New Trends in Observer-Based Control: An Introduction to Design Approaches and Engineering Applications 0128170387, 9780128170380

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New Trends in Observer-based Control: An Introduction to Design Approaches and Engineering Applications

NEW TRENDS IN OBSERVER-BASED CONTROL An Introduction to Design Approaches and Engineering Applications VOLUME 1 EDITED BY

Olfa Boubaker Quanmin Zhu Magdi S. Mahmoud José Ragot Hamid Reza Karimi Jorge Dávila

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-817038-0 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Mara Conner Acquisition Editor: Sonnini R. Yura Development Editor: Gabriela D. Capille Production Project Manager: Anitha Sivaraj Cover Designer: Miles Hitchens Typeset by SPi Global, India

Contributors Carlos-Manuel Astorga-Zaragoza Electronic Engineering Department, Tecnológico Nacional de México/CENIDET, Cuernavaca, Morelos, Mexico Fazia Bedouhene Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria Cherifa Bennani Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria Talel Bessaoudi Department of Electrical Engineering, National Higher Engineering School of Tunis, University of Tunis, Tunis, Tunisia Hamza Bibi Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria Faouzi Bouani National Engineering School of Tunis, LR11ES20, Laboratory of Analysis, Conception and Control of Systems, University of Tunis El Manar, Tunis, Tunisia Olfa Boubaker National Institute of Applied Sciences and Technology, University of Carthage, Tunis, Tunisia Latifa Boutat-Baddas Research Center for Automatic Control of Nancy, UMRCNRS 7039, Université de Lorraine/IUT de Longwy, Cosnes-et-Romains, France Jérôme Cieslak IMS-Lab, Automatic Control Group, University of Bordeaux, Talence, France Jorge Dávila School of Mechanical and Electrical Engineering, IPN Instituto Politécnico Nacional, Mexico City, Mexico Mohamed Darouach Research Center for Automatic Control of Nancy, UMRCNRS 7039, Université de Lorraine/IUT de Longwy, Cosnes-et-Romains, France David Gómez-Gutiérrez Tecnologico de Monterrey, School of Engineering and Science, Jalisco, Mexico Stefano Di Gennaro University of L’Aquila, Department of Information Engineering, Computer Science and Mathematics, Center of Excellence DEWS, L’Aquila, Italy David Henry IMS-Lab, Automatic Control Group, University of Bordeaux, Talence, France Fayçal Ben Hmida Department of Electrical Engineering, National Higher Engineering School of Tunis, University of Tunis, Tunis, Tunisia Chien-Shu Hsieh Department of Electrical and Electronic Engineering, Ta Hwa University of Science and Technology, Qionglin, Hsinchu, Taiwan, ROC Dalil Ichalal IBISC, Univ Evry, Université Paris Saclay, 91020 Courcouronnes, France

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CONTRIBUTORS

Bin Jiang College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China Hamid Reza Karimi Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy Houria Kheloufi Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria Khaled Laboudi Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France Mihai Lungu Faculty of Electrical Engineering, University of Craiova, Craiova, Romania Magdi S. Mahmoud Systems Engineering Department, KFUPM, Dhahran, Saudi Arabia Noureddine Manamanni Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France Zehui Mao College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China Didier Maquin CRAN, Université de Lorraine-CNRS, 54000 Nancy, France. Benoît Marx CRAN, Université de Lorraine-CNRS, 54000 Nancy, France. Mamadou Mboup Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France Nadhir Messai Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France Mokhtar Mohamed Instrumentation, Control and Embedded Systems Research Group, School of Engineering and Digital Arts, University of Kent, Canterbury, United Kingdom Rodolfo Orjuela IRIMAS, Université de Haute-Alsace, 68093, Mulhouse, France Gloria-Lilia Osorio-Gordillo Electronic Engineering Department, Tecnológico Nacional de México/CENIDET, Cuernavaca, Morelos, Mexico José Ragot

CRAN, Université de Lorraine-CNRS, 54000 Nancy, France.

Antonio Ramírez-Teviño CINVESTAV, Jalisco, Mexico Souad Bezzaoucha Rebaï Automatic Control Research Group, Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg City, Luxembourg Hichem Salhi National Engineering School of Tunis, LR11ES20, Laboratory of Analysis, Conception and Control of Systems, University of Tunis El Manar, Tunis, Tunisia Abdulaziz Sherif Electrical Engineering Department, University of Tripoli, Tripoli, Libya Hieu Trinh School of Engineering, Faculty of Science Engineering and Built Environment, Deakin University, Geelong, VIC, Australia Carlos Renato Vázquez Tecnologico de Monterrey, School of Engineering and Science, Jalisco, Mexico

CONTRIBUTORS

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Holger Voos Automatic Control Research Group, Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg City, Luxembourg Seifeddine Ben Warrad National Institute of Applied Sciences and Technology, University of Carthage, Tunis, Tunisia Xing-Gang Yan Instrumentation, Control and Embedded Systems Research Group, School of Engineering and Digital Arts, University of Kent, Canterbury, United Kingdom Ali Zemouche CRAN UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France; EPI Inria DISCO, Laboratoire des Signaux et Systèmes, CNRSCentrale Supelec, Gif-sur-Yvette, France Quan Min Zhu Department of Engineering Design and Mathematics, University of the West of England, Bristol, United Kingdom

Foreword For more than half a century, considerable efforts have been made in the field of system control and monitoring. These actions are, obviously, crucial in the presence of technological risks directly impacting human health and the environment. Techniques to be developed aim, in general, to better understand, at each moment, the state of a system. The estimation phase is, of course, insufficient, and should be completed by state analysis to evaluate its normal or abnormal character. In the latter case, the analysis is further refined to accurately localize where the anomaly is to be found, to specify which part of the system, sensor, or actuator, is faulty. In order to judge the importance of the anomaly, its magnitude should be estimated. The ultimate phase of diagnosis seeks to specify the cause of the anomaly. In some cases, although this remains marginal because of great difficulty at the moment, the future evolution estimation of the anomaly is made. All these steps can contribute to considering how to react to anomalies in order to reduce their effects by means of appropriate control laws. This book does not attempt to address all these problems, but it can be a good introduction to some of the techniques designed to estimate the system states in different situations. In particular, the design of an observer to reconstruct the system states from partial measurements, how to use a state observer to detect and locate anomalies, and how to adjust a control law to counter the effect of anomalies on the behavior of a system will be discussed. In addition, a number of difficulties resulting from realistic physical constraints are considered: the presence of uncertainties and delays in system models; the influence of unmeasured exogenous inputs on system dynamics; the nonlinear behavior of systems; switching systems, or systems with several operating modes and interconnected systems. This diversity of objectives, systems, and constraints is covered in complementary chapters, and addresses a large part of the problem of observer state estimation and its applications. This book is a timely and comprehensive reference guide for graduate students, researchers, engineers, and practitioners in the areas of control theory. The content has been written for investigators acting in the fields of electrical, mechanical, aerospace, or mechatronics engineering. With contributions by eminent scientists in the field of control theory and systems engineering from 22 countries, this book covers the latest advances in observer-based control, from new design approaches to control engineering applications. Readers will find the fundamentals and applications

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FOREWORD

related to this topical issue. The book contains examples that make it ideal for advanced courses, as well as for researchers starting to work in the field, or engineers wishing to enter the field quickly and efficiently. The authors of the various chapters have tried to clearly present the theoretical concepts underlying the proposed solutions, and to illustrate them with pedagogical examples of modest dimensions; but allowing us to clearly see the implementation of these solutions and to assess their relevance through numerical data. In some cases, short Matlab programs complete the formulation. The book is structured in 13 chapters, and the organization is given as follows. Chapter 1 is dedicated to the class of descriptor systems. After some reminders on proportional and proportional-integral observers, the authors propose a new dynamical observer called “general observer structure,” and its extension to the case of systems with disturbing input. The stability analysis of the observer is proved via a Lyapunov method and solved via a set of linear matrix inequalities (LMIs). Several academic examples illustrate the performance of the proposed structure. Chapter 2 proposes an observer design technique for nonlinear interconnected systems with uncertain variable parameters. The observer has adaptive parameters that are adjusted from a stability study of the reconstruction error. The two examples that are given, coupled reverse pendulums and a quarter vehicle system, illustrate the implementation. The case of a linear switching system is discussed in Chapter 3, incorporating two difficulties: the presence of unknown inputs, and a lack of knowledge of the switching law. The observer is then designed to estimate the continuous and discrete states of the system. The proposed technique is applied to a modulation/demodulation procedure in a secure communication system with chaotic behavior. Another interesting situation is the subject of Chapter 4: state estimation for linear systems with unknown inputs and delays affecting their states and inputs. A first method proposes the design of a delay-dependent unknown input observer (UIO), whereas a second one suggests the design of a delay-independent UIO. The numerical example of the quadruple-tank benchmark is used to illustrate the efficiency of the two proposed methods for the case study. Chapter 5 presents the basics, progress, and outlook for the observerbased control design problem in dynamical systems. After reviewing the roots and needs of the problem, the authors have provided complete analytical results pertaining to dynamic modeling, control design, and computer simulation of several distinct approaches. The authors have also investigated issues regarding robust stability and robust performance of control design for different system configurations. In Chapter 6, the authors have developed new sufficient LMI conditions for the problem of stabilization of discrete-time uncertain switched linear

FOREWORD

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systems under arbitrary switching rules. Different scenarios of the use of Finsler’s lemma are proposed to reduce the conservatism of existing results in the literature. Numerical examples and simulation results are presented to demonstrate the effectiveness of the proposed methods. Model predictive control (MPC) based on state observers for nonlinear multivariable systems is the subject of Chapter 7. To overcome classical limits, the authors developed an adaptive MPC-based observer for nonlinear multivariable systems. The implementation of the proposed approach to a three-tank benchmark system is performed, with a comparison between linear and nonlinear predictive controllers. The authors of Chapter 8 propose a new decentralized observer-based controller design method for nonlinear discrete-time interconnected systems with nonlinear interconnections. An enhanced linear matrix inequality design condition is provided to guarantee asymptotic stability for systems with both known and unknown interconnection bounds. Two numerical examples illustrate the effectiveness of the design approach. Chapter 9 presents results of the polytopic model (PM) approach to cope with the modeling, stability analysis, state feedback control, state, and unknown input estimation, and finally, fault-tolerant control of nonlinear systems. The backbone of all presented results is the capacity of the PM structure to represent nonlinearities in a selected operating range of the system. It is proposed to design a fault-tolerant controller fed with the simultaneous state and unknown input estimates. The benefits of the active fault-tolerant controller based on the PM approach are illustrated in an example consisting of the stabilization of the lateral dynamics of a vehicle. Chapter 10 studies the application of high-order sliding mode observers for the estimation of faults and their later compensation in linear systems. The study is restricted to the systems with strongly observable faults. The main idea is to exploit the finite-time convergence of the high-order sliding mode-based observers to estimate the states, and also the dynamic effects, of the faults, showing that these are powerful tools not only to estimate states, but also unknown signals. The methodology is illustrated with the design of a fault-tolerant control of the roll autopilot for a missile mode. The authors of Chapter 11 consider the problem of simultaneous state and fault estimation of linear descriptor and nonlinear descriptor discretetime stochastic systems with arbitrary unknown disturbances. The study is based on input filtering and the use of a robust two-stage Kalman filter. Chapter 12 investigates the problem of the simultaneous estimation of discrete state, continuous state, and faults of a class of switched linear systems with measurement noise. A new algebraic approach is developed in order to estimate, in real time, and with a negligible delay, the switching times, and to reconstruct the discrete state. The proposed strategy is illustrated by a system with three operating modes whose dynamics are affected by two faults.

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Chapter 13 introduces an appropriate model associating paradigms from control theory and computer science to deal with the system subject to both physical attacks and sensor/actuator attacks via the connected network. Inspired by a combination of the classical fault-tolerant control approach and the event-triggered control, an observer-based, attacktolerant control solution is proposed. The control design is applied to a laboratory benchmark including a three-tank system subject to physical attacks. Finally, on behalf of all the editors, I would like to express my gratefulness to all the authors of the book for their valuable contributions, and all reviewers for their helpful and professional efforts to provide valuable comments and feedback.

José Ragot Université de Lorraine, CNRS, CRAN, F54000 Nancy, France

C H A P T E R

1 On Dynamic Observers Design for Descriptor Systems Gloria-Lilia Osorio-Gordillo*, Mohamed Darouach † , Latifa Boutat-Baddas † , Carlos-Manuel Astorga-Zaragoza* *Electronic

Engineering Department, Tecnológico Nacional de México/CENIDET, Cuernavaca, Morelos, Mexico † Research Center for Automatic Control of Nancy, UMR-CNRS 7039, Université de Lorraine/IUT de Longwy, Cosnes-et-Romains, France

1 INTRODUCTION Descriptor systems were introduced by Luenberger in 1977 to represent the systems for which the results obtained in the domain of control and observation of standard systems cannot be applied. This class of systems can represent the physical phenomena that the model by ordinary differential equations cannot describe. Descriptor systems have found their application in modeling the motion of aircraft, and in chemical processes, the mineral industry, electrical circuits, economic systems, and robotics [1]. Some control techniques, such as those that are based on state-feedback control, do not have all the states available for their measurement. Either by technical or economic reasons it is difficult, or even impossible, to measure all the system state variables. Therefore, it is necessary to estimate the state of the system. Observers are one of the principal tools to provide the estimation of system state variables. The first work on the problem of reconstruction of state variables was devoted to standard linear systems [2]. Since then, many theoretical results have been presented, and they are widely used in control and fault diagnosis. Observers can be classified by order: full-order, reduced-order, and partial-order observers, and by structure: proportional, proportionalintegral, and generalized observer. In the estimation by a proportional

New Trends in Observer-based Control https://doi.org/10.1016/B978-0-12-817038-0.00001-9

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© 2019 Elsevier Inc. All rights reserved.

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1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

observer (PO), there always exists a static estimation error in the presence of disturbance. In order to deal with this disadvantage of the PO, proportional-integral observers (PIOs) were introduced with an integral gain of the output error (difference between the estimated output and the measured output) in their structure, this form of observer achieves steadystate accuracy in the state estimation. Then, it is apparent that a modified structure with additional degrees of freedom in the observer can provide the best estimation and robustness in the presence of parameter variation and disturbances. A new structure of observer was developed by Goodwin and Middleton [3] and Marquez [4], known as the generalized dynamic observer (GDO). This structure presents an alternative state estimation that can be considered to be more general than the PO and the PIO.

2 BASIC THEORY OF DESCRIPTOR SYSTEMS 2.1 Definition of Descriptor Systems A linear descriptor system with constant coefficients can be represented by the following set of equations: E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t)

(1.1)

where x(t) ∈ Rn is the semistate vector, u(t) ∈ Rm is the input, and y(t) ∈ Rny is the measured output. E is a singular matrix with constant parameters, it is assumed that rank(E) ≤ n. Descriptor systems, also known as singular systems or differentialalgebraic systems, are a class of systems that can be considered a generalization of dynamical systems. The descriptor system representation is a powerful modeling tool because it can describe processes governed by both differential equations and algebraic equations. So it represents the physical phenomena that the model, by ordinary differential equations, cannot describe. These systems were introduced by Luenberger [5] from a control theory point of view, and since, great efforts have been made to investigate descriptor systems theory and its applications.

2.2 Some Properties of Descriptor Systems Consider the following descriptor system E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t)

(1.2)

where matrices E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rny ×n are real, and x(t), u(t), and y(t) are vectors of appropriate dimensions.

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BASIC THEORY OF DESCRIPTOR SYSTEMS

5

2.2.1 Regularity of Descriptor Systems The regularity property in descriptor systems guarantees the existence and uniqueness of solutions. We can give the following definition for regularity. Definition 1 (Yip and Sincovec [6]). System (1.2) is said to be regular if there exists a constant scalar s ∈ C such that det(sE − A) = 0 or equivalently, the polynomial det(sE − A) is not identically zero. In this case, we also say that the pair (E, A), or the matrix pencil sE − A, is regular. Remark 1. In [7, 8] the authors show that the regularity of the matrix pair (E, A) is a property not needed for observer design, instead of the preceding regularity  property for squaresystems, this property is replaced by normal − rank sE − A BU(s) Ex(0) = normal − rank(sE − A) for the rectangular descriptor systems. Where the normal − rank of the matrix pencil sE − A is defined as the rank of (sE − A) for almost all s ∈ C and U(s) is the Laplace transformation of u(t) (see [7, 9] and references therein). 2.2.2 Stability of Descriptor Systems Stability of a dynamical system describes the response behavior of the system at infinity time with respect to initial condition disturbances, and is well regarded as one of the most important properties of dynamical systems. Definition 2 (Duan [10]). The regular descriptor linear system (1.2) is asymptotically stable if, and only if, eig(E, A) ⊂ C− = {s|s ∈ C, Re(s) < 0} where eig(E, A) is defined as the roots of det(sE − A) = 0, which must lie in the stable region, that is, the open left-half plane for the continuous-time systems. 2.2.3 Impulse-Free Behavior Definition 3 (Duan [10]). If the state response of a descriptor linear system, starting from an arbitrary initial value, does not contain impulse terms, then the system is called impulse-free. The following statements are equivalent: • The pair (E, A) is impulse-free. • deg(det(sE −   A)) = rank(E). E 0 • rank = n + rank(E). A E

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1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

2.2.4 Admissibility Definition 4. The pair (E, A) is said to be admissible if it is regular, impulse-free, and stable. Theorem 1. System (1.2) or the pair (E, A) is admissible if and only if there exists a nonsingular matrix Θ such that ET Θ = Θ T E ≥ 0 and AT Θ +Θ T A < 0.

2.2.5 C-Observability It is assumed that system (1.2) has the following slow and fast subsystems x˙ 1 (t) = A1 x1 (t) + B1 u(t) y1 (t) = C1 x1 (t)

(1.3)

N x˙ 2 (t) = x2 (t) + B2 u(t) y2 (t) = C2 x2 (t)

(1.4)

and

where x1 (t) ∈ Rn1 , x2 (t) ∈ Rn2 , and N are a nilpotent matrix. The relations between the coefficient matrices of the two systems are given by QEP = diag(In1 , N ) QAP = diag(A   1 , I n2 ) B QB = 1 B  2  CP = C1 C2

(1.5)

where the matrices Q and P are the left and right transformation matrices, respectively. Definition 5 (Duan [10]). The regular system (1.2) is called completely observable (C-observable), if the initial condition x(0) of the system can be uniquely determined from the output data y(t), 0 ≥ t ≥ ∞. Alternatively, the system (1.2) is C-observable if the zero output y(t) ≡ 0 with u(t) ≡ 0 implies that the system has only the trivial solution x(t) ≡ 0. Consider the regular system (1.2), with its slow subsystem (1.3) and fast subsystem (1.4). 1. The slow subsystem (1.3) is C-observable if and only if   sE − A rank = n, ∀s ∈ C, s finite C 2. The fast subsystem (1.4) is C-observable if and only if   E rank =n C

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BASIC THEORY OF DESCRIPTOR SYSTEMS

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3. System (1.2) is C-observable is and only if conditions of statements 1 and 2 hold or,   αE − βA rank = n∀(α, β) ∈ C2 \{(0, 0)} C 2.2.6 R-Observability The system (1.2) is called observable within the reachable set (R-observable) if any state in the reachable set can be uniquely determined by y(t) and u(t) for t ≥ 0. Definition 6 (Duan [10]). The regular descriptor system (1.2) is Robservable if and only if   sE − A rank = n, ∀s ∈ C, s finite C 2.2.7 Impulse Observability Definition 7 (Boukas [11]). Impulse observability guarantees the ability to uniquely determine the impulse behavior in x(t) from information of the impulse behavior in the output y(t). System (1.2) is called impulse observable if ⎡ ⎤ E A rank ⎣ 0 E ⎦ = n + rank(E) 0 C 2.2.8 Detectability Definition 8 (Dai [1]). The system (1.2) is detectable if and only if all its output’s transmission zeros are stable, that is   sE − A = n, ∀s ∈ C+ , s finite rank C 2.2.9 Stabilizability Definition 9 (Duan [10]). System (1.2) is stabilizable if there exists a state feedback controller u(t) such that the resulted closed-loop system is stable. The regular system (1.2) is stabilizable if and only if   rank sE − A B = n, ∀s ∈ C+ , s finite

2.3 Practical Examples In the following sections some descriptor models of practical systems are shown.

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1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

FIG. 1.1 Hydraulic system.

2.3.1 Hydraulic System Model of the change of the height of water in three tanks, with an input flow in the first tank and with the third tank leaking (Fig. 1.1). The pressures at the bottom of the tanks 1, 2, and 3 are represented as p1 , p2 , and p3 , respectively. The pipe from tank 1 branches off to tanks 2 and 3. The pressure at the pipe branch is given as pB . By using the Hagen-Poiseuille equation, the flow rates between the tanks and the pipe branch can be written as: F1B = (p1 (t) − pB (t)) FB2 = (pB (t) − p2 (t)) FB3 = (pB (t) − p3 (t))

π d4p 128ηLB π d4p 128ηL1 π d4p 128ηL2

(1.6a) (1.6b) (1.6c)

where η is the dynamic viscosity, Li , ∀i ∈ [1, 2, B] are the lengths of pipes, and dp is the diameter of the pipe. All the fluid leaving tank 1 should enter into tanks 2 and 3. This is presented as a constraint. F1B = FB2 + FB3

(1.7)

The pressure in each tank is given by: p1 (t) = ρgh1 (t) p2 (t) = ρgh2 (t) p3 (t) = ρgh3 (t)

I. OBSERVER DESIGN

(1.8a) (1.8b) (1.8c)

BASIC THEORY OF DESCRIPTOR SYSTEMS

9

where hi , ∀i ∈ [1, 2, 3] is the depth in each tank, ρ is the density of the liquid, and g is the gravity acceleration. The rate at which the fluid leaves or enters the tank is directly proportional to the rate of change of the height of the fluid in the tank. This relation comes through the analysis of mass conservation of incompressible fluids: (1.9a) a1 h˙ 1 (t) = −F1B + Fin ˙ (1.9b) a2 h2 (t) = FB2 a 3 a3 h˙ 3 (t) = FB3 − h3 (t) (1.9c) 10 where a1 , a2 , and a3 are the cross-sectional areas of tanks 1, 2, and 3, a3 h3 (t) represents respectively. Fin is the input flow in tank 1, and the term 10 the leaking in tank 3. Using Eqs. (1.6)–(1.9) the state-space representation of the three interconnected tanks can be constructed as ⎤ ⎡ ⎤ ⎡˙ h1 (t) a1 0 0 0 0 0 0 ⎢ 0 a2 0 0 0 0 0⎥ ⎢ h˙ 2 (t) ⎥ ⎥ ⎢ ⎥⎢ ⎢ 0 0 a3 0 0 0 0⎥ ⎢ h˙ (t) ⎥ ⎢ ⎥⎢ 3 ⎥ ⎢ 0 0 0 0 0 0 0⎥ ⎢p˙ (t)⎥ = ⎢ ⎥⎢ B ⎥ ⎢ 0 0 0 0 0 0 0⎥ ⎢ p˙ (t) ⎥ ⎢ ⎥⎢ 1 ⎥ ⎣ 0 0 0 0 0 0 0⎦ ⎣ p˙ 2 (t) ⎦ 0 0 0 0 0 0 0 p˙ 3 (t) ⎤ ⎡ ⎤ ⎤⎡ ⎡ −k1 0 0 0 0 0 k1 1 h1 (t) ⎥ ⎢ h2 (t) ⎥ ⎢0⎥ ⎢ 0 0 −k 0 0 0 k 2 2 ⎥ ⎢ ⎥ ⎥⎢ ⎢ a3 ⎥ ⎢ ⎥ ⎢ 0 ⎢ k3 0 0 −k3 ⎥ 0 − 10 ⎥ ⎢ h3 (t) ⎥ ⎢0⎥ ⎢ ⎥ ⎢−ρg ⎢ ⎥ ⎢ 0 0 0 1 0 0 ⎥ ⎢pB (t)⎥ ⎥ + ⎢0⎥ Fin ⎢ ⎥ ⎥ ⎢ 0 ⎢ ⎥ −ρg 0 0 0 1 0 ⎥ ⎢ p1 (t) ⎥ ⎢ ⎢ ⎢ 0⎥ ⎦ ⎦ ⎣ 0 ⎣ ⎣ p2 (t) 0⎦ 0 −ρg 0 0 0 1 p3 (t) 0 k2 k3 0 0 0 −(k1 + k2 + k3 ) k1 (1.10) where k1 =

πd4p 128ηLB , k2

=

πd4p 128ηL1 ,

and k3 =

πd4p 128ηL2 .

2.3.2 Mechanical System Model of the mass-spring-damper system, which includes a rigid bar that can prevent the motion of the second mass. The exciting force is applied to mass 1 (Fig. 1.2). The positions of masses m1 and m2 are represented by x1 and x2 , respectively. The spring coefficients are given as k1 and k2 . The damper coefficients are given as b1 and b2 .

I. OBSERVER DESIGN

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1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

FIG. 1.2 Mechanical system.

Using Newton’s second law, the forces acting on each mass can be written as: m1 x¨ 1 (t) = −b1 x˙ 1 (t) − k1 x1 (t) + b2 (˙x2 (t) − x˙ 1 (t)) + k2 (x2 (t) − x1 (t)) + u(t) (1.11) m2 x¨ 2 (t) = −b2 (˙x2 (t) − x˙ 1 (t)) − k1 (x2 (t) − x1 (t)) + αμ(t)

(1.12)

and the constraint equation 0 = α(x1 (t) + x2 (t)) + (1 − α)μ(t)

(1.13)

where α represents the state of the switch 1 = closed and 0 = open, and μ(t) is the force absorbed. Using Eqs. (1.11)–(1.13), the state-space representation of the mechanical system can be constructed as ⎤⎡ ⎤ ⎡ x˙ 1 (t) 1 0 0 0 0 ⎢ ⎥ ⎢0 1 0 0 0⎥ ⎥ ⎢x˙ 2 (t)⎥ ⎢ ⎢0 0 m1 0 0⎥ ⎢x¨ 1 (t)⎥ ⎥⎢ ⎥ ⎢ ⎣0 0 0 m2 0⎦ ⎣x¨ 2 (t)⎦ 0 0 0 0 0 μ(t) ˙ ⎤⎡ ⎤ ⎡ ⎤ ⎡ x1 (t) 0 0 0 1 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 1 0 ⎥ ⎥ ⎢x2 (t)⎥ ⎢0⎥ ⎢ ⎢ ⎥ ⎢ ⎥ 0 ⎥ =⎢ ⎥ ⎢x˙ 1 (t)⎥ + ⎢1⎥ u(t) ⎢−(k1 + k2 ) k2 −(b1 + b2 ) b2 ⎦ ⎣ ⎣ −k1 b2 −b2 α k1 x˙ 2 (t)⎦ ⎣0⎦ α α 0 0 (1 − α) μ(t) 0 (1.14)

I. OBSERVER DESIGN

OBSERVERS FOR DESCRIPTOR SYSTEMS

11

FIG. 1.3 Electric system.

2.3.3 Electric System Model of an electric system controlled by the voltage v (Fig. 1.3). The currents i1 and i2 are measured through the resistors R1 and R2 . The electric charge in the capacitor C is denoted as q, and L represents the inductance. The following relations are obtained by using current and voltage Kirchhoff’s laws q˙ (t) = i3 (t) 1 L˙i2 (t) = q(t) − i2 (t)R2 C and the constraint equation

(1.15) (1.16)

1 q(t) − v(t) (1.17) C Using Eqs. (1.15)–(1.17), the state-space representation of the electrical circuit can be expressed as ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ 0 0 1 q˙ (t) 0 q(t) 1 0 0 1 ⎣0 L 0⎦ ⎣˙i2 (t)⎦ = ⎣ 0 ⎦ ⎣i2 (t)⎦ + ⎣ 0 ⎦ v(t) (1.18) C −R2 1 ˙i3 (t) (t) −1 i 0 0 0 R R 3 1 1 C 0 = R1 i2 (t) + R1 i3 (t) +

3 OBSERVERS FOR DESCRIPTOR SYSTEMS A descriptor system is a set of equations that are the result of modeling a system. These equations represent a general class of phenomena that is evolving in time, where some of the variables are related in a dynamical way, whereas others are purely static [5]. The so-called state observer is reconstructing the state variables of the system asymptotically. Based on this, an observer should satisfy the following two necessary conditions [1]: 1. The inputs of the observer should be the control input and the measured output of system. I. OBSERVER DESIGN

12

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

FIG. 1.4 Observer’s general scheme.

2. Its output should satisfy the asymptotically condition limt→∞ e(t) = 0, where e(t) is the estimation error, that is, xˆ (t) − x(t), the difference between the state and its estimate. The general scheme of an observer is shown in Fig. 1.4. Some applications of observers for descriptor systems can be found in [12], where a nonlinear observer for descriptor systems is designed to estimate the state variables and unknown inputs in a wastewater treatment plant. Discrete-time descriptor systems are used in [13] to design observers for state estimation in an experimental hydraulic tank system. The observers design for descriptor systems have been widely studied in [8, 14–16]. As for observers, design for standard and descriptor systems with unknown inputs has been treated in [17–21]. The design of unknown input observers is a crucial problem because, in many practical cases, all input signals cannot be known. Moreover, this class of observers is widely used in the area of fault diagnosis, even if all the inputs are known (see [19, 22], and references therein). All these results use the PO. In the estimation by a PO there always exists a static error estimation in the presence of disturbances. In order to deal with the disadvantage of PO, PIOs were introduced with an integral gain in their structure, which achieves steadystate accuracy in their estimations. The first results on the PIO were presented by Wojciechowski [23] for single-input, single-output (SISO) systems. Its extension to multivariable systems was presented in [24–26], where the authors show the performances of the PIO compared with PO in the presence of disturbances and uncertainties. Some recent results on the PIO for systems with unknown inputs are presented in [17, 27]. As shown, numerous works have focused on the design and implementation of algorithms for estimation and control for descriptor systems. Different techniques for state estimation have been developed, and these depend largely on the structure of the model, the information available in the process, and the relations that can be established between them.

3.1 Descriptor State Observer Descriptor state observers are typically derived from the system representation. An additional term is included in order to ensure that the estimated state converges to the real state. Specifically, the additional term

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OBSERVERS FOR DESCRIPTOR SYSTEMS

13

consists of the difference between the system output and the estimated output, and is then multiplied by a matrix of correction. This is added to the state of the observer to produce the so-called descriptor state observer. Some results of descriptor state observers in descriptor systems are found in [28–30]. In [1] a singular observer for singular systems is presented. The following system is considered: E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t)

(1.19)

where E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rny ×n are constant matrices. It is assumed that system (1.19) is regular and nr = rank(E) < n. Considering that system (1.19) is detectable, the following full-order observer is proposed: Ex˙ˆ (t) = Aˆx(t) + Bu(t) + L(y(t) − yˆ (t)) yˆ (t) = Cˆx(t)

(1.20a) (1.20b)

such that limt→∞ (ˆx(t) − x(t)) = 0. Let e(t) = xˆ (t) − x(t) be the estimation error between real and estimated states. Its derivative is described by: E˙e(t) = (A − LC)e(t)

(1.21)

Thus, if limt→∞ e(t) = 0 the estimated state xˆ (t) converges asymptotically toward x(t). If rank(E) < n, observer (1.20) is called descriptor state observer. Otherwise, when rank(E) = n, E = In is assumed without loss of generality, so the observer (1.20) is called a Luenberger observer.

3.2 Proportional Observers Commonly, the estimation of the different variables of interest is performed through a Luenberger observer, also known as a PO. Some results on PO for descriptor systems are shown in [8, 20, 28, 31]. In [8], a functional PO design for linear descriptor systems is presented. The approach is based on a new definition of partial impulse observability. The following system is considered: E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t) z(t) = Lx(t)

(1.22)

where z(t) ∈ Rp is the vector to be estimated. E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , C ∈ Rny ×n , and L ∈ Rp×n are known constant matrices.

I. OBSERVER DESIGN

14

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

Definition 10. The system (1.22) with u(t) = 0, or the triplet (C, E, A) is said to be partially impulse observable with respect to L if y(t) is impulse free for t ≥ 0, only if Lx(t) is impulse free for t ≥ 0. The reduced-order observer for the singular system (1.22) is given by:   ⊥ −E Bu(t) + Hu(t) (1.23a) ζ˙ (t) = Nζ (t) + G y(t)  ⊥  −E Bu(t) ˆz(t) = Pζ (t) + Q (1.23b) y(t) where ζ (t) ∈ Rq0 is the state of the observer, zˆ (t) ∈ Rp is the estimate of z(t). Let rank(E) = nr and E⊥ ∈ Rnr1 ×n be a full row rank matrix such that E⊥ E = 0, in this case nr1 = n − nr . Now, consider the following conditions  ⊥  E A (i) NTE − TA + G =0 C ⎡ ⎤   TE (ii) P Q ⎣E⊥ A⎦ = L C (iii) H = TB so, there exists a matrix T of appropriate dimension such that ε(t) = ζ (t) − TEx(t), and its derivative is given by:  ⊥  E A ε˙ (t) = Nε(t) + NTE − TA + G x(t) + (H − TB)u(t) (1.24) C and from Eq. (1.23b) we get:

⎤ TE z(t) = Pε(t) + P Q ⎣E⊥ A⎦ x(t) C 



⎡

(1.25)

If conditions (i)–(iii) hold, and N is Hurwitz, then limt→∞ (ˆz(t) − z(t)) = 0 for any initial conditions. The observer matrices are obtained by solving Sylvester equations (i)–(iii). In this way, the author proposes an algorithm to obtain a functional reduced-order observer for descriptor systems; likewise, some particular cases are discussed in this chapter. Sufficient conditions for the existence and stability of these observers are given.

