CONTROLO 2020: Proceedings of the 14th APCA International Conference on Automatic Control and Soft Computing, July 1-3, 2020, Bragança, Portugal [1st ed.] 9783030586522, 9783030586539

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Lecture Notes in Electrical Engineering 695

José Alexandre Gonçalves Manuel Braz-César João Paulo Coelho   Editors

CONTROLO 2020 Proceedings of the 14th APCA International Conference on Automatic Control and Soft Computing, July 1–3, 2020, Bragança, Portugal

Lecture Notes in Electrical Engineering Volume 695

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Naples, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Bijaya Ketan Panigrahi, Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, Munich, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, Humanoids and Intelligent Systems Laboratory, Karlsruhe Institute for Technology, Karlsruhe, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Università di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Sandra Hirche, Department of Electrical Engineering and Information Science, Technische Universität München, Munich, Germany Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Stanford University, Stanford, CA, USA Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martín, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Sebastian Möller, Quality and Usability Laboratory, TU Berlin, Berlin, Germany Subhas Mukhopadhyay, School of Engineering & Advanced Technology, Massey University, Palmerston North, Manawatu-Wanganui, New Zealand Cun-Zheng Ning, Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Graduate School of Informatics, Kyoto University, Kyoto, Japan Federica Pascucci, Dipartimento di Ingegneria, Università degli Studi “Roma Tre”, Rome, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, Universität Stuttgart, Stuttgart, Germany Germano Veiga, Campus da FEUP, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Beijing, China Junjie James Zhang, Charlotte, NC, USA

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José Alexandre Gonçalves Manuel Braz-César João Paulo Coelho •

•

Editors

CONTROLO 2020 Proceedings of the 14th APCA International Conference on Automatic Control and Soft Computing, July 1–3, 2020, Bragança, Portugal

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Editors José Alexandre Gonçalves Escola Superior de Tecnologia e Gestão Instituto Politécnico de Bragança Bragança, Portugal

Manuel Braz-César Escola Superior de Tecnologia e Gestão Instituto Politécnico de Bragança Bragança, Portugal

João Paulo Coelho Escola Superior de Tecnologia e Gestão Instituto Politécnico de Bragança Bragança, Portugal

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-3-030-58652-2 ISBN 978-3-030-58653-9 (eBook) https://doi.org/10.1007/978-3-030-58653-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Going to a conference is always an opportunity to establish contacts with other researchers and to grow culturally. Often, from informal conversations fostered by the conference’s social programme, synergies are generated, leading to new ideas and partnerships. Indeed, networking is the fundamental reason for being present at any conference. Moreover, every time we visit a new country, distinct realities are observed and a kind of mild acculturation happens. Experiences gathered from travelling shape our vision of the world and promote cultural enrichment. The CONTROLO 2020 Conference was intended not only to be a scientific event, but also to give the opportunity for the control community to know Bragança, the Polytechnic Institute of Bragança, and to network face-to-face in friendly, safe, stimulating and international environment. The city was prepared to receive this scientific community, with a welcome reception at the Castle of Bragança and a gala dinner at the gardens of the Abade de Baçal Museum, providing all the participants to get in touch with some of the most important landmarks of the Bragança cultural heritage. The ideal situation would be to communicate in person but, due to the current pandemic situation, the CONTROLO 2020 Conference was forced to change to an online event. In spite of all these difficulties, the decision to maintain the date of this event has demonstrated the resilience of APCA in adapting to extreme conditions. In addition, it was also important in this decision-making, to ensure that the dissemination of the research and development works, submitted and accepted for publication, would be made public in due time. A total of 93 papers, originating from 39 different countries, were submitted for publication, from which 76 were accepted, after an evaluation process that totalled more than 300 reviews. Besides its six special sessions and a MATLAB course, the CONTROLO 2020 Conference had the honour of having five of the most relevant personalities in automatic control community in its five plenary sessions, namely

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Preface

professors André Preumont, Eduardo Camacho, Karl Johansson, Kevin Passino and Sebástian Dormido. We hope that all interested readers can benefit scientifically from this conference proceedings. Best Regards, CONTROLO 2020 Local Organizing Committee

Contents

Concepts of Threshold Assessment for a First Course in Control Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Anthony Rossiter

1

Predictive Functional Control for Unstable First-Order Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Muhammad Saleheen Aftab, John Anthony Rossiter, and Zhiming Zhang

12

Bridging Theory to Practice: Feedforward and Cascade Control with TCLab Arduino Kit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. B. de Moura Oliveira, John D. Hedengren, and José Boaventura-Cunha

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Cooperative Circumnavigation for a Mobile Target Using Adaptive Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joana Fonseca, Jieqiang Wei, Tor Arne Johansen, and Karl Henrik Johansson Performance Enhancement of a Neato XV-11 Laser Scanner Applied to Mobile Robot Localization: A Stochastic Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . José Gonçalves, João Paulo Coelho, Manuel Braz-César, and Paulo Costa

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Prototyping and Control of a Conveyor Belt: An Educational Experiment in Mechatronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . José Gonçalves, João Ribeiro, and Paulo Costa

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Welding Process Automation of Aluminum Alloys for the Transport Industry: An Industrial Robotics Approach . . . . . . . . . . . . . . . . . . . . . . João Ribeiro, José Gonçalves, and Nuno Mineiro

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Sub-Riemannian Geodesics on Nested Principal Bundles . . . . . . . . . . . . Mauricio Godoy Molina and Irina Markina

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A Comparative Study of Dynamic Mode Decomposition (DMD) and Dynamical Component Analysis (DyCA) . . . . . . . . . . . . . . . . . . . . . Moritz Kern, Christian Uhl, and Monika Warmuth

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An Approach to Model Validation for Model Predictive Control Based on Dvurecenska’s Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Victor D. Reyes Dreke, Manuel A. Pérez Serrano, and Claudio Garcia Cloud-Based Framework for Robot Operation in Hospital Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Nuno M. Fonseca Ferreira and José Boaventura-Cunha Identification and Control of Precalciner in the Cement Plant . . . . . . . . 126 Jakub Osmic, Emir Omerdic, Edin Imsirovic, Tima O. Smajlovic, and Edin Omerdic Rolling on Affine Tangent Planes: Parallel Transport and the Associated Sub-Riemannian Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Verlimir Jurdjevic The Numerical Control of the Motion of a Passive Particle in a Point Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Carlos Balsa and Sílvio Gama Gas Network Hierarchical Optimisation—An Illustrative Example . . . . 159 T.-P. Azevedo Perdicoúlis and P. Lopes dos Santos Calibration-Free HCPV Sun Tracking Strategy . . . . . . . . . . . . . . . . . . . 170 Manuel G. Satué, Manuel G. Ortega, Fernando Castaño, Francisco R. Rubio, and José M. Fornés Geometric Algorithm to Generate Interpolating Splines on Grassmann and Stiefel Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Luís Machado, Fátima Silva Leite, and Ekkehard Batzies Dynamic Model for the pH in a Raceway Reactor Using Deep Learning Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Pablo Otálora, José Luis Guzmán, Manuel Berenguel, and Francisco Gabriel Acién Double Back-Calculation Approach to Deal with Input Saturation in Cascade Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Marta Leal, Ángeles Hoyo, José Luis Guzmán, and Tore Hägglund State-Space Estimation Using the Behavioral Approach: A Simple Particular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Lorenzo Ntogramatzidis, Ricardo Pereira, and Paula Rocha A State-Space Model Inversion Control Method for Shake Table Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 José Ramírez-Senent, Jaime H. García-Palacios, and Iván M. Díaz

Contents

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Minimum Energy Control of Passive Tracers Advection in Point Vortices Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Carlos Balsa, Olivier Cots, Joseph Gergaud, and Boris Wembe Cubic Splines in the Grassmann Manifold . . . . . . . . . . . . . . . . . . . . . . . 243 Fátima Pina and Fátima Silva Leite Distributed Adaptive Predictive Control Based on Switched Multiple Models and ADMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Margarida Nabais and João M. Lemos Optimization of the Clinker Production Phase in a Cement Plant . . . . . 263 Silvia Maria Zanoli, Lorenzo Orlietti, Francesco Cocchioni, Giacomo Astolfi, and Crescenzo Pepe An Extrinsic Approach to Sub-Riemannian Geodesics on the Orthogonal Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Knut Hüper, Irina Markina, and Fátima Silva Leite Assessment of the Nutritional State for Olive Trees Using UAVs . . . . . . 284 Pablo Cano Marchal, Diego Martínez Gila, Sergio Illana Rico, Javier Gómez Ortega, and Javier Gámez García Temperature Control on Double-Pipe Heat-Exchangers: An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 S. J. Costa, R. Ferreira, and J. M. Igreja A DOBOT Manipulator Simulation Environment for Teaching Aim with Forward and Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . . 303 Thadeu Brito, José Lima, João Braun, Luis Piardi, and Paulo Costa An IIoT Solution for SME’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Bruno Cunha, Elder Hernández, Rui Rebelo, Cristóvão Sousa, and Filipe Ferreira Existence and Uniqueness for Riemannian Cubics with Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Margarida Camarinha, Fátima Silva Leite, and Peter Crouch Home Energy Management System in an Algarve Residence. First Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 A. Ruano, K. Bot, and M. Graça Ruano LSLOCK: A Method to Estimate State Space Model by Spatiotemporal Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Tsuyoshi Ishizone and Kazuyuki Nakamura Passive Particle Dynamics in Viscous Vortex Flow . . . . . . . . . . . . . . . . . 352 Gil Marques, Maria João Rodrigues, and Sílvio Gama

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A Decentralized Strategy for Variational Collision Avoidance on Complete Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Leonardo J. Colombo and Jacob R. Goodman Optimal Route Planning in Steady Planar Convective Flows . . . . . . . . . 373 Roman Chertovskih, Maxim Staritsyn, and Fernando Lobo Pereira A Fuzzy Based Model to Assess the Influence of Project Risk on Corporate Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Ricardo Santos, Antonio Abreu, João M. F. Calado, José Miguel Soares, José Duarte Moleiro Martins, and Vitor Anes Study on the Isolator-Structure Interaction. Influence on the Supporting Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 J. Pérez-Aracil, E. Pereira, Iván M. Díaz, and P. Reynolds Fallback Approximated Constrained Optimal Output Feedback Control Under Variable Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Christian Kallies, Mohamed Ibrahim, and Rolf Findeisen Teaching Neural Control with an Arduino Based Control Kit . . . . . . . . 415 Ramiro S. Barbosa Differential Observation and Integral Action in LTI State-Space Controllers and the PID Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Paulo Garrido Endpoint Geodesics on the Set of Positive Definite Real Matrices . . . . . 435 Maximilian Stegemeyer and Knut Hüper Decentralized Control for Multi-agent Missions Based on Flocking Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Rafael Ribeiro, Daniel Silvestre, and Carlos Silvestre Model Predictive Control of a Pusher Type Reheating Furnace . . . . . . . 455 Silvia Maria Zanoli, Francesco Cocchioni, Chiara Valzecchi, and Crescenzo Pepe Distributed LQ Control of a Water Delivery Canal Based on a Selfish Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 João P. Belfo, João M. Lemos, and A. Pedro Aguiar A Comparative Performance Study of Inertial Vibration Controllers for an Ultra-lightweight GFRP Footbridge . . . . . . . . . . . . . . . . . . . . . . . 477 Carlos Martín Concha Renedo, José Manuel Soria, Christian Gallegos, and Iván M. Díaz MPC Framework for Supply Chain Management Integrating On-Time Delivery and Transport Management . . . . . . . . . . . . . . . . . . . 487 Eduardo Araújo, João Lemos Nabais, and Miguel Ayala Botto

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Control of the Depth of Anesthesia Using a New Model for the Action of Propofol and Remifentanil on the BIS Level . . . . . . . . . . . . . 497 Jorge Silva, Teresa Mendonça, and Paula Rocha Using Multi-UAV for Rescue Environment Mapping: Task Planning Optimization Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Ricardo Rosa, Thadeu Brito, Ana I. Pereira, José Lima, and Marco A. Wehrmeister Robustness Issues in Event-Based PI Control Systems: Internal Model Control Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 R. Vilanova, C. Pedret, M. Barbu, M. Beschi, and A. Visioli A Fractional Order Predictive Control for Trajectory Tracking of the AR.Drone Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Ricardo Cajo, Shiquan Zhao, Douglas Plaza, Robain De Keyser, and Clara Ionescu Practical Validation of a Dual Mode Feedforward-Feedback Control Scheme in an Arduino Kit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 P. B. de Moura Oliveira and Damir Vrančić On the Use of a Maximum Correntropy Criterion in Kalman Filtering Based Strategies for Robot Localization and Mapping . . . . . . 548 Matheus F. Reis, Hamed Moayyed, and A. Pedro Aguiar Extrinsic Sensor Calibration Methods for Mobile Robots: A Short Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 Ricardo B. Sousa, Marcelo R. Petry, and António Paulo Moreira CDM Controller Design of a Grid Connected Photovoltaic System . . . . 570 João Paulo Coelho, Wojciech Giernacki, José Gonçalves, and José Boaventura-Cunha Classification of Car Parts Using Deep Neural Network . . . . . . . . . . . . 582 Salik Ram Khanal, Eurico Vasco Amorim, and Vitor Filipe Soiling Monitoring Modelling for Photovoltaic System . . . . . . . . . . . . . . 592 Vitor H. Pagani, Nelson A. Los, Wellington Maidana, Paulo Leitão, Marcio M. Casaro, and Claudinor B. Nascimento Vision-Based Object Detection and Localization for Autonomous Airborne Payload Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 James Sewell, Theo van Niekerk, Russell Phillips, Paul Mooney, and Riaan Stopforth Stabilization Using In-domain Actuator: A Numerical Method for a Non Linear Parabolic Partial Differential Equation . . . . . . . . . . . . 616 Thérèse Azar, Laetitia Perez, Christophe Prieur, Emmanuel Moulay, and Laurent Autrique

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Direct Power Control of a Doubly Fed Induction Generator Using a Lyapunov Based State Space Approach . . . . . . . . . . . . . . . . . . . . . . . 628 Yassine Boukili, A. Pedro Aguiar, and Adriano Carvalho Model of a DC Motor with Worm Gearbox . . . . . . . . . . . . . . . . . . . . . . 638 Vítor H. Pinto, José Gonçalves, and Paulo Costa Trajectory Planning for Landing with a Direct Optimal Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 Bertinho A. Costa and João M. Lemos Accelerated Generalized Correntropy Interior Point Method in Power System State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 Hamed Moayyed, Diyako Ghaderyan, Yassine Boukili, and A. Pedro Aguiar Your Turn to Learn – Flipped Classroom in Automation Courses . . . . 668 Filomena Soares, P. B. de Moura Oliveira, and Celina P. Leão Modeling of an Elastic Joint: An Experimental Setup Approach . . . . . . 676 Vítor H. Pinto, José Lima, José Gonçalves, and Paulo Costa On the Control Models in the Trajectory Tracking Problem of a Holonomic Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 Aleksandr Andreev and Olga Peregudova LMI-Based Sliding Mode Controller Design for an Uncertain Single-Link Flexible Robot Manipulator . . . . . . . . . . . . . . . . . . . . . . . . 696 José Manuel Andrade and Christopher Edwards Control of Bio-Inspired Multi-robots Through Gestures Using Convolutional Neural Networks in Simulated Environment . . . . . . . . . . 707 A. A. Saraiva, D. B. S. Santos, Nuno M. Fonseca Ferreira, and José Boaventura-Cunha Addendum to “Seeking a Unique View to Control of Simple Systems” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 Mikulas Huba, M. Hypiusová, P. Ťapák, and A. Serbezov Open Hardware and Software Robotics Competition for Additional Engagement in ECE Students - The Robot@Factory Lite Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 Vítor H. Pinto, Armando Sousa, José Lima, José Gonçalves, and Paulo Costa Engine Labels Detection for Vehicle Quality Verification in the Assembly Line: A Machine Vision Approach . . . . . . . . . . . . . . . . 740 Sílvio Capela, Rita Silva, Salik Ram Khanal, Ana Teresa Campaniço, João Barroso, and Vítor Filipe

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Mitigation of Earthquake-Induced Structural Pounding Between Adjoining Buildings – State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . 752 Pedro Folhento, Rui Barros, and Manuel Braz-César Prototyping of a Low-Cost Stroboscope to Be Applied in Condition Maintenance: An Open Hardware and Software Approach . . . . . . . . . . 762 Laiany Brancalião, Caio Camargo, José Gonçalves, and José Lima Dynamic Survey of a Telecommunication Tower by Interferometric Radar Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Fábio Paiva, Rui Barros, Jorge Henriques, Tiago Cunha, and Pierre Feyfant Motion-Based Design of Semi-active Tuned Mass Dampers to Control Pedestrian-Induced Vibrations in Footbridges Under Uncertainty Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 Javier Fernando Jiménez-Alonso, José Manuel Soria Herrera, Carlos Martín de la Concha Renedo, and Francisco Guillen-González Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

Concepts of Threshold Assessment for a First Course in Control Engineering John Anthony Rossiter(B) Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UK [email protected]

Abstract. This paper focuses on the combined challenge of encouraging students to engage with learning and assessment of their competence levels. A core challenge for many staff is the need to distinguish different levels of learning, and to evidence core competence clearly, especially for students with lower marks. This paper proposes a novel assessment strategy which separates core competencies from the more challenging application of the learning in engineering problem solving. The assessment design is efficient for staff and students and allows a reduction in student stress levels while simultaneously giving them strong incentives to adopt good working practices. Evaluation evidence is given to demonstrate the efficacy of the approach. Keywords: Assessment Student engagement

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· Learning outcomes · Student stress ·

Introduction

The most stressful part of a student journey is the assessment. Potentially future career options and a sense of self-worth are tied into the marks that students achieve. Nevertheless, despite the huge importance of good performance in assessment, the majority of students are quite poor in study skills and time management and the consequence is an increased stress at the end of year exams and a prevalence of cramming as opposed to real learning. Many strategies have considered how to support students in better learning models and a particularly popular one [2,4,9,12,18] from the early 2000s is the use of many low weighted computer quizzes. The idea is that the small marks available in each quiz give students an incentive to engage with the quiz content and thus, implicitly, to learn the core requirements of a course in regular manner. However, even though largely successful, these quizzes create stresses of their own: 1. Despite the low weighting (often 2–3% per quiz) students get fixated on the actual mark for each quiz and even argue about something that, in effect may be worth only 0.02% of the module mark. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 1–11, 2021. https://doi.org/10.1007/978-3-030-58653-9_1

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J. A. Rossiter

2. Because the actual mark is recorded, students are under pressure (stress) to perform high quality work every couple of weeks in modules where this is deployed and thus may feel under continuous stress. 3. Some students complain that the quizzes are simplistic as they do not reward working and penalise incorrect answers with a zero mark. So, while successful to some extent in encouraging engagement, a typical implementation can add significantly to student stress. An alternative model may provide the quizzes as formative feedback opportunities only e.g. [13,19]; that is, students take the quizzes, or indeed use other resources [5,7], to self-assess their progress but receive no marks for doing so. The author has found through experience that this model does not work for many students, irrespective of the quality of the resources, with two main negatives. 1. Most students do not do what is considered as optional [9] and thus fall behind in their studies with a knock on effect of increasing dissatisfaction and student struggles/disengagement. 2. Irrespective of the support provided, students seem unhappier with ‘formative feedback’ provision, even though this is intended to remove stress by allowing practice and learning without the performance being assessed formally. They want a more tangible reward for doing something. Hence, this paper aims to improve on the model of regular computer assessment by considering mechanisms for reducing the stress in this process. It so happens that accreditation processes [1,6,20] also form a strong motivation for supporting the proposal. Accreditors put a lot of emphasis on evidence that students who pass a course have met all the learning outcomes; without this evidence that cannot accredit the degree as they argue they do not have evidence that the students have achieved the requisite skills. One downside of this mode of thinking, certainly within the UK, is that accreditors increasingly require that students achieve a good mark in every assessment component of a module (courseworks, exams, laboratories) because, if these contain distinct learning outcomes, then we can only demonstrate the student is competent, if they pass them all. One can argue about the truth or not of the accreditors’ assumptions, but they make the rules and, in simple terms, we must provide clear evidence that students have passed all the learning outcomes for a (every) module if they are to be accredited. So, in summary this paper focuses on two parallel requirements: 1. Encourage students to engage with module material on a regular basis and develop good study skills. 2. Demonstrate, efficiently, that students have met all the learning outcomes for a module to meet accreditation requirements. It is emphasised that this paper does not discuss assesssments used to distinguish between different levels of passing performance. Section 2 introduces the concept of learning outcomes and their relevance to degree education and from this makes an assessment design proposal. Section 3 discusses the proposal and its link to

Threshold Assessment

3

accreditation and Sect. 4 gives some evaluation data for the proposal of this paper based on a pilot.

2

Assessing Learning Outcomes

It is noted at the outset that accreditating bodies are inconsistent in their requirements for evidence of learning outcome (LO) achievement. For example, if a module is assessed by examination only, they are happy to accept that a student receiving a pass mark on the examination has met all the LO, as long as the examination assesses all the LO. However, all academics know that many students question spot and do not answer many parts of an examination paper, so a pass mark does not mean that have passed all the different elements but rather, in all likelihood they have done quite well on some questions. Despite this inconsistency where this is a single module assessment, in the UK accrediting bodies [20] (whose policies are sanctioned in liaison with other international accrediting bodies) will put undue focus on the need to pass all individual module assessments, where there are several components. This is probably because it may be clearer how the LO are separated into the different assessment components. This requirement puts huge stresses on both students and examination boards as they now have to, not only consider the student overall mark but also the component marks; in many cases students with a high overall module mark such as 15% over a pass mark, are still recorded as fail due to a single component being just a few percent below the pass mark. The context of this paper takes account of this accreditation issue and thus, some aspects may not be fully applicable elsewhere. A module leader would like an efficient mechanism for demonstrating that passing students have met all the LO and, the argument here, is that can usefully be separated from performance assessment whereby we aim to distinguish different levels of passing performance. In summary then, the proposal is to develop a two fold assessment strategy. 1. Pass/fail (or threshold) components which students MUST pass to demonstrate they have met all the LO for the module. 2. Classification components which are used to distinguish performance of different levels; a student with bare pass may score zero on these components as long as they have passed all the threshold components. The design of the latter assessments is not discussed here, suffice to say they need not, and indeed should not, contain any elementary parts.

3

Assessing Threshold Competence

A competent chartered engineer needs to be skilled in applying their knowledge to solve problems and indeed, for many challenging problems, may also need to demonstrate imagination and creativity. Conversely, if one considers a single topic (for example electrical circuits), it may be sufficient for most engineers to

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have only foundational knowledge of core components and circuit analysis as this topic may have very little relevance to their job. For topics that are not their speciality, it is not reasonable to expect engineers to undertake problem solving and thus, for accreditation purposes, basic competence only is sufficient. 3.1

What Is Threshold Competence?

A core question for academics is to decide: what constitutes a basic level of knowledge and/or to what extent must students be able to apply that knowledge to meet accreditation requirements? This paper assumes a graduated level of ability, something like the following (it is understood that such things often cannot be tied down precisely so this list is intended to be indicative). – A bare pass can be achieved by demonstrating knowledge or recall of core information and the selection and application of simple but fundamental computations linked to a topic. – For a medium level of pass, in addition to the above core skills, students should demonstrate the ability to perform straightforward applications of some of the module knowledge for problem solving. – For a strong pass, students should accurately solve most straightforward applications they have seen before within a module, and also tackle more difficult computations/analysis for some but not all topics in the module. – For a first class mark, say top 10–15%, students should accurately solve all applications they have seen before within a topic to a high degree of technical depth and also some slightly more challenging applications with novel aspects. – For a excellent mark (top few students only), in addition to the above, students should be able to select appropriate techniques and apply creativity in problem solving for scenarios they have not encountered before. This paper forcuses on the first item in the list and thus requires a definition of the core information and computations that underpin a module and suffice to demonstrate a pass level of performance. 3.2

Illustration of Threshold Competence

This paper focuses on a first course in control [17] and thus one may wish to consider, what would constitute a basic level of competence for this type of topic? Some suggestions for discussion are given in Table 1, although clearly benchmarking this is context/institution/degree dependent in general. The list is not intended to be complete but rather is illustrative. 3.3

Assessment of Threshold Competence and Staff Loading

Most modules will have tutorial sheets, quizzes and assignments and these are often designed with the performance assessment in mind, that is to distinguish different levels of student ability. Hence a tutorial sheet will begin with threshold

Threshold Assessment

5

Table 1. Examples of threshold skills in an introductory control course (not comprehensive). Topic

Threshold concepts

Advanced concepts

1st order responses Time constant, steady-state gain, curve sketching, modelling from step responses

Interpretation, repercussions, parameter design, impact of parameter changes, etc.

Laplace transforms Core signals, partial fractions of low order transfer functions, final value theorem, application to simple ODEs

High order transfer functions, application in unknown context, interpretation, ...

System behaviours LHP/RHP, stability, speed of response, decays rates, damping ratio

Parameter design, impact of parameter changes, novel applications, interpretations, ...

Feedback

Impact of proportional design, PI design, performance specifications, application to new scenarios, ...

Closed-loop transfer functions, closed-loop poles, closed-loop responses

questions and then gradually move into harder and more open-ended questions. Similarly, an assignment may have a straightforward introductory part followed by increasingly challenging application components. The main proposal here is to separate out the threshold components entirely, that is to have entirely distinct threshold tutorial sheets/quizzes/laboratories. In all likelihood these already exist so to some extent it is simply a reorganisation of how material is presented to the students. However, herein lies the major change which will facilitate a reduction in staff marking time. The aim of threshold assessment is to award pass or fail, that is, not to award a mark. Hence, marking of these components is a binary decision and can be done very efficiently. – A student attending a laboratory is judged either to have met the required skills or not. No mark is awarded. – A student taking a tutorial/test on threshold concepts (Table 1) is judged to pass or fail only, again no mark is awarded. Indeed, most straightforward skills can be assessed with simple binary decisions: (i) correct calculation; (ii) multi-choice questions; (iii) etc. In Sheffield we intend to use a pass mark of 70% on a large number of straightforward questions.

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Remark 1. During the trial it was agreed that students scoring less than 70% would receive the actual mark, that is, they still receive some credit for the knowledge and skills they have demonstrated. In summary, as part of the planning for the module delivery, the module leader ensures the core LO (at a pass/fail level only) are included in some threshold assessments. Students are expected to pass these assessments and, should they pass them all they will pass the module. Assessment will largely be automatic/binary as considering pass/fail competence only and thus can be done very efficiently. 1. Hand written tutorial sheets with binary marking schemes can be assessed quickly by a teaching assistant or a staff member or peer or an automated system, especially if the answers are entered on a standard proforma. 2. Computer quiz environments are very efficient at assessing binary decisions and once written, can be used in subsequent years. These environments will usually link to the student database adding another efficiency gain, that is, no manual handling of the assessment. 3.4

Managing Student Expectations and Stress Levels

A second motivation for so-called threshold assessment is to reduce student stress levels. One might think this sounds perverse: how does adding several small assessments reduce overall stress? In simple terms the argument is that the assessments are pass/fail, that is, students are not expected to do their very best work and rather need only show that they have mastered the basics; the more advanced application of knowledge is not included. This reduces the pressure to over-prepare and any concerns about whether the mark should be 80% rather than 76% as both are awarded a pass, that is, the same mark. The positive aspect of this is that students have marks in the bag which they have merited by doing what all conscientious students should be doing anyway, keeping up with lectures and tutorial sheets at least to a basic level. In truth, we would expect students to be working harder than is required to just pass the threshold assessments. Students can then approach their more challenging assignments and end of year exams with the confidence that they have already passed the module, can be accredited and can progress. Moreover, by keeping up with basic level skills one would expect students to progress better and be more content with the more challenging aspects. A secondary aspect is how the threshold assessments are delivered. Clearly the most important aspect is that the students develop and demonstrate basic competence, but without feeling overly stressed by the process. One stress alleviator is to emphasise that it does not matter whether it takes a student numerous attempts to achieve this, as these assessments do not influence classification. Consequently, the author gives students multiple attempts, typically 3–5, so that students who fail at the first attempt need not be stressed in the knowledge they can revise a bit more and try again. It should be emphasised that this multiple

Threshold Assessment

7

attempt approach is most efficiently delivered through a computer quiz environment as this entails no extra load to the staff member and moreover, these environments allow random number allocation and random question selection so that the student will get a similar but different test each time. Alternatives favoured by some colleagues, and used by the author for one class, are the use of straightforward short in-class tests. These can be marked, by peers, at the end of the class as the marking scheme is very simple, thus saving marking time. Students who fail this test have a week to hand in a perfectly correct solution to the test. Again, as perfection is now expected, these later submissions will be very quick to handle and forces students to, at the very least, write down and engage with perfect solutions.

4

Evaluation of Pilot

A pilot of the proposed strategy was undertaken in the Autumn of 2019 and data was collected about both student perceptions and performance across 3 modules: (i) year 2 chemical engineers; (ii) year 2 general engineers and (iii) year 1 aerospace, bioengineering and systems engineers. This section summarises the quantitative data and gives some conclusions. We will add some qualitative data to the final paper, if the students provide this in the end of module feedback forms. Some early face to face feedback suggests that the ability to demonstrate basic competence was hugely important to self-confidence for some students, which also ameliorates stress significantly. From a staff perspective, using a virtual learning environment and colours, it is easy to get an overview of how the class is doing as a whole and also to pick out individuals who may be struggling. Figure 1 shows progress in week 8 for the smallest cohort, from which it is clear which student is totally disengaged (row 24) and needs following up: – Assessments 1, 2, 3 and 5 should be complete and the RED squares show students who did not pass this. – Assessments 4 and 6 are in progress so you can see which students have still to complete. The remainder of this section presents a subset of the data collected from the students of the three different modules in the trial. It is noted that the profiles are similar across all modules. Questions used a typical Likert scale, that is: 1. definitely agree, 2. agree, 3. neutral, 4. disagree, 5. definitely disagree. The questions are given in the figure legends, see Figs. 2, 3, 4, 5 and 6. A basic summary is: 1. Figure 2 indicates that the assessment regime helped the large majority of students with their time management and keeping on top of their studies. 2. Figure 3 reinforces the evidence seen in previous years; if the quizzes do not carry a mark, most students will not use them despite their formative value.

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Fig. 1. Interim overview of student progress by week 8: Green means complete, Red means no attempt or zero and dark Red/Brown indicates a mark between 0 and 70.

Fig. 2. Having the threshold assessments gave me an incentive to keep up with my studies and clarity on how much I should be working and thus helped my time management.

Fig. 3. If the quizzes did not count, I would still have used them just as much to aid my learning.

3. Figures 4 and 5 indicate that, despite having an assessment every 1–2 weeks, actually most students felt the approach reduced their stress levels. 4. Figures 6 and 7 suggest that students like this assessment approach and would like it to be retained.

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Fig. 4. Having multiple attempts along with the need to only score 70% helped reduce the stress of these assessments so I could focus on learning rather than getting stressed about trying to achieve the highest score possible.

Fig. 5. Allowing me to achieve a significant percentage of the module mark by Christmas has helped reduce my overall stress linked to assessment and progress at University.

Fig. 6. I liked having the threshold assessments (quizzes and laboratories).

Fig. 7. I would prefer that there were no threshold assessments, that is the quizzes did not count in the module mark, and the end of year examination was worth more.

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5

Conclusions

This paper has proposed an assessment regime which clearly separates basic learning outcomes from more advanced ones and thus allows students to demonstrate they deserve to pass a module/course in a less stressful manner, with only classification assessments being done at the end of a module. The preliminary evaluation demonstrates that the students appreciate the approach, even though, by their own tacit admission, it means they are working harder than they would if all interim assessment and feedback was formative. A core advantage is that the students are engaging earlier with mastering foundational concepts and the implicit hope is that this will enable them to engage better with the more challenging aspects of the module later in term. Indeed, the anecdotal evidence is that lecture attendance later in term is higher this year than in previous years, and this is likely linked to students still being able to follow what is going on. We are waiting for a discussion with accreditors (scheduled for May 2020 but postponed due to COVID-19) on their reception to this strategy, for example, do they welcome it and can this make their quality assurance easier? In fact, recent announcements from accrediting bodies suggests that this approach is likely to be actively encouraged and indeed the author’s University has adopted it in year 1 for the academic year 2019–20 in response to the COVID crisis.

References 1. ABET: Accreditation in the USA. http://www.abet.org 2. Croft, A.C., Danson, M., Dawson, B.R., Ward, J.P.: Experiences of using computer assisted assessment in engineering mathematics. Comput. Educ. 37(1), 53–66 (2001) 3. Crouch, C.H., Mazur, E.: Peer instruction: ten years of experience and results. Am. J. Phys. 69, 970–977 (2001) 4. Lawson, D.A.: The effectiveness of a computer assisted learning programme in engineering mathematics. Int. J. Math. Educ. Sci. Technol. 26, 4 (1995) 5. Egerstedt, M.: MOOC on control of mobile robots (2016). https://www.coursera. org/course/conrob 6. ENAEE: European network for accreditation of engineering education. http:// www.enaee.eu 7. Khan academy. https://www.khanacademy.org/ 8. Lynch, S., Becerra, V.: MATLAB assessment for final year modules. In: The Use of MATLAB Within Engineering Degrees, HEA Workshop and Seminar Series (2011) 9. Rossiter, J.A., Gray, L., Rossiter, D.: Case studies of the resources students use. In: IFAC World Congress (2005) 10. Rossiter, J.A., Rossiter, D., O’Brien, G.D.: Experiences in the use of web-based delivery for first year engineers. In: WBE (2004) 11. Rossiter, J.A., Croft, A.C.: Engaging engineers in learning. In: HEA Annual Conference (2005) 12. Rossiter, D., Rossiter, J.A.: Applications of online pedagogy to a first year blended learning module using a VLE. In: International Conference on Innovation, Good Practice and Research in Engineering Education (2006)

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13. Rossiter, J.A., Rossiter, D.: A blended learning module design approach to engage new students. In: Second International Blended Learning Conference, Bedford (2007) 14. Rossiter, J.A., Giaouris, D., Mitchell, R., McKenna, P.: Typical control curricula and using software for teaching/assessment: a UK perspective. In: IFAC World Congress (2008) 15. Rossiter, J.A.: Using quizzes instead of paper based exams to assess control topics. In: Control 2018 (2018) 16. Rossiter, J.A., Jones, B.L., Murray, R.M., Vlacic, L., Dormido, S.: Opportunities and good practice in control education: a survey. In: IFAC World Congress (2014) 17. Rossiter, J.A., Zakova, K., Huba, M., Serbezov, A., Visioli, A.: A first course in feedback, dynamics and control: findings from an online pilot survey for the IFAC community. In: IFAC Symposium on Advances on Control Education (2019) 18. Sim, G., Holifield, P., Brown, M.: Implementation of computer assisted assessment: lessons from the literature. J. ALT-J Res. Learn. Technol. 12, 3 (2004) 19. STACK: Mathematics stack exchange, December 2017. https://math. stackexchange.com/ 20. UK-SPEC: Engineering council. http://www.engc.org.uk/ukspec.aspx

Predictive Functional Control for Unstable First-Order Dynamic Systems Muhammad Saleheen Aftab1(B) , John Anthony Rossiter1 , and Zhiming Zhang2 1

2

Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, UK {msaftab1,j.a.rossiter}@sheffield.ac.uk State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou, China [email protected] Abstract. Predictive functional control (PFC) has emerged as a popular industrial choice owing to its simplicity and cost-effectiveness. Nevertheless, its efficacy diminishes when dealing with challenging dynamics because of prediction mismatch in such scenarios. This paper presents a proposal for reducing prediction mismatch and thus improving behaviour for simple unstable processes; a two-stage design methodology prestabilises predictions via proportional compensation before introducing the PFC component. It is demonstrated that pre-stabilisation reduces the dependency of the closed-loop pole on the coincidence point and also improves robustness to uncertainty. Simulation results verify the improved performance as compared to conventional PFC. Keywords: PFC · Coincidence horizon Proportional compensation

1

· Pre-stabilisation ·

Introduction

Predictive functional control (PFC) offers numerous beneficial attributes such as trivial coding, easy implementation and simple handling without needing sophisticated knowledge, software or specialised personnel. These qualities, along with systematic handling of constraints and dead-times compared to other conventional methods, say proportional-integral-derivative (PID) control, make PFC a popular alternative in industry, with numerous successful applications [1]. Conventional PFC [1–3] matches the plant output prediction to a desired first-order target trajectory at only one future point, the so-called coincidence point, by keeping the predicted input constant. One may ask if there exists a reliable criterion for selecting the desired target dynamics and coincidence point? Researchers have established generic guidelines for systems with relatively benign dynamics. For example, it is recommended [2] to use a one-step ahead model prediction for first-order plant as this guarantees target behaviour for first-order systems [4]. Alternatively, one recommendation for higher-order systems is to choose the point of inflection (where the gradient is maximum) on the c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 12–22, 2021. https://doi.org/10.1007/978-3-030-58653-9_2

PFC for Unstable First Order Systems

13

step response curve as the coincidence point although it is arguable whether this would work well for systems with challenging dynamics. Moreover, for monotonically convergent higher-order systems, a coincidence point where the open-loop step response has risen to approximately 40–80% of the steady-state is often a better choice [4]. Nevertheless, matching underdamped, unstable and nonminimum phase dynamics with target first-order behaviour does not make sense and coincidence point selection for such systems is not straight-forward. Challenging dynamics demand a different parametrisation of the degrees-of-freedom [5], as the typical constant input assumption within the prediction horizon may be inappropriate. One recent attempt [6] parametrised the input with first-order Laguerre polynomial, which improves prediction consistency and convergence rate as compared to the original PFC for systems with simple dynamics; however, this approach is not really tailored to systems with difficult dynamics. The main objective of this paper is to build on the ideas in [5,7] and indeed conventional wisdom in PFC [2] which is to modify unstable dynamics before applying the PFC design. Accepted practice in the mainstream MPC community uses pre-stabilisation [8,9], so this paper proposes a a two-stage PFC design methodology by integrating pre-stabilised dynamics with PFC decision making. Initially we restrict our study to first-order unstable plants focusing on the effects of a pre-stabilising structure on closed-loop performance, sensitivity and constraint handling. Specifically this paper analyses the relationship between the target pole, pre-stabilising gain and coincidence horizon and establishes guidelines for systematic and effective tuning. Generally a trade-off between closedloop performance and sensitivity is observed, which signifies the importance of offline sensitivity analysis for proper selection of tuning parameters; something not in the conventional PFC literature. With pre-stabilisation, numerical simulations show improved closed-loop performance as compared to conventional PFC. Extensions for systems with higher-order dynamics constitutes future work. The remainder of this paper is organised as follows: Sect. 2 succinctly formulates the control problem. Section 3 proposes the two-stage PFC and discusses sensitivity analysis, tuning procedures and constraint handling. Section 4 presents the numerical illustrations. Finally the paper concludes in Sect. 5.

2

Problem Statement

Consider an unstable first-order plant given by: Gp (z) =

bp z −1−w 1 + ap z −1

(1)

where ap and bp are the plant parameters, w is the system delay and |ap | ≥ 1 represents the open-loop unstable pole. The system Eq. (1) is subject to input, input rate and output constraints i.e. umin ≤ u(k) ≤ umax

Δumin ≤ Δu(k) ≤ Δumax

ymin ≤ y(k) ≤ ymax

(2)

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where Δ = 1−z −1 is the difference operator. The objective is to design a PFC by first stabilising the prediction dynamics. Furthermore the controller is expected to show some degree of robustness against measurement noise, disturbances and multiplicative uncertainty.

3

Two-Stage Predictive Functional Control

This section proposes a two-stage design approach to controlling the unstable system with PFC. In stage one, the prediction model is stabilised offline through proportional compensation before employing PFC. It should be noted that although open-loop PFC may stabilise unstable systems in an unconstrained environment, pre-stabilisation is necessary for accurate constraint handling. Denote the system model representing (1) as Gm (z), (am = ap and bm = bp if Gm = Gp ): bm z −1 Gm (z) = (3) 1 + am z −1 The dead-time w is excluded from the prediction model and is added separately in the PFC control law. Next we discuss two alternatives to stabilise system Eq. (3).

Fig. 1. Pre-stabilisation with proportional compensation

3.1

Stage-1: Model Pre-stabilisation

The delay-free model (3) can be stabilised with proportional compensation either in the feedback path (Fig. 1(a)) or in the forward path (Fig. 1(b)). The closedloop transfer function for both cases has the form: Tm (z) =

βz −1 ym (z) = v(z) 1 + αz −1

(4)

where β = bm and β = Kbm for compensation in feedback and forward paths respectively and α = am + Kbm . Evidently Tm (z) is stable if 0 ≤ |α| < 1. Moreover, the input uk for feedback path compensation is parameterised as: uk = vk − Kym,k

(5)

PFC for Unstable First Order Systems

15

and for forward path compensation as: uk = K(vk − ym,k )

(6)

The implementation of PFC with Fig. 1(a) for integral systems only was reported verbally in [7]. The current study generalises this concept for unstable dynamics and analyses the potential merits and demerits against the structure of Fig. 1(b). The expectation is to gain useful insights for generalising pre-conditioning with more advanced compensation for more complex plants.

Fig. 2. PPFC structure—PFC on pre-stabilised model with proportional gain in (a) feedback path, and (b) forward path

3.2

Stage-2: PFC Design

The pre-stabilised PFC (PPFC) structure employing a PFC loop on the stabilised model is shown in Fig. 2. In PFC, the output prediction, yp,k is required to follow target first-order dynamics such that: yp,k+i = R − (R − yp,k )ρi

(7)

where R is the steady-state set-point value and ρ is the target closed-loop pole. The PFC control law matches the output prediction yp,k+i and target output R − (R − yp,k )ρi at a single point in future, known as the coincidence point h, while assuming a constant predicted input, i.e. vk = vk+i|k , ∀i > 0. Hence, after recursion on model (4), an i-step ahead model prediction is obtained [1,3]: ym,k+i = (−α)i ym,k + [(−α)i−1 β + (−α)i−2 β + · · · + β]vk

(8)

The prediction Eq. (8), requires correction from bias due to uncertainties with the offset term dk where dk = yp,k − ym,k . Thus PFC is defined from: yp,k+i = ym,k+i + dk = R − (R − yp,k )ρi

(9)

Substituting from (8), the solution to (9), or PPFC law, is given as: vk =

R − (R − yp,k )ρh − (−α)h ym,k − dk h h−j β j=1 (−α)

(10)

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Theorem 1. For a given ρ and h either pre-stabilisation technique results in the same control law provided equal proportional gain is used. Proof. First using Eq. (10) in Eq. (5) with β = bm , gives: ufk back =

R − (R − yp,k )ρh − (−α)h ym,k − dk − Kym,k h h−j b m j=1 (−α)

(11)

Now using Eq. (10) in Eq. (6) with β = Kbm :   R − (R − yp,k )ρh − (−α)h ym,k − dk f orward =K − ym,k = ufk back uk h h−j K j=1 (−α) bm Thus same control law results irrespective of the pre-stabilisation technique.   Remark 1. Theorem 1 shows there is no obvious advantage of either prestabilisation method. Thus for complex systems, pre-conditioning in the feedback path is expected to give same performance as in the forward path. Remark 2. System delays can be easily incorporated into PFC control law [3] by noting that E(yp,k+w ) = yp,k + ym,k − ym,k−w . Therefore Eq. (10) becomes: vk =

R − [R − E(yp,k+w )]ρh − (−α)h ym,k − dk h h−j β j=1 (−α)

(12)

where dk = yp,k − ym,k−w . When w = 0, Eq. (10) and Eq. (12) are no different. 3.3

Sensitivity Analysis

The ability of a feedback loop to reject unwanted perturbations in the form of noise, disturbance and multiplicative uncertainty can be assessed with frequency domain sensitivity analysis [10]. Control law (11) can be re-arranged as: uk = F (z)R − M (z)yp,k − N (z)ym,k

(13)

where F (z), M (z) and N (z) are appropriate polynomials. Note further: {ym,k = Gm (z)uk , (13)} ⇒ D(z)uk = F (z)R − M (z)yp,k

(14)

with D(z) = 1+N (z)Gm (z). Eq. (14) is represented in the block diagram of Fig. 3 where disturbance dy,k and measurement noise nk are also shown; the effective control law is C(z) = M (z)D−1 (z). Consequently, PC (z) = 1 + C(z)Gp (z) = D(z)A(z)+M (z)B(z) is the closed-loop pole polynomial. From Fig. 3, sensitivity of the plant input to noise is found to be: Sun (z) = C(z)[1 + C(z)Gp (z)]−1 = M (z)PC−1 (z)A(z)

(15)

PFC for Unstable First Order Systems

17

whereas sensitivity of the plant output to disturbance is: Syd (z) = [1 + C(z)Gp (z)]−1 = A(z)PC−1 (z)D(z)

(16)

Sensitivity Sδ (z) of the closed-loop pole to multiplicative uncertainty uses: PC (z) = 1 + C(z)[Gp (z) + δGp (z)]   = [1 + C(z)Gp (z)] 1 + δC(z)Gp (z)[1 + C(z)Gp (z)]−1 where δ is possibly a frequency dependent scalar. Thus: Sδ (z) = C(z)Gp (z)[1 + C(z)Gp (z)]−1 = M (z)PC−1 (z)B(z)

(17)

Fig. 3. PPFC block diagram for sensitivity analysis

3.4

Tuning

There are two tuning parameters for a given target pole ρ: the pre-stabilising gain K and the coincidence point h. K determines the position of the pole |α| in z-plane which logically should be restricted between ρ and 1. Therefore K can be tuned within a range where KU and KL are upper and lower limits:     1 + am ρ + am (18) 0, 

T +τ

f (t)f (t) dt > ε1 In .

τ

T will be termed an excitation period of f . d Indeed, in this case dt (ˆ r − r) = −γ(D˙ 1c )2 (ˆ r − r) − ϑr˙ , where ϑr˙ = r(γ ˙ D˙ 1c (r − + 1) is bounded by M1 ε2 . Indeed all its elements are bounded by M1 and recall that |r| ˙ ≤ ε2 . Note that r − D1c is bounded because r and D1c are bounded ˙ (r− as well. Furthermore, as it will be clear soon, ϑr˙ can be replaced by ϑr˙ = r(γV D1c ) + 1) using Eq. (8), where V is the bounded estimate of D˙ 1c . However, the implementation of (5) needs the derivative of D1b and D1c which is not desired. It would require explicit differentiation of measured signals with accompanying noise amplification. Thus, for some positive constant α we adopt the state variable filtering and then design the estimator as follows

D1c )

1 η(t) = z˙1 = −αz1 (t) + (D1b )2 2 1 m(t) = z˙2 = −αz2 (t) + (D1c )2 2 V (t) = z˙3 = −αz3 (t) + D1c

(6) (7) (8)

with initial conditions z1 (0) = z2 (0) = z3 (0) = 0. Now together the above dynamics, the estimator for r is given as   rˆ˙ = −γV η − m + V rˆ . (9) Now we need to know c(t) but we only know D1c (t) and D1b (t). Thus, we must use again adaptive estimation for the centre c(t) of the target. Observe that d ˙  (p1 − c). (Dc )2 = 2(p˙ 1 − c) dt 1

(10)

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39

Assume the estimation of c is denoted as ˆ c, we have 1 d d 2 (D1c )2 − p1  + p˙  c = p˙  c − c) + c˙  (c − p1 ). 1ˆ 1 (ˆ 2 dt dt

(11)

Then the dynamic 1 d  d 2 ˆ c˙ = −γ p˙ 1 (Dc )2 − p1  + p˙  c 1ˆ 2 dt 1 dt

(12)

can estimate the parameter c under some persistent excitation condition on p˙ 1 . Indeed, in this case d 2 (ˆ c − c) = −γp˙ 1  (ˆ c − c) − ϑc˙ , dt

(13)

where ϑc˙ = γ c˙  p˙ 1 (c − p1 ) + c˙ is bounded by M2 ε1 . Indeed all its elements are ˙ ≤ ε1 . Note that c − p1 is bounded because c bounded by M2 and recall that |c| and p1 are within a finite map. Furthermore, as it will be clear soon, ϑc˙ can be replaced by ϑc˙ = γ c˙  V2 (c − p1 ) + c˙ using Eq. (15), where V2 is the estimate of p˙ 1 and it is bounded. However, the implementation of (12) needs the derivative of p1 (t) and D1c (t) which is not desired. Therefore we use the previously defined Eq. (7) for D1c (t) and redefine it as η2 (t) = z˙2 and add the following filter 1 m2 (t) = z˙4 = −αz4 (t) + p1 (t)pT1 (t) 2 V2 (t) = z˙5 = −αz5 (t) + p1 (t)

(14) (15)

with initial conditions z4 (0) = z5 (0) = 0. Now together the above dynamics, the estimator for c is given as   ˆ c˙ = −γV2 η2 − m2 + V2T ˆ c . (16) 3.2

Control Algorithm

This subsection relates to the protocol followed by the USVs for control. Recalling Fig. 3, we will construct the USV control block. Therefore, we want to obtain the desired control input ui (t) using the previously measured and estimated variables. The total velocity of each USV comprises of two sub-tasks: approaching the target and circumnavigating it. Therefore we define the direction of each USV towards the estimated centre of the target as the bearing ψ i (t), ψ i (t) =

ˆ c(t) − pi (t) ˆ c(t) − pi (t) . = c ˆ ˆ c(t) − pi (t) Di (t)

(17)

The first sub-task is related to the bearing ψ i (t) and the second  one is related 0 1 to its perpendicular, Eψ i (t). We define a rotation matrix E = . −1 0

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Then, let us first consider the control law ui where δ is a parameter to be defined. 1 ui = ˆ c˙ + ((Dˆic − rˆ) − rˆ˙ )ψ i + βi Dˆic Eψ i (18) δ The control actuation of a USV is limited, therefore we have to make sure that the implemented control is within the actuation bounds and so we introduce ui = δui

(19)

where δ is the same as before. For a specific ui it is possible to have ui within some specified bounds.

4

Convergence Results

In this section we prove that the estimator and control algorithm proposed in the previous section converge to the desired behaviour. Theorem 1. The initial condition satisfies Dˆic (0) > rˆ(0) > 0. Suppose p˙ 1 (t) ˙ ≤ ε1 , and |r| ˙ ≤ ε2 . Consider the system (1) with the and D˙ 1c (t) are p.e., c c(0) > 0, then control protocol (19), and the initialisation satisfying pi (0) − ˆ there exists K1 , K2 and K3 such that circumnavigation of the moving circle with equally spaced USVs can be achieved asymptotically up to a bounded error, i.e. lim sup ˆ c(t) − c(t) ≤ K1 ε1 ,

(20)

r(t) − r(t)| ≤ K2 ε2 , lim sup |ˆ

(21)

ˆ c (t) − rˆ(t)| ≤ K3 ε2 , lim sup |D i

(22)

t→∞ t→∞ t→∞

lim βi (t) =

t→∞

2π . n

(23)

Proof. The proof is divided into three parts. In the first part, we prove that (20) and (21) hold. In the second part, we prove that the estimated distance Dˆic converges to the estimated radius rˆ, or in other words, that (22) holds. In the last part, we show that the angle between the USVs will converge to the average consensus for n USVs, βi = 2π n , meaning (23) holds. We will assume the implementable controller is given by Ui = δui . 1. Firstly, we prove that (20) and (21) hold. The proof for boundedness of the centre (20), can be found on [12], Proposition 7.1. The proof for boundedness of the radius, however, needs to be derived in this paper. Then, we have that     r˜˙ = rˆ˙ = −γV η − m + V rˆ = −γV η − m + V (˜ r + r)   = −γV 2 r˜ − γV η − m + V r = −γV 2 r˜ + G(t)

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41

  where G(t) = −γV η − m + V r . We know that |G(t)| ≤ k1 2 for some k1 , 2 ≥ 0 because V is bounded and that |η − m + V r| < k2 we can prove that for a Lyapunov function Wr = 12 r˜2 we get ˙ r = r˜r˜˙ = r˜(−γV 2 r˜ + G(t)) = −γV 2 r˜2 + r˜G(t) W ≤ −γV 2 r˜2 + k1 2 r˜ ˙ r ≤ 0 to hold, −γV 2 r˜2 + k1 2 r˜ ≤ 0 must hold. So, we then we get that for W k 1 2 1 2 1 2 ˙ r| is within ± kγV have that when r˜ ≥ γV 2 or r˜ ≤ − kγV 2 , Wr ≤ 0 so that |˜ 2. c ˙ This error r˜ is then proved to converge asymptotically to a ball since D1 is p.e. 2. We prove that all USVs reach the estimate of the boundary of the moving c(t) = limt→∞ Dˆic (t) = rˆ(t), so circles asymptotically, i.e., limt→∞ pi (t) − ˆ (22) holds. Consider the function Wi (t) := Dˆic (t) − rˆ(t) whose time derivative for t ∈ [0, +∞) is given as c˙ − p˙ i ) ˙ c − pi ) (ˆ ˙ i = (ˆ W − rˆ Dˆic =−

1 (ˆ c − pi ) δ((Dˆic − rˆ − rˆ˙ )ψ i + βi Dˆic Eψ i ) − rˆ˙ δ Dˆic

 (ˆ c − pi ) ˆc − rˆ − 1 rˆ˙ ) − (c − pi ) Eψ i δβi Dˆc − rˆ˙ ψ δ( D i i i δ Dˆic Dˆic 1 = − δ(Dˆic − rˆ − rˆ˙ ) − rˆ˙ = −δWi . δ

=−

Hence for t ∈ [0, +∞), we have Dˆic (t) = δWi (0)e−t + rˆ(t) which implies Wi is converging to zero exponentially. 3. Finally, we show that the angle between the USVs will converge to the average consensus for n USVs, βi = 2π n , so (23) holds. Firstly, note that we can write an angle between two vectors βi = ∠(v2 , v1 ) as βi = 2 atan2((v1 × v2 ) · z, v1 v2  + v1 · v2 ) and its derivative as

where z =

v1 ×v2 v1 ×v2  , vˆi

vˆ1 × z vˆ2 × z β˙ i = v˙1 − v˙2 v1  v2  =

v1 vi  , i

= 1, 2.

(24)

(25)

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Then, for v1 = pi − ˆ c and v2 = pi+1 − ˆ c we get vˆ1 × z vˆ2 × z v˙1 − v˙2 β˙i = v1  v2  vˆ1 × z δ((Dˆic − rˆ − rˆ˙ )ψ i + βi Dˆic Eψ i ) = v1  vˆ2 × z ˆ c − rˆ − rˆ˙ )ψ i+1 + βi+1 Dˆc Eψ i+1 ) δ((D − i+1 i+1 v2  = δ(−βi + βi+1 ), i = 1, . . . , n − 1 β˙ n = δ(−βn + β1 ). which can be written in a compact form as β˙ = −δB  β

(26)

where B is the incidence matrix of the directed ring graph from v1 to vn . First, we note that the system (26) is positive (see e.g., [16]), i.e., βi (t) ≥ 0 if βi (0) ≥ 0 for all t ≥ 0 and i ∈. This proves the positions of the USVs are not interchangeable. Second, noticing that B  is the (in-degree) Laplacian of the directed ring graph which is strongly connected, then by Theorem 6 in [17], β converges to consensus 2π n 1. Remark 1. Note how the USV Ai will necessarily maintain its relative position pi throughout the circumnavigation mission. In fact, we can prove that USV Ai is always in position pi . Remark 2. Note that the p.e. condition is assumed for Theorem 1 and not proved. However, in the results section we will verify if the p.e. assumptions are true for our simulations, within the simulation time.

5

Numerical Results

In this section, we present simulations for the protocol designed in Sect. 3. We use the derived method for estimation of the target (9) and (16) and the controlling protocol for the USVs (18). For this section, we discretize the whole algorithm to be able to use it computationally. The first subsection takes into account the persistent excitation condition and the second subsection analyses what happens when this condition is not verified. 5.1

Simulations with p.e. Guarantees

In this subsection, we simulate a moving target with initial position (x(0), y(0)) = (25, 25), radius r(0) = 10 and dynamic according to x(t + 1) = x(t) + α1 (t) + 0.5 y(t + 1) = y(t) + α2 (t) + 0.5 r(t + 1) = r(t) + α3 (t).

(27)

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Fig. 4. Time-lapse of four USVs (blue rectangles) circumnavigating a moving target (red) with representation of their paths (green)

Fig. 5. First and second row: real and estimated target’s centre c : x, y and radius r. Third row: tracking error of USV A1, D1b and angle β1 . Fourth row: control input of USV A1, u1 : x, y

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However, we simulate that the UAV will provide as an initial noisy estimate of (ˆ x(0), yˆ(0)) = (25, 25), radius rˆ(0) = 20. Note that at time t = 0 the radius estimate is double the real radius. Here, αi (t) is a random scalar drawn from the uniform distribution within the interval of [−0.5, 0.5] for i = 1, 2, 3. For this generated target we got the following results. In Fig. 4 the USVs circumnavigate the moving target. In Fig. 5 we have 7 plots. On the first and second row we compare the real and estimated target. Note that the estimate of the centre ˆ c(ˆ x, yˆ) has an estimation error of up to 2 units. Also note that the estimate of the radius rˆ is composed of two instances. In the first, the initial estimate provided by the UAV was very noisy and so we can see the estimate converging rapidly to a more accurate estimation. In the second we can see an estimation error of up to 2 units. On the third row left column, we can see the distance Dib of each target to the boundary of the target - the perfect tracking would result in a distance Dib of 0 for all USVs, for every time step. Here we have an error of up to 0.5 units, except for the very beginning where the error can reach 10 units. This is merely because in the beginning the USVs are far away from the target. On the third row right column, we have the angle between USV A1 and A2, β1 . Having 4 USVs, the perfect tracking would result in 2π/4 = π/2 ≈ 1.57 for all USVs, for every time step. We can see this reference as the red line in the plot so we see that, for USV A1, the error is up to 0.2 radians. Finally, on the fourth row we have the control input of USV A1, both in x and y in blue. Recall Remark. 2 where we stated that, for a practical implementation, there should be a maximum velocity umax . For this case study we defined that umax = 1.5 and we plotted this limit in red. Note how the control input stays within the limit values 1.5 and −1.5. Since we considered as an assumption that p˙ 1 (t) and D˙ 1c (t) are p.e., we will evaluate whether they hold for this simulation example. According to [18], we can adapt Definition 2. to the discrete time case so we obtain the functions fp˙ 1 (t) =

t+m

˙ 1 (k), p˙  1 (k)p

fD˙ c (t) =

t+m

1

k=t

D˙ 1c (k)2 ,

(28)

k=t

which must fulfill ρ2 > fp˙ 1 (t) > ρ1 and ρ4 > fD˙ c (t) > ρ3 for positive ρi . 1 As seen in Fig. 6, these conditions are fulfilled for ρ1 = 1.1026, ρ2 = 6.8371, ρ3 = 0.2443 and ρ4 = 8.8497. Then, for these results in this simulating time span, the p.e. conditions hold. 5.2

Simulations Without p.e. Guarantees

In this subsection, we simulate a static target with position (x(0), y(0)) = (25, 25) and radius r(0) = 10. As in the previous subsection, we simulate that the UAV provides an estimate of (ˆ x(0), yˆ(0)) = (25, 25) and radius rˆ(0) = 20. As seen in Fig. 7, the estimation of the position seems correct but the estimation of the radius seems wrong.

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Fig. 6. First row: fp˙ 1 (t) is bounded by strictly positive bounds. Second row: fD˙ c (t) is 1 bounded by strictly positive bounds.

Fig. 7. Four USVs (blue rectangles) circumnavigating a moving target (red) with representation of their paths (green)

From the first row Fig. 8 we can see how the estimates for the centre c(x, y) are correct for all the simulation time. However, from the second row we can see a steady state error for the estimation of r. Recall that the estimators derived in Sect. 3 for c and r rely on the p.e. conditions for p˙ 1 and D˙ 1c , respectively. Then, it seems that the p.e. condition on D˙ 1c does not hold, and, therefore, the estimation of r does not convergence to the real r. In fact, observing Fig. 9 we can conclude that, for this simulation time, even though the p.e. condition is verified for p˙ 1 , it is not verified for D1c since for some time t the minimum bound is not strictly positive.

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Fig. 8. First and second row: real and estimated target’s centre c : x, y and radius r. Third row: tracking error of USV A1, D1b and angle β1 . Fourth row: control input of USV A1, u1 : x, y

Fig. 9. First row: fp˙ 1 (t) is bounded by strictly positive bounds. Second row: fD˙ c (t) is 1 bounded by a strictly positive bound and zero.

6

Conclusions

We designed an algorithm that guarantees circumnavigation of an irregular shape approximated by a circle up to a bounded error. The algorithm relies on one UAV and a number of USVs according to the size of the target and to the importance

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of monitoring its fronts. Then, the proposed control protocol was proven to converge up to a bounded error assuming p˙ 1 (t) and D˙ 1c (t) to be p.e. As future work, we would like to exploit the circumnavigating USVs as in-site measuring vehicles. In order to do so, we are studying the hypothesis of using USVs capable of measuring the concentration of algal. This applies for the case in which we wish to monitor harmful algal blooms.

References 1. Sivertsen, A., Solbø, S., Storvold, R., Tøllefsen, A., Johansen, K.-S.: Automatic mapping of sea ice using unmanned aircrafts. In: ReCAMP Flagship Workshop, p. 30 (2016) 2. Lucieer, A., Turner, D., King, D.H., Robinson, S.A.: Using an unmanned aerial vehicle (UAV) to capture micro-topography of antarctic moss beds. Int. J. Appl. Earth Obs. Geoinf. 27, 53–62 (2014) 3. Zolich, A., Palma, D., Kansanen, K., Fjørtoft, K., Sousa, J., Johansson, K.H., Jiang, Y., Dong, H., Johansen, T.A.: Survey on communication and networks for autonomous marine systems. J. Intell. Rob. Syst. 95, 789–813 (2018). https://doi. org/10.1007/s10846-018-0833-5 4. Egerstedt, M., Hu, X.: Formation constrained multi-agent control. IEEE Trans. Robot. Auto. 17(6), 947–951 (2001) 5. Dimarogonas, D.V., Johansson, K.H.: On the stability of distance-based formation control. In: 2008 46th IEEE Conference on Decision and Control, pp. 1200–1205. IEEE (2008) 6. Cao, M., Morse, A.S., Yu, C., Anderson, B., Dasgupta, S.: Controlling a triangular formation of mobile autonomous agents. In: 2007 46th IEEE Conference on Decision and Control, pp. 3603–3608. IEEE (2007) 7. Matveev, A.S., Ovchinnikov, K.S.: Distributed communication-free control of multiple robots for circumnavigation of a speedy unpredictably maneuvering target. In: 2018 European Control Conference (ECC), pp. 1797–1802 (2018) 8. Sun, Z.: Cooperative Coordination and Formation Control for Multi-agent Systems. Springer Theses. Springer International Publishing, Cham (2018) 9. Franchi, A., Stegagno, P., Oriolo, G.: Decentralized multi-robot encirclement of a 3D target with guaranteed collision avoidance. Auton. Robots 40, 07 (2015) 10. Ani Hsieh, M.-Y., Kumar, V., Chaimowicz, L.: Decentralized controllers for shape generation with robotic swarms. Departmental Papers (MEAM), vol. 26, Sept. 2008 11. Li, G., St-Onge, D., Pinciroli, C., Gasparri, A., Garone, E., Beltrame, G.: Decentralized progressive shape formation with robot swarms. Auton. Robots 43, 1–17 (2018) 12. Shames, I., Dasgupta, S., Fidan, B., Anderson, B.D.O.: Circumnavigation using distance measurements under slow drift. IEEE Trans. Autom.Control 57(4), 889– 903 (2012) 13. Fonseca, J., Wei, J., Johansson, K.H., Johansen, T.A.: Cooperative decentralized circumnavigation with application to algal bloom tracking. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (2019) 14. Anderson, B.: Exponential stability of linear equations arising in adaptive identification. IEEE Trans. Autom. Control 22(1), 83–88 (1977)

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15. Shimkin, N., Feuer, A.: Persistency of excitation in continuous-time systems. Syst. Control Lett. 9(3), 225–233 (1987) 16. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. In: Series in A Wiley-Interscience Publication, Wiley (2000) 17. Wei, J., Yi, X., Sandberg, H., Johansson, K.H.: Nonlinear consensus protocols with applications to quantized communication and actuation. IEEE Trans. Control Netw. Syst. 6, 598–608 (2018) 18. ˚ Astr¨ om, K.J.: Adaptive Control, pp. 437–450. Springer, Heidelberg (1991). https:// doi.org/10.1007/978-3-662-08546-2 24

Performance Enhancement of a Neato XV-11 Laser Scanner Applied to Mobile Robot Localization: A Stochastic Modeling Approach Jos´e Gon¸calves1(B) , Jo˜ ao Paulo Coelho1 , Manuel Braz-C´esar2 , and Paulo Costa3 1

2

Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal {goncalves,jpcoelho}@ipb.pt Instituto Polit´ecnico de Bragan¸ca & CONSTRUCT R&D Unit, FEUP, Porto, Portugal [email protected] 3 FEUP and INESC-TEC, Porto, Portugal [email protected]

Abstract. Laser scanners are widely used in mobile robotics localization systems but, despite the enormous potential of its use, their high price tag is a major drawback, mainly for hobbyist and educational robotics practitioners that usually have a reduced budget. The Neato XV-11 Laser Scanner is a very low cost alternative, when compared with the current available laser scanners, being this fact the main motivation for its use. The modeling of a hacked Neato XV-11 Laser Scanner allows to provide valuable information that can promote the development of better designs of robot localization systems based on this sensor. This paper presents, as an example, the performance enhancement of a Neato XV11 Laser Scanner applied to mobile robot self-localization, being used as case study the Perfect Match Algorithm applied to the Robot@Factory competition. Keywords: Laser scanner

1

· Modeling · Neato XV-11 · Perfect Match

Introduction

In mobile robotics applications the most common tasks comprise mapping, localization, navigation and obstacle avoidance. In order to perform them efficiently, the robot needs to sense, calculate the distances to the obstacles and to build the map for robot navigation. To achieve that, laser scanners are widely used in mobile robotics localization systems [1,2] but, despite the enormous potential of its use, their high price tag is a c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 49–62, 2021. https://doi.org/10.1007/978-3-030-58653-9_5

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major drawback, mainly for hobbyist and educational robotics practitioners that usually have a reduced budget. The Neato XV-11, shown in Fig. 1, is a robot sold to vacuum domestic rooms [3], that includes a low cost 360◦ laser distance scanner. The laser scanner can be removed from the XV-11, allowing robotics practitioners to use it in their projects, being a very low cost alternative. The XV11 laser can be bought on-line at ebay for less than e150, without controller. The proposed controller increases the price in less than e50, being only necessary an Arduino due and some discrete electronics, while, as an example, the alternative low cost RPLIDAR 360 costs, at this moment, more than e400 [4]. A comparison between three laser rangefinders (URG-04LX, XV-11 laser scanner, Kinect derived) was developed [5], the XV-11 laser demonstrated to be reasonable accurate and precise with the more competitive cost. In [7] Neato XV-11 was used for Simultaneous Localization and Mapping, being modeled and simulated using V-Rep software with satisfactory results.

Fig. 1. Neato XV-11 [4]

The laser scanner hardware approach differs from the previous work [8], because the laser scanner motor is now controlled in closed loop. The presented approach is a much more reliable hardware implementation, when compared with the typical open loop approaches, being the presented model specific for the presented hack. The laser scanner was modeled concerning the parameters noise and error as a distance function. The Knowledge of the referred parameters, presented in [9], allows to provide valuable information that can promote the development of better designs of robot localization systems based on this sensor. This paper presents the performance enhancement of a Neato XV-11

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Laser Scanner applied to mobile robot self-localization, being used as case study the Perfect Match Algorithm applied to the Robot@Factory competition [10]. The Robot@factory competition attempts to recreate a problem similar to the one that an autonomous robot will face during its use in a plant. This scaled plant has a supply warehouse, a final product warehouse and eight processing machines. The competition arena, where the robot has to self-localize and navigate, is shown in Fig. 2.

Fig. 2. Robot@Factory competition arena

This paper is structured as follows. Sect. 2 describes the main features of the Hacked Neato XV-11 Laser Scanner. In Sect. 3 the laser scanner modeling is described. Section 4 presents the Perfect Match Algorithm description. Section 5 presents the enhancement of the Laser Scanner performance applied to the Perfect Match Algorithm. Finally, Sect. 6 presents the conclusions.

2

Hacked Neato XV-11 Laser Scanner Description

As described by Konolige et al. in [5], the Neato XV-11 laser scanner, shown in Fig. 3a, is a low-cost laser scanner equipped with features like eye-safe, fully functional in standard indoor lighting conditions and some outdoor conditions, it is small sized and has a low power consumption. Instead of using time of flight measurement, like the more expensive laser scanners, it uses triangulation to determine the distance to the target, using a fixed-angle laser, a CMOS imager and a DSP for subpixel interpolation [11]. The sensor establishes a serial communication with a 115200 bps baudrate, sending data with a 5 Hz acquisition frequency. Its power consumption without

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motor is relatively low: ∼145 mA @ 3.3 V, which is a very important factor in order to increase the autonomy of a mobile robot with its power based only on the on-board batteries. It provides a 360◦ range of measurements, with an angular resolution of 1◦ , with its range from 0.2 m up to 6 m with an error inferior to 0.03 m. When the laser scanner is removed from the Neato XV-11 robot, its motor has to be controlled by the user, being necessary to be powered with 3.0 V continuous voltage (∼60 mA), in order to produce a turn rate of 240 rpm. Typically it is used a voltage regulator to obtain the 3.0 V. Although this approach is the most popular, it is not the most efficient, because it is an open loop control, being observed oscillations in the motor velocity. An alternative to the referred approach is the use of the turn rate information contained in the data to close the loop [3,5]. In this project the motor was controlled in closed loop. To control and to obtain measurements of the hacked Neato laser scanner, it was used an Arduino Due, which provides the 3.3 V requested by the laser scanner and can establish the needed serial communication. The data packet sent by the sensor is composed by a start header, an index byte, the motor speed (Vn ), the laser measured data and a checksum. A full revolution will yield 90 packets, containing 4 consecutive readings each. The length of a packet is 22 bytes. This amounts to a total of 360 readings (1 per degree) on 1980 bytes. Each packet is organized as follows: < start >< index >< speedL >< speedH > [Data0] [Data1][Data2][Data3] < checksumL >< checksumH > By this way there is available information to close the loop up to 450 hz, having in mind that the Laser spins at 5 Hz frequency. Using the received motor speed data the control loop is closed by calculating the error relative to the speed (V ), needed to maintain the laser frequency up to 5 Hz (5 Hz @ 240 rpm). Posteriorly the error is passed by an integrative like filter, resulting in a PWM control signal, which actuates on a N-Channel Mosfet powering the motor. In Fig. 3b it is shown the control loop diagram.

(a) Hacked Neato XV-11 laser scanner.

(b) Laser scanner motor control.

Fig. 3. Hacked Neato XV-11 laser scanner and its motor control

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Laser Scanner Modeling Experimental Setup

To model the hacked Neato XV-11 Laser Scanner an experimental setup was developed in order to obtain several measurement datasets. The data was obtained with the goal of extracting information about the sensor minimum and maximum ranges and its measurement error. Usually in order to model distance sensors the authors use industrial robot to ensure repeatability in the process of acquiring data, as shown in the example found in [6]. In this particular case the measurement range was too big to develop the refereed tests. In this case the target was always maintained perpendicular to the wall and parallel to the sensor. Initially it was printed an A4 sheet with a target with three areas, the center area white and the two sides areas in black, shown in Fig. 4a, as a way to also assert the color influence in the measure chain. It also can be seen in Fig. 4a the robot prototype. This has a square shape with the laser (hacked from Neato XV-11) centered in the front side. To ensure that the laser angle stayed in the same position, the prototype was placed parallel and against a wall, distanced from the corner 6.2 m. This way the laser remained static during the experiments, while the target moved perpendicular to the wall and parallel to the prototype front. As for the target positioning, it was centered with the laser scanner.

(a) Alignment between robot and target.

(b) Experimental setup layout example.

Fig. 4. Laser Scanner experimental setup

During the experiment it were obtained 44 datasets with different distances from the target to the laser scanner. The measurements were taken from 0.15 to 6 m, with the step sizes listed in Table 1. In each dataset it were obtained 128 samples of the 360◦ scans. An example of a dataset extraction experimental layout is shown in Fig. 4b.

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J. Gon¸calves et al. Table 1. Measure datasets distance and respective step size Distances (m) Step size (m)

3.2

0.15–0.6

0.05

0.6–2

0.1

2–6

0.2

Laser Scanner Model

The datasets obtained via the experimental setup previously described, show the minimum measure range is lower than the described in [5], being of 0.15 m instead of the 0.2 m. Nevertheless, despite the use of closed loop control of the motor to ensure the rotation speed to obtain the frequency required by the laser scanner, measurements above 5 m suffer frequent data loss, which results on inconclusive data to support the model, so it could not be considered in the model development. The maximum range described in [11] is 6 m, despite sometimes data being retrieved, suffer from frequent missed measures, reducing the usable range to 5 m. For each dataset, 128 samples were taken and the mean value and standard deviation of each dataset were calculated in order to obtain the laser scanner model and the noise estimation. It was observed that obstacle color did not influenced the measured distances, this becomes one advantage when compared with laser scanners based on different technologies [12]. In Fig. 5a are shown the mean values of the samples of each dataset relative to the real distance. As it can be seen in Fig. 5b the laser scanner measurements tend to increase the error with the distance, reaching values up to 0.54 m at 5 m measurements to the object. This brings the need for sensor calibration, which will be presented at Sect. 5, in order to retrieve a more accurate measure of the distance to an obstacle.

(a) Distance measured by the laser scanner.

(b) Distance error

Fig. 5. Laser Scanner distance error

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Using a second order polynomial regression, as shown in Fig. 5a represented by a red line, a viable function is obtained to model the sensor measurements (Dl ) relative to the real distance (D). This way the model can be defined by Eq. 1, in order to retrieve the Neato XV-11 laser scanner distances. Dl (D) = a1 ∗ D2 + a2 ∗ D + a3

(1)

with a1 = 0.0323533713 a2 = 0.9420719264 a3 = 0.0177078126 The standard deviation of each dataset, shown in Fig. 6a, has an exponential behavior, increasing with the distance, getting more noticeable after 2.4 m, and reaching values up to 0.072 m at an obstacle distance of 5 m. The standard deviation obtained for each distance is applied to model the sensor noise. In Fig. 6b is shown the noise using a logarithmic scale, being applied an exponential regression to the standard deviation values (σ), obtaining an approach of the measures noise, represented by the Eq. 2.

(a) Standard deviation from each dataset. (b) Measurement noise standard deviation.

Fig. 6. Laser Scanner measurement noise

σ(D) = b1 ∗ eb2 ∗D

(2)

with b1 = 0.0001523985 b2 = 1.3102842636 An example of the laser scanner measure histogram is shown in Fig. 7a. In order to demonstrate that the sensor provides data with a gaussian probability distribution it was used the normal probability plot, which is a graphical technique for assessing whether or not a data set is approximately normally distributed. The data is plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line. Departures from

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this straight line indicate departures from normality [13]. The points in Fig. 7b form a nearly linear pattern, which indicates that the normal distribution is a good model for this data set. The effect of the discretization on the measurement is also observable.

(a) Laser scanner measure probability distribution.

(b) Normal probability plot.

Fig. 7. Laser scanner measurement probability distribution and normal probability plot

4

Perfect Match

The absolute localization estimation uses the information gleaned from the laser range finder. From its data, the robot position and orientation are extracted using the Perfect Match algorithm [14]. This algorithm minimizes the error between the measured and expected distances. Its implementation can be very efficient and its results can be obtained in real time. Map. The Perfect Match algorithm, requires a local map to calculate the expected distance measures. As the map of the Robot@Factory arena is known, it can be created off-line. A matrix where each cell represents the presence or absence of an obstacle (walls, processing machines or other element present in the field) implements the known map, shown in Fig. 8. The chosen grid has a resolution of 1 cm. Measurements Mapping. To compare the expected and the measured distances it is necessary to convert the relative distance measures to absolute coordinates. Assuming s1 ...sn measurements vector, where si is the ith measure from a full laser range finder (Fig. 9) sweep, and αi is the ith measure angle, then the absolute position can be calculated using Eq. 3.         sxi x cos(θr ) sin(θr ) cos(αi ) si (3) = r + syi yr −sin(θr ) cos(θr ) sin(αi )

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Fig. 8. Map example

Error Minimization. As it was already mentioned, the Perfect Match algorithm minimizes the error between the measured and expected distances. To speed up the error calculation, a distance map is precomputed. It is a matrix where each cell holds the distance to the closest obstacle. It can be obtained from the original map matrix by applying a Distance Transform with coefficients given by Eq. 4. This mask performs an approximation to the Euclidean distance. According to Pythagoras’ theorem, the distance between two diagonal points in a grid √ √ should be 1 + 1 = 2 which can be approximated to 1.4142. Further adjusting this value to 1.5 allows to, by multiplying all values by 2, have only integer distances, as can be seen in 4. There is an error of about 6% when applying this simplification. ⎡ ⎤ 323 ⎣2 0 2⎦ (4) 323 An example of a distances map is shown in Fig. 10. With the distance matrix it is very fast to evaluate the error between the measured and expected distances. As the squared error can lead to severe bias from outliers, a modified error equation, as shown in 5, is applied. error = 1 −

c2 c2 + e2

(5)

This error function has the advantage that if a given measurement error is very high, its influence is bounded. Assuming that the estimation of the robot position is done for both x, y and θ, it is necessary to calculate the partial derivative for each of the variables, in

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Fig. 9. Robot in the world and the measurement point relating to the robot

order to minimize the estimated error, as it is suggested by [14]. This algorithm takes into account the partial derivatives for all points. The calculated partial derivatives indicate the position updating direction.

5

Perfect Match Algorithm Performance Enhancement

As seen in Sect. 3, the error for the hacked Neato XV-11 laser scanner increases when the distance to the object increases. So in order to increase the laser measurement accuracy a non linear calibration function was used. For this calibration step, the function parameters are estimated using the data obtained from the experimental setup. This way it was developed a function, which is the inverse of Eq. 1, to convert the sensor measurements to more accurate distances (Dcal ). The laser calibration is shown in Eq. 6. (6) Dcal (Dl ) = c1 ∗ Dl2 + c2 ∗ Dl + c3 with c1 = −0.0242594 c2 = 1.03703951 c3 = −0.0086895 After applying the calibration equation to the laser data the measurement error was reduced, From the one that was shown in Fig. 5b, the error was reduced to a maximum of 0.025 m, as shown in Fig. 11. The robot in the Robot@Factory arena and the distance measures can be seen in Fig. 12. The match between the expected measure and the real location

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Fig. 10. Example of a distances map with values divided by 2

of the obstacle, in the case the walls, is better for nearby objects. Distant ones are less precise mainly due to the increased noise present on those measures. It is intuitive to optimize the localization algorithm to give less importance to the less reliable measures. Having the noise model for the laser scanner measures, allows to modify the error equation, by incorporating the expected noise standard deviation. A weighting factor can be applied to each measure, as shown in Eq. 7. errorCompi =

errori stdev(noise)

(7)

The noise standard deviation can be estimated from the measured distance, applying Eq. 2. The overall effect is to consider less important measures made from greater distance, because they will be less reliable. The position update will be less influenced by those measures and it can be more accurate.

Fig. 11. Distance error after applying the calibration.

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Fig. 12. Robot and distance measures.

Two different cases are presented to illustrate that effect. In the first case the localization algorithm is performed for a few seconds with the robot stopped. In the second case the robot travels, more or less, in a straight line from the bottom to the top of the map. For both cases, the mean squared error between the position of the robot and the estimate obtained by the Perfect Match algorithm is calculated for x, y and the robot orientation. As it can be seen in Table 2 there is a significant improvement on the localization precision for almost all components, the Mean Squared Errors (MSE) for the classic and the new proposed approach are compared, for both described situations. Table 2. Localization precision Approach Classic New Case First First

6

Difference Classic New Difference % Second Second %

x M SE

0.00125 0.000712 −0.430

y M SE

0.00125 0.0012

−0.040

0.0327 0.033

θ M SE

0.12877 0.0679

−0.473

1.232

0.003

0.00233 −0.223 1.1002

0.00917 −0.107

Conclusions

Neato XV-11 is a robot that includes a low cost 360◦ laser scanner, this sensor can be extracted from the robot, allowing robotics practitioners to use it in their

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projects. The Neato XV-11 laser scanner is a very low cost alternative, when compared to the current available laser scanners. This paper presents a study concerning the accuracy, the hacking and modeling of the Neato XV-11 laser scanner. The modeling of a hacked Neato XV11 Laser Scanner allows to provide valuable information that can promote the development of better robot localization systems based on this sensor. It was presented the performance enhancement of a Neato XV-11 Laser Scanner applied to mobile robot self-localization, being used as case study the Perfect Match Algorithm applied to the Robot@Factory competition. Both the sensor calibration and the noise error knowledge contributed to a better performance of the presented localization algorithm. Acknowledgements. This work has been supported by FCT – Funda¸ca ˜o para a Ciˆencia e Tecnologia within the Project Scope: UIDB/05757/2020.

References 1. Arras, K., Tomatis, N.: Improving robustness and precision in mobile robot localization by using laser range finding and monocular vision. In: 1999 Third European Workshop on Advanced Mobile Robots (Eurobot 1999), pp. 177–185. IEEE (1999) 2. Surmann, H., Nuchter, A., Joachim, H.: An autonomous mobile robot with a 3D laser range finder for 3D exploration and digitalization of indoor environments. Robot. Auton. Syst. 45(3), 181–198 (2003) 3. Hacking the Neato XV-11 (2015). https://xv11hacking.wikispaces.com/ 4. Neato Robotics XV-11 Tear-down (2015). https://www.sparkfun.com/news/490 5. Konolige, K., Augenbraun, J., Donaldson, N., Fiebig, C., Shah, P.: A low-cost laser distance sensor. In: 2008 IEEE International Conference on Robotics and Automation, ICRA 2008, pp. 3002–3008, 19–23 May 2008 6. Malheiros, P., Goncalves, J., Costa, P.: Towards a more accurate model for an infrared distance sensor. In: International Symposium on Computational Intelligence for Engineering Systems, ISEP-Porto Portugal, 18–19 November 2009 7. Bajracharya, S.: BreezySLAM: A Simple, Efficient, Cross-Platform Python Package for Simultaneous Localization and Mapping. Washington Lee University (2014) 8. Lima, J., Goncalves, J., Costa, P.: Modeling of a low cost laser scanner sensor. In: CONTROLOX014 Proceedings of the 11th Portuguese Conference on Automatic Control (2014) 9. Campos, D., Santos, J., Goncalves, J., Costa, P.: Modeling and simulation of a hacked Neato XV-11 laser scanner. In: Robot2015 - Second Iberian Robotics Conference (2015) 10. Gon¸calves, J., Lima, J.,Costa, P., Moreira, A.: Manufacturing education and training resorting to a new mobile robot competition. In: 2012 Flexible Automation Intelligent Manufacturing (FAIM 2012) Ferry Cruise Conference HelsinkiStockholm-Helsinki, June 2012. https://doi.org/10.1109/ICNN.1993.298623 11. Shah, P., Konolige, K., Augenbraun, J., Donaldson, N., Fiebig, C., Liu, Y., Khan, H., Pinzarrone, J., Salinas, L., Tang, H., Taylor, R.: Distance sensor system and method, US 2010/0030380 A1 (2010) 12. Lima, J., Goncalves, J., Costa, P., Moreira, A.: Modeling and simulation of a laser scanner: an industrial application case study. In: FAIM (2013)

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13. Chambers, J., Cleveland, W., Kleiner, B., Tukey, P.: Graphical Methods for Data Analysis. Bellmont, Wadsworth (1983) 14. Lauer, M., Lange, S., Riedmiller, M.: Calculating the perfect match: an efficient and accurate approach for robot self-localization. In: Robocup 2005: Robot Soccer World Cup, vol. 4020, pp. 142–153 (2006)

Prototyping and Control of a Conveyor Belt: An Educational Experiment in Mechatronics Jos´e Gon¸calves1(B) , Jo˜ ao Ribeiro2 , and Paulo Costa3 1

Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal [email protected] 2 Instituto Polit´ecnico de Bragan¸ca and CIMO, Bragan¸ca, Portugal [email protected] 3 FEUP and INESC-TEC, Porto, Portugal [email protected] Abstract. In this paper it is presented an educational experiment, that consists of a mechatronic system applied to demonstrate concepts such as prototyping and control. The described mechatronic system is based on a conveyor belt, that was integrated with a manipulator, being physical devices commonly used in the industry. The conveyor Belt was prototyped from scratch, using 3d print technology. Its movement is based on the closed loop control of a DC Motor, based on a PID. The Conveyor Belt was integrated with a Braccio Manipulator from Arduino, using the ZMQ communication library, which is a high-performance asynchronous messaging library. Keywords: Control education

1

· Conveyor Belt · Manipulator · PID

Introduction

Mechatronics is a multidisciplinary subject that deals with many concepts such as Mechanics, Electronics, Control and Computer Science [1]. It is very important to have experimental kits in the classroom, both for training as well as for demonstration. In this paper it is described an educational experiment that consists of a mechatronic system applied to demonstrate concepts such as prototyping and to enhance the study of control methods. The described mechatronic system is based on the control of a Conveyor Belt, that was integrated with a manipulator, as shown in Fig. 1, being physical devices commonly used in the industry. The manipulator was applied to remove parts from the Conveyor Belt, performing different operations, depending on the part characteristics. For the Communication between the Conveyor and the Manipulator it was used ZeroMQ (also spelled ØMQ, 0MQ or ZMQ) which is a high-performance asynchronous messaging library zmq, aimed at use in distributed or concurrent applications [2]. The c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 63–71, 2021. https://doi.org/10.1007/978-3-030-58653-9_6

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Fig. 1. Conveyor Belt and the manipulator

Conveyor Belt is equipped with a distance sensor that identifies the presence and dimensions of a part, and using bilateral communication, requests a manipulator to remove a part from the Conveyor, receiving an acknowledge when the part was removed. The paper is structured as follows, initially the conveyor prototyping is described, then its control is described and finally some conclusions and future work are presented.

2

Conveyor Belt Prototyping

The Conveyor Belt Components can be observed in Fig. 2, it use an Arduino Board connected to a Laptop, being its Mechanics, sensors and actuators described in the next subsections. 2.1

Mechanics

The conveyor Belt mechanics was developed using 3d printing technology [9]. This technology was chosen because it allows mechanical rapid prototyping at a low cost. Some examples of the models a shown in the next Figures, such as the conveyor structure (Fig. 3), the part that allows to adjust the conveyor tension (Fig. 4a), a gear (Fig. 4b) and the some conveyor moving parts (Fig. 5).

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Fig. 2. Conveyor Belt components

Fig. 3. Conveyor Belt structure

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a) Conveyor Belt tension adjust part

b) Conveyor Belt Gear

Fig. 4. Conveyor Belt parts

Fig. 5. Conveyor Belt moving parts

2.2

Sensor

The chosen sensor to detect the parts is the V L53L0, shown in Fig. 6, which uses ST’s FlightSense technology to precisely measure how long it takes for emitted pulses of infrared laser light to reach the nearest object and be reflected back to a detector, so it can be considered a tiny, self-contained lidar system. This time-of-flight (TOF) measurement enables it to accurately determine the absolute distance to a target without the object’s reflectance greatly influence the measurement. A median filter was applied in order to remove outliers, increasing, significantly, the robustness of the obtained measurements.

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Fig. 6. Time-of-flight sensor

2.3

Actuator

The applied DC motors have built in encoders which are very important in mechatronics in order to obtain the closed velocity control and also to estimate the calculation of the traveled distance by the part in the conveyor, the applied drive is the DFRobot L293 Arduino Shield.

3 3.1

Mechatronic System Control Conveyor Belt Control

The proposed laboratory control experiment has as goal the study, by the students, of PID controllers design [4–6]. Students applied tuning control loops based on the Internal Model Control (IMC) tuning method [7]. This method has many advantages, such as its robustness and the ability for the user to specify the closed-loop time constant, being its major drawback the fact that when the process has a very long time constant, the error will be integrated for a very long time. The IMC tuning method was developed for use on self-regulating processes. A self-regulating process stabilizes at some point of equilibrium, which depends on the process design and the controller output. If the controller output is set to a different value, the process will respond and stabilize at a new point of equilibrium, as shown in Fig. 7. The first step that students must do is to apply a step change in the controller output (CO), as shown in the example of Fig. 7. The observed process output provides the parameters that allow an approximate estimate of the process as a first order system with time delay, as shown in Eq. 1, where C(s) is the process output and R(s) is the process input [3].

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Fig. 7. Self-regulating process step response [8]

Fig. 8. Conveyor Belt open loop step response

K C(s) = e−td s R(s) τs + 1

(1)

where: td is the dead time, τ is the time constant and K is the process gain, given by the total change in PV (Process Variable) [in %] divided by change in CO [in %]. From the DC motor step response (Fig. 8), changing the controller output

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from 2 to 10 Volt, the process gain and time constant are estimated, being the dead time considered null. The process parameters are shown in Table 1. Table 1. Process parameters. Parameters Value K

5.5

τ

0.172

td

0

Fig. 9. Conveyor Belt closed loop control

Then students calculate the PID controller, based on the IMC tuning method, where the derivative time is null, the integral time is given by τ and the controller gain is given by Eq. 2, where τcl parameter allows to choose the desired closedloop response time. Students applied τcl = τ , in order to the closed-loop system dynamics remains similar as its open loop response. τ (2) Kc = K(τcl + td ) In Fig. 9 the closed loop response can be observed, it is noticeable that even in closed loop the fact that the system friction is not constant, it causes some disturbances in steady state.

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Conveyor Control Integrated with the Manipulator

Initially the conveyor is stopped until it is initiated by an operator, then the conveyor speed becomes constant being controlled in closed loop. When the distance sensor detects a part the conveyor reduces its speed and places the part in a position that can reached by the manipulator. The part placement is based on the concept of odometry, enhancing the fact that the use of incremental encoders in mechatronics is important, not only to apply closed loop velocity control, but also for position control and estimate. When the part is placed in the correct position and stopped, the Conveyor communicates with the application that controls the manipulator, notifying that a part has to be removed from the conveyor and if that part is tall or short. The manipulator performs different operations, depending on the part characteristics. When the manipulator removes the part the conveyor notifies the conveyor that it is safe to initiates the movement, and the conveyor starts its movement again. This behavior can be observed in Fig. 10 and a demonstrative video can be seen in [10].

Fig. 10. Conveyor behavior when integrated with the manipulator

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Conclusions and Future Work

In this paper presented an educational experiment, that consists of a mechatronic system applied to demonstrate concepts such as prototyping and control. The experimental setup prototype mechanics was 3d printed, this technology was chosen because it allows mechanical rapid prototyping at a low cost. The proposed laboratory control experiment had as goal the study, by the students, of PID controllers design. Students applied, as an example, tuning control loops based on the Internal Model Control (IMC) tuning method. During the experiment it was also enhanced the importance of communications, in order to integrate different equipments that must work cooperatively and also the importance of encoders in mechatronics application that not only allow the closed loop velocity control but also the position control and estimate. As future work the authors intend to apply different control methods to the described mechatronic system. Acknowledgements. The authors are grateful to the Foundation for Science and Technology (FCT, Portugal) for financial support by national funds FCT/MCTES to CIMO (UIDB/00690/2020).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Bishop, R.: The Mechatronics Handbook. CRC Press, New York (2002) ZeroMQ: An open-source universal messaging library (2020). https://zeromq.org/ Ramsey, A.: Dynamics. Cambridge Library Collection - Mathematics (2009) ˚ Astr¨ om, K.J., H¨ agglund, T.: Advanced PID Control, ISA - The Instrumentation, Systems, and Automation Society, Research Triangle Park, NC 27709 (2006) Vilanova, R., Visioli, A.: PID Control in the Third Millennium: Lessons Learned and New Approaches, Advances in Industrial Control book series (2012) Oliveira, P.M.: Modern heuristics review for PID control optimization: a teaching experiment. In: IEEE International Conference on Control and Automation (2005) Rivera, D., Morari, M., Skogestad, S.: Internal model control: PID controller design. Ind. Eng. Chem. Process Des. Dev. 24, 252–265 (1986) Dataforth Corporation: Tuning control loops with the IMC tuning method, Application Note AN124 (2016). http://www.dataforth.com Yeole, S., Madhav, C.V., Sri, R., Kesav, N.H.: Importance and utilization of 3D printing in various applications. Int. J. Mod. Eng. Res. (2016) Author: Conveyor Belt control (2020). https://www.youtube.com/watch? v=qDtJAQsNQEs

Welding Process Automation of Aluminum Alloys for the Transport Industry: An Industrial Robotics Approach João Ribeiro1(B) , José Gonçalves2 , and Nuno Mineiro3 1 CIMO and Instituto Politécnico de Bragança, Campus de Santa Apolónia,

5300-253 Bragança, Portugal [email protected] 2 Instituto Politécnico de Bragança and CeDRI, Bragança, Portugal [email protected] 3 Roboplan, R. Da Paz 292, 3800-587 Cacia, Portugal [email protected]

Abstract. The materials used in the transport industry have been changing in the last decades. The traditional and heavy steel have been switching by the light alloys like aluminum alloys. However, despite their advantages as low density and high corrosion resistance, the manufacturing process, especially fusion welding, is very demanding and challenging. In the transport industry, most of the hyperstatic components made in aluminum alloys are welded manually with the associate financial costs as well as the lack of quality and repeatability. For these reasons, it is urgent to develop new methodologies to automate this process. The present work intends to show a scientific method to automate the welding process of hyperstatic frames, very common in bicycles, made in aluminum alloy. This methodology involves two steps, the first one in which is performed numerical simulations to determine the optimal welding parameters to minimize the distortion and residual stresses. The second step is experimental one, and it is created an automated welding cell with a robot to weld the frames. It has been proved that it is possible to obtain welding aluminum frames with acceptable quality in agreement with the ASME IX standard. Keywords: Automation · Robotic · Welding · Numerical simulation · Optimization · Aluminum alloys

1 Introduction The transport industry is one of the most critical sectors in the global economy, and any change in this sector has an impact on the costs of virtually all other economic areas. Due to the ever-higher energy prices and progressively more restrictive environmental requirements, there was a need, on the part of this industry, to adapt to a new reality. Thus, the different sectors of the transport industry have been adapted to the new requirements, © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 72–81, 2021. https://doi.org/10.1007/978-3-030-58653-9_7

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at different levels, ranging from the development of more efficient propulsion systems, commonly known as engines [1, 2] to the selection of lighter materials [3, 4]. The use of aluminum alloys in the transport industry has been growing, particularly in the light and heavy automotive industry, both in terms of structure [5] and sheet metal components [6]. According to the European Aluminum Association, the replacement of steel by aluminum alloys in vehicles allowed a 30% saving in energy consumption and a significant reduction in CO2 emissions [7], this material also has the advantage of being fully recycled. Due to their low density and resistance to corrosion, these materials are also widely used in non-motorized vehicles, namely in bicycles [8] and whose use has grown significantly [9]. Despite all the listed advantages of aluminum alloys, there are some disadvantages compared to iron-carbonic alloys, namely the higher price, more demanding manufacturing processes in terms of quality control [10], more difficult welding junction processes [11, 12] and a lower fatigue life [13]. In the case of the transport industry, these limitations are more significant when welding structural elements such as chassis or bicycle frames. These elements correspond to hyperstatic structures with high rigidity that, associated with the residual stresses generated during the welding process, may cause cracking [14] or distortion of the structure [5, 14] and, in the case of no cracking or distortions, the level of residual stresses is so high that it will significantly decrease the fatigue life of these structures [15, 16]. Specific characteristics intrinsic to aluminum introduce difficulties for welding. The most important of these corresponds to the formation of a surface oxide layer, resulting from the reaction of aluminum with atmospheric oxygen. This film protects the metal against corrosion, but it also has a considerably higher melting point than the original Al element. Consequently, successful welding depends, in part, on the technique applied for the disintegration of this oxide layer, which will remain stable even after the melting of the aluminum [17]. In the case of MIG welding, the flow of the shielding gas over the consumable electrode removes the oxide from the melting bath. However, there are other essential limitations in the welding process for aluminum alloys used in the manufacture of hyperstatic structures, the most relevant is the susceptibility to cracking in welding [17], the distortions and thermally induced stresses and post-weld heat treatment. Despite the difficulties mentioned above, it is possible to weld hyperstatic structures in aluminum alloys with conventional techniques (MIG or TIG), as long as chosen the appropriate welding parameters, the correct joint preparation, the fixation of the elements to be welded with the degree of adjusted stiffness and a welding sequence that limits distortion and residual stresses [18, 19]. However, it is only possible to accommodate all these parameters during the welding process; so, it is necessary to adjust and balance some of these elements. In contrast, welding is being carried out and, as such, requires that these processes be manual and depend on the qualification and sensitivity of the welder. Therefore, it is challenging to automate this process; accordingly, there is a need for an in-depth study and development for the automation of welding systems for hyperstatics structures made of aluminum alloys. The proposed approach to overcome the limitations of automated or robotic welding to weld hyperstatic aluminum alloy structures will be carried out in two distinct phases: numerical simulation using finite element programs dedicated to welding and

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experimental tests with a welding robot based on an optimization method, Taguchi, to determine the optimal welding parameters for the type of welding under analysis.

2 Numerical Simulation In the last decades, numerical simulation methods have been progressively replacing the performance of experimental tests in several manufacturing processes [20–23], which have higher costs and the time to prepare and perform the tests. Much higher than the computation time required to simulate the behavior of processed materials. In the case of welds, simulations must foresee distortions [23], residual stresses [24], and, in some cases, changes in the microstructure of the weld bead [25]. In these simulations, it is necessary to define the geometry of the elements to be connected, the welded materials, the welding parameters (current, voltage, travel speed, contact-to-work-distance), and the rigidity degree of the element’s fixation. In this work was implemented a preliminary numerical simulation to minimize the residual stresses in welding of 6082-T6 aluminum alloy. To reach this goal, the Taguchi optimization method was used, determining which combination of parameters minimizes the residual stresses in the 6082-T6 aluminum alloy. The welding parameters selected were the welding current, the torch angle, and the welding travel speed. For each parameter, three levels were used, combined in an L27 orthogonal array. The 27 combinations established by the orthogonal array were simulated in SymufactWelding® software. After all, combinations were simulated, carried out the data treatment, reaching the combination of levels for each parameter, which minimizes the level of residual stresses, as well as the most influential parameter in its appearing. In the work presented in this section, it was analyzed the influence of the essential welding parameters on the residual stress amplitude in the welding corner joint. For this, it was developed a Taguchi orthogonal array of tests, L27, where it is possible to combine different levels for the defined parameters. Table 1 shows the levels used for each parameter. Table 1. Welding parameters and levels. Number

Parameters

Units

Levels 1

2

3

1

Welding current

A

163

181

202

2

Welding travel speed

mm/s

10

13

16

3

Torch angle

º

30º

45º

60º

Based on the combination of parameters and levels defined by L27 array, were implemented 27 simulations with the SymufactWelding® software, which uses the numeric model of double ellipsoid for distribution of energy transferred by the heat source to the

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part. The previously experimental specimen (Fig. 1) calibrated the dual ellipsoid model and transversal section cord, which, in this case, was used as a robot. This measurement was performed with metallographic samples of the welding cord. After all simulations completed, the values of the maximum principal stress (MaxPS) were withdrawn at various points along the x-axis, Fig. 2. In each simulation are determined the maximum values of residual stresses. Analyzing the obtained results was established by the combination that gives the less amount of residual stresses. So, for MaxPS combination which minimizes the residual stress is the welding current of 202 A, welding speed of 10 mm/s and a torch angle with 30º, on the contrary combination that maximize residual stress is the welding current of 163 A, welding speed of 16 mm/s and a torch angle with 60º. This data will be used as preliminary parameters in the experimental procedures.

Fig. 1. Experimental specimen of a corner joint.

Fig. 2. Line of reading of the MPS computed.

3 Experimental Procedures Experimental methods are usually performed in controlled laboratory settings [26], although they can also be implemented in the field [27]. In the work developed within the

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scope of this project, experimental tests were carried out in a laboratory with a controlled environment, which consisted of a robot with a welding apparatus and a fixation system for the parts (gig or gabarit). In addition to this base structure, it was used macrography equipment to analyze the weld beads [28] and apparatus for dimensional control to measure distortions [29]. During this study, it was measured the residual stresses using the incremental hole method [30] or the contour method [31], and the internal defects were controlled with ultrasound [32]. Some mechanical properties of the welding were also determined [33], namely, the yield strength, the tensile strength, and the elongation. A significant part of the quality control elements of the welds is based on the ASME IX standard [34]. The experimental tests allowed us to determine the optimal welding parameters for obtaining good quality welds, in this sense and to reduce the number of tests, it was decided to use an optimization method widely used in quality control in the industry, called the Taguchi method [35, 36]. This method allows the definition of experimental test plans, where various combinations of welding parameters are indicated. Between these combinations, it is possible to obtain the optimum value for a given control characteristic. The advantages of this method have popularized it in the definition of experimental test plans in various manufacturing processes [37, 38], namely, in welding [39, 40]. Before implementing the experimental tests, several Taguchi orthogonal arrays were created with different combinations of control factors and levels. For this purpose, eight factors were chosen (current intensity, travel speed, electrode wire speed, type of transfer, welding sequence, thickness of base material, base material and addition material) and the levels varied between 2 and 4 and were chosen, in a first phase, according to the database obtained in the numerical simulations. The experimental tests focused, primarily, on the welding of tubular hyperstatic triangular structures, very common in bicycle frames. Figure 3 shows the type of hyperstatic triangular structure used in this project.

Fig. 3. Welded tubular hyperstatic triangular structure.

The selected aluminum alloy was the 6061-T6 alloy, which is the most common in the manufacture of bicycles. To carry out the experimental tests it was necessary to design and manufacture a system (gig or gabarit) that would guarantee, with high

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accuracy and repeatability, the positioning and fixation of the tubes to be welded. The tubes were always placed in the same position on the gig that is attached to a positioner. Several programs were carried out, according to the previously defined test arrays. After the first set of tests, it was found that some levels of parameters used resulted in a very high level of defects and that they were observed visually. In other cases, it was proved, in macrographic exams, that there were defects of incomplete penetration and undercut. This first evaluation made it possible to select new levels of parameters. The experimental tests were repeated, iteratively, to determine the optimal combination, in Fig. 4, it is possible to observe the assembly used in one of the welding tests performed. During the period of the experimental trials, more than two hundred welding tests were carried out (tubular triangles). As a result of these tests, some frames showed visible defects, that invalidated them immediately. In contrast, others were subjected to a set of destructive and non-destructive tests, defined by the standards mentioned above.

Fig. 4. The set-up used in the experimental tests of welding.

Non-destructive tests were carried out with ultrasound technology. The ultrasound device uses the “phase array” method, this new method allows a more global and accurate assessment in the detection of internal defects. Thus, a high number of welded joints was analyzed with this technology, and it determined that some of them contained pores inside and incomplete fusion, the frames that provided these defects were also rejected and evaluated, in greater detail, with macrographic techniques. Some of the pictures that did not present internal defects were also subjected to macrographic analysis to assess the geometry of the cord. The destructive tests were performed to determine some mechanical properties of the welded joints and to assess the geometry of the bead or the occurrence of internal defects. This classification also includes the technique used to measure residual stresses, with the incremental hole method. For the mechanical characterization, the dimensions and geometry of the welded specimens follow the ISO 4136. The tensile strength of the joint was determined, as well as the location of the fracture. The test pieces were welded with the welding parameters of the frames that did not present unacceptable defects. The macrographic tests were carried out on welded joints of frames with defects detected in the ultrasound exams. The macrographic tests allowed to evaluate the geometry of joints, as well as, to observe and measure the heat-affected zone. It is possible,

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also, to observe the most important detects like porosity, incomplete fusion and penetration, solidification structure, segregations, and, in some cases, it was possible to observe cracks after cooling. Figure 5 shows a macrographic image of one of the analysed welded joints. Another factor that was analyzed to assess the quality of welded joints was residual stress. For the measurement of residual stresses, a classic method was used, which is widely used in industrial and academic environments, called the incremental holedrilling. For this method is used a special rosette of electric strain gauges in the center of which a small hole (diameter 0.6–1.8 mm) is made up to 1 mm deep, with an increment of 0.1 mm. The incremental hole-drilling method allows measuring with high resolution and accuracy the residual stresses installed in the welded joint. The measurements of residual stresses happened in welded joints without defects detected by the previously described techniques, and it found that the value of residual stresses lies between 30% to 60% of the yield stress of the base material.

Fig. 5. Macrography of one of the joints analyzed in this work.

In addition to the evaluation of welded joints mechanical characteristics, the bead geometry and defects, and the level of residual stresses, it was also measured the distortions that occurred in the frame after welding and its removal from the gig. The distortion measurement was performed using a C-Track stereoscopic vision device, which uses the Metrolog Software, using the HandyProbe to capture control points. The statistical treatment was carried out using the technique of analysis of variance (ANOVA), in this report will be presented some results related to control factors more relevant to obtain quality welding. After several welding tests and analysis of results, it was found that the most critical welding parameters were the addition material (A), the current intensity (B), the thickness of the base material (C), the travel speed (D), and the speed of the electrode wire (E). For most welding parameters, three levels were considered, except for the addition material and, as mentioned above, only the two most suitable materials for welding 6061-T6 alloy. Considering the referred parameters and levels, the most appropriate Taguchi array is L18. After conducting the 18 tests (welding 18 frames), was carried out the quality control described above. Subsequently evaluating the results, was used the ANOVA technique to determine which parameters were most influential for each of the control factors

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evaluated. This scientific article will present the analysis of variance for the most relevant control factor, which is distortion. Table 2 shows the ANOVA analysis for maximum distortion. Observing the Table 2 is possible to claim that the travel speed is the most influential welding parameter (29.46%) for the frame distortion follow by the addition material (20.44%). The least influential parameter is the current intensity with a contribution of 2.22%. Table 2. ANOVA analysis for maximum distortion. Source DF SQ

MS

F-Value Contribution (%)

A

1

33.135 33.135 7.89

B

2

1.804 0.43

2.22

C

2

27.110 13.555 3.23

16.72

D

2

47.764 23.882 5.69

29.46

3.607

20.44

E

2

16.92

8.460 2.01

10.44

Error

8

33.595

4.199

20.72

Total

17 162.131

4 Conclusions The presented work has implemented a methodology applied to automate with a robotic welding cell to weld hyperstatic bicycle frame of aluminum alloy. This methodology involved two phases, where the first step involved numerical simulations to minimize the distortion and residual stresses. Using this data as preliminary parameters were implemented in the experimental tests. For the numerical simulations was used the SymufactWelding® software to simulate the welding process in which the parameters were chosen after the implementation of the Taguchi method. It was simulated 27 combinations of welding parameters and levels. In the experimental procedures, a robotic welding cell was assembled to test different welding parameters to obtain a hyperstatic frame in aluminum alloy. The first values of welding parameters used in the experimental tests were based on the numerical simulations. The experimental procedure also used the optimization method of Taguchi, and the ASME IX standard indicated the quality parameters. So, were implemented destructive and non-destructive tests to qualify the quality of the welding frames. It was possible to obtain welding aluminum frames with acceptable quality in agreement with the ASME IX standard. Acknowledgements. The authors are grateful to the Foundation for Science and Technology (FCT, Portugal) for financial support by national funds FCT/MCTES to CIMO (UIDB/00690/2020).

