99.1 Introduction Optimal design is of great importance for safe and reliable operation of structures under various physical and chemical influences [6–8]. Simultaneous action of mechanical stresses and active environment may initiate various stress-assisted processes of deterioration of structure materials [3, 14, 16, 18, 27, 28]. Such processes also affect the flow regimes along the damaged surfaces due to increasing their roughness [17]. Strength analysis of pipes subjected to stress-assisted uniform corrosion was carried out by many authors, e.g. [4, 5, 7, 9, 15, 19, 21, 26]. In contrast to the case of uniform wear when a closed-form solution can often be obtained, M. Elaeva · Y. Pronina (B) · S. Kabrits St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] M. Elaeva e-mail: [email protected] S. Kabrits e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_99

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spatially non-uniform corrosion is mainly modeled by numerical methods [2, 6, 29], although there are some exceptions presenting approximate solutions [10, 23, 24]. The problem of optimal design of a pipe exposed to stress-assisted spatially uniform corrosion under pressure is considered in [20], and that article deals with minimizing material consumption for the pipe with a limited service life. (The mentioned paper contains typos—see the comment at the end of Sect. 99.3.) In the present paper, we consider the question of minimization of the average material consumption per unit of the service life of a pipe, which can help to minimize material costs for a pipeline with unlimited service life with the possibility of pipes replacement. Moreover, we take into account the effect of a protective film on the vessel surface, which was not addressed in our previous works.

99.2 Formulation of the Problem Consider a linearly elastic isotropic long circular cylindrical pipe under internal pr and external p R pressures and subjected to uniform external corrosion. The outer surface of the pipe is covered with a protective coating of the known life t p . Due to corrosion, the outer radius R(t) of the pipe decreases with time t with the rate [4, 16, 28]: dR =0, at t ≤ t p (99.1) vR = − dt vR = −

dR = [a + mσ1 (R)] exp[−b(t − t p )] , dt

at t > t p ,

(99.2)

where σ1 (R) is the principal stress with the maximum absolute value, on the outer surface of the pipe; a, m, and b are experimentally determined constants (b is the corrosion inhibition coefficient). The inner radius remains constant: r (t) = r = const. Maximum allowable stress, σ ∗ , of the pipe material is considered to be known. The purpose of the study is to find an optimal initial thickness of the pipe, which allows us to minimize the average metal consumption per unit of the service life of the pipe. In other words, for a given inner radius r , we need to compute the initial outer radius R(0) = R0 , which can help to minimize material costs for unlimited operation of the pipeline with the possibility of the pipes replacement and without reuse of the material of the old parts replaced. The cost of production, installation, replacement, and coating of pipes is not taken into account.

99.3 Solution of the Problem In this work, we investigate the case when the maximum principal stress on the pipe surfaces is the hoop stress: σ1 = σθθ . According to the Lame formulas, the hoop

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stress on the inner and outer surfaces is defined by the following equations: σ1 (r ) = σθθ (r ) =

pr r 2 − p R R 2 pr − p R 2 + 2 R , 2 2 R −r R − r2

(99.3)

σ1 (R) = σθθ (R) =

pr r 2 − p R R 2 pr − p R 2 + 2 r . 2 2 R −r R − r2

(99.4)

Substituting (99.4) into (99.2) yields dR r 2 (a + mp R − 2mpr ) − R 2 (a − mp R ) exp[−b(t − t p )] . = dt R2 − r 2

(99.5)

Integrating (99.5) with taking into account conditions R(0) = R0 , R(t ∗ ) = R ∗ , and (99.1), one can obtain    R 2mr ( pr − p R ) 1 ln 1− b − +√ t = tp − b (a − mp R ) (a + mp R − 2mpr )(a − mp R )3/2 √   R ∗   R (a − mp R )  × atanh √ at b > 0, r (a + mp R − 2mpr )  R0 (99.6) ∗

t∗ = tp −

R ∗ − R0 2mr ( pr − p R ) +√ (a − mp R ) (a + mp R − 2mpr )(a − mp R )3/2 √  R ∗   R (a − mp R )  × atanh √ at b = 0 . r (a + mp R − 2mpr )  R0 (99.7)

Here, R ∗ = R(t ∗ ) is the outer radius at time t ∗ , at which the maximum allowable stress, σ ∗ , is reached at any point in the pipe. The value σ ∗ can be related to either a strength limit (taking into account safety factors), or any other critical stress, e.g. corresponding to stability loss if p R > pr . Notice that some numerical methods for stability problems of thin-walled structures are addressed in [1, 11–13]. An algorithm for assessing the durability of pressure vessels subjected to corrosion in the presence of several failure mechanisms is presented in [22, 25]. It is known that the hoop stress reaches its absolute maximum value on the inner surface of the pressurized pipe. Then, equating (99.3) to σ ∗ , we can express R ∗ from the equation thus obtained in the following form: R∗ = r

pr + σ ∗ . σ ∗ + 2 p R − pr

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Equations (99.6) and (99.7) set one-to-one correspondence between the durability t ∗ and the initial outer radius R0 . Since the initial weight of the pipe is proportional to its initial cross-sectional area, π(R02 − r 2 ), we consider the objective function F=

R02 − r 2 , t∗

which is proportional to the average material consumption per unit of pipeline life. Here, t ∗ is defined by expressions (99.6) or (99.7). Finding the minimum of the function F will allow us to determine the required optimal initial thickness of the pipe: h 0 = R0 − r . Comment on [20]. Formulae (99.6) and (99.7) are the generalization of the solution for external corrosion presented in [20] to the case of pipes covered with a protective coating. Unfortunately, since they published draft article (but not the corrected variant), the mentioned solution contains a typo: arc tangent must be replaced by the hyperbolic arc tangent, and accordingly the figures should be slightly different. For unknown reason, publishers rearranged some of the other formulae without the consent of the author as well.

99.4 Computational Results Figures 99.1 and 99.2 show the dependencies of the objective function F on the initial thickness h 0 and the corresponding life t ∗ of the pipe, respectively, for various lives of the protective coating: t p = 0 (blue dash-dotted lines); t p = 5 [tc ] (red dashed lines); t p = 7 [tc ] (orange solid lines); and t p = 15 [tc ] (violet dotted lines). The black solid line near the ordinate axis in Fig. 99.2 represents the initial part of the objective function F from 0 to t p , common to all the cases with t p > 0. The following parameters are used for these examples: b = 0, a = 0,16 [lc /tc ], m R = 0,0008 [lc /(tc pc )], pr = 10 [ pc ], p R = 3 [ pc ], σ ∗ = 500 [ pc ], and r = 90 [lc ]. Here, lc , tc , and pc are the conventional units of length, time, and stress, correspondingly. The minimum of each curve in Figs. 99.1 and 99.2 corresponds to the minimum average cost of the material per unit of the pipe lifetime. As expected, for not very thick uncoated pipes (blue dash-dotted lines), it is more advantageous to set the initial thickness of the pipe as large as possible for a given manufacturing technology. A slight increase in these and other curves in Figs. 99.1 and 99.2 at very large thicknesses/lives is due to the relative increase in the weight of the layers of the pipe material of the same thickness with an increase in their radius. The minimum of the objective function F in this example (t p = 0) corresponds to the values h 0 = 39, 03 [lc ] and t ∗ = 196.62 [tc ] (see Figs. 99.1 and 99.2), which can hardly be used in practice. For pipes with coatings, one or two local minimums can be observed on the corresponding curves: at the minimal allowable thickness and/or at a larger

99 On Minimization of Metal Costs for a Pipeline Exposed … Fig. 99.1 Dependency of the objective function F on the initial thickness h 0 at b=0

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100

F

80 60 40 20 0 0

20

40

60

80

h0 Fig. 99.2 Dependency of the objective function F on the life t ∗ of the pipe at b=0

100

F

80 60 40 20 0 0

100

200

300

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t*

one. For relatively long t p , the minimal initial thickness h 0 = R ∗ − r , corresponding to the case when σ (r )|t=0 = σ (r )|t=t p = σ ∗ , may be optimal in terms of material savings (see dotted violet lines). But the life of such pipes is equal to the life of the coating: t ∗ = t p ; therefore, the total cost of the pipeline will be greatly affected by the cost of manufacturing and replacing the pipes, as well as protective coating. For relatively short t p , the global minimum of F corresponds to the life t ∗ >> t p , e.g. at t p = 5 [tc ] (red dashed lines), the mentioned minimum corresponds to h 0 = 34.3 [lc ] and t ∗ = 173.65 [tc ]. For t p = 7 [tc ] (orange solid lines), the second minimum corresponding to h 0 = 32.73 [lc ] and t ∗ = 166.40 [tc ] is approximately the same as the first minimum at t ∗ = t p . Similar dependencies for inhibited corrosion at b = 0.007 [1/tc ] are shown in Figs. 99.3 and 99.4, other parameters being the same as in Figs. 99.1 and 99.2. As one can see from Fig. 99.3, at a certain value h 0 (slightly less than 30), all the curves drop sharply to zero. This value is the minimum initial thickness, at which corrosion has time to stop before any critical state is reached, and the corresponding

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Fig. 99.3 Dependency of the objective function F on the initial thickness h 0 at b>0

100 80

F

60 40 20 0 0

5

10

15

20

25

30

h0 Fig. 99.4 Dependency of the objective function F on the life t ∗ of the pipe at b>0

100 80

F

60 40 20 0 100

200

300

400

500

t*

lifetime t ∗ tends to infinity (Fig. 99.4). This is the most optimal initial thickness of the pipe. However, if (for technological reasons) the pipe thickness is limited to a value less than the specified optimum, then for relatively short t p and for uncoated pipes, it is reasonable to set an initial thickness as large as possible, while for relatively long t p (violet dotted line), minimum metal consumption may correspond to minimal possible initial pipe thickness (and most frequent pipe replacement as t ∗ = t p ). The program used here was created in MATLAB® software ver. R2019a.

99.5 Conclusion Thus, we considered the problem of minimization of the average material consumption per unit of the service life of a pipe covered with a protective film and exposed to mechanochemical corrosion. The presented solution can help to minimize material

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costs for unlimited operation of the pipeline with the possibility of pipes replacement. The cost of production, installation, replacement, and coating of pipes is not taken into account. This problem is reduced to finding the minimum of the corresponding objective function. For an uncoated pipe, this function has only one minimum at a relatively large initial thickness, which cannot always be set in practice for technological reasons. In such cases, it is more advantageous to set the initial thickness of the pipe as large as possible for a specified manufacturing technology. For pipes with coatings, the objective function may have two points of the local minima: at the minimal allowable thickness and at a relatively large one, and we need to choose the global minimum. Of course, the total cost of the pipeline is greatly affected by the cost of manufacturing, replacing, coating the pipes, and so on.