3.3 Proportional-Integral Observers Compared with PO, PIOs provide more robust estimation against model uncertainties and better perturbation attenuation. As in the conventional

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OBSERVERS FOR DESCRIPTOR SYSTEMS

15

systems, in view of the advantages of integral actions, some research has introduced the integral term in observer design for descriptor systems [17, 32, 33]. In [34], the authors present a method to design a PIO for unknown input descriptor systems. The following system is considered: E˙x(t) = Ax(t) + Bu(t) + Nd(t) y(t) = Cx(t)

(1.26)

where d(t) ∈ Rnd is the unknown input. E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , N ∈ Rn×nd , and C ∈ Rny ×n are known constant matrices. Considering rank(N) = nd , rank(C) = ny , and a nonsingular matrix P, the following restricted system equivalent (r.s.e.) can be derived [1]:         B E0 0 x˙ (t) A0 N0 x(t) + 0 u(t) = ˙ 0 0 d(t) 0 0 I d(t) (1.27)     x(t) y0 (t) = C0 0 d(t)     −B1 u(t) A1 where E0 ∈ Rnr ×n , y0 = ∈ ∈ Rm+n−nr , and C0 = C y(t) R(m+n−nr )×n . The following PIO for the descriptor system (1.27) is considered: ˆ ζ˙ (t) = Gζ (t) + L1 y0 (t) + L2 y0 (t) + Ju(t) + T1 Nd(t)

(1.28a)

˙ˆ d(t) = L3 (y0 (t) − yˆ 0 (t))

(1.28b)

xˆ (t) = M1 ζ (t) + T2 y0 (t)

(1.28c)

yˆ 0 (t) = C0 xˆ (t)

(1.28d)

Eq. (1.28b) describes the integral loop added to the proportional equaˆ the tion (1.28a). Now, defining e(t) = x(t) − xˆ (t) and ed (t) = d(t) − d(t) estimation error system expressed in terms of e(t) and ed (t) is:      G T1 N e(t) e˙(t) = 0 ed (t) e˙d (t) −L3 C0    T1 B0 − J T1 A0 − GT1 E0 − L1 C0 − L2 C0 x(t) + + u(t) (1.29) 0 0 

where G = T1 A0 − L2 C0 . is obtained. EstiLetting L1 = GT2 , J = T1 B0 , and an autonomous system   T1 A0 − L2 C0 T1 N mation errors converge asymptotically to 0 if matrix 0 −L C    3 0   L2  T1 A0 T1 N C0 0 , and it − is Hurwitz. This matrix becomes L3 0 0

I. OBSERVER DESIGN

16

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

  L can be stabilized by setting the gain matrix 2 , if and only if the pair L3

   T1 A0 T1 N  , C0 0 is detectable. 0 0 In this research, the authors propose a PIO for descriptor systems with an unknown input, and the particular case of reduced-order PIO is also described.

4 GENERALIZED DYNAMIC OBSERVER Consider the following descriptor system E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t)

(1.30)

where E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rny ×n are known constant matrices. Now, consider the following GDO for system (1.30)   ⊥ −E Bu(t) + Ju(t) (1.31a) ζ˙ (t) = Nζ (t) + Hv(t) + G y(t)  ⊥  −E Bu(t) v˙ (t) = Sζ (t) + Lv(t) + M (1.31b) y(t)  ⊥  −E Bu(t) xˆ (t) = Pζ (t) + Q (1.31c) y(t) where ζ (t) ∈ Rq0 represents the state vector of the observer, v(t) ∈ Rq1 is an auxiliary vector, and xˆ (t) ∈ Rn is the estimate of x(t). Let rank(E) = nr < n and E⊥ ∈ Rnr1 ×n , in this case nr1 = n − nr . Remark 2. The order of the observer is q0 ≤ n, when q0 = n − ny , the reduced order observer is obtained, and for q0 = n the full order one is obtained. The observer (1.31) is in a general form and generalizes the existing ones. In fact: • For H = 0, S = 0, M = 0, and L = 0 the observer reduces to the PO for descriptor systems (see, e.g., [8] and references therein).   ⊥ ˙ζ (t) = Nζ (t) + G −E Bu(t) + Ju(t) (1.32a) y(t)  ⊥  −E Bu(t) xˆ (t) = Pζ (t) + Q (1.32b) y(t)    • For H = 0, S = 0, M = 0, L = 0, G = 0 Ga , and Q = 0 the following observer is obtained:

I. OBSERVER DESIGN

 Qa , then

17

GENERALIZED DYNAMIC OBSERVER

ζ˙ (t) = Nζ (t) + Ga y(t) + Ju(t) xˆ (t) = Pζ (t) + Qa y(t)

(1.33a) (1.33b)

which is the form used for the unknown input PO for descriptor systems [35].   • For L = 0, S = −C, and M = −CQ + 0 I , then the following observer is obtained  ⊥  −E Bu(t) ζ˙ (t) = Nζ (t) + Hv(t) + G + Ju(t) (1.34a) y(t) v˙ (t) = y(t) − Cˆx(t)  ⊥  −E Bu(t) xˆ (t) = Pζ (t) + Q y(t)

(1.34b) (1.34c)

which is the form used for the unknown input PIO for descriptor systems. In the sequel we assume that: Assumption 1.

⎤ E rank ⎣E⊥ A⎦ = n C ⎡

This assumption is equivalent to the impulse observability from Definition 7 (see [16]). Assumption 2.   sE − A rank = n, ∀s ∈ C+ , s finite C The relation between Assumption 1 and the impulse observability is given by the following lemma. Lemma 1. The following conditions are equivalent: 1. System ⎡ (1.30) ⎤ is impulse observable. E 2. rank ⎣E⊥ A⎦ = n. C ⎡ ⎤ E A 3. rank ⎣ 0 E ⎦ = n + rank(E). 0 C

 E⊥ is of full EE+ column rank, where E+ is any generalized inverse of matrix E, such that EE+ E = E, then Proof (Darouach et al. [36]). Consider that matrix

I. OBSERVER DESIGN



18

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

⎤ ⎤ ⎡ ⊥ ⎡ ⎤ ⎤ 0 0 ⎡ E 0 E⊥ A E A E A + ⎥ ⎢EE+ 0 0⎥ ⎢ ⎥ ⎣ 0 C⎦ = rank ⎢E EE A⎥ rank ⎣ 0 C⎦ = rank ⎢ ⎦ ⎣ 0 ⎣ I 0 0 C ⎦ 0 E 0 E 0 0 I 0 E ⎡ ⎡ ⎤ ⎤ ⊥ 0 E⊥ A 0 E A   ⎢E ⎢E EE+ A⎥ I −E+ A 0 ⎥ ⎥ ⎥ = rank ⎢ = rank ⎢ ⎣ ⎣0 ⎦ 0 I C 0 C ⎦ 0 E 0 E ⎤ ⎡ E = rank ⎣E⊥ A⎦ + rank(E) C ⎡

4.1 Generalized Dynamic Observer Design Now, we can give the following lemma. Lemma 2. There exists an observer of the form (1.31) for the system (1.30) if the following two statements hold. 1. There exists a matrix T of appropriate dimension such that the following conditions are satisfied:  E⊥ A (a) NTE + G −TA = 0 C (b) J = TB  E⊥ A + STE = 0 (c) M C ⎡ ⎤   TE (d) P Q ⎣E⊥ A⎦ = In C   N H 2. The matrix is Hurwitz. S L Proof . Let T ∈ Rq0 ×n1 be a parameter matrix and define the error ε(t) = ζ (t) − TEx(t), then its derivative is given by:  ⊥ 

E A ε˙ (t) = Nε(t) + Hv(t) + NTE + G −TA x(t) + (J − TB)u(t) (1.35) C by using the definition of ε(t), Eqs. (1.31b), (1.31c) can be written as:  ⊥ 

E A v˙ (t) = Sε(t) + Lv(t) + M + STE x(t) (1.36) C ⎤ ⎡   TE (1.37) xˆ (t) = Pε(t) + P Q ⎣E⊥ A⎦ x(t) C

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GENERALIZED DYNAMIC OBSERVER

19

Now, if conditions (a)–(d) of Lemma 2 are satisfied, the following observer error dynamics are obtained from Eqs. (1.35), (1.36)      ε˙ (t) N H ε(t) = (1.38) v˙ (t) S L v(t) and from Eq. (1.37) we get:  in this case if

N S



xˆ (t) − x(t) = e(t) = Pε(t)

(1.39)

H is Hurwitz, then limt→∞ e(t) = 0. L

4.1.1 Determination of the Observer Parameters Before giving the solution to the constraints (a)–(d) of Lemma 2, the following definitions should be considered. Definition 11 (Generalized Inverse [37]). Let A ∈ Rn×m . If rank(A) = m, then A+ is a left inverse of A, it satisfies A+ A = I. If rank(A) = n, then A+ is a right inverse of A, it satisfies AA+ = I. Both left and right inverses satisfies AA+ A = A. Lemma 3. Consider the following equation of a nonhomogeneous system: AX = B

(1.40)

where A ∈ Rn×m is a constant matrix, B ∈ Rn×p is a constant matrix, and X ∈ Rm×p is the matrix to determine. Eq. (1.40) admits a solution, if and only if   rank(A) = rank A B in this case, the general solution to Eq. (1.40) is given by: X = A+ B − (I − A+ A)Z

(1.41)

where Z is an arbitrary matrix of appropriate dimension. Equivalently, for a system with the form X A = B, the necessary and sufficient condition for the existence of a solution is:   A rank(A) = rank B in this case, the general solution is given by: X = BA+ − Y(I − AA+ )

(1.42)

where Y is an arbitrary matrix of appropriate dimension. Now, the parameterization of the all solutions to the algebraic constraints (a)–(d) of Lemma 2 are given.

I. OBSERVER DESIGN

20

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

q0 ×n be a full row rank matrix, such that the matrix Σ = ⎡ Lemma ⎤ 4. Let R ∈ R R ⎣E⊥ A⎦ is of full column rank, then under Assumption 1, the general solution to C constraints (a)–(d) of Lemma 2 is given by:

T = T1 − Z1 T2 N = N1 − Z1 N2 − Y1 N3 G = G1 − Z1 G2 − Y1 G3 S = −Y2 N3 M = −Y2 G3 P = P1 − Y3 N3 Q = Q1 − Y3 G3

(1.43) (1.44) (1.45) (1.46) (1.47) (1.48) (1.49)

where Z1 , Y1 , Y2 , and Y3 are arbitrary matrices of appropriate dimensions. q0 ×n be a full row rank matrix, such that the matrix Proof ⎡ ⎡ . Let ⎤R ∈ R ⎤ E R Σ = ⎣E⊥ A⎦ is of full column rank and let Ω = ⎣E⊥ A⎦. Conditions (c) C C and (d) of Lemma 2 can be written as: ⎡ ⎤   TE   S M ⎣ ⊥ ⎦ 0 (1.50) E A = P Q In C the necessary and sufficient condition for Eq. (1.50) to have a solution is: ⎡ ⎤ TE ⎤ ⎡ ⎢E⊥ A⎥ TE ⎢ ⎥ ⊥ ⎥ rank ⎣E A⎦ = rank ⎢ ⎢ C ⎥=n ⎣ 0 ⎦ C In ⎡ ⎤ TE Now, because rank ⎣E⊥ A⎦ = n, there always exist matrices T ∈ Rq0 ×n and C q0 ×(nr1 +ny ) such that: K∈R  ⊥  E A TE + K =R (1.51) C which can be written as:



T

 K Ω=R

(1.52)   Ω by using Lemma 3, rank = rank(Ω) or equivalently RΩ + Ω = R. Then R the general solution of Eq. (1.52) is given by:

I. OBSERVER DESIGN

GENERALIZED DYNAMIC OBSERVER



T

 K = RΩ + − Z1 (In+nr1 +ny − ΩΩ + )

21 (1.53)

where Ω + is the generalized inverse of Ω (see Definition 11). Eq. (1.53) is equivalent to: T = T1 − Z1 T2 K = K1 − Z1 K2

(1.54) (1.55)  0 ,

     I In , T2 = (In+nr1 +ny − ΩΩ + ) n , K1 = RΩ + Inr1 +ny 0 0   0 K2 = (In1 +nr1 +ny − ΩΩ + ) , and Z1 is an arbitrary matrix of Inr1 +ny appropriate dimension. Now, define the following matrices:       I I I N1 = T1 AΣ + q0 , N2 = T2 AΣ + q0 , N3 = (Iq0 +nr1 +ny − ΣΣ + ) q0 , 0 0 0     0 0 K˜ 1 = T1 AΣ + , K˜ 2 = T2 AΣ + , Inr1 +ny Inr1 +ny   0 + ˜ , K3 = (Iq0 +nr1 +ny − ΣΣ ) Inr1 +ny     K K + + G1 = T1 AΣ , G2 = T2 AΣ , Inr1 +ny Inr1 +ny   K and G3 = (Iq0 +nr1 +ny − ΣΣ + ) Inr1 +ny

where T1 = RΩ +

By inserting the equivalence of TE from Eq. (1.51) into condition (a) of Lemma 2 it leads to:  ⊥   ⊥  E A E A N R−K +G = TA (1.56a) C C  ⊥  E A NR + K˜ = TA (1.56b) C where K˜ = G − NK, and Eq. (1.56b) can be written as:   N K˜ Σ = TA

(1.57)

The general solution of Eq. (1.57) is given by:   N K˜ = TAΣ + − Y1 (Iq0 +nr1 +ny − ΣΣ + )

(1.58)

by replacing matrix T from Eq. (1.54) into Eq. (1.58) it gives: N = N1 − Z1 N2 − Y1 N3 K˜ = K˜ 1 − Z1 K˜ 2 − Y1 K˜ 3 where Y1 is an arbitrary matrix of appropriate dimension. I. OBSERVER DESIGN

(1.59) (1.60)

22

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

As matrices N, T, K, and K˜ are known, we can deduce the form of matrix F as: G = K˜ + NK = K˜ 1 + N1 K − Z1 (K˜ 2 + N2 K) − Y1 (K˜ 3 + N3 K)

(1.61a) (1.61b)

= G1 − Z1 G2 − Y1 G3

(1.61c)

On the other hand, from Eq. (1.51) we obtain: ⎡ ⎤   TE −K ⎣E⊥ A⎦ = Iq0 Σ 0 Inr1 +ny C inserting Eq. (1.62) into Eq. (1.50) we get:      −K 0 S M Iq0 Σ= 0 Inr1 +ny In P Q

(1.62)

(1.63) 

Iq0 0

−K

−1

Because matrix Σ is of full column rank and = Inr1 +ny   Iq 0 K , the general solution to Eq. (1.63) is given by: 0 Inr1 +ny  

      Iq 0 K Y S M 0 Σ + − 2 (Iq0 +nr1 +ny − ΣΣ + ) (1.64) = 0 Inr1 +ny Y3 P Q In where Y2 and Y3 are arbitrary matrices of appropriate dimensions. Then matrices S, M, P, and Q can be determined as: S = −Y2 N3 M = −Y2 G3 P = P1 − Y3 N3

where P1 = Σ +



Iq 0 0



Q = Q1 − Y3 G3   K and Q1 = Σ + . Inr1 +ny

(1.65) (1.66) (1.67) (1.68)

4.2 GDO Stability Analysis Now, by using Eqs. (1.59), (1.65), (1.67) the observer error dynamics (1.85), (1.86) can be written as ϕ(t) ˙ = (A1 − YA2 )ϕ(t) e(t) = Pϕ(t)

I. OBSERVER DESIGN

(1.69)

GENERALIZED DYNAMIC OBSERVER

23

      N1 − Z1 N2 0 N3 0 where A1 = , P = P1 0 , Y = , A2 = 0 0 0 −Iq1     ε(t) Y1 H , and ϕ(t) = . Y3 = 0 is taken for simplicity to satisfy v(t) Y2 L condition (d) of Lemma 2. From the preceding results we can give the following lemma. Lemma 5. The following statements are equivalent:   sE − A (1) rank = n, ∀s ∈ C, Re(s) ≥ 0. C    N2 , N1 is detectable. (2) The pair N3 ⎤ ⎡ sR − T1 A ⎢ E⊥ A ⎥ ⎥ = rank(Σ), ∀s ∈ C, Re(s) ≥ 0. (3) rank ⎢ ⎦ ⎣ C T2 A Proof . The proof starts by showing that condition (1) is equivalent to condition (3). In fact, we have ⎤ ⎡ ⎤ ⎡    sE − A I 0  sE − A sE − A = rank ⎣ E⊥ A ⎦ rank = rank ⎣−E⊥ 0⎦ C C 0 I C ⎤ ⎡   ⎤ ⎡ sE − A A ⎢ sE⊥ A ⎥ ⎥ ⎢ ⎢sΩ − 0 ⎥ ⎥ ⎥ ⎢ = rank ⎢ ⎢ sC ⎥ = rank ⎣ E⊥ A ⎦ ⎣ E⊥ A ⎦ C C ⎤ ⎤ ⎡⎡  ⎤ ⎡ R ⎢⎣E⊥ A⎦ Ω + 0⎥ sΩ − A ⎥⎢ ⎢ 0 ⎥ ⎥⎢ ⎥ = rank ⎢ ⎥ ⎣ E⊥ A ⎦ ⎢ C+ ⎣ (ΩΩ − I) 0⎦ C 0 I  ⎤ ⎤ ⎡ ⎡ T1 A + A    ⊥  sR− ⊥ sΣ − ΣΩ ⎥ ⎢ ⎢ E A A ⎥ E A ⎥ ⎢ ⎢s  0 ⎥ − Ω+ ⎥ ⎢ ⎢ 0 ⎥ C C + ) A ⎥ = rank ⎢ ⎥ (I − ΩΩ = rank ⎢ ⎥ ⎢ ⎢ 0 ⎥ E⊥ A ⎥ ⎥ ⎢ ⎢ ⊥ ⎦ ⎦ ⎣ ⎣ E A C C T2 A ⎤ ⎡  sR − T1 A  ⎤ ⎡ ⎥ ⎢ E⊥ A sR − T1 A + A ⎥ ⎢ Ω ⊥ ⎥ ⎢ ⎢ C 0 ⎥ ⎥ = rank ⎢ E A ⎥ = rank ⎢ ⊥ ⎥ ⎢ ⎦ ⎣ C E A ⎥ ⎢ ⎦ ⎣ T2 A C T2 A I. OBSERVER DESIGN

24

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

we have used the fact that Ω + Ω = I, the last inequality results from the fact that    ⊥  E A + A ⊂ R(T2 A) R Ω 0 C Now, we can show that condition (3) is equivalent to condition (2). From the definition of matrix Σ we have the following equality ⎡ ⎤  ⎤ ⎡  sR − T1 A s I 0 Σ − T1 AΣ + Σ ⎢ E⊥ A ⎥   ⎥ = rank ⎣ ⎦ rank ⎢ I 0 Σ ⎣ ⎦ C + T2 AΣ Σ T2 A  ⎡  ⎤ s I 0 0 − T1 AΣ + ⎢ ⎥ 0 I 0 ⎥Σ = rank ⎢ ⎣ ⎦ 0 0 I + T2 AΣ By to rank ⎤ of Lemma 4, condition (3) is ⎡equivalent ⎤ ⎡  using    the results + s I 0 − T1 AΣ + s I 0 − T1 AΣ     ⎢ ⎥ ⎥ ⎢ 0 I 0 I ⎦ Σ = rank(Σ) if and only if matrix ⎣ ⎦ ⎣ + + T2 AΣ T2 AΣ   ⎤ ⎡ I +

sI − T1 AΣ ⎢ ⎥ 0 ⎢ ⎥ ⎢ 0   I⎥ ⎢ ⎥ ⎥ I is of full column rank or equivalently the matrix ⎢ ⎢ T2 AΣ +

⎥ if ⎢ ⎥ 0 ⎢ ⎥   ⎣ ⎦ I (I − ΣΣ + )

0 of full column rank, where represent matrices without any importance. ⎤ ⎡ sI − N1 This condition is equivalent to the matrix ⎣ N2 ⎦ of full column rank, N3 which proves the lemma. From Eq. (1.69) we must determine matrices Z1 and Y such that (A1 − YA2 ) is Hurwitz. The following lemma gives the condition to guarantee this stability. Lemma 6. There exists a parameter matrix Y such that (A1 −YA2 ) is Hurwitz  sI − A1 = if and only if the pair (A2 , A1 ) is detectable or equivalently rank A2 q0 + q1 , ∀s ∈ C, Re(s) ≥ 0. Now, we have ⎤ ⎡ 0 sI − N1 + Z1 N2   ⎢ sI − A1 0 sIq1 ⎥ ⎥ rank = rank ⎢ ⎣ A2 N3 0 ⎦ 0 −Iq1 I. OBSERVER DESIGN

25

GENERALIZED DYNAMIC OBSERVER

 = rank

 sI − N1 + Z1 N2 + q1 N3

= q 0 + q1 ,

∀s ∈ C, Re(s) ≥ 0

This is exactly the condition of the detectability of the pair (N3 , N1 −Z1 N2 ), which proves the lemma. The following theorem gives the linear matrices inequality’s (LMI’s) conditions to the determination of all GDO matrices for the descriptor system (1.30). Theorem 2. Under Assumptions 1 and 2 there exist two parameter matrices stable if there Z1 and Y such that observer error system (1.69)   is asymptotically X1 X2 such that the following exists a symmetric positive definite matrix X = X2T X3 LMI is satisfied. N3T⊥ (N1T X1 + X1 N1 − N2T W1T − W1 N2 )N3T⊥T < 0

(1.70)

In this case matrix W1 = X1 Z1 and matrix Y are parameterized as follows: √ Y = −X−1 (−σ B T + σ LΓ 1/2 )T (1.71) where Γ = σ BB T − Q > 0





(1.72)

X1 (N1 − Z1 N2 ) + (N1 − Z1 N2 )T X1 (N1 − Z1 N2 )T X2 with Q = , B = (*) 0   T N3 0 and matrix L is any matrix such that L 2 < 1 and σ > 0 is 0 −Iq1 any scalar such that Γ > 0. Proof . Let V(ϕ(t)) = ϕ(t)T Xϕ(t) be a Lyapunov function, thus its derivative along the trajectory of observer error system (1.69) is: ˙ V(ϕ(t)) = ϕ(t)T [(A1 − YA2 )T X + X(A1 − YA2 )]ϕ(t)

(1.73)

The asymptotic stability of observer error system (1.69) is guaranteed if ˙ and only if V(ϕ(t)) < 0. This leads to the following LMI: (A1 − YA2 )T X + X(A1 − YA2 ) < 0

(1.74)

which can be rewritten as: BX + (BX )T + Q < 0 AT1 X,

Q = XA1 + and B = A2 where X = et al. [38] the inequality (1.75) is equivalent to: −YT X,





B ⊥ QB T⊥ < 0

(1.75) T.

According to Skelton (1.76)

with B ⊥ = N3T⊥ 0 . By using the definition of matrices Q and W1 inequality (1.76) becomes:

I. OBSERVER DESIGN

26

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

N3T⊥ (N1T X1 + X1 N1 − N2T W1T − W1 N2 )N3T⊥T < 0

(1.77)

From the elimination lemma, if condition (1.76) is satisfied, then the parameter matrix Y, is obtained as in Eqs. (1.71), (1.72).

Algorithm 1 1. Chose a matrix R ∈ Rq0 ×n such that matrix Σ is of full column rank. 2. Compute matrices N1 , N2 , N3 , T1 , T2 , K1 , K2 , and P1 defined in the proof of Lemma 4. 3. Solve the LMI 1.70 to find matrices X and Z1 . 4. Determine arbitrary matrix L, such that L < 1, and a scalar σ > 0, such that Γ > 0, then obtain the parameter matrix Y as in Eq. (1.71). 5. Compute the observer matrices N, H, G, J, S, L, M, P, and Q, by using Eq. (1.59) to compute N, Eq. (1.71) to compute H and L, Eqs. (1.65)–(1.68) to compute S, M, P, and Q taking Y3 = 0 for simplicity, matrix G is given in Eq. (1.61c) and J is given by equation (b) from Lemma 2.

4.3 Particular Cases of the GDO In this section we consider two particular cases of our results. 4.3.1 Proportional Observer Consider the following descriptor system: E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t) with the PO: ζ˙ (t) = Nζ (t) + Ga y(t) + Ju(t) xˆ (t) = Pζ (t) + Qa y(t) and the error dynamics (1.69) become: ¯A ¯ 2 )ε(t) ¯1 −Y ε˙ (t) = (A ¯ e(t) = Pε(t) ¯ 2 = N3 , P¯ = P1 , and Y ¯ = Y1 . Consequently, ¯ 1 = N1 − Z1 N2 , A where A matrices Q, B, and X of Theorem 2 become: ¯ TX Q = X(N1 − Z1 N2 ) + (N1 − Z1 N2 )T X, B = N3T and X = −Y     R E Matrices Σ and Ω are defined as Σ = and Ω = . C C

I. OBSERVER DESIGN

GENERALIZED DYNAMIC OBSERVER

27

4.3.2 Proportional-Integral Observer Consider the following descriptor system: E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t) with the PIO:

  ⊥ −E Bu(t) + Ju(t) ζ˙ (t) = Nζ (t) + Hv(t) + G y(t) v˙ (t) = y(t) − Cˆx(t)  ⊥  −E Bu(t) xˆ (t) = Pζ (t) + Q y(t)

and the error dynamics (1.69) become: ¯¯ A ¯¯ − Y ¯¯ )ϕ(t) ϕ(t) ˙ = (A 1 2 ¯ ¯ e(t) = Pϕ(t)

      0 ¯¯ = N3 ¯¯ = ¯¯ = N1 − Z1 N2 0 , A , P¯¯ = P1 0 , and Y where A 1 2 0 −CP1 0 −Iq1    I  Y1 H . 0 Consequently, matrices Q, B, and X of Theorem 2 become: Q= 

X1 (N1 − Z1 N2 ) + (N1 − Z1 N2 )T X1 − X2 CP1 − (CP1 )T X2T (*)

 B=

N3T 0

 0 ¯¯ T X, such that Y and X = −Y 1 −Iq1

(N1 − Z1 N2 )T X2 − (CP1 )T X3 0



  +  I H = X X. 0

4.4 Simulation Example Consider the electronic circuit system shown in Fig. 1.5, where C stands for capacitor. R1 and R2 are resistors, L is an inductor, Vs (t) is the voltage resource, V1 (t) is the capacitor voltage, Vout is the voltage in R2 , and i1 (t) and iL (t) are the amperages of the currents flowing over them, respectively. Then, the circuit can be characterized by Eq. (1.78) according to the Kirchhoff law dV1 (t) = i1 (t) C dt diL (t) (1.78) = V1 (t) − R2 iL (t) L dt Vs (t) = V1 (t) + R1 (iL (t) + i1 (t))

I. OBSERVER DESIGN

28

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

FIG. 1.5 Electronic circuit.

 T The measurable output is y(t) = iL (t) i1 (t) . Let the semistate variables  T  T x1 (t) x2 (t) x3 (t) = V1 (t) iL (t) i1 (t) and the input variable u(t) = Vs (t). Then the preceding circuit system can be characterized as E˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t) where

⎡

⎤ C 0 0 E = ⎣ 0 L 0⎦ , 0 0 0   C= 0 0 1

⎡

0 0 A = ⎣1 −R2 1 R1

⎤ 1 0 ⎦, R1

(1.79) ⎡

⎤ 0 B = ⎣ 0 ⎦ , and −1

The parameters are given as follows: C = 100 mF, R1 = 1000 , R2 = 10 , L = 0.5H, and Vs (t) = 4sin(1.5t). 4.4.1 Generalized Dynamic Observer The objective here is to design a GDO such that the actual states of the circuit system can be estimated. So that, consider E⊥ =  preceding  0 0 1 to verify Assumptions 1 and 2 ⎤ ⎡   E sE − A ⊥ ⎦ ⎣ =3 rank E A = 3 and rank C C   1 1 1 For the GDO we have chosen matrix R = such that rank(Σ) = 3. 0 0 1 By using the YALMIP toolbox, we solve the LMI (1.70) to find matrices X and Z1 ⎡ ⎤   19.01 −2.60 6.51 1.18 0.63 1.18 1.18 0.91 ⎣ ⎦ X = −2.60 19.53 6.51 and Z1 = 1.16 1.08 1.16 1.16 1.12 6.51 6.51 10

I. OBSERVER DESIGN

29

GENERALIZED DYNAMIC OBSERVER

⎡

0.10 0.10 ⎢0.10 0.10 ⎢ Now, considering σ = 1000 and L = ⎢ ⎢0.10 0.10 ⎣0.10 0.10 0.10 0.10 Eqs. (1.71), (1.72) we get: ⎡ 3.28 22.35 3.28 −15.79 40.96 −0.26 −41.49 Y = ⎣ −0.26 −12.38 −51.64 −12.38 26.87

⎤ 0.10 0.10⎥ ⎥ 0.10⎥ ⎥ and by solving 0.10⎦ 0.10 ⎤ 83.67 78.24 ⎦ −215.82

Finally, we can get all the matrices of the observer as:       −0.92 −18.63 83.67 0 39.25 N= , S= , H= , −1.74 −49.82 78.24      1.18 −1.19 17.32 L = −215.82, J = , G= , M = 0.03 1.18 −1.18 26.42 ⎡ ⎤ ⎤ ⎡ 0.04 −45.55 1 0 P = ⎣0 −0.5⎦ , and Q = ⎣0.05 −56.97⎦ × 10−3 0.04 57.02 0 0.5

 −33.74 ,

In order to provide a comparison of the GDO with the PO and PIO, the latter are also designed. 4.4.2 Proportional Observer  1 1 Consider matrices R = 0 1

⎡ ⎤  0.1 0.1 1 , L = ⎣0.1 0.1⎦, and σ = 1000 the 0 0.1 0.1 following PO matrices are obtained:       −99.85 0.11 1.89 −131.74 , J= , N= , Ga = −98.64 0.10 1.90 −120.54 ⎡ ⎡ ⎤ ⎤ 1 −1 0 P = ⎣0 1 ⎦ , and Qa = ⎣0⎦ 0 0 1

4.4.3 Proportional-Integral Observer

⎡

0.1 ⎢0.1   ⎢ 1 1 0 By considering matrices R = ,L = ⎢ ⎢0.1 0 0 1 ⎣0.1 0.1 σ = 1000 the following PIO matrices are obtained:

I. OBSERVER DESIGN

0.1 0.1 0.1 0.1 0.1

⎤ 0.1 0.1⎥ ⎥ 0.1⎥ ⎥, and 0.1⎦ 0.1

30

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

        −0.51 17.70 56.43 −1.63 1.63 7.30 , H= , J= , G= , 0.26 −9.53 16.69 0.54 −0.54 1.27 ⎡ ⎡ ⎤ ⎤ 1 0.5 0 −0.19 P = ⎣0 −0.50⎦ , and Q = ⎣0 −1.06⎦ 0 0.50 0 1.06

N=

4.4.4 Simulation Results The initial conditions for the system are x(0) = [0.05, 0.01, 0.01]T , for the GDO, PIO, and PO null initial conditions are considered. To evaluate the performance of the observers, uncertainties ρ1 = 10 and ρ2 = 0.5 are added in the parameters R1 and R2 , then we obtain parameters (R1 + ρ1 ) and (R2 + ρ2 ). The results of simulation are depicted in Figs. 1.6–1.12. Fig. 1.6 shows the input u(t). Figs. 1.7–1.12 show the system states and their estimations by the GDO, PIO, and PO; also, these figures show the estimation error of each observer. From these results, we can see that the GDO presents a new structure that can improve the performances of the existing observers for

Voltage (V)

4 2 0 –2 –4 0

5

10

15

20

25

30

35

40

30

35

40

Time (s)

FIG. 1.6 Input u(t). 0.06

Amplitude

0.04 0.02 0 –0.02 –0.04 0

5

10

15

20

Time (s)

FIG. 1.7 Estimate of x1 (t). I. OBSERVER DESIGN

25

0.02

Amplitude

0 –0.02 –0.04 –0.06

0

5

10

15

20

25

30

35

40

Time (s)

FIG. 1.8 Estimation error of x1 (t). 10–3

Amplitude

5

0

–5

0

5

10

15

20

25

30

35

40

25

30

35

40

35

40

Time (s)

FIG. 1.9 Estimate of x2 (t). 10–3

1

Amplitude

0.5 0 –0.5 –1

0

5

10

15

20

Time (s)

FIG. 1.10 Estimation error of x2 (t).

Amplitude

5

10–3

0

–5 0

5

10

15

20

Time (s)

FIG. 1.11 Estimate of x3 (t).

25

30

32

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

10–3

1

Amplitude

0.5 0 –0.5 –1

0

5

10

15

20

25

30

35

40

Time (s)

FIG. 1.12 Estimation error of x3 (t).

TABLE 1.1 Error Evaluation IAE Observer

GDO

PIO

PO

x1 (t)

0.0827

0.1140

0.2620

x2 (t)

0.0025

0.0036

0.0048

x3 (t)

3.22 × 10−17

0.0017

0.0026

descriptor systems. In order to show the different performances of these observers, Table 1.1 is presented, in which the integral absolute error (IAE) is considered.