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References 1. Demirdöven, N., Deutch, J.: Hybrid cars now, fuel cell cars later. Science 305(5686), 974–976 (2004) 2. Williamson, S., Emadi, A., Rajashekara, K.: Comprehensive efficiency modeling of electric traction motor drives for hybrid electric vehicle propulsion applications. IEEE Trans. Vehicular Technol. 56(4), 1561–1572 (2007) 3. Ghassemieh, E.: New Trends and Developments in Automotive Industry. InTech, Rijeka (2011) 4. Chawla, N., Chawla, K.: Metal-matrix composites in ground transportation. J. Minerals, Metals Mater. Soc. 58(11), 67–70 (2006) 5. Carle, D., Blount, G.: The suitability of aluminium as an alternative material for car bodies. Mater. Des. 20, 267–272 (1999) 6. Banhart, J.: Aluminium foams for lighter vehicles. Int. J. Vehicle Des. 37(2–3), 114–125 (2005) 7. Aluminium in transport. European Aluminium Association Report (2014) 8. Lin, C., Huang, S., Liu, C.: Structural analysis and optimization of bicycle frame designs. Adv. Mech. Eng. 9(12), 1–10 (2017) 9. IBF. http://www.ibike.org/library/statistics-data.htm. Accessed 27 Jan 2020 10. Toros, S., Ozturk, F., Kacar, I.: Review of warm forming of aluminum–magnesium alloys. J. Mater. Process. Technol. 207(1–3), 1–12 (2008) 11. Lakshminarayanan, A.K., Balasubramanian, V., Elangovan, K.: Effect of welding processes on tensile properties of AA6061 aluminium alloy joints. Int. J. Adv. Manuf. Technol. 40, 286–296 (2007). https://doi.org/10.1007/s00170-007-1325-0 12. Malarvizhi, S., Balasubramania, V.: Effect of welding processes on AA2219 aluminium alloy joint properties. Trans. Nonferrous Metals Soc. China 21(5), 962–973 (2011) 13. Avalle, M., Belingardi, G., Cavatorta, M., Doglione, R.: Casting defects and fatigue strength of a die cast aluminium alloy: a comparison between standard specimens and production components. Int. J. Fatigue 24(1), 1–9 (2002) 14. Barnes, T., Pashby, I.: Joining techniques for aluminium spaceframes used in automobiles Part I - solid and liquid phase welding. J. Mater. Process. Technol. 99, 62–71 (2000) 15. Maddox, S.: Review of fatigue assessment procedures for welded aluminium structures. Int. J. Fatigue 25(12), 1359–1378 (2003) 16. James, F., et al.: Residual stresses and fatigue performance. Eng. Failure Anal. 14(2), 384–395 (2007) 17. Gourd, L.: Fundamentals of Welding - Principles of Welding Technology, 3rd edn. Butterworth-Heinemann, Paris (1995) 18. Teng, T., Fung, C., Chang, P., Yang, W.: Analysis of residual stresses and distortions in T-joint fillet welds. Int. J. Pressure Vessels Piping 78(8), 523–538 (2001) 19. Teng, T., Chang, P., Tseng, W.: Effect of welding sequences on residual stresses. Comput. Struct. 81(5), 273–286 (2003) 20. Umbrello, D.: Finite element simulation of conventional and high-speed machining of Ti6Al4V alloy. J. Mater. Process. Technol. 196(1–3), 79–87 (2008) 21. Lewis, R., Ravindran, K.: Finite element simulation of metal casting. Int. J. Numerical Methods Eng. 47(1–3), 29–59 (2000) 22. Guo, L., Yang, H., Zhan, M.: Research on plastic deformation behaviour in cold ring rolling by FEM numerical simulation. Modell. Simulat. Mater. Sci. Eng. 3(7), 1029–1046 (2005) 23. Deng, D., Murakawa, H., Liang, W.: Numerical simulation of welding distortion in large structures. Comput. Methods Appl. Mech. Eng. 196(45–48), 4613–4627 (2007)

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24. Zeng, Z., Wang, L., Du, P., Li, X.: Determination of welding stress and distortion in discontinuous welding by means of numerical simulation and comparison with experimental measurements. Computat. Mater. Sci. 49, 535–543 (2010) 25. Toyoda, M., Mochizuki, M.: Control of mechanical properties in structural steel welds by numerical simulation of coupling among temperature, microstructure, and macro-mechanics. Sci. Technol. Adv. Mater. 5(1–2), 255–266 (2004) 26. Janosch, J.: International institute of welding work on residual stress and its application to industry. Int. J. Pressure Vessels Piping 85(3), 183–190 (2008) 27. Assunção, E., Quintino, L., Miranda, P.: Comparative study of laser welding in tailor blanks for the automotive industry. Int. J. Adv. Manuf. Technol. 49(1–4), 123–131 (2010) 28. Peel, M., Steuwer, A., Preuss, M., Withers, P.: Microstructure, mechanical properties and residual stresses as a function of welding speed in aluminium AA5083 friction stir welds. Acta Mater. 51(16), 4791–4801 (2003) 29. Sattari-Far, I., Javadi, Y.: Influence of welding sequence on welding distortions in pipes. Int. J. Pressure Vessels Piping 85(4), 265–274 (2008) 30. Ribeiro, J., Monteiro, J., Lopes, H., Vaz, M.: Moiré interferometry assessement of residual stress variation in depth on a shot peened surface. Strain 47(S1), e542–e555 (2011) 31. Richter-Trummer, V., Moreira, P., Ribeiro, J., Castro, P.: Moiré Interferometry The contour method for residual stress determination applied to an AA6082-T6 friction stir butt weld. Mater. Sci. Forum 681, 177–181 (2011) 32. Arone, M., Cerniglia, D., Nigrelli, V.: Defect characterization in Al welded joints by noncontact Lamb wave technique. J. Mater. Process. Technol. 176(1–3), 95–101 (2006) 33. Kumar, A., Sundarrajan, S.: Optimization of pulsed TIG welding process parameters on mechanical properties of AA 5456 Aluminum alloy weldments. Mater. Des. 30(4), 1288–1297 (2009) 34. Part QW Welding - Welding General Requirements. ASME IX (2013) 35. Ross, P.: Taguchi Techniques for Quality Engineering, 2nd edn. McGraw-Hill, New York (1988) 36. Wu, Y., Wu, A.: Taguchi Methods for Robust Design. American Society of Mechanical Engineers, New York (2000) 37. Ribeiro, J., César, M.B., Lopes, H.: Optimization of machining parameters to improve the surface quality. Procedia Struct. Integrity 5, 355–362 (2017) 38. Syrcos, G.: Die casting process optimization using Taguchi methods. J. Mater. Process. Technol. 135(2), 68–74 (2003) 39. Sapakal, S., Telsang, M.: Parametric optimization of MIG welding using taguchi design method. Int. J. Adv. Eng. Res. Stud. 1(4), 28–30 (2012) 40. Pal, S., Malviya, S., Pal, S., Samantaray, A.: Optimization of quality characteristics parameters in a pulsed metal inert gas welding process using grey-based Taguchi method. Int. J. Adv. Manuf. Technol. 44(11–12), 1250–1260 (2009)

Sub-Riemannian Geodesics on Nested Principal Bundles Mauricio Godoy Molina1 and Irina Markina2(B) 1

Departamento de Matem´ atica y Estad´ıstica, Universidad de La Frontera, Temuco, Chile [email protected] 2 Department of Mathematics, University of Bergen, Bergen, Norway [email protected]

Abstract. We study the interplay between geodesics on two nonholonomic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.

Keywords: Sub-Riemannian geometry Submersions · Hamiltonian formalism

1

· Principal bundles ·

Introduction

Our paper is related to the geometric control theory of mechanical systems with symmetries. To be precise, we consider a configuration space M together with a Lie group H acting on M which preserves some constraints on the velocities. Of particular importance are non-holonomic constraints, which are restrictions that can not be reduced to position constraints. In our model these restrictions are modelled as a smooth distribution D inside the tangent bundle T M of the configuration space which is transverse to the infinitesimal action of H. All these data are combined in a geometric structure called principal bundle. We also assume that there exists a Lie subgroup K < H such that the restriction of the action of K is also a principal bundle. This leads to an interaction of two non-holonomic systems. By making use of the Hamiltonian formalism we study the interplay of the geodesic curves in these non-holonomic systems. Geometric examples of this construction include the quaternionic Hopf fibration [2], the Gromoll-Meyer exotic sphere [6] and the twistor bundle of S 4 [1]. M. G. Molina—Partially supported by grants Fondecyt #1181084 by the Chilean Research Council and DI20-0023 by Universidad de La Frontera. I. Markina—Partially supported by grant # 262363/O70 of the Norwegian Research Council. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 82–92, 2021. https://doi.org/10.1007/978-3-030-58653-9_8

Geodesics on Nested Principal Bundles

2

83

Nested Principal Bundles

In this paper, all manifolds and Lie groups are assumed to be connected. Definition 1. Let H be a Lie group. A submersion πH : M → N is a principal −1 H-bundle if H acts freely and transitively from the right on the fibers πH (n), n ∈ N. π

H We denote a principal H-bundle as H  M −→ N . Note that N is diffeomorphic to the quotient M/H. The vertical bundle V → M is a vector bundle defined by Vm = ker dm πH , m ∈ M . Let h be the Lie algebra of the group H. Given a vector ξ ∈ h, we define the fundamental vector field on M by  d  m. expH (tξ), m ∈ M, σm (ξ) = dt t=0

where expH : h → H is the group exponential, and m.h denotes the action of h ∈ H on m ∈ M . Let K < H be a closed Lie subgroup of H with the Lie algebra k. We say that the triplet (M, H, K) is a nested principal bundle if the restriction πH M/H of the action to K is also a principal bundle. In this case, if H  M −→ πK M/K are the principal H- and K-bundles respectively, we and K  M −→ have the vertical bundles VH = ker dπH ∼ =h×M

and VK = ker dπK ∼ = k × M.

Consider two Ehresmann connections DH → T M and DK → T M for πH and πK respectively, that is ker dπH ⊕ DH = ker dπK ⊕ DK = T M . Assume that the distributions DH and DK are invariant under the action of H and K respectively. The aim of the present paper is to study the sub-Riemannian structure of the triplet (M/K, D, gD ), where the distribution D → T (M/K) is defined by D = dπK (DH ) and the metric gD will be defined later. Observe that π : M/K → M/H is a submersion where a fiber is the homogeneous space H/K, so that we have the following diagram =H zz z z zz . zzz * K M

/ H/K

(1)

 / M/K w ww πH wπw w w  {ww M/H πK

where the two triangles commute. Notice that, in principle, the space H/K is just a homogeneous space and H/K → M/K → M/H is just a fibration. Lemma 1. The distribution D is an Ehresmann connection for π.

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Proof. By a dimension counting argument, it is enough to show that D is transverse to ker dπ. If v ∈ D ∩ ker dπ, then v = dπK (w) for some w ∈ DH , and dπ(v) = 0. This implies that dπ(dπK (w)) = d(π ◦ πK )(w) = 0, by the chain rule. Since diagram (1) commutes, we know that π ◦ πK = πH , and thus dπH (w) = 0. From this, we conclude that w ∈ DH ∩ ker dπH . By the assumption, DH is an Ehresmann connection for πH , and therefore w = 0. This implies that v = 0.  Remark 1. Note that dπK gives the isomorphisms DH ∼ = D and DK ∼ = T (M/K). It follows that the vector w such that v = dπK (w) is unique for any v ∈ D.

3

Hamiltonians in Nested Principal Bundles

Let g be a Riemannian metric on M , such that DH = VH⊥ and DK = VK⊥ with respect to g, and such that g is invariant under the action of H (thus also invariant under K). It follows that DH is H-invariant and DK is K-invariant. For each m ∈ M , the restriction g|V H defines the positive definite symmetric bilinear form on the Lie algebra h   IH m ∈ M, ξ, η ∈ h. m (ξ, η) = g|V H σm (ξ), σm (η) , We require that IH m does not depend on m ∈ M . According to the terminology in [10, Chapter 11], the metric g satisfying all of these hypotheses is called of constant bi-invariant type with respect to both group actions. The metrics gD K and gD H , obtained by restriction, are sub-Riemannian metrics on M for the distributions DK and DH , respectively. Define the Riemannian metrics gM/K and gM/H by the equalities g(v, w) = gM/K (dm πK (v), dm πK (w)),

v, w ∈ DK ,

m ∈ M,

(2)

g(v, w) = gM/H (dm πH (v), dm πH (w)),

v, w ∈ DH ,

m ∈ M.

(3)

Proposition 1. The map π : M/K → M/H is a Riemannian submersion, with respect to the metrics (2) and (3), respectively. Remark 2. The map dπK |D K : DK → T (M/K) is an isometry on each fiber with respect to the Riemannian metrics gD K and gM/K . Given a smooth subbdundle D of T M and a metric tensor gD on M defined only for vectors belonging to Dm , m ∈ M , the gD sharp map gD : T ∗ M → T M , ∗ M , then gD (λ) ∈ Dm is the unique map satisfying im gD = D and if λ ∈ Tm is the unique vector for which λ(w) = gD (gD (λ), w), for all w ∈ Dm , m ∈ M . ∗ ∗ ∗ ∗ The  cometric gD :T M × T M → R associated to gD is defined by gD (λ, μ) = gD gD (λ), gD (μ) . Every cometric defines a function H ∈ C ∞ (T ∗ M ), called the Hamiltonian associated to g ∗ , by the formula H(m, λ) =

1 ∗ g (λ, λ), 2

∗ λ ∈ Tm M.

Geodesics on Nested Principal Bundles

85

Using the metrics g, gD H , gD K , gV H = g|V H and gV K = g|V K on M , we define the respective Hamiltonian functions on T ∗ M . Note that g = gD H + gV H = gD K + gV K . This implies the equalities HM = HD H + HV H = HD K + HV K . Similarly, considering the metrics gM/K , gD = gM/K |D and gV = gM/K |V on M/K, one has the respective Hamiltonian functions on T ∗ (M/K) and a decomposition HM/K = HD + HV . ∗ : T ∗ (M/K) → T ∗ M the induced map of cotangent Let us denote by πK ∗ bundles defined by πK (λ)(v) = λ(dm πK (v)), for λ ∈ T ∗ (M/K), v ∈ Tm M , m ∈ M. Proposition 2. The following identities take place: (a) (b) (c)

DH ∗ D H H  ,∗  D ◦ πK = DH K ◦ πK = HV . −H H   ∗ ∗ = HD K ◦ πK = HM/K . As a consequence, we have that HM − HV K ◦ πK

Proof. Let n = dim M , r = rk DH and s = rk DK . Consider X1 , . . . , Xr , . . . , Xs , V1 , . . . , Vn−s a local frame of vector fields of X(M ) orthonormal with respect to g, where the vector fields X1 , . . . , Xr span the local sections of DH , the vector fields X1 , . . . , Xs span the local sections of DK and the vector fields V1 , . . . , Vn−s span the local sections of VK . Denote by Yj = dπK (Xj ), for j = 1, . . . , s. It follows from this choice that Y1 , . . . , Ys are orthonormal with respect to gM/K and that Y1 , . . . , Yr span D at each p ∈ M/K. Given the Riemannian metric g, we have a canonical isomorphism between T M and T ∗ M , thus we have the dual frames pX1 , . . . , pXs , pV1 , . . . , pVn−s defined on M and pY1 , . . . , pYs defined on M/K. With all of these notations, we have the Riemannian Hamiltonians

HM (λ) =

n−s s s 1 ∗ 1 ∗ 1 ∗ g (λ, pXi )2 + g (λ, pVj )2 , HM/K (μ) = g (μ, pYi )2 , 2 i=1 2 j=1 2 i=1 M/K

where λ ∈ T ∗ M and μ ∈ T ∗ (M/K), the horizontal Hamiltonians 1 ∗ g (λ, pXi )2 , 2 i=1 r

HD H (λ) =

1 ∗ g (λ, pXi )2 , 2 i=1 s

HD K (λ) =

1 ∗ g (μ, pYi )2 , 2 i=1 M/K r

HD (μ) = and the vertical Hamiltonians HV K (λ) =

n−s 1 ∗ g (λ, pVj )2 , 2 j=1

HV (μ) =

s 1  ∗ g (μ, pYj )2 . 2 j=r+1 M/K

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M. Godoy Molina and I. Markina

Before we start computing, it is convenient to note that the diagram T ∗ (M/K) 

∗ πK

/ T ∗M

gM/K

 T (M/K) o

(4)

g

 TM

dπK

∗ = gM/K . To see this, commutes, that is, we have the equality dπK ◦ g ◦ πK ∗ observe that dπK (Xj ) = Yj implies that πK (pYj ) = pXj . Indeed ∗ πK (pYj )(Xi ) = pYj (dπK (Xi )) = pYj (Yi ) = δji ,

where δji is the Kronecker delta. Also note that g pXi = X i , which implies  ∗   g ∗ that  πK (pYj ) = Xj . Finally, we can conclude that dπK g πK (pYj ) = gM/K pYj . By linearity, the commutativity of the diagram dπK (Xj ) = Yj =  follows. To prove (a), let μ ∈ T ∗ (M/K), then we compute r r   DH 1 ∗ ∗ 1 g ∗ ∗ (μ) = H ◦ πK g (πK (μ), pXi )2 = g( πK (μ), g pXi )2 2 i=1 2 i=1

=

1 g ∗ 1 ∗ g( πK (μ), Xi )2 = gM/K (dπK g πK (μ), dπK Xi )2 2 i=1 2 i=1

=

1 1 ∗ gM/K (gM/K μ, Yi )2 = g (μ, pYi )2 = HD (μ). 2 i=1 2 i=1 M/K

r

r

r

r

In the fourth equality we used the fact that πK is a Riemannian submersion, and in the fifth one, the commutativity of diagram (6). A similar computation can be performed for (b). Equality (c) can be obtained adding (a) and (b).  Remark 3. Proposition 2 is a special case of the so-called lifted Hamiltonian ∗ −1 ∗ ) in [7]. In our case, the map π 2 defined in [7] corresponds to (πK |im πK and ∗ extended by zero to the orthogonal complement of im πK . We observe that since ∗ ∼ T M = DK ⊕⊥ VK , then the metric produces the two isomorphisms DK = ∗ ∼ ∗ Ann(VK ) and VK = Ann(DK ). Here Ann(E) ⊂ T M is the annihilator of the vector subbundle E ⊂ T M .

4

Sub-Riemannian Geodesics

Definition 2. A sub-Riemannian manifold is a triplet (M, D, gD ), where D → T M is a smooth (integrable/non-integrable) vector subbundle of T M and gD is a metric tensor on M defined only for vectors belonging to Dp for all p ∈ M .

Geodesics on Nested Principal Bundles

87

Given a Riemannian metric g = gT M on M , we denote by HM the Hamiltonian associated to g ∗ . Given a sub-Riemannian metric gD on M , we denote by HD ∗ the Hamiltonian associated to gD . Definition 3. Let (M, D, gD ) be a sub-Riemannian manifold. The image of the → −D → − projection ΠM : T ∗ M → M of the flow et H of the Hamiltonian vector field H D D associated to H is called a sub-Riemannian geodesic. The aim of this section is to relate the sub-Riemannian geodesics in the subRiemannian manifolds (M, DH , gD H ) and (M/K, D, gD ). Notice that we have the following commutative diagram of cotangent bundles ∗ πK

T ∗ (M/K) T ∗M o s9 i4 s i ∗ i s i πH s i iiii ss ssiiiiiiπi∗ s s i ΠM/K ΠM T ∗ (M/H)

ΠM/H

 M/H

 / M/K i ss iiii πH sss iiii i i s i ss iii π ystisiiiii  M

πK

Let (m, λ) ∈ T ∗ M , then the sub-Riemannian geodesic starting at m ∈ M ∗ M and tangent to DH is given by with covector λ ∈ Tm → −DH   (m, λ), γDsRH (t; m, λ) = ΠM ◦ et H

for t > 0 sufficiently small. An analogous definition is valid for sub-Riemannian geodesics γDsR (t; n, μ), (n, μ) ∈ T ∗ (M/K). Let ωM and ωM/K be the canonical symplectic forms on T ∗ M and T ∗ (M/K), respectively. Theorem 1. Let (n, μ) ∈ T ∗ (M/K), and consider m ∈ M with n = πK (m) and ∗ ∗ ∗ μ = λ ∈ Tm M . If the map πK : T ∗ (M/K) → T ∗ M is a symplectomorphism, πK then   (5) πK γDsRH (t; m, λ) = γDsR (t; n, μ). Proof. For any w ∈ T T ∗ (M/K), we have   ∗   → − ∗ (w) = dHD H dπK (w) ωM/K ( H D , w) = dHD (w) = d HD H ◦ πK −  → ∗ (w) , = ωM H D H , dπK from the definition of the Hamiltonian vector fields, the chain rule and Propo∗ sition 2. The map πK : T ∗ (M/K) → T ∗ M is a symplectomorphism, that is ∗ ∗ (β) , for all α, β ∈ T T ∗ (M/K). Then for any ωM/K (α, β) = ωM dπK (α), dπK

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 ∗  ∗ w ∈ T T ∗ (M/K) we also have ωM/K (HD , w) = ωM dπK (HD ), dπK (w) . Since → − → ∗ − the symplectic form ωM is non-degenerate, we deduce that H D H = dπK (HD ) from  ∗ − −   → → ∗ ∗ ( H D ), dπK (w) = ωM H D H , dπK (w) . ωM dπK This implies that the flows of the Hamiltonian vector fields are related by −  → − → −D ∗ → tdπK HD tHD H ∗ (m, λ) = πK e (m, λ) = e ◦ et H (n, μ). To complete the proof, observe that the natural diagram T ∗M o

∗ πK

(6)

ΠM/K

ΠM

 M

T ∗ (M/K)

πK

 / M/K

commutes, therefore → −D  → −D      H ∗ (t; m, λ) = πK ◦ ΠM ◦ πK πK γDsRH (t; m, λ) = πK ◦ ΠM ◦ et H ◦ et H (t; n, μ) → −D   = ΠM/K ◦ et H (t; n, μ) = γDsR (t; n, μ),

which is the equality sought after.



Corollary 1. The sub-Riemannian geodesics γDsRH (t; m, λ) and γDsR (t; n, μ) in Theorem 1 have the same projection to M/H, that is     πH γDsRH (t; m, λ) = π γDsR (t; n, μ) .

5 5.1

Examples The Quaternionic Hopf Fibration from Sp(2)

An important special case of the case in which K  H is a normal subgroup is the quaternionic Hopf fibration, as constructed in [2]. A comprehensive introduction to Hopf fibrations, one can find in [3]. Recall that the 10-dimensional compact symplectic group Sp(2) = U(4) ∩ Sp(4, C) can be defined through quaternionic matrices as follows  

ab Sp(2) = Q = ∈ M (2 × 2, H) : Q∗ Q = QQ∗ = id , cd where Q∗ denotes the transpose  (quaternion) conjugate of Q. Let us consider

 μ0 the 6-dimensional subgroup H = ∈ Sp(2)  Sp(1)×Sp(1) of diagonal 0ν quaternionic matrices in Sp(2). The right multiplication





μ0 μ ¯0 , Q → Q , (7) 0ν 0 ν¯

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defines a left group action of H on Sp(2). The homogeneous space Sp(2)/H of this action is diffeomorphic to the usual 4-dimensional sphere S 4 by means of the “stereographic projection”

ab πH Sp(2)/H −→ S4, H a, |c|2 − |a|2 ), → (2d¯b, |b|2 − |d|2 ) = (−2c¯ cd where S 4 = {(q, x) ∈ H × R : |q|2 + x2 = 1}. Let K be the 3-dimensional subgroup of H such that ν = 1. Restricting the left action (7) to the subgroup K determines the homogeneous space Sp(2)/K which is diffeomorphic to the usual 7-dimensional sphere S 7 by means of the

a b πK projection map K −→ (b, d), where S 7 = {(b, d) ∈ H×H : |b|2 +|d|2 = 1}. cd In this way, we have two maps as in the diagram

Sp(2) πK

S

πH

7

S4

As a direct consequence of this definition, we see that the quaternionic projective line HP1 is diffeomorphic to the sphere S 4 under the map [b : d] → (2d¯b, |b|2 − |d|2 ). Obviously this diffeomorphism is invariant under right multiplication by an element in Sp(1). Using these identifications, the projectivization map H2 − {(0, 0)} → HP1 ,

(b, d) → [b : d]

induces a map h : S 7 → S 4 called the quaternionic Hopf map. Since Sp(1) is diffeomorphic to the 3-dimensional sphere S 3 , we have the diagram

Sp(2) πH

πK

S3

S7

h

S4.

Observe that the quaternionic Hopf map provides a principal bundle with a typical fiber S 3 , called the quaternionic Hopf fibration. The map h corresponds to the submersion π : Sp(2)/K → Sp(2)/H. It is known [2,5] that the distribution D = dπK (DH ) on S 7 is bracket generating of step 2. We endow Sp(2) with a bi-invariant Riemannian metric g defined by g(u, v) = Re tr(u · v ∗ ),

u, v ∈ sp(2) ⊂ M (2, H),

where v ∗ denotes the transposed conjugate of v. The Ehresmann connections DH and DK are chosen as the left-translations of the orthogonal complements to h ∼ = sp(1) × sp(1) and k ∼ = sp(1) × {0} in sp(2), with respect to g.

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Gromoll-Meyer Exotic Sphere

The Gromoll-Meyer sphere [6] is constructed in a similar fashion as the quaternionic Hopf fibration, but does not fit in the scheme of Subsect. 5.1, in the relation of sub-Riemannian geodesics, see [2]. Consider M = Sp(2), and the subgroup of M : H = Sp(1) × Sp(1), acting on the right by









¯ ¯ ¯0 xy λxμ λy λ xy μ0 .(λ, μ) = ¯ = , (8) ¯ ¯ zw λzμ λw 0λ zw 01

xy where (λ, μ) ∈ H and ∈ M . Consider the restriction of the action (8) to zw the subgroup Δ = {(λ, λ) ∈ Sp(1) × Sp(1)} < H, which is not normal in H. As before, the maps πH and πΔ are the quotient maps with respect to the action of H and Δ respectively. In a similar way as before, it can be shown that the homogeneous space M/H is diffeomorphic to the sphere S 4 with respect to the action (8). On the other hand, the homogeneous space ΣGM := M/Δ, called the Gromoll-Meyer exotic sphere, is a seven dimensional manifold homeomorphic, but not diffeomorphic, to the sphere S 7 , see [9]. The corresponding submersion π : ΣGM → S 4 is an S 3 -bundle over S 4 which is not a principal bundle. The distribution D = dπΔ (DH ) on ΣGM has been recently shown to be bracket generating of step 2, see [2]. Endowing M with the Riemannian metric g from Subsect. 5.1, we define the Ehresmann connections DH and DΔ as the orthogonal complements to VH = ker dπH and VK = ker dπΔ , with respect to g. In this case, the bilinear form IH m does depend on m ∈ M , therefore it is necessary to consider more general formulas for sub-Riemannian geodesics, see [4]. 5.3

Twistor Space of S 4

Let N be a four dimensional Riemannian manifold. The twistor space T(N ) of N is the fiber bundle of almost complex structures on N that are compatible with the Riemannian metric. In the case of N = S 4 this yields to a well known construction where T(N ) = CP 3 and the bundle map is given by T CP 3 e −→ eH = e ⊕ ej ∈ HP 1 ∼ = S4,

∼ H2 . The fibers correspond to where e ∈ CP 3 is thought of as a line in C4 = 1 ∼ 2 spheres CP = S endowed with its unique complex structure. The map T is sometimes referred as the twistor projection. For more details, see [1,8].

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Consider the inclusion S 1 → S 3 ⊂ C2 given by eiθ → (eiθ , 0). The twistor projection T fits in the following diagram 3

>S }} } } }} . }} * 7 S1 S

H1

/ CP 1

 / CP 3 oo ooo o o h oo  o ooo T 4 w S H3

where H1 : S 3 → CP 1 and H3 : S 7 → CP 3 are the classical Hopf fibrations and h : S 7 → S 4 is the quaternionic Hopf fibration from Subsect. 5.1.

6

Conclusions and Future Work

In the paper, an interplay between two principal bundles is studied. It leads to a fiber bundle, called nested bundle, that is not principal in general. We described the relation between natural distributions, Hamiltonians, and geodesics on all three involved fiber bundles. The motivation for the study were some examples from geometry and physics. Similar systems can appear in the rolling problems, where, in a local chart, we can consider a subgroup K of the group H of isometric transformations acting on a configuration space of two rolling bodies. In the future we consider different rolling systems appearing in the robotics, or spline constructions related to the approximation of curves on Grassmann and/or Stiefel manifolds.

References 1. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond, Ser. A 362(1711), 425–461 (1978) 2. Bauer, W., Furutani, K., Iwasaki, C.: A co-dimension 3 sub-Riemannian structure on the Gromoll-Meyer exotic sphere. Differ. Geom. Appl. 53, 114–136 (2017) 3. Gluck, H., Warner, F., Ziller, W.: The geometry of the Hopf fibrations. Enseign. Math. 32(2), 173–198 (1986). no 3–4 4. Godoy Molina, M., Grong, E.: Riemannian and sub-Riemannian geodesic flows. J. Geom. Anal. 27(2), 1260–1273 (2017) 5. Godoy Molina, M., Markina, I.: Sub-Riemannian geodesics and heat operator on odd dimensional spheres. Anal. Math. Phys. 2(2), 123–147 (2012) 6. Gromoll, D., Meyer, W.: An exotic sphere with nonnegative sectional curvature. Ann. Math. 100(2), 401–406 (1974) 7. Grong, E.: Submersions, Hamiltonian systems, and optimal solutions to the rolling manifolds problem. SIAM J. Control Optim. 54(2), 536–566 (2016) 8. Heller, S.: Conformal fibrations of S3 by circles. Harmonic maps and differential geometry, pp. 195–202. Contemp. Math., 542, Amer. Math. Soc., Providence, RI (2011)

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9. Milnor, J.W.: On manifolds homeomorphic to the 7-sphere. Ann. Math. 64(2), 399–405 (1956) 10. Montgomery, R.: A tour of Subriemannian geometries, their geodesics and applications. In: Mathematical Surveys and Monographs, vol. 91. American Mathematical Society (2002)

A Comparative Study of Dynamic Mode Decomposition (DMD) and Dynamical Component Analysis (DyCA) Moritz Kern(B) , Christian Uhl, and Monika Warmuth Center for Signal Analysis of Complex Systems, Ansbach University of Applied Sciences, Residenzstr. 8, 91522 Ansbach, Germany {moritz.kern,christian.uhl,monika.warmuth}@hs-ansbach.de

Abstract. Two dimensionality reduction methods, dynamic mode decomposition (DMD) and dynamical component analysis (DyCA), are briefly introduced and compared by application on epileptic EEG data. DMD approximates a linear operator whose eigendecomposition describes the underlying system in frequency space. A reduction in dimensionality is achieved by retrospectively selecting relevant DMD modes. DyCA, on the other hand, naturally provides a dimensionality reduction by projecting onto a relevant subspace in time domain during the process. Keywords: Dimensionality reduction · Multivariate signal processing · Data analysis · Dynamic mode decomposition · DMD Dynamical component analysis · DyCA

1

·

Introduction

Complex systems can be found across many scientific domains like medical sciences, weather and climate, financial markets, epidemiology, fluid mechanics, sociology, and physics. Measurements taken on complex systems by several sensors are typical sources for multivariate time series. One particular example are measurements of electrical brain activity, also known as electroencephalography (EEG). Often the recorded data is of enormous scale and data analysis is therefore resource intensive. For this reason dimensionality reduction of highdimensional time series data is of significant interest. The most common dimensionality reduction method is principal component analysis (PCA) [6], which ranks its modes according to their energy content or variance in the data. But this stochastic model assumption is not optimal for high-dimensional time series with a coherent deterministic structure. Methods like dynamic mode decomposition (DMD) [9] and dynamical component analysis (DyCA) [10] aim at reducing the dimensionality while preserving the underlying dynamics of the complex system. The results obtained with these methods not only provide a reduced c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 93–103, 2021. https://doi.org/10.1007/978-3-030-58653-9_9

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representation of the data, but also allow further insights into the nature of these systems. Although both DMD and DyCA have been used for comparison with PCA in the past, DMD and DyCA have not yet been juxtaposed. This paper is structured as follows: in Sects. 2 and 3 we provide a brief summary of the DMD and the DyCA algorithms. In Sect. 4 we apply both methods on epileptic EEG data and compare the results and further algorithmic properties of DMD and DyCA in Sect. 5.

2

Dynamic Mode Decomposition (DMD)

Dynamic mode decomposition was originally developed in the field of fluid dynamics to extract dynamical features of flow fields describing the motion of the flow [9]. But the fact that DMD is an equation-free method that solely operates on the given measurement data made it a popular tool in the data analysis of high-dimensional nonlinear dynamic systems. The modes computed by the DMD algorithm are considered as spatial locations identifying coherent structures in the data by their magnitude and phase. Each DMD mode is associated with a corresponding eigenvalue that represents a growth/decay rate and an oscillation frequency. Therefore, DMD is strongly connected to the Koopman operator theory [8,11]. Since its first introduction in 2008, the DMD algorithm has been adapted and generalized in different ways, allowing its applicability to various classes of data sets. The DMD algorithm is often formulated using either an Arnoldi-like approach [9] or an approach based on singular value decomposition (SVD). As the SVD-based approach offers better numerical stability and has therefore gained acceptance for practical implementations, we will now describe a brief summary of this algorithm following [5,11] as well as [7,9]. Let {q(t1 ), ..., q(tT )} be a multivariate time series sampled at regularly spaced time points t = t1 , ..., tT . Each data point q(t) ∈ IRN consists of N values e.g.. measured features. Combining all time series elements into a matrix yields the N × T -data matrix   ⎤ ⎡     Q = ⎣ q(t 1 ) q(t 2 ) · · · q(tT ) ⎦ ∈ IRN ×T . (1)    For further derivation of the DMD algorithm we define the following matrices   ⎤   ⎤ ⎡  ⎡        ⎦ ⎣ ⎣ (2) X = q(t 1 ) q(t 2 ) · · · q(tT−1 ) , Y = q(t 2 ) q(t 3 ) · · · q(tT ) ⎦       with X, Y ∈ IRN ×(T −1) and assume that there exists a linear operator in form of an unknown matrix A ∈ IRN ×N such that Y = AX

(3)

holds. In vector notation this reads q(tj+1 ) = Aq(tj ), j = 1, ..., T − 1. To detect the underlying dynamical information, the DMD modes and eigenvalues should

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approximate the eigenvectors and eigenvalues of A. If the data is generated by an underlying nonlinear system, this assumption yields a linear approximation Y ≈ AX. The operator A is to be chosen such that it yields a least-squares solution of (3), i.e such that A minimizes the Frobenius norm of AX − Y F . This condition is fulfilled for choosing A as follows: A = Y X+

(4)

with X + being the Moore-Penrose pseudoinverse of X. The dynamic mode decomposition of (X, Y ) is then given by the eigendecomposition of A. The exact DMD algorithm reads as follows: Algorithm 1. 1. Arrange the data into matrices X and Y as in (2). 2. Compute the (reduced) singular value decomposition of X by X = U ΣV ∗

(5)

with U ∈ IRN ×r , V ∈ IRT −1×r and Σ ∈ IRr×r . Here, r stands for the rank of X and V ∗ denotes the transpose of V . The pseudoinverse X + is then given by X + = V Σ −1 U ∗ .  ∈ IRr×r as a 3. Equation (4) then reads A = Y X + = Y V Σ −1 U ∗ . Compute A rank-r approximation of A by projecting A onto PCA modes contained in U :  = U ∗ AU = U ∗ Y V Σ −1 A

(6)

 by 4. Compute the eigendecomposition of A  = WΛ AW

(7)

 The Matrix Λ is a diagonal where the columns of W are the eigenvectors of A. matrix containing the corresponding DMD eigenvalues λ1 , ..., λr . 5. Reconstruct the eigendecomposition of A from W and Λ by computing Φ = Y V Σ −1 W.

(8)

 0 the corresponding column ϕi in Φ is called DMD mode of A. For For λi = λi = 0 the corresponding DMD mode is computed by ϕi = U wi . It can be shown that each pair (λi , ϕi ) of eigenvalue and corresponding DMD mode is an eigenvalue/eigenvector of A [11]. A simple model to reconstruct the time series q(t) is then given by q(tj ) = ΦΛtj /Δt Φ+ q(t1 ).

(9)

with q(t) being the reconstructed version of q(t). Per construction the operator A is computed such that it minimizes q(t) − q(t). Dimensionality reduction can be achieved by considering only a subset of all DMD modes consisting of the most dominant modes for the reconstruction in (9). However, the selection of the most important modes is non-trivial. It is often assumed that the DMD modes

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with large norms correspond to the important dynamic features. Misleading modes with large norms but rapidly decaying contributions to the dynamics can be penalized by additionally weighting the norms of the DMD modes with the magnitude of the corresponding eigenvalue, i.e. |λi |p ϕi , where p is some number of iterating steps [7,11]. Other well-known methods for mode selection are for example optimized DMD [2] or sparsity promoting DMD [3].

3

Dynamical Component Analysis (DyCA)

Dynamical component analysis has recently been introduced in [4,10] as an alternative dimensionality reduction method to PCA that preserves the underlying dynamics of a complex system and provides the user with an additional parameter estimate for that system. The method assumes that a multivariate time series can be split into a deterministic part which can be described by a set of linear and nonlinear differential equations and independent noise components. By projecting the data onto a low-dimensional subspace, DyCA aims to approximate the underlying dynamics by seeking the best-fit solution for the system with respect to the linear differential equations. This leads to a generalized eigenvalue problem, whose eigenvalues describe the quality of the fit. In this section we will present the assumptions for DyCA as well as the DyCA algorithm by following [4,10]. Let {q(t1 ), ..., q(tT )} be a multivariate time series with q(t) ∈ IRN , t = t1 , ...tT as in (1) with T ≥ N . Additionally let the data matrix Q and its time derivative Q˙ be of full rank N . It is assumed that the signal q(t) can be decomposed into deterministic and noise components q(t) =

n 

xi (t)wi +

i=1

p 

ξj (t)ψj

(10)

j=1

with linearly independent wi , ψj ∈ IRN and n + p ≤ N . We would like to note at this point that we know neither the wi , xi (t), ψj , ξj (t) nor the parameters n and p which means that DyCA, like DMD, is a purely data-driven method for dimensionality reduction. Whereas the amplitudes ξj (t) are assumed to be of stochastic character, the deterministic amplitudes xi (t) are required to obey a set of m unknown linear ordinary differential equations x˙ i (t) =

n 

ai,k xk (t) for i = 1, ..., m

(11)

k=1

with m ≥

n 2

and n − m nonlinear differential equations x˙ i (t) = fi (x1 (t), ..., xn (t))

for i = m + 1, ..., n

(12)

with fi being unknown nonlinear smooth functions. By applying the DyCA we obtain estimates for the components wi , xi (t) and the parameter n as well as

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for the also unknown ai,k ∀i, k and the parameter m. Furthermore we define the coefficient matrix ⎤ ⎡ a1,1 · · · a1,m a1,m+1 · · · a1,n ⎢ .. . ⎥ m×n .. A := [A1 , A2 ] = ⎣ ... . . . ... (13) . .. ⎦ ∈ IR . am,1 · · · am,m

am,m+1 · · · am,n

from the coefficients ai,k of the linear differential equations and assume that A2 has full rank n − m. In terms of matrix notation we can rewrite (10) as Q = WX + ΨΞ

(14)

with Q ∈ IRN ×T being the data matrix as in (1) and ⎤ ⎡ ⎤ ⎡ x1 (t1 ) · · · x1 (tT ) x1 (t) ⎢ .. ⎥ = ⎢ .. ⎥ ∈ IRn×T .. W = [w1 , ..., wn ] ∈ IRN ×n , X = ⎣ ... . . ⎦ ⎣ . ⎦ xn (t1 ) · · · xn (tT )

xn (t) (15)

and

⎡

Ψ = [ψ1 , ..., ψp ] ∈ IRN ×p ,

⎤ ⎡ ⎤ ξ1 (t1 ) · · · ξ1 (tT ) ξ1 (t) ⎢ .. ⎥ = ⎢ .. ⎥ ∈ IRp×T (16) Ξ = ⎣ ... . . . . ⎦ ⎣ . ⎦ ξp (t1 ) · · · ξp (tT )

ξp (t)

Our goal now is to separate the deterministic part W X of the signal Q from the stochastic part Ψ Ξ by using the assumption that the amplitudes xi (t) of the matrix X can be expressed by a system of differential equations (11) and (12). Thus we finally obtain a noise-free reconstruction of the signal Q, which we will  To achieve this goal we first consider a generalized left inverse refer to as Q. W − ∈ IRn×N of W with (17) X = W − Q. Due to the assumption of linearly independent modes wi , the rows {u∗1 , ..., u∗n } of W − can be understood as linearly independent projection vectors with the help of which the amplitudes xi (t) can be calculated by xi (t) = q(t)∗ ui .

(18)

Differentiating (18) with respect to t yields x˙ i (t) = q(t) ˙ ∗ ui .

(19)

DyCA now aims at estimating these projection vectors ui and approximating the n corresponding modes wi in a second step. Therefore we define vi = k=1 ai,k uk for i = 1, ..., m and write (11) as follows: q(t) ˙ ∗ ui =

n  k=1

ai,k q(t)∗ uk = q(t)∗ vi

(20)

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To obtain estimations u i , vi for the projection vectors ui , vi we define D(ui , vi ) := q(t) ˙ ∗ ui −q(t)∗ vi 2  with · denoting the time average and compute u i , vi , i = q(t) ˙ ∗ ui 2  1, ..., m as solutions of the least squares problem arg min

ui ,vi ∈IRN

m  q(t) ˙ ∗ ui − q(t)∗ vi 2

i=1

q(t) ˙ ∗ ui 2

(21)

subject to u1 , ..., um being linearly independent. After performing some basic computations which can be found in [10] (21) reduces to the solution of a generalized eigenvalue problem (see (23)). For the generalized eigenvalues λi we obtain ui , vi ) = 1 − λi (22) min D(ui , vi ) = D( ui ,vi ∈IRN

i.e. the linear approximation of (11) is well suited for λi ≈ 1. The DyCA algorithm then reads as follows: Algorithm 2. 1. Compute N × T matrix Q˙ of time derivatives q(t) ˙ as well as ˙ ∗ and C2 = 1 Q˙ Q˙ ∗ . the N × N correlation matrices C0 = T1 QQ∗ , C1 = T1 QQ T 2. Solve the generalized eigenvalue problem C1 C0−1 C1∗ u = λC2 u

(23)

and obtain N eigenvalues λi with corresponding eigenvectors u i . 3. Sort all eigenvalues in descending order and name them as follows: λ1 ≥ λ2 ≥ · · · ≥ λN

(24)

The corresponding eigenvectors are denoted u 1 , ..., u N . 4. To select all eigenvalues close to 1, choose a threshold α > 0 and obtain  (25) m  = |{λi  λi ≥ α}| as an estimate of m, i.e. the number of linear differential equations driv, ..., λN and corresponding eigenvectors ing the system. The eigenvalues λm+1  , ..., u  can be discarded. u m+1 N   by 5. Compute the vi , i = 1, ..., m vi = C0−1 C1∗ u i

(26)

m 1 , ..., vm 6. Consider span{ u1 , ..., u ,v  }. Due to the condition that A2 has full rank n − m and m ≥ n2 , n  = dim (span{ u1 , ..., u m 1 , ..., vm ,v  })

(27)

is an estimate of n. i ) and rename 7. Choose a minimal subset of vectors vi (linearly independent to u , ..., u n such that them u m+1  span{ u1 , ..., u m m+1 , ..., u n } = IRn . ,u 

(28)

− ∈ IRn×N , the estimated generalized left The vectors u ∗i are the rows of W − inverse of W .

A Comparative Study of DMD and DyCA

 =W − Q, C  = 8. Compute X X

1 T

99

X  ∗ and X

= 1 QX  ∗ C −1 W  X T

(29)

∈ IRN טn is the least squares solution as an estimate for W . The matrix W 2 X  . of arg min Q − W F  W

∗  = [ 9. Define U u1 , ..., u m  ] and compute

= 1U  Q˙ X  ∗ C −1 A  X T

(30)

˜ n  ∈ IRmט as an estimate for A with A being the least squares solution of 2  ˙   arg min U Q − AXF .  A

˙  Q˙ as X.  Note that in the last step one could also interpret U The time series can then be reconstructed by =W X.  Q (31) are called DyCA modes. Even without the condition to The columns w i of W i , vi , which can be used the rank of A2 , the DyCA provides projection vectors u to estimate the underlying dynamics of the data. In this case, however, we do not get a reliable estimate for n.

4

Application to EEG-Data

In this example we consider scalp EEG data recorded during epileptic seizures. Since EEG data during an epileptic event is known to be of deterministic structure [12], while the EEG data during normal activity is of stochastic nature, we consider this example suitable to apply DyCA and DMD. The seizure event was cut out of the original data. The sequence has a length of 512 samples (2 s) and was recorded with 25 sensors at 256 samples per second. Both algorithms were implemented using MATLAB R2018a (The MathWorks, Inc.).

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DMD: The DMD-algorithm as described in Sect. 2 was applied to a 2 s sample of the seizure sequence. In a first step the data matrix X was augmented from 25 × 512 to 100 × 509 by stacking up shifted copies of X h-times (h = 4) according to (32) (method adopted from [1]): ⎡ ⎤ q(t1 ) q(t2 ) . . . q(tT −h ) ⎢ q(t2 ) q(t3 ) · · · q(tT −h+1 )⎥ ⎢ ⎥ (32) Xaugmented = ⎢ . ⎥ .. .. .. ⎣ .. ⎦ . . . q(th ) q(th+1 ) · · · q(tT −1 )  as in (7), the DMD-modes were After calculating the eigendecomposition of A calculated according to (8). The resulting 100 DMD modes (φi ) and corresponding eigenvalues (λi ) were weighted according to |λi |p ϕi  (see Fig. 1). The 54 largest were chosen to reconstruct the signal using (9). The reconstruction can be seen in Fig. 3.

1 0.5 0

0

10

20

30

40

50

60

70

80

90

100

Fig. 1. Resulting 100 DMD modes weighted with |λi |p ϕi  here: p = 30

As can be observed in Fig. 1 the resulting values come as pairs. This observation can easily be explained by looking at equation |λi |p ϕi , indicating that eigenvalues λi appear as complex conjugate pairs. DyCA. The DyCA-algorithm was applied 1 to the same 2 s long seizure sample used 0.8 for the DMD. In contrast to DMD no 0.6 augmentation was done to the (25 × 512) 0.4 data matrix Q. The 25 sorted eigenvalues 0.2 obtained by solving the eigenvalue prob0 lem (see (23)) can be seen in Fig. 2. To 0 10 20 select all eigenvalues close to 1, a threshold of α = 0.92 was chosen (see (25)). The two largest eigenvalues and their corFig. 2. DyCA eigenvalues 2 were used responding vectors u 1 and u to calculate v1 and v2 with (26). A pro =W − Q, were the rows of W − jection of the data Q to IR4 was achieved by X

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are u ∗1 , u ∗2 , v1∗ and v2∗ . To reconstruct the time series, the inverse-projection was  utilized according to (31), leading to the reconstructed signal Q. In Fig. 3 the reconstructed signals using DMD and DyCA are displayed together with the raw signal for comparison.

200 100 0 −100 22

22.2

22.4

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Fig. 3. Comparison of 2 s long reconstruction of Fz electrode signal using DyCA and DMD, raw signal data is displayed for reference

5

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The previous sections show that both methods, DMD and DyCA, are suitable tools to analyze high-dimensional data while preserving the dynamics of the underlying system. Both algorithms are purely data-driven methods that do not require knowledge of the underlying system matrix. Whereas DMD minimizes q(t) − q(t) by finding a linear operator A as least squares solution for AX − Y F , DyCA minimizes q(t) − q(t) by finding n the best fit of the data to a set of linear differential equations, i.e. x˙ i (t) − k=1 ai,k xk . One could say that both methods work with a linear model assumption and reduce the minimization problems to the solution of an eigenvalue problem for special matrices. Furthermore, the assumptions of the DMD algorithm can be interpreted as less strict than those of DyCA, but the DyCA is more specific and will probably yield better results than DMD if the data follow the given assumptions well enough. This is also shown by the previous Sect. 4 as can be seen in Fig. 3. Despite the fact DyCA needs only 4 modes to reconstruct the signal, the reconstruction result is better than that of DMD, where 54 modes were considered for the reconstruction. Although DMD and DyCA share many similarities from an algorithmic perspective, they differ in the way of how the dimensionality reduction is performed. The singular value decomposition in step 2 of the DMD algorithm does indeed

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offer a possibility for dimensionality reduction, but carries the danger of unintentional removal of spatial functions that are important for the dynamics of the system. Therefore, this step is often only used to reduce the size of large matrices to enable the computation of an eigenvalue decomposition at all. The actual dimensionality reduction is done by subsequent selection of the DMD modes most important for the system. On the contrary, the DyCA algorithm naturally includes a dimensionality reduction and does not require the user to identify the most interesting DyCA modes as a post processing step. Additionally, it can be difficult to choose the right DMD modes for an application, a task which is simply non-present in DyCA once an appropriate threshold α has been chosen. While DyCA performs the dimensionality reduction in time domain by projecting the signal onto a lower-dimensional subspace, DMD yields results in form of a frequency spectrum, hence the dimensionality reduction operates in frequency domain. Executing the dimensionality reduction in different domains also explains the different additional information both algorithms provide to the user. DMD generates information about the system's oscillatory behavior, whereas DyCA offers a parameter estimation of the linear differential equations driving the system. In conclusion DyCA and DMD might be interpreted as being tools answering different questions. Which one performs better is highly dependent on the data and the task itself. Acknowledgments. This work was supported in part by the German Federal Ministry of Education and Research (BMBF, Funding number: 05M20WBA) and in part by the European Regional Development Fund (ERDF, project: TZM).

References 1. Brunton, B.W., Johnson, L.A., Ojemann, J.G., Kutz, J.N.: Extracting spatialtemporal coherent patterns in large-scale neural recordings using dynamic mode decomposition. J. Neurosci. Methods 258, 1–15 (2015) 2. Chen, K.K., Tu, J.H., Rowley, C.W.: Variants of dynamic mode decomposition: boundary condition. Koopman and Fourier analyses. J. Nonlinear Sci. 22, 887–915 (2012) 3. Jovanovi´c, M.R., Schmid, P.J., Nichols, J.W.: Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26, 024103 (2014) 4. Korn, K., Seifert, B., Uhl, C.: Dynamical Component Analysis (DyCA) and its application on epileptic EEG. In: ICASSP, pp. 1100–1104. IEEE (2019) 5. Kutz, J.N., Fu, X., Brunton, S.L.: Multiresolution dynamic mode decomposition. SIAM J. Appl. Dyn. Syst. 15, 713–735 (2016) 6. Pearson, K.: On lines and planes of closest fit to a system of points in space. Philos. Mag. 2, 559–572 (1901) 7. Proctor, J.L., Eckhoff, P.A.: Discovering dynamic patterns from infectious disease data using dynamic mode decomposition. Int. Health 7(2), 139–145 (2015) 8. Rowley, C., Mezic, I., Bagheri, S., Schlatter, P., Henningson, D.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009) 9. Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)

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10. Seifert, B., Korn, K., Hartmann, S., Uhl, C.: Dynamical Component Analysis (DyCA): dimensionality reduction for high-dimensional deterministic time-series. In: 2018 IEEE 28th International Workshop on MLSP (2018) 11. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014) 12. van Veen, L., Liley, D.T.J.: Chaos via Shilnikov’s saddle-node bifurcation in a theory of the electroencephalogram. Phys. Rev. Lett. 97, 208101 (2006)

An Approach to Model Validation for Model Predictive Control Based on Dvurecenska’s Metric Victor D. Reyes Dreke(B)

, Manuel A. Pérez Serrano , and Claudio Garcia

Escola Politécnica, University of São Paulo, São Paulo, SP, Brazil {victord.reyesdreke,mperezs9308}@usp.br, [email protected]

Abstract. In this paper, Dvurecenska’s validation metric is proposed as a criterion for selecting a suitable model to be used by a Dynamic Matrix Control algorithm. As part of the work developed in this paper, it is analyzed how this algorithm performs when using different models. Besides, a comparison between the validation of these model results is performed. Additionally to the above metrics, the Theil Inequality Coefficient index and the FIT index are employed too. This paper was developed on a simulated plant based on Clarke’s benchmark, which is controlled with a Dynamic Matrix Control. As a result, it can be seen that Dvurecenska’s metric accomplishes its objective and in some cases, gives better validation criteria than the other metrics analyzed. Keywords: Validation metrics · Model predictive control · Dvurecenska metrics · Dynamic Matrix Control

1 Introduction The term Model Predictive Control (MPC) designates a variety of control methods that explicitly use the process model to predict the output at future time instants (horizon) and calculate a control signal by minimizing an objective function. Since its appearance in the 1970s, the MPC has proven to be a suitable solution for optimization control problems and in some cases increasing the efficiency and the profit of the industry [1, 2]. Papers like [2, 3] and [4] have shown that success rates of this applications are uneven and still have room to improve. Several factors can influence MPC controller performance. However, to all of the aforementioned authors, a key factor is the use of a model that accurately describes the process dynamics. Consequently, model validation is a crucial part of the MPC commissioning stage that requires continuous revision and improvement. To obtain the process model, most industrial MPC controllers employ a model identification software included in their packages. These software can be complex to use; therefore, it is more likely for the user to choose a model based on visual analysis more than quantitative methods [4]. Albeit accepted, visual analysis is a qualitative method, which has different results depending on the person. This feature is the main © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 104–113, 2021. https://doi.org/10.1007/978-3-030-58653-9_10

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drawback of this analysis and the reason why other types of methods are employed in model validation. System identification experts commonly give more importance to quantitative validation methods, because their results are robust, reproducible and more straightforward. One of the most popular methods is the FIT index. Papers like [5, 6], and [7] are fair examples of this metric being employed to validate models; although, these models are not used in MPC applications. Another validation metric is the Theil Inequality Coefficient (TIC) index. Despite being more employed in economic model validation; its capacity for handling process model validation is remarkable as seen in [8, 9] and [10]. Even when FIT and TIC indexes prove to be a useful metric for System Identification and MPC controller applications, these metrics are not failsafe. [11] is an example of how these two metrics can induce the user to choose an inappropriate model. In that case, the MPC controller did not follow the setpoint when models with reasonably good FIT index were used. In this paper, it is used Dvurecenska’s metrics as a validation criterion for models employed by an MPC controller. To assess this metric performance, a set of models is validated using the three aforementioned metrics and their results are compared. Also, to assess the model’s suitability to be used by a DMC algorithm, a comparison of the Clarke’s benchmark behaviour for different models is performed. The outline of the paper is as follows. Simulated plant based on Clarke’s benchmark is discussed in Sect. 2. Validation metrics are presented in Sect. 3. Simulation results are shown in Sect. 4. Conclusions are drawn in Sect. 5.

2 Simulated Plant Clarke’s benchmark was proposed in [12] and it was used to introduce the Generalized Predictive Control (GPC). The simulated plant used in this paper is based on this benchmark and it is composed of five elements, as shown in Fig. 1. The block “Process” contains the process transfer function seen in (1) that describes the open-loop process behavior. A PI controller tuned with proportional gain Kc = 1.5 and integral time constant Ti = 10 s/rep is also employed. The block “MPC” represents a Dynamic Matrix Control (DMC) algorithm.

Fig. 1. Plant diagram.

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G(z) =

0.00768z −1 + 0.02123z −2 + 0.00357z −3 1 − 1.9031z −1 + 1.1514z −2 − 0.2158z −3

(1)

To simulate a more realistic condition where measurement noise and perturbations exist, the block called “Perturbation” represent a white noise signal with zero mean and variance of 2 * 10−6 plus an integrator, respectively and the block called “Measurement Noise” another white noise signal with zero mean and variance 10−8 . All this plant’s elements were implemented using Matlab®’s Simulink®. 2.1 Dynamic Matrix Control Algorithm Cutler and Ramaker (1980) developed the DMC algorithm aiming to solve complex control problems that are not solvable with traditional PID control concepts. It has been widely employed in the industry, mainly in the petrochemical area [13]. This algorithm was designed to use Finite Step Response (FSR) to predict future outputs; therefore, the disturbance is considered constant along the prediction horizon. The predicted values along the prediction horizon are obtained as shown in (2). yˆ (t + k|t) =

k i=1

gi u(t + k − i) + f (t + k)

(2)

where k is the prediction horizon; u is the input variable rate, gi are the step response coefficients; f (t + k) is the free-response; which is the part of the response that corresponds to the past inputs; and is kept constant and equal to the last value of the manipulated variable in future time instants; and is described as follows: ∞ f (t + k) = ym (t) + (3) (gk+i − gi )u(t − i) i=1

where ym (t) is the measured value of the output. Hence, the predicted process values for p prediction horizon and m control horizon can be estimated as follows ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ g1 0 · · · 0 u(t) f (t + 1) ⎥ ⎢ g g ··· 0 ⎥ ⎢ 2 1 ⎢ u(t + 1) ⎥ ⎢ f (t + 2) ⎥ ⎥ ⎢ . ⎥ ⎥ ⎢ ⎢ .. . . .. ⎥ ⎢ . ⎥ ⎥ ⎢ ⎢ .. .. . . . ⎥ ⎢ . ⎥ ⎥ ⎢ ⎢ ∗⎢ +⎢ (4) yˆ = ⎢ . . ⎥ ⎥ ⎥ ⎢ gm gm−1 · · · g1 ⎥ ⎥ ⎥ ⎢ ⎢ . . ⎥ ⎢ . . ⎦ ⎦ ⎣ ⎣ .. . . .. ⎥ ⎢ .. . . . ⎦ ⎣ . . . u(t + m − 1) m×1 f (t + p) p×1 gp gp−1 · · · gp−m+1 p×m yˆ = G ∗ u + f

(5)

where G is called the system’s dynamic matrix, f is the free-response vector, yˆ is a p-dimensional vector containing the system predictions along the horizon, u represents the m-dimensional vector of control increments. The unconstrained DMC controller objective is to estimate a control action that minimizes the following cost function p

2 m

2 δ yˆ (t + j|t) − r(t + j) + λ u(t + j − 1) (6) J = j=1

j=1

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where r(t + j) is the setpoint value, λ is the weighting factor for control increments, δ is the weighting factor for predicted error. The solution to this optimization problem is presented in (7). −1  G T δIp (r − f ) u = G T δIp G + λIm

(7)

Finally, u is the vector composed of the future control increment, I m and I p are identity matrixes with p and m dimension respectively, and r is setpoint vector. In this paper, the DMC controller has a 1 s sample time and a control and prediction horizon of 2 and 10, respectively. The nominal values of the setpoint (SP) and the process variable (PV) are equal to 0. Its weight values are manipulated variable (MV) rate weight, λ = 2.22553; output variable (OV) weight, δ = 0.44933.

3 Validation Metrics 3.1 Dvurecenska’s Validation Metrics Dvurecenska’s metric was proposed in [14] to perform validations of computational solid mechanical models. The main feature of this metric is to combine a threshold based on the uncertainty in the measurement data with a normalized relative error. This characteristic makes this metric robust in the presence of large variations in the data. Another aspect is that the result from this metric is the probability that the predictions of a model represent well the process [14], making it easier to understand the result. Dvurecenska’s metric is evaluated in four steps: (i) compute a normalized relative error ek for each time instant as follows:    yˆ (t) − y(t)    (8) ek (t) =  max|y(t)|  where yˆ (t) and y(t) are the predicted and the original process output, respectively. (ii) Compute a weight we for each error; ek (t) ∗ 100 we (t) = n t=1 ek (t)

(9)

(iii) Define an error threshold eth using the next equation eth =

2uexp max|y(t)|

(10)

where uexp is the total uncertainty in the measured data. In the original paper, this uncertainty is calculated as the square root of the sum of the square of both calibration uncertainty ucal and decomposition uncertainty udeco . The first one can be obtained using calibration procedures for the measurement instrument, but the second one is the mean square error (MSE) between an original image and a reconstructed image. As this paper is focused on validating models that describe time series and not images, the original

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definition is not useful. To overcome this fact, different udeco and uexp are defined as follows:  1 2 2 u + ucal (11) uexp = 2 deco 1 N 2 udeco = (12) (r(t) − y(t))2 t=1 N where r(t) is the setpoint value at time t. Having a threshold that depends on the mean squared error between the output process and the setpoint, ensures to penalize every value with a relative normalized error greater than this MSE. Consequently, a lower MSE means a stricter threshold. (iv) Calculate the validation metric, VM as the sum of those weighted errors we less than the error threshold eth .  VM (%) = we (t)||ek (t)100 kg); 16 ultrasonic sensors for safe navigation; Kinect v2 Time-of-Flight camera, ideal for SLAM; Omnidirectional drive mechanism with four independent high-power and highresolution encoders; 6 degree of freedom robotic manipulator; Internal CPU for high-level programming Intel ® NUC D54250WYK; Internal controller for low-level programming Arduino MEGA ADK; More than 4 h of energy autonomy in full operation;

Therefore, FORTE was designed to tackle the several challenges inherent to social robotics, with focus on both research and educational markets as it is fully open source: from Arduino (https://www.arduino.cc/) low-level programming to ROS (www.ros.org/) mid-level programming.

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The proposed testing environment allows the validation of the proposed framework, from communication, through security and storage issues. The technologies used are open-source, allowing full configuration. Our approach is similar to the implementation of an Extraction, Transformation and Loading (ETL) system used in big data context. In this way, data is retrieved from heterogeneous sources, using computational methods to extract, transform and loading data, such as laser sensor and others, to a common format into a staging area (cloud storage platforms) for posterior analysis. Figure 6 shows a real “piece” of meta-data extracted during the performed experiments:

Fig. 6. Meta-data extracted during tests realized with Robot FORTE.

As it can be observed, different sensors are retrieving data from the environment, creating a data deluge, in our case: ~3.3 GB of data retrieved for 54 s, that includes all robot data, like uncompressed images and laser data. Our problem grows exponentially, when we have multiple robots retrieving, communicating and uploading data into the Cloud, which requires an upload bandwidth over end-user normal applications. So, in this work a methodology to assure that data transmission will be near real-time applications is proposed. Some issues could be mitigated by: (1) if the building is mapped (first time) and afterwards only the environment changes are upload to Cloud; (2) the retrieved images are compressed to decrease the amount of data. After this step, we have transformed ROS data into a CSV file, using a Python routine, that will be sent via web services to a cloud staging area (Influx DB), reducing upload time slot (uncompressed images were discarded). Therefore, all data is stored in a common format, to be processed by Grafana. Figure 7 shows the data processed by a MATLAB routine in the Cloud Resources Layer, representing the cloud of points produced by scanners, like lasers or cameras, used to acquire information of the external surfaces of the objects around. Therefore, this figure represents the environment that surrounds the robot FORTE-RC. The obtained results reveal that cloud computing services can help the robotic field by using processing and storage capabilities. In this work, the obtained results encouraged to improve data acquisition, transformation and storing for posterior analyses. To support the computation methods, the cloud machine used have the following capabilities: Memory (47.1 Gb); Processor (Intel @ CoreTM i7-3970X @ 3.50 GHz × 12); Graphics (GeForce GTX 660 Ti/PCIe/SSE2 and HDD (3,9 TB). Achieving 63 s to process data obtained from robot sensors to develop the SLAM operation represented in Fig. 7.

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Fig. 7. SLAM operation to retrieve the point cloud from a hospital environment.

The procedure of retrieving the data from the environment with robot FORTE-RC sensors, that were sent by web service, and therefore processed by software routines to perform SLAM task outside robot “brain”, provides a decreasing in time consumption and make data available to posterior analysis [24–30]. Our approach to decrease timeprocessing data and upload time to a cloud service, based on approaches implemented on Big Data context. Were data are retrieved, transformed and loaded into a common staging area in a standard data schema, different computational methods can process them. The establishment of information between the robot and the brain, allows robots to be less expensive to perform the same tasks, reducing computational time. Also will depend on the processing capacity installed in the system.

5 Conclusions and Future Work Aforementioned, the robot uses the Internet backbone infrastructure to interact with Cloud, which enables the Cloud services for robot applications. Therefore, the Cloud is used as a “brain”, taking leverage of computation capabilities for storage and processing tasks. This paper discussed some related works focusing on cloud robotics and robotics applied to the medical domain. The proposed approach for a cloud-based framework applied to the hospital environment, using cloud resources in a robot environment, will improve their processing and storage capabilities. Issues regarding security could be treated outside robots “brains”, were several tested software’s such as proxy servers or encryption programs could be applied. Other issues related to data transmission could be questionable if we were dealing with multiple robots in concurrent environments. For future work, it is planned to evaluate the proposed framework to validate a robot scenario emerged into medical environments (hospitals), using exhaustively the technologies enumerated in Sect. 4. Other technologies, such as data mining methods that could be applied to complex path navigation, may also be added to support the framework. Using the defined framework layers and by upgrading them, different tasks will be tested, such as object recognition and complex path navigation.

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Acknowledgment. This work is financed by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project UIDB/50014/2020.

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Identification and Control of Precalciner in the Cement Plant Jakub Osmic1(B)

, Emir Omerdic2 , Edin Imsirovic2 and Edin Omerdic3

, Tima O. Smajlovic2

,

1 University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina

[email protected] 2 Cement Factory Lukavac, 75300 Lukavac, Bosnia and Herzegovina 3 University of Limerick, Limerick, Ireland

Abstract. Identification and control of precalciner in the cement production process with the increased use of alternative fuels is presented in this paper. The model of the system consists of the precalciner model as well as of pulverized coal transport model and alternative fuel transport model. The manipulated variable in the precalciner system is pulverized coal, while the alternative fuel calorific value change and the alternative fuel mass flow change are the main disturbances. Outputs of the precalciner model are the precalciner output gas temperature, as a controlled variable, and concentration of CO as measured variable. During the identification process, raw meal mass flow and primary secondary and tertiary air flows are considered to be constant. Based on identified model of the precalciner and the fuel transport model, the mixed logical dynamical controller is designed and implemented at Cement Factory Lukavac (Bosnia and Herzegovina). The mixed logical dynamical controller is realized on the cRIO 9014 real-time controller using LabVIEW (G) programming language. The real-time responses indicate improvements in precalciner temperature control while keeping reasonable low frequency and intensity of CO peaks. Keywords: Precalciner · Identification · Control

1 Introduction At the present dry process cement production including preheater tower with cyclone stages and a precalciner, with the increased use of alternative fuels, become the most prevalent type of technology in the cement production. The main reasons for the use of mentioned technology are to increase energy efficiency and decrease the cost of cement production. By using conveyor belts and elevator raw meal (CaCO3 ) is transported to the top of the cyclone preheater stages. Each cyclone preheater stage consists of a cyclone separator (C1 … C5) and a heat exchange pipe (see Fig. 1). Raw meal, driven by gravitational force, travels from top cyclone preheater stage to the bottom of the cyclone preheater stage and enters into a precalciner. At the same time, hot gasses travel from the exit of precalciner through cyclone preheater upwards. In cyclone preheater, © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 126–135, 2021. https://doi.org/10.1007/978-3-030-58653-9_12

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Raw meal

Fan

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Precalciner Conveyor belts

Alternative fuel Tertiary air

Rotary kiln

Fig. 1. Schematic representation of cyclone preheater and precalciner

exchange of heat between hot gasses and raw meal and separation of solid from gas takes place. In this way, the raw meal is preheated to around calcination temperature. Most of the calcination (decarbonization) process is performed in precalciner according to the endothermic calcination reaction [1, 2]. CaCO3 + Heat → CaO + CO2

(1)

To achieve calcination temperature between 860 °C and 950 °C, according to (1), additional heat must be provided which is mostly (up to 70%) achieved by burning a primary fuel (such as pulverized coal) and alternative fuel inside the precalciner. The rest of the heat is provided from the hot exhaust gasses of the kiln. The decarbonized hot meal and gasses from precalciner are separated in additional cyclone stage situated at the output of the precalciner. The decarbonized hot meal then enters the rotary kiln where the sintering is performed. The hot gasses travel upward through cyclone preheater stages mostly driven by a fan, situated at the upper outlet of cyclone preheater. The chemical composition of the coal (ash, carbon, hydrogen, nitrogen, oxygen, and sulfur) is considered to be constant and can be computed by analysis of fuel. As alternative fuel can be used different kinds of waste, residues from distillation processes, used tires, carcass meal etc. The composition, water content, calorific values and burnability of alternative fuel can vary significantly. All these factors can significantly affect the quality of the hot meal and sintering condition of the clinker. To achieve a satisfactory quality of clinker, it is necessary to stabilize the quality of the hot meal entering the rotary kiln. This stabilization is mainly achieved by proper control of temperature within the precalciner [3]. To achieve complete combustion of primary fuel (pulverized coal) and alternative fuel inside precalciner adequate of air, i.e. O2 must be supplied which is

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performed by primary (pulverized coal/air duct) and tertiary airflow ducts [4]. With constant production rate, airflow is usually constant but it can be changed by tertiary air duct valve. Providing there is adequate oxygen present in the mixture, the following reaction describes the overall result of the oxidation (combustion) of carbon. C + O2 → CO2 + 394 kJ mol−1

(2)

In the case of incomplete combustion, not all of the carbon in the fuel will be oxidized to carbon dioxide, but some will be partially oxidized to carbon monoxide according to the following reaction 2C + O2 → 2CO + 221 kJ mol−1

(3)

Heat losse

“The main effect of CO production on the combustion process is to reduce the heat release from the fuel” [4]. Carbon monoxide is a significant pollution factor that, under certain circumstances, may even lead to explosive gas mixtures, so CO is usually legally regulated [3, 5]. Usually, whenever combustion conditions are worsening, the level of O2 will decrease and there are CO peaks. In the case of severe CO peaks, some part of the production process related to the fuel input could be automatically switched off. As a consequence of an excess of oxygen/air flows through the system are heat losses (see Fig. 2) [6]. Except this, in this situation, sulfur trioxide and nitrogen oxides are generated (see Fig. 3) [6].

Unburned fuel Air lack

λ optimal

Heat in exhaust gasses Air excess

λ - air / fuel ratio

Environment pollution

Fig. 2. Combustion energy losses versus air/fuel ratio

SO3, NOx

CO Air lack

λ optimal

Air excess

Fig. 3. Pollution versus air/fuel ratio

λ - air / fuel ratio

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2 Control Problem Formulation

Quality range

Poor quality

Temp. range

fCaO

To achieve a satisfactory quality of clinker, quality of hot meal leaving precalciner must be stabilized, and this stabilization can be mainly achieved by proper control of temperature within precalciner. The relation between temperature T within precalciner and degree of calcination f CaO as a fraction of CaO in the hot meal is shown in Fig. 4 [3]. From Fig. 4, it can be seen that above a certain level of temperature (860 °C) level of calcination (decarbonisation) is increased with rising temperature. Above a certain temperature, level of calcination saturates. Except this, at high temperature, there is a higher probability of capital cyclone blockages which can increase downtime and production loss. As a consequence, there is an optimal region of temperature within precalciner when the satisfactory level of calcination is achieved without wasting additional heat by increasing temperature within precalciner (see Fig. 4.). In this way, the temperature at the output of the precalciner is the controlled variable. In the optimal region of temperature, sensitivity of calcination is high so it is a challenging problem to keep the good and constant quality of degree of calcination. As mentioned in the introduction section, to achieve good combustion, proper mixing of fuel and O2 must be achieved. At Cement Factory Lukavac (CFL) airflow can be changed manually by changing openness of valve of the tertiary air duct, but in normal operation conditions, airflow is kept constant. To adapt to different kind of disturbances, such as change of raw meal mass flow and composition, change of alternative fuel mass flow and burnability etc., slightly excess of airflow (i.e. flow of O2 ) must be provided in normal operating conditions. At the same time, violation of optimum operational conditions, according to Fig. 2 and Fig. 3, should be as low as possible. Transport delay of the primary fuel (e.g. pulverized coal) is relatively small (of the order of few seconds) due to use of the pneumatic means of transportation. Also, the calorific value of the primary fuel is usually rather constant. These are reasons that the primary fuel is used as a manipulated variable. Transportation of the alternative fuel from an alternative fuel storage facility to the inlet of precalciner is usually performed by using long conveyor belts. As result transport delay of alternative fuel is of the order of a hundred seconds. Consequently, the mass flow of the alternative fuel is not used as a manipulated variable. To decrease the cost of the clinker production, it is desirable to increase the ratio (Rfuel ) of the mass flow of the alternative fuel over the mass flow of the primary fuel.

Poor energy efficiency

T Fig. 4. Degree of calcination versus temperature within a precalciner [3]

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In some cases, it can lead to lack of controllability due to primary fuel actuators mass flow low limits. For example, in a situation when the mass flow of primary fuel is close to the low limit, the calorific value of alternative fuel can suddenly increase to a big extent, which can lead to severe increase of temperature inside the precalciner (off course with transport delay close to the alternative fuel transport delay). This effect cannot be compensated by decreasing primary fuel mass flow since it is already at the low limit. To prevent this situation, it is desirable to keep Rfuel reasonably bellow maximum allowable level. This situation can also be mitigated by the installation of simple alternative fuel mass flow controller based on the two-level hysteresis controller which is activated in an emergency when the temperature reaches an undesirable high level. Besides, since the variability of parameters of the alternative fuel is much bigger than the variability of parameters of primary fuel, increasing of Rfuel makes the control task more difficult. Consequently, variations of temperature within the precalciner increase with increasing Rfuel .

3 Precalciner Model Identification and Controller Design Some of the earliest results in clinker production control using statistical methods are presented in [7]. One of the first steps towards introducing a systematic approach for the precalciner control is presented in [8]. In [8] structure of the general model of precalciner is presented. The main concepts of control of precalciner including adaptive and fuzzy logic control have been also presented. In [9] indirect adaptive neural control for precalcination in cement plants is presented. Precalciner control in the cement plant production using model predictive control is presented in [3]. In [3] the main results of temperature control within precalciner using a model predictive controller, which was implemented in the Expert Optimizer Advanced Process control platform of ABB, are presented. Numerical modelling of flow and transport processes in precalciner, based on Navier-Stokes equations and Lagrangean dynamics is presented in [10]. Numerical models that are presented are not suitable for controller design. For the purpose of the controller design a first principle model of cement precalcination system is developed in [11]. Model of the precalcination system is divided into two submodels: preheater submodel and precalciner submodel. The mathematical model consists of energy and mass flow first-order differential equations for solid, dust and gas. Simplified modelling of a precalciner in the form of in series-connected first-order transfer functions terms and PID parameterization based on the simplified model is presented in [12] and [13]. Due to the simplicity of the model, the tuning results could be used as initial values of parameters of the PID controller in a realistic situation. Identification of model of precalciner using Matlab System Identification Toolbox and fuzzy-based PID for precalciner temperature control is presented in [14]. The real-time response shows that the fuzzy-based PID outperforms the classical PID regarding Integral Square Error (ISE). Block diagram of the precalciner and transport model of pulverized coal and alternative fuel used for process identification in CFL are shown in Fig. 5. Inputs into the model are the mass flow of pulverized coal – F c (ton/h) (manipulated variable) and mass flow of alternative fuel – F a (ton/h) (disturbance). The outputs of the system are the temperature of exhaust gasses of the precalciner – T (controlled variable) and concentration of CO C CO (measured variable).

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To simplify controller design usual practice is to reduce a model of the system as much as possible while keeping the main characteristics of the system preserved. Besides, if we suppose that variation of the temperature, controlled by a well-designed controller, will be small, then the use of a linear model between inputs and temperature will be satisfactory. By using least square identification method, transfer function in ARX form between the mass flow of the pulverized coal and the temperature is of the form   H1 z −1 =

1 + a1

z −1

b10 + a2 z −2 + a3 z −3

(4)

while transfer function between the mass flow of alternative fuel and temperature is of the form   H2 z −1 =

b20 + b21 z −1 + b22 z −2 1 + a1 z −1 + a2 z −2 + a3 z −3

(5)

where z represents a forward shift operator. In the case of alternative fuel conveyor belt failure, the mass flow of alternative fuel into precalciner suddenly changes to 0 ton/h. In this situation model of the system changes which is modelled by using “alternative fuel start/stop switch” shown in Fig. 5. If the load of the conveyor during stop state is not changed, states of “alternative fuel transport model” must be preserved and used after the restart of the conveyor. By using real-time verification set of inputs (deviations around nominal point) to the process that is shown in Fig. 6a, real-time responses (deviations around nominal point) of the temperature and simulated response of the temperature are shown in Fig. 6b. Figure 6b shows satisfactory matching between the two responses (over 75%). alternative fuel conveyor start/stop switch stop

0 Fa

Fc

H2(z -1)

Alternative fuel transport model Pulverized coal transport model

start u

+ H1(z -1)

Regressors u(t), u(t-1), ... CCO(t-1), ...

+ Sigmoid function network

Fig. 5. Block diagram of precalciner and fuel transport models

T CCO

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Fig. 6. a) Real-time inputs of precalciner, b) Real-time and simulated responses of the precalciner temperature

Mathematical model of C CO as a function of F C is highly nonlinear. This model is approximated by using a sigmoid function network whose inputs are current and past values of u (output of the “Pulverized coal transport model” block in Fig. 5) and past values of C CO . This type of model is called “nonlinear ARX model” in Matlab – System Identification Toolbox. The Trust-region-reflective algorithm, as a numerical search method, has been used for iterative parameter estimation of the model. Block diagram of the mixed logical dynamical controller (MLDC) is shown in Fig. 7. In the automatic mode, and when the alternative fuel conveyor belt is operative, the command signal F c_cmd of pulverized coal is calculated by controllers which discrete transfer functions are H c1 (z−1 ) and H c2 (z−1 ) while controller which discrete transfer function H c3 (z−1 ) is inactive. These controllers are realized by using classical control algorithms (PID, Feedforward). In the case of alternative fuel conveyor belt failure, controller H c2 (z−1 ) is disabled while controller H c3 (z−1 ) is active. During this period, the states of the controller H c2 (z−1 ) are preserved. In the manual mode, pulverized coal command signal F c_cmd equals manual command signal F c_man while it is ensured that transition from automatic to manual mode and vice versa is performed bump-less. The controller output is calculated every 3 s. The MLDC is realized on the cRIO 9014 real-time controller using LabVIEW (G) programming language.

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0

stop

Hc3(z-1)

Fa

Hc2(z-1)

start Tref

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alternative fuel conveyor start/stop switch

stop

start -

+

Terror -

Hc1(z-1)

+

auto Fc_cmd

Fc_man

T

manual

Fig. 7. Block diagram of a mixed logical dynamical controller of precalcinator

In Fig. 7 F a is mass flow of alternative fuel, T and T ref are measured value and reference value of the temperature of output gasses of the precalciner respectively. The initial values of the parameters of the controllers was determined by minimization of ISE performance index     t dCCO 2 2 2 + rc · Fc d τ rT · Terror + rCO · (6) ISE = dτ 0 by using Genetic Algorithm as (global) optimization method. Fine tuning of the controllers parameters was performed manually during normal operation of the process. In (6) rT , rCO and rc are weighting factors, F c is mass flow of pulverized coal and C CO is CO concentration.

4 Real-Time Measurements and Responses Real-time responses of the precalciner temperature in the case of manual control and the case of automatic control are presented in Fig. 8 and Fig. 9, respectively. It is obvious that, regarding of variation of precalciner temperature, automatic control outperforms manual control. This improvement in performance is especially present in the situation of alternative fuel conveyor belt fault, that is when the mass flow of alternative fuel suddenly drops to 0 ton/h (at 170th minute in Fig. 8). In the case of manual control, the operator is usually occupied by other tasks so his reaction is often delayed or inadequate. The other critical situation is when alternative fuel conveyor belt restarts fully loaded (at 350th min in Fig. 8). The operator usually is not able to react properly and the temperature rises too high. The alternative fuel conveyor belt fault is shown in Fig. 9 (at around 135th min). In this case, the controller manages to keep the temperature deviations quite small. From Fig. 9 it is obvious that, if the ratio (Rfuel ) of the mass flow of alternative fuel over the mass flow of primary fuel is increasing, then the control problem becomes more challenging and deviations of the temperature are bigger. Since the controller outperforms manual control, it follows that this ratio, in the case of automatic control,

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can be bigger than in the case of manual control. In this way, by using the proposed controller, production costs can be decreased. Another benefit stems from the fact that, by keeping the temperature more stable and in the optimal range, the quality of the clinker is increased while the reduction of energy losses is achieved.

Fig. 8. Real-time signals from the process in the case of manual control

Fig. 9. Real-time signals from the process in the case of automatic control

5 Conclusions and Future Work The results presented show that MLDC outperforms manual control of precalciner temperature. The better stabilization of temperature can be achieved and consequently better and constant quality of raw meal calcination can be achieved. The ratio of the mass flow of alternative fuel over the mass flow of primary flue can be increased, which leads to a decrease in production costs. By keeping the temperature within the optimal range, energy losses are minimized while the frequency and intensity of CO peaks are reasonably low. The future works will include implementation of control algorithms to improve the other parts of the cement production process.

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References 1. Peray, K.E.: The Rotary Cement Kiln, 2nd edn. Chemical Publishing Co., Inc, New York (1986) 2. Deolalkar, S.P.: Handbook for Designing Cement Plants. BSP Publications, Hyderabad (2009) 3. Stadler, K.S., Wolf, B., Gallestey, E.: Precalciner control in the cement production process using MPC. In: IFAC Proceedings, vol. 40, pp. 201–206 (2007). Elsevier 4. Mullinger, F., Jenkins, B.: Industrial and Process Furnaces – Principles, Design and Operation. 1st edn. Elsevier, Oxford OX2 8DP, UK (2008) 5. Chen, Q.D.: The Principle and Application of New Dry Cement Technique. China Building Industry Press, Beijing (2004) 6. Zupancic, B.: Vodenje sistemov. Univerza v Ljubljani – Fakulteta za elektrotehniko, Zalozba FE in FRI, Ljubljana (2013) 7. Otomo, T., Nakagawa, T., Akaike, H.: Statistical approach to computer control of cement rotary kilns. Automatica 8(1), 35–48 (1972) 8. Griparis, M.K., Koumboulis, F.N., Vlachos, N.S., Marinos, I.: Precalcination in cement plants (system description and control trends). In: IFAC Manufacturing, Modelling, Management and Control, pp. 273–278, Patras, Greece (2000) 9. Koumboulis, F.N., Kouvakas, N.D.: Indirect adaptive neural control for precalcination in cement plants. Mathematica Comput. Simulat. 60, 325–334 (2002). Elsevier 10. Fidaros, D.K., Baxevanou, C.A., Dritselis, C.D., Vlachos, N.S.: Numerical modelling of flow and transport processes in a calciner for cement production. Power Technol. 171, 81–85 (2007). Elsevier 11. Wang, Z., Yuan, M., Wang, B.W., Wang, T.: Dynamic model of cement precalcination process. In: Proceedings of the 27th IASTED International Conference on Modeling, Identification, and Control, pp. 160–165, Innsbruck, Austria (2008) 12. Tsamatsoulis, D.C.: Simplified modelling of cement kiln precalciner. Int. J. Mater. 3, 69–73 (2016) 13. Tsamatsoulis, D.C., Zlatev, G.: PID parameterization of cement kiln precalciner based on simplified modelling. Int. J. Neural Netw. Adv. Appl. 3, 41–45 (2016) 14. Mohankrishna, P.B., Vigneshwaran, S., Padmadarshan, M.B., Brijet, Z.: Fuzzy based PID for calciner temperature control. Int. J. Pure Appl. Math. 119(12), 14563–14570 (2018)

Rolling on Affine Tangent Planes: Parallel Transport and the Associated Sub-Riemannian Problems Verlimir Jurdjevic(B) Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada [email protected]

Abstract. This paper addresses the natural optimal control problem associated with the kinematic equations generated by the rollings of symmetric Riemannian spaces on their affine tangent spaces at a fixed point. This optimal problem can be viewed as a certain left-invariant sub-Riemannian problem on a Lie group G associated with a three-step distribution F on the Lie algebra g of G. We will use the Maximum Principle of optimality to obtain the appropriate Hamiltonian, and then we will show that the corresponding Hamiltonian system of equations admits an isospectral representation, and hence, is completely integrable. As a byproduct, this representation reveals intriguing connections with mechanical tops, while at the same time, it sheds additional light on the discovery in [7] that the elastic curves, elasticate in Euler’s terminology, can be obtained entirely by the rolling sphere problems on spaces of constant curvature. Keywords: Rolling distributions · Riemannian and sub-Riemannian geodesics · Horizontal distributions · Grassmann manifolds · Lie group actions · Pontryagin maximum principle · Extremal curves

1

Introduction

Mathematical considerations of objects rolling on one another have a long and diverse history. The interest in the rolling phenomena very likely began with the wheel rolling on a flat surface without slipping and the discovery of the cycloid, the crown jewel of all curves. More generally, one dimensional rollings of curves rolling along other curves resulted in many beautiful curves, in which the cardioid and the astroid stand out as the particular cases. In the case of rolling surfaces, the sphere rolling on its affine plane was an inspirational example for Levi-Civita in his pioneering study of parallelism [13] where he introduced parallelism through rolling, and noted that parallelism depends not only on the points in question but also on the path that connects these points. Assuming that a surface S can be rolled without slipping along a curve α(t) on the tangent plane P at a point P1 on S, then during the rolling, c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 136–147, 2021. https://doi.org/10.1007/978-3-030-58653-9_13

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α(t), as the point of contact with the stationary plane P, traces a curve β(t) in P, called the development of α. In this study, Levi-Civita declared tangential directions u1 and u2 at the points P1 and P2 to be parallel along α(t) if they are parallel, in the ordinary sense, along the developed curve β(t). Over the time parallelism got absorbed into the theory of connections and the interest in the proximity to rolling waned to the point of becoming just a footnote in its history [4]. Rolling, on the other hand, evolved along different lines depending on the context, and ultimately led to two different kinds of rolling, intrinsic and extrinsic. The extrinsic rollings, like the ones in Levi-Civita’s book, are defined relative to an ambient Euclidean space in which the rolling manifold and its tangent affine space are embedded [10,12,14]. In contrast, intrinsic rollings are defined as the rollings that map parallel vectors along the rolled curve onto parallel vectors along the developed curve. In this definition, rolling is defined in terms of parallelism, rather than the other way around as suggested by the extrinsic rollings [2]. Leaving these subtle differences aside, we are left with a general question: what is the geometric significance of rolling? With this question in mind, we will focus on the fundamental case, a Riemannian manifold rolling on its affine tangent plane, and address the natural optimal control problem associated with the corresponding kinematic equations, with the hope that its geodesics may shed light on this question. We will show that the extremal equations associated with rollings of general symmetric Riemannian spaces admit an isospectral representation [6] and hence are completely integrable, that is, in principle, solvable by quadratures. We will also show the extremal equations reveal intriguing overlaps with mechanical tops, and, along the way, we will shed additional light on the discovery in [7] that the elastic curves on spaces of constant curvature can be extracted entirely through from the rolling sphere problems. To be more explicit, let us consider the quintessential case induced by the kinematic equations of an n-dimensional sphere S n rolling on its affine plane {x0 + v : (v, x0 ) = 0} at a point x0 ∈ Rn+1 . Here ( , ) denotes the standard inner product in Rn+1 . To begin with, x0 defines a closed subgroup K = {g ∈ SO(n + 1) : gx0 = x0 } in SO(n + 1). If k denotes the Lie algebra of K, let p denote its orthogonal complement in so(n + 1) relative to the trace metric − 12 T r(AB). Then the kinematic equations of S n rolling on the affine tangent plane at x0 are given by the following system of differential equations ⎧ dR ⎪ ⎪ ⎪ ⎨ dt = R(t)U (t), U (t) ∈ p R(t)(0) = I, v(0) = 0 , (1) ⎪ ⎪ dv ⎪ ⎩ = U (t)x0 , dt The decomposition so(n+1) = k⊕p can be be described more explicitly in terms of the wedge product in Rn+1 defined by v ∧ w := v ⊗ w − w ⊗ w = vw − wv  . An easy calculation shows that p = {u ∧ x0 : u ∈ Tx0 S n }, k = {u ∧ v : (u, x0 ) = (v, x0 ) = 0},

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and that − 12 Tr(AB) = A, B = (u, v) for any A = v ∧x0 , B = w∧x0 . Moreover, [u ∧ x0 , w ∧ v] = (u, v)w ∧ x0 − (u, w)v ∧ x0 ; [u ∧ x0 , v ∧ x0 ] = u ∧ v. Consequently, k and p satisfy the following Lie algebraic conditions (known as the Cartan decomposition) [p, p] ⊆ k, [p, k] ⊆ p, [k, k] ⊆ k.

(2)

Therefore, kinematic equations of the rolling sphere are defined entirely in terms of the Cartan decomposition. These conditions guarantee that any initial configuration (v0 , R0 ) can be connected to an arbitrary final configuration (v1 , R1 ) in Rn × SO(n + 1) by a trajectory (v(t), R(t)) in (1). The associated sub-Riemannian problem consists of finding a trajectory (v(t), R(t)) in (1) along which the energy T of transfer E = 0 ||u(t)||2 dt is minimal. It was shown in [7] that the Hamiltonian system associated with this sub-Riemannian problem is completely integrable, and shares its integrals of motion with a certain affine-quadratic Hamiltonian H=

1 Lk , Lk  + A, Lp , Lk ∈ p, Lp ∈ p, 2

where A is a fixed element in p. This connection to the affine-quadratic Hamiltonian revealed unexpected presence of elastic curves and mechanical tops among the sub-Riemannian geodesics [7]. In this paper we will show that these discoveries extend to the kinematic equations of arbitrary symmetric Riemannian spaces rolling on their affine tangent spaces with only minor modifications. These investigations originated in a joint project with K. H¨ uper, I. Markina and F. S. Leite and K. Krakowski aimed at bringing various notions of rolling under a common theoretic framework.

2

Symmetric Spaces and Their Kinematic Distributions

We will now assume g is an m-dimensional Lie algebra of a semisimple Lie group G with a Lie algebra decomposition g = p ⊕ k subject to the conditions g = p ⊕ k,

[p, k] = p, and k = [p, p].

(3)

Evidently k is a Lie subalgebra of g, while p is in general only a vector subspace of g. We will also assume that g is endowed with a non-degenerate quadratic form  ,  that is positive-definite on p, and additionally satisfies A, [B, C] = [A, B], C,

(4)

for all elements A, B, C of g. Finally, we assume that p and k are orthogonal relative to  , . Typically,  ,  is a suitable scalar product of the Killing form T r((adA)(adB)), A, B ∈ g. In addition, we will assume that there is an isometric mapping π : p → V , where V is a fixed vector space of the same dimension as

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p that is endowed with a Euclidean metric ( , ). Here, isometric means that (π(A), π(B)) = A, B, for all A, B in p. We will then consider the following optimal control problem in the configT uration space Q = V × G: Minimize 12 0 U (t), U (t) dt over the trajectories of dg dx = π(U (t)), = g(t)U t), (5) dt dt generated by a bounded and measurable curve U (t) in p, that satisfy fixed two point boundary conditions: x(0) = 0, x(T ) = x1 , g(0) = I, g(T ) = g1 . Note that the configuration space Q = V × G is a Lie group ( V is an abelian Lie group, hence Q is a product of Lie groups). The Lie algebra of Q consists of pairs (a, A) ∈ V × g with the Lie bracket [(a, A), (b, B)] = ([a, b], [A, B]) = (0, [A, B]). As such Q admits a global frame of left invariant vector fields Xi (x, g) = (π(ρ(Bi ), g(Bi )) defined by any basis B1 , . . . , Bm in g, where ρ(B) denotes the projection of B onto p. We will identify V with (V, 0) and g with (0, g) so that V × g is identified with V + g. Then any basis A1 , . . . , An in p induces a family of vector fields F = {Xi (x, g) = ei + gAi , ei = π(Ai ), i = 1, . . . , n} on Q such that (5) can be written as n n dx  dg  = = ui (t)ei , ui (t)g(t)(Ai ), (6) dt dt i=1 i=1 for some bounded and measurable control functions u1 (t), . . . , un (t). When n 2 A1 , . . . , An is an an orthonormal basis in p, then U (t), U (t) = i=1 ui (t). Thus the solutions of our optimal control problem can be equated with the subRiemannian geodesics associated with the metric U, U  on the Cartan space V × p in V × g. The distribution F generated by X1 , . . . , Xn is highly non-integrable in the language of distributions, In fact, [F, F] is the linear span of {gA : A ∈ k}. Hence, [[F, F], F] is the linear span of {gA; A ∈ p}. All of these observations follow from the Cartan conditions. But then F + [[F, F], F] includes the translations {a : a ∈ V }, which shows that Lie(F) is of full rank on V × G, and hence any two points of V × G can be connected by a trajectory of our control system. The distribution defined by F (or by equations (5)) will be called a rolling distribution for the following reasons. Let K be the connected Lie subgroup in G whose Lie algebra is k, and let M = G/K denote the space of left cosets {xK : x ∈ G}. Then G acts transitively on M by the left action φg (xK) = gxK. Let π : G → G/K be the natural projection: π(g) = φg (x0 ), and let k be the Lie algebra of K. In the case that (1) M is isometrically embedded in some N -dimensional Euclidean space EN (in which case its affine tangent space at x0 defined by Txaff M = x0 + Tx0 M 0 is also embedded in EN ); (2) The action of G ⊂ SO(EN ) on M is transitive, and extends linearly to an action on EN ; (3) The Euclidean metric on EN is G-invariant,

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then we have the following proposition [11]: = Txaff M Proposition 1. The kinematic equations for the rolling of M on M 0 along the curve x(t) = φg(t) (x0 ) = g(t)x0 ∈ M are: ⎧ dg ⎪ ⎪ = g(t)U (t), U (t) ∈ p ⎪ ⎨ dt (7) ⎪ ⎪ dv ⎪ ⎩ = de π(U (t)) dt Hence the kinematic equations associated with the rollings covered by this propoM . In sition can be seen as the rolling distributions on V × G with V = Txaff 0 particular, the following symmetric spaces conform to the above situation. Rolling Hyperboloids and Spheres. As demonstrated in [7], hyperboloids can be regarded as the “spheres” in the hyperbolic geometry, hence their rolling distributions can be obtained in the same manner as in the case of rolling Euclidean spheres. To deal with these two cases in a unified manner, let (x, y) denote the Euclidean metric when  = 1, and the Lorentzian metric when  = −1. More precisely, n  xi yi ,  = ±1. (8) (x, y) = x0 y0 +  i=1

Then Sn will denote the Euclidean sphere for  = 1, and the hyperboloid of one sheet {x ∈ Rn+1 : (x, x) = 1, x0 > 0} for  = −1. Additionally, SO (n + 1) will denote SO(n + 1) when  = 1, and SO0 (1, n), the connected component of SO(1, n) that contains the group identity, when  = −1. Then so (n + 1) will denote the Lie algebra of SO (n + 1). Analogous to the Euclidean case, a ⊗ b, a, b ∈ Rn+1 , will denote the rank-one matrix defined by (a⊗ b)x = (b, x) a, x ∈ Rn+1 , and then a∧ b will denote the matrix a⊗ b−b⊗ a. Since ((a ∧ b)x, y) + (x, (a ∧ b)y) = 0, a ∧ b belongs to so (n + 1) for any a, b in Rn+1 . M = {x0 +v : Consider now the rollings of Sn on the affine tangent plane Txaff 0 v ∈ Rn+1 , (v, x0 ) = 0}. Let K denote the subgroup of SO (n + 1) that leaves x0 fixed, and let k denote its Lie algebra. It is easy to show that the Lie algebra k , and its orthogonal complement p in so (n + 1) are given by the following expressions: p = {A = u ∧ x0 : u ∈ Rn+1 , (u, x0 ) = 0}, k = {B = v ∧ w : v ∈ R

n+1

,w ∈ R

n+1

, (v, x0 ) = (w, x0 ) = 0},

(9) (10)

In the canonical case x0 = e0 , the preceding matrices can be also written as



 0 −u∗ 0 0 A= ,B = , u, v, w in Rn . u 0 0v∧w

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By an argument formally identical to the one used in the Euclidean case, one can show that the rolling distribution associated with Sn rolling on its affine tangent plane V = {x0 + v : v ∈ Rn+1 , (v, x0 ) = 0} is given by the following equations dv dR = U (t)x0 = u(t), = RU (t), U (t) = u(t) ∧ x0 , (u(t), x0 ) = 0. dt dt

(11)

See [9] or [7]. Rolling Compact Lie Groups. Let H be a compact Lie group, such as, for instance, SO(n), SU (n), or Sp(n), and let h denote its Lie algebra. Then H is a Riemannian symmetric space, with its metric induced by a bi-invariant quadratic form  ,  on h. This implies that both the right and the left actions of H on itself are isometries. Hence, the action ((g, h), m) → gmh−1

(12)

of G = H × H on H is an isometry for each (g, h) ∈ G. We will think of H as the orbit of H × H through the group identity e, that is, we will identify H with the quotient (H × H)/K, where K = {(h, h) : h ∈ H} is the isotropy subgroup of the group identity e. This realization induces an orthogonal splitting p ⊕ k of the Lie algebra h × h of H × H with k = {(A, A) : A ∈ h}, p = {(A, −A) : A ∈ h}

(13)

that will be relevant for the rolling equations obtained below. In the particular cases that H = SO(n), SU (n), Sp(n), H can be embedded in the vector space V of n × n matrices with real, complex, or quaternionic entries, depending upon the case, endowed with a Euclidean metric M1 , M2  = T r(M1 M2∗ ). This metric is G invariant for the extended action G to V . Hence M = H satisfies the conditions of Proposition 1, in which case equations (7) take on the following form dx dg 1 dh 1 = U (t), = g(t)U (t), = − h(t)U (t), U (t) ∈ h. dt dt 2 dt 2 So in this case, V = h and G = H × H. 2.1

(14)

Rolling Oriented Grassmannians

Recall that a Grassmannian Grnk is the set of all k-dimensional oriented vector subspaces of an n-dimensional vector space Vn . We will think of it as a subset of the orthogonal group O(n) by identifying each k-dimensional vector space W in Grnk with the orthogonal reflection RW defined by x if x ∈ W, RW (x) = −x, if x ∈ W ⊥ .

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Orthogonal reflections satisfy g 2 = Id, or g ∗ = g, hence g belongs to the vector space V of symmetric n × n matrices. The orthogonal group acts on the Grassmann manifolds Grnk under the action (g, RW ) → gRW g ∗ ,

g ∈ Gn .

(15)

It is easy to verify that this action is transitive. Therefore, Grnk can be realized as the orbit of O(n) through the space W spanned by the standard basis e1 , . . . , ek . Then the space of oriented Grasmannians Grkn + is the orbit through W generated by the action of SO(n), that is Grkn + = {gRW g ∗ : g ∈ SO(n)}. It then follows that Grkn + = SO(n)/K where



 A 0 K = g ∈ SO(n), g = , A ∈ O(k), C ∈ O(n − k) 0 C

 I 0 is the isotropy group of RW = In,n−k = k . Then 0 −In−k 

A1 0 ∈ so(n) : A∗1 = −A1 , A∗2 = −A2 } k={ 0 A2 is the Lie algebra of K, and 

0 B : B ∈ Mk,n−k (V )}. p={ −B ∗ 0

(16)

(17)

is its orthogonal complement relative to the trace form. The subspaces p and k satisfy Cartan relations g = p ⊕ k,

[p, k] ⊆ p,

[p, p] = k,

[k, k] ⊆ k.

It is well known that there is a left-invariant metric on Grkn , induced by the trace form, relative to which G = SO(n) is the isometry group. The action of G on Grkn + extends to V with (g, v) → gvg −1 , for v ∈ V . Therefore, the rolling distribution associated with rollings of Grassmannians Grkn + on the affine tangent spaces at a point x0 = Ik,n−k conforms to the conditions of Proposition 1, and its kinematic equations (7) are given by ⎧

 dg 0 u(t) ⎪ ⎪ = g(t)U (t), U (t) = ∈p ⎪ ⎪ −u (t) 0 ⎨ dt (18)

 ⎪ ⎪ dv 0 u ⎪ ⎪ = [U (t), Ik,n−k ] = 2 ∗ ⎩ u 0 dt Additional details about these three cases can be found in [9].

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143

Hamiltonians and the Extremal Curves

We now return to the equations (5) and the associated optimal control problem. Our immediate aim is to use the Maximum Principle to obtain the equations for the extremal curves, that is, curves in the cotangent bundle T ∗ Q of the configuration space Q that project onto the sub-Riemannian geodesics. To take advantage of the left-invariant symmetries, we will regard the cotangent bundle as the product Q × q∗ , where q∗ is the dual of the Lie algebra q of Q. Thus, points ξ ∈ Tq∗ Q will be identified with (q, ), ∈ q∗ via the formula ξ(qX) = (X), X ∈ q. The Maximum Principle tells us that the extremal curves can be either normal or abnormal. Normal extremals are the integral curves of the Hamiltonian 1 2 H (ξ), 2 i=1 i n

H(ξ) =

(19)

where Hi (ξ) are the Hamiltonian lifts Hi (ξ) = ξ(Xi (q) = (ei + Ai ), ξ ∈ Tq∗ Q, associated with vector fields X1 , . . . , Xn that define nequation (6). The abnormal extremals are the integral curves ξ(t) of H(ξ) = i=1 ui (t)Hi (ξ) subject to the constraints that Hi (ξ(t)) = 0, i = 1, . . . , n. Based on the results in [7] it is very likely that every geodesic is the projection of a normal extremal curve, in which case the abnormal extremals could be ignored. In any event we will assume that to be the case, and will proceed with the normal extremals only. Since our control system (6) is left-invariant, each Hamiltonian lift Hi is a function on the dual q∗ of q. This implies that each Hi = ξ(Xi (q) = (ei + Ai ) becomes a linear function in q∗ . Then the normal extremals (q(t), (t)) are the solutions of n  dq d = q(t) = −ad∗ dH( (t))( (t)) Hi ( (t)), dt dt i=1

(20)

Equations (20) are in “quadrature” form thanks to the left-invariant symmetries: the extremal controls ui = Hi are determined by the pivotal equation d = −ad∗ dH( (t))( (t). dt

(21)

Once the solutions of (21) are known, the remaining equation for the projection q(t) on the configuration space Q is determined by the extremal control, and the problem is reduced n to solving a fixed, time varying system of differential equations dq = q(t) i=1 Hi ( (t)). dt In our specific situation, however, there are additional symmetries that simplify the solutions; to begin with, our optimal problem is bi-invariant over V , since V is an abelian group. To reveal the practical significance behind this remark, we need to take a more detailed look at (21). First, note that q∗ = V ∗ × g∗ , where V ∗ = { ∈ q∗ : (X) = 0, X ∈ g}. Hence each ∈ q∗ can be written as = 1 + 2 , 1 ∈ V ∗ and 2 ∈ g∗ . It follows

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that (x˙ + X) = 1 (x) ˙ + 2 (X), for each tangent vector (x, ˙ X) ∈ V × g. In this ( ) = (e ) + (A ), and hence, the differential dH of H is given notation, H i 1 i 2 i n ( (e ) + (A ))(e + A ). The extremal equation (20), when by dH = 2 i i i i=1 1 i evaluated in the direction x˙ + X becomes n  d 2 d 1 (x) ˙ + (X) = −( 1 + 2 )[dH, (x, ˙ X)] = − (l1 (ei ) + 2 (Ai ) 2 [Ai , X], dt dt i=1

for each (x, ˙ X) ∈ V × g. It follows that 1 (t) is constant and that n  d 2 (X) = − (l1 (ei ) + 2 (Ai ) 2 [Ai , X], X ∈ g. dt i=1

(22)

The fact that 1 is constant is a direct result of the bi-invariance of our system relative to V . To uncover other symmetries it will be convenient to identify q∗ with q via the natural quadratic forms on each of the factors, and then recast the preceding equations on the tangent bundle Q × q. More precisely, each 1 in V ∗ will be ˙ = (p, x), ˙ x˙ ∈ V , in which case, p will be identified identified with p ∈ V via 1 (x) with p = (p1 , . . . , pn ), where p1 , . . . , pn denote the coordinates of p relative to the basis e1 , . . . , en . Similarly, ∈ g∗ will be identified with L ∈ g via the formula (X) = L, X, X ∈ g. But then each L ∈ g can be decomposed into the sum L = P + K with P ∈ p and K ∈ k. With these identifications at our disposal, Hi (ξ) = (ei + Ai ) = 1 (ei ) + (Ai ) = pi + Pi , since Ai ∈ p. Therefore the Hamiltonian H is given 2 n ∂H nby H = n 1 ∂H 2 (p + P ) . But then dH = e + A is equal to i i=1 i i=1 ∂pi i i=1 (pi + 2 ∂Pi i Pi )ei + (pi + Pi )Ai . Then the extremal Eq. (22) satisfy 

n n   dL , X = −L, [ (pi + Pi )Ai , X] = −[L, (pi + Pi )Ai ], X dt i=1 i=1

(23)

for any vector field X on Q. Since X is arbitrary, n  dL = [ (pi + Pi )Ai , L] dt i=1

(24)

Recall now that A1 , . . . , An is an orthonormal basis in p, and that V and p are the coordinate vector vector spaces of the same dimension. Hence P1 , . . . , Pn is  n of P relative to this basis. We will identify P ∈ p with i=1 Pi Ai , and write n i=1 pi Ai = A. Since each pi is constant along an extremal curve, A is a constant element in p, and each A in p can be attained by some p1 , . . . , pn . All this shows that (24) can be written as dL dt = [A + P, L], or dP dK = [A + P, K], = [A, P ], dt dt

(25)

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since A + P ∈ p, [P, K] ∈ p, and [A, P ] ∈ k. Coupled with dx  = (pi + Pi )ei , dt i=1 n

(26)

in V , these equations constitute the extremal equations for our sub-Riemannian rolling problem. Equations (25) may  be regarded as the Poisson equations on g generated by n the Hamiltonian H = 12 i=1 (pi +Pi )2 parametrized by the constants p1 , . . . , pn . Proposition 2. Equations dK dP = [A + P, K], = [A, P ], dt dt are completely integrable on each adjoint orbit {g(P0 + K0 )g −1 , g ∈ G} in g. Consequently, the complete system dx  dP dK dg = = g(A + P ), = [A + P, K], = [A, P ] (pi + Pi )ei , dt dt dt dt i=1 n

(27)

is completely integrable on the manifold p = constant, g(P + K)g −1 = constant. Proof. Equations (25) are of the same form as the equations for the rolling spheres in [7], and hence admit an analogous isospectral representation 1 dLλ = [Mλ , Lλ ], Mλ = (A + P ), Lλ = P + λK + (1 − λ2 )A. dt λ

(28)

The proof is straightforward and will be omitted. Integrability then follows from the general theory of isospectral systems ( see the discussion in [6]). Consequently, each sub-Riemannian rolling system is completely integrable (in the sense of Liouville). Equations (28) reveal a curious and mysterious connection with the AffineQuadratic Hamiltonian 1 ˆ H(L) = Lk , Lk  + A, Lp , L = Lk + Lp 2

(29)

on g where and Lp and Lk are the factors in the Cartan decomposition g = p ⊕ k. Its Hamiltonian equations are given by dLp dLk = [A, Lp ], = [Lk , Lp ] + [A, Lk ] = [A − Lp , Lk ] dt dt

(30)

This case is called canonical in [6]. Apart from the notations, the canonical equations are the same as our equations (25). To see this, let Lp = −P, and Lk = −K, where P and K are the variables in (25). Then, dK dP dLk dLp =− = −[A, P ] = [A, Lp ], =− = −[A + P, K] = [A − Lp , Lk ]. dt dt dt dt

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Under this correspondence, Lp +λLk +(λ2 −1)A = −(P +λK +(1−λ2 )A), hence the two Hamiltonian systems. and H = 12 ||Lk ||2 + A, Lk  share the isospectral integrals of motion (as originally stated in [7]). The above observation suggests mechanical interpretations for the rolling geodesics, since the affine-quadratic Hamiltonian systems are traditionally associated with mechanical tops [15]. For the rolling distributions induced by the rolling spheres on spaces of constant curvature, the affine-quadratic Hamiltonian H = 12 ||Lk ||2 + A, Lk  on the isometry groups SO(n + 1), SO(1, n) and SE(n) generates the elastic curves on M = G/K [6]. The fact that the equations for ˆ equathe rolling spheres, equations (25), and the corresponding equations for H, tions (30), are essentially the same reveals the presence of elastic curves in the equations associated with the rolling spheres [7]. Beyond the spaces of constant curvature, the affine-quadratic Hamiltonian is no longer associated with elastic curves, and the connection with the elastic curves seems to disappear, which then raises the question about the geometric significance of the rolling geodesics for the geometry of symmetric Riemannian spaces. At this point, there are only partial answers, and the general question remains an open problem. Acknowledgement. The author is grateful to the referees for the helpful comments on the earlier version of the paper.

References 1. Brockett, R.W., Dai, L.: Non-holonomic kinematics and the role of elliptic functions in constructive controllability. Motion Planning, pp. 1–22. Kluwer Academic, Dordrecht (1993) 2. Chitour, Y., Kokkonen, P.: Rolling Manifolds: Intrinsic Formulation and Controllability (2011). arXiv:1011.2925v2 3. Chitour, Y., Molina, M.G., Kokkonen, P.: The rolling problem: overview and challenges. Geometric Control theory and Sub-Riemannian Geometry. INdAM Series 5, pp. 103–123. Springer, New York (2014) 4. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978) 5. Jurdjevic, V.: Geometric Control Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, New York (1997) 6. Jurdjevic, V.: Optimal Control and Geometry: Integrable Systems. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016) 7. Jurdjevic, V., Zimmermann, J.: Rolling sphere problems on spaces of constant curvature. In: Mathematical Proceedings of Cambridge Philosophical Society, pp. 729–747 (2008) 8. Jurdjevic, V.: The geometry of the Ball-Plate Problem. Arch. Rational Anal. 124, 305–328 (1993) 9. H¨ uper, K., Silva, L.F.: On the geometry of rolling and interpolation curves on S n , SOn , and Grassmann manifolds. J. Dyn. Control Syst. 13(4), 467–502 (2007) 10. H¨ uper, K.M., Kleinsteuber, M., Silva, L.F.: Rolling Stiefel manifolds. Int. J. Syst. Sci. 39(9), 881–887 (2008)

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11. Krakowski, K.A., Silva, L.F.: Controllability of Rolling Symmetric Spaces. In: Proceedings of International Conference on Automatic Control and Soft Computing, 4–6 June. IEEE Explorer, Azores (2018) 12. Krakowski, K.A., Machado, L., Silva, L.F.: Rolling symmetric spaces. In: Geometric Science of Information, pp. 550–557. Springer (2015) 13. Levi-Civita, T.: The Absolute Differential Calculus. Blackie and Son Ltd., London (1929) 14. Sharpe, R.W.: Differential geometry. GTM, 166. Springer, New York (1997) 15. Reyman, A.G., Semenov-Tian Shansky, M.A.: Group-theoretic methods in the theory of finite-dimensional integrable systems. In: Arnold, A.I., Novikov, S.P. (eds.) Encyclopaedia of Mathematical Sciences. Springer, Heidelberg (1994)

The Numerical Control of the Motion of a Passive Particle in a Point Vortex Flow Carlos Balsa1(B) and S´ılvio Gama2 1

2

Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Bragan¸ca, Portugal [email protected] CMUP - Centro de Matem´ atica da Universidade do Porto, Porto, Portugal [email protected] Abstract. This work reports numerical explorations in the advection of one passive tracer by point vortices living in the unbounded plane. The main objective is to find the energy-optimal displacement of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The direct formulation of the corresponding control problems is presented for the case of N = 1, N = 2, N = 3 and N = 4 vortices. The restrictions are due to (i) the ordinary differential equations that govern the displacement of the passive particle around the point vortices, (ii) the available time T to go from the initial position z0 to the final destination zf , and (iii) the maximum absolute value umax that is imposed on the control variables. The resulting optimization problems are solved numerically. The numerical results show the existence of nearly/quasi-optimal controls.

Keywords: Point Vortex Numerical optimization

1

· Passive tracer · Control problem ·

Introduction

Point vortex can be seen as a mathematical model used to describe the dynamics of vortex-dominated flows [1]. This model is based on a low dimensional description of the flow features [2]. This work is concerned with the dynamics of a passive particle advected by a two-dimensional point vortex flow. A passive particle is small enough not to perturb the velocity field, but also large enough not to perform a Brownian motion. Particles of this type are the tracers used for flow visualization in experiments in fluid mechanics [3]. We consider also that passive particles have the same density of the fluid in which they are immersed. We want to drive a passive particle from an initial starting point to a final terminal point, both given a priori, in a prescribed finite time. The flow is originated by the displacement of a certain number, say N, of point vortices. This problem has some similarities with the fish-like locomotion problem [4,5]. Here, the vortex dynamics is governed by N point vortices and the control is due c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 148–158, 2021. https://doi.org/10.1007/978-3-030-58653-9_14

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to the possibility of impulsion in any direction of the two dimensional plane. Of course we want to minimize the total amount of energy spent in the impulsion. This issue can also be seen as part of the set of general open control problems identified in Protas [6]. There are two main classes of control methods: direct and indirect. Generically speaking, the direct approach consists of first discretizing and then optimizing, while the indirect approach, first optimizes and then discretizes. Direct methods discretize the problem with respect to time in order to obtain a nonlinear programming problem (NLP) that can be solved by an optimization method. These methods are generally handy for singular or constrained arcs of the trajectory, but their accuracy can be affected by the discretization [7,8]. Indirect methods use the Pontryagin’s Maximum Principle to derive optimal conditions, where it is necessary to maximize the Hamiltonian, which can be achieved by collocation or shooting methods [9]. Indirect methods are fast and accurate, but they are also sensitive to the starting guesses of the adjoint problems. The approach followed in this work is the direct one. The displacement of the passive particle is transformed in a control problem. The time disposable to preform the displacement is divided in a fixed number n of subintervals, where the control variables are constant. The discretized problem is solved numerically by an optimization method. In each subinterval, the vortex dynamics is integrated by the fourth order Runge-Kutta numerical scheme. The first tests, presented in the previous work [10], point to the viability of this approach. Comparing with the indirect approach, where the derivation of optimal conditions needs some simplifying assumptions, our approach enables to work with more realistic problems as, for instance, a large number of vortices. One of these simplifications is to consider that the vortex dynamics is induced just by two point vortices [11]. In the present study, the passive particle moves in a two dimensional flow whose dynamics is given, at any time interval, by N point vortices. Four different problems are considered. Each one corresponds to a different value of N , ranging from 1 up to 4. The nonlinear programming problem, corresponding to each of these cases, is formulated and solved numerically by using the Matlab Optimization Toolbox [12]. In Sect. 2, the equations that give the motion of passive particles advected by N point vortices in the infinite real plane are introduced, followed by the formulation of the control problems. Section 3 is devoted to the numerical solutions of these control problems. This paper ends in Sect. 4 with the presentation of some final considerations.