References 1. Atavin, I.V., Melnikov, B.E., Semenov, A.S., Chernysheva, N.V., Yakovleva, E.L.: Influence of stiffness of node on stability and strength of thin-walled structure. Mag. Civil Eng. 80(4), 48–61 (2018) 2. Awrejcewicz, J., Krysko, A.V., Krylova, E.Y., (...), Zhigalov, M.V., Krysko, V.A.: Analysis of flexible elastic-plastic plates/shells behaviour under coupled mechanical/thermal fields and one-sided corrosion wear. Int. J. Non-Linear Mech. 118, 103302 (2020) 3. Butusova, Y.N., Mishakin, V.V., Kachanov, M.: On monitoring the incubation stage of stress corrosion cracking in steel by the eddy current method. Int. J. Eng. Sci. 148, 103212 (2020) 4. Dolinskii, V.M.: Calculations on loaded tubes exposed to corrosion. Chem. Pet. Eng. 3(2), 96–97 (1967) 5. Elishakoff, I., Ghyselinck, G., Miglis, Y.: Durability of an elastic bar under tension with linear or nonlinear relationship between corrosion rate and stress. J. Appl. Mech., Trans. ASME, 79(2), 021013 (2012) 6. Fridman, M.M.: Optimal design of compressed columns with corrosion taken into account. J. Theor. Appl. Mech. 52(1), 129–137 (2014) 7. Fridman, M.M., Elishakoff, I.: Design of bars in tension or compression exposed to a corrosive environment. Ocean Syst. Eng. 5(1), 21–30 (2015) 8. Gasratova, N.A., Zuev, V.S.: On the design of deep-sea optical elements made of PMMA. IOP Conf. Ser.: Mater. Sci. Eng. 734(1), 012069 (2020) 9. Gutman, E.M., Zainullin, R.S., Shatalov, A.T., Zaripov, R.A.: Strength of gas industry pipes under corrosive wear conditions. Nedra, Moscow (1984) (in Russian) 10. Gutman, E.M., Haddad, J., Bergman, R.: Stability of thin-walled high-pressure cylindrical pipes with non-circular cross-section and variable wall thickness subjected to non-homogeneous corrosion. Thin Walled Struct. 43(1), 23–32 (2005) 11. Kabrits, S.A., Kolpak, E.P.: Finding bifurcation branches in nonlinear problems of statics of shells numerically. In: 2015 International Conference on “Stability and Control Processes” in Memory of V.I. Zubov, SCP 2015 - Proceedings 7342171, pp. 389–391 (2015) 12. Kabrits, S.A., Kolpak, E.P.: Quasi-static axisymmetric eversion hemispherical domes made of elastomers. AIP Conf. Proc. 1738, 160006 (2016) 13. Kabrits, S.A., Kolpak, E.P.: Stability of an arch type shock absorber made of a rubber-like material. AIP Conf. Proc. 1959, 070015 (2018) 14. Kostyrko, S.A., Shuvalov, G.M.: Surface elasticity effect on diffusional growth of surface defects in strained solids. Contin. Mech. Thermodyn. 31(6), 1795–1803 (2019)

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15. Ovchinnikov, I.G., Pochtman, Yu.M.: Calculation and rational design of structures subjected to corrosive wear (review). Mater. Sci. 27(2), 105–116 (1992) 16. Pavlov, P.A., Kadyrbekov, B.A., Kolesnikov, V.A.: Strength of steels in corrosive media. Nauka, Alma Ata (1987) (in Russian) 17. Pavlovsky, V.A., Chistov, A.L., Kuchinsky, D.M.: Modeling of pipe flows. Vestnik SanktPeterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya 15(1), 93–106 (2019) 18. Poluektov, M., Freidin, A.B., Figiel, L.: Modelling stress-affected chemical reactions in nonlinear viscoelastic solids with application to lithiation reaction in spherical Si particles. Int. J. Eng. Sci. 128, 44–62 (2018) 19. Pronina, Y., Sedova, O.: Analytical solution for the lifetime of a spherical shell of arbitrary thickness under the pressure of corrosive environments: The effect of thermal and elastic stresses. J. Appl. Mech. 88(6), 061004 (2021) https://doi.org/10.1115/1.4050280 20. Pronina, Y.: Design of pressurised pipes subjected to mechanochemical corrosion. In: Zingoni, A. (ed.) Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications - Proceedings of the 7th International Conference on Structural Engineering, Mechanics and Computation, pp. 644–649 (2019) 21. Pronina, Y.G.: Thermoelastic stress analysis for a tube under general mechanochemical corrosion conditions. In: Proceedings of the 4th International Conference on Computational Methods for Coupled Problems in Science and Engineering, COUPLED PROBLEMS 2011, pp. 1408– 1415 (2011) 22. Pronina, Y.G.: Estimation of the life of an elastic tube under the action of a longitudinal force and pressure under uniform surface corrosion conditions. Russ. Metall. (Metally) 2010(4), 361–364 (2010) 23. Pronina, Y.: An analytical solution for the mechanochemical growth of an elliptical hole in an elastic plane under a uniform remote load. Eur. J. Mech. A-Solid 61, 357–363 (2017) 24. Pronina, Y.G., Khryashchev, S.M.: Mechanochemical growth of an elliptical hole under normal pressure. Mater. Phys. Mech. 31(1–2), 52–55 (2017) 25. Pronina, Y., Sedova, O., Grekov, M., Sergeeva, T.: On corrosion of a thin-walled spherical vessel under pressure. Int. J. Eng. Sci. 130, 115–128 (2018) 26. Sedova, O., Pronina, Y.: The thermoelasticity problem for pressure vessels with protective coatings, operating under conditions of mechanochemical corrosion. Int. J. Eng. Sci. 170, 103589 (2022) https://doi.org/10.1016/j.ijengsci.2021.103589 27. Shuvalov, G.M., Kostyrko, S.A.: Surface self-organization in multilayer film coatings. AIP Conf. Proc. 1909, 020196 (2017) 28. Yang, H.-Q., Zhang, Q., Tu, S.-S., Wang, Y., Li, Y.-M., Huang, Y.: A study on time-variant corrosion model for immersed steel plate elements considering the effect of mechanical stress. Ocean Eng. 125, 134–146 (2016) 29. Zhao, S., Pronina, Y.: On the stress state of a pressurised pipe with an initial thickness variation, subjected to non-homogeneous internal corrosion. E3S Web Conf. 121, 01013 (2019)

Chapter 100

Interaction of Misfit Dislocations with Perturbated Surface in Epitaxial Thin Film Sergey Kostyrko, Mikhail Grekov, and Takayuki Kitamura

Abstract The analytical method for the analysis of the stress fields arising as a result of the interaction of misfit dislocations with an undulated surface in an epitaxial thin film is considered. The boundary-value problem of the plane theory of elasticity for an infinite isotropic half-space containing an infinite row of line edge dislocations parallel to the undulated boundary is formulated assuming that the elastic properties of the film and substrate materials are approximately equal. The depth of dislocations beneath the surface is equal to the film thickness, and dislocations are spaced with the distance equal to the surface perturbation wavelength. The solution is based on Goursat–Kolosov’s complex potentials, Muskhelishvili’s representations, the boundary perturbation method, and superposition technique. Using the first-order approximation of the boundary perturbation method, the hoop stress distribution along the sine-curved surface is analyzed by varying the distance of dislocations to the unperturbed boundary.

100.1 Introduction In recent years, the development of micro- and nanoelectronic devices has stimulated extensive studies in the field of thin solid films. It has been observed that during the epitaxial growth of thin films they are submitted to the large stresses caused by the lattice mismatch between film and substrate materials [2]. Due to reduced stability of the surface atomic layers, the elastic field of misfit stresses serves as a driving force S. Kostyrko (B) · M. Grekov St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] M. Grekov e-mail: [email protected] T. Kitamura Department of Mechanical Engineering and Science, Graduate School of Engineering, Kyoto University, Kyoto-daigaku-katsura, Nishikyo-ku, Kyoto 615-8540, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_100

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for surface roughness formation via surface diffusion mechanism [8, 12]. Another inelastic deformation mechanism associated with the relaxation of the misfit stresses is the dislocation nucleation [1]. Since surface and interface are the source and sink of dislocations besides stress concentration, the investigation of elastic interaction of dislocations with such topological defects is of considerable importance in thin film materials. The mobility of dislocations plays a crucial role in the behavior of material and depends on the internal forces acting on them [14]. A great number of theoretical works have been presented during the last decades discussing the dislocations of different types of near free surface of the circular and elliptical holes; planar surface and interface of one- and two-phase solids, inside and outside of mono- and multilayered thin films; and circular and elliptical inclusions, inside hollow and bulk particles [6, 10]. However, most of the studies have been limited to an ideal shape of considered surfaces and interfaces. Since the surface/interface of most engineering materials is unavoidably rough and contains topological irregularites with feature sizes ranging from micro- to nanometers, it is of great interest to investigate the elastic interaction of dislocations with surface/interface roughness. Motivated by this challenge, the present work is focused on the analysis of stress fields induced by misfit dislocations interacting with the perturbated surface of thin film. Supposing that the elastic properties of film and substrate materials are approximately equal, we formulate the boundary value problem (BVP) of plane theory of elasticity for an infinite isotropic half-space containing an infinite row of line edge dislocations. The depth of dislocations beneath the unperturbed surface is equal to the film thickness, and they are spaced with the distance equal to the surface perturbation wavelength. The formulation is valid for an arbitrary Burgers vector and surface shape. The analytical solution of the formulated BVP is based on Goursat– Kolosov’s complex potentials, Muskhelishvili’s representations [11], boundary perturbation method (BPM) [3, 7, 9], and superposition technique [4, 5]. A similar approach has been developed in [13] to obtain Green’s functions for a bimaterial with a local undulation of the interface. In the present work, using the first-order approximation of BPM, the hoop stress distribution along the sine-curved surface is analyzed varying the distance of dislocations to the unperturbed boundary.

100.2 Problem Formulation We consider a two-dimensional BVP for an infinite isotropic half-space  with undulated surface profile  described by the continuous periodic function f :  = {z : x2 < ε f (x1 )} ,  = {z : z ≡ ζ = x1 + iε f (x1 )} ,

(100.1)

where z = x1 + i x2 is the complex variable, (x1 , x2 ) are the global Cartesian coordinates, f (x1 ) = f (x1 + a), i.e. a is the perturbation wavelength, the maximum

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Fig. 100.1 Schematic of the presented problem formulation for an epitaxial film with misfit dislocations beneath the curved surface

deviation of the surface from a flat configuration is A = aε, i.e. max | f (x1 )| = a, and ε is a small parameter, i.e. 0 < ε  1 (Fig. 100.1). The plane strain conditions are assumed to be satisfied by the presence of the periodic set of straight edge dislocations with the Burgers vector b defined by the components b1 and b2 in the Cartesian coordinates (x1 , x2 ), respectively. The depth of dislocations beneath the unperturbed surface is equal to h > A, and they are spaced along x1 direction with the equal intervals a: z k = ak − i h, k = 0, ±1, ±2, . . .

(100.2)

We assume that the boundary  is free from traction: σ (ζ ) = σnn (ζ ) + iσnt (ζ ) = 0, ζ ∈ ,

(100.3)

where σnn and σnt are the components of stress tensor defined in the local Cartesian coordinates (n, t), and n and t are normal and tangential to the interface. As the initial mismatch stresses in film are relaxed by array of misift dislocations, the conditions at infinity can be formulated as follows: lim σαβ = lim ω = 0, α, β = {1, 2},

x2 →−∞

x2 →−∞

(100.4)

where σαβ are the stresses in coordinates (x1 , x2 ) and ω is the rotation angle.

100.3 Superposition Technique In order to define the stress state of considered structure due to elastic interaction of the curved surface and array of edge dislocations, we employ the superposition technique [4, 5]. Assuming linear elastic behavior, the solution of BVP (100.1)– (100.4), specifically the stress complex vector σ = σnn + iσnt , can be obtained as a superposition of the solutions of two auxiliary problems: σ (z) = σ 1 (z) + σ 2 (z), z ∈ ,

(100.5)

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where the stress σ 1 corresponds to the infinite plane with elastic properties of the half-space  under periodically distributed edge dislocations and stress σ 2 arises in half-space similar to  under the unknown surface load p distributed along the curved boundary . The solution of the first auxiliary problem σ 1 is known and could be written in terms of Goursat–Kolosov complex potentials 1 and  1 as it follows [5]:   σ 1 (z) = 1 (z) + 1 (z) + z 1 (z) +  1 (z) e−2iα ,

(100.6)

where iμ(b1 + ib2 ) λ + 3μ π(z − z 0 ) , H= , κ= , a a(κ + 1) λ+μ π(z − z 0 ) π(z − z 0 ) π H  1 (z) = (H − H ) ctg − (z + 2i h) cosec2 , (100.7) a a a

1 (z) = −H ctg

α is the angle between t and x1 axes and prime denotes the derivative with respect to argument. Following Eq. (100.7), the values of 1 and  1 at infinity are equal to lim 1 (z) = ±i H,   lim  1 (z) = lim z 1 (z) +  1 (z) = ±i(H − H ). x2 →±∞

x2 →±∞

x2 →±∞

The solution of the second auxiliary problem is related to the complex potentials  = {z : x2 > −ε f (x1 )}, respectively:

2 and ϒ 2 holomorphic in the region  and    σ 2 (z) = 2 (z) + 2 (z) − ϒ 2 (z) + 2 (z) − (z − z) 2 (z) e−2iα .