5 H∞ GENERALIZED OBSERVER DESIGN Consider the following descriptor system with disturbances E˙x(t) = Ax(t) + Bu(t) + Dw(t) y(t) = Cx(t) + Fw(t)

(1.80)

where w(t) ∈ Rnw is the disturbance vector of finite energy and matrices E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , D ∈ Rn×nw , C ∈ Rny ×n , and F ∈ Rny ×nw . Let rank(E) = nr and E⊥ ∈ Rnr1 ×n , in this case, nr1 = n − nr . The GDO for system (1.80) is given by   ⊥ ˙ζ (t) = Nζ (t) + Hv(t) + G −E Bu(t) + Ju(t) (1.81a) y(t)  ⊥  −E Bu(t) v˙ (t) = Sζ (t) + Lv(t) + M (1.81b) y(t)  ⊥  −E Bu(t) xˆ (t) = Pζ (t) + Q (1.81c) y(t) I. OBSERVER DESIGN

H∞ GENERALIZED OBSERVER DESIGN

33

so that, the dynamic error becomes:  ⊥ 

E A ε˙ (t) =Nε(t) + Hv(t) + NTE + G − TA x(t) + (J − TB)u(t) C  ⊥ 

E D + G − TD w(t) (1.82) F by using the definition of ε(t), Eqs. (1.81b), (1.81c) can be written as:  ⊥ 

 ⊥  E A E D v˙ (t) = Sε(t) + Lv(t) + M + STE x(t) + M w(t) (1.83) C F ⎤ ⎡  ⊥    TE E D xˆ (t) = Pε(t) + P Q ⎣E⊥ A⎦ x(t) + Q w(t) (1.84) F C Now, if conditions (a)–(d) of Lemma 2 are satisfied, the following observer error dynamics is obtained from Eqs. (1.82), (1.83) ⎤ ⎡  ⊥  E D      − TD⎥ ⎢G F ε˙ (t) N H ε(t)  ⊥  ⎥ w(t) (1.85) = +⎢ ⎦ ⎣ v˙ (t) S L v(t) E D M F and from Eq. (1.84) we get:



 E⊥ D xˆ (t) − x(t) = e(t) = Pε(t) + Q w(t) (1.86) F   N H in this case, if w(t) = 0 and is Hurwitz, then limt→∞ e(t) = 0. S L

5.1 Parameterization for the Robust Case When a system is subject to disturbances, as is shown in Eq. (1.80), a bilinearity in the matrix G is involved in the stability analysis of the observer. By developing matrix G we obtain:     + K1 − Z1 K2 + K1 − Z1 K2 − Z1 T2 AΣ G =T1 AΣ Inr1 +ny Inr1 +ny   K − Z1 K2 − Y1 (Iq0 +nr1 +ny − ΣΣ + ) 1 Inr1 +ny where the unknown matrices are Z1 and Y1 . In order to avoid this bilinearity, an adaptation in the  parameterization is carried. ⊥D E Let K¯ 2 = K2 and Z1 = Z(In1 +nr1 +ny − K¯ 2 K¯ 2+ ), where Z is an F  ⊥  E D arbitrary matrix of appropriate dimensions, so that G , from the F observer error system (1.85) becomes: I. OBSERVER DESIGN

34

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS



 E⊥ D (1.87) = Gd1 − ZGd2 − Y1 Gd3 F   ⊥  K1 E D + where Gd1 = T1 AΣ , Gd2 = (In1 +nr1 +ny − K¯ 2 K¯ 2+ )T2 AΣ + Inr1 +ny F    ⊥   ⊥  K1 K1 E D E D , and Gd3 = (Iq0 +nr1 +ny − ΣΣ + ) . Inr1 +ny Inr1 +ny F F In ⊥the same way,  ⊥ we  obtain the following expressions for T, K, N, E D E D M , and Q F F G

T = T1 − ZT2 K = K1 − ZK2 N = N1 − ZN2 − Y1 N3  ⊥  E D M = −Y2 Gd3 F  ⊥  E D Q = Qd1 − Y3 Gd3 F

(1.88) (1.89) (1.90) (1.91) (1.92)

where T2 = (In1 +nr1 +ny − K¯ 2 K¯ 2+ )T2 , K2 = (In1 +nr1 +ny − K¯ 2 K¯ 2+ )K2 , N2 =   ⊥  K1 E D (In1 +nr1 +ny − K¯ 2 K¯ 2+ )N2 , and Qd1 = Σ + . Where we have Inr1 +ny F used the fact that K¯ 2 K¯ 2+ K¯ 2 = K¯ 2 (see Definition 11). In order to study the observer stability, the observer error dynamics (1.85)–(1.86) can be written as: ϕ(t) ˙ = (A1 − YA2 )ϕ(t) + (B1 − YB2 )w(t) e(t) = Pϕ(t) + Qw(t) where N1 − ZN2 A1 = 0   Gd3 ,P = B2 = 0   ε(t) ϕ(t) = . v(t)

(1.93)

     N3 G − T1 D + Z(T2 D − Gd2 ) 0 0 , B 1 = d1 , A2 = , 0 0 −Iq1 0     Y1 H P1 − Y3 N3 0 , Q = Qd1 − Y3 Gd3 , Y = , and Y2 L

5.2 H∞ GDO Stability Analysis In this section we present a method for designing an H∞ GDO given by Eq. (1.81). This design is obtained from the determination of matrices Z and Y such that the worst estimation error energy e 2 is minimum for all bounded energy disturbance w(t). This problem is equivalent to the minimization problem minγ s.t. Gwe ∞ < γ , where Gwe is the transfer

I. OBSERVER DESIGN

35

H∞ GENERALIZED OBSERVER DESIGN

function from the disturbance to the estimation error and γ is a given positive scalar. The solution to this problem is given by the following theorem. Theorem 3. Under Assumptions 1 and 2, there exists an H∞ GDO (1.81) such that the error dynamics inEq. (1.93) is stable and Gwe ∞ < γ , if there  X1 X1 > 0, with X1 = X1T satisfying the following exists a matrix X = X1 X2 LMIs. ⎡ ⎤ Π2 (P1 − Y3 N3 )T Π1 N1T X1 − NT2 W1T ⎢ (*) ⎥ T⊥T 0 Π2 0 ⎥C < 0 (1.94) C T⊥ ⎢ ⎣ (*) (*) −γ¯ Inw (Qd1 − Y3 Gd3 )T ⎦ (*) 0 (*) −In where Π1 = X1 N1 − W1 N2 + N1T X1 − NT2 W1T Π2 = X1 (Gd1 − T1 D) + W1 (T2 D − Gd2 ) and

 −γ¯ Inw (*)

(1.95a) (1.95b)

 (Qd1 − Y3 Gd3 )T 0 R−1 − R−1 BlT [ϑ − ϑCrT (Cr ϑCrT )−1 Cr ϑ]Bl R−1

(1.98b)

ϑ= S=

⎡

⎤

(1.98c)

Π2 (P1 − Y3 N3 )T Π1 N1T X1 − NT2 W1T ⎢ (*) ⎥ 0 Π2 0 ⎥, B = where Q = ⎢ T ⎣ (*) (*) −γ¯ Inw (Qd1 − Y3 Gd3 ) ⎦ (*) 0 (*) −In ⎡ ⎤      −Iq0 +q1 N3 Gd 3 0 ⎣ 0 ⎦, C = 0 , and matrices Π1 and Π2 are 0 0 −Iq1 0 defined in Eq. (1.95), and matrices L, R, and Z are arbitrary matrices of appropriate dimensions satisfying R > 0 and L 2 < 1. Matrices Cl , Cr , Bl , and Br are any full rank matrices such that C = Cl Cr and B = Bl Br . Proof . The bounded real lemma guarantees that the observer error system (1.93) is stable and Gwe ∞ < γ if and only if there exists a matrix X = XT > 0 such that:

I. OBSERVER DESIGN

36

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

⎡

(A1 − YA2 )T X + X(A1 − YA2 ) X(B1 − YB2 ) ⎣ (*) −γ 2 Inw (*) (*)

⎤ PT QT ⎦ < 0 −In

(1.99)

which can be written as: BX C + (BX C)T + Q < 0 (1.100) ⎡ T ⎤ T A1 X + XA1 XB1 P where X = XY, γ¯ = γ 2 , Q = ⎣ (*) −γ¯ Inw QT ⎦, C = (*) (*) −In ⎡ ⎤ −Iq0 +q1   A2 B2 0 , and B = ⎣ 0 ⎦. 0 From the elimination lemma, the solvability conditions of inequality (1.100) are: C T⊥ QC T⊥T < 0 ⊥

(1.101a)

⊥T

(1.101b) B QB < 0 ⎡ ⊥ ⎤   AT2 0 Inw 0 0 ⎦. By using the definition with B ⊥ = and C T⊥ = ⎣ BT 2 0 0 In 0 In of matrices C, Q, and W1 the inequality (1.101a) becomes: ⎡ ⎤ Π2 (P1 − Y3 N3 )T Π1 N1T X1 − NT2 W1T ⎢ (*) ⎥ T⊥T 0 Π2 0 ⎥C C T⊥ ⎢ 0 in Eq. (1.98b), then obtain the parameter matrix Y as in Eq. (1.97). 5. Compute the observer matrices N, H, G, J, S, L, M, P, and Q, by using Eq. (1.90) to compute N, Eq. (1.97) to compute H and L, Eqs. (1.65)–(1.68) to compute S, M, P, and Q, matrix G is given in Eq. (1.61c), and J is given by equation (b) from Lemma 2.

5.3 Particular Cases of the H∞ GDO 5.3.1 Proportional Observer Consider the following descriptor system: E˙x(t) = Ax(t) + Bu(t) + Dw(t) y(t) = Cx(t) + Fw(t) with the PO: ζ˙ (t) = Nζ (t) + Ga y(t) + Ju(t) xˆ (t) = Pζ (t) + Qa y(t) and the error dynamics (1.93) become: ¯A ¯ 2 )ε(t) + (B¯ 1 − Y ¯ B¯ 2 )w(t) ¯1 −Y ε˙ (t) = (A ¯ ¯ e(t) = Pε(t) + Qw(t) ¯ 2 = N3 , B¯ 1 = Gd − T1 D + Z(T2 D − Gd ), B¯ 2 = Gd , ¯ 1 = N1 − ZN2 , A where A 1 2 3 ¯ = Qd − Y3 Gd , and Y ¯ = Y1 . Consequently, matrices D, ¯P = P1 − Y3 N3 , Q 1 3 C, B, and X of Theorem 3 become: Q= 

(N1 − ZN2 )T X + X(N1 − ZN2 ) (*) (*)

⎡

X(Gd1 − T1 D + Z(T2 D − Gd2 )) −γ¯ Inw (*)

(P1 − Y3 N3 )T (Qd1 − Y3 Gd3 )T −In



⎤ −Iq0 +q1 ¯ with γ¯ = γ 2 . Matrices Gd3 0 , B = ⎣ 0 ⎦, X = XY 0     R E Σ and Ω are defined as Σ = and Ω = . C C

 C = N3



5.3.2 Proportional-Integral Observer Consider the following descriptor system: E˙x(t) = Ax(t) + Bu(t) + Dw(t) y(t) = Cx(t) + Fw(t)

I. OBSERVER DESIGN

38

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

with the PIO:

  ⊥ ˙ζ (t) = Nζ (t) + Hv(t) + G −E Bu(t) + Ju(t) y(t) v˙ (t) = y(t) − Cˆx(t)  ⊥  −E Bu(t) xˆ (t) = Pζ (t) + Q y(t)

and the error dynamics (1.93) become: ¯¯ A ¯¯ − Y ¯¯ B¯¯ )w(t) ¯¯ − Y ¯¯ )ϕ(t) + (B ϕ(t) ˙ = (A 1 2 1 2 ¯¯ ¯¯ e(t) = Pϕ(t) + Qw(t)

  N1 − ZN2 0 ¯¯ ¯ ¯ , A2 = Y3 = 0 is taken for simplicity, and matrices A1 = 0 −CP   1      Gd1 − T1 D + Z(T2 D − Gd2 ) ¯¯ F 0 N3 , B¯¯ 1 = , B2 = d3 , P¯¯ = P1 0 , 0 0 −Iq1 F − CQd1     I ¯ ¯ = Qd , and Y = Y1 H . Consequently, matrices Q, B, and X of Q 1 0 Theorem 3 become: ⎤ ⎡ Π2 PT1 Π1 (N1 − ZN2 )T X1 − PT1 CT X2 ⎢ (*) 0 Π3 0 ⎥ ⎥ Q=⎢ ⎣ (*) (*) −γ¯ Inw QT ⎦ 0

(*)

(*)

d1

−In

with

    Π1 = (N1 − ZN2 )T − PT1 CT X1 + X1 (N1 − ZN2 ) − CP1   Π2 = X1 Gd1 − T1 D + F − CQd1 + Z(T2 D − Gd2 )     Π3 = X1 Gd1 − T1 D + Z(T2 D − Gd2 ) + X2 F − CQd1 ⎡ ⎤      −Iq0 +q1 N3 Gd 3 0 ¯¯ with γ¯ = γ 2 , C= 0 , B = ⎣ 0 ⎦, and X = XY 0 0 −Iq1 0   +   I X. such that Y1 H = X 0

5.4 Simulation Example Consider the electronic circuit system shown in Fig. 1.5, which now is going to be characterized as E˙x(t) = Ax(t) + Bu(t) + Dw(t) y(t) = Cx(t) + Fw(t)

I. OBSERVER DESIGN

(1.104)

H∞ GENERALIZED OBSERVER DESIGN

where

39

⎡ ⎤ ⎤ ⎡ ⎤ 0 C 0 0 0 0 1 E = ⎣ 0 L 0⎦ , A = ⎣1 −R2 0 ⎦ , B = ⎣ 0 ⎦ , −1 1 R1 R1 0 0 0 ⎡ ⎤     0 0 1 0 0 D = ⎣1⎦ , C = , and F = 0 0 1 1 0 ⎡

The parameters are given as follows C = 100 mF, R1 = 4 , R2 = 0.05 , L = 0.1H, and Vs (t) = 4sin(1.5t). The disturbance w(t) is of finite energy.

5.4.1 H∞ Generalized Dynamic Observer   Consider E⊥ = 0 0 1 to verify Assumptions 1 and 2 ⎡ ⎤   E sE − A ⊥ ⎣ ⎦ =3 rank E A = 3 and rank C C   1 1 0 For the H∞ GDO we have chosen matrix R = such that 0 1 1 rank(Σ) = 3. Using YALMIP √toolbox to solve the LMIs (1.94), (1.96) to find matrices X, Z, Y3 , and γ = γ¯ ⎡ ⎤ 125.28 0 125.28 0 ⎢ 0 125.28 0 125.28⎥ ⎥, X=⎢ ⎣125.28 0 375.83 0 ⎦ 0 125.28 0 375.83   −0.05 0.01 0 0.01 0 −0.01 Z= , 0 −0.06 0 0 0 0.01 ⎡ ⎤ −3.96 1.22 −16.05 −12.23 −16.19 3.87 3.54 4.66 ⎦ , and γ = 0.03 Y3 = ⎣ 1.13 −0.37 −0.13 0.06 0.13 −0.45 −0.58 ⎡ ⎤ 2 3 1 4 9 7 9 ⎢3 2 6 3 3 5 4⎥ ⎥ Now, considering matrices Z = ⎢ ⎣6 1 4 2 6 2 6⎦, L = 9 5 2 7 4 3 2 ⎡ ⎤ 0.10 0.10 0.10 0.10 ⎢0.10 0.10 0.10 0.10⎥ ⎢ ⎥ ⎣0.10 0.10 0.10 0.10⎦ and R = 0.0001 × I4 and solving Eqs. (1.97), 0.10 0.10 0.10 0.10 (1.98) we get:

I. OBSERVER DESIGN

40

1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

⎡

9.98 ⎢ 33.86 ⎢ Y=⎣ −2.43 −10.38

32.27 122.90 −7.60 −37.81

−9.98 −33.79 2.45 10.39

−2.27 −21.32 0.25 6.62

7.75 40.18 12.54 −10.23 −2.19 −42.65 −3.80 0.76

⎤ −2.33 30.32 ⎥ ⎥ −1.89 ⎦ −39.38

Finally, we compute all the matrices of the observer as:       −10.54 −32.24 2.44 7.59 40.18 −2.33 N= , S= , H= , −33.64 −122.92 10.40 37.80 −10.23 30.32       0 −42.65 −1.89 0.20 −0.58 1.38 J= ,L = , G= , 0 0.76 −39.38 0.25 −0.10 −1.59 ⎤ ⎡   2.86 −5.87 −0.09 0.30 0.35 , P = ⎣−0.67 1.62 ⎦ , M = −0.05 −0.08 0.19 −0.02 −0.09 ⎡ ⎤ 0.03 0 −0.13 and Q = ⎣0.24 0 −0.96⎦ 0 0 1 In order to provide a comparison of the GDO with the PIO and the PO, the latter are also designed. 5.4.2 H∞ Proportional Observer     0 1 0 2 3 1 4 By considering matrices R = , Z = , L = 1 0 1 3 2 6 3   0.1 0.1 , R = 0.0001 × I2 , and γ = 1 the following PO matrices are 0.1 0.1 obtained:       0 0.01 0 0 0.10 , J= , N= , Ga = −9.75 0.17 2.46 −4.92 −2.29 ⎡ ⎡ ⎤ ⎤ 0 1 0.02 0 P = ⎣0.5 0⎦ , and Q = ⎣0.99 0⎦ 0 0 0 1

5.4.3 H∞ Proportional-Integral Observer  By considering matrices R = ⎡

0.1 ⎢0.1 L = ⎢ ⎣0.1 0.1

0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.1

1 0

0 1

⎡ 2  ⎢3 0 ,Z =⎢ ⎣6 1 9

3 2 1 5

1 6 4 2

4 3 2 7

9 3 6 8

7 5 2 4

⎤ 9 4⎥ ⎥, 6⎦ 9

⎤ 0.1 0.1⎥ ⎥, R = 0.0001 × I4 , and γ = 0.10 the following 0.1⎦ 0.1

I. OBSERVER DESIGN

41

H∞ GENERALIZED OBSERVER DESIGN

PIO matrices are obtained:     −38.45 −154.64 −352.54 −78.83 N= , H= , −122.70 −506.16 −121.41 −507.16 ⎡     0.92 2.91 34.57 −1.79 −1.84 G= ,J = , P = ⎣−0.11 4.73 117.73 −9.79 −2.44 −0.11 ⎡ ⎤ 0.76 −3.07 −3.04 and Q = ⎣0.03 0.88 −0.12⎦ 0.03 −0.12 0.88

⎤ −0.21 0.05 ⎦ , 0.05

5.4.4 Simulation Results The initial conditions for the system are x(0) = [0.2, 0.2, 0.2]T , for the GDO, PIO, and PO null initial conditions are considered. To evaluate the performance of the observers, an uncertainty ρ = 0.05 is added in the parameters R1 and R2 , then we obtain parameters (R1 + ρ) and (R2 + ρ). The results of simulation are depicted in Figs. 1.13–1.20. Fig. 1.13 shows the input u(t), and Fig. 1.14 represents the disturbance behavior.

Voltage (V)

4 2 0 –2 –4 0

5

10

15

20

25

30

35

40

25

30

35

40

Time (s)

FIG. 1.13 Input u(t).

Voltage (V)

0.5

0

–0.5 0

5

10

15

20

Time (s)

FIG. 1.14 Disturbance w(t).

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1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

1

Amplitude

0.5 0 –0.5 –1 –1.5

0

5

10

15

20

25

30

35

40

25

30

35

40

25

30

Time (s)

FIG. 1.15 Estimate of x1 (t). 1

Amplitude

0.5 0 –0.5 –1 –1.5

0

5

10

15

20

Time (s)

FIG. 1.16 Estimation error of x1 (t).

Amplitude

1 0.5 0 –0.5 –1

0

5

10

15

20

35

40

Time (s)

FIG. 1.17 Estimate of x2 (t).

Figs. 1.15–1.20 show the system states and their estimations by the GDO, PIO, and PO, also these figures show the estimation error of each observer. In order to show the difference in the estimations, Table 1.2 is presented where the IAE is considered as criteria of performances evaluation.

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43

H∞ GENERALIZED OBSERVER DESIGN

Amplitude

0.4

0.2

0

–0.2

0

5

10

15

20

25

30

35

40

25

30

35

40

25

30

35

40

Time (s)

FIG. 1.18 Estimation error of x2 (t). 0.2

Amplitude

0.1 0

–0.1 –0.2 0

5

10

15

20

Time (s)

FIG. 1.19 Estimate of x3 (t).

Amplitude

0.05

0

–0.05 0

5

10

15

20

Time (s)

FIG. 1.20 Estimation error of x3 (t). TABLE 1.2 Error Evaluation IAE Observer

GDO

PIO

PO

x1 (t)

0.8655

10.61

5.235

x2 (t)

0.44

2.639

2.788

x3 (t)

7.80×10−4

0.69

1.24×10−3

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1. ON DYNAMIC OBSERVERS DESIGN FOR DESCRIPTOR SYSTEMS

6 CONCLUDING REMARKS This chapter presents some basic concepts and properties for descriptor systems with constant coefficients. The observer design for this class of systems is studied by using the structure of the GDO; it contains an additional variable that is a generalization of the integral of the output error (difference between the estimated output and the output). To analyze the stability of the observer, the Lyapunov method is used, and is guaranteed by solving a set of LMIs. The numerical examples demonstrate that the proposed approach to design dynamic observers can be useful to expand the perspectives to practical applications, such as process monitoring, observer-based control, fault diagnosis, and fault estimation strategies. These perspectives, and other theoretical aspects to be treated (e.g., observers for discrete and/or nonlinear descriptor systems, unknown input estimation), make evident that there are still open problems to be solved in this field.

References [1] L. Dai, Singular Control Systems, Springer-Verlag, New York, NY, 1989. [2] D.G. Luenberger, An introduction to observers, IEEE Trans. Autom. Control 16 (1971) 596–602. [3] G.C. Goodwin, R.H. Middleton, The class of all stable unbiased state estimators, Syst. Control Lett. 13 (1989) 161–163. [4] H.J. Marquez, A frequency domain approach to state estimation, J. Franklin Inst. 340 (2003) 147–157. [5] D.G. Luenberger, Dynamic equations in descriptor form, IEEE Trans. Autom. Control AC-22 (1977) 312–321. [6] E.L. Yip, R.F. Sincovec, Solvability, controllability, and observability of continuous descriptor systems, IEEE Trans. Autom. Control 26 (1981) 702–707. [7] J.Y. Ishihara, M.H. Terra, Impulse controllability and observability of rectangular descriptor systems, IEEE Trans. Autom. Control 46 (6) (2001) 991–994. [8] M. Darouach, On the functional observer for linear descriptor systems, Syst. Control Lett. 61 (2012) 427–434. [9] V. Ionescu, C. Oarˇa, M. Weiss, Generalized Riccati Theory and Robust Control: A Popov Function Approach, 1 ed., Wiley, 1999. [10] G.R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, New York, NY, 2010. [11] E.K. Boukas, Control of Singular Systems With Random Abrupt Changes, Springer, New York, NY, 2008. [12] B. Boulkrone, M. Darouach, M. Zasadzinski, S. Gillé, D. Fiorelli, A nonlinear observer design for an activated sludge wastewater treatment process, J. Process Control 19 (2009) 1558–1565. [13] J.M. Araujo, P.R. Barros, C.E.T. Dorea, Design of observers with error limitation in discrete-time descriptor systems: a case study of a hydraulic tank system, IEEE Trans. Control Syst. Technol. 20 (2012) 1041–1047. [14] P.C. Müller, M. Hou, On the observer design for descriptor systems, IEEE Trans. Autom. Control 38 (1993) 1666–1671.

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[15] L. Zhou, G. Lu, Detection and stabilization for discrete-time descriptor systems via a limited capacity communication channel, Automatica 45 (2009) 2272–2277. [16] M. Darouach, H∞ unbiased filtering for linear descriptor systems via LMI, IEEE Trans. Autom. Control 54 (2009) 1966–1972. [17] D. Koenig, Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation, IEEE Trans. Autom. Control 50 (2005) 212–217. [18] M. Hou, P.C. Müller, Design of observers for linear system with unknown inputs, IEEE Trans. Autom. Control 37 (1992) 871–875. [19] M. Darouach, M. Zasadzinski, S.J. Xu, Full-order observers for linear systems with unknown inputs, IEEE Trans. Autom. Control 39 (1994) 606–609. [20] M. Darouach, M. Boutayeb, Design of observers for descriptor systems, IEEE Trans. Autom. Control 40 (1995) 1323–1327. [21] M. Hou, P.C. Müller, Observer design for descriptor systems, IEEE Trans. Autom. Control 44 (1999) 164–168. [22] M. Darouach, Complements to full order observer design for linear system with unknown inputs, Appl. Math. Lett. 22 (2009) 1107–1111. [23] B. Wojciechowski, Analysis and Synthesis of Proportional Integral Observers for Single-Input-Single-Output Time-Invariant Continuous Systems (Ph.D. thesis), Gliwice, Poland, 1978. [24] D. Söffker, T. Yu, P.C. Müller, State estimation of dynamical systems with nonlinearities by using proportional-integral observer, Int. J. Syst. Sci. 26 (1995) 1571–1582. [25] T. Kaczorek, Proportional-integral observers for linear multivariable time-varying systems, Regelungstechnik 27 (1979) 359–362. [26] S. Beale, B. Shafai, Robust control systems design with proportional integral observer, Int. J. Control 50 (1989) 97–111. [27] Z. Gao, T. Breikin, H. Wang, Discrete-time proportional and integral observer and observer-based controller for systems with both unknown input and output disturbances, Optimal Control Appl. Methods 29 (2008) 171–189. [28] M. Hou, P.C. Müller, Design of a class of Luenberger observers for descriptor systems, IEEE Trans. Autom. Control 40 (1995) 133–136. [29] Z. Gao, D.W.C. Ho, State/noise estimator for descriptor systems with application to sensor fault diagnosis, IEEE Trans. Signal Process. 54 (2006) 1316–1326. [30] G.R. Duan, D. Howe, R.J. Patton, Robust fault detection in descriptor linear systems via generalized unknown input observers, Int. J. Syst. Sci. 33 (2010) 369–377. [31] W. Wang, Y. Zou, Analysis of impulsive modes and Luenberger observers for descriptor systems, Syst. Control Lett. 44 (2001) 347–353. [32] A.G. Wu, G.R. Duan, Generalized PI observer design for linear systems, IMA J. Math. Control Inf. 25 (2008) 239–250. [33] M.M. Share, M. Aliyari, H.D. Taghirad, Unknown input-proportional integral observer for singular systems: application to fault detection, in: 18th Iranian Conference on Electrical Engineering, Isfahan, Iran, 2010, pp. 662–667. [34] D. Koenig, S. Mammar, Design of proportional-integral observer for unknown input descriptor systems, IEEE Trans. Autom. Control 47 (2002) 2057–2062. [35] M. Darouach, M. Zasadzinski, M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs, IEEE Trans. Autom. Control (41) (1996) 1068–1072. [36] M. Darouach, L. Botat-Baddas, M. Zerrougui, H∞ observers design for a class of nonlinear singular systems, Automatica 47 (2011) 2517–2525. [37] D.S. Bernstein, Matrix Mathematics. Theory, Facts and Formulas, Princeton University Press, Princeton, NJ, 2009. [38] R.E. Skelton, T. Iwasaki, K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, London, 1998.

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C H A P T E R

2 Adaptive Observer Design for Nonlinear Interconnected Systems With Applications Mokhtar Mohamed*, Xing-Gang Yan*, Zehui Mao † , Bin Jiang † , Abdulaziz Sherif ‡ *Instrumentation,

Control and Embedded Systems Research Group, School of Engineering and Digital Arts, University of Kent, Canterbury, United Kingdom † College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China ‡ Electrical Engineering Department, University of Tripoli, Tripoli, Libya

1 INTRODUCTION The advancement of modern technologies has produced many complex systems. An important class of such systems, which is frequently called a system of systems, or large-scale system, can be expressed by sets of lower-order ordinary differential equations that are linked through interconnections. Such models are typically called large-scale interconnected systems (see, e.g., [1–4]). In order to achieve the desired performance of the closed-loop system in the presence of uncertainties, robust control methods are needed. In recent decades, much of the literature has focused on designing advanced robust controllers for such systems using H∞ control [5], backstepping techniques [6], robust adaptive control [7], and sliding mode control [8–10]. Increasing requirements for system performance have resulted in increasing complexity within systems modeling, and thus, it becomes interesting to consider nonlinear, large-scale interconnected systems. Such models are then used for controller design. In order to obtain the required levels of performance from the controllers, it is desirable to

New Trends in Observer-based Control https://doi.org/10.1016/B978-0-12-817038-0.00002-0

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© 2019 Elsevier Inc. All rights reserved.

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2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

have knowledge of all the system states for use by the control scheme. This state information may be difficult or expensive to obtain, therefore, it would be advantageous to design an observer to estimate all the system states using only the subset of information available from the measured and known inputs and outputs of the system. If the uncertainties in the system are in the form of unknown parameters, then adaptive techniques can be applied such that these unknown parameters are estimated by designing adaptive observers, which is more challenging [11]. This motivates efforts to use an adaptive scheme to design adaptive observers that simultaneously estimate the unknown parameters, and the unavailable states of a dynamical system.

1.1 Interconnected Systems Large-scale interconnected systems have been studied since the 1960s (see [12] and references therein) due to their relevance in a number of practical application areas, and the availability of pertinent theoretical results. Large-scale interconnected systems widely exist in the real world, for example, power networks, ecological systems, transportation networks, biological systems, and information technology networks [2, 13]. A largescale system is composed of several subsystems with interconnections, whereby the dynamics interact [14]. The application of centralized control [15] to prescribe stability of an interconnected system, particularly when the system is spread over a wide geographical area, may require additional costs for implementation, and careful consideration of the required information sharing between subsystems. This motivates consideration of the design of decentralized control strategies, whereby each subsystem has a local control based only upon locally available information. Early work focused on linear systems [16, 17]. However, due to the uncertainties and disturbances present in large-scale interconnected systems, study of the stability of such systems is a very challenging task [18]. Subsequent results used decentralized control frameworks for nonlinear large-scale interconnected systems. The study of such decentralized controllers has stimulated a great deal of literature (e.g., [19–21]), and recently, [22, 23]. In much of this work, however, it is assumed that all the system state variables and the system parameters are available for use by the controller [1, 2, 24, 25]. However, this assumption can limit practical application, as usually only a subset of state variables may be available/measurable [26]. Moreover, many practical systems have unknown parameters. It becomes of interest to establish adaptive observers to estimate the system states and the system parameters simultaneously. It should also be noted that such adaptive observer design has been applied for fault detection and isolation [26–28]. This further motivates the study of adaptive observer’s design for nonlinear large-scale interconnected systems.

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INTRODUCTION

49

1.2 Adaptive Observer The concept of an observer was first introduced by Luenberger [29], in which the difference between the output measurements from the actual plant and the output measurements of a corresponding dynamical model were used to develop an injection signal to force the resulting output error to zero. In the 1970s, the problem of designing observers for estimating system states for large-scale interconnected systems was addressed in [16]. Subsequently, many methods have been developed to design observers for large-scale interconnected linear systems [30–33]. In the real world, many practical control systems involve unknown parameters due to the mechanical wear and modeling errors. Therefore, adaptive observers have been developed to estimate the unavailable states and the unknown parameters simultaneously. Over the past few decades, much literature has been devoted to the design of adaptive observers for linear and nonlinear systems. The early results are mainly for linear systems [34, 35]. In the case of nonlinear systems with unknown parameters, many adaptive observers have been developed (see e.g., [36–38]). Adaptive observers for nonlinear systems have been published in [36], based on the fact that the nonlinear systems can be transformed to a particular observable canonical form. The authors in [37, 38] proposed adaptive observers for nonlinear systems that can be transformable by a global state space transformation to other coordinates, with some extra constraints and conditions imposed on the system. These proposed adaptive observers have been extended in [39, 40] to deal with a general class of nonlinear systems. However, the convergence of the parameters’ estimation errors depends on persistence of the excitation condition. More recently, adaptive observers using different techniques have been proposed in, for example, [41–43], where the unknown parameters are limited to be constant. Compared with much existing work in adaptive observer design with unknown constant parameters, the corresponding observation results for unknown time varying parameters (TVPs) are very limited. The authors in [44] proposed a sampled output, high-gain observer for a class of uniformly observable nonlinear systems in which the unknown parameters are bounded. An adaptive estimator is proposed in [45] to estimate TVPs for nonlinear systems. However, all the system states are assumed to be available. The H− /H∞ fault detection observer in the finite frequency domain has been designed in [46] for a class of linearparameter, varying descriptor systems. Boizot et al. [47] developed an adaptive observer by using an extended Kalman filter to reduce the effect of perturbations. However, in terms of the parameter estimation for nonlinear systems, it is usually very difficult to analyze the stability of the extended Kalman filter. It should be noted that unknown parameters considered in these papers are constant. An adaptive redesign of reduced order nonlinear observers is presented in

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2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

[48], in which the solution of a partial differential equation is required, which may not be possible in most cases. In order to improve the quality of the current drawn from the utility grid, an adaptive nonlinear observer is designed in [49] to estimate the inductor current, which is required in the closed-loop control system of power factor correction as an essential part of AC/DC converters. An adaptive observer is designed for a class of MIMO uniformly observable nonlinear systems with linear and nonlinear parameterizations in [50], and the exponential convergence of the error dynamics for both types of parameterization are guaranteed under the persistent excitation condition. Tyukin et al. [51] considered the problem of asymptotic reconstruction of the state and parameter. However, in both [50, 51], it is required that the unknown parameters are constant. The literature in [52] proposed an adaptive state estimator for a class of multiinput and multioutput nonlinear systems with uncertainties in the state and the output equations, in which the systems considered are not interconnected systems. The work in [53] proposed an adaptive observer that expands the extended state observer to nonlinear disturbed systems. However, the adaptive extended state observer is linear, and requires that the error dynamics can be transformed into a canonical form. An adaptive observer applying sliding mode techniques has been developed in [54] to enhance the performance of the adaptive observer proposed by Yan and Edwards [55]. Adaptive sliding mode, observer-based fault reconstruction for nonlinear systems with parametric uncertainties is considered in [56]. However, the unknown parameters considered in these papers are constant. Many adaptive observers have been developed using sliding mode techniques for particular applications and for particular purposes (see e.g., [57–60]); and thus, corresponding specific conditions need to be imposed on the systems considered. Sliding mode techniques with super twisting algorithms are used in [61] to design adaptive observers for nonlinear systems in which the unknown parameter vector is assumed to be constant.

1.3 Contribution In this chapter, observers are designed for a class of nonlinear interconnected systems with uncertain TVPs, in which both the isolated subsystems and the interconnections are nonlinear. The designed observers are variable structure interconnected systems, but may not result in sliding motion. Under the condition that the difference between the unknown TVPs and the corresponding uncertain nominal values are bounded by constants, adaptive updating laws are proposed to estimate the parameters. The persistence of excitation conditions is not required. A set of sufficient conditions are proposed such that the error dynamics formed by the system states and the designed observers are asymptotically stable, while the

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SYSTEM DESCRIPTION AND PROBLEM FORMULATION

51

parameters’ estimation errors are uniformly, ultimately bounded using LaSalle’s theorem. The results obtained are applied to a coupled inverted pendulum system, and simulation results are presented to demonstrate the effectiveness and feasibility of the developed results. The main contribution includes: (i) Both the interconnections and isolated subsystems take nonlinear forms. (ii) The unknown parameters considered in the system are time varying, and the corresponding nominal values are not required to be known. (iii) The asymptotic convergence of the observation error between the states of the considered systems and the states of the designed observers is guaranteed; while the estimate errors of the TVPs are uniformly, ultimately bounded.

1.4 Notation For a square matrix A, A > 0 denotes a symmetric positive definite matrix, and λmin (A)(λmax (A)) denotes the minimum (maximum) eigenvalues of A. The symbol In represents the nth-order unit matrix and R+ represent the set of nonnegative real numbers. The set of n×m real matrices will be denoted by Rn×m . The Lipschitz constant of the function f will be written as f . Finally, · denotes the Euclidean norm, or its induced norm.

2 SYSTEM DESCRIPTION AND PROBLEM FORMULATION Consider a nonlinear interconnected system composed of N subsystems described as follows x˙ i = Ai xi + fi (xi , ui ) + Bi θi (t)ξi (t) +

N 

Hij (xj )

(2.1)

j=1

j=i

yi = Ci xi

(2.2)

where xi ∈ Rni , ui ∈ Ui ∈ Rmi (Ui is the admissible control set), and yi ∈ R are the state variables, inputs, and outputs of the ith subsystem, respectively. The functions fi (·) are known to be continuous, the scalars θi (t) ∈ R are unknown TVPs, and ξi (t) ∈ R are known regressor signals. The matrices Ai ∈ Rni ×ni , Bi ∈ Rni ×1 , and Ci ∈ R1×ni are constants, and Ci is full column rank. The terms N 

Hij (xj )

j=1

j=i

are the known interconnections of the ith subsystems for i = 1, . . . , N.