2

Statement of the Point Vortex Control Problem

The two-dimensional incompressible Euler equations are ∂u + (u · ∇) u = −∇p , ∂t

(1)

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and ∇ · u = 0,

(2)

plus initial and boundary conditions, wherein u=(u1 , u2 ) , ui =ui (X, t) (i=1, 2) is the two-dimensional incompressible velocity field, X = (x, y) ∈ R2 the space coordinates, t is the time variable, ∇ = (∂x , ∂y ) is the gradient, and p is the pressure. Due to the incompressible condition in two spatial dimensions, i.e. ∇ · u = 0, one can express the velocity field u in terms of the so-called stream-function Ψ : u = (u1 , u2 ) = (∂y Ψ, −∂x Ψ ) .

(3)

Introducing the vorticity vector (consider u ∈ R3 , where the third component is zero): w = ∇ × u = (0, 0, ∂x u2 − ∂y u1 ) , the two-dimensional scalar vorticity, given by ω = (w)3 = ∂x u2 − ∂y u1 ,

(4)

is linked with the stream-function through the Poisson equation ∇2 Ψ = −ω .

(5)

Taking the curl in both sides of Eq. (1) leads us to the vorticity formalism of the Euler equations [13] ∂t ω + (u · ∇) ω ≡

dω = 0. dt

(6)

Equation (6) enables us to get the time evolution of the vorticity. Once we have the vorticity, the stream-function is computed through the solution of the Poisson Eq. (5). In turn, the velocity field is obtained by Eq. (3). This is the typical procedure used to solve the Euler equation. Effectively, it can be shown (see [14]) that the solution of the Poisson Eq. (5), in the whole plane, is given by  1 Ψ= ln |X − X  | ω(X  ) dX  , (7) 2π R2 and that u is equal to the convolution between the kernel K(·, ·) and the given vorticity: u = (∂y Ψ, −∂x Ψ ) = K ∗ ω , (8)   y 1 x , − . (9) 2π x2 + y 2 x2 + y 2 The point vortices correspond to the particular case where the vorticity is given by the weighted sum

with

K(x, y) =

ω = ω (t, X) =

N  α=1

kα δ (X − Xα (t)) ,

(10)

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where N is the number of vortices and δ(·) is the δ−Dirac function. The quantity kα is the circulation of the vortex α (α = 1, 2, · · · , N ) that, at time t, is located in Xα (t) = (xα (t), yα (t)). The vortex equations are N k y −y dxα = − β=1 2πβ αr2 β , β=α αβ dt N kβ xα −xβ dyα = , (11) β=1 2 rαβ β=α 2π dt 2 = (xα − xβ )2 + (yα − yβ )2 . together by appropriate initial conditions. Here, rαβ The Eqs. (11) can be written using the complex variables N 1  kβ dzα∗ = dt 2πi β=1 zα − zβ

(α = 1, 2, . . . , N ) ,

(12)

β=α

  where zα = xα + yα i i 2 = −1 and zα∗ is its complex conjugate. A passive particle is by definition a point vortex with zero circulation. Therefore, the dynamics of a system with P passive particles advected by a set of N point vortices is given by (12) together with the equations for the passive particles N 1  kβ dzα∗ = dt 2πi zα − zβ

(α = N + 1, N + 2, . . . , N + P ) ,

(13)

β=1

with the respective initial conditions. The abstract formulation of the problem we want to address can be stated as follows [6]. Consider the state vector (M = N + P ) T

X(t) = [x1 (t) y1 (t) x2 (t) y2 (t) · · · xM (t) yM (t)] ∈ R2M ,

(14)

and the evolution systems (12)–(13) concisely written as dX = f(X) , dt

X(0) = X0 ,

(15)

where f : R2M → R2M is the function describing the advection velocities of the vortices and particles due to the induction of all the vortices. In this work, we will consider P = 1. If we denote U : [0, +∞[→ Rm a time dependent input control with m degree of freedom, the system (15) can be upgraded to dX = f(X) + b(X) U(t) , dt

X(0) = X0 .

(16)

where b(X): Rm → R2M is the control operator describing the action of the control, U(·) , in the system dynamics. The control problem can then be stated as follows: given the initial state X0 of the system and the prescribed XT = X(T ) terminal state, determine U(·) that drives X0 to XT during the time interval [0, T ] and minimizing, for instance, T , or some cost function, etc.

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Solving the Control Problem by Direct Approach

The numerical approach used to solve the control problem presented above is based on a direct approach. It consists in discretizing the problem and solving it using an optimization method. In this way, the control function U(·) is replaced by n control variables u0 , u1 , · · · , un−1 . The calculations were performed in Matlab, thanks to the nonlinear programming solver fmincon. This solver provides some constrained optimization algorithms, such as the Interior Point or the Active-Set (see [12]). We begin, in Subsect. 3.1, by solving this problem for the case of a single passive particle in a single vortex flow and, after that, in Sect. 3.2, we address the cases involving up to four vortices. We would like to point out that the cases N = 2 and N = 3 correspond to integrable point vortex dynamics, whereas N = 4 (or higher) corresponds (generically) to chaotic point vortex dynamics [3,15]. 3.1

Flow Created by One Single Vortex

From a practical point of view, the control problem introduced in Sect. 2 can be illustrated for the case of one single passive particle (P = 1) which moves thanks to the presence of a single vortex (N = 1) as follow: P Minimize : subject to :

T 0

2

|u(t)| dt

1 z˙  = 2πi × z(0) = z0 z(T ) = zf |u| ≤ umax

k z

+u

with u ∈ C, and z0 , zf ∈ C, T > 0 and umax > 0 given. In this optimization problem, the objective (cost) function represents, for instance, the energy spent by the control u(·) to drive the passive particle from the starting point z0 to the final point zf . The first restriction corresponds to the state equation that governs the position z of the particle as a function of the time. The control function u is introduced in this equation in order to move the particle from z0 to zf in a fixed value of time T > 0 . The points z0 and zf are previously defined, as well as the time T available to reach the destination zf . Additionally, in the fourth restriction, we impose that the absolute value of the control is not greater than a prescribed value umax .

The Numerical Control of the Motion of a Passive Particle

153

To address this problem, we proceed to the discretization of the control function. We replace u(·) by n (discrete) variables defined as (t0 = 0, tn = T ) u(t) = u0 u(t) = u1 u(t) = u2

if t0 ≤ t < t1 , if t1 ≤ t < t2 , if t2 ≤ t < t3 , .. .

u(t) = un−1 if tn−1 ≤ t ≤ tn . Thus, each variable ui (i = 1, 2, · · · , n) corresponds to a constant value of the control exercised in the sub-interval [ti−1 , ti ) . All of these subintervals have amplitudes equal to Δt = (tn − t0 ) /n . The discretization of the objective function by the rectangle method lead to the approximation T



2 2 2 2 2 |u| dt ≈ Δt |u0 | + |u1 | + |u2 | + · · · + |un−1 | ≡ fn .

(17)

0

The control problem (P) is then replaced by its discretized version: DPn Minimize : Δt

n−1 k=0

2

|uk |

subject to : k 1 z˙  = 2πi z + u0 , z(0) = z0 , |u0 | ≤ umax , t0 ≤ t < t1 k 1  z˙ = 2πi z + u1 , z(t1 ) = zt1 , |u2 | ≤ umax , t1 ≤ t < t2 .. .

k 1 z˙  = 2πi z + un−1 , z(tn−1 ) = ztn−1 , |un−1 | ≤ umax , tn−1 ≤ t < tn z(tn ) = zf

In the numerical results obtained with the optimization problem DPn , where n is the number of control variables, ranging from 1 to 4, it is considered one single vortex with circulation k = 10 , located at the origin, and it is sought a vector u ∈ Cn that drives the passive particle from z0 = −1 − i to zf = 2 + 2 i , in exactly T = 10 (natural) units of time, and minimizes the objective function defined by (17). This optimization problem is solved numerically with the Interior Point optimization algorithm [16], included in the fmincon Matlab solver. The results obtained with one vortex are presented in Table 1. We can observe that value of the objective function decreases with the number of control variables. A significant reduction is observed, when going from n = 1 to n = 2, but for n ≥ 3 the variations are small. The Fig. 1 a) depicts the trajectories corresponding to each optimal control, presented in Table 1. The trajectories obtained with n = 3 and 4 are close to being overlapping. 3.2

Flow Created by Several Vortices

In this Subsection, we address the problem of a single passive particle (P = 1) displaced by multiple vortices (N ). As before, we want to find

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u = [u0 , u1 , . . . , un−1 ] ∈ Cn that drags the particle from z0 = −1 − i to the final destination zf = 2 + 2 i . We consider also the time of displacement T = 10 , and equal circulation for the all vortices ki = 10 , for i = 2, 3, 4 . To solve the different optimization problems, we used the fmincon Matlab solver. This solver offer different optimization methods, e.g.. the Interior Points, Sequential Quadratic Problem (SQP), and Trust Region (see [12]). We tried all these methods and came to the conclusion that the best results where given by the Interior Points optimization method. Two vortices (N = 2) and one particle (P = 1). In the two vortices and one particle problem, the vortices positions are given by [17]:

 1 (k1 z1 (0) + k2 z2 (0)) + (z1 (0) − z2 (0)) k2 ei Ω t  z1 (t) = k1 +k 2 (18) 1 (k1 z1 (0) + k2 z2 (0)) + (z2 (0) − z1 (0)) k1 ei Ω t z2 (t) = k1 +k 2 where Ω = k1+k2 2πD 2 , D = |z2 (0) − z1 (0)| , and z1 (0) and z2 (0) are the initial position. In this case, the initial vortex positions are z1 (0) = 0.5 + 0.5 i , and z2 (0) = 1.5 − 0.5 i . The passive particle position is given by the equation   k1 k2 1 ∗ + z˙ = + u, (19) 2πi z − z1 (t) z − z2 (t) with the given initial condition z(0) = z0 . The results obtained with two vortex are presented in Table 1 and Fig. 1 b). Three vortices (N = 3) and one particle (P = 1). In the problem with three vortices (N = 3) and one particle (P = 1), the vortices equations are ⎧

 k2 k3 1 ∗ ⎪ z ˙ = + ⎪ 2πi z1 −z2 z1 −z3  ⎪ ⎨ 1 k1 1 ∗ 3 (20) z˙2 = 2πi z2 −z1 + z2k−z 3 ⎪

⎪ ⎪ k k 1 ∗ 1 2 ⎩ z˙3 = 2πi z3 −z1 + z3 −z2 with the initial conditions z1 (0) = 0.5 + 0.5 i , z2 (0) = 1.5 − 0.5 i , and z3 (0) = 1 + i . The passive particle equation is   k1 1 k2 k3 + + + u, (21) z˙ ∗ = 2πi z − z1 z − z2 z − z3 with the initial condition z(0) = z0 = −1 − i . The results obtained with three vortex are presented in Table 1 and Fig. 1 c). Four vortices (N = 4) and one particle (P = 1). We address now the case of four vortices (N = 4) and one particle (P = 1). This is an interesting case, because the dynamics of the vortices is nonintegrable [3]. (The cases with five or more vortices will be the subject of a further study.) For N = 4, the vortices equations are

The Numerical Control of the Motion of a Passive Particle

155

Table 1. Optimal controls and objective function values obtained with N point vortices. n u0

u1

u2

u3

fn

N =1 1 0.189 − 0.179i 2 0.012 − 0.249i

0.68 0.109 − 0.015i

0.37

3 −0.039 − 0.257i 0.122 − 0.077i

0.077 + 0.000i

4 −0.081 − 0.248i 0.106 − 0.130i

0.110 − 0.047i

0.31 0.077 + 0.002i

0.29

N =2 1 −0.203 + 0.206i

0.84

2 −0.214 + 0.195i −0.192 + 0.215i

0.84

3 −0.117 − 0.138i −0.033 + 0.013i 0.087 − 0.062i

0.15

4 0.096 − 0.128i

−0.041 + 0.013i −0.013 + 0.014i 0.110 − 0.079i

0.12

2 0.651 − 0.539i

0.601 − 2.789i

44.26

3 0.054 − 0.151i

−0.384 + 0.190i −0.140 − 2.175i

4 0.008 + 0.033i

−0.218 + 0.120i 0.345 + 0.583i

N =3 1 0.127 − 2.254i

50.10 16.53 −0.289 − 1.863i 10.19

N =4 1 −3.928 + 1.387i

173.48

2 −1.237 − 1.288i −1.107 + 3.285i

79.02

3 −0.845 − 1.851i −1.037 − 0.460i −1.129 − 3.233i

57.18

4 −0.971 + 1.260i 2.042 − 1.957i

⎧ ⎪ z˙1∗ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ z˙2∗ = ⎪ z˙3∗ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z˙4∗ =

1 2πi 1 2πi 1 2πi 1 2πi

−0.628 + 0.311i −2.006 + 2.778i 56.51



k2

z1 −z2 k1

z2 −z1 k1

z3 −z1 k1 z4 −z1

+ + + +

k3 z1 −z3 k3 z2 −z3 k2 z3 −z2 k2 z4 −z2



+ + + +

k4 z1 −z4  k4 z2 −z4  k4 z3 −z4  k3 z4 −z3

(22)

with the respective initial positions z1 (0) = 0.5 + 0.5 i, z2 (0) = 1.5 − 0.5 i, z3 (0) = 1 + i and z4 (0) = −1 − 2 i. The passive particle equation is   k1 1 k2 k3 k4 + + + + u, (23) z˙ ∗ = 2πi z − z1 z − z2 z − z3 z − z4 with the initial condition z(0) = −1 − i . The results obtained with four vortex are also presented in Table 1 and Fig. 1 d). The results presented in Table 1 show that, in general, for any number of vortices, the values of the objective function decrease with the number of control variables. Figure 1, that depicts the trajectories corresponding to each optimal

156

C. Balsa and S. Gama a) N = 1

b) N = 2

c) N = 3

d) N = 4

Fig. 1. Trajectories corresponding to the optimal controls.

control presented in Table 1, shows that these trajectories become more complex with the increase in the number of vortices. In most of the case where N > 1, the displacement of the passive tracer between the point of departure and arrival is not direct, but in epicycles in order to take advantage of the displacement induced by the vortices to move without spending energy.

4

Conclusions

This work presents the formulation of a set of control problems related with the advection of one passive tracer by N point vortices in the unbounded plane. These control problems result from the necessity of displacing the particle between two points within a fixed time period. The control strategy used is based on direct approach, which is unusual in this kind of problems. This approach enables to work with vortex dynamics resulting from the interaction of several point vortices. In this work it is considered a dynamics induced by N vortices, for N = 1, 2, 3, 4. The period of time allowed for the movement to take place is discretized in a certain number of subintervals, wherein the control variable is assumed to be

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constant, resulting in a nonlinear programming problem, which is solved numerically. It was observed that all problems presented here have a nearly/quasioptimal control, whatever the number of control variables. The greater the number of control variables, the smaller the amount of energy needed to perform the displacement. In general, the energy needed to make the displacement increases with the number of points vortices. Acknowledgements. SG was partially supported by CMUP, which is financed by national funds through FCT – Funda¸ca ˜o para a Ciˆencia e a Tecnologia, I.P., under the project with reference UIDB/00144/2020; by Project STRIDE NORTE-01-0145FEDER-000033, funded by ERDF NORTE 2020; and by project MAGIC POCI01-0145- FEDER-032485, funded by FEDER via COMPETE 2020 - POCI and by FCT/MCTES via PIDDAC.

References 1. Aref, H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15(1), 345–389 (1983) 2. Vainchtein, D., Mezi´c, I.: Vortex-based control algorithms. Control of Fluid Flow, pp. 189–212. Springer, Berlin (2006) 3. Babiano, A., Boffetta, G., Provenzale, A., Vulpiani, A.: Chaotic advection in point vortex models and two-dimensional turbulence. Phys. Fluids 6(7), 2465–2474 (1994) 4. Pereira, F.L., Grilo, T., Gama, S.: Optimal multi-process control of a two vortex driven particle in the plane. IFAC-PapersOnLine 50(1), 2193–2198 (2017) 5. Chertovskih, R., Karamzin, D., Khalil, N.T., Pereira, F.L.: Regular pathconstrained time-optimal control problems in three-dimensional flow fields. Eur. J. Control (2020). https://doi.org/10.1016/j.ejcon.2020.02.003 6. Protas, B.: Vortex dynamics models in flow control problems. Nonlinearity 21(9), R203 (2008) 7. Betts, J.T., Kolmanovsky, I.: Practical methods for optimal control using nonlinear programming. Appl. Mech. Rev. 55, B68 (2002) 8. Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. Cambridge University PR, New York (2010) 9. Pontryagin, L.S.: Mathematical Theory of Optimal Processes (Classics of Soviet Mathematics), vol. 4, Gordon and Breach Science Publishers (1986) 10. Balsa, C., Gama, S.M.A., Braz C´esar, M.: Control problem in passive tracer advection by point vortex flow: a case study. In: Papadrakakis, M., Fragiadakis, M. (eds.) COMPDYN 2019 - 7th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (2019) 11. Pereira, F.L., Grilo, T., Gama, S.: Optimal power consumption motion control of a fish-like vehicle in a vortices vector field. In: OCEANS 2017 - Aberdeen. IEEE (2017) 12. MathWorks: Matlab Optimization Toolbox: User’s Guide (R2020a). The MathWorks, Inc., Natick, Massachusetts, United State (2020). https://www.mathworks. com/help/pdf doc/optim/ 13. Chorin, A.J.: Vorticity and Turbulence. Springer, New York (1994) 14. Newton, P.K.: The N-vortex Problem: Analytical Techniques. Springer Science & Business Media, New York (2013)

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15. Gama, S., Milheiro-Oliveira, P.: Statistical properties of passive tracers in a positive four-point vortex model. Phys. Rev. E 62(1), 1424 (2000) 16. Waltz, R.A., Morales, J.L., Nocedal, J., Orban, D.: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math. Program. 107(3), 391–408 (2006) 17. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2000)

Gas Network Hierarchical Optimisation—An Illustrative Example T.-P. Azevedo Perdico´ ulis1(B) and P. Lopes dos Santos2 1

2

Engineering Department at ECT UTAD, ISR, University of Coimbra, Coimbra, Portugal [email protected] INESC TEC & EE Department at FEUP, University of Porto, Porto, Portugal [email protected] Abstract. In this work, we describe the application of hierarchical coordination to gas networks optimisation, required by the problem decomposition. The problem is decomposed geographically and then special attention is given to the coordination of an iterative algorithm where every agent finds its best settings based on the decisions of its nearestneighbours. The result is a decentralised fixed-point algorithm whose structure is described in an example network and using an algebraic setting of the kind of the transient optimisation problem, owing to the case study complexity. The method performance is assessed on the examplenetwork. Keywords: Coordination · Fixed-point algorithm optimisation · Graph theory · Hierachical theory

1

· Gas network

Introduction

Dynamic optimisation of networks are high dimension problems, whose successful design often involves decomposition into smaller subsystems, each one of those with its own goal and constraints [6,11]. Modelling networks as a noncooperative game, where every player makes decisions independently, takes advantage of the physical dispersion of the problem [8–10]. Also, the network steady-state behaviour during a certain period of time suggests the substitution of the transient problem by a sequence of steady-state ones [2,4]. However, splitting the problem into subsystems, even when interconnections between these are minimised, does not guarantee that a global or overall ‘optimum’ will be attained. In fact, optimising a single subsystem as part of a larger system, without regarding the interaction effects, can lead to such degraded performance in the other subsystems that the final outcome will be worse than without any optimisation at all. Consequently, system coordination needs to be considered, to ensure that

c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 159–169, 2021. https://doi.org/10.1007/978-3-030-58653-9_15

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all the subsystems acting together will achieve a global/overall goal [13]. Hierarchical coordination is one of the most common forms for comprising connectivity properties between subsystems. In other words, once the system has been decoupled in separable problems, the interconnections are taken into account at a higher-level, that checks for global/overall feasibility, and may be controlled by the levels above [5,14]. In this work we describe a 2-stage approach of the problem in Sect. 2; at every stage, a two-level decomposition of the system, according to both time and space, leads to a three-level hierarchy structure. In Sect. 3, temporal and spatial subsystems, as well as the respective coordinators, are defined for both stages. The decomposition-coordination method to be adopted—the goal coordination—is also characterised. In Subsect. 3.4, the approach to decomposition is proven correct on a algebraic linear problem, which is tested on an example networks. In Sect. 4, we withdraw some conclusions and directions for future work.

2

Space-Time Decomposition

The transient optimisation problem is decomposed into a sequence of timerelated steady-state problems with some inventoring between time-levels. Hence, the inventoring relates to time and is represented by the first equation of the following model: B0 x0 + B1 x1 + · · · + B24 x24 = b0 period coordination G1 x1 = b1 Gτ xτ = bτ , τ = 2, . . . , 24

1st hour τ th hour

(1)

P 1 P 2 P 3 P 4 P 5 Q1 Q2 Q3 Q4 At every time-level the network, underlying structure is represented by ∗ ∗ ⎞ P1 ⎛ ∗ ∗ ∗ ∗ matrix G—the connectivity matrix. P2 ⎜ ⎟ ∗ ∗ ∗ P3 ⎜ ⎟ This is an incidence matrix for the ⎟ ⎜ ∗ ∗ P 4 ⎜ ⎟ whole network, that after some ele∗ ∗ ∗ ⎟ G = P5 ⎜ ⎟ ⎜ ∗ ∗ mentar algebraic operations shows a Q1 ⎜ ⎟ ⎟ ⎜ ∗ ∗ Q2 ⎝ diagonal shape. As a matter of fact, ⎠ ∗ ∗ ∗ ∗ Q 3 the ∗ are the local sub-incidence matri∗ ∗ Q4 ces. This diagonal structure highlights the existence of two different types of elements: (i) independent controllable units, Pi ; (ii) its connecting pipe-legs that we call network-components, Qj . Also, that elements of one set only connect to elements of the other. Such network decomposition can then be represented by an adjacency graph (and matrix) [12]. Taking advantage of this underlying structure, every steady-state problem is decomposed and then conceptualised as a noncooperative game, that is solved by a 2-phase algorithm. To take into account time-level inventory, every steadystate problem is initialised with the solution found at its previous (Fig. 1).

Gas Network Hierarchical Optimisation

161

Fig. 1. Initial-problem decomposition: at every time-level there is a dependency graph.

In this work, global convergence refers to the verification of the network connectivity constraints, and relates to spatial decomposition. Overall convergence refers to the verification of the state-equation which models the coupling between consecutive time-levels, and relates to temporal decomposition [12]. We approach the problem in two stages. At the first stage— initial-problem—we prime network integrity and consequently the problem is computed along the period with global convergence being required at various points of the control interval. At the second stage—general-problem—if overall convergence has not been verified in the end of the period, we rectify the solution in the most efficient manner. That is, the solution Fig. 2. General-problem decom- obtained at the first stage initialises the recposition. tifying iterative problem at the second stage, which iterates until global and overall convergence are verified (Fig. 2). Problem 1 (Separable global equilibrium problem). simultaneously, each of the optimisation problems

Find x1 , . . . , xN to solve,

min fi (xi ) xi

(2)

s.t. ci (xi , x ¯i ) = 0 xi ∈ Rni , and n1 + · · · + nN . The connectivity constraints are incorporated using Lagrangian multipliers: yj , j = 1, . . . , M, where M is the number of network-components. This formulation relates to both every static problem at the initial-problem and the transient problem (and every local distributed OCP) at the general-problem. Hence: Problem 2 (Separable optimal control problem). For every i, solve simultaneously each of the dynamic optimisation problems

162

T.-P. A. Perdico´ ulis and P. L. dos Santos τf −1

min J= τ

{ui }



τ =τ0

τ

τ

I(xτi , uτi , τ ) + F (xi f , ui f , τf )

s.t. xτi +1 = ζ(xτi , uτi , τ )

τ = τ0 , τ0 + Δτ, . . . , τf − Δτ

x(τ0 ) = x0 , x(τf ) = xf and

{uτi }

(3) (4)

∈ U,

where uτi is the control-vector at time-level τ ; xτi is the state-vector at time-level τ τ ; xτi 0 is the initial state; xi f is the final state; J{uτi }τ =τ0 ,...,τf is the objective functional [1]. Global problem means optimisation over the whole physical network, and overall problem means optimisation over the whole control interval.

3

Hierachical Coordination Theory

The principle behind hierarchical coordination is to decompose a problem, whose objective is to extreme a criterion associated with a complex system, into a certain number of simpler subsystems. Once the solutions of the subproblems are found, they have to imply the solution of the whole system. Therefore Solution [P1 (y), P2 (y), . . . , PN (y)]y=y∗ =⇒ Solution of P   Solution P0 (Y ), P1 (Y ), . . . , Pτf−1 (Y ) Y =Y ∗ =⇒ Solution P

(5)

Initial-problem: P represents every steady-state problem and Pi (i = 1, . . . , N ) the local subproblems. P is the overall optimisation problem and Pτ = P (τ = 0, . . . , τf −1) the steady-state problems. General-problem: P is decomposed into a set of local OCPs, i.e., Pi = P, and every Pi is locally further decomposed into temporal subproblems Pτ [13]. Indeed, relations (5) cannot, generally, be satisfied because of the existence of the interactions between the Pi s (Pτ s) and the resulting conflicts. For this reason, it is necessary to introduce a ‘coordination parameter’, in such a way that these relations will be satisfied. Thus, y are the Lagrangian multipliers corresponding to the network connectivity constraints of Problem 1, and Y are the costate-variables corresponding to the state-equation of Problem 2. Hence, hierarchical methods involve choosing y (or Y ) from some initial value y (0) (or Y (0) ) and iterating to the final values y ∗ (or Y ∗ ).

3.1

Spatial Subsystem Pi

Every local subsystem—a network active element (or player)—has a certain number of composed-inputs coming from other subsystems, through the network-components. At iteration-k, every subsystem finds its best settings x∗ = (x∗1 , . . . , x∗N ) [2,9], that are passed to the network-components. Based on this information, the network feasibility is assessed and the local subsystems

Gas Network Hierarchical Optimisation

163

α1 N 

(0) (k+1)

(k+1)

xi

cjk x∗k = αj

k=1

k=i

x∗i (k+1) = zi

the i-th system

.. .

ui

αM

Fig. 3. The i-th subsystem.

informed about the quality of their settings. Accordingly, the players compute again—iteration-(k + 1). Hence: (0) (k+1)

xi

=

M  j=1

(k+1)

αj

=

M  N  j=1

(k)

cjk zk

(6)

k=1

k=i

⎛ ⎞ M  (k+1) ⎠ zi = x∗i = Ξi ⎝ . αj

(7)

j=1

αj =

N

k=1

k=i

cjk x∗k represents the local numeric processing at each network–

M (k+1) (0) (k+1) component j; αj are the network feasibility indicators; xi = j=1 αj expresses the computation of every subsystem optimal settings depending on the values supplied by its adjacent network-components. Equation (7) represents local constraints of the subsystem. Assuming the objective N function, f , for the global system to be additive-separable, we have f = i=1 fi (xi , ui ), where fi is the every subsystem constrained optimising function. Hence, we define the Lagrangian function as: ⎛ ⎞ N N N  N M     ⎜ ⎟ L= fi (xi , ui ) + Yi (zi − Ξi ) + yi j ⎝αj − cjk zk ⎠ , (8) i=1

i=1

i=1 j=1

k=1

k=i

Yi and yi are the costate variables and the Lagrangian multipliers of the i-th subsystem, respectively. Other constraints, such as operational levels, are neglected in this analysis. Assuming constraints (7) independent and the functions fi and Ξi piecewise continuous and piecewise continuously differentiable at the first order, the i-th subproblem is: ⎛ ⎞ M N  ⎜  ⎟ Li = fi (xi , ui ) + Yi (zi − Ξi ) + yj ⎝αj − cjk zk ⎠ . (9)

  k=1

dynamic problem



:=Hi (xi ,ui ,Yi )



j=1

k=i

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T.-P. A. Perdico´ ulis and P. L. dos Santos

Hi is the Hamiltoniam for subsystem i. Thus, the optimal solution must satisfy ∂H ∂H ∂H = 0, ∂u = 0, ∂Y = the stationary conditions for L(xi , ui , Yi , yi ). Namely, ∂x i i i ∂L 0, ∂yi = 0, and L and H are, respectively, the Lagrangian/ Hamiltonian for the global problem (Fig. 3).

3.2

Non-feasible, or Goal Coordination Method

Since L has a separable form, each subsystem is defined by the Lagrangian (9): ⎛ ⎞ N M ⎜  ⎟ min Hi (xi , ui ) + yij cjk zk ⎠ (10) ⎝αj − j=1

k=1

k=i

As the connectivity constraints are adjusted, the objective function is accordingly modified, and once coordination has been accomplished, changes are either null or negligible, which means that the connectivity constraints   have been satisfied. N N M N That is, ΔHi = yi αj − k=1 cjk zk ∼ = 0, x∗ (y) will be the i=1

i=1

j=1

j

k=i

solution of Problem (10). This means that: αj −

N

k=1

k=i

i

cjk zk = εj ; if εj < some

established tolerance for ∀j, i.e., the global solution has been achieved. The whole procedure starts with the subsystems optimising locally, given an initial value for the Lagrangian multipliers, y 0 . In other words, at the lower level Eq. (3.1) are solved and x, u, Y are determined. Next, given x, u, Y , the coordinator at the higher level fixes a new y (see Figs. 5 and 6). That is, from Eq. (8), we can write: N   T  Lxi dxi + LTui dui + LTYi dYi + LTyi dyi (11) dL = i=1

When an optimum is found at level 2, we have: Lxi = 0 Lui = 0 LYi = 0, N T and Eq. (11) reduces to dL = i=1 Lyi dyi , which means that the global optimum will be achieved when LTyi dyi also becomes zero, which is done at the upper level [13].

Space Coordinator Problem. Although the stationary conditions characterise a solution, they do not constitute a method for determining it. Thus, to solve the problem we need to apply a solution algorithm for unconstrained nonlinear programming [3], and quasi-Newton methods, can be a good choice [3]. We are seeking a solution which, according to our assumptions, corresponds to a saddle-point of L. Thus, we choose dyi = KLyi , K > 0 that, after discretisation, leads to the following Newton-method type coordinator algorithm—that corresponds to the update-phase in the iterative 2-phase algorithm, where D is a matrix of conjugated directions [3]:

Gas Network Hierarchical Optimisation

xτ +1

x∗τ

(τ +1)-th system

165

x∗τ +1 = zτ +1

Fig. 4. The (τ + 1)-th subsystem.

⎤−1

⎡

⎢ Ly(k) ⎥ ⎥ y (k+1) = y (k) − C ⎢ ⎣ dy ⎦  

Ly(k)

0 < C < 2.

(12)

D

3.3

Temporal Subsystem Pt

Every steady-state problem depends on its previous, with xτ +1 as the intermediate input supplied by the previous system, and x∗τ +1 = zτ +1 is the output into the next one. Equation (13) represents the state-equation and ζ is the linear transformation corresponding to matrix H—the matrix that represents the underlying dynamics of the network. Equation (14) represents other constraints of the subsystem and T is a function which expresses the output dependence on the constraints imposed to the subsystem (Fig. 4): xτ +1 = ζ(x∗τ , u∗τ ) x∗τ +1 = Tτ +1 (xτ +1 ).

(13) (14)

Assuming the objective function for the overall system to be τf −1



τf −1

It (xτ , uτ )+F (xτf , uτf )+

τ =τ0



τ =τ0

τf −1

Yτ (xτ +1 − ζ(xτ , uτ ))+



yτ (zτ +1 − Tτ +1 ) ,

τ =τ0

where zτ +1 are defined as in Fig. 4, and yτ represents the space connectivity Lagrangian multipliers at time-level τ . We define the Hamiltonian for subsystem τ as: H(xτ , uτ , yτ , Yτ , τ ) = I(xτ , uτ , yτ , τ ) + YτT ζ(xτ , uτ , τ ). Assuming constraints (14) independent and functions Iτ , F , and Tτ piecewise continuous and piecewise continuously differentiable at first order, then the optimality conditions must satisfy adequate stationary conditions. Similarly to what happens for the spatial subsystems, time is resolved at the higher level and all the rest the lower level. Initial -problem 1. Level-0 gives Y 0 to every steady-problem at level-1, in order to calculate xτ , uτ , yτ ;

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2. Level-1 gives to level-2 the initial values Y 0 and y 0 ; 3. Every unit optimises, and (x∗i , u∗i ) is devolved to level-1;    ∂H   < ε1 is not verified, then go to step 5, otherwise 4. At level-1, if condition  ∂y  go to step 6; 5. A new y is fixed and devolved to level-2. Go to step 3; 6. Level-1 devolves y ∗ to level-0;    ∂H   < ε2 is not verified, the overall optimum has not 7. At level-0, if condition  ∂Y 

∂H ∂H been obtained. Hence, we have conditions ∂H ∂x = ∂u = 0 (level-2), ∂y ≈ 0 (level-1) and ∂H ∂Y ≈ 0 (level-0), and the general-problem follows (see Fig. 5). ∂H ≈ 0 at level-0, which is not very likely to happen. Otherwise, condition ∂Y

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General -problem 1. Master receives its initialisation values from initial-problem and communicates them to lower levels; 2. At level-2 every time-level a solution (x∗ , u∗ ) and devolves it to level-1;   determines  ∂H   < ε2 is not verified then go to step 4, else go to step 3. At level-1, if condition  ∂Y  5; 4. A new Y is fixed and devolved to level-2. Go to step 2; 5. A local minimum has been achieved, therefore every unit devolves (x∗i , u∗i , Y ∗ ) to level-0;    ∂H   < ε1 is not verified then go to step 7, else go to step 6. At level-0, if condition  ∂y  8; 7. A new y is fixed and communicated to level-1. Go to step 3; 8. The whole process finishes, which means that an overall global optimum has been ≈ 0 (level-1) and ∂H ≈ 0 (level-0) are verified obtained. That is, conditions ∂H ∂Y ∂y (see Fig. 6).

Spatial decomposition of the optimisation problem gives two parallel sets, whose dependencies are represented by the dependency graph. Setting the processes into the decomposition-hierarchical context: (i) for the initial-problem, the optimise-processes compute at level–2 and the update-processes compute at level–1. (ii) for the general-problem, the update-processes compute at level–0, and the optimise-processes should compute at level–1. However, because every OCP is time decomposed, the time-processes compute at level–2 and time coordination of the overall solution is performed at level–1.

3.4

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Q1 ↓ P2 Q2

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↓ P3 Off-take

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To validate the structure of the distributed mathematical model, important points are: (i) to examine the performance of the 2-phase Jacobian iterations against a single one; (ii) to discuss the importance of a good starting value

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for the steady-state problems (first stage) and for the transient problem (second stage); (iii) to appraise the time-decomposition of every local OCP (second stage); (iv) to illustrate the iterative scheme used to find the local optimum [12]. For the network of the figure, the number of iterations needed to reach the system solution with the 2-phase iterative scheme was less than with a canonical one. Also, other networks were tested obtaining equal results. This is very convenient since the large amount of numeric processing can be done separately at every network-component. Furthermore, the 2-phase iterative scheme naturally establishes synchronisation and communication phases in view of a distributed design. When using the vectorial version of the algorithm to analyse temporal decomposition, although the same number of iterations is obtained, the problems locally computed are smaller, and therefore the time needed to compute the second phase of iteration is expected to be shorter. Thus, we may conclude that the vectorial case is more efficient. Goult et al. mention the relevance of a good starting approximation to the application of iterative algorithms to nonlinear problems [7]. At the first stage, x may be initialised with the optimum at the previous time period of operation, which is reasonable when the network does not change drastically from one period of operation to the next one; another possibility is to find a starting value in precomputation, using for instance steady-state optimisation. To improve the quality of the initialisation at the 2nd stage, every steady-state problem (1st stage) is initialised with its previous optimal solution, since this procedure takes into account some time-level connectivity. To illusT  trate iterative scheme x(τ +1) = Hx(τ ) , x = x0 , x1 , . . . , x23 , we consider some discontinuities between time levels and iterate until the discontinuities fall within some tolerance. Every subinterval is initialised with its mid-point and  then follows the iteration scheme x(τ +1) = g x(τ ) , where g holds the problem structure [12, Chapter 6 and 7].

4

Conclusion and Future Work

We depict the decomposition of the mathematical model according to time/ space. As a result, a sequence of steady-state problems decomposed according to their geographical dispersion computes along the period. This solution is rectified by a decentralised dynamic problem computed in the end of the period also decomposed according to time. The spatial and temporal subsystems are characterised, as well as a Newtonian coordinator. Among hierarchical-coordination techniques, the goal coordination method allows a bigger choice for the decomposition, permitting any physical properties of the system to be taken into account. It is chosen because since the system is partitioned into independent entities, it is our prime concern to provide for global feasibility. The work concludes with the validation of the 2-stage decomposition on a linear algebraic setting. In the future, we would like to implement other coordinators and compare their performance. It still remains to validate the problem using a small real network.

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Acknowledgments. We would like to thank the reviewers for the valuable suggestions. Work financed by FCT - Funda¸ca ˜o para a Ciˆencia e a Tecnologia under project: (i) UID/EEA/00048/2019 for the first author and (ii) UID/EEA/50014/2019 for the third author.

References 1. Anandalingam, G., Friesz, T.L.: Hierarchical optimization: an introduction. Ann. Oper. Res. 34, 1–11 (1992) 2. Ba¸sar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. Academic Press, London (1982) 3. Bazaraa, M.S., Shetty, C.M.: Nonlinear Programming Theory and Algorithms. Wiley, New York (1979) 4. Costantino, N., Dotoli, M., Falagario, M., Fanti, M.P., Mangini, A.M., Sciancalepore, F., Ukovich, W.: A hierarchical optimization technique for the strategic design of distribution networks. Comput. Ind. Eng. 66, 849–864 (2013) 5. Dell’Olmo, P., Lulli, G.: A new hierarchical architecture for air traffic management: Optimisation of airway capacity in a free flight scenario. Eur. J. Oper. Res. 144, 179–193 (2003) 6. Fallside, F., Perry, P.F.: Hierarchical optimisation of a water-supply network. IEEIERE Proceedings - India 13, 98–99 (1975) 7. Goult, R.J., Hoskins, R.F., Milner, J.A., Pratt, M.J.: Computational Methods in Linear Algebra. Stanley Thornes (Publishers) Ltd., London (1974) 8. Abou-Kandil, V.I.H., Freiling, G., Jank, G.: Matrix Riccati Equations in Control and Systems Theory. Birkh¨ auser Verlag, Basel (2003) 9. Jank, G., Kun, G.: Optimal control of disturbed linear-quadratic differential games. Eur. J. Control 8(2), 152–162 (2002) 10. Nagurney, A.: An algorithm for the single commodity spatial price equilibrium problem. Reg. Sci. Urban Econ. 16, 573–588 (1986) 11. Osiadacz, A.J., Salimi, M.A.: Hierarchical dynamical simulation of gas flow in networks. Civ. Eng. Syst. 5, 199–205 (1988) 12. Perdico´ ulis, T.-P.A.: A distributed model for dynamic optimisation of networks, PhD thesis, University of Salford, UK (1999) 13. Singh, D., Titli, A.: Systems: Decomposition Optimisation and Control. Pergamon Press, Oxford (1978) 14. Wei, Q., Kappes, M., Prehofer, C., Zhong, S., Farnham, T.: A hierarchical general purpose optimisation framework for wireless networks. In: 11th European Wireless Conference 2005 - Next Generation Wireless and Mobile Communications and Services, pp. 1–7 (2006)

Calibration-Free HCPV Sun Tracking Strategy Manuel G. Satu´e(B) , Manuel G. Ortega, Fernando Casta˜ no, Francisco R. Rubio, and Jos´e M. Forn´es Escuela Superior de Ingenieros, Universidad de Sevilla, Sevilla, Spain {mgarrido16,mortega,fercas,rubio,jfornes}@us.es

Abstract. When using high concentration photovoltaics modules, sun trackers must meet severe specifications in order to keep sun pointing error within a very small angle. Those required specifications are not only mechanical (misalignments in the structure itself, clearances of the joints, etc.) but also regards the installation (misalignments of the platform with respect to geographical north). These uncertainties are error sources that make necessary to calibrate the system after installation and possibly recalibrate it from time to time because of aging. This paper presents a control strategy that avoids the necessity of any kind of calibration by using, indirectly, the produced electric power as feedback. The control strategy is valid as far as the sun tracker is able to perform movements in the azimuth and elevation coordinates independently. Experimental results with a two axes solar tracker are exposed showing the validity of the proposed control strategy under sunny conditions.

Keywords: Sun tracking

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Introduction

High concentration photovoltaic (HCPV) modules use Fresnel lenses to concentrate the sun rays on to the solar cells. The lenses are characterized by their half-acceptance angle, α, which ideally is the maximum angle at which incoming sun rays can be captured by the photovoltaic cells [1]. In practice, due to imperfections, the half-acceptance angle is defined as the angle for which efficiency drops to 90% of its maximum (located where incidence angle is zero) as shown if Fig. 1. HCPV modules have a small half-acceptance angle, typically around 1◦ . This fact is a very important restriction for the control system of the HCPV sun trackers, which must have a very high precision in order to keep the solar modules performance around its maximum. To achieve the aiming precision requirement, HCPV sun trackers use encoders of thousand of pulses per turn and gearboxes but there are some uncertainties that decrease the accuracy and do not depend on the mechanical design or the different components that constitute the solar tracker, but of its installation, its assembly, etc. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 170–179, 2021. https://doi.org/10.1007/978-3-030-58653-9_16

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Fig. 1. Concentrator model. On the left, definition of half-acceptance angle. In the center, efficiency surface. On the right, efficiency values on the x − y plane of the virtual cell.

Fig. 2. (a) Two axis sun tracker. The frame associated to the platform of the sun tracker is not aligned with the geographical reference system. (b) Detail of solar modules not aligned with the wing of the sun tracker.

The control strategy presented in this paper is applied to the two axis (azimuthelevation) sun tracker shown in Fig. 2. In order to track the Sun, the positioner orientates its wing by determining the coordinates θ1 and θ2 . The sources of uncertainty that can be present in a sun tracking system are the following: – errors related to foundation, calibration and mechanical assembly: • misalignment of the sun tracker azimuth zero with respect to the geographical north, which does not correspond to magnetic north. It can be caused by a bad installation or because an inaccurate calibration of the sun tracker (or both) (Fig. 2a). • misalignment of the sun tracker elevation zero with respect to the skyline. As in the previous case it can be caused due to a bad installation or by a bad calibration. • misalignment of the modules with respect to the wing (Fig. 2b). – errors related to solar equations: • controller unit clock error. It may have a large drift over time. • geographical positioning error of the sun tracker. • accuracy of the algorithm itself. It may have different magnitude errors depending on date and time.

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The methods to mitigate the sources of uncertainty related to the installation are to perform a precise installation of the solar tracker by specialized personnel, which is costly, or to perform a system calibration after installation [2] (and periodically), which is complex. The open loop control strategies [3] rely exclusively on Solar Equations to determine the position of the Sun and are greatly affected by installation errors and, obviously, error introduced by Solar Equations themselves. On the other hand, closed-loop control strategies [7] use sensors (typically electro-optical sensors) capable of providing the position of the Sun with respect to the reference system of the sensor itself [4]. These closed loop control strategies are not affected by installation errors, but they are affected by the error in the assembly of the sensor on to the wing of the sun tracker, so that, in order to not lose pointing precision, it is necessary to carry out a precise (and expensive) assembly or a calibration of the orientation of the sensor with respect to the wing. In this work, the design and implementation of a control algorithm for HCPV sun trackers which allows the auto-correction of the different uncertainties that affect the entire system is presented. This means that the need to perform an accurate, and therefore expensive, installation of the equipment or the need to perform an initial (and even periodic) calibration of it is eliminated. The control strategy is valid as far as the sun tracker is able to perform movements in the azimuth and elevation coordinates independently. The idea is to use the photovoltaic modules as a sensor to estimate the position of the Sun, and the Solar Equations to predict the future position of the sun with the purpose of maintaining the pointing vector of the sun tracker ahead of the position of the sun. In addition, the algorithm allows to reduce the number of movements that the sun tracker performs throughout the day. This is achieved by maximizing the path of the solar beam projection on the plane of the solar cell, so that all the space within the limits of the maximum efficiency zone is used. To do this, it is necessary to predict the position of the Sun in the future and to use of a simple model of the lens of the solar collector together with the kinematic model of the solar tracker. Until the Sun projects outside the region of maximum efficiency, there will not be a new pointing movement. The tracking of the position of the Sun is carried out by means of a closed loop control strategy in which the Solar Equations provide approximate coordinates for the pointing of the wing of the sun tracker and a controller applies a correction on these coordinates by feed-backing the instantaneous DC power produced to estimate the real position of the Sun. This work is inspired by the algorithm proposed in [5] and provides test results performed with a high concentration solar tracker of larger dimensions and nominal electric power produced, such as those found in solar plants. The equipment used in [5] did not have HCPV solar modules, so the authors emulated its behavior using conventional modules with tubes perpendicular to the modules cells. Furthermore, the solar tracker that was used was a low-cost tracker designed for domestic equipment.

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Sun Tracker Description

The sun tracker used in this work is shown in Fig. 2. The mechanical structure has two degrees of freedom, one to follow the movement of the Sun in Azimuth and another to follow it in elevation. It consists of a fixed pole on which the azimuthal rotation mechanism (worm drive) is mounted, which in turn supports the elevation mechanism (linear actuator) and the wing, which houses the HCPV modules. The total catchment area of the HCPV modules is 9.3 m2 . To orientate the wing there are two three-phase asynchronous motors commanded by two variable frequency drives which are managed by a programmable logic controller. The measure of the orientation coordinate, θ1 , is provided by an encoder and the measure of the elevation coordinate, θ2 , is provided by an inclinometer mounted on the wing. The plant has 24 solar modules connected in series, so that the voltage that can be reached in terminals will be 443 V open circuit and the maximum intensity will be 5.73 A. The power inverter has a nominal power of 2500 W. In order to implement the proposed control strategy the sun tracker is equipped with an optical sensor which provides measures of direct normal solar irradiance and a DC power sensor. In the tests, the control strategy executes in a PLC and a PC which share information.

3

Tracking Strategy

The sun tracking strategy computes azimuth and elevation set-points (which are references for the low level azimuth and elevation motor controllers) by using the Solar Equations as an open loop feed-forward control, and applying a correction on them by using power measurements in a closed loop as outlined in Fig. 3a. Solar Equations allows to compute the position of the sun given time, longitude and latitude. This work uses the PSA algorithm to estimate the position of the Sun [6]. The need to use the closed loop is due to the fact that there are still errors in the estimation of the position of the Sun.

Fig. 3. (a) Control scheme. Solar Equations act as a feed-forward open-loop. The loop is closed with two PI controllers. (b) Idealized representation of the trajectory of the sun rays projected over the photovoltaic cell and its associated electric power trajectory during a sun tracker orientation movement.

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The instantaneous produced electric power is used indirectly as a measure of the sun position. To do so, a simple model of a squared concentrator is used in conjunction with the Solar Equations (corrected with the azimuth and elevation offsets, which start at zero degrees) and the sun tracker kinematic model. These models are used to predict both the next set-point for the sun tracker and the time instant on which the sun rays will exceed a certain angle, β (β < α), while the sun tracker is motionless at the previously calculated set-point. This prediction consist of two stages: 1) Motionless sun - mobile sun tracker: in this stage, the previously mentioned models are used in order to obtain a sun tracker pointing vector which matches the future position of the Sun. Specifically, the objective is to obtain the sun tracker pose at which the concentrated solar beam projects in the border of the squared maximum efficiency region defined in the virtual cell. The values of sun tracker coordinates (θ1 , θ2 ) corresponding to the time instant just before said condition is fulfilled are stored as the next set-point for the sun tracker, (θ1 , θ2 )ref . 2) Mobile sun - motionless sun tracker: in this stage it is considered that the solar tracker maintains the previously calculated pose while the simulation time continues to advance from the time instant at which the first stage finishes. The objective is to find the time instant at which the concentrated ray leaves the squared maximum efficiency region again. This time instant is stored as the time of the next execution of the ‘control sequence’, tnext . The main idea behind the previous prediction is that the sun tracker will move ahead of the Sun and then will wait for the sun to travel the maximum possible distance over the photovoltaic cell in order to be able to estimate the position of the Sun in sun tracker coordinates by analyzing the measured electric power. It also allows to minimize the number of movements that the sun tracker performs throughout a day, when comparing with a fixed time movement control scheme. The next step is to give the computed set-point to the low level azimuth and elevation controllers. The movement is not performed on both axes at the same time, but one at a time. First only the orientation changes and then only the elevation. While these movements are carried out the following measures are sampled in separated tables for orientation and elevation: – – – –

time, ts sun tracker coordinate, θ1 or θ2 (θ1/2 ) produced DC electric power, P irradiance (DNI), Irr. η = P/(Irr · Ac )

(1)

Figure 3b shows an idealized representation of the trajectory of the sun concentrated beam projected on to the photovoltaic cell when the sun tracker performs

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an orientation movement. The trajectory starts at point A and ends at point B, and during the movement of the sun tracker from A to B, the electric power describes the curve shown (ideally). After the completion of these movements, the controller processes the trajectories using a non-causal FIR filter [8] and construct the instantaneous performance trajectories, [ts, η], of both separated movements using (1). These performance trajectories along with the sun tracker coordinate trajectories, [ts, θ1/2 ], are analyzed in order to estimate a measure of the sun position in sun tracker coordinates, (θ1 m , θ2 m ), during the sequence of movements as well as an associated time-stamp, tsm , of the time instant in which it happened. Encoder, inclinometer and power sensor provide very noisy signals, so it is necessary to filter them, but without changing their phase, as ultimately the objective is to determine certain points over the trajectories. The same analysis can be performed using the power trajectory, but the use of the performance mitigates the effect of the clouds on the shape of the trajectory. The estimation of the sun position in the analysis stage will be studied later in Sect. 3.1. The time-stamp tsm allows to evaluate the Solar Equations, so that the controller can compute the errors in azimuth and elevation that exist between the Solar Equation and the measurements at the time instant given by tsm . These errors are the inputs of two asynchronous PI controllers whose outputs are the offsets that will correct the Solar Equations the next time the controller decides to perform a sun pointing movement, which will happen when the controller clock reaches the time tnext . It may happen that in the analysis stage, the trajectories associated to the azimuth and/or elevation movements is discarded. It may be due either because it has not enough samples due to a very small associated movement or because it is not classified within the expected types of trajectory due to a cloudy period, or even because the power inverter is off. In that case the associated PI controller is not executed, and instead the last computed offset is used. If both trajectories associated to the azimuth and elevation movements are accepted in the analysis stage, the time stamp tsm is computed as the time instant in the middle of the time interval [t(θ1m ), t(θ2m )]. If only one of the trajectories is accepted, tsm will

Fig. 4. Trajectory classification. The trajectory shapes depicted are just simplifications for the sake of clarity.

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be the time instant corresponding to the value of the coordinate at which the m ). Sun position is estimated, t(θ1/2 3.1

Trajectory Analysis

The analysis uses the trajectory of the efficiency in the conversion of light energy into electricity (1), [ts, η], and its associated movement trajectory, [ts, θ1/2 ]. Figure 4 shows six types of expected trajectories. Most of them are based in the different shapes which can be obtained when slicing the virtual power surface of the cell. The analyzer module will try classify the trajectory [η, θ1/2 ] into one of the classes from C1 to C6 by simple checks. If the trajectory does not correspond to any of the classes, it will be classified as rejected trajectory. The variable threshold, thr, is defined as a percentage of the maximum performance, max. Notice that points A and B can be initial and final point or vice versa, depending of the sense of the movement and the path that the projection of the concentrated sun beam describes in the plane of the photovoltaic cell during the said movement. (2) θ∗ = (θA + θB )/2 – Class C1: Centered trajectory. This is the type of path expected when the controller is not in a start-up phase. Initial and final points are over the threshold, thresh. The point with maximum performance, max, is not A nor B and is greater than the performance of A and B. The value of the sun tracker coordinate where the sun is located, θ∗ , is chosen with expression (2). – Class C2: Centered trajectory. Initial and final points are below the threshold, thresh. The point with maximum performance, max, is not A nor B and is greater than the performance of A and B. The value of θ∗ is chosen with expression (2). – Class C3: Half-centered trajectory. One of the extreme points is below the threshold and the other is over it. The value of θ∗ is chosen with expression (2). – Class C4: Off-center trajectory. This kind of trajectory arises when the controller is in a start-up phase. The maximum is located on one end of the trajectory. The value of θ∗ is the corresponding to the maximum. – Class C5: Off-center trajectory. The maximum is located on one end of the trajectory. The value of θ∗ is the corresponding to the maximum. – Class C6: Low performance trajectory. This type of trajectory is obtained when the inverter is off due to low radiation, or when the inverter is in the start-up phase (so there is enough solar radiation) after a while without receiving radiation. The objective of this step is to estimate the sun position in sun tracker coordinates. The resulting measured sun coordinates will not always be a good estimation because of the lack of information in the trajectories, but at least must point in the right direction. If this requirement is met, the control strategy will correct the discrepancies between Solar Equations and measurements over time, and the trajectories will have better information which will allow calculating better estimates.

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Experimental Results

Figure 5 and Fig. 6 show the results of an experiment carried out on 9/24/2019. In order to test the tracking strategy, the sun tracker was not precisely calibrated. The estimated error in the orientation coordinate is approximately 1◦ . Another known uncertainty of the tracker is the assembly of the solar modules to the wing, as they do not lie all on the plane of the wing. The estimated alignment error is approximately equal to 0.5◦ (cone). Figure 5a and b illustrate the sampled performance trajectories for the orientation and elevation movements respectively. Figure 6a shows the computed azimuth and elevation offsets which are used to correct the Solar Equations. Figure 6b depicts the electric power produced in the DC side of the power inverter. Figure 5a shows how at the beginning of the day, the orientation trajectories have a poor performance or even negative. The cause of the negative performance is related to a bad adjustment of the power sensor zero, which is specially evident when the inverter is off due to a low solar radiation or inaccurate sun pointing. It can also be observed in Fig. 5b that there are no elevation trajectories at the beginning of the day. The reason is that the controller has a software limit in the elevation coordinate, θ2 , equal to 20◦ in order to avoid collisions with the devices attached to the pole of the sun tracker. Therefore, until the Sun elevation reaches 20◦ the sun tracker is stuck at said elevation and can not perform elevation movements. In fact, the initial low performance orientation trajectories are not just due to a low solar radiation at the beginning of the day but to the fact that the sun tracker is not performing the corresponding elevation movements to point to the Sun, and for a while, it is pointing bad. A comparison between the proposed closed-loop control strategy and an fixed time interval open-loop by means of Solar Equations is depicted in Fig. 7. The operation of the open-loop consist in pointing to the position of the Sun provided by the Solar Equations, which is not necessarily the real position of the Sun, every two minutes. The time interval between pointing movements of the openloop strategy is approximately half of the mean time it takes for the closed loop strategy, being the later interval not constant. In this case, the system is uncalibrated on purpose and therefore the openloop strategy clearly performs way below the maximum capability, while the proposed control strategy is able to adapt to the uncertainties to improve the performance. The degree of un-calibration in the azimuth coordinate is estimated to be 1.5◦ approximately for this test.

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Conclusions

A closed loop control strategy that allows to adapt to the different sources of uncertainty that affect the sun tracker system as a whole has been developed and tested. Experimental tests show that the proposed sun tracking strategy is suitable for low levels of misalignment of the platform of the sun tracker with respect to the geographical reference system. This continuous adaptation eliminates the need to perform a very precise installation or to calibrate the system, which is highly recommended when using classic control strategies, either in closed or open loop, so that the performance of the solar plant does not decrease. Experimental results that prove the correct functioning of the proposed algorithm for low levels of misalignment of the platform of the sun tracker with respect to the geographical reference system. Also the benefits with respect to a not accurately calibrated open-loop control strategy are shown. The following questions will be addressed in future work: making trajectory classification more robust, more accurate estimating of the sun’s position in trajectory analysis, testing behavior under changing weather conditions. Acknowledgment. This work was supported by the Spanish Ministry of Economy and Competitiveness [grant DPI2016-79444-R]. This support is gratefully acknowledged.

References 1. Hern´ andez, M., Cvetkovic, A., Ben´ıtez, P., Mi˜ nano, J.C.: High-performance K¨ ohler concentrators with uniform irradiance on solar cell. In: Nonimaging Optics and Efficient Illumination Systems V, vol. 7059, pp. 70–78 (2008). https://doi.org/10. 1117/12.794927 2. Satu´e, M.G., Casta˜ no, F., Ortega, M.G., Rubio, F.R.: Auto-calibration method for high concentration sun tracker. Sol. Energy 198, 311–323 (2020). https://doi.org/ 10.1016/j.solener.2019.12.073 3. Abdallah, S., Salem, N.: Two axis sun tracking system with PLC control. Energy Convers. Manag. 45, 1931–1939 (2004). https://doi.org/10.1016/j.enconman.2003. 10.007 4. http://www.solar-mems.com/solar-tracking/ 5. Rubio, F., Ortega, M., Gordillo, F., L´ opez-Mart´ınez, M.: Application of new control strategy for sun tracking. Energy Convers. Manag. 48, 2174–2184 (2007). https:// doi.org/10.1016/j.enconman.2006.12.020 6. Blanco-Muriel, M., Alarc´ on-Padilla, D.C., L´ opez-Moratalla, T., Lara-Coira, M.: Computing the solar vector. Sol. Energy 70, 431–441 (2001). https://doi.org/10. 1016/S0038-092X(00)00156-0 7. Roth, P., Georgiev, A., Boudinov, H.: Design and construction of a 590 system for sun-tracking. Renew. Energy 29, 393–402 (2004). https://doi.org/10.1016/S09601481(03)00196-4 8. Gustafsson, F.: Determining the initial states in forward-backward filtering. IEEE Trans. Signal Process. 44, 988–992 (1996). https://doi.org/10.1109/78.492552

Geometric Algorithm to Generate Interpolating Splines on Grassmann and Stiefel Manifolds Lu´ıs Machado1,2(B) , F´ atima Silva Leite1,3 , and Ekkehard Batzies4 1

2

3

Institute of Systems and Robotics, DEEC – UC, 3030-290 Coimbra, Portugal Department of Mathematics, University of Tr´ as-os-Montes e Alto Douro (UTAD), 5000-801 Vila Real, Portugal [email protected] Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal [email protected] 4 Faculty of Computer and Electrical Engineering, Hochschule Furtwangen University, 78120 Furtwangen, Germany [email protected]

Abstract. In this paper, we present a simplified geometric algorithm to generate interpolating splines on Grassmann and Stiefel manifolds, where position and velocity are required to change smoothly. In this construction, each spline segment is computed using local data only. It turns out that this algorithm does not require a recursive procedure and it is based on the explicit expressions for geodesics or quasi-geodesics on those manifolds. Keywords: Grassmann and Stiefel manifolds Quasi-geodesics · Geometric splines

1

· Geodesics ·

Introduction

Since we are living in a highly computational age there are more and more abstract notions that are becoming extremely valuable in various applications. In this paper, our focus goes to the Stiefel and Grassmann manifolds that have proven to be perfect scenarios for countless applications in computer vision. One can mention pattern recognition applications in the context of still imagery, videos, image-sets or 3D imagery, visual tracking, texture dynamics, age estimation, clustering, modeling human activities, classification, segmentation and domain adaptation (for more details see [10] and [13] and references therein). From a mathematical point of view, a point in the Grassmann manifold is a k−dimensional vector subspace of a n−dimensional vector space, while a point in a Stiefel manifold is a k−orthonormal frame that generates that subspace [4]. These two manifolds provide geometric structures for subspaces and are intimately related with each other. The big difference is that there is no unique c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 180–189, 2021. https://doi.org/10.1007/978-3-030-58653-9_17

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order or basis of a matrix representing an element in the Grassmann manifold, while the order of the basis is important for elements in the Stiefel manifold. Stiefel and Grassmann manifolds can be naturally endowed with Riemannian metrics and the rich framework of differential geometry provides more elegant and computationable methods than those obtained via conventional approaches, usually dealing with data quantified in a vector space [1,4]. Following these considerations, we propose in this paper a simple geometric algorithm to solve an interpolation problem posed on the Stiefel and Grassmann manifold, where not only position, but also velocity are required to change smoothly. This kind of problems is frequently used in joint interpolation motions of robots. In this case, working in the joint space facilitates the adjustment of the trajectory according to the design requirements, thus allowing to avoid problems arising with kinematics singularities [11]. The key to presenting this algorithm relies on the knowledge of explicit expression for the geodesic or quasi-geodesic joining two given points in the Grassmann [3] or Stiefel manifolds [9]. Explicit formulas for geodesics connecting two arbitrary points in the Stiefel manifold are not known. For that reason, geodesics have been replaced by other simple curves also having constant, but not necessarily zero, geodesic curvature. These curves fit the definition of quasigeodesics in [1] and have been used successfully to implement a modified De Casteljau algorithm in [9]. In the construction used in the present paper, each spline segment is obtained by the analogue to a convex combination of two components, that we call left and right components, parametrized by a smoothing real function that is responsible for guaranteeing the required degree of smoothness of the desired spline. It turns out that the proposed algorithm is simple to implement and only requires the computation of three geodesics or quasigeodesics and does not involve any recursive procedure. The paper is organized as follows. In Sect. 2, one gathers the main definitions and properties associated to the geometric structure of the Grassmann and Stiefel manifolds. The main results of the paper appear in Sect. 3. We start with the formulation of the interpolation problem. Then we prove the main results and present the algorithms that completely solve the interpolation problem in the Grassmann and Stiefel manifolds. Finally, in Sect. 4 we present the conclusions of the paper.

2

The Geometry of Grassmann and Stiefel Manifolds

In this section we recall the main definitions associated to the geometry of Stiefel and Grassmann manifolds and several properties that will be useful later. Our main references for these introductory definitions are [3] and [4]. 2.1

The Grassmann Manifold

The real Grassmann manifold (or Grassmannian) Gn,k is a smooth and compact manifold of dimension k(n − k). The diffeomorphism between Gn,k and the set

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of all symmetric projection operators of rank k ([5]) allows us to look at Gn,k as an embedding submanifold of the manifold s(n), consisting of all real symmetric matrices of order n. We adopt the following definition for the Grassmannian   Gn,k := P ∈ s(n) : P 2 = P and rank(P ) = k . (1) Let gl(n) denotes the vector space of all n × n real matrices and so(n) the Lie algebra of all n × n skew-symmetric real matrices. Given P ∈ Gn,k , define glP (n) := {M ∈ gl(n) : M = P M + M P }; sP (n) := s(n) ∩ glP (n); soP (n) := so(n) ∩ glP (n). The following, will play an important role in the derivation of our main results. Proposition 1 [3, Lemma 2.1; Lemma 3.2]. Let P ∈ Gn,k , M ∈ glP (n) and k ∈ N. Then 1. P M 2k−1 P = 0; 3. P M 2k = P M 2k P = M 2k P ; 5. eM P − P e−M = sinh M .

2. P M 2k−1 + M 2k−1 P = M 2k−1 ; 4. [P, [P, M ]] = M ;

If P ∈ Gn,k then ΘP ΘT ∈ Gn,k , for any Θ ∈ O(n). Thus α : (−ε, ε) → Gn,k given by α(t) = Θ(t)P ΘT (t), where Θ is a curve in O(n) satisfying Θ(0) = I, is a curve in the Grassmann manifold passing through P at t = 0. The tangent space to a point P ∈ Gn,k is therefore given by    TP Gn,k = X, P : X ∈ soP (n) , (2) which is proved to be isomorphic to sP (n) [3]. The Grassmann manifold will be equipped with the metric inherited from the Euclidean space Rn×n , which coincides with the Frobenius metric, cf. [6]    (3) [X1 , P ], [X2 , P ] = tr X1T X2 .   The geodesic γ in Gn,k satisfying γ(0) = P and γ(0) ˙ = X, P is given by γ(t) = etX P e−tX .

(4)

An explicit formula for a geodesic γ : [0, 1] → Gn,k joining a point P (at t = 0) to a point Q (at t = 1) was derived in [3] and gives the initial velocity vector in terms of the initial and final points only. This geodesic is given explicitly by γ(t) = etX P e−tX , where X =

 1 log (I − 2Q)(I − 2P ) . 2

(5)

Here ‘log’ stands for the matrix principal logarithm, which is uniquely defined if the orthogonal matrix (I − 2Q)(I − 2P ) has no negative real eigenvalues [7].

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2.2

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The Stiefel Manifold

The Stiefel manifold of orthonormal k-frames in Rn has the following matrix representation:   Stn,k := S ∈ Rn×k : S T S = Ik . This is a submanifold of Rn×k having dimension nk − (k + 1)k/2. Note that if S ∈ Stn,k then clearly SS T ∈ Gn,k . The following properties, when S ∈ Stn,k and M ∈ glSS T (n), are immediate consequences of those in Proposition 1, and will be useful later on. 1. S T M S = 0; 2. SS T M 2 S = M 2 S. Next, we present a parametrization of the tangent space TS Stn,k that will be very useful throughout the paper. Proposition 2 [9, Proposition 5]. Let S ∈ Stn,k , so that P = SS T ∈ Gn,k . Then, (6) TS Stn,k = {XS + SΩ : X ∈ soP (n) and Ω ∈ so(k)} . Moreover, if V = XS + SΩ ∈ TS Stn,k , then X = V S T − SV T + 2SV T SS T and Ω = S T V.

(7)

The Stiefel manifold will be equipped with the so called canonical metric, cf. [4]

    V1 , V2 = tr V1T I − 12 SS T V2 , where V1 , V2 ∈ TS Stn,k . According to [4], geodesics in the Stiefel manifold are solutions of the following second order differential equation,  2 γ¨ + γ˙ γ˙ T γ + γ γ T γ˙ + γ˙ T γ˙ = 0.

(8)

In Edelman et al. [4] can be found an explicit formula for a geodesic on the Stiefel manifold that starts at a given point with a prescribed velocity vector. However, the one that is presented below takes the advantage of the parametrization of the tangent space of the Stiefel manifold given in Proposition 2 and is much simpler than the one given in [4]. This result will be useful to solve the interpolation algorithm in the Stiefel manifold and the proof uses some of the properties of the Stiefel manifold stated in [9]. Proposition 3. Let S ∈ Stn,k and V = XS + SΩ ∈ TS Stn,k . Then, the curve defined explicitly by T γ(t) = et(X+SΩS ) S, (9) is a geodesic in Stn,k (satisfies (8)), satisfying γ(0) = S and γ(0) ˙ =V.

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Up to now, no explicit formula for the geodesic joining two arbitrary points in the Stiefel manifold is known. However, so far we only have been able to derive a simple expression for quasi-geodesics that join two arbitrary points in the Stiefel manifold. Quasi-geodesics can be seen as generalizations of geodesics and are curves with very interesting geometric properties [1]. The next result characterizes quasi-geodesics on the Stiefel manifold and its proof can be found in [9]. Proposition 4 (cf. Proposition 6 in Proposition 2. Then, the mapping R : TS Stn,k is defined by RS (V ) = eX SeΩ and β : t → etX SetΩ is a quasi-geodesic  ˙ 1. β(t) = etX XS + SΩ etΩ ;

[9]). Let S, X, and Ω be as in the T Stn,k → Stn,k whose restriction to is a retraction on the Stiefel manifold, in Stn,k that satisfies β(0) = S and

 ¨ = etX X 2 S + 2XSΩ + SΩ 2 etΩ . 2. β(t) Given two distinct points S1 and S2 in the Stiefel manifold, the next theorem exhibits X ∈ soS1 S1T (n) and Ω ∈ so(k) so that the quasi-geodesic defined by β(t) = etX S1 etΩ joins the point S1 (at t = 0) to the point S2 (at t = 1). Theorem 1 (cf. Theorem 7 in [9]). Let S1 and S2 be two distinct points in Stn,k . Then, the quasi-geodesic defined by β(t) = etX S1 etΩ , where   1 log (I − 2S2 S2T )(I − 2S1 S1T and Ω = log S1T e−X S2 , 2 satisfies β(0) = S1 and β(1) = S2 and has the following properties:   ˙ 2 = − 1 tr X 2 − 1 tr Ω 2 (constant speed); 1. β(t) 2 2  2 T 22 ˙ = tr Ω S X S1 (constant covariant acceleration). 2. Dt β(t) 1 X=

(10) (11)

The second property listed in Theorem 1 is an improvement of the same property proved in [9].

3

Formulation of the Interpolation Problem

Throughout this section, M is used to denote one of the manifolds Gn,k or Stn,k , and TP M is the tangent space of M at a point P ∈ M . The general problem to be studied here is formulated as follows: Problem (P): Given m distinct points P1 , P2 , . . . , Pm in M , m tangent vectors V1 , V2 , . . . , Vm , (Vi ∈ TPi M, 1 ≤ i ≤ m), and a partition of the unit interval 0 = t1 < t2 < · · · < tm = 1, find a C k −smooth curve (k ≥ 1), s : [0, 1] → M , ˙ i ) = Vi , 1 ≤ i ≤ m. satisfying the interpolation conditions s(ti ) = Pi and s(t The approach to solve Problem (P) closely follows the one developed in [8] for Euclidean spaces, spheres and the Lie group of rotations. The idea is

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  that each spline segment si in the interval ti , ti+1 is constructed by using the analogue to a convex combination of two geodesic curves li and ri , that we call left and right component, respectively. Geometrically, the point si (τ ) is obtained by considering the geodesic arc that joins li (τ ) (at t = 0) to ri (τ ) (at t = 1), parameterized by a real valued function φ, called smoothing function. So, let φ be a real valued smooth function satisfying the boundary conditions φ(0) = 0

φ(1) = 1

˙ ˙ φ(0) = φ(1) = 0.

(12)

Remark 1. It can be easily checked that the smooth function φ defined by φ(t) = 3t2 − 2t3 , satisfies the required boundary conditions (12). If a greater degree of smoothness of the spline curve at the knots was desired, then it would be suffice to increase the degree of the polynomial φ. According to Lemma 3.2 of [8], if φ is defined by φ(t) = γ

k  

k (−1)i k+1+i t , i k+1+i i=0

k k (−1)i k where γ −1 = i=0 i k+1+i , then the spline curve s is C and satisfies the required boundary conditions. In order to prove some of the statements in the current section, one needs the expression for the derivative of the matrix exponential. Lemma 1 (Sattinger and Weaver [12]). If t → A(t) is a matrix real valued smooth function, then   −u   A(t)  d A(t) eu − 1  A(t) 1 − e ˙ ˙  e A(t) e A(t) . = =e   dt u u u=adA(t) u=adA(t) 3.1

Interpolation Algorithm on the Grassmannian

According to the characterization of the tangent space to a point on the Grassmannian  by (2), for each Vi ∈ TPi Gn,k , there exists Xi ∈ soPi (n) such that  given V i = X i , Pi . Theorem 2. Let φ be a real valued smooth function satisfying (12), and li , ri be the Grassmannian valued smooth functions defined by li (t) = etXi Pi e−tXi and ri (t) = e(t−1)Xi+1 Pi+1 e−(t−1)Xi+1 . Then the curve t → si (t) = eφ(t)Yi (t) li (t)e−φ(t)Yi (t) , where   Yi (t) = 12 log I − 2ri (t) I − 2li (t) , is smooth and satisfies the interpolation conditions si (0) = Pi

si (1) = Pi+1

s˙ i (0) = Vi

s˙ i (1) = Vi+1 .

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Proof. Since φ(0) = 0 it is immediate to see that si (0) = li (0) = Pi . Now, using φ(1) = 1, it turns out that si (1) = eYi (1) li (1)e−Yi (1) , which is exactly the end point of the geodesic joining li (1) to ri (1). Thus, si (1) = ri (1) = Pi+1 . In order to prove the last two boundary conditions, differentiate si (t) with respect to t and write   eu − 1  φ(t)Yi (t) ˙ ˙ ˙ φ(t)Y li (t)e−φ(t)Yi (t) s˙ i (t) = i (t) + φ(t)Yi (t) si (t) + e  u u=adφ(t)Yi (t)   eu − 1  ˙ ˙ φ(t)Y − si (t) i (t) + φ(t)Yi (t)  u u=adφ(t)Yi (t)    u  e − 1  φ(t)Yi (t) ˙ ˙ ˙ φ(t)Y li (t)e−φ(t)Yi (t) = i (t) + φ(t)Yi (t) , si (t) + e u u=adφ(t)Y (t) i

(13) ˙ Using conditions φ(0) = φ(0) = 0, one concludes immediately that s˙ i (0) = l˙i (0) = Vi . To prove the last assertion, notice that ri (t) = eYi (t) li (t)e−Yi (t) , and therefore,     eu − 1  ˙ Yi (1) , ri (1) + eYi (1) l˙i (1)e−Yi (1) , r˙i (1) = u u=adY (1) i

which, according to (13) and the boundary conditions (12), enable us to conclude that s˙ i (1) = r˙i (1) = Vi+1 . Based on the previous results, we next present an algorithm to generate an interpolating C 1 -smooth spline on the Grassmanian with prescribed points and velocities. 3.2

Interpolation Algorithm on the Stiefel Manifold

According to the characterization of the tangent space to a point on the Stiefel manifold given in Proposition 2, for each Vi ∈ TPi Stn,k , there exist Xi ∈ soPi PiT (n) and Ωi ∈ so(k), such that Vi = Xi Pi + Pi Ωi . The next result describes the way of constructing a spline curve t → s(t) on the Stiefel manifold that completely solves problem (P). We notice that, in this case, the left and right components of each spline segment have been joined using the quasi-geodesic expression derived in Theorem 1. Theorem 3. Let φ be a real valued smooth function satisfying (12) and li , ri be the smooth functions taking values on Stn,k , defined by   T T li (t) = et Xi Pi +Pi Ωi Pi Pi and ri (t) = e(t−1) Xi+1 Pi+1 +Pi+1 Ωi+1 Pi+1 Pi+1 . Then the curve t → si (t) = eφ(t)Xi (t) li (t)eφ(t)Ωi (t) , where   Xi (t) = 12 log I − 2ri (t)riT (t) I − 2li (t)liT (t)  Ωi (t) = log liT (t)e−Xi (t) ri (t) ,

(14) (15)

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Algorithm 1: compute a C 1 − smooth curve t → s(t) ∈ Gn,k , for t ∈ [0, 1], ˙ i ) = Vi = [Xi , Pi ], for 1 ≤ i ≤ m. such that: s(ti ) = Pi and s(t Input: 0 = t1 ≤ t2 ≤ · · · ≤ tm = 1

(partition of the unit interval [0, 1])

P1 , . . . , P m

(m points in Gn,k )

X1 , . . . , Xm , (Xi ∈ soPi (n)) (m vectors) φ(t) = 3t2 − 2t3

(smoothing function)

Output: t → s(t) 1

2

Compute the left and right components of each segment of the spline: li (t) = etXi Pi e−tXi ri (t) = e(t−1)Xi+1 Pi+1 e−(t−1)Xi+1 Compute the velocity vector of the geodesic joining li (t) to   ri (t): Yi (t) = 12 log I − 2ri (t) I − 2li (t)

3

Compute the spline segment si at t, t ∈ [0, 1], for 1 ≤ i ≤ m − 1:

4

si (t) = eφ(t)Yi (t) li (t)e−φ(t)Yi (t) Designe the spline curve s, by using  t−t  i , t ∈ [ti , ti+1 ], 1 ≤ i ≤ m − 1. s(t) = si ti+1 − ti return t → s(t)

is smooth and satisfies the interpolation conditions si (0) = Pi

si (1) = Pi+1

s˙ i (0) = Vi

s˙ i (1) = Vi+1 .

Proof. It is immediate to check that si (0) = li (0) = Pi . To prove that si (1) = ri (1) = Pi+1 , note that Xi (1) given by (14) is the velocity vector of the geodesic that joins li (1)liT (1) to ri (1)riT (1) on the Grassmann manifold. So, si (1) = eXi (1) li (1)eΩi (1) = eXi (1) li (1)liT (1)e−Xi (1) ri (1) = ri (1)ri (1)T ri (1) = ri (1). In order to prove the derivative conditions, differentiate si with respect to t and write   eu − 1  φ(t)Xi (t) ˙ ˙ ˙ φ(t)X s˙ i (t) = li (t)eφ(t)Ωi (t) i (t) + φ(t)Xi (t) si (t) + e  u u=adφ(t)Xi (t)   1 − e−u  ˙ ˙ φ(t)Ω + si (t) i (t) + φ(t)Ωi (t) .  u u=adφ(t)Ω (t) i

According to the boundary conditions (12) of function φ, it is immediate to check that s˙ i (0) = l˙i (0) = Xi Pi + Pi Ωi PiT Pi = Xi Pi + Pi Ωi = Vi .

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In order to prove the remaining condition, first notice that T Pi+1 = Xi+1 Pi+1 + Pi+1 Ωi+1 = Vi+1 , r˙i (1) = Xi+1 Pi+1 + Pi+1 Ωi+1 Pi+1

and secondly, according to (15), notice also that eXi (t) li (t)eΩi (t) = eXi (t) li (t)liT (t)e−Xi (t) ri (t) = ri (t)riT (t)ri (t) = ri (t). Finally,

  eu − 1  X˙ i (1) ri (1) + eXi (1) l˙i (1)eΩi (1) r˙i (1) =  u u=adXi (1)   1 − e−u  Ω˙ i (1) = s˙ i (1). + ri (1)  u u=adΩ (1) i

We now proceed with the algorithm to generate the interpolation curve on the Stiefel manifold that completely solves problem (P). Algorithm 2: compute a C 1 −smooth curve t → s(t) ∈ Stn,k , for t ∈ [0, 1], ˙ i ) = Vi = Xi Pi + Pi Ωi , for 1 ≤ i ≤ m. such that: s(ti ) = Pi and s(t Input: 0 = t1 ≤ t2 ≤ · · · ≤ tm = 1

(partition of the unit interval [0, 1])

P1 , . . . , P m

(m points in Stn,k )

X1 , . . . , Xm , (Xi ∈ soPi P T (n)) (m vectors in soPi P T (n)) i

i

Ω1 , . . . , Ωm , (Ωi ∈ so(k))

(m vectors in so(k))

φ(t) = 3t2 − 2t3

(smoothing function)

Output: t → s(t) 1

Compute the left and right components of each segment of the spline:   T i Ωi Pi li (t) = et Xi +P Pi   T ri (t) = e(t−1) Xi+1 +Pi+1 Ωi+1 Pi+1 Pi+1

2

Compute the velocity vector of the quasi-geodesic joining li (t) to ri (t):    Xi (t) = 12 log I − 2ri (t)riT (t) I − 2li (t)liT (t)   Ωi (t) = log liT (t)e−Xi (t) ri (t)

3

Compute the spline segment si at t, t ∈ [0, 1], for 1 ≤ i ≤ m − 1: si (t) = eφ(t)Xi (t) li (t)eφ(t)Ωi (t)

4

Design the spline curve s, by using  t−t  i , s(t) = si ti+1 − ti return t → s(t)

t ∈ [ti , ti+1 ],

1 ≤ i ≤ m − 1.

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189

Conclusion

In this paper we proposed a simple geometric algorithm to solve an interpolation problem on the Grassmann or Stiefel manifolds, where not only the position, but also the velocity, are required to change smoothly. According to the explicit expression for geodesics or quasi-geodesics that join two given points, the algorithm is simple to implement and it has a reduced complexity when compared with other geometric algorithms like, for instance, the De Casteljau algorithm [2,9]. The main drawback is that it requires frequently the computation of logarithms of orthogonal matrices. We sometimes have to face the problem of having orthogonal matrices with eigenvalue −1. Overcoming this issue, using for instance local approximations of the matrix logarithm, is the focus of our current research. Acknowledgements. Work supported by OE – national funds of FCT/MCTES (PIDDAC) under project UID/EEA/00048/2019.

References 1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008) 2. Batista, J., Krakowski, K.A., Machado, L., Martins, P., Silva Leite, F.: Multi-source domain adaptation using C 1 -smooth subspaces interpolation. In: IEEE International Conference on Image Processing (ICIP), pp. 2846–2850 (2016) 3. Batzies, E., H¨ uper, K., Machado, L., Silva Leite, F.: Geometric mean and geodesic regression on Grassmannians. Linear Algebra Appl. 466, 83–101 (2015) 4. Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998) 5. Helmke, U., Moore, J.: Optimization and Dynamical Systems. Springer-Verlag, London (1994) 6. Helmke, U., H¨ uper, K., Trumpf, J.: Newton’s method on Grassmann manifolds (2007). arXiv:0709.2205 7. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991) 8. Jakubiak, J., Silva Leite, F., Rodrigues, R.: A two-step algorithm to smooth spline generation on Riemannian manifolds. J. Comput. Appl. Math. 194, 177–191 (2006) 9. Krakowski, K.A., Machado, L., Silva Leite, F., Batista, J.: A modified Casteljau algorithm to solve interpolation problems on Stiefel manifolds. J. Comput. Appl. Math. 311, 84–99 (2017) 10. Lui, Y.M.: Advances in matrix manifolds for computer vision. Image Vis. Comput. 30, 380–388 (2012) 11. Perumalsamy, G., Visweswaran, V., Jose, J.: Joseph Winston, S., Murugan, S.: Quintic interpolation joint trajectory for the path planning of a serial two-axis robotic arm for PFBR steam generator inspection. In: Badodkar, D., Dwarakanath, T. (eds.) Machines, , Mechanism and Robotics. Lecture Notes in Mechanical Engineering, pp. 637–648. Springer, Singapore (2019) 12. Sattinger, D.H., Weaver, O.L.: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. Springer, Heidelberg (1980) 13. Srivastava, A., Turaga, P.: Riemannian Computing in Computer Vision. Springer, Heidelberg (2016)

Dynamic Model for the pH in a Raceway Reactor Using Deep Learning Techniques Pablo Ot´ alora1 , Jos´e Luis Guzm´an1(B) , Manuel Berenguel1 , and Francisco Gabriel Aci´en2 1

2

Department of Informatics, University of Almer´ıa, Ctra. Sacramento s/n, ceiA3, CIESOL, 04120 Almer´ıa, Spain {p.otalora,joseluis.guzman,beren}@ual.es Department of Chemical Engineering, University of Almer´ıa, Ctra. Sacramento s/n, ceiA3, CIESOL, 04120 Almer´ıa, Spain [email protected]

Abstract. This paper presents a black-box dynamic model for microalgae production in raceway reactors. The black-box model, developed using Deep Learning techniques, allows the estimation of the pH in a 100 m2 raceway reactor. The model has been created using only and exclusively data, what gives a high ease of use. The results obtained verify the effectiveness of this type of techniques for the modelling of complex dynamic processes. The model was validated for different weather conditions obtaining satisfactory results. Thus, the obtained model is fairly useful for simulation purposes or for the implementation of model-based control techniques. Keywords: Deep Learning Raceway reactor

1

· Neural network · Microalgae production ·

Introduction

Microalgae production is a process with an increasing interest due to the high variety of its applications. Examples can be found in derived products for cosmetics, animal food or human nutrition. Moreover, the production process is useful for wastewater treatment, eliminating pollutants such as phosphorus or nitrogen, or to mitigate CO2 emissions from other industrial facilities. Typically, microalgae cultivation can be accomplished in two different ways: in tubular photobioreactors and in open reactors or “raceways”. The first ones take place in an environment where microalgae conditions are strongly controlled [4], while the second ones are carried out in large open ponds. This last type of photobioreactor, despite being susceptible to external contaminants and incapable of controlling their temperature, has the advantage of being less expensive and more easily scalable, making them the most commonly used at commercial scale. However, conventional raceway reactors are unable to maximize biomass production c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 190–199, 2021. https://doi.org/10.1007/978-3-030-58653-9_18

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capacity mainly because of inadequate fluid-dynamic and mass transfer capacity. Thus, an optimization of the process is required to reduce costs as much as possible and to increase the production, what is closely related to reach optimal values for pH, dissolved oxygen, temperature, light integration, and CO2 injections [3]. For this purpose, the development of models and the implementation of advanced control strategies is essential. In literature, a lot of effort has been made to develop nonlinear models to describe the dynamics of the microalgae system variables [2,3,5,6]. These models are extremely relevant, as they are a key element to optimize the system design and its operation mode. Most of the available nonlinear models are based on first principles balances, which are very useful tools for the process understanding. However, these models have a high complexity and are subject to parameter uncertainties, since many of the biological parameters are very complicated to be perfectly calibrated all the time. In the last decades, because of the increasing computer capacity, Machine Learning, and more specifically, Deep Learning or Neural Networks, are becoming more relevant in the development of models for different fields [1]. These algorithms are capable of developing a model based solely on the data, without any physical meaning and without being explicitly programmed for it [9]. So, this paper deals with the development of a “black box” model for the pH of a raceway reactor making use of this type of techniques [17]. The core idea consists in obtaining a robust dynamic model that is easily updatable, based only on data, and well adapted to any circumstance that may take place in the system. As described above, the microalgae production process depends on solar irradiance and many other variables, such pH, dissolved oxygen, or medium temperature. Since the light requirements and temperature cannot be manipulated during normal operation, the pH and DO are the typical variables to be controlled and kept close to given optimal values. Among all the variables, the pH is the most important one in the process [13,15,16]. For that reason, the model presented in this paper is focused on the pH estimation based on the rest of the variables, which are assumed to be measured in the system.

2 2.1

Materials and Methods Microorganism and Culture Medium

The microalgae strain modelled in this work was Scenedemus almeriensis (CCAP 276/24). This strain is resistant to temperatures up to 45 ◦ C and pH values up to 10, although the optimal values for its growth are 35 ◦ C and a pH of 8. The medium used in the experiments was Arnon, prepared by fertilizers instead of pure chemicals. 2.2

Raceway Reactor

All the data used were taken from the raceway reactor located at the Research Centre “Las Palmerillas” (36◦ 48’ N–2◦ 43’ W), property of the Cajamar Foundation (Almer´ıa, Spain), in the year 2016 [5]. The reactor, as shown in Fig. 1,

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is composed of two 50 m long channels connected at their ends by 180◦ bends. It also has a 0.59 m3 pit 1 m away from the curve of one of the two channels, where air or CO2 is injected through a diffuser in order to control the variables of interest (dissolved oxygen and pH). The liquid is propelled by a wheel with 8 blades of 1.2 m in diameter, driven by a speed-controllable electric motor. The reactor can be divided into 3 main parts depending on these elements: the paddle wheel, the pit and the channels. Each of these points has a different pH and dissolved oxygen value, which are measured separately.

Fig. 1. Real view of the raceway photobioreactor.

2.3

Variables of Interest

In the performed tests, the following variables were measured with a sampling period of one minute: – – – –

Dissolved oxygen, pH, and medium temperature. Medium level. Air, CO2 and medium flow rates being injected. Solar radiation and ambient temperature.

Thus, all the previous variables have been used to develop the proposed black-box model in order to estimate the pH variable. 2.4

Deep Learning for Dynamic Modelling

Computational learning algorithms are able to “learn” from data to obtain models with different purposes without being explicitly programmed for it [1]. The use of one or another type of algorithms will depend on the problem and the available data.

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Neural networks are within this set of algorithms [7]. These are intended to emulate the functioning of biological neurons through a structure formed by layers of nodes and connections between them, predominantly in parallel. Each node or neuron has a series of inputs and a dependent output whose relationship is expressed in the following equation [10]:    Wk xk + b (1) y=φ k

where y symbolizes the output of the node, xk the value of each input k of the node, Wk is the weight of each input k, b is the bias of the node, and φ is its activation function. Thus, if the values of Wk and b are known for all the nodes, as well as their activation function, it is possible to obtain the network output for any combination of inputs. The objective of the learning algorithm is therefore to determine the Wk and b values for each neuron that minimize the difference between the predicted and actual process output. The process to solve this problem can be divided into two main stages: data processing and network training. Data Processing. In any computational learning algorithm, the quality of the resulting model is directly proportional to the quality of the data. The more data we have and the higher the quality, the better predictions we can expect. Likewise, if the data are erratic or insufficient, it will be impossible to obtain reasonable results no matter how much the network is trained. Thus, this stage is critically important, besides taking up most of the time of network development. Usually, the raw data records available for any problem are not suitable to be directly assumed by the network, either due to sensor noise, wrong samples, or data gaps. Therefore, it is necessary to standardize the data and mitigate any irregularities in them. Some techniques used for this purpose are data interpolation, filtering or directly removing too poor sets. How this work has been done in this paper will be described later on. Once the data have been treated, it will be separated into two sets: one set destined to train the network, which will cover around 70% of the total data, and a second set whose purpose is to test the network trained by the first set, in order to ensure that the model is not exclusively focused on memorizing the training data, but also has the ability to generalize to other different situations. Network Training. The training of the network will be carried out once the processed data are available. For this purpose, it is necessary to define a series of elements and parameters that will shape it. Firstly, it is essential to determine a proper network architecture, that is, the layers that will constitute the network. This is frequently a not deterministic process, as several iterations are necessary to compare between them and to select the appropriate layers. A higher number of layers will give us a more complex network that can better adapt to different behaviours, but it is also possible to produce overfitting, which is the over memorization of training data.

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Besides the number of layers, it is necessary to determine their type, as well as their number of nodes. This depends on the kind of problem that is being faced: regression or classification, involvement of temporary component, the need to avoid overfitting etc. The architecture developed for this paper and its justification will be explained later on. Furthermore, there are certain parameters that will affect the learning process, regardless of the network architecture. These are the number of epochs and the batch size. One epoch equals one network pass through the entire data set, while the batch size defines the number of samples the network passes through before its parameters are adjusted. Therefore, if there is a set of 100 samples and it is trained during 500 epochs with a batch size of 20, the network will recalculate its parameters 5 times per epoch and 2500 times overall. These parameters are fundamental not only for the time it takes to train the net, but also to improve learning and prevent overfitting. After the architecture and parameters of the network have been selected, the next step is the training stage. It is important to subsequently validate the network obtained using the test set in order to achieve the most accurate and robust prediction possible.

3 3.1

Results Model Development

The proposed model is a black-box model, which is not intended to demonstrate or represent the physical interactions between the variables, but rather to obtain the correlation between each of the system variables and the pH at the desired point (at the end of the channel). Since it is a dynamic system, time plays a crucial role and the model must reflect this issue. For this purpose, a LSTM (Long Short Term Memory) layer was selected [14], which stores the ‘network state” at each instant. So, the model output at a given time does not only depend on the inputs at that time, but also on the network state. The data processing, model development and model validation have been done in the MATLAB environment and using of the Deep Learning Toolbox [8,10–12]. Due to the use of this type of layer, it is necessary to guarantee the continuity of the data. Therefore, the data processing stage begins by discarding the data sets of days in which a large amount of data is missing. If the data gaps last only a few moments, it is not necessary to discard the day, but if the gap is long enough to make interpolation illogical, the dataset must be deleted. In the case of small data gaps, the interpolation of each sample instant shall be made between the nearest previous and next instants whose measurement is correct. To improve the network training and performance, the mean and standard deviation of each of the variables shall be calculated to normalize them according to the following expression: X −μ (2) Xn = σ where Xn symbolizes the already normalized variable, μ is the arithmetic mean, σ the standard deviation, and X the raw variable.

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Fig. 2. Diagram of the selected network architecture.

A data set of 105 days was originally used to develop the proposed model. After the data processing stage and analysis, 27 days were discarded because of errors and gaps in the measurements. Thus, a data set of 78 days was finally considered, where 60 of them were randomly selected to constitute the training set, and the remaining 18 for validation purposes. As previously mentioned, the core of the proposed network is the LSTM layer. Thus, the first layer set will be a sequential input layer, having as many nodes as input variables are used to make the predictions. The purpose of this layer is to serve as the data entry. The second layer will be the LSTM. A sequential output mode will be configured for it, as well as a number of nodes between 200 and 300, This parameter was selected after several tests with different values, picking the one that had the best performance. After that, a fully connected layer of 50 nodes will be established, which will work as an intermediate layer to provide more depth to the network. Following this, the use of a dropout layer is optional. The utility of this layer is to ignore a percentage of the data in each iteration. In normal circumstances, this is not positive for the network, but in case of overfitting, it is very helpful. Finally, another fully connected layer will be introduced with a single node for the output, and a last regression layer. In Fig. 2 a diagram of the selected architecture for this work is shown, where all the stages described above are summarized. The number of nodes in each layer has been selected after several iterations, in order to make it as low as possible to accelerate the training of the network, but large enough so that it doesn’t deteriorate the prediction. Regarding the learning parameters, different numbers of epochs have been tested. It has been demonstrated that a higher number of epochs leads to better results, and that overfitting does not take place with less than 3000 epochs. Since above a certain number of epochs the difference is not significant, the final value given to this parameter has been 2000 epochs. Besides, as there are 60 days available for training, the selected batch size has been 20, so that three iterations are carried out in each period and no data is left out. The optimizer ‘Adam’ was used, with an initial learning ratio of 0.01.

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3.2

Model Results

In this section, validation results of the proposed model are presented. The performance of the network has been evaluated for two different purposes: onestep prediction and multi-step prediction. In the first case, the network is used as a regression model, where the network state is updated with the real value of the pH variable at each instant time. Thus, only predictions for one step ahead are performed. On the other hand, in the multi-step prediction case, the network is used as an independent model, where the network state is updated by using the own model predictions. For both cases, the Root Median Square Error (RMSE) metric was used to analyze the goodness of fit of the model:  (3) RM SE = μ((Yreal − Ypred )2 )

Fig. 3. Radiation and CO2 flow rate profiles for four representative days.

In this paper, four different days have been randomly selected to show the model results. Figure 3 shows some of the variables for these days, which are used as inputs for the proposed model. Notice that days with many different input profiles are considered for the validation process. First of all, the performance of the network will be checked by performing the one-step prediction. The test data of all the variables are available for each instant and the aim is to predict the pH value in the following instant. In this test, the results are really promising with RMSE value of 0.1082. Figure 4 shows the obtained results for the data set presented in Fig. 3, where it is observed that the model behaves really well for all the data sets. This solution for one-step prediction can be very useful for fault detection techniques or real time estimation. However, for simulation or control purposes the prediction horizon of a single sample instant is insufficient.

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Fig. 4. Experimental and predicted data of pH for four representative days with a single-step prediction.

For this reason, the multi-step prediction was subsequently implemented. The basis of this test is the same as the previous one, with the difference that for the prediction of an instant k + 1 the pH value at the instant k will not be the real one, but the one predicted at the previous instant. This makes it possible to predict the pH value indefinitely. However, there are some limitations to be considered. Notice that the resulting network has a certain error in the predictions such as observed in the one-step prediction case. This means that, since future predictions depend on values with a slight error, that error is fed back, and therefore increases proportionally to the number of sample instants that are intended to be known. Therefore, the prediction was made for a whole day, which corresponds to 1440 samples. Despite the aforementioned, the results obtained are highly satisfactory, as can be seen in the Fig. 5. As expected, the prediction error is increased, reaching a value of 0.3720, but even so the network is able to predict a complete day with considerable accuracy, especially in the instants when the control was taking place during the real experiments. This performance is mainly related to the use of dissolved oxygen as an input to the network, since this variable is highly correlated to the pH. Facing future works, the aim is to develop a network with a similar structure capable of predicting both variables, in order to achieve a complete simulation of the system. Notice that even in time lapses when a small offset appear, the dynamics of the system are well represented during all the daily operation.

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Fig. 5. Experimental and predicted data of pH for four representative days with a multi-step prediction.

4

Conclusions

In this paper, a dynamic model based on neural networks has been developed and validated for a raceway photobioreactor. The model allows the prediction of the pH value in the channels with a time horizon up to one day based only on measurable data inputs. The methodology followed in the article grants the possibility of obtaining similar models for the prediction of other variables of interest, such as dissolved oxygen or biomass concentration. This type of models opens a wide variety of possibilities for future works in the field of photobioreactor modelling and control, providing highly reliable non-linear models that are easily updateable and whose calibration is fully automated, depending only on measurable data. Acknowledgements. This work has been partially funded by the following projects: DPI2017 84259-C2-1-R (financed by the Spanish Ministry of Science and Innovation and EU-ERDF funds) and the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 727874 SABANA.

References 1. Amini, M., Chang, S.: A review of machine learning approaches for high dimensional process monitoring. In: IISE Annual Conference and Expo 2018, Orlando, USA, pp. 390–395 (2018) 2. del Rio-Chanona, E.A., Wagner, J.L., Ali, H., Fiorelli, F., Zhang, D., Hellgardt, K.: Deep learning-based surrogate modeling and optimization for microalgal biofuel production and photobioreactor design. AIChE J. 65(3), 915–923 (2019)

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3. Fern´ andez, I., Aci´en, F.G., Berenguel, M., Guzm´ an, J.L.: First principles model of a tubular photobioreactor for microalgal production. Ind. Eng. Chem. Res. 53(27), 11121–11136 (2014) 4. Fern´ andez, I., Aci´en, F.G., Berenguel, M., Guzm´ an, J.L., Andrade, G.A., Pagano, D.J.: A lumped parameter chemical-physical model for tubular photobioreactors. Chem. Eng. Sci. 112, 116–129 (2014) 5. Fern´ andez, I., Aci´en, F.G., Guzm´ an, J.L., Berenguel, M., Mendoza, J.L.: Dynamic model of an industrial raceway reactor for microalgae production. Algal Res. 17, 67–78 (2016) 6. Garc´ıa-Ma˜ nas, F., Guzm´ an, J.L., Berenguel, M., Aci´en, F.G.: Biomass estimation of an industrial raceway photobioreactor using an extended Kalman filter and a dynamic model for microalgae production. Algal Res. 37(June 2018), 103–114 (2019) 7. Gupta, A.: Introduction to deep learning: part 1. Chem. Eng. Prog. 114(6), 22–29 (2018) 8. Hudson, M., Martin, B., Hagan, T., Demuth, H.B.: Deep Learning ToolboxTM User’s Guide (1992) 9. Kim, B.S., Kang, B.G., Choi, S.H., Kim, T.G.: Data modeling versus simulation modeling in the big data era: case study of a greenhouse control system. Simulation 93(7), 579–594 (2017) 10. Kim, P.: MATLAB Deep Learning (2017) 11. Mathworks: Deep learning in MATLAB. The MathWorks, Inc. (2018). https:// www.mathworks.com/help/nnet/deep--learning-12. Mathworks: Practical Deep Learning Examples with MATLAB, p. 33 (2018) 13. Pawlowski, A., Guzm´ an, J.L., Berenguel, M., Aci´en, F.G.: Control system for pH in raceway photobioreactors based on wiener models. IFAC-PapersOnLine 52(1), 928–933 (2019) 14. Pon Kumar, S.S., Tulsyan, A., Gopaluni, B., Loewen, P.: A deep learning architecture for predictive control. IFAC-PapersOnLine 51(18), 512–517 (2018) 15. Posadas, E., Morales, M.D.M., Gomez, C., Aci´en, F.G., Mu˜ noz, R.: Influence of pH and CO2 source on the performance of microalgae-based secondary domestic wastewater treatment in outdoors pilot raceways. Chem. Eng. J. 265, 239–248 (2015) 16. Wu, Z., Zhu, Y., Huang, W., Zhang, C., Li, T., Zhang, Y., Li, A.: Evaluation of flocculation induced by pH increase for harvesting microalgae and reuse of flocculated medium. Biores. Technol. 110, 496–502 (2012) 17. Zhang, S., Zaiane, O.R.: Comparing deep reinforcement learning and evolutionary methods in continuous control. In: NIPS 2017 Deep Reinforcement Learning Symposium, Long Beach, USA (2017)

Double Back-Calculation Approach to Deal with Input Saturation in Cascade Control Problems ´ Marta Leal1 , Angeles Hoyo1 , Jos´e Luis Guzm´an1(B) , and Tore H¨ agglund2 1

2

Department of Informatics, ceiA3, CIESOL, University of Almer´ıa, Ctra. Sacramento s/n, 04120 Almer´ıa, Spain {angeles.hoyo,joseluis.guzman}@ual.es Department of Automatic Control, Lund University, Box 118, 22100 Lund, Sweden [email protected]

Abstract. This paper presents a solution to the saturation problem in cascade control schemes. When cascade control approaches work in linear mode without saturation influence, important improvements can be achieved in industrial control loops. The effect of disturbances and/or nonlinear actuator behaviours on the main process variables can be considerably reduced. However, when saturation arises in the inner loop, these improvements cannot be reached and even sometimes the saturated cascade control scheme gives worse results than a single control loop. Thus, this work analyzes this situation and introduces an alternative solution to solve this problem and to reduce the impact of the saturation effect. Keywords: Cascade control control · Process control

1

· Anti-windup · Back-calculation · PID

Introduction

Cascade control is a very common control structure in process control [1,2]. There are two different cascade control approaches, series and parallel, but the series one is dominating in process control and is the one treated in this paper [3]. There are two major reasons for the wide use of cascade control. The first one is that it provides a fast and efficient compensation of load disturbances entering the inner loop with respect to a single control loop. The second one is that it provides an easy way to compensate for nonlinearities in the inner loop, typically nonlinear actuators [8]. However, a problem with the cascade control structure is that antiwindup is non-trivial and must be treated properly. Integrator windup may occur in all controllers with integral action when a signal in the control loop becomes saturated. This is a well-known problem, and practically all PID controllers are equipped with antiwindup features to avoid windup when the control signal saturates. However, windup may also occur for c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 200–209, 2021. https://doi.org/10.1007/978-3-030-58653-9_19

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other reasons, e.g. when limiters or selectors are used outside the controller or the controller block. In cascade control, windup may occur in the outer controller because of limitations in the inner control loop. These limitations may be caused by mode switches, from automatic to manual mode or from external to internal setpoint. However, the most common reason is that the control signal of the inner controller gets saturated. This is the problem treated in this paper. When the control signal of the inner controller becomes saturated, integral action in the outer controller must be inhibited to avoid windup. This is a problem that has not been well studied in the literature and only a few contributions mentioned this issue [4–6]. Ad-hoc solutions can be found to solve the saturation problem in specific practical applications. One approach that is sometimes used in industry is to make the control signal in the outer loop tracking the process output of the inner loop when the inner loop saturates. This requires some logic information to perform this mode switch. Some other solutions are based on reference governor approaches, where the outer loop is based on MPC or a similar control algorithm that deals with constraints [7]. In this paper, we propose an alternative solution that is based on back calculation. Back calculation is one of the most common antiwindup methods used in PID controllers [2]. It has the advantages that no logical signal or mode switches are needed, and that a tracking-time constant can be set to tune the properties of the antiwindup. In the following sections, the problem is explained in more detail, the proposed solution is presented, and finally it is compared with other solutions.

2

Cascade Control

Figure 1 shows the classical cascade control approach that is used in this paper to analyze the problem. The control approach is composed by two control loops, the inner loop and the outer loop. The signals, controller parameters, and process transfer functions in the control scheme are represented with sub-indexes 1 and 2 to refer the outer and the inner loop, respectively. So, the process outputs are represented by yj , the control signals by uj and the set-points by rj ; where j = {1, 2}. Notice that the control signal of the outer control loop, u1 is the set-point value for the inner loop, u1 = r2 . A load disturbance in the inner loop, d has been also considered. In both control loops, a PI controller is used as feedback controller that is represented by the following transfer function:   sTij + 1 (1) Cj (s) = Kj Tij s where Kj is the proportional gain and Tij is the integral time. Notice that as the paper is focused on the analysis of the saturation problem, the derivative term is omitted for the sake of simplicity.

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Fig. 1. Cascade control scheme with classical back-calculation approach.

Both PI controllers are implemented with an antiwindup scheme based on the back-calculation approach. So, the corresponding tracking constants, Ttj are included to each PI control algorithm such as shown in Fig. 1. The process dynamics, P1 (s) and P2 (s), with the following transfer functions have been considered: P1 (s) =

Kp1 e−sL1 , sT1 + 1

P2 (s) =

Kp2 e−sL2 s

(2)

The process dynamics in the inner loop is an integrator with delay in order to better show the effect of the saturation problem.

3

Saturation Problem Solutions

The saturation problem in a control loop arises when the control variable reaches the limits of the actuator. When this happens, the feedback loop is broken and the actuator will remain at its limit regardless the control error. So, when the integral action is used in both control loops within a cascade control scheme, it is necessary to have an approach to reduce the saturation effect or windup phenomena. Such as commented above, this issue has not received so much attention in the literature. The main problem with the saturation in a cascade control architecture appears when the inner loop is saturated. In that case, the windup effect occurs in the inner loop and is propagated to the outer loop. Thus, antiwindup techniques, such as the back-calculation solution (see Fig. 1), can be included in the inner loop to reduce the saturation effect. This solution allows to reduce the saturation time in the inner loop, but the windup effect is still transmitted to the outer loop and the control performance of the main process variable is deteriorated. The key point in this problem is that the outer loop has not information about the saturation in the inner loop. Therefore, the solution comes by somehow

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Fig. 2. Cascade control scheme with the industrial switching approach.

Fig. 3. Cascade control scheme with the proposed double back-calculation approach.

notifying the outer loop when the inner loop goes into saturation. In this sense, a common industrial approach is shown in Fig. 2. A switch mode is added to the outer loop. Now, the tracking signal in the outer loop will be switched between the control signal of the outer loop, u1 , and the process output of the inner loop, y2 . When, the inner loop is not saturated, the tracking signal will be u1 to keep the anti-windup properties in the outer loop. Otherwise, when the inner loop saturates, the outer-loop tracking signal is switched to y2 , what will make the control error in the inner loop zero and thus reducing the saturation effect [2]. Note that this approach requires logic that takes care of the switching. In this paper, a new solution to the problem is proposed, which is shown in Fig. 3. The idea is to avoid using any logical signal, and just to modify the classical back-calculation scheme to deal with the windup problem. Specifically, the proposed approach consists in adding a new tracking signal to the outer loop

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within the back-calculation control scheme. Thus, the integral term in the outer loop will be modified when any of both control signals, u1 or u2 , are saturated. The important issue in this approach is that a new tuning parameter, Tt3 , is available as an extra tracking parameter, that will allow to look for a tradeoff between the saturation time in the inner loop and the performance of the process output of the outer loop (the main process variable).

4

Results

This section presents a simulation study to show the saturation problem in cascade control and the analysis of the different solutions described in the previous section. For this study, the following process transfer functions are considered: P1 (s) =

1e−10s , 100s + 1

P2 (s) =

2e−2s s

(3)

The SIMC tuning method has been selected to tune the PI controllers in both control loops [9]. So, the inner PI controller is first tuned for a closedloop time constant of 10 s, what results in the following controller parameters: K1 = 0.04167 and Ti1 = 48. Once the inner loop is designed, the outer controller is tuned considering the closed-loop inner dynamics plus the dynamics of P2 (s). Then, the SIMC method is used to tune the PI controller of the outer loop for a closed-loop time constant of 90 s, obtaining K2 = 0.9804 and Ti2 = 100 as controller parameters. In the following sections, simulation results for the tracking and load disturbance rejection cases are evaluated for the proposed example. 4.1

Reference Tracking Example

The saturation problem is first excited by applying a large setpoint change in the outer loop. For this first study, the saturation limits for the control signal in the inner loop were set to [−1 0.04], in order to show the simulation results clearly. Figure 4 shows a comparison for the cascade control scheme without antiwindup and the control approach shown in Fig. 1 where the basic backcalculation solution was considered. As observed, the saturation problem in the inner loop generates the windup effect that is propagated to the outer loop. When the back-calculation approach is used, the saturation time is considerably reduced and the performance of the process output is improved. Table 1 shows performance results using the IAE (Integral Absolute Error) for the process outputs and the saturation time for both control approaches. For the backcalculation approach, a tracking time constant Tt2 = 0.05 was used and no better results were obtained for smaller values. Therefore, no further improvement can be reached with this approach.

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Fig. 4. Saturation problem for a large setpoint change in r1 . The control schemes without antiwindup approach and with classical back-calculation technique are shown. Table 1. IAE for y1 and y2 and saturation time for u2 for results in Fig. 4. No antiwindup Classical antiwindup IAE y1

208.54

IAE y2

289.75

Saturation time 268 s

143.86 154.10 144.6 s

Fig. 5. Saturation problem for a large setpoint change in r1 . The control schemes with classical back-calculation, industrial solution, and the proposed approach are shown.

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Then, the proposed solutions presented in Sect. 3 were simulated and compared with the classical back-calculation scheme. Figure 5 shows the graphical results and Table 2 the performance indices. It can be seen that the industrial solution (control scheme from Fig. 2) and the proposed double back-calculation approach (control scheme from Fig. 3) provide very similar results. Both solutions reduce the saturation time with respect to the classical back-calculation scheme, and the performance of the process outputs are substantially improved. For the proposed approach, the new tracking time constant was set to a value of Tt3 = Tt2 /10 = 0.005. Notice that in the case of the industrial solution, there is no chance to modify the response as we can just apply the control scheme. However, with the approach proposed in this paper, a new degree of freedom is available through the tracking time constant Tt3 . Figure 6 shows a new example for three different values of Tt3 (namely, 0.005, 0.0005, 0.00005) and where the result is compared with the industrial approach. In this figure, only the main process output y1 and the inner control signal u2 are displayed to show the results better. As observed, it is possible to reduce the saturation time below the industrial solution. From the obtained results, it can also be deduced that it is possible to tune Tt3 to look for a tradeoff between saturation time in the inner loop and performance of the process outputs, being this the main advantage of the new proposed approach. Table 2. IAE for y1 and y2 and saturation time for u2 for results in Fig. 5. Classical Antiwindup Industrial solution Proposed solution

4.2

IAE y1

143.86

117.4788

111.86

IAE y2

154.10

44.32

66.15

Saturation time 144.6 s

78.5 s

102.1 s

Disturbance Rejection Example

In this case, the reference in the outer loop is kept constant and a load disturbance is entered in the inner loop. So, the saturation problem arises because of the load disturbance. In this case, the saturation limits for the control signal u2 were modified to [−2, 2] in order to show the results better. The same controller parameters and tracking constants as in the previous examples were used. For the proposed approach, a tracking constant of Tt3 = 0.005 was considered. Figure 7 shows the graphical simulation results, and the IAE values and the saturation times are given in Table 3. Notice how when the antiwindup scheme is not considered, the performance is considerably deteriorated because of a long saturation time in the inner loop. In this case, it is interesting to see that the classical back-calculation approach and the industrial solution give the same result. However, the proposed control scheme provides much better results reducing the saturation time and improving the performance of the process outputs.