(100.8)

Assuming α = 0 and α = π/2 in Eq. (100.8) when x2 → −∞, we obtain lim 2 (z) = lim ϒ 2 (z) = 0.

x2 →−∞

x2 →+∞

Taking into account the boundary condition (100.3) and formulation of superposition principle (100.5), we express the unknown surface load p in terms of the functions 1 and  1 . After that we transform the boundary condition σ 2 (ζ ) = p(ζ ) into following boundary equation for the unknown functions 2 and ϒ 2 passing to the limit in Eq. (100.8) when z → ζ and α = α0 :    

2 (ζ ) + 2 (ζ ) − ϒ 2 (ζ ) + 2 (ζ ) − ζ − ζ 2 (ζ ) e−2iα0 = p(ζ ),   p(ζ ) = − 1 (ζ ) − 1 (ζ ) − ζ 1 (ζ ) +  1 (ζ ) e−2iα0 . (100.9)

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100.4 Boundary Perturbation Method In accordance with BPM [3, 7, 9, 13], we seek the unknown functions 2 and ϒ 2 as power series in the small parameter ε:

2 (z) =

∞ ∞ εm 2 εm 2

(m) (z), ϒ 2 (z) = ϒ (z). m! m! (m) m=0 m=0

(100.10)

2 at the curvilinear boundary The values of the functions 1 ,  1 , 2(m) and ϒ(m)  are represented by Taylor series in the vicinity of the line x2 = 0 treating the real variable x1 as a parameter:

1 (ζ ) =

∞ [iε f (x1 )]l

l!

l=0

2(m) (ζ ) =

l=0

∞ [iε f (x1 )]l l=0

∞ [iε f (x1 )]l

l!

1(l) (x1 ),  1 (ζ ) =

2

2(l) (m) (x 1 ), ϒ(m) (ζ ) =

l!

 1(l) (x1 ),

∞ [−iε f (x1 )]l l=0

l!

2(l) ϒ(m) (x1 ).

(100.11) In view of the equality ε f  (x1 ) = tg α0 , one can write e−2iα0 = 1 + 2

∞  m+1 −iε f  (x1 ) .

(100.12)

m=0

Substituting Eqs. (100.10)–(100.12) into Eq. (100.9) and equating the polynomial coefficients of the power εm (m = {0, 1, . . . }), we arrive at the sequence of 2 the Riemann–Hilbert problems on the boundary values of functions 2(m) and ϒ(m) holomorphic outside the boundary : − + (m) (x 1 ) − (m) (x 1 ) = Fm (x 1 ),

where the auxiliary functions (m) are defined as follows: ⎧ 2 ⎨ ϒ(m) (z), Im z > 0 (m) (z) = , ± (m) (x 1 ) = lim (m) (z). z→x1 ±i0 ⎩ 2

(m) (z), Im z < 0 The functions Fm are known for each mth-order of approximation and depend on complex potentials 1 and  1 , and the solution derived at the previous step. Assuming that the magnitude of the surface undulation is sufficiently small, we concern further with the first-order approximation analysis of the stress fields. The expressions for the known functions Fm in the case m = {0, 1} can be written as F0 (x1 ) = 1 (x1 ) + 1 (x1 ) + x1 1 (x1 ) +  1 (x1 ),

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  2 2 F1 (x1 ) = −i f (x1 ) 2 (0) (x 1 ) + ϒ(0) (x 1 ) + 2 (0) (x 1 )     2 + 2i f  (x1 ) 2(0) (x1 ) + ϒ(0) (x1 ) + i f (x1 ) 1 (x1 ) − x1 1 (x1 ) −  1 (x1 )   − 2i f  (x1 ) 1 (x1 ) +  1 (x1 ) .

(100.13) According to [11], the solutions of Riemann–Hilbert BVP can be obtained in terms of Cauchy-type integrals: 1 (m) (z) = 2πi

∞ −∞

Fm (ξ ) dξ + Cm , z−ξ

(100.14)

where C0 = i H and C1 = 0. Based on the properties of Cauchy-type integrals, we express the functions 2(0) 2 and ϒ(0) as follows: ⎧

(z), Im z > 0 i H (κ − 1) ⎨ 1 . (100.15) (0) (z) = C0 + + ⎩ 2 − 1 (z) − z 1 (z) − 1 (z), Im z < 0 2 , we substitute Eq. (100.15) into To derive the expressions for 2(1) and ϒ(1) (100.13) and represent it by the Fourier series: +∞

1 F1 (x1 ) = B j E j (x1 ), B j = a j=−∞

a/2 F1 (t)E − j (t)dt,

(100.16)

−a/2

where E j (x1 ) = exp (λ j x1 ), λ j = 2π j/a. Substituting (100.16) into (100.14) and using the properties of Cauchy-type integrals, we obtain ⎧ +∞ ⎪ ⎪ ⎪ − B j E j (z), Im z > 0 ⎪ ⎪ ⎪ ⎨ j=1 . (1) (z) = ⎪ −∞ ⎪ ⎪ ⎪ ⎪ B j E j (z), Im z < 0 ⎪ ⎩ j=−1

Thus, the solution of the second auxiliary problem and, as a consequence, of the BVP (100.1)–(100.4) is obtained in the first-order approximation of BPM. The expressions for the stress tensor components defined in the local Cartesian coordinates (n, t) can be found from Eqs. (100.5), (100.6), and (100.8) using the results for

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angles α and α + π/2:   σnn (z) + iσnt (z) = 1 (z) + 1 (z) + z 1 (z) +  1 (z) e−2iα   + 2 (z) + 2 (z) − ϒ 2 (z) + 2 (z) − (z − z) 2 (z) e−2iα , σtt (z) + σnn (z) = 4Re[ 1 (z) + 2 (z)].

100.5 Numerical Results As a numerical example, we analyze the hoop stress distribution along the sinusoidally shaped surface: f (x1 ) = (−1)s a cos

2πa , s = {0, 1}. x1

It is worth to note that such surface morphology can be developed in epitaxial thin film under various growth conditions [2, 8]. Figures 100.2 and 100.3 show the distribution of σtt along the undulated surface with amplitude-to-wavelength ratio ε = 0.1 as a result of interaction of the surface with the edge dislocations located beneath the peaks and valleys (s = 0 and 1 in Figs. 100.2 and 100.3, respectively) for different Burgers vector orientations b = (b1 , 0) and b = (0, b2 ) (Figures (a) and (b), respectively) and different distance of dislocations to the unperturbed boundary h = {0.2a, 0.5a, a}. To truncate the series (100.16) and use a finite number of terms for the approximation of function F1 , we employ the integral criterion described in [4, 5]. For comparison, we present here the solution for a flat surface when ε = 0. As it follows from the presented figures, the maximum level of stresses is observed for dislocations with Burgers vector orientation parallel to the surface. In this case,

Fig. 100.2 The distribution of the normalized hoop stress along the undulated surface as a result of interaction of the surface with the edge dislocations located beneath the peaks

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Fig. 100.3 The distribution of the normalized hoop stress along the undulated surface as a result of interaction of the surface with the edge dislocations located beneath the valleys

the effect of the curved surface is most evident. However, the effect of the distance of dislocations to the boundary is most visible for Burgers vector orientation perpendicular to the surface. Acknowledgements The work is supported by the Russian Foundation for Basic Research under grant 18-01-00468.

References 1. Freund, L.B.: Dislocation mechanisms of relaxation in strained epitaxial films. MRS Bull. 17, 52–60 (1992) 2. Freund, L.B., Suresh, S.: Thin Film Materials: Stress, Defect Formation and Surface Evolution. University Press, Cambridge (2003) 3. Grekov, M.A., Kostyrko, S.A.: Surface effects in an elastic solid with nanosized surface asperities. Int. J. Solids Struct. 96, 153–161 (2016) 4. Grekov, M.A., Sergeeva, T.S.: Interaction of edge dislocation array with bimaterial interface incorporating interface elasticity. Int. J. Eng. Sci. 149, 103233 (2020) 5. Grekov, M.A., Sergeeva, T.S., Pronina, Y.G., Sedova, O.S.: A periodic set of edge dislocations in an elastic solid with a planar boundary incorporating surface effects. Eng. Fract. Mech. 186, 423–435 (2017) 6. Hirth, J.P.: A brief history of dislocation theory. Metall. Trans. A 16, 2085–2090 (1985) 7. Kostyrko, S.A., Grekov, M.A.: Elastic field at a rugous interface of a bimaterial with surface effects. Eng. Fract. Mech. 216, 106507 (2019) 8. Kostyrko, S., Shuvalov, G.: Surface elasticity effect on diffusional growth of surface defects in strained solids. Contin. Mech. Thermodyn. 31, 1795–1803 (2019) 9. Kostyrko, S., Grekov, M., Altenbach, H.: Stress concentration analysis of nanosized thin-film coating with rough interface. Contin. Mech. Thermodyn. 31, 1863–1871 (2019) 10. Lubarda, V.A.: Dislocation Burgers vector and the Peach-Koehler force: a review. J. Mater. Res. Technol. 8, 1550–1565 (2019) 11. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Leiden (1977) 12. Umeno, Y., et al.: Multiphysics in Nanostructures. Springer, Japan (2017)

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13. Volkov, I.D., Grekov, M.A.: Green’s functions for bonded dissimilar materials with a slightly curved interface. Vestn. St. Petersburg State Univ. Ser. 1. Math. Mech. Astron. 3, 126–136 (2007) (in Russian) 14. Yan, Y., et al.: Criterion of mechanical instabilities for dislocation structures. Mater. Sci. Eng. A 534, 681–687 (2012)

Chapter 101

Wave Motions of Liquid with Consideration of the Density Diffusion Sergey Peregudin, Elena Peregudina, and Svetlana Kholodova

Abstract Waves of small amplitude in a stratified liquid are considered. The original physical problem is described by the system of partial differential equations with corresponding boundary-value conditions. We further study linearized free-wave problems in a stratified liquid, problems of internal waves in a rotating stratified liquid, problems of forced internal waves in a rotating stratified liquid, and problems of free internal waves in the presence of horizontal density diffusion. The results obtained can be applied in problems of hydrodynamics, theory of waves, geophysics, applied mathematics, marine construction at the design stage, and in marine wave suppression problems.