I. OBSERVER DESIGN

52

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

Assumption 1. The matrix pairs (Ai , Ci ) are observable for i = 1, . . . , N. From Assumption 1, there exist matrices Li such that Ai − Li Ci are Hurwitz stable. This implies that, for any positive-definite matrices Qi ∈ Rni ×ni , the Lyapunov equations (Ai − Li Ci )T Pi + Pi (Ai − Li Ci ) = −Qi

(2.3)

have unique positive-definite solutions Pi ∈ Rni ×ni . Assumption 2. There exist matrices Fi ∈ R such that solutions Pi to the Lyapunov equations (2.3) satisfy the constraints BTi Pi = Fi Ci ,

i = 1, . . . , N

(2.4)

Remark 1. To solve the Lyapunov equations (2.3) in the presence of the constraints, Eq. (2.4) is the well-known constrained Lyapunov problem (CLP) [62]. Although there is no general solution available for this problem, associated discussion, and an algorithm, can be found in [63], which may help to solve the CLP for a specific system. Assumption 3. The uncertain TVPs θi (t) satisfy |θi (t) − θ0i | ≤ 0i

(2.5)

where θ0i are unknown constants, and 0i are known constants for i = 1, . . . , N. Remark 2. Assumption 3 is to specify a class of uncertainties tolerated in the observer design. The unknown constants θ0i given in Eq. (2.5) are called the nominal value of the uncertain TVPs θi (t) throughout this chapter. Different from the existing work (see e.g., [64, 65]), the unknown parameters θi (t) are time varying, and the nominal values θ0i are not required to be known. For further analysis, the terms Bi θi (t)ξi (t) in system (2.1) are rewritten as Bi θi (t)ξi (t) = Bi [θ0i + i (t)]ξi (t)

(2.6)

where the scalers i (t) = θi (t) − θ0i . Assumption 4. The nonlinear terms fi (xi , ui ), with respect to xi ∈ Rni , for ui ∈ Ui ∈ Rmi for i = 1, 2, . . . , N and Hij (xj ) satisfy the Lipschitz condition. Assumption 4 implies that there exists nonnegative function fi and constant Hij such that fi (ˆxi , ui ) − fi (xi , ui ) ≤ fi (ui )ˆxi − xi 

(2.7)

Hij (ˆxj ) − Hij (xj ) ≤ Hij ˆxj − xj 

(2.8)

for i = 1, 2, . . . , N and i = j. Remark 3. Assumption 4 is the limitation to the nonlinear terms, and the interconnections that are necessary to achieve the asymptotic stability of the observation error dynamics. It should be noted that in Assumption 4, it is required that fi (xi , ui ) satisfies Lipschitz condition, with respect to only the variable xi . I. OBSERVER DESIGN

53

ADAPTIVE OBSERVER DESIGN WITH PARAMETERS ESTIMATION

For nonlinear interconnected system (2.1)–(2.2) satisfying Assumptions 1–4, the objective of this chapter is to design an observer with appropriate adaptive laws such that the states of the system (2.1)–(2.2) can be estimated asymptotically, and the estimation errors of the unknown parameters θi (t) in Eq. (2.1) are uniformly bounded.

3 ADAPTIVE OBSERVER DESIGN WITH PARAMETERS ESTIMATION In this section, an asymptotic observer is designed, and the proposed adaptive laws are presented. From Eq. (2.6), system (2.1) can be rewritten as x˙ i = Ai xi + fi (xi , ui ) + Bi [θ0i + i (t)]ξi (t) +

N 

Hij (xj )

(2.9)

j=1

j=i

yi = Ci xi

(2.10)

For systems (2.9)–(2.10), construct dynamical systems T x˙ˆ i = Ai xˆ i + fi (ˆxi , ui ) + Li (yi − yˆ i ) + Bi θˆi (t)ξi (t) − 2P−1 i (Fi Ci ) |ξi (t)|0i

× ψi (ˆyi , yi ) − Bi ˆi (t)ξi (t) +

N 

Hij (ˆxj )

(2.11)

j=1

j=i

yˆ i = Ci xˆ i

(2.12)

where Pi and Ci satisfy Eqs. (2.3), (2.4),  ψi (ˆyi , yi ) =

Fi (ˆyi −yi ) , Fi (ˆyi −yi )

0,

Fi (ˆyi − yi ) = 0 Fi (ˆyi − yi ) = 0

(2.13)

for i = 1, 2, . . . , N, and θˆi (t) is given by the adaptive law as follows θ˙ˆi (t) = −2δi (Fi (ˆyi − yi ))T ξi (t)

(2.14)

where δi is a positive constant that is a design parameter, the known constant 0i satisfies the inequality in Assumption 3, and ˆi (t) is defined by 1 ˆi (t) = − θˆi (t) δi

(2.15)

for i = 1, 2, . . . , N. Let exi = xˆ i − xi . Then, from systems (2.9), (2.10) and (2.11), (2.12), the error dynamical systems can be described by

I. OBSERVER DESIGN

54

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

e˙xi = (Ai − Li Ci )exi + [fi (ˆxi , ui ) − fi (xi , ui )] +

N  [Hij (ˆxj ) − Hij (xj )] + Bi θ˜i (t)ξi (t) j=1

j=i T − Bi ˆi (t)ξi (t) − Bi i (t)ξi (t) − 2P−1 yi , yi ) i (Fi Ci ) |ξi (t)|0i ψi (ˆ

(2.16)

where θ˜i (t) is defined by θ˜i (t) = θˆi (t) − θ0i

(2.17)

for i = 1, 2, . . . , N. For the convenience of further analysis, let ˜i (t) = ˆi (t) − 0i

(2.18)

where the known constant 0i satisfies the inequality in Assumption 3 and ˆi (t) is defined in Eq. (2.15), for i = 1, 2, . . . , N.

4 STABILITY OF THE ERROR DYNAMICAL SYSTEMS The following result is ready to be presented: Theorem 1. Under Assumptions 1–4, the error dynamical systems (2.16) with adaptive law (2.14) are uniformly ultimately bounded if the matrix W T + W is positive definite, where the matrix W = [wij ]N×N and its entries wij are defined by  λmin (Qi ) − 2fi Pi , i = j wij = (2.19) i = j −2Pi Hij , where Pi and Qi satisfy Lyapunov equation in Eq. (2.3) and λmin (Qi ) represents the minimum eigenvalue of the matrix Qi for i = 1, 2, . . . , N. Further, the errors exi given in Eq. (2.16) satisfy lim exi (t) = 0,

t→∞

i = 1, 2, . . . , N

(2.20)

Proof . For systems (2.14), (2.16), consider the candidate Lyapunov function  N N   1 1 2 T 2 ˜ exi Pi exi + (2.21) V= θ (t) + ˜i (t) 2 δi i i=1

i=1

where δi > 0 is a design parameters given in Eq. (2.14) for i = 1, 2, . . . , N. Note that, in Eq. (2.21), ˜i (t) is dependent on θ˜i (t). From Eqs. (2.15), (2.17), (2.18), it can be seen that the relationship between ˜i (t) and θ˜i (t) is given by

I. OBSERVER DESIGN

55

STABILITY OF THE ERROR DYNAMICAL SYSTEMS

˜i (t) = ˆi (t) − 0i 1 = − θˆi (t) − 0i δi 1 = − (θ˜i (t) + θ0i ) − 0i δi Then, from Eq. (2.16)  N N    1 ˙ T T ˙ ˜ ˜ (˙exi Pi exi + exi Pi e˙xi ) + θi (t)θi (t) + ˜i (t)˜i (t) δi i=1 i=1 ⎧ ⎪ N ⎪ ⎨  = eTxi [(Ai − Li Ci )T Pi + Pi (Ai − Li Ci )]exi + 2eTxi Pi [fi (ˆxi , ui ) − fi (xi , ui )] ⎪ ⎪ i=1 ⎩

˙ = V

+ 2eTxi Pi

N  [Hij (ˆxj ) − Hij (xj )] + 2eTxi Pi Bi θ˜i (t)ξi (t) − 2eTxi Pi Bi i (t)ξi (t) j=1

j=i

1 θ˜i (t)θ˙˜i (t) + ˜i (t)˙˜i (t) δi ⎫ ⎪ ⎪ ⎬ −1 T T − 4exi Pi Pi (Fi Ci ) |ξi (t)|0i ψi (ˆyi , yi ) ⎪ ⎪ ⎭

− 2eTxi Pi Bi ˆi (t)ξi (t) +

(2.22)

By using condition (2.4) and Ci exi = yˆ i − yi , eTxi Pi Bi = ((Pi Bi )T exi )T = (BTi Pi exi )T = (Fi Ci exi )T = (Fi (ˆyi − yi ))T

(2.23)

Substituting Eq. (2.23) into Eq. (2.22), it follows that ⎧ ⎪ N ⎪ ⎨  ˙ = V eTxi [(Ai − Li Ci )T Pi + Pi (Ai − Li Ci )]exi + 2eTxi Pi [fi (ˆxi , ui ) − fi (xi , ui )] ⎪ ⎪ i=1 ⎩ + 2eTxi Pi

  N  1 [Hij (ˆxj ) − Hij (xj )] + 2(Fi (ˆyi − yi ))T ξi (t) + θ˙˜i (t) θ˜i (t) δi j=1

j=i

− 2(Fi (ˆyi − yi ))T i (t)ξi (t) − 2(Fi (ˆyi − yi ))T ˆi (t)ξi (t) ⎫ ⎪ ⎪ ⎬ T ˙ + ˜i (t)˜i (t) − 4(Fi (ˆyi − yi )) |ξi (t)|0i ψi (ˆyi , yi ) ⎪ ⎪ ⎭ I. OBSERVER DESIGN

(2.24)

56

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

From Eq. (2.17), it can be seen that θ˙˜i (t) = θ˙ˆi (t) because θ0i is constant. By substituting Eqs. (2.13), (2.14) into Eq. (2.24) gives ⎧ ⎪ ⎪ N ⎨ ˙ = eTxi [(Ai − Li Ci )T Pi + Pi (Ai − Li Ci )]exi + 2eTxi Pi [fi (ˆxi , ui ) − fi (xi , ui )] V ⎪ ⎩ i=1 ⎪ + 2eTxi Pi

N 

[Hij (ˆxj ) − Hij (xj )] − 2(Fi (ˆyi − yi ))T i (t)ξi (t)

j=1

j=i

− 2(Fi (ˆyi − yi ))T ˆi (t)ξi (t) + ˜i (t)˙˜i (t) − 4Fi (ˆyi − yi ) |ξi (t)|0i

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

From Eq. (2.18), it can be seen that ˙˜i (t) = ˙ˆi (t). ⎧ ⎪ N ⎨ ⎪ ˙ = V eTxi [(Ai − Li Ci )T Pi + Pi (Ai − Li Ci )]exi + 2eTxi Pi [fi (ˆxi , ui ) − fi (xi , ui )] ⎪ ⎩ i=1 ⎪ + 2eTxi Pi

N 

[Hij (ˆxj ) − Hij (xj )] − 2(Fi (ˆyi − yi ))T i (t)ξi (t)

j=1

j=i

− [2(Fi (ˆyi − yi ))T ξi (t) − ˙˜i (t)]ˆi (t) − 0i ˙˜i (t) − 4Fi (ˆyi − yi ) |ξi (t)|0i

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(2.25) Substituting Eq. (2.15) into Eq. (2.25) yields ⎧ ⎪ ⎪ N ⎨ ˙ = eTxi [(Ai − Li Ci )T Pi + Pi (Ai − Li Ci )]exi + 2eTxi Pi [fi (ˆxi , ui ) − fi (xi , ui )] V ⎪ ⎩ i=1 ⎪ + 2eTxi Pi

N 

[Hij (ˆxj ) − Hij (xj )] − 2(Fi (ˆyi − yi ))T i (t)ξi (t)

j=1

j=i

− 20i (Fi (ˆyi − yi ))T ξi (t) − 4Fi (ˆyi − yi ) ξi (t)0i

I. OBSERVER DESIGN

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

57

STABILITY OF THE ERROR DYNAMICAL SYSTEMS

It is clear from Eq. (2.3) that ˙ ≤ V

N  

−eTxi Qi exi + 2exi  Pi [fi (ˆxi , ui ) − fi (xi , ui )]

i=1

+ 2exi  Pi 

N 

[Hij (ˆxj ) − Hij (xj )]

j=1

j=i

−2(Fi (ˆyi − yi ))T ξi (t)[i (t) + 0i ] − 4Fi (ˆyi − yi ) ξi (t)0i

≤

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

⎧ ⎪ N ⎪ ⎨ 

−eTxi Qi exi + 2exi  Pi [fi (ˆxi , ui ) − fi (xi , ui )] + 2exi  Pi 

i=1

⎪ ⎪ ⎩

j=1

j=i

−Hij (xj )] + 4Fi (ˆyi − yi ) ξi (t)0i − 4Fi (ˆyi − yi ) ξi (t)0i

≤

⎧ ⎪ N ⎪ ⎨ 

−eTxi Qi exi + 2exi  Pi [fi ˆxi − xi ] + 2exi  Pi 

i=1

≤−

N  [Hij (ˆxj )

⎪ ⎪ ⎩

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

N  [Hij ˆxj − xj ] j=1

j=i

⎧ ⎪ N ⎨ ⎪

N 

⎪ ⎩ i=1 ⎪

j=1

(λmin (Qi ) − 2Pi fi )exi 2 −

⎪ ⎪ ⎭

⎫ ⎪ ⎪ ⎬

(2Pi Hij exi exj )

j=i

⎫ ⎪ ⎪ ⎬

⎪ ⎪ ⎭

(2.26)

Then, from the definition of the matrix W in Eq. (2.19) and the preceding inequality, it follows that ˙ ≤ − 1 XT [W T + W]X V 2

(2.27)

where X = [ex1 , ex2 , . . . , exN ]T . From the LaSalle’s theorem (see e.g., [66]), all the solutions of Eq. (2.16) are globally, uniformly bounded and satisfy lim XT [W T + W]X = 0

t→∞

Further, from the facts λmin (W T + W)X2 ≤ XT (W T + W)X

I. OBSERVER DESIGN

(2.28)

58

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

and X2 = ex1 2 + ex2 2 + · · · + exN 2 it is straightforward to see from Eq. (2.28) and the condition W T + W > 0 that lim exi (t) = 0,

t→∞

i = 1, 2, . . . , N

Hence the conclusion follows. Remark 4. It should be noted that the constructed Lyapunov function (2.21) is a function of variables exi , θ˜i , and ˜i while the right-hand side of inequality (2.27) is a function of variables exi only. Therefore, Theorem 1 ˙ is semipositive definite instead of positive definite. implies that V Remark 5. Theorem 1 shows that the augmented systems formed by Eq. (2.16) and the adaptive law (2.14) are uniformly ultimately bounded. It should be noted that the estimated states xˆ i given by the observer (2.11) converge to the system states xi in Eq. (2.1) asymptotically, although the estimate error for the parameters may not be asymptotically convergent. As the uncertain parameters θi in system (2.1) are time-varying, the approaches developed in [28, 65] cannot be applied to the systems considered in this chapter. Remark 6. The designed observer is a variable structure interconnected system, but it may not produce a sliding motion, which is different from the work in [65]. In addition, the unknown parameters are considered constants in [65], while in this chapter they are TVPs.

5 CASE STUDY EXAMPLES In order to illustrate the method developed in this chapter, case study examples on a coupled pendulum system and a quarter-car suspension are carried out in this section.

5.1 A Coupled Inverted Pendulum Consider a system formed by two inverted pendulums connected by a spring, as given in Fig. 2.1. There are two balls that are attached at the end of two rigid rods, respectively. The symbols u1 and u2 denote external torques imposed on the two pendulums, respectively, which are the control inputs. The distance b between the two pendulums is assumed to be changeable, with respect to time t. Let ϕ1 = x11 , ϕ2 = x21 , ϕ˙1 = x12 , and ϕ˙2 = x22 . The coupled inverted pendulums can be modeled as (see e.g., [66, 67])

I. OBSERVER DESIGN

59

CASE STUDY EXAMPLES

m1

Spring (interconnection)

m2

ϕ2

ϕ1

r

u2

u1 b

FIG. 2.1 Coupled inverted pendulums.

   0 1 x11 0  = + m1 gr kr2 0 0 x12 J1 − 4J1 sin(x11 ) +     0 0 + kr (l − b) + kr2 2J1 4J1 sin(x21 )     x11 = 1 1 x12     0 1 x21 0  = + m2 gr kr2 0 0 x22 J2 − 4J2 sin(x21 ) +     0 0 + kr (l − b) + kr2 2J2 4J2 sin(x11 )     x21 = 1 1 x22





x˙ 1

y1 x˙ 2

y2

1 J1 u1

(2.29) (2.30)  1 J2 u2

(2.31) (2.32)

The end masses of pendulums are m1 = 0.7 kg and m2 = 0.6 kg, the moments of inertia are J1 = 5 kg and J2 = 4 kg, the constant of connecting spring is k = 90 N/m, the pendulum height is r = 0.25 m, and the gravitational acceleration is g = 9.81 m/s2 . In order to illustrate the developed theoretical results, it is assumed that (l − b(t)) = θ1 (t) = θ2 (t) is an unknown TVP for i = 1, 2; where l is the natural length of the spring, and b(t) is the distance between the two pendulum hinges. The aim is to estimate the unknown TVP (l − b(t)) = θ1 (t) = θ2 (t) for i = 1, 2, where l is the natural length of the spring, and b(t) is the distance between the two pendulum hinges. In order to avoid system states going to infinity, and for simulation purposes, the following feedback transformation is introduced

I. OBSERVER DESIGN

60

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

ui = −ki xi + vi ,   k1 = 10 15   k2 = 8 12

i = 1, 2

(2.33) (2.34) (2.35)

Then, with the given preceding parameters, the systems (2.29)–(2.32) can be rewritten as      0 0 1 x11 x˙ 1 = + −2 −3 x12 0.06215 sin(x11 ) + 15 v1       A1

f1 (x1 ,u1 )

    0 0 + (l − b(t)) + 3    0.2813 sin(x21 )   θ1 (t)   B1

H12 (x2 )

   x11 y1 = 1 1    x12 



(2.36)

(2.37)

C1

    0 0 1 x21 + x˙ 2 = −2 −3 x22 0.01632 sin(x21 ) + 14 v2       A2

f2 (x2 ,u2 )

 0 0 + (l − b(t)) + 3.75    0.352 sin(x11 )     θ2 (t)   





B2

(2.38)

H21 (x1 )

   x21 y2 = 1 1    x22 

(2.39)

C2

Choose

 Li = 0

 0

and

Qi = 4I

for i = 1, 2. It follows that the Lyapunov equations (2.3) have unique solutions:   5 1 , i = 1, 2 (2.40) Pi = 1 1 satisfying the condition (2.4) with F1 = 3,

and

F2 = 3.75

For simplicity, it is assumed that ξi (t) = 1,

0i = 1,

and

for i = 1, 2.

I. OBSERVER DESIGN

δi = 2

61

CASE STUDY EXAMPLES

By direct computation, it follows that the matrix W T + W is positive definite. Thus, all the conditions of Theorem 1 are satisfied. This implies that the following dynamical systems are the asymptotic observers of the nonlinear interconnected systems (2.36)–(2.39):      0 1 xˆ 11 ˙xˆ 1 = 0 + −2 −3 xˆ 12 0.06215 sin(ˆx11 ) + 15 v1     0 (ˆy1 − y1 ) 0 ˆ + θ (t) − 0.4 ˆy1 − y1  3 1     0 0 (2.41) − ˆ1 (t) + 0.2813 sin(ˆx21 ) 3     xˆ 11 yˆ 1 = 1 1 (2.42) xˆ 12        0 0 1 xˆ 21 0 x˙ˆ 2 = + + θˆ (t) −2 −3 xˆ 22 3.75 2 0.01632 sin(ˆx21 ) + 14 v2       0 0 0 (ˆy2 − y2 ) (2.43) − ˆ2 (t) + − 0.352 sin(ˆx11 ) 3.75 0.5 ˆy2 − y2      xˆ 21 yˆ 2 = 1 1 (2.44) xˆ 22 The designed adaptive laws are given by θ˙ˆ1 (t) = −4(2.25(ˆy1 − y1 ))T θ˙ˆ (t) = −4(2.8125(ˆy − y ))T 2

2

2

(2.45) (2.46)

For simulation purposes, the unknown parameters θ0i and θi (t) are chosen as 0 and 0.6 sin t, respectively, for i = 1, 2. Simulation in Figs. 2.2 and 2.3 shows that the estimation error between the states of the system (2.29)–(2.32) and the states of the observer (2.41)–(2.44) converges to zero asymptotically. Fig. 2.4 shows that the estimation of the parameters is uniformly bounded with satisfactory accuracy.

5.2 A Quarter-Car Suspension Consider a vehicle (car, bus, etc.) divided into four parts. In the case of the four wheels, each part is a composite mechanical spring-damper system consisting of the quarter part of the mass of the body (together with passengers), and the mass of the wheel. The vertical positions are described by upward directed x1 and x2 , see Fig. 2.5. The distance between the road surface and the wheel’s contact point is the disturbance w, varying

I. OBSERVER DESIGN

62

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

FIG. 2.2 The time response of the first subsystem states x1 = col(x11 , x12 ) and their estimation xˆ 1 = col(ˆx11 , xˆ 12 ).

together with the road surface. The suspension is active, which means that the actuator produces the force F (control signal). A good suspension system should have satisfactory road holding stability, while providing good traveling comfort when riding over bumps and holes in the roads.

I. OBSERVER DESIGN

CASE STUDY EXAMPLES

63

FIG. 2.3 The time response of the second subsystem states x2 = col(x21 , x22 ) and their estimation xˆ 2 = col(ˆx21 , xˆ 22 ).

Denote m1 and m2 as the mass of the quarter body and the wheel, respectively, while the flexible connections are described by the viscous damping factor b1 and the spring constants (k1 , k2 ). The motion equations can be described as follows (see e.g., [68]): m1 x¨ 1 = F − b1 (˙x1 − x˙ 2 ) − k1 (x1 − x2 ) m2 x¨ 2 = −F + b1 (˙x1 − x˙ 2 ) + k1 (x1 − x2 ) − k2 (x2 − w)        0 1 0 0 x11 x˙ 1 = −k1 −b1 + 1 + 1 x12 m1 F m1 (k1 x21 + b1 x22 ) m m 1

1

I. OBSERVER DESIGN

(2.47) (2.48) (2.49)

64

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

FIG. 2.4 Upper: The time response of θˆ1 (t) (dashed line) and θ1 (t) (solid line); bottom: the time response of θˆ2 (t) (dashed line) and θ2 (t) (solid line).

I. OBSERVER DESIGN

65

CASE STUDY EXAMPLES

FIG. 2.5 A quarter-car suspension.

    x11 y1 = 1 1 x12     0 1 0 x21 x˙ 2 = −k2 −b1 + −k1 x22 m2 m2 m2 x21 −   0 + 1 m2 (k1 x11 + b1 x12 )     x21 y2 = 1 1 x22

(2.50)  1 m2 F

 +

0 k2 m2

 w (2.51) (2.52)

where m1 = 500 kg, m2 = 300 kg, b1 = 900 N/m/s, k1 = 900 N/m, and k2 = 600 N/m. In order to avoid system states going to infinity, and for simulation purposes, the following feedback transformation is introduced ui = −ki xi   ki = 3.18 4.18 ,

(2.53) i = 1, 2

(2.54)

Then, with the given parameters, the systems (2.49)–(2.52) can be rewritten as        0 1 0 x11 0 + (2.55) + x˙ 1 = −1.8 −1.8 x12 1.8(x21 + x22 ) 0.002 u          A1

   x11 y1 = 3 3    x12

f1 (x1 ,u1 )

H12 (x2 )



(2.56)

C1

I. OBSERVER DESIGN

66

2. NONLINEAR INTERCONNECTED SYSTEMS WITH APPLICATIONS

        0 1 0 0 x21 0 + w + + −2 −3 x22 2  3(x11 + x12 ) −3x21 − 0.0033 u        θ(t)    

x˙ 2 =

A2

B2

f2 (x2 ,u2 )

H21 (x1 )

(2.57)

    x21 y2 = 1.8 1.8    x22

(2.58)

C2

In order to illustrate the developed theoretical results, it is assumed that all the system states are available, and the aim is to estimate the distance between the road surface and the wheel’s contact point w, which is varying together with the road surface. Choose   L= 0 0 and Q = 4I It follows that the Lyapunov equations (2.3) have unique solutions:   5 1 P= (2.59) 1 1 satisfying the condition (2.4) with F = 0.667 For simplicity, it is assumed that ξ(t) = 1,

0 = 1,

and

δ=5

The following dynamical systems are the observer of the nonlinear interconnected system (2.55)–(2.58):        0 1 xˆ 11 0 0 ˙xˆ 1 = (2.60) + + −1.8 −1.8 xˆ 12 0.002 u 1.8(ˆx21 + xˆ 22 )     xˆ 11 yˆ 1 = 3 3 (2.61) xˆ 12          0 ˆ 0 (ˆy2 − y2 ) 0 1 xˆ 21 0 + θ (t) − + x˙ˆ 2 = 2 1.2 ˆy2 − y2  −2 −3 xˆ 22 −3ˆx21 − 0.0033 u     0 0 (2.62) − ˆ (t) + 3(ˆx11 + xˆ 12 ) 2 2     xˆ 21 yˆ 2 = 1.8 1.8 (2.63) xˆ 22 The designed adaptive law is given by θ˙ˆ (t) = −10(0.667(ˆy2 − y2 ))T

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(2.64)

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ˆ (dashed line) and θ (t) (solid line). FIG. 2.6 The time response of θ(t)

For simulation purposes, the unknown parameters θ0 and θ (t) are chosen as 0 and 0.5 sin t, respectively. Fig. 2.6 shows that the estimation of the parameter is uniformly, ultimately bounded with satisfactory accuracy. Remark 7. For a real system, the positions and/or the velocities are usually chosen as system outputs. However, sometimes, the linear combination of the position and velocity are taken as system outputs. Physically, such an aggregation of the output might arise in some real systems [69, 70], for example, certain remote-control applications where the number of transmission and receive lines/frequencies are limited [69].

6 CONCLUSION In this chapter, an adaptive observer design for a class of nonlinear, large-scale interconnected systems with unknown TVPs has been proposed based on the Lyapunov direct method. The unknown parameters vary within a given range. A set of sufficient conditions has been developed to guarantee that the observation error system, with the proposed adaptive laws, is globally, uniformly bounded. The states of the designed observer are asymptotically convergent to the original system states. Therefore, from the state estimation point of view, the designed observers are asymptotic observers. Case study examples on a coupled inverted pendulum system and a quarter-car suspension show the practicability of the developed observer scheme for nonlinear interconnected systems.

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Acknowledgments The authors gratefully acknowledge the support of the National Natural Science Foundation of China (61573180) for this work.

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C H A P T E R

3 On the Observability and Observer Design in Switched Linear Systems David Gómez-Gutiérrez*, Carlos Renato Vázquez*, Antonio Ramírez-Teviño † , Stefano Di Gennaro ‡ *Tecnologico de Monterrey, School of Engineering and Science, Jalisco, Mexico † CINVESTAV, Jalisco, Mexico ‡ University of L’Aquila, Department of Information Engineering, Computer Science and Mathematics, Center of Excellence DEWS, L’Aquila, Italy

1 INTRODUCTION Hybrid dynamical systems (HS) are formed by a collection of continuous models switching among them, according to a discrete rule. This class of models captures the continuous and discrete interaction that appears in complex systems, such as biological systems [1], automotive systems [2], networked control systems [3], and fault-tolerant systems [4], among many others. Frequently, when properties on the continuous dynamics of the HS are investigated, a model with a higher level of abstraction, known as a switched system, is considered by assuming an exogenous and unpredictable switching signal, dropping any information about the rules that describe the behavior of the discrete evolution. The most studied of these models are the switched linear systems (SLS), consisting of a collection of linear systems (LS) and an exogenous switching signal that is determined at each time instant during the evolving LS. SLS are suitable for representing and approximating practical systems with complex dynamics, such as electric circuits with discrete components, continuous systems under a discrete specification, and systems with impacts [5]. Moreover, highly nonlinear behaviors, such as chaos, can be generated by SLS (see, e.g., [6–10]).

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Many contributions have been reported in the literature concerning the analysis of basic system properties of SLS, such as stability, controllability, and observability. With respect to the latter, different observability problems for this kind of system arises depending on the considered assumptions. For example, the dwell time (i.e., the time that an LS remains active), which could be unconstrained or restricted by a fixed minimum/ maximum value, the inputs that could be known or partially unknown (because of noisy inputs, disturbances, etc.), and the switching signal that could be known or unknown. In this chapter, we present an observability analysis for SLS under partially unknown inputs and unknown switching signals, without maximum dwell time. For the known switching signal case, we refer an interested reader to [11, 12] for observability analysis, and to [11, 13] for observer design. The considered setting allows tackling of a broad class of control problems. For example, SLS under these assumptions arises in the fault detection of systems in which the safety is of paramount importance (chemical/nuclear reactors, avionic systems, etc.) and the control reconfiguration, necessary to face such events, needs to act in a short time. In such case, because the SLS is subject to arbitrary and unpredictable switches, it is necessary to ensure the estimation of the SLS state before another switching occurrence, or even if no switching occurs at all. Because the conditions for the observability with unknown inputs of the individual LS have been already established [14], the main problem is then to infer, from the knowledge of the continuous measurements, which LS is evolving. This is usually referred to as the distinguishability problem. This problem, together with the SLS observability, has been extensively studied in the literature for the case of known inputs [15–19]. On the other hand, the observability of continuous-time SLS subject to unknown disturbances and unknown switching signals has been less explored. In the context of unknown switching signals, the concept of observability of SLS becomes more complex than in the LS case. The complexity comes from two main reasons: first, because the autonomous case [15] is not equivalent to the nonautonomous case [16–18], because the input greatly affects the observability. This gives rise to different observability notions, for example, it could be interesting to ensure the computation of the state for every input [18]; on the other hand, it could be desirable to find an input [16] or a generic set of inputs [17] that allow us to infer the state (these two latter cases result in equivalency [17]). A second reason for the complexity appears because in SLS with unknown switching signals, the set of initial conditions for which the state cannot be recovered from the input-output information lies on a subspace. Thus, if this subspace is not the complete state space, then there exist state trajectories for which the evolving system can still be inferred. This gives rise to different

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observability notions: the recovering of the system state (or part of it) for every nonzero state trajectory [15, 18], or for a generic set of state trajectories [17]. Furthermore, under the assumption of an unknown switching signal, the observability in SLS can be studied separately for continuous and discrete states [17]. A geometric characterization of the main results presented in [15–18] for SLS with unknown switching signals and no maximum dwell time is reported in [20]. If, on the contrary, a maximum dwell time is considered, then the SLS state does not need to be recovered before the first switching time, but after a finite number of switches, either by taking advantage of the underlying discrete event system [16] or by investigating the distinguishability between LS [18]. This problem has been considered in [21–23] for autonomous systems, and in [16, 18] for the known inputs case. Similar problems have been considered for the observability analysis of discrete-time SLS, considering the presence of unknown switching signals and unconstrained dwell time, for both cases with known inputs [24], and with unknown but bounded disturbances [25–27]. Nevertheless, as pointed out in [17], continuous- and discrete-time SLS have some notable differences, which require examination of these two classes of systems separately. A different set of observability problems arises when the switching signal is known [28] or observable from the discrete measurements [29]. In this situation, the main problem is to compute the continuous state. If no constraint on a maximum dwell time is given, then this problem reduces to the observability of each LS. On the other hand, if a maximum dwell time is set, then the problem is translated into the recovering of the continuous state after a finite number of switches; in this case, it has been shown that the observability of each LS is not necessary [28]. As far as the observer design is concerned, if the switching signal is unknown, the observer for SLS requires us to estimate the switching signal from the measurements via a location observer (see, e.g., [30]). In [30], the location observer uses a residual generator, similar to those used in fault detection [31], in order to infer a change in the continuous dynamics. In [32], an algorithm for computing the switching signal in real-time has been proposed. This algorithm, however, requires the numerical computation of derivatives of inputs and outputs, and has been only presented for monovariable SLS. An extension of this algorithm, considering structured perturbations (i.e., disturbances with known derivatives), has been presented in [33]. In [34], a super twisting-based step-by-step observer for switched autonomous nonlinear systems that can be transformed into the normal form representation using Lie derivatives (see [35]) has been presented. In this form, the systems differ only for the last dynamic equation, which is identified by the robust observer, and used to estimate the currently evolving system, and simultaneously, the continuous states.

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In [36], the location observer is formed by a set of Luenberger observers with an associated robust differentiator used to obtain the exact error signal that updates the estimate. Using the results on distinguishability for every nonzero state trajectory, the authors show the convergence of the observer. Unfortunately, as discussed before, this distinguishability notion requires the strong observability of the extended LS. The observer design for the following systems has been addressed based on robust finite-time [37] and fixed-time differentiators [38], which allows recovering the continuous and discrete state, as well as the disturbance affecting the system [39–42]. The observability and the observer design when the discrete dynamic is governed by a discrete event system has been addressed in [18, 23]. In the forthcoming discussion, the observability property is analyzed for SLS considering unknown inputs and unknown switching signals. Here, no constraints are considered on the minimum/maximum value of the dwell time, apart from the assumption of non-Zeno behavior. The discussion herein presented is based on a geometrical approach, leading to a general framework that includes several results reported in the literature, characterizing the observability property for different assumptions on the initial condition and the applied inputs. Moreover, the construction of a global finite-time observer scheme is presented.