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Fig. 6. Effect of tracking constant Tt3 for the proposed control approach.

Fig. 7. Saturation problem for an incoming load disturbance d in control signal u2 . The control schemes without antiwindup, with classical back-calculation technique, industrial solution, and the proposed control approach are shown.

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Table 3. IAE and saturation time comparisons between the system with no antiwindup, classical antiwindup, industrial solution, and the new proposed solution. No

5

Classical Industrial Proposed

IAE y1

1248.80 674.05

674.05

383.12

IAE y2

2414.70 1293.70

1293.70

457.34

Saturation time 699.9 s 281.7 s

281.7 s

149.9 s

Conclusion

The saturation problem for the cascade control scheme has been evaluated for the setpoint tracking and load rejection cases. It was observed that the performance of the main process variable is considerably deteriorated when the inner loop goes into saturation when no antiwindup is applied. Then, the classical back-calculation approach was used to reduce the saturation effect, but it was demonstrated that the improvement is limited. Therefore, an industrial solution was implemented that is based on adding a switch mode in the tracking signal of the outer loop. This control approach provides the same result as the classical back-calculation scheme for the load disturbance rejection case, and it improves the response for the tracking case. However, there is no possibility to tune the desired response. Finally, a new control approach was introduced that consists in adding an extra tracking term to the outer loop. The new control algorithm obtains better results for the tracking and load disturbance cases, and also it provides a new tuning parameter that allows to a tradeoff between the saturation-time reduction and the performance of the process outputs. Acknowledgements. This work has been partially funded by the project DPI2017 84259-C2-1-R, financed by the Spanish Ministry of Science and Innovation and EUERDF funds.

References 1. Franks, R.G., Worley, C.W.: Quantitative analysis of cascade control. Ind. Eng. Chem. 48(6), 1074–1079 (1956) 2. ˚ Astr¨ om, K.J., H¨ agglund, T.: Advanced PID Control. Systems and Automation Society, ISA-The Instrumentation (2006) 3. Brosilow, C., Joseph, B.: Techniques of Model-Based Control. Prenticel Hall (2002) 4. Rehan, M., Ahmed, A., Iqbal, N.: Static and low order anti-windup synthesis for cascade control systems with actuator saturation: an application to temperaturebased process control. ISA Trans. 49, 293–301 (2010) 5. Sundari, S., Nachiappan, A.: Simulation analysis of Series Cascade control Structure and anti-reset windup technique for a jacketed CSTR. J. Electr. Electron. Eng. 7(2), 93–99 (2013) 6. Nudelman, G., Kulessky, R.: New approach for anti-windup in cascade control system. Technical report, The Israel Electric Corporation Ltd. (2002)

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7. Martin, K., Michal, K.: MPC-Based Reference Governors: Theory and Case Studies. Springer (2019) 8. Vilanova, R. and Arrieta, O.: PID tuning for cascade control system design. In: Universitat Aut Canadian Conference on Electrical and Computer Engineering, Niagara Falls, ON, pp. 1775–1778 (2008) 9. Skogestad, S.: Simple analytic rules for model reduction and PID controller tuning. J. Process Control 13(4), 291–309 (2003)

State-Space Estimation Using the Behavioral Approach: A Simple Particular Case Lorenzo Ntogramatzidis1 , Ricardo Pereira2(B) , and Paula Rocha3 1

Department of Mathematics and Statistics, Curtin University, Perth, WA, Australia [email protected] 2 CIDMA, Department of Mathematics, University of Aveiro, Aveiro, Portugal [email protected] 3 SYSTEC, Faculty of Engineering, University of Porto, Porto, Portugal [email protected]

Abstract. In this paper we apply the behavioral estimation theory developed in Ntogramatzidis et al. (2020) to the particular case of statespace systems. We derive new necessary and sufficient conditions for the solvability of the estimation problem in the presence of disturbances, and provide a method to construct an estimator in case the problem is solvable. This is a first step to investigate how our previous results, derived within the more general behavioral context, compare with the results from classical state space theory.

Keywords: Behavior

· Estimation · Disturbance · State-space

Dedicated to F´ atima Leite on the occasion of her 70th birthday.

1

Introduction

The estimation of an unmeasurable system variable from another which can be measured is a standard problem in linear state-space systems theory. A simple case is the estimation of the state from the output, but other situations can be of interest, such as the estimation of a linear function of the state, under the presence of unknown disturbances, based on the sole knowledge of the output (in case the input is taken to be zero). These problems have been addressed, among others, using the geometric approach to state-space systems, see, for instance, [1,2,6]. More recently, in [4], the estimation problem in the presence of disturbances has been considered from the standpoint of the behavioral approach introduced by J.C. Willems [7], allowing to consider more general classes of linear systems. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 210–220, 2021. https://doi.org/10.1007/978-3-030-58653-9_20

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Nevertheless, it is interesting to investigate how the results obtained within the behavioral framework apply for the case of state-space systems.This is precisely the goal of this paper, where some preliminary results are presented for the particular case of the system with one unknown disturbance, one measured output and one variable to be estimated. In Sect. 2, we present some background material on the behavioral estimation problem with unknown disturbances. Section 3 is devoted to the estimation in state-space systems using the behavioral approach. Finally, some conclusions are drawn in Sect. 4.

2

Preliminaries: The Behavioral Estimation Problem with Disturbances

In this paper we define a behavior Bw as the solution set of a system of linear differential equations with constant coefficients:  d  d w = 0 = ker R dt (1) Bw := w ∈ U : R dt where w is the behavior variable, U = C ∞ (R, Rw ), w ∈ N is the number of components of w, and R(s) is a suitably sized polynomial matrix (i.e., a matrix with entries in the ring of polynomials R[s]). For the sake of simplicity, if no confusion d and the indeterminate s, arises, we sometimes omit the differential operator dt d and write simply R instead of R dt or R(s). In the sequel, the behavior variable is often partitioned into sub-variables. If,   for instance, w is partitioned as w = w1 w2 the behavior Bw will bedenoted  by B(w1 ,w2 ) , and the matrix R will accordingly be partitioned as R = R1 R2 . In this way, the equation Rw = 0 becomes R1 w1 = −R2 w2 . The following behavioral estimation problem in the presence of unknown disturbances is a slight modification of the one introduced in [4]. (BED) problem: Given a behavior B(w0 ,w1 ,w2 ,d) described by:  d d d R0 ( dt )w0 = R1 ( dt )w1 + E( dt )d

(2)

d w2 = K( dt )w0

where w1 is the only measured variable, d is an unknown disturbance, w2 is the variable to be estimated, and, moreover, K(s) is a polynomial matrix with full row rank, design, if possible, a system d d N ( dt )w 2 = P ( dt )w1

that produces an estimate w 2 for w2 from w1 , in the sense that the estima 2 (t) − w2 (t) is asymptotically zero, i.e., lim e2 (t) = 0, tion error e2 (t) = w t→+∞

independently from d.

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Remark 1. The difference between this problem and the one considered in [4] is that here we assume K(s) to have full row rank, which is a reasonable assumption to make as it means that the components of w2 are unrelated. The results obtained in [4] allow to formulate the next theorem. Theorem 1. The BED problem for the behavior described by (2) is solvable if and only if there exist a polynomial matrix T (s), of suitable size, and a stable polynomial q(s) such that T (s)L(s)R0 (s) = q(s)K(s), where L(s) is a minimal left-annihilator of E(s). In this case, an estimator for w2 from w1 is given by d d d d q( dt )w 2 = T ( dt )L( dt )R1 ( dt )w1 .

Remark 2. • Recall that a polynomial matrix L(s) is said to be a minimal left annihilator (mla) of a given polynomial matrix M (s) if L(s)M (s) = 0 and, moreover, all the other left annihilators N (s) of M (s) are left multiples of L(s), i.e., if N (s)M (s) = 0, then N (s) = X(s)L(s) for some polynomial matrix X(s). • Recall also that a polynomial q(s) is said to be stable if all its zeros have d )w = 0 satisfy negative real part. As is well-known, the solutions w of q( dt lim w(t) = 0 if and ony if q(s) is stable. t→+∞

3

State-Space Estimation Using the Behavioral Approach

In order to illustrate the use of the results from the behavioral approach to state-space estimation, we consider the following problem. Given a state-space system: ⎧ ⎪ ⎨

d dt x(t)

= Ax(t) + Bd(t)

y(t) = Cx(t) ⎪ ⎩ z(t) = Hx(t)

(3) (4) (5)

  where A ∈ Rn×n , B ∈ Rn×1 , C, H ∈ R1×n , x(t) = x1 (t) · · · xn (t) ∈ Rn is the state, d(t) ∈ R is an unknown disturbance, y(t) ∈ R is the measured system output, and z(t) ∈ R is a variable to be estimated, find, if possible, and estimator for z from y, i.e., design a state-space system 

d (t) dt x

x(t) + By(t) = A

x z (t) = C (t) + Dy(t),

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that produces and estimate z for z such that the estimation error e(t) := z (t) − z(t) converges to zero a t tends to +∞, independently from the disturbance d and the initial conditions x(0) and x (0). Assumption: In order to study this problem, we shall assume that B = en , where   ej is the j-th vector from the canonical basis of Rn , i.e., B = 0 · · · 0 1 . This assumption entails no loss of generality, since it is always possible to make a change of variable x = Sx that brings B into this form. In the sequel I denotes the identity matrix of suitable size. (5) can be written as: ⎧       d ⎪ ⎪ ⎨ dt I − A x = 0 y + B d 1 0 C ⎪ ⎪ ⎩ z = Hx, d  I −A which is in form (2) with R0 (s) = dt , R1 (s) = C K(s) = H, w0 = x, w1 = y and w2 = z.

Note that Eqs. (3)–

(6) (7)     0 B , E(s) = , 1 0

Following the same line of reasoning as in [4], as a first step we eliminate the disturbance from Eq. (6), by pre-multiplying both of its sides by an mla of [ B0 ]. As is well-known in the behavioral theory, [7,8], this yields an equation in x and y with the same (x, y) solutions as the original one. Taking the special form of B into account, it is not difficult to see that   I 0 L(s) = 0 1 is an mla of [ B0 ], and hence (6)–(7) are equivalent to: ⎧   d    ⎪ I 0 dt I −A 0 ⎨ x= y 1 C ⎪ ⎩ z = Hx

(8) (9)

from the point of view of the (x, y, z) solutions. Therefore, the estimation of z from y can be made based on Eqs. (8)–(9). In order to simplify the notation from now on we denote the differential d d operator dt by σ, i.e., σ := dt . Clearly, I 0 (σI − A) consist of the first n−1 rows of σI −A. Thus, letting A = [aij ], C = [cj ], and H = [hj ], i, j = 1, . . . , n, (8)–(9) can be written as ⎧ ⎪ ⎨ (σI − A)x − Bu = 0 y = Cx + Du ⎪ ⎩ z = Hx + Ju

(10) (11) (12)

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where A = [aij ], B = [ain ], C = [cj ], and H = [hj ], i, j = 1, . . . , n − 1, D = cn ,   J = hn , x = x1 · · · xn−1 , and u = xn ; or, equivalently, as: ⎧     σI − A −B x 0 ⎪ ⎪ = y ⎪ ⎪ u 1 C D ⎪ ⎪ ⎪     ⎪ ⎨ R0 (σ)   R1 (σ) ⎪   ⎪ x ⎪ ⎪ =z H J ⎪ ⎪ ⎪   u ⎪ ⎩  K(σ) Now, taking into account that R0 (s) = L(s)R0 (s), R1 (s) = L(s)R1 (s) and K(s) = K(s), it follows from Theorem 1 that the behavioral estimation of z from y is possible if and only if there exist a stable polynomial q(s) and a (row) matrix T (s) such that: q(s)K(s) = T (s)R0 (s), i.e., such that:



 sI − A −B q(s) H J = T (s) . C D 



(13)

In the sequel we analyse the existence of solution q(s), T (s) to Eq. (13) (with the desired property of stability for q(s)) by considering two separate cases: the case where   sI − A −B R0 (s) := (14) C D is nonsingular, and the case where this matrix is singular. 3.1

The Nonsingular Case

Assume that R0 (s) is nonsingular, i.e., that it has nonzero determinant. Then, (13) is equivalent to:  −1 H J R0 (s) = q −1 (s)T (s).    =: G(s)



(15)

Denoting the proper rational matrix on the left-hand side of (15) by G(s), this equality means that q −1 (s)T (s) is a left factorization of G(s). Moreover, it is not difficult to show that this factorization can be taken as a left-coprime factorization of G(s). Thus q(s) is stable if and only if the poles of G(s), i.e., the zeros of q(s), are stable. Consequently, there exists a behavioral estimator for z from y if and only if G(s) is a stable transfer function. In this case, the behavioral estimator is given by:   ⎧ 0 ⎪ ⎨ T (σ)R0 (σ) x = T (σ) y 1 ⎪   ⎩ , z = H J x

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  x where x = , which, by (13), is equivalent to: u   ⎧   0 ⎪ ⎨ q(σ) H J x  = T (σ) y 1 ⎪   ⎩ z = H J x , leading to: q(σ) z = T (σ)

  0 y. 1

(16)

In order to obtain a state-space model for the behavioral estimator, note that q(s)−1 T (s) [ 01 ] = G(s) [ 01 ] is a proper rational matrix. Therefore, one can real B, C and D, so as to obtain ize (16) in state-space form, for suitable matrices A, an estimator:  x + By σ x = A x z = C + Dy. Example 1. Consider the state-space system (3)–(5) with         −2 3 0 A= , B= , C= 01 and H = 1 2 . 5 4 1 Defining, as explained above, the matrices   s + 2 −3 R0 (s) = and 0 1

  K(s) = 1 2

we have that (13) holds with the stable polynomial q(s) = s + 2 and the matrix T (s) = 1 2s + 7 as   q(s)K(s) = T (s)R0 (s) = s + 2 2s + 4 . Since R0 (s) is non singular and q(s)−1 T (s) [ 01 ] = 2s+7 s+2 is a proper rational matrix, a state-space realization of the behavioral estimator for z from y is given by  σ x = −2 x+y z = 3 x + 2y. 3.2

The Singular Case

Assume now that R0 (s) is singular, i.e., that det R0 (s)= 0. Note that R0 (s) is the Rosenbrock matrix, [5], of the state-space system A, B, C, D . Since the transfer function of this system is given by: det(R0 (s)) F(s) = , det(sI − A)

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this means that F(s) = 0, i.e., the input u does not influence the output y. In other words, the y trajectories corresponding to an initial condition x(0) and to an input u coincide with the y trajectories corresponding only to the initial condition x(0). So, y is the output of the system  (σI − A)x = 0 y = Cx whereas z is the output of the system 

(σI − A)x = Bu z = Hx + Ju.

Therefore, estimating z from y is possible only if u does not influence z either, i.e., the transfer function  −1 F (s) = H sI − A B+J (17) is also zero. In this case, the z trajectories are given by:  (σI − A)x = 0 z = Hx, and our estimation problem becomes the following. Given the system ⎧ ⎪ ⎨ (σI − A)x = 0 y = Cx ⎪ ⎩ z = Hx

(18) (19) (20)

where y is the only measured variable, find an estimator for z from y. Now, assume without loss of generality that A, C is in Kalman observability decomposition form [3], i.e.,     A11 0 A= , C = C1 0 , A21 A22       with A11 , C 1 observable, and x = x . 1 x2   Partitioning H = H 1 H 2 accordingly, Eqs. (18)–(20) become: ⎧      ⎪ σI − A11 0 0 x1 ⎪ ⎪ ⎪ = (21) ⎪ ⎪ x2 0 −A21 σI − A22 ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎨  x1  =y (22) C1 0 ⎪ x2 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪  x1  ⎪ ⎪ ⎪ (23) =z H1 H2 ⎪ ⎩ x 2

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The corresponding behavioral estimation problem (of estimating z from y) is solvable if and only if there exist a row-matrix T (s) and a stable polynomial q(s) such that: ⎤ ⎡ sI − A11 0 ⎥ ⎢ ⎥ ⎢   ⎥ = q(s) H 1 H 2 . ⎢ (24) T (s) ⎢ C 1 0 ⎥ ⎦ ⎣ sI − A22  Partition T (s) = T1 (s) T2 (s) T3 (s) ; then (24) is equivalent to ⎧   ⎪   sI − A11 ⎪ ⎪ ⎨ T1 (s) T2 (s) − T3 (s)A21 = q(s)H 1 C1 ⎪ ⎪ ⎪ ⎩ T (s) sI − A  = q(s)H . 3 22 2 

−A21

(25) (26)

Note that (26) is equivalent to:  −1 q −1 (s)T3 (s) = H 2 sI − A22 =: G2 (s). This implies that a solution q(s), T3 (s) with q(s) stable exists  if and only if the poles of G2 (s), that coincide with the observable modes of A22 , H 2 are stable. Remark 3. G2 (s) is the transfer function from the initial condition x2 (0) of x2 to z for the system  (σI − A22 )x2 = 0 (27) z = H 2 x2 . Moreover, this system is obtained by setting in (21)–(23) x1 = 0, i.e., it corre  sponds to the restriction of the system A, 0, H, 0 , given by Eqs. (21) and (23)   to the unobservable subspace of A, 0, C, 0 .   As for Eq. (25), note that, since A11 , C is an observable pair, there exists a unimodular matrix V (s) such that     sI − A11 I = V (s) . 0 C1 Hence, (25) is equivalent to: T 1 (s) − T3 (s)A21 = q(s)H 1 , where T 1 (s) = T1 (s)V (s), which yields: T 1 (s) = T3 (s)A21 + q(s)H 1 .

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So, whenever G2 (s) is stable, a solution to (24) can be found as follows. Take q(s)−1 T3 (s) to be a left-coprime factorization of G2 (s). Then the zeros of q(s) coincide with the poles of G2 (s) and hence q(s) is a stable polynomial. Define T1 (s) := T 1 (s)V −1 (s), i.e.,   T1 (s) = T3 (s)A21 + q(s)H 1 V −1 (s). For reasons that will become clear below, choose T2 (s) such that q −1 (s)T2 (s) is proper, for instance  take T2 (s) = 1.  Then T (s) = T1 (s) T2 (s) T3 (s) and q(s) stable are such that (24) holds, and an estimator for z from y is given by: ⎡ ⎤ ⎡ ⎤ ⎧   σI − A11 0 0 ⎪  ⎪ x1 ⎪ ⎣ ⎦ ⎣ ⎪ 1⎦ y T (σ) = T (σ) C 0 1 ⎪  ⎪ x ⎨ 0 −A21 σI − A22  2     ⎪   ⎪ ⎪  q(σ) H 1 H 2 x ⎪ ⎪ ⎪   ⎩  z = H 1 H 2 x 0 z = T2 (σ)y or still implying that q(σ) z = T (σ) 1 y, which is equivalent to q(σ) 0 to q(σ) z = y. (28) This can be realized in state-space form as: ⎧   ⎨ σx =A x  + By ⎩   z = C x  + Dy,     for suitable matrices A, B, C and D. Example 2. Consider again the state-space system (3)–(5) now with         −2 0 0 A= , B= , C= 60 and H = 2 0 . 5 4 1 With the notations introduced in (10)–(12) we have that the matrix R0 (s) defined in (14) is singular and given by   s+20 R0 (s) = . 6 0 Applying the previous procedure, in this case, we have that A11 = A = −2, C 1 = C = 6 and H 1 = H = 2 and hence Eq.(25) holds with  the  stable  polynomial q(s) = s + 5 and the matrix T (s) = T1 (s) T2 (s) = 2 1 . From (28) a behavioral estimator for z from y is given by (σ + 5) z = y which can be realized in state-space form as  σx  = −5x +y z = x .

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From the previous considerations we can derive the following result. Theorem 2. Consider the state-space system defined and the  by Eqs. (3)–(5)  problem of estimating z from y. Moreover, define A, B, C, D , H and J as in (10)–(12), R0 (s) as in (14), G(s) as in (15), and F (s) as in (17). The estimation problem is solvable if and only if one of the following two conditions is satisfied. 1. R0 (s) is nonsingular and G(s) is stable.   2. R0 (s) is singular, F (s) is zero, and the restriction of the system A, 0, H, 0   to the unobservable subspace of A, 0, C, 0 has a stable transfer function from the state initial condition to the output z. In case the estimation problem is solvable, the construction of the corresponding estimator can be done as explained before.

4

Conclusions

In this paper we applied the behavioral estimation theory developed in [4] to the particular case of state-space systems with an unknown disturbance, one measured output and one variable to be estimated. Although the considered state-space systems were very simple, the preliminary results that we derived encourage us to consider more complex cases. The connection between the results obtained with the behavioral approach and the ones obtained with pure statespace methods is currently under investigation and will be reported in a future contribution. Acknowledgement. This work was supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology, references UIDB/04106/2020 and UIDP/04106/2020, by UIDB/00147/2020 – SYSTEC - Research Center for Systems and Technologies funded by national funds through the FCT/MCTES (PIDDAC) and by the Australian Research Council under the grant DP190102478. Moreover, it was also supported by R&D Unit UIDB+P/00147/2020 funded FCT/MCTES and projects: STRIDE – NORTE-01-0145-FEDER-000033, funded by N2020, ERDF, MAGIC PTDC/EEIAUT/32485/2017 funded by FEDER funds through COMPETE2020 – POCI and by national funds, PDMA - NORTE-08-5369-FSE-000061, through Programa Operacional. Regional do Norte (NORTE 2020).

References 1. Basile, G., Marro, G.: Controlled and conditioned invariant subspaces in linear system theory. J. Optim. Theory Appl. 3(5), 306–315 (1969). https://doi.org/10.1007/ BF00931370

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2. Basile, G., Marro, G.: On the observability of linear, time-invariant systems with unknown inputs. J. Optim. Theory Appl. 3(6), 410–415 (1969). https://doi.org/10. 1007/BF00929356 3. Chen, C.T.: Linear System Theory and Design, 3rd edn. Oxford University Press Inc., Oxford (1998) 4. Ntogramatzidis, L., Pereira, R., Rocha, P.: A behavioral approach to estimation in the presence of disturbances. IEEE Trans. Autom. Control 66(6) (2021) 5. Rosenbrock, H.: State-Sace and Multivariable Theory. Studies in Dynamical System Sries. Wiley Interscience Division, Hoboken (1970) 6. Trentelman, H.L., Stoorvogel, A.A., Hautus, M.: Control Theory for Linear Systems. Springer, London (2001) 7. Willems, J.: Models for dynamics. In: Dynamics Reported. Series Dynamic Systems Application, vol. 2, pp. 171–269 Wiley, Chichester (1989) 8. Zerz, E.: Topics in Multidimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences, vol. 256. Springer, London (2000)

A State-Space Model Inversion Control Method for Shake Table Systems José Ramírez-Senent(B)

, Jaime H. García-Palacios , and Iván M. Díaz

Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, Spain [email protected]

Abstract. Shake tables used to perform laboratory dynamic testing of structures are commonly powered by hydraulic servoactuators. Due to the inherent nonlinearity of servohydraulic systems, advanced control algorithms are required to achieve accurate reference tracking. Traditionally, these algorithms have been based on estimating, at a first stage, the Frequency Response Function of the system and inverting it to yield the Impedance Function. The Impedance is then operated with the desired system output and the result transformed into time domain to obtain an initial estimate of the drive to be fed to the system. This estimate is improved during the test in an iterative fashion. One of the main drawbacks of this approach is the fact that references which are not known beforehand and change in real time, cannot be directly addressed. In this work, a control method based on the inversion of a state-space model of the servo-actuator is presented. First off, a non-linear model of the testing system is developed. Next, a Feedback Linearization procedure to minimize servovalve non linearities is explained and the statespace identification procedure is described. Then, the model inversion method is addressed, together with the state variables estimator required to perform real time control. Finally, the simulation results obtained are discussed. The suggested method shows an excellent performance in numerical experiments; nevertheless, further studies must be undertaken to ensure its successful implementation in actual shake table systems. Keywords: Shake table · Feedback linearization · System identification · Dynamics inversion · State-space modeling

1 Introduction Despite the advances in simulation power of computers and material constitutive models, there is still a common agreement in the need for dynamic structural testing. In particular, great effort has been made to understand the performance of structures under exceptional loads, among which, earthquakes constitute a representative example. One of the most widespread methods to perform laboratory dynamic testing of structures is shake table testing [1]. A shake table is a very rigid platform onto which scaled or full-scale specimens under test (SUTs) are installed. These platforms are able to move © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 221–231, 2021. https://doi.org/10.1007/978-3-030-58653-9_21

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in one or several degrees of freedom (DoFs) according to prescribed motion profiles, such as accelerograms. There is a wide variety of shake table morphologies depending on their specific purposes. Very often, shake tables are powered by hydraulic servoactuators which are set up in a configuration capable of reproducing the desired motion DoFs (see Fig. 1). This work will focus on a one horizontal DoF shake table powered by a single hydraulic actuator.

Fig. 1. (a) One DoF shake table (b) Two DoF shake table (c) Three DoF shake table (d) Six Dof shake table. Figure courtesy of Vzero Engineering Solutions, SL.

The inherent non-linearity of servo-hydraulic actuation systems requires advanced control systems to satisfy the demanding accuracy criteria usually required. The classical control approach, which will be described in Sect. 2, essentially consists in identifying the Frequency Response Function (FRF) for the complete system at an initial stage, inverting it to obtain the impedance function (Z) and then operating Z with the desired system response to estimate the initial drive to be fed to the system after an inverse Fourier transformation process. This drive is updated as the test proceeds, in an iterative fashion, with the subsequently acquired system inputs-outputs pairs [2]. In this study, a new control method, based on the inversion, in real time, of a statespace model of the servo-actuator is presented. In order to assess the performance of the new method, a fully non-linear model of a one DoF shake table, loaded with a one-story building has been developed. Details on this model are given in Sect. 3. In order to derive the inversion algorithm, the first step to take is to identify a proper state-space model of the actuation system. Prior to the identification process, which is explained in Subsect. 4.2, a feedback linearization scheme used to minimize the effects of servo-valve non-linearity is explained in Subsect. 4.1. The inversion algorithm itself is dealt with in Sect. 5; as it will be seen, a simple algebraic relation together with a state variables estimator are required to perform the proposed dynamic inversion. Results obtained with the new suggested procedure are shown in Sect. 6. Finally, the main conclusions are outlined in Sect. 7, emphasizing on the one hand, the advantages of the new control method over the classical and outlining further steps that must be taken for successful actual implementation on the other.

2 The Classical Shake Table Control Approach A commonly adopted control architecture for servohydraulic shake table systems consists in employing two nested controllers; i.e. the Outer Loop Controller (OLC) and the Inner Loop Controller (ILC), see Fig. 2. The OLC is in charge of overall system’s

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dynamics identification, DoF motion control and solution of inverse and direct kinematics in case of multiaxial shake tables; however, this study is limited to the one DoF case. The ILC usually consists in a Proportional-Integral-Derivative (PID) controller which uses the displacement of servoactuator rod as the primary feedback signal. Other signals like rod acceleration or differential pressure across actuator chambers, or features such as feed-forward action, may also be used in the ILC algorithm [3]. OLC {aref}

Iterative algorithm

FFT

+ FFT

FFT-1

Inverse kinematics

{d}

ILC

{u}

{Δpact}

{e}

Z estimator

{aDoF}

{dact} ; {aact}

Direct kinematics

{dact} ; {aact}

Fig. 2. Shake table classical control architecture: test execution (control mode).

There are two differentiated stages when performing a shake table test according to classical approach: (a) system identification and (b) test execution, see Fig. 2. In order to identify the dynamics of system under control, several blocks of random stimuli are fed to the ILC and the response of the system (normally acceleration signals), is measured. Inputs and outputs are then transformed into frequency domain and the FRF of the system is estimated [4]. With a view to predict an approximation to the initial drive to be injected to the system, d 0 , the FRF is inverted to yield the impedance function, Z(ω), which is post-multiplied by the desired responses vector aref (ω): d 0 (ω) = Z(ω) · aref (ω).

(1)

Once the initial drive is calculated, it is transformed into time domain and the system is ready to start target waveform reproduction (control mode). During test execution, an iterative algorithm is executed in OLC according to the following equation [2]:   (2) d n+1 (ω) = d n (ω) + K · Z(ω) · an (ω) − aref (ω) , where K represents a correction gain and aN represents the accelerations obtained during the n-th iteration; the impedance function may be updated between iterations. Several considerations can now be made on the classical control approach: 1. Tests of considerably different SUTs require an identification run at the beginning of the test due to significative changes in overall system dynamics. 2. The identified impedance function constitutes a poor estimate of system dynamics and leads to insufficient control performance when dealing with marked non-linear behavior such as that of servohydraulics or SUTs undergoing plastic deformation or failure.

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3. Identification and control algorithms run in frequency domain and therefore require that complete blocks are acquired and processed before an updated drive block is output. The duration of this blocks depends on the chosen frequential resolution. Consequently, this algorithm is best suited for non-changing references. 4. Direct and inverse Fourier transformation processes and impedance function storage imply considerable computational and memory requirements for the controller. 5. Generally, many iterations at low level are required until a satisfactory drive is obtained. Moreover, when ramping up to full level test, the scaled drive may not cause an appropriate response due to eventual non-linearities showing up.

3 Shake Table System Non-linear Model In order to predict the behavior of the new suggested control algorithm and assess its potential, a non-linear model of a one DoF shake table loaded with a one-story building has been implemented. Figure 3 shows a schematic of the modeled elements, main variables implied and employed sign criteria. , ̇ , ̈

Servo-valve Ps PR usv

Q1

MS

SUT

ysv

C

Q2

K/2

K/2

Actuator Q1

Q2

Q1-B

Q1-2

Q2-B P2

P1

Shake table FT

MT ,

,

Ffr

Fig. 3. One DoF shake table system schematic.

The flow through servovalve ports has been modeled with the following flow-pressure expressions for a critically lapped servovalve [5]:  √ Cs · ysp · sgn(Ps − P1 ) |Ps − P1 |; if ysp ≥ 0 √ , (3) Q1 = Cs · ysp · sgn(P1 − Ps ) |P1 − PR |; if ysp < 0  √ Cs · ysp · sgn(PB − PR ) |PB − PR |; ifysp ≥ 0 √ Q2 = , (4) Cs · ysp · sgn(PS − PB ) |PS − PB |; ifysp < 0 in which Cs is a constant which depends on servovalve geometry, rated flow and fluid used, ysp is the displacement of main spool, Ps and PR are the pressures at inlet and outlet ports of servoactuator’s manifold and P1 , Q1 and P2 , Q2 are the pressures and flow rates associated to servoactuator’s chambers. More complex expressions accounting for laminar to turbulent flow transition in servovalve orifices could have been used [6, 7]. The modelled servovalve features a nominal flow rate of 400 L per minute with a pressure drop of 35 bar per land. In this work, the dynamics of the spool of the servovalve

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have been neglected; therefore, the relationship between voltage sent to servovalve and spool motion can be stated as: usv = Csp · ysp .

(5)

The evolution of the pressures in the chambers of servoactuators can be modelled with the Continuity Equation applied to each chamber of the servoactuator: (v01 + Aw x) ˙ P1 + Aw x˙ = Q1 − Q1−2 − Q1B β1

(6)

(v02 − Aw x) ˙ P2 − Aw x˙ = −Q2 + Q1−2 − Q2B β2

(7)

where v0i and βi are the dead volumes and Bulk moduli of each chamber respectively, Aw is the wet area of piston, Q1−2 is the leakage flow between chambers and QiB represent the leakage flows from actuator chambers to their respective bearing. In this work these leakage flows have been neglected. The acceleration of servo-actuator is governed by Newton’s Second Law: mp x¨ p = P1 Aw − P2 Aw − Ffr − FT ,

(8)

where mp and x¨ p represent rod mass and acceleration respectively (the latter has been assumed to be identical to shake table acceleration), Ffr is the friction force acting on the rod, which has been modeled as a purely viscous friction and FT is the force exerted on the table by the actuator. Finally, the dynamics of the shake table and the building are modelled with:            x¨ T C −C K −K FT MT + mp 0 x˙ T xT + + = , (9) x¨ S x˙ S xS 0 0 MS −C C −K K where MT is shake table mass, MS is the mass of the only building story, C is the damping coefficient which has been calculated specifying a modal damping ratio of 2% and K is the resultant stiffness of the building pillars. Table 1 summarizes the values of the main physical parameters used in the implemented model. Table 1. Values of parameters used in the model. Parameter Value

Parameter Value

Ps [bar]

280

MT [kg]

PR [bar]

0

MS [kg]

Aw [m2 ]

0.0019 C [Ns/m]

2183.7

mp [kg]

80

3.9478e+06

K [N/m]

3000 1000

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4 Description of the Proposed Control Method In this section, the proposed new control method is explained. Figure 4 shows a schematic diagram which will be explained from hereon. aref

Model inversion

u

Feedback linearization

Servovalve

Q1 ;Q2

ysp Ps Pr

{x}

P1 P2

State variables estimator

̈

Fig. 4. Block diagram of the proposed control method

4.1 Servovalve Flow Feedback Linearization The main source of non-linearity in the servo-hydraulic system is the square root present in the flow-pressure equation of the servovalve (see Eq. 3 and Eq. 4). In order to minimize this non-linearity, a feedback linearization algorithm has been implemented [8]. The objective is to cancel out the square root by injecting an appropriate voltage to the servovalve. For that purpose, a transformation of the output of the controller is sought, so that the overall behavior of the controller-servovalve system is linear. The main assumption made to design the transformation is that servovalve orifices and absolute values of flow rates to and from actuator chambers are completely symmetric. Voltage to be applied is then calculated as:  ζ√ ; if ysp ≥ 0 sgn(Ps −P1 ) |Ps −P1 | , (10) usv = ζ√ if ysp < 0 sgn(P −P ) |P −P |; 1

R

1

R

where ζ represents the output from the model inversion block. Calculation of usv implies measuring with transducers the pressures both at servoactuator chambers and at the inlet and outlet of servovalve manifold as well as knowing spool position sign. If all the hypotheses from previous paragraph were verified, flow-pressure equation non-linearity cancellation would be perfect, and the application of the voltage determined by Eq. 10 would lead the servovalve to deliver a flow rate proportional to controller output. The more formal Exact Linearization Method [9] could have also been used for this purpose instead of the direct method employed here. Figure 5 shows a comparison of the achieved system identification results with and without this feedback linearization implementation. 4.2 State-Space Model Identification Since the aim of the new method is to cancel out the dynamics of the servoactuator system, it is clear, that first of all, an accurate system model is required to later implement a dynamics inversion with reasonably good performance.

A State-Space Model Inversion Control Method for Shake Table Systems Accelerations vs time

8

shake table first storey

6

Accelerations [m/s 2]

227

4 2 0 -2 -4 -6 -8

0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 5. Accelerations of shake table and first story as a response to low level drive.

The Subspace Identification Method via Principal Component Analysis [10, 11] has been used to perform the identification of the state-space model of the servoactuator system. The inputs to the target system are the voltage applied to the servovalve and the force exerted on servoactuator’s rod tip which can be estimated as: Fp = mr ar − (P1 − P2 )Aw ,

(11)

where mr is the rod mass and ar is the rod acceleration which can be measured using an accelerometer, alternatively a load cell can also be used for this purpose. Friction forces have not been considered in Eq. 10, though they are accounted for in the model. The system output is the rod acceleration; therefore, a Multiple Input Single Output (MISO) system is considered. In order to estimate the state-space model, a low level linear swept sine (± 0.02 V), with initial and final frequencies of 1 and 100 Hz respectively, and a sweep time of 20 s has been used as input. Figure 5 shows simulated system response. As it can be appreciated, two resonance peaks can be distinguished at around 9.8 and 19.1 Hz, corresponding to the mode of the building and to the oil column resonance of the shake table-SUT system respectively. This oil column resonance is typical in shake tables and its value usually falls within frequency operational range. After performing the numerical identification experiment, the discrete state-space model obtained for an inputs/output sampling rate of 1 ms and a target number of states of 3, are shown in Fig. 6. This discrete State-Space model has been converted into a continuous time one making use of the Tustin Algorithm in order to perform numerical simulations of the overall system. Figure 7(a) shows the response of the non-linear system and the one obtained with the linear approximation. Figure 7(b) shows the difference between the non-linear model and linear approximation outputs, in absolute value, with and without the feedback linearization feature. Both figures have been zoomed in the area of interest since errors are much less outside.

5 System Dynamics Inversion Algorithm The main idea behind the inversion of the previously identified state-space model is to factor out the input to the system required to cause the desired output. In the case under

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0.995919 x(k+1)= 0.0723687 0 y(k) =

-0.0723687 0.995919 0

0 0 -0.462539

-6.42895 -0.466983 0.128788

x(k) +

-1.86568E-9 x(k) + 3.75031E-10 -2.73745E-7

-1.72497E-7 9.20771E-7 u(k) + 2.52849E-7

-0.1968 0.564478 -0.0130104

e(k)

1.73475E-8 -0.000333197 u(k) + e(k)

fs = 1000Hz

Fig. 6. Identified discrete time State-Space model.

(a)

(b)

Fig. 7. (a) Non-linear and linear responses. (b) Identification error.

study, the output equation of the identified state-space model has the form: ⎧ ⎫   ⎨ x1 ⎬   ζ  at = c1 c2 c3 , x2 + d1 d2 ⎩ ⎭ Fp x3

(12)

where xi are the state variables, which in principle have no direct physical meaning. It should be possible to calculate the drive ζ by factoring it out from Eq. 12 as follows:   ζ = at − c1 x1 − c2 x2 − c3 x3 − d2 Fp /d1 (13) and injecting it into the feedback linearization block at every control cycle. In Eq. 13 at is the desired acceleration time history, Fp is the force acting on rod tip, which can be estimated making use of Eq. 11 and ci and di are the elements of the identified C and D matrices. In order to obtain a suitable estimate of the state values to be used in Eq. 13, a state estimator block as the one shown in Fig. 8 needs to be implemented (Fig. 4), using the previously identified A and B continuous matrices. With this control approach, all the relevant dynamic information of the complete system is completely summarized in A, B, C and D matrices and, theoretically, arbitrarily changing acceleration profiles, unknown before the test, can be reproduced.

A State-Space Model Inversion Control Method for Shake Table Systems {u}

[B]

+ +

229

∫ [A]

Fig. 8. State variables estimator block

(a)

(b)

Fig. 9. Reference and acceleration. (a) complete. (b) zoomed.

6 Simulation Results In this section, simulation results obtained when trying to reproduce a synthesized windowed gaussian random acceleration time history are shown. This waveform features a frequency content between 0.5 and 50 Hz and peak values of approximately 1.5 g, 0.3 m/s and 42 mm. Figure 10 shows the achieved tracking error. As it can be appreciated, acceleration tracking is excellent excepting for an initial section at the beginning of the profile, very likely due to numerical simulation parameters issues (Fig. 9).

Fig. 10. Acceleration error with the new control method.

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7 Conclusions and Further Work In this work a new method to control acceleration output of a one DoF shake table motion has been presented. First off, a brief review of the shake table classical control method has been given. With this approach, an estimate of the FRF of the overall system is obtained by measuring its response to random stimuli. Then, impedance is computed and operated with the desired response yielding an initial estimate of the drive to be injected in the system, which is corrected iteratively during the test. In order to assess the proposed algorithm, a non-linear model of a one DoF shake table has been developed. To minimize the non-linearity of the servovalve, a feedback linearization scheme has been implemented. This algorithm requires measuring pressures at chambers and manifold of the servoactuator as well as servovalve spool position. A state-space model has then been derived. The inputs to this model are the control order and the force exerted on actuator rod, which summarizes all the external actions on the actuator. The output of the model is the acceleration of the rod. With an accurate estimate of system dynamics, inversion of system dynamics is straightforward by factoring out the required drive from the output equation of the state-space model. Simulated results show excellent results in acceleration reproduction. Therefore, the suggested procedure appears promising due to its simple implementation and advantages over traditional approach, being the main of them the facts that (i) no iterations are required, (ii) acceleration reference does not need to be known before the test and (iii) its reduced computational load requirements. However, many factors need to be accounted for before actual implementation: acquisition and processing of signals with noise, delays in transducers acquisition, latency of control loop, state-space model update in real-time and the addition of a parallel controller to account for non-modelled dynamics. Besides these, the influence in results of different tested SUTs must be studied, even though, the developed algorithm, theoretically, does not require identification runs since all the information coming from the SUT is summarized in the input force to the model. Acknowledgements. The authors acknowledge the financial support from the Spanish Ministry of Science, Innovation and Universities through the project SEED-SD (RTI2018-099639-B-I00). The authors are also grateful to Vzero for providing pictures.

References 1. Williams, M.S., Blakeborough, A.: Laboratory testing of structures under dynamic loads: an introductory review. Philos. Trans. R Soc. Lond. A 359(1786), 1651–1669 (2001) 2. Underwood, M.A.: Digital control systems for vibration testing machines (chap. 26). In: Shock and Vibration Handbook, 6th edn. McGraw Hill, New York (2009) 3. MTS: 793.xx Software System. MTS Systems Corporation, Eden Prairie, Minnesota (2003) 4. de Silva, C.W.: Vibration Testing in Vibration and Shock Handbook, 1st edn. CRC Press, Boca Raton (2005) 5. Merrit, H.E.: Hydraulic Control Systems, 1st edn. Wiley, New York (1991)

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6. Sivaselvan, M.V., Hauser, J.: Trajectory exploration approach to hybrid simulation. In: Hybrid Simulation: Theory, Implementation and Applications, pp. 15–24. Taylor and Francis (2008) 7. Belikov, J., Petlenkov, E.: Model based control of a water tank system. In: Proceedings of the 19th IFAC World Congress, pp. 10838–10843 (2014) 8. Slotine, J.E., Li, W.: Applied Nonlinear Control, 1st edn. Prentice-Hall Inc., Englewood Cliffs (1991) 9. Bisták, P., Huba, M.: Three-tank virtual laboratory for dynamical feedforward control based on Matlab. In: Proceedings of the 19th International Conference on Electrical Drives and Power Electronics (EDPE), pp. 318–323 (2017) 10. Instruments, N.: LabVIEW System Identification VIs Algorithm References. National Instruments, Austin (2014) 11. Ljung, L.: System Identification Theory for the User, 2nd edn. Prentice Hall, Upper Saddle River (1999)

Minimum Energy Control of Passive Tracers Advection in Point Vortices Flow Carlos Balsa1 , Olivier Cots2 , Joseph Gergaud2 , and Boris Wembe2(B) 1 Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal [email protected] 2 Toulouse Univ., INP-ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France {olivier.cots,joseph.gergaud,boris.wembe}@irit.fr

Abstract. In this work we are interested in controlling the displacement of particles in interaction with N point vortices, in a two-dimensional fluid and neglecting the viscous diffusion. We want to drive a passive particle from an initial point to a final point, both given a priori, in a given finite time, the control being due to the possibility of impulsion in any direction of the plane. For the energy cost, the candidates as minimizers are given by the normal extremals of the Pontryagin Maximum Principle (PMP). The transcription of the PMP gives us a set of nonlinear equations to solve, the so-called shooting equations. We introduce these shooting equations and present numerical computations in the cases of N = 1, 2, 3 and 4 point vortices. In the integrable case N = 1, we give complete quadratures of the normal extremals. Keywords: Helhmoltz-Kirchhoff N vortices model · Energy minimization · Pontryagin maximum principle · Indirect shooting method

1

Introduction

This work is concerned with the control of the displacement of particles in interaction with point vortices, in a two-dimensional fluid, where the viscous diffusion is neglected, which is equivalent to using the Euler equation instead of the Navier-Stokes equation as the mathematical model of the fluid flow. We refer to Ref. [12] for details about vortex theory. In most of the control problems, concerning realistic flows, the solution is achieved by means of simplified models such as point vortex [17]. There is a special interest in the use of control methods applied to vortex dynamics in the fields of geophysical fluid dynamics, aeronautic and hydrodynamic [16]. In the context of hydrodynamics, the fish-like locomotion and autonomous underwater vehicles are applications of point vortex that have received some attention in the c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 232–242, 2021. https://doi.org/10.1007/978-3-030-58653-9_22

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last years due to the necessity of data sampling in the oceans water [8,10,13]. Moreover, from the dynamics of such systems there is an intense activity research initiated by Poincar´e [14] to compute periodic trajectories avoiding collisions and such techniques lead to the concept of choregraphy developed by [6] for the Nbody problem and [5] for the N-vortex system, showing the relations between both dynamics in the Hamiltonian frame [11]. From the control point of view, there is a lot of development related to space navigation for the N-body problem, see [3], valuable in our study for ship navigation in the N-vortex problem. In this work we apply the Pontryagin Maximum Principle (PMP) to control the displacement of a passive particle (which is by definition a zero circulation vortex), influenced by vortex points located around it, in an optimal way (we consider the energy cost, see [2] for time minimization). This problem is a particular case of the challenging “Problem 2” included in [16]. A passive particle is small enough not to perturb the velocity field, but also large enough not to perform a Brownian motion. Particles of this type are the tracers used for flow visualization in fluid mechanics experiments [1]. We consider also that the passive particles have the same density of the fluid in which it is embedded. Explicitly we want to drive a passive particle from an initial starting point to a final terminal point, both given a priori, in a given finite time. Here, the vortex dynamics is governed by N point vortices and the control is due to the possibility of impulsion in any direction of the two dimensional plane (a sufficiently long time is considered so that the optimal control remains bounded and its amplitude is small enough). The article is structured as follows: Sect. 2 is devoted to the statement of the control problem. The maximum principle is stated in Sect. 3, with some results in the one vortex case. The cases of 2, 3, 4 vortices are treated in Sect. 4.

2

Vortex Dynamics and Statement of the Problem

We give in this section a short description of the vortex dynamics and we refer to [12] for more details. This description is followed by the formulation of the control problem addressed in this work. Let us consider the case of a two-dimensional fluid, for which the incompressible Euler equations are given by ∂ν + (ν · ∇) ν = −∇p, ∂t

∇ · ν = 0,

(1)

where ν stands for ν(X, t) := (ν1 (X, t), ν2 (X, t)) and represents the velocity field and p is the pressure of the fluid. Due to ∇·ν = 0 (the incompressibility equation) from (1), one can write ν = (ν1 , ν2 ) =: (∂y Ψ, −∂x Ψ ) where Ψ is called the streamfunction. Besides, let w denote the viscosity vector and introduce ν˜ := (ν, 0), then w is given by the relation w = ∇ ∧ ν˜ = (0, 0, ∂x ν2 − ∂y ν1 ) =: (0, 0, ω), and with the two previous formulas, one can deduce the Poisson equation satisfied by Ψ , that is ∇2 Ψ = −ω. The resolution of this equation leads to  1 Ψ (X, t) = ln(X − Y ) ω(Y, t) dY, 2π R2

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where · is the Euclidean norm. On the other hand, considering a finite number N of point vortices, then the viscosity vector can be written in the form ω(X, t) = N i=1 ki δ(X −Xi (t)), where δ is the Dirac mass and where ki is the circulation of the ith-vortex. These two previous relations allow us to write the vortex dynamics as follows: N N   kj yi − yj kj xi − xj dxi dyi =− = , , 2 2 dt 2π r dt 2π rij ij j=1 j=1 j=i

j=i

2 := (xi − xj )2 + (yi − where (xi , yi ) is the position of the ith-vortex and where rij 2 yj ) is the square distance between the vortices i and j. As mentioned above, the aim of this paper is not to control the vortices but the displacement of particles. The idea is therefore to consider a particle (or passive tracer) as a point vortex with zero circulation and to apply a small amplitude control acting only on the passive tracer [16]. The control system is then written as follows: q˙ = F0 (q) + u1 F1 (q) + u2 F2 (q) where q := (x, y, x1 , y1 , · · · , xN , yN ) ∈ R2(1+N ) is the vector of positions of the particle ∂ ∂ and F2 (q) = ∂y , and and the vortices, where the control fields are F1 (q) = ∂x where the drift F0 is given by N N   kj y − yj ∂ kj x − xj ∂ + F0 (q) = − 2 2π r ∂x 2π rj2 ∂y j j=1 j=1 ⎛

+

⎞

N N N   ⎜  kj yi − yj ∂ kj xi − xj ∂ ⎟ ⎜− ⎟, + 2 2 ⎝ ⎠ 2π r ∂x 2π r ∂y i i ij ij i=1 j=1 j=1 j=i

j=i

with ri2 := (x − xi )2 + (y − yi )2 , i = 1, · · · , N . The optimal control problem of interest is then defined as follows: minimize the transfer energy J(u) := T u(t)2 dt to drive a passive particle from an initial point (x0 , y0 ) ∈ R2 to a 0 target point (xf , yf ) ∈ R2 , both given a priori, in a given finite time T > 0. The initial positions of the vortices being also given.

3 3.1

Case of One Vortex Pontryagin Maximum Principle and Shooting Function

In the single vortex case one has (x˙ 1 , y˙ 1 ) = (0, 0), that is the vortex is static and can be fixed to the origin of the reference frame. Hence, the control system may be reduced to q(t) ˙ = F0 (q(t)) + u1 (t)F1 (q(t)) + u2 (t)F2 (q(t)),

(2)

where q = (x, y) ∈ R2 (by a slight abuse of the notation since (x1 , y1 ) = (0, 0) is constant) is the position of the particle and where the drift and the control fields are given by

Minimum Energy Control of Passive Tracers Advection

μ F0 (q) = 2 x + y2



∂ ∂ +x −y , ∂x ∂y

F1 =

∂ , ∂x

F2 =

∂ , ∂y

μ :=

235

k . 2π

Explicitly the control problem in this case is written: x(t) ˙ =−

μ y(t) + u1 (t), x2 (t) + y 2 (t)

y(t) ˙ =

μ x(t) + u2 (t), x2 (t) + y 2 (t)

Lemma 1. For any q0 , qf there exists a control joining q0 to qf in time T > 0. Proof. Considering the polar coordinates (r cos θ, r sin θ) = (x, y) and an adapted rotating frame for the control, v = u e−iθ , the control system (2) ˙ = μ/r(t)2 + v2 (t)/r(t). From q0 , we can apply a becomes r(t) ˙ = v1 (t), θ(t) constant control v(t) = (α1 , 0) until the distance qf  is reached and then apply a constant control v(t) = (0, α2 ) until the target xf is reached, where α1 , α2 ∈ R are suitably chosen according to the time T . Let q0 ∈ R2 denote the initial condition, qf ∈ R2 the target and T > 0 the transfer time. Let u ∈ L∞ ([0, T ], R2 ) be an optimal solution (assuming its existence) and let q denote the associated optimal trajectory. According to the Pontryagin maximum principle [15], then there exists an absolutely continuous function p : [0, T ] → R2 satisfying the adjoint equation a.e. over [0, T ]: p(t) ˙ = −∇q H(q(t), p(t), u(t)),

(3)

where H(q, p, u) := p · (F0 (q) + u1 F1 (q) + u2 F2 (q)) + p0 u2 is the pseudoHamiltonian.1 Besides, we have: p0 ≤ 0, the pair (p, p0 ) never vanishes

(4)

and the optimal control satisfies the maximization condition a.e. over [0, T ]: H(q(t), p(t), u(t)) = max2 H(q(t), p(t), w). w∈R

(5)

Any quadruplet (q, p, p0 , u) solution of (2) and (3)–(5) is called an extremal and is said to be abnormal if p0 = 0 otherwise it is said to be normal. In the normal case, when p0 = 0, we can fix by homogeneity p0 = −1/2. Definition 1. An extremal is a called BC-extremal if q(0) = q0 and q(T ) = qf . Proposition 1. Let (q, p, p0 , u) be an extremal, then, the extremal is normal (that is there are no abnormal extremals) and the control u is smooth. Proof. Let us introduce the Hamiltonian lifts Hi (z) := p · Fi (q) with z := (q, p). If p0 = 0, then the maximization condition leads to H1 = H2 = 0, that is p1 = p2 = 0, which contradicts the PMP. Hence p0 < 0 and we can fix p0 = −1/2. With this normalization, the control may be written in the feedback form u(t) = Φ(z(t)) with Φ(z) := (H1 (z), H2 (z)) smooth. It is clear and well-known that in this case, the control law t → u(t) is smooth. 1

The standard inner product is written a · b, for a, b in R2 .

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Since we have only normal extremals we fix p0 = −1/2 and define the following true Hamiltonian (plugging u in feedback form into the pseudo-Hamiltonian): 1 1 H(z) := H(z, Φ(z)) = H0 (z) + (H12 (z) + H22 (z)) = H0 (z) + p2 . 2 2 #— Let’s introduce the Hamiltonian system H(z) := (∇p H(z), −∇q H(z)) and #— the pseudo-Hamiltonian system H(z, u) := (∇p H(z, u), −∇q H(z, u)). Then, we have: Proposition 2. Let (z, u) be an extremal (p0 = −1/2 is implicit). Then: #— #— #— z(t) ˙ = H(z(t), u(t)) = H(z(t)) = H0 (z(t)) + (p(t), 0) . According to this proposition, we can define the exponential mapping: #— expt,q0 (p0 ) := π ◦ exp(tH)(q0 , p0 ), #— where π(q, p) := q is the canonical projection on the state space and exp(tH)(z0 ) #— is the solution at time t of z˙ = H(z) with the initial condition z0 . Finally, let us introduce the shooting function: S : R2 −→ R2 p0 −→ S(p0 ) := expT,q0 (p0 ) − qf . Then, we have the classical following relation between BC-extremals and zeros of the shooting function. Proposition 3. Let (q, p, u) be a BC-extremal (p0 = −1/2 is implicit), then, S(p(0)) = 0. Conversely, let p0 ∈ R2 s.t. S(p0 ) = 0. Then, defining z(t) := #— exp(tH)(q0 , p0 ) over [0, T ] and u(t) := Φ(z(t)), the pair (z, u) is a BC-extremal. 3.2

Integration of the Extremal Solutions

Writing the system in polar coordinates, the Hamiltonian becomes: H(r, θ, pr , pθ ) = pθ

μ 1 2 p2θ (p + ). + r2 2 r r2

Proposition 4. The system is Liouville integrable and, in polar coordinates, the extremals are given by: if c := pθ (2μ + pθ ) ≥ 0, then  (c4 t + c5 )2 + c3 (c4 t + c5 ) , , pr (t) = r(t) = c4 r(t)

c6 c5 c4 t + c5 θ(t) = √ − arctan √ arctan √ , c3 c4 c3 c3

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and if c < 0, then 

(c4 t + c5 )2 + c3 (c4 t + c5 ) , , pr (t) = c4 r(t)     √ √  c5 − −c3   c4 t + c5 ) − −c3  c6    ,  √ √ − log  θ(t) = √ log  c4 −c3 c4 t + c5 ) + −c3  c5 + −c3  r(t) =

with c1 = pr (0), c2 = pθ , c3 = c, c4 = c21 +

c3 , r02

c5 = r0 |c1 |, c6 = c4 (μ + c2 ).

Proof. The θ coordinate is cyclic, so pθ defines a second first integral with H. This guarantees the Liouville integrability of the system. For the integration of the extremals, one has to solve the following system: r(t) ˙ = pr (t),

p˙r (t) =

p2θ 2μpθ + , r3 (t) r3 (t)

˙ = θ(t)

μ r2 (t)

+

pθ , 2 r (t)

p˙θ (t) = 0,

with (r(0), θ(0), pr (0), pθ (0)) =: (r0 , θ0 , c1 , c2 ). Since pθ = c2 is constant, we ˙ = have from the two first equations that r¨(t) = c3 /r3 (t), whence r¨(t)r(t) 3 ˙ (t), with c3 := c2 (μ + c2 ). Solving this last equation we obtain r(t) = c 3 r(t)/r ((c4 t + c5 )2 + c3 )/c4 with c4 := c21 + c3 /r02 and c5 = r0 |c1 |. One deduces pr (t) by differentiating this relation. The integration of θ depends on the sign of c3 . ˙ = c6 /((c4 t + c5 )2 + c3 ), with c6 := c4 (μ + c2 ), thus, if c3 ≥ 0 Indeed one has θ(t) one has the first case, otherwise one has the second. 3.3

Numerical Methods and Results

The HamPath2 code [4,9] is used to compute the BC-extremals. A Newton-like algorithm is used to solve the shooting equation S(p0 ) = 0. Providing H and S to HamPath, the code generates automatically the Jacobian of the shooting function. To make the implementation of S easier, HamPath supplies the exponential #— mapping. Automatic Differentiation is used to produce H and is combined with Runge-Kutta integrators to assemble the exponential mapping. We present here, in the one vortex case, two examples. For the examples, we fix the initial condition to q0 = (2, 0), the transfer time to T = 10 and we consider two targets: qf = (−2, 0) and qf = (3, −3), to emphasize the influence of the vortex circulation (the strength of the drift being dependent to the distance between the particle and the vortex). The circulation is fixed to k = 2πμ, with μ = 2q0 . The two results are detailed in the caption of Fig. 1, where we represent the projection of the two BC-extremals in the state space, that is the trajectories, together with the control laws.

4

Cases of 2, 3 and 4 Vortices

4.1

Pontryagin Maximum Principle and Shooting Function

In the case of N > 1 vortices, the vortices are not static and have to be considered in the dynamics. Let us recall that the control system has the form 2

www.hampath.org.

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2

u 1 (t)

0.03

y

1

u 2 (t)

0.02 0.01

0

0

-1

-0.01 -0.02

-2 -2

-1

0

1

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0

x

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5

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t 0.3 u 1 (t)

2

u 2 (t)

0.2

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y

0 0

-1

-0.1

-2

-0.2

-3 -4

-2

0

2

-0.3

x

0

5

10

t

Fig. 1. One vortex. (Top) qf = (−2, 0) and the cost is J ≈ 4.2675e−3. (Bottom) qf = (3, −3) and the cost is J ≈ 5.5177e−1. In both cases, the shooting equation S = 0 is solved with a good accuracy of order 1e−12. On the left, we have the trajectory (in blue), the point vortex is represented by a red dot, the initial condition by a blue dot. The two control components are given on the right.

q˙ = F0 (q) + u1 F1 (q) + u2 F2 (q), where F0 , F1 , and F2 are given in Sect. 2, and that the state q contains the position of the particle together with the positions of the N vortices. In the control problem, see Sect. 2, the initial positions of the particle and the vortices are fixed. Let us denote by q0 ∈ R2(1+N ) the vector of these positions. On the other hand, the final position of the particle is fixed while the final positions of the N vortices are free. Let us denote by (xf , yf ) ∈ R2 the target for the particle. The transfer time is also fixed and denoted T > 0. In order to apply the maximum principle, we define the pseudo-Hamiltonian: H(q, p, u) := p · F0 (q) + u1 p · F1 (q) + u2 p · F2 (q) + p0 (u21 + u22 ), where p0 will be fixed to −1/2 according to: Proposition 5. Let (q, p, p0 , u) be a BC-extremal. Then, the extremal is normal and the control u is smooth. Proof. If p0 = 0, then the maximization condition leads to H1 = H2 = 0, that is px = py = 0 all along the extremal. Decomposing the adjoint vector as p := (px , py , px1 , py1 , · · · , pxN , pyN ), then, the transversality conditions implies that at

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the final time, px1 (T ), py1 (T ), . . . , pxN (T ), pyN (T ) are zero, which contradicts the PMP. Hence p0 < 0 and we can fix p0 = −1/2. With this normalization, the control may be written in the feedback form u(t) = Φ(z(t)) with Φ(z) := (H1 (z), H2 (z)) smooth. Here again, the control law t → u(t) is smooth. In the same spirit as in the one vortex case, one can define the following shooting function (with adapted and clear notation) to compute the BC-extremals: S : R2(1+N ) −→ R2(1+N )  p0 −→ S(p0 ) := x(T, q0 , p0 ) − xf , y(T, q0 , p0 ) − yf ,  px1 (T, q0 , p0 ), · · · , pyN (T, q0 , p0 ) 4.2

Numerical Results

In the following examples, we fix the initial particle position to (x0 , y0 ) = (2, 0) and the target to (xf , yf ) = (3, −3). To make a comparison with respect to the number of vortices we impose the condition ΣN i=1 |ki | = k, where k is the circulation in the one vortex case, that is k = 2πμ with μ = 2q0 . The initial positions of the vortices are taken arbitrarily. In general, the main difficulty to solve the shooting equations is the initialization of the adjoint vector due to the sensitivity of the underlying Newton method. In order to overcome this difficulty, we use the following algorithm: – For two vortices, we first solve the subproblem where we set one of the circulation to zero, that is we solve a one vortex problem; – Then, we use a path following algorithm [4] implemented in the HamPath software where the homotopic parameter is the value of the circulation which increase from 0 to the desired value; – We repeat this procedure to obtain solutions for problems with more vortices. We summarize in Table 1 the results in the one vortex case together with the new results for N = 2, 3 and 4. The trajectories and the controls are given in Figs. 2, 3 and 4, respectively for N = 2, 3 and 4. Table 1. In the two first columns are given the number of vortices with their circulations. In the third is given the initial positions of the particle and the vortices, the target for the particle being given in the fourth. The fifth column gives the cost associated to the computed BC-extremal, whose trajectory and control are plotted in the figure given by the last column. Vortices Circulations

Initial positions (q0 )

Target Cost Figure

N =1

k = 8π

(2, 0)

(−2, 0) 0.004 1

N =1

k = 8π

(2, 0)

(3, −3) 0.552 1

N =2

k1 = k2 = 4π

(2, 0, 0, 1, 0, −1)

(3, −3) 1.947 2

N =3

k1 = k2 = k3 = 8π/3

(2, 0, 0, 1, 1, −1, −1, −1)

(3, −3) 0.406 3

N =4

k1 = k2 = k3 = k4 = 2π (2, 0, 1, 0, 0, 1, −1, 0, 0, −1) (3, −3) 0.335 4

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0.4 u 1 (t)

3

u 2 (t)

0.3

2 0.2

1

u2

(x,x i )

0.1

0 0

-1 -0.1

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-4 -4

-2

0

2

-0.4

4

0

(y,y i )

2

4

6

8

10

t

Fig. 2. Two vortices. Trajectory and control in the two vortices case. The initial adjoint vector being p0 = (0.28, 0.15, 0.097, −0.47, −0.11, 0.088) (see Table 1 for details). 4

0.2 u 1 (t)

3

u 2 (t)

0.1

2 0

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(y,y i )

1

-1

-0.2

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-3 -0.4

-4 -4

-2

0

2

-0.5

4

0

(x,x i )

2

4

6

8

10

t

Fig. 3. Three vortices. Trajectory and control in the three vortices case. The initial adjoint vector being p0 = (−0.37, −0.037, 0.088, 0.34, 0.69, −0.48, −0.36, −0.026) (see Table 1 for details). 4 0.35 u 1 (t)

3

0.3

2

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1

0.15

0

u2

(y,y i )

u 2 (t)

0.25

0.1

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-4 -4

-2

0

(x,x i )

2

4

-0.15 0

2

4

6

8

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t

Fig. 4. Four vortices. Trajectory and control in the four vortices case. The initial adjoint vector is p0 = (0.17, −0.0015, 0.041, −0.18, −0.025, −0.26, 0.11, 0.11, −0.18, 0.22) (see Table 1 for details).

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Conclusion

In this article, we have solved the control problem in the cases N = 1, 2, 3, and 4 vortices. In the case of a single vortex, where it is trivial to show the integrability of the resulting Hamiltonian system, we have provided analytic expressions of extremals. In the other cases, we have limited ourselves to a numerical study of the problem by presenting some solutions obtained thanks to the HamPath software. Beyond 4 vortices, it is well known in the literature that the vortex system is chaotic (see for instance [7,16]), so controlling the particle in an environment containing more than 4 vortices no longer guarantees a solution to the problem. On the other hand, we have arbitrarily considered the initial positions of the vortices and we propose in future experiments to optimize this initial configuration in order to improve the results we have obtained.

References 1. Babiano, A., Boffetta, G., Provenzale, A., Vulpiani, A.: Chaotic advection in point vortex models and two-dimensional turbulence. Phys. Fluids 6(7), 2465–2474 (1994) 2. Bonnard, B., Cots, O., Wembe, B.: A Zermelo navigation problem with a vortex singularity. ESAIM Control Optim. Calc. Var. (2020). https://doi.org/10.1051/ cocv/2020058 3. Bonnard, B., Faubourg, L., Tr´elat, E.: M´ecanique c´eleste et contrˆ ole des v´ehicules spatiaux. In: Math´ematiques & Applications, vol. 51, p. 276. Springer, Berlin (2006) 4. Caillau, J.-B., Cots, O., Gergaud, J.: Differential continuation for regular optimal control problems. Optim. Meth. Softw. 27(2), 177–196 (2011) 5. Calleja, R.C., Doedel, E.J., Garc´ıa-Azpeitia, C.: Choreographies in the n-vortex problem. Regul. Chaot. Dyn. 23(5), 595–612 (2018) 6. Chenciner, A., Gerver, J., Montgomery, R., Sim´ o, C.: Simple choreographic motions of n bodies: a preliminary study. In: Newton, P., Holmes, P., Weinstein, A. (eds.) Geometry, Mechanics, and Dynamics, pp. 287–308. Springer, New York, NY (2002) 7. Chen, Y., Kolokolnikov, T.: Collective behaviour of large number of vortex in the plane. Proc. R. Soc. A. 469(2156), 12 (2013) 8. Chertovskih, R., Karamzin, D., Khalil, N.T., Lobo Pereira, F.: Regular pathconstrained time-optimal control problems in three-dimensional flow fields. Eur. J. Control. https://doi.org/10.1016/j.ejcon.2020.02.003 9. Cots, O.: Contrˆ ole optimal g´eom´etrique: m´ethodes homotopiques et applications, Ph.d. Thesis. Institut Math´ematiques de Bourgogne, Dijon, France (2012) 10. Liu, J., Hu, H.: Biological inspiration: from carangiform fish to multi-joint robotic fish. J. Bionic Eng. 7(1), 35–48 (2010) 11. Meyer, K., Hall, G., Offin, D.C.: Introduction to Hamiltonian dynamical systems and the n-body problem. In: Applied Mathematical Sciences, vol. 90, p. 399. Springer, New York (2009) 12. Newton, P.K.: The n-vortex problem: analytical techniques. In: Applied Mathematical Sciences, vol. 145, p. 420. Springer (2013) 13. Pereira, F.L., Grilo, T., Gama, S.: Optimal multi-process control of a two vortex driven particle in the plane. IFAC-PapersOnLine 50(1), 2193–2198 (2017) 14. Poincar´e, H.: Œuvres, Gauthier-Villars (1952)

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15. Pontryagin, L.S., Boltyanski˘ı, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York, London (1962). Translated from the Russian by Trirogoff, K.N. Edited by Neustadt, L.W 16. Protas, B.: Vortex dynamics models in flow control problems. Nonlinearity IOP Sci. 21(9), R203–R250 (2008) 17. Vainchtein, D., Mezi, I.: Vortex-based control algorithms. In: Control of Fluid Flow, pp. 189–212. Springer, Berlin (2006)

Cubic Splines in the Grassmann Manifold F´ atima Pina1,2 and F´ atima Silva Leite1,2(B) 1

Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal [email protected] 2 Institute of Systems and Robotics, DEEC-UC, 3030-290 Coimbra, Portugal [email protected]

Abstract. We present a detailed implementation of the De Casteljau algorithm to generate cubic splines that solve certain interpolation problems in the Grassmann manifold. Keywords: Cubic splines · De Casteljau algorithm Grassmann manifolds · Interpolating curves

1

· Geodesics ·

Introduction

Interpolating nonlinear data arises in many different areas, ranging from robotics and computer vision to industrial and medical applications (see, e.g., [2]). A particular illustrative example is the classical problem of reconstructing a scene from several snapshots taken at different time instants. This can be formulated as an interpolation problem in the Grassmann manifold. This manifold, consisting of all k-dimensional subspaces in IRn , plays an important role in many computer vision applications where subspace methods are used to represent an image set by a linear subspace that is spanned by all the images in the set. In Euclidean spaces, cubic splines, which are C 2 -smooth curves obtained by piecing together cubic polynomials, are particularly important since they minimise the average acceleration. A well-known recursive procedure to generate interpolating polynomial curves in Euclidean spaces is the classical De Casteljau algorithm [4]. Generalizations of such curves to non-Euclidean manifolds is particularly useful in many engineering applications and were first introduced in [8]. In the present paper we implement the generalised De Casteljau algorithm for the Grassmann manifold, following closely the work in [3] concerning the reinterpretation of the De Casteljau algorithm for connected and compact Lie groups. The main feature of the algorithm is based on recursive geodesic interpolation in order to find a polynomial curve that solves a 2-boundary value problem. These boundary conditions might be of Hermite type, i.e., consisting of initial and final points together with initial and final velocities, or consists of initial and final points, initial velocity and initial intrinsic acceleration. While the first conditions, together with other interpolation requirements are more natural in c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 243–252, 2021. https://doi.org/10.1007/978-3-030-58653-9_23

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applications, they pose difficulties that do not arise if the second type of boundary conditions is prescribed. After presenting some preliminary results in Sect. 2 and the geometry of the Grassmann manifold in Sect. 3, we dedicate the main section to the implementation of the De Casteljau algorithm for that important manifold. At the end of this section we derive relations between the two types of boundary conditions and explain how to generate a geometric cubic spline from cubic polynomials obtained by using the proposed algorithm.

2

Notations and Auxiliary Results

In the sequel, gl(n) denotes the Lie algebra of n × n real matrices, equipped with the Lie bracket defined by the commutator, i.e., [A, B] := AB − BA. The adjoint operator in gl(n) is defined by adA B := [A, B]. The vector space of n × n symmetric matrices is denoted by s(n) and the Lie subalgebra of gl(n), consisting of the skewsymmetric n × n matrices is denoted by so(n). The rotation group SO(n), having so(n) as its Lie algebra, will also play an important role here. +∞ Ak Given A ∈ gl(n), the matrix exponential of A is defined by eA = k=0 k! . For X ∈ gl(n) not having eigenvalues in the closed negative real line, i.e., σ(X) ∩ IR− 0 = ∅, there exists a unique real logarithm of X whose spectrum lies in the infinite horizontal strip {z ∈ C : −π < Im(z) < π}. This logarithm is usually called principal logarithm of X and is the only logarithm that we consider here. When X ∈ SO(n), log X ∈ so(n). When X − I < 1, log X is uniquely defined k +∞ k+1 (X−I) by the convergent power series: log X = . Properties of k=1 (−1) k these matrix functions can be found in [6] and [5], but we emphasise the formula 1 eA B e−A = eadA (B) = B +[A, B]+ [A, [A, B]]+· · · , which plays an important 2! role here. In the sequel, we also assume the following notations: f (z) =

ez −1 z

denotes the sum of the series

+∞

zk k=0 (k+1)! ;

(1)

(z − 1)k , if |z − 1| < 1. k+1 (2) Note that f (z)g(ez ) = 1. The next result can be found in [10]. g(z) =

log z z−1

denotes the sum of the series

+∞

k k=0 (−1)

Lemma 1. Let t −→ A(t) be a differentiable matrix valued function. Then, d A(t) A(t) e = ΔL , where A(t) (t) e dt  1    L ˙ ˙ ΔA(t) (t) = . eu adA(t) (A(t))du = f adA(t) A(t)

(3) (4)

0

The following lemma contains important properties of the operator defined in (4). Proof details can be found in Sect. 2 of [9].

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Lemma 2. Let t −→ A(t) be a differentiable matrix valued function. Then −A(t) ΔL = −ΔL −A(t) (t) e A(t) (t);   L = A(0) and Δ(t−1)A(t) (t) = A(1);   t=0 t=1  d  d ˙ ˙  ΔL (t) = 2A(0) and ΔL (t) = 2A(1). d t t=0 tA(t) d t t=1 (t−1)A(t)

e  L ΔtA(t) (t)

3

A(t)

(5)

The Geometry of the Grassmann Manifold

The Grassmann manifold is the set of all k-dimensional subspaces in IRn . Here, similarly to [7] and [1], we use the following matrix representation of the Grassmann manifold

(6) Gk,n := P ∈ s(n) : P 2 = P and rank(P ) = k , which is compact and connected, of real dimension k(n − k), 0 < k < n. For an arbitrary point P ∈ Gk,n , define the following sets of matrices glP (n) := {A ∈ gl(n) : A = P A + AP };

soP (n) := so(n) ∩ glP (n).

The tangent space at a point P ∈ Gk,n can be defined by



TP Gk,n = [P, Ω] : Ω ∈ soP (n) = ad2P (S) : S ∈ s(n) .

(7)

(8)

We consider the Grassmann manifold equipped with the Riemannian metric, induced by the Euclidean inner product on each tangent space, given by   (9) [P, Ω1 ], [P, Ω2 ] = −tr Ω1 Ω2 . Using the last description of the tangent space at P , the normal space at P , with respect to the Riemannian metric (9), can be defined by

(10) (TP Gk,n )⊥ = S − ad2P (S) : S ∈ s(n) . The proof of the next two lemmas can be found in [9]. Lemma 3. Let P ∈ Gk,n and Ω ∈ soP (n). Then, [Ω, [Ω, P ]] ∈ (TP Gk,n )⊥ . Lemma 4. Let P ∈ Gk,n , A, B ∈ glP (n), and t ∈ IR. Then, [A, P ] = [B, P ] ⇐⇒ A = B;

(11)

e2tA (I − 2P ) = eadtA (I − 2P ).

(12)

Now, we present some results about geodesics in the Grassmann manifold with respect to the Riemannian metric in (9). Lemma 5 ([1]). The unique geodesic t → γ(t) in Gk,n , satisfying the initial conditions γ(0) = P and γ(0) ˙ = [Ω, P ], where Ω ∈ soP (n), is given by γ(t) = etΩ P e−tΩ .

(13)

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The next result gives an explicit formula for the minimising geodesic arc connecting two points in Gk,n . Proposition 1 ([1]). Let P, Q ∈ Gk,n be such that the matrix (I − 2Q)(I − 2P ) has no negative real eigenvalues. Then, the minimising geodesic arc in Gk,n that joins P (at t = 0) to Q (at t = 1), is parameterised explicitly by (13), with Ω = 12 log((I − 2Q)(I − 2P )).

4

De Casteljau Algorithm in the Grassmann Manifold

4.1

An Interpolation Problem in Gk,n

Problem 1. Given  + 1 distinct points pi ∈ Gk,n , with i = 0, 1, . . . , , a discrete sequence of +1 fixed times, t0 < t1 < · · · < t−1 < t , and vectors ξ0 , η0 tangent to Gk,n at p0 , and ξ tangent to Gk,n at p , solve the following problem: Find a C 2 -smooth curve γ : [t0 , t ] → Gk,n , satisfying interpolation conditions γ(ti ) = pi ,

1 ≤ i ≤  − 1,

(14)

and boundary conditions (of Hermite type): γ(t0 ) = p0 , γ(t ) = p , ˙  ) = ξ ∈ Tp Gk,n , γ(t ˙ 0 ) = ξ0 ∈ Tp0 Gk,n , γ(t

(15)

or, alternatively, boundary conditions: γ(t0 ) = p0 , γ(t ˙ 0 ) = ξ0 ∈ Tp0 Gk,n , where

γ(t ) = p , Dγ˙ (t0 ) = η0 ∈ Tp0 Gk,n , dt

(16)

Dγ˙ stands for the covariant derivative of the velocity vector field γ. ˙ dt

Without loss of generality, in the sequel we consider t0 = 0 and t = 1, since the reparametrisation (t → s) defined by s = t(t − t0 ) + t0 maps [0, 1] to [t0 , t ]. The solution of this problem is a geometric cubic spline, obtained by piecing together geometric cubic polynomials generated by the generalised De Casteljau algorithm. For geodesically complete manifolds, this algorithm consists of three successive geodesic interpolation steps that first appeared in [8]. 4.2

Implementation of the De Casteljau Algorithm in Gk,n

Although the Grassmann manifold is geodesically complete, we have seen that an explicit formula for the geodesic that joins two points may be unknown in some particular situations. So, in this case, the implementation of the algorithm is restricted to a convex open subset of the manifold where the expression to compute geodesic arcs is known. The generation of a geometric cubic polynomial

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in Gk,n , starts from four given points x0 , x1 , x2 , x3 in Gk,n . The superscripts in the operators Ωij below are chosen according to the corresponding step. Algorithm Step 1 - Construct three geodesic arcs 1

1

β1 (t, xi , xi+1 ) = etΩi xi e−tΩi = e

t adΩ 1

xi ,

i

i = 0, 1, 2,

1 log((I − 2xi+1 )(I − 2xi )) ∈ soxi (n). 2 Step 2 - Construct two families of geodesic arcs Ωi1 :=

2

2

t adΩ 2 (t)

2

2

t adΩ 2 (t) 1

β2 (t, x0 , x1 , x2 ) = etΩ0 (t) β1 (t, x0 , x1 ) e−tΩ0 (t) = e β2 (t, x1 , x2 , x3 ) = etΩ1 (t) β1 (t, x1 , x2 ) e−tΩ1 (t) = e

0

(18)

β1 (t, x0 , x1 ),

(19)

β1 (t, x1 , x2 ),

(20)

1 log((I − 2β1 (t, x1 , x2 ))(I − 2β1 (t, x0 , x1 ))) ∈ soβ1 (t,x0 ,x1 ) (n), 2 1 Ω12 (t) := log((I − 2β1 (t, x2 , x3 ))(I − 2β1 (t, x1 , x2 ))) ∈ soβ1 (t,x1 ,x2 ) (n). 2 Step 3 - Construct a family of geodesic arcs Ω02 (t) :=

3

3

β3 (t, x0 , x1 , x2 , x3 ) = etΩ0 (t) β2 (t, x0 , x1 , x2 ) e−tΩ0 (t) = e

(17)

t adΩ 3 (t) 0

(21) (22)

β2 (t, x0 , x1 , x2 ),

(23) Ω03 (t) :=

1 log((I −2β2 (t, x1 , x2 , x3 ))(I −2β2 (t, x0 , x1 , x2 ))) ∈ soβ2 (t,x0 ,x1 ,x2 ) (n). (24) 2

As a result, and taking into consideration (17), (19), (20) and (23), we obtain the geometric cubic polynomial in the Grassmann manifold. Definition 1. The curve t ∈ [0, 1] → β3 (t) := β3 (t, x0 , x1 , x2 , x3 ), defined by t adΩ 3 (t)

β3 (t) = e

0

e

t adΩ 2 (t) 0

e

t adΩ 1 0

x0 ,

(25)

with Ω01 , Ω02 and Ω03 given by (18), (21), and (24), is called a geometric cubic polynomial in the Grassmann manifold, associated to the points xi , i = 0, 1, 2, 3. Remark 1. Notice that Ω02 (0) = Ω03 (0) = Ω01 , Ω12 (0) = Ω11 , Ω12 (1) = Ω03 (1) = Ω21 , Ω02 (1) = Ω11 , and that the curve β3 , just defined, joins the point x0 (at t = 0) to x3 (at t = 1). However, it doesn’t pass through x1 and x2 , which are called control points since they are responsible for the shape of the curve. Lemma 6 ([9]). Let Ωij be defined by (18), (21), (22) and (24). Then, 2

1

1

e2Ω0 (t) = e2tΩ1 e2(1−t)Ω0 , 2 1 1 e2Ω1 (t) = e2tΩ2 e2(1−t)Ω1 , 3 2 2 e2Ω0 (t) = e2tΩ1 (t) e2(1−t)Ω0 (t) ; (t−1) ad

2

t ad

1

t ad

2

(t−1) ad

(26) 1

Ω0 (t) Ω0 e e Ω1 = e Ω0 (t) e , (t−1) adΩ 2 (t) t adΩ 1 t adΩ 2 (t) (t−1) adΩ 1 1 2 1 1, e =e e e (t−1) adΩ 3 (t) t adΩ 2 (t) t adΩ 3 (t) (t−1) adΩ 2 (t) 0 1 0 0 e =e e . e

(27)

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We are now in conditions to state the following result which contains an alternative way of defining the geometric cubic polynomial β3 in Gk,n . This will be particularly useful in the computation of the derivatives of the cubic polynomial at the endpoint (t = 1). Theorem 1. Let t ∈ [0, 1] → β3 (t) be the geometric cubic polynomial in Gk,n defined in Definition 1. Define another curve t ∈ [0, 1] → γ(t) in Gk,n , by (t−1) adΩ 3 (t)

γ(t) = e

0

e

(t−1) adΩ 2 (t) 1

e

(t−1) adΩ 1 2

x3 ,

(28)

where Ω03 , Ω12 and Ω21 are as defined at the beginning of Subsect. 4.2. Then, β3 (t) = γ(t),

t ∈ [0, 1].

Proof. The proof results from simple computations, attending to Remark 1 and applying formulas (27) in Lemma 6. QED Lemma 7 ([9]). For j = 2, 3, let i = 3 − j. Then, the following holds.    d  2Ω j (t)  ˙ j (0) e2Ω01 , where χ := f (ad2Ω 1 ); 0 Ω e = 2 χ 0 0 0  0 dt t=0    d  2Ω j (t)  1 e i = 2 χ1 Ω˙ ij (1) e2Ω2 , where χ1 := f (ad2Ω21 );  dt t=1     1 ˙ j (0) ; Ω (t) = −2 Ω + 2 χ ΔL  j 0 0 0 2(1−t)Ω (t) t=0

0

  ΔL (t)  j 2tΩ (t) i

t=1

=

2 Ω21

 + 2 χ1

 Ω˙ ij (1) .

(29) (30)

(31)

Remark 2. In what follows, we must guarantee that the operators χ0 and χ1 have inverse. From the definition of f and g in (1) and (2) respectively, we know that f (A)g(eA ) = I, for eA − I < 1. So, if this restriction holds for A = ad2Ω01 and for A = ad2Ω21 , taking into account the definitions of χ0 and χ1 above, we immediately obtain := g(e χ−1 0

ad2Ω 1 0

) and χ−1 := g(e 1

ad2Ω 1 2

).

Lemma 8 ([9]). For j = 2, 3, let i = 3 − j. Then  1  Ω1 − Ω01 , Ω˙ 0j (0) = (j − 1) χ−1 0   1 2Ω21 1 −2Ω21 Ω˙ ij (1) = (j − 1) χ−1 Ω . − e Ω e 2 1 1

(32)

(33) (34)

Theorem 2. The curve t ∈ [0, 1] → β3 (t) in Gk,n defined in (25) satisfies the following boundary conditions: β3 (0) = x0 ,

  1  Dβ˙ 3 (0) = 6 χ−1 Ω1 − Ω01 , x0 ; 0 dt

   Dβ˙ 3 1 2Ω21 1 −2Ω21 (1) = 6 χ−1 − e Ω e Ω , x . 3 2 1 1 dt

β˙ 3 (0) = [3Ω01 , x0 ], β˙ 3 (1) = [3Ω21 , x3 ],

β3 (1) = x3 ;

(35) (36) (37)

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249

Proof. We have already pointed out in Remark 1 that β3 satisfies the first set of boundary conditions. To prove the first condition in (36), we differentiate both sides of (25), use t ad 1 the identity e Ω0 Ω01 = Ω01 , and Lemma 2, with A(t) replaced by tΩ0j (t), for j = 2, 3, as follows. 3 3 β˙ 3 (t) = ΔL (t)β3 (t) + etΩ0 (t) ΔL (t) e−tΩ0 (t) β3 (t) tΩ tΩ02 (t) 

03 (t)3 2 2 3 + etΩ0 (t) etΩ0 (t) Ω01 e−tΩ0 (t) e−tΩ0 (t) , β3 (t) 3

2

2

3

+ β3 (t) etΩ0 (t) etΩ0 (t) ΔL (t) e−tΩ0 (t) e−tΩ0 (t) −tΩ 2 (t)

(38)

0

tΩ03 (t)

+ β3 (t) e ΔL (t) e −tΩ03 (t) = [Ω(t), β3 (t)] ,

−tΩ03 (t)

where t adΩ 3 (t)

Ω(t) := ΔL tΩ 3 (t) (t) + e 0

0

t adΩ 3 (t)

ΔL tΩ 2 (t) (t) + e 0

0

e

t adΩ 2 (t) 0

Ω01 ∈ soβ3 (t) (n).

Evaluating at t = 0, and taking into consideration Lemma 2 and Ω02 (0) = Ω03 (0) = Ω01 , we obtain Ω(0) = 3Ω01 , which implies β˙ 3 (0) = [3Ω01 , x0 ]. To prove the second condition in (36), differentiate β˙ 3 (t) to get ˙ β3 (t)] + [Ω(t), [Ω(t), β3 (t)]]. β¨3 (t) = [Ω(t), The first term above belongs to Tβ3 (t) Gk,n and, according to Lemma 3, the second term belongs to (Tβ3 (t) Gk,n )⊥ . So, since the covariant derivative of β˙ 3 is the tangential projection of β¨3 , it follows that

 Dβ˙ 3 ˙ (t) = Ω(t), β3 (t) . dt Now, it is enough to evaluate at t = 0. This involves using the results in Lemmas 2 and 8, and several computations that are omitted due to lack of space, but can be found in [9]. Finally, the proof of identities (37) is similar to that of identities (36), but using instead of (25) the alternative expression for β3 in Theorem 1, and evaluating at t = 1. QED Obtaining the Control Points from the Boundary Conditions. In this subsection we will show how to get the control points from the boundary conditions, in order to implement the De Casteljau algorithm. • Case 1 - The Boundary Conditions are of Type (15) The boundary conditions (15) in Problem 1 can be written as: β3 (0) = x0 ,

β3 (1) = x3 ,

β˙ 3 (0) = [V0 , x0 ],

β˙ 3 (1) = [V3 , x3 ],

(39)

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where x0 , x3 ∈ Gk,n , V0 ∈ sox0 (n), and V3 ∈ sox3 (n). According to the implementation of the algorithm, in order to generate the cubic polynomial that satisfies (39), we must be able to choose the control points x1 and x2 from those boundary conditions. The following theorem answers this question. Theorem 3. The control points x1 and x2 , used in the De Casteljau algorithm to generate the geometric cubic polynomial that satisfies the boundary conditions (39), are given by:   2 2 1 1 (40) I − e 3 V0 (I − 2x0 ) , x2 = I − (I − 2x3 ) e 3 V3 . x1 = 2 2 1

Proof. We know that e2Ω0 = (I − 2x1 )(I − 2x0 ), or, equivalently, I − 2x1 = 1 e2Ω0 (I − 2x0 ). Therefore, solving the last equation for x1 , and using Ω01 = V0 /3, which follows immediately by comparing the value of β˙ 3 (0) in Theorem 2 with  1 2 the condition β˙ 3 (0) = [V0 , x0 ] above, we obtain x1 = I − e 3 V0 (I − 2x0 ) . 2 The second identity is obtained with similar arguments, using the definition of Ω21 and the obvious relation Ω21 = V3 /3. QED • Case 2 - The Boundary Conditions are of Type (16) The boundary conditions (16) in Problem 1 can be written as: β3 (0) = x0 ,

β3 (1) = x3 ,

β˙ 3 (0) = [V0 , x0 ],

Dβ˙ 3 (0) = [W0 , x0 ], dt

(41)

where x0 , x3 ∈ Gk,n , and V0 , W0 ∈ sox0 (n). Theorem 4. The control points x1 and x2 , used in the De Casteljau algorithm to generate the geometric cubic polynomial that satisfies the boundary conditions (41), are given by:   2 1 2 2 1 1 I − e 3 V0 (I − 2x0 ) , x2 = I − e 3 χ0 (W0 )+ 3 V0 e 3 V0 (I − 2x0 ) . (42) x1 = 2 2 Proof. It is enough to obtain the control point x2 . Taking into account (41) and  1  1 2 Theorem 2, we must have 6χ−1 Ω1 − Ω01 = W0 , i.e., 2Ω11 = χ0 (W0 ) + V0 . 0 3 3 1 On the other hand, from the definition of Ω11 , e2Ω1 = (I − 2x2 )(I − 2x1 ). So,  1 1 solving for x2 , one gets x2 = I − e2Ω1 (I − 2x1 ) . Now, using the expression 2 1 2 2 of Ω11 and of x1 in terms of x0 and V0 , we obtain x2 = (I −e 3 χ0 (W0 )+ 3 V0 e 3 V0 (I − QED 2x0 ))/2, as required. The Case 1, corresponding to the Hermite boundary conditions, can be considered simpler than the Case 2, since it doesn’t involves the computation of covariant derivatives. However, the Case 2, where the data is not symmetrically specified, has computational advantages over the Case 1, namely whenever the goal is to generated cubic splines, i.e., piecing together several geometric cubic polynomials so that the overall curve is C 2 -smooth.

Cubic Splines in the Grassmann Manifold

251

Corollary 1. The relationship between the boundary conditions of types (15) and (16) is the following:    2  2 2 −1 V V 3 0 W0 = 3χ0 log (I − 2x3 ) e 3 e 3 (I − 2x0 ) − V0 , 3   3 1 2 2 V3 = log (I − 2x3 ) e 3 χ0 (W0 )+ 3 V0 (I − 2x0 ) e− 3 V0 . 2

5

Generating Cubic Splines in Gk,n

We now explain how to solve the interpolation Problem 1 for the boundary conditions of type (16). The objective is to generate a geometric cubic spline, i.e., a C 2 -smooth curve that satisfies the interpolation and the boundary conditions and such that when restricted to each subinterval is a geometric cubic polynomial. The crucial procedure is the generation of the first cubic polynomial, denoted by γ1 , joining p0 to p1 and having prescribed initial velocity [V0 , p0 ] and initial covariant acceleration [W0 , p0 ]. Although this has already been described in the previous section, we summarise the results here for the sake of completeness. We also adapt the notations so that the curve is given in terms of the data. The interpolation curve γ of Problem 1 may be generated by piecing together cubic polynomials defined on each subinterval [ti , ti+1 ], i = 0, 1, . . . ,  − 1. Without loss of generality, we assume that all spline segments are parameterised in the [0, 1] time interval. 5.1

Generating the First Spline Segment

Apply the De Casteljau algorithm to obtain the first spline segment t adΩ 3 (t)

γ1 (t) = e

0

e

t adΩ 2 (t) 0

e

t adΩ 1 0

(43)

p0 ,

where Ω01

=

Ω11

=

Ω21

=

Ω02 (t) = Ω12 (t) = Ω03 (t) =

1 log((I 2 1 log((I 2 1 log((I 2 1 log((I 2 1 log((I 2 1 log((I 2

− 2x1 )(I − 2p0 )); − 2x2 )(I − 2x1 )); − 2p1 )(I − 2x2 )); − 2e

t adΩ 1

x1 )(I − 2 e

t adΩ 1

− 2e

t adΩ 1

x2 )(I − 2 e

t adΩ 1

− 2e

t adΩ 2 (t)

1

2

1

t adΩ 1

e

1

0

1

p0 )); x1 ));

x1 )(I − 2 e

t adΩ 2 (t) t adΩ 1 0

e

0

p0 )),

and the control points are given by x1 =

1 2 (I − e 3 V0 (I − 2p0 )); 2

x2 =

1 1 2 2 (I − e 3 χ0 (W0 )+ 3 V0 e 3 V0 (I − 2p0 )). 2

252

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Generating Consecutive Spline Segments

After having generated the first spline segment, one continues in a similar way for the second spline segment. Since the cubic spline is required to be C 2 -smooth, the initial velocity and initial covariant acceleration for this second spline segment must equal the end velocity and the end covariant acceleration of the previous spline segment, which are given by the formulas in Theorem 2. The other  − 2 consecutive segments are generated similarly. The solution of Problem 1 is the cubic spline curve resulting from the concatenation of the  consecutive segments.

6

Conclusion

We have presented all the details to implement the De Casteljau algorithm in the Grassmann manifold. For practical applications, one still has to rely on computing stable matrix exponentials and logarithms of structured matrices. This is out of the scope of our work, but efficient numerical methods to compute matrix functions have been developed along the years (see, for instance, [5]), and are expanding at a fast rate. Acknowledgements. The authors acknowledge Funda¸ca ˜o para a Ciˆencia e a Tecnologia (FCT) and COMPETE 2020 program for financial support to project UIDB/00048/2020.

References 1. Batzies, E., H¨ uper, K., Machado, L., Silva Leite, F.: Geometric mean and geodesic regression on grassmannians. Linear Algebra Appl. 466, 83–101 (2015) 2. Bressan, B.: From Physics to Daily Life: Applications in Informatics, Energy, and Environment. Wiley-Blackwell, Germany (2014) 3. Crouch, P., Kun, G., Silva Leite, F.: The de Casteljau algorithm on Lie groups and spheres. J. Dynamical Control Syst. 5(3), 397–429 (1999) 4. de Casteljau, P.: Outillages M´ethodes Calcul. Technical Report, Andr´e Citro¨en Automobiles SA (1959) 5. Higham, N.: Functions of Matrices: Theory and Computation. SIAM (2008) 6. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991) 7. H¨ uper, K., Silva Leite, F.: On the geometry of rolling and interpolation curves on S n , SOn , and Grassmann manifolds. J. Dynamical Control Syst. 13(4), 467–502 (2007) 8. Park, F., Ravani, B.: B´ezier curves on Riemannian manifolds and Lie groups with kinematics applications. ASME J. Mech. Des. 117, 36–40 (1995) 9. Pina, F., Silva Leite, F.: Cubic splines in the Grassmann manifold generated by the De Casteljau algorithm. In: Pr´e-Publica¸co ˜es do Departamento de Matem´ atica - Universidade de Coimbra, Portugal, number 20-07 (2020) 10. Sattinger, D.H., Weaver, O.L.: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. Applied Mathematical Sciences, vol. 61. Springer, New York (1986)

Distributed Adaptive Predictive Control Based on Switched Multiple Models and ADMM Margarida Nabais and Jo˜ ao M. Lemos(B) INESC-ID, IST, Univ Lisboa, Lisbon, Portugal [email protected], [email protected]

Abstract. This article presents a novel distributed adaptive predictive controller based on supervised multiple models. The local control agents embed a linear model based predictive control (MPC) algorithm and a coordination algorithm that relies on the Alternating Direction Method of Multipliers (ADMM) distributed optimization algorithm. In order to illustrate the controller proposed, this one is applied to the coordinated control of steam flow and pressure in a biomass thermal power plant. A comparison is made with two other distributed adaptive controllers, with coordination based on a game procedure, one with LQG, and the other with local MPC controllers. Keywords: Distributed control · Adaptive control · Supervised multiple model · Predictive control · ADMM · Thermal power plant

1

Introduction

As systems are becoming increasingly complex and connected in large networks, the growth in complexity requires the use of distributed decision-making, in which a number of local agents interact with their neighbours in order to stabilize the overall network, while approximating global optimality. Furthermore, adaption is important when the process is unknown or when its dynamics changes in time, e.g. due to non-linear effects that imply a modification in the response with the operating point, or when there are changes due to unpredictable factors, such as ageing or wearing, or due to varying characteristics associated to low cost manufacturing. In a distributed scheme, the local manipulated variables are obtained using local measurements and small-scale models of the local dynamics, that are easier to design and implement [1]. Furthermore, distribution provides flexibility to the system structure, allows the system to be fault-tolerant by being redundant This work was supported by national funds through FCT, Funda¸ca ˜o para a Ciˆencia e a tecnologia, Portugal, under project UIBD/50021/2020 and by POR Lisboa, LISBOA01-0145-FEDER-031411 (project HARMONY). c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 253–262, 2021. https://doi.org/10.1007/978-3-030-58653-9_24

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and reconfigurable, and does not require information of the whole system, thus requiring less computational effort [2]. The coordination between the local controllers can be made using several techniques. A possible approach consists in using a distributed LQG control algorithm based on game theory, where controllers communicate with their neighbours in order to reach a consensus on the control action [3,4]. In this case, if the convergence condition is attained, the whole system will arrive at a Nash equilibrium. Another solution consists of using MPC (model predictive control), instead of LQG as local controllers [5]. Examples od distributed MPC based on game concepts can be found in [6,7]. A robust D-MPC algorithm has been developed in [8] to deal with coupled constraints. An alternative, and a more powerful coordination algorithm to tackle the consensus problem, is a special augmented Lagrangian-based algorithm called ADMM [9]. A distributed form of this algorithm is presented in [10], namely D-ADMM. On the other way, although there are several possible adaptation mechanisms, the Switched Multiple Model Adaptive Controller (SMMAC) [11,12] is the one considered hereafter. According to this technique, the controller to apply to the plant is selected by a supervisor from a bank. The supervisor selects the controller that corresponds to the model that has the most similar observed behaviour to the plant. There is a rich literature on SMMAC that cannot be reviewed here for lack of space. We only cite [13], that proposes a robust SMMAC architecture to deal with highly uncertain exogenous plant disturbance environments. Coupling distributed MPC with adaptive control seems however to be an unexplored subject. Therefore, the contribution of this work is to develop a distributed adaptive controller based on supervised multiple models as the adaptation mechanism, MPC as local control agents, and coordination based on ADMM. This controller is applied to the steam/flow coordinated control of a biomass thermal power plant, for which simulation results are presented. The article is organised as follows: Sect. 2 describes the elements used for SMMAC, and Sect. 3 describes the MPC algorithm used in local controllers, as well as the D-ADMM coordination algorithm; Sect. 4 addresses the case study in steam flow/pressure coordination and, finally, Sect. 5 draws conclusions.

2

Switched Multiple Model Adaptive Control

It is assumed that the process to be controlled is represented by the linear, time invariant (LTI), and finite dimensional state model  x(k + 1) = Ax(k) + Bu(k) (1) y(k) = Cx(k), where u, y ∈ R, x ∈ Rn and k ∈ N. The SMMAC aims to control the plant by switching between a finite set of controllers, each matching the models in the set M = ∪N i=1 Mi [11,12], that correspond to different instances of the matrices in (1). In this structure there are two main blocks, a bank of controllers and a supervisor, as represented in Fig. 1.

Distributed Adaptive Control

255

The bank of controllers is composed by a set of controllers C = ∪N i=1 Ci , where each local controller Ci is designed to match model Mi . The switch between two different models occurs when there is a need to adjust and find a model that better fits the plant dynamics. This task is performed by the supervisor.

Fig. 1. Simplified SMMAC Architecture.

The local models are obtained by drawing a grid obtained by varying some parameters, locating the models such as to cover the parameter space. The choice of how many models are considered must be a compromise since, if this number is not sufficient, there might be some dynamics that are not considered by any of the existent models. On the contrary, it is not desirable to have too many models since the difference between model dynamics might not be relevant, which could lead to stability problems as the supervisor would constantly change its choice. Bumpless transfer between different controllers is ensured by inserting an integrator common to all controllers [14], resulting in a redesign of the controllers with the augmented state models of the form        Ai Bi x ˆi (k) 0 x ˆi (k + 1) = + δui (k), (2) 0 I I ui (k + 1) ui (k) 

    x ˆi (k) yi (k) = Ci 0 , ui (k) 

(3)

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with identity matrix I with the same dimensions of vector ui . An anti-windup mechanism is also included. The supervisor decides, at each discrete time k, the value of the index σ of the controller that best matches the plant, so that the best possible performance is achieved [15,16]. The supervisor comprises three blocks: Model Bank; Performance index generator; Switching logic. The model bank is made of local models Mi , each one representing a possible outcome for the plant dynamics. This block is responsible for determining the output estimation yˆi , which is obtained by a Kalman filter designed from the corresponding model. The prediction error ei (k) = yˆi (k) − yi (k) is computed in order to compare each model i to the actual plant output y and squared to be always positive, which results in the signal wi (k) = |ei (k)|2 . The performance index πi is generated according to (1 − λπ )2 wi (k), (4) πi (k) = (¯ q − λπ )2 where λπ is a parameter defined between 0 and 1 and q¯ is the forward shift operator. The switching logic chooses the index σ associated with the model that has the most similar plant behaviour, that is σ = argmin πi (k).

(5)

Fast switching may be a problem due to the possibility of the overall system becoming unstable. This situation can be avoided by imposing a condition [15, 17], according to which, once a controller is applied to the process, it has to remain selected for a minimum amount of time τD , the dwell-time. Additionally, another condition is enforced, where there has to be a minimum of samples with the same controller index τS before the supervisor changes its decision.

3

Distributed MPC Based on D-ADMM

Consider a class of distributed systems composed by M interconnected subsystems coupled through the control inputs, the state-space model of each subsystem is defined as   xi (k + 1) = Ai xi (k) + j∈(Ni ,i) Bij Δuj (k) (6) yi (k) = Ci xi (k) where xi ∈ Rqi , i = 1, ..., M , are the states of each subsystem and Δuj ∈ Rrj , j = 1, ..., M , are the different incremental manipulated inputs. The set Ni comprises the neighbours of agent i, i.e., it indicates which agents control input Δuj affect xi . To each subsystem i, associate a cost Ji and define the total cost to be minimized, J, by M  J= Ji . (7) i=1

Distributed Adaptive Control

257

The predictive model of agent i is ¯i (k) + Wi,i ΔUi + Yi = Πi x



Wi,j ΔUj ,

(8)

j∈Ni

with

⎡

⎤ Δyi (k + 1) ⎢ Δyi (k + 2) ⎥ ⎢ ⎥ Yi = ⎢ ⎥, 1 : .. ⎣ ⎦ . Δyi (k + N )

⎡

Δui (k) Δui (k + 1) .. .

⎤

⎢ ⎥ ⎢ ⎥ ΔUi = ⎢ ⎥, ⎣ ⎦ Δui (k + N − 1)

(9)

where x ¯i (k) represents the augmented state at time k as in Eq. (2). The predictor matrices are ⎡

Wi,p

¯i,p C¯i B 0 ¯i,p ¯i,p ⎢ C¯i A¯i B C¯i B ⎢ =⎢ .. .. ⎣ . . N −1 ¯ N −2 ¯ Bi,p C¯i A¯i Bi,p C¯i A¯i

... ... .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥, ⎦

¯i,p ... C¯i B

⎡ ¯ ¯ ⎤ Ci Ai ⎢ C¯i A¯i 2 ⎥ ⎢ ⎥ Πi = ⎢ . ⎥ , ⎣ .. ⎦ N C¯i A¯i

(10)

with p ∈ i ∪ Ni . The D-ADMM algorithm [10] aims to solve separable optimisation problems in networks of interconnected nodes that can be formulated as follows M i (11) minimise i=1 Ji (z ) z

subject to zji = zlj l ∈ Si ∩ Sj , (i, j) ∈ E xi (k + 1) = Ai xi (k) + j∈(Ni ,i) Bij zli (k), in which, for j > i, z i denotes all control input copies that have an effect on agent i, including its own ui , such that each agent has its own copies of the manipulated variables. Finally, zli represents the copy of control signal l in agent i and Ji (z i ). The augmented Lagrangian of (11) is L(z, λ) =

 i

⎡ ⎣Ji (z i ) +





j∈Ni l∈Si ∩Sj

j i λij l (zl − zl ) + ρ



⎤ 2    i j ⎦ − z zl l

j∈Ni l∈Si ∩Sj

2

(12)

where z is the set of all copies in the set of node, λij l is the dual variable of element l associated to nodes i and j for all j > i, λ is the collection of all λij l and ρ > 0 is the predefined augmented Lagrangian parameter. Problem (11) can be solved using the method of multipliers by minimising the augmented Lagrangian (12) with respect to every copy of the control signal, keeping λ fixed at time k, iterating on k zqp (k + 1) = arg min L(z, λ), zqp

(13)

where q is any control input and p is any subsystem. The dual variable λ is also updated according to ij j i λij l (k + 1) = λl (k) + ρad (zl − zl ).

(14)

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During each sampling time, every agent must communicate its solution of (13) to all its neighbours. The last step after the communication between agents consists of each agent applying its own control input, i.e, making zii = ui . This process is repeated until some stopping criteria are met. Since this solution requires too much computational time, an analytical alternative is preferable. For each control agent, Ji is defined as T

Ji = (Yi − Ri )T (Yi − Ri ) + ρi ΔU¯i ΔU¯i

(15)

or, considering the predictive models, 2

2

Ji = Πi x ¯i (k) + Wi,i ΔUi + Wi,j ΔUj − Ri  + ρi ΔUi  ,

i = j

(16)

where Ri is the reference vector and x ¯i (k) is the augmented state as presented in Eq. (2). It is necessary to change the control variables so that each agent has a replica of all manipulated variables, resulting in T

ΔU¯i = [ΔU1i

T

ΔU2i ,

...

T

i T , ΔUM ] .

The predictor models may thus be written in a compact way as ⎤ ⎡ xˆi (k) Yi = Πi ⎣ ui (k) ⎦ + Wi ΔU¯i , uj (k)

(17)

(18)

with Wi being a matrix where each element is defined as in (10) and zero otherwise. For the purpose of applying D-ADMM, we consider local cost functions, associated to each subsystem, given by Ji,ad = Ji + (γi −

 j∈Ni

 2 1 ΔU¯j )T ΔU¯i + ρad ΔU¯i  , 2

(19)

where γi is a Lagrange multiplier parameter. The gradient of the cost functions Ji,ad is ∂Ji,ad = 2ΔU¯i Mi + Φi . ∂ΔU¯i

(20)

where Φi is a vector and M is a matrix whose entries depend on the predictive ∗ model parameters and the optimum value for each ΔU¯i is given by 1 ∗ ΔU¯i = − Mi Φi . 2

(21)

Since the D-ADMM algorithm is highly dependent on the network used, it is more relevant to show the algorithm directly applied to our use case with two agents, resulting in Algorithm 1 hereafter.

Distributed Adaptive Control

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Algorithm 1. D-ADMM based D-MPC 1: Initialisation: γ = [γ1 γ2 ]T ; ΔU¯i = 0 2: repeat I 3: M1 = W1T W1 + ρ¯1 + ρad 2 ¯1 − R1 ) − γ − ρad ΔU¯2 4: Φ1 = 2W1 (Π1 x 5: ΔU¯1 = − 12 M1−1 Ψ1 6: M2 = W2T W2 + ρ¯2 + ρad I 2 ¯2 − R2 ) − γ − ρad ΔU¯1 7: Φ2 = 2W2 (Π2 x 8: ΔU¯2 = − 12 M2−1 Ψ2 9: γ = γ + ρad (ΔU¯1 − ΔU¯2 ) 10: until pre-defined maximum number of iterations nI or stopping criteria are met.

The outputs ΔU¯1 (1) and ΔU¯2 (N + 1) become the manipulated variables u1 and u2 respectively.

4

Case Study - Biomass Thermal Power Plant

To illustrate the distributed adaptive controller proposed, a biomass thermal power plant is studied. The goal is to control both the steam flow and pressure through the valve position(u1 ) and the power grid velocity (u2 ). In Fig. 2, CF and CP represent the controller of the steam flow and steam pressure respectively. FS indicates the steam flow output and PS the steam pressure output.

Fig. 2. Distributed control system of the biomass power plant.

The simulations were conducted using a biomass thermal power plant model identified from plant data. The parameters that configure the controllers are saturation limits of [−20%, 20%] for both control inputs, a dwell-time of τD = 150 and τS = 50 samples. For the D-ADMM an horizon of N = 20, ρ1 = 400 and ρ2 = 50 were used. Figure 3 shows the results with the adaptive distributed MPC based on DADMM, proposed in this article, as well as two other distributed controllers in which coordination is based in a game, one in which the underlying control law is MPC (identified as DMPC), and another in which the underlying control is LQG (identified as DLQG). In Fig. 3(b) it is possible to verify that all the distributed

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Grid Velocity (%) Valve Position (%)

algorithms have a similar response, even when there was a change in the plant dynamics, and they were all capable of tracking the reference. In general, the D-ADMM based D-MPC algorithm was the fastest one to reach the reference. Besides this fact, none of the manipulated variables in fig. 3(a) saturated, making it possible to track the reference without static error.

4

DLQG DMPC DADMM

2 0 20

0

500

1500

2000

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2000

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2000

1000

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Time (min)

DLQG DMPC DADMM

10

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0 0

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Time (min)

Steam Pressure (bar) Steam Flow (t/h)

(a) Manipulated variables.

ref DLQG DMPC DADMM

2 1 0 -1

0

500

0

500

2 1 0

Time (min) (b) Outputs.

Fig. 3. Response of the thermal power plant with distributed adaptive control.

The supervisor decisions for both local controllers, for steam flow and pressure, are shown in Fig. 4. In both cases, all the algorithms were able to make the right choice of controllers, although with a small delay.

10

25

True DLQG DMPC DADMM

261

True DLQG DMPC DADMM

Decision 2

Decision 1

Distributed Adaptive Control

20 15 10

5

5 0

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1500

Time (min) (a) Decision σ1 .

2000

0

500

1000

1500

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Time (min) (b) Decision σ2 .

Fig. 4. Supervisor decisions.

5

Conclusion

A novel distributed adaptive controller is presented that combines adaptation based on switched multiple models (SMMAC), local predictive controllers and coordination based on a distributed version of the alternating direction method of multipliers (D-ADMM). The version of D-MPC used refers to linear plants with no constraints, but the same ideas can be used for nonlinear D-MPC and in the presence of constraints. Although no claim is made on general stability results based on a theoretical analysis, the simulation experience shows evidence that the algorithm is able to stabilize networked plants, provided that there is a non-adaptive distributed controller that stabilizes the network, and for which the conditions known for the application of SMMAC hold. In this respect, in addition to the dwell time condition to prevent fast switching, the imposition of a minimum of samples with the same controller index to allow controller switching shows a significant improvement. To illustrate the algorithm, simulations on the coordinated control of steam flow and pressure in a biomass power plant are presented.