101.1 Main Equations and Boundary Conditions By a model of the liquid one means a hypothetical medium [1] which does not take into account certain physical properties of the real liquid, which are immaterial for the class of motions under consideration. Moreover, the applicability range of each model depends on the degree of conformity of the model-based results with experimental findings. So, a model of a liquid may prove quite applicable in the study of one type of motions, but it can be completely unfit in the analysis of motions of a different type. The problem of waves generated by oscillations of a flat plate was considered in [2]. The problems considered above can be also treated numerically by constructing the corresponding difference scheme [3] for further visualization and analysis of S. Peregudin (B) Saint Petersburg State University, 7/9, Universitetskaya nab., Saint Petersburg 199034, Russia e-mail: [email protected] E. Peregudina Saint Petersburg Mining University, 2, 21st Line, Saint Petersburg 199106, Russia e-mail: [email protected] S. Kholodova ITMO University, 49, Kronverksky Pr., Saint Petersburg 197101, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_101

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the hydrodynamic parameters. The article [4, 5] are devote to a mathematical model of distribution of the long waves above the deformable bottom. The paper [6] considers the problem of constructing asymptotics describing the far fields of internal gravitational waves from an oscillating point source of perturbations moving in a stratified vertically semi-infinite medium of variable buoyancy. The article [7] considers the model distribution of the shear flow over depth and obtains an analytical solution to the problem in the form of the characteristic Green’s function, which is expressed in terms of the modified Bessel functions of the imaginary index. Let us consider the motion of an incompressible perfect liquid, whose density (in the unperturbed state) depends only on the vertical coordinate [1], i.e. ρ(x, y, z, t) = ρ0 (z) + ρ1 (x, y, z, t), where ρ0 (z) is the motion-free density distribution and ρ1 (x, y, z, t) is the dynamic supplement. A liquid whose density in the unperturbed state depends only on one parameter (the depth (ρ = ρ0 (z))) is called stratified. It is assumed that the stratifidρ0 < 0. We also represent the pressure as the sum cation is stable, i.e. dz 0 p=g

ρ0 (z)dz + p1 (x, y, z, t),

z < 0,

z

where the first term determines the hydrostatic pressure at the point with coordinate z, and p1 is the dynamic supplement of the pressure, the z-axis is vertically upward, the plane z = 0 coincides with the unperturbed free surface of the domain occupied by the liquid. The problem of waves of small amplitude in a rotating layer of a perfect incompressible stratified liquid of constant depth can be reduced to the solution of the system of partial differential equations [1] ∂v y 1 ∂ p1 1 ∂ p1 ∂vx − 2ωv y = − , + 2ωvx = − , ∂t ρ0 ∂ x ∂t ρ0 ∂ y ∂v y ∂vz ρ1 ∂ρ1 ∂vx ∂vz dρ0 1 ∂ p1 = −g − , + vz = 0, + + =0 ∂t ρ0 ρ0 ∂z ∂t dz ∂x ∂y ∂z (101.1) with the boundary conditions  vz = 0 for z = −H,

  1 ∂ p1  = 1 ∂ p0 . − v z  0 gρ ∂t gρ 0 ∂t z=0

(101.2)

Note that the profile of the  surface  free  ζ can be found from the known function p1  1 by the formula ζ = gρ 0 p1  − p0 . z=0

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101.2 Forced and Internal Waves in a Rotating Liquid Let us study free ( p0 = 0) waves in a stratified perfect liquid of constant depth. We consider a motion periodic in the time and in the horizontal coordinates. The solution of system (101.1), (101.2) will be sought in the form   vx , v y , vz , p1 , ρ1 = v˜ x (z), v˜ y (z), v˜ z (z), p˜ 1 (z), ρ˜1 (z) eiθ , θ = mx + ny − σ t. (101.3) Substituting (101.3) into (101.1) and (101.2), we arrive at the boundary-value problem for the system of ordinary differential equations with respect to v˜ x , v˜ y , v˜ z , p˜ 1 , ρ˜1 . By excluding the sought-for functions with respect to vz , we get the following second-order linear homogeneous differential equation: m 2 + n2  2 1 dρ0 dvz d 2 vz − σ − N 2 vz = 0 + dz 2 ρ0 dz dz σ 2 − 4ω2

(101.4)

with the boundary conditions vz = 0 for z = −H,

g(m 2 + n 2 ) dvz − 2 vz = 0 for dz σ − 4ω2

z = 0,

(101.5)

g dρ0 is the squared Brunt–Väisälä frequency. The tildes over symρ0 dz bols are dropped for clarity. We write Eq. (101.4) as where N 2 = −

m 2 + n2  2 1 dρ0 , β(z) = − 2 σ − N2 . 2ρ0 dz σ − 4ω2 (101.6)

Inserting vz = u(z) exp( α(z)dz) in Eq. (101.6) and in the boundary condition (101.5), we get the equation vz + 2α(z)vz + β(z)vz = 0, α(z) =



u(z) + q u(z) = 0, 2

g(m 2 + n 2 ) 1 q =− 2 + 2 σ − 4ω2 4ρ0 2



dρ0 dz

2 −

1 d 2 ρ0 2ρ0 dz 2 (101.7)

with the boundary conditions u(z) = 0 for z = −H,

  g(m 2 + n 2 ) u(z) = 0 for z = 0. u(z) + α − 2 σ − 4ω2 

If q 2 = const, then the equation for R(ρ0 ) =

dρ0 , ρ0 = ρ0 (z) assumes the form dz

  dR 1 2 m 2 + n2 m 2 + n2 2 2 R = R − 2g 2 R−2 q + 2 σ ρ0 . dρ0 σ − 4ω2 σ − 4ω2 2ρ02

(101.8)

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This equation is a second-order Abel equation. Note that Eq. (101.8) is also satisfied for ρ0 (z) = ρ 0 e−kz (k > 0, ρ 0 = ρ0 (0)). For q 2 = const, the general solution of Eq. (101.7) reads as u(z) = D1 sin qz + D2 cos qz.

(101.9)

It has a nontrivial solution if the following dispersion relation is satisfied: q(σ 2 − 4ω2 ) , tgq H = g(m 2 + n 2 ) − α(σ 2 − 4ω2 )

 1 dρ0  α= . 2ρ 0 dz z=0

(101.10)

If the liquid is exponentially stratified (ρ0 (z) = ρ 0 e−kz , k > 0), the dispersion Eq. (101.10) assumes the form 

 g(m 2 + n 2 ) k th(b0 H ) + b0 = 0, b0 = − σ 2 − 4ω2 2



m 2 + n2 k2 + 2 (σ 2 − 4ω2 ). 4 σ − 4ω2 (101.11) In the case n = 0, the dispersion relation (101.11) coincides with Cherkesov’s dispersion relation (see [1]). So, we have completely determined the vertical component of the velocity of the wave motion in the case when the wave vector (m, n) and the frequency σ are related by the dispersion relation (101.10) or (101.11). Let us study free internal waves under the two assumptions from [1]. The first one is the Boussinesq assumption to the effect that in the equation m 2 + n2  2 1 dρ0 dvz d 2 vz − σ − N 2 vz = 0 + 2 2 2 dz ρ0 dz dz σ − 4ω one can neglect the second term, i.e. here one assumes that    2   1 dρ0 dvz   d vz   .   , max  2   max  dz ρ0 dz dz  1 dρ0 is a small quantity in the actual sea condiρ0 dz tions. The second assumption (the rigid-lid condition) is that the boundary condition on the free surface is replaced by the more simple condition vz = 0 for z = 0. This means that the surface waves are filtered out, because the free surface is replaced by the fixed rigid surface. So, in the oceanographic approximation, the free internal wave problem reduces to the solution of the ordinary differential equation According to [1], the quantity

m 2 + n2 d 2 vz − dz 2 σ 2 − 4ω2



 g dρ0 2 + σ vz = 0 ρ0 dz

(101.12)

101 Wave Motions of Liquid with Consideration of the Density Diffusion

885

with the boundary condition vz = 0 for z = 0; vz = 0 for z = −H. For exponentially distributed density, Eq. (101.12) assumes the form gk − σ 2  2 d 2 vz m + n 2 vz = 0. + 2 2 2 dz σ − 4ω For 2ω < σ
Mt Eq. (102.3) is so complicated that it is solved only numerically. If M = P(L − x) we have Eq. (102.3) in another form. We consider the bending of the vertical beam by influence of the concentrated load. The load P simulates the iceberg (see Fig. 102.2 on the left). The formulas (102.2) do not change. We have an essential difference between the case with a constant moment. The thicknesses of the plastic zones z 1 , z 2 becomes functions of x, and all three stressed states take place in the beam. There are three different types of stress states in the beam cross section, depending on the value of the longitudinal coordinate x. There are the elastic case (a) with x1 < x < L, the case (b) with one plastic zone x2 < x < x1 , and the case (c) with two plastic zones 0 < x < x2 (see Fig. 102.2 on the left). After transformations of formula (102.1) and equilibrium equation M(κ) = P(L − x) we have in the elastic case (a) (P − 4bhγ ν)(L − x) d 2 w 4wbh(L − x)γ1 = . + 2 dx EI EI This equation is also the Airy equation and its solution is w(ξ ) = c1 Ai(K ξ ) + c2 Bi(K ξ ) −

P − 4bhνγ . 4bhγ1

The integrating constants are the solution of the system c1 Ai(K L) + c2 Bi(K L) =

P − 4bhγ ν L , 4bhγ1 L

c1 K Ai  (K L) + c2 K Bi  (K L) = 0.

(102.4)

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In the case (b) with one plastic zone we have the equation d 2w σt = dx2 Eh



2Mt . Mk (d + 1)d−1 − P(L − x) − 4wbh(L − x)γ1 + 4bhγ ν(L − x)

In the case (c) the equilibrium equation has a form d 2w σt (d + 1)3/2 = dx2 4Eh

 Mt . Mk − P(L − x) − 4wbh(L − x)γ1 + 4bhγ ν(L − x)

The solutions of these equations are possible only numerically. Now we will consider the solution of the assigned problems using FEM. In Figs. 102.3 and 102.4, some results of numerical solution with help of software package ANSYS 13.0 are shown. For example, we give the Mises effective stress distribution (in Russian terminology, stress intensity) near the termination, modeled using the three-dimensional model of the FEM in the ANSYS 13.0 software package taking into account the weight of the vertical beam, hydrostatic lateral pressure, and load P = Mt /L or P = 0.98Mk /L. Note that in the indicated model, the full calculation of the problem for a beam 10 m long is about 40 min of computer time, for a beam 25 m long—90 min, and for

Fig. 102.3 The effective stress in cross section x = 0, P = Mt /L. Only elastic state is present

Fig. 102.4 The effective stress in cross section x = 0, P = 0.98Mk /L. The cross section of the beam is completely in a plastic state

102 Elastic–Plastic Bending of Vertical Supports of Drilling Platforms

897

Fig. 102.5 3D network for FEM calculation. The number of elements is 4000 per meter of beam length

a beam 100 m—about 4 h. Of course, this once again confirms the importance and necessity of constructing analytical solutions. To prevent calculation defects, we smooth the corners (see Fig. 102.5).

102.4 The Numerical Example for Deflections Calculation Let us study a numerical example. We take the following data for steel A40X L = 10 m, h = 0.1 m, b = 1.0 m, E = 212 GPa, σt = 760 MPa, γ1 = 78000 N/m3 , γ = 1030 N/m3 , ν = 0.33, d = 1.3. Values of moments Mt , M∗ , Mk depending on the parameter d can be seen in Table 102.1. Values of deflection w with weight and hydrostatic pressure can be seen in Table 102.2. In this table P = Pt = 5, 07 · 105 N, w1 —deflection without weight and pressure, w2 —deflection without pressure, and w3 —deflection w with weight and hydrostatic pressure. Deflection values are indicated in meters. There are results of the solution Airy’s equation (102.4). The results of calculation according to FEM are indicated in brackets. Analysis of the results allows us to conclude: when designing, it is imperative to calculate the supports of

Table 102.1 The values of the moments d 1.0 1.1 Mt · 106 H m M∗ · 106 H m Mk · 106 H m

5.07 5.07 7.61

5.07 5.55 7.97

Table 102.2 The values of the deflections L 10 15 w1 w2 w3

1.1950 (1.1900) 1.2001 1.1904 (1.1890)

2.6887 2.7307 2.708

1.2

1.3

5.07 5.99 8.30

5.07 6.39 8.60

20

50

4.780 4.958 4.887

7.690 (7.580) 7.868 7.705 (7.800)

898

G. Pavilaynen et al.

hydraulic structures, taking into account the hydrostatic effect of water, which has a reinforcing effect when the structure is stretched. Note the good agreement between the results of the analytical solution and the numerical calculation.

102.5 Conclusions In this paper, we considered two cases of an elastic–plastic bending vertical beam. One is under the action of concentrated load and weight—another is under the action the constant bending moment, weight, and hydrostatic pressure. The problems are considered analytically or numerically. The results may be used to compare the solutions using FEM and experimental results. Acknowledgements This investigation was supported financially by the Russian Foundation for Basic Researches (grants no. 18-01-00523). The authors are grateful to S.M. Kovalev, an employee of the Arctic and Antarctic Research Institute (St. Petersburg, Russia) for the photos provided.