2 SWITCHED LINEAR SYSTEMS AND BASIC BEHAVIOR An LS is represented by the dynamic equation x˙ (t) = Ax(t) + Bu(t) + Sd(t), x(t0 ) = x0 y(t) = Cx(t)

(3.1)

where x ∈ X = Rn is the state vector, u ∈ U = Rg is the control input, y ∈ Y = Rq is the output signal, d ∈ D = Rm is the disturbance, and A, B, C, and S are constant matrices of appropriate dimensions. Here Σ(A, B, C, S) denotes the LS (3.1) with matrices A, B, C, and S. In the case of unperturbed systems, meaning that all the inputs are known (i.e., S = 0) the LS (3.1) is denoted by Σ(A, B, C). To refer to scalar systems (i.e., those where U , Y, and D are one-dimensional spaces) we write Σ(A, b, c, s). When the matrices are clear from the context, an LS is simply denoted by Σ. The input function space, denoted by Uf , is considered to be Lp (U). Throughout this chapter B stands for  B, S for  S, and K for Ker C. A subspace T ⊂ X is called A-invariant if AT ⊂ T . The supremal  A-invariant subspace contained in K is N = ni=1 Ker(CAi−1 ). The subspace N is known as the unobservable subspace of the LS Σ. A subspace V ⊂ X is said to be (A, B)-invariant if there exists a state feedback u = Fx such that (A+BF)V ⊂ V or, equivalently, if AV ⊂ V+B. The set of maps F for

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which (A + BF)V ⊂ V holds is denoted as F(V). The set of (A, B)-invariant subspaces contained in a subspace L ⊂ X is denoted by (A, B; L). It is well known that this set is closed under addition, then it contains a supremal element [43, 44] denoted sup (A, B; L). Throughout this chapter, the concept from measure theory of “almost everywhere” or for “almost every” is used to express that the observability property is virtually certain to hold. When referring to Rn , a property is said to hold “almost everywhere” or “for almost every” if it only fails on a set of Lebesgue measure zero, also known as a “null set” [45]. An extension of these concepts to function spaces has been proposed in [46], where a property on a function space is said to hold “for almost every” (in the sense of [46]) if the set of exceptions is a shy set, also known as a Haar null set [46, 47]. Let V be a complete metric linear space, possible infinite-dimensional (e.g., Rn and Lp (U )1 ). Although we do not report here the formal definition of a shy set S ⊂ V, the following important properties give an insight on its meaning [46]: • A set S ⊂ Rn is shy, if, and only if, it has a Lebesgue measure of zero. Thus, the concept of a shy set is a natural extension of the concept of the null set, because both concepts coincide in Rn . • Every countable set in V and every proper subspace of V are shy with respect to V. In particular, this last property will be used to demonstrate that under the conditions derived in Section 3.1, the observability property holds for almost every control input and for almost every state trajectory. The following result will be used along the manuscript. Lemma 1 (Basile and Marro [43]). Any state trajectory x(t), t ∈ [t0 , τ ] of Σ belongs to a subspace L ⊆ X if, and only if, x(t0 ) ∈ L and x˙ (t) ∈ L almost everywhere in [t0 , τ ]. Definition 1. An SLS is described as a tuple Σσ = F, σ , where F = {Σ1 , . . . , Σl } is a collection of LS and σ : [t0 , ∞) → {1, . . . , l} is the switching signal determining, at each time instant the evolving LS Σσ ∈ F. The SLS’s state equation is represented by x˙ (t) = Aσ (t) x(t) + Bσ (t) u(t) + Sσ (t) d(t), y(t) = Cσ (t) x(t), σ (t0 ) = σ0

x(t0 ) = x0

(3.2)

We use xi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) to denote the state trajectory x(t) obtained when σ (t) = i, ∀t ∈ [t0 , t), the inputs u(t) and d(t) are applied, and the initial condition is x(t0 ) = x0 . In a similar way, yi (t, x0 , u(t), d(t)) represents 1

Lp (U ), 1 ≤ p ≤ ∞ is the set of all piecewise continuous functions u: R≥0 → U satisfying  1/p p < ∞ [43].

u(·) p = 0∞ u(t) p dt

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the output trajectory provided by xi (t, x0 u(t), d(t)), that is,    t Ai t −Ai τ xi (t, x0 , u(t), d(t)) = e e (Bi u(τ ) + Si d(τ ))dτ x0 + 0

and

   t yi (t, x0 , u(t), d(t)) = Ci eAi t x0 + e−Ai τ (Bi u(τ ) + Si d(τ ))dτ 0

When we want to stress that the state trajectory is restricted to an interval [τ1 , τ2 ], we write yi (t, x0 , u[τ1 ,τ2 ] , d[τ1 ,τ2 ] ) where u[τ1 ,τ2 ] , d[τ1 ,τ2 ] are the restriction of the functions u(t), d(t) to [τ1 , τ2 ]. Even though an SLS is formed by a collection of LS, the classical results on the fundamental properties of LS, such as stability, observability, and controllability, do not hold straightforwardly for SLS. For example, given an SLS, the stability of each LS in the collection is neither necessary nor sufficient for the stability of the SLS [48]. That is, on one hand, an inappropriate switching among stable LS can lead to instability. On the other hand, it may be possible to stabilize an SLS formed by unstable LS by means of a switching control [49]. Moreover, although composed of LS, an SLS can exhibit highly nonlinear dynamics, such as bifurcation and chaos. Consider, for example, the Chua’s circuit, which is composed of linear storage elements and a piecewise-linear resistor (Chua’s diode) [50, 51]. Therefore, the study of the fundamental properties of hybrids and SLS, such as observability [17], controllability [28], and stability [52, 53], as well as the development of control algorithms [54, 55] and hybrid observers becomes an important and challenging topic. The analysis must consider the complex behavior that SLS may exhibit, frequently using tools developed for continuous control systems, nonsmooth systems, and discrete event systems. Unless otherwise is stated, the following assumptions on the SLS dynamic are made. Assumption 1. The signals y(t) and u(t) are measurable, while x(t), d(t), and σ (t) are nonmeasurable. Furthermore, σ (t) is assumed to be generated exogenously, that is, we do not assume any knowledge of an underlying discrete event system. Assumption 2. Only a finite number of switches can occur in a finite interval, that is, Zeno behavior is not possible. The initial condition of the SLS is bounded, that is, x0 < δ with a known constant δ. Assumption 3. A minimum dwell time in each discrete state is assumed, that is, if tk−1 and tk are two consecutive switching times, then tk − tk−1 > τdk . However, only the dwell time for the first switching time τd1 is assumed to be known. No maximum dwell time is set.

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Assumption 4. For simplicity, the state x(t) is assumed to be continuous, that is, at each switching time tl , x(tl ) = x(t− l ). Assumption 5. The disturbance d(t) satisfies for all t ≥ t0 , |d(t)| < D, ˙ and |d(t)| < L with known constants D and L.

3 OBSERVABILITY ANALYSIS FOR SWITCHED LINEAR SYSTEMS UNDER DISTURBANCE The forthcoming observability discussion regards SLS subjects and unknown switching signals, partially unknown inputs, with unconstrained dwell time. Considering three cases, namely observability for every nonzero state trajectory, observability for “almost every” control input, and observability for “almost every” state trajectory, which will be presented separately for continuous and discrete states. The concept from the measure theory of “almost everywhere” or for “almost every” will be used here to express that the observability property is practically certain to hold. When referring to Rn , a property is said to hold “almost everywhere” or “for almost every” if it only fails on a set of Lebesgue measuring zero, also known as a “null set” [45].2 The approach herein subsumes some of the well-known results on observability of SLS under known inputs. In particular, if the disturbance is absent, the results herein presented reduce to those presented in [15–18] for SLS with unconstrained dwell time. When considering unknown inputs, the results of those papers remain only necessary. First, because no maximum dwell time is set, the SLS can remain at any LS of the collection during infinite time. Thus, for the observability of the SLS, each LS must be observable under the assumption of unknown inputs, and it must be possible to know which LS is currently evolving based on the input and output information. This possibility regards the property of “distinguishability.” Definition 2. Given an SLS, an LS Σi is said to be distinguishable from another LS Σj , by measuring u[t0 ,τ ] , y[t0 ,τ ] , if for every x0 , d[t0 ,τ ] , x0 , and d[t0 ,τ ] yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] )

(3.3)

Thus, if Eq. (3.3) does not hold for some x0 , d[t0 ,τ ] , x0 , and d[t0 ,τ ] and the output provided by the SLS is equal to yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ), then it is impossible to determine from the measured signals whether the state trajectory was generated by Σi or by Σj . In a similar way, it is impossible to determine if the continuous initial state is 2

An extension of these concepts to function spaces has been proposed in [46], where a property on a function space is said to hold “for almost every” (in the sense of [46]) if the set of exceptions is a “shy set,” also known as a “Haar null set” [46, 47].

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x0 or x0 . Clearly, Eq. (3.3) does not hold for some x0 and d (t) if x0 , u[t0 ,τ ] , and d[t0 ,τ ] are all zero, however, in this study we are only interested on nonnull trajectories. Different problems may arise when investigating the distinguishability property. For example, one may be interested in the fact that Eq. (3.3) holds for every x0 , u[t0 ,τ ] , and d[t0 ,τ ] (except when they are all zero), or in finding a set of inputs u[t0 ,τ ] ensuring that Eq. (3.3) holds despite x0 and d[t0 ,τ ] . Moreover, as it will be shown later, Eq. (3.3) does not hold only when x0 and x0 belong to the so-called indistinguishability subspace, a proper subspace hereinafter introduced. Then, the LS can be frequently distinguished during the evolution of an SLS, even if they are not distinguishable for every state trajectory. Thus, in order to solve the observability problem, we mainly focus on solving the following set of distinguishability problems. Problem 1 (Distinguishability for Every Nonzero State Trajectory). Given the LS Σi and Σj and observing the signals y[t0 ,τ ] and u[t0 ,τ ] , under which conditions can Σi be distinguished from Σj for every nonzero state trajectory? Namely, under which conditions, the set {x0 : ∃ x0 , u[t0 ,τ ] , d[t0 ,τ ] , d[t0 ,τ ] , yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] )} (3.4) contains only the zero initial state? A solution to this problem provides the basis for determining the SLS state for every state trajectory of the system. This notion ensures the computation of the SLS state in every situation. However, in most applications, this can be rather restrictive. Thus, the study can be focused on finding a control input allowing the estimation of the SLS state. We go further by showing that if such an input exists, then the SLS’s state can be estimated for almost every control input. The solution to the following problem provides the basis for this observability notion. Problem 2 (Distinguishability for Almost Every Control Input). Given the LS Σi and Σj and observing the signals y[t0 ,τ ] and u[t0 ,τ ] , under which conditions can Σi be distinguished from Σj for almost every control input? Namely, under which conditions is {u[t0 ,τ ] : ∃ x0 , x0 , d[t0 ,τ ] , d[t0 ,τ ] , yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] )} (3.5) a null set? Even if the state of the SLS cannot be determined in the sense of Problems 1 and 2, it still may be possible to determine the state for almost every state trajectory, because, as it will be shown later, the set of initial conditions for which the evolving LS cannot be determined forms a subspace R ⊆ X . Thus, if this is a proper subspace, then almost every state trajectory (except for those starting in R, which is a shy set) will allow the estimation of the evolving system.

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The following problem provides the basis for the characterization of the observability for almost every state trajectory. Problem 3 (Distinguishability for Almost Every State Trajectory). Given the LS Σi and Σj and observing the signals y[t0 ,τ ] and u[t0 ,τ ] , under which conditions can Σi be distinguished from Σj for almost every state trajectory? Namely, under which conditions are R = {x0 : ∀u[t0 ,τ ] , d[t0 ,τ ] ∃x0 , d[t0 ,τ ] , yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] )}

(3.6)

a null set? Definition 3. The discrete state σ0 and σ (t) of an SLS Σσ (t) is said to be observable if σ0 and σ (t) can be uniquely determined from the SLS structure and the input-output information. Definition 4. The continuous state x0 and x(t) of an SLS Σσ (t) is said to be observable if x0 and x(t) can be uniquely determined from the SLS structure and the input-output information.

3.1 Geometrical Analysis The observability analysis presented here is based on the geometrical observability approach for LS under unknown inputs (disturbances). Let us recall first some concepts and results regarding the geometrical observability analysis for LS. Definition 5. A subspace T ⊂ X is called A-invariant if AT ⊂ T . Denoting as K the kernel of the output matrix C, the supremal A-invariant n i−1 ). The subspace N is subspace contained in K is N = i=1 Ker(CA known as the unobservable subspace of the LS Σ. A subspace V ⊂ X is said to be (A, B)-invariant3 if there exists a state feedback u = Fx such that (A + BF)V ⊂ V or, equivalently, if AV ⊂ V + B. The set of maps F for which (A + BF)V ⊂ V holds is denoted as F(V). The set of (A, B)-invariant subspaces contained in a subspace L ⊂ X is denoted by (A, B; L). The observability of an LS under unknown inputs is characterized as follows. Theorem 1 (Basile and Marro [14]). Let Σ be an LS with associated matrices A, B, C, and S. Then the LS Σ is observable under partially unknown inputs (also referred as strongly observable) if, and only if, the supremal (A, S)-invariant subspace contained in K ⊂ X , denoted as sup (A, S; K), is trivial.4 3

When referring to unknown inputs we write (A, S)-invariant subspace. In [43] (A, B)invariant subspaces are known as “(A, B)-controlled invariants” or “(A, S)-controlled invariants when referring to unknown inputs.” 4

This theorem subsumes the classical result on observability of LS with known inputs, because sup (A, 0; K) = N . Observability under unknown inputs is also referred in the literature as “strong observability” [56].

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The invariant sup (A, S; K) = V (ν) can be computed as follows: V (0) = K V (i) = K ∩ A−1 (S + V (i−1) ),

(3.7)

i = 1, . . . , ν

where ν = dim(K) [43, 44]. Now, in order to solve the different distinguishability problems between Σi and Σj introduced in the previous section, we define an auxiliary extended LS, which contains the dynamics of both constituting systems, but the outputs of the two systems are subtracted. This extended LS is denoted by Σij and is formed with the matrices

Ai 0 B Aij = Bij = i Bj 0 Aj

 S 0 Cij = Ci −Cj Sij = i (3.8) 0 Sj The state space of Σij is denoted by Xij . We denote by xˆ 0 the initial state, xˆ (t) ˆ the disturbance signal of the extended LS Σij . the state trajectory, and d(t) Lemma 2. Let Σi and Σj be two LS. Σi and Σj are indistinguishable in the interval [t0 , τ ], that is, the equation yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) holds if, and only if, the extended LS Σij produces a zero T  , output for all t ∈ [t0 , τ ], that is, yij (t, xˆ 0 , u[t0 ,τ ] , dˆ [t0 ,τ ] ) = 0 with xˆ 0 = xT0 xT 0 T  T T ˆ u(t) and d(t) = d (t) d (t) . Proof . Assume that Σi and Σj are indistinguishable in the nonzero interval [t0 , τ ], that is, the equation yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) holds, or equivalently

 t e−Ai τ [Bi u(τ ) + Si d(τ )]dτ Ci eAi t x0 + 0

= Cj e

Aj t

x0

 +

t

e 0

−Aj τ





[Bj u(τ ) + Sj d (τ )]dτ

this equality can be written as

 t eAi t  eAi (t−τ )  0 x0 Ci −Cj Ci −Cj + 0 eAj t x0 0 0





S 0 Bi d(τ ) dτ = 0 u(τ ) + i Bj 0 Sj d (τ )

eAj (t−τ )

 writing Eq. (3.9) in terms of the matrices (3.8), and with xˆ 0 = xT0

 t Aij t −Aij τ ˆ Cij e e [Bij u(τ ) + Sij d(τ )]dτ = 0 xˆ 0 + 0

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0

(3.9) xT 0

T

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Because all the steps can be written in the reverse order, then the converse is also true. Definition 6. Given two LS Σi , Σj , the indistinguishability subspace Wij of Σi , Σj is defined as 



x0 ∃u[t0 ,τ ] , d[t0 ,τ ] , d[t0 ,τ ] Wij = : x0 yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) The indistinguishability subspace represents the set of initial conditions that, under a particular input, makes it impossible to determine which is the evolving system, and the current discrete state. Let Qi : Xij → X be the  T  = x, then if the state projection of Xij over X such that Qi xT xT trajectory xi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) evolves inside Qi Wij then it is impossible to determine, from the measurements, if the evolving state trajectory is xi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) or xj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ), thus it is not possible to infer either the continuous initial state, which could be x0 or x0 or the discrete state, which could be σ (t) = i or σ (t) = j ∀t ∈ [t0 , τ ]. The following result characterizes the indistinguishability subspace.  Lemma 3. Let Σi and Σj be two LS and let Bij = Bij Sij . Then the indistinguishability subspace Wij is equal to the supremal (Aij , Bij )-invariant subspace contained in Kˆ ij = Ker Cij , denoted as sup (Aij , Bij ; Kij ).  T Proof . Assume that xT0 xT ∈ Wij . Hence, there exist signals u(t), 0 d(t), and d (t) such that Eq. (3.3) is satisfied, which implies, by virtue of Lemma 2, that yij (t, xˆ 0 , u[t0 ,τ ] , dˆ [t0 ,τ ] ) = 0, that is,

 t e−Aij τ Bij u(τ )dτ = 0 ∀t ≥ t0 (3.10) Cij eAij t xˆ 0 + t0

 T T  T (t) d ˆ T (t) . S B , and u(t) = , B = xT u ij ij ij 0 Note that, if xˆ 0 ∈ Wij then there exists u(t) such that xˆ (t) ∈ Wij for all t ≥ t0 . Now, according to Lemma 1, for an arbitrary t = ¯t, x˙ˆ (¯t) = Aij xˆ (¯t) + Bij u(¯t) ∈ Wij , hence Aij Wij ⊂ Wij + Bij . Furthermore, because for every t ≥ t0 the extended system Σij produces a zero output, then Wij ⊂ Kˆ ij showing that Wij ∈ (Aij , Bij ; Kij ). In order to show that Wij is the supremal in (Aij , Bij ; Kij ), let Vij ∈ (Aij , Bij ; Kij ) and xˆ 0 ∈ Vij , then there exists u(t) = Fˆx(t) with F ∈ F(Vij ) such that (Aij + Bij F)Vij ⊂ Vij making the state trajectory xˆ (t) to completely belong to Vij . Because Vij ⊂ Kˆ ij then yˆ (t) = Cij xˆ (t) = 0 for all t ≥ t0 . Thus, Eq. (3.10) holds and Vij ⊂ Wij . By means of the algorithm described by Eq. (3.7), the indistinguishability subspace can be computed. This subspace is fundamental in our approach, because the different observability notions will be derived from  with xˆ 0 = xT0

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it. For example, the conditions for solving Problem 1 are given in the following proposition. Proposition 1. Let Σi and Σj be two LS and let Wij be as in Definition 6. Then Σi can be distinguished from Σj for every nonzero state trajectory if, and only if,  T  = x. Furthermore, Qi Wij = 0, where Qi : Xij → X is such that Qi xT xT Σi and Σj are distinguishable from each other for every nonzero state trajectory if, and only if, Wij = 0. Proof . (Sufficiency) Notice that Qi Wij is equal to the set in Eq. (3.4). Because Qi Wij = 0, then Eq. (3.3) only holds when x0 , u(t), and d(t) are all equal to 0 (without any restriction on x0 and d (t)), thus Σi is indistinguishable from Σj only for the zero state trajectory. Clearly, if Wij = 0, then Eq. (3.3) only holds with x0 , x0 , u(t), d(t), and d (t) all equal to 0, and Σi and Σj are distinguishable from each other for every nonzero state trajectory. (Necessity) It follows by noticing that Qi Wij is equal to the set in Eq. (3.4). Notice that if the conditions of Proposition 1 hold, then the continuous state can also be determined, because Qi Wij = 0 implies the observability with partially unknown inputs of the LS Σi , because otherwise there exists x0 ∈ sup (Ai , Si ; Ki ) such that yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = 0, and therefore  T T x0 0 ∈ Wij and x0 ∈ Qi Wij . However, neither Wij = 0 nor Qi Wij = 0 are necessary for recovering the continuous initial and current states. Definition 7. Let Σi and Σj be two LS. The indistinguishability subspace T  Wij is said to be symmetric if every vector xT xT ∈ Wij is of the form x = x . The following proposition presents the conditions for recovering the continuous initial state. Proposition 2. Let Σi and Σj be two LS. The continuous initial state can be uniquely determined for every state trajectory if, and only if, their indistinguishability subspace Wij is symmetric. Proof . Because the continuous initial state can be uniquely determined for every state trajectory if, and only if, yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) ⇒ x0 = x0 . Thus, the result follows trivially from Definition 6. Because the indistinguishable trajectories are generated from the same continuous initial state, then it can be recovered, even though the evolving system cannot. Based on the previous results, we can easily characterize the following observability notions. Theorem 2. Let G = F, σ  be an SLS with partially unknown inputs and maximum dwell time τ = ∞. Then 1. The discrete state σ0 and σ (t) of G are observable for every nonzero state trajectory if, and only if, ∀Σi , Σj ∈ F i = j, Wij = 0.

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2. The continuous state x0 and x(t) of G are observable for every state trajectory if, and only if, ∀Σi , Σj ∈ F i = j their indistinguishability subspace Wij is symmetric. The proof follows trivially from Propositions 1 and 2, respectively. Next, we focus on solving Problem 2, that is, we are interested in deriving the conditions under which the SLS’s state can be estimated for almost every control input. In order to be able to distinguish Σi from Σj , these inputs need to be capable of steering the state trajectory outside Qi Wij , that is, the projection of the indistinguishability subspace Wij over X . The following lemma will be used such conditions.  in deriving  ˆ ˆ Lemma 4. Let Wij = sup (Aij , L1 L2 ; Kij ) where Lˆ 1 : Rr1 → Xij and Lˆ 2 : Rr2 → Xij with Lˆ k =  Lˆ k , k = 1, 2. Then Wij coincides with sup (Aij , Lˆ 1 ; Kij ) if Lˆ 2 ⊆ Wij + Lˆ 1 .   Proof . Because  Lˆ 1 Lˆ 2 = Lˆ 1 + Lˆ 2 then Aij Wij ⊂ Wij + Lˆ 1 + Lˆ 2 . Hence, for every w1 ∈ Wij there exist w2 ∈ Wij , z1 ∈ Lˆ 1 , and z2 ∈ Lˆ 2 such that Aij w1 = w2 + Lˆ 1 z1 + Lˆ 2 z2 . Set g = z1 − h with h such that Lˆ 1 h + Lˆ 2 z2 = w3 ∈ Wij , such a h always exists for every z2 because by assumption Lˆ 2 ⊆ Wij + Lˆ 1 , thus Aij w1 = w2 + w3 + Lˆ 1 g and Aij Wij ⊂ Wij + Lˆ 1 . Hence Wij ∈ (Aij , Lˆ 1 ; Kij ). Because, trivially, every element of (Aij , Lˆ 1 ; Kij ) is contained in Wij (because Wij is supremal) then Wij coincides with sup (Aij , Lˆ 1 ; Kij ). In order to solve Problem 2, we first derive the conditions for the existence of an input, allowing it to distinguish Σi from Σj , and then we show that if these conditions hold, then Eq. (3.5) is a shy set. Proposition 3. Let Σi and Σj be two LS. Then there exists a control input ensuring the distinguishability between the LS Σi and the LS Σj , that is, there exists a control input u[t0 ,τ ] such that ∀x0 , x0 , d[t0 ,τ ] , d[t0 ,τ ] yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] )

(3.11)

if, and only if, Bij  Wij + Sij or equivalently Bij  sup (Aij , Sij ; Kij ) + Sij

(3.12)

Proof . (Necessity) We next prove that if Eq. (3.12) does not hold, Eq. (3.11) does not hold as well, by showing that for every possible u[t0 ,τ ] there exists an initial condition xˆ 0 and an unknown input dˆ [t0 ,τ ] such that the extended LS Σij produces a zero output for all t ∈ [t0 , τ ]. ˆ = Fˆx(t) + v(t) with F ∈ F(V * ) and Let V * = sup (Aij , Sij ; Kij ) and set d(t) v(t) such that for every t ∈ [t0 , τ ] Bij u(t) + Sij v(t) ∈ V * ⊂ Wij

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Notice that such a v(t) always exists because Eq. (3.12) does not hold. Thus, if xˆ 0 ∈ V * , then ∀t ∈ [t0 , τ ] x˙ˆ (t) ∈ V * , and xˆ (t) ∈ V * ⊂ Wij ∀t ∈ [t0 , τ ] by Lemma 1, showing that Eq. (3.11) does not hold. (Sufficiency) Suppose that Eq. (3.11) does not hold, then by Lemma 2 ˆ such that Σij produces a zero output for every u(t), there exist xˆ 0 and d(t)  T  for all t ∈ [t0 , τ ]. Now, let Bij = Bij Sij , u = uT dˆ T , and Wij = sup (Aij , Bij ; Kij ), and set u(t) = Fˆx(t) + vˆ (t), with F ∈ F(Wij ) and v(t) =  T T v1 (t) vT2 (t) , notice that this is not a restriction over u(t) because there are no constraints on v1 (t). Now, because for every v1 (t) there exist xˆ 0 and v2 (t), such that xˆ (t) ∈ Wij , then by Lemma 1 x˙ˆ (t) = (Aij + Bij F)ˆx(t) + Bij v1 (t) + Sij v2 (t) ∈ Wij This implies that for every v1 (t) there exists v2 (t), such that Bij v1 (t) + Sij v2 (t) ∈ Wij . Hence, Bij ⊂ Wij + Sij and by Lemma 4, Wij coincides with sup (Aij , Sij ; Kij ) and Bij ⊂ sup (Aij , Sij ; Kij ) + Sij , which completes the proof. Thus, if Bij  Wij + Sij holds, not all the admissible values of x˙ˆ (t) for the state trajectories starting at Wij belong to Wij , thus it is possible, by a smooth input, to maintain x˙ˆ (t) out of Wij for a finite time, and reach points not belonging to Wij , despite any disturbance. We show next that, if this condition holds, then almost every input can be used to distinguish the LS. To this aim, recall first that if the distinguishability property only fails to hold in a shy set of inputs, the distinguishability property is said to hold for almost every input. Based on the fact that every proper subspace of Uf is shy, we show that, if the conditions of Proposition 3 hold, then the set of exceptions Eq. (3.5) is a proper subspace of Uf . Thus, almost every input can be used to distinguish the LS. Notice that due to the superposition property of the extended LS, the set of inputs in Eq. (3.5) is a subspace of Uf , in which u(t) takes values, because if u[t0 ,τ ] and u[t0 ,τ ] are in Eq. (3.5). Then, according to Eq. 3.5, there exist combinations of initial conditions and inputs such that the extended LS produces zero output when these inputs are applied, that is, yij (ˆx0 , u[t0 ,τ ] , dˆ [t0 ,τ ] ) = yij (ˆx0 , u[t0 ,τ ] , dˆ [t0 ,τ ] ) = 0 for some xˆ 0 , xˆ 0 , dˆ [t0 ,τ ] , and dˆ [t0 ,τ ] . Due to the superposition property of the extended LS Σij , for each α, α  ∈ R yij (α xˆ 0 + α  xˆ 0 , αu[t0 ,τ ] + α  u[t0 ,τ ] , α dˆ [t0 ,τ ] + α  dˆ [t0 ,τ ] ) = 0 Hence, by Lemma 2, Eq. (3.11) does not hold and therefore αu[t0 ,τ ] +α  u[t0 ,τ ] is in Eq. (3.5) and thus, Eq. (3.5) is a subspace of Uf .

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Because we have shown the existence of a smooth input steering the state trajectory outside Wij , then Eq. (3.5) is a proper subspace of Uf , and thus is a shy set with respect to Uf [46]. Hence, almost every input u[t0 ,τ ] can be used to guarantee the distinguishability between Σi and Σj . The following remark relates the set of inputs in Eq. (3.5) with the system zeros [57, 58] of the extended LS Σij , because by Lemma 2 if the LS are indistinguishable then the extended LS produces a zero output. Remark 1. Let U = U1 + U2 with U1 = {u: Bij u ∈ Wij + Sij } and U1 ∩ U2 = 0. Clearly, by an argument similar to the necessity in the proof of Proposition 3, every input u: R≥0 → U1 belongs to Eq. (3.5) for a suitable ˆ d(t). However, the inputs u: R≥0 → U2 that are contained in Eq. (3.5) are strictly linear combinations of vectors of the form uzi tmk −1 ezi t , where zi is such that Ker P(zi ) = 0, where

zi I − Aij −Bij −Sij P(zi ) = Cij 0 0 and the maximum value of mk is the algebraic multiplicity of zi . The values of zi such that P(zi ) drops its normal rank are system zeros of the extended LS Σij . For more details on system zeros, the reader may refer to [57, 58]. In [39], an observer design for the case when Si = 0, i = 1, . . . , m, has been proposed. From the previous proposition we can state the conditions for the observability for almost every control input, which is formally presented in the following theorem. Theorem 3. Let G = F, σ  be an SLS with partially unknown inputs and maximum dwell time τ = ∞. Then 1. The discrete state σ0 and σ (t) of G are observable for almost every control input if, and only if, ∀Σi , Σj ∈ F i = j, Bij  Wij + Sij . 2. The continuous state x0 and x(t) of G are observable for almost every control input if, and only if, every LS Σi ∈ F is observable with partially unknown inputs and ∀Σi , Σj ∈ F i = j Bij  Wij + Sij or Wij is symmetric. Now, we focus on deriving the conditions for the observability for almost every state trajectory. There are two ways of achieving this. The first one, by avoiding the existence of a set of initial states R given by Eq. (3.6), such that Σi is indistinguishable from Σj for every state trajectory starting from that set. The necessary and sufficient condition for this case is    Bij Sˆ i  Wij + Sˆj  T  T with Sˆ i = STi 0 and Sˆ j = 0 STj . The second one, by requiring that if this set exists, then it is a shy set. In the former, almost every input (control input or disturbance) allows us to

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distinguish the LS, while in the latter, almost every state trajectory, except for those starting in R, will be distinguishable. Lemma 5. Let V = sup (Aij , Sˆj ; Kij ). If  [Bij Sˆ i ] ⊆ Wij + Sˆj , then the set R coincides with the subspace Qi V.  Proof . Because  Bij Sˆ i ⊆ Wij +Sˆj then any trajectory xi (t, x0 , u(t), d(t)) starting in Wij will remain in Wij regardless of the input u(t) and disturbance d(t) applied. Thus, for every x0 ∈ Wij and every u[t0 ,τ ] and d[t0 ,τ ] there exist x0 and d[t0 ,τ ] such that yi (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) = yj (t, x0 , u[t0 ,τ ] , d[t0 ,τ ] ) holds. Thus Qi Wij coincides with R. It follows from Lemma 4 that, if    Bij Sˆ i ⊆ Wij + Sˆj then Wij = V. Proposition 4. Let Σi and Σj be two LS and let V = sup (Aij , Sˆj ; Kij ). Then the system Σi is distinguishable from Σj for almost every state trajectory (i.e., R is a shy set) if, and only if, one of the following conditions hold     1.  Bij Sˆ i  Wij + Sˆj (equivalently  Bij Sˆ i  V + Sˆj ) or 2. dim(Qi Wij ) < dim(X ) (equivalently dim(Qi V) < dim(X )). Proof . (Sufficiency) Two cases are possible:   (a)  Bij Sˆ i  V + Sˆj , then it does not exist d (t) such that the effect of u(t) and d(t) does not show up at the output of the extended systems, thatis, the set  R is empty. (b) If  Bij Sˆ i ⊆ V + Sˆj but dim(Qi V) < dim(X ) then by Lemma 5 Qi V = R, because Qi V is a subspace of X and dim(Qi V) = dim(R) < dim(X) then R is clearly a shy set. (Necessity) Assume that  Bij Sˆ i ⊆ V + Sˆj and dim(Qi V) = dim(X ) then by Lemma 5, Qi V = R = X . Thus, R is not a shy set. Based on these results, we can easily characterize the following observability notions. Theorem 4. Let G = F, σ (t) be an SLS with partially unknown inputs and maximum dwell time τ = ∞. Then 1. The discrete states σ0 and σ (t) of the SLS G are observable for almost every state trajectory if, and only if, every pair of different LS are distinguishable for almost every state trajectory, according to Proposition 4. 2. The continuous states x0 and x(t) of the SLS G are observable for almost every state trajectory if, and only if, every pair of different LS are either distinguishable for almost every state trajectory according to Proposition 4, or their distinguishability subspace Wij is symmetric.

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Example 1. Consider the LS shown in the following table: A ⎡ Σi

Σj

−3 ⎢ ⎢ ⎢ 1 ⎣ 0 ⎡ −2 ⎢ ⎢ ⎢ 1 ⎣ 0

0 −1 1 0 0 −3

⎤ 0 ⎥ ⎥ 0⎥ ⎦ −2 ⎤ 0 ⎥ ⎥ −1⎥ ⎦ 0

C ⎡ ⎤T 0 ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 1 ⎡ ⎤T 0 ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎣ ⎦ −1

B ⎡ ⎤ 0 ⎢ ⎥ ⎢ ⎥ ⎢2⎥ ⎣ ⎦ 0 ⎡ ⎤ 0 ⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎣ ⎦ 1

S ⎡ ⎤ 1 ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 0 ⎡ ⎤ 1 ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 0

The indistinguishability subspace with partially unknown inputs of Σi and Σj is ⎡ ⎤ 1 0 0 0 ⎢0 2 0 0 ⎥ ⎢ ⎥ ⎢0 0 2 0 ⎥ ⎢ ⎥ Wij = ⎢ 0 2 ⎥ ⎢0 0 ⎥ ⎣0 1 −3 −1 ⎦ 0 1 −5 1 Thus, according to Proposition 1, for every x0 ∈ Qi Wij = X there exist u(t) and d(t), such that the equation yi (t, x0 , u(t), d(t)) = yj (t, x0 , u(t), d (t)),

∀t ≥ 0

(3.13)

holds for some x0 and d (t), making impossible to infer the evolving LS. For example, suppose that the evolving LS is Σi , with ⎡ ⎤ 0 2 4 x0 = ⎣2⎦ , u(t) = e−3t + and d(t) = 0 3 3 0 then the output is 4 −3t 2 e + 3 3 Because the LS Σj produces this same output with ⎡ ⎤ 0 4 2 20 38 16 x0 = ⎣1⎦ , u(t) = e−3t + and d (t) = − + e−3t − te−3t 3 3 9 9 3 1 yi (t, x0 , u(t), d(t)) =

then it is impossible to determine the evolving LS and the continuous initial state for the current state trajectory. Hence, the LS Σi and Σj cannot be distinguished for every state trajectory.

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Moreover, because

⎡ ⎤ 0 ⎢2⎥ ⎢ ⎥ ⎢0⎥ ⎥ Bij =  ⎢ ⎢0⎥ ⊂ Wij + Sij ⎢ ⎥ ⎣1⎦ 1

then according to Proposition 3 there is no input u(t) allowing us to distinguish Σi from Σj , that is, the effect of the input u(t) does not show up at the output of the extended   system. Furthermore, because  Bij Sˆ i ⊂ Wij + Sˆj and Qi Wij = X , then for every state trajectory of Σi , there exists a state trajectory of Σj producing the same output, therefore it is always impossible to distinguish the LS Σi from the LS Σj , that is, ∀x0 , u(t), d(t)∃x0 , d (t), such that Eq. (3.13) holds.

x In particular, Eq. (3.13) holds with x0 such that 0 ∈ Wij and d (t) = x0  Fˆx(t) such that F ∈ F(Wij ) (e.g., F = 1 −4 4 0 2 0 ). The existence of this state feedback implies that for every state trajectory of the LS Σi , there exist x0 and d (t) (not necessarily constructed as a state feedback) in Σj , such that it is impossible to determine the evolving system and the initial continuous state, because there exist state trajectories in both LS producing the same input-output information. Now, consider the same LS, but with ⎡ ⎤ ⎡ ⎤ 1 0 Bi = ⎣2⎦ , Bj = ⎣1⎦ 0 0 In this case, almost every input u(t) allows the determination of the evolving LS, because Bij  sup (Aij , Sij ; Kij ) + Sij Thus, the SLS Σσ = {F, σ } with these input matrices is observable for almost every input u(t). Example 2. Consider the unperturbed SLS Σσ (t) where σ (t) is an exogenous switching signal and F = {Σ1 , Σ2 } is the continuous dynamics where Σ1 (A1 , B1 , C1 , 0) and Σ2 (A2 , B2 , C2 , 0) have system matrices as shown in  T  T 0 0 2 and B2 = 0 0 −1 and = Table 3.1, input matrices B 1



1 0 −1 1 0 −1 and C2 = . output matrix C1 = 0 −1 0 0 −1 0 The unobservable subspace of the extended system Σˆ 12 is T  N12 =  eT3 eT3 ⊆ W12

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TABLE 3.1 System Matrix of Linear Systems

A

Σ1 ⎡ −2 ⎢ ⎢ ⎢2 ⎣ 3

⎤ 0 −1 0

0

⎥ ⎥ 0⎥ ⎦ −3

Σ2 ⎡ −1 ⎢ ⎢ ⎢ 6 ⎣ 5

−1 −5 −2

⎤ 0

⎥ ⎥ 0⎥ ⎦ −3

where ek ∈ R3 is the kth unit vector, with 1 in the kth entry and 0 elsewhere. Because Bˆ12 is not contained in N12 , then according to LS theory, there exist inputs (in fact, almost every input) that produces a nonzero output in the extended LS regardless of the initial conditions, which implies, by virtue of Lemma 2, that Σ1 and Σ2 produce a different input-output information in such cases, and it is possible to determine the evolving LS using such information. Hence, the SLS Σσ (t) is observable for almost every control input. However, if we consider the inputs to be unknown, that is, Σ1 (A1 , 0,  T C1 , S1 ) and Σ2 (A2 , 0, C2 , S2 ) with disturbance matrices S1 = 0 0 2 and  T S2 = 0 0 −1 , then the evolving LS and the continuous state should be determined from the continuous output. Unfortunately, Si ⊆ Wij + Sj , which implies that for every x0 ∈ Qi Wij and d(t) applied to Σi there exist x0

x and d (t) applied to Σj , such that yi (x0 , 0, d(t)) = yj (x0 , 0, d (t)) with 0 ∈ x0 Wij and d (t) = −2d(t), becoming impossible to distinguish Σi from Σj for every state trajectory starting in x0 ∈ Qi Wij ; in a similar way Σj cannot be distinguished from Σi for states’ trajectories starting in x0 ∈ Qj Wij .