References 1. Zhang, Y., Li, S.: Networked model predictive control based on neighbourhood optimization for serially connected large-scale processes. J. Process Control 17(1), 37–50 (2007) 2. Zhang, Y., Li, S., Li, N.: Distributed model predictive control over network information exchange for large-scale systems. Control Eng. Practice 19(7), 757–769 (2011) 3. Maestre, J., Pe˜ na, D., Camacho, E., Alamo, T.: Distributed model predictive control based on agent negotiation. J. Process Control 21, 685–697 (2011) 4. Lemos, J.M., Pinto, L.: Distributed linear-quadratic control of serially chained systems. IEEE Control Syst. 19(7), 757–769 (2012) 5. Sanchez, G., Murillo, M., Giovanini, L., Limache, A.: Distributed model predictive control based on Dynamic Games. In: 2008 Chinese Control and Decision Conference (2008)

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6. Maestre, J.M., De La Pe˜ na, D.M., Camacho, E.F.: A distributed MPC scheme with low communication requirements. In: Proceedings of the American Control Conference, vol. 1, pp. 2797–2802 (2009) 7. Maestre, J.M., Mu˜ noz De La Pe˜ na, D., Camacho, E.F.: Distributed model predictive control based on a cooperative game. Adv. Astronaut. Sci. 143, 1933–1950 (2012) 8. Richards, A., How, J.P.: Robust distributed model predictive control. Int. J. Control 80(9), 1517–1531 (2007) 9. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Machine Learn. 3(1), 1–122 (2010) 10. Mota, J.F., Xavier, J.M., Aguiar, P.M., P¨ uschel, M.: D-ADMM: A communicationefficient distributed algorithm for separable optimization. IEEE Trans. Signal Process. 61(10), 2718–2723 (2013) 11. Narendra, K.S., Driollet, O.A.: Adaptive control using multiple models, switching and tuning. In: Adaptive Systems for Signal Processing, Communications, and Control Symposium, AS-SPCC 2000, pp. 159–164. IEEE (2000) 12. Lourenco, J., Lemos, J.: Learning in multiple model adaptive control switch. IEEE Instrum. Measur. Mag. 9, 24–29 (2006) 13. Fekri, S., Athans, M., Pascoal, A.: Issues, progress and new results in robust adaptive control. Int. J. Adapt. Control Signal Process. 20(10), 519–579 (2006) 14. Lemos, J.M., Neves-Silva, R., Igreja, J.M.: Adaptive Control of Solar Energy Collector Systems. Springer (2012) 15. Morse, A.S.: supervisory control of families of linear set-point controllers: Part I: Exact matching. IEEE Trans. Automat. Control 41(10), 1413 (1996) 16. Hespanha, J.P., Liberzon, D., Morse, A.S.: Overcoming the limitations of adaptive control by means of logic-based switching. Syst. Control Lett. 49(1), 49–65 (2003) 17. Borrelli, D., Morse, A.: Discrete-time supervisory control of families of linear setpoint controllers. IFAC Proc. Volumes 29(1), 5150–5155 (1996)

Optimization of the Clinker Production Phase in a Cement Plant Silvia Maria Zanoli1(B) , Lorenzo Orlietti2 , Francesco Cocchioni2 , Giacomo Astolfi2 , and Crescenzo Pepe2 1 Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, AN, Italy

[email protected] 2 Alperia Bartucci S.p.A., Corso Vittorio Emanuele II, 37038 Soave, VR, Italy

Abstract. In this paper, the control and the optimization of the clinker production phase of an Italian dry cement industry is described. A tailored Advanced Process Control architecture has been proposed, based on a two-layer Model Predictive Control strategy. The critical process variables have been included in the controller setup through linear models with delays. The controller moves are computed through two subsequent constrained optimization problems with cooperation and consistency assumptions. The developed controller replaced a previous control system based on operators’ conduction. Thanks to the predictive approach, more profitable operating points have been guaranteed, significantly improving the process control performances. Energy consumption and environmental emissions have been reduced, taking into account quality and production requirements. Keywords: Advanced Process Control · Model Predictive Control · Clinker production · Cement rotary kiln · Energy efficiency

1 Introduction Italian cement industries are among the major cement producers in the European panorama. In the last decades, because of the increasingly stringent specifications on environmental impact and energy saving, technology in the cement plant has been significantly improved [1]. In this context, Advanced Process Control (APC) systems have become the dominant stratagem for the improvement of the energy efficiency of the cement plants with no or with slight hardware modifications [2]. An important incentive in the Italian industry has been given by the introduction of the energy efficiency certificates [1]. The most significant process of the cement production chain, in terms of energy efficiency and quality of the product, is represented by the production of the main component of the cement, called clinker. Clinker production process involves different phases; the last phases of this critical process take place in a rotary kiln [3]. The multivariable and nonlinear nature of the clinker production process has triggered a control challenge that, during the last years, has attracted the attention of researchers and practioneers. In the control literature, different solutions for the clinker production © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 263–273, 2021. https://doi.org/10.1007/978-3-030-58653-9_25

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phase control and optimization are present; different ways to model and manage the process nonlinearities have been proposed. For example, in [2], a Model Predictive Control (MPC) approach based on neural network models of the individual sub-processes is proposed; the efficiency of the adopted control solution has been proved through the effective reduction of the emissions and the improvement of clinker consistency. In [4], an adaptive MPC approach is proposed for controlling a white cement rotary kiln, exploiting a constrained generalized predictive controller. Simulation results prove the reliability of the proposed approach. In [5], an intelligent fuzzy predictive controller is proposed. A generalized predictive control is adopted to realize the nonlinear multivariable system adaptive predictive control. The application on cement rotary kiln control is discussed in detail as a simulation example. In [6], a first principles model of a cement kiln is used to control and optimize the burning of clinker. An MPC strategy is used to stabilize a temperature profile along the rotary kiln, guaranteeing good combustion conditions and maximizing production. In this paper, an APC architecture based on a two-layer linear MPC strategy is designed for the clinker production phase control and optimization. The approach described in [7] is further detailed and extended. In particular, through a crossfertilization procedure, the approach proposed in [8] is suitably customized for the process at issue. The validity of the proposed control strategy has been confirmed through implementation on the clinker production phase of an Italian cement industry. The paper is organized as follows: the process description is reported in Sect. 2, together with the control requirements and modellization. Section 3 resumes some details on the APC architecture, while Sect. 4 reports some field results. Section 5 summarizes the conclusions.

2 Clinker Production Phase Description In the analyzed cement industry, hydraulic binders are produced through a dry process: the raw materials are converted into a finely ground powder through the action of mills; then, the obtained raw meal is treated with a baking procedure in a rotary kiln (internal temperature reaches about 1600 [°C]), giving rise to clinker. Subsequently, clinker is grinded and combined with other components, such as calcium sulfate or pozzuolan, obtaining the desired type of cement. The clinker production phase at issue processes about 899000 tons per year of raw meal, using 59100 tons per year of fuel (coal); about 584000 tons per year of clinker are produced. Figure 1 depicts the process at issue. Initially, raw meal is entered into a suspension pre-heater (Fig. 1, left side), composed by four cyclones stages. The cyclones stages are disposed one above the other, giving rise to a tower. Here raw meal is subject to a preheating/drying phase (typical temperatures range: 650 [°C]–900 [°C]), while it is up in the air with exhaust gas from downstream zones, that is pulled by an induced draft (ID) fan (Fig. 1, left side). The suspension preheater increases the heat transfer rate, allowing a full and efficient heat exchange. In each cyclone stage, there is a separation between raw meal and exhaust gas, and then their reunification before next cyclone stage. This cyclical process of blending, separation, and remixing is repeated until the material is discharged from the last cyclone stage to the rotary kiln. A rotary kiln is a steel cylinder, horizontally slightly sloped, that rotates

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Fig. 1. The considered clinker production phase.

around its axis: its structure allows processed mixture to move along it. To withstand the required high temperature, the kiln is equipped with refractory materials. Different phases take place, e.g. calcination and clinkering. An air/fuel (coal) burner placed at its end (Fig. 1, right side) primarily supplies the furnace combustion. When the clinker leaves the kiln, it is characterized by a temperature of about 1200 [°C]; for this reason, the clinker is subjected to a cooling phase that is performed by a cooler, in order to obtain suitable clinker temperatures (e.g. 100 [°C]) for the subsequent grinding phase. Furthermore, the cooler recovers the clinker excess heat and supplies it to the combustion air. Furthermore, there is a precalciner between the suspension pre-heater and the kiln. The precalciner burner triggers additional combustion reactions, which exploit additional hot air provided through heat recovery. An APC system in a clinker production unit of a cement plant must lead to productivity and efficiency increase, while assuring the desired quality of byproducts; in addition, pollution impact should be kept within given limits and fuel specific consumption should be minimized. In a cement rotary kiln, the main thermodynamical constraints regard cyclones and kiln zones temperatures, together with oxygen concentration, while environmental ones refer to carbon monoxide concentration and nitrogen oxides levels. Mechanical constraints involve kiln motor power. Furthermore, “quality constraints” are related to free lime analysis, performed on clinker samples, collected at the end of the cement rotary kiln. All the mentioned crucial process variables to keep under control are measured through sensors and/or laboratory analysis. In order to satisfy the control specifications just reported, an accurate study of chemical and physical phenomena involved in the considered cement rotary kiln has been conducted. From this study, the fundamental Controlled Variables (CVs, y) were identified: fan, cyclones and kiln oxygen, nitrogen oxides and carbon monoxide, cyclones temperatures and pressures, kiln motor power, and clinkering temperature (y[my ×1] ). An aspect that has to be highlighted is the redundancy of some classes of CVs, i.e. oxygen concentrations. The main oxygen concentration to be kept under control is the cyclones

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oxygen; however, the analyzer related to this variable can clog and its measurement can be subjected to drifts. In these situations, the cyclones oxygen control can be indirectly performed by controlling the other oxygen concentrations (e.g. fan oxygen). As control inputs (Manipulated Variables, MVs, u[lu ×1] ), ID fan speed, meal flow rate and kiln and precalciner fuel flow rates (coal) have been selected. Furthermore, as measured input Disturbance Variables (DVs) have been considered (d[ld ×1] ): kiln speed, kiln and precalciner tertiary air flows, and radial air pressure (measured at the furnace outlet). Tables 1, 2 and 3 report the selected MVs, DVs and the main CVs together with their typical constraints boundary setting values and range values. A black-box approach has been adopted for the identification phase, obtaining linear time invariant asymptotically stable strictly proper minimum phase models with delays [11]. Table 1. Manipulated Variables (MVs). Variable name

Constraints example

Meal flow rate

130 [t/h]–140 [t/h]

Kiln coal

3900 [kg/h]–4500 [kg/h]

Precalciner coal 3800 [kg/h]–4100 [kg/h] ID fan speed

660 [rpm]–690 [rpm]

Table 2. Disturbance Variables (DVs). Variable name

Range example

Rotation kiln speed

1 [rpm]–2.5 [rpm]

Kiln tertiary air

4 [%]–40 [%]

Precalciner tertiary air 10 [%]–50 [%] Radial air pressure

100 [mbar]–250 [mbar]

In the considered cement plant, laboratory analysis on clinker samples oriented to free lime monitoring, collected at the end of the cement rotary kiln, are carried out four times a day. Within the APC system, the sporadic feedback from free lime analysis is taken into account through a correction of fuels and meal flow rate constraints, based on the desired free lime range that for the process at issue is 0.8 [%]–1.2 [%]. The customized APC system has to guarantee an optimal trade-off between fuels minimization and meal flow rate maximization, while meeting constraints related to MVs and CVs. The defined control specifications required the constrained control of some CVs to be performed using only a defined set of MVs. At this regard, the following specifications have been defined: • meal flow rate has not to be used for fan and cyclones oxygen control and for carbon monoxide control;

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Table 3. Main Controlled Variables (CVs). Variable name

Constraints example

Cyclones oxygen

1.5 [%]–3 [%]

Fan oxygen

5.3 [%]–7 [%]

Kiln nitrogen oxides

1100 [ppm]–1230 [ppm]

Fan nitrogen oxides

700 [ppm]–800 [ppm]

Cyclones carbon monoxide

0 [%]–0.06 [%]

First stage right temperature

830 [°C]–860 [°C]

First stage left temperature

830 [°C]–860 [°C]

Second stage exit temperature 705 [°C]–730 [°C] Kiln motor power

105 [kW]–125 [kW]

Clinkering temperature

1100 [°C]–1300 [°C]

Table 4. Decoupling matrix related to the main CVs. Variable name

Meal F. R. Kiln coal Prec. coal ID fan sp.

Cycl. ox.

0

1

1

1

Fan ox.

0

1

1

1

Kiln nitr. ox.

1

1

1

0

Fan nitr. ox.

1

1

1

1

Cycl. C. monox.

0

1

1

1

F. S. R. exit temp.

0

1

1

0

F. S. L. exit temp.

0

1

1

0

S. S. exit temp.

1

0

1

0

Kiln mot. pow.

1

1

0

0

Clink. temp.

1

1

1

0

• ID fan speed has not to be used for kiln nitrogen oxides control; • meal flow rate and ID fan speed have not to be used for first stage left/right temperatures control; • kiln fuel and ID fan speed have not to be used for second stage exit temperature control; • precalciner fuel and ID fan speed have not to be used for kiln motor power control; • ID fan speed has not to be used for clinkering temperature control. For the inclusion of these specifications in the controller design (see next section), the approach reported in [8] has been exploited. Taking into account the identified steady-state gain matrix of the considered process, the Decoupling Matrix DE ([10 × 4])

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reported in Table 4 is defined [8]. The generic (i, j) element of DE refers to the relationship between the ith CV of Table 3 and the jth MV of Table 1. In particular, the zero elements represent the relationships to inhibit according to the previous specifications. For example, for the 1st CV of Table 3 (cyclones oxygen), meal flow rate must not be used to perform any control action: this specification is represented by the zero element on the position (1, 1) (see next section for more details).

3 APC Architecture Figure 2 depicts the architecture of the developed APC system. At each control instant k, updated plant measurements (u(k − 1), d (k), y(k)) are provided by a Supervisory Control and Data Acquisition (SCADA) system. The inclusion of each process variable within the controller formulation is defined by its status value: the initial status values are provided to Data Conditioning & Decoupling Selector (DC&DS) block (Fig. 2, right side, u-d-y Status). The initial status values are set through a cooperative action of plant operators and local control loops logics (Fig. 2, right side, Plant Signals & Parameters). For each CV or MV, two main status values have been defined, active and inactive. The controller can use only active MVs in order to satisfy the specifications related to the only active CVs [8]. DC&DS block performs different operations, e.g. abnormal situations checks, and computes the final status value for each process variable (Fig. 2, left side, u-d-y Status). In addition, DC&DS block updates the initial Decoupling Matrix DE with the final status values information [8]. The initial DE related to the main CVs has been reported in Table 4. The control strategy is based on a two-layer linear MPC system that computes the updated MVs values (Fig. 2, u(k)) exploiting a receding horizon idea [12]. 3.1 MPC Formulation The MPC scheme, more detailed with respect to the scheme reported by the authors in [7], is based on the solution of two subsequent constrained optimization problems. These problems are formulated exploiting process variables predictions on a prediction horizon Hp . In this work, Hp is assumed to be set allowing steady state reaching to the obtained models. Considering a controller sampling time equal to 1 [min] (in accordance to the obtained process model and to the computational load required by the overall control algorithm), Hp = 150 [min] has been set. The CVs predictions are composed by a known part, i.e. the CVs free response (when no future moves are performed on the input variables), that is provided by a Predictions Calculator module (see Fig. 2). The forced components of MVs and CVs dynamic predictions are formulated on Hu (control horizon) dynamic control moves (assumed in the first prediction steps) [12]. In this work, Hu = 20 has been set. The dynamic future control moves are indicated with ˆu(k + i − 1|k)(i = 1, . . . , 20) terms. Only the first move ˆu(k|k) is really applied to the plant at each control instant. The optimal sequence ˆu(k + i − 1|k) is computed by the lower layer of the scheme, represented by a Dynamic Optimizer (DO in Fig. 2); the optimization problem of this module has been formulated as a Quadratic Programming (QP) problem, represented

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Fig. 2. APC system architecture.

by the cost function (1), subject to the linear constraints (2). DO QP problem is solved using standard techniques, i.e. active-set or interior point algorithms. VDO (k) =

my  Hp   j=1 i=Hwj

+ +

 2  Q(j,j) (i) · yˆ j (k + i|k) − rj (k + i|k)

Hp   

2 uˆ (k + i − 1|k) − ur (k + i − 1|k)S(i)

i=1 Hu  

 ˆu(k + i − 1|k)2

i=1

R(i)

(1)

+ εDO (k)2ρDO

subject to i. lbdu_DO (i) ≤ ˆu(k + i − 1|k) ≤ ubdu_DO (i), i = 1, . . . , Hu ii. lbu_DO (i) ≤ uˆ (k + i − 1|k) ≤ ubu_DO (i), i = 1, . . . , Hu iii. lby_DOj (i) − γlby_DOj (i) · εDO (k) ≤ yˆ j (k + i|k) ≤ uby_DOj (i) + γuby_DOj (i) · εDO (k), j = 1, . . . , my , i = Hwj , . . . , Hp iv. εDO (k) ≥ 0

(2)

In these expressions, · is the Euclidean norm. r(k + i|k) and ur (k + i − 1|k) represent the CVs and MVs reference trajectories values. The tracking errors between the predicted CVs and MVs values (ˆy(k + i|k) and uˆ (k + i − 1|k)) and the related reference trajectories are weighted by positive semi-definite diagonal weight matrices S(i) and Q(i). The original weight matrices (Fig. 2, DO Tuning Parameters) are processed by DO module taking into account the information provided by the final Decoupling Matrix DE [8] and by the obtained models through tailored weights zeroing policies. At this regard, for the generic jth active CV, a parameter Hwj is defined. It indicates the first prediction instant on which the first MVs future move (contained in ˆu(k|k) vector) related to at least one active MV tied to the jth active CV and not inhibited for controlling the jth active CV will be active. For example, considering the first CV of

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Table 3, i.e. cyclones oxygen, when this CV and all MVs are active, it will be Hw1 = 1, in fact the cyclones oxygen has not time delays in its relationship with ID fan speed (see Table 1 and Table 4). The original weight matrices have been tuned in order to minimize the error with respect to the steady-state targets provided by the upper layer (Targets Optimizing and Constraints Softening, TOCS in Fig. 2) for active process variables. R(i) positive definite matrices weight the DO MVs moves. The DO moves and magnitudes (ˆu(k + i − 1|k), uˆ (k + i − 1|k)) of active MVs are constrained by lbdu_DO (i), ubdu_DO (i), lbu_DO (i) and ubu_DO (i) (hard constraints). For the generic jth active CV, its predictions on the window Hwj − Hp (these predictions take into account status values and DE information) are constrained by lby_DOj (i) and uby_DOj (i) terms. CVs constraints have been considered as soft constraints: violations are admitted in critical situations through a εDO (k) nonnegative slack variables vector, introduced in (1) by ρDO positive definite diagonal matrix and in (2) by γlby_DOj (i) and γuby_DOj (i) vectors of positive elements. In this work, for the main CVs (Table 3), an example among the adopted grouping policies for constraints relaxations is represented by the presence of six groups. Table 5 reports the group number associated to each CV. For example, cyclones oxygen and fan oxygen share the same slack variable. A joint setting of γlby_DOj (i), γuby_DOj (i) and ρDO terms allow to suitably rank the importance of constraints relaxations, giving a priority order on CVs constraints. For example, the oxygen concentrations are more important than temperatures. Table 5. Grouping policy for the CVs constraints relaxations. Variable name

DO constraints group

Cyclones oxygen

1

Fan oxygen

1

Kiln nitrogen oxides

3

Fan nitrogen oxides

3

Cyclones carbon monoxide

6

First stage right temperature

2

First stage left temperature

2

Second stage exit temperature 2 Kiln motor power

4

Clinkering temperature

5

TOCS module performs the first constrained optimization; solving a Linear Programming (LP) the end points of the CVs and MVs reference   it computes  problem, trajectories (r k + Hp |k and ur k + Hp − 1|k , u-y Target in Fig. 2). Furthermore, it guarantees a feasible steady-state configuration for DO CVs constraints (y Constraints in Fig. 2) [7, 8].

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4 Field Results The study and design phases of the project for the development of the APC system began in September 2014 and ended in November 2014. Since February 2015, the controller has been installed on the considered Italian cement industry for the optimization of the clinker production phase, replacing plant operators’ manual conduction. With respect to the previous control system, the designed controller assured improved control performances related to oxygen and nitrogen oxides levels, together with an improved management of the cyclones temperatures. In this context, energy saving has been obtained, through the attainment of a profitable trade-off between kiln fuels minimization and meal flow rate maximization. In terms of plant behavior, these improvements are explained by the reduction of the standard deviation of the main controlled variables, which allows a more safe approach to process operating limits. Figure 3 and 4 show a comparison between the previous control system and the developed APC system related to the cyclones oxygen ([%]) and kiln nitrogen oxides ([ppm]) control; the comparison has been performed through a performance test during the commissioning phase: an about three weeks total period has been considered. Thanks to an optimized management of the MVs that have to be used for their control (see Table 4), a reduction on their standard deviation is observed. In particular, about 39 [%] reduction has been registered on cyclones oxygen standard deviation, together with about 3 [%] increase on its mean value. Furthermore, about 32 [%] reduction has been registered on kiln nitrogen oxides standard deviation, together with about 15 [%] decrease on its mean value. This result has been very meaningful, because it implies a significant emissions reduction, thus allowing environmental impact decreasing.

Fig. 3. Field results: cyclones oxygen trends before and after APC activation.

The activation of the APC system reduced the kiln average specific consumption. Figure 5 shows the results obtained after about two years since the first start-up of the proposed APC system. Blue line indicates the monthly energy saving ([%]) with respect to the computed baseline, while red line reports the cumulative energy saving (about 2.9 [%] saving). In Fig. 5, January data have not been reported, due to maintenance operations on the plant. The average specific consumption reduction and, more generally, the energy efficiency achievement (that involves also emissions reduction) allowed obtaining Italian energy efficiency certificates (Italian acronym TEE).

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Fig. 4. Field results: kiln nitrogen oxides trends before and after APC activation.

Fig. 5. Field results: monthly energy saving and cumulative energy saving after APC system activation.

5 Conclusions In this work, an Advanced Process Control architecture based on a Model Predictive Control strategy, has been designed for the clinker production phase control and optimization. The research activity has been motivated by the fact that, in the considered Italian cement industry, high profitability margins have been observed through the preliminary process study. Before the installation the APC system, the considered process was regulated through operators’ manual conduction. Thanks to its constrained multivariable mathematical formulation based on two subsequent optimization problems, a better trade-off between energy and production specifications is guaranteed. Improvements on the control of the critical process variables, e.g. oxygen concentrations and nitrogen oxides levels, have been observed after the installation on the real plant. The consequent reduction of their standard deviation and variance allowed a safe approach to process operating limits, leading to fuel specific consumption reduction and energy efficiency certificates achievement. Thanks to the tailored hardware and software configuration, characterized by a continuous communication between the APC system PC and the plant, the proposed control solution has been certified as Industry 4.0 compliant.

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References 1. ENEA: L’ottenimento dei Certificati Bianchi – Settore Cemento. Certificati Bianchi (2014) 2. Boe, E., McGarel, S.J., Spaits, T., Guiliani, T.: Predictive control and optimization applications in a modern cement plant. In: Cement Industry Technical Conference, pp. 1–10 (2005) 3. Alsop, P.A.: Cement Plant Operations Handbook, 3rd edn. International Cement Review (2001) 4. Ziatabari, J., Fatehi, A., Beheshti, M.T.H.: Cement rotary kiln control: a supervised adaptive model predictive approach. In: Annual IEEE India Conference, pp. 371–376 (2008) 5. Li, Z.: Intelligent fuzzy predictive controller design for multivariable process system. J. Comput. Inf. Syst. 6(9), 3003–3011 (2010) 6. Stadler, K.S., Poland, J., Gallestey, E.: Model predictive control of a rotary cement kiln. Control Eng. Pract. 19, 1–9 (2011) 7. Zanoli, S.M., Pepe, C., Rocchi, M.: Improving performances of a cement rotary kiln: a model predictive control solution. J. Autom. Control Eng. 4(4), 262–267 (2016) 8. Zanoli, S.M., Pepe, C.: Two-layer linear MPC approach aimed at walking beam billets reheating furnace optimization. J. Control Sci. Eng. 2017, 1–15 (2017). Article ID 5401616 9. Arad, S., Arad, V., Bobora, B.: Advanced control schemes for cement fabrication processes. Robot. Autom. Constr. 23, 381–404 (2008) 10. Martin, G., Lange, T., Frewin, N.: Next generation controllers for kiln/cooler and mill applications based on model predictive control and neural networks. In: IEEE/PCA 42nd Cement Industry Technical Conference (2000) 11. Zhu, Y.: Multivariable System Identification for Process Control. Elsevier Science, Oxford (2001) 12. Maciejowski, J.: Predictive Control with Constraints. Prentice-Hall, Harlow (2002)

An Extrinsic Approach to Sub-Riemannian Geodesics on the Orthogonal Group Knut H¨ uper1(B) , Irina Markina2 , and F´ atima Silva Leite3,4 1

3

Institute for Mathematics, Julius-Maximilians-Universit¨ at, W¨ urzburg, Germany [email protected] 2 Mathematical Institute, University of Bergen, Bergen, Norway [email protected] Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal [email protected] 4 Institute of Systems and Robotics, DEEC-UC, 3030-290 Coimbra, Portugal

Abstract. In this paper we use a variational approach, combining holonomic and nonholonomic constraints, to find an equation for sub-Riemannian geodesics on the orthogonal group. This approach is extrinsic in nature and makes the paper fully self-contained and possibly more accessible for a wide audience. The problem is formulated in the vector space of real square matrices, subject to two side conditions, and solved using a Langrange multiplier approach. The nonholonomic constraint corresponds to the requirement that the curves are tangent to a leftinvariant distribution. This distribution is defined by the vector space that shows up in a Cartan decomposition of the Lie algebra associated to the orthogonal group. Keywords: Calculus of variations · Cartan decomposition · Lagrangian multipliers · Nonholonomic constraints · Orthogonal Group · Sub-Riemannian geodesics

1

Introduction

The control of nonholonomic mechanical systems, i.e., systems whose motions are subject to nonintegrable differential constraints, has attracted growing attention over the years. We refer to [2,3,6], and [7] for details concerning the geometry of these problems and to [4] and [11] for an engineering point of view. One can find many examples of nonholonomic systems in physics, mechanical engineering and robotics. These nonholonomic constraints arise naturally in the presence of underactuated systems, which are characterised by having less inputs than degrees of freedom and include robot manipulators with flexible links and joints, mobile cars, robotic cars with trailers, aircrafts and underwater vehicles, among c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 274–283, 2021. https://doi.org/10.1007/978-3-030-58653-9_26

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others. The configuration space of most mechanical systems has components that are Lie groups, and in this context the orthogonal group plays an important role. One important objective in the study of underactuated systems is the possibility of controlling their motion, so that, in spite of the reduced dimension of the input space, it is still possible to reach any configuration by properly steering the input. This corresponds to the ability of connecting an initial state to a final state by a trajectory of the system. For the problem studied here, all possible trajectories of the system correspond to curves that are tangent to the distribution defined by the nonholonomic constraints. In spite of that, the bracket generating property of the system ensures that one can move continuously between any two given admissible configurations. Among the connecting curves we distinguish an optimal one which minimises a certain cost functional induced by the natural sub-Riemannian metric on the nonintegrable distribution. More precisely, in this paper we consider the group of orthogonal matrices, endowed with the trace metric, and look for curves that minimise the length functional and have their velocity vector restricted to be tangent to a distinguished subspace of the tangent space at each point. This problem is known as a sub-Riemannian problem on the Lie group On of orthogonal matrices and can be considered as a particular interesting case usually tackled by geometric optimal control theory methods involving the Pontryagin Maximum Principle and dealing with the nonholonomic constraints. However, in this paper we propose a simpler alternative approach which is extrinsic and variational in nature and does not require additional knowledge besides some variational and matrix calculus. For that, we embed the orthogonal group On into the vector space IRn×n of square matrices and reformulate the minimising problem on this bigger space with additional constraints to ensure that the solution will stay in the group of orthogonal matrices. This variational view point is combined with a Lagrange multiplier approach, where the multipliers are associated to the holonomic and nonholonomic constraints. The organisation of the paper is the following. In Sect. 2 we set the terminology and notations that will be used later. The statement of the problem and some convenient reformulations appear at the beginning of Sect. 3. In Subsect. 3.1 we derive necessary conditions for a curve in On to minimise the given energy functional and to satisfy the prescribed nonholonomic constraints, subject to some initial conditions. Finally, we comment about the importance of the sub-Riemannian geodesics in the orthogonal group to derive geodesics on homogeneous spaces that are quotients of On by certain subgroups of On , such as Stiefel and Grassmann manifolds.

2

Terminology and Notations

In what follows, IRn×n stands for the Lie algebra of n × n real matrices, the Lie bracket being the matrix commutator, i.e., [A, B] := AB − BA. The orthogonal group On in its standard representation is denoted by On := {X ∈ IRn×n | X  X = In },

(1)

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where In is the n × n-identity matrix. The Lie algebra of On is the set of real skewsymmetric n × n-matrices, denoted by son , and so defined by son := {A ∈ IRn×n | A = −A }.

(2)

The tangent space of On at X is then TX On ∼ = X son . = son X ∼

(3)

In the sequel we will mainly use the second isomorphism in (3). In particular, to each A ∈ son one associates a left-invariant vector field in On defined by XA, with X ∈ On . We also need the vector space of symmetric (n × n) matrices symn := {A ∈ IRn×n |A = A }.

(4)

The vector space IRn×n is also considered as a Riemannian manifold endowed with the Euclidean metric, i.e., with the usual Frobenius inner product ·, · on each tangent space TX IRn×n ∼ = IRn×n ·, · : IRn×n × IRn×n → IR,

A, B := tr(A B).

(5)

It is well known that IRn×n is a direct sum of son and symn and, moreover, son , symn  = 0. Consequently, we write: IRn×n = son ⊕⊥ symn .

(6)

Accordingly, any matrix A ∈ IRn×n can be uniquely decomposed as the sum of a skewsymmetric and a symmetric matrix, here denoted by Askew and Asym respectively, that is, (7) A = Askew + Asym , where Askew :=

A−A , 2

and

Asym :=

A+A . 2

(8)

Furthermore, we consider any Cartan decomposition of son : son = k ⊕ p,

(9)

where k is a Lie subalgebra of son and p is a vector space satisfying the additional Lie algebraic relations [k, k] ⊆ k,

[p, k] ⊆ p,

k ⊆ [p, p].

(10)

The restriction of ·, · to son is a scalar multiple of the Killing form and it also happens that p⊥ = k. So, instead of (9), we write son = k ⊕⊥ p.

(11)

Putting together (6) and (11), we have IRn×n = k ⊕⊥ p ⊕⊥ symn ,

(12)

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i.e., every matrix A ∈ IRn×n can be uniquely decomposed as A = Ak + Ap + Asym .

(13)

The inner product ·, · satisfies A, [B, C] = B, [C, A],

A, B, C ∈ son .

(14)

The space p induces a family Hp of left-invariant vector fields, and the associated distribution Hp (X) = {XA ∈ TX On |X ∈ On , A ∈ p} is called horizontal. The left-invariant vector fields XB, with B ∈ k, are called vertical. A smooth curve ˙ ∈ p is called horizontal. t → X(t) ∈ On that satisfies X  (t)X(t) Conditions (10) imply that p + [p, p] = son , which in turn implies that Hp is a two step bracket generating distribution and, consequently, any two points in the same connected component of On can be joined by a horizontal curve [1,5,13].

3

A Sub-Riemannian Geodesic Problem on On

We are interested in the solutions of the following optimal control problem:  1 min ui ∈L2 ([0,1],IRm ) 2

m 1

0

u2i (t) dt

(15)

i=1

subject to: ˙ X(t) = X(t)

m 

 ui (t)Ai

,

X ∈ On ,

m < n,

(16)

i=1

X(0) = X0 , X(1) = X1 , where {A1 , . . . , Am } is an orthonormal basis for the subspace p ⊂ son , and X0 and X1 are given matrices belonging to the same connected component of On . The above optimal control problem can be reformulated as the following sub-Riemannian geodesic problem on On .  1   ˙ min 1 tr X˙  (t)X(t) dt (17) X∈On 2

0

subject to ˙ X  (t)X(t) ∈ p, (nonholonomic constraint); X(0) = X0 , X(1) = X1 , (boundary conditions).

(18)

Solutions of this problem are called sub-Riemannian geodesics. These problems can be tackled using the geometric optimal control theory involving the Pontryagin Maximum Principle and consequently Hamiltonian methods. This approach

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has been addressed before, for instance in [8] and [10], but here we propose an alternative variational approach that is extrinsic in nature. The sub-Riemannian geodesic problem on On is formulated as a Riemannian problem on IRn×n over the space of smooth curves S := C ∞ ([0, 1], IRn×n ), subject to holonomic and nonholonomic constraints:  1   ˙ tr X˙  (t)X(t) dt (19) min 12 X∈S

0

subject to: X  (t)X(t) = In , (holonomic constraint); ˙ X  (t)X(t) ∈ p, (nonholonomic constraint); X(0) = X0 , X(1) = X1 , (boundary conditions). 3.1

(20)

A Lagrange Multiplier Approach to Solve the Sub-Riemannian Geodesic Problem

To solve this problem we use a Langrange multiplier approach, by extending the Lagrangian in (19) by adding extra terms corresponding to the holonomic and nonholonomic constraints. So, instead of minimising the functional (19) we minimise the functional defined by F : S → IR,  1   (21)

1 ˙ S  X  X −In +Q X  X˙ dt =: F (X, S, Q). (X, S, Q) → 2 tr X˙  X+ 0

Here S := C ([0, 1], V ), where V := IRn×n × symn × (k × symn ). The first summand in (21) is the energy functional, i.e., the squared norm of the velocity of the smooth curve X(t) ∈ IRn×n integrated over the interval [0, 1]. The second summand in (21) incorporates the holonomic constraint in (20). Here the matrix S ∈ symn serves as a matrix valued Langrange multiplier to ensure that the curve X will stay on the orthogonal group. Clearly, by taking derivatives this will imply ˙ = −X˙  (t)X(t) ∈ son for all t ∈ [0, 1]. The third summand in (21) X  (t)X(t) will ensure the nonholonomic constraint in (20). Here the matrix Q ∈ k ⊕⊥ symn ˙ when will serve as a second Lagrange multiplier, ensuring that the velocity X, left-translated back to the identity In ∈ On , will stay in the subspace p. According to the theory of calculus of variations, a neccessary condition for a critical point of the functional (21) is the vanishing of the first derivative of the following smooth function, evaluated at 0. ∞

Fε : [−δ, δ] × S → IR,  1   (22)



 ˙ 1 ε, (Xε , Sε , Qε ) → 2 tr X˙ ε X˙ ε + Sε Xε Xε − In + Q ε Xε Xε dt, 0

where δ > 0, and Xε , Sε , Qε are admissible variations of X, S, Q, respectively, defined by Xε := X + εY ∈ IRn×n ,

Y (0) = Y (1) = 0,

(23)

Sub-Riemannian Geodesics

Sε := S + εT ∈ symn ,

T (0) = T (1) = 0,

Qε := Q + εR ∈ k ⊕⊥ symn ,

R(0) = R(1) = 0.

By computing the first derivative of Fε at 0, we obtain:  1 

tr X˙  Y˙ + 12 T X  X − In + SX  Y Fε (0) = 0

 + 12 R X  X˙ + 12 Q Y  X˙ + X  Y˙ dt  1 

¨  Y + 1 T X  X − In + (XS) Y = tr − X 2 0



˙  R + 1 (XQ ˙  ) Y − + 12 (X  X) 2 

1

=

tr 0



+

¨ + XS + −X

1

tr



0

X  X−In 2

˙  XQ 2

−



1 d  2 dt (XQ)

˙ ˙  XQ+X Q Y 2



 T dt +

1

0

tr



 Y dt

279

(24) (25)

(26)

 dt

X  X˙  R 2

 dt.

For the second equality in (26) we have integrated by parts, respecting the boundary conditions which Y has to fullfil. Hence, F  (0) = 0 holds for all Y, T, R in their respective spaces, if and only if ¨ + XS + −X

 ˙ X(Q −Q) 2

−

˙ XQ 2 

= 0,

X X = I, X˙  X = −X  X˙ ∈ p.

(27)

The objective now is to simplify the first equality in (27) to get rid of the Lagrange multipliers, eventually to solve the corresponding Euler-Lagrange-type differential equation under the holonomic and nonholonomic constraints. We proceed as follows. First, we rewrite the first equality in (27) exploiting S = S  , respecting the skewsymmetry of X  X˙ ∈ son and the holonomic constraint X  X = In , to arrive at ¨− S = X X Thus,

 ˙ X  X(Q −Q) 2

+

˙ Q 2

¨ X − =X

(Q−Q )X˙  X 2

¨ −X ¨  X + [X  X, ˙ Q−Q ] + S − S = X X 2

+

˙ Q 2

˙ Q ˙ Q− 2

= S.

= 0.

(28)

(29)

We see that the latter equality depends only on the skewsymmetric part of the  Lagrange multiplier Q. Consequently, exploiting Q−Q = Qk and accordingly 2 ˙ Q ˙ Q− ˙ = Qk , we obtain 2

¨ −X ¨  X + [X  X, ˙ Qk ] = −Q˙ k . X X

(30)

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Recall the special structure of the terms in (30) and the consequences, i.e., ¨ −X ¨  X ∈ son , X X X  X˙ ∈ p, ˙ Qk ] ∈ p. [X  X,

(31)

The holonomic constraint, i.e., the second condition in (27), implies, by taking derivatives ¨ +X ¨  X = 0. X  X = In =⇒ X  X˙ + X˙  X = 0 =⇒ 2X˙  X˙ + X  X

(32)

The nonholonomic constraint, i.e., the third condition in (27), implies, by taking derivatives 

d ¨ ∈ p. (33) X  X˙ ∈ p =⇒ dt X X˙ ∈ p ⇐⇒ X˙  X˙ + X  X The last condition in (33), respecting X˙  X˙ ∈ symn together with (32) implies ¨ k = 0. (X  X)

(34)

Last but not least, from (30) and (31) we get ¨ p= (X  X)

˙ [Qk ,X  X] . 2

(35)

Finally, the last equality in (30) together with (34) imply Q˙ k = 0

=⇒ k Qk = const,

(36)

i.e., the skewsymmetric part Qk of the Lagrange multiplier Q is a constant. In summary, we have proved the following result Theorem 1. The necessary condition for the matrix X ∈ IRn×n to be a critical point of the functional (21) is given by the system of differential-algebraic equations ¨ p = [Qk ,X (X  X) 2  ¨ (X X)k = 0, X  X = In , X  X˙ ∈ p.



˙ X]

, (37)

Corollary 1. System (37) in Theorem 1 implies that a critical point of the functional (21) satisfies the one-parameter family of ordinary differential equations

¨ = X (X  X) ˙ 2 + [Qk , X  X] ˙ . X (38)

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Proof. According to the decomposition (13), ¨ = (X  X) ¨ sym + (X  X) ¨ k + (X  X) ¨ p. X X

(39)

˙ the symmetric From the last equality in (32) and the skewsymmetry of X  X,  ¨ part of X X can be written as ¨ sym = (X  X)

¨ X ¨ X X  X+ 2

˙ 2. = −X˙  X˙ = −X˙  XX  X˙ = (X  X)

(40)

So, using the first two equations in (37) together with (40), and also absorbing the constant 12 in the value of the constant matrix Qk , the decomposition (39) reduces to the family of ordinary differential equations (38).   Let us rewrite Eq. (38) in a way that involves the data from both summands of the Lie algebra son = p ⊕ k. We define ˙ U (t) := X  (t)X(t) ∈ p.

(41)

Therefore, by making use of (40) and (38) we obtain ¨ U˙ = X˙  X˙ + X  X ˙ 2 + X X ¨ = −(X  X)

(42)

= −U + U + [Qk , U ] = [Qk , U ]. 2

2

That is, the horizontal tangent vector U (t) ∈ p fulfils a linear time-invariant matrix valued ordinary differential equation of first order, with the solution U (t) = etQk U (0) e−tQk ,

U (0) = X0 X˙ 0 ,

˙ X˙ 0 := X(0).

(43)

In the sequel we will simply use Q instead of Qk , since, as explained earlier, the symmetric part Qsym of the Lagrange multiplier Q could be chosen equal to zero. As the main result of the paper we state the following. Theorem 2. For fixed Q ∈ k, the unique solution of the initial value problem

¨ X(t) = X(t) U 2 (t) + [Q, U (t)]

= X(t) etQ P 2 +[Q, P ] e−tQ , (44) P := U (0) = X  X˙ 0 ∈ p, 0

X0 ∈ On , is the sub-Riemannian geodesic tangent to the left-invariant distribution defined by p, and starting from the point X0 ∈ On with initial velocity X˙ 0 . The solution is given by (45) X(t) = X0 et(P +Q) e−tQ .

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Proof. The ordinary differential equation in Theorem 2 is linear time variant which can be solved by means of the following time variant coordinate transformation (46) Y (t) := X(t) etQ . By making use of the substitution (46) into (44) we reduce the initial system to the Cauchy problem of the linear second order equation with constant coefficients

Y¨ − 2Y˙ Q + Y Q2 − P 2 − [Q, P ] = 0, P ∈ p, (47) Q ∈ k, Y (0) ∈ On . Although the corresponding characteristic second order equation

Γ 2 − 2Γ Q + In Q2 − P 2 − [Q, P ] = 0

(48)

for the ordinary differential equation in (47) is matrix valued, and therefore not defined over a commutative ring, we solve it by the following Ansatz. Define the matrix Γ := αP + βQ, where α and β are real numbers. Then, by substituting this Ansatz into (48), we find α = β = 1. It leads to the unique solution of the initial value problem (47), given by Y (t) = Y (0) et(P +Q) .

(49)

The previous equation, together with (46), finishes the proof of Theorem 2.

 

Formula (45) is in accordance to the results in [8] and [10], but here this formula was derived using an approach that is possibly more suitable for a wide audience. ˙ Remark 1. Formula (45) defines a curve in On having velocity vector X(t) ∈ ˙ ∈ p ⊂ son . This formula can be used Hp (X(t)), or analogously X  (t)X(t) to find the Riemannian geodesics on homogeneous spaces On /K, where K is the isotropy subgroup at a point in On , having k as its Lie algebra. We get a Riemannian metric on On /K by pushing forward the bi-invariant metric on On , when restricted to the horizontal distribution Hp ⊂ T On . The Riemannian geodesic on On /K is the projection of (45) under the projection map π : On → On /K, when Q = 0 and P ∈ p. This idea was already exploited when the homogeneous space is the Stiefel manifold Stn,k consisting of all k-orthonormal frames in IRn , or the Grassmann manifold Grn,k of all k-dimensional subspaces in IRn . In the Stiefel case On /K ∼ = Stn,k , and in the Grassmann = On /On−k ∼ ∼ ∼ case On /K = On /(On−k × Ok ) = Grn,k . The reader can find more details in [10], Chapter 11, [12], Chapter 11, and in [9].

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Acknowledgements. This work was initiated when the first and the third authors were visiting ICMAT in Madrid in November 2019. The hospitality of our hosts, David Mart´ın de Diego and Leonardo Colombo, is greatly appreciated. This work was continued when the authors met in 2020, at the Institute for Mathematics in W¨ urzburg, Germany. The first author has been supported in part by the German Federal Ministry of Education and Research (BMBF-Projekt 05M20WWA: Verbundprojekt 05M2020 - DyCA). The second author was partially supported by the project Pure Mathematics in Norway, funded by the Trond Mohn Foundation. The third author acknowledges Funda¸ca ˜o para a Ciˆencia e a Tecnologia (FCT) and COMPETE 2020 program for the financial support to the project UIDB/00048/2020.

References 1. Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. Springer, Berlin (2004) 2. Bloch, A., Colombo, L., Gupta, R., de Diego, D.M.: A geometric approach to the optimal control of nonholonomic mechanical systems. In: Analysis and Geometry in Control Theory and its Applications. Springer INdAM Series, vol. 11., pp. 35–64. Springer, Cham (2015) 3. Bloch, A.M.: Nonholonomic Mechanics and Control, 2nd edn. Springer, New York (2015) 4. Choukchou-Braham, A., Cherki, B., Djema¨ı, M., Busawon, K.: Analysis and Control of Underactuated Mechanical Systems. Springer, Cham (2014) ¨ 5. Chow, W.L.: Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117, 98–105 (1939). in German ´ 6. Cushman, R., Duistermaat, H., Sniatycki, J.: Geometry of Nonholonomically Constrained Systems. World Scientific, Hackensack (2010) 7. Hussein, I.I., Bloch, A.M.: Optimal control of underactuated nonholonomic mechanical systems. IEEE Trans. Automat. Control 53(3), 668–682 (2008) 8. Jurdjevic, V.: Optimal Control and Geometry: Integrable Systems. Cambridge University Press, Cambridge (2016) 9. Jurdjevic, V., Markina, I., Silva Leite, F.: Extremal curves on Stiefel and Grassmann manifolds. J. Geom. Anal. 1–31 (2019). https://doi.org/10.1007/s12220-01900223-1 10. Montgomery, R.: A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications. American Mathematical Society, Providence (2002) 11. Moreno-Valenzuela, J., Aguilar-Avelar, C.: Motion Control of Underactuated Mechanical Systems. Springer, Cham (2018) 12. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983) 13. Rashevskii, P.K.: About connecting two points of a complete nonholonomic space by an admissible curve. Uch. Zapiski Ped. Inst. K. Liebknecht 2, 83–94 (1938). (in Russian)

Assessment of the Nutritional State for Olive Trees Using UAVs Pablo Cano Marchal1(B) , Diego Mart´ınez Gila1 , Sergio Illana Rico2 , Javier G´omez Ortega1 , and Javier G´amez Garc´ıa1 1

Robotics, Automation and Computer Vision Group, University of Ja´en, Ja´en, Spain [email protected] 2 Integraci´ on Sensorial y Rob´otica, Ja´en, Spain

Abstract. This work presents the prediction of nutritional state for olive trees based on multispectral information gathered by a camera mounted on an Unmanned Aerial Vehicle (UAV). The results show that this data gathering technique is very promising for this application. The models were constructed using support vector machines for regression with a gaussian kernel and five-fold validation, and provided Mean Absolute Errors (MAE) around 15% for all the nutrients considered, and particularly satisfactory results for N, K, Ca, Mg and B. Keywords: Precision agriculture · UAV · Multispectral images

1 Introduction Precision agriculture, defined as “a management strategy that uses information technology to bring data from multiple sources to bear on decisions associated with crop production” by the American National Research Council [11], is a key trend in modern agriculture, as it enables to obtain higher or equal yields than traditional practices while optimizing the use of the resources, thus improving the profitability and sustainability of the activity [14]. Two major sources of information for their use in precision agriculture are currently under active research: Internet of Things (IoT) devices that can be installed in the groves and continuously measure and upload data coming from different types of sensors remotely to the cloud, and unmanned aerial vehicles (UAV) that typically are used to obtain aerial images of the crops employing cameras that provide information in different wavelengths. Since IoT devices are typically installed and remain in the field, they have the advantage that they can provide a continuous stream of data, while UAV only provide information when the flights take place, thus supplying data only at specific time instants. On the other hand, IoT devices provide information specific of their particular location, which may require the installation of several devices in large groves, while UAV naturally provide information of the whole grove simply by planning the flight trajectory accordingly. The assessment of the nutritional state of crops is fundamental for making informed fertilization decisions, both in which elements to provide and their amount. Olive c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gonc¸alves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 284–292, 2021. https://doi.org/10.1007/978-3-030-58653-9_27

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orchards can present a significant spatial variability in the fertilization needs [7], so employing UAV to assess these nutritional state emerges as an interesting application, given their ability to provide individual information for each tree of the grove. A previous work by G´omez Casero et al. [5] reported the assessment of N and P deficiency in olive trees using an hyperspectral camera with pictures taken manually from above the tree. In turn, the potential of using UAV in predicting the nutritional state and health state of crops has been object of recent research for rice [14] and vineyards and tomatoes [2]. Several other works have reported different results relating nitrogen content in different crops employing various vegetation indexes computed from multispectral and hyperspectral sources [1, 8, 12, 13]. This paper presents preliminary results of the prediction of several nutrition elements using multispectral information collected with an UAV. The data was obtained under a publicly founded joint project with other entities in Andaluc´ıa, Spain. The rest of the paper is organized as follows: next Section provides some details on the location of the orchards and olive varieties used in the study, as well as the hardware employed. Section 3 introduces the data analysis procedure and obtained prediction models, while Sect. 4 shows the obtained results. Finally, Sect. 5 completes the paper with the conclusions.

2 Groves, Hardware and Reference Data The project was carried out in five olive orchards located in five different provinces of Andaluc´ıa (Almer´ıa, C´ordoba, Granada, Ja´en and M´alaga) with three different cultivar varieties: picual, hojiblanca and arbequina. Table 1 includes the reference location, variety and planting layout of each of the orchards. The UAV employed was a DJI Matrice 600 [3], where two cameras were mounted: a thermographic Flir Duo Pro R [4] and a multispectral Micasense RedEdge-M [10], which provides information in five wavelengths bands: blue (B), green (G), red (R), red-edge (RE) and near infrared (NIR). Figure 1 shows a picture of the UAV with the cameras mounted onboard. For the results presented in this paper, only data provided by the Micasense camera were used. In order to obtain the reference target data for the models, four sampling areas were chosen for each of the orchards. Each sampling area was composed of six trees, and olive leaves were taken from each of those and pooled together. The reference analysis of the olive leaves were carried out by the Olivarum Laboratory of the Caja Rural Foundation in Ja´en. The elements analyzed were N, P, K, Ca, Mg, Mn, Cu, Zn, Na, S and B. Table 1. Location of the olive orchards studied in the project. Almer´ıa

C´ordoba

Granada

J´aen

M´alaga

Latitude

36.903818

38.085914

37.314252

38.362235

37.179805

Longitude

−2.080262 −4.263042 −3.353941 −2.741775 −4.681253

Variety

Arbequina

Picual

Planting layout (m) 3.75 × 1.25 Irregular

Picual

Picual

Hojiblanca

–

8×8

7×5

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Fig. 1. Picture of the DJI Matrice 600 UAV used to capture the images.

The analysis of N was performed using an element analyzer, while the rest of elements were incinerated in a Mufla oven, with the P being analyzed using an spectrophotometer and the rest of elements with an ICP. The total scope of the project included seven flight campaigns, performed from July 2018 to November 2019. This paper reports results from the first 4 flight campaigns, for a total of 282 data points corresponding to 282 olive trees.

3 Image Pre-processing, Feature Selection and Prediction Models The images taken with the RedEdge-M were preprocessed using the open-source library developed by Micansense [9] in order to transform the raw pixel intensity value into reflectance values and to remove the optical and imager effects. The procedure requires an image of a known irradiance panel to be taken before each flight, thus allowing to correct for the influence of the solar radiation intensity and time of the year. Figure 2 shows the original raw images as captured by sensors of the camera mounted on the drone and the resulting reflectance images after the preprocessing process that was carried out. The regions of the images containing the olive trees of the sampling areas were manually selected, and the reflectance values of the pixels stored into arrays of variable length, according the different sizes of the trees in the images. As commented before, the Micasense camera provides images of five different spectral bands. Using this information, the five different vegetation indexes included in Table 2 where computed for each pixel. The use of vegetation indexes for the prediction of different features of the crops has been extensively used in the literature, see for instance [1, 8, 12, 13]. This way, the final feature vector associated with each tree was formed, by using the mean value of the pixel arrays for the original five channels provided by the Micasense

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Table 2. Vegetation indexes (VI) used as features together with the mean values of each of the five bands. More information of the VI can be found in [14]. Name

Acronym Formula

Normalized Difference Vegetation Index NDVI

(NIR − R)/(NIR + R)

Green Normalized Difference VI

gNDVI

(NIR − G)/(NIR + G)

Normalized difference red-edge index

NDRE

(NIR + RE)/(NIR + RE)

Green cholorphyll index

CIG

NIR/G − 1

Normalized NIR index

NNIR

NIR/(NIR + RE + G)

camera and the five vegetation indexes, thus having a dimension of 10. The ratio of feature dimension to number of points provided some confidence that overfitting should not be a problem, as was corroborated by the results obtained in the five-fold validation. The data treatment and model training was performed using the scikit-learn module in Python [15]. The first step in the data analysis procedure was to map both the features and the targets to the [0, 1] interval. Then, some exploratory endeveours using linear models were carried out using ridge regression [6], but the obtained results were not successful due to underfitting. Thus, a model with a greater capacity was called upon, so the use of support vector machine for regression with a gaussian kernel was evaluated. Support vector machines for regression (SVR) solve the optimization problem [6]:

Fig. 2. Images captured by the AUV with the MicaSense RedEdge multispectral camera. Upper row shows the original raw images the lower row shows the reflectance values. Each column represents a channel, left to right: blue (B), green (G), red (R), red-edge (RE) and near infrared (NIR).

minimize w,b,ξ ,ξˆ

n 1 T w w +C ∑ (ξi + ξˆi ) 2 i=1

subject to yi − wT φ (xi ) ≤ ε + ξi , wT φ (xi ) − yi ≤ ε + ξˆi ,

ξi ≥ 0, i = 1, . . . , n, ξˆi ≥ 0, i = 1, . . . , n.

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Table 3. Average mean absolute error and maximum absolute error and their standard deviations computed using 5-fold cross validation. Nutrient Mean absolute error % Max. absolute error % N

12.05 (2.74)

49.13 (13.20)

P

15.78 (2.05)

58.05 (6.82)

K

13.62 (1.50)

45.84 (6.29)

Ca

13.10 (1.11)

52.33 (6.54)

Mg

11.79 (1.31)

56.89 (12.88)

Mn

13.99 (1.44)

58.05 (10.07)

Cu

17.51 (1.32)

67.11 (5.42)

Zn

17.55 (1.94)

62.15 (8.88)

Na

11.03 (2.00)

83.80 (9.40)

B

14.17 (1.81)

55.51 (10.99)

Here, C is the parameter that defines the penalization of the deviations of each sample beyond the insensitive region defined by ε . This formulation of the penalty function offers the advantage that does not penalize errors smaller than ε and is also not too affected by outliers, as the deviation is penalized linearly and not quadratically. Five-fold validation was used for the construction and validation of the models, with all the parameters for the preprocessing steps being computed using exclusively the training data, and then applied to the validation fold. As expected, this type of model provided a substantial improvement compared the results obtained with the linear model, as will be shown by the results presented in the following Section.

4 Results and Discussion A summary of the obtained results is included in Table 3, where the mean absolute error of prediction (MAE) and the maximum absolute error for the models of each nutrient are detailed. The values are expressed as percentage of the data range, and are computed as the average of the values obtained with the validation subset in each of the five folds, with the standard deviation included in brackets. A first inspection of these results shows that the MAE lies around 15% for every nutrient, with a minimum of 11.04% for Na and a maximum of 17.5% for Zn. In turn, the spread of the maximum absolute error is wider, with values ranging from 49% for N to 83% for Na. A greater insight of the results can be obtained examining Figs. 3 and 4, where true vs. predicted values and residuals plots are depicted. These plots correspond to the first fold of data, but are representative of the behavior obtained in the others as well. As seen in the plots, there are some nutrients, namely N, K, Ca, Mg and B that present fairly satisfactory results. For these nutrients, the residual plots show no major concerns but for some points that seem to have a different behavior than the rest. These appear in the upper parts of the actual value axis (y − axis) for K, Ca, Mg and B and form almost a

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Fig. 3. Plots of the distribution of training and test points for one of the cross-validation sets for each of the studied elements.

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Fig. 4. Plots of the distribution of training and test points for one of the cross-validation sets for each of the studied elements.

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horizontal line. Since the target values are shared for subsets of 6 points, corresponding to the 6 trees that share the same reference analysis, these points probably correspond to some anomaly for a particular sampling instance. Further work will explore in detail this behavior, however, due to the choice of SVR and its robustness to outliers, the obtained results are not too penalized by these points. A second set of nutrients including P, Mn, Zn and Cu offer a slightly worse performance, but their predictions could still be useful for screening purposes. Finally, for Na, the predictive model is not too good, probably due to the limited number of different target values, as many samples actually share the same value. For all these elements, the inclusion of additional samples could probably be useful to improve the results.

5 Conclusions The results obtained for the nutrients N, K, Ca, Mg and B show a very interesting potential for the application of UAV equipped with multispectral cameras for the assessment of the nutritional state of olive tress. These results open the door to tailored fertilization plans that take into account the spatial variability in the distribution of nutrients in the groves. The results presented employed the data from all the different groves to build a model common to all the locations. A possible via to improve the accuracy of the predictions is to build specific models for each of the three different olive varieties, as the variety provokes differences in the leaves that could introduce some unwanted variability in the feature vectors. However, more data is likely to be needed in order to avoid the risk of overfitting the models, particularly for Hojiblanca and Arbequina, where only one grove for each variety was available. Other lines of work to improve the accuracy of the predictions include exploring the use of different models, such as neural networks or partial least squares, and ensemble approaches, thus employing several trained models to combine their predictions into a single final value. Finally, having a good segmentation of the olive trees is an important step to assure the quality of the feature vector, so this task was performed manually to minimize variability due to including varying amounts of surrounding land pixels. However, this step is quite time consuming, so another research line that would easen the adoption of the technique is the automatic segmentation of the trees, so that a robust algorithm can be used that prevents from having to perform this task manually. Acknowledgments. This work was developed under the activities of the project “Precision Agriculture in olive groves using unmanned aerial vehicles” (reference GOP3I-JA-16-0015) and partially supported by it. The project was carried out jointly with “Fundaci´on Caja Rural de Ja´en”, “Fundaci´on Andaluza de Desarrollo Aeroespacial (FADA)”, the “Centro Provincial de Agricultores de ASAJA Ja´en”, the “Instituto de Investigaci´on y Formaci´on Agraria y Pesquera (IFAPA)” and the regional delegations of ASAJA in Almer´ıa, C´ordoba, Granada and M´alaga.

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References 1. Ag¨uera, F., Carvajal, F., P´erez, M.: Measuring sunflower nitrogen status from an unmanned aerial vehicle-based system and an on the ground device. ISPRS Int. Arch. Photogram. Remote Sens. Spat. Inf. Sci. XXXVIII-1, 33–37 (2012). https://doi.org/10.5194/ isprsarchives-xxxviii-1-c22-33-2011 2. Candiago, S., Remondino, F., De Giglio, M., Dubbini, M., Gattelli, M.: Evaluating multispectral images and vegetation indices for precision farming applications from UAV images. Remote Sens. 7(4), 4026–4047 (2015). https://doi.org/10.3390/rs70404026 3. DJI: DJI Matrice 600 Pro - DJI. https://www.dji.com/es/matrice600-pro 4. FLIR: FLIR Duo Pro R HD Dual-Sensor Thermal Camera for Drones — FLIR Systems. https://www.flir.com/products/duo-pro-r/ 5. G´omez-Casero, M.T., L´opez-Granados, F., Pe˜na-Barrag´an, J.M., Jurado-Exp´osito, M., Garc´ıa-Torres, L., Fern´andez-Escobar, R.: Assessing nitrogen and potassium deficiencies in olive orchards through discriminant analysis of hyperspectral data. J. Am. Soci. Hortic. Sci. 132(5), 611–618 (2007). https://doi.org/10.21273/jashs.132.5.611 6. Hastie, T., Tibshirani, R., Friedman, J.H.J.H.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edn. Springer, New York (2009). https://doi.org/10. 1111/j.1751-5823.2009.00095 18.x ´ 7. L´opez-Granados, F., Jurado-Exp´osito, M., Alamo, S., Garc´ıa-Torres, L.: Leaf nutrient spatial variability and site-specific fertilization maps within olive (Olea europaea L.) orchards. Eur. J. Agron. 21(2), 209–222 (2004). https://doi.org/10.1016/j.eja.2003.08.005 8. Mahajan, G.R., Pandey, R.N., Sahoo, R.N., Gupta, V.K., Datta, S.C., Kumar, D.: Monitoring nitrogen, phosphorus and sulphur in hybrid rice (Oryza sativa L.) using hyperspectral remote sensing. Precis. Agric. 18(5), 736–761 (2017). https://doi.org/10.1007/s11119-016-9485-2 9. Micasense: GitHub - micasense/imageprocessing: MicaSense RedEdge and Altum image processing tutorials. https://github.com/micasense/imageprocessing 10. MicaSense: RedEdge-MX — MicaSense. https://www.micasense.com/rededge-mx 11. National-Research-Council: Precision Agriculture in the 21st Century (1997). https://doi. org/10.17226/5491 12. P¨ol¨onen, I., Saari, H., Kaivosoja, J., Honkavaara, E., Pesonen, L.: Hyperspectral imaging based biomass and nitrogen content estimations from light-weight UAV. In: Remote Sensing for Agriculture, Ecosystems, and Hydrology XV, vol. 8887, p. 88,870J, October 2013. https://doi.org/10.1117/12.2028624 13. Shi, T., Wang, J., Liu, H., Wu, G.: Estimating leaf nitrogen concentration in heterogeneous crop plants from hyperspectral reflectance. Int. J. Remote Sens. 36(18), 4652–4667 (2015). https://doi.org/10.1080/01431161.2015.1088676 14. Stavrakoudis, D., Katsantonis, D., Kadoglidou, K., Kalaitzidis, A., Gitas, I.Z.: Estimating rice agronomic traits using drone-collected multispectral imagery. Remote Sens. 11(5) (2019). https://doi.org/10.3390/rs11050545 15. Varoquaux, G., Buitinck, L., Louppe, G., Grisel, O., Pedregosa, F., Mueller, A.: Scikit-learn. GetMobile Mob. Comput. Commun. 19(1), 29–33 (2015). https://doi.org/10.1145/2786984. 2786995

Temperature Control on Double-Pipe Heat-Exchangers: An Application S. J. Costa1(B) , R. Ferreira2 , and J. M. Igreja3 1

ISEL-IPL, R. Conselheiro Em´ıdio Navarro, 1, Lisbon, Portugal [email protected] 2 ISEL-IPL and IST-ULisboa, Lisbon, Portugal 3 ISEL-IPL and INESC-ID, Lisbon, Portugal

Abstract. This paper begins with the development of first principles models for Heat-exchangers. The found results are then used to improve a laboratory temperature control solution for a dual mode Heat-exchanger apparatus. In particular, the co-current and counter-current transfer functions are obtained from the partial differential equations that explain the heat balances for both, cold and hot fluid traveling through the pipes. The Pole-Zero maps are also depicted from the transfer functions for further dynamical analysis. Finally the PID control for the laboratory apparatus, which can work in both co- and counter-current mode, is presented along with some preliminary results. The hot water recycle control scheme for the laboratory apparatus and the adoption of the PID solution is justified by the number and location of poles and zeros found near in the origin vicinity, after beeing calculated for the open-loop transfer function. Keywords: Partial differential equations · Irrational transfer functions · Heat-exchangers · PID control

1

Introduction

The study of irrational transfer functions is of paramount importance in infinitedimensional systems theory because they play the same role as the rational transfer functions for finite dimensional systems. In the majority of the cases they can be written as an infinite series by partial fraction expansion, producing by truncation, strictly proper finite linear time invariant systems. Then high accuracy approximate solutions, both in time and in frequency, can be used for dynamics study and control design purposes. Distributed parameter process systems (DPS), typically fluid flow systems with momentum, mass and heat transport phenomena [3], like tubular, reactors and bioreactors, heat exchangers, pulp digesters, piping and valves networks, and to some degree of extent water pipe distribution systems and water open canals, are modelled by coupled PDEs [7]. In this broad class of systems of interests, meaningfully irrational c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 293–302, 2021. https://doi.org/10.1007/978-3-030-58653-9_28

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transfer functions can be systematically obtained from first-principles physical models. Similar dynamical solutions for other infinite-dimensional systems obtained from coupled partial differential equations (PDE) using transfer functions can be found in [9] and [4]. In the first one, a book about modelling and control of irrigation canals, transfer functions, the poles and zeros location are obtained to analytically solve the linear Saint-Venant Equations for water canals. In the second one, a tutorial paper, transfer functions for a some number of representative infinite-dimensional examples are derived along with the poles, zeros and partial fraction expansions, transfer functions. In this paper the focus is on finding the transfer function of a Heat-exchanger model in co- and counter-current resorting to a general solution for PDE systems using Laplace transform. After that, a control solution for a laboratory apparatus is explained and justified by the transfer functions found. Roughly speaking, PID tuning for stable transfer functions is typically done by putting the controller zeros near the system dominant poles and then adjusting the controller gain until a satisfactory performance be achieved. The rest of the paper is organized as follows: Sect. 2 details the mathematical assumptions in order to explain how to systematically find transfer functions for the studied class of linear PDEs. Sections 3 and 4, along with Sect. 2 results, present the transfer functions derived for the double-pipe Heat-exchanger and for the co-current flow design, respectively. In Sect. 5 a simple PID control [1] is used in an application to a real laboratory apparatus and some results are presented. Section 6 draws conclusions.

2

Mathematical Preliminaries

Consider the class of distributed parameters dynamical systems, linear invariant in relation to time and space translation, that can be written as ∂z = Az(t) + Bud (t) ∂t ˘ ˘ b (t) 0 = Az(t) + Bu

(1)

(z(0), t ≥ 0) where A is a differential space operator, and B is a bounded linear space map. Boundary conditions are included using a point-wise non dynamic equation, with space operators A˘ and B˘ acting on z(0, t) and z(L, t), eventually, through the inputs ub (t). Initial conditions are given by z(x, 0). Both x and t, respectively space and time, have the domain (x, t) ∈ [0, L] × R+ . Notice that there are two kinds of inputs: spatially distributed Bud (t) and acting point-wise on the boundaries ub (t). The model given in (1) can be rewritten in the form of an augmented set of partial differential equations, given by J

∂η(x, t) ∂ n η(x, t) = M η + N (x)ud (t), +A ∂tn ∂x

(2)

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where J, A, and M are real matrices, and N (x) is a spatial dependent matrix of integrable bounded real functions. Furthermore, η(x, t) is a vector of extended state variables, that includes z(x, t) and its space derivatives until a certain ∂ 2 z(x,t) maximal order ( ∂z(x,t) ∂z , ∂z 2 , ...), ud (t) is a distributed input function, and n is the order of the derivative with respect to time. Using the Laplace transform in time, one obtains A +J

n 

∂η(x, s) = −(sn J − M )η(x, s) ∂x

(3)

(i−1)

sn−i η (x, 0) + J N (x)ud (s),

i=1

to which the solution is given by, the solution equation (SE) η(x, s) = Φ(x, s)η(0, s)  x + Φ(x, s) Φ−1 (ν, s)A−1 [J η0 + N (ν)ud (s)] dν,

(4)

0

with

Φ(x, s) = e−A

and η0 (x, s) =

n 

−1

(sn J−M )x

,

(i−1)

sn−i η (x, 0),

(5)

(6)

i=1

where Φ(x, s) is the so-called solution matrix (SM) and η0 ≡ η0 (x, s) represents the initial conditions sum. The model is well-posed if A is diagonal and by so invertible. The overall solution for η(x, s) comprises three terms: a left-hand side boundary conditions term, one term for the initial condition, and another for the distributed inputs N (x)ud (t). Each term can be analyzed and solved separately so that, in a certain sense, there are three different cases or problems to address. In this paper, the initial conditions solution is not included, so η(x, 0) = 0 (column of zeros) is considered from now on. If the initial conditions are zero, and if there is no distributed input term ( N (x)ud (t) = 0), the solution is reduced to a left-hand side boundary conditions problem, with (7) η(x, s) = Φ(x, s)η(0, s) = eB(s)x η(0, s) and B(s) = −A−1 (sn J − M ). In general, dynamical systems must satisfy left-hand side and right-hand side boundary conditions related by η(L, s) = Φ(L, s)η(0, s). Assuming that B(s) can be diagonalizable, then B(s) = V (s)Λ(s)V −1 (s), where V (s) = [v1 v2 · · · ] collects its eigenvectors in columns, Λ(s) is a diagonal matrix of its eigenvalues as Λ(s) = diag[λ1 , λ2 . . .], and Φ(x, s) = eB(s)x = V (s)eΛ(s)x V −1 (s),

(8)

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where eΛ(s)x = diag[ eλ1 x , eλ2 x , . . .]. It can also be written as Φ(x, s) = eB(s)x = V (s)eΛ(s)x W T (s),

(9)

with W T (s) = V (s)−1 . Yet, collecting the left eigenvectors of B(s) in W (s) = [w1 w2 · · · ], it gives   Φ(x, s) = vk (s)wkT (s)eλk (s)x = φk (s)eλk (s)x . (10) k

k

Note that Φ(x, s) is a finite sum of square complex matrices, multiplied by (scalar) space dependent complex exponential functions. Recall that the eigenvalues are given by the solution of |λk (s)I −B(s)| = 0, the right and left eigenvectors by B(s)vk (s) = λk (s)vk (s), and wkT (s)B(s) = wkT (s)λk (s). In the next sections, the Heat-exchanger equations can be solved applying the SE, Eq. (4), and consequently the transfer functions for co- and counter-current calculated.

3

Double-Pipe Heat-Exchanger Model in Counter-Current Mode

In this section, a full 4 × 4 model for a double-pipe heat exchanger is introduced. By making a simplifying assumption, a 2×2 hyperbolic model is written and the its transfer function derived. Poles and zeros are computed and two numerical examples depicted. Consider the full model as introduced in [5], detailed in [6], and given by ∂θ1 ∂θ1 ±u ∂t ∂x ∂θ2 ∂θ2 +v ∂t ∂x 2 ∂φ1 2 ∂ φ1 −c ∂t ∂x2 ∂φ2 ∂ 2 φ2 − c2 ∂t ∂x2

= a1 (φ1 − θ1 )

(11)

= a2 (φ1 − θ2 ) + a2 (φ2 − θ2 ) = b1 (θ1 − φ1 ) + b2 (θ2 − φ1 ) = b3 (θ2 − φ2 )

where c2 is the thermal diffusivity (m2 s−1 ) for conduction along the metal parts. Parameters a1 to b3 (s−1 ) are related with liquids-metal heat transfer. Fluid velocities are v, u > 0 (ms−1 ). Temperature deviations from θ1 to φ2 are, respectively, for shell and tube fluids, tube and shell walls (o C) and the tube length is given by L (m). Considering that the second order space derivative in the metal 2 parts is negligible in the transient response, ∂∂xφ2i ≈ 0 in the third and fourth equations of (11). The counter-current mode for the heat exchanger is chosen when the minus signal in u is considered, the opposite sign in relation to v. Applying the Laplace Transform and then neglecting the shell metal dynamics it yields, for the SM (12) Φ(x, s) = eB(s)x the matrix

Temperature Control on Heat-Exchangers



where f1 (s) = s+b u , f2 (s) = b = a21 . Using (9) it becomes 

where α(s) =

f B(s) = 1 g2

 −g1 , −f2

s+a v ,

b u

g1 (s) =

and g2 (s) =

(13) a v,

with a =

Φ(x, s) = e−αx

cosh(βx) + βγ sinh(βx) −2 b v sinh(βx) 2 a u sinh(βx) cosh(βx) − βγ sinh(βx)

a2 2

and

 (14) ,

(u−v)s+au−bv , 2uv

 (u + v)2 s2 + 2(u + v)(au + bv)s + (au − bv)2 , β(s) = 2uv and γ(s) = is given by

297

(u+v)s+au+bv . 2uv

(15)

The solution for a left-hand side boundary conditions θ(x, s) = Φ(x, s)θ(0, s),

(16)

but counter-current mode implies choosing one left-hand side boundary condition for one fluid and the right-hand side boundary condition for the other. Particularly, in this case θ1 (L, t) = uh (t) is the input is the hot liquid inlet temperature in the right-hand side, for control proposes, and θ2 (0, t) = uc (t), referring to the cold water inlet temperature in the left-side as a disturbance input. The transfer matrix is then given by   Φ11 (x,s) ΔΦ1     uh (s) θ1 (x, s) (L,s) Φ11 (L,s) = ΦΦ11 (17) ΔΦ2 21 (x,s) θ2 (x, s) uc (s) Φ11 (L,s) Φ11 (L,s)

where ΔΦ1 = Φ12 (x , s)Φ11 (L, s) − Φ11 (x , s)Φ12 (L, s) and ΔΦ2 = Φ11 (x , s)Φ22 (L, s)− Φ21 (x , s)Φ12 (L, s), which is identical to the transfer matrix presented in [2]. The transfer function from the hot liquid inlet and the cold liquid outlet is then 2 a u γ sinh(βL) θ2 (L, s) γ β cosh(βL) = . (18) θ1 (L, s) 1 + γ sinh(βL) β cosh(βL)

Notice that the transfer function has no delay terms, γ(s) is a polynomial in s, and contains a feedback in P (s) given by P (s) =

γ(s) sinh(β(s)L) β(s) cosh(β(s)L)

(19)

which is a irrational proper transfer (meromorphic) function, as seen in [8], in particular, a quotient of entire or holomorphic functions with exact poles and zeros. This allows that partial fraction expansion can be performed and a model approximation can be obtained. The global transfer function (18) can be reconstructed by the feedback of (19).

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The poles of (19) are just the cosh(β(s)L) zeros, because β(s) zeros, in the denominator, are cancelled by the two first sinh(β(s)L) zeros in the numerator. The polynomial γ(s) adds an additional zero to P (s). Figure 1 depict the PoleZero maps for two parameters sets, one efficient (a = b = 10) and one inefficient (a = b = 1) heat exchanger. At this point, it is worth mentioning that knowing the exact poles and zeros location can be very valuable for exploiting different PID designs, in alternative to the use of empirical methods. 250

250 Poles Zeros

200

150

150

100

100 Imaginary Axis

Imaginary Axis

200

50 0 −50

50 0 −50

−100

−100

−150

−150

−200

−200

−250 −8

−6

−4 −2 Real Axis

0

−250 −40

−30

−20 −10 Real Axis

0

Fig. 1. Pole-zero maps for the model with N = 100.

4

Co-current Heat-Exchanger

Keeping in mind the same type of logic and reasoning used in Sect. 3, a 2 × 2 co-current Heat-exchanger linear temperature dynamics can be written as ∂w ∂w +u = a(z − w) ∂t ∂x ∂z ∂z +v = a(w − z), ∂t ∂x

(20)

where u > 0 and v > 0 are fluid velocities, a is a heat transfer coefficient and state variables x and w are temperatures. Applying the Laplace transform with zero initial conditions, it yields       s+a a   d z(x, s) z(x, s) 0 v −v = , (21) + − ua s+a w(x, s) 0 dx w(x, s) u with the SM given by Φ(x, s) = eB(s)x .

(22)

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The B(s) matrix eigenvalues are λ1,2 = −α(s) ∓ β(s), with α(s) = (u+v) 2uv (s + a), and  (u − v)2 s2 + 2(u − v)2 as + (u + v)2 a2 . (23) β(s) = 2uv The SM is given by   a cosh(βx) − βγ sinh(βx) vβ sinh(βx) Φ(x, s) = e−αx (24) γ a cosh(βx) + β sinh(βx) uβ sinh(βx) ,

and γ(s) = (u−v) 2uv (s + a). Looking into the transfer matrix Φ(x, s), it can be concluded that each entry has an infinite number of zeros and no poles. The only exception is when u = v, in which case there are no zeros neither poles. Consider now the case when v = u and x = L, the outlet transfer function  ∞

aL sinh (β(s)L) −α(s)x aL −θa z(L, s) β(s)2 L2 = e e = 1+ e−θs . w(0, s) v β(s)L v k2 π2

(25)

k=1

√ uv(a2 +uvk2 π 2 ) This function is a holomorphic, with zeros at zk1,2 = −a ± 2j . (u−v)2 Figure 2 shows the transfer function (25) zeros obtained numerically and analytically, in the same 3-D plot and for the following values u = 0.25, v = 1, a = 5, L = 1. In closed-loop, the poles are given by Zeros 25 20 15 10

Im(s)

5 0 −5 −10 −15 −20 −25 −6

−5.5

−5

−4.5

−4

Re(s)

Fig. 2. Heat-exchanger 3-D numerically obtained zeros (left). Analytically calculated zeros. (right).

 ∞

aL β(s)2 L2 1+K 1+ e−α(s) = 0, v k2 π2

(26)

k=1

where K is a proportional gain. In this case, there are infinite zeros and no poles. Note that causality is ensured by θ = u+v 2uv L, in (25). The closed-loop poles can be numerically computed using (26). Computing the open-loop time response

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will imply being able to perform a partial fraction decomposition to the transfer function (25), which is not possible because this function is not meromorphic. Another way is to try to directly apply the inverse Laplace transform integral to z(L, s), which proves in practice to be quite difficult or even impossible. Exact frequency response plots (Bode diagrams) can otherwise be easily computed. It is not very realist to consider (26) as a physically implementable closed-loop system, because it is not possible to change instantaneously the inlet temperature w(0, s). In practice, the use of a hot water heating system, with recycle, is quite common. Including a heating system, it can be modelled by ˙ t) + w(0, t) = kh q(t) + w(L, t), τh w(0,

(27)

V where τh = Au (V is the volume of the heating tank; Au is the flow rate in the heating system; A tube area), kh = ρC1p V (ρ and Cp specific mass and specific heat respectively), and q(t) is the heating power that can be directly manipulated. The last equation constitutes a point-wise differential equation that acts as a left-side boundary condition for w(z, t). Using (7) and (27), it yields the transfer matrix     z(0, s) z(x, s) −1 , (28) = Φ(x, s)F (L, s)B0 q(s) w(x, s)



with F (L, s) = and

1 0 −Φ21 (L, s) τh s + 1 − Φ22 (L, s)

 (29)



 1 0 . B0 = 0 Kh

(30)

The poles are given by: τh s+1 − Φ22 (L, s) = 0, and the transfer function between q(s) and z(L, s) is kh Φ12 (L, s) z(L, s) = . (31) q(s) τh s + 1 − Φ22 (L, s) Now, q(s) can be used to control the cold fluid temperature using the transfer function (31) for design purposes. Note that the heating system recycle of hot water introduces a feedback dynamics responsible for the poles existence. Figure 3 (left and right) reveals the poles and zeros approximated location, through the coloured zones, hottest for poles and coldest for zeros. In this case, co-current mode, simplifying assumptions lead to a dynamical systems without poles. Realizing that the hot water recycle introduces a feedback that in its turn highlights the combined dynamics between the exchanger and the recycle, also revealing a dominant pole near to the origin and two complex-conjugated pairs, before the zeros appearance, can be of great importance in tuning the PID controller.

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20 15

Imaginary axis

10 5 0 −5 −10 −15 −20 −20

−15

−10

−5

0 Real axis

5

10

15

20

Fig. 3. Heat-exchanger with recycle, zeros located in coldest colours zones and poles in zones of hot colours (left). Heat-exchanger with recycle, poles and zeros approximated location (right).

5

Application

In this section the laboratory apparatus including the hot water recycle and the control system is detailed. Some results from preliminary tuning experiments are depicted in Fig. 4. The apparatus used was a ISI Impianti Heat-exchanger with a concentric double-pipe format. The inner piping has a internal cross-section of 10 mm and an external cross-section of 12 mm. The outer pipe has an internal cross-section of 16 mm and an external cross-section of 18 mm. The total length is approximately 3 m. The heat exchange hot fluid is in a closed circuit that includes a tank with a maximum capacity of approx. 20 L, with an electric heating resistance power of 2000 W connected to solid state relay and four-speed RS25/60R class F Wilo pump. The PID controller used is a West 3500. The data acquisition was performed using K and J-type thermocouples connected to FEMA and Industrial Interface transducers. The signals were acquired in two data acquisition board setups using Advantech PCL-813 and National Instruments model 9481 analog cards for the hardware-based PID. Figure 4 depict set-point variations, for both co-current (left) and counter-current (right) modes, respectively, at a nominal cold fluid flow rate of approximately 1.25 l/min and a nominal hot

Fig. 4. Temperature control to a constant set-point of 22 ◦ C in co-current mode (left) and counter-current mode (right).

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fluid flow rate of 3.5–4 l/min (equivalent to the 1300 rpm in the pump (minimum rotation value). The PID controller parameters were kept between co and counter-current tests and are as follows: proportional band (PB) of 4, a integral time of 120 s and a derivative time of 45 s. Note that the PID control law adjusts the duty-cycle of the power relay, which might explain some difficulty to reach the set-point with zero off-set.

6

Conclusions

Solving PDE models for distributed systems is a complex task, but being able to obtain their solutions and the respective transfer functions should provide new and more exact methods for dynamical analysis and control purposes. These methods can then be used as an alternative to more empirical approaches or narrow approximations that use finite-dimensional models, having a potential impact in control design. In this paper, the irrational transfer functions of a dual mode double-pipe Heat-exchange was derived and later used in conjunction with a hot water recycle assembly to establish guidelines for the control system design. Then, a PID controller scheme was successfully implemented to control the cold fluid outlet temperature of the Heat-exchanger around an imposed setpoint for the two modes. Acknowledgements. This work was supported by national funds through FCT, Funda¸ca ˜o para a Ciˆencia e Tecnologia, under project UIDB/50021/2020.

References 1. Astr¨ om, K., H¨ agglund, T.: PID Controllers: Theory, Design, and Tuning. The Instrumentation, Systems, and Automation Society, ISA, Pittsburgh (1995) 2. Bartecki, K.: A general transfer function representation for a class of hyperbolic distributed parameter systems. Int. J. Appl. Math. Comput. Sci. 23(2), 291–307 (2013). https://doi.org/10.2478/amcs-2013-0022 3. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, Revised, 2nd edn. Wiley, New York (2006) 4. Curtain, R., Morris, K.: Transfer functions of distributed parameter systems: a tutorial. Automatica 45(5), 1101–1116 (2009). https://doi.org/10.1016/j.automatica. 2009.01.008 5. Hsu, J.P., Gilbert, N.: Transfer functions of heat exchangers. AIChE J. 8(5), 593– 598 (1962). https://doi.org/10.1002/aic.690080506 6. Igreja, J.M., Lemos, J.M.: On modal decomposition and model uncertainty bounds for linear distributed parameter transport systems. In: 2017 25th Mediterranean Conference on Control and Automation (MED), pp. 93–98 (2017). https://doi.org/ 10.1109/MED.2017.7984101 7. Jovi´c, V.: Analysis and Modelling of Non-Steady Flow in Pipe and Channel Networks. Wiley, New York (2013). https://doi.org/10.1002/9781118536896 8. Kawano, Y., Ohtsuka, T.: Input-output linearization for transfer functions of inputaffine meromorphic systems. SICE J. Control Meas. Syst. Integr. 5(3), 133–138 (2012). https://doi.org/10.9746/jcmsi.5.133 9. Litrico, X., Fromion, V.: Modeling and Control of Hydrosystems, 1st edn. Springer, London (2009). https://doi.org/10.1007/978-1-84882-624-3

A DOBOT Manipulator Simulation Environment for Teaching Aim with Forward and Inverse Kinematics Thadeu Brito1(B) , Jos´e Lima1,2 , Jo˜ ao Braun3 , Luis Piardi1 , and Paulo Costa2,4 1

Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal {brito,jllima,piardi}@ipb.pt 2 INESC TEC - INESC Technology and Science, Porto, Portugal 3 Federal University of Technology - Paran´ a, Toledo, Brazil [email protected] 4 Faculty of Engineering, University of Porto, Porto, Portugal [email protected]

Abstract. Industrial Manipulators were becoming used more and more at industries since the third industrial revolution. Actually, with the fourth one, the paradigm is changing and the collaborative robots are being accepted for the community. It means that smaller manipulators with more functionalities have been used and installed. New approaches have appeared to teach students according to the new robot’s capabilities. The DOBOT robot is an example of that since it captivates the student’s attention with an uncomplicated programming front-end, tools, grippers and extremely useful for teaching STEM. This paper proposes a dynamic based simulation environment that can be used to teach, test and validate solutions to the DOBOT robot. By this way, the student can try and validate, at their homework without the real robot, the developed solutions and further test them at the laboratory with the real robot. Currently, remote testing and validation without the use of a real robot is an advantage. The comparison of the provided simulation environment and the real robot is presented in the approach. Keywords: Simulation · Teaching · Problem based learning based learning · Manipulator robots

1

· Project

Introduction

Robotics is an interdisciplinary science that great potential for education. It requires several areas of knowledge to understand its concept fully. In the last few years, according to [1], robots’ presence increased in the education market, both as tools to motivate students to explore STEM (Science, Technology, Engineering and Mathematics) disciplines and as curriculum materials for teaching content. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 303–312, 2021. https://doi.org/10.1007/978-3-030-58653-9_29

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In fact, several attempts have been made worldwide to introduce robotics in science and technology subjects in schools, from kindergarten to high school [2]. It can be taught in class, as projects or even by competition as it encourages the students to develop solutions to tasks [3]. To motivate robotics in education, it is necessary to make this technology accessible to students. In this way, there are some approaches to do this - for instance, educational programs from governments, organizations, and enterprises to bring robotics to schools. Alternatively, reduced costs of robotics components and the development of low-cost educational kits helps to spread the technology to students. Finally, there are simulations. Although they don’t offer the opportunity to experience the robotics’ components at hand, it provides a realistic experience to a real scenario without costs. Moreover, simulations software are nowadays capable of simulating non-idealities and dynamic constraints of the real scenario such as friction, uncertainty, measurement noise, among others. It is a tool that allows students, at their homework, to develop a solution and validate it on the real scenario at the classroom later. Besides, they offer a peaceful transition from the simulation environment to the real world, minimizing development costs. Hence, they became a mandatory step in research and development projects. This robotic arm was designed for practical education. For this reason, it has several advantages such as being small, low-cost and easy to learn (it can be coded with visual programming language). Moreover, it is easy to change the tools that can be attached to the arm. However, one considerate disadvantage is that this robot does not have a simulation platform. For this reason, students are obliged to purchase the robot or, if their education institution has one, they need to remain inside the institution’s laboratory to study. Consequently, simulations must be the most realistic possible to permit that the simulation’s code works in the real world, allowing students to study robotics. Thus, this work presents a dynamic based simulation tool, based on SimTwo, that helps to develop, test, and validate programs that after will be run in the DOBOT Magician manipulator. This work is structured as follows: after the introduction, a brief state of the art is presented in Sect. 2. After, in Sect. 3, the DOBOT manipulator is explained. The results are shown in Sect. 4. The conclusion and future work are discussed in Sect. 5.

2

State of Art

According to [2], research in robotics in education focused on the interaction between the invention of new technologies and the development of new ways of learning. Furthermore, with the purpose to make children active participants in education, research since the ’60s has been made to develop robotic construction kits [2]. This concept is emphasised by Jeffrey Johnson [4], claiming that robotics provides an effective way for children to learn many of the things on the national curriculum for science, technology, and mathematics. He also

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states that robotics is arguably the most effective way of motivating and supporting the study of the many areas of the curriculum. As stated by [5], an underwater robotics program for middle and high schools and was developed by the Stevens Center for Innovation in Engineering & Science Education at The Stevens Institute of Technology. She states that its purpose is to teach students engineering design and STEM concepts and at the same time increasing awareness in engineering, and IT careers. Besides, organisations such as AAAI (Association for the Advancement of Artificial Intelligence) and IEEE (Institute for Electrical and Electronics Engineers) and others hold tournaments for their students members at their significant conferences [1]. He states that it motivates them to pursue relevant degrees by sharpening the students’ skills by the robot’s construction-related activities, getting them excited about technologies and businesses linked to that organisation. Regarding education robot platforms, many research institutions and universities continue to develop custom platforms such as Honda’s Asimo or even research platforms such as iCub and Willow Garage’s PR2. Moreover, he states that low-cost kits and platforms are becoming widely used in primary, secondary and higher-level education throughout the world. In [6], is presented a low-cost printable bots that could be incorporated in university engineering disciplines as educational mechanisms. Therefore, it is possible to explain the hardware and programming tools of an educative robotic platform based on low-cost open source technologies.

3

DOBOT Manipulator

After significant advances in robotics over the years, there has been an explosive growth in the range of robotic applications in the industrial sector, in particular the use of the palletizing structure. This type of robotic manipulators is commonly used in manufacturing, handling, process operation, assembly, inspection and other processes [7]. Due to this vast increase in industrial applications, implementing an educational robot model that resembles that used in industry is extremely important for students’ learning in schools, institutes and universities.

Fig. 1. DOBOT Magician working in education method [8].

According to [8], DOBOT Magician is a multifunctional desktop robotic arm for practical education, as can be seen in Fig. 1. It can perform 3D printing,

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laser engraving, writing, and drawing depending on the tool attached to the endeffector. This robot supports 13 extensible interfaces and over 20 programming languages for secondary development. Being low-cost, small in size, and with easy operation, every student can practice on their own. Moreover, it can be controlled by APP, Bluetooth, Wi-Fi, mouse, among others. By supporting multi-robot collaboration, it is possible to control several DOBOT Magician using the same device. It was successfully applied by many institutions and enterprises such as Tsinghua University and Volkswagen. Figure 2 shows illustrations of the DOBOT Magician scheme, where it is possible to view the point of origin of the robot (Fig. 2a), with the orientations of each joint represented by J1 . . . J4 in the Fig. 2b, and in the Fig. 2c, it shows each of the necessary distances for performing forward and inverse kinematics.

(a) Point of origin in the Cartesian (b) Connections be- (c) Distances between plane. tween the joints and each joint and the height from the point their orientations. of origin.

Fig. 2. DOBOT Magician scheme. Adapted from [8].

Compared to industrial manipulators, this model is relatively inexpensive. However, equipping a classroom with a model for each student would become an additional investment. On the other hand, keeping only a few copies at the disposal of students can cause long queues between them. Therefore, a possible solution to this problem is to create a simulated environment that approximates the real characteristics of DOBOT Magician. In this way, students can test their scripts with the simulation tool before putting them into practice in real situations, and consequently, avoiding numerous inconveniences. 3.1

Forward and Inverse Kinematics of DOBOT

Some techniques must be developed to identify the positions of specific points on the robotic arm, as well as to determine the Cartesian Coordinates from which the tool (end-effector) is acting. This identification depends on the structure of the joints and links (parts connected through the joints) in reference to some point, which is usually the basis of the manipulator itself. Determining where is the end-effector is of great importance for industrial processes, therefore

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demonstrating this study to students through DOBOT Magician can become a promising and engaging approach. In this way, the kinematics for manipulators is responsible for studying movements without considering the causes that originate them [9]; therefore, the kinematics will deal with distances, angles, accelerations and translation and angular speed [10]. To determine kinematics in manipulators is necessary a careful study, even for robotic arms with few Degrees of Freedom (DoF) such as the DOBOT Magician (assuming the configuration with 3 DoF, because this work approach does not need the J4 ). Figure 3a shows the number of variables and the complexity of kinematics.

(a) Example of palletizer robot connecting-bar [7].

(b) Representation of DOBOT Magician in 3D vectors.

Fig. 3. Demonstration of DOBOT Magician geometry.

To simplify the geometric analysis, it is possible to transform each robot link into a three-dimensional vector and then carry out the projections of these vectors through trigonometric functions. Figure 3b shows the DOBOT Magician vector analysis. Where each dimensional variable has the same configurations that the manufacturer provides (Fig. 2), that is, the same number of joints (Jn ) and lengths (Ln ). The base angle is called “alpha (α)” (generated by the movement of J1 ), whereas the angle formed by the joint J2 is declared as “beta (β)”, and consequently, the angle formed by the last joint J3 is defined by “gamma (γ)”. Therefore, it is possible to obtain the coordinates of the end-effector in XY Z through the angles of each joint, that is, the forward kinematics is defined by the following expressions: X = radius. cos(α)

(1)

Y = radius. sin(α)

(2)

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Z = L1 − L2 . sin(β) + L3 . sin(−γ)

(3)

radius = (L4 + L2 . cos(β) + L3 . cos(γ))

(4)

where

This means that with the use of Eqs. 1–4, it is possible to determine where the end-effector will be when the angles are known. Hence, it can be used to configure the simulated environment to synchronize the motion with the real robot. Similar to that performed for forward kinematics, that is, by making the projections of the vectors indicated in the Fig. 3b, it is possible to identify the angles that each joint forms (α, β and γ) to reach with the end-effector a certain known XY Z point. In other words, it is possible to determine the inverse kinematics. To do this, simply apply the following expressions:   Y α = arctan (5) X π − (Q1 + Q2 ) (6) β= 2  2  π L2 + L32 − hypotenuse2 − arccos γ= (7) 2 2.L2 .L3 where  Q1 = arctan

Z radius

 (8)

and  Q2 = arccos

L22 − L33 + hypotenuse2 2.L2 .hypotenuse

 (9)

In this approach, the radius needs to be written according to the known XY coordinates: radius = (X 2 + Y 2 ) − L4 And the hypotenuse variable is defined by:  hypotenuse = Z 2 + radius2

(10)

(11)

It is commonly seen that inverse kinematics can generate infinite values for some angles when knowing the XY Z point of the end-effector, as seen in [11]. However, the DOBOT Magician manufacturer specifies in the manual that some joints are limited, and this problem does not happen [8]. For this reason, this work does not address the treatment of infinite values at angles. Therefore, Eqs. 1–11 can be inserted in the simulator settings, so that the simulated robotic arm can be moved equally with the arm used in the real situation.

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Results

Simulation environment was based on SimTwo [12]. The environment model will be available to help students for the off-line development and training. The STL files (provided by the manufacturer) were converted into 3DS format and the entire assembly of DOBOT Magician in SimTwo is shown in the Fig. 4, with the suction tool fixed at the end-effector.

Fig. 4. SimTwo scenario with a 3D model of the DOBOT Magician manipulator.

The tests have the as main objective to compare the performance of the simulated robot in SimTwo with the real robot. For this, the first step is to determine the reading of each value. In Fig. 5, it is possible to see the options panel of both, that is, in Fig. 5a, the panel showing the official DOBOT Magician software informs the tool’s coordinate and the angle values of each joint also. Therefore, in Fig. 5b, it is shown how the values are displayed during the simulation, resembling as close as possible to the official version.

(a) The official Option Panel.

(b) The SimTwo panel.

Fig. 5. Comparison between the official option panel and the simulated panel.

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Based on information from the official options panel, nine points are chosen within the real robot’s workspace. As mentioned earlier, this work does not consider DOBOT Magician’s last DoF due to the application of the suction tool does not need to use this degree to perform actions. Therefore, the informed values R and J4 will not be necessary for the tests of this work. One by one of the nine points are inserted in SimTwo to perform the movement analysis between the real and simulated scenario, as shown in Fig. 6. Therefore, the first point is inserted in the software of the real robot (Fig. 6a), and then the same point is inserted in the simulation, shown by Fig. 6b. To avoid doubts about the tool’s positioning in the simulated environment, a yellow cube is placed in the same coordinates. In this way, it is possible to check if the point of the simulated suction tool is positioned in the expected coordinate.

(a) Real DOBOT Magician.

(b) Simulated DOBOT Magician.

Fig. 6. The first point that compares the pose between real and virtual robot.

Since the values obtained with the software of the real robot can serve as parameters, the inverse and forward kinematics can be tested. During the tests of forward kinematics, it is established that SimTwo must generate the XY Z coordinates as close as possible to those of DOBOT Studio through the inserted angles. In this way, Table 1 describes the nine values of J1 , J2 and J3 (from DOBOT Studio), and the values that SimTwo generated from the tool coordinates (defined as X  , Y  and Z  ). Therefore, the error values are made according to the difference between the coordinates generated by SimTwo with the coordinates of DOBOT Studio. For example, the error of the X coordinate (EX ) is produced by EX = X  − X, and consequently, the same for the other coordinates. Similarly, tests with the inverse kinematics configured in SimTwo are based on the values provided by DOBOT Studio. However, during the tests, the comparison of the simulation with the real environment is made with the generated

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Table 1. Results of forward kinematics applied in SimTwo. X  [mm] Y  [mm]

Z  [mm]

From SimTwo

J1 [◦ ]

J2 [◦ ]

J3 [◦ ]

Y−Y

Z − Z

206.7014 0.0036

134.991

0.0010

0.0035 0.0006

0.0000

0.0001

206.7006 0.0036

0.0041

0.0010

27.3328 54.6673 0.0007

0.0000

−0.0001

140.2823 0.0024

0.0048

0.0015

8.2131

−0.0012 0.0000

325.3315 0.0057

0.0058

0.0006

69.4926 18.7652 0.0007

0.0021

0.0002

34.5805

45.1669 40.3497 0.0006

0.0004

−0.0002

220.2139 151.8048 0.0056

0.0003

EX [mm] EY [mm] EZ [mm] X − X

From DOBOT Studio

65.3559 0.0007

150.5779 −162.442 −65.1879 −47.1706 56.9885 70.6978 0.0006

−0.0006 0.0000

245.597

−48.3805 −64.5283 −11.1441 61.7856 60.8264 0.0007

0.0000

−0.0001

−68.2998 51.5912

55.3750 80.5549 0.0004

0.0005

0.0001

246.4169 154.7887 −65.9097 32.1353

72.9032 45.9187 0.0006

0.0004

−0.0001

121.0937 152.734

joints. In other words, in these tests, the objective is to insert the XY Z coordinates (from DOBOT Studio) in the SimTwo panel and analyze which angles are generated, namely J1 , J2 and J3 . Table 2 has each of the nine probe points, as well as the values of the errors found. These are defined by the difference between the angles generated by SimTwo and those provided by DOBOT Studio. For example, the Joint 1 error (EJ1 ) is characterised by EJ1 = J  − J, so the same method is used in the other joints. Table 2. Results of inverse kinematics applied in SimTwo. X[mm]

Y [mm]

Z[mm]

From DOBOT Studio

J1 [◦ ]

J2 [◦ ]

J3 [◦ ]

From SimTwo

206.7008 0.0036

134.9909 0.0010

206.6999 0.0036

0.0042

0.0010

140.2816 0.0036

0.0048

0.0015

325.3308 0.0036

0.0056

220.2133 151.8044

0.0058

0.0000

EJ1 [◦ ] J1

−

EJ2 [◦ ]

J1 J2

EJ3 [◦ ]

− J2 J3 − J3

0.0035 0.0000

−0.0003 0.0000

27.3326 54.6674 0.0000

−0.0002 0.0001

8.2129 65.356

0.0000

−0.0002 0.0001

0.0006

69.4922 18.7657 0.0000

−0.0004 0.0005

34.5805

45.1666 40.3499 0.0000

−0.0003 0.0002

150.5773 −162.4414 −65.1879 −47.1706 56.9884 70.6981 0.0000

−0.0001 0.0003

245.5963 −48.3805

−64.5282 −11.1441 61.7854 60.8266 0.0000

−0.0002 0.0002

121.0933 152.7335

−68.2999 51.5912

55.375

0.0000

246.4163 154.7883

−65.9096 32.1353

72.9029 45.919

80.5552 0.0000 0.0000

0.0003

−0.0003 0.0003

To validate the proposed approach, it is expected that the SimTwo simulator would have the same performance as performing the applications through DOBOT Studio. During the analysis, it is noted that the developed forward kinematics obtained error averages of 0.0006, 0.0002 and 0 for XY Z, respectively. With these average error values, it is possible √ to make the error in terms of the Euclidean distance, that is, EEuclidean = 0.00062 + 0.00022 = 0.000 632 4 mm.

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For the inverse kinematics, the average error obtained for J1 was 0, and in J2 and J3 had 0.0002.

5

Conclusion

In this paper, a simulation environment for the DOBOT commercial platform was presented. The forward and inverse kinematics of such robot are addressed and implemented on the simulator. The developed tool will allow students to do the work as if they had a real robot. Results demonstrated that the simulator is capable to execute the trajectories in a similar way as the real one and with an error of positioning of 0.0006 mm (Euclidean distance) compared with the real platform IDE for the forward kinematics and almost zero degrees for all joints on the inverse kinematics. As future work, it is planned to develop an interpreter for the DOBOT files in order to increase the compatibility between real and simulation, that means, students can use the same files at both robots. Acknowledgements. This work is financed by National Funds through the Portuguese funding agency, FCT - Funda¸ca ˜o para a Ciˆencia e a Tecnologia, within project UIDB/50014/2020.

References 1. Miller, D.P., Nourbakhsh, I.: Robotics for education. In: Springer Handbook of Robotics, pp. 2115–2134. Springer, Cham (2016) 2. Alimisis, D., Kynigos, C.: Constructionism and robotics in education. Teacher Educ. Robot.-Enhanced Constructivist Pedagogical Methods 11–26 (2009). ISBN 978-960-6749-49-0 3. Braun, J., Fernandes, L.A., Moya, T., Oliveira, V., Brito, T., Lima, J., Costa, P.: Robot@ factory lite: an educational approach for the competition with simulated and real environment. In: Iberian Robotics conference, pp. 478–489. Springer, Cham, November 2019 4. Johnson, J.: Children, robotics, and education. Artif. Life Robot. 7, 16–21 (2003) 5. Eguchi, A.: Robotics as a learning tool for educational transformation. In: Proceeding of 4th International Workshop Teaching Robotics, Teaching with Robotics & 5th International Conference Robotics in Education Padova (Italy), July 2014 6. Curto, B., Moreno, V.: Robotics in education. J. Intell. Robot. Syst. 81(1), 3 (2016) 7. Yu, H., Sun, Q., Wang, C., Zhao, Y.: Frequency response analysis of heavy-load palletizing robot considering elastic deformation. Sci. Prog. (2019). https://doi. org/10.1177/0036850419893856 8. DOBOT Magician - Lightweight Intelligent Training Robotic Arm (n.d.). https:// www.dobot.cc/dobot-magician/product-overview.html. Accessed 8 Feb 2020 9. Groover, M.P.: Automation, Production Systems, and Computer-Integrated Manufacturing. Pearson Education India, Bengaluru (2016) 10. Todd, D. J.: Fundamentals of Robot Technology: An Introduction to Industrial Robots, Teleoperators and Robot Vehicles. Springer, Heidelberg (2012) 11. Brito, T.: Intelligent collision avoidance system for industrial manipulators. Master Thesis (2019). http://hdl.handle.net/10198/19319 12. Paulo, C., Jos´e, G., Jos´e, L., Paulo, M.: Simtwo realistic simulator: a tool for the development and validation of robot software. Theory Appl. Math. Comput. Sci. 1(1), 17–33 (2011)

An IIoT Solution for SME’s Bruno Cunha1(B)

, Elder Hernández1 , Rui Rebelo1 and Filipe Ferreira1

, Cristóvão Sousa1,2

,

1 INESC TEC, 4200-465 Porto, Portugal

[email protected] 2 CIICESI-ESTG, Politécnico do Porto, 4610-156 Felgueiras, Portugal

Abstract. The innovation and digitalization of the industry is happening triggered by the Industry 4.0 and Industrial Internet-of-Things (IIoT) paradigm. Enterprises are following the trend of digital transformation and are fostering projects that enable a higher comprehension of I4.0 solutions to answer their needs. The IIoT platforms have been a central component for industrial systems architectures to enable interoperability and data flow within industrial settings. However, the digitalization process has all sorts of shortcomings associated to them, and in the SME’s this transformation has been slow to none. In this work we showcase a proof of concept of an IIoT platform that intends to simplify the digitalization process in SME’s, based on the Portuguese footwear industry cluster. Keywords: IIoT · I4.0 · SME · Digital twin

1 Introduction The industry 4.0 was born from industry needs to fulfill the challenges of a globalized world focused on customer requirements, where the customization, flexibility, decentralization, rapid development and resources optimization plays a big role in achieving that goal [1]. The Industry 4.0 have been key to foster the matureness of new technologies emerging from the concepts of Cloud computing, Internet-of-Things, Industrial-Internetof-Things, Cyber-Physical systems and Artificial Intelligence [2]. In order to standardize the interconnection of this concepts, it was defined a reference architecture. Platform Industrie 4.0 proposes a Reference architectural Model Industrie 4.0 (RAMI4.0), which is a guide for transformation and implementation of industry 4.0. RAMI4.0 encloses a number of highly diverse aspects as concepts, processes and practices, in a three dimensional model showing the most important aspects in industry 4.0, sharing a common understanding of the underlying conceptual framework, representing an adaptable roadmap for I4.0 [3]. The model proposed by RAMI4.0 is split in three axis. The vertical axis comprehends the IT layers, which are the layers that differs the digital from the real world in the solution architecture. The left-hand horizontal axis represents the Product life cycle and Value Stream point of view, enclosing the development phase until the maintenance/usage of the product. The right-hand horizontal axis represents the functional hierarchy levels in a factory [3]. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 313–321, 2021. https://doi.org/10.1007/978-3-030-58653-9_30

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Focusing on the vertical axis, there is six layers. The asset layer focus both human and physical resources of the shop floor. The Integration layer decodes aspects of information gathering from assets and human interaction. This layer represents a bridge between the physical and the digital world. The communication layer is responsible to empower the standardization of the data flow between digital systems and the Information layer. The information layer ensures the pre-processing, data integrity and consistency, and, in the end, the data is provisioned to be consumed by other applications. The last layer, the Business layer, is where the business processes are defined and linked in combination with management of the Functional layer [3]. The I4.0 paradigm brings new IIoT solutions that enable industry to access data from their resources, assets and systems. Whereas IIoT solutions are being increasingly adopted by large enterprises, SME’s are struggling to follow this paradigm shift towards digital transformation [4]. IIoT projects are multidisciplinary and social. Technical in nature, where technology is just a part of this whole setup. This demands for high skilled human resources, openness to investment in new technological assets, innovative mindset, awareness of their own capabilities, and above “all”, a clear understanding of I4.0 technology in context of use [5, 6]. In order to overcome some of SMEs challenges and provide a showcase for I4.0 technology, we are discussing state of the art IIoT architecture for SMEs to embrace digital transformation. This paper is organized as follows: Sect. 2 presents the IIoT platforms and their characteristics. In Sect. 3, a proposal for the IIoT platform architecture for SMEs is shown with the related technological stack. In Sect. 4, the use cases used to validate the IIoT Platform functionalities. In Sect. 5, the results of the implementation are presented. In Sect. 6, the conclusion of the work is presented, and future works proposed.

2 IIoT Platforms Our stance is that the IIoT platforms are a way to fulfill the challenges in implementation of I4.0, being considered as essential structures for the future industrial architectures for the industry. The IIoT platform must be RAMI4.0 compliant, narrowing the scope to comprehend three layers of vertical axis, which are the Integration, Communication and Information layer. Establishing the fundamental pillars for the vertical integration of the industry and the creation of cyber-physical systems [7]. An IIoT platform is a mediator between the assets, that is the connected physical devices and persons in the asset layer, and the business applications, which correspond to those organization information systems in the functional layer [8, 9]. As an example, to portray the IIoT platforms function is the scenario of obtaining data from the lowest level of vertical integration, the assets layer. This data is obtained from devices with communication capabilities (i.e.: PLC’s) existing on the shop floor, and it is processed and stored by the IIoT platform, making it available for other applications or systems in others levels of integration (control, production, production planning or even management). Associated with these capabilities of the IIoT platform, concerns regarding performance, obtaining data in real time, integrity and security are also focus of these solutions. IIoT platforms focus on interoperability among devices (hardware) and services (software), managing, monitoring and controlling a network of components, linked to a

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common architecture and platform. One of the main functions is the exchange of data between various devices, allowing the extraction and forwarding of data between OT and IT systems [10]. This setup must be agnostic not only to the communication channels used, but also to the types of devices it connects to. This will allow the development of a flexible and adaptable IIoT platform and at the same time ensure its relevance in the future. The IIoT platform must be able to communicate with different types of equipment, from different manufacturers and with different communication protocols, and thus offer independence from the shop floor and integration with management and supervision systems of companies such as a MES or ERP system, and therefore solve the interoperability problem between systems [10]. Other functionalities for the IIoT platform consist on: Connectivity and normalization – encompassing the management of IIoT data and the efficient sending of commands to heterogeneous IIoT devices; Device management – ensures the discovery and registration of new IIoT devices; Processing and action management – makes use of a set of functions that operate on the data flows received from different IIoT devices; Data storage – ensures the collection of data from IIoT devices; Data visualization – the data visualization components allow to explore the data of the IIoT platform; Analytics –set of API’s that allow knowledge to be generated; Additional tools - Additional tools such as user management, interfaces [11]. These top-level functionalities or requirements serves as guidelines for the development of the architecture that will support the IIoT platform. For the IIoT platform proof of concept, features such as Analytics, Additional Tools and Security and Privacy are not given high priority in the development of the IIoT platform and are not within the scope of the project, since the technology user, already provide security features.

3 Architecture and Implementation This work aimed to build a proof of concept IIoT Platform for the Footwear sector in Portugal, based on affordable and customizable technology, so that it can fit the needs and requirements of SMEs. As the Footwear SMEs equipment landscape is heterogeneous in terms of automation and sensing. The application of the IIoT platform developed in this work was modeled based on I3.01 , which is, prepared to use the equipment already existing in the companies, but obeying the I4.0 paradigm in terms of communication, allowing in the future, the connection to other I4.0 technologies. Since IIoT implies standardization, the OPC UA communication protocol will support the communication standardization from all connected devices, enabling to retrieve data from the shop floor, which is then forwarded to a message broker for persistence and access. The IIoT platform will also receive data from the Logistics layer, providing context to data from the Production layer. Another crucial aspect within IIoT architectures relates to the digital twin concept. Digital twin refers to the ability of the IIoT architecture to, virtually, represent the physical systems, not only in terms of its data structures, but also, in terms of its relationships, either with other systems or devices [12]. This ease the process of obtaining and process 1 I3.0: Industry 3.0 [2].

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real time data. For this, it is necessary to interconnect the various systems available, ranging from systems related to the logistics layer, such as MES or ERP, to the various PLC’s and sensors in the Production layer that are on the shop floor [13]. The data provided by the IIoT platform will then be presented in dashboards in order to simplify their visualization. 3.1 Technological Stack The problem inherent in the project and the underlying technology it, falls within the layer of Information, Communication and Integration, of the vertical axis of the RAMI4.0 architecture. This is a problem fundamentally at the level of interoperability between systems, and this depends on obtaining data from several systems/equipment in the lower layers and the correct treatment and aggregation of that data into a single source that is available for the upper layers of the architecture to consume (Fig. 1).

Fig. 1. Technological stack of the IIoT platform

According to the afore mentioned motivation, the main requirements, and the RAMI model, an adequate technological stack is established, identifying the following components, from protocols, to gateways, databases, servers and brokers (Fig. 2): 1. 2. 3. 4. 5. 6.

Automation Driver Integrator; Edge gateway Node-Red; Main Message Broker Apache Kafka; MQTT Message Broker RabbitMQ; MySQL and InfluxDB Databases; Grafana.

The following Table 1 portrays the chosen technologies that will enable the required functions of the IIoT platform. All technologies, except Kepware, are Open source software, and so accomplish the requirements of affordable and customizable technology.