References 1. Shhinek, K., Loset, S., Hoyland, K.: Physics and Mechanics of Ice. UNIS (1999) 2. Politko, V.A., Kantargi, I.G.: Calculation of marine ice-resistant structures on the effect of ice loads, taking into account Russian building standards. Hydraulic engineering. No. 1. St. Petersburg, pp. 27–35 (2017) 3. Borodkin, V.A., Gavrilo, V.P., Kovalev, S.M., Lebedev G.A.: Influence of Structural Anisotropy of Sea Ice on Its Mechanical and Electrical Properties. Proc. of the Second Int. Offshore and Polar Eng. conf., San Francisco, California, U.S.A., 14–19 June 1992, vol. II. pp. 670-674 (1992) 4. Lachugin, D.V., Pavilaynen, G.V.: On the bending of structural materials with plastic anisotropic effect. In: Kustova, E., Leonov, G., Morosov, N., Yushkov, M., Mekhonoshina, M. (eds.) AIP Conference Proceedings, p. 070019 (2018) 5. Pavilaynen, G., Naumova N.: Elastic-plastic deformations of SD-beams. In: Seventh Polyakhov’s Reading 2015, Saint-Petersburg, Russia, p. 7106764 (2015) 6. Pavilaynen, G.V.: Mathematical model for the bending of plastically anisotropic beams. Vestnic of the St. Petersburg University: Mathematics 48(4), 275–279 (2015) 7. Pavilaynen, G.V., Bembeeva, A.I., Canin M.S.: Elastic-plastic bending SD-beams. Vestnic of the St. Petersburg University: Mathematics 1(2), 284–291 (2014) 8. Pavilaynen, G.V.: Elastic-plastic deformations of a beam with the SD-effect. In: AIP Conference Proceedings 2014 ICNAAM, Greese, Rodos (2014)

Chapter 103

Sommerfeld Effects in Two-Mass Crusher with Three Degrees of Freedom Serge E. Miheev and Petr D. Morozov

Abstract The crushing process in a vibration two-mass crusher with mechanical restrictions of degrees of freedom up to three and with an asynchronous engine as an unbalance drive is considered. The system of ordinary differential equations that accurately describes the dynamics of the device at idle is applied. To simulate the stroke, this system was transformed into a system of stochastic differential equations. The analysis of the crushing process parameters was made at various rotational frequencies of the magnetic field of the asynchronous drive. The frequencies at which the Sommerfeld effect is manifested are established.

103.1 Introduction The simplicity of single-mass vibration crushers with fixed mortars cannot compensate their main drawback: poor balance, which leads to the leakage of vibration into the environment in unacceptable volumes. In a two-mass scheme, the suspension of the mortar in elastic connection with the transfer of the vibrator from striker to mortar appeared to be not only a good balance, but also, a good ratio of crushing forces to vibration forces. Then the use of a single rotating unbalance as a vibrator would lead to the appearance of parasitic-side vibrations. Therefore, two unbalances are installed symmetrically with separate drives, with parallel axes and opposite rotation. The physical model of the crusher with such a scheme and with a restriction of the degree of freedom of the striker movement relative to the mortar movement to one, i.e. only nine degrees, was created in [1]. The same rotational speed of magnetic fields of asynchronous drives leads to synchronization of rotation of unbalances, which theoretically extinguishes to zero the total lateral force transmitted by them to S. E. Miheev (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] P. D. Morozov Mekhanobr-Tekhnika, St. Petersburg, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_103

899

900

S. E. Miheev and P. D. Morozov

Fig. 103.1 General scheme of a two-mass vibration cone crusher with three degrees of freedom

the mortar. This synchronized mode is quite accessible to analysis, however, there is a few problems. Any actual installation and the chaotic presence of ore in the device will create asymmetry. This will lead to the mandatory appearance of rotational vibrations of the striker and the mortar. Therefore, instead of complete synchronization the steadiness and stability must be considered [2]. Here, a similar scheme is researched with an additional restriction of the degree of freedom of the mortar to one, alongside vertical axis. Since horizontal efforts from unbalances in this scheme do not affect its dynamics, for theoretical research it is possible to limit oneself to one unbalance, and for a physical model, the vibrator to execute in the form of unbalances paired with identical gears, so that they rotate in opposite directions with one speed. Then there are three degrees of freedom (Fig. 103.1). The results of this article are based on [5].1 They are obtained in numerical experiments with this scheme and also may be applied to the mentioned scheme with nine degrees of freedom when its unbalances are synchronized, despite possible small errors caused by asymmetry. As in [3], energy consumption has become a good mark to research the device.

1

Based on [5] originally published open access under a CC BY 4.0 license, https://doi.org/10. 21595/vp.2020.21542.

103 Sommerfeld Effects in Two-Mass Crusher with Three Degrees of Freedom

901

103.2 Depiction of Crushing in Differential Equations The differential equations for the crusher at idling were published in [4]. The function  simulating the crushing process was introduced in [5]. ⎧ z˙ = z  ⎪ ⎪ ⎪ ⎪ z˙ s = z  s ⎪ ⎪ ⎪ ⎪ ϕ˙ = ϕ  ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ κ2 (z s (t) − z(t)) − κ1 z (t) − P cos2 ϕ(t) + I J −1 sin2 ϕ(t) ⎪ ⎪ ⎪ − Z (t) = ⎪ ⎪ m b + m u − m 2u r 2 J −1 sin2 ϕ(t) ⎪ ⎪ ⎪ 2  ⎪ ⎨ m u ϕ  (t) r cos ϕ(t) + m u J −1 r sin ϕ(t) μ (ω − ϕ1(t)) − ) . − ⎪ m b + m u − m 2u r 2 J −1 sin2 ϕ(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z˙  = Z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z˙ s = (κ2 (z − z s ) − )/m s ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ϕ˙ = (μ ω − ϕ  − m u r Z sin ϕ − Pr sin ϕ )/J ⎪ ⎪ ⎪   ⎩ w˙ = μ ω − ϕ  ϕ  (103.1) The physical sense of the system parameters is the same as in [5]: z s —the striker displacement, in mm; z s —the striker displacement rate, in mm/s; z—the mortar displacement, in mm; z  —the mortar displacement rate, in mm/s; ϕ—the rotational angle of the unbalance gravity center, counted from its lowest point counterclockwise, in rad; ϕ  —the angular velocity of unbalance rotation, in rad/s; I —the unbalance inertia moment, in kg·m2 ; J = I + m u r 2 —the unbalance inertia moment relative to the rotational axis, in kg·m2 ; r —the distance between the shaft center and the gravity center of the unbalance, in mm; m u , m b , m s —the masses of the unbalance, the mortar and the striker, respectively, in kg; κ1 , κ2 —the stiffness coefficients of the elastic bonds of the mortar with the base and the striker with the mortar, respectively, in N/m; ω—the magnetic field rotational speed of the engine, in rad/s; w—the engine power at current moment, in watts; and μ—the coefficient characterizing the potential power of the asynchronous engine with dimension N· m ·s2 = J·s2 = W·s. The physical parameters in the system (103.1) were taken close to the physical model ones in [1]. That is, m b = 59 kg, m s = 7.4 kg, m u = 1.57 kg, κb = 840000 N/m, κs = 120000 N/m. The inertia moment of the unbalance was calculated as I = 0.001 kg·m2 . We appoint the parameter μ as 0.005 W· s.

902

S. E. Miheev and P. D. Morozov

The system (103.1) has been solved by fourth-order Runge–Kutta method with a constant step h. Its value was chosen experimentally by comparing the engine work w (t) and the energy accumulated at idling E (t), which should coincide according to , the difference the law of energy conservation. At the step h = 5 · 10−4 s, ω = 100 rad s between them at the 60th second was 9.08 · 10−3 % of the value of the engine work, and with a step of h = 2.5 · 10−4 s it was 7.26 · 10−3 %. This is quite satisfactory. The following experiments were with h = 2.5 · 10−4 s. Note that such a small discrepancy between the work and the energy also indicates the correctness of the system (103.1) and the correctness of its transfer to the program code.

103.3 Model of Crushing Process We have used our ideas from [5]. That is, the time of one unbalance revolution destruction is significantly bigger than the time of ore piece destruction. Therefore, it is advisable to describe speed changes of the mortar and the striker by delta functions. So,  contains delta functions. As ordinary integration methods demand much better right sides of the equations, we do instant speed changes of the mortar and the striker at the moment of crushing by hand. So, let (t) = Aδ(t − t  ), where A is some number. Then, excluding  from system (103.1), replenish it with non-differential equations reduced from the equality of the forces produced by the mortar and the striker to crush a piece of ore2 ⎧ ⎪ ⎪ ⎨

def

z  (t  ) = z  (t  − 0) + A

 m b + m u − m 2u r 2 J −1 sin2 ϕ(t  − 0) def

z s (t  ) = z s (t  − 0) − A/m s   def ϕ  (t  ) = ϕ  (t  − 0) − Am u r sin ϕ(t  − 0) J m b + J m u − m 2u r 2 sin2 ϕ(t  − 0) . (103.2) Here, random variable A is the force of the impact of a piece of ore on the body and the striker, and random variable t  is the time of this impact. As per the construction, they both are non-negative. We assume the act of crushing leaves a noticeable share (1 − σ ) of convergence speed of the mortar and the striker just before this act. That is, ⎪ ⎪ ⎩

(1 − σ )(˙z s (t  − 0) − z˙ (t  − 0)) ≤ z˙ s (t  ) − z˙ (t  ) ≡

 ≡ z˙ s (t  − 0) − A/m s − z˙ (t  − 0) + A m b + m u − m 2u r 2 J −1 sin2 ϕ(t  − 0) ⇐⇒ A ≤ σ (˙z s (t  − 0) − z˙ (t  − 0))

 1

−1 −1 m2 r 2 def + m b + m u − u sin2 ϕ(t  − 0) = B(t  ). ms J

So, at random times t  , the speed jumps of the striker, the body, and the unbalance are delivered by the system (103.2). The magnitude of the jumps depends on the

2

The weight of the piece is negligible in comparison to these forces.

103 Sommerfeld Effects in Two-Mass Crusher with Three Degrees of Freedom

903

random non-negative parameter A(t  ), which does not exceed the value of B(t  ). def Therefore, setting A(t  ) = v B(t  ), where v is a random value distributed on [0, 1], does not conflict with the logic of the construction. We pay attention to the uniform distribution for v. The approach of the striker and the mortar, which began at some moments τ , will last approximately half the period of unbalance rotation, if it would further def ˙ ). Assume that crushing episodes rotate with constant speed ϕ(τ ˙ ), i.e. T = π/ϕ(τ occur during time intervals [τ + T /4, τ + 3T /4]. And also assume that in each such interval exactly one episode of crushing occurs at a random moment t  with a uniform distribution on it.

103.4 Experiments A series of numerical integrations of the system of stochastic differential equations (103.1) + (103.2) with different power frequency magnetic field ω = 90, 95, . . . , 225 were performed. During each experiment, this frequency was unchanged. After starting with zero initial values of the variables the construct accumulated energy at idle during one second. Then the working stroke was switched on. Further, the process was investigated only 10 s, since practical stabilization of the stochastic process was observed after 3 s at each given ω. On the time interval [3, 4] the maximum and minimum of the following quantities were calculated: unbalance angular speed ϕ, ˙ full energy E accumulated in the crusher, unbalance energy E u , E − E u , z, and z − z s . In the table for the differences, the functions max and min were denoted by overline and underline, respectively. The same time interval was used to calculate three integral characteristics of the engine efficiency: usage, which is the ratio of real power take-off to the possible maximum for given engine and ω, w being the performed work, and cw being the ratio of w to the force on the unbalance shaft. We understand that Sommerfeld effect (SE) appears, when these parameters change tenfold for a little frequency variation. For σ = 0.01, two SEs were detected in frequency intervals [125, 130] and [185, 190] (Table 103.1). After that, the refinement of these intervals was performed in ω step of 1. We have named the left sides of such refined intervals as Sommerfeld thresholds (ST). So, we have got two ST ω1 = 126 rad/s and ω2 = 188 rad/s. Just above ω2 , SE was much brighter than just above ω1 . When both ST passed, the parameters usage, w and cw change tenfold (Fig. 103.2). Just above the thresholds the crusher efficiency drops in all respects. In particular, the usage and the amplitude of the oscillations of the clearance between the mortar and the striker decrease and the load on the unbalance axis increases.