3.2 Distinguishability Conditions for Perturbed SISO Switched Affine Systems Under Bounded Disturbances Switched affine systems (SAS) are a subclass of SLS derived by assuming ∀t ≥ t0 , u(t) = 1. SAS are interesting models for applications in different engineering areas. For example, nonlinear continuous dynamics are frequently approximated by AS linearized at different operation points, thus the active AS depends on the continuous state, leading to autonomous switching as a piecewise affine system [59, 60]. The analysis and control of these kinds of systems, based on SAS, are frequently affected by disturbances and parametric variations. For this reason, such analysis should be performed by taking into account the disturbances affecting the system. Moreover, SAS are frequently used to generate chaotic behavior (see, e.g., [50, 61, 62]) for chaotic modulation. In particular, in the nonautonomous chaotic modulation [63] (also known as message-embedded modulation [64–66]) using chaotic attractors generated by SAS, a message is embedded as a disturbance affecting the chaotic behavior.

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A wide class of chaotic attractors are generated by SAS to be used in nonautonomous chaotic modulations. To the best of our knowledge, the nonautonomous modulation uses general classes of SAS with chaotic behavior. In this section, it is shown that any pair of SISO-perturbed AS are indistinguishable from the output. For this reason, a new distinguishability condition taking advantage of the known disturbance bound (Assumption 5) is introduced to allow inference of the evolving AS. Proposition 5. Any two strongly observable SISO AS Σi and Σj are outputindistinguishable under disturbances. Otherwise stated, for every initial condition x0 and disturbance d(t) applied to Σi there exists an initial condition x0 and a disturbance d (t) (not necessarily equal to x0 and d(t)) applied to Σj such that the corresponding output trajectories are equal, that is, yi (t, x0 , d(t)) = yj (t, x0 , d (t)) for all time t ≥ 0, thus making it impossible to infer from the output which is the evolving AS. −1 Proof . Consider a similar transformation x = Ti x˘ such that Ti = OΣ i and OΣi are the observability matrix of the AS Σi (recall that a nonsingular coordinate transformation does not change the input-output behavior of the AS). Because Σi is strongly observable, such a similarity transformation is well defined, and the transformed system is in the observability canonical form with the unknown disturbance d(t) affecting only the last state variable [67]. In the new coordinates, the AS is represented by x˙˘ 1 = x˘ 2 + b˘ i1 x˘˙ 2 = x˘ 3 + b˘ i2 .. . x˙˘ n = −ai x˘ 1 − ai n

˘2 n−1 x

− · · · − ai1 x˘ n + b˘ in + β i d(t)

y(t) = x˘ 1 where sn + ai1 sn−1 + · · · + ain−1 s + ain

(3.14)

is the characteristic polynomial of Ai . Now, let us introduce the variable transformation x¯ 1 = x˘ 1 and x¯ k = x˘ k + b˘ ik−1 , k ∈ {2, . . . , n}, thus x¯ = Ti−1 (x + bi ). Then, the AS Σi can be represented as x˙¯ 1 = x¯ 2 x˙¯ 2 = x¯ 3 .. . x˙¯ n = α i (¯x) + β i d(t) y(t) = x¯ 1

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(3.15)

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where i

α (¯x) =

−ain x¯ 1

−

n 

ain−k+1 (¯xk − b˘ ik−1 ) + b˘ in

(3.16)

k=2

Notice that the new state variables are the output and their derivatives, that is, x¯ k =

dk−1 y(t) dtk−1

,

∀k ∈ {1, . . . , n}

(3.17)

Now, applying the analogous transformation procedure to the AS Σj (Aj , bj , cj , sj ), with x¯ = Tj−1 (x + bj ), Σj can be represented as x˙¯ 1 = x¯ 2 x˙¯ 2 = x¯ 3 .. . x˙¯ n = α j (¯x) + β j d(t) y(t) = x¯ 1

(3.18)

Notice that if the disturbance d(t) is applied to Σi and the signal d (t) =

1 i (α (¯x) − α j (¯x) + β i d(t)) βj

(3.19)

is applied to Σj as a disturbance, then Eqs. (3.15), (3.18) have the same output behavior. Therefore, the output trajectory obtained when the disturbance d(t) is applied to the system Σi with the initial condition x0 is equal to that obtained when the disturbance d (t) in Eq. (3.19) is applied to Σj with the initial condition x0 = Tj Ti−1 (x0 +bi )−bj . Notice that it is possible to compute such x0 and d(t) for any pair x0 and d(t). Thus, it is impossible to determine from the output whether the evolving subsystem is Σi or Σj . Similarly, it is impossible to determine from the output whether the initial condition is x0 or x0 . Therefore, the AS Σi is output-indistinguishable from Σj . The previous proposition establishes that any pair of perturbed SISO AS are always output-indistinguishable. The next proposition additionally establishes that Σi and Σj become output-indistinguishable only with disturbances of the form (3.19). Proposition 6. Let Σi and Σj be two strongly observable perturbed AS. Suppose the disturbance d(t) is applied to the system Σi with the initial condition x0 and the signal d (t) is applied to Σj as a disturbance with the initial condition x0 . Both AS produce the same output trajectories if d (t) fulfills Eq. (3.19) and x0 = Tj Ti−1 (x0 + bi ) − bj , where Tk−1 = OΣk with OΣk being the observability matrix of Σk , k = i, j.

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Furthermore, the generated state trajectories xi (t, x0 , d(t)) and xj (t, x0 , d (t)) fulfill with xj (t, x0 , d(t)) = Tj Ti−1 (xi (t, x0 , d (t)) + bi ) − bj Proof . The sufficiency has been demonstrated herein. To prove the necessity, assume that ∃x0 , x0 , d(t), d (t) such that yi (t, x0 , d(t)) = yj (t, x0 , d (t)). Let us consider the coordinate transformations x¯ i = Ti−1 (xi +bi ) and x¯ j = Tj−1 (xj +bj ) for Σi and Σj , respectively (which do not affect the input-output behavior of the AS). Because yi (t, x0 , d(t)) = yj (t, x0 , d (t)), dk i y (t, x0 , d(t)) = dtk i x¯ = x¯ j and α i (¯xi )

∀t ≥ t0 , then

dk j y (t, x0 , d (t)), for all k ≥ dtk + β i d(t) = α j (¯xj ) + β j d (t).

0, which

implies that Based on these equations, it is easy to see that d (t) must be equal to Eq. (3.19) in order to make Σi output-indistinguishable from Σj and that, in such case, xj (t) = Tj Ti−1 (xi (t) + bi ) − bj , which particularly holds for xj (0) = x0 = Tj Ti−1 (x0 + bi ) − bj . Lemma 6. Consider a SAS and suppose the evolving system is either i or j . By using the knowledge of the disturbance bound D imposed by Assumption 5, it can be inferred whether the system evolves in Σi or Σj if the signal d (t) that, if applied to Σj as a disturbance, would make Σj output-indistinguishable from Σi does not satisfy the disturbance bound condition for a proper time interval, that is, if for a proper time interval it holds that   1  (3.20) |d (t)| =  j (α i (¯x) − α j (¯x) + β i d(t)) > D β Proof . The proof follows directly from the previous propositions. Notice that, in some cases, this bound condition does not provide additional information, for example, when β i /β j < 1 and the system is evolving in a certain state region where α i (¯x) − α j (¯x) is relatively small with respect to β i d(t). The following theorem formalizes this new distinguishability condition. Lemma 7. Consider an SAS and two AS, Σi and Σj , of the collection. Consider the disturbance bound D imposed by Assumption 5. Suppose that the ij system is evolving in either Σi or Σj . Let Bx¯ ⊆ Rn be the set of vectors x ∈ Rn that fulfills       1 β     i (3.21)  (α i (x ) − α j (x )) > D 1 +     βj  βj  where the polynomial functions α i (x ) and α j (x ) are given by Eq. (3.16). During ij the evolution of the SAS, if for a proper time interval x¯ (t) ∈ Bx¯ , where x¯ k (t) = dk−1 y(t) for k ∈ [1, . . . , n], then it can be inferred whether the system evolves in Σi dtk or Σj . In such case, it is said that the pair Σi and Σj is bound-distinguishable.

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Proof . Considering |d(t)| ≤ D, the condition (3.21) implies the condition (3.20). Thus, whenever the system is evolving in a state region such that ij x¯ (t) ∈ Bx¯ , the conditions of Lemma 6 hold. Then it is possible to determine whether Σi or Σj is evolving. Example 3. Let us consider, for example, the multiscroll attractor Σσ (x) proposed in [62]5 given by the following collection of LS. A ⎡ Σ1

⎤ 0

1

−1

−0.2461

−8.0521

−2.0060

0

1

−1

−0.2461

−6.8438

−2.0060

⎢ ⎢ ⎢ ⎣ ⎡

Σ2

⎢ ⎢ ⎢ ⎣ ⎡

Σ3

0

1

−1

−0.2461

−8.0521

−2.0060

⎢ ⎢ ⎢ ⎣

0

⎥ ⎥ 1 ⎥ ⎦ −1.1102 ⎤ 0 ⎥ ⎥ 1 ⎥ ⎦ −1.1102 ⎤ 0 ⎥ ⎥ 1 ⎥ ⎦ −1.1102

b ⎡

⎤ 0 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎣ ⎦ 5.5355 ⎡ ⎤ 0 ⎢ ⎥ ⎢ ⎥ ⎢0 ⎥ ⎣ ⎦ 0 ⎡ ⎤ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦ −5.5355

and the switching rule

⎧ ⎨1 if x1 ≥ 1/3 σ (x) = 2 if −1/3 < x1 < 1/3 ⎩ 3 if x1 ≤ −1/3

If y = x2 then Σ1 and Σ3 can be proved to be indistinguishable by showing that both ASs have the same set of equations when written in the form of Eq. (3.15), with α 1 (¯x) = α 3 (¯x) = −9.1623¯x1 −3.2792¯x2 −1.3563¯x3 . Moreover, α 2 (¯x) − α 1 (¯x) = 1.2083¯x1 and the condition of Lemma 7 is not satisfied for x¯ 1 = x2 ∈ (−1, 1). On the contrary, if y = x3 and the disturbance d(t) = m(t) is bounded by D = 1 then the condition of Lemma 7 is satisfied. In detail, β 1 = β 2 = 1 and α 1 (¯x) = −9.1623¯x1 − 3.2792¯x2 − 1.3563¯x3 + 5.5355 α 2 (¯x) = −7.9540¯x1 − 3.2792¯x2 − 1.3563¯x3 α 3 (¯x) = −9.1623¯x1 − 3.2792¯x2 − 1.3563¯x3 − 5.5355 Thus, |α 1 (¯x) − α 3 (¯x)| = 11.071 > 2D and for x¯ 1 = x3 ∈ (−2.5, 2.5) |α 2 (¯x) − α 1 (¯x)| = |−5.5355 + 1.2083¯x1 | > 2D 2

and

3

|α (¯x) − α (¯x)| = |5.5355 + 1.2083¯x1 | > 2D 5 In order to transmit high-frequency signals, the state velocity of the chaotic system x ˙ can be multiplied by ζ , translating the eigenvalues of each LS ζ times. In such a case, the observer state needs to be multiplied by ζ in order to maintain the same behavior at any time scale.

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FIG. 3.1 Preservation of the multiscroll attractors under small disturbances.

It can be seen by numerical simulation that the evolution inside the basin of attraction satisfies x3 ∈ (−2.5, 2.5) as shown in Fig. 3.1.

4 OBSERVER DESIGN FOR SLS UNDER DISTURBANCES In this section we show that if the continuous and the discrete states of the SLS are observable according to Theorem 4, then the SLS admits an observer based on a multiobserver structure, in which a finite-time observer is designed for each LS composing the SLS. It will be shown that, based on the observability results previously presented, a set of finite-time observers allow us to decide the evolving LS from the output estimation error. The observability of the SLS implies that the observer associated with the evolving LS will produce a zero output estimation error, whereas the rest of the observers, whose associated with LS that are distinguishable from the evolving one, cannot produce a zero output estimation error. Thus, the evolving LS is the one associated with the observer with zero output estimation errors, whereas an exact estimation of the continuous state is provided by such an observer. This observer extends the design proposed in [39], in order to consider unknown inputs. Definition 8. Let u[τ1 ,τ2 ] , y[τ1 ,τ2 ] be a measurable input-output behavior of the SLS Σσ (t) = F, σ  in the time interval [τ1 , τ2 ] and let Σi , Σj ∈ F, then the LS Σi and Σj are said to be (u[τ1 ,τ2 ] , y[τ1 ,τ2 ] )-indistinguishable, if there exist state trajectories (not necessarily starting from the same initial

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condition) xi (t, x(τ1 ), u[τ1 ,τ2 ] , d[τ1 ,τ2 ] ) of Σi and xj (t, x (τ1 ), u[τ1 ,τ2 ] , d[τ1 ,τ2 ] ) of Σj , such that yi (t, x(τ1 ), u[τ1 ,τ2 ] , d[τ1 ,τ2 ] ) = yj (t, x (τ1 ), u[τ1 ,τ2 ] , d[τ1 ,τ2 ] ) = y[τ1 ,τ2 ] Otherwise, Σi and Σj are said to be (u[τ1 ,τ2 ] , y[τ1 ,τ2 ] )-distinguishable. The construction of the observer for each LS is based on the multivariable observer form (see, e.g., [67]). Let us first introduce some results about the transformation of an observable LS into such a form. Lemma 8. If the LS is observable under partially unknown inputs, that is, if sup (A, S; K) is the trivial subspace, then there exists a set of integers {r1 , . . . , rq } such that ∀i ∈ {1, . . . , q}∀k ∈ {0, . . . , ri − 2}, ci Ak S = 0, and rank(O{r1 ,...,rq } ) = n where ⎛ ⎞ c1 .. ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ ⎜ c1 Ar1 −1 ⎟ ⎜ ⎟ ⎜ ⎟ .. O{r1 ,...,rm } = ⎜ ⎟ . ⎜ ⎟ ⎜ cq ⎟ ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . cq Arq −1 .

Proof . The proof follows a similar development as the one shown in [43, Section 4.3.1]. Consider the following iterative process; for simplicity, let B = 0, as the control input does not affect the observability under unknown inputs in LS. Let q0 (t) = Y0 x(t)

(3.22)

with q0 (t) = y(t) and Y0 = C. Differentiating Eq. (3.22) yields q˙ 0 (t) = Y0 Ax(t) + Y0 Sd(t). Let P0 be a projection matrix such that Ker P0 = (Y0 S). Thus (3.23) P0 q˙ 0 (t) = P0 Y0 Ax(t)

q0 Y0 then Eqs. (3.22), (3.23) can be combined and Y1 = Let q1 = P0 q˙ 0 P0 Y0 A to get q1 (t) = Y1 x(t). It is easy to see that



Ker Y1 = Ker Y0 ∩ Ker(P0 Y0 A) = K ∩ A−1 (K + S) The kth iteration of the procedure yields qk (t) = Yk x(t) such that Ker Yk = Ker Yk−1 ∩ A−1 (Ker Yk−1 + S) = K ∩ A−1 (Ker Yk−1 + S)

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Notice that such an iteration process coincides with EQ. 3.7, and therefore converges to sup (A, S; K). Reordering Yk yields the matrix O{r1 ,...,rq } . Because, after a sufficient number of iterations k, Ker Yk = sup (A, S; K) = 0 then rank(O{r1 ,...,rq } ) = n. The values {r1 , . . . , rq } are known as the observability indices of Σ(A, B, C, S) [67]. An LS observable under unknown inputs can be transformed by means of the similarity transformation, into the multivariable ¯ B, ¯ S) ¯ 6 (see [67, 68]) with matrices ¯ C, observer form Σ(A, ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ C1,r1 · · · 01,rq Ar1 · · · Or1 ,rq Sr1 ,m ⎢ . ⎥ ⎥ . .. . .. ⎥ .. .. ¯ =⎢ ¯ =⎢ A ⎣ .. ⎣ .. ⎦ , S¯ = ⎣ .. ⎦ , C . . . . ⎦ Srq ,m Orq ,r1 · · · Arq O1,r1 · · · C1,rq where 01,ri is a 1 × ri zero matrix, Ori ,rj is a ri × rj zero matrix with possible nonzero entries at the first column, Ari is a ri × ri matrix, and C1,ri is a 1 × ri matrix of the form ⎡ ⎡ ⎤ ⎤ * 1 0 ··· 0 0 ··· 0 ⎢* 0 1 · · · 0 ⎥ ⎢0 · · · 0⎥ ⎢ ⎢ ⎥ ⎥  ⎢ .. .. .. . . ⎥ . . .⎥ .. ⎥ , Sri ,m = ⎢ Ari = ⎢ . . . ⎢ .. . . . .. ⎥ , C1,ri = 1 0 · · · 0 . ⎢ ⎢ ⎥ ⎥ ⎣* 0 0 · · · 1 ⎦ ⎣0 · · · 0⎦ * 0 0 ··· 0 * ··· * where the entries marked by “*” represent possible nonzero values. The matrix B has no particular structure. Notice that Σ is formed by q single output subsystems of dimensions r1 , . . . , rq coupled only by the measured variables. Notice that the transformed LS is composed of blocks coupled only through measured variables (¯x1 , x¯ r1 +1 , . . .). Each ith block is of dimension ri and is of the form x˙¯ i1 = ai1 (y) + x¯ i2 + B¯ i1 u(t) x¯˙ i2 = ai2 (y) + x¯ i3 + B¯ i2 u(t) .. . x¯˙ ir = air (y) + B¯ ir u(t) + d¯ i (t) i

i

(3.24)

i

j = 1, . . . , ri is a linear combination of the output variables x¯ 1 , x¯ r1 +1 , . . . and d¯ i (t) = S¯ ri ,m d(t). It follows from Assumption 5 that the disturbances d¯ i (t) are differentiable and satisfy L ≥ |d˙¯ (t)| with known constants L , i = 1, . . . , m. where the

aij (y)

i

i

i

The observer form can be obtained taking T−1 = O{r1 ,...,rq } as a baseline. The procedure for computing this transformation can be obtained as the dual of the controller form presented in [67, Example 6.4-7, p. 436].

6

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Thus, we can add to Eq. (3.24) the dynamic of d¯ i , that is, d˙¯ i = υi (t) Then, an observer based on the robust differentiator described in [37] can be designed for each block in order to estimate the state of the system in the presence of disturbances. The observer is designed in such a way that its error dynamics coincide with the differentiation error of [37], for which convergence has already been demonstrated. Thus, each block admits the following observer x˙˜ i1 = ai1 (y) + x˜ i2 + B¯ i1 u + l1 ρ|xi1 − x˜ i1 |ri /(ri +1) sign(xi1 − x˜ i1 ) x˜˙ i2 = ai2 (y) + x˜ i3 + B¯ i2 u + l2 ρ 2 |xi1 − x˜ i1 |(ri −1)/(ri +1) sign(xi1 − x˜ i1 ) .. . x˜˙ ir = air (y) + B¯ ir u(t) + ξi + lr ρ ri |xi − x˜ i |1/(ri +1) sign(xi − x˜ i ) i

i

i

i

ξ˙i = lri +1 ρ ri +1 sign(xi1 − x˜ i1 )

1

1

1

(3.25)

1

Defining the error dynamics as eij = xij − x˜ ij j = 1, . . . , m and eir +1 = d¯ i − ξ , i yields to e˙i1 = ei2 − l1 ρ|ei1 |ri /(ri +1) sign(ei1 )

e˙i2 = ei3 − l2 ρ 2 |ei1 |(ri −1)/(ri +1) sign(ei1 ) .. . e˙iri

=

e˙ir +1 i

=

(3.26)

− lri ρ ri |ei1 |1/(ri +1) sign(ei1 ) υi (t) − lri +1 ρ ri +1 sign(ei1 ) eir +1 i

Because the m subsystems are only coupled through the measured variables, which are fed into the observer by output injection through the terms aij (y), the error dynamics of the subsystems are independent from each other and coincide with the form of the differentiation error of the highorder sliding mode differentiation presented in [37]. Thus, with the proper choice of the observers’ parameters ρ and li , a finite-time convergence to the continuous variables can be obtained in the presence of the disturbance d(t) [37], with an arbitrary small convergence time. Thus, an estimation of the variables x¯ i1 , . . . , xiri is obtained in finite time. By designing an observer (3.25) for each block, the continuous state x(t) of the LS can be estimated in finite time. Proposition 7. Let the initial conditions of the observer (3.25) be taken as zero, and let the continuous initial condition of the SLS (3.2) be bounded by δ, that is,

x0 < δ, with a known constant δ, as in Assumption 2. Then, for every constant τk the gains of Eq. (3.25) can be designed in such a way that the estimation error (3.26) converges to the origin with an upper convergence bound of less than τk . Proof . Consider the error dynamics given in Eq. (3.26). Take ρ ≥ 1 and consider the time-scaling ˜t = tρ together with the coordinate change I. OBSERVER DESIGN

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  ek = P˜ek with P = diag(1, ρ, . . . , ρ rk ) and e˜ k = e˜k1 · · · e˜krk ξ˜ k . These transformations and time scaling take Eq. (3.26) into the following form d(˜ek1 ) d˜t d(˜ekr ) k d˜t d(ξ˜ k ) d˜t

= e˜k2 − k1 ˜ek1 α1 .. . = ξ˜ k − krk ˜ek1 αrk

(3.27)

= −krk +1 ˜ek1 αrk +1

which does not depend on ρ. Therefore, due to the finite-time stability of Eq. (3.27), ∀δ > 0, exists τ˜k such that it holds e˜ (˜t, e˜ 0 ) = 0,

∀˜e0 such that ˜e0 < δ and ∀˜t ≥ τ˜k

where e˜ 0 = e˜ (t0 ). Going back to original coordinates e and real time t, the preceding implies that e(t, e0 ) = 0,

∀e0 such that e0 < δ and ∀t ≥ τ˜k /ρ

Indeed, the preceding implication is correct, as the inequality e0 < δ clearly implies that ˜e0 < δ due to the straightforward inequality

˜e ≤ e ,

∀ρ ≥ 1

Therefore, ∀δ, τk > 0 there exists ρ(δ, τk ) such that e(t, e0 ) = 0, ∀e0 such that

e0 < δ and ∀t ≥ τk . So, one really has to make the assumption that the initial conditions are bounded by some δ > 0, as in Assumption 2, but for every bound δ and every time τk > 0, there exist gains such that the error dynamics (3.26) goes to zero in finite time less than τk . Another alternative is to design the individual observers such that the convergence error coincides with the error of the uniformly convergent, robust-exact differentiator proposed in [38]. The main advantage of that algorithm is that it has a fixed time-convergence bound that is independent of the initial condition, where a similar argument can be made to show that a uniformly convergent observer can also be designed with an upper convergence bound less than τk . Finally, assuming that each pair of LS Σi and Σj are (u[τ1 ,τ2 ] , y[τ1 ,τ2 ] )distinguishable under unknown inputs, if an observer is designed for each LS with time convergence bound τ < τ1 < τ2 , then the observer associated with the evolving LS will be the only one maintaining ey (t) = y(t) − y˜ (t) = 0 ∀t ∈ [τ1 , τ2 ]. Lemma 9. Let Σσ (t) be an SLS and Σˆ σˆ (t) its observer with time convergence bound τ and let σ (t) = i, ∀t ∈ [t0 , t1 ). Then, if Σi and Σj are distinguishable under unknown inputs, according to Proposition 4, then for almost every input j j u(t), ey (t) = 0 almost everywhere in [τ , t1 ) (where ey (t) is the output error signal ey (t) = y − y˜ j of the jth observer).

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101

FIG. 3.2 Observer for SLS under disturbances.

Proof . The proof follows by noticing, from Eq. (3.25), that whenever the j

j

output error signal ey satisfies ey = 0, then the observer becomes a copy of its associated LS, driven by a perturbation signal ξ and producing the same output of Σi , because due to finite-time convergence of Eq. (3.25), the error correction terms in Eq. (3.25) associated with each block becomes equal to 0. j

However, as the Σi and Σj are distinguishable, then ey = 0 cannot occur on a nonzero interval. Thus, it follows by the assumption that Σi and Σj are distinguishable under unknown inputs that whenever σ (t) = i, ∀t ∈ [τ , t1 ) only the observer associated with Σi satisfies eiy = 0 in the interval [τ , t1 ). Hence, a multiobserver structure depicted in Fig. 3.2 can be used to estimate the continuous state x(t) of the SLS and to compute the switching signal σ (t). Notice that, because by Lemma 9 only the observer associated with the evolving LS gives ey = 0, then by analyzing this error signal, the evolving LS can be ascertained. In a similar way, once the evolving LS Σj has been detected, the switching occurrence to another LS, say Σj , can be detected by the time when the error signal no longer satisfies ey = 0, because by the pairwise distinguishability, when switching from Σi to Σj , the signal eiy can no longer be maintained at zero from a nonzero interval. Proposition 8. Let the continuous and discrete state of Σσ (x) be observable under unknown inputs, that is, each LS Σi ∈ F is observable under unknown inputs, and pairwise, the LS in F is distinguishable under unknown inputs, and I. OBSERVER DESIGN

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let Tk be the similarity transformation taking the LS Σk into the multivariable observer form. Then the state x(t) of Σσ (x) and the switching signal σ (x) can be estimated by the following procedure: 1. Design an observer (3.25) with time convergence bound τ  τd for each Σi ∈ F. 2. The current state of σ (x) is k if eky (t) = 0, ∀t ∈ [τ , τ + ε] with τ + ε  τd . That is, the evolving LS is Σk if its associated observer is the only one satisfying ey = 0 for a small time interval. If the LS Σk is detected x˜ (t) = Tk x(t) with ξ˜ (t) = ξ˜ k (t). 3. A switching is detected when the observer associated with Σk no longer satisfies ey = 0. 4. After the switching tj is detected the state of the observer i, i ∈ {1, . . . , l}, is i − reinitialized as x˜ i (tj ) = Ti−1 Tk x˜ k (t− j ) where x (tj ) is the estimated state of the LS Σi at the switching time t− j . 5. The next state of σ (x) is l such that the observer associated with l is the only one satisfying ely (t) = 0, ∀t ∈ [tj , tj + ε].

Proof . The proof follows by the previous argument, and from the observability result, which requires each pair of LS to be distinguishable for almost every control input. Proposition 9. Let Σσ (t) be an SLS and let Σˆ σˆ (t) be the proposed observer. Then if the switching signal of Σσ (t) is observable for almost every input then only the observer of Σi will give eiy (t) = 0 for all t ∈ [τk , t1 ]. Furthermore only eiy (t) can be 0 in a nonzero interval, and the index of the evolving system can be obtained by the only observer, giving ey (t) = 0 for all t ∈ [τk , τk + ε], with δ ≈ 0. Next, let us explain how each switching can be detected, and how to completely estimate the switching signal. For this, let us assume that the LS Σi is evolving, and it is detected by the proposed observer, and that at time t1 a switching occurs from Σi to Σj . Then, the output of the evolving state trajectory is y(t) = yi (t, x0 , u[t0 ,t1 ) , d[t0 ,t1 ) ) for t ∈ [t0 , t1 ) and y(t) = yj (t, x(t1 ), u[t1 ,t2 ) , d[t0 ,t1 ) ) for t ∈ [t1 , t2 ). Thus, the switching time t1 can be detected if yj (t, x(t1 ), u[t1 ,t2 ),d[t

0 ,t1 )

) = yi (t, x(t1 ), u[t1 ,t2 ) , d[t0 ,t1 ) )

(3.28)

This inequality holds because Σi and Σj are distinguishable. Thus, Σi and Σj cannot produce the same input-output information during [t1 , t2 ), and thus Σi can no longer provide ey (t) = 0 during a nonzero interval. Thus, the switching time t1 can be detected as established in the following lemma. Lemma 10. Let Σσ (t) be an SLS and let Σˆ σˆ (t) be the proposed observer and let i ey (t) = be the signal ey of the observer associated with the observable subsystem of Σi . Then if the switching signal of Σσ (t) is observable for almost every input, then

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103

the switching time is detected as the time t > τd for which eiy (t) is no longer zero. In fact, eiy (t) = 0 almost everywhere in [t1 , t2 ). In a similar way, any subsequent switching time can be detected. Theorem 5. Let Σσ (t) be an SLS and let Σˆ σˆ (t) be the proposed observer scheme. If the continuous state of Σσ (t) is observable for almost every state trajectory, then for almost every state trajectory σˆ (t) = σ (t) and xˆ (t) = x(t) for all t > τk , with τk a finite time. Remark 2. Notice that, for the case of unperturbed SLS, whenever T  T ey (t) ξ T (t) = 0 the observer becomes an exact copy of its associated LS, producing the same input-output information as the evolving LS. Thus, for the case of unperturbed systems, which has been considered in [39], the previously discussed observer can be applied by substituting ey by  T (t) = eTy (t) ξ T (t) in the analysis. Remark 3. Because a finite-time estimation is achieved, and no finite escapes can occur in SLS, the proposed observer can be designed separately from a global controller. Example 4. Now, if we consider the SLS composed of LS with system  T matrices as in Table 3.1, input matrices B1 = 1 2 0 and B2 =  T   0 1 0 , output matrices C1 = 0 0 1 and C2 = 0 1 −1 , and T  T  disturbance matrices S1 = 1 0 0 and S2 = 1 − 12 − 12 , then the indistinguishability subspace W12 is ⎡ ⎤ −1 0 0 0 0 ⎢ 0 1 0 0 0⎥ ⎢ ⎥ ⎢ 0 0 0.8165 0 0⎥ ⎥ W12 =  ⎢ ⎢ 0 0 0 −1 0⎥ ⎢ ⎥ ⎣ 0 0 0.4082 0 0.7071⎦ 0 0 −0.4082 0 0.7071 It is easy to verify that both Σ1 and Σ2 are observable under unknown inputs, in addition, Σ1 and Σ2 are distinguishable from each other for almost every input as B12  W12 . Thus, according to Theorem 4, the continuous and discrete state of the SLS Σσ (t) are observable, in infinitesimal time, for almost every control input u(t), and the proposed observer design can be applied to estimate the continuous and discrete state of Σσ (t) . Fig. 3.3 illustrates the application of Lemma 9 to infer the evolving LS. It can be seen that the output estimation error e1y , related to the observer of Σ1 , converges to zero in finite-time, whereas the output estimation error e2y , related to the observer of Σ2 , cannot be zero (when the system evolves in Σ1 ) for a nonzero interval, allowing us to infer the evolving LS and the discrete state σ . In this example, the detection of the switching time is trivial, as the continuous output is discontinuous. At the switching

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Time (s) FIG. 3.3 Inferring the evolving LS from the output estimation error.

instants, the observers are reinitialized, as described in Proposition 8. After the reinitialization at 3.5 s, only the output estimation error of the observer associated with the evolving LS can be maintained as 0, in this case, the signal e2y is the only one that remains at 0. Hence, the proposed observer determines the evolving LS and the switching signal, using only the output information y(t) in spite of the unknown disturbance affecting the system. Fig. 3.4 shows the estimation of the continuous state using the procedure described in Proposition 8. The continuous and discrete states of the SLS are estimated in finite-time by the proposed observer, using only the output information y(t) in spite of the unknown disturbance d(t). It is worth noticing that, in the case where the disturbance is scalar, the proposed observer also estimates the unknown disturbance d(t). However, in general, the unknown disturbance is not estimated. The design of a unknown input observer for SLS can be derived in a straightforward manner, if in addition to the observability under unknown inputs, we require each LS to be invertible (i.e., two different disturbance signals cannot produce the same output).

5 OBSERVER DESIGN FOR SAS UNDER BOUNDED DISTURBANCES WITH NONAUTONOMOUS CHAOTIC MODULATION An SAS may exhibit a complex nonlinear behavior, such as chaos, under a suitable selection of the affine subsystems and the switching

I. OBSERVER DESIGN

BOUNDED DISTURBANCES WITH NONAUTONOMOUS CHAOTIC MODULATION

FIG. 3.4 Estimation of the continuous state x(t).