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Table 1. Technology stack and functionalities Component

Layer

Function

Kepware [14]

Integration

Allows communication with PLC’s that do not natively include the OPC UA protocol and has the possibility to forward the data obtained from these automata by OPC UA

Node-Red [15]

Communication Gateway between the OPC UA protocol and the Message Broker

Apache Kafka [16] Communication Main Message Broker of the IIoT Platform RabbitMQ [17]

Communication Complement the main broker of the IIoT platform Allow the IIoT platform to communicate with other devices with different protocols like MQTT and AMQP

MySQL [18]

Information

Aims to persist data and make it available to the IIoT platform

InfluxDB [19]

Information

Aims to persist data, as a time series, and make it available to the IIoT platform

Grafana [20]

Information

Enable the visualization of data from the IIoT platform

Fig. 2. IIoT platform architecture

Dependencies

Apache Zookeeper

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3.2 Implemented Architecture Figure 3 depicts the proposed IIoT Platform Architecture populate with the technology stack. In order to standardize the communication protocols from connected devices (PLCs) that do not use OPC UA natively, a solution to bridge industrial communication drivers such as Kepware is proposed. This solution can function as a means to translate different communication protocols to obtain data from the PLC and forward it via OPC UA to the Node-Red. Node-Red was used in conjunction with an OPC UA library. In this way, Node-Red started to function as an integration gateway, obtaining the data by OPC UA from the PLC’s and persisting it in relational databases with specific libraries. The data gathered from the PLC’s by Node-Red it is then published to a Topic2 in the Apache Kafka Message Broker of the IIoT Platform. Using the Connect API from Apache Kafka, the broker is able to monitor the arrival of new data into a specific Topic and then persist the data in a Sink3 . As a Sink, it was used InfluxDB, a time series database, which manages the data as FIFO, according to the insert time. This type o Database offers a flexible and simple way to integrate with Grafana, used to create dashboards. In this implementation, Docker containers were used to develop the platform, and Rancher to enable the containers management. To obtain data from enterprise systems such as ERPs, an application was developed to connect the Information System database and collect the data within. This data is then “shipped” to the Message Broker, using MQTT Protocol. The MQTT Message Broker was implemented using RabbitMQ. This tool was chosen, for this specific use case, once we wanted to implement a queue behavior, which was much more complex to implement with Apache Kafka. Besides, the RabbitMQ empower the IIoT Platform to connect to devices with different communication protocols like MQTT and AMQP. Within this implementation, Node-Red consumes the data from the MQTT Message Broker, and then populate a MySQL database. The loading process rewrites all the data in the database. Finally, this “enriched” data might be used for simulation purposes, assisting the decision-making process on validating (re)planning scenario. Figure 3 demonstrates how the technology stack transmits the data. To test the proof of concept IIoT Platform, three use cases are considered in terms of data sources, PLC’s in the shop floor, that are commonly used as way to monitor and control other machines and sensors, two use cases are considered, a PLC device with OPC UA implemented and a PLC without OPC UA. The third case considered an ERP system, a commonly used Information System in the business environment.

2 Topic: Entity whose function is to aggregate and store the various data of the same category

published to the message broker. 3 Sink: IIoT platform data collector, where data is replicated.

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Fig. 3. Component integration diagram

4 Use Cases The evaluation scenarios took place on a real industrial setting in the scope of the project, where three cases were considered for the platform evaluation: i) the first one aimed at validate the communication to a PLC without OPC UA implementation; ii) the second addressed the communication to a PLC supporting the OPC UA protocol, and; iii) the third, intended to assess the interoperability between the proposed platform and the company’s Enterprise Resource System (ERP). In the first setup scenario, the IIoT platform intended to collect some process variables from a reactive oven. This kind of equipment is often used in the footwear industry, whose main function is to dry glue from the soles. For this setup, some sensors were installed on the PLC, in order to get needed data, such as, air velocity, temperatures and humidity. Concretely, it was intended to evaluate the IIoT platform ability to monitor, persist and represent historical data (gathered from the sensors), trace back the production context and, thus, preventing quality issues with the footwear. For the second use case, it was considered a more complex setup including a CNC machine, which is used to cut a variety of materials in different shapes, and due to this fact, the generated data is more complex. Once the CNC machine is already equipped with a PLC implementing OPC-UA protocol, it was possible to retrieve its data model with more 1000 variables, allowing the IIoT platform to monitor and control of the machine. As mentioned above, the third use case aimed at validating the integration of an Information System, specifically, an ERP from a footwear company. This ERP manages the soles and shoes in the shop floor. This integration was used to facilitate the availability of the data to be consumed by other applications. In the use case a simulation model that simulate the production line in the shop floor was the consumer of data originated from the ERP system.

5 Results We share the view that IIoT platforms are an important asset on the decision-making process once it follows a cross-functional approach, connecting several data sources

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within an organization, orchestrating the information flow related to both business and operational processes. Additionally, we intended to achieve an affordable solution for SMEs with highly customized features. The aforementioned use case scenarios were designed to assess the platform flexibility and interoperability, ensuring a vertical integration solution and offering a common enterprise information bus. The IIoT platform was able to operate and deliver information from a bottom-up data flow either scenario or a top-down information flow. From the first use case, we were able to deliver the realtime OEE4 data. The second use case allow a bi-directional information flow, which is possible to compute real-time OEE data and, at the same time, to control the CNC machine due to its OPC-UA readiness. In the third situation it was tested successfully, the integration with an ERP data model. At this level and having access to the current overall data state of the organization, it was possible to perform simulation models offering advanced predicted data and/or validate action plans. The IIoT platform delivers useful data, providing different views on information according to the different needs at the different layers, enabling the digital twin representation. This continuum of real-time data is a benefit resource for SMEs management and business in general.

6 Conclusion In conclusion, an IIoT platform proof of concept that was discussed and success-fully completed. This included the design of an architecture and the definition of a technological stack, delivering an I4.0 vertical integration solution, allowing bi-directional data interoperability from OT systems to the cloud and third-party applications. The IIoT solution was grounded on literature, prototyping, industry specifications and open-source tools, in order to develop an adaptive and affordable IIoT platform with a short implementation period. IIoT platforms prove to be a vehicle capable of bringing innovation from industry 4.0 in a sector that is becoming more dynamic. For future works, it is necessary to focus on security and privacy issues of IIoT setups. Also, link new asset requirements such as plug-and-produce to enable the potential behind I4.0. In relation to the IIoT platforms concept, in our opinion and considering that it is still an area in development, there is still resistance when it comes to its adoption and implementation by the SME industry. However, demystifying its complexity, realizing the functionalities and the impact on the decision support capacity of companies, the importance of IIoT platforms is commonly understood as an important vehicle for the introduction of industry 4.0 in industrial environments. IIoT platforms are the future of the global industry, allowing the encapsulation of the various layers that make up the current industry, leading to the understanding of these layers and promoting their scalability, which is RAMI 4.0 compliant.

4 OEE: Overall Equipment Effectiveness.

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Acknowledgements. This work is financed by National Funds through the Portuguese funding agency, FCT – Fundação para a Ciência e a Tecnologia within project UIDB/50014/2020.

References 1. Lasi, H., Fettke, P., Kemper, H.-G., Feld, T., Hoffmann, M.: Industrie 4.0. WIRTSCHAFTSINFORMATIK 56(4), 261–264 (2014) 2. Schwab, K.: The Fourth Industrial Revolution (2016) 3. VDI STATUS REPORT Reference Architecture Model Industrie 4.0 (RAMMI 4.0), June 2015 4. Rojas, R.A., Ruiz Garcia, M.A.: Implementation of industrial Internet of Things and cyberphysical systems in SMEs for distributed and service-oriented control. In: Matt, D.T., Modrák, V., Zsifkovits, H. (eds.) Industry 4.0 for SMEs: Challenges, Opportunities and Requirements, pp. 73–103. Springer, Cham (2020) 5. Cruz-Cunha, M.M. (ed.): Enterprise Information Systems for Business Integration in SMEs: Technological, Organizational, and Social Dimensions. IGI Global, Hershey (2010) 6. Olson, D.L., Johansson, B., De Carvalho, R.A.: Open source ERP business model framework. Robot. Comput.-Integr. Manuf. 50, 30–36 (2018). https://doi.org/10.1016/j.rcim.2015.09.007 7. Barros, A.C., Azevedo, A., Rodrigues, J.C., Marques, A., Toscano, C., Simões, A.C.: Implementing cyber-physical systems in manufacturing. Presented at the CIE47, Lisbon (2017) 8. Kagermann, H., Wahlster, W., Helbig, J.: Securing the future of German manufacturing industry: recommendations for implementing the strategic initiative industrie 4.0 (2013) 9. Ferreira, F., Faria, J., Azevedo, A., Marques, A.L.: Industry 4.0 as enabler for effective manufacturing virtual enterprises. In: Afsarmanesh, H., Camarinha-Matos, L.M., Lucas Soares, A. (eds.) Collaboration in a Hyperconnected World, vol. 480, pp. 274–285. Springer, Cham (2016) 10. Leveraging industrial software stack advancement for digital transformation. Digital McKinsey 11. Darmois, E., Elloumi, O., Guillemin, P., Moretto, P.: IoT standards – state-of-the-art analysis. In: Vermesan, O., Friess, P. (eds.) Digitising the Industry Internet of Things Connecting the Physical, Digital and Virtual Worlds. River Publishers, Gistrup (2016). ISBN 978-87-9337981-7 12. Gill, H., Baheti, R.: Cyber-physical Systems. In: The Impact of Control Technology. IEEE Control Systems Society (2011) 13. Filipe, F., José, F., Américo, A., Luísa, M.A.: Product lifecycle management enabled by industry 4.0 technology. In: Advances in Transdisciplinary Engineering, pp. 349–354 (2016). https://doi.org/10.3233/978-1-61499-668-2-349 14. Semle, A.: IIoT Protocols to Watch - Kepware Whitepaper (2015) 15. Leki´c, M., Gardaševi´c, G.: IoT sensor integration to node-RED platform. In: 2018 17th International Symposium INFOTEH-JAHORINA (INFOTEH), East Sarajevo, pp. 1–5 (2018) 16. Garg, N.: Apache Kafka. Packt Publishing, Birmingham (2013) 17. Dossot, D.: RabbitMQ Essentials. Packt Publishing, Birmingham (2014) 18. Suehring, S.: MySQL Bible. Wiley, New York (2002) 19. Naqvi, S.N.Z., Yfantidou, S.: Time Series Databases and InfluxDB. Universite libre de Bruxelles (2017) 20. “Grafana.” https://grafana.com/grafana/. Accessed 02 Oct 2020

Existence and Uniqueness for Riemannian Cubics with Boundary Conditions Margarida Camarinha1(B) , F´ atima Silva Leite2,3 , and Peter Crouch4 1

2

University of Coimbra, CMUC, Department of Mathematics, 3001-501 Coimbra, Portugal [email protected] University of Coimbra, Department of Mathematics, 3001-501 Coimbra, Portugal [email protected] 3 Institute of Systems and Robotics, DEEC-UC, 3030-290 Coimbra, Portugal 4 Arlington College of Engineering, University of Texas at Arlington, Arlington, TX 76019-0019, USA [email protected]

Abstract. We study local existence and uniqueness for Riemannian cubics satisfying boundary conditions. We define the biexponential map and use it to relate initial and boundary data. We also describe biconjugate points along cubics by means of the biexponential map.

Keywords: Riemannian cubics points · Bi-Jacobi fields

1

· Biexponential map · Biconjugate

Introduction

Riemannian manifolds, in particular Riemannian Lie groups and symmetric spaces, are the configuration spaces used to model many systems arising in engineering, physics and medicine. Applications such as path planning of the rigid body in robotics, spacecraft control in aeronautics, quantum control in quantum information processing, 3D animation in computer graphics and regression schemes for computational anatomy in medical imaging have been the main motivation for extending interpolation and approximation methods to the Riemannian setting. Several alternative approaches have been investigated in the last decades, some of them using optimal control and calculus of variations (see for instance [1–4,6,9,12,14,18] for an overview of the most recent literature on the subject). Among all, the ones based on the so-called Riemannian cubics are of particular interest to this paper. It is well know that, in Euclidean space, cubic polynomials are the minimizers of the integrated squared norm of the acceleration. This notion can be extended to Riemannian manifolds by considering the stationary paths of a natural Riemannian version of this optimization problem with Lagrangian given by c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 322–331, 2021. https://doi.org/10.1007/978-3-030-58653-9_31

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the squared norm of the covariant acceleration [13,16]. Based on this generalization, P. Crouch and F. Silva Leite developed in [7,8] the so-called dynamic interpolation method, which led to the Riemannian cubic splines. In this paper we address the question of existence and uniqueness of Riemannian cubics under boundary conditions on position and velocity. Although the existence of Riemannian cubics and cubic splines has been thoroughly investigated (see for instance [11] and all the references therein), even today no local results establishing uniqueness of Riemannian cubics with prescribed boundary data are available. The organization of the paper is as follows. In Sect. 2, we describe Riemannian cubics, summarize some of their properties and study local existence and uniqueness of cubics with initial data (Proposition 1). The main results are proved in Sect. 4, where we establish local existence and uniqueness of cubics satisfying boundary conditions. To attain this goal we make use of Proposition 1. A generalization of the exponential map to this new situation, the biexponential map, plays the main role here (Sect. 3). In Sect. 5, the relation between the biexponential map and biconjugate points is also explored.

2

Riemannian Cubics

In what follows M is an n-dimensional smooth manifold equipped with a Riemannian metric < . , . >, ∇ denotes the Levi-Civita connection on M . The covariant derivative of a vector field X along a curve c in M is denoted by DX dt , Di X the ith-order covariant derivative of X by dti , with i = 1, 2, 3, . . ., and the ithi order covariant derivative of c by Ddtic (i ≥ 2). Moreover, R denotes the curvature tensor field and ∇R the covariant  differential of R. The i-tangent bundles T i M = p∈M (Tp M )i of M , i = 1, 2, 3, . . ., are useful along the paper, in particular, the tangent bundles T M , T 2 M and T 3 M . They have the fibre bundle structure defined by the natural projection σ from T i M to M . Let (p, v) and (q, w) ∈ T M , T a positive real number and Ω the set of all C 1 piecewise smooth curves c : [0, T ] → M verifying the boundary conditions c(0) = p,

dc dc (0) = v, c(T ) = q, (T ) = w. dt dt

(1)

The Riemannian cubics are the critical points of the functional defined in Ω by  J(c) =

T

< 0

D2 c D2 c , > dt, dt2 dt2

which are given by the Euler-Lagrange equation   DV D3 V , V V = 0, +R dt3 dt

(2)

(3)

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with V the velocity vector field along c ∈ Ω. Trajectory planning in robotics, with SO(3) as configuration space, was the main motivation for the first studies of this variational problem (see [7,8,16] for details). Riemannian cubics can be seen as a second order version of geodesics, being also called 2-geodesics or biharmonic curves (see for instance [15,17]). Geodesics are, in particular, cubics; however, our interest is to study proper cubics, that is, cubics that are not geodesics. Clearly the initial data are the key to make this distinction. A cubic c is uniquely defined by its initial conditions c(0) = p,

D2 c D3 c dc (0) = v, (0) = y, (0) = z, 2 dt dt dt3

(4)

with (p, v, y, z) ∈ T 3 M . Such a cubic will be denoted by cp,v,y,z . It is important to observe that cubics depend on the choice of the parametrization. Linear parametrizations preserve cubics, that is, the cubic cp,v,y,z satisfies the property cp,v,y,z (as) = cp,av,a2 y,a3 z (s), a ∈ IR.

(5)

Simple examples of cubics are cubic reparametrizations of geodesics which have the initial tangent vectors in (4) with the same direction. The property (5) is the key to establish the following local existence and uniqueness conditions for Riemannian cubics, based on existence and uniqueness of initial value problem. Proposition 1. Let (U, ϕ) be a coordinate system in M , q ∈ U and T a positive real number. Then there exists a neighborhood D of q, D ⊂ U , and a real number  > 0 such that, if p ∈ D and v, y, z ∈ Tp M with  v < ,  y <  and  z < , then there exists an unique cubic c = cp,v,y,z : (−T, T ) → U satisfying the initial conditions (4). Furthermore, this cubic depends smoothly on the point (p, v, y, z). Proof. Consider a coordinate system (U, ϕ) in M and the corresponding coor˜ , ϕ) dinate system (U ˜ in T 3 M . The Eq. (3) can be written as the following system ˜, on U ⎡ ⎤ ⎡ ⎤ c V ⎥ ⎥ ⎢ d ⎢ Y − τc (V, V ) ⎢V ⎥ = ⎢ ⎥, (6) ⎦ Z − τc (V, Y ) dt ⎣ Y ⎦ ⎣ −τc (V, Z) + Rc (Y, V, V ) Z where τp : (Tp M )2 → Tp M and Rp : (Tp M )3 → Tp M are the operators defined by the Christoffel symbols and the curvature tensor field, for each p ∈ M . ˜ , there exists a neighborhood B of (q, 0, 0, 0), B ⊂ U ˜, For (q, 0, 0, 0) ∈ U and δ > 0 such that for each (p, v, y, z) ∈ B, there exists a unique cubic cp,v,y,z : (−δT, δT ) → U satisfying the initial conditions (4) and depending smoothly on the point (p, v, y, z) ∈ B. Now, we take a neighborhood D of q and a number θ > 0 such that the open set B(D, θ) = {(p, v, y, z) ∈ T 3 M : p ∈ D,  v < θ,  y 
0 such that, for each (p0 , v0 , y0 , z0 ) ∈ B, there exists a unique cubic cp0 ,v0 ,y0 ,z0 : (−δ, δ) → M . Let C be a neighborhood of (y, z) in (Tp M )2 such that {(p, v)} × C ⊂ B, and define the map biexpt(p,v) , 0 < t < δ, which assigns to each point (y0 , z0 ) ∈ C the point biexpt(p,v) (y0 , z0 ) = (c(t), dc dt (t)) ∈ T M , where c = cp,v,y0 ,z0 . This map is well defined and smooth on C. The map biexpt(p,v) establishes the connection between the initial conditions (4) and the boundary conditions (1) and permits to ensure local existence and uniqueness of cubics satisfying the boundary conditions. This result follows straightforwardly from Proposition 1 and the inverse function theorem. Proposition 2. Let (p, v, y, z) ∈ T 3 M , biexpT(p,v) the biexponential map defined in a neighborhood D of (y, z) in (Tp M )2 and (q, w) = biexpT(p,v) (y, z), with T > 0. If the biexponential map biexpT(p,v) is not critical at (y, z), then there exist two neighborhoods W1 and W2 of (p, v) and (q, w), respectively, and a neighborhood V of (p, v, y, z) such that: 1. For each point (p1 , v1 ) ∈ W1 , (q1 , w1 ) ∈ W2 there exists a unique cubic b satisfying db db b(0) = p1 , b(T ) = q1 , (0) = v1 , (T ) = w1 , (7) dt dt db D2 b D3 b (8) (0), 2 (0), 3 (0)) ∈ V. dt dt dt 2. This cubic b depends smoothly on the points (p1 , v1 ) and (q1 , w1 ), in the sense that the map [0, T ] × W1 × W2 → M (t, (p1 , v1 ), (q1 , w1 )) → b(t) (b(0),

is smooth. 3. biexpT(p,v) maps an open set C ⊂ D in (Tp M )2 diffeomorphically onto an open set Z ⊃ W2 in T M .

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Now, we are interested in establishing conditions under which the biexponential map biexpt(p,v) at (y, z) is not critical at (y, z), for t > 0. From now on, for the sake of simplicity, we denote biexpt(p,v) by biexp. We define the covariant variation of biexp at (y, z), which we denote by δ(y,z) biexp, as the linear map from (Tp M )2 to (Tq M )2 given by δ(y,z) biexp = ψ ◦ biexp∗|(y,z) , where (q, w) = biexpt(p,v) (y, z) and ψ is the canonical isomorphism between T(q,w) T M and (Tq M )2 defined by the Levi-Civita connection (see for instance [19] for more details). The value of the covariant variation δ(y,z) biexp at the vector (x1 , x2 ) in (Tp M )2 may be expressed as δ(y,z) biexp(x1 , x2 ) = (

D2 α ∂α (0, t), (0, t)), ∂r ∂r∂t

where α is the one-parameter variation of the cubic cp,v,y,z that verifies 1. αr is a cubic, r ∈ (−δ, δ); ∂α (r, 0) = v, r ∈ (−δ, δ); 2. α(r, 0) = p, ∂t 3 2 D α D α 3. (0, 0) = y, (0, 0) = z; ∂t2 2 ∂t3 D3 α DD α D 4. (0, 0) = x1 , (0, 0) = x2 . 2 ∂r ∂t ∂r ∂t3 We call bi-Jacobi variation to such a variation of c. The biexponential map is not critical at (y, z) if and only if the covariant variation δ(y,z) biexp is bijective. Now it is enough to evaluate explicitely δ(y,z) biexp so that we may obtain conditions ensuring its bijectivity. As before, we denote D2 c briefly by c the cubic cp,v,y,z and by V , Y and Z the vector fields dc dt , dt2 and 3 D c 1 2 2 dt3 along c. Let (x , x ) ∈ (Tp M ) and α be the variation of c defined as before by conditions (1)–(4). We also employ the following notation. δc(t) =

∂α D D3 α D2 α D D2 α (0, t), δZ(t) = (0, t). (0, t), δV (t) = (0, t), δY (t) = ∂r ∂r∂t ∂r ∂t2 ∂r ∂t3

Since αr is a cubic, it verifies Eq. (3). So, we get immediately that ∂α D4 α D2 α ∂α (r, t)) (r, t), r ∈ (−δ, δ). (r, t) = −R( (r, t), ∂t4 ∂t2 ∂t ∂t If we differentiate this equation, we see that D D4 α (0, t) = −(∇δc(t) R)(Y (t), V (t))V (t) − R(δY (t), V (t))V (t) ∂r ∂t4 −R(Y (t), δV (t))V (t) − R(Y (t), V (t))δV (t).

(9)

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Now, if we compute the covariant derivatives of the vector fields δc, δV , δY and δZ along c, we obtain the following differential system. D δc = δV dt D δV = δY − R(δc, V )V dt D δY = δZ − R(δc, V )Y dt

.

(10)

D δZ = −R(δc, V )Z − (∇δc R)(Y, V )V dt −R(δY, V )V + 2R(δV, Y )V − R(δV, V )Y According to the previous notation, the value of the covariant variation δ(y,z) biexp at (x1 , x2 ) in (Tp M )2 may be expressed as δ(y,z) biexp(x1 , x2 ) = (δc(t), δV (t)), where (δc(t), δV (t)) is given by the solution of the differential system (10) verifying the initial conditions δc(0) = 0, δV (0) = 0, δY (0) = x1 , δZ(0) = x2 .

(11)

Let us consider an orthonormal parallel frame field {X1 , . . . , Xn } along c. Making use of the same notation for vector fields along c and their coordinate vectors in the previous frame field, the system (10) may be written as ⎡ ⎤ ⎡ 0 δc 0 In d ⎢ δV ⎥ −R(V, V ) 0 In ⎢ ⎣ δY ⎦ = ⎣ −R(V, Y ) 0 0 dt δZ −(∇R)(Y, V, V ) − R(V, Z) 2R(Y, V ) − R(V, Y ) −R(V, V )

⎤⎡ ⎤ 0 δc 0 ⎥ ⎢ δV ⎥ . In ⎦ ⎣ δY ⎦ 0 δZ

(12)

According to Proposition 1 the biexponential map at (p, 0) is defined for an arbitrary value of t. With initial data (p, 0, 0, 0) the cubic c(p,0,0,0) is just the point p and the system (12) reduces to ⎡ ⎡ ⎤ ⎤ δc

δc ⎢ ⎥ ⎥ d ⎢ ⎢ δV ⎥ = 0 I3n ⎢ δV ⎥ . (13) 0 0 ⎣ δY ⎦ dt ⎣ δY ⎦ δZ δZ Thus we conclude that the map δ(y,z) biexp is bijective. It follows that Proposition 2 holds. Corollary 1. Let p ∈ M and T > 0. There exists a neighborhood W of (p, 0) in T M and a positive real number  so that: 1. For each points (p1 , v1 ) and (q1 , w1 ) ∈ W , there exists a unique cubic c such that dc dc (0) = v1 , (T ) = w1 , c(0) = p1 , c(T ) = q1 , dt dt

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D2 c D3 c (0) < ,  3 (0) < . 2 dt dt 2. This cubic c depends smoothly on the points (p1 , v1 ) and (q1 , w1 ). 3. biexp = biexpT(p,0) maps the open set B (0, 0) = {(y, z) : y < , z < } in (Tp M )2 diffeomorphically onto an open set Z ⊃ W in T M . 

Proof. Since the differential of biexp at (0, 0) is bijective, Proposition 2 holds for (p, v, y, z) = (p, 0, 0, 0). The cubic c(p,0,0,0) reduces to the point p, W1 and W2 are both neighborhoods of (p, 0) in T M and V is a neighborhood of (p, 0, 0, 0). So, we may choose W = W1 ∩ W2 , a neighborhood U of p and  > 0 such that V = B(U, ) = {(p1 , v1 , y1 , z1 ) : p1 ∈ U, v1  < , y1  < , z1  < } and the result follows. 

5

Biconjugate Points and the Biexponential Map

Bi-Jacobi fields along a cubic c are the variational vector fields obtained by moving cubics around c. If there exists a family of cubics through c while also satisfying the boundary conditions (1), then the points in T M which give rise to these boundary conditions play an important role in the study of the minimizing properties of cubics (see [5] for more details). The points t1 and t2 are said to be biconjugate along a cubic c, t1 , t2 ∈ [0, T ], t1 = t2 , if there exists a non-zero bi-Jacobi field W along c satisfying W (t1 ) = 0,

DW DW (t1 ) = 0, W (t2 ) = 0, (t2 ) = 0. dt dt

The multiplicity of t1 and t2 as biconjugate points is the dimension of the vector space consisting of all such bi-Jacobi fields. We may also say that the points dc (c(t1 ), dc dt (t1 )) and (c(t2 ), dt (t2 )) are biconjugate along c, if there is no ambiguity dc (there exists ambiguity if the points (c(t1 ), dc dt (t1 )) and (c(t2 ), dt (t2 )) coincide). Proposition 3. [5] If t = 0 and t = T are not biconjugate along c, then any bi-Jacobi field along c is uniquely determined by its values and its covariant derivative values at t = 0 and t = T . Now, we shall interpret biconjugate points in terms of the biexponential map. We shall assume that the map biexpt(p,v) , which we will denote briefly by biexpt , is defined in a neighborhood of (y, z), for each t ∈ [0, T ]. Proposition 4. The points t = 0 and t = t0 , t0 ∈ (0, T ], are biconjugate along c if and only if the map biexpt0 is critical at (y, z). Proof. We suppose that the biexponential map biexpt0 is critical at (y, z), that is, the map δ(y,z) biexpt0 is not bijective. Thus, there exists (x1 , x2 ) ∈ (Tp M )2 ,

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(x1 , x2 ) = (0, 0), such that it is possible to construct a bi-Jacobi variation α of c satisfying ∂α (s, 0) = v, s ∈ (−, ) α(s, 0) = p, ∂t D2 α D3 α (0, 0) = y, (0, 0) = z, ∂t2 ∂t3 D D2 α D D3 α (0, 0) = x1 , (0, 0)) = x2 , 2 ∂s ∂t ∂s ∂t3

(14)

∂α D2 α (0, t0 ) = 0, (0, t0 ) = 0. ∂s ∂s∂t We consider the bi-Jacobi field X along c defined by the bi-Jacobi variation α. DX Equations (14) imply that X(0) = X(t0 ) = DX dt (0) = dt (t0 ) = 0, but X is not identically zero, since D2 X D3 X 1 (0) = x and (0) = x2 . dt2 dt3 Hence t = 0 and t = t0 are biconjugate along c. Now, suppose that the map biexpt0 is not critical at (y, z). Consider 2n linearly independent vectors (x1i , x2i ) ∈ (Tp M )2 , i = 1, . . . , 2n. It is possible to construct the bi-Jacobi variations αi as above, for each (x1i , x2i ). The correi sponding bi-Jacobi fields X i along c verify X i (0) = 0 and DX dt (0) = 0, for each i = 1, . . . , 2n and DX i (t0 )), i = 1, . . . , 2n, (X i (t0 ), dt are linearly independent. So, no non-trivial linear combination of (X i (t),

DX i (t)), i = 1, . . . , 2n dt

can vanish and it follows that the 2n bi-Jacobi fields Xi , i = 1, . . . , 2n, are linearly independent. Since the dimension of the vector space Jc (0), of the biJacobi fields X along c which verify X(0) = 0 and DX dt (0) = 0 is 2n, clearly no non-zero bi-Jacobi field along c vanishes at both t = 0 and t = t0 , that is, t = 0  and t = t0 are not biconjugate along c. Proposition 4 shows how biconjugate points are related with boundary data conditions for existence and uniqueness of cubics. Now, suppose that the points t = 0 and t = T are not biconjugate along c. According to Proposition 3, a bi-Jacobi field is determined by its values and its covariant derivative values at t = 0 and t = T . Thus, it is natural to be able to construct a bi-Jacobi field in terms of these values. We present such a construction.

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Let w0 , z0 ∈ Tp M and w1 , z1 ∈ Tq M . Since t = 0 and t = T are not biconjugate along c, we know, from the previous result, that Proposition 2 holds. Hence, let W1 and W2 be the neighborhoods in T M of (p, v) and (q, w), respectively, and  the positive real number, all of them given by that proposition. Consider the curves a : (−, ) −→ W1 and b : (−, ) −→ W2 verifying a(0) = (p, v), ψ(

db da (0)) = (w0 , z0 ), b(0) = (q, w), ψ( (0)) = (w1 , z1 ), dr dr

where ψ is the linear isomorphism between T(q,w) T M and (Tq M )2 defined as before. According to Proposition 2, we may construct a bi-Jacobi variation α of c, by setting αr the cubic verifying (αr (0),

dαr dαr (0)) = a(r), (αr (T ), (T )) = b(r), dt dt

for each r ∈ (−, ). The variational vector field W associated to α corresponds to the bi-Jacobi field along c verifying the boundary conditions W (0) = w0 , W (T ) = w1 ,

6

DW DW (0) = z0 , (T ) = z1 . dt dt

Conclusions

In this paper, we obtain local existence and uniqueness conditions for Riemannian cubics satisfying boundary conditions. For this purpose we consider boundary data with sufficiently short length. In a future work we aim to study Riemannian cubics with more general boundary data. The biexponential map has an important role in this study since it gives a correspondence between initial and boundary data. The relation between this map, biconjugate points and local minimizing properties of Riemannian cubics is far from being explored. A global analysis of the existence and multiplicity of Riemannian cubics has been developed in R. Giamb` o et al. ([10]), but from a local viewpoint there is still much to be understood. There is a renewed interest in designing geometric variational integrators for optimal control problems with boundary conditions [6]. The correspondence between initial and boundary data is useful in the construction of these integrators for optimal control of fully actuated mechanical systems. In the future we are interested in developing similar methods for optimal control of nonholonomic systems. Acknowledgments. The research of M. Camarinha was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. The second author acknowledges Funda¸ca ˜o para a Ciˆencia e a Tecnologia (FCT) and COMPETE 2020 program for the financial support to the project UIDB/00048/2020.

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References 1. Abrunheiro, L., Camarinha, M., Clemente-Gallardo, J., Cuch´ı, J.C., Santos, P.: A general framework for quantum splines. Int. J. Geom. Methods Mod. Phys. 15(9), 1850147 (2018) 2. Arroyo, J. , Garay, O. J., Menc´ıa, J. J.: Unit speed stationary points of the acceleration, J. Math. Phys. 49(1), 013508, 16 pp. (2008) 3. Batzies, E., H¨ uper, K., Machado, L., Silva Leite, F.: Geometric mean and geodesic regression on Grassmannians. Linear Algebra Appl. 466, 83–101 (2015) 4. Bloch, A., Camarinha, M., Colombo, L. J.: Dynamic interpolation for obstacle avoidance on Riemannian manifolds. Int. J. Control, 1–13 (2019).https://doi.org/ 10.1080/00207179.2019.1603400 5. Camarinha, M., Silva Leite, F., Crouch, P.: On the geometry of Riemannian cubic polynomials. Differ. Geom. Appl. 15, 107–135 (2001) 6. Colombo, L., Ferraro, S., de Diego, D.M.: Geometric integrators for higher-order variational systems and their applications to optimal control. J. Nonlinear Sci. 26(6), 1615–1650 (2016) 7. Crouch, P., Silva Leite, F.: Geometry and the dynamic interpolation problem. In: Proceedings of American Control Conference, Boston, pp. 1131–1137 (1991) 8. Crouch, P., Silva Leite, F.: The dynamic interpolation problem on Riemannian manifolds, Lie groups and symmetric spaces. J. Dyn. Control Syst. I(2), 177–202 (1995) 9. Gay-Balmaz, F., Holm, D.D., Meier, D.M., Ratiu, T.S., Vialard, F.: Invariant higher-order variational problems II. J. Nonlinear Sci. 22(4), 553–597 (2012) 10. Giamb` o, R., Giannoni, F., Piccione, P.: Optimal control on Riemannian manifolds by interpolation. Math. Control Signals Syst. 16, 278–296 (2002) 11. Heeren, B., Rumpf, M., Wirth, B.: Variational time discretization of Riemannian splines. IMA J. Numer. Anal. 39(1), 61–104 (2019) 12. Hussein, I.I., Bloch, A.M.: Dynamic coverage optimal control for multiple spacecraft interferometric imaging. J. Dyn. Control Syst. 13(1), 69–93 (2007) 13. Jiang, G.Y.: 2-harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A 7, 389–402 (1986) 14. Machado, L., Silva Leite, F., Krakowski, K.: Higher-order smoothing splines versus least squares problems on Riemannian manifolds. J. Dyn. Control Syst. 16(1), 121– 148 (2010) 15. Montaldo, S., Oniciuc, C.: A short survey on biharmonic maps between Riemannian manifolds. Rev. Union Mat. Argent. 47(2), 1–22 (2006) 16. Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inform. 6, 465–473 (1989) 17. Popiel, T.: Higher order geodesics in Lie groups. Math. Control Signals Syst. 19, 235–253 (2007) 18. Pottmann, H., Hofer, M.: A variational approach to spline curves on surfaces. Comput. Aided Geom. Des. 22(7), 693–709 (2005) 19. Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)

Home Energy Management System in an Algarve Residence. First Results A. Ruano1,2(B)

, K. Bot1

, and M. Graça Ruano1,3

1 CINTAL, Universidade do Algarve, 8005 Faro, Portugal

[email protected] 2 IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal 3 CISUC, University of Coimbra, Coimbra, Portugal

Abstract. Home Energy Management Systems (HEMS) are becoming progressively more researched and employed to invert the continuously increasing trend in (electrical) energy consumption in buildings. One of the critical aspects of any HEMS is the real-time monitoring of all variables related to the management system, as well as the real-time control of schedulable electric appliances. This paper describes a data acquisition system implemented in a residential house in the South of Portugal. With the small amount of data collected, a Radial Basis Function (RBF) model, designed by a Multi-objective Genetic Algorithm (MOGA) framework, to forecast total electric consumption was developed. Results show that, even with these little data, the model can be used in a predictive control scheduling mechanism for HEMS. Keywords: Home energy management systems · Electric energy consumption forecasting · Neural networks · Multi-objective optimization · IoT acquisition systems · Non-intrusive load monitoring

1 Introduction The goal of a Home Energy Management System (HEMS) is to manage the flow of electricity efficiently in the house, to reduce the electricity consumption (and consequently the energy bill) while increasing or maintaining the comfort of its occupants. Despite the substantial interest of the research community, due to the complexity and diversity of the systems, as well as the use of suboptimal control strategies, energy consumption is still higher than necessary, and users are unable to yield full comfort in their homes [1]. Energy monitoring is a key point of a HEMS; it can be done installing measuring devices at every load of interest or using Non-Intrusive Load Monitoring (NILM) methods, which desegregate the overall electricity usage, using a measure of the load at the utility service entry [2]. The authors would like to acknowledge the support of Programa Operacional Portugal 2020 and Operational Program CRESC Algarve 2020 grant 01/SAICT/2018. Antonio Ruano also acknowledges the support of Fundação para a Ciência e Tecnologia grant UID/EMS/50022/2020, through IDMEC, under LAETA. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 332–341, 2021. https://doi.org/10.1007/978-3-030-58653-9_32

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A first step in reducing energy use is to create awareness of the energy consumption rate among the occupants. This can be done much more effectively using the disaggregation of energy achieved by the NILM algorithms. Classifying the appliances as schedulable or non-schedulable enables the use of scheduling algorithms to improve the efficiency of a HEMS. Assuming that renewable sources of energy and/or battery storage are available in the home, several Demand Response strategies can be used to further improve the efficiency of a HEMS, such as time-of-use pricing, critical peak pricing, real-time pricing and peak-time pricing. Furthermore, if forecasts of the consumption profiles as well as electrical energy data are available, Model-Based Predictive Control (MBPC) can be employed to derive optimal schedules for the schedulable appliances (for each HVAC systems), as well as to manage efficiently the flow of electricity between PV installation, battery storage, grid and house, in such a way that the electricity usage or its bill will be minimized. Recent approaches of the use of MBPC in HEMS can be found in [3–5]. All these aspects will be considered in a Portuguese FCT-funded project, NILMforIHEM, which is currently being executed. This paper aims to introduce the first steps of the project, having as objectives to describe the case study considered, comprised of the selected home and the technical description of the acquisition system implemented, and to present the model developed to forecast the total electricity consumption of the household, based on the preliminary data obtained,. The forecasting model developed is a RBF neural network, designed using by a Multi-Objective Genetic Algorithm (MOGA) framework. The case study is described in Sect. 2. Section 3 describes the model design methodology and the results obtained are discussed in Sect. 4. Conclusions and guidelines for future work are drawn in Sect. 5.

2 Case Study Description This work uses data collected from a residential house, situated in Gambelas, Faro, in the south of Portugal. It is a detached house, with two floors and with 20 different spaces (including garden, halls, and so on). The house has a PV installation, composed of 20 Sharp NU-AK panels [6], arranged in two strings, each panel with a maximum power of 300 W. The inverter is a Kostal Plenticore Plus converter (KI) [7], which also controls a BYD Battery Box HV H11.5 (with a storage capacity of 11.5 kWh) [8]. Several electrical appliances exist in this house, and a json file was created according to the format used by the NILM Toolkit [9]. The house electric panel is a Schneider panel consisting of 16 monophasic circuit breakers, plus a triphasic one. The house also has available a few TP-Link HS100 Wi-Fi Smart Plugs (SP) [10], one Intelligent Weather Station (IWS) [11], and a few Self-Powered Wireless Sensors (SPWS) for measuring room climate variables [12]. An acquisition system was afterwards implemented to monitor several electric variables. The data that will be used for NILM identification is supplied by a Carlo Gavazzi (EM340) 3 phase energy meter [13]. This meter is a class X certificated device, and electrical measurement is done using a 2 wires Modbus RTU connection. EM340 supplies 45 different electric variables, sampled at 1 Hz. Additional electric variables are

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measured for every circuit breaker to provide approximate ground truth for the NILM identification. The measurement devices, in this case, are Circutor Wibeees (WB) [14], which are plug and play wireless devices to measure electric consumption. Each one provides a hotspot to perform the first configuration using a mobile app from the manufacturer. By default, a WB sends data acquired to a free manufacturer web service. This behaviour can be disabled, and the data acquired is still available using an internal web interface/service or using the Modbus IP protocol. The devices use Hall Effect technology, and, because of that, some calibrations are required for correct measurements. Measurements of voltage, current, frequency, active reactive and apparent power, power factor, active inductive reactive and capacitive reactive energy are obtained every second for the 16 monophasic circuit breakers, the same number for each phase of the triphasic one, together with totalized values. In total, 198 variables are sampled by the WBs every second. Please note that as the measurement devices are not synchronized, the instants of time for each breaker are different. Variables related with the energy produced by the PV, stored in the battery and injected in the grid are available either from the inverter or from a Kostal smart energy meter (KEM) [15]. Measurements of home electrical consumption are also available in the inverter. In total, 21 variables are obtained by KEM and 47 by KI, at a sampling interval of 1 min. The data access is done using a cable IP network using the Modbus IP protocol. Smart Plugs are used for on/off control of some equipment. Additionally, they allow sockets belonging to the same CB to be measured individually. This type of devices connects to an existing wireless network using an initial access point and a manufacturer mobile app. They can be read/controlled using a cloud API or directly using an internal web service, which is the case here. At present 3 SPs are employed, enabling the measurement of 6 variables every second for each plug. The IWS a device that measures the air temperature and relative humidity, and global solar radiation, and predicts their evolution within a self-defined prediction horizon. To enable these forecasts, a two stages strategy is employed. When small measured data are available, a nearest neighbor algorithm is employed. When enough data are available, neural networks predictive models are automatically off-line designed and uploaded to the IWS, for real-time use. More details of the IWS can be found in [11]. Finally, SPWS devices are used for measuring room data, such as air temperature and relative humidity, status (open/close) of doors and windows, walls temperature, light and room movement. They are Ultra-Low-Power devices and communicate via ISM radio band working on 2.4 GHz or 868 MHz frequencies. They will be used to measure thermal comfort for usage in predictive control of some air-conditioners in the house. For more information on the use of SPWS for HVAC predictive control, please see [16]. Gateways and a Technical Network are responsible for the data transmission from/to the measurement devices. A technical IP-cabled and a wireless network have been created using a network router located inside an extension of the electric board case. This router separates the home network from the technical network. All devices, except the SPs and SPWS, are connected to that network. To perform the data acquisition from the existing devices EasyGateway devices are used. EasyGateway [12] is a fault-tolerant IoT gateway that supports a variety of reception/acquisition protocols such as Modbus, SNMP, Easy modules and serial http, as well as a set of Data Delivered Connectors (DDC)

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commonly used on IoT environments, such as Mqtt, and Ampq. To try to guarantee the acquisition rates required, five of these gateways are used within the electric board case, and the WBs, EM340, KI and KEM measurements were distributed among them. The weather station has its own internal gateway. An additional EasyGateway is located on a centralized position in the house, to enable communications of the SPWS. The data transmission of the EasyGateways is always performed at a 1-min base, which means that the data acquired by the measurement devices related to each gateway are packed and transmitted at that rate. The gateways can also send data to up to three different DDCs. Here, 2 DDCs and 2 IoT Platforms are used. One platform is used inside home, employing an EasyMqs DDC and another, using a Generic Ampq DDC, is in the cloud. The same IoT platform is used inside home and in the cloud. It receives data from the configured message queue servers. The data arrived pass by a set of configured plug-ins for each type of entities configured on the platform. The application provides a web page where the end-user can configure a set of definitions related to data storage and management. It also allows data visualization using plots grouped by sensors category, and data download in 4 standard formats, csv, xlsx, mat and npz. For a more detailed description of the IoT platform, please see [12]. A diagram of the acquisition system is presented in Fig. 1. On the right side of the figure images of the IoT platform are displayed.

Fig. 1. Data acquisition diagram.

3 Home Consumption Model Design Data collected from 27 November 2019 to the 31st of January was available for model design. The electric data employed was the active power, obtained from the EM340 device, which has been averaged in 15 min periods. This time step is employed due

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to the technical requirements for interchanging energy information between prosumer and energy suppliers [17] in the Portuguese market. In addition to electric power, two exogenous variables were employed: a codification of each day, within a week, considering holidays and their position within the week [18], and the occupation within the period considered. Future work will also employ external weather data. The objective of the model is to forecast the home electrical consumption, within a period of 12 h (48 steps-ahead). For such, a 1-step model will be designed, and the forecasts within the desired Prediction Horizon (PH) are obtained in a multi-step fashion (Fig. 2).

Fig. 2. Data for model design: top) active power, bottom left) day code, right) occupation

3.1 Methodology for Model Design The models employed are RBF neural networks, in a Nonlinear AutoRegressive with eXogenous inputs (NARX) configuration. As mentioned above, two exogenous variables (day code and occupation) and their delays are used as inputs to the RBF, together with delays of the modelled variable (electric power). For data selection and dataset construction, the ApproxHull algorithm proposed in [19] is employed. ApproxHull is an incremental randomized approximate convex hull algorithm applicable to high dimension data that treats memory and computation time efficiently. The convex hull vertices

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obtained are compulsorily introduced in the training set so that the model can be designed with data covering the whole operational range. According to user partition preferences, the rest of the training set is obtained by a random selection of the rest of the design data, as well as the testing and the validation sets. These data sets are supplied to a Multi-Objective Genetic Algorithm (MOGA) design framework. According to the user-specified set of objectives (which will be minimized or met as restrictions), range of neurons and range of inputs, the genetic algorithm part of MOGA will search for the best set of inputs and neurons of model individuals that solve the optimization problem. Each model in the population is trained by an improved version of the Levenberg-Marquardt algorithm. After MOGA execution a model is selected from the non-dominated solutions obtained, considering the values of the objectives and possible additional performance criteria. For a detailed explanation of the MOGA framework please consult [20]. In this work, the objectives to minimize are the RMSEs of the training set (εtr ), the testing set (εte ), the model complexity (O) and the forecasting error (εp ). This last criterion is useful as we intend to design a predictive model and uses for its computation a subset (D), with p data points, of the existing time-series. It is computed as: ε(D, PH ) = ⎡ ⎢ ⎢ E(D, PH ) = ⎢ ⎣

PH 

RMSE(E(D, PH ), i)

i=1

e[1, 1] e[2, 1] .. .

e[1, 2] e[2, 2] .. .

··· ··· .. .

e[1, PH ] e[2, PH ] .. .

      e p − PH , 1 e p − PH , 2 · · · e p − PH , PH

(1) ⎤ ⎥ ⎥ ⎥ ⎦

(2)

In this work, MOGA is executed with 100 generations, population size of 100, proportion of random emigrants of 0.10 and a crossover rate of 0.70. The training set is used for model parameter estimation, the test set for early-stopping and the validation set used in final model selection, using its RMSE (εv ). Each model design involves typically two iterations of MOGA. The first uses an unconstrained approach, while in the second some objectives are used as restrictions, taking into consideration the unconstrained results.

4 Results and Discussion This section presents the data concerning the data set creation and model design, as well as the performance of the forecasting model. 4.1 ApproxHull During the available 66 days of data, in three periods there was lack of data: 18-Dec-2019, 14:37:30 to 17:52:30, 13-Jan-2020, 12:52:30 to 16:07:30 and 18-Jan-2020, 02:37:30 to 12:52:30. Therefore, fours periods of data were available for model design. Additionally,

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as we shall use lags of power consumption immediately before the current sample (20), lags centred on the sample 24 h before (9) and one week before (9), the largest lag is 676. As the four periods of available data had 1978, 2475, 718 and 587 samples, only the first two were employed. As the first 676 cannot be used from each period (they are the necessary lags), 3005 records were employed by ApproxHull. 802 convex hull vertices were found and incorporated in the training set. As we considered 60%, 20% and 20% of the data for training, testing and validation, these three sets had 1803, 601 and 601 samples, respectively. All the data are scaled in the range [−1, 1]. 4.2 Moga As the work aims to use the model for forecasting, a prediction set D was also supplied to MOGA, comprising data from 21-Dec-2019, 20:52:30 to 05-Jan-2020, 03:37:30. In the first MOGA run the 4 objectives were minimized, and the ranges of neurons and inputs allowed were [2, 10] and [1, 20], respectively. There were 315 models in the non-dominated set. The minimum values of εtr , εte , εv and εp are shown in the first line of Table 1. Analysing the results, model 3163 was chosen. It has a structure shown in Eq. 3, which means that this model did not employ any of the exogenous variables and incorporated lags of the modelled variable around 24 h and one week before. The errors obtained by this model are shown in the 1st line of Table 2. ⎛ ⎞ y(k − 1), y(k − 2), y(k − 3), y(k − 5), y(k − 6), y(k − 11), y(k) = f ⎝ y(k − 13), y(k − 14), y(k − 15), y(k − 17), y(k − 93), y(k − 96), ⎠, (3) y(k − 97), y(k − 670), y(k − 673), y(k − 675), y(k − 676)

Table 1. Non-dominated sets Problem εtr

εte

εva

εp

1st

0,152 0,139 0,147 9,997

2nd

0,156 0,139 0,146 9,836

Table 2. Selected models result. Problem Features Neurons εtr

εte

εva

εp

1st

17

10

0,163 0,147 0,1784 10.62

2nd

19

9

0,163 0,145 0,1175 9,83

The predictive performance of the chosen model was evaluated in a different segment of the active power time series, with samples from 8-Dec-2019, 09:52:30 to 18-Dec2019, 02:22:30. Figure 3a) illustrates the evolution of RMSE, over the prediction horizon, for the period considered, and the 1-step-ahead forecast, compared with the real values

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of active power. After having analyzed the results, MOGA was executed again, this time using the values of 0.17 and 0.16 as goals for εtr and εte , respectively, and reducing the maximum allowable value for input terms to 18. The non-dominated set had 256 models, and the number of prefered models (that also met the restrictions) was 127. The 2nd row of Table 1 illustrates performance values for the prefered set. It can be concluded that a smaller value of the prediction error was obtained. Having analysed the results, model 5371 was chosen. It has a structure very similar to the one chosen in the previous MOGA iteration:

a)

b)

c) Fig. 3. Left: evolution of the RMSE over PH; right: predicted and real active power series. a) 1st MOGA execution; b) 2nd MOGA execution; c) ensemble approach

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⎛

⎞ y(k − 1), y(k − 2), y(k − 3), y(k − 4), y(k − 6), y(k − 11), ⎠ y(k) = f ⎝ y(k − 14), y(k − 92), y(k − 93), y(k − 97), y(k − 98), y(k − 99), y(k − 670), y(k − 671), y(k − 675), y(k − 676)

(4)

The performance results obtained with this model are presented in the 2nd row of Table 2. As before, the forecasting performance was assessed in the same period employed for the previous MOGA iteration. The RMSE evolution and the 1-step-ahead forecast are presented in Fig. 3b). As can be seen, slightly better results are obtained. Finally, an ensemble approach was experimented. Making use of the preferred set, the results were obtained as the median of the values of the preferred set models. The graphs equivalent to Figs. 3a) and b) are presented in Fig. 3c). This last approach significantly improved the forecasting performance.

5 Conclusions This paper describes the first steps of the development of a HEMS for an existing house, in the South of Portugal, under normal usage. The first step to achieve this was the development of an IoT acquisition system and platform, which was the first focus of this paper. With the small amount of data collected so far, a model to forecast the total electric consumption was also developed. Its performance will be improved, as more data become available. Notice that none of the exogenous variables considered, day code and occupation, was selected in the chosen models. Additionally, we assume that the use of weather information, mainly the air temperature, will bring an increased forecasting performance. Subsequently, models to forecast the consumption of schedulable electric appliances will be developed, as well as models for predicting the electric energy produced by the PV installation. These predictive models will be used to design a predictive control HEMS solution, which is the final goal of the current project.

References 1. Pau, G., Collotta, M., Ruano, A., Qin, J.: Smart home energy management. Energies 10, 382 (2017) 2. Ruano, A., Hernandez, A., Ureña, J., Ruano, M., Garcia, J.: NILM techniques for intelligent home energy management and ambient assisted living: a review. Energies 12, 2203 (2019) 3. Wang, J., Li, S., Chen, H., Yuan, Y., Huang, Y.: Data-driven model predictive control for building climate control: three case studies on different buildings. Build. Environ. 160, 106204 (2019) 4. Guo, X., Bao, Z., Yan, W.: Stochastic model predictive control based scheduling optimization of multi-energy system considering hybrid CHPs and EVs. Appl. Sci. 9, 356 (2019) 5. Schmidt, M., Åhlund, C.: Smart buildings as cyber-physical systems: data-driven predictive control strategies for energy efficiency. Renew. Sustain. Energy Rev. 90, 742–756 (2018) 6. Europe Solar-Store: Sharp NU-AK300. https://www.europe-solarstore.com/sharp-nu-ak300. html. Accessed 03 May 2020 7. Kostal: Plenticore Plus. https://www.kostal-solar-electric.com/en-gb/products/hybrid-invert ers/plenticore-plus. Accessed 03 May 2020

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8. Eft-Systems: Battery Box HV. https://www.eft-systems.de/en/TheB-BOX/product/BatteryBo xHV/3. Accessed 03 May 2020 9. Kelly, J., Batra, N., Parson, O., Dutta, H., Knottenbelt, W., Rogers, A., Singh, A., Srivastava, M.: Nilmtk v0. 2: a non-intrusive load monitoring toolkit for large scale data sets: demo abstract. In: Proceedings of the 1st ACM Conference on Embedded Systems for Energyefficient Buildings, pp. 182–183 (2014) 10. Tp-link: Wi-Fi Smart Plug. https://www.tp-link.com/pt/home-networking/smart-plug/hs100/. Accessed 03 May 2020 11. Mestre, G., Ruano, A., Duarte, H., Silva, S., Khosravani, H., Pesteh, S., Ferreira, P.M., Horta, R.: An intelligent weather station. Sensors 15, 31005–31022 (2015) 12. Ruano, A., Silva, S., Duarte, H., Ferreira, P.M.: Wireless sensors and IoT platform for intelligent HVAC control. Appl. Sci. 8, 370 (2018) 13. Carlo Gavazzi Automation Company: EM340 utilises touchscreen technology. https://www. carlogavazzi.co.uk/blog/carlo-gavazzi-energy-solutions/em340-utilises-touchscreen-techno logy. Accessed 03 May 2020 14. Circutor: Consumption analyzers. http://circutor.com/en/products/measurement-and-control/ fixed-power-analyzers/consumption-analyzers. Accessed 03 May 2020 15. Kostal: Kostal Smart Energy Meter. https://shop.kostal-solar-electric.com/en/kostal-smartenergy-meter.html. Accessed 03 May 2020 16. Ruano, A.E., Pesteh, S., Silva, S., Duarte, H., Mestre, G., Ferreira, P.M., Khosravani, H.R., Horta, R.: The IMBPC HVAC system: a complete MBPC solution for existing HVAC systems. Energy Build. 120, 145–158 (2016) 17. Presidência do Conselho de Ministros: Decreto-Lei n.o 162/2019 de 25 de outubro (2019) 18. Ferreira, P.M., Ruano, A.E., Pestana, R., Kóczy, L.T.: Evolving RBF predictive models to forecast the Portuguese electricity consumption. IFAC Proc. Vol. 2 (2009). https://doi.org/10. 3182/20090921-3-TR-3005.00073 19. Khosravani, H.R., Ruano, A.E., Ferreira, P.M.: A convex hull-based data selection method for data driven models. Appl. Soft Comput. 47, 515–533 (2016) 20. Ferreira, P.M., Ruano, E.: Evolutionary Multiobjective neural network models identification: evolving task-optimised models. In: New Advances in Intelligent Signal Processing, pp. 21–53 (2011)

LSLOCK: A Method to Estimate State Space Model by Spatiotemporal Continuity Tsuyoshi Ishizone1(B) and Kazuyuki Nakamura1,2 1 2

Meiji University, 4-21-1, Nakano, Nakano-ku, Tokyo, Japan [email protected] JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, Japan

Abstract. Model estimation from spatio-temporal data is important topic since it helps us to extract useful information from big data in recent years. In this paper, we introduce an estimation algorithm of the linear Gaussian state space model with focusing on the real-time property. The proposed algorithm uses two key ideas, localization and spatial uniformity, to reduce the number of the parameters. Thanks to this, we obtain stable method to estimate the parameters regarding state transition and states. In addition, the proposed algorithm is quicker and more accurate than existing methods, therefore, it suffices the requirement of the rapid response for the alternation of the fields. Keywords: Kalman filter · Online learning Noise reduction · Short-term prediction

1

· Real-time calculation ·

Introduction

A quick feedback for noise reduction and short-term prediction is indispensable in areas such as weather forecasting and adjusting a scanning probe microscope (SPM). In the weather forecast, the engineers need a speedy model construction method for instantaneous forecasting. Since SPMs require adjustment of some machine parameters, a method for real-time adjustment is helpful in carrying out experiments. A number of methods for noise reduction and prediction have been proposed for recent 50 years. In particular, deep learning methods for short-term prediction achieve a breakthrough (e.g. [1,6–9]) since ConvLSTM[7], which is the pioneering work. Although these methods have high predictive performance, these are lack of interpretability due to the complicated nonlinear calculation. In order to guarantee the intepretability, we focus on the state space model, which is widely used in time-series modeling. In particular, the Kalman filter (KF) is a powerful tool to estimate the states in the linear Gaussian scenario. Moreover, an expectation maximization (EM) algorithm functions as a parameter estimator if the parameters are unknown. This method, however, spends a great deal of calculation time and is difficult for application to high-dimensional data. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 342–351, 2021. https://doi.org/10.1007/978-3-030-58653-9_33

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We previously developed two methods, referred to as LLOCK and SLOCK, to estimate the states and the state transition for spatio-temporal data in realtime (see [4]). The former method uses the idea of localization, which means estimation of the elements of the transition matrix in local space. The latter method uses the idea of spatial uniformity, which assumes the elements are spatially uniform in the global space. However, the two methods have a tradeoff relationship. The former method is suitable for data with different dynamics among regions. The latter method is appropriate for data with rapid change in a short time span. On the other hand, the former method is unsuitable for the latter data and the latter method is inappropriate for the former data. This paper, thus, propose a method to overcome the trade-off relationship by combining the two key ideas: localization and spatial uniformity. Applied to two synthetic data, the proposed method was found to be superior to the existing methods and to be efficient in terms of calculation and memory cost. This paper is structured as follows: after a quick review of the Kalman filter, we describe our previously developed and new methods in Sect. 3. Then, we state the experimental results for application to two synthetic data in Sect. 4. Finally, we conclude the present paper in Sect. 5.

2

Kalman Filter

In this section, we begin the state space model, then state the Kalman filter, which is the basis of our method. 2.1

State Space Model

The state space model is a time-series model described by two equations xt = ft (xt−1 ) + vt ,

(1)

yt = ht (xt ) + wt , Nx

Ny

Nx

(2) Ny

and wt ∈ IR represent the state where xt ∈ IR , yt ∈ IR , vt ∈ IR of the system, the observation, the system noise, and the observation noise, respectively, at a given time t. This model considers the observations {yt } are emission from the unknown states {xt }, whose transition is controlled by the transition operator. However, the transition operator ft and the observation operator ht are imperfect, i.e., the model considers the noises vt and wt . In the linear scenario, using ft (xt−1 ) = Ft xt−1 , ht (xt ) = Ht xt , we can obtain xt = Ft xt−1 + vt , yt = Ht xt + wt ,

(3) (4)

from (1) and (2). Suppose the system noise and the observation noise follow zero-mean Gaussian distribution, i.e., vt ∼ N (0, Qt ) and wt ∼ N (0, Rt ), we can obtain the linear Gaussian formulation for the Kalman filter.

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Kalman Filter

The Kalman filter (KF) is the most powerful tool to estimate the states of the state space model in the linear Gaussian scenario (see [5]). In this situation, if the initial state follows a Gaussian distribution, the state xt at a time t given the observations {yu }su=1 also follows a Gaussian distribution. Thus, we can describe conditional distribution p(xt |y1:s ) = p(xt |y1 , · · · , ys ) by the mean and the covariance matrix, which are denoted to xt|s and Vt|s . Thanks to the Gaussianity, we can obtain update formula xt|t−1 = Ft xt−1|t−1 , Vt|t−1 =

Ft Vt−1|t−1 FTt + Qt , Vt|t−1 HTt (Ht Vt|t−1 HTt

(5) (6) −1

+ Rt ) Kt = xt|t = xt|t−1 + Kt (yt − Ht xt|t−1 ),

,

Vt|t = Vt|t−1 − Kt Ht Vt|t−1 ,

(7) (8) (9)

which are called the Kalman filter. Using this, the expectation maximization (EM) algorithm enables us to estimate the parameters θ = (x1|1 , V1|1 , F, H, Q, R) as an approximately most likelihood estimator (see [3]). However, the algorithm needs some iterations and tremendous calculation time for high-dimensional states or observations.

3

Proposal Method

With this section, we review our previously developed method to estimate the parameters in real-time, referred to as linear operator construction with the Kalman filter (LOCK, see detail in [4]). Then, we propose a new method in order to apply to spatio-temporal data, referred to as locally and spatially uniform LOCK (LSLOCK). 3.1

Linear Operator Construction with Kalman Filter (LOCK)

To realize real-time estimation of the parameters, we previously develop the method referred to as linear operator construction with the Kalman filter (LOCK). The method uses three ideas: assumption of observation transition, a time-invariant interval, and an online learning framework. In addition, the method focuses on the estimation problem of the transition matrix Ft . The first idea means we introduce the assumption of the linear Gaussian observation transition yt = Gt yt−1 + ut , ut ∼ N (0, St ), Gt = Ht Ft H− t−1 ,

(10) (11)

− T T T T St = (H− t−1 ) Ft Ht Rt−1 Ht Ft Ht−1 + Ht Qt Ht + Rt ,

(12)

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where for a matrix A ∈ IRn×m , A− denotes to the Moore-Penrose type pseudo inverse matrix. From (3), (4), and (10), we can obtain yt = Ht (Ft xt−1 + vt ) + wt = Gt (Ht−1 xt−1 + wt−1 ) + ut ,

(13)

then, by taking expectation, we can obtain unbiased estimator of the transition matrix ˆ t = H− Gt Ht−1 . F (14) t In addition, from the assumption the transition matrices and the observation matrices are time-invariant in interval τ , we can obtain ˆ t = Yt Y− , Yt = (yt−τ +1 , · · · , yt ), G t−1

(15)

thus, we can obtain the estimation of the transition matrix ˆ t = H− Yt Y− H, F t−1

(16)

through the observation transition. Besides, lending the idea from online learning (e.g. see [2]), ˆ t , −c}, c}, Ft = Ft−τ − η min{max{Ft−τ − F

(17)

where the operators min and max are element-wise, η and c represent the learning rate and cutoff distance, respectively, which control the maximum update amount of the estimated matrix. In [4], we also proposed the two methods, referred to as local LOCK (LLOCK) and spatially uniform LOCK (SLOCK), for spatio-temporal data. Introducing localization and spatial uniformity, respectively, the two methods provided better performance than KF for some synthetic data (see detail in [4]). 3.2

Locally and Spatially Uniform LOCK (LSLOCK)

We propose a new method, referred to as locally and spatially uniform LOCK (LSLOCK), for spatio-temporal data. The proposed method uses the two key ideas: localization and spatial uniformity. The former one means an i-th observation variable at time t is affected only by the i-th neighbor observation variables at time t − 1. For example, if image sequence data are given, an (i, j)-th pixel at time t is represented by regression from the surrounding 3 × 3 pixels at time t − 1 (Fig. 1). The idea, thus, solves the estimation problem of the transition matrices in the local space. The latter one represents the assumption of spatial uniformity of the elements in the local space. For example, in the image sequence data again, the effect from the (i, j)-th pixel to the (i, j + 1)-th pixel is close to that from the (i + 1, j)-th pixel to the (i+1, j +1)-th pixel. In the other words, the effects of same direction in the vicinity of the target pixel are similar. Using this, the method consider the effects of same direction are same in each local space.

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Fig. 1. An example of the idea of localization

For mathematical formulation, we introduce a matrix P ∈ ({0}∪IN)Ny ×({0}∪ IN) , referred to herein as parameter matrix, whose each element represents a parameter number. If two elements of the matrix are same, the corresponding two elements of the observation transition matrix are same, i.e., Ny

Pi1 j1 = Pi2 j2 ⇒ Gi1 j1 = Gi2 j2 .

(18)

In the previous example, for the image sequence data, the elements corresponding to each direction have same number. The only exception is zero; the zero elements of P correspond to the zero element of G. Since using numbers in ascending order is without loss of generality, we consider the maximum number α (≤ Ny2 ) as the number of the unique values, excluding zero, of the matrix. Since the maximum number also corresponds to the number of parameters of the observation transition matrix, the method use a vector θ ∈ IRα , referred to herein as parameter vector, whose i-th element corresponds to the i-th parameter of the observation transition matrix. Using the non-zero elements of the parameter matrix, we define three sets N (i) = {j ∈ INNy | Pij = 0} (∀i ∈ INNy ),

(19)

N2 (i) = {k ∈ INNy | ∃j ∈ N (i), k ∈ N (j)} (∀i ∈ INNy ),

(20)

B(i, j, k) = {l ∈ N (i) | Pjl = k} (∀i ∈ INNy , j ∈ INN2 (i) , k ∈ INα ),

(21)

where INn represents set {1, · · · , n}. Using these sets, the parameter matrix, and the parameter vector, LSLOCK estimates the observation transition matrix by  (yt−1 )l , (22) (Ξi )jk = l∈B(i,j,k)

θ i = (Ξi )− (yt )N (i) , ˆ i,N (i) = (θ i )P G , i,N (i)

(23) (24)

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347

for ∀i ∈ INNy , where for the set N (i) ⊂ INNy and the vector yt ∈ IRNy , (yt )N (i) denotes to (yt,N (i)1 , · · · , yN (i)|N (i)| )T ∈ IR|N (i)| , here N (i)j represents j-th lowest number of N (i). For the set N (i) ⊂ INNy and a matrix M ∈ IRNy ×Ny , Mi,N (i) means the vector (MiN (i)1 , · · · , MiN (i)|N (i)| )T ∈ IR|N (i)| . Moreover, for the vectors θ i ∈ IRα and u = Pi,N (i) ∈ INm (m ≤ α), θ u represents the vector (θu1 , · · · , θum )T ∈ IRm . This algorithm is summarized in Algorithm 1 and described in Fig. 2. Algorithm 1. Estimation algorithm of the observation transition matrix used by LSLOCK Input: an observations {ys }ts=t−τ , a parameter matrix P, sets N (i) and B(i, j, k) ˆ Output: an estimated observation transition matrix G for i ∈ INNy do Construct the matrix Ξi , whose each element are calculated by (Ξi )jk =  l∈B(i,j,k) (yt−1 )l Calculate the parameter vector by θ i = (Ξi )− (yt )N (i) ˆ by G ˆ i,N (i) = (θ i )P Calculate the i-th column vector of G i,N (i) end for

Fig. 2. An overview of LSLOCK

4

Experiment

We applied KF, LLOCK, and LSLOCK to two synthetic image sequence data: global flow and circular wave1 . In the former data, objects move toward each direction in each interval, as explained detail in [4]. In the latter data, a wave propagates concentrically from a source point (14, −10), where pixel size is one and the left upper pixel is (0, 0), as shown in Fig. 3. After generating the true data, adding Gaussian noise N (0, 202 ) for global flow and N (0, 0.12 ) for circular wave, we obtained the pseudo observations. 4.1 1

Global Flow

Our code is available at https://github.com/ZoneTsuyoshi/lslock.

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(a) The initial data

(b) Time transition at a source point

Fig. 3. Generation of the circular wave data. (a) The initial data at time t = 0. (b) Time transition of values at a source point

Experimental Condition. We set the parameters of LSLOCK to τ = 10, η = 0.8, c = 1.0, and the state space model given by F0 = H = V0 = I and Q = R = 0.22 × I, and the parameter matrix whose elements correspond to each nine direction: surrounding eight directions and identity. The parameters of LLOCK are almost same as those of LSLOCK; the only exception is update interval τ = 50. We also adjusted hyper-parameters and network architecture of the ConvLSTM [7], which is a benchmark predictive method used by deep learning technique. Experimental Result. We calculated the root mean squared error (RMSE) of the observations, the estimated results used by KF, LLOCK, ConvLSTM, and LSLOCK, as shown in Fig. 4. The left one shows the time transition of the

(a) Time transition of RMSE

(b) RMSE of short-term prediction

Fig. 4. The estimated results for global flow data. (a) Time transition of the RMSE of the observations (black solid line), the estimated results used by KF (light-gray dashed line), LLOCK (light-gray solid line), and LSLOCK (gray solid line). (b) Time transition of the RMSE of the observations, the predicted results used by KF, LLOCK, ConvLSTM (gray dashed line), and LSLOCK when we obtain data until t = 200 (black dotted line) and perform prediction thereafter

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RMSEs; the right one shows the RMSE of the predicted results when we obtain data until t = 200 and perform prediction thereafter. These figures demonstrate the proposed method provides better performance than KF, LLOCK, ConvLSTM, and the observations in terms of noise reduction and short-term prediction. We also applied classical noise reduction methods such as median filter and Gaussian filter to global flow data. Although parameters of these methods are adjusted, they underperformed LSLOCK in terms of RMSE. 4.2

Concentric Circle Wave

Experimental Condition. Almost all parameters are same as global flow setting. The exceptions are learning rate η = 1.0 of LLOCK and LSLOCK, and update interval τ = 30 of LLOCK. Experimental Result. We calculated the RMSE of the observations, the estimated results used by KF, LLOCK, ConvLSTM, and LSLOCK, as shown in Fig. 5. The left one shows the time transition of the RMSEs; the right one shows the RMSE of the predicted results when we obtain data until t = 200 and perform prediction thereafter. The left figure indicates the proposed method is superior to the observations, KF, and LLOCK in all time span. The right figure demonstrates the proposed method provides better predictive performance than KF, LLOCK, ConvLSTM, and the observations, although this data are highly nonlinear. Same as global flow data, the classical noise reduction methods are inferior to the proposed method in terms of RMSE. Moreover, we measured the calculation time regarding the EM algorithm, LLOCK, and LSLOCK in the cases when images are 10×10, 20×20, and 30×30,

(a) Time transition of RMSE

(b) RMSE of short-term prediction

Fig. 5. The estimated results for circular wave data. (a) Time transition of the RMSE of the observations (black solid line), the estimated results used by KF (light-gray dashed line), LLOCK (light-gray solid line), and LSLOCK (gray solid line). (b) Time transition of the RMSE of the observations, the predicted results used by KF, LLOCK, ConvLSTM (gray dashed line), and LSLOCK when we obtain data until t = 200 (black dotted line) and perform prediction thereafter

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and the update interval of LLOCK and LSLOCK is 10. Because the estimated values by EM diverged at second iteration, we measured the time at first iteration and multiplied the time by 10. From Fig. 6, although the calculation cost of the proposed method is higher than that of LLOCK, the cost is much lower than that of the EM algorithm. In addition, because the memory cost of LSLOCK is almost same as that of LLOCK, the proposed method is more efficient than the EM algorithm in terms of the memory cost as described in [4].

Fig. 6. The calculation cost of the EM algorithm (black line), LLOCK (light-gray line), and LSLOCK (gray line)

5

Conclusion

In this paper, we propose a new method to estimate the states and the state transition for spatio-temporal data in real-time. Thanks to two key ideas, localization and spatial uniformity, the method provides better predictive performance than KF, LLOCK, ConvLSTM, and the observations for two synthetic data. Notably, the proposed method works well for concentric circle data although the data has highly nonlinearity and our model is just linear. We consider this is because the true transition is approximately linear in each spatio-temporally local space. Conversely, LLOCK and ConvLSTM require uniformity in wider space. The former method could not capture the nonlinear transition due to requirement of the larger intervals at the estimation step. The latter method could not capture the local transition owing to sharing kernels in global space. Moreover, the calculation and memory cost of the proposed method are much less than that of the EM algorithm. The proposed method, thus, has potential to real-time estimation in areas such as weather forecasting and object tracking.

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Nevertheless, the proposed method has two main drawbacks: difficulty in estimating long-time correlation, and tuning of hyper-parameters. Although the proposed method use the online learning framework, the updating algorithm is too simple to capture long time dependent correlation. The latter one means an automatic selection or elimination of hyper-parameters helps practitioners to use the proposed method.

References 1. Ballas, N., Yao, L., Pal, C., Courville, A.: Delving deeper into convolutional networks for learning video representations (2015). arXiv preprint arXiv:1511.06432 2. Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, Heidelberg (2006) 3. Ghahramani, Z., Hinton, G.E.: Parameter Estimation for Linear Dynamics Systems. Technical Report CRG-TR-96-2, Department of Computer Science, University of Toronto (1996) 4. Ishizone, T., Nakamura, K.: Real-time Linear Operator Construction and State Estimation with the Kalman Filter (2020). arXiv preprint arXiv:2001.11256 5. Kalman, R.E.: A new approach to linear filtering and prediction problem. Trans. ASME-J. Basic Eng. 82(D), 35–45 (1960) 6. Lotter, W., Kreiman, G., Cox, D.: Deep predictive coding networks for video prediction and unsupervised learning (2016). arXiv preprint arXiv:1605.08104 7. Shi, X., Chen, Z., Wang, H., Yeung, D.: Convolutional LSTM network: a machine learning approach for precipitation nowcasting. In: Advances in Neural Information Processing Systems, pp. 802–810 (2015) 8. Shi, X., Gao, Z., Lausen, L., Wang, H., Yeung, D., Wong, W., Woo, W.: Deep learning for precipitation nowcasting: a benchmark and a new model. In: Advances in Neural Information Processing Systems, pp. 5617–5627 (2017) 9. Wang, Y., Zhang, J., Zhu, H., Long, M., Wang, J., Yu, P.H.: Memory in memory: a predictive neural network for learning higher-order non-stationarity from spatiotemporal dynamics. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2019)

Passive Particle Dynamics in Viscous Vortex Flow Gil Marques1(B) , Maria Jo˜ ao Rodrigues1,2 , and S´ılvio Gama1,2 1

2

Faculdade de Ciˆencias, Universidade do Porto, Porto, Portugal [email protected] Centro de Matem´ atica, Universidade do Porto, Porto, Portugal

Abstract. We focus on the description of point vortices both in inviscid and viscous environments and try to extend the idea of quantification of chaos in inviscid vortex systems of Babiano et al. [1] to viscous environments. In particular, we notice that viscosity can disrupt stable dynamics and cause initially stable trajectories to go through chaotic behavior, before succumbing to viscosity completely. We also find that the logarithms of the duration of this chaotic behavior and the kinematic viscosity coefficient seem to be connected by a linear relationship. Most approaches to control of point vortices don’t take viscosity into account. However, viscosity exists in most real fluids. As such, it is important to take it into account to try to obtain better descriptions of real world problems.

Keywords: Point vortices exponents

1

· Viscous point vortices · Chaos · Lyapunov

Introduction

The motion of fluids is critical to understanding a wide range of phenomena in our world. Furthermore, it should be noted that fluid dynamics is behind one of the most relevant problems of nowadays: climate change. It is impossible to deny that our climate has been changing drastically in last few decades and that it is affecting the living conditions of our planet, as well as the planet as we know. Fluid dynamics’ most important equations are the Navier-Stokes equations, which (except in very particular cases) have no general analytical solution and are usually very difficult equations to solve, even numerically. One alternative description of fluids which evades the direct resolution of the Navier-Stokes equations is the point vortices description [2,3]. The idea is to describe fluids through a quantity that measures the local rotation of the flow (the vorticity) by modeling the objects that in fact generate that rotation. The n-vortex problem has been thoroughly studied throughout the years for inviscid environments, however, there is yet much to be done in this alternate description when we add viscosity into the equations. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 352–362, 2021. https://doi.org/10.1007/978-3-030-58653-9_34

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In this work, we intend to study the dynamic behavior of the movement of free/passive particles dragged by viscous vortices. The case of inviscid vortices has already been considered in the literature in several problems of optimal control. For instance, in [4,5] the need to introduce viscous dissipation was made through an exponential decay of the vortex circulations. Thus, with this study, we want to gain a better insight on how particles behave when they pass in the vicinity of viscous vortices, having in mind to enrich the works [4,5]. We follow the idea of Babiano et al. [1] on using Lyapunov exponents to quantify chaos on inviscid point vortex models and extend the study to a viscous environment. Protas [6] turned his attention to some optimization and control problems on vortex-dominated flows using inviscid point vortices, and we find it worthwhile to try to tackle those same problems using the multi-Gaussian model, thus accounting for the viscosity of the flow. In particular, Problems 1 and 4 of his work seem the most interesting to analyze using this approach.

2

Point Vortices

In the vorticity formulation, the 2D incompressible Navier-Stokes equations can be written as ∂ω + (u · ∇) ω = ν∇2 ω, (1) ∂t where ω is the vorticity, u is the velocity field and ν is the kinematic viscosity. In an inviscid flow (ν = 0), the solution of this equation with the initial condition ω (r, t = 0) = Γ δ (r − r0 ) is what is usually called an inviscid point vortex. This vortex, if left alone, will stay in the position r0 forever and its’ circulation Γ will induce a velocity field in the plane. In a system of n of these vortices, the velocity field caused by each of them will affect the remaining n − 1 vortices. The equations of motion of such a system can be written in a compact way by using complex variables z (t) = x (t) + iy (t) as (see [7,8]) z˙α∗

n 1  Γβ = , 2πi zα − zβ

α = 1, . . . , n

(2)

β=α

and the velocity field generated by the n vortices will be z˙ ∗ =

n 1  Γβ . 2πi z − zβ

(3)

β=1

In systems of n ≥ 4 inviscid vortices, it is known that, generically, the movement of test particles within the flow is chaotic. However, point vortices create regions of stability around them; i.e., while they move chaotically within the fluid, the movement of particles within those regions of stability is predictable with respect to the vortex center [1]. These are usually called stability islands. Furthermore, particles within these islands cannot be ejected from them and particles outside them cannot enter the stability regions.

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This inviscid problem has been thoroughly studied over the years. However, flows with zero viscosity are unrealistic and we still lack a consistent theory for viscous environments. In an environment with non-zero viscosity, the solution of the Navier-Stokes equations for an initial condition ω (z, t = 0) = Γ δ (z − z0 ) corresponds to the Lamb-Oseen vortex(see [9,10]): a vorticity field that decays and is diffused radially outward with a gaussian profile. One important result on the Lamb-Oseen vortex is that it is an asymptotically stable attracting solution of the vorticity equation for any integrable initial vorticity configuration, as was proven byGallay and Wayne [11]. The n superposition of n point vortices, ω (z, t = 0) = α=1 Γα δ (z − zα ), is one such configuration. This idea gave rise to the multi-Gaussian model of point vortices, that assumes that the vorticity field is a superposition of Lamb-Oseen vortices at all times: each of the initial point vortices spread axisymmetrically as if it they were isolated, modeling the diffusive term in the vorticity equation (ν∇2 ω); the center position of each vortex, however, is affected by all the other vortices and moves according to the velocity field created by them, as it is expected due to the convective term (u · ∇) ω. In a more recent work, Gallay [12] showed that under certain conditions, the solution of the Navier-Stokes equation in the inviscid limit ν → 0 for δDirac initial conditions converges to a superposition of Lamb-Oseen vortices. This provides yet another reason for the viability of the multi-Gaussian model in the description of fluids. The vorticity field for a system of n vortices located at zα , α = 1, . . . , n is ω (z, t) =

  n 1  |z − zα |2 Γα exp − , 4πνt α=1 4νt

(4)

the velocity field that arises from the vortices is z˙ ∗ =

   n 1  Γβ |z − zβ |2 1 − exp − , 2πi z − zβ 4νt

(5)

β=1

and the equations of motion for each of the n vortices are z˙α∗

   n 1  Γβ |zα − zβ |2 = 1 − exp − , 2πi zα − zβ 4νt

α = 1, . . . , n

(6)

β=α

3

Lyapunov Exponents

A chaotic system is essentially characterized by its’ non predictability, meaning that it is impossible to know its’ time evolution for some generic initial condition. Lyapunov exponents are quantities that can help to quantify chaos. They measure the rates of divergence or convergence of nearby trajectories in phase space: a negative Lyapunov exponent means that trajectories approach one another,

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while a positive one indicates the divergence of the nearby orbits and therefore chaotic behavior (a zero Lyapunov exponent means that the nearby trajectories maintain their distance). Consider a system ruled by some equation x˙ = f (x), where x = (x1 , . . . , xn ) and some trajectory x (t) of this system. Furthermore, consider a second trajectory, x ˜ (t), close to the latter, such that |δ| 0 ∀ t ∈ [0, T ∗ ]; ∂ ¯ ∗ ˙ (t), u∗ (t), ψ(t), μ(t), λ) for a.a. t ∈ [0, T ∗ ]; ψ(t) = − H(x ∂x ¯ ∗ (t), u, ψ(t), μ(t), λ) = H(x ¯ ∗ (t), u∗ (t), ψ(t), μ(t), λ) for a.a. t ∈ [0, T ∗ ]; max H(x u∈U . ¯ (x∗ (T ∗ ), u, ψ(T ∗ ), μ(T ∗ ), λ) = ¯ (B, u, ψ(T ∗ ), μ(T ∗ ), λ) = 0. max H max H u∈U

u∈U

Furthermore, function μ is non-increasing component-wise, and satisfies the   ∗  g(x (t)), dμ(t) = 0. complementary slackness condition: [0,T ∗ ]

Control processes satisfying the Maximum Principle are usually called extremals. Remark that, as it follows from Theorem 1, for any extremal process ¯ u, ψ(t), μ(t), λ) = 0 holds for all σ = (x, u, T ), the conservation law max H(x(t), u∈U

t ∈ [0, T ]. Moreover, if σ satisfies the conditions of the Maximum Principle with certain Lagrange multipliers (λ, ψ, μ), then it also satisfies the same conditions with the multipliers (λ, ψ(t) + (Dg(x(t))) a, μ(t) + a) , for any a ∈ Rk . 2.2

Further Problem Data Specification and the Structure of Normal Extremals

A further analysis of problem (P ), based on Theorem 1, requires some concretizations. We assume that • | · | is the Euclidean norm, • ϕ(u) = |u|2 − 1, i.e., controls are constrained in the unit ball of Rn , and • g(A) < 0, i.e. x(0) belongs to the interior of the admissible domain. The case α = 0 boils down to a time-optimal control problem subject to additional state constraints, which is analyzed in [5,6]. We, thus, shall focus on the ¯ can be written as case α > 0. By supposing that λ = 0, the Hamiltonian H   1 ¯ H(x, u, ψ, μ, λ) = αλ ξ, v(x) + u − |u|2 + 1 − λ, 2

. 1 ψ − (Dg(x)) μ . The (unique) where we abbreviate ξ = ξα (x, ψ, μ, λ) = αλ ¯ and the maximized Hamiltonian are then easily computed as maximizer of H ⎧ ⎨ ξ, |ξ| ≤ 1, ξ u α (x, ψ, μ, λ) = ProjB (ξ) = ⎩ , |ξ| > 1; |ξ| ⎧ 2 ⎪ ⎨ |ξ| + ξ, v + 1, |ξ| ≤ 1, ¯ + λ = αλ 2 max H u∈U ⎪ ⎩ |ξ| + ξ, v + 1 , |ξ| > 1. 2

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Let σ = (x, u, T ) be extremal and (λ = 0, ψ, μ) be associated Lagrange multipliers. Suppose that (P ) is regular w.r.t. the state constraints, and J(x(t)) ≤ 1 for all t ∈ [0, T ], i.e., Theorem 1 is applicable. Then, the conservation law entails: ⎧ 2 ⎪ ⎨ |ξ| + ξ, v + 1, |ξ| ≤ 1, 1 2 = along [0, T ]. α ⎪ ⎩ |ξ| + ξ, v + 1 , |ξ| > 1, 2 Without loss of generality, we can take μ(0) = 0. Given the data (α, v(A)), the conservation law grants an estimate on the values of ψ(0). For example, if v(A) = 0, then the initial value of ψ lies on the sphere |ψ(0)| = (1 − α2 )λ. By the complementarity condition, μ can vary only on intervals of motion along the boundary {x ∈ Rn : g(x) = 0} of the admissible domain. A driving signal providing this motion regime can be expressed in a feedback form . u(x) ensuing from 0 = Γ (x, u(x)) = Dg (x) (v (x) + u(x)), and the respective expression for μ as a function of (x, ψ, λ) can be finally found from the relation u (x, ψ, μ, λ) = u(x). For example, consider the planar case n = 2, assume that v(A) = 0, and let the state be constrained in the horizontal strip: g(x1 , x2 ) = (−x2 , x2 − 1) . Then Γ (x, u) = (−(v2 (x) + u2 ), (v2 (x) + u2 )) . Clearly, at any point x ∈ R2 only one of the conditions x2 = 0 or x2 = 1 can hold, that is J(x) ≤ 1 for any x ∈ R2 . For i = 1, 2, and all x, u ∈ R2 , the equality Γi (x, u) = 0 implies u2 = . −v2 (x). Then, by assuming that |v(x)| < 1, the condition 0 = ϕ(u) = |u|2 − 1 ensures that u1 = 0. We observe that, for any i = 1, 2 such that gi (x) = 0, the vectors ∇gi = (0, ±1) and ∇ϕ = 2u are never collinear. Thus, Theorem 1 is indeed applicable. The adjoint equation writes: ∂ ¯ = −Dv(x) ψ + ν ∇v2 (x), ψ˙ = − H ∂x

(5)

. 1 where we set ν = μ2 −μ1 . In this notation, ξ = αλ (ψ1 , ψ2 −ν) , and the expression 2 , takes the form (recall that v22 < 1) for ν, implied by the equality −v2 (x) = u  ⎧ 1 |ψ1 | ≤ αλ, |v2 (x)| ≤ αλ α2 λ2 − ψ12 , ⎨ ψ2 + αλ v2 (x), |v2 (x)| ν= |ψ1 |, otherwise. ⎩ ψ2 ±  1 − v22 (x) This formula is applicable when x2 = 0 or x2 = 1. Finally, we have at hand all the ingredients of the following conceptual approach for the numerical computation of the field of normal extremals, based on the shooting algorithm, which is described in details in [5]. For initialization of the shooting, we set μ(0) = 0 and choose λ = 1, which implies |ψ(0)| = 1 − α/2. Letting ψ(0) = (1 − α/2)(cos(θ), sin(θ)) , the sphere

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in R2 is described by a scalar parameter θ ∈ [0, 2π]. Fixed θ, the Hamiltonian system (1), (5) is integrated numerically forward in time with the initial condition

 (x, ψ) (0) = A, (1 − α/2)(cos(θ), sin(θ)) . If the x-component of such a numeric solution meets the boundary of the admissible domain, the “feedback” control (u1 (x, ψ), −v2 (x)) should be switched on, where, for α > 0, ⎧  1 ⎪ ψ1 , |ψ1 | ≤ α 1 − v22 (x), ⎪ α ⎨  u1 (x, ψ) = 1 − v22 (x), ψ1 > α 1 − v22 (x), and ⎪ ⎪ ⎩  − 1 − v22 (x), otherwise. Integrating further in time, under the action of this control, the “cost of leaving” the boundary of the admissible domain should be calculated at each time instance by computing the value min{|x(t) − B| : t ∈ [0, T ]} for some a priori fixed sufficiently large time horizon T (we used T = 50 in computations).

3

Application to Route Planning in Convective Flows

In order to study optimal route planning in a convective flow field, we consider the following model of convection. A fluid or gas in a plane horizontal layer is heated from below. The Boussinesq approximation is assumed: buoyancy depends linearly on temperature, density variation is neglected in the mass conservation equation, and the flow is incompressible. This is the so-called Rayleigh– B´enard convection (see, e.g., [12]). Rayleigh–B´enard convection in a plane layer has been well studied. It is characterized in the dimensionless form by the Rayleigh number, R (indicating the magnitude of thermal buoyancy forces) and the Prandtl number, P (the ratio of kinematic viscosity to thermal diffusivity). When the Rayleigh number is larger than the critical value Rc (this value is independent of P ) a flow sets in. In a large range of parameters, the fluid flow takes the form of steady planar (twodimensional) rolls. We consider such flows rolls in the present study. For larger values of R the planar flows loose stability, undergo a very complex sequence of local [13,14] and global [15] bifurcations giving rise to turbulent, essentially three-dimensional convective flows. The bottom (x2 = 0) and the upper (x2 = 1) boundaries of the layer are maintained at constants temperature (the upper bottom is colder), and assumed to be fluid impermeable and stress-free, i.e. ∂v1 = v2 = 0. ∂x2

(6)

Periodicity in the horizontal direction x1 is assumed: v(x1 , x2 ) = v(x1 + nL, x2 ), ∀n ∈ Z, where L > 0 is given. For the boundary conditions (6) and R > Rc = 657.5, the motionless state of the hydrodynamic system looses stability. According to the linear stability

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theory [16], the most unstable mode can be represented in the form v(x) = (−β1 sin(2πx1 /L) cos(πx2 ), β2 cos(2πx1 /L) sin(πx2 )),

(7) √ where 2π/L is the critical wavenumber, length of the convective cell is L = 2 2, parameters β1 and β2 define the magnitude of the flow. In order to satisfy the ∂v1 ∂v2 incompressibility condition, + = 0, we take β2 = 2β1 /L. In the present ∂x1 ∂x2 study, the one-parameter (β1 ) family of flows (7) is considered. By using the code from [17], we checked that, for P = 1 and 658 ≤ R ≤ 1700, the flow (7) contains the most of the kinetic energy of the solution to the (nonlinear) equations governing the convective system (the Navier-Stokes and the heat transfer equations): all other harmonics present in the solution for the flow are at least two orders of magnitude smaller. For the same Prandtl number and R ≥ 1760, the flow is three-dimensional and time dependent (see [13] for further details on transition to turbulent convection). 3.1

Computational Results

By using the method described in the previous section we solved the optimal control problem (P) numerically for some sample convective flows. Both starting and the terminal point are in the horizontal midline, A = (L/4, 0.5) and B = (7L/4, 0.5). The flow is given by (7) and the parameter β1 is varied. For the optimal solution we also measure the total energy spent by the  1 T controlled object: E = |u(t)|2 dt. 2 0 For β1 = 0.1, the flow is weak and, hence, the optimal trajectory for the time optimal problem (α = 0) is a weak perturbation of the straight line connecting points A and B (see the red curve √ in Fig. 1(a)). The traveling time is T = 4.20 time units (cf. traveling time 3 2 ≈ 4.24 in the absence of the flow). The energy spent is E = T /2 = 2.10. The optimal trajectory for the same flow, but for the cost functional I(α) with α = 0.9 (i.e. the “weight” of the kinetic energy is high) is shown in Fig. 1(a) by the blue curve; the corresponding traveling time is larger – T = 9.20, however, traveling along this trajectory is much more economic from the “energetic” point of view – the energy spent is E = 0.91. For β1 = 0.5, the flow is stronger and, hence, the optimal trajectories are notably different from the straight line, in the contrary, in follows upward and downward fluid flows (see Fig. 1(b)). The optimal traveling time for the time optimal problem is T = 3.81 time units (shorter in comparison with the previous “weak convection” case), the energy spent is E = 1.91 (see the red curve in Fig. 1(b)). For the same flow, the trajectory minimizing the cost functional with α = 0.9 (see the blue line in Fig. 1(b)) is longer to follow – T = 8.71, but the energy looses are significantly smaller, E = 0.50. For a stronger convective flow, β1 = 0.9 (see Fig. 1(c)), the time optimal problem yields T = 3.43 and E = 1.72. For the same flow, the problem with α = 0.9 has an optimal solution corresponding to T = 7.29 and E = 0.39.

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Fig. 1. Optimal routes for the problem (P) and flows (7) for β1 = 0.1 (a), 0.5 (b) and 0.9 (c). Red curves represent optimal solutions of the time optimal problem, α = 0, the blue ones stand for α = 0.9.

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Discussion

As a matter of comparison of our numeric approach with the existing algorithms, we find important to note the following. For nonconvex problems with state constraints, methods based on Pontryagin’s Maximum Principle demonstrate the potential of finding global solutions (if the structure of the field of extremals is relatively simple, which is typical for applications). In contrast, the direct methods involving the standard NLP solvers like IPOPT, APOPT, BPOPT etc. (such are, e.g., the academic packages BOCOP and GEKKO) give only local extrema. An alternative could be algorithms, based on the dynamic programming or feedback necessary optimality conditions, but their elaboration for state constrained problems remains an open question.

4

Conclusion

In the paper, we present a deterministic approach for the global search in optimal route planning problems with state constraints, based on some recent advances in the theory of Pontryagin’s Maximum Principle. Solving numerically a two-point boundary value problem, it lets one to find a global minimum by choosing the optimal one out of all extremals, and, hence, presents a coherent methodology for operating with state constraints. Practical usefulness of the proposed method is shown by considering its application to the problem of route planning in convective flows. Using three planar convective flows, we show the optimal routes for the time optimal control problem as well as for the problem where the cost functional involves also the energy spent to move the controlled object. In the present study, we are focused on the simplest problem of optimal route planning, where the controlled object is represented by a material point influenced by a convective flow. Extension of the proposed approach to similar control problems for more complex and more realistic models, such as autonomous robots, is a next step we plan to perform in future. Acknowledgements. Support from FCT (Portugal): R&D Unit SYSTEC – POCI01-0145-FEDER-006933/SYSTEC funded by ERDF | COMPETE2020 | FCT/MEC | PT2020 extension to 2018, UID/EEA/00147/2020 and NORTE-01-0145-FEDER000033 supported by ERDF | NORTE 2020 is highly acknowledged. Results of Sect. 3 were supported (during visits of the first author) by the Russian Science Foundation in the framework of the project no. 19-11-00258 carried out in the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russian Federation.

References 1. Guervilly, C., Hughes, D., Jones, C.: Large-scale vortices in rapidly rotating Rayleigh-B´enard convection. J. Fluid Mech. 758, 407–435 (2014)

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2. Arutyunov, A.V., Karamzin, D.Yu.: On some continuity properties of the measure Lagrange multiplier from the maximum principle for state constrained problems. SIAM J. Control Optim. 53(4), 2514–2540 (2015) 3. Karamzin, D., Pereira, F.L.: On a few questions regarding the study of stateconstrained problems in optimal control. J. Optim. Theor. Appl. 180(1), 235–255 (2019) 4. Arutyunov, A., Karamzin, D.: A survey on regularity conditions for stateconstrained optimal control problems and the non-degenerate maximum principle. J. Optim. Theor. Appl. (in press) 5. Chertovskih, R., Karamzin, D., Khalil, N., Pereira, F.L.: An indirect method for regular state-constrained optimal control problems in flow fields. IEEE Trans. Autom. Control (in press). https://doi.org/10.1109/TAC.2020.2986179 6. Chertovskih, R., Karamzin, D., Khalil, N., Pereira, F.L.: Regular path-constrained time-optimal control problems in three-dimensional flow fields. Eur. J. Control (in press). https://doi.org/10.1016/j.ejcon.2020.02.003 7. Chertovskih R., Khalil N.T., Karamzin D., Pereira F.L.: An indirect numerical method for a time-optimal state-constrained control problem in a steady twodimensional fluid flow. In: Proceedings of the 2018 IEEE OES Autonomous Underwater Vehicle Symposium, pp. 1–6 (2019). https://doi.org/10.1109/AUV.2018. 8729750 8. Chertovskih, R., Khalil, N.T., Karamzin, D., Pereira, F.L.: Path-constrained trajectory time-optimization in a three-dimensional steady flow field. In: Proceedings of the 2019 European Control Conference, pp. 3746–3751 (2019). https://doi.org/ 10.23919/ECC.2019.8796083 9. Chertovskih, R., Khalil, N.T., Karamzin, D., Pereira, F.L.: Optimal path planning of AUVs operating in flows influenced by tidal currents. In: Proceedings of the 9th International Scientific Conference on Physics and Control (PhysCon2019), pp. 40–43. LLC Publishing House “Pero”, Moscow (2019) 10. Chertovskih, R., Khalil, N.T., Pereira, F.L.: Time-optimal control problem with state constraints in a time-periodic flow field. In: Ja´cimovi´c, M., Khachay, M., Malkova, V., Posypkin, M. (eds.) 10th International Conference on Optimization and Applications, OPTIMA 2019. CCIS, vol. 1145, pp. 340–354. Springer (2020). https://doi.org/10.1007/978-3-030-38603-0 25 11. Filippov, A.F.: On certain problems of optimal regulation. Bull. Moscow State Univ. Ser. Math. Mech. 2, 25–38 (1959) 12. Getling, A.V.: Rayleigh-B´enard Convection: Structures and Dynamics. World Scientific, Singapore (1998) 13. Podvigina, O.M.: Magnetic field generation by convective flows in a plane layer. Eur. Phys. J. B Condens. Matter Complex Syst. 50(4), 639–652 (2006). https:// doi.org/10.1140/epjb/e2006-00171-4 14. Podvigina, O.M.: Magnetic field generation by convective flows in a plane layer: the dependence on the Prandtl numbers. Geophys. Astrophys. Fluid Dyn. 102(4), 409–433 (2008). https://doi.org/10.1080/03091920701841945 15. Chertovskikh, R., Chimanski, E.V., Rempel, E.L.: Route to hyperchaos in Rayleigh-B´enard convection. Europhys. Lett. 112, 14001 (2015). https://doi.org/ 10.1209/0295-5075/112/14001 16. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961). Dover Publications Inc., New York (1981) 17. Chertovskih, R., Gama, S.M.A., Podvigina, O., Zheligovsky, V.: Dependence of magnetic field generation by thermal convection on the rotation rate: a case study. Phys. D 239(13), 1188–1209 (2010). https://doi.org/10.1016/j.physd.2010.03.008

A Fuzzy Based Model to Assess the Influence of Project Risk on Corporate Behavior Ricardo Santos1,2(&) , Antonio Abreu2,3 , João M. F. Calado2,4 José Miguel Soares5 , José Duarte Moleiro Martins6 , and Vitor Anes2,4

,

1

GOVCOPP - University of Aveiro, Aveiro, Portugal [email protected] 2 ISEL- Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Lisbon, Portugal 3 CTS Uninova, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisbon, Portugal 4 IDMEC-IST-UL, Lisbon, Portugal 5 ADVANCE - Centro de Investigação Avançada em Gestão do ISEG, ISEG - Lisbon School of Economics and Management, Universidade de Lisboa, Lisbon, Portugal 6 ISCAL - Instituto Politécnico de Lisboa (ISCAL), Instituto Universitário de Lisboa (ISCTE-IUL), Business Research Unit (BRU-IUL), Lisbon, Portugal

Abstract. Nowadays, the competitiveness on industry, requires a good preparation from the organizations, considering all events that may occur, which brings new challenges for the corporation’s management. To address such challenges, risk management’s models have been used to give a sense of threat prevention, by assessing each project’s risk and the risk from the corporation itself as well. However, such models are normally based on human perception, which brings some subjectivity around the risks involved, making their definition less accurate. Additionally, there is a lack of models that allows to better define the corporation’s risk, by exploring the influence from the risk’s project. To address these issues, this paper presents an approach, supported by fuzzy logic, to analyze the risk’s level in an organization, by considering the influence of their projects. A case study will be used to assess the model robustness and to discuss the benefits and challenges found. Keywords: Corporate risk management
Fuzzy logic


Project risk
Risk management
Project

1 Introduction Nowadays, the dynamics existed with markets and industry in general, pose major challenges for companies, forcing them to be innovative and flexible in order to achieve a competitive advantage over others. To make this possible, it is necessary to put into

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 383–393, 2021. https://doi.org/10.1007/978-3-030-58653-9_37

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practice the strategy outlined by top management, which is implemented through projects, programs and portfolios, which companies use to successfully achieve their strategic objectives [1]. Additionally, projects emerge as a response to a need, opportunity, or threat, which may be unique and isolated or as components of programs and/or portfolios [2] following a defined organizational strategy [3]. Risk Management (RM), arises as a process of risk identification, assessment, monitoring and reporting, allowing the management of the uncertainties that the organization, by preventing threats, or even, by identifying opportunities, through the analysis of the risk’s involved, in terms of their impacts and probabilities of occurrence [2, 3]. Through their identification and evaluation, it’s possible the elaboration of a set of actions to minimize any negative effects that may occur [4]. Additionally, the Project Management (PM), has gained greater notoriety and relevance in recent decades, as an application of knowledge, skills, techniques, and tools to meet the requirements caused by the unpredictability of each project [3]. The unpredictability rises with the complexity of the projects involved [4, 5]. Project risk management (PRM) is currently considered as one of the main success factors of projects, developed by organizations [6]. The greater the effectiveness of this management, the greater the likelihood of PRM’s success [2–4]. Corporate Risk Management (CRM) focuses on the overall performance of the organization as a whole by creating, preserving and realizing value through holistic integration across all domains by identifying and addressing risks that have impacts on the strategic objectives defined by the organization [7, 8]. As such, RMP should not be practiced independently of CRM as the risks that are associated with a project may condition the performance of the organization. According to several authors [9], there are currently several methods (e.g. Decision Trees, Delphy’s Analysis, Failure Modes and Effects Analysis (FMEA), Risk Matrix, among others) that can be used to identify and analyze the risk. However, the potential existing relationships between PRM and CRM, have not been conveniently explored in the existing literature [7]. In this paper, it will be presented an approach to explore the relation between PRM and CRM in organizations, followed by a case study to assess its robustness. Thus, Sect. 2 describes the research method, where it’s presented the proposed approach, followed by its deployment. Section 3, describes the case study used to validate the proposed model, followed by the correspondent obtained results, with Sect. 4 ending the work, with some concluding remarks.

2 Research Method 2.1

Proposed Approach

The proposed approach intends to present a model to assess the organization’s risk level, by exploring the possible influence of each project to be considered by the organization, through the assessment of the correspondent risk. In this sense, the risks normally associated with PRM, as well as with CRM, can be organized, according to a taxonomy based on the works from [9], such as the one presented on Fig. 1.

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Fig. 1. Wheel of corporation and project risks considered in this work (adapted from [9]).

The completion of a project, with or without success, will have an impact on the various domains of the organization since the development of projects leads to changes in the internal organizational environment [7, 8], so the expected impact on a project might bring impact in one or even more, organization’s domains, since that Project Risk Management (PRM), is an integral element of Corporate Risk Management (CRM) (Fig. 2).

Fig. 2. Relationship between the PRM and CRM

A project risk (PR) with the possibility of affecting the identified project domain (s), may influence one of the corporate risk (CR) metrics, since that this variable predicts the effect that a CR can originate in the organization with the transition from the current state of the organization to a future state reached through the project’s deployment [2]. Each project, in this way, can impact the corporate risk of the organization, through their associated domains with it (e.g. Strategy (S), Operations (O), Finance (F), Information Systems (IS) and Environment (E)).

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In this work, it was considered the following organization domains; Strategy (S), Marketing (M) and Operations (O). 2.2

Models’s Architecture

The approach developed here, uses fuzzy logic’s, in order to integrate both issues of risk assessment, namely quantitative and qualitative, into a single approach, pretending to be compatible with the uncertainty and ambiguity of human perception, regarding risk assessment. Thus, the approach presented here, is based on two hierarchical levels, namely, project and corporative risk levels. The first one, has the purpose of assessing the project risk level (PRn), by considering a given project n, while the second one, intends to assess the corporative risk level (CR), considering the possible influence CRn of a project n in the organization’s risk (Fig. 3).

Fig. 3. Proposed model.

Based on the risk’s taxonomy, presented on Fig. 1, there is an individual risk (R-n), regarding each organization’s project n, which is obtained from each expected impact (I-n) and the correspondent probability of occurrence (P-n) considered (Fig. 2), i.e.: RSPRn ¼ PSPRn  ISPRn

ð1Þ

RTPRn ¼ PTPRn  ITPRn

ð2Þ

RCPRn ¼ PCPRn  ICPRn

ð3Þ

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Based on Figs. 2 and 3, the Project Risk (PRn) level, regarding to a given project n, can be obtained by aggregating all individual risk category considered, i.e.: PRn ¼ xRSn  RSn þ xRTn  RTn þ xRCn  RCn

ð4Þ

The weights ðxRn Þ, corresponding each one, to a project risk category considered in this work, allows to satisfy the following condition: 1 ¼ xRSn þ xRTn þ xRCn

ð5Þ

The possible influence, regarding a project risk (PRn) on corporate risk (CR), is also  included, by considering an existence of an expected impact variable (IPRn!Dom Corp ), which will affect one or more, domains of the organization/corporation. In this work, it was considered the following organizational domains: Strategic (S), Marketing (M) and Operations (O). Based on the project domains (i.e.: Scope (S), time (T) and Cost (C), it is obtained a  set of impacts values, for each organizational domain impact value (IPRn!Corp Dom Þ considered. The perception of the risk manager on selecting the influence of the project
on each organization domain Proj InfPRn!CorpDom , is considered, by applying fuzzy logic, based on a set of linguistic rules. The case with more impact, is then considered, by considering the maximum value of the 3 values obtained before, i.e.: Proj InfPRn ¼ maxfProj InfPRn!S ; Proj InfPRn!M ; Proj InfPRn!O g

ð6Þ

The influence of the impact on each project domain, is also considered, since that the resources used in the project, could affect the available resources in the organization. Therefore, the case with more impact, defines the project impact ðPIPRn Þ, which is obtained, by considering the maximum value of the 3 values referred before, i.e.: PIPRn ¼ maxfISPRn ; ITPRn ; ICPRn g

ð7Þ


Thus, the total impact from the project on corporation domain IPRn!Corp Dom will be resulted from the sum of Proj InfPRn with the PIPRn , i.e.: IPRn!Corp Dom ¼ Proj InfPRn þ PIPRn

ð8Þ

A weighting factor F ðIeRP ! Dom OrgÞ is then applied to this value, i.e.: FðIPRn!Corp Dom Þ ¼ ðIPRn!Corp Dom
0; 1Þ þ 1

ð9Þ

Thus, the corporate risk for each project n (CRn), is resulted from the combination of probability of occurrence of CR event with the expected impact of CR in the organization, i.e.:

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CRn ¼ PCR
ICR
F IPRn!Corp Dom




ð10Þ

The Corporative Risk, is therefore achieved, based on the contribution of each Project Risk (PRn) considered, and according to its relevance ðxPRn Þ, i.e.: CR ¼ xPR1  CR1 þ xPR2  CR2 þ xPRn  RCn

ð11Þ

This approach allows the risk manager, to know, assess and prioritize each project considered, by knowing its correspondent level of PRn and CRn respectively. 2.3

Fuzzy Deployment

The levels of a Fuzzy Inference System (FIS) based on a set of IF-THEN heuristic rules that is used to obtain the risk levels (by impact category). Based on Fig. 3 and with regards to the Project Level, the fuzzy system “Fuzzy 1”, is based on the expressions (1 and 3), regarding the inputs (P-PRn) and (I-PRn), i.e.: RPRn ¼ P PRn  IPRn

ð12Þ

The inference rules, regarding the FIS “Fuzzy 1” (Fig. 4), are formulated as follows: “IF probability (P-PRn) is P AND impact (I-PRn) is I THEN project risk level (R-PRn) is R. Similar approach, can be defined, regarding the Corporate Risk level

(CRn), based on (10), although with the project influence variable F IPRn!Corp Dom included, i.e.: CRPRn ¼ PCRn
ICRn
F IPRn!Corp Dom




ð13Þ

Therefore, the inference rules, regarding the FIS “Fuzzy 3” (Fig. 4), can be formulated as follows: “IF probability (P-CRn) is P AND impact (ICRn) is I AND Project Influence

F IPRn!Corp Dom is F, THEN the corporate risk level (CRPRn) is R (Fig. 3). Regarding the possible influence of the project n; on each organization domain
Proj InfPRn!CorpDom , can be considered through FIS “Fuzzy 2” (Fig. 3), based on a set of linguistic rules, which can be formulated as follows: “IF impact (ISPRn!Corp Dom ) is AND impact (ITPRn!Corp Dom ) is It AND impact (ITPRn!Corp Dom ) is Ic, THEN the influence of the project n, on each organization domain Proj InfPRn!Corp Dom is Inf. 2.4

Linguistic Variables

Based on [9], it’s usually suggested from the literature, that linguistic levels should not exceed nine levels, since it represents the limits of human perception in terms of discrimination. Based on such recommendations, the linguistic variables were defined in order to not exceed 5 levels, with their corresponding membership functions, being of triangular type.

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Table 1. Linguistic variables, regarding the probability (P) and membership functions

Table 2. Linguistic variables and membership functions, regarding each impact (I), for each project domain (scope, cost and time)

On Table 1, it’s presented the linguistic variables, regarding the project’s probability of occurrence (P-n), considering each project domain. The same table was also considered for the probability of occurrence of the organization (PCR). On Table 2, it’s presented the linguistic variables, regarding the different project impacts considered for the project risk level and regarding each project domain. The same table was also considered, for the project impacts on each organizational  domain impact value (IPRn!Corp Dom ) considered, i.e., Marketing (M), Operations (O) and Strategy (S). On Table 3, it’s presented the linguistic variables, regarding the project individual risk (R-n) for each project domain considered. The same table was used for CRn.

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R. Santos et al. Table 3. Linguistic variables, regarding corporate level risk

2.5

Model’s Implementation Process

In order to build the rules set, the definition of the membership functions, and to analyze the behavior of a fuzzy inference system (FIS), it was used MATLAB® software, though its Fuzzy Logic ToolboxTM (or Fuzzy Logic Designer in version R2017a). The three FIS considered here, were implemented in this toolbox, by using the inference mechanism approach proposed by Mamdani (Fig. 4), given its intuitive use and better adaptation to the inputs from human reasoning, and its wide acceptance [9]. Regarding the project level, the FIS “Fuzzy 1”, has on its inputs, the Probability of occurrence (Pp) and the expected Impact (I-), while, and regarding the output, is the Project Risk Level by category (RPn). The influence of a project n on each organizational domain, was implemented on FIS “Fuzzy 2”, which has on its inputs, the project expected impact in the organization’s domain, regarding each project risk category (Scopus, Time and Cost) considered. As the output the variable, “Fuzzy 2” has the project influence on each organization domain (Proj_InfPRn). Regarding “Fuzzy 3”, the FIS considered here, has similar inputs regarding “Fuzzy 1”, i.e., the probability of corporate risk occurrence (Pc) and the correspondent expected impact (Iec). Additionally, it also has the project risk influence variable (PR_Inf) with the output, being the Corporate Risk Level.

Fig. 4. Fuzzy Inference System (FIS) deployment on Matlab

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Between the defuzzification methods available on literature, it was used the centroid method as the defuzzification approach, considering the FIS proposed in this work, given its widely application on several works of this type [9].

3 Case Study In order to assess the robustness of the proposed model, a case study was developed, based on an electricity distribution and commercialization company, which has a portfolio of 5 investment projects, related to the strategic business area, production of electricity. Each one of the projects considered, refers to the realization of an electric power plant, associated with a given production technology. For this purpose, 3 technologies were considered, namely, wind, photovoltaic and water. Each investment project encompasses a set of risks related to the scope (e.g. type of technology used in the production of electricity and tariffs for the sale of energy), time (e.g. execution time of the plant’s electrical installation project) and costs involved (e.g. accidents during the execution of the work). Based on Fig. 3, and regarding the models’ application, a group of experts evaluated the 17 parameters that constitute the inputs of the model based on the three identified risks (Table 5), namely: ISRPn, ITRPn, ICRPn, PsRPn, PtRPn, PcRPn (Project level) and ISRPn!S, ITRPn!S,ICRPn!S, ISRPn!M, ITRPn!M, ICRPn!M, ISRPn!O, ITRPn!O, ICRPn!O, ICR, PCR (Corporate level) regarding each one of the 5 projects considered. The model inputs are presented on Table 4. The inputs, regarding the different weights considered, are presented on Table 5.

Table 4. Model inputs

Table 5 illustrates the results obtained, considered the 5 projects’ risks and the overall corporate risk as well.

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R. Santos et al. Table 5. Model outputs, including the weights considered

The results presented above shows that it’s possible not only to achieve the project individual risk, as well as the organization’s global risk. According to the results presented on Table 5, it’s possible to prioritize the 5 projects, based on its individual risk (PRn), where the one with the highest risk is the Project nr. 4, followed by the Project nrs. 5, 2, 3 and 1. However, Project nr. 4 presents more risk for the organization, in terms of its individual contribution for the corporate risk (CRPRn), which shows the influence of the project in the organizational risk.

4 Conclusions In this work it was presented an approach, that allows to assess the risk from an organization, not only through its different domains, but also based on each project’s influence, through its correspondent impacts. In this context, the influence of each project in the organization’s domain, was also considered and assessed, which have allowed to achieve not only the project’s level risk, but as well as it’s contribution for the organization’s level risk, regarding each domain considered. The approach presented here, also allows to assess and prioritize each project considered, based on its individual risk. Furthermore, the use of Fuzzy inference systems to assess the different risks involved, has allowed to mitigate the analysis subjectivity, associated with the uncertainty and ambiguity, that characterizes the risk assessment, given its dependence on human perception. As a future work, this work could also include in the risk assessment, not only the threats, but the opportunities involved at a project and a corporate level. Acknowledgement. This work was partially supported by the Polytechnic Institute of Lisbon through the Projects for Research, Development, Innovation and Artistic Creation (IDI&CA), within the framework of the project IEOMAB—internationalization of companies operating in the Angolan and Brazilian markets, IPL/2019/IEOMAB_ISCAL.