92.09

97.08

101.99

104.19

105.38

106.29

107.02

107.64

148.88

143.19

146.41

150.73

153.40

155.60

157.84

161.59

165.80

169.45

175.81

178.30

185.49

206.36

208.96

211.74

216.21

220.50

223.92

229.35

95

100

105

110

115

120

125

130

135

140

145

150

155

160

165

170

175

180

185

190

195

200

205

210

215

220

225

max ϕ˙

90

ω

219.69

214.69

208.62

202.53

196.38

189.51

180.41

107.39

115.17

116.44

122.61

126.92

132.01

134.34

136.42

138.05

138.21

134.11

126.81

108.66

95.85

96.25

96.78

97.32

97.72

97.42

93.72

88.11

min ϕ˙

35.10

34.11

33.28

32.71

32.61

33.50

37.50

253.66

227.11

200.99

174.44

147.34

120.57

95.03

69.83

46.44

27.11

17.51

16.19

27.22

98.04

79.97

61.54

43.52

26.48

12.31

6.70

5.56

max E

34.81

33.77

32.87

32.18

31.86

32.25

35.05

192.95

173.36

153.34

133.24

113.08

93.39

74.44

56.17

38.96

24.85

16.94

15.12

20.64

73.96

60.85

47.32

34.16

21.75

11.41

6.67

5.48

min E

5.219

5.804

6.492

7.181

8.740

11.151

17.498

241.478

212.818

187.777

161.517

134.077

105.636

81.302

56.453

33.429

14.657

6.051

5.976

19.103

90.571

73.122

54.179

36.665

20.058

6.157

1.154

0.642

E − Eu

1.557

1.714

2.150

2.841

3.310

4.606

8.418

176.772

151.904

133.524

115.235

96.696

77.803

57.084

40.231

23.665

10.233

3.263

2.348

7.743

65.320

52.645

39.284

26.633

14.547

4.729

0.755

0.199

E − Eu

0.000

0.002

0.003

0.002

0.005

0.003

0.024

0.673

0.628

0.588

0.538

0.474

0.399

0.322

0.228

0.133

0.042

0.006

0.013

0.074

0.616

0.531

0.430

0.309

0.172

0.036

0.000

0.004

usage

Table 103.1 The parameters of the steady-state crushing process at different frequencies

0.04

0.30

0.30

0.21

0.52

0.32

2.26

60.7

53.7

47.6

41.2

34.3

27.2

20.6

13.7

7.5

2.2

0.3

0.6

3.1

24.1

19.1

14.2

9.4

4.7

0.9

0.0

0.1

w

4.0E-05

3.1E-04

3.2E-04

2.3E-04

6.0E-04

3.8E-04

2.7E-03

9.1E-02

8.7E-02

8.0E-02

7.4E-02

6.4E-02

5.4E-02

4.3E-02

2.9E-02

1.6E-02

5.0E-03

7.0E-04

1.5E-03

7.3E-03

1.1E-01

8.6E-02

6.5E-02

4.3E-02

2.2E-02

4.4E-03

1.4E-05

5.1E-04

cw

2.54

2.62

2.72

2.97

2.78

3.28

4.14

16.11

14.60

14.03

12.34

11.36

9.97

8.47

7.31

5.79

4.17

2.81

2.63

5.70

12.30

11.06

9.53

7.83

5.79

3.18

1.22

0.72

max z

7.32

7.85

8.01

8.77

9.22

11.12

13.73

56.50

54.13

50.07

47.53

42.36

37.21

33.39

27.37

22.13

14.98

9.43

7.94

14.09

20.96

18.73

16.17

13.06

9.53

4.98

1.60

0.89

z − zs

−7.25

−7.56

−8.15

−8.96

−10.07

−11.18

−13.45

−57.78

−54.23

−51.33

−46.20

−41.42

−38.11

−33.04

−27.85

−21.85

−14.92

−9.36

−8.08

−14.32

−21.00

−18.79

−16.06

−13.17

−9.51

−4.96

−1.58

−0.86

z − zs

904 S. E. Miheev and P. D. Morozov

103 Sommerfeld Effects in Two-Mass Crusher with Three Degrees of Freedom

905

Fig. 103.2 Maximal increasing and decreasing of primary clearance between striker and mortar at different frequencies, σ = 0.1

103.5 Conclusion The studies have revealed that asynchronous driver of the unbalanced vibrator in the vibratory crusher with three degrees of freedom has two effective frequencies of the magnetic field rotation. Each of these frequencies corresponds to the sharp upper threshold of the Sommerfeld effect. A very small increase of the frequencies leads to a strong deterioration in the working process. Sommerfeld higher thresholds are more effective than the lower one. Therefore, in the design of the crusher, the parameters corresponding to the operation at a higher Sommerfeld threshold, which can be calculated according to the above algorithm, should be laid down. After the installation is made, the proximity of its operation to the threshold frequency can be provided by a controller that changes the rotation speed of the magnetic field according to information on the difference between it and the unbalance rotational speed. The first threshold of Sommerfeld in our model is close to the frequency of natural vibrations of the mortar with a fixed unbalance and a stationary striker (125.9 rad/s). Hence, the phenomena can be regarded as a special behavior of the system in quasiresonant mode [6].

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Sommerfeld effects can also be detected on other crushing models. If the latter uses simple friction, the behavior of the corresponding mechanical systems can be subjected to a qualitative analysis [7], as well as our approach. It is shown that to prevent vibration leak the elastic coupling between the shafts of engine and unbalance introduces significant features in the dynamics [8]. So, the next problem to research by considered instrument is the influence of the elastic coupling on Sommerfeld thresholds. Then the crusher will have four degrees of freedom.

References 1. Shishkin, E.V., Kazakov, S.V.: Dynamics of vibratory crusher in resonance frequency range. J. Obogashchenie Rud (6), 29–33 (2014) 2. Shishkin, E.V., Kazakov, S.V.: Vibrational dynamic system for the reduction of solid materials. Vibroengineering PROCEDIA 25, 65–69 (2019) 3. Samukov, A.D., Cherkasova, M.V., Rzhankova, N.B., Dmitriev, S.V.: Energy consumption in the manufacture of metal powders by vibratory disintegration methods. Vibroengineering PROCEDIA 32, 52–57 (2020) 4. Morozov, P.D., Miheev, S.E.: Mathematical model of the vibration cone crusher with three degrees of freedom. Vibroengineering PROCEDIA 25, 42–47 (2019) 5. Morozov, P.D., Miheev, S.E.: Stochastic model of the stroke of a two-mass cone crusher. Vibroengineering PROCEDIA 32, 45–51 (2020) 6. Vaisberg, L.A., Kazakov, S.V., Shishkin, E.V.: Vibrational disintegration of solid materials in quasiresonant modes. In: Congress Proceedings. International Mineral Processing Congress, pp. 297–305 (2019) 7. Murashko, A.Y., Orlov, V.B., Zubov, A.V., Bondarenko, L.A., Petrova, V.A.: Qualitative analysis of the behavior of one mechanical system. Int. J. Innov. Technol. Explor. Eng. 8(7), 653–658 (2019) 8. Blekhman, I.I., Blekhman, L.I., Yaroshevich, N.P.: Upon drive dynamics of vibratory machines with inertia excitation. J. Obogashchenie Rud (4), 49–53 (2017)

Chapter 104

On the Account of Transverse Young–Laplace Law Under Stability of a Rectangular Nano-Plate Anatolii Bochkarev

Abstract As known, the constitutive relations of the Gurtin–Murdoch model of the surface elasticity take into account both elastic properties of a solid and surface tension of a liquid. Previously, most authors believed that, surface tension does not significantly affect the mechanical properties of a thin-walled elastic nano-objects and therefore it either was not or only in-plane taken into account, ignoring the Young–Laplace law in the transverse direction. In the present work, on the basis of the nonlinear representation of the surface tension obtained earlier, the structure of the strain energy of a nano-plate is substantiated, taking into account the surface tension in both tangential and transverse directions. Using the example of compressive buckling of a nano-plate, it is shown that the accounted surface tension terms can increase the size effect of the critical load up to 80%.

104.1 Introduction The continuum model of surface stresses or surface elasticity has been developed by Gurtin and Murdoch (GM) [1, 2]. The GM constitutive relations take into account both elastic properties of a solid (through the strain-consistent terms) and surface tension of a liquid (through the displacement gradient). The atomistic modeling in [3, 4] has supported such continuum approach. Steigmann and Ogden [5, 6] have generalized this model by introducing surface couple stresses that allowed the surface layer to also resist bending. Now, the two-dimensional theories with surface stresses taken into account have been an important tool modeling the mechanical behavior of nano-plates and shells. So in [7], the large deflections of nano-plates have been offered to describe with the von Kármán plate theory with the surface stresses taken into account on the base of the linearized constitutive relations of the GM surface elasticity. However, in this work, the total energy has been written out in terms of “stress–strain”, while surface A. Bochkarev (B) St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_104

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tension is not expressed through strain. As a result, the Young–Laplace conjugation condition in the transverse direction has remained unfulfilled. An attempt to correct this inaccuracy has been made in [8] using a hypothesis on the σ33 -distribution made in [9]. This refinement gave the expectedly insignificant size effect for the plate theory. Anyway, the widely used approach has been the transition to the symmetric membrane forces on base of the simplified (strain-consistent) model of the surface elasticity: for studying the effective properties of nanomaterials and the thin-walled structures in [10–13], with induced residual stress in [14]; for modeling a laminated thin plate in [15], with an effect of the surface integrity in [16]; and for the buckling of a nano-plate in the Kirsch problem [17]. In models covering related physical processes, the authors usually have taken into account the gradient terms, for example, under the elastic buckling of the currentcarrying double-nanowire systems in a magnetic field [18] or in the viscoelastic behavior of a nano-plate with the simultaneous effects of the microstructure rotation and surface energy [19]. Another approach proposed in [20, 21] in relation to the third-order shear deformation theory of Reddy was to move to symmetric membrane forces artificially, while remaining within the complete constitutive relation. In order to study the effect of surface tension on the stiffness properties of a nano-plate, the author compared the results of modeling the compressive buckling of a rectangular nano-plate under different flexural boundary conditions according to the GM constitutive relations without the gradient terms [22] and with them [23]. The preliminary result was that the in-plane gradient terms were not included in the strain energy of the midplane and therefore did not significantly affect the critical load, and the correction could be described by a correction factor. This factor corrects the dimensionless critical load up to 1.5% and the end critical load up to 3.5%, what it would seem that justifies using the simplified strain-consistent GM model. In [24], it is shown the structure of surface tension with nonlinear terms equal the von Kármán strains taken into account. Based on this result, in this study, the surface tension is fully incorporated into the deformation energy of the nano-plate, including the terms responsible for the Young–Laplace condition in the transverse direction. It is shown that this term can significantly influence on the size effect under buckling. So, in the case of the compressive buckling of a rectangular nano-plate, this size effect can increase the rigidity of a nano-plate already up to 80% compared to the strain-consistent model.

104.2 A Nano-Plate with Gurtin–Murdoch Surface Stresses A homogeneous linearly elastic plate is considered, having a nanoscale thickness h and occupying in the Cartesian coordinate system the region {(xα ; x3 ) ∈  × [−h/2, +h/2]},  ⊂ R2 . The axis x3 is orthogonal to a midplane of the plate and the axis xα (α = 1, 2) is in the midplane.

104 On the Account of Transverse Young–Laplace Law …

909

Let on the facial surfaces of the plate there are the surface stresses τ ± (under x3 = ±h/2) according to the linearized constitutive relation [1, 2] τ ± = τ0 A + 2(μs − τ0 )˜ε ± + (λs + τ0 )A tr ε˜ ± + τ0 ∇s u± .