I. OBSERVER DESIGN

105

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3. ON THE OBSERVABILITY AND OBSERVER DESIGN IN SWITCHED LINEAR SYSTEMS

rule. Chaotic SAS exhibit properties, such as wide spread spectrum, dense periodic orbits, and strong dependence on the initial conditions. These features make them suitable for communication applications, mainly because broadband information carriers enhance the robustness of communication channels against interferences with narrow-band disturbances, which is the basis of spread-spectrum communication techniques. In chaos-based communications, the broadband coding signal is generated at the physical layer, rather than algorithmically, as in code division multiple access [69]. Additionally, the irregular signals and seeming randomness of chaotic systems make them useful to hide information, enhancing software-based encryption, to achieve privacy in the communication [63, 64]. One of the chaos-based modulation methods is the nonautonomous chaotic modulation [63] (also known as message-embedded modulation [64–66, 70]), which has been previously considered using the Lorenz system [70] and the generalized Lorenz system [64] as the chaotic attractors. In this modulation method, a message is embedded by means of a nonlinear function that is then fed to the chaotic system as an input. The modulated signal (which is an analog signal) to be transmitted is obtained as the output of the chaotic system. The receiver recovers the message from the transmitted signal by synchronization with the emitter. From a control theory perspective, the original message is estimated from the modulated signal by means of a disturbance observer that takes advantage of the knowledge of the nominal system. In our approach, the chaotic attractor for chaos-based nonautonomous modulation is assumed to be generated by an SAS. One of the main advantages of using SAS is the simple circuitry required for the implementation of the chaotic system, for example, using Chua’s circuit [50], DC-to-DC converters [71], or the general jerk circuit [61, 72]. In this section, we show that the proposed observer design for perturbed SISO SAS can be applied for the nonautonomous chaotic modulation using general chaotic attractors generated by SAS (see, e.g., [7, 9, 10, 50, 62, 73, 74]). This modulation/demodulation process is depicted in Fig. 3.5. Consider an SAS composed by strongly observable AS and let Ti such that ⎤ ⎤⎡ ⎡ 1 0 ··· 0 ci ⎥ ⎢ ai ⎢ 1 ··· 0 ⎥ ⎥ ⎢ ci Ai ⎥ ⎢ 1 −1 Ti = ⎢ . (3.29) ⎥ ⎢ . . . .. . . 0 ⎦ ⎣ .. ⎥ ⎦ ⎣ .. ain−1 ain−2 · · · 1 ci An−1 i

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107

FIG. 3.5 Chaotic modulation/demodulation process.

be the transformation taking Σi (Ai , bi , ci , si ) into the observable canonical form, where ai1 , . . . , ain are the coefficients of the characteristic polynomial ˜ equation (3.14) of Ai . Then, with d(t) = ξ(t), the observer (3.25) becomes x˙˜ 1 = −ai1 y + x˜ 2 + bˇ i1 + l1 ρ|y − x˜ 1 |n/(n+1) sign(y − x˜ 1 ) .. . (3.30) ˜ + ln ρ n |y − x˜ 1 |1/(n+1) sign(y − x˜ 1 ) x˙˜ n = −ain y + bˇ in + d(t) ˙ d˜ = ln+1 ρ n+1 sign(y − x˜ 1 ) y˜ = x˜ 1 where l1 , . . . , ln and ρ are observer parameters to be adjusted and bˇ ij is the jth element of the vector bˇ i = Ti −1 bi . The state and the disturbance estimates are given by xˆ (t) = Ti x˜ and ˆd(t) = 1 d(t), ˜ respectively. In other words, if the AS Σi is evolving with βi the state trajectory x(t) and the affecting disturbance d(t) then, after the ˆ = d(t) observer converges in finite-time, the estimates xˆ (t) = x(t) and d(t) will be produced. Let us provide a couple of comments regarding practical issues during the observer’s synthesis. • If the gains are appropriately adjusted, the observer (3.30) corresponding to the evolving AS will accurately estimate the state and the disturbance in finite-time. See [37] for a selection of the observer gains for up to the fifth order. See [75] for the gain

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selection of a second-order observer, together with a tight estimation of the convergence time bound. For a third-order observer, a proper gain selection is l1 = 9.5608, l2 = 6.8681, l3 = 0.0219, and ρ n−1 > |βi |L/0.0081 [76], for a convergence time bound see [76]. • In our framework, it is expected that the observer corresponding to the evolving AS converges before the first switching. Thus, the time convergence must be lower bounded by the dwell time τd imposed in Assumption 3. It is known that such a convergence bound can be obtained with an appropriate selection of the observer gains, as the initial condition lies in a known bounded set according to Assumption 2. It is shown in Proposition 7 that such convergence bound can be obtained with an appropriate selection of the observer gains, provided that the initial condition is bounded as required by Assumption 2. In particular, assume that using the gain selection ¯l1 , ¯l2 , . . . , ¯ln+1 , ρ, ¯ a time convergence bound T¯ f is obtained, then the convergence time bound for a gain selection l1 = ρ¯¯l1 , l2 = ρ¯ 2¯l2 , . . . , ln+1 = ρ¯ n+1¯ln+1 and ρ ≥ 1 is given by Tf = T¯ f /ρ. The next proposition demonstrates that if the observer of another AS converges, then the estimate of the disturbance will be equal to that of Eq. (3.19). Thus, by means of such estimated disturbance and Lemma 6, the estimations provided by such an observer can be discarded. Proposition 10. Let Σi be the evolving AS with x(t) as the state trajectory and d(t) as the affecting disturbance. If the output estimation error of the observer j associated with Σj , denoted as ey = y − y˜ j , becomes 0, then the observer will ˆ making Σi outputproduce an estimate of the state xˆ (t) and the disturbance d(t) ˆ indistinguishable from Σj , where d(t) has the form of Eq. (3.19). Proof . For the sake of simplicity, assume that the AS Σi and Σj are represented in the observer canonical form [67]. Let Σi be the evolving AS with x(t) being the state trajectory, and let x˜ j (t) be the state of the observer j j associated with Σj . Denote the entries of the error vector as eˆk = xk − x˜ k , j

j

k = 1, . . . , n. Thus, if y − x˜ 1 = eˆ1 = 0 then the dynamic behavior of  T eˆj = eˆj1 · · · eˆjn becomes j j j 0 = eˆ2 + (−ai1 y + bˇ i1 ) − (−a1 y + bˇ 1 ) j j j j e˙ˆ2 = eˆ3 + (−ai2 y + bˇ i2 ) − (−a2 y + bˇ 2 ) .. .

j j j e˙ˆn = (−ain y + bˇ in + β i d(t)) − (−an y + bˇ n + d˜ j (t))

I. OBSERVER DESIGN

(3.31)

BOUNDED DISTURBANCES WITH NONAUTONOMOUS CHAOTIC MODULATION

109

Differentiating the first equation and combining it with the second j equation in Eq. (3.31), we get eˆ3 = −(−ai1 y˙ − ai2 y + bˇ i2 ) + (−ak1 y˙ − ak2 y + bˇ i2 ). j

j

Differentiating eˆ3 and combining it with Eq. (3.31), we get eˆ4 = −(−ai1 y¨ − j ai2 y˙ + ai3 y + bˇ i2 ) + (−ak1 y¨ − ak2 y˙ + a3 y + bˇ i2 ). Following this procedure, we get j

(n−2)

that eˆn = −(−ai1 y j

(n−1) j − · · · − ain−1 y + bˇ i2 ) + (−ak1 y − · · · − an−1 y + bˇ i2 ).

Differentiating eˆn and combining it with the last equation of Eq. (3.31), we get that dˆ j (t) = 1j d˜ j (t), with d˜ j (t) = (α(i, x¯ ) − α(j, x¯ ) + β i d(t)), thus dˆ j (t) is β

j

equal to Eq. (3.19). Because with ey = y− y˜ j = 0 the observer (3.30) becomes a copy of Σj represented in the canonical observer form, producing the same output information as Σi , then the state estimate obtained by the observer associated with Σj is the one given in Proposition 6. Following the observer scheme depicted in Fig. 3.2, the evolving AS ˜ detects the only observer that satisfies |d(t)| ≤ D and e(t) = y(t) − y˜ (t) = 0 for a proper time interval. The state estimate is given by the observer of the evolving AS, once it is determined. This is formally stated in the following proposition. In the sequel, let us denote as xˆ i (t), yˆ i (t), and dˆ i (t) the estimates of the state, the output, and the disturbance provided by the observer of the ith AS, respectively. Similarly, let us denote as ei (t) = y(t) − yˆ i (t) the estimation output error provided by such an observer. Proposition 11. Let Σσ (t) be an SAS and consider a collection of observers of the form (3.30), one for each AS in the SAS, evolving in parallel, as depicted in Fig. 3.2. Suppose that the system remains in the initial AS a time longer than τd , and suppose that each observer has a time convergence bound τ  τd . Suppose that the system evolves inside a region in which the output and their derivatives ij fulfill x¯ ∈ Bx¯ , for every pair of ASs Σi and Σj , as defined in Lemma 7. Then, the state of the switching signal σ (t) can be detected by the index k of the only kth observer satisfying |dˆ k (t)| ≤ D

and

eky (t) = 0

∀t in a proper interval [τ , τ + Δt],

Δt > 0 (3.32) Once it is inferred that the evolving AS is Σk , an exact estimate of the continuous state of the SAS and the affecting disturbance is provided by the observer associated with Σk . Proof . If Σi is evolving with x(t) as the state trajectory and d(t) as the affecting disturbance, then the observer associated with the AS Σj either gives ey = 0 ∀t > τ , from which it can be asserted that Σk is not the evolving AS, or it precisely produces an estimate of the state xˆ k (t) and the disturbance dˆ k (t) when Σi is output-indistinguishable from Σk , according to Proposition 10. However, the conditions of Lemma 7 are satisfied, thus

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if i = k (i.e., for an observer not associated with the evolving AS) then the condition (3.32) cannot hold for a proper time interval. Consequently, |dˆ k (t)| > D and thus it can be asserted that Σk is not the evolving AS. On the contrary, if i = k (i.e., for the observer associated with the evolving AS) then, exact estimates of the evolving state trajectory x(t) and the affecting disturbance d(t) are obtained by the observer, which clearly is consistent with the knowledge on the disturbance bound, that is, Eq. (3.32) is satisfied. According to the previous proposition, the evolving AS can be detected as the index k of the only observer satisfying Eq. (3.32). After a switching occurrence, the same observer can no longer maintain the condition (3.32). Thus, the switching occurrence is detected when such a condition no longer holds. Based on the previous propositions, the observer structure is presented in the next algorithm. The observer structure estimates the switching signal, the continuous state, and the affecting disturbance. Proposition 12. Suppose that the state of Σσ (x) evolves in a state region such that the output and their derivatives fulfill the condition on Lemma 7. Then the state x(t) of Σσ (x) , the disturbance d(t) and the switching signal σ (t) are estimated by the proposed observer, depicted in Fig. 3.2 by using the condition (3.32) to infer the evolving AS and detect the switching time when such a condition no longer holds. Proof . Proposition 11 guarantees that if the AS Σk is evolving, then only its associated observer will satisfy condition (3.32) for a proper time interval, thus the evolving system is detected, and exact estimates of the continuous state x(t) and the affecting disturbance d(t) are given by xˆ k (t) ˆ and d(t), respectively. Next, after a switching occurrence, the condition of Proposition 11 can no longer hold, thus the switching time is detected. Because no jumps occur in the continuous state of Σσ (x) , by reinitializing each observer with the estimated value at the switching time, all the observers have accurate estimates of the state and the disturbance, but only the observer associated with the new evolving system will satisfy condition (3.32). Consequently, no additional time is required for the convergence of the observer scheme. Following in this way, the switching signal σ (t), the continuous state x(t) and the affecting disturbance d(t) are continuously estimated. Example 5. Let us consider, for example, the multiscroll attractor Σσ (t) proposed in [62] given in Example 3. Notice that the state x is unknown at the receiver, thus, if the switching depends on an unmeasured state, then the switching signal σ (t) is unknown at the receiver. Because of the message-embedding method of Fig. 3.5 using the nonlinear function g(x, m), the estimates of both the continuous state and

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the disturbance are required for recovering the hidden message m(t). Moreover, once x(t) and d(t) are estimated, the recovery of the hidden ˆ is straightforward. For this message by means of the function h(ˆx, d) reason, in this example, we focus on the estimation of both x(t) and d(t) and for simplicity the function g(x, m) of Fig. 3.5 is assumed to be such that g(x, m) = m, hence d(t) = m(t) (such method is known as nonautonomous chaotic modulation); however, different g(x, m) functions enhancing the security of the communication can be straightforwardly addressed. Although measurement (or channel) noise is not affecting the system, this example has practical applications in fiber-optic and visible-light communications, for example, visible light communication systems in indoor spaces have a very high signal-to-noise ratio (SNR), in the range of 40–70 dB [77, 78]. Under such SNR, the effect of noise is negligible. Let us analyze the fulfilling of assumptions and the condition of Lemma 7 for this application example. • First, it is always possible to impose a suitable bound for the message, and hence, on the disturbance. • The evolution inside the basin of attraction of a chaotic attractor is confined inside an invariant bounded set [79]. For example, such a bounded set can be obtained by using the methodology proposed in [80], where piecewise quadratic Lyapunov functions are used to derive tight bounds for the chaotic oscillations, and for the evolution of each individual AS. • Regarding the assumption on the minimum dwell time for the first switching, regions for suitable initial conditions guaranteeing that no switching may occur before τd can be found by using minimum-time control methods, which allow us to obtain, by computational methods, the set of states that can be reached from x0 by a bounded control with time t ≤ τd (in our case a bounded disturbance), see [43, Algorithm 3.5.1 and Problem 3.5.1]. Thus, appropriate bounds for the initial conditions can be provided by the system designer. Now, let us report the results. The demodulation process, that is, the chaotic synchronization and the estimation of the signal d(t), for the chaotic system Σσ (t) described herein is shown in Figs. 3.6–3.8. In Fig. 3.6, in time point ❶ it is shown that only the observer associated with the evolving AS Σ3 is able to satisfy the conditions of Proposition 11 for a proper time interval before the first switching. Thus, it is asserted that Σ3 is evolving. Next, the switching occurrence ❷ is detected when the

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FIG. 3.6 From top to bottom: SAS output versus output estimation of each observer, disturbance versus estimation of the disturbance of each observer, switching signal. Estimation process: ❶ Estimation of the evolving AS. ❷ Switching occurrence. ❸ Switching detection. ❹ Reinitialization of the observer. ❺ Detection of the subsequent evolving AS.

observer associated with Σ3 no longer maintains |d˜ k (t)| < D with eky (t) = 0, as indicated by ❸. Once the switching occurrence is detected, then each observer is reinitialized, as indicated in ❹, and after the reinitialization, only the observer associated with Σ2 is able to maintain the condition |dˆ k (t)| < D with eky (t) = 0 as shown by ❺. Consequently, the switching signal σ (t), the continuous state x(t), and the signal d(t) can be estimated, as shown in Fig. 3.7.

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FIG. 3.7 Chaotic synchronization; estimation of the continuous state x(t), the switching signal σ (t), and the information signal d(t).

The discontinuities in the estimated variables that appear in Figs. 3.7 and 3.8 occur because the initial condition of the estimated switching signal was σˆ (t0 ) = 1, thus the continuous state and the disturbance were estimated as ˆ = dˆ 1 (t), respectively. Once the evolving AS is detected, xˆ (t) = xˆ 1 (t) and d(t) this value was updated to σˆ (t0 ) = 3, and the estimates of the continuous ˆ = state and the affecting disturbance were updated to xˆ (t) = xˆ 3 (t) and d(t) ˆd3 (t). The estimate of the signal d(t) by the SAS observer is also shown in Fig. 3.8.

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FIG. 3.8 Estimation of the signal d(t) by the SAS observer.

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C H A P T E R

4 On Unknown Input Observer Design for Linear Systems With Delays in States and Inputs Seifeddine Ben Warrad*, Olfa Boubaker*, Mihai Lungu † , Quan Min Zhu ‡ *National Institute of Applied Sciences and Technology, University of Carthage, Tunis, Tunisia † Faculty of Electrical Engineering, University of Craiova, Craiova, Romania ‡ Department of Engineering Design and Mathematics, University of the West of England, Bristol, United Kingdom

1 INTRODUCTION Time-delay systems are frequently encountered in various engineering systems, such as chemical and biological processes, hydraulic systems, and manufacturing processes. In the past several decades, the analysis and control of continuous-time delay systems has become more widely documented in the literature [1–3]. In this framework, the emergence of delays in dynamical systems was diagnosed as one of the major causes of loss of stability and degradation of performance, and significant research activities have been developed to deal with this fact, which is considered an inevitable phenomenon in stability and control systems [4]. Systems represented by a set of delayed ordinary differential equations are called time-delay systems, and, in many cases, they are considered to be distributed parameter systems. The literature shows that interest has grown significantly in the past decade in regard to the observer’s design for such systems [4], in which two types of models are found: delay-dependent observers [5, 6] and delay-independent observers [7, 8].

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Delay-dependent observers have more complex models, as they use the information about the time delay to estimate the desired functions; whereas the delay-independent observers do not require any information on the time delay to reach the same objective. In this framework, different approaches have been suggested, namely the infinite dimensional approach [9], the polynomial approach based on the ring theory [7, 8], the finite spectrum assignment theory [10], the Lyapunov functional approach based on the Lyapunov-Krasovskii theory [6], and the LyapunovRazumikhin theory [11]. In the same context, a great deal of attention has also been paid to systems with multiple time delays, for which the well-known limitations are the presence of delays, either in the states of the plant or in the plant’s inputs and outputs [12]. The problem of estimating the state variables with delays is of much significance in many applications. For linear timeinvariant (LTI) multivariable systems with multiple time delays, only a few elaborated results can be found in this framework [12–15]. On the other side, researchers have shown interest, for many decades, in the design of unknown input observers (UIOs) for LTI systems without delays [16–21]. However, when it comes to including multiple delays in the model, which might appear in the states, the inputs, or the outputs, the observer design becomes more complicated. In this work, we investigate two different approaches for designing fullorder observers with unknown inputs and delays in states and inputs: a delay-dependent UIO, and a delay-independent UIO. The delayed system has instantaneous control inputs and delayed control inputs, which is a problem that has never been solved until now, to the best of our knowledge. The necessary and sufficient conditions for the asymptotic stability and the existence conditions of the two observers are provided, and a comparative analysis is also established. The chapter is organized as follows: Section 2 states the problem related to the delay-dependent and delay-independent observers’ design; Sections 3 and 4 propose the novel design approaches, and the proof of the theorems related to their asymptotic stability, as well as the existence conditions of the two robust observers. Section 5 shows the efficiency of the proposed approaches through simulation results in the test of the quadruple-tank benchmark. Finally, some conclusions are shared in Section 6.

2 PROBLEMS STATEMENT AND PRELIMINARIES Consider the class of LTI multivariable systems described by:  x˙ (t) = A0 x(t) + A1 x(t − τ1 ) + B0 u(t) + B1 u(t − τ2 ) + Dd(t) y(t) = Cx(t)

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where x(t) ∈ Rn is the system state vector, u(t) ∈ Rm the system known input vector, d(t) ∈ Rq the system unknown input vector, and y(t) ∈ Rp the output vector; the known matrices A0 ∈ Rn×n , A1 ∈ Rn×n , B0 ∈ Rn×m , B1 ∈ Rn×m , C ∈ Rp×n , and D ∈ Rn×q are constant matrices with appropriate dimensions, while τ1 and τ2 are the known and constant time delays of the system (4.1). Consider for the infinite dimensional system (4.1), an approximated finite dimensional system, obtained by using one of the approximation methods [22, 23] and described by:  ˜ ˜ ˜ χ(t) ˙ = Aχ(t) + Bu(t) + Dd(t) (4.2) ˜ y˜ (t) = Cχ(t) 

where χ(t) ∈ Rn and y˜ (t) ∈ Rp are the new state vector and the new output ˜ ∈ Rn ×n , B˜ ∈ Rn ×m , C ˜ ∈ Rp×n , and D ˜ ∈ Rn ×q are vector, respectively. A known constant matrices. The objective of this chapter is to design two different design approaches of UIOs for the system (4.1): (1) A delay-dependent UIO described by:  ˙ = Nζ(t) + Mζ(t − τ1 ) + Ly(t) + Jy(t − τ1 ) + G0 u(t) + G1 u(t − τ2 ) ζ(t) xˆ (t) = ζ(t) − Ey(t) (4.3) with ζ(t) ∈ Rn ; xˆ (t) ∈ Rn as the system estimated state vector. N, M, L, J, G0 , G1 , and E are unknown matrices of appropriate dimensions to be calculated such that xˆ (t) asymptotically converges to x(t). The following assumptions have been considered: (A1) rank(D) = q; (A2) rank(C) = p; (A3) the pairs (A0 , C) and (A1 , C) are observable (detectable). (2) A delay-independent UIO described by:  ˙˜ = N ˜ ˜ + L˜ ˜ ζ(t) ˜ y(t) + Gu(t) ζ(t) (4.4) ˜ − E˜ ˜ y(t) − Qr(t) χ(t) ˆ = ζ(t) 

˜ ∈ Rn , r(t) = χ(t) with ζ(t) ˆ − Tx(t) is an approximation vector,  ×n  n is a constant full column rank matrix, while χ(t) ˆ ∈ Rn is the T∈R ˜ and E˜ are unknown matrices of ˜ L, ˜ G, estimated vector of χ(t). N, appropriate dimensions to be designed such that χ(t) ˆ asymptotically converges to χ(t). Q is a constant parameter chosen in order to enhance performance of the designed observer. For the design of the observer (4.4), the following assumptions have been considered: (A4) ˜ = p; (A6) the pair (A, ˜ C) ˜ is observable ˜ = q; (A5) rank(C) rank(D) (detectable).

I. OBSERVER DESIGN

122

4. LINEAR SYSTEMS WITH DELAYS IN STATES AND INPUTS

Lemma 1 (Magdi [2]). Consider the following linear system with a single constant delay described by: ˙ = Av ξ(t) + Ad ξ(t − τ) (4.5) ξ(t) where Av and Ad are matrices with appropriate dimensions and τ is the time-delay. The time-delay system (4.5) is asymptotically stable for any τ ≥ 0 if there exist matrices R > 0, S > 0, and W verifying: RAv + ATv R + RAd S−1 ATd R + S + W = 0 or equivalently, the LMI



RAv + ATv R + S •

 RAd 0, S > 0 satisfying the matrix inequality:   RN + NTR + S RM 0 called the Lipschitz constant. The fault pattern is modeled by f (t), which also describes its time evolution. Here, we assume a network without time delays, and packet dropouts during the data transfer and sampling period is s. Then, at the sampling time sk , k ∈ N, the process output y(sk ) is sent to the controller and the control command u(sk ) is sent back to the process to be used as soon as it arrives, until the next control command arrives. We assume that the controller computing time can be omitted. The resulting system can be rewritten as x˙ (t) = Ax(t) + Bu(t+ ) + g(x, t) + φ(x, y, u) + Ef (t) y(t+ ) = Cx(t) where

u(t+ )

= u(sk ),

y(t+ )

(5.35) = y(sk ),

t+

∈ [sk , sk+1 ).

3.2 Basis for Fault Estimation For the purpose of fault estimation, the system given in Eq. (5.35) can be rewritten in a form such that the unobservable part is decoupled from the states that are affected from the fault [11]. The system is given as follows: x˙ 1 (t) = A1 x1 (t) + A2 x2 (t) + B1 u(t+ ) + g1 (x, t) + φ1 (x, y, u) y1 (t+ ) = C1 x1 (t) x˙ 2 (t) = A3 x1 (t) + A4 x2 (t) + B2 u(t+ ) + g2 (x, t) + φ2 (x, y, u) + Ef (t) y2 (t+ ) = C2 x2 (t)

(5.36)

where x1 ∈ Rn−q , x2 ∈ Rq , y1 ∈ Rr−q , y2 ∈ Rq , and C2 is full rank, where q represents the number of observable states. Only x1 needs to be estimated.

3.3 Observer Design The state estimation in this case is carried out using an LO, where an adaptive algorithm is proposed for parameter estimation, and a fault estimation technique is given. The following assumptions are recalled from Mao et al. [12]. Assumption 1. There exists a matrix B1 ∈ (n−q)×l such that φ(x, u, y) = B1 φ 1 (x, u, y), where |φ 1 | ≤ q(x, u, y) ≤ q0 for a function q and a scalar q0 > 0. Assumption 2. There exist matrices P1 = PT1 ∈ (n−q)×(n−q) , R1 ∈

(r−q)×l and LO gain L ∈ (n−q)×(r−q) such that (A1 − LC1 ) is stable, and (A1 − LC1 )T P1 + P1 (A1 − LC1 ) = −Q P1 B1 = CT1 R1 −λmin (Q) + 2γ λmax (P1 ) < 0

II. OBSERVER-BASED CONTROL DESIGN

(5.37) (5.38) (5.39)

OBSERVER-BASED FAULT-TOLERANT CONTROL

157

For the unobservable subsystem in Eq. (5.36), the observer is given in the following equation: x, t) + B1 u(t+ ) x˙ˆ 1 (t) = A1 xˆ 1 (t) + A2 C−1 2 y2 (t) + g1 (ˆ + φ1 (ˆx, y, u)θˆ (t) − L(C1 xˆ 1 − y1 ) y1 (t) = C1 x1 (t)

(5.40)

The adaptive algorithm for parameter estimation is given as T θ˙ˆ = Γ φ 1 (ˆx, u, y)RT1 (y1 − yˆ 1 )

(5.41)  T where xˆ := xˆ T1 xT2 is the state vector with estimated part and Γ = Γ T > 0 is the adaptation gain, which is selected by a trial and error method, so as to ensure that θ˙ˆ does not grow unbounded. In the sequel, we let ex (t) = x(t) − xˆ (t), e1 (t) = x1 (t) − xˆ 1 (t), ey (t) = y1 (t) − yˆ 1 (t), eθ (t) = θ (t) − θˆ (t). Theorem 2. Under Assumption 1, the observer described by Eq. (5.40) together with the parameter adaptive algorithm (5.41) can realize limt→∞ e1 = 0 and a bounded eθ ∈ L2 . Furthermore, limt→∞ eθ = 0 under the persistent excitation condition. Proof . The Lyapunov function considered here is 

V(t) = eT1 (t)P1 e1 (t) + eTθ (t)Γ −1 eθ (t) The derivative of this function will be + T ˙ = eT (t)[P1 (A1 + LC1 ) + (A1 + LC1 )T P1 ]e1 (t) − 2eθ (t)φ T V(t) 1 (ˆx, y, u(t ))R1 ey (t) 1

T + + ˆ + 2eT 1 (t)P1 [g1 (x) − g1 (ˆx)] + 2e1 (t)P1 [φ1 (x, y, u(t ))θ(t) − φ1 (x, y, u(t ))θ(t)]

+ − 2eT 1 (t)P1 B1 sgn(R1 )[θ0 (q0 + q(ˆx, u(t ), y)sgn(y1 (t) − yˆ (t)))]

(5.42)

and in view of Assumption 1: + T ˙ ≤ −eT (t)Qe1 (t) − 2eθ (t)φ T V(t) 1 (ˆx, y, u(t ))R1 ey (t) 1

T + ˆ + 2eT 1 (t)P1 [g1 (x) − g1 (ˆx)] + 2e1 (t)C1 R1 φ1 (ˆx, y, u(t ))[θ(t) − θ(t)]

T + + 2eT 1 (t)C1 R1 [φ 1 (x, y, u(t ))

+ − φ 1 (x, y, u(t+ ))]θ (t) − 2eT 1 (t)C1 R1 sgnR1 [θ0 (q0 + q(ˆx, u(t ), y)sgn(ey (t)))]

≤ −λmin (Q)e1 2 + 2γ λmax (P1 )e1 2 ≤ 0

(5.43)

Inequality (5.43) implies the stability of the origin, that is, eθ = 0 and e1 = 0 and the uniform boundedness of eθ and e1 . According to Barbalat’s lemma, limt→∞ e1 (t) = 0. Moreover, the persistent excitation condition means that

II. OBSERVER-BASED CONTROL DESIGN

158

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

there exists two positive constants, σ and t0 , such that for all t, the following inequality holds:  t+t0 φ1T (x(s), y(s), u(s))φ1 (x(s), y(s), u(s))ds ≥ σ I (5.44) t

One can conclude now that limt→∞ eθ (t) = 0. This completes the proof. For fault estimation, suppose E is invertible, then the fault estimation can be obtained from Eq. (5.36) as fˆ (t) = E˙x2 (t) − A3 xˆ 1 (t) − A4 x2 (t) − B2 u(t+ ) + g2 (ˆx, t) − φˆ 2 θˆ

(5.45)

Remark 4. Note that the fault estimation requires differentiation of the state x2 , which is not easy, in practice, due to the presence of noise.

3.4 Control Design The fault-tolerant control law used here utilizes feedback linearization, along with a feedback controller gain designed using the LQ technique. Consider the following system: x˙ (t) = fk (x(t), u(t), t), x(sk ) =

t = sk , ∀k ∈ N

− gk (x(s− k ), u(sk ), sk )

t = sk ,

∀k ∈ N

(5.46)

where fk and gk are functions from n × to n such that fk (0, t) = 0, gk (0, t) = 0, ∀t ≤ 0. The impulse time sequence sk forms an increasing sequence from some initial time s0 ≥ 0 to ∞. β: [0, ∞) × [0, ∞) → [0, ∞) belongs to class KL if β(·, t) is class K function. Definition 1. Suppose that a sequence of impulse times sk is given. We say that the impulsive system (5.46) is input-to-state practically stable (ISpS) over [0, t) if there exists functions β ∈ KL, γ ∈ K∞ and a constant ς > 0, such that for any bounded input u and any initial condition x(s0 ), we have x(t) ≤ β(x(s0 ), t) + γ (u[s0 ,t) ) + ς ,

∀t ≥ s0

(5.47)

It is preferred to define ISpS over classes of impulse sequence. A system defined as Eq. (5.46) is ISpS over the class S if for any sk ∈ S the condition is satisfied. This class is characterized as S := sk | ≤ sk − sk−1 ≤ τMATI

(5.48)

where τMATI is the upper bound on the sampling interval. Also ρ(t) := t−sk , t ∈ [sk , sk+1 ) indicates the amount of time that has passed because the last impulse and 0 ≤ ρ(t) ≤ τMATI .

II. OBSERVER-BASED CONTROL DESIGN

OBSERVER-BASED FAULT-TOLERANT CONTROL

159

Lemma 1. A smooth function V: n × → is an ISpS-Lyapunov function for the system (5.46) if for any impulse sequence sk ∈ S and any t ≥ s0 , the corresponding solution x(·) to Eq. (5.46) satisfies: 1. V is proper, positive definite, that is, there exist functions α1 , α2 of class K∞ such that α1 (x) ≤ V(x, ρ) ≤ α2 (x),

∀x, ∀ρ ∈ [0, τMATI ]

(5.49)

2. There exist a positive-definite function α3 , a class K function X , and a nonnegative constant c, such that the following implication holds: x ≥ X (u) + c ˙ ⇒ V(x, ρ) ≤ −α3 (|x|), V(x(sk ), 0) ≤ lim V(x(t), ρ(t)), t→sk

∀t = sk , ∀t ∈ N ∀t ∈ N

(5.50) (5.51)

When Eq. (5.50) holds with c = 0, V is called an Input-to-State Stable (ISS)-Lyapunov function for the system (5.46). Proof . From inequalities in Eqs. (5.49), (5.50), we have ˙ ρ) ≤ −α3 (x) ≤ −α3 α −1 (V(x, ρ)) V(x, 2 ≤ −α3 α2−1 (V(x(s0 ), 0))

(5.52)

There exists some KL-function β˜ which only depends on α2 and α3 , such ˜ that V(x(t)) ≤ β(V(x(s 0 ), 0)) for x ≥ X (u) + c, from which it follows that x(t) ≤ β(x(s0 ), t)

(5.53)

˜ 2 (r), t). We can now say that x ≥ β(x(s0 ), t) + where β(r, t) = α1−1 β(α X (u) + c for all x. This completes the proof. Now, the control law is given as follows: u(t+ ) = −W xˆ (t+ ) − B* g(ˆx(t+ )) − B* Efˆ (t+ ) − B† φ(ˆx(t+ ), y(t+ ), u(t+ )) (5.54) where W is the controller gain determined using LQR and B† is the right inverse of B. This control law gives the closed-loop system as follows: x˙ (t) = (A − BW)x(t) + g(x(t)) − g(ˆx(t+ )) + M1 e(t) + M2 z(t) + Υ   M1 := BW φ(x, y, u) E     M2 := M21 M22 M23 = BW φ(ˆx(t+ ), y(t+ ), u(t+ )) E Υ = [φ(x, y, u) − φ(ˆx(t), y(t+ ), u(t+ ))]θˆ (t)

T e(t):= eTx (t) eTθ (t) eTf (t)

II. OBSERVER-BASED CONTROL DESIGN

(5.55)

160

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

zx (t) = xˆ (t) − xˆ (t+ ), zθ (t) = θˆ (t) − θˆ (t+ ),

T z(t):= zTx (t) zTθ (t) zTf (t)

zf (t) = fˆ (t) − fˆ (t+ )

Theorem 3. The system in Eq. (5.55) is ISpS over the class S of impulse sequences, if there exist matrices P > 0, R > 0, X > 0, matrices W, G, N, and scalars μi , i = 1, . . . , 7, with μ1 = 1/μ1 + 1/μ2 + 1/μ3 , that satisfy the following inequalities Ξ1 + τMATI Ξ2 < 0  Ξ1 τMATI N 0, if Ξ < 0, ∀ρ ∈ 0 τMATI , whose necessary and sufficient condition is

Ξ1 + τMATI Ξ2 < 0,

Ξ1 + τMATI Ξ3 < 0

(5.64)

Using Schur complements, the inequalities in Eq. (5.64) can be written as the inequalities given by Theorem 3. Now, let μ1 = 1/μ1 + 1/μ2 + 1/μ3 . When τMATI → 0 the conditions (5.56), (5.57) reduce to   Ξ11 Ξ12 0 Similarly, by concatenating the output variables and the state of the controller, we get T  ˜ Z(k) = z(k)T u(k − 1)T u(k − 2)T · · · u(k − d)T     sc ca ˜ ˜ ˜ + 1) = W ˜ dsc , dca sc y˜ (k) Z(k k −1 k−dk −1 Z(k) + M dk , dk−dsc k     ca ˜ ˜ sc ca ˜ (k) ˜ (5.71) u(k) = L˜ dsc k , dk−dsc −1 Z(k) + H dk , dk−dsc −1 y k

k

II. OBSERVER-BASED CONTROL DESIGN

166 where

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

  ca , d W dsc sc ⎢  k k−dk −1 ⎢ sc ca   ⎢ L dk , dk−dsc −1 k ⎢ sc ca ˜ W dk , dk−dsc −1 = ⎢ 0 k ⎢ ⎢ .. ⎣ . 0 ⎤ ⎡ ca ) M(dsc k dk−dsc k −1   ⎥ ⎢ ca ⎥ H dsc  ⎢  k , dk−dsc −1 ⎥ ⎢ k ⎥ ⎢ ˜ dsc , dca sc = M . k ⎥ ⎢ k−dk −1 .. ⎥ ⎢ ⎦ ⎣ 0 0 ⎧    ⎨ L dsc , dca sc  ca k k−dk −1 = L˜ dsc  k , dk−dsc k −1 ⎩ 0      ca H dsc k , dk−dsc ˜ dsc , dca sc H k −1 k k−dk −1 = 0 ⎡

⎤ 0 0 ⎥ ⎥ 0 · · · 0 0⎥ ⎥ 0 · · · 0 0⎥ ⎥ .. . . .. .. ⎥ . . .⎦ . 0 ··· I 0 0

···

0 ···

0

0 ···

0 ···

I

0

for

···

0 for dca k =0  0 for dca k >0

dca k =0

for dca k >0

By augmenting Eqs. (5.70), (5.71), and defining the state variable as   ˜ T T ˜ T Z(k) X(k) = X(k) Then the closed-loop equation is obtained as  

ca ca ¯ sc ) X(k) + Υ˜ w(k) X(k + 1) = φ¯ + Γ¯ K dsc k , dk−dsc −1 , dk κ(d k k

y(k) = κX(k) ˜

(5.72)

where

     0 φ˜ 0 0 Γ˜ ) = φ¯ = , Γ¯ = , κ(d ¯ sc k κ(d ˜ sc ) 0 0 I 0   ⎤ k ⎡  ˜ dsc , dca sc ˜ dsc , dca sc   W M ca ca ⎣  k k−dk −1  k k−dk −1 ⎦ , d , d = K dsc sc k k−dk −1 k sc ca ca Υ˜ dsc L˜ dk , dk−dsc −1 k , dk−dsc k k −1   κ˜ = κ 0 · · · 0 0 0 0 · · · 0 0 T  Υ˜ = Υ T 0 · · · 0 0 0 0 · · · 0 0



I 0

Matrices φ¯ and Γ¯ are of the form ˜ u (k)U1 IT φ¯ = φAUG + ΔφAUG = φAUG + GΔ 2 ˜ ¯ Γ = ΓAUG + ΔΓAUG = ΓAUG + GΔu (k)U1 IT 2

II. OBSERVER-BASED CONTROL DESIGN

(5.73)

ROBUST CONTROL OF NCS WITH PARTIALLY KNOWN TRANSITION MATRIX

Actuator

Plant Embedded processor

~

uk

167

Sensor

yk

dk–1

Random sensor-to-controller delay tk

Random controller-to-actuator delay dk

Controller

uk

~

yk

FIG. 5.9 Setup of NCS.