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References 1. PMBOK Guide: A Guide to the Project Management Body of Knowledge: PMBOK Guide, 6th ed, vol. 53 (2017). International Journal of Production Research 2. Project Management Institute: Success in Disruptive Times - Expanding the Value Delivery Landscape to Address the High Cost of Low Performance. PMI’s Pulse of the Profession, 35 (2018). https://www.pmi.org/learning/thought3. Hoyt, R.E., Liebenberg, A.P.: The value of enterprise risk management. J. Risk Insur. 78(4), 795–822 (2011) 4. Chapman, R.J.: Simple Tools and Techniques for Enterprise Risk Management, 2nd edn. Wiley, Hoboken (2011) 5. Abreu, A., Calado, J.M.F.: A risk model to support the governance of collaborative ecosystems. In: Proceedings of 20th World Congress of the International Federation of Automatic Control, Toulouse, França, 9–14 July 2017, vol. 50, no. 1, pp. 10544–10549 (2017) 6. Cagliano, A.C., Grimaldi, S., Rafele, C.: Choosing project risk management techniques. A theoretical framework. J. Risk Res. 18(2), 232–248 (2015) 7. Grace, M.F., Leverty, J.T., Phillips, R.D., Shimpi, P.: The value of investing in enterprise risk management. J. Risk Insur. 82(2), 289–316 (2015) 8. Smith, C.W.: Corporate Risk Management: Theory and Practice, (2015) (June), 637 9. Abreu, A., Martins, J., Calado, J.M.F.: Fuzzy logic model to support risk assessment in innovation ecosystems. In: 2018 13th APCA International Conference on Automatic Control and Soft Computing (CONTROLO), Ponta Delgada, pp. 104–109 (2018)

Study on the Isolator-Structure Interaction. Influence on the Supporting Structure J. P´erez-Aracil1(B) , E. Pereira2 , Iv´ an M. D´ıaz3 , and P. Reynolds1 1

Vibration Engineering Section, College of Engineering, Mathematics and Physical Sciences, North Park Road, Exeter EX4 4PY, UK [email protected] 2 Department of Signal Processing and Communications, Universidad de Alcal´ a, 28805 Alcal´ a de Henares, Madrid, Spain 3 E.T.S. Ingenieros de Caminos, Canales y Puertos, Universidad Polit´ecnica de Madrid, 28040 Madrid, Spain https://emps.exeter.ac.uk/engineering/research/ves

Abstract. Vibration isolation techniques allow reduction in base movement transmitted to a platform to be isolated. There are different techniques that can be applied, but in many scenarios, the most suitable control system is active vibration isolation. This applies an active force on a platform, and hence on the supporting structure. In some applications, the base structure can be considered as an infinitely stiff system. However, the use of the supporting structure for other purposes, such as human activities or different isolators, leads to a requirement to study how the active vibration isolation affects the base response in case it is not infinitely stiff. In this work, the general vibration isolation scenario is presented and the influence of active vibration isolation on the response of the supporting structure is analysed and compared with that of the alternative passive vibration isolation technique. Keywords: Vibration isolation · Active vibration control Isolator-structure interaction · Flexible structures

1

·

Introduction

Vibration isolation is the set of techniques used to reduce the vibration transmitted from a base to the platform of an isolator. Depending on the way in which the controlled force is generated, it is possible to distinguish three different techniques: (1) passive vibration isolation (PVI), (2) semi-active vibration isolation (SAVI), (3) active vibration isolation (AVI). In both techniques PVI [1] and SAVI [2], the force is generated as a consequence of the relative movement between the base and the platform. Vibration isolation performance is limited and some applications demand the application of an active force leading to the c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 394–403, 2021. https://doi.org/10.1007/978-3-030-58653-9_38

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use of AVI techniques [3–5]. With the complementary use of a sensor and a controller, this strategy allows to reach zero static displacement in the platform, to follow trajectories of target devices and to add active damping in the system to reach higher vibration isolation performance, among others [6]. There are many applications in which it is possible to consider that the base structure is a completely rigid system [7–10]. Then, the interaction phenomenon is negligible. However, the current trend towards slender and lighter structures constructed with light materials such as fiber-reinforced polymer (FRP) flooring is increasing the importance of the interaction phenomenon [11,12]. Therefore, this work studies the consequences of using PVI and AVI when the base structure is considered as flexible system, showing thus that the vibration isolation performance may be compromised due to isolator-structure interaction. The remainder of this paper is organised as follows. In the next section, a general vibration isolation framework is presented, in which the interaction phenomenon is considered. Section 3 presents the analysis of the influence on the supporting structure response that the use AVI technique has, and the controller used in this work is presented. In Sect. 4, the experimental identification of the isolator is shown, and two scenarios are analysed. Finally, the conclusions are given and discussed in Sect. 5.

2

Vibration Isolation Framework

The general vibration isolation problem is shown in Fig. 1, in which the mass to be isolated is represented by mp . The dynamic properties of the isolator are its stiffness and viscous damping, which are represented by kp and cp , respectively. The active force imparted by the isolator is denoted by fa (t), which depends on the chosen controller cf (t). The supporting structure is modelled as a single degree of freedom system, with mass mb , stiffness kb and viscous damping cb . The perturbation force is applied on the base, and it is denoted by fd (t). The equation of motion of the mass mp is: ¨p (t) − cp (x˙ p (t) − x˙ b (t)) − kp (xp (t) − xb (t)) = mp x ¨p (t), cf (t) ∗ x

(1)

in which ∗ is the convolution operator. The equation of motion of the supporting structure mass mb is: ¨p − cb x˙ b − kb xb = mb x ¨b . fd (t) − mp x

(2)

The transfer function (TF) that relates the movement of the platform and the base Xp (s)/Xb (s), which is usually called transmissibility, is derived: Gi (s) =

2ζp ωp s + ωp2 s2 + 2ζp ωp s + ωp2 −

, 1 2 mp Cf (s)s

(3)

where s represents the Laplace variable and the upper cases are used to denote the Laplace transform of the variables. The dynamic parameters ωp and ζp are

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c f (t)

x p (t) mp

kp

fa (t)

cp

xb (t) mb

cb

kb

fd (t)

Fig. 1. General perspective of the experimental setup.

the natural frequency and the damping ratio of the isolator, respectively, and they can be calculated from ωp2 = kp /mp and ζp = cp /(2mp ωp ). The general system, considering the interaction between the isolator and the supporting structure, can be ordered in the block diagram of Fig. 2.

Fig. 2. General scheme of the isolation problem.

The TF T (s) represents the passive transmissibility of the isolator, and it represents the ratio Xp (s)/Xb (s) when no active control is working in the isolator. It can be expressed as: T (s) =

s2

2ζp ωp s + ωp2 . + 2ζp ωp s + ωp2

(4)

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The relation between the platform acceleration and the force applied on it is denoted by Ga (s). Its expression is defined by: s2 /mb . s2 + 2ζp ωp s + ωp2

Ga (s) =

(5)

Finally, the base structure TF without considering any interaction is represented by Gb (s): s2 /mb . (6) Gb (s) = 2 s + 2ζb ωb s + ωb2 The dynamic parameters of the base, ωb and ζb , are obtained from ωb2 = kb /mb and ζb = cb /(2mb ωb ). If the interaction phenomenon is considered, the base structure TF between its acceleration, x ¨b (t), and the applied disturbance force, fd (t), is: Gb (s) , 1 + mp Gb (s)T (s) Gb (s) I , (s) = GAV b 1 + mp Gb (s)Gi (s) VI (s) = GP b

(7) (8)

VI The TF GP (s) represents the base acceleration against perturbation forces b I (s) when the isolator is working in passive mode. On the other hand, the TF GAV b represents the case when the AVI system is working in the isolator, as independent vibration system. They facilitate analysis of the response of the base structure for different isolator dynamic properties and controllers, Cf (s).

3

Influence of AVI Technique on the Supporting Structure

In this section, the influence that the AVI technique has on the base response is analysed. Firstly, a parameter to study the influence on the base structure is defined. Secondly, the controller used in this work is explained. 3.1

Influence Parameter

The vibration isolation performance is generally improved with the use of AVI techniques, however to understand how the different vibration isolation techniques affect the base response, the next parameter is defined: γ=

I (s)∞ GAV b , P V Gb I (s)∞

(9)

I which is the ratio between the H-infinity norm of both TFs, GAV (s) and b PV I Gb (s). If the value of γ is lower than one, it implies that the use of AVI control system produces a reduction on the base response respect to the use of PVI control. If γ > 1, the base response is increased when AVI control is working in the isolator with respect to the PVI control. If γ is approximately one, both control strategies generate the same response on the base structure.

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3.2

Controller Design

The control design used in this work is based on a direct velocity feedback (DVF) system. The aim is to introduce damping to the isolator system, with the aim to reduce the peak response at its natural frequency. The active force is proportional to the velocity of the platform, which allows to virtually emulate an sky-hook damper [13]. The ideal controller can be expressed as Cf (s) = Kv /s, where Kv is the control gain. As Kv increases, the active damping imparted by the active force Fa (s) increases too. However, in a real scenario, the implementation of this controller implies some issues, as it is very sensitive to low-frequency noise. This motivates the use of a lossy integrator, defined by: Cf (s) =

Kv , s/ωc + 1

(10)

in which ωc is the cut-off frequency of the controller. The selection of this frequency depends on the dynamic properties of the isolator and it plays a key role in the stability of the system. Based on the dynamic properties of the isolator used in this work, the cut-off frequency has been chosen to be 0.1ωp . As occurs with the ideal DVF controller, when Kv increases, the damping of the system does too. The controller gain Kv is chosen so that the TF Gi (s)  reaches −3 dB, at the damped frequency of the isolator, defined by ωpd = ωp

4

1 − ζp2 .

Case Studies: Numerical Results

In this section, two different scenarios are presented. The same isolator is used in both of them, which is based on an APS Dynamics Model 400 electrodynamic actuator. To identify its dynamic properties, it was situated on a beam (Fig. 3a) which was excited with a random perturbation. Both platform and base accelerations were measured while no controller was activated. The TF T (s) then was identified (Fig. 3b) and the natural frequency of the isolator ωp = 8.79 rads−1 (1.4 Hz) and its damping ratio ζp = 0.075, extracted from the peak-picking method [14]. The TF was identified to be: T (s) =

1.32s + 77.37 . s2 + 1.32s + 77.37

(11)

The identification of the dynamic parameters of the isolator allows to determine the transfer function between the platform acceleration and the force applied to it Xp (s)/Fa (s), since the mass to be isolated is known, mp = 31 kg. It is given by: 0.032s2 . (12) Ga (s) = 2 s + 1.32s + 77.37 The controller used in this work is based on Eq. 10, so its expression is: Cf (s) =

0.8796 . s + 0.8796

(13)

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7 6 5 4 3 2 1 0

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Fig. 3. General view of the experimental set-up (a) and experimental and theoretical FRFs of the TF T (s) (b).

The results are divided in two subsections. In the first one, the base response increases when the AVI control system is used with respect to the PVI technique. In the second subsection, the response of the supporting structure is reduced for AVI control with respect to PVI. The frequency response functions (FRFs) of the base structure are analysed and the platform time responses are shown for two cases: with and without isolator-base structure interaction between the isolator and the base structure. In both cases, the base is assumed to have a damping ratio, ζb , of 0.005. To compare the effect of the interaction phenomenon on the response of the platform, two scenarios are studied. In the first one, the isolator is situated on the flexible supporting structure, and a chirp signal is applied on the base system from 0 to 10 Hz, with an amplitude of 50 N. Both platform and base accelerations are registered. This platform acceleration is influenced by the base movement, then the interaction phenomenon is considered. In the second case, the registered base movement of the first scenario is used as input of Gi (s). The dynamics of the base Gb (s) are not considered, then the interaction phenomenon is avoided and the vibration isolation problem considers the base as a rigid body system. 4.1

Case 1: The Base Response Is Worsened for AVI Control

This scenario is presented as a case in which the use of AVI control increases the base response with respect to the use of PVI control. The relation of the dynamic properties between the supporting structure and the isolator is established by rm = mp /mb = 0.01, and rω = ωp /ωb = 0.9, which represent the mass and frequency ratios. The TFs of the supporting structure are shown in Fig. 4. As can be observed, the incorporation of an isolator on the base structure implies a reduction in its response. However, if both techniques PVI and AVI are compared, the use of this second one implies an increment of 93% in the response respect the passive system. In this scenario, the task developed on the supporting structure would be considerably affected by the vibration isolation work.

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Fig. 5. Time response of the platform against a chirp signal applied on the supporting structure of 50 N of amplitude for a worsened base response using AVI control system.

The time domain response of the platform is shown in Fig. 5. In this scenario, in which the base response is worsened for the use of AVI control system respect the PVI control, the interaction phenomenon lightly affects the platform movement.

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Fig. 6. Base FRFs without isolator (dotted line), with PVI control system (black line), and with AVI control system (grey line).

4.2

Case 2: The Base Response Is Improved for AVI Control

In this second scenario, the use of AVI control increases the base response respect to the use of PVI control. The mass and frequency ratios between the isolator and the base structure are rm = 0.1 and rω = 1.2, respectively. The value of γ is 0.47, then the peak response of the FRF in which the AVI control is used (grey 1.5

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Fig. 7. Time response of the platform against a chirp signal applied on the supporting structure of 50 N of amplitude for an improved base response using AVI control system.

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line) is lower than those of PVI control (black solid line), see Fig. 6. The vibration isolation work in this scenario does not imply any problem for a hypothetical task developed on the supporting structure. The time platform response considering the interaction phenomenon is shown in Fig. 7. As can be observed, the effect of the interaction phenomenon is more important than in the first case. This motivates the analysis of the effect that the vibration isolation task has in the response of the supporting structure.

5

Conclusions

Some vibration isolation tasks can be analysed under the hypothesis of a rigid support. However, when the base is flexible, the interaction phenomenon between the isolator and the supporting structure needs to be considered, especially if there are other activities on the base, such as human activities or other isolators. In this work, the general vibration isolation framework has been studied, which facilitates analysis of the base response for any controller working an isolator. The influence on the base response has been analysed for both techniques PVI and AVI. The dynamic properties of a real isolator have been extracted, and its transfer functions have been used to simulate the results in two scenarios. In the first one, the use of AVI technique worsens the response of the base respect to the use of PVI control. In the second case, the use of AVI control improves the base response respect to the use of PVI technique. The time response of the platform has been analysed in two scenarios. In the first one, the interaction is considered, while in the second one the base is considered as an infinitely rigid system. Future works will study the range of rm and rω in order to know when the interaction should be considered and when AVI improve the vibration performance respect to PVI. In addition, the results will be validated with experiments. Acknowledgements. This work is funded by the College of Engineering, Mathematics and Physical Sciences at the University of Exeter. Iv´ an M D´ıaz also acknowledges the financial support provided by the Ministry of Science, Innovation and Universities (Government of Spain) by funding the Research Project SEED-SD (RTI2018-099639B-I00).

References 1. Rivin, E.I.: Passive Vibration Isolation. ASME Press [u.a.], New York (2003). OCLC: 635787366 2. Liu, Y., Waters, T., Brennan, M.: A comparison of semi-active damping control strategies for vibration isolation of harmonic disturbances. J. Sound Vibr. 280, 21–39 (2005) 3. Preumont, A., et al.: A six-axis single-stage active vibration isolator based on Stewart platform. J. Sound Vibr. 300, 644–661 (2007) 4. Beijen, M.: Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation. Ph.D. thesis, Technische Universiteit Eindhoven (2018). OCLC: 8086831301

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5. Li, M., Zhang, Y., Wang, Y., Hu, Q., Qi, R.: The pointing and vibration isolation integrated control method for optical payload. J. Sound Vibr., September 2018 6. Ruzicka, J.E.: Active vibration and shock isolation. In: National Aeronautic and Space Engineering and Manufacturing Meeting, February 1968 7. Kaplow, C., Velman, J.: Active local vibration isolation applied to a flexible space telescope. J. Guidance Control Dyn. 3, 227–233 (1980) 8. Wang, C., Xie, X., Chen, Y., Zhang, Z.: Active vibration isolation through a Stewart platform with piezoelectric actuators. J. Phys. Conf. Ser. 744, 012006 (2016) 9. Wang, C., Xie, X., Chen, Y., Zhang, Z.: Investigation on active vibration isolation of a Stewart platform with piezoelectric actuators. J. Sound Vibr. 383, 1–19 (2016) 10. Mikhailov, V.P., Bazinenkov, A.M.: Active vibration isolation platform on base of magnetorheological elastomers. J. Magn. Magn. Mater. 431, 266–268 (2017) 11. Bastaits, R., Rodrigues, G., Mokrani, B., Preumont, A.: Active optics of large segmented mirrors: dynamics and control. J. Guidance Control Dyn. 32, 1795– 1803 (2009) 12. Zheng, G.T., Tu, Y.Q.: Analytical study of vibration isolation between a pair of flexible structures. J. Vibr. Acoust. 131, 021006 (2009) 13. Preumont, A.: Vibration Control of Active Structures. Springer, New York (2018) 14. Bendat, J., Piersol, A.: Engineering Applications of Correlation and Spectral Analysis. Wiley, New York (1993)

Fallback Approximated Constrained Optimal Output Feedback Control Under Variable Parameters Christian Kallies(B) , Mohamed Ibrahim, and Rolf Findeisen Laboratory for Systems Theory and Automatic Control, Otto von Guericke University, Magdeburg, Germany {christian.kallies,mohamed.ibrahim,rolf.findeisen}@ovgu.de http://www.control.ovgu.de

Abstract. Safety critical control problems often require the availability of fallback strategies, in case of failure of the main control scheme, sensors or actuators. Those controllers should provide safe operation or emergency shut down of the system under all circumstance. They should also be able to operate subject to reduced information, and limited computation power. We propose a verifiable and efficiently implementable output feedback controller based on an approximated explicit solution of a constrained optimal control problem. The control law is derived by solving an infinite horizon optimal control problem utilizing Al’brekht’s Method to obtain power series expansions. The feedback control law is a polynomial in terms of the measurements and estimated parameters, thus the online evaluation can be done efficiently. We provide conditions for convergence and existence of the optimal control law and the corresponding value function. Simulation results for the control of a non-linear quadcopter example show the effectiveness of the proposed strategy. Keywords: Approximated optimal control · Non-linear adaptive control · Al’brekht’s method · Parametric uncertainties · Sensor failure

1

Introduction

Many safety critical control tasks require the availability of a fallback controller [1]. This controller can be activated in case of the failure of the main control system, e.g. due to computational problems, sensors or actuators failures. A specific example is the control of the autonomous aerial vehicle which is subject to the failure of the sensor or actuator. The fallback controller then needs to be able to operate with reduced state information and should be also implemented on a reduced embedded computation power. To improve the reliability and prevent the system failure, the control system, e.g. in an aircraft, needs to be combined with a fault diagnosis module, e.g. a state/parameter observer [2], an unknown c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 404–414, 2021. https://doi.org/10.1007/978-3-030-58653-9_39

Approximated Constrained Optimal Output Control pest

Parameter Estimation pest (y, umin )

Control System yref

ysaf ety

Primary Controller umin (y1 , pest ) Fallback Controller umin (y2 , pest )

Measurements δs umin

System x˙ = f (x, u, p)

x

Primary Sensor y1 = h1 (x, p)

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Faults

δs

y

Standby Sensor y2 = h2 (x, p)

Fig. 1. Overall structure for the considered fault tolerant control case. The explicit solution of an optimal control problem umin (y, pest ) is calculated once offline for both controllers, and used as fallback controller. The control parameters pest are updated online via parameter estimation.

input observer [3], and an iterative learning observer [4], and suitable detection units, c.f. [5] for more details. Many different control schemes have been proposed as fallback controllers, e.g. gain adaptation schemes based on sliding mode controller [1,6], adaptive gain controller [7] or dual-layer control approach [8]. Often the primary control system is designed to be optimal with respect to a certain performance criterion, e.g. minimize the energy consumption, while taking the vehicle dynamics and constraints into account. Implementing such optimal controllers, as fallback strategy, is often challenging, e.g. in case of limited computational power. Furthermore, a priory verification of the stability of the resulting closed-loop is often not possible as the controller involves the numerical solution of an optimal control problem. This can be overcome by the use of an explicit solution of the optimal control problem. To approximate the solution of the Hamilton–Jacobi–Bellman equations for non-linear systems, several approaches have been proposed, e.g. based on inverse optimal design approaches [9,10] or the use of approximated controllers based on neural networks [13–15]. A formal iterative procedure was developed in [16] to solve an optimal control problem via power series expansions. Another approach was proposed in [17], which is applicable to a wider class of non-linear systems by expanding both the optimal cost and the non-linear dynamics as a power series in terms of the states. Al’brekht’s Method is used in [18] for discrete-time systems and also includes uncertain parameters as a subsystem with known dynamics. In [18], output variables are controlled using the complete state information. In this paper, we propose an explicit optimal control as a fallback controller for non-linear systems based on the power series expansion method, presented in [16,18–20], which can directly be based on a non-linear continuous-time model. We consider the case of the output feedback with constraints on the output values and input in presence of varying parameters. The proposed feedback control law is expressed as a power series expression of the current measurements y and the estimated parameters pest , see Fig. 1. Therefore, the optimal control strategy not only minimizes the performance index, but also considers the variable - uncertain

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- but estimated parameters pest . This allows to enhance the system performance and robustness.

2

Approximated Explicit Solutions

This section outlines Al’brekht’s Method to derive approximated solutions for constraint optimal control problems subject to variable parameters without full state information. The controller is based on the approximated solution of the following optimal control problem. Note that we first consider the unconstrained case before showing how constraints can be included.   V y(0), p = min

∞

u(·)

   y(t), u(t) dt

(1a)

0

s.t. x˙ = f (x, u, p), y = h(x, p),

(1b) (1c)

y(0) = y0 ∈ Rny ,

(1d)

where p ∈ Rnp is an uncertain parameter vector, which is assumed to be constant in theory but might vary in reality. The parameters p can also be seen as measured states with vanishing derivatives. The basic idea is to approximate the solution of this optimal control problem, which is a static output feedback controller via a suitable series expansion. To do so, we assume the functions f : Rnx × Rnu × Rnp → Rnx , h : Rnx × Rnp → Rny and  : Rny × Rnu → R are analytic in all variables such that they can be expanded into power series. f (x, u, p) = F x + Gu +

∞ 

f [i] (x, u, p),

h(x, p) = Hx +

i=2

∞ 

h[i] (x, p),

i=2

∞  1 1 (y, u) = y T yy y + y T yu u + uT uu u + [i] (y, u) 2 2 i=3

Here f [i] , h[i] , [i] denote all terms homogeneous with degree i in (x, u, p), (x, p) resp. (y, u). The matrices F ∈ Rnx ×nx , G ∈ Rnx ×nu , H ∈ Rny ×nx , yy ∈ Rny ×ny , yu ∈ Rny ×nu and uu ∈ Rnu ×nu need to fulfill conditions which will be addressed later. The dimension ny of the output should be smaller or equal to the state dimension nx . Furthermore it is crucial that  does not contain a linear part and ∀p ∈ Rnp : f (0, 0, p) = 0

and h(0, p) = 0.

(2)

The control law umin (·, ·) as well as the value function V (·, ·) are assumed to be analytic in a domain which contains the origin. In contrast to existing works [19], both the control law and the value function are expanded into power series in dependence of the parameters p and output variables y instead of states x.

Approximated Constrained Optimal Output Control

umin (y, p) = Ky +

∞ 

[i]

umin (y, p)

407

(3a)

i=2

and V (y, p) =

∞  1 T y Vyy y + V [i] (y, p) 2 i=3

(3b)

Note that (2) implies that ∀p ∈ Rnp : umin (0, p) = 0 and V (0, p) = 0. Therefore the linear part of the control law and the quadratic part of the value function do not depend on the parameters. To find the power series (3), the HamiltonJacobi-Bellman (HJB) equation is solved degree-wise. The HJB equation and the first order optimality criteria for the unconstrained problem are given by   0 = ∇y V (y, p) · ∇x h(x, p) · f x, umin (y, p), p   +  y, umin (y, p) , (4a)   0 = ∇y V (y, p) · ∇x h(x, p) · ∇u f x, umin (y, p), p   + ∇u  y, umin (y, p) (4b) and also include the output function h(·, ·). Considering y = x leads to the well known state feedback case [21,22]. Taking all terms that are linear in (y, p) from (4b) one obtains 0 = y T Vyy · H · G + y T yu + y T K T uu . Since this formula is valid in a neighborhood of the origin, y can be neglected and a formula for K is obtained if uu is invertible.   T T T K = −−1 uu · G H Vyy + yu Collecting all terms of degree two in (y, x) from (4a) 1 1 0 = y T Vyy · H · (F x + GKy) + y T yy y + y T yu Ky + y T K T uu Ky 2 2

(5)

is obtained. One could replace the output y with Hx but doing so would lead to a Riccati like equation which is not solvable under the usual assumptions. Therefore it is better to replace x in terms of y. Therefore we assume that H has full row rank resp. ˜ ∈ Rnx ×ny : H · H ˜ = In . (I) ∃H y Thus there exists an ny dimensional subspace of Rnx , where x can be replaced ˜ This replacement contains all the information that is needed to calculate by Hy. Vyy and thus K. Substituting K and x in (5) and again leaving out y results in the Riccati equation     T T T ˜ +H ˜ T F T H T Vyy + yy − Vyy HG + yu · −1 0 = Vyy HF H uu · G H Vyy + yu which is solvable if the conditions (II)–(IV) are holding.

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    ˜ := HF H, ˜ GH is stabilizable. (II) The pair F˜ , G  (III) The pair F˜ , yy ) is detectable. (IV) [2] (y, u) is convex and uu is positive definite. In further calculation steps terms which are homogeneous with degree i from (4b) and terms homogeneous with degree i + 1 from (4a) are collected to obtain [i] ˜ such that the equations for umin (y, p) and V [i+1] (y, p). x is substituted by Hy polynomials depend on (y, p). The resulting equations are linear in the unknown [i] polynomials umin and V [i+1] resp. their coefficients and yield a unique solution ˜ ˜ if F + GK is invertible, which is the case since Vyy is a solution of a Riccati equation and thus K is stabilizing the linearized output dynamics ˜ y˙ = F˜ y + Gu. Numerically these linear equations are obtained using symbolic differentiation which gives one equation for each coefficient. The calculation of the power series is ceased at a desired degree to obtain an explicit approximation of the control law and the value function. Note that the approximation error does not necessarily decrease with every further degree of approximation [22]. The question arises under which conditions both power series in (3) exist and converge. Theorem 1 (Existence of the power series). Consider the optimal control problem ∞ min u(·)

1 T 1 y yy y + y T yu u + uT uu u dt 2 2

(6a)

0

s.t. x˙ = f (x, p) + Gu, y = Hx, y(0) = y0 ∈ Rny .

(6b) (6c) (6d)

If the conditions (I)–(IV) and (2) hold then the power series in (3) exist locally around the origin and solve (4). Remark 1. The main idea of the proof can be found in the original paper by E.G. Al’brekht [16]. The proof is based on finding a converging power series which upper bounds (3b). Treating the parameters as measured states with vanishing ˜ are the main changes that one derivatives and replacing the states with x = Hy has to make. Theorem 1 only considers a quadratic cost function and system dynamics that are linear with respect to the input. The convergence in the [i] general case remains unclear even though umin (y, p) and V [i] (y, p) can be found. Remark 2. As mentioned before, the linearized system is stabilized by applying u = Ky. Thus local stability of the output is always achieved.

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Including Inequality Constraints

In practice it is important to take constraints (output and input) into account. Similar to [22] we propose to handle the constraints via adding logarithmic barrier functions to the cost function. This idea is well established in optimal control, e.g. [23,24]. In the following we consider the m ∈ N inequality constraints g(y, u) = g0 +

∞ 

g [i] (y, u) ≤ 0,

i=1

where g : R ×R → R is assumed to be analytic in all variables. Furthermore g0 is assumed to be component-wise negative such that the origin is a feasible point. We use a Taylor series expansion and a component-wise logarithm to get ny

nu

m



g(y, u) − g0 − log 1 − −g0



 i ∞  g(y, u) − g0 1 · . = i −g0 i=1

The linear part of this series will be neglected to keep the properties of the quadratic part of the cost function. To tune the cost function and and the behaviour close to the constraints, all terms are multiplied with a vector c ∈ Rm >0 of penalty constants. ˜ u) = (y, u) + cT · (y,

 i ∞  g(y, u) − g0 1 · i −g0 i=2

Remark 3. If the constraints g are convex, then ˜[2] is also convex. Furthermore the conditions (III) and (IV) stay valid. Calculating the control law and value function stays the same and solvability can be guaranteed for every choice of c. Remark 4. The stability of the closed loop system can be checked afterwards by using the polynomial approximation of V as a local Lyapunov function candidate, c.f. [20], or by searching of suitable candidates. The proposed strategy to include inequality constraints does not guarantee strict constraint satisfaction since it is usually not possible to calculate and use the whole series of umin . In fact if the control law is only calculated up to a certain degree d then from the cost function ˜[2] , . . . ˜[d+1] are used which implies that the cost at g(x, u) = 0 is finite. We note that similar approaches [18–20] omit constraints or they add vector norms of x, y, and u with even degrees to the cost function. Other approaches [12] are usually restricted to convex state constraints.

3

Quadcopter Example

Quadcopters are widely used in many applications as they are highly maneuverable and capable of diverse tasks such as hovering, vertical takeoff and landing. We consider a 10D scaled quadrotor model [25]:

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p˙y = vy ,

θ˙ = −d1 · θ + vθ , v˙ θ = −d0 · θ + n0 · uθ , v˙ y = g · tan(φ) + ωy , φ˙ = −d1 · φ + vφ , v˙ φ = −d0 · φ + n0 · uφ ,

p˙z = vz ,

v˙ z = −g + kt · uz

p˙x = vx , v˙ x = g · tan(θ) + ωx ,

The states are defined as: x = [px , py , pz , vx , vy , vz , θ, φ, vθ , vφ ] . Herein, px , py , pz define the position coordinates, vx , vy , vz are translational velocities. φ, θ, vφ , vθ are roll and pitch angles and rates respectively. While the control vector u = [uz , uφ , uθ ] includes the adjustable vertical thrust, roll and pitch angles. g = 9.81, kt = 0.91, n0 = 10, d0 = 10, d1 = 8 represent the known parameters. We consider two output vectors, i.e. different sensor information: y1 = [px , py , pz , vx , vy , vz , θ, φ, vθ , vφ ] ,

y2 = [pz , vx , y , vz , vθ , vφ ]

Thus we can measure the full state in the first scenario, while in the failure case, some states can not be measured or even estimated, e.g. position px , py and angles θ, φ due to loss of the GPS signal and gyro faults. While we can measure the altitude pz using i.e. a pressure altimeter. Compared to the model used in [25], we consider state-dependent uncertainties (ωx , ωy ) which is typically the case for aerial vehicles [21]. ωx = pxx · vx + pyx · vy + pzx · vz ,

ωy = pxy · vx + pyy · vy + pzy · vz

Here pij represent the effect on the j axes due to the velocity component i. The parameters can vary and we assume that they can be obtained via a suitable estimation approach. As cost function we use (y, u) =

1 1 T y yy y + uT uu u 2 2

with uu = I3 , yy = I10 for full state measurements and yy = I6 in case of sensor failure. In both scenarios the input constraints 0 ≤ uz ≤ 2g, |uφ | ≤ π π 9 , |uθ | ≤ 9 are used. To fulfill the assumptions made in Sect. 2, the input uz will ˜z + kgt . The outlined approach is used to derive a parametric be replaced by uz = u approximated explicit solution of (1). The resulting control law umin (y, p) only requires the evaluation of the power series expansion of the current output values y and the uncertain parameters p. This allows for real-time implementation even in case of limited computational power. For safety and reliability verification the resulting control law can furthermore be validated under different circumstances (e.g. sensor faults and wind disturbance). 3.1

Simulation Results

In this section, the effectiveness of the proposed strategy to mitigate the effect of the external disturbances is validated via simulation results for a quadcopter. The simulations consider different scenarios of the external disturbances. In the simulations, the control objective is to guide the quadcopter to the origin, i.e.

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yref = 0 ∈ R10 resp. ysafety = 0 ∈ R6 . For reducing the disturbance effect, the controller is parametrized in terms of the uncertain variable parameters: umin (y, p) =

d 

[i]

umin (y, p).

i=1

We first analyze the influence of increasing the approximation order d of the optimal control law. In the sensor fault scenarios, see Fig. 2, the third order approximation of the control law achieves the best, fast and safe landing despite the wind disturbances. This is achieved by a sufficiently high penalty value in the objective function. The third order approximation achieves fast landing at closest position (16, 16) comparing to second order at (21.5, 21.5) and first order at (30.5, 30.5). Furthermore, one can see that considering the disturbance estimation improves the control performance, see Fig. 4 and increase the safety by stoping the vehicle motion despite the wind disturbance, see Fig. 3. In this work, we proposed a real-time processor-in-the-loop simulation, where the main nonlinear simulation is carried out in the ground station, Intel CoreT M i7-6700 CPU @ 3.40 GHz, while the controller is running on the onboard computer Intel NUC/CoreT M i7-8559U 4 × 2.7 GHz. Both computations are performed via two different Matlab environment, where the communication architecture between the two computers is realized through Ethernet network via user datagram protocol. Using the explicit solution reduces the computational time significantly, see Table 1, which allows to implement the optimal controller in real-time even on computationally limited embedded systems.

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4

Conclusion and Future Work

In this paper, an approximated optimal output feedback control approach is proposed as a fallback controller. This controller is based on an approximated optimal control approach using Al’brekht’s Method in order to achieve a computational implementable solution. The strategy obtains an approximated explicit solution of the optimal control problem allowing for variable parameters. This allow to capture uncertainties and the mismatch between the model and the

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system being controlled. The efficiency and performance of the proposed approximated optimal control is evaluated via a quadcopter simulation study. The simulation studies underline the capability of the approach with respect to efficient implementation, constraint handling despite sensor faults and disturbance rejection. Future work will focus on validating the approach in real experiments.

References 1. Hu, H., Liu, L., Wang, Y., Cheng, Z., Luo, Q.: Active fault-tolerant attitude tracking control with adaptive gain for spacecrafts. Aerospace Sci. Technol., 105706 (2020) 2. Mousavi, M., Rahnavard, M., Haddad, S.: Observer based fault reconstruction schemes using terminal sliding modes. Int. J. Control 93(4), 1–8 (2018) 3. Cristofaro, A., Johansen, T.A.: Fault tolerant control allocation using unknown input observers. Automatica 50(7), 1891–1897 (2014) 4. Hu, Q., Niu, G., Wang, C.: Spacecraft attitude fault-tolerant control based on iterative learning observer and control allocation. Aerosp. Sci. Technol. 75, 245– 253 (2018) 5. Yin, S., Xiao, B., Ding, S.X., Zhou, D.: A review on recent development of spacecraft attitude fault tolerant control system. IEEE Trans. Ind. Electron. 63(5), 3311–3320 (2016) 6. Wang, X., Sun, S., van Kampen, E.J., Chu, Q.: Quadrotor fault tolerant incremental sliding mode control driven by sliding mode disturbance observers. Aerosp. Sci. Technol. 87, 417–430 (2019) 7. Tian, B., Yin, L., Wang, H.: Finite-time reentry attitude control based on adaptive multivariable disturbance compensation. IEEE Trans. Ind. Electron. 62(9), 5889– 5898 (2015) 8. Edwards, C., Shtessel, Y.: Adaptive dual-layer super-twisting control and observation. Int. J. Control 89(9), 1759–1766 (2016) 9. Pan, Z., Ezal, K., Krener, A.J., Kokotovic, P.V.: Backstepping design with local optimality matching. IEEE Trans. Autom. Control 46(7), 1014–1027 (2001) 10. Margaliot, M., Langholz, G.: Some nonlinear optimal control problems with closedform solutions. Int. J. Rob. Nonlinear Control IFAC-Affiliated J. 11(14), 1365–1374 (2001) 11. Xiao, B., Huo, M., Yang, X., Zhang, Y.: Fault-tolerant attitude stabilization for satellites without rate sensor. IEEE Trans. Ind. Electron. 62(11), 7191–7202 (2015) 12. Xin, M., Balakrishnan, S.: A new method for suboptimal control of a class of non-linear systems. Optimal Control Appl. Methods 26(2), 55–83 (2005) 13. Dalamagkidis, K., Valavanis, K.P., Piegl, L.A.: Nonlinear model predictive control with neural network optimization for autonomous autorotation of small unmanned helicopters. IEEE Trans. Control Syst. Technol. 19(4), 818–831 (2010) 14. Liu, Y., Zhang, H., Yu, R., Qu, Q.: Data-driven optimal tracking control for discrete-time systems with delays using adaptive dynamic programming. J. Franklin Inst. 355(13), 5649–5666 (2018) 15. Kiumarsi, B., Lewis, F.L.: Actor–critic-based optimal tracking for partially unknown nonlinear discrete-time systems. IEEE Trans. Neural Netw. Learn. Syst. 26(1), 140–151 (2014) 16. Al’brekht, E.: On the optimal stabilization of nonlinear systems. Appl. Math. Mech. 25(5), 836–844 (1961)

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17. Garrard, W.L., Enns, D.F., Antony Snell, S.: Nonlinear feedback control of highly manoeuvrable aircraft. Int. J. Control 56(4), 799–812 (1992) 18. Krener, A.: Adaptive horizon model predictive control. IFAC-PapersOnLine 51(13), 31–36 (2018a) 19. Krener, A.: Adaptive horizon model predictive regulation. IFAC-PapersOnLine 51(20), 54–59 (2018b) 20. Lucia, S., Rumschinski, P., Krener, A., Findeisen, R.: Improved design of nonlinear model predictive controllers. IFAC-PapersOnLine 48(23), 254–259 (2015) 21. Ibrahim, M., Kallies, C., Findeisen, R.: Learning-supported approximated optimal control for autonomous vehicles in the presence of state dependent uncertainties. In: Proceedings of the 19th European Control Conference, Saint Petersburg, Russia (2020) 22. Kallies, C., Ibrahim, M., Findeisen, R.: Approximated explicit infinite horizon constraint optimal control for systems with parametric uncertainties. In: Proceedings of 21st IFAC World Congress. Berlin, Germany (2020) 23. Feller, C., Ebenbauer, C.: Input-to-state stability properties of relaxed barrier function based MPC. IFAC-PapersOnLine 48(23), 302–307 (2015) 24. Wu, Z., Christofides, P.D.: Optimizing process economics and operational safety via economic MPC using barrier functions and recurrent neural network models. Chem. Eng. Res. Des. 152, 455–465 (2019). https://doi.org/10.1016/j.cherd.2019. 10.010 25. K¨ ohler, J., Soloperto, R., M¨ uller, M.A., Allg¨ ower, F.: A computationally efficient robust model predictive control framework for uncertain nonlinear systems. Submitted to IEEE Trans. Autom. Control (2019)

Teaching Neural Control with an Arduino Based Control Kit Ramiro S. Barbosa(B) GECAD - Research Group on Intelligent Engineering and Computing for Advanced Innovation and Development, Department of Electrical Engineering, Institute of Engineering – Polytechnic of Porto (ISEP/IPP), Porto, Portugal [email protected]

Abstract. This paper presents an Arduino based control kit used to reinforce concepts on neural networks. The kit is used in the Unit Course of Advanced Control Systems taught at Institute of Engineering of Polytechnic of Porto. It describes some experiments regarding modeling, identification and control using neural networks. The education control kit proved to be a useful tool to consolidate the theoretical concepts as well to make the classes more interesting, participatory and motivating for students. Keywords: Neural control · Arduino · Control education

1 Introduction One of the best ways to learn about control is to design a controller, implement it on a system and visualize its operation experimentally. Usually the control concepts are taught almost on a theoretical point of view. Although this approach is necessary to create a strong mathematical basis about the course subjects, for the students is often difficult to see its connection to the reality. One way to overcome these difficulties is to use computer simulations. However, a model of the system used in this kind of simulations is only an approximate model of the real system. In addition, the simulations do not reflect all phenomena involved in the physical system. In this work, we describe a set of experiments using an experimental lab kit based on Arduino to reinforce the concepts of neural networks applied to control. The kit consists essentially on two heaters, two actuators and two temperature sensors. The goal is to control the actuators to maintain a temperature setpoint. Students perform experiments regarding system modelling, system identification and control. The lab kit is used for the Unit Course (UC) of Advanced Control Systems (SISCA) at Institute of Engineering of Polytechnic of Porto (ISEP/IPP). We can find other similar take home laboratory kits [1–3]. Temperature Control Lab (TCLab) is an application of feedback control with an Arduino. It was developed at Brigham Young University, and is a pocket-size lab with software in Python, MATLAB, and Simulink for basic and advanced control theory with applications in modeling, estimation, and control [4]. This resource is used worldwide © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 415–424, 2021. https://doi.org/10.1007/978-3-030-58653-9_40

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by numerous Universities for process control instruction. The UTAD University used the TCLab on the Biomedical Engineering course to teach introductory feedback control. Students using TCLab addressed topics like model identification based on step response, design of PID controllers, and digital implementation of PID controllers. The feedback from students was positive motivating them to pursue further experiments [5]. The structure of the paper is as follows. Section 2 introduces the experimental control kit, describing its main components and functionalities. Section 3 presents the lab experiments applied to the control kit illustrating concepts of modeling, identification and control using neural networks. Finally, Sect. 4 addresses the main conclusions.

2 Arduino Based Lab Control Kit The experimental lab control kit used in the UC of SISCA held at ISEP/IPP consists of an Arduino UNO (or compatible) development board and a shield with the thermal processes to be controlled. In addition, it is necessary a USB cable, a PC/laptop with a compatible USB port, and a 12 V 3 A power supply. Figure 1 presents the complete setup of the system. USB Port 12 V 3 A Power supply

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Fig. 1. System setup for Arduino based control kit.

The Arduino shield is shown in Fig. 2. It contains two TMP36 temperature sensors (S1 and S2), two TIP31A power transistors (T1 and T2), and two power resistors of 33 , 5 W (R1 and R2). The temperature sensors generates an output voltage proportional to the measured temperature. The transistor actuators (T1 and T2) allows to adjust the power dissipated in the power resistors (R1 and R2) in order to maintain the temperature at a prescribed desired value. This board also contains two LEDs (LED 1 and LED 2), which can be controlled by ON/OFF or by PWM, and a push button that can be used as an I/O input or as an external interrupt. The lab works are realized usually in groups of two students (to be defined according to the number of students enrolled in the UC). The lab kit is provided to each group, which is responsible for it until the end of the semester. The shield PCB layouts are also made available to students develop and implement their own temperature control kit. This hardware kit is suitable for students to take home and use on their home facilities.

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In the experiments outlined in this work, the temperature controller is running in Simulink on a host computer. This was accomplished employing the Arduino IO Package from the Mathworks that includes a specialized blockset and a server program running on board the Arduino for communicating between the board and the host computer running Simulink. This IO package allows us to communicate with the board in real time, and observe and plot the measured temperature (and other variables) during the experiment [7]. The lab kit may be used to reinforce concepts on a broad range of topics such as modeling techniques (step and doublet testing), simple control algorithms (On-Off, On-Off with hysteresis), PID control, feedforward and cascade control, and intelligent control techniques (fuzzy and neural control), among others.

3 Experiments 3.1 Modeling One of the first experiments that students must perform is to derive a model for the dynamics of the thermal system based on the step response. The observation of the temperature curve responses indicate that the dynamics of the system can be approximated by a first order transfer function with a time delay: G(s) =

K e−τd s τs + 1

(1)

where K is the DC gain, τ the time constant, and τd the time delay. With a sampling period of T , the discrete-time equation is (with no time delay):   y(k) = e−T /τ y(k − 1) + K 1 − e−T /τ u(k − 1) (2) y(k) = ay(k − 1) + bu(k − 1)

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  where a = e−T /τ , b = K 1 − e−T /τ . Table 1 shows the estimated parameters for thermal model S1 using the two-point method and a linear neuron as represented in (2) [8]. Table 2 presents the model parameters for thermal model S2 obtained by the twopoint method. As can be seen, S1 represents a first-order system while S2 is modeled by a first-order system with a time delay (FOPDT). In this experiment, the students may try different methods for obtaining the model parameters, namely the neural network, tangent method, tangent-and-point method, or an optimization method based on a leastsquares fit [8–10]. The SSE value represents the sum of squared errors between the model data (or network data) and the step response of thermal system. Table 1. Parameters of thermal model S1. Method

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Two-point method 0.73 107 – Linear neuron

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Two-point method 0.65 129 12

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3.2 System Identification System identification is an integral part of any control architecture design. The system identification is necessary to establish a model based on which the controller can be designed, and it is useful for tuning and simulation before applying the controller to the real system [8]. The objective is to train a neural network to represent the forward dynamics of the process. The prediction error between the process output and the neural network output is used as the neural network training signal, as represented in Fig. 3. The model used for nonlinear identification is the Nonlinear Autoregressive-Moving Average (NARMA) model [11, 12]:   y(k + d ) = f y(k), . . . , y(k − n + 1), u(k), u(k − 1), u(k − m + 1) (3) where u(k) is the system input, y(k) is the system output and d is the system delay. The neural network is trained to approximate the nonlinear function f . The neural network model is a two-layer feedforward network with hyperbolic tangent hidden units and linear output units. The structure of the neural network process model is given in Fig. 4. The equation for the process model is given by:  N  M   vl σ wli ϕi (k) + wl0 + v0 (4) yˆ (k|θ) = l=1

i=1

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where yˆ is the predicted value of the output y at sampling instant t = kT , θ = [vl wli ] is the vector containing all network weights and biases, ϕ is the regression vector which contains past outputs and past inputs:  T ϕ(k) = y(k − 1) · · · y(k − n)u(k − 1) · · · u(k − m) = [ϕ1 (k)ϕ2 (k) · · · ϕN (k)]T

(5)

Fig. 3. System identification.

Fig. 4. Neural network process model.

 

Given a set of collected data, u(k), y(k) , k = 1, . . . , P] , the network is trained to minimize the mean square error criterion: J =

P 2 1  y(k) − yˆ (k|θ) 2P

(6)

k=1

In this way, a gradient (descent) iterative search algorithm can be applied such that, at each iteration, the weights are modified along the opposite direction of the gradient: θ(i+1) = θ(i) − η(i)

∂J ∂θ(i)

(7)

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where η is the learning rate of the algorithm. For the identification phase, a doublet type signal is applied to thermal system S1 (Fig. 5). The model structure of network is selected such that m = n = 2. With a batch training procedure, the network is trained to achieve a mean square error (MSE) < 1 × 10−4 . We use the Levenberg-Marquardt algorithm [12, 13], which uses a backpropagation procedure that is very similar to the backpropagation algorithm, and 10 neurons in the hidden layer. Figure 5 show the responses of the thermal system and of the neural network model. We verify that both curves are almost identical assessing the performance of the neural model to represent the system dynamics. In this experiment, we ask students to iterate on number of hidden layers, number of neurons in each layer, activation functions, MSE goal, learning rate, and training algorithms in order to assess the learning capability of network to predict the nonlinear output. This gives students a better understanding of neural network operation and the function of each part of network structure. After this, the trained network is tested on a different input (i.e. unknown) from that used in the training. Figure 6 shows the responses of neural network and thermal system to a step 80 System Model

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input. Once more, we see a good accordance between the system response and the neural network output proving the good generalization capability of the neural network. 3.3 Control This section introduces two simple neural control structures, the direct inverse neural control and the feedforward-feedback control structure. The goal is to give to students the fundamentals of application of neural networks in control, and at same time to present control structures that yield good performance in practice. In general, simple solutions can solve most of the problems. Besides the schemes presented here, the students are encouraged to apply most complex structures such as neural predictive control and model reference control. Direct Inverse Neural Control. In this control structure, the neural network is trained to act as the inverse of the system, and use this as a controller. Figure 7 shows the control scheme [8].

Fig. 7. Direct inverse neural control.

Assume that the system can be described by:   y(k) = f1 y(k − 1), . . . , y(k − n), u(k − 1), u(k − m)

(8)

Perform an experiment on the thermal system to collect the set of data {[u(k), y(k)], k = 1, . . . , P]}. An inverse model of the system can be inferred from the data set {[y(k), u(k)], k = 1, . . . , P]}. The output of the inverse model is u(k):   u(k) = f2 y(k + 1), y(k), . . . , y(k − n + 1), u(k − 1), . . . , u(k − m + 1) (9) Substitute in Eq. (9), the output y(k + 1) by the desired output, the reference r(k). If the neural network represents the exact inverse, the system output at time k + 1 will track the input reference r(k) (see Fig. 7).

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The training of the network is done analogously to the system identification problem of Subsect. 3.2 by changing the regressors and network output. In this case, the network is trained to minimize the criterion: J =

P 2 1  u(k) − uˆ (k|θ) 2P

(10)

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Figure 8 (upper graph) shows the reference signal and the response of the thermal system S1 for the direct inverse neural control. The control action is shown in Fig. 8 (lower graph). The model structure of network is selected such that m = n = 1. We use the Levenverg-Marquardt algorithm and 10 neurons in the hidden layer. We observe an adequate transient response performance as well a good steady-state behavior. In this experiment, we propose to students to test the tracking capability of the neural network based controller by applying different arbitrary specified trajectories. A simulated feedback system of Fig. 7 is used to compare the responses of the real system and of the simulated one. In this way, the accuracy of the model and existing nonlinearities of the system can be assessed. 80 Output Reference

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Feedforward-Feedback Control. This control structure is shown in Fig. 9. The feedforward control is used to improve the reference tracking, while feedback is used for stabilizing the system and for suppressing disturbances [8, 12]. In this structure, an inverse model is trained as in (9). The feedforward component of the control input is then given by replacing all system outputs by corresponding reference values:   (11) uff (k) = f r(k + 1), . . . , r(k − n + 1), uff (k − 1), . . . , uff (k − m + 1)

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Figure 10 shows the reference signal and the response of thermal system S1 for the feedforward-feedback control. The inverse neural network model was trained using m = n = 1, the Levenverg-Marquardt algorithm, and 10 neurons in the hidden layer. The PID controller was tuned using the IMC method and model parameters of transfer function (1) [9, 14]. The output response shows reasonable performance in both transient and steady-state behaviors. 60

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4 Conclusions This paper has given an overview on the use of neural networks in teaching control systems. A low-cost Arduino lab control kit was used to reinforce concepts in modelling, system identification and control. A set of experiments were established to fill the gap

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between the theory and practice. These experiments are carried out as a module in the UC of Advanced Control Systems of the MSc. on Electrical and Computer Engineering held at the Institute of Engineering of Polytechnic of Porto (ISEP/IPP). The student’s feedback comments about the use of the lab kit were quite positive. In student’s perspective, the laboratorial classes were more motivating and interesting. On the instructor’s side, it was noted an increase in student attendance and participation, not only in the laboratorial classes but also in other components of the UC. For the success of this methodology also contributed the easily plug and play kit via USB, adequate lab support resources, and MATLAB/Simulink software with the free installation support hardware packages. Other advantages referred by students include the easy to use and affordable control kit. Acknowledgements. This work is supported by FEDER Funds through the “Programa Operacional Factores de Competitividade - COMPETE” program and by National Funds through FCT “Fundação para a Ciência e a Tecnologia” under the project UIDB/00760/2020.

References 1. Rossiter, J.A., Jones, B.L., Pope, S., Hedengren, J.D.: Evaluation and demonstration of take home laboratory kit. In: 12th IFAC Symposium on Advances in Control Education, Philadelphia, PA, USA, 7–9 July 2019, pp. 56–61 (2019) 2. Rossiter, J.A., Dormido, S., Vlacic, L., Jones, B.L.I., Murray, R.M.: Opportunities and good practice in control education: a survey. In: Proceedings of the 19th World Congress, the International Federation of Automatic Control, Cape Town, South Africa, 24–29 August 2014, pp. 10568–10573 (2014) 3. Park, J., Abraham Martin, R., Kelly, J.D., Hedengren, J.D.: Benchmark temperature microcontroller for process dynamics and control. Comput. Chem. Eng. 135, 1–13 (2020) 4. Hedengren, J.D.: Temperature Control Lab. http://apmonitor.com/pdc/index.php/Main/Ard uinoTemperatureControl. Accessed 10 Feb 2020 5. Moura Oliveira, P., Hedengren, J.D.: An APMonitor temperature lab PID control experiment for undergraduate students. In: 24th IEEE Conference on Emerging Technologies and Factory Automation (ETFA), Zaragoza, Spain, 10th–13th September 2019, pp. 790–797 (2019) 6. Barbosa, R., Mendes, H.: Sistema Eletrónico de Regulação Térmica para Arduino. Laboratory tutorial (in Portuguese) (2019). available upon request to author email: [email protected] 7. Legacy MATLAB and Simulink Support for Arduino. https://www.mathworks.com/mat labcentral/fileexchange/32374-legacy-matlab-and-simulink-support-for-arduino?s_cid=src htitle. Accessed 10 Feb 2020 8. Gopal, M.: Digital Control and State Variable Methods: Conventional and Intelligent Control Systems, 3rd edn. McGraw-Hill, Singapore (2010) 9. Seborg, D.E., Edgar, T.F., Mellichamp, E.A., Doyle, F.J.: Process Dynamics and Control, 3rd edn. Wiley, NY (2011) 10. Jesus, I.S., Barbosa, R.S.: Genetic optimization of fuzzy fractional PD+I controllers. ISA Trans. 57, 220–230 (2015) 11. Hagan, M., Demuth, H., De Jesus, O.: An introduction to the use of neural networks in control systems. Int. J. Robust Nonlinear Control 12(11), 959–985 (2002) 12. Hagan, M., Demuth, H., Beale, M.: Neural Network Design. PWS Publishing, Boston (1996) 13. Haykin, S.: Neural Networks and Learning Machines, 3rd edn. Pearson, New Jersey (2009) 14. Åström, K.J., Hägglund, T.: PID Controllers: Theory, Design, and Tuning. Instrument Society of America, Pittsburgh (1995)

Differential Observation and Integral Action in LTI State-Space Controllers and the PID Special Case Paulo Garrido1,2(B) 1 CAR RG, Algoritmi Center, Guimarães, Portugal

[email protected] 2 LIAAD, INESC TEC, Porto, Portugal

Abstract. This paper makes the case that practical differentiation of measured state variables may be seen as an observation or estimation scheme for linear time invariant state space controllers. It is shown that, although not having the separation property, the estimation error of this scheme converges to zero if the resulting closed loop system is strictly stable. On the basis of this concept, it is shown that PID controllers may be interpreted as a special case of state space controllers endowed with differential observation. The interesting consequences of this interpretation are discussed. Keywords: State observation · Differential observation · State-space controllers · PID controllers

1 Introduction The goal of this paper is two fold. First, one wants to present the concept of differential observation as embodying an approach – along full state [1] and Luenberger observers [2] – to estimate states in state space controllers. Second, one wants to show that with the concept of differential observation and the well-known implementation of integral action in state space controllers, one is able to understand PID controllers [3] as a particular case of state space controllers. This understanding has relevant consequences for control theory, the understanding of control practice, the design of PID controllers and control teaching. These are better appreciated after presenting the main ideas in this introductory section. Section 2 develops the concept of differential observation in continuous time assuming a flat frequency response above the upper differentiation frequency. Properties of this observation scheme are given as well as the general expression of a state space controlled system with this observation scheme. Section 3 gives a review of integral action in state space controllers. In Sect. 4 one presents the interpretation of a PID controller as a state space controller with differential observation. Section 5 concludes with some notes about further research. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 425–434, 2021. https://doi.org/10.1007/978-3-030-58653-9_41

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1.1 Differential Observation While not quite referred as such, practically differentiating a signal can be understood as estimating the derivative of the signal. This is so because differentiation is a non-causal operation. As it is believed that non-causal processes cannot be physically implemented, it follows that if one uses an electronic device, say a properly configured op-amp, to calculate the derivative of its input what one gets is at most an approximation of the values that would be output by an ideal (non-causal) differentiator. This can also be seen in the frequency domain as the amplitude frequency response of an ideal differentiator should tend to infinity as ω tends to infinity. A practical differentiator must be band limited or must have an upper or corner frequency ωc above which amplitude ceases to grow. The higher this frequency is, the smaller is the approximation error. Assume that in a control system, plant variable x 1 is the derivative of variable x 2 . One measures x 2 and practically differentiates the measure to obtain x s1 . The approximation error es1 = x 1 –x s1 cannot be calculated on-line (otherwise x 2 could be differentiated) but under appropriate conditions it can be bound and shown to converge to zero. This allows one to understand x s1 as an observation or estimate of x 1 . Of course, there are other causes of uncertainty, as disturbances or measurement errors, but the differentiation error es1 is intrinsic to the process. Full-order or reduced order state observers are supposed to perform only causal operations, but they also have an intrinsic and irreducible cause of output error: the uncertainty about the plant model used in the observer. This is not the case of practical differentiators as they are model free. Therefore, we can interpret the operation of practical differentiators as differential observation of state variables at par with reduced-order observers. The output of practical differentiators, as the output of full or reduced order observers, can be seen as estimates of non-measured state variables. While practical differentiation has an intrinsic error in being a causal approximation to a non-causal operation, full or reduced order observers present an estimation error whenever their model parameters do not match the plant parameters – furthermore, in this condition, they will lose the separation property. Studying differential observation as a reduced order observation scheme, gives the following results further developed in Sect. 2: – In principle, differential observation can be used together with full state feedback and partial only state measurement as full or reduced order observation can. – Differential observation does not have the separation property [3]. The poles resulting from upper or corner frequencies ωc1 , ωc2 , …, do displace the closed loop poles from the positions intended assuming full-state measurement. This effect grows with the lowering of the corner frequencies and diminishes with their increase. – The displacement effect can be easily assessed for strict stability of the regulated system modes and of estimation errors, therefore it turns out that differential observation can be incorporated into full state feedback designs as an appropriate observation scheme to consider.

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1.2 Integral Action In order to counter the effect of disturbances and to diminish steady-state errors, one may use integral action or feedforward schemes. These ones rely upon disturbance measurement or estimation through disturbance observers while integral action is model free, giving good results under uncertainty or changing plant parameter values. A full state feedback does not warrant zero steady-state errors to references or disturbances that do not tend to zero. Among these, steps and ramps are of special interest to consider. Steps are often good models not only for many references and disturbances but also, given linearity, to constant components of any input. Obtaining zero error in steady state to a step reference or disturbance for some state variable can be accomplished adding to the system, at most, one integral of the variable error. Obtaining zero error in steady state to a ramp reference or disturbance for some state variable can be accomplished adding to the system, at most, two integrals of the variable error. In order to get zero steady state error to a time power kt n reference or disturbance the system transfer function must be of type n – which means that the loop transfer function must have n pure integrators. Integral action is readily inserted into state-space controllers by associating to each variable for which a requirement of zero steady state error exists one or more states which will be an integral of the variable error, an integral of the integral of the variable error, etc. and making the controller calculate them. Understanding these integrals as additional state variables in an augmented system allows one to establish the closed loop poles of the augmented system through a full-state feedback vector. Applying integral action is not a free lunch and the price to pay lies somewhere between slow modes of the error integrals and significant worsening of gain and phase margins. Furthermore, use of integral action requires some form of anti-reset windup to counter the effect of actuator saturation. 1.3 Interpreting PID Controllers as Slots of State-Space Controllers For generality, let one assume that a system has n state variables, of which n/2 are measured, let us call them the P-variables. The other n/2 variables are derivatives of the P-variables and we will call them the D-variables – to be estimated through differential observation. Assume that it is meaningful to specify independent reference trajectories for the P-variables as well as finite or null steady-state errors to be attained through integral action. Then, the controller must calculate n/2 error integrals, which one will call the I-variables. The D, P and I-variables make up the augmented system state upon which a full feedback will be applied. Metaphorically, one can say that the control system has three layers of variables: D, P and I. Now it can be recognized that a usual PID controller makes up a “vertical” slot of such a state space LTI controller. Actually, one can see that a PD controller can be interpreted as a full state feedback with differential observation for a second order system with the property of one state variable being the derivative of the other. A PID controller will be the version with integral action of such a controller.

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1.4 Consequences of the Interpretation From a theoretical viewpoint, understanding PID controllers as special cases of statespace controllers allows one to unify the theory of state space controllers and PID controllers, actually subsuming the somehow scattered and intuitive theory of PID controllers into the rigorous, elegant, systematic mathematical formalism of state space theory underlying a deeper understanding of control. One will not need to refer anymore to PID and state space controllers as if they were different kinds of entities. A PID controller is what results when one designs a state space controller for a system that can be controlled assuming a second (or first) order linear model, through using differential observation and integral action. This is a kind of most searched theoretical result in any branch of science: to make the theory simpler by diminishing the number of distinct entities to be considered. Also from a theoretical viewpoint, this result is most satisfactory in another way. It is a well-known fact that the vast majority of actually working controllers are of the PID type. Often this is commented with an overtone of regret, as it seems that more “mathematically advanced” control algorithms do not find their way into practice. Face to the results of this paper, the overtone of regret may end as whenever one applies a PID controller one is in fact applying state space control methods! The question of why Luenberger observation did not displace differentiation to estimate the unmeasured state and also of why feedforward schemes did not displace integral action to compensate for disturbances has a convincing answer in operational and implementation terms: both Luenberger observers and feedforward schemes require i) a well-known linear model that ii) represents the system well and iii) whose parameters are known with great precision and iv) are time invariant. Considering the uncertainty of model knowledge, the difficulty of parameter measuring and, even if this could be satisfactorily solved, the need to go into an adaptive controller rather than a time invariant one for systems whose parameters change in time (most of them), one concludes on the grounds of state space theory that a most attractive solution in terms of implementation cost for many systems must be a PID controller experimentally tuned. From the viewpoint of design, the results of this paper suggest to design PID controllers in a state-space based approach. Several possibilities exist. One may first assume a full-state feedback to arbitrate closed loop poles leading to P and D gains (and the I gain) and then check that the pole shift resulting from practical differentiation stays within acceptable bounds. Another possibility (requiring further research) will be to develop experimental procedures to tune PID controllers on a state space basis. For control teaching the results of the paper are also to consider. If the view that PID controllers are a special case of state space controllers becomes usual then it may also be recognized that teaching PID controllers before state space ones – and these as a kind of an “advanced” type – does not make much sense. In an introductory control course, one will start with the state space approach and derive the PID controllers as the special case that they are. It is to expect that such a change in teaching will lead to a significant positive change in practice.

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2 Differential Observation If one could make an ideal differentiator, its transfer function would be Hi (s) =

Y (s) =s U (s)

(1)

Equation (1) implies that the modulus of H i (jω) must grow with ω without limit. For a practical differentiator, the condition exists that the modulus of H p (jω) must cease to grow above some corner frequency ωc . Here one will assume that the practical differentiator has as transfer function the series of (1) with a first-order low pass whose corner frequency is ωc : Hp (s) =

ωc Y (s) =s U (s) s + ωc

(2)

  This implies that lim Hp (jω) = ωc . Therefore the condition is fulfilled with ω→∞ |H p (jω)| becoming constant for frequencies above ωc . In the time domain, the output and input of such practical differentiator are related as: dy du + ωc y = ωc dt dt

(3)

Now, let one assume that in a plant one wants to control, state variable x 1 is the derivative of state variable x 2 : dx2 = x1 dt

(4)

One can measure x 2 and differentiate it to obtain an estimate x s1 of x 1 with a practical differentiator as in (3). The relation between x s1 and x 2 becomes: dx2 dxs1 + ωc xs1 = ωc dt dt

(5)

dxs1 + ωc xs1 = ωc x1 dt

(6)

Therefore:

So that x s1 equals x 1 passed through a first order low pass with corner frequency ωc . The estimation error e1 decreases with the value of ωc : e1 = x1 − xs1 =

1 dxs1 ωc dt

(7)

Let one assume that the plant is single input nth -order: dx = Ax + Bu dt

(8)

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Also, a full-state feedback –K = –[k 1 k 2 … k n ] has been calculated to set the eigenvalues of the regulated plant to the n desired values P = [p1 p2 … pn ] assuming full-state measurement. But, instead of full-state measurement, one measures x 2 to x n and one substitutes x 1 by its estimate above to generate the feedback: u = −[k1 k2 . . . kn ][xs1 x2 . . . xn ]T

(9)

Because x s1 = x 1 – e1 , the state derivative becomes:   dx = Ax + B −[k1 k2 . . . kn ][xs1 x2 . . . xn ]T dt   = Ax + B −[k1 k2 . . . kn ][x1 − e1 x2 . . . xn ]T = (A − BK)x + Bk1 e1

(10)

Let one define A1 and B1 as row matrices equaling the first row of A and B, respectively: A1 = A(1, :) B1 = B(1, :)

(11)

By (6) and (8) it follows that the derivative of the estimation error e1 can be written: dx1 dxs1 de1 = − dt dt dt = A1 x + B1 u − (ωc x1 − ωc xs1 )   = A1 x + B1 −[k1 k2 . . . kn ][xs1 x2 . . . xn ]T − ωc e1   = A1 x + B1 −[k1 k2 . . . kn ][x1 + e1 x2 . . . xn ]T − ωc e1 = (A1 − B1 K)x + B1 k1 e1 − ωc e1 = (A1 − B1 K)x + (B1 k1 e1 − ωc )e1

(12)

One considers that the regulated system has an augmented state x a = [x e1 ]T . It follows that the state equation of the regulated system with full-state feedback and differential estimation of x 1 becomes: dxa =A x dt dx  r a   A − BK Bk1 x dt de1 = A − B K B k − ω e 1 1 1 1 c 1 dt

(13)

One may conclude that: i) If the eigenvalues of Ar are strictly stable, the system is strictly stable and the estimation error converges to zero after an impulse disturbance in the state. ii) The eigenvalues of Ar will be different from the intended P values. This amounts to differential estimation not enjoying the separation property: the estimator does move the regulated system poles from the intended P positions. This effect diminishes with increasing ωc , so the amount of measurement noise in x 2 will basically determine its impact. The above analysis can readily be generalized for m variables to be estimated through practical differentiation of m measured variables, equivalent results holding.

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3 Review of Integral Action in State Space Controllers Let one assume again an nth -order plant: dx = Ax + Bu dt

(14)

One applies to (14) a full-state feedback and a servo signal for state variable x i : u = −Kx + ki xir = −k1 x1 − . . . + ki (xir − xi ) − . . . kn xn

(15)

In (15) it is K =[k 1 ,…, k i ,…,k n ], such that the closed loop eigenvalues are at the intended positions given by a vector P. Also, x ir is the reference variable for x i and the feedback of variable x i has been substituted by the error feedback k i (x ir –x i ). The closed loop system is written: dx = (A − BK)x + Bki xir dt

(16)

To add integral action for variable x i , one makes the controller to calculate a variable x n+1 equal to an integral of the error for some arbitrary t = 0: t xn+1 (t) =

(xir (τ ) − xi (τ ))d τ

(17)

0

One interprets (17) as making up together with (14) an augmented system with state x a = [x x n+1 ]T: ⎡  dxa dt

=

dx dt dxn+1 dt



= Aa xa + Ba u

⎢ ⎢ =⎢ ⎢ ⎣

.. . dxi dt

.. .

dxn+1 dt + Bar xir

⎡

⎡.⎤ .. ⎤⎡ .. ⎤ . . . ⎥ ⎢ ⎥⎢ ⎥   ⎢.⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ A 0⎥ B ⎥=⎢ ⎥⎢ xi ⎥ ⎢ ⎥ ⎥ ⎢ .. ⎥⎢ .. ⎥ + 0 u + ⎢ .. ⎥xir (18) ⎣ ⎦ ⎣ ⎦ ⎦ ⎣.⎦ . . · · · −1 · · · 0 xn+1 1 ⎤

For this augmented system, a gain vector K a , with ith entry k ai that places the closed loop eigenvalues at desired positions Pa can be calculated giving the closed loop system: dxa = (Aa − Ba Ka )xa + [Ba kai + Bar ]xir dt

(19)

It follows that integral action can be seamlessly integrated in a state space controller for any number of variables and with any number of integrators for each variable. A practical limitation to this will be the degradation in gain and phase margins implied by increasing the number of pure integrators in the control loop.

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4 PID Controllers as State Space Controllers with Differential Observation The above analysis allows one to interpret a PID controller as a state space controller with differential observation and integral action. Let one assume a second order plant where state variable x 1 is the derivative of state variable x 2 :  dx1       a1,1 a1,2 x1 b dt + 1 u (20) dx2 = 1 0 x2 0 dt Together with (20), one will assume that only x 2 is measured: y = x 2 . Let one assume that one wants: i) A full state regulatory feedback for the above system with a servo signal for y to follow a reference yr . ii) Differential observation of x 1 . iii) Integral action on the error e = yr – y. The equations defining the state space controller will be:

dxs1 (t) dt

2 (t) + ωc xs1 (t) = ωc dxdt t t x3 (t) = (yr (τ ) − y(τ ))d τ = (yr (τ ) − x2 (τ ))d τ

0

0

u(t) = −k1 xs1 (t) − k2 x2 (t) − k3 x3 (t) + k2 yr (t) t = −k1 xs1 (t) + k2 (yr (t) − y(t)) − k3 (yr (τ ) − y(τ ))d τ

(21)

0

It may be recognized that one can summarily describe the above controller by the PI on error D on output rule t u(t) = Kp e(t) + Ki

e(τ )d τ − Kd

dy(t) dt

(22)

0

In Eq. (22), K p = k 2 , K i = –k 3 , K d = k 1 and practical differentiation is abbreviated as ideal differentiation. Some remarks are in order. First, it is clear that single PID control rules can only be interpreted as full state feedbacks for second or first order systems – in the latter, derivative action will not be used. Application of single PID control rules to third or higher order systems is a kind of “term-deficient” state feedback. Second, any of the possible variations of PID control rules can be accommodated in this interpretation. For example, an also derivative on error PID can be obtained by adding to the command signal in (21) the result of practically differentiating the reference signal.

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Interpretation of state space controllers with differential observation as “stacks” of PID control rules is also possible. For an example, one may refer the many instances in the literature where an inverted pendulum on a cart is stabilized with “PID control”. Stabilization is obtained by differentiating the displacement signals – therefore obtaining estimates of velocities – and computing the command variable as a value that represents a full state feedback. Integral action may as well be added.

5 Conclusions and Further Research In this paper, one has presented practical differentiation as a scheme to observe or estimate state variables that are derivatives of measured variables in linear time invariant state space controllers. It was shown that, although this scheme does not enjoy the separation property – intended regulatory closed loop eigenvalues are displaced by the observation eigenvalues –, the estimation error converges to zero if the resulting closed loop is strictly stable. One has also shown that understanding practical differentiation as a scheme to observe or estimate state variables allows one to interpret PID control rules as full state feedback controllers – with reference following and integral action – for first and second order systems or as “vertical slots” of state-space controllers. The consequences of this understanding were discussed in the introductory section. There are several aspects which one would like to investigate further. First, it will be in order to extend the approach to discrete time controllers. The fact that these are band limited to the Nyquist frequency makes one expect differences with respect to the continuous time version presented here. An interesting one is that, differently from the continuous time case, high-frequency behavior of discrete differentiation is unique. The above analysis is predicated on a constant high frequency response of practical differentiators and it will be more complicated if more than one pole is considered in the model of a practical differentiator. Second, the displacement of intended eigenvalues by the observation eigenvalues may be subject to scrutiny in order to quantify its effects in behavior as a function of the corner frequency value of practical differentiators. Of course, the practical application of differential observation must be assessed. As PID rules are ubiquitously applied, it follows that according to the interpretation developed in this paper people do apply – maybe unwittingly – differential observation. In relation with a full state and Luenberger observers, a differential observer presents the advantage of being model free – one does not to have a system model to get the needed estimates of variables. On the other hand, to design a full state feedback for intended closed loop regulatory eigenvalues one needs a system model – the observation scheme not withstanding. So, if a system model is necessary after all, why not to use a Luenberger observer, with better performance? It may be the case that PID control rules allow for a simple and effective way to tune experimentally the majority of control loops without much theory and numeric calculations. Another aspect of LTI control design that this results pose for research are “defectiveorder” controllers, i.e. controlling a system that clearly requires an nth -order model through a feedback with less than n terms.

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As a last remark, one may presume based on state variables basis change that for second order systems the condition of one state variable being the derivative of the other may be lifted without loss of the possibility to interpret a PD controller as a stabilizing full state feedback. Acknowledgements. This work has been supported by FCT – Fundação para a Ciência e Tecnologia within the R&D Units Project Scope: UIDB/00319/2020.

References 1. Åström, K.J., Wittenmark, B.: Computer-Controlled Systems: Theory and Design. Courier Corporation, North Chelmsford (2013) 2. Luenberger, D.: An introduction to observers. IEEE Trans. Autom. Control 16(6), 596–602 (1971) 3. Åström, K.J., Hägglund, T., Astrom, K.J.: Advanced PID control. Research Triangle Park, vol. 461. ISA-The Instrumentation, Systems, and Automation Society (2006) 4. Friedland, B.: Control System Design: An Introduction to State-Space Methods. Courier Corporation, North Chelmsford (2012)

Endpoint Geodesics on the Set of Positive Definite Real Matrices Maximilian Stegemeyer1 and Knut H¨ uper2(B) 1

Max-Planck-Institut f¨ ur Mathematik in den Naturwissenschaften, Leipzig, Germany [email protected] 2 Institute for Mathematics, Julius-Maximilians-Universit¨ at, W¨ urzburg, Germany [email protected]

Abstract. In this paper we study the endpoint geodesics problem on the differentiable manifold of real positive definite matrices. The objective is to connect two points given as initial data on that space by a unique geodesic. We first recall partially well-known facts about the differential geometry of this manifold. Then we consider further features, namely the property of being an extrinsic symmetric space together with associated Riemannian isometries. This paper is essentially self-contained and therefore accessible to a wide audience. Keywords: Endpoint geodesics · Extrinsic symmetric space · Reductive homogeneous spaces · Symmetric positive definite matrices

1

Introduction

The differentiable manifold Posn of real symmetric positive definite (n × n)matrices plays a prominent role in many applications. From a purely mathematical point of view it is a nice example of a homogeneous space, i.e., it is an orbit of a smooth group action. However, the differential geometric structure is much richer. Some of those aspects are recalled here and not so well-known facts are exploited to solve the endpoint geodesics problem more or less explicitly. As for this space all calculations, most of them important for computer implementations, boil down to ordinary matrix computations, the formulas will be well-suited to real world applications. In recent years Posn has drawn new interest from many different perspectives. After E. Cartan classified symmetric spaces [5,6], every monograph about Lie theory and homogeneous spaces deals with interesting parts of that theory, cf. [12,19,20], to cite some classical treatises. However, crosslinks to certain areas of applied mathematics have been obtained as well, see e.g., [3] (symmetric spaces, Jordan algebras and Lie triple systems) or [22] (ODE-related numerical analysis), to mention only very few. For Posn , in particular, we refer to [17] (theory of buildings), [1] (novel Lie group structure) and [18] (Riemannian geometry and Cholesky decomposition). c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 435–444, 2021. https://doi.org/10.1007/978-3-030-58653-9_42

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From an engineering point of view, e.g., medical image processing, Posn , for small n, is of particular interest. For instance, in diffusion tensor imaging, this includes statistics on Pos3 , one is interested in real time computations to support medical imaging such as magnetic resonance imaging, cf. [25]. In robotics, namely in dextrous hand grasping, real time calculations, including control on Posn , seem to be now fairly standard, cf. [13,14,21]. Furthermore, Riemannian calculus on Posn has recently penetrated computer vision and image classification, cf. [7,11]; moreover, there are machine learning applications as well, e.g. [27]. The endpoint geodesics problem can be quickly stated as follows. Given as initial data two points on a Riemannian manifold, calculate the geodesic which connects them, based on the given two-point data only. This is a boundary value problem, clearly to be distinguished from an initial value problem. It is wellknown that the former has a unique solution, e.g., if the points are sufficently close together. However, closed or explicit formulas for such a geodesic are rare to find. The situation for the initial value problem of finding a geodesic for a given initial point together with an initial velocity might be simpler to solve. Usually, one at least has the corresponding Euler-Lagrange equation available, a (numerical) solution of which can be found. To solve the endpoint geodesics problem is of particular importance if one is interested in interpolation problems. This happens in statistics on manifolds (incl. medical imaging), but in computer added geometric design as well. For the latter simply think of the calculation of smooth splines, or Bezier curves or equivalently of the de Casteljau algorithm, all generalised to the manifold setting, see [8,16,23,26]. In particular, the methodology behind the design of de Casteljau algorithms on manifolds is inherently based on a control theoretic approach. A purely numerical approach to the endpoint geodesics problem on Stiefel manifolds has been recently addressed, see [4] or [29].

2

A Reductive Metric and Geodesics in Posn

Here we recall the space of symmetric positive definite matrices as a noncompact homogeneous space. Many results we present are scattered over the literature, well-known to the mathematical community. But this is definitely less the case in engineering areas. We will see that it becomes a reductive homogeneous space. See [24] for an introduction to reductive homogeneous spaces. After that, we will show that it is even a symmetric space. It is then possible to give explicit formulas for the geodesics and to solve the endpoint geodesics problem. Definition 1. We denote the symmetric positive definite (n × n)-matrices by Posn := {P ∈ Symn | P  0},

Symn := {S ∈ IRn×n | S = S  }.

(1)

This is an open subset of the vector space of symmetric matrices Symn , therefore it is a submanifold. There is a natural group action of the general linear group GLn (IR) on Posn . In order to work with a connected Lie group we restrict this

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action to the component of the identity in GLn (IR), i.e., the subgroup of all invertible matrices with positive determinant, denoted by GL+ n (IR). The action is given by Φ : GL+ n (IR) × Posn −→ Posn ,

(A, P ) −→ AP A .

(2)

Since all symmetric positive definite matrices can be diagonalised and, moreove, have positive eigenvalues, it is clear that this action is transitive. It is also smooth, since matrix multiplication and transposition are smooth operations. As Posn ⊆ Symn is an open submanifold, the tangent space at every point P ∈ Posn is TP Posn ∼ = Symn . Hence, we could define a Riemannian metric on Posn by using the Frobenius scalar product on Symn . Geodesics for this metric are straight lines in Symn . However, the GL+ n (IR)-action is not isometric with respect to this metric. In order to connect the transitive GL+ n (IR)-action to Riemannian geometry on Posn , we need to find another metric. We show that Posn becomes a reductive space with respect to the GL+ n (IR)-action. Lemma 1. The space of symmetric positive definite matrices Posn is diffeomorphic to the homogeneous space GL+ n (IR)/SOn . Proof. We have already seen that the GL+ n (IR)-action on Posn is transitive, hence we need to determine the stabiliser group at one point. Let A ∈ GL+ n (IR) be a matrix such that the identity matrix I ∈ Posn is mapped to itself by the group action, which means (3) I = AIA , i.e., A is forced to be orthogonal, A ∈ On . Since by assumption det(A) > 0, we even have A ∈ SOn . Conversely, every matrix A ∈ SOn clearly stabilises the identity matrix. Hence, the stabiliser group of I is equal to the group SOn . We conclude that Posn and GL+ n (IR)/SOn are diffeomorphic, i.e.,  Posn ∼ = GL+ n (IR)/SOn . By the theory of homogeneous spaces, we know that the projection π : GL+ n (IR) → Posn

(4)

is a surjective submersion. Explicitly, it is given by π(A) = AA ,

(5)

+ for A ∈ GL+ n (IR). We now want to describe the quotient GLn (IR)/SOn as a + reductive homogeneous space. The Lie algebra of GLn (IR) is the Lie algebra of all real (n × n)-matrices gln (IR). This Lie algebra can be decomposed into the skew-symmetric matrices son and the symmetric matrices

gln (IR) = son ⊕ Symn .

(6)

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Lemma 2. The space of symmetric positive definite matrices considered as the quotient Posn ∼ = GL+ n (IR)/SOn is reductive with respect to the decomposition gln (IR) = son ⊕ Symn . Furthermore, the following bracket relations hold [son , son ] ⊆ son ,

[son , Symn ] ⊆ Symn ,

[Symn , Symn ] ⊆ son .

(7)

Proof. We need to show that AdSOn (Symn ) ⊆ Symn . Let Θ ∈ SOn and P ∈ Symn , then (8) (ΘP Θ ) = ΘP  Θ = ΘP Θ , so AdΘ P ∈ Symn . This shows reductivity. The bracket relations can easily be checked, the computations are therefore omitted.  The tangent map of the projection (4) at I ∈ GL+ n (IR) gives an isomorphism between the complement of the Lie algebra of the stabiliser group, which is Symn , and the tangent space at π(I) ∈ Posn . Explicitly, the tangent map at I is dπI (ξ) = ξ + ξ  ,

(9)

for ξ ∈ gln (IR). This shows that ker(dπI ) = son . I.e., if we restrict dπI to Symn we obtain the isomorphism dπI : Symn −→ TI Posn ,

A −→ 2A.

(10)

The idea is now to specify an SOn -invariant scalar product on Symn and then to declare the map dπI to be an isometry. Using the transitivity of the GL+ n (IR)action on Posn , we can then define a smooth Riemannnian metric. On Symn we choose the scalar product to be X, Y := 4tr(XY )

(11)

for X, Y ∈ Symn , which is obviously SOn -invariant. If P = AA = Φ(A, I) ∈ Posn for some A ∈ GL+ n (IR), then we obtain an isomorphism (dπ)−1 ◦ (dΦA )−1 : TP Posn → Symn ,

X −→ 12 A−1 X(A )−1 .

(12)

We define a metric on Posn by demanding the above map to be an isometry.   X, Y P := 12 A−1 X(A )−1 , 12 A−1 Y (A )−1 (13)   = tr X(AA )−1 Y (AA )−1 = tr(XP −1 Y P −1 ). It defines a smooth Riemannian metric on Posn for which the GL+ n (IR)-action is isometric. In order to describe geodesics with respect to this metric, we will now show that this is even a symmetric space. Proposition 1. The space of symmetric positive definite matrices is a symmetric space. The global symmetry at a point P ∈ Posn is given by sP : Posn −→ Posn ,

S −→ P S −1 P.

(14)

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Proof. A global symmetry is an isometry that satisfies sP (P ) = P and (dsP )P = −idTP Posn , so we need to check these properties. For P ∈ Posn , we have sP (P ) = P P −1 P = P and the tangent map at an arbitrary point S ∈ Posn for X ∈ TS Posn is (15) (dsP )S (X) = −P S −1 XS −1 P. Here we used the formula for the derivative of the inverse. For S = P we get (dsP )P (X) = −P P −1 XP −1 P = −X,

(16)

so (dsP )P = −idTP Posn . Finally, we need to show that sP is an isometry. Indeed, let S ∈ Posn and X, Y ∈ TS Posn . Then (dsP )S (X) ∈ TP S −1 P Posn , so we have     (dsP )S X,(dsP )S Y P S −1P = tr P S −1XS −1P (P S −1P )−1P S −1YS −1P (P S −1P )−1 = tr(XS −1 Y S −1 ) = X, Y S .

This shows that sP is a global symmetry at P . Since P was arbitrary, there is a  global symmetry at every point. Consequently, Posn is a symmetric space. Corollary 1. The geodesic in Posn starting at P = AA where A ∈ GL+ n (IR) with initial velocity X ∈ TP Posn , is for all t ∈ IR given by   γ(t) = A · exp tA−1 X(A )−1 · A . (17) Proof. In general, this is a property of symmetric spaces, cf. [24], Chap. 11, 31. proposition. Using Eq. (12), we see that tangent vector X is mapped to ξ = 12 A−1 X(A )−1 ∈ Symn . The theory for symmetric spaces then tells us that the geodesic starting at P with intial velocity X is for all t ∈ IR given by γ(t) = π(A exp(tξ)) = A · exp(tξ) exp(tξ  ) · A   = A · exp(2tξ) · A = A · exp tA−1 X(A )−1 · A .

(18) 

Theorem 1. Let P, S ∈ Posn such that P = AA for some A ∈ GL+ n (IR). The unique geodesic γ : IR → Posn with γ(0) = P and γ(1) = S has initial velocity   γ(0) ˙ = A · log A−1 S(A )−1 · A . (19) Explicitly, it is given for all t ∈ IR by    γ(t) = A · exp t log A−1 S(A )−1 · A .

(20)

Proof. Let P = AA ∈ Posn and S ∈ Posn be the endpoint of a geodesic that starts at P , i.e., there is a geodesic γ : IR → Posn such that γ(0) = P and γ(1) = S. Such a geodesic always exists, since Posn is a symmetric space by Proposition 1 and therefore complete and connected. Hence, any two points can be linked by a geodesic. By Corollary 1, the geodesic is of the form γ(t) = A · exp(2tξ) · A

(21)

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for all t ∈ IR and some ξ ∈ Symn . Consequently, S = A · exp(2ξ) · A and exp(2ξ) = A−1 S(A )−1 .

(22)

Clearly, the expression A−1 S(A )−1 is a symmetric positive definite matrix. Since the exponential map exp : Symn → Posn is a global diffeomorphism, there is a unique ξ ∈ Symn such that Eq. (22) holds. This is given by   ξ = 12 log A−1 S(A )−1 (23) and the remaining explicit formulas can be checked straight-forwardly.



In summary, we have solved the endpoint geodesics problem on Posn globally. Note that in general, endpoint geodesics problems can only be solved locally, e.g., in spaces of positive sectional curvature there can be closed geodesics and uniqueness of geodesics is only achieved locally.

3

Posn as an Extrinsic Symmetric Space

In this section, we want to see that Posn can even be realised as an extrinsic symmetric space in a vector space with indefinite inner product. This is useful in many applications, since the metric is then obtained by just restricting the inner product to the respective tangent spaces. For the notion of extrinsic symmetric spaces one might consult [9,10,15]. The strategy in this section is to adapt to Posn the approach taken in [2] for Grassmannians. Eventually, by restriction we recover the above results. Definition 2. Let V be a real vector space with an indefinite inner product ·, · . Let M be a Riemannian submanifold of V and for p ∈ M define the extrinsic symmetry at p to be the affine linear map Rp : V → V that fixes p and satisfies dRp |Tp M = −id|Tp M and dRp |Np M = id|Np M . If Rp (M ) = M for all p ∈ M , then M is called an extrinsic symmetric space. Extrinsic symmetric spaces are always symmetric. This is because the restriction of the extrinsic symmetries to the manifold M gives the global symmetries of a symmetric space. The embedding Posn → Symn that we studied in Sect. 2 is clearly not extrinsic symmetric. The idea is to embed Posn in a larger vector space such that the embedding can be extrinsic symmetric. We define the vector space 

(24) p := [ B0 A0 ] ∈ IR2n×2n  A, B ∈ Symn ∼ = Symn × Symn . The Lie group GL+ n (IR) acts on p in the following way. Define the Lie group

    A 0 +  K := (25) 0 (A )−1 ∈ GL2n (IR) A ∈ GLn (IR) . This is a closed subgroup of GL2n (IR) being isomorphic to GL+ n (IR). If k ∈ K, then its inverse is given by Sk  S, where S is the matrix S := [ I0 I0 ] ∈ p. Here

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again I equals the (n × n)-identity matrix.  The group K acts on p by the adjoint  A 0 representation, i.e., for k = 0 (A )−1 ∈ K and X = [ C0 B0 ] ∈ p, we have kXk −1 =



0

 −1

(A )

CA

−1

ABA 0



∈ p.

(26)

We define an inner product on p by setting X, Y p := − 12 tr(SX  SY ).

(27)

It is clear that this is a bilinear and symmetric map. For the non-degeneracy, define the subspaces  

 0 A 

2n×2n  A ∈ Symn , p := [ B0 B0 ] ∈ IR2n×2n B ∈ Symn . (28) p := −A 0 ∈ IR Then p = p ⊕ p and explicit computations show that ·, · p is positive definite on p and negative definite on p , respectively. Hence, ·, · p is non-degenerate. One can also check that ·, · p is K-invariant. Proposition 2. The map ι : Posn → p,

ι(P ) :=



0 P P −1 0



(29)

is an isometric embedding of the symmetric positive definite matrices into the inner product space (p, ·, · p ). Proof. The map ι is smooth and an injective immersion. On the image, we can define an inverse and see that this is smooth as well. Consequently, ι(Posn ) → p is an embedded submanifold. To see that it is isometric, let P ∈ Posn and X, Y ∈ TP Posn . Then the metric at P on Posn is X, Y P = tr(XP −1 Y P −1 ). The tangent map of ι is

 dιP (X) =

0 X −P −1 XP −1 0



,

(30)

(31)

where we used the formula for the tangent map of the inverse. Consequently,   −1 −1 0 dιP (X), dιP (Y ) p = − 12 tr −XP 0 Y P −P −1 XP −1 Y (32) −1 −1 = tr(XP Y P ) = X, Y P . This shows that the embedding is even isometric.



Note that we can also think of this embedding as the orbit  of the point S ∈ p A 0 under the K-action on p. This is because for k = 0 (A )−1 ∈ K, we have kSk −1 =



0 AA (AA )−1 0



∈ ι(Posn ).

This action of K is equivalent to the GL+ n (IR)-action on Posn .

(33)

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Lemma 3. Let P ∈ Posn and consider the embedding ι : Posn → p. Define M := ι(Posn ). The tangent space at p = ι(P ) to the submanifold M is  

 0 A  (34) Tp M =  A ∈ Symn . −P −1 AP −1 0 Accordingly, the normal space is

 0 Np M = P −1 AP −1

    A ∈ Symn .

A 0

(35)

The extrinsic symmetry at p is given by Rp (X) = pXp for all X ∈ p. Proof. The claim about the tangent space is clear by the proof of Proposition 2. A computation shows that the subspace that we claim to be equal to Np M is indeed orthogonal to Tp M with respect to ·, · p . For dimensional reasons  be equal to this subspace. For the extrinsic symmetry, let X = Np M must then 0 A −P −1 AP −1 0

pXp =

∈ Tp M . Then

 P

0

−1

P 0

 −P

0 A AP −1 0



−1

P

0

−1

P 0



 =

P

−1

0 −A AP −1 0



= −X.

(36)

A similar computation gives pXp = X for all X ∈ Np M . This shows that the normal space involution is Rp (X) = pXp for X ∈ p.  Theorem 2. In the embedding ι : Posn → p the symmetric positive definite matrices are an extrinsic symmetric space. Proof. We set M = ι(Posn ). Let p, q ∈ M and we need to  show that Rp (q) ∈ M .  0 P 0 Q There are P, Q ∈ Posn such that p = P −1 0 and q = Q−1 0 , so Rp (q) =

 P

0

−1

P 0



0 Q Q−1 0

 P

0

−1

P 0



 =

P

−1

0 P Q−1 P QP −1 0



.

(37)

Now, P Q−1 P is a symmetric positive definite matrix, so Rp (q) = ι(P Q−1 P ) ∈ M . Since p and q were arbitrary, this shows that M is closed under all extrinsic symmetries, in fact, it is extrinsic symmetric.  Finally, we want to draw the connection between the extrinsic symmetric embedding and the endpoint geodesics problem. To describe the geodesics in the extrinsic symmetric embedding, we define 

A 0  2n×2n  A ∈ Symn . (38) m := 0 −A ∈ IR ∼ GL+ (IR) (see Eq. (25)). This is a subspace of the Lie algebra of the group K = n Corollary 1 shows that the geodesics in the extrinsic symmetric embedding starting at a point p = kSk −1 where k ∈ K are of the form γ(t) = Adexp (tkξk−1 ) p = Adexp(tΩ) p = exp(t adΩ)p

(39)

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for all t ∈ IR where ξ ∈ m and where we defined Ω := kξk−1 . This formula for the geodesics holds true in general for extrinsic symmetric spaces that are in the form of such an orbit space as we described it in this section. Recall that we realised Posn as the orbit (40) Posn ∼ = AdK S, see Eq. (33). Many interesting extrinsic symmetric spaces show up in a similar manner as such orbits. It can be shown abstractly that for these extrinsic symmetric orbits the endpoint geodesics problem can be solved in the following unified way. Consider two points p and q in the extrinsic symmetric orbit and a geodesic γ which satisfies γ(0) = p and γ(1) = q. Then γ is described by Eq. (39) and the element Ω satisfies Rq ◦ Rp = exp(2 adΩ).

(41)

This equation can locally be solved for adΩ which uniquely determines the geodesic by Eq. (39). In the case of the symmetric positive definite matrices, one can see that we recover the global results of Sect. 2. For the Grassmannian this method yields the results that were already discussed in [2]. Hence, Eq. (41) yields a unified framework for solutions of the endpoint geodesics problem in extrinsic symmetric spaces, for more details we refer to [28]. Acknowledgements. This work has been supported in part by the German Federal Ministry of Education and Research (BMBF-Projekt 05M20WWA: Verbundprojekt 05M2020 - DyCA).

References 1. Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29(1), 328–347 (2007) 2. Batzies, E., H¨ uper, K., Machado, L., Silva Leite, F.: Geometric mean and geodesic regression on Grassmannians. Linear Algebra Appl. 466, 83–101 (2015) 3. Bertram, W.: The Geometry of Jordan and Lie Structures. Springer, Berlin (2000) 4. Bryner, D.: Endpoint geodesics on the Stiefel manifold embedded in Euclidean space. SIAM J. Matrix Anal. Appl. 38(4), 1139–1159 (2017) 5. Cartan, E.: Sur une classe remarquable d’espaces de Riemann. Bull. Soc. Math. France 54, 214–264 (1926). In French 6. Cartan, E.: Sur une classe remarquable d’espaces de Riemann. II. Bull. Soc. Math. France 55, 114–134 (1927). In French 7. Cherian, A., Sra, S.: Positive definite matrices: data representation and applications to computer vision. In: Minh, H.Q., Murino, V. (eds.) Algorithmic Advances in Riemannian Geometry and Applications. Advances in Computer Vision and Pattern Recognition, pp. 93–114. Springer, Cham (2016) 8. Crouch, P., Kun, G., Silva Leite, F.: The De Casteljau algorithm on Lie groups and spheres. J. Dyn. Control Syst. 5(3), 397–429 (1999) 9. Eschenburg, J.-H., Heintze, E.: Extrinsic symmetric spaces and orbits of srepresentations. Manuscripta Math. 88(4), 517–524 (1995)

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10. Eschenburg, J.-H., Heintze, E.: Erratum to: “Extrinsic symmetric spaces and orbits of s-representations”. Manuscripta Math. 92(3), 408 (1997) 11. Harandi, M., Basirat, M.K., Lovell, B.C.: Coordinate coding on the Riemannian manifold of symmetric positive-definite matrices for image classification. In: Turaga, P.K., Srivastava, A. (eds.) Riemannian Computing in Computer Vision, pp. 345–361. Springer, Cham (2016) 12. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Reprint with Corrections of the 1978 Original. American Mathematical Society, Providence, RI (2001) 13. Helmke, U., H¨ uper, K., Moore, J.B.: Quadratically convergent algorithms for optimal dextrous hand grasping. IEEE Trans. Robot. Autom. 18(2), 138–146 (2002) 14. Jia, P., Wu, L., Wang, G., Geng, W.N., Yun, F., Zhang, N.: Grasping torque optimization for a dexterous robotic hand using the linearization of constraints. Math. Probl. Eng. 2019(1), 17 (2019). Art. ID 5235109 15. Kobayashi, S.: Isometric imbeddings of compact symmetric spaces. Tohoku Math. J. 2(20), 21–25 (1968) 16. Krakowski, K.A., Machado, L., Silva Leite, F., Batista, J.: A modified Casteljau algorithm to solve interpolation problems on Stiefel manifolds. J. Comput. Appl. Math. 311, 84–99 (2017) 17. Lang, S.: Bruhat-Tits-R¨ aume. Elem. Math. 54(2), 45–63 (1999). In German 18. Lin, Z.: Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition. SIAM J. Matrix Anal. Appl. 40(4), 1353–1370 (2019) 19. Loos, O.: Symmetric Spaces. I: General Theory. W. A. Benjamin Inc., New YorkAmsterdam (1969) 20. Loos, O.: Symmetric Spaces. II: Compact Spaces and Classification. W. A. Benjamin Inc., New York-Amsterdam (1969) 21. Mu, X., Zhang, Y.: Grasping force optimization for multi-fingered robotic hands using projection and contraction methods. J. Optim. Theory Appl. 183(2), 592– 608 (2019) 22. Munthe-Kaas, H.Z., Quispel, G.R.W., Zanna, A.: Symmetric spaces and Lie triple systems in numerical analysis of differential equations. BIT 54(1), 257–282 (2014) 23. Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inf. 6, 465–473 (1989) 24. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press Inc., New York (1983) 25. Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006) 26. Popiel, T., Noakes, L.: B´ezier curves and C 2 interpolation in Riemannian manifolds. J. Approx. Theory 148(2), 111–127 (2007) 27. Sra, S., Hosseini, R.: Geometric optimization in machine learning. In: Minh, H.Q., Murino, V. (eds.) Algorithmic Advances in Riemannian Geometry and Applications. Advances in Computer Vision and Pattern Recognition, pp. 73–91. Springer, Cham (2016) 28. Stegemeyer, M.: Endpoint geodesics in symmetric spaces. Master’s thesis, Institut f¨ ur Mathematik, Julius-Maximilians-Universit¨ at, W¨ urzburg, Germany, February 2020 29. Zimmermann, R.: A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric. SIAM J. Matrix Anal. Appl. 38(2), 322–342 (2017)

Decentralized Control for Multi-agent Missions Based on Flocking Rules Rafael Ribeiro1 , Daniel Silvestre2(B) , and Carlos Silvestre2 1

2

Instituto Superior T´ecnico, Universidade de Lisboa, Lisbon, Portugal [email protected] Department of Electrical and Computer Engineering, Faculty of Science and Technology, University of Macau, Macau, China [email protected], [email protected]

Abstract. This paper addresses the problem of having a multi-agent system converging to multiple dynamic rendezvous areas for networks of agents with no localization sensors and limited communication capabilities. The localization is performed by measurement/communication towers that determine the noisy position and velocity of each relevant agent and transmit them using a directional antenna. There is no assumption on the connectedness of the network topology, which is unknown to all agents. The proposed solution consists of an improved flockingbased movement algorithm tailored to the proposed scenario, with mechanisms to prevent collisions. The performance of the solution is presented through multiple simulations for a multitude of environments. By an appropriate selection of an utility function, these empirical results hint to the possibility of realizing different types of missions intended for multi-agent systems using swarm behavior.

Keywords: Decentralized control algorithms

1

· Multi-agent systems · Flocking

Introduction

The rendezvous problem, defined as a group of agents attempting to converge to the same area in the mission plane, is a central issue when establishing a swarmlike network of agents to perform a given assignment. Since centralized solutions D. Silvestre—is also with the Institute for Systems and Robotics, Instituto Superior T´ecnico, Universidade de Lisboa, Lisbon, Portugal. C. Silvestre—is on leave from Instituto Superior T´ecnico, Universidade de Lisboa, Lisboa, Portugal. This work was partially supported by the project MYRG2018-00198-FST of the University of Macau; by the Portuguese Funda¸ca ˜o para a Ciˆencia e a Tecnologia (FCT) through Institute for Systems and Robotics (ISR), under Laboratory for Robotics and Engineering Systems (LARSyS) project UIDB/50009/2020. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 445–454, 2021. https://doi.org/10.1007/978-3-030-58653-9_43

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require additional computational and communication capabilities, this paper focuses on decentralized control rules (i.e., no communication exists between nodes) that can rendezvous agents with limited connectivity along with an utility function values defining the mission. We assume all agents are identical and follow the same algorithm to avoid the need for leader selection protocols, specialized messages, or the existence of single points-of-failure. Moreover, the local control law can cope with asynchronous and delayed communication, agents leaving and entering the network, packet losses, among other problems. The envisioned scenario consists of towers equipped with localization sensors and directional antenna, while agents only have limited communication capabilities. Agent movement is therefore asynchronous since it is triggered by the information broadcast from a tower with the position and velocity of their neighbors. In the literature for control systems, the rendezvous problem is typically solved using linear iterative consensus algorithms that converge to some weighted average of the initial state. However, these works assume a connected topology since running them in a network containing clusters results in different rendezvous locations for each partition. The literature has several examples addressing different scenarios: switching topology with time-delays [1]; networks with stochastic and asymmetric communications [2,3]; and networks with communication link failures [4,5]. In this paper, we propose flocking rules [6] to address the problem of non-connected network topologies. Interesting features for rendezvous algorithms include noisy measurements, delays in the communication channels, uncertainty in the agents dynamics and collision-free guarantees. In [7], position measurements are defined as ball of radius r containing the true state. The presented consensus algorithm is later generalized for convex polytopes in [8], showing how agents can compute worstcase predictions for the positions of their neighbors. The authors resorted to the concept of Set-Valued Observers (SVOs) to propagate estimations of position for each agent when receiving asynchronous messages. A direct result of the theoretical properties proved in [9–12] for SVOs is that it can be used to devise a guaranteed collision avoidance method. Furthermore, according to [13], if the agent dynamics have no uncertainty, the set-valued estimates for the position have no added conservatism, i.e., are optimal. The inclusion of an SVO-based collision avoidance is a major difference to the work in [8]. In [14], an algorithm is presented to have a group of mobile agents locate a maximum of a scalar field. We avoid posing the assumption of an utility function with a single stationary point, and the ability to have communication between all pair of close-by agents. The main contributions of this paper are as follows: – A rendezvous algorithm for multiple dynamic rendezvous targets that accounts for sensor noise, asynchronous directional communication, unknown network topologies - possibly disconnected - and a mission plane defined by a utility function, while guaranteeing no collisions between the agents; – Extensive simulations depicting the behavior in a variety of missions.

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These contributions are achieved while relaxing the assumptions from the literature of knowing the connected network topology and the number of agents [7], or having a connected topology [3].

2

Problem Statement

In this paper, it is addressed the problem of a multi-agent system (MAS) composed of n nodes converging and following multiple dynamic rendezvous points using a decentralized control law. Nodes have no localization sensors and receive position and velocity updates from a directional broadcast by the communication towers - which can be a fixed structure or a more expensive and sophisticated mobile agent. The mission is defined by a (possibly time-varying) utility function, assigning a numerical value to signal the quality of each location. Directional communication entails that broadcasts can be received by all agents in a strip of the mission plane, which forms a circular segments with a fixed angle radiating from the tower. Sent data consists of position and velocity estimates for the agents in the strip. In doing so, the overall system complexity is reduced and power can be saved at the individual nodes. In a multi-tower environment, each of can be in charge of a portion of the terrain and iterate in a round-robin fashion, centering its strip on each agent. It is left for future work the addition of heuristics to select tower updates based on the current configuration of the multi-agent system or on event- and self-triggering rules. More precisely, the network topology is defined by the set V of all agents and the directed links of the form (i, j) representing i and j to be neighbors. The set E(k) contains all (i, j) at time instant k and form the edge set of a directed graph defined as Gk = (V, E(k)). Given the existence of noise, a tower transmits, at time k, a polytopic set-valued estimate Xi (k) of the true position of node i, xi (k). Figure 1 illustrates the neighbor definition: any two agents i and j are considered neighbors if they are in the same strip, while others, such as l, are unknown to the agents. The set of all neighbors of i is denoted by Ni (k).

Fig. 1. Strip centered at node i with i being neighbors with agents j and l (left). Strip centered at node j with i’s only guaranteed neighbor j (right).

This paper addresses three main issues associated with directional antennae: i) absence of a connected and known network topology;

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ii) limited unidirectional communication; iii) position and velocity estimates received at different time instants. In the context of this paper, each agent dynamics is given by a double integrator: qi (k + 1) = qi (k) + vi (k + 1), vi (k + 1) = vi (k) + ui , (1) i = 1, 2, . . . , n where qi , vi and ui represent agent i’s position, velocity and control input, respectively.

3

Proposed Solution

Given that the network can be composed of multiple clusters of agents spread over the mission plane, the problem cannot be solved with a purely consensusbased approach. In [8], it was shown that this results in convergence to multiple clusters, depending on the initial configuration of agents and broadcast sequence chosen by the measurement towers. In our proposed solution, agent movement is based on flocking, initially developed in [6], with additional components tailored for the envisioned scenario. An update of agent i is triggered by the reception of a message containing the set-valued estimate of the position Xj (k) and velocity vj (k) for all neighbor agents of i in Ni (k). The actuation law comprises: the three original flocking rules - separation, cohesion, alignment - and three additional ones - attraction, utility, and randomness. In the remainder, the center for the polytope associated with agent i is labeled as ci (k). Separation Component. As defined in [6], the separation movement component maintains the agents at a minimum distance - used to maximize the area covered by the flock and as a simple collision avoidance mechanism. The separation vector, Us , is constructed as: Us =

1 |Ni (k)|



ci (k) − cj (k)

(2)

j∈Ni (k)

Cohesion Component. The cohesion vector, Uc , simulates attraction forces to maintain the agents at a maximum distance. Uc =

1 |Ni (k)|

 j∈Ni (k)

cj (k) − ci (k)

(3)

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Alignment Component. The alignment component, Ua , maintains the agents heading in the same direction, to simulate the coordinated movement of a real flock.  1 vj (k) − vi (k) (4) Ua = |Ni (k)| j∈Ni (k)

Fig. 2. Left: Rays being cast from both the moving agent’s polytope and an obstacleagent’s polytope. The red Xs represent the first intersection point between a ray and the opposite polytope. Right: Example of the allowed movement for an agent that would otherwise collide.

Attraction Component. The attraction component simulates an attraction/repulsion force towards other agents based on the difference of utility in their corresponding positions. These forces motivate the agents to get closer to higher-utility neighbors. Formally, Uattr is defined as:  1 Uattr = (hj − hi )(cj (k) − ci (k)) (5) |Ni (k)| j∈Ni (k)

where hi and hj are the utility function’s values in agent i’s centroid position and each neighbor j’s centroid position, respectively. Utility Component. Similar to the attraction component, as the secondary objective is to converge around an area with the highest found utility value, the utility movement component, Uu , moves the agent in the direction of the utility function gradient, assuming it is differentiable. The vector is defined as: Uu = ∇hi (k)

(6)

Randomness Component. In order to avoid movement deadlocks, the agents’ movement behavior has an added random component, defined as a unit vector with a random angle. An agent’s acceleration vector, u, is calculated as the weighted average of all components defined as: u = α · Us + β · Uc + γ · Ua + δ · Uattr +  · Uu + ζ · Ur

(7)

where all weights α, β, γ, δ, , ζ ∈ R, and all the component vector were previously normalized. The agent’s velocity is then modulated by our proposed collision avoidance strategy defined in Subsect. 3.1.

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Collision Avoidance Strategy

The intuition behind the proposed collision avoidance mechanism is that, given the property that xi (k) ∈ Xi (k), one can propagate the set-valued estimates and check if it intersects with any other polytopes for the neighboring nodes. The proposed strategy uses ray tracing and line intersection checks to detect collisions. From each vertex of the moving polytope Pi , rays are cast in the direction of the velocity vector with length equal to the maximum distance an agent can move per iteration. Each of the rays is checked for intersections against the edges of agent j’s polytope, Pj . This is repeated with rays from the vertices of Pj , in the opposite direction of the velocity vector and the edges of Pi . Figure 2(a) illustrates this method. Our strategy consists of repeating this pair-wise process for the moving agent, i, and each neighboring agent, j ∈ Ni (k). The movement magnitude will be equal to the distance to the most imminent collision, depicted in Fig. 2(b).

4

Simulation Results

In this section, we simulate the behavior of 10 agents in various scenarios, considering utility functions with: single maximum or minimum and multiple maxima and minima. Across all simulations, the size of the mission plane is 100 × 100 and the strips have an 8 degree half-angle. Videos of the simulations can be seen at Github. The scenarios included are: 1. Single maximum utility function to account for rendezvous missions: (a) Static rendezvous point; (b) Moving rendezvous point; 2. Single minimum utility function to illustrate escape missions: (a) Static minimum point; (b) Moving minimum point; 3. Multiple maxima and minima to depict the case of conflicting objectives: (a) Static rendezvous areas; (b) Static rendezvous areas in a mission plane with illegal zones; (c) Rendezvous areas whose utility value decreases when explored, and increases when not observed for surveillance missions; (d) Rendezvous areas whose utility value decreases when explored, and increases when not observed with illegal zones; (e) Random creation and destruction of rendezvous and minimum points; Due to space constraints we omit the results for the static case (although it can be seen in our Github archive). Figures 3(a) and 3(b) illustrate the case of simulation type 1b - single moving rendezvous point - which initially is at the center of the mission plane, moving randomly over time. The proposed decentralized algorithm reaches the destination while avoiding collisions.

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Fig. 3. Initial configuration for a single moving maximum at the center (left) and their final configuration (right) using one communication tower (marked by the red X).

Fig. 4. Initial configuration for a single moving minimum at the center (left) and their final configuration (right) using one communication tower (marked by the red X).

Figures 4(a) and 4(b) illustrate the simulation scenarios for type 2 - single minimum. A direct consequence of not having a maximum is that nodes do not converge to a point. This simulation shows that the agents dynamically move away from a low-utility area. The simplest scenario when there are multiple rendezvous points and locations to avoid is represented in Figs. 5(a) and 5(b), where the agents converge to the multiple static rendezvous targets while avoiding the low-utility areas. Figures 6(a) and 6(b) illustrate a similar scenario but with illegal zones. Once again, the algorithm is capable of leading agents to converge to the multiple static rendezvous targets, while avoiding both low-utility and illegal areas. Illustrating the simulated scenarios with dynamic-valued rendezvous areas, in Figs. 7(a) to 8(b), the final configurations of the agents appear to have the clusters outside the desired rendezvous areas. This is attributed to the utility values of the rendezvous areas being dynamic: these configurations represent the agents in a intermediary state between having fully explored a rendezvous area and having converged to a new one. Figures 7(a) and 7(b) represent the simulation type 3c, in which, the utility value of the rendezvous areas is decreasing when visited and increasing while unexplored. Additionally, at each discrete time step, there is a probability that

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Fig. 5. Initial configuration for multiple static rendezvous and minima areas (left) and their final configuration (right) using one communication tower (marked by the red X).

Fig. 6. Initial configuration for a mission plane with illegal zones (black squares) and a utility function with multiple static maxima (rendezvous areas) and multiple static minima (left) and their final configuration (right).

Fig. 7. Initial configuration for a mission plane with multiple static rendezvous areas whose utility value decreases when an agent explores it, and multiple static minima (left) and their final configuration (right) using two communication towers (marked by red X)

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Fig. 8. Initial configuration for a mission plane with multiple moving rendezvous zones and multiple static minima (left) and their final configuration (right). At each discrete time step, there is a probability that a utility function maximum or minimum is created or removed.

a utility function maximum or minimum is created or removed. In Figs. 8(a) and 8(b), represents a random creation and destruction of desired areas. In both cases, the nodes adapt to the disappearance or loss of utility of the areas and search for newer ones.