(104.1)

Here μs , λs are the Lamé surface constants; τ0 is the residual surface stress (all are assumed to be equal on both plate faces); A is the two-dimensional unit tensor; ε˜ = (A · ε · A) is the in-plane projection of strain tensor ε ; ∇s is the two-dimensional nabla operator in the midplane; u is a displacement; w = n · u is a deflection; and n is a normal vector to the midplane. Equilibrium of faces and bulk phase is provided by the Young–Laplace law [1] ± n · σ ± = ∇s · τ ± .

(104.2)

The constitutive relation of bulk phase (−h/2 < x3 < +h/2) is expressed by the usual Hooke’s law for plates, neglecting σ33 compared to σαβ σ˜ = (A · σ · A) =

 E  (1 − ν)˜ε + νA tr ε˜ . 2 1−ν

(104.3)

Here E is Young’s modulus and ν is Poisson’s ratio of bulk phase. Let us consider a potential energy of this nano-plate with accounting surface tension described by Eqs. (104.1)–(104.3).

104.2.1 Potential Energy of a Nano-Plate with Surface Stresses The strain energy of this nano-plate consists of a strain energy of bulk phase E U= 2(1 − ν 2 )

   +h/2  ν tr 2ε˜ + (1 − ν)˜ε · ·˜ε d x3 d  −h/2

and surface energies of faces ± [1] V

±

1 = 2

     τ0 2 + 2tr ε˜ + (λs + τ0 )tr 2ε˜ + 2(μs − τ0 )˜ε · ·˜ε + τ0 ∇s u · ·∇s u d. ±

Accepting the kinematic Kirchhoff hypothesis, the total energy of a nano-plate can be expressed in the form of the well-known classic structure of the potential energy of a macroplate as sum of effective strain energies of the midplane and bending [22, 23] W = U + V + + V − = W1∗ + W2∗

910

A. Bochkarev

W1∗ =

1 2

 

  4τ0 (1 + tr ε 0 ) + C ∗ νt∗ tr 2ε 0 + (1 − νt∗ )εε 0 · ·εε 0 + 2τ0 (∇s u)0 · ·(∇s u)0 +





1 2τ0 (∇s w)2 d; W2∗ = 2

 

(104.4)



  D ∗ νf∗ tr 2 (∇s ∇s w) + (1 − νf∗ )∇s ∇s w · ·∇s ∇s w d,



(104.5)     0   where ε = A · ε · A x3 =0 and (∇s u) = ∇s u · A x3 =0 are the in-plane projections of the strain tensor and the displacement gradient in the midplane; and the effective tangential and flexural elastic moduli and stiffnesses ε0

νt∗ = ν ∗f =

 1  Eh νC + 2(λs + τ0 ) , C ∗ = C + 4μs + 2λs − 2τ0 , C = ∗ C 1 − ν2

 1  Eh 3 . ν D + 0.5h 2 (λs + τ0 ) , D ∗ = D + h 2 μs + 0.5h 2 λs , D = ∗ D 12(1 − ν 2 )

Let us consider the nonlinear strains and surface tension of von Kármán type [24] ε0 =

 1 1 ∇s u0 + (∇s u0 )T + ∇s w ⊗ (∇s w)T = e0 + ∇s w ⊗ (∇s w)T 2 2   τ0 (∇s u)0 = τ0 ∇s u0 − w∇s ∇s w ,

 where e0 is a linear part of this strain and u0 = A · ux3 =0 is the in-plane displacement in the midplane. Neglecting the terms w of the fourth order of smallness, the effective energy of deformation of the midplane (104.4) can be represented as a sum W1∗ = 1 2

1 2

 

 

   4τ0 (1 + tr e0 ) + C ∗ νt∗ tr 2 e0 + (1 − νt∗ ) e0 · ·e0 + 2τ0 ∇s u0 · ·∇s u0 d +



   (∇s w)T · 2τ0 A + C ∗ (1 − νt∗ )e0 + νt∗ A tr e0 + 2τ0 ∇s u0 · ∇s w + 2τ0 (∇s w)2 −



  2τ0 (∇s w)T · ∇s u0 · ∇s w + 2w∇s ∇s w · ·∇s u0 d.

(104.6)

In the case of the Saint-Venant bending theory, the in-plane strains do not depend on the deflection and then the first integral is a constant. The second integral, in the presence of the deflection, will give an increment of this strain energy. However, its summands in the third line of Eq. (104.6) have the fourth order of smallness, because ∇s u0 ∼ (∇s w)2 as accepted in von Karman theory. Therefore, the final increment of the strain energy of the median plane takes the form

W1∗ =

1 2

  

 (∇s w)T · T∗ · ∇s w + 2τ0 (∇s w)2 d

(104.7)

104 On the Account of Transverse Young–Laplace Law …

911

expressed through effective membrane forces   T∗ = 2τ0 A + C ∗ (1 − νt∗ )e0 + νt∗ A tr e0 + 2τ0 ∇s u0 . How strongly this can affect the stiffness properties is illustrated by the example of the compressive buckling of a rectangular nano-plate, complementing the previously obtained results in [23].

104.3 Compressive Buckling of a Rectangular Nano-Plate The influence of surface stresses on the compressive buckling of a rectangular nanoplate is studied under the uniaxial compression p with the biaxial pretension 2τ0 ∗ ∗ ∗ ∗ T11 = 2τ0 − ph, T12 = 0; T22 = 2τ0 , T21 =0

(104.8)

for three classical cases of the flexural boundary conditions: • All four edges are Simple supported (SSSS) w(0, x2 ) = w(a1 , x2 ) = 0, w,11 (0, x2 ) = w,11 (a! , x2 ) = 0; w(x1 , 0) = w(x1 , a2 ) = 0, w,22 (x1 , 0) = w,22 (x1 , a2 ) = 0.

(104.9)

• Three edges are Simple supported and one is Free (SSSF) w(0, x2 ) = w(a1 , w(x1 , 0) = 0, w,22 (x1 , 0) = 0,

x2 ) = 0, w,11 (0, x2 )= w,11 (a1 , x2 ) = 0; ∗ ∗ w −D (104.10) ,22 + νf w,11 (x 1 , a2 ) = 0,   2τ0 w,2 − D ∗ w,222 + (2 − νf∗ )w,112 (x1 , a2 ) = 0.

• Two edges are Simple supported, two other are Claimed and Free (SCSF) w(0, x2 ) = w(a1 , w(x1 , 0) = 0, w,2 (x1 , 0) = 0,

x2 ) = 0, w,11 (0, x2 ) = w,11 (a1 , x2 ) = 0; ∗ ∗ w −D (104.11) ,22 + νf w,11 (x 1 , a2 ) = 0,   ∗ 2τ0 w,2 − D w,222 + (2 − νf∗ )w,112 (x1 , a2 ) = 0.

The Euler critical load is sought by the Ritz method.

104.3.1 Ritz Method The strain energy increment of the midplane (104.7) for a uniform stress (104.8) and the bending energy (104.5) have the form in the Cartesian coordinate system

W1∗

1 = 2

a1 a2   2 2 2 4τ0 (w,1 d x1 d x2 + w,2 ) − phw,1 0

0

(104.12)

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W2∗

1 = 2

a1 a2  0

  2 D ∗ (w,11 + w,22 )2 + 2(1 − νf∗ )(w,12 − w,11 w,22 ) d x1 d x2 .

0

(104.13) Using the deflection approximation as a series w(x1 , x2 ) =



Ak wk (x1 , x2 )

(104.14)

k

each term of which satisfies the kinematic boundary conditions from (104.9), (104.10), or (104.11), and considering the coefficients Ak as the generalized coordinates, the Ritz method leads to the matrix eigenvalue problem   4τ0 a 2 ∗

W (δ ) A, s W1∗ (δ1 j )A = W2∗ + i j 1 D∗

(104.15)

where s = pha 2 /D ∗ is a searching dimensionless eigenvalue; A is a generalized coordinate vector; W2∗ is the matrix of the bending energy (104.13); and W1∗ is the matrix of the increment of strain energy of the midplane (104.12). The deflection approximation (104.14) and the critical load are as follows: —SSSS (exact value) π x1 π x2 w(x1 , x2 ) = A11 sin sin a1 a2 a2  a12 2 4τ0  π 2 D∗  s 1 + 12 . pcr = 2 ł 1 + 2 + h a1 h a2 a2

(104.16)

—SSSF (the zero approximation) x2

π x1  kπ x2  A10 + A1k sin a1 a2 k=1 a2 ∗  2 2 ∗ ) 6(1 − ν 3 a12  4τ D a π f 0 1 s 1 + + . 1+ = 2 pcr π2 h π 2 a22 a1 h a22 w(x1 , x2 ) = sin

(104.17)

—SCSF (calculated numerically) w(x1 , x2 ) = sin

x2

π x1 x2  kπ x2  A10 + . · A1k sin a1 a2 a2 k=1 a2

(104.18)

104.3.2 Numeric Results To study the nature of the size effect according to the complete GM model, the numeric results of solving the eigenvalue problem (104.15) were made for a square

104 On the Account of Transverse Young–Laplace Law … Al[111]

2.5

913 Al[100]

2.5

SSSS SSSF SCSF

2

2

1.5

1.5

s min

/

min

SSSS SSSF SCSF

1

1 0

50

100

150

200

0

50

(a)

150

200

(b)

2.5

2.5 with trans. Y-L law without trans. Y-L law

p scr / p cr

100

with trans. Y-L law without trans. Y-L law

2

2

1.5

1.5

1

1 0

50

100

150

200

0

50

a, nm (c)

100

150

200

a, nm (d)

Fig. 104.1 Relative (to macro) minimal eigenvalue (a), (b) and Euler’s critical load (c), (d)—for Al, Al—with transverse Young–Laplace law and without it in dependence on the side a

nano-plate with the side a = 10h from aluminum with the bulk moduli λ = 58.2, μ = 26.1 GPa and with the surface moduli, obtained by a simulation method in [3] • Al[111]: λs = 6.85, μs = −0.38, τ0 = 0.91 (N/m) and • Al[100]: λs = 3.49, μs = −5.43, τ0 = 0.57 (N/m). The behavior of the relative (to macro) minimal eigenvalue and Euler’s critical load (104.16)–(104.18) in dependence on the side a is displayed in Fig. 104.1. As can see, the results with the transverse Young–Laplace law take values up to 80% more than without it.

104.4 Conclusions Surface tension ensures that the balance of surface stress and bulk is satisfied according to the Young–Laplace law [23]. Taking into account only the in-plane surface tension in [22] enhances the size effect when buckling only up to 3.5% compared with the strain-consistent model [22]. The present study showed that the surface tension in the transverse direction enhances this size effect up to 80%. So, the surface stresses can make a significant contribution to the rigidity.

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Acknowledgements It was supported by the Russian Foundation for Basic Research, grant no. 18-01-00468.