Remark 5. The closed-loop system in Eq. (5.72) cannot be converted into normal MJLS by the application of the two-mode-dependent controller proposed in Eq. (5.68). This is due to the fact that the closed-loop system ca ca in Eq. (5.72) is a function of dsc k , dk−dsc −1 , dk as shown in Fig. 5.9. The delay k

ca dca is associated with both dsc k and dk , which makes the analysis more k−dsc −1 k

complicated and challenging. Because the delay dca is amalgamate k−dsc −1 k

into the controller design, this chapter takes advantage of a multistep jump of the Markov chain, rather than concatenating the delay values. This kind of two-mode-dependent controller for a class of discrete-time Markovian jump linear systems with partially unknown transitions has not been completely explored in the literature.

4.2 Analytical Results In this section, we first deal with the necessary and sufficient condition to guarantee the stochastic stability of system (5.72), which is shown in = i, dca = Theorem 1. For notational simplicity, we denote dsc sc      k  k−dk −1  ca sc ca sc ca sc ca m, W dsc k , dk−dsc −1 , M dk , dk−dsc −1 , L dk , dk−dsc −1 , and H dk , dk−dsc −1 k

k

k

k

will be represented as W(i, m), M(i, m), L(i, m), and H(i, m), respectively. The definitions of stochastic stability, H2 and H∞ norms used for further investigation, are following: Definition 2. If for every finite X0 = X(0), initial mode dsc 0 ∈ S1 and ca − 1) ∈ S , there exists a finite Q > 0 such that the dk−dsc −1 = d(k − dsc 2 0 0

condition:

 E

∞  k=0

 X(k)2 |X0 ,dsc0 ,dca

k−dsc −1 0

< X0T QX0

holds. Then the system (5.72) is said to be stochastically stable.

II. OBSERVER-BASED CONTROL DESIGN

(5.74)

168

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

Definition 3. The H2 norm of the system (5.72) can be obtained as follows: Hyw 2 =

d1  d2 d1  

α(i0 , m0 )yn,i0 ,m0 2

(5.75)

n=1 i0 =0 m0 =0

Definition 4. Consider a system with energy-bounded disturbance. Then the H∞ norm of such a system is defined as a measure of degree of robust stability of the system that signifies the energy attenuation in the worst case scenario. The H∞ norm is defined as Hyw 2∞ = sup

sup

ca dsc 0 ∈S1 dk−dsc −1 0

y ∈S2 w∈l2 (0,∞) w sup

(5.76)

Theorem 4. System (5.72) is said to be stochastically stable under the proposed control law (5.69), with zero exogenous disturbance signal (w(k) = 0), if, and only if, their exist matrices P(i, m) > 0 such that the following matrix inequality satisfies: L(i, m) = P(i, m) +

d2  d1  d2  j=0 r2 =0 r1 =0

1+i−j j ¯ πij λmr1 λr2r1 × [φ¯ + λK(i, m, r1)κ(i)] ¯ T P(j, r2)

¯ × [φ¯ + λK(i, m, r1)κ(i)] ¯ 0 is calculated from the following equation R(i, m) = κ˜ T κ˜ +

d2  d1  d2 

πij λmr2 λr2 r1 × [φ¯ + Γ¯ K(i, m, r1 )κ(i)] ¯ T R(j, r2 ) 1+i−j j

j=0 r1 =0 r2 =0

¯ × [φ¯ + Γ¯ K(i, m, r1 )κ(i)] for i ∈ S1 , m ∈ S2 .

II. OBSERVER-BASED CONTROL DESIGN

(5.79)

ROBUST CONTROL OF NCS WITH PARTIALLY KNOWN TRANSITION MATRIX

169

4.3 Robust H2 Control In this section, an H2 controller is designed for the special MJLS system. The designed controller is two-mode-delay-dependent and robust. The main aim is to develop a control law (5.69) such that the H2 norm of the system (5.72) as proposed in the preceding definition is minimized. Due to this, the control law design procedure will be changed into an optimization-solving problem. Theorem 6. The closed-loop system (5.72) with stabilizing controller (5.69) is stochastically stable and Hyw  < γ , if there exist matrices W(i, m), M(i, m), ¯ r2 ) > 0, P(i, m) > 0 and a set of L(i, m), H(i, m), and symmetric matrices X(j, scalar ε1 (i, m) > 0, ε2 (i, m) > 0, . . . , ε(d1 +1)(d2 +1)(d2 +1) (i, m) > 0 satisfies the following inequality: d2  d2 d1  d1   i0 =0 m0 =0 j0 =0 r02 =0

⎡

1+i −j0

α(i0 ,m0 ) π(i0 j0 ) λm0 r002

−P(i, m) + κ˜ T κ˜ ⎣ Vκ(i, m) ΔVu(i, m)

˜ < γ2 × tr{J˜T P(j0 , r02 )J}

⎤ • • ⎦ 0, P(i, m) > 0 and a set of scalars ε1 (i, m) > 0, ε2 (i, m) > X(j, 0, . . . , ε(d1 +1)(d2 +1)(d2 +1) (i, m) > 0 satisfying d1  d2  d1  d2  i0

m0

j0

1+i −j ˜ < β2 α(i0 ,m0 ) πi0 m0 λm0 r002 0 tr{J˜T P(j0 , r02 )J}

r02

⎤ − P(i, m) • • • ˆ m) − X(i, m) • ⎥ ⎢ Vκ(i, m) M(i, • ⎥ 0, P(i, m) > 0 and a set of scalar ε1 (i, m) > 0, ε2 (i, m) > X(j, 0, . . . , ε(d1 +1)(d2 +1)(d2 +1) (i, m) > 0 such that the inequality (5.87) satisfies. Proof . We know that the system is stochastically stable and Hyw  < γ if Eq. (5.80) satisfies. ¯ r) = P(i,r) ∪ P(i,r) Eq. (5.80) can be rewritten as We also know that P(i, K UK ⎤ ⎡ T −P(j, s) + κ˜ κ˜ • •   ˆ s) E(i, r) = λrs πij ⎣ Vκ(j, s) −X(j, s) + M(j, • ⎦ r i s∈I Vu(j, s) 0 −ˆε(j, s)I j∈IK K ⎤ ⎡ T −P(j, s) + κ˜ κ˜ • •   ˆ s) + λrs πij ⎣ Vκ(j, s) −X(j, s) + M(j, • ⎦ r i Vu(j, s) 0 −ˆε (j, s)I j∈IU K s∈IU K (5.86) ⎡ (i,r) (i,r) + PK κ˜ T κ˜ −P ⎢ K(i,r) E(i, r) = ⎣ PK Vκ(i, m) (i,r) PK Vu(i, m)

⎤ • • ⎥ (i,r) (i,r) ˆ −PK X(i, m) + PK M(i, m) • ⎦ (i,r) 0 −PK εˆ (i, m)I

II. OBSERVER-BASED CONTROL DESIGN

ROBUST CONTROL OF NCS WITH PARTIALLY KNOWN TRANSITION MATRIX

⎡

−P(j, s) + κ˜ T κ˜ ⎣ + λrs πij Vκ(j, s) r i Vu(j, s) s∈IU j∈IU K K 



• ˆ s) −X(j, s) + M(j, 0

173

⎤ • • ⎦ −ˆε (j, s)I

Therefore, if we have ⎤ ⎡ (i,r) (i,r) −PK + PK κ˜ T κ˜ • • ⎥ ⎢ (i,r) (i,r) (i,r) ˆ −PK X(i, m) + PK M(i, m) • ⎦ 0 and a set of scalars ε1 (i, m) > 0, ε2 (i, m) > matrices X(j, 0, . . . , ε(d1 +1)(d2 +1)(d2 +1) (i, m) > 0 such that the inequality (5.90), (5.91) are satisfied. Proof . We know that the system is stochastically stable and Hyw  < β, Hyw ∞ < γ if Eq. (5.85) satisfies. ¯ r) = P(i,r) ∪ P(i,r) Eq. (5.85) can be rewritten as We also know that P(i, K

UK

⎤ − P(j, s) • • •   ˆ s) − X(j, s) • ⎢ Vκ(j, s) M(j, • ⎥ ⎥ E(i, r) = λrs πij ⎢ ⎣ 1 J˜T P(j, s) 0 −I • ⎦ r γ i s∈ I j∈IK K Vu(j, s) 0 0 −ˆε (j, s)I ⎤ ⎡ T κ˜ κ˜ − P(j, s) • • •   ˆ s) − X(j, s) • ⎢ Vκ(j, s) M(j, • ⎥ ⎥ λrs πij ⎢ + ⎣ 1 J˜T P(j, s) 0 −I • ⎦ r γ i j∈IU K s∈IU K Vu(j, s) 0 0 −ˆε (j, s)I (5.89) ⎡ (i,r) T ⎤ (i,r) • • • PK κ˜ κ˜ − PK ⎢ P(i,r) Vκ(j, s) P(i,r) M(j, ⎥ ˆ s) − P(i,r) X(j, s) • • ⎢ ⎥ K K E(i, r) = ⎢ K (i,r) 1 ˜T ⎥ i λr I ⎣ ⎦ PK γ J 0 −πK • K ⎡

κ˜ T κ˜

(i,r)

PK Vu(j, s)

0

II. OBSERVER-BASED CONTROL DESIGN

0

(i,r)

−PK εˆ (j, s)I

174

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

⎡

κ˜ T κ˜ − P(j, s)   ⎢ Vκ(j, s) + λrs πij ⎢ ⎣ 1 J˜T P(j, s) r γ i j∈IU K s∈IU K Vu(j, s) ⎡

⎤ • • ⎥ ⎥ • ⎦ −ˆε(j, s)I

• • ˆ s) − X(j, s) • M(j, 0 −I 0 0

Therefore, if we have

(i,r) T P(i,r) K κ˜ κ˜ − PK ⎢ P(i,r) Vκ(j, s) ⎢ K ⎢ (i,r) ⎣ PK γ1 J˜T (i,r)

• • (i,r) ˆ (i,r) PK M(j, • s) − PK X(j, s) i 0 −πK λrK I

PK Vu(j, s)

0 ⎡ T κ˜ κ˜ − P(j, s) ⎢ Vκ(j, s) ⎢ ⎣ 1 J˜T P(j, s) γ Vu(j, s)

0

• • •

⎥ ⎥ ⎥ 0 such that for all admissible uncertainties satisfying Eqs. (5.96), (5.97) and arbitrary switching rule σ (·) activating subsystem

II. OBSERVER-BASED CONTROL DESIGN

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5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

j ∈ N at instant k + 1 and subsystem i at instant k, the Lyapunov functional 

difference V(xk , k) satisfies V(xk , k) = V(xk+1 , k+1)−V(xk , k) ≤ −εxTk xk , ∀xk = 0. Definition 6. Given a scalar γ ≥ 0, the L2 gain G of switched system (ΣJo ) over S is 

G = inf{γ ≥ 0 : zk  < γ 2 wk , ∀σ ∈ S, ∀λj ∈ Λj , j ∈ N} Definition 7. Switched system (ΣJo ) is said to be UQS with an L2 gain G < γ if for all switching signal vector σ ∈ S and for all admissible uncertainties satisfying Eqs. (5.96), (5.97) it is UQS and ∀wk = 0, zk  < γ 2 wk . In the sequel, we consider the following quadratic Lyapunov functional 

V σ (xk , k) = xTk Pσ xk ,

0 < PTσ = Pσ , σ ∈ S

(5.100)

Theorem 11. The following statements are equivalent: (A) There exists an SLF of the type Eq. (5.100) with σ ∈ S and a scalar γ > 0 such that switched system (5.98), (5.99) is UQS with an L2 gain G < γ . (B) There exist matrices 0 < PTi = Pi , 0 < XjT = Xj , i ∈ N, j ∈ N and a scalar γ > 0 satisfying the LMIs ⎡ ⎤ −Pi 0 ATip CTip ⎢ T⎥ −γ 2 I ΓipT Φip ⎢ • ⎥ ⎢ ⎥ < 0, (i, j) ∈ N × N, p ∈ {1, . . . , Mi } ⎣ • • −Xj 0 ⎦ • • • −I (5.101)

5.1 Switched Dynamic Output-Feedback We direct attention to the more general case, and employ at every mode i ∈ N a switched dynamic output-feedback scheme of the form: (ΣC ): ζk+1 = Aci ζk + Bci yk uk = Cci ζk

(5.102)

Augmenting controller (5.102) to switched system (5.92)–(5.94) and defin ing the composite vector ξkT = xTk ζkT , we get the closed-loop system (ΣJC ) : ξk+1 = Ai ξk + Γ¯i wk zk = Ci ξk + Φi wk

II. OBSERVER-BASED CONTROL DESIGN

(5.103)

181

SWITCHED DISCRETE-TIME SYSTEMS

where the respective matrices are given by     Γj Ai Bj Cci ¯ Ai = , Γi = , Bci Φj Bci Li Aci

 Ci = Ci

Di Cci



(5.104)

Application of Theorem 11 shows that switched system (5.104) is UQS with an L2 gain G < γ if there exist matrices 0 < PiT = Pi , 0 < YjT = Yj , j ∈ N and a scalar γ > 0 satisfying the LMIs ⎡

−Pi ⎢ • ⎢ ⎣ • •

0 −γ 2 I • •

ATi Γ¯iT −Yj •

⎤ CiT ΦiT ⎥ ⎥ < 0, 0 ⎦ −I

Introducing the shorthand    Psi 0 Xsi  Pi = , Xi = Pi−1 = 0 Pci 0

 0 , Xci

(i, j) ∈ N × N

 Yj =

Ysj 0

(5.105)

0 Ycj

 (5.106)

we have the following result. Theorem 12. Consider switched  (5.103), (5.104) with output matrix  system Li having the SVD form Li = Ui Λpi , 0 ViT , Λpi ∈ Rp×p . This system is UQS T = X , 0 < XT = X , with an L2 gain G < γ if there exist matrices 0 < Xsiu siu siv siv T T T 0 < Xci = Xci , 0 < Ysi = Ysi , 0 < Yci = Yci , Ωci , Πcj , Υci , Ψci , (i, j) ∈ N × N and a scalar γ > 0 satisfying the systems LMIs ⎡ ⎤ 0 0 Xsi ATi LTi ΩcjT Xsi CTi −Xsi ⎢ ⎥ −Xci 0 ΛTci BTi ΥcjT ΠciT DTi ⎥ ⎢ • ⎢ ⎥ ⎢ • • −γ 2 I ΓiT ΨciT ΦiT ⎥ ⎢ ⎥ < 0, (i, j) ∈ N × N ⎢ • • • −Ysj 0 0 ⎥ ⎢ ⎥ ⎣ • 0 ⎦ • • • −Ysj • • • • • −I (5.107) Moreover, the gain matrices are given by Aci = Υci Xci−1 ,

−1 Bcj = Ωcj Ui−1 Λ−1 pi Xsiu Λpi Ui ,

Cci = Πci Xci−1

(5.108)

Proof . By applying the congruent transformation [Xi , I, I, I] to LMIs (5.105) with Xi = Pi−1 and expanding the result using Eq. (5.106) along with the matrix substitutions Aci Xci = Υci , Li Xsi = Ξsi Li , Ωci = Bci Ξci , −1 Ξi = Ui Λpi Xsiu Λ−1 pi Ui , and Cci Xci = Πcj . Remark 7. The optimal switched dynamic output-feedback with L2 gain for system (5.103), (5.104) subject to the polytopic representation (5.96),

II. OBSERVER-BASED CONTROL DESIGN

182

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

(5.97) can be determined by solving the following convex minimization problem over LMIs: Minimize γ wrt Xsiu > 0, Xsiv > 0, Xcj > 0, Ysj > 0, Ycj > 0, Ωcj , Υcj , Πcj , γ > 0 ⎡ ⎤ −Xsi 0 0 Xsi AT LT Ω T Xsi CT ip ip cj ip ⎢ ⎥ ⎢ • −Xci 0 Π T BT ΥcjT ΛT DT ⎥ ci ip ci ip⎥ ⎢ ⎢ T ⎥ • −γ 2 I ΓiT ΨciT Φip ⎢ • ⎥ < 0, ∀(i, j) ∈ N × N, p ∈ {1, . . . , Mi } ⎢ ⎥ ⎢ • ⎥ • • −Y 0 0 sj ⎢ ⎥ ⎣ • 0 ⎦ • • • −Ysj • • • • • −I

6 DISTURBANCE OBSERVER-BASED CONTROL FOR NONLINEAR SYSTEMS This section proposes a DOBC approach for nonlinear systems under disturbances, namely, nonlinear DOBC or NDOBC. Within this framework, instead of considering the control problem for a nonlinear system under disturbances as a single one, it is divided into two subproblems, each with its own design objectives. The first subproblem is the same as the control problem for a nonlinear system without disturbances, and its objective is to stabilize the nonlinear plant and achieve performance specifications such as tracking or regulation. The second subproblem is to attenuate disturbances. A nonlinear disturbance observer is designed to deduce external disturbances, and then to compensate for the influence of the disturbances using proper feedback. A block diagram of the proposed NDOBC is shown in Fig. 5.16. x˙ (t) = f (x(t)) + g1 (x(t))u(t) + g2 (x(t))d(t) y(t) = h(x(t))

(5.109)

Rn ,

where x ∈ u ∈ R, and x ∈ R are the state vector, input, and external disturbance, respectively. It is assumed that f (x), g1 (x), and g2 (x) are smooth functions in terms of x. A general design procedure for system (5.109) is proposed in the following NDOBC design procedure. (S1) Design a nonlinear controller for system (5.109) to achieve stability and other performance specifications under the assumption that the disturbance is measurable. This can be achieved using linear/nonlinear control techniques; for example, dynamic inversion control, feedback linearization, gain scheduling, or sliding mode control. (S2) Design a nonlinear disturbance observer to estimate the disturbance.

II. OBSERVER-BASED CONTROL DESIGN

DISTURBANCE OBSERVER-BASED CONTROL FOR NONLINEAR SYSTEMS

183

d yd

u

Nonlinear controller dˆ

Nonlinear system

y

x

Nonlinear disturbance observer

FIG. 5.16 Schematic of nonlinear disturbance observer-based control.

(S3) Integrate the disturbance observer with the controller by replacing the disturbance in the control law with its estimation yielded by the disturbance observer.

6.1 Nonlinear Disturbance Observer Let the disturbance be generated by a linear exogenous system σ˙ (t) = Aσ ,

d = Cσ

(5.110)

where σ ∈ Rm and d ∈ R. The exogenous system (5.110) is assumed to be neutral stable, which implies that a persistent disturbance is imposed on the system (5.109). At this point, we need to estimate the unknown disturbance d. We do this using a basic disturbance observer of the form ˆ σ˙ˆ (t) = Aσˆ + α(x)(˙x − f (x) − g1 (x)u + g2 (x)d) dˆ = Cσˆ

(5.111)

where α(x) is the nonlinear gain function of the observer. However, the disturbance observer (5.111) cannot be implemented because the derivative of the state is required. An effective nonlinear disturbance observer is suggested to have the form r˙(t) = (A − α(x)g2 (x)C)r + Aq(x) − α(x)(g2 (x)Cq(x) + f (x) + g1 (x)u) σˆ (t) = r + q(x) dˆ = Cσˆ

(5.112)

II. OBSERVER-BASED CONTROL DESIGN

184

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

where r ∈ Rm is the internal observer state and q(x) ∈ Rm is a nonlinear function to be designed. The nonlinear observer gain α(x) is then determined by ∂q(x) (5.113) ∂x It can be shown that the following results [22] hold: Theorem 13. Let the estimation error e be defined as σ − σˆ . Consider system (5.109) under the disturbance generated by the exogenous system (5.110). The disturbance observer (5.112) can exponentially track the disturbance if the nonlinear gain function α(x) is chosen such that α(x) =

e˙(t) = (A − α(x)g2 (x)C)e(t)

(5.114)

is globally exponentially stable regardless of x. Theorem 14. The estimation yielded by the nonlinear disturbance observer (5.112) converges to the disturbance d globally exponentially if there exists a gain K such that the transfer function H(s) = C[sI − (A − KβC)]−1 K

(5.115)

where β is a constant to selected by the designer.

6.2 Composite Controller Given that the nonlinear disturbance observer is designed by Eq. (5.112), this observer can then be integrated with a separately designed controller to yield the closed-loop system depicted in Fig. 5.16. The main result of this section is stated in Theorem 15. Theorem 15. Consider nonlinear system (5.109) with a well-defined disturbance-to-output relative degree, and under the disturbance (5.110). The closed-loop system under the nonlinear composite controller, namely, NDOBC, designed by the procedure in steps (S1)–(S3), as shown in Fig. 5.16, is semiglobally exponentially stable in the sense that for an initial state x and σ satisfying x(0) ≤ ω1 ,

σ (0) ≤ ω2

(5.116)

where ω1 and ω2 are given scalars, could be arbitrarily large lim x(t) −→ 0,

t−→∞

lim e(t) −→ 0

t−→∞

(5.117)

if the following conditions are satisfied: (1) when the disturbance d is measurable, there exists a control law u(x, d) = δ(x) + γ (x)d such that the closed-loop system is globally exponentially stable for any disturbance; (2) the nonlinear disturbance observer (5.112) is designed with appropriately selected function q(x) and there exists a gain function K such the transfer function (5.115) is asymptotically stable and strictly positive real. II. OBSERVER-BASED CONTROL DESIGN

NCS WITH QUANTIZATION AND NONSTATIONARY RANDOM DELAYS

185

7 NCS WITH QUANTIZATION AND NONSTATIONARY RANDOM DELAYS Recently, there have been three main methods to deal with control input data loss for real-time NCS, that is to use zero control input, keep the previous one, or use the predictive control sequence [23]. In the NCS, it is assumed that the measurement signals are quantized before being communicated. In this section, we provide new results on NCS with nonstationary packet dropouts and quantized feedback. We extend the work of [24, 25] and elaborate on [26] by developing an improved quantized observer-based stabilizing control algorithm to estimate the states and control input through the construction of an augmented system where the original control input is regarded as a new state. Due to a limited bandwidth communication channel, the simultaneous occurrence of measurement and actuation delays are considered using nonstationary random processes modeled by two mutually independent stochastic variables. Several properties of the developed approach are delineated. The OBC is designed to exponentially stabilize the networked system and solve within the LMI framework. The theory is illustrated by simulation experiments on a lab-scale four-tank system.

7.1 Problem Setup Consider the NCS with random communication delays, where the sensor is clock driven and the controller and the actuator are event driven. The discrete-time linear time-invariant plant model is as follows: xp (k + 1) = Axp + Bup ,

yp = Cxp

n

(5.118)

m

where xp (k) ∈ is the plant’s state vector and up (k) ∈ and yp (k) ∈ p are the plant’s control input and output vectors, respectively. A, B, and C are known as real matrices with appropriate dimensions. With reference to Fig. 5.17, the measured output yp (k) is transmitted through a logarithmic quantizer that yields yq (k). Let the set of quantized levels be described as V = {±vj , vj = !j v0 , j = 0, ±1, ±2, . . .} ∪ {0}, 0 < ! < 1,

v0 > 0

where the parameter ! is called the quantization density, and the logarithmic quantizer q(·) is defined by ⎧ if vmj < ν ≤ vMj ⎨ vj , (5.119) q(ν) = 0, if ν = 0 ⎩ −q(−ν), if ν < 0

II. OBSERVER-BASED CONTROL DESIGN

186

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

FIG. 5.17 System block diagram.

where ω = (1 − !)/(1 + !), vmj = vj /(1 + ω), and vMj = vj /(1 − ω). Note that the quantization effects can be transformed for a given quantization density ! into sector bounded format as q(ν) − ν = ν,

 ≤ ω

(5.120)

Based on the quantized signals, the controller will be designed such that the desired dynamic performance of system (5.118) is achieved while the data packet dropout arises. Toward our goal, we assume, for a more general case, that the measurement with a randomly varying communication delay is described by  y (k), δ(k) = 0 yc (k) = q yq (k − τkm ), δ(k) = 1 which in view of yq (k) = q(yp (k)) = (1 + )yp (k), it becomes  (1 + )yp (k), δ(k) = 0 yc (k) = (1 + )yp (k − τkm ), δ(k) = 1

(5.121)

where τkm stands for measurement delay, the occurrence of which satisfies the a Bernoulli distribution, and δ(k) is Bernoulli distributed white sequence. It order to capture the current practice of computer communication management that experiences different time-dependent operational modes, we let Prob{δ(k) = 1} = pk

II. OBSERVER-BASED CONTROL DESIGN

NCS WITH QUANTIZATION AND NONSTATIONARY RANDOM DELAYS

187

TABLE 5.1 Pattern of pk pk

q1

q2

···

qn−1

qn

Prob(pk = q)

r1

r2

···

rn−1

rn

where pk assumes discrete values, see Table 5.1. Two particular classes can be considered: Class 1: pk has the probability mass function where qr − qr−1 = constant for r = 2, . . . , n. This covers a wide range of cases including uniform discrete distribution, symmetric triangle distribution, decreasing linear function, or increasing linear function. Class 2: pk = X/n, n > 0, and 0 ≤ X ≤ n is a random variable that follows the binomial distribution B(q, n), q > 0. ¯ where Remark 8. It is significant to note that the case Prob{δ(k) = 1} = δ, δ¯ is a constant value, is widely used in the majority of results on NCS. In this chapter, we focus on nonstationary dropouts. Remark 9. It is worth mentioning that the present methodology can be easily applied to the class of systems xp (k + 1) = (A + δA)xp + (B + δB)up yp = (C + δC)xp where δA, δB, and δC are parametric perturbations that belong to the class of unknown-but-norm bounded perturbations. The same arguments apply to the class xp (k + 1) = Axp + Bup + f (xp ) yp = Cxp where f (xp ) is a vector-bounded nonlinearity. Taking into consideration the time delay that occurs on the actuation side, we proceed to design the following OBC: Observer: xˆ (k + 1) = Aˆx + Bup (k) + L(yc (k) − yˆ c (k))  Cˆx(k), δ(k) = 0 yˆ c (k) = Cˆx(k − τkm ), δ(k) = 1 Controller: uc (k) = Kˆx(k)  u (k), up = c uc (k − τka ),

α(k) = 0 α(k) = 1

II. OBSERVER-BASED CONTROL DESIGN

(5.122)

(5.123)

188

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

TABLE 5.2 Pattern of sk sk

w1

w2

···

wn−1

wn

Prob(sk = w)

x1

x2

···

xn−1

xn

where xˆ (k) ∈ n is the estimate of the system (5.118), yˆ c (k) ∈ p is the observer output, and L ∈ n×p and K ∈ m×n are the observer and controller gains, respectively, and τka is the actuation delay. The stochastic variable α(k), mutually independent of δ, is also a Bernoulli distributed white sequence with Prob{α(k) = 1} = sk where sk assumes discrete values. By similarity, a particular class is that sk has some probability mass function as in Table 5.2, where sr − sr−1 = constant for r = 2, . . . , n. In this chapter, we assume that τka and τkm are time-varying and have the following bounded condition: m + τ− m ≤ τk ≤ τm ,

τa− ≤ τka ≤ τa+

Define the estimation error by e(k) = xp (k) − xˆ (k). Then, it yields ⎧ (A + BK)xp (k) ⎪ ⎪ ⎨ −BKe(k), α(k) = 0 xp (k + 1) = α) (k) + BKx (k − τ Ax ⎪ p p k ⎪ ⎩ −BKe(k − τkα ), α(k) = 1

(5.124)

(5.125)

e(k + 1) = xp (k + 1) − xˆ (k + 1) ⎧ (A − LC)e(k) ⎪ ⎪ ⎨ δ(k) = 0 −LCxp (k), (5.126) = m) Ae(k) − LCe(k − τ ⎪ k ⎪ ⎩ m −LCxp (k − τk ), δ(k) = 1

T In terms of ξ(k) = xTp (k) eT (k) , system (5.125), (5.126) can be cast into the form: ξ(k + 1) = Aj ξ(k) + Bj ξ(k − τkm ) + Cj ξ(k − τka )

(5.127)

where {Aj , Bj , Cj , j = 1, . . . , 4} and j is an index identifying one of the following pairs {(δ(k) = 1, α(k) = 1), (δ(k) = 1, α(k) = 0), (δ(k) = 0, α(k) = 0), (δ(k) = 0, α(k) = 1)}:     A 0 A + BK −BK , A2 = , A1 = 0 A 0 A     A + BK −BK A 0 , A4 = , A3 = −LC A − LC −LC A − LC     0 0 0 0 , B2 = , B1 = −LC −LC −LC −LC II. OBSERVER-BASED CONTROL DESIGN

189

NCS WITH QUANTIZATION AND NONSTATIONARY RANDOM DELAYS

 0 0  0 C3 = 0 B3 =

    0 0 0 BK , B4 = , C1 = 0 0 0 0    0 BK −BK , C4 = 0 0 0

  −BK 0 , C2 = 0 0

 0 , 0 (5.128)

Remark 10. It is remarked for simulation processing that we can express Eqs. (5.125), (5.126) in the form xp (k + 1) = sk [Axp (k) + BKxp (k − τkα ) − BKe(k − τkα )] + (1 − sk )[(A + BK)xp (k) − BKe(k)] e(k + 1) = pk [Ae(k) − LCe(k − τkm ) − LCxp (k − τkm )] + (1 − pk )[(A − LC)e(k) − LCxp (k)]

(5.129) (5.130)

where the values of the random variables pk , sk are generated in the manna r discussed earlier. Remark 11. It is important to note from Eq. (5.128) that   A + BK −BK , j = 1, . . . , 4 (5.131) Aj + Bj + Cj = −LC A − LC The interpretation of this result is that Aj + Bj + Cj represents the fundamental matrix of the delayed system (5.127), which must be independent of the mode of operation. This will help in simplifying the control design algorithm. The aim of this section is to design an observer-based feedback stabilizing controller in the form of Eqs. (5.122), (5.123) such that the closed-loop system (5.127) is exponentially stable in the mean square. Our approach is based on the concepts of switched time-delay systems [21]. For simplicity in exposition, we introduce σ1 (k) = Prob{δ(k) = 1, α(k) = 1}, σˆ 1 = E[σ1 ], σ2 (k) = Prob{δ(k) = 1, α(k) = 0}, σˆ 2 = E[σ2 ] σ3 (k) = Prob{δ(k) = 0, α(k) = 0}, σˆ 3 = E[σ3 ], σ4 (k) = Prob{δ(k) = 0, (5.132) α(k) = 1}, σˆ 4 = E[σ4 ] where E[σi ] is the expected value of σi , i = 1, . . . , 4. Because we assume that δ(k) and α(k) are independent random variables, then it follows from Eq. (5.132) that σˆ 1 = E[pk ]E[sk ], σˆ 2 = E[pk ]E[1 − sk ], σˆ 3 = E[1 − pk ]E[1 − sk ], σˆ 4 = E[1 − pk ]E[sk ] (5.133)

7.2 Design Results In the sequel, we will thoroughly investigate the stability analysis and controller synthesis problems for the closed-loop system (5.127). We II. OBSERVER-BASED CONTROL DESIGN

190

5. OBSERVER-BASED CONTROL DESIGN: BASICS, PROGRESS, AND OUTLOOK

initially derive a sufficient condition under which the closed-loop system (5.127) with the given controller (5.122), (5.123) is exponentially stable in the mean square. Extending on [26], the following Lyapunov function candidate is constructed to establish the main theorem: V(ξ(k)) =

5 

Vi (ξ(k))

(5.134)

i=1

V1 (ξ(k)) =

4 

σˆ j ξ T (k)Pξ(k),

P>0

j=1

V2 (ξ(k)) =

4 

σˆ j

4 

σˆ j

V4 (ξ(k)) =

V5 (ξ(k)) =

j=1

k−1 

ξ T (i)Qj ξ(i)

−τm− +1

σˆ j



k−1 

ξ T (i)Qj ξ(i)

=−τm+ +2 i=k+−1

j=1 4 

Qj = QTj > 0

i=k−τka

j=1 4 

ξ T (i)Qj ξ(i),

i=k−τkm

j=1

V3 (ξ(k)) =

k−1 

−τa− +1

σˆ j



k−1 

ξ T (i)Qj ξ(i)

(5.135)

=−τa+ +2 i=k+−1

It is not difficult to show that there exist real scalars μ > 0 and υ > 0 such that μξ 2 ≤ V(ξ(k)) ≤ υξ(k)2

(5.136)

Remark 12. Note that the Lyapunov functional Eqs. (5.134), (5.135) is constructed to deal with the measurement and actuation delay terms. The first term in Eq. (5.135) is standard to the delay-less nominal systems, while the second term and the fourth term together correspond to the delay-dependent conditions, with respect to the measurement delay. In a similar way, the third term and the fifth term together correspond to the delay-dependent conditions, with respect to the actuation delay. This construction serves in reducing the number of manipulated variables (LMI variables and other unknown variables) and limiting the basic storage requirements, features that improve the performance of the developed criteria. It has been demonstrated that a special version of Eqs. (5.134), (5.135) renders less conservative results, see [21, 26]. We now present the analysis result for system (5.127) to be exponentially stable. Theorem 16. Let the controller and observer gain matrices K and L be given. The closed-loop system (5.127) is exponentially stable if there exist matrices 0 < P,

II. OBSERVER-BASED CONTROL DESIGN

191

NCS WITH QUANTIZATION AND NONSTATIONARY RANDOM DELAYS

0 < QTj = Qj , j = 1, . . . , 4 and matrices Ri , Si , and Mi , i = 1, 2, such that the following matrix inequality holds   Λ1j Λ2j 0 for all i, j ∈ Λ, matrices K˜ i ∈ Rp×n , Lˆ i ∈ Rn×m , and positive P˜ 11 − G j

i

scalars ξi , ai , νi , bi , κi , ci , so that the following LMI holds for all i, j ∈ Λ: min Trace(Di1 ) subject to  ij

Ξ1 Si1