References 1. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291–323 (1975); Addenda to our paper 59, 389–390 2. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978) 3. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000) 4. Shenoy, V.B.: Atomic calculations of elastic properties of metallic FCC crystal surfaces. Phys. Rev. B 71, 94–104 (2005) 5. Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond. A 453, 853–877 (1997) 6. Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. R. Soc. Lond. A 455, 437–474 (1999) 7. Lim, C.W., He, L.H.: Size-dependent nonlinear response of thin elastic films with nano-scale thickness. Int. J. Mech. Sci. 46, 1715–1726 (2004) 8. Huang, D.W.: Size-dependent response of ultra-thin films with surface effects. Int. J. Solids Struct. 45, 568–579 (2008) 9. Lu, P., He, L.H., Lee, H.P., Lu, C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43, 4631–4647 (2006) 10. Altenbach, H., Eremeyev, V.A., Morozov, N.F.: On equations of the linear theory of shells with surface stresses taken into account. Mech. Solids 45, 331–342 (2010) 11. Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49, 1294–1301 (2011) 12. Eremeyev, V.A.: On effective properties of materials at the nano- and microscales considering surface effects. Acta Mech. 227, 29–42 (2016) 13. Eremeyev, V.A., Lebedev, L.P.: Mathematical study of boundary-value problems within the framework of Steigmann-Ogden model of surface elasticity. Contin. Mech. Thermodyn. 28, 407–422 (2016) 14. Ru, C.Q.: A strain-consistent elastic plate model with surface elasticity. Contin. Mech. Thermodyn. 28, 263–273 (2016) 15. Xu, M.: Effect of surface and interface energies on the nonlinear bending behaviour of nanoscale laminated thin plates. Mech. Compos. Mater. 52(5), 673–686 (2016) 16. Shaat, M.: Effects of surface integrity on the mechanics of ultra-thin films. Int. J. Solids Struct. 136–137, 259–270 (2018) 17. Bochkarev, A.O., Grekov, M.A.: Influence of surface stresses on the nanoplate stiffness and stability in the Kirsch problem. Phys. Mesomech. 22(3), 209–223 (2019) 18. Kiani, K.: Elastic buckling of current-carrying double-nanowire systems immersed in a magnetic field. Acta Mech. 227, 3549–3570 (2016) 19. Attia, M., Mahmoud, F.: Size-dependent behavior of viscoelastic nanoplates incorporating surface energy and microstructure effects. Int. J. Mech. Sci. 123, 117–132 (2017) 20. Ansari, R., Gholami, R.: Size-dependent modeling of the free vibration characteristics of postbuckled third-order shear deformable rectangular nanoplates based on the surface stress elasticity theory. Compos. Part B Eng. 95, 301–316 (2016) 21. Gholami, Y., Ansari, R., Gholami, R., Rouhi, H.: Analyzing primary resonant dynamics of functionally graded nanoplates based on a surface third-order shear deformation model. ThinWalled Struct. 131, 487–499 (2018)

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22. Bochkarev, A.O.: Influence of boundary conditions on stiffness properties of a rectangular nanoplate. Procedia Struct. Integr. 6, 174–181 (2017) 23. Bochkarev, A.O.: Compressive buckling of a rectangular nanoplate. In: AIP Conference Proceedings, vol. 1959, p. 070007 (2018) 24. Bochkarev, A.O.: On the account of surface tension nonlinearity under of nano-plate bending. Mech. Res. Commun. 106, 103521 (2020)

Chapter 105

Stress Analysis of a Spherical Pressure Vessel with Multiple Notches Olga S. Sedova and Daria D. Okulova

Abstract Finite element analysis of a thin-walled spherical vessel with multiple pits on its outer surface under internal pressure is conducted. The pits are modeled as spherical notches of the same size. Uniform and random arrangement of notches along the equator of the sphere is considered. The distribution of the maximum stress near defects is obtained for the different numbers of defects.

105.1 Introduction Steel structures are widely used in engineering. Loading and severe operating conditions often cause various defects on the surface of structural elements [1–3]. Corrosion may lead to general wear [4–6] or initiate the appearance of local damaged areas [7]. Once appeared, surface defects act as stress concentrators [7–9], causing high local stresses and resulting in acceleration of destruction of structural elements [10]. To assess the strength and reliability of the structure, it is necessary to carefully analyze the stress concentration near defects. Many authors investigated the effect of surface defects on strength, fatigue life, and buckling behavior of structures experimentally and numerically. Finite element analysis is one of the powerful tools for such problems, often used to estimate stresses in the vicinity of defects and in the whole structures [11]. A large number of studies concerning the stress distribution analysis of structure elements weakened by surface defects consider rounded bars and plates under different loading conditions [12–14]. Fewer articles focus on pressure vessels with surface cracks or pits [15, 16]. Paper [17] studies post-buckling behavior and ultimate strength of uniaxially compressed plates weakened by corrosion pittings from one side. Numerical investigation of stress fields in an elastic plate perforated by corrosion under uniaxial and O. S. Sedova (B) · D. D. Okulova St. Petersburg State University, 7/9, Universitetskaya nab., St. Petersburg 199034, Russia e-mail: [email protected] D. D. Okulova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Smirnov and A. Golovkina (eds.), Stability and Control Processes, Lecture Notes in Control and Information Sciences - Proceedings, https://doi.org/10.1007/978-3-030-87966-2_105

917

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shear loading is presented in [18]. Experimental and numerical investigation of the ultimate strength of simply supported pitted platings under uniaxial compression load is discussed in [19]. Numerical assessment of the ultimate strength of simply supported loaded platings with the random pitting corrosion wastage is performed in [20]. Paper [21] focused on the stochastic modeling of surface defects in a plate through experiments and finite element method. Different approaches to reliability assessments of pressurized corroded pipelines are discussed in [22]. Paper [23] is concerned with stress distribution near-surface cracks on shells of revolution under internal pressure and loaded bars with U- and V-shaped notches. Pipeline with a single corrosion-induced defect under internal pressure and axial compressive stress is considered in [16], and the effect of varying parameters (defect size and compressive stress value) is addressed. In [24], a thick pressurized spherical shell with a single cylindrical defect on the outer surface is considered. Surface defects are usually approximated in research by the following shapes: • elliptical and semi-elliptical holes often used for modeling cracks in plates, bars, and vessels [15, 23, 25]; • rectangular cuts which model local corrosion in pipes [25]; • hemispherical cavities as a model of pits on the surface of structures [21]; and • circular cone representing local corrosion of plates [17]. Experiments [26] showed that the pits have a circular-segment shape with constant depth/diameter ratio. Some problems of uniform corrosion of thin- and thick-walled spherical vessels have been solved analytically in papers [27–29]. However, the stress–strain state of shells of revolution with multiple surface defects remains one of the insufficiently studied areas. In this paper, we consider a thin-walled sphere under internal pressure with defects on its outer surface. The defects are assumed to have a form of hemispherical notches of the same size. Uniform and random arrangement of defects along the equator of the sphere is modeled. Different numbers of notches are considered. The stress state of a shell is investigated by finite element analysis in the ANSYS Workbench package. Calculations are made in the framework of the linear theory of elasticity.

105.2 Problem Formulation Consider a linearly elastic thin-walled sphere subjected to internal pressure p. The inner and outer radii of the shell are denoted by r and R, respectively. It is assumed that the shell is weakened with notches on the outer surface. All notches have the same size. In this paper, we assume that defects are located on one of the equators of the sphere. Two cases are investigated: when notches are evenly spaced along the equator of the sphere, and when notches are arranged randomly along the equator of the sphere. Different numbers of defects n are considered.

105 Stress Analysis of a Spherical Pressure Vessel with Multiple Notches Fig. 105.1 The model of a pressurized hollow sphere with a hemispherical notch

919

δ h = δ/ 2

p O

r

R

We consider hemispherical cavities with the depth h equal to half of its radius δ (Fig. 105.1). It is required to analyze stress distribution in the vicinity of the notches depending on their number for two cases: defects evenly spaced and randomly arranged along the shell’s equator.

105.3 Computational Model Parameters To perform a finite element analysis, we built an array of 3D CAD models of geometries of a thin notched sphere. The inner and outer radii of the shell were set to r = 340 mm and R = 350 mm, respectively. The radius of all the notches is δ = 6 mm, and their depth is h = 3 mm. Various numbers of defects were considered: n ∈ [16; 260]. For each value of n two types of geometry were built: when notches are evenly spaced and randomly distributed on the equator of a sphere. IronPython scripts were created to automate the modeling of random defects pattern. For the case of evenly distributed defects, only one-eighth part of the whole sphere was considered due to the symmetry of the model, while for randomly distributed notches, a half of the sphere was considered. A series of finite element simulations were carried out by ANSYS Workbench package. A ten-node element SOLID187 was selected from the element library. Structural steel with Young’s modulus E = 200 GPa and Poisson’s ratio μ = 0.3 was used in the model. The sphere is subjected to internal pressure p = 1 MPa. The boundary conditions include frictionless support on the surfaces of symmetry. The finite element meshing automatically provided in ANSYS for damaged zones was rather coarse. Hence, to get fine mesh for notched regions, sizing options and extra levels of mesh refinement were used. To ensure the convergence of solution, multiple simulations with different element sizes were performed for each CAD model.

Fig. 105.2 The maximum stress σ in the vicinity of the defects for the different number of defects. Red stars indicate evenly distributed notches, green circles, blue boxes, and maroon diamonds correspond to the random pattern of notches location

O. S. Sedova and D. D. Okulova

σ, MPa

920

n

105.4 Results and Discussion The maximum stress occurs in the vicinity of the notches σ for the different number of notches n shown in Fig. 105.2. Here and below the red stars are built for notches evenly distributed on the equator of a sphere, while the green circles, blue boxes, and maroon diamonds show results for the random pattern of notches location. It is seen from Fig. 105.2 that for the evenly distributed notches, the stress increases with n growing up to 236, then the stress decreases with the further increase in n. This could be explained by the redistribution in the stresses due to the interaction between the stress fields of neighboring notches. Note that, given sizes of the shell and the defects, for n ≥ 216, neighboring notches overlap. For the case of the randomly distributed defects, an interaction between the stress fields near neighboring notches is observed regardless of n, resulting in the non-monotonous change of the maximum principal stress σ as n grows. Moreover, the interaction between the stress fields around neighboring notches for the random pattern provides stress concentrations that are greater than the stresses in the shell with evenly distributed notches for the same number of defects n. The maximum principal stress σ in the vicinity of the notches for the maximum dmax and minimum dmin distances between two neighboring notches is shown in Figs. 105.3 and 105.4. The maximum dmax and minimum dmin are calculated as the maximum and minimum distances between the centers of two neighboring spheres served as the basis for corresponding notches, respectively. It is seen from Figs. 105.3 and 105.4 that the results for the evenly and randomly distributed notches are not close to each other. Hence, the maximum stress in the sphere appears to be significantly influenced by other factors, rather than the distance

Fig. 105.3 The dependencies of maximum stress σ in the vicinity of the defects on the maximum distance between two neighboring notches dmax . Red stars indicate evenly distributed notches, green circles, blue boxes, and maroon diamonds correspond to the random pattern of notches location

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σ, MPa

105 Stress Analysis of a Spherical Pressure Vessel with Multiple Notches

Fig. 105.4 The dependencies of maximum stress σ in the vicinity of the defects on the minimum distance between two neighboring notches dmin . Red stars indicate evenly distributed notches, green circles, blue boxes, and maroon diamonds correspond to the random pattern of notches location

σ, MPa

dmax , mm

dmin , mm

between two neighboring notches. The whole combination of the distribution of the defects impacts the stress concentration in the shell. It should be noted that the maximum stress values for evenly and randomly distributed notches are approximately the same, namely, the difference between the maximum values is 8.6% (Fig. 105.2). However, for evenly spaced defects, this maximum is observed for the relatively small range of varying the numbers of defects, while for the random distribution, this maximum is observed for the larger range of the numbers of defects.

105.5 Conclusion The numerical investigation of the stress state of a pressurized thin-walled sphere with hemispherical notches on its outer surface was carried out. Two cases of the notches distribution were considered: when the notches are evenly spaced and randomly

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distributed on the equator of the outer surface of the sphere. Stress distribution in the vicinity of the defects depending on their number was analyzed. The calculation results showed that • For evenly distributed notches, the maximum stress occurred near defects first increases slightly, then rises sharply and then drops slightly as the number of the defects increases. • The significant increase in the stress concentration was observed when the number of defects grows such that notches get closer to each other. However, when neighboring defects are close to overlap, the stress starts slightly decreasing with growing number of defects. • For the random pattern of defects, overlaps between neighboring notches took place for all considered numbers of defects. • The stress concentrations calculated for the randomly arranged notches were greater than stresses in the shell with evenly distributed notches for the same number of defects. • The maximum stress values for evenly and randomly distributed notches are close to each other; however, for evenly spaced defects, this maximum is observed for the relatively small range of varying numbers of defects, while for the random distribution, this maximum is observed for the larger range of the numbers of defects.

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