5

Conclusions and Future Work

This paper addressed the problem of having a group of mobile agents following a mission in a decentralized manner. Agents have no localization sensors, but are equipped with limited directional communication capabilities, receiving noisy positioning data from fixed towers. The proposed solution avoids the need for a connected topology or synchronous communication by defining improved flocking rules a guaranteed collision-free mechanism. The efficacy of the algorithm is shown using various simulated environments with a sparse node occupation. We envisioned two main directions for future work: i) the propagation of the set-valued estimates of the agent position to allow the nodes to move without recent information; ii) adding a formation component to the movement to tackle more complicated tasks not achievable by a swarm of vehicles.

References 1. Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520– 1533 (2004) 2. Antunes, D., Silvestre, D., Silvestre, C.: Average consensus and gossip algorithms in networks with stochastic asymmetric communications. In: 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pp. 2088– 2093 (December 2011) 3. Silvestre, D., Hespanha, J.P., Silvestre, C.: Broadcast and gossip stochastic average consensus algorithms in directed topologies. IEEE Trans. Control Netw. Syst. 6(2), 474–486 (2019)

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4. Patterson, S., Bamieh, B., El Abbadi, A.: Convergence rates of distributed average consensus with stochastic link failures. IEEE Trans. Autom. Control 55(4), 880– 892 (2010) 5. Fagnani, F., Zampieri, S.: Average consensus with packet drop communication. SIAM J. Control Optim. 48, 102–133 (2009) 6. Reynolds, C.W.: Flocks, herds, and schools: a distributed behavioral model. ACM SIGGRAPH Comput. Graph. 21, 25–34 (1987) 7. Sadikhov, T., Haddad, W.M., Goebel, R., Egerstedt, M.: Set-valued protocols for almost consensus of multiagent systems with uncertain interagent communication. In: American Control Conference (ACC), pp. 4002–4007 (June 2014) 8. Silvestre, D., Rosa, P., Hespanha, J.P., Silvestre, C.: Set-consensus using set-valued observers. In: American Control Conference (ACC), Chicago, Illinois, USA (July 2015) 9. Silvestre, D., Rosa, P., Hespanha, J.P., Silvestre, C.: Stochastic and deterministic fault detection for randomized gossip algorithms. Automatica 78, 46–60 (2017) 10. Silvestre, D., Rosa, P., Hespanha, J.P., Silvestre, C.: Set-based fault detection and isolation for detectable linear parameter-varying systems. Int. J. Robust Nonlinear Control 27(18), 4381–4397 (2017) 11. Silvestre, D., Rosa, P., Hespanha, J.P., Silvestre, C.: Fault detection for LPV systems using set-valued observers: a coprime factorization approach. Syst. Control Lett. 106, 32–39 (2017) 12. Silvestre, D., Rosa, P., Hespanha, J.P., Silvestre, C.: Self-triggered and eventtriggered set-valued observers. Inf. Sci. 426, 61–86 (2018) 13. Shamma, J.S., Tu, K.Y.: Set-valued observers and optimal disturbance rejection. IEEE Trans. Autom. Control 44(2), 253–264 (1999) 14. Turgeman, A., Datar, A., Werner, H.: Gradient free source-seeking using flocking behavior. In: American Control Conference (ACC), pp. 4647–4652 (July 2019)

Model Predictive Control of a Pusher Type Reheating Furnace Silvia Maria Zanoli1(B) , Francesco Cocchioni2 , Chiara Valzecchi2 , and Crescenzo Pepe2 1 Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, AN, Italy

[email protected] 2 Alperia Bartucci S.p.A, Corso Vittorio Emanuele II, 37038 Soave, VR, Italy

Abstract. This paper focuses on the design of an Advanced Process Control system for the control and the optimization of a pusher type billets reheating furnace. The controller formulation is based on a two-layer Model Predictive Control strategy. The formulated linear model of the furnace variables and the billets temperature behavior is constituted by a combination of Linear Time Invariant and Linear Parameter Varying models. The basic control mode is based on an adaptive approach. Simulation and field results on a plant located in an Italian steel industry have shown the reliability of the proposed control system and its optimality with respect to the previous control system, based on local PID controllers managed by operators. Significant improvements on process control have been obtained, together with a major and safer approach to process operating limits. Keywords: Model Predictive Control · Advanced Process Control · Billets reheating furnace · Energy efficiency

1 Introduction Steel industries process raw materials (e.g. waste steel products) to obtain small steel bars at an intermediate stage of manufacture, e.g. billets or slabs. To produce the final products, billets are introduced in reheating furnaces, where their reheating process takes place at high increasing temperatures. After their path along the furnace, billets are transported to the rolling mill stands, where they are plastically deformed in order to obtain the finished products like tube rounds or iron rods [1–3]. The middle phase of the steel industries workflow, i.e. the Reheating Phase, represents the most important subpart from an energy efficiency and quality of product point of view. The control efficiency of this subpart is of strategic importance due to the high energy amount required: an optimal trade-off between conflicting requirements, i.e. environmental impact decreasing, energy saving, and production and product quality increasing has to be ensured.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 455–465, 2021. https://doi.org/10.1007/978-3-030-58653-9_44

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In a steel industry reheating furnace there are many aspects to be taken into account. The multivariable and nonlinear time varying nature of the process together with the presence of many conflicting specifications, call for Advanced Process Control (APC) solutions. In the billets reheating furnaces control literature, different approaches have been proposed. The potential of model-based control and optimization is reported in [4]; here a proposed approach is based on a transient nonlinear furnace model and the related control system is developed. In [5], a nonlinear model predictive controller is designed for a continuous reheating furnace for steel slabs. Based on a first principles mathematical model, the controller defines local furnace temperatures so that slabs reach their desired final temperatures. An integrated method of intelligent decoupling control based on a recurrent neural network for zones temperature estimation and a heat transfer model for billets temperature prediction is proposed in [6]. In [7], a double model for the heating process of the reheating furnace is presented, together with the related double model slab tracking control system. In [8], the reheating furnace process is modeled as a nonlinear optimization problem (genetic algorithms approach) with the goal of minimizing fuel cost while satisfying a desired discharge temperature. The APC system is based on a global furnace linear model composed by Linear Time Invariant and Linear Parameter Varying models [9]. Two control modes have been designed. In [10] the control mode that does not include the billets temperature behavior has been discussed. In this paper, further details on the APC system are provided with respect to [9] and [10]; the main control mode is represented by an adaptive model predictive controller. Significant simulation and field results are discussed, focusing on energy efficiency aspects. The process description and the controller features are resumed in Sect. 2. Simulation and real results are reported in Sect. 3 and in Sect. 4, respectively. Section 5 provides the conclusions.

2 Process and APC Features 2.1 Process Modelling and Control Requirements The considered reheating furnace can contain up to 136 billets (mb = 136) with rectangular section; typical billets features have been reported in Table 1. Billets inlet temperature is measured by an optical pyrometer located near the inlet of the furnace. The billets outlet temperature is measured by an optical pyrometer located in the rolling mill area. Billets movement along the furnace depends on the designated furnace production rate (up to 120 [t/h]) and it is carried out by pushers. Billets are pushed through different furnace areas (Preheating, Heating and Soaking) at increasing temperatures. The combustion reactions that produce furnace zones and billets heating are triggered by air/fuel (natural gas) burners; the combustion air supplied to each burner is preheated by a heat exchanger (smoke-exchanger). Measurements of the temperature of each furnace zone and of the smoke-exchanger are acquired by thermocouples. Fuel (natural gas) and air flow rates are measured through flowmeters. Air and furnace pressures are measured by manometers. Billets transition at the furnace inlet and outlet is detected by photocells. No temperature measurements for the billets that are within the furnace are available. A control system for billets reheating furnaces must ensure correct triggering

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Table 1. Billets features. Feature

Value

Section

0.2 [m] × 0.16 [m]

Length

9 [m]

Mass

2.2815 [t]

Inlet temperature

20 [°C]–700 [°C]

Outlet temperature 1000 [°C]–1100 [°C]

Table 2. Virtual sensor inputs vector. Input name

Acronym [Units]

Tunnel temperature

Tun [°C]

Zone 6 temperature

Temp6 [°C]

Zone 5 temperature

Temp5 [°C]

Zone 4 temperature

Temp4 [°C]

Zone 3 temperature

Temp3 [°C]

Mean Zones 2–1 Temp. TempM 21 [°C]

of the involved thermodynamic and physical reactions, in order to guarantee a safe furnace conduction and the desired billets outlet temperature. The minimization of the fuel specific consumption, together with the furnace production rate maximization, clearly represents a crucial factor for energy efficiency achievement and improvement. In addition to these requirements, there is the need to meet stringent quality standards of the finished products and to comply with rigorous environmental standards (CO2 emissions reduction). To overcome the absence of temperature sensors inside the furnace to measure billets temperature, a virtual sensor has been developed [9]. The virtual sensor formulation has been based on a first principles adaptive nonlinear model that includes heat phenomena and billets movement. The related inputs are represented by linear combinations of the furnace zone temperatures (see Table 2). In Table 2, the Mean Zones 2–1 Temperature variable indicates the mean between the temperatures of the last two zones that are vertically disposed (zone 1 and zone 2, Soaking Area). The virtual sensor model has been suitably linearized for its inclusion in a linear control strategy, obtaining a Linear Parameter Varying (LPV) model. Among the selected Controlled Variables (CVs) for the APC system design, the billets temperatures have been included and grouped in the bCVs (b) group. A second CVs group that has been defined is represented by the zones Controlled Variables (zCVs, y) group. This includes all furnace zone temperatures, temperature differences between adjacent furnace zones (Table 2), smoke-exchanger temperature, total air flow rate, and opening position (percentage) of the valves related to fuel flow rates. As Manipulated

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Variables (MVs, u), the fuel flow rate and the stoichiometric ratio of each furnace zone equipped with its own burners set (all zones except tunnel) have been selected. Finally, in the Disturbance Variables (DVs, d) group, furnace and air pressures, together with furnace production rate, have been included [9]. zCVs-MVs/DVs models have been obtained through step test procedures: Linear Time Invariant (LTI) asymptotically stable strictly proper models without delays have been derived. Through these models, bCVs have been directly tied to the MVs and DVs. 2.2 APC Architecture and Control Modes Two control modes have been formulated: • adaptive APC mode: this mode makes use of both identified zCVs-MVs/DVs linear time invariant models and first principles bCVs LPV model and exploits billets virtual sensor information. An adaptive two-layer linear Model Predictive Control (MPC) strategy has been formulated [9]; • zones APC mode: zCVs-MVs/DVs identified linear time invariant models are used within a two-layer linear MPC strategy [10, 11]. Figure 1 schematically depicts the APC architecture. The two-layer MPC strategy is based on the solution of two cascaded optimization problems, solved by a Dynamic Optimizer module (Fig. 1, DO) and a Targets Optimizing and Constraints Softening module (Fig. 1, TOCS). DO module, i.e. the lower layer of the proposed MPC structure, has been based on the minimization of the quadratic cost function (1), subject to the linear constraints (2).

Fig. 1. Schematic representation of the APC system.

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Hp −1  Hp    uˆ (k + i|k) − ut (k + i|k)2 + yˆ (k + i|k) − yt (k + i|k)2 S(i) Q(i) i=0 i=1 2 Hu  mb  2  2  ˆ    ˆu(k + Mi |k) + bi (k + ej k) − lbb_DOj  R(i) + εy (k) ρ +

VDO (k) =

i=1

y

j=1

+ εb (k)2ρb

Tj

(1)

subject to i. lbdu_DO (i) ≤ ˆu(k + Mi |k) ≤ ubdu_DO (i), i = 1, . . . , Hu ii. lbu_DO (i) ≤ uˆ (k + Mi |k) ≤ ubu_DO (i), i = 1, . . . , Hu iii. lby_DO (i) ≤ γlby_DO (i) · εy (k) ≤ yˆ (k + i|k) ≤ uby_DO (i) + γuby_DO (i)· εy (k), i = 1, . . . , Hb  iv. lbb_DOj − γlbb_DOj · εbj (k) ≤ bˆ i (k + ej k) ≤ ubb_DOj + γubb_DOj · εbj (k), j = 1, . . . , mb v. εy (k) ≥ 0; εb (k) ≥ 0

(2)

DO module main task is the computation, at each control instant k, of Hu (control horizon) MVs future moves ˆu(k + Mi |k), taking into account of MVs and zCVs specifications and predictions (ˆu(k + i|k), yˆ (k + i|k)) over a prediction horizon Hp (receding horizon strategy [12]). Furthermore, predictions on bCVs are taken into account in order to guarantee specifications at the related furnace exit instants (ej in (1)–(2)). The specifications related to the process variables consist in hard constraints (lbdu_DO , ubdu_DO , lbu_DO , ubu_DO in (2)), soft constraints (lby_DO , uby_DO , lbb_DO , ubb_DO in (2)) and targets (ut (k + i|k), yt (k + i|k), lbb_DO in (1)). In soft constraints one slack variable for each CV has been included (εy (k) and εb (k) vectors). S, Q, R, Tj , ρy , ρb are positive semi-definite (or definite) diagonal matrices, while γlby_DO , γuby_DO , γlbb_DO , γubb_DO are vectors composed by positive elements. The end terms of ut (k + i|k) and yt (k + i|k), together with zCVs constraints, are provided by the upper layer steady- state module (TOCS) through the minimization of a linear cost function subject to linear constraints [11]. The main difference between TOCS and DO modules formulation is that TOCS module does not include bCVs specifications given their LPV model. The APC basic control mode is constituted by the adaptive mode. Switching between adaptive and zones APC modes depends on the status value of bCVs. The bCVs status value is defined by a cooperative action between SCADA, virtual sensor and Data Conditioning & Decoupling Selector block (Fig. 1, b Status) and determines the inclusion of bCVs terms in (1)–(2). When the specifications of at least one bCV must be taken into account, the status value of at least one bCV is active and the adaptive APC mode is activated (see (1)–(2) for DO formulation); otherwise (for example, in case of bad estimations of the virtual sensor), the control system switches to the zones APC mode [10]. 2.3 Some Tuning and Parameters Details In the following sections, simulation (Sect. 3) and field results (Sect. 4) related to the adaptive APC mode will be reported.

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Taking into account an MPC block sampling time of 1 min (according to the developed process model), in the adaptive APC mode the parameters Hp , Hu and Mi are adapted taking into account the actual furnace production rate. In the simulation case study proposed in Sect. 3, the time of billets movement is equal to 120 [s], which is related to a furnace production rate of about 70 [t/h]. Accordingly, a value of 272 [min] is set for Hp , time needed for the exit of the billet located in the 1st place (the 1st place is assumed to be the closest place to the furnace inlet) from the furnace. In a parametric way, Hu = 55 moves has been set and the Mi MVs movement instants have been suitably distributed over Hp . The constraints related to fuel flow rates (MVs) and to the zone temperatures have been reported in Tables 3, 4. Furthermore, the differences between the adjacent zone temperatures of Table 2 are only upper constrained by a value of 0 [°C]. The zCVs specifications require a constrained control, so the related tracking terms in (1) are not present. As previously described, TOCS module formulation does not take into account the information provided by the developed billets temperature overall linear model; for this reason, the steady-state targets supplied by TOCS module may be unreachable in the adaptive APC mode and a profitable trade-off between MVs tracking and bCVs constraints/targets satisfaction must be ensured. Table 3. Fuel flow rates constraints. Fuel flow rate Acronym [Units] Constraints Zone 6

Fuel 6 [Nm3/h]

0–800 [Nm3/h]

Zone 5

Fuel 5 [Nm3/h]

0–1600 [Nm3/h]

Zone 4

Fuel 4 [Nm3/h]

0–650 [Nm3/h]

Zone 3

Fuel 3 [Nm3/h]

0–650 [Nm3/h]

Zone 2

Fuel 2 [Nm3/h]

0–250 [Nm3/h]

Zone 1

Fuel 1 [Nm3/h]

0–250 [Nm3/h]

Table 4. Zone temperatures constraints. Acronym Constraints Tun

550–950 [°C]

Temp6

800–1150 [°C]

Temp5

800–1150 [°C]

Temp4

800–1200 [°C]

Temp3

1000–1250 [°C]

Temp2

1000–1250 [°C]

Temp1

1000–1250 [°C]

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3 Simulation Results In order to show the performances of the adaptive APC mode, a simulated plant operating condition is proposed. The identified zCVs-MVs/DVs model is exploited as zCVsMVs/DVs plant model and no measurement noise is assumed. The developed billets temperature nonlinear model that is exploited by the virtual sensor is used as plant model for simulating the relationships between billets temperature and the related input vector (Table 2). At the initial control instant, the activation of the adaptive APC mode is requested. The estimation performed by the virtual sensor gives reliable results and bCVs are considered in the control problem. Tables 3, 4 list a selection of the MVs and of the zCVs considered in the simulation; the differences between the adjacent zone temperatures of Table 2 are included in the control problem. DVs and the MVs not listed in Table 3 have been assumed constant, not influencing the proposed example. As already described in Sect. 2.3, the furnace production rate is equal to about 70 [t/h] and the constraints related to the considered MVs and zCVs are reported in Tables 3, 4. The 136 billets that are initially present within the furnace are characterized by different temperatures (in the range 20 [°C]–1140 [°C]) and the billets that enter the furnace are assumed characterized by a furnace inlet temperature of about 170 [°C]. The Rolling phase specifications require that the billets reach the rolling mill stands with a temperature in the range 1020 [°C]–1040 [°C]. This specification, through the online virtual sensor action, is online converted in a temperature range at the furnace outlet (constant at 1040 [°C]–1060 [°C] for the case at issue). As starting point of the simulation, all the considered process variables are within the assigned constraints. The TOCS-DO cooperative action, together with the information provided by the billets temperature virtual sensor and by the LPV model within DO module, leads the process to a more profitable operating point. As can be observed in Fig. 2a), the exited billets that reach the pyrometer placed on the rolling mill area are characterized by temperatures decreasing towards the assigned lower bound. This is obtained through a coordinated control action of the fuel flow rates of the different furnace zones (Fig. 2b– c): only two fuel flow rates have been reported, for brevity). The fuel flow rates control action is forwarded to the billets temperatures through the variables of Table 2.

4 Field Results Study and design phases of the project for the development of the APC system began in January 2015 and ended in May 2015. In early June 2015, the APC system, has been installed on the considered Italian steel plant for the optimization of the Reheating phase, substituting the local PID temperature controllers managed by plant operators. Figure 3 shows a field plant condition under the control of the developed APC system. A five hours period is taken into account and the APC performances are shown. The results of the estimation performed by the virtual sensor are shown in Fig. 3a) (green stars). The actual furnace production rate (Fig. 3b)) is equal to about 50 [t/h]. All 12 MVs can be exploited by the controller; all main zCVs and all DVs are included in the control problem (some process variables have not been shown for brevity). The constraints related to the MVs and to some zCVs are summarized in Tables 3, 4. The temperature differences between the variables of Table 2 are upper constrained by 0 [°C].

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Fig. 2. Simulation results: a) bCVs; b) Fuel1 ; c) Fuel5

The 136 billets that are initially present within the furnace are characterized by different temperatures (in the range 30 [°C]–1100 [°C]) and the billets that successively enter the furnace are characterized by different inlet temperatures (in the range 15 [°C]– 80 [°C], see Fig. 3c)). The furnace production rate varies in the range 30 [t/h]–105 [t/h] (Fig. 3b)), while the furnace pressure and the air pressure (DVs) are characterized by average values of about 0.75 [mmH2O] and 84 [mbar], respectively. Furthermore, the Rolling phase specifications require that the billets reach the rolling mill stands with a temperature in the range 1060 [°C]–1105 [°C]. As starting point of the proposed plant condition, all the considered process variables are within the assigned constraints. The TOCS-DO cooperative action, together with the information provided by the billets temperature virtual sensor and by the LPV model within DO module, allows maintaining the process at a profitable operating configuration, despite the not constant furnace

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Fig. 3. Field results: a) bCVs; b) furnace production rate; c) billets inlet temperatures.

production rate and billets inlet temperatures. Exited billets reaching rolling mill stands are characterized by temperatures that approach the assigned lower bound (Fig. 3 a)), obtained through a coordinated MVs control action. The control action is forwarded on the billets temperatures through the variables of Table 2. Figure 4 shows the monthly energy saving (blue line) and the cumulative energy saving (red line) obtained by the APC system related to the last three years. August and December months have not been reported due to the plant maintenance. The energy saving is evaluated with respect to the computed fuel specific consumption baseline, that varies with the furnace hot charge. The cumulative energy saving related to the last three years is about 3.76 [%].

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Fig. 4. Field results: monthly energy saving and cumulative energy saving (2017–2019).

5 Conclusions In this paper, an Advanced Process Control system based on a two-layer linear Model Predictive Control strategy has been designed for the optimization of a pusher type billets reheating furnace, located in an Italian steel plant. The developed control method exploits the action of a virtual sensor for the achievement of the estimation of billets temperature. Furthermore, a global linear model has been developed which is partially adaptive. Performances on the steel plant have shown the major efficiency of the proposed control solution with respect to the previous one, based on local PID controllers managed by plant operators. When the process was controlled through the local PID temperature controllers managed by plant operators, the simultaneous fulfillment of energy and product quality specifications was difficult to achieve. With the previous control system, operators safely ensured the desired billets exit temperatures and neglected aspects related to fuel minimization, thus achieving a low energy efficiency. The cumulative energy saving obtained in the last three years is about 2.82 [%] (with respect to the defined fuel specific consumption baseline). Thanks to the tailored hardware and software configuration, characterized by a continuous communication between the APC system PC and the plant, the proposed control solution has been certified as Industry 4.0 compliant.

References 1. Energy Efficiency Office: Continuous steel reheating furnaces: Operation and Maintenance, Good Practice Guide 77. Harwell, Oxfordshire (1993) 2. Trinks, W., Mawhinney, M.H., Shannon, R.A., Reed, R.J., Garvey, J.R.: Industrial Furnaces. John Wiley & Sons, New York (2004) 3. Glinkov, M.A.: General theory of furnaces. J. Iron Steel Inst. 584–594 (1968) 4. Steinboeck, A.: Model-based control and optimization of a continuous slab reheating furnace. Shaker Verlag GmbH, Aachen (2011) 5. Steinboeck, A., Wild, D., Kugi, A.: Nonlinear model predictive control of a continuous slab reheating furnace. Control Eng. Pract. 21(4), 495–508 (2013) 6. Liao, Y.X., She, J.H., Wu, M.: Integrated Hybrid-PSO and Fuzzy-NN decoupling control for temperature of reheating furnace. IEEE Trans. Ind. Electr. 56(7), 2704–2714 (2009)

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7. Yi, Z., Su, Z., Li, G., Yang, Q., Zhang, W.: Development of a double model slab tracking control system for the continuous reheating furnace. Int. J. Heat Mass Transf. 113, 861–874 (2017) 8. Santos, H.S.O., Almeida, P.E.M., Cardoso, R.T.N.: Fuel costs minimization on a steel billet reheating furnace using genetic algorithms. Modelling and Simulation in Engineering, Article ID 2731902, p. 11 (2017) 9. Astolfi, G., Barboni, L., Cocchioni, F., Pepe, C., Zanoli, S.M.: Optimization of a pusher type reheating furnace: an adaptive model predictive control approach. In: 6th International Symposium on Advanced Control of Industrial Processes, pp. 19–24 (2017) 10. Zanoli, S.M., Pepe, C., Barboni, L.: Application of advanced process control techniques to a pusher type reheating furnace. In: Journal of Physics: Conference Series, vol. 659, no. 1 (2015) 11. Zanoli, S.M., Pepe, C.: Two-layer linear MPC approach aimed at walking beam billets reheating furnace optimization. J. Control Sci. Eng., Article ID 5401616, p. 15 (2017) 12. Maciejowski, J.: Predictive Control with Constraints. Prentice-Hall, Harlow (2002)

Distributed LQ Control of a Water Delivery Canal Based on a Selfish Game Jo˜ao P. Belfo1(B) , Jo˜ ao M. Lemos1 , and A. Pedro Aguiar2 1

2

INESC-ID, Instituto Superior T´ecnico, Lisbon, Portugal [email protected] SYSTEC, Faculdade de Engenharia da Universidade do Porto, Porto, Portugal

Abstract. This article describes the design of distributed LQ controllers based on a linear model of a water delivery canal. According to a distributed strategy, a local control agent is associated to each gate of the canal. The main goal is to control the position of the gates in order to drive the water levels for each pool to variable references. For this purpose, local optimization problems are presented, that will be solved at each local controller. In order to achieve a consensus between agents, a coordination method, based on game theory, is presented, including its convergence analysis. The overall controller architecture is described. Simulations, illustrating scenarios in which the convergence conditions are satisfied, are presented. Keywords: Distributed optimal control · DLQ · Selfish game theory · Small gain theorem · Water delivery canal

1

· Game

Introduction

Water delivery canals [1,2] provide an interesting example of an application for which distributed control may be used with advantage. These systems are spread over wide-space areas, and transport water from the sources (e.g. containers) to the user locations, being equipped with sensors, actuators and local controllers in isolated spots. In order to control this kind of network, one may consider decentralized or even centralized architectures. However, due to their physical characteristics and complexity of the transmission network involved, those architectures may lead to poor performance or even instability. These features motivate the use of distributed optimal control strategies, in which each local controller acts in a coordinated way by communicating with their neighbors to reach a consensus [3]. These strategies comprise an optimization algorithm used to solve local dynamic optimization problems associated to This work was performed within the framework of the project HARMONY, Distributed Optimal Control for Cyber-Physical Systems Applications, financed by FCT under contract AAC n2/SAICT/2017 - 031411, project IMPROVE - POCI-01-0145-FEDER031823, and pluriannual INESC-ID funding UIDB/50021/2020. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 466–476, 2021. https://doi.org/10.1007/978-3-030-58653-9_45

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each of the local controllers, and a coordination method that allows the local controllers to communicate among them to achieve coordinated action. The optimization algorithm considered here is the LQ control algorithm [4,5]. Other optimization algorithms may be considered, in particular MPC control, that has the advantage of allowing the incorporation of constraints in an easy way, either to ensure stability or related to process operation [6–9], but requires a higher computational load. The coordination method defines how the controllers communicate and also the information that must be communicated. Game theory methods, such as the selfish game considered here, [5] are often used as coordination methods, since they are easy to implement and require low computational time. However, they have the disadvantage of converging to a Nash equilibrium, that may be far from the global optimal solution given by the centralized system. To circumvent this issue, methods based on duality, such as the ADMM algorithm [10], may be considered, although they present a more complex optimization structure as well as a higher computational load. Although a network with only three agents is considered in this work, results may be found in [10] and [11] about larger networks. The convergence analysis of the coordination method used here, the selfish game, uses small gain conditions [12], together with input-to-state stability conditions [13,14]. 1.1

Contributions and Organization

The contribution of this paper consists on the design of distributed LQ controllers based on linear models, considering the selfish game as the coordination method, and its application to the water delivery canal problem. A result on the convergence of the coordination procedure is presented. The structure of this paper is as follows: the water delivery canal is described in Sect. 2, including the linearized model obtained through mathematical identification of the nonlinear canal plant, and the control problem is formulated. In Sect. 3, the solution of the local optimization problems is presented, and the selfish game is introduced, together with the convergence analysis. Next, in Sect. 4, simulation results are shown, illustrating scenarios in which the convergence conditions are satisfied.

2 2.1

Water Delivery Canal Canal Description

The canal considered in this article is composed of four pools interconnected by four gates, as Fig. 1 suggests. Each pool i is equipped with two sensors to measure the upstream, Mi , and the downstream, Fi , level of the water, and a local controller Ci , for i = 1, 2, 3. The canal has also a container that injects water to pool 1 with flow Q0 . More details regarding the physical system can be found in [5] and [15].

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The water level of each pool i is controlled by the gate ui , where ui represents the position of the gate. The first three pools are terminated by undershoot gates and the last is terminated by an overshoot gate.

Fig. 1. Schematic of the water delivery canal

The main goal is to control the water levels at each pool by operating the gate positions ui . However, instead of directly considering the gate positions as the input signals, a change of variable is considered in order to yield a linear dependence on the manipulated variable. The manipulated variable considered is the water flow drawn by the gates, vi , obtained by  vi (t) = ui (t)W 2g(Fi − Mi+1 ), (1) where W = 0.49 m is the width of the gates and g = 9.81 m/s2 is the gravity acceleration. 2.2

State-Space Model

Using the nonlinear plant of the canal, the following linearized model is obtained by identification methods x(t + 1) = Ax(t) + Bv(t),

(2)

y(t) = Cx(t), where

⎤ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ A1 0 0 B11 B12 0 v1 (t) x1 (t) x(t) = ⎣x2 (t)⎦ , A = ⎣ 0 A2 0 ⎦ , B = ⎣B21 B22 B23 ⎦ , v(t) = ⎣v2 (t)⎦ , 0 B32 B33 x3 (t) 0 0 A3 v3 (t) (3) for the situation in which only the first three gates are controlled and the fourth gate is kept at a fixed position. Note that the interaction between controllers is effected only through their manipulated variables vi , which explains why matrix A is diagonal. Moreover, the matrix B translates the interconnected structure of the controllers, the controllers 1 and 3 being not directly connected.

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Since the states are not accessible, a Kalman filter is used for state estimation, through ˆi (t − 1|t − 1) + Bi v(t − 1) (4) x ˆi (t|t − 1) = Ai x x ˆi (t|t) = x ˆi (t|t − 1) + Wi (Fi (t) − Ci x ˆi (t|t − 1)) , where Bi is the i-th row of matrix B defined in (11) and matrix Wi is given by −1

Wi = Di Ci (Ci Di Ci + gi )

,

(5)

with Di the solution of the algebraic Riccati equation  −1 Ai + qi , Di = Ai Di I + Bi ri−1 Bi Di

(6)

where gi and qi are the covariance matrices of the measurement and process noise, respectively. In order to assure that the water level of pool i given by the linear model, yi , follows a reference ri , integral action is embedded in the controller design. The integrator (I) (I) (7) xi (t + 1) = xi (t) + Ti (ri (t) − Ci xi (t)) , where Ti is a constant parameter, is added to each controller as an additional state, yielding the augmented linear model ¯ij vj (t) + Ψi ri (t), B ¯i (t) + (8) x ¯i (t + 1) = A¯i x j∈Si

where

x ¯i (k) =

 



 xi (t) Ai 0 Bij 0 ¯ ¯ , A , B , Ψ = = = , i ij i (I) 0 Ti −Ti Ci 1 xi (t)

(9)

and Si is the set of agents with which agent i interacts including itself. For instance, S2 = {1, 2, 3}, since agent 2 interacts with the agents 1 and 3. Similarly, S1 = {1, 2} and S3 = {2, 3}. 2.3

Problem Formulation

Consider the three-agent system, where each agent is composed by a local controller with the dynamics given by (8). The main goal is to minimize a global cost function J(¯ x, v) = J1 (¯ x1 , v1 , v2 ) + J2 (¯ x2 , v1 , v2 , v3 ) + J3 (¯ x3 , v2 , v3 ),

(10)

where Ji is the local cost known only at the agent i, for i = 1, 2, 3, defined by Ji =

∞ t=0

x ¯i (t) Qi x ¯i (t) + vi (t) Ri vi (t),

(11)

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for Qi =

  Ci Ci 0 , 0 1

and Ri a constant parameter. The local optimization problem that each local controller must solve is min vi

s. t.

∞

x ¯i (t)Qi x ¯i (t) + vi (t)Ri vi (t), ¯ Bij vj (t) + Ψi ri (t), ¯i (t) + x ¯i (t + 1) = A¯i x

t=0

j∈Si

x ¯i (0) = x ¯i0 ,

(12)

where vj is given for j ∈ Si \ {i}. Since the solution of the optimization problem (12), for i = 1, 2, 3, does not yield the value of (v1 , v2 , v3 ) that minimizes the global cost J, a coordination method is needed.

3

Distributed LQ Control

The solution of the local optimization problem (12) is given by the control law [5] ¯ij vj (t), B ¯i (t) − Ki Ψi ri (t) − Ki (13) vi∗ (t) = −KLQi x j∈Si \{i}

in which the state feedback gain is   ¯  Pi B ¯ii −1 R−1 B ¯  Pi A¯i , KLQi = 1 + Ri−1 B ii ii i

(14)

where the matrix Pi satisfies the algebraic Riccati equation   ¯ii B ¯  Pi −1 A¯i + Qi . Pi = A¯i Pi I + Ri−1 B ii

(15)

The Ki gain is given by 

    ¯ii ¯ii −1 R−1 B ¯ii Pi + Mi−1 Γi , Pi B Ki = 1 + Ri−1 B i with

 −1   ¯ii B ¯ii ¯ii R−1 B ¯ii Mi = I + A¯i Pi I + Ri−1 B B Pi − A¯i , i

and

  ¯ii B ¯  Pi −1 . Γi = A¯i Pi I + Ri−1 B ii

3.1

(16)

Coordination Method

The coordination method considered here is a selfish game, based on game theory. For each sampling time t, define Vit (k), i = 1, 2, 3, as the signal that captures the sequence of decisions (i.e. the solutions of the local optimization problem) of each local controller. The method is summarized in the following Algorithm 1.

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Algorithm 1. DLQ control 1: Initialization: x ¯i (0) = x ¯i0 , Vi0 (0) = Vi0 , for all i = 1, 2, 3 2: for t ≤ simulation time do 3: for k = 0, ..., Ni − 1 do 4: Agent i, for all i = 1, 2, 3, solve (12):  ¯ij Vjt (k) ¯i (t) − Ki Ψi ri (t) − Ki j∈Si \{i} B 5: Vit (k + 1) = −KLQi x t 6: Broadcast Vi (k + 1) to its neighbors 7: end for 8: vi (t) ← Vit (Ni ) 9: Vit+1 (0) ← Vit (Ni ) for all i = 1, 2, 3 10: Agent i applies the control signal vi (t) 11: end for

It is remarked that, during the coordination process (i.e., lines 3–7 of the Algorithm 1, where Ni is the total number of iterations), each agent optimizes its own local cost function, without considering what is optimal to its neighbors. Besides the evolution of the dynamics of each agent for each sampling time t, there is also an evolution of each control signal indexed by k, for each t, given by the signals Vit (k). This internal dynamic (internal loop in Algorithm 1) is dictated by the sequence of solutions of the local optimization problems, which are different for each k. 3.2

Convergence of the Coordination Method

The dynamics of the coordination game consists of iterating in k the equations ⎧ t ¯12 V t (k) ⎪ ¯1 (t) − K1 Ψ1 r1 (t) − K1 B 2 ⎨V1 (k + 1) = −KLQ1 x   ¯21 V t (k) + B ¯31 V t (k) , (17) ¯2 (t) − K2 Ψ2 r2 (t) − K2 B V2t (k + 1) = −KLQ2 x 1 3 ⎪ ⎩ t ¯32 V t (k) ¯3 (t) − K3 Ψ3 r3 (t) − K3 B V3 (k + 1) = −KLQ3 x 2 Note that each equation in (17) can be considered as a discrete time linear dynamic system. For instance, observing the first equation, the system output corresponds to the variable V1t (k) and the input corresponds to the variable ¯1 (t) and K1 Ψ1 r1 (t) are fixed V2t (k). It is remarked that first two terms KLQ1 x terms for each game executed at each t, since they depend on constants and on signals that vary in t. To analyze the convergence of the selfish game, the following proposition is used. Proposition 1. Assume that all the sub-systems in (17) are ISS in the sense that there exists some KL-function βi and some K-function γij (1 ≤ i ≤ 3 and for each i, j ∈ Si \ {i}) such that |V1t (k + 1)| ≤ β1 (|V1t (0)|, k) + γ12 (V2t ), |V2t (k + 1)| ≤ β2 (|V2t (0)|, k) + γ21 (V1t ) + γ23 (V3t ), |V3t (k + 1)| ≤ β3 (|V3t (0)|, k) + γ32 (V2t ),

(18)

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where

¯ij |V t (k). γij (Vjt ) = |Ki B j

(19)

If the following set of small-gain conditions holds ¯12 ||K2 B ¯21 | < I, |K1 B

¯23 ||K3 B ¯32 | < I |K2 B

(20)

where I is the identity matrix, then the interconnected system (17) is input-to-state stable (ISS) and, consequently, the game converge, at each sampling time t. Proof. The first step is to prove that the dynamics of the systems presented in (17) taking as output the variables Vit (k + 1) and as input Vjt (k) for all j ∈ Si \ {i}, are ISS [13], using the definition of an input-to-state stable (ISS) system presented in [14] and in [13]. The second step is to show that the overall system composed by the interconnected systems in (17) possesses the global stability property, since the ISS conditions in (18) implies that each system in (17) is globally stable. 

4

Results

Figure 2 represents the controller diagram, including the Kalman filter, used for state estimation, the integrator and a block used for changing the manipulated variable, according to (1).

Fig. 2. Local controller diagram, where fi is the function that translated the change of variable, according to (1).

Note that the linear model was identified around an equilibrium point. The equilibrium values must thus be subtracted from the measurements of the levels in the nonlinear model, yielding the signals ri , Fi , and ui presented in Fig. 2. Furthermore, the equilibrium value must be added to the optimization results, yielding the signal ui ∗ in Fig. 2. It is also remarked that the nonlinear plant of the canal contains a saturation and slew rate blocks through which the signal ui ∗ passes before being applied to the canal. The signal ui corresponds to the measurement made right after these two blocks.

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It is remarked that the identification of the linearized model around a specific equilibrium point presents low degradation for different equilibrium points, meaning that the same linearized model may be used for a wide operation range. To define the controller, there are several constant parameters that need to be defined, in particular Ri , Ti , gi , and qi used in the Kalman filter. Note that the convergence conditions in (20) depend on the parameters Ri and Ti and, thus, their value must be such that the coordination converge. Furthermore, an anti-windup strategy for the integrator may be considered. Setting R1 = R2 = R3 = 3000 and T1 = T2 = T3 = 0.3, (20) yields ¯12 ||K2 B ¯21 | = 0.0086 < 1, |K1 B

¯23 ||K3 B ¯32 | = 0.0069 < 1, |K2 B

(21)

and, thus, the coordination converge. Furthermore, it is possible to verify that the convergence of the game, for this particular application, is very robust to the parameters Ri and Ti . Figure 3 shows the nonlinear response of the water levels in the three pools, considering the parameters R1 = R2 = R3 = 3000 and T1 = T2 = T3 = 0.1, for Ni = 10, including the position of the three gates (Fig. 4) that is applied to the nonlinear plant. The results obtained here present a performance similar to the ones shown in [5] and [6].

Fig. 3. Water levels in the three pools following the references.

It is remarked that the convergence conditions only address the convergence of the coordination method and not the stability of the overall system. To illustrate this fact, let R1 = R2 = R3 = 100 and T1 = T2 = T3 = 2, which yield the conditions ¯12 ||K2 B ¯21 | = 2.97 × 10−04 < 1, |K2 B ¯23 ||K3 B ¯32 | = 1.01 × 10−04 < 1, (22) |K1 B meaning that the coordination converge. However, the simulation result in Fig. 5 shows an unstable and undesired behavior of the water levels. This behavior may

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Fig. 4. Position of each gate of the simulation presented in 4

also exist due to differences between the nonlinear plant and the linearized model used to design the controllers, that can be addressed by decreasing the integrator gain Ti and increasing the control cost Ri , as the simulation in Fig. 3 suggests.

Fig. 5. Water levels in the three pools following the references, for R1 = R2 = R3 = 100 and T1 = T2 = T3 = 2.

5

Conclusions

The water delivery canal used for the case study is described and its linearized model is presented, in which an integral action is added, together with an

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anti-windup strategy, to allow the controllers to follow non-zero references. Through simulations, it was possible to conclude that the existence of an upper/lower bound limit of charging of the integrator is fundamental for the stability of the water levels, when the integrator gain Ti is increased. This is due to the fact that, when the gain Ti increases, the controller will notice the error between the reference and the actual water level at a faster rate and, thus, will tend to act in a more abrupt way, leading to the creation of reflected waves and unstable behaviors. In addition, it was possible to conclude that, when the value of Ri increases and Ti decreases, the water level of pool i presents a smoother and more stable evolution. It is remarked that the convergence conditions only address the convergence of the coordination method, and not the stability of the overall system. This fact means that it is possible for the coordination method to converge and the overall system to be unstable. The stability of the distributed control system is addressed in other works.

References 1. Mareels, I., Weyer, E., Ooi, S.K., Cantoni, M., Li, Y., Nair, G.: Systems engineering for irrigation systems: successes and challenges. In: Proceedings of the 16th IFAC World Congress (2005) 2. Litrico, X., Fromion, V.: Modeling and Control of Hydrosystems. Springer (2009) 3. Cantoni, M., Weyer, E., Li, Y., Ooi, S.K., Mareels, I., Ryan, M.: Control of largescale irrigation networks. In: Proceedings of the IEEE, vol. 95, no. 1, pp. 75–91 (2007) 4. Lemos, J.M., Pinto, L.F., Rato, L.M., Rijo, M.: Distributed LQG control of a water delivery canal with feedforward from measured consumptions. In: 20th Mediterranean Conference on Control & Automation, pp. 722–727 (2012) 5. Lemos, J.M., Pinto, L.: Distributed linear-quadratic control of serially chained systems. IEEE Control Syst. Mag., 26–38 (2012) 6. Igreja, J.M., Lemos, J.M., Cadete, F.M., Rato, L.M., Rijo, M.: Control of a water delivery canal with cooperative distributed MPC. In: American Control Conference, pp. 3346–3351 (2012) 7. Mosca, E.: Optimal, Predictive, and Adaptive Contro. Prentice Hall (1995) 8. Maciejowski, J.M.: Predictive Control with Contraints. Prentice Hall (2002) 9. Maestre, J.M., Mu˜ noz de la Pe˜ na, D., Camacho, E.F.: Distributed model predictive control based on a cooperative game. Optim. Control Appl. Meth. 32, 153–176 (2011) 10. Mota, J.F.C., Xavier, J.M.F., Aguiar, P.M.Q., P¨ uschel, M.: Distributed ADMM for model predictive control and congestion control. In: 51st IEEE Conference on Decision and Control, pp. 5110–5115 (2012) 11. Mota, J.F.C.: Communication-Efficient Algorithms For Distributed Optimization. Ph.D. Thesis. Instituto Superior T´ecnico and Carnegie Mellon University (2013) 12. Zhongping, J., Yuandan, L., Yuan, W.: Nonlinear small-gain theorems for discretetime large-scale systems. In: Proceedings of the 27th Chinese Control Conference, pp. 16–18 (2008) 13. Jiang, Z.P., Wang, Y.: Small gain theorems on input-to-output stability. In: Proceedings of the 3rd International DCDIS Conference, pp. 220–224 (2003)

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14. Jiang, Z., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001) 15. Sampaio, I.G.C.: Fault Tolerant Control of a Water Delivery Canal. Master Thesis. Instituto Superior T´ecnico (2012)

A Comparative Performance Study of Inertial Vibration Controllers for an Ultra-lightweight GFRP Footbridge Carlos Mart´ın Concha Renedo(B) , Jos´e Manuel Soria, Christian Gallegos, and Iv´ an M. D´ıaz ETSICCP, Universidad Polit´ecnica de Madrid, 28040 Madrid, Spain [email protected] Abstract. Currently, the use of advanced lightweight materials in construction (as aluminium or fibre-reinforced polymer (FRP) composites) are enabling to create aesthetically pleasing structural forms by means of reducing their weight as much as possible. The results of this are ultra-lightweight structures (usually with low both mass and inherent damping) which difficultly comply with the vibration serviceability limit state (VSLS) under human-induced dynamic loading. Moreover, when assessing VSLS, the existing human-structure interaction (HIS) should be considered as it tends to be remarkable. Hence, smart damping strategies can be seen as a good solution to preserve lightweight nature of these structures in the most efficient way. This work aims to investigate whether the efficiency of three types of inertial controllers applied to an ultra-lightweight FRP footbridge degrades or not, when considering the HSI as another inherent element of the dynamic system to be controlled. Keywords: Vibration control controllers

1

· Lightweight structures · Inertial

Introduction

Nowadays, advanced materials with higher stiffness and lower density as fiber reinforced polymers (FRP) or aluminium have enabled to develop even more efficient structures. In this context, strength or stiffness are not constraints for designers to create slender structural forms. However, their low weight and inherent damping have led them to not satisfy the Vibration Serviceability Limit State (VSLS) under human-induced dynamic loading [1]. Usually, vibrations in lightweight structures are due to resonant responses when the first or the second harmonic of the human loading excites the fundamental structural resonance. However, ultra-lightweight structures can also present undesired vibrations due to either non-resonant responses excited by lower harmonics, or due to resonant loading related to higher harmonics. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 477–486, 2021. https://doi.org/10.1007/978-3-030-58653-9_46

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Furthermore, whilst, in classic steel lightweight structures, the mass of human users is not comparable with the structural modal mass, in ultra-lightweight structures, this relation may be significant. This fact might cause a HumanStructure Interaction (HSI) when the structural acceleration overcome a certain limit. Thus, in order to properly asses the VSLS in ultra-lightweight structures, it is required to use an interactive loading model for the human [2]. Inertial controllers can be a good solution to overcome the vibration problem in ultra-lightweight structures. They can be passive, semi-active or active according to the nature of the feedback control implemented. Passive ones, most commonly known as Tuned Vibration Absorbers (TVAs), are able to damp resonant vibrations when tuned to a single problematic vibration mode. Semi-active versions of the TVA (STVAs) are able to change the properties of the controller in real time, by means of smart dampers, achieving a certain grade of adaptability and making them more robust. Thirdly, active controllers (AVAs) make use of actuators to apply counteract control forces in real time, allowing to mitigate both resonant and non-resonant multi-harmonic responses. The three inertial controller types can be designed as a feedback system. Up to now, many strategies based on feedback control have been proposed and developed involving both the controller design and its optimal placement in the structure [3]. However, from the authors knowledge, HSI has not been yet considered in the design process. This paper intends to assess whether or not the HSI might play an important role when designing future inertial controllers applied to an upcoming laboratory ultra-lightweight structure which is being designed now. The remainder of the paper is organized as follows, the second section describes the structural layout of the studied FRP footbridge. Section 3 outlines the closed-loop system model with HSI and an inertial controller. In Sect. 4 the design parameters of three inertial controllers (TVA, STVA and AVA) are provided and some results about their efficiency with and without considering HSI are included. Finally, some conclusions are given in the last section.

2

Structure Description

The footbridge proposed in this study is a simply supported all-GFRP structure. It is 10.0 m long and 1.50 m wide, formed by planks placed onto two main beams that are laterally restrained with square tubes, as shown in Fig. 1a and 1c. All pultruded elements described in Fig. 1b are manufactured by Fiberline A/S [4]. It has been designed according to [5] and [6]. The total weight of the footbridge is 1.66 ton, including handrails and other non-structural elements. A linear elastic FE model of the footbridge was developed in Abaqus software [7], based on the information aforementioned. Figure 2a shows the numerical model developed and Fig. 2b displays the first vibration mode of the footbridge FE model, with a natural frequency of 4.01 Hz. In a recent research, a mean damping ratio of 2.5% was obtained for 8 FRP footbridges located in Europe [1]. Thus, a reasonable damping ratio of 2% was assumed here.

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Fig. 1. Footbridge: (a) frontal view, (b) cross-section of elements, (c) side view.

(a) Bottom view.

(b) Vibration mode.

Fig. 2. FE model of the footbridge.

A deflection design limit of L/350 (L being the length of the span) was assumed. According to [8], an uniformly distribute live load of 2.5 kN/m2 was considered to check deflection. As a result, the total considered load for the analysis was 3.6 kN/m2 , and the deflection obtained at midspan was 27 mm (L/370).

3

Inertial Vibration Controllers and Human-Structure Interaction

Although ultra-lightweight bridges are likely to display at the same time dynamic responses composed of many different resonant and non resonant harmonics, this paper only studies the most responsive first vertical vibration mode, to illustrate the concepts. The excitation considered is one person bouncing at midspan for 35 s for two cases: 1) bouncing at 2 Hz which causes a resonant response due to the second harmonic of the human loading and 2) bouncing out of resonance at 2.5 Hz. Firstly, the dynamic problem is studied assuming that the person does not interact with the structure. Given the high accelerations, it would be reasonable to expect a remarkable human-footbridge interaction. For this reason, the dynamic

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analysis is repeated accounting for HSI. The footbridge has been modelled as an equivalent mass-spring-damper system (which represents the vertical dynamic behaviour at midspan) with a mass of 772 kg, a natural frequency of 4.01 Hz, and a damping ratio of 2%. For the first preliminary dynamic analysis, a non interactive model is used. Here the human is represented just as an external force following the dynamic loading model for bouncing proposed in ISO 10137 [9]. Thus, the human force has been computed as a Fourier series composed of three harmonics with different amplitudes, as Eq. (1) and Fig. 3 displays:   3  αn sin(2πf t + φn ) , (1) Fh (t) = Q 1 + n=1

where Q is the weight of an average human, usually taken as 700 N and α1 , α2 and α3 represent the amplitude coefficients of each harmonic with values of 0.5, 0.25 and 0.15, respectively. 600

Input force (N)

400 200 0 -200 -400 -600 10

10.2

10.4

10.6

10.8

11 Time (s)

11.2

11.4

11.6

11.8

12

(a) Time domain. Zoom view.

Amplitude (abs)

150

100

50

0 0

1

2

3

4 Frequency (Hz)

5

6

7

8

(b) Frequency domain.

Fig. 3. Vertical human force excitation, Fh (N).

The second dynamic analysis accounts for HSI. This model represents the human body dynamics by means of a Mass-Spring-Damper-Actuator system,

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(a) Uncontrolled structure.

(b) Uncontrolled structure with HSI.

(c) Controlled structure with an inertial controller.

(d) Controlled structure with HSI and with an inertial controller.

Fig. 4. Simplified control schemes used to evaluate the control performance.

attached to the structure at mid span [10]. Hence, the human’s dynamics are defined by body’s natural frequency, damping ratio and mass, whilst the force generated by the human legs is applied as a pair of forces acting on both the pedestrian and the footbridge. The human parameters considered are: mass of 66 kg, natural frequency of 2.4 Hz, damping ratio of 20% and an actuator force equal to the one used for the non interactive model. The inertial controllers (passive, semi-active or active) are designed for the original structure without HSI and considering only the fundamental vibration

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modes. Then, the HSI is included and the controller performance is assessed. Figure 4 illustrates the control scheme considered. The transfer function of the structural model without HSI between the acceleration response and the input force is Gs (s) =

1/m1 s2 s2 X(s) = 2 , F (s) s + 2 ω1 ζ1 s + ω12

(2)

where s = jω, ω1 = 2πf1 being the circular natural frequency of the structure (rad/s), f1 its respective natural frequency and m1 , k1 and ζ1 being the mass (kg), stiffness (N/m) and damping ratio of the equivalent structural system described above. Secondly, both the HSI and the inertial controller are modelled as two separated closed-loops as Figs. 4b and 4c depict. Each one of them feedbacks the structural system response with an interactive force FHSI or Fc . Finally, the overall model shown in Fig. 4d can be seen as two simultaneous closed-loops accounting for HSI and the inertial controller interaction at the same time.

4

Simulations

An overall amount of sixteen different dynamic simulations have been performed. Four different control configurations were studied: uncontrolled structure and controlled via TVA, STVA and AVA. Indeed, each control case was analyzed with and without considering HSI at the two different bouncing frequencies aforementioned. Inertial controller’s parameters have been obtained without taking into account the influence of the HSI. This section will cover the design of the proposed inertial controllers. 4.1

TVA

The TVA has been designed using the approximate solution provided by Asami and Nishihara [11], based on H∞ optimization for primary structural systems with vanishing damping. These expressions minimize the structure acceleration considering a single vibration mode and harmonic excitation:  1 (3) η= 1+μ   3μ 27 ζT = (4) · 1 + μ, 8(1 + μ)3 32 μ = mT /mS being the mass ratio between the absorber and the primary system. A mass ratio of 2% of the structural equivalent mass has been selected, this is 15 kg. Parameter η = ωT /ωS is the frequency ratio between both systems.

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4.2

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STVA

A phase control strategy for the STVA damping has been considered [12,13]. The phase control presented has been adopted since this is clearly geared to practical implementation due to the measured real-time parameters employed: the structure acceleration instead of displacement and the inertial mass velocity instead of the relative velocity. The lowest structural response is achieved when the velocity of the inertial mass and structure acceleration have opposing phases, so the semi-active device objective is to bring the inertial mass motion as close as possible to this phase. This is equivalent to a phase lag of 90◦ between the structure acceleration and the control force. The control law adopted is of the ON/OFF type due to its simplicity. Thus, the adopted control law when ideal viscous damping is assumed is defined as:  cT = cmin (normal functioning) x ¨S · x˙ T ≤ 0 ⇒ (5) cT = cmax (blocking functioning), x ¨S · x˙ T > 0 ⇒ in which cmax is the maximum damping achieved by the damper, cmin is the optimal damping obtained for the TVA, x ¨S is the structure acceleration (measured by an accelerometer) and x˙ T is the velocity of the inertial mass (which might be obtained from the integration of an accelerometer signal installed on the inertial mass). Finally, both signals are low-pass filtered in order to avoid control instabilities associated to the ON/OFF control rule. It is important to highlight that the stiffness and mass of the STVA are the same as those calculated for the TVA. 4.3

AVA

Direct Velocity Feedback (DVF) control has been implemented in the AVA. Although DVF is, by its nature, unconditionally stable, when actuator dynamics are included, the closed-loop system becomes conditionally stable and the stability margin has to be studied previously to the implementation. The velocity estimated is multiplied by a control gain KC producing a command signal to the actuator. Firstly, the limit control gain for stability is derived as KClim  100. Finally a control gain KC equal to KClim /2 has been chosen providing enough stability gain margin and leading to a safe implementation [14]. The actuator transfer function between the voltage and the force developed can be described as a linear second-order system: Ga (s) =

s2

150 s2 , + 4.021 s + 101.12

(6)

where the natural frequency of the shaker is ωA = 10.05 rad/s (1.60 Hz). This model corresponds to an APS 113 Electrodynamic shaker. The AVA has an inertial mass of 13 kg, lightly smaller than the one used for the TVA and STVA.

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20

5

15

Acceleration (m/s 2 )

10

0

10

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VDV(m/s 1. 75 )

Peak = 12.024 m/s2 , MTVV = 8.179 m/s2 , VDV = 20.674 m/s 1.75

5

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15

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25 Time (s)

30

35

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(a) Uncontrolled. 2

Peak = 2.328 m/s , MTVV = 1.367 m/s2 , VDV = 3.421 m/s 1.75

3.5 3

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2.5

1

2 0 1.5 -1

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Peak = 1.849 m/s , MTVV = 1.007 m/s2 , VDV = 1.819 m/s 1.75

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Fig. 5. Resonance response structure without HSI.

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A Comparative Performance Study of Inertial Vibration Controllers

5

485

Results

Figure 5 provides the resonant response without HSI at the four cases of study. Three indicators are provided, the acceleration Peak value, the Peak value of the running root mean square acceleration computed using 1 s windows (named as maximum transient vibration value or MTVV) and the cumulative Vibration Dose Value (VDV). Figure 6 provides a summary for all the cases. The VDV is quite convenient for comparing the vibration controllers performance since it provides an accumulative value of the acceleration over the time. 20.7

Resonance

3.42 1.82 1.27 8.52

Resonance + HSI

No resonance

No resonance + HSI 0

3.49 2.23 1.57 1.55 1.61 1.64 1.32

Active Semi-active Passive Uncontrol

2.24 2.35 2.49 1.86

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VDV (m/s 1.75 ) Fig. 6. VDV of the structural response for all cases.

As Fig. 6 clearly depicts, considering HSI implies to intrinsically change the dynamic characteristic of the system to be controlled, changing its natural frequencies. In the example analyzed here the inertial controllers are a bit detuned reducing slightly their efficiency. To explain the dynamic behaviour of the inertial controllers out of resonance is quite more complex, however it seems to deteriorate also when considering HSI.

6

Conclusions

This research is the first step towards a new way of designing inertial controllers applied to future ultra-lightweight structures excited by human-induced vibrations. The results obtained will lead to further comprehension on how inertial controllers should be designed. Future works should asses whether or not the HSI influence their performance, and how this could be accounted for. In this sense, ultra-lightweight structures are challenging for vibration control due to: multi-modal and non-resonant dynamic responses and the HSI phenomenon.

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Acknowledgements. The authors acknowledge the financial support provided by the Ministry of Science, Innovation and Universities (Government of Spain) by funding the Research Project SEED-SD (RTI2018-099639-B-I00). Carlos M. C. Renedo would like to thank Universidad Polit´ecnica de Madrid for the financial support through a PhD research grant.

References ˇ 1. Wei, X., Russell, J., Zivanovi´ c, S., Mottram, J.T.: Measured dynamic properties for FRP footbridges and their critical comparison against structures made of conventional construction materials. Composite Struct. 223, 110956 (2019) 2. Jim´enez-Alonso, J.F., S´ aez, A.C., Caetano, E., Magalh˜ aes, F.: Vertical crowd– structure interaction model to analyze the change of the modal properties of a footbridge. J. Bridge Eng. 21(8) (2016) 3. Wang, X., Pereira, E., Garc´ıa-Palacios, J.H., D´ıaz, I.M.: A general vibration control methodology for human induced vibration. Struct. Control Health Monit. 26, e2406 (2019) 4. Fiberline A/S. Fiberline Design Manual, Middelfart, Denmark (2003) 5. Ascione, L., Caron, J.F., Godonou, P., Van IJselmuijden, K., Knippers, J., Mottram, T., et al.: Prospect for new guidance in the design of FRP structures. Joint Research Centre, JRC (2016) 6. AASHTO: Guide specifications for design of FRP pedestrian bridges. American Association of State Highway and Transportation Officials, Washington, DC. (2008) 7. SIMULIA: Abaqus v.6.14 Analysis User’s Guide, Dassault Syst`emes Simulia Corporation (2014) 8. EN 1991-2: Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges (2003) 9. International Standards Organization: ISO 10137 -Bases for design of structures Serviceability of buildings and walkways against vibrations (2007) 10. Dougill, J.W., Wright, J.R., Parkhouse, J.G., Harrison, R.E.: Human structure interaction during rhythmic bobbing. Struct. Eng. 84(22), 32–39 (2006) 11. Asami, T., Nishihara, O.: Closed-form exact solution to H1 optimization of dynamic vibration absorbers: application to different transfer functions and damping systems. J. Vib. Acoust. 125(3), 398–405 (2003) 12. Soria, J.M., D´ıaz, I.M., Garc´ıa-Palacios, J.H.: Further steps towards the tuning of inertial controllers for broadband-frequency-varying structures. Struct. Control Health Monit. 27(1), e2461 (2020) 13. Moutinho, C.: Testing a simple control law to reduce broadband frequency harmonic vibrations using semi-active tuned mass dampers. Smart Mater. Struct. 24(5), 964–1726 (2015) 14. D´ıaz, I.M., Reynolds, P.: Robust saturated control of human-induced floor vibrations via a proof-mass actuator. Smart Mater. Struct. 18(12) (2009)

MPC Framework for Supply Chain Management Integrating On-Time Delivery and Transport Management Eduardo Ara´ ujo2 , Jo˜ ao Lemos Nabais1,2(B) , and Miguel Ayala Botto2 1

2

School of Business Administration, Instituto Polit´ecnico de Set´ ubal, Campus do IPS, Estefanilha, 2910-761 Set´ ubal, Portugal [email protected] IDMEC, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal {eduardo.araujo,ayalabotto}@tecnico.ulisboa.pt

Abstract. Currently, there is a challenge in Supply Chain Management (SCM) which consists on the coordination of efforts to correspond to clients demands in the presence of conflicting goals of the various chain actors. On the one hand, suppliers, manufacturers, and retailers are interested in providing to customers the right product, in the right amount, at the right time, for the right price, and at the right place. On the other hand, transport providers (TP) want to efficiently allocate their fleets and minimise the number of movements. To solve the trade-off between (i) on-time delivery, and (ii) efficient transportation management, this work presents a novel dynamic approach for real-time SCM integrating transportation operations, based on a model predictive control framework. Focusing on the discrete time case, a flow perspective is taken in which the material flow is decomposed into flow of goods and flow of transportation vehicles to model the conflicting views of the various actors. The framework performance is illustrated with simulation studies considering two-echelon vertical integrated chain with manufacturing. Keywords: MPC · Multi-product Inventory control · Transportation

1

· Multi-transport · Production ·

Introduction

Industry and trade are the driving forces behind economical growth and the improvement of living standards. Over the last decades, the increasing consumption and globalization have created the need for more transportation, strengthening the internal competition in Supply Chains (SC) [1]. To meet the growing demands in terms of product customization, price and service levels from the customer side, companies are urged to lower their costs, while still maintaining high quality standards. Furthermore, global warming and other environmental c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 487–496, 2021. https://doi.org/10.1007/978-3-030-58653-9_47

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concerns, have been pressuring even more transport providers (TP) towards reducing movement costs [1] . This strong interaction between shippers (i.e. suppliers, manufacturers, and retailers) and TP have created the need for higher levels of efficiency, quality of service, timeliness, and responsiveness across supply chains [1]. Various scientific communities have devoted attention to the optimisation and management of operations in SC. Operations research methods are the most widely used when modelling these systems. Over the past years, modern control theory has been attracting the attention from the scientific community as a powerful alternative method to analyse and design SC from a dynamical system point of view [2–4]. However, even though transport management is an integral part of SCM, these domains have either been treated independently from each other [5–7], or integrated but assuming that goods and/or transportation resources are readily available whenever and wherever needed [8,9]. Furthermore, research on SCM has focused on cost, which hampers network planning integration, for it sharpens differences between the actors, pushing them to take individualistic and local views of the value chain [5]. To fulfil the needs of all SC actors and achieve a higher quality of service based on the consistency of timely and efficient delivery, companies must manage their transportation strategies from a holistic SC view [5,10]. Consequently, such an approach entails the need for new SC models that are able to deal with both perspectives at the same time. That is, models which enable suppliers, manufacturers and retailers to (1) deal with multi-products, (2) monitor and manage stocks, (3) schedule production activities, (4) monitor work-in-progress (WIP), and (5) coordinate activities with predefined time-windows, while, at the same time, enabling TP to (1) monitor different transportation types, (2) deal with costs associated with the different resources’ capacities, and (3) monitor the location and state of the vehicles composing their transport fleet. This paper sets forth a new approach for real-time SCM based on model predictive control (MPC). The proposed methodology is based on a flow perspective and focuses on the discrete time case. It integrates ideas from operations research and control theory, resulting in interpretable, tractable and flexible dynamic models. The outlined modelling framework produces linear, time-invariant, state-space supply chain representations that are both controllable and reachable. The presented approach was initially based on the work of Nabais et al. [11,12], and evolved into an extension of the works of Perea-Lopez et al. [13] and Braun et al. [14]. While their work focused on coordinating production and inventory activities across the SC, this work generalizes and integrates both the perspectives of the shippers, and the transport providers, enabling the proposed managing tool to be employed by any SC member, regardless of its role. The remainder of the paper is structured as follows. Section 2 presents the proposed modelling approach, which will then be used to devise the control structure, which is presented in Sect. 3. The model is implemented and applied to a real-world SC in Sect. 4 and the analysis is concluded in Sect. 5.

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2

489

Modelling Supply Chains

In SC, actors (e.g. suppliers, manufacturers, retailers, transport providers) cooperate to move commodities from the point of origin to the point of consumption in order to meet customers’ requirements. However, different SC members may have different goals. Whilst shippers (e.g. suppliers, manufacturers, retailers) strive to deliver the right product, in the right amount, at the right time, for the right price, and at the right place, TP want to efficiently allocate their fleets, minimise the number of movements and move at full capacity. Thus, from a modelling perspective a SC is a network of independent entities or nodes, connected by arcs (or links) representing the flows of material and information, as shown in Fig. 1.

Fig. 1. Supply chain dynamics—nodes, arcs, and flows of material and information.

The movement of materials and resources across the SC creates flows that have impact on the contents of each node. Therefore, the dynamics of a SC can be modelled resorting to mass balances at each node. The SC dynamics can be given as a superposition of two fundamental layers, in which the movement of a commodity is interpreted as a synchronised, superimposed flow of material in both the commodity and the transportation layer (Fig. 2). From an operations management point of view, there are a variety of operations that can be find in any SC. Namely, one or more SC members may focus on the procurement of raw material, others in processing and transforming those materials into finished goods, others, still, may focus only on storing and/or distributing commodities. Taking a holistic, SC locations can be categorized into: Sink nodes: are defined as the most down-stream members of the network, usually representing the retailing level. Thus, they are composed of an unloading zone, followed by a reception zone. The unloading zone is the area where commodities and transportation agents are decoupled. Source nodes: source nodes receive (and/or transform) commodities that will be dispatched and flow through the SC until they arrive at the sink nodes; Figure 3 presents the model of a SC composed of two manufacturers and three retailers, employing the presented modelling approach. Denoting by n the number of nodes, nm the number of resources, nu the number of links between adjacent nodes, and nz the number of output nodes,

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(a) Shippers and TP interactions.

(b) Material flow results in two independent flows.

(c) SC modelling taking a layered approach.

(d) Synchronized flow of material from nodes 3 and 7 to nodes 1 and 5, respectively.

Fig. 2. SC as a system composed of multiple layers.

(a) Commodity Layer.

(b) Transportation Layer.

(c) SC layers.

Fig. 3. Modelling supply chains. (Dashed squares are transport nodes.)

one can represent the model by making use of a state-space representation as follows, x(k + 1) = Ax(k) + Bu(k) + Bd d(k), y(k) = Cy x(k), z(k) = Cz x(k),

(1)

where x(k) is the state space vector, u(k) is the control action, d(k) is the exogenous input vector, y(k) is the vector of measured outputs and z(k) is the

MPC Framework for SCM Integrating On-Time Delivery

vector of outputs which are to be controlled. Matrices A, B, Bd , Cy , are matrices of appropriate size given as follows, ⎡ ⎡ ⎡ ⎤ ⎤ B1 0 · · · 0 Bd1 0 · · · A1 0 · · · 0 ⎢ 0 B2 · · · 0 ⎥ ⎢ 0 Bd2 · · · ⎢ 0 A2 · · · 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ A=⎢ . . . . ⎥ , B = ⎢ .. .. . . . ⎥ , Bd = ⎢ .. .. . . ⎣ . . ⎣ . ⎣ .. .. . . .. ⎦ . .. ⎦ . . 0

0 · · · Anm

0 0 · · · Bnm

0

491

and Cz 0 0 .. .

0 · · · Bdnm

⎤ ⎥ ⎥ ⎥, ⎦ (2)

Cy = I, and Cz is defined by the user.

3

Controlling Supply Chains

The SCM problem is here resorted as a model predictive control (MPC) problem. Thus, a dynamic model, a performance index (or cost function) and a set of constraints must be defined. Regarding the cost function, the problem at hand is formulated as a reference tracking problem, in which the controller must minimise the tracking error between a given reference, r, and the system output, ˜ z, over a given prediction horizon, Hp . The cost function to be minimised is defined as follows, Hp  2 2 J= [r(k + i) − ˜ z(k + i)]Q + ρ [u(k + i)] , (3) i=1

where u is the collection of decision variables that minimise J, and Q and ρ are weighing parameters of appropriate dimensions. The constraints are defined as follows: Synchronised: for the controller to be able to produce a synchronised, superimposed flow of material, the following constraint is imposed:  uijm (k) ≤ λuijm (k), (4) m

where ujim is the incoming stream of a given resource m to be processed at node i coming from node j and λ > 0 represents the maximum load capacity. Resource pulling: To guarantee that a given resource is at a given node at the time of pulling, one could define the following constraint:  uijm (k) ≤ xim (k − τi ), (5) i

where xim is the inventory level of resource m at node i, τi is the time-delay produced by the processing-time of node i.

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The MPC optimisation problem can then be written as follows: min J = u ˜

Hp  i=1

2

2

[r(k + i) − ˜ z(k + i)]Q + ρ [u(k + i)] ,

s.t. x(k) ≥ 0, z ≤ z(k) ≤ z, u ≤ u(k) ≤ u,  uijm (k) ≤ xim (k − τi ),

(6) (7) (8) (9) (10)

i



uijm (k) ≤ λ uijm (k).

(11)

m

Constraints (7)–(11) impose the SC structural features. Equation (7) assures all states are positive at every time-step. Equation (8) imposes the minimum and maximum nodes’ capacities, denoted by z and z, respectively. Equation (9) limits the material flow between nodes, imposing minimum and maximum admissible values as well, denoted by u and u, respectively. Equation (10) guarantees the pulled resource is available at the node at the time of pulling, where τ represents the time node i requires to process resource m. Equation (11) assures the maximum transport loading capacity, λ, is respected.

4 4.1

Simulation Experiment Scenario Description and Initial Set up

The chosen network is a two-echelon vertical integrated chain dedicated to the manufacture of one type of product (P1), as shown in Fig. 4(a). Commodities must be delivered to three different retailers (R1, R2, and R3) while respecting the following constraints. Commodity Layer: manufacturing activities take place in two different manufacturing sites (M1 and M2) with different production capacities. While M1 may produce batches of up to 253 units of P1, M2 is limited to batches of up-to 120 units. The batch production takes 8 hours for each site. In turn, retailers present different levels of demand and must not wait more than 12 hour between placing a purchase order (PO) and receiving goods; Transportation Layer: a fleet of 10 transportation vehicles is ready to pickup and dispatch commodities as soon as they become available. Vehicles are homogeneous in terms of capacity, up-to 45 units of final products (FP), and take 1 hour to travel between adjacent nodes. The problem to be solved consists in controlling i) the transportation of goods from source to demand nodes, and ii) the position and status of the available transport agents over time, while assuring goods are delivered on time.

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493

A 20 h period is considered, in which each time-instant represents 1 h. The main objective consists in delivering the right amount of finished goods at the right place (253 units to R1, 45 units to R2, 75 units to R3), and at the right time (12 h after the order placing) (see Table 1). The simulation assumes the SC to be empty of commodities at starting time. It is further assumed that the maximum capacity of each node is much larger than the amounts of commodities being transported. More, at each time-step it can be transported as many commodities as needed, the only limitation being the availability of transportation. Table 1. Demand, production and disposition of vehicles at starting time. Demand [units] Max batch capacity [units] Vehicles [units] M1 —

253

6

M2 —

120

4

R1 253

—

0

R2 45

—

0

R3 75

—

0

(a) Manufacturing 1 (M1).

site

(b) Retailer 1 (R1).

Fig. 4. Final model of the Supply Chain of the case study.

To solve this problem, the MPC controller was set to have a prediction horizon, Hp , equal to 8 time-instants (hours). Moreover, matrix Q was defined as follows Q = diag(1000, 100, · · · , 100, 1, · · · , 1), and ρ = 1. For optimisation purposes, it was used the OPTI Toolbox’s SCIP solver. 4.2

Results

Figure 5 presents in and out flows of material (i.e. commodities and transportation) between M1 and R1.

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Commodity Layer: the batch of FP 8 h to produce. Goods enter the loading zone 10 h and they are dispatched at 11 h, as shown in Fig. 5(a). In turn, R1 receives the goods at 12 h, as desired (see Fig. 5(b)). Transportation Layer: six vehicles wait for transportation assignment at M1 (Fig. 5(a)). Since each vehicle may move 45 units of P1, the whole fleet is assigned to transport the whole batch of 253 units to R1. Vehicles 1 h to arrive at the retailer and unload the cargo at 12 h, as shown in Fig. 5(b). Transportation vehicles return to M1, where they rest ready for further transportation assignments.

(a) Manufacturing site 1 (M1).

(b) Retailer 1 (R1).

Fig. 5. R1 places a purchasing order (PO) to M1. To cope with the delivery date constraints, 6 vehicles are assigned to move goods from M1 to R1. FP are unloaded and stocked at the desired time.

Analogously, Fig. 6 presents the exchange of material flows between M2 and R2 and R3. Commodity Layer: M2 receives two purchasing orders (PO) of different dimensions. Whilst R2 orders a batch of 45 units, R3 requires a 75 unit order of P1. Thus, M2 is required to produce 120 units totally. As shown in Fig. 6(a), the desired amount of FP is produced in 8 h (between the PO reception and the stocking process, at 9 h). Goods are loaded 10 h and 11 h. After, goods are dispatched to two different locations, R2 and R2. The finished products arrive at the desired locations at 12 h, where they are unloaded and stocked, as shown in Fig. 6(b) and (c); Transportation Layer: Four vehicles wait for transportation assignment at M1. Three of the four vehicles are assigned to transport both orders to the respective customers, as shown in Fig. 6(a). Vehicles 1 h to arrive at the retailer and unload the cargo at 12 h, as shown in Fig. 6(b) and (c). Transportation vehicles return to M1, where they rest ready for further transportation assignments.

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(a) Manufacturing site 2 (M2).

(b) Retailer 2 (R2).

(c) Retailer 3 (R3).

Fig. 6. R2 and R3 place PO of different sizes to M1. To cope with time constraints, 2 vehicles move goods from M2 to R2 and 1 vehicle is assigned to transport commodities to R3. There is one vehicle that stays awaiting assignment in M2.

5

Conclusion and Future Work

To answer to the increasing demand for higher levels of efficiency, quality of service, timeliness, and responsiveness across SC, a new dynamic approach for realtime supply chain management integrating transportation operations, based on an MPC framework was proposed. The outlined modelling framework is based on a flow perspective and builds on the fact that there are two flows in SC. Namely, information and material flows, where the material flow may be further divided into flow of goods and flow of transports. Thus, the SC dynamics can be considered as a superposition of two fundamental layers: i) the layer of commodities, and ii) the layer of transportation, where each layer consists of (possibly different) networks across which material is allowed to flow. In the proposed modelling framework, the movement of goods is represented by a synchronised, superimposed flow of goods and transportation vehicles. The devised notation was then used to develop a centralised, constrained MPC scheme, where the variables’ mapping from the MPC framework to the SC domain was accomplished by representing inventories as states, and flows of material as control actions. In the illustration example it was possible to coordinate the purchasing order, production and transport to deliver on-time. Acknowledgements. This work is financed by national funds through FCT - Foundation for Science and Technology, I.P., through IDMEC, under LAETA, project UIDB/50022/2020.

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References 1. Hoff, A., Andersson, H., Christiansen, M., Hasle, G., Løkketangen, A.: Industrial aspects and literature survey: fleet composition and routing. Comput. Oper. Res. 37(12), 2041 (2010) 2. Ivanov, D., Sokolov, B., Kaeschel, J.: Integrated supply chain planning based on a combined application of operations research and optimal control. Cent. Eur. J. Oper. Res. 19(3), 299 (2011) 3. Sarimveis, H., Patrinos, P., Tarantilis, C.D., Kiranoudis, C.T.: Dynamic modeling and control of supply chain systems: a review. Comput. Oper. Res. 35(11), 3530 (2008) 4. Subramanian, K., Rawlings, J.B., Maravelias, C.T., Flores-Cerrillo, J., Megan, L.: Integration of control theory and scheduling methods for supply chain management. Comput. Chem. Eng. 51, 4 (2013) 5. van Riessen, B., Negenborn, R.R., Dekker, R.: Synchromodal container transportation: an overview of current topics and research opportunities. In: Corman, F., Voss, S., Negenborn, R. (eds.) Computational Logistics, ICCL 2015. Lecture Notes in Computer Science, vol. 9335. Springer, Cham (2015) 6. Mosca, A., Vidyarthi, N., Satir, A.: Integrated transportation - inventory models: a review. Oper. Res. Perspect. 6, 100101 (2019) 7. Min, H., Zhou, G.: Supply chain modeling: past, present and future. Comput. Ind. Eng. 43(1–2), 231 (2002) 8. Khurana, A., Adlakha, V., Lev, B.: Multi-index constrained transportation problem with bounds on availabilities, requirements and commodities. Oper. Res. Perspect. 5, 319 (2018) 9. Pinho, T.M., Coelho, J.P., Veiga, G., Moreira, A.P., Boaventura-Cunha, J.: A multilayer model predictive control methodology applied to a biomass supply chain operational level. Complexity (2017) 10. Dong, C., Boute, R., McKinnon, A., Verelst, M.: Investigating synchromodality from a supply chain perspective. Transp. Res. Part D Transp. Environ. 61, 42 (2018) 11. Nabais, J.L., Negenborn, R.R., Botto, M.A.: A novel predictive control based framework for optimizing intermodal container terminal operations. In: Voß, S., Hu, H., Shi, X., Stahlbock, R. (eds.) Computational Logistics, ICCL 2012. Lecture Notes Computer Science, vol. 7555, pp. 53–71. Springer, Heidelberg (2012) 12. Nabais, J., Negenborn, R., Carmona Ben´ıtez, R., Ayala Botto, M.: Achieving transport modal split targets at intermodal freight hubs using a model predictive approach. Transp. Res. Part C Emerg. Technol. 60, 278 (2015) 13. Perea, E., Grossmann, I., Ydstie, E., Tahmassebi, T.: Dynamic modeling and classical control theory for supply chain management. Comput. Chem. Eng. 24(2–7), 1143 (2000) 14. Braun, M.W., Rivera, D.E., Flores, M., Carlyle, W.M., Kempf, K.G.: A model predictive control framework for robust management of multi-product, multi-echelon demand networks. Ann. Rev. Control 27(2), 229 (2003)

Control of the Depth of Anesthesia Using a New Model for the Action of Propofol and Remifentanil on the BIS Level Jorge Silva1(B) , Teresa Mendon¸ca2 , and Paula Rocha1 1

Research Center for Systems and Technologies (SYSTEC), Faculdade de Engenharia da Universidade do Porto (FEUP), Rua Dr. Roberto Frias, 4200-465 Porto, Portugal {jmpps,mprocha}@fe.up.pt 2 Research Center for Systems and Technologies (SYSTEC), Faculadade de Ciˆencias da Universidade do Porto (FCUP), Rua do Campo Alegre, 4169-007 Porto, Portugal [email protected]

Abstract. This paper presents a simple individualized open-loop control scheme for the automatic delivery of anesthetics in order to achieve and maintain a desired depth of anesthesia. This control scheme is based on a new model for the action of the hypnotic propofol and the analgesic remifentanil, that is more suitable for parameter identification according to the usual clinical practice. The model parameters are estimated on-line (with a short estimation time), allowing a quick tuning of the controller.

Keywords: Modelling and identification anesthesia · TCI

1

· Bispectral index · Depth of

Introduction

The introduction of automated process in anesthesia can be a helpful tool for the anesthesiologists, and has therefore attracted the attention of several researcher groups, [2,3,5,7]. In this context, the design and implementation of patient individualized schemes to determine the dosing of different anesthetics assumes particular importance as an alternative to populational model based protocols. In this contribution we focus on the study of the depth of anesthesia (DoA), which is related to the degrees of hypnosis and analgesia induced in a patient. J. Silva—This work was supported by UID/EEA00147/2019 – Research Center for Systems and Technologies funded by national funds through the FCT/MCTES through national fund (PIDDAC). The author Jorge Silva acknowledges the support from FCT, under the PDMA-NORTE2020-CCDRN-NORTE-08-5369-FSE-000061. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 497–506, 2021. https://doi.org/10.1007/978-3-030-58653-9_48

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This component of anesthesia is usually measured through the bispectral index (BIS) level, a parameter that can be obtained by an EEG. Here, we shall consider the administration of hypnotic propofol and analgesic remifentanil in order to achieve a desired BIS level. As is well-known, the combined effect of two drugs can be mathematically described by means of two dynamical models that relate the dose of each of the drugs with the corresponding effect concentration, together with a static nonlinearity that takes the interaction of the two drugs into account in order to yield the final effect. The “traditional” description of the relation between the drug dose and the corresponding effect concentration is done via pharmacokinetic/pharmacodynamic (PK/PD) models [4]. Although these models have a clear physiological meaning, they present many parameters, which is inconvenient for parameter identification procedures. To overcome this drawback, here we consider the parameter parsimonious models introduced in [9] to describe the linear dynamic relation between the dose of propofol/remifentanil and the corresponding effect concentration. Each of these models only involves one patient dependent parameter. In order to model the combined effect of propofol and remifentanil, we follow [4] to obtain a static nonlinear relation that yields the BIS level as a function of the effect concentration of each drug. This static nonlinearity is a generalization of Hill’s equation and involves, in turn, two patient dependent parameters. It should be noticed that our Hill equation differs from the ones used in [6] and [9] in order to encompass the usual drug administration protocol in clinical practice. This general scheme is presented in Fig. 1, where up , ur , Cep and Cer respectively denote the doses of propofol and remifentanil, and Gp (s) and Gr (s) are the corresponding transfer functions. These. together with the Hill equation, will be specified in the sequel.

Fig. 1. General scheme for the effect of propofol and remifentanil on the BIS level.

An online procedure for the identification of the four parameters of our model was proposed in [8]. This procedure takes advantage of drug administration patterns that are compatible with the clinical practice, and allows a fast identification of the parameters.

Control of the Depth of Anesthesia Using a New Model

499

Here, we use this identification procedure as a preliminary step to obtain estimates of a patients’s individual model parameters. Such parameters are used to tune a simple controller (to be started right after the estimation phase) in order to achieve and maintain a desired BIS level. The organization of this paper is as follows. Section 2 is devoted to the presentation of our model as well as of the parameter estimation procedure. The automatic control scheme to be used is introduced in Sect. 3, where also simulations are shown to illustrate the obtained results. Section 4 contains simulations to illustrate the obtained identification and control results. Section 5 contains some concluding results.

2 2.1

The Action of Propofol and Remifentanil on the BIS Level: Modelling and Identification The Model

The models introduced in [9] for the effect concentrations of propofol and remifentanil are given by the following transfer functions

and

Cep (s) 90α3 = G (s) = p U p (s) (s + α)(s + 9α)(s + 10α)

(1)

Cer (s) 6η 3 = Gr (s) = r U (s) (s + η)(s + 2)(s + 3η)

(2)

where Cep (s) and Cer (s) are the Laplace transforms of the propofol and the remifentanil effect concentrations, U p (s) and U r (s) are the Laplace transforms of the propofol and the remifentanil drug doses, and α and η are patient dependent parameters. The coefficients that affect the parameters (1, 9 and 10 for propofol, and 1, 2 and 3 for remifentanil) were determined by optimized fitting using the information of a large database of real cases collected during surgeries. According to [4], the combined effect E of two drugs A and B is given by the generalized Hill equation: E(t) =

E0 1 + (UA (t) + mUB (t))γ

(3)

where E0 corresponds to the case where no drugs are administered; UA and UB stand for the potency of A and B, respectively, and are given by: UA (t) =

CeA (t) CeB (t) ; U (t) = B A B EC50 EC50

(4)

A B and EC50 are the effect concentrations of A and B, respectively, where EC50 associated with 50% drug effect. Finally, m and γ are patient dependent parameters.

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Here, we consider propofol to be drug A, while drug B is remifentanil. Moreover, p A B r = EC50 = 10 and EC50 = EC50 = 0.01. according to [9], we take EC50 This yields to following Hill equation: z(t) =

97.7 1 + (mUp (t) + Ur (t))γ

(5)

The complete model (1), (2) and (5) possesses four patient dependent parameters, namely α, η, γ and m. This is the same nomenclature as the one used in [9]. However our parameters have a different meaning due to the different structure of the two models, since in [9] drugs A and B are respectively taken to be remifentanil and propofol. 2.2

On-Line Identification of the Model Parameters

For the sake of completeness we summarize the on-line identification method proposed in [8]. This method comprises two main steps. First the parameters α and γ are estimated based on the patient’s response to a bolus of 600 µg/kg of propofol. After these parameters are estimated, a constant dose of 3 µg/kg/min of propofol together with a constant dose of 0.002 µg/kg/min of remifentanil is administered. This allows to estimate the parameters m and η as explained in the sequel. It is not difficult to show that, according to (1), the administration of a bolus of 600 µg/kg of propofol produces the following time domain response for Cep : Cep (t) = 750αe−αt − 6750αe−9αt + 6000αe−10αt , t ≥ 0

(6)

By (5), this produces a BIS level response z p (t) = 1+

97.7  p γ Ce (t) 10

(7)

taking into account that no remifentanil has been administrated. Thus, when z p (t) is equal to half its maximum value, i.e., at the time instant p when t = T50 97.7 = 48.85 (8) z p (t) = 2 p the value of Cep (T50 ) satisfies: p Cep (T50 ) = 10

(9)

p Now, since the value of T50 can be obtained by inspection of the BIS level p p ) response (recall that z50 = 48.85), it is enough to solve (9) for α (with C˜ep (T50 given by (6)), i.e.: p p p C˜ep (t) = 750αe−αT50 − 6750αe−9αT50 + 6000αe−10αT50 = 10

(10)

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in order to obtain an estimate α ˆ for this parameter. The order to estimation of γ can be performed by analyzing the BIS level z(T ∗ ) response at a time instant p . At this time instant, an estimate T ∗ ≥ T50 ˆ ˆ ∗ ˆ ∗ ˆ ∗ C˜ep∗ = C˜ep (T ∗ ) = 750ˆ αe−αT − 6750ˆ αe−9αT + 6000ˆ αe−10αT

(11)

of C˜epˆ(T ∗ ) is available and the estimate γˆ for γ can be computed by solving for γ equation: z(T ∗ ) = 1+

97.7  ˜ pˆ

γ

Ce (T ∗ ) 10

(12)

which yields  γˆ =

log log

97.7 z p (T ∗ )

 ˜ pˆ

 −1  ∗

(13)

Ce (T ) 10

In order to obtain estimates for the parameters η and m, a constant dose of 0.0003 µg/kg/min of remifentanil is administered from the time instant T ∗ on. The corresponding time domain response for the effect concentration of remifentanil is then, obtained from 2 followed by inverse Laplace transform. yielding:  Cer (t)

=

0,

3 −3e

−η(Δt)

−2η(Δt)

+3e 1000

−η(Δt)

−e

+1

t < T∗



(14)

, t ≥ T∗

where Δt = t − T ∗ . Together with the constant dose of remifentanil, a constant dose of 3 µg/kg/min of propofol is administered (according to clinical practice). The corresponding time domain response for the effect concentration of propofol is given by: C˜ep (t) =



0,

−15 −α(Δt) 4 e

+

15 −9α(Δt) 4 e

−10α(Δt)

− 3e

t < T∗ + 3, t ≥ T ∗

(15)

This must be added to the effect concentration of propofol induced by bolus, given by (6), in order to obtain the joint response:

Cep (t) =

⎧ 750αe−αt − 6750αe−9αt + 6000αe−10αt , ⎪ ⎪ ⎨ 750αe−αt − 6750αe−9αt + 6000αe−10αt − ⎪ ⎪ ⎩ 15 −9α(Δt) + 4e − 3e−10α(Δt) + 3,

The resulting BIS level response is then given by:

t < T∗ 15 −α(Δt) 4 e

(16) t ≥ T∗

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⎧ p t < T∗ ⎨ z (t), 97.7  γ , t ≥ T ∗ z(t) = ⎩ 1+ Cep (t) +m Cep (t) 10

(17)

0.01

Now, reading out the values z(t ) and z(t ) of this response for two time instants t and t ≥ T ∗ we get: z(t ) ≈ 1+

 ˆp

97.7

Ce (t ) 10

+

Cer (t ) m 0.01

γˆ

(18)

and z(t ) ≈ 1+

 ˆp

97.7

Ce (t ) 10

+m

Cer (t ) 0.01

γˆ

(19)

where Cˆep (t ) and Cˆep (t ) are the estimates for the effect concentration of propofol obtained from (16) by replacing the parameter α by its estimate α ˆ. Equations (18) and (19) allow to compute approximate values a and b for ˆ for m and ηˆ for mCer (t ) and mCer (t ), respectively, based on which estimates m η can be obtained. Indeed, if mCer (t )

≈ 0.01

 γ1ˆ  Cˆep (t ) 97.7 −1 := a − z(t ) 10

(20)

and mCer (t ) ≈ 0.01

 γ1ˆ  Cˆep (t ) 97.7 −1 := b − z(t ) 10

(21)

it follows from that a Cer (t ) ≈ r  Ce (t ) b

(22)

where the expressions for Cer (t ) and Cer (t ), that can be computed from (14), only involve the unknown parameter η. Thus, solving the Eq. (22) for η yields an estimate ηˆ for this parameter. Finally, using (for instance) (20), one obtains the estimate: m ˆ =

a r ˆ Ce (t )

(23)

where Cˆer (t ) is given by (14) with η = ηˆ and t = t , and the estimation procedure is complete at time t∗ = max{t , t }.

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Control Scheme

Having in mind the design of a simple individualized automated drug delivery system for the BIS level, we consider a target controlled infusion (TCI) scheme. In our case, this consists in administrating constant doses of propofol and remifentanil in order to achieve and maintain a desired BIS level z ∗ . The controller is only put in action after the identification procedure is completed and estimates of the patient parameters are available. Assume that a BIS level of z ∗ (determined by clinical practice) is to be tracked. This can be achieved by the administration of constant (steady-state) doses u∗p and u∗r of propofol and of remifentanil, respectively, as explained next. Here we consider that these doses are related by: u∗r = ρu∗p

(24)

where ρ is a constant to be chosen according to clinical criteria, and the steadystate dose up∗ is to be determined. Note that such doses respectively correspond to following steady-state values of the effect concentrations of Cep∗ and Cer∗ = ρCep∗ of propofol and remifentanil: Cer∗

=

Cep∗ = u∗p

(25)

=

(26)

u∗r

ρu∗p

since the steady state gains of the transfer functions (1) and (2) are both equal to 1. Now, from Eqs. (4) and (5), it follows that, for the estimated patient: z∗ =

97.7

1+

u∗ p 10

u∗ p

γˆ

(27)

+ mρ ˆ 0.01

yielding the propofol steady-state dose: u∗p



 97.7 − 50 1/ˆγ 1 = 0.1 + 100ρm ˆ 50 

(28)

The steady-state dose of remifentanil u∗r is given by (24) for the value of u∗p computed in (28) and a properly chosen value of ρ. Taking into account that previous steps of 3 µg/kg/min of propofol and 3 × 10−4 µg/kg/min of remifentanil have been administrated in the identification phase, the extra propofol dose to be given in the control phase must be equal to: ∗ (29) up∗ control (t) = up − 3 whereas the extra remifentanil to be administrated in this phase is equal to: ∗ ∗ ur∗ control (t) = ur − 0.0003 = ρup − 0.0003

(30)

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Simulation Results

In order to validate the proposed model and parameter estimation procedures, the following strategy is used. We consider the model proposed in [9], for which a table of parameters, identified from a database of real cases in the framework of Galeno project, is available. This serves as basis for creating simulated patients, considered to be the “real” patients, to which our modelling and identification procedure is applied. The parameters of the model proposed in [9], taken from [1], and the identified parameters for our model, for two representative cases, are displayed in Table 1. As expected, the parameters of the two models do not coincide, due to the different structure of these models. Table 1. Parameters of the model proposed in [9], taken from [1], and estimated parameters for our model. Real patient

α

γ

η

m

Case 1

0.0321 3.3903 0.0666 1.9085 Case 1

0.0392 3.9575 0.1934 6.1798

Case 2

0.0860 0.9780 0.0212 1.4202 Case 2

0.0988 1.0332 0.8548 4.4354

Estimated patient

α ˆ

γ ˆ

ηˆ

m ˆ

The time instants T ∗ , where the estimation of α and γ are achieved, and t∗ , where the estimates of η and m are obtained are respectively T ∗ = 4.1397 min. and t∗ = 5.6397 min. for Case 1, and T ∗ = 0.8019 min. and t∗ = 2.6019 min. for Case 2. The corresponding results of the proposed on-line estimation and subsequent application of the proposed control scheme are illustrated in Fig. 2. Here a value of z ∗ = 50 for the BIS reference level was considered, whereas a value of ρ = 10−4 was taken for the ratio between the steady-state doses of remifentanil and propofol. In order to assess the performance of our procedure, the BIS level response corresponding to the estimated drug doses is compared with the one resulting from the administration of the drug doses computed with basis on the real patient model.

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BIS level (Case 1)

100 80 60 40 20 0 0

50

100

150

200

250

200

250

Time (minutes)

BIS level (Case 2)

100 80 60 40 20 0 0

50

100

150

Time (minutes)

Fig. 2. BIS responses for Case 1 and Case 2 corresponding to the administration of the drug dose resulting from our identification and control procedure (in red) and to the drug dose resulting from a similar administration profile, but based on the parameters of the real patient (in black).

5

Conclusions

The good simulation results suggest that our model and on-line identification methods are suitable for control. Although here only a very simple control strategy (TCI) was used, one can think of other control schemes to be applied after on-line identification, for instance, in order to accelerate convergence to the desired reference for the BIS level. This questions will be considered in future research.

References 1. Almeida, J., Mendon¸ca, T., Rocha, P.: A simplified control scheme for the depth of anesthesia. IFAC PapersOnLine 49, 230–235 (2016) 2. Dumont, G.: Closed-loop control of anesthesia – a review. In: Proceedings of the 8th IFAC Symposium on Biological and Medical Systems (2012) 3. Ionescu, C.M., De Keyser, R., Torrico, B.C., De Smet, T., Struys, M.M., NormeyRico, J.E.: Robust predictive control strategy applied for propofol dosing using BIS as a controlled variable during anesthesia. Comput. Methods Programs Biomed. 55, 2161–2170 (2008)

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4. Minto, C.F., Schnider, T.W., Short, T.G., Gregg, K.M., Gentilini, A., Shafer, S.L.: Response surface model for anesthetic drug interactions. Anesthesiology 92, 1603– 1616 (2000) 5. Nogueira, F., Mendon¸ca, T., Rocha, P.: Automatic control of the depth of anesthesia - clinical results. IFAC PapersOnLine 48, 540–544 (2015) 6. Nogueira, F., Mendon¸ca, T., Rocha, P.: Positive state observer for the automatic control of the depth of anesthesia - clinical results. Comput. Methods Programs Biomed. 171, 99–108 (2019) 7. Sawaguchi, Y., Furutani, E., Shirakami, G., Araki, M., Fukuda, K.: A modelpredictive hypnosis control system under total intravenous anesthesia. IEEE Trans. Biomed. Eng. 55, 874–887 (2008) 8. Silva, J.M., No´e, A.S., Mendon¸ca, T., Rocha, P.: Modeling and identification for the action of propofol and remifentanil on the BIS level. In: 21st IFAC World Congress (2020) 9. Silva, M.M., Mendon¸ca, T., Wigren, T.: Online nonlinear identification of the effect of drugs in anaesthesia using a minimal parameterization and BIS measurements. In: Proceedings of the American Control Conference, vol. 2, pp. 4379–4384 (2010)

Using Multi-UAV for Rescue Environment Mapping: Task Planning Optimization Approach Ricardo Rosa1,5(B) , Thadeu Brito2 , Ana I. Pereira2,3 , Jos´e Lima2,4 , and Marco A. Wehrmeister5 1

Federal Institute of Education, Science and Technology - Paran´ a, Cascavel, Brazil [email protected] 2 Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal {brito,apereira,jllima}@ipb.pt 3 Algoritmi Center, University of Minho, 4710-057 Braga, Portugal 4 INESC Technology and Science, Porto, Portugal 5 Federal University of Technology - Paran´ a (UTFPR), Curitiba, Brazil [email protected] Abstract. Rescuing survivors in unknown environment can be extreme difficulty. The use of UAVs to map the environment and also to obtain remote information can benefit the rescue tasks. This paper proposes an organizational system for multi-UAVs to map indoor environments that have been affected by a natural disaster. The robot’s organization is focused on avoiding possible collisions between swarm’s members, and also to prevent searching in locations that have already discovered. This organizational approach is inspired by bees behavior. Thus, the multiUAVs must search, in a collaborative way, in order to map the scenario in the shortest possible time and, consequently, to travel the shortest reasonable distance. Therefore, three strategies were evaluated in a simulation scenario created in the V-REP software. The results indicate the feasibility of the proposed approach and compare the three plans based on the number of locations discovered and the path taken by each UAV.

Keywords: Unmanned Aerial Vehicles environment mapping · Path planning

1

· Multiple UAV · Collaborative

Introduction

Unmanned Aerial Vehicles (UAVs), commonly known as drones, are continually evolving; every year, more sophisticated UAVs are available on the market. This technological advance allows researchers and professionals to use UAVs in several activities, such as search and rescue [1], surveillance, inspection, forest monitoring, agriculture, among others [2,3]. The variety in these applications makes them a tool with broad potential, as can be seen in infrastructure inspections. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 507–517, 2021. https://doi.org/10.1007/978-3-030-58653-9_49

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When performing inspection tasks, UAVs bring a better three-dimensional understanding of the environment [4]. This has a large importance when scanning an unknown region after a disaster. UAV can be used to search for victims in places where access is steep or inaccessible. A single UAV is not able to cover a large area in a short time, but multiple UAVs are. These robots acquire data and images remotely and send them immediately to an operations center. Thus, the rescue team can better plan its actions, making the most appropriate decisions on how to perform the rescue tasks to save the survivors as quickly as possible and to decrease the risk of rescuers life. However, when multiple robots perform a task as a team, they can interfere with each other in carrying out individual (sub-)tasks. Coordination mechanisms are mandatory to prevent collisions between robots [5]. Such a coordination can be based on local decisions (i.e. the robot makes a decision based on its local information) or global decisions (i.e. the individually robot or a global entity uses information from all robots to decide what to do). The current literature presents a variety of heuristics to manage the multi-robot region exploration. For instance, [5] proposes to choose the target exploration region based on some relevant feature, e.g. distance to the next communication device - the likelihood for a region to be chosen is higher if it is closer to a communication element. This work proposes a multi-UAV management system to map indoor environments that have been affected by a natural disaster. The proposed approach is inspired in bees organization and how they build their hives. That is, several UAVs navigate and explore the unstructured internal environment in the same way that bees travel through space when they build a honeycomb. This method employs multiple (homogeneous) aerial robots; ground robots are not considered. The size of the hexagons is fixed at a radius size of 0.5 m. Some procedures to plan and coordinate the set of UAVs need to be applied and tested before used in real situations. In this way, the focus of this work is to create a simulated environment to develop a coordination strategy for these robots, and thus accomplish the exploration of the simulated scenario. One UAV is enough to map the unstructured and unknown environment. Although, more UAVs will map faster but more complex planning is needed. In our work, a set of three UAVs is used to map the unstructured environment before the team rescuers begin their exploration. All procedures are implemented and tested in the V-REP robot simulator, allowing to validate the proposed approach. Three strategies were tested, and two cases were used to map the unstructured environment before rescue teams entered the environment. In addition, it is essential to highlight the main contributions of this work. First, a comparison between different strategies to identify the best task planning to map an internal dynamic environment. Second, is to analyze the efficiency of using one or more UAVs in a collaborative procedure. The results of these two contributions can determine the quality of the path planning schedule of the multi-UAVs that are exploring an unstructured simulated environment. That is, it is expected that the proposed procedure will manage the multi-UAVs in order to, as quickly as possible, explore all the unknown scenario.

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The paper is organized as follows. Section 2 presents the related work. Then, the developed algorithms and the simulation environment are explained in Sect. 3. Results are presented in Sect. 4, and finally, the last section draws the conclusions and points out future work directions.

2

Related Work

The control of one UAV is treated by some researchers as complex, due to the multiple directions it may have, so controlling multiple UAVs has a higher complexity considering all the possibilities. Therefore, [6], defines that the cooperative control is divided into three areas: complexity, information structure, and uncertainty. The difficulty in moving in unknown environments and with dynamic obstacles is pointed out by [7] and demonstrates the importance of communication between the UAVs to optimize the discovery of the environment through a path planner. Recent surveys, such as [13–15], highlight the challenges and issues on coverage and path planning using UAVs. In [9], the problem of controlling multiple UAVs is addressed as algebraic and dynamic constraints, formulating the problem as optimal control. That work goal is to treat reconfiguration of UAVs with bio-inspired algorithms, combining Particle Swarm Optimization (PSO) and Genetic Algorithm (GA). Sometimes UAVs can be shot down because of the events in the hazardous environments. In [8], an algorithm is proposed to regroup the UAVs when a member of the group is knocked over. In this way, the UAVs team carry out flight training in order to always carry on the mission. Considering a low-lying urban environment, [10] presents two safety systems for UAVs. The first system is developed to identify the static objects on the map, through geographic data of the environment. The second method makes the relation between the static map with unknown objects found, identifying them as dynamic objects. The work presented in [11] merges the Dijkstra algorithm and Simulated Annealing (SA) to streamline the process aiming to decrease the search time for survivors in a real earthquake situation. Even if the developed system has failures in the mathematical model, a reduction in search time and obstacles avoidance is demonstrated. Another approach [12] to the problem of co-operation between drones assigns basic problems to each element of a UAV team by determining Pareto dominate the front. Similar to cited previous works, this paper focuses on offering a task planning method for multi-UAVs. The difference between the studies listed is in the definition of a search strategy to command the UAVs not to perform the paths already found. Hence, the main goal is to optimize the search of the UAVs to map all the environment. In this way, the rescue teams can reduce the time of service or decision making in a catastrophe environment.

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Multi-UAV Indoor Environment Mapping

A simulation environment in the V-REP platform is used to validate the proposed approach and the implementation of the proposed algorithms was performed in MATLAB. Therefore, the following subsections are designed to demonstrate the work developed. 3.1

Overview

The proposed approach, to develop a UAVs management system, is inspired by how the honey bees build their hive. Honey bees use hexagonal pattern cylinders to construct a complex structure progressively by adding waxes produced and manipulated by several bees [16]. The proposed indoor mapping approach aims to produce a 3D occupancy map (based on the ideas of the Octomap [17]) that may be used by the rescue teams in the victims rescue planning. To improve the search, multi- UAVs will create the 3D map from the scenario collaboratively, i.e. several UAVs will explore/navigate the indoor environment to inform the rescue teams. Consequently, the team can detect the obstacles that may hinder their movements. The 3D maps provide not only the information on the space occupancy (i.e. the possible accesses to the places with the indoor environment) but also other information such as photos and thermal images of the region. In this context, the main objective is to map the neighbour cells as faster as possible. Case the cell has neighbours (up to six cells), the UAV informs if the neighbour cells are reachable/unobstructed. Therefore, a maximum of six new cells is included in the global list of unvisited cells which indicates the places that need to be visited/explored by the UAVs team. While the global list contains someplace to search, the management system of the multi-UAVs is enabled. Then, the task manager starts to select an available cell from a list based on a given criterion. This selection is addressed in Subsect. 3.3. Besides, the Point Cloud is used to build the 3D occupancy map, but a discussion on this is out of the scope of this paper due to space limitations. On the other hand, it is essential to highlight that the UAVs will not use such a 3D map for localization, navigation, or obstacle avoidance. 3.2

Simulation Environment

A simulation scenario was used as a case study to analyze the behavior of the management system of multi-UAV and also validate the proposed approach using V-REP [18] platform. The V-REP has APIs that allow communication with many programming languages. Due to this, the proposed management system is implemented in MATLAB because of its robust library kits. Figure 1 shows the simulated environment in V-REP software. The scenario is an area of 10 m × 10 m, with four rooms divided by walls, demonstrated by Fig. 1a. Three similar UAVs (i.e. the same configurations) are used for the simulation. The selected architecture is composed by four-rotor

Using Multi-UAV for Rescue Environment Mapping: Task Planning

(a) Full scenario with three UAVs waiting for start

511

(b) Isometric view of one UAV searching.

Fig. 1. Simulated scenario in V-REP software.

UAV arranged on a “X” frame type, as shown in Fig. 1b. The UAVs can perform Vertical Take-Off and Landing (VTOL) and are equipped with a front laser sensor which produces a Point Cloud that represents the environment within the sensor range at a given position. The simulation starts with the first UAV, called U1 , performing the first cell exploration, i.e. the construction of the first hexagonal cell, which is identified with an ID number, e.g. C1 for the first cell. When any UAV is chosen to explore one cell given by the management system, it rotates in order to detect six faces of the hexagon using the laser sensor. Considering a radius of 0.5 m (this is the size of each UAV structure), the UAV assesses each face in order to check whether it can reach the center of a possible adjacent hexagon, i.e. a new neighbour cell. If possible, it means that adjacency exists and a new neighbour cell is identified and its identifier is placed in the global list of unvisited cells, SC. Once the first UAV (U1 ) finishes the first cell (C1 ) exploration, it navigates to one neighbour cell. This decision is performed by the management system, and it is the work’s focus explained in the next subsection. In this way, the indoor environment is released for other UAVs to be explored. For this, each UAV requests a cell to explore from the global list of unvisited cells. At the end of the cell exploration, its identifier is removed from the unvisited cells list and added into the visited cells list. The indoor environment mapping process continues until the unvisited cells list is empty. By ensuring that all UAVs begin their exploration from cell C1 , it is possible to maintain the uniformity of the topological map, avoiding the area overlap of two cells with distinct identifiers. 3.3

Strategies to Select the Cell to Visit

An important issue is how the set of UAVs will deal with the cluster of the global list of unvisited cells, SC, and which cell should be chosen to be explored first. In this work, is used the simulation scenario in V-REP with three different

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strategies to evaluate the best approach to select the new cell to search. To implement the management system is used the MATLAB and its libraries, and the command to coordinate the multi-UAV is sent to V-REP. The management system implementation establish a set of of n UAVs, SU = {U1 , ..., Un }, and a set of nc of unvisited cells SC = {C2 , ..., Cnc }. All UAVs elaborate the set SC in a collaborative way, where each UAV, Uj , collaborates for the global list SC through its particular list of unvisited cells SCj in a given time when exploring a given cell. To perform the path between each selected UAV and the chosen cell, it is applied the Dijkstra’s Method [19]. It is crucial to highlight that Dijkstra’s is a common and established method for path planning. Therefore, it is used only to execute the trajectories when any UAV need to navigate between the cells. In this work, three strategies to selected the cell to be explored were tested as detailed. Strategy 1: based on First In First Out (FIFO) order. In this case, the oldest cell inserted in the global list of unvisited cells, SC, will be the first to visit and mapped. Strategy 2: based on Euclidean distance with respect to the first cell C1 . Considering that each cell has associated GPS coordinates, the cell that will be chosen to visit is the cell Cj = arg min ||C1 − Ci ||, for i = 2, ..., |SC|. Strategy 3: based on Relative Euclidean distance with respect to the first cell C1 and the location of each UAV. In this case, the selected cell that will be visited by UAV, Ul , located in the cell Cj = |SC| arg min i=1 ||C1 − Ci || + ||Cl − Ci ||.

4

Simulations and Results

The proposed approach to manage the search tasks of UAVs to map a catastrophic environment has been validated via simulation. The V-REP simulator runs on a computer with 6-Core Intel Xeon 3,33 GHz CPU, 6 GB of RAM, and a GPU ATI Radeon HD 5770. The three proposed strategies were analysed on how they affect the UAVs behavior in the simulation of two different situations: Case 1 - the mapping is done by one UAV; Case 2 - three UAVs accomplish the same mapping. Strategy 1 - FIFO. Mapping the simulated scenario always begins in Cell 1. In Fig. 2, the red lines represent the order of UAV task planning and not the actual path. The red lines indicate the way the UAV received the cells from the global list of positions to be searched. For example, it is possible to verify that during the execution of Strategy 1, the UAV starts in Cell 1, and it explores the scenario in a sequential manner, that is, going through all the cells obeying the increasing numerical order (Fig. 2a). Next, the same Strategy 1 is tested with the three UAVs. The operation of this approach can be seen in Fig. 2b. To differentiate each task planning, the red line indicates the task schedule of U1 ,

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the yellow line is a U2 mark, and the blue line is the third UAV. In the same way, as in the previous test, the marking of these three lines represents only the decisions of the task manager to supply the cells that are in the list of not researched.

(a) Strategy 1 - One UAV

(b) Strategy 1 - Three UAVs

Fig. 2. Task planning with Strategy 1 for environment mapping

Strategy 2 - Euclidean Distance. The tests of the second approach take place under the same conditions as the previous test, that is, the same simulation scenario, the same number of robots in each analysis situation, the same computational configuration. The result obtained is shown in Fig. 3, where, once again, the objective is to verify the behavior of UAV task planning. Therefore, Fig. 3a represents the Case 1, when the system management determines the next cells to be explored by the shortest distance from the initial cell. For example, C3 is chosen before C2 because it has the center point closest to C1 . In the next stage, Strategy 2 is applied to Case 2, that is, with 3 UAVs exploring the scenario with the second proposed method (illustrated by Fig. 3b).

(a) Strategy 2 - One UAV

(b) Strategy 2 - Three UAVs

Fig. 3. Task planning with Strategy 2 for environment mapping

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Strategy 3 - Relative Euclidean Distance. The third test is shown in Fig. 4 with Strategy 3 for both proposed situations. It is performed in a similar way to that presented in Strategy 2, however, in this test the decision of the task management system is based not only on the distance between the chosen cell and the initial cell. In this approach, two distances are considered, the first one is from the cell selected to be explored with the initial cell, and the second one is the distance from the idle UAV with the chosen cell. Both Cases are executed with Strategy 3 and shown in Fig. 4a and Fig. 4b, respectively. To compare the different strategies it was considered that each movement between two neighbour cells needs 10 s and to map the environment of a given cell a UAV needs 60 s. The Table 1 presents the obtained results, where T N represents the time needed to map all the environment (in seconds) and N C represents the number of total Cells identified by the UAVs. Table 1. Time needed by the set of UAVs to mapping the environment (seconds). Strategy I Strategy II Strategy III TN NC TN NC TN NC Case 1 (1 UAV) 3870 43 Case 2 (3 UAVs) 3440 38

(a) Strategy 3 - One UAV

3550 39 3580 39

3150 41 3260 41

(b) Strategy 2 - Three UAVs

Fig. 4. Task planning with Strategy 3 for environment mapping

Considering the rate between the total T N and the value of N C, it is possible to conclude that, in average, the Strategy I has 90 s, Strategy II has 91 s and Strategy III has 70 s. Therefore, considering Strategy I, it is possible to verify that the time increases 1% for Strategy I and, for Strategy III, the time decreases 15%.

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As it can be identified in Table 1, the different number of cells for different strategies is due to the dynamic and real simulation environment. Considering the Case 2, with three UAVs, the best strategy was the III, where it was possible to identify 41 cells. In terms of rate, the Strategy I obtains 91 s, Strategy II 92 s and Strategy III obtains 80 s, that means a decrease time of 12% with respect to Strategy I. The Fig. 4 presents the final result in the simulator environment. It is possible to conclude that Strategy III obtains better results with a lower time and will be the selected one for further developments and implementations. Discussion. To validate the proposed approach, it is expected that UAVs will be able to spread in the best possible way. During the analysis, it is noted that the use of each task management method changes the identification number of each cell found (shown in Sect. 3.1). This behavior is explained by the fact that each method makes the UAVs spread more and more (or less) from the initial point, so the order of the cells can be different when compared between the proposed Strategies. So, for the comparison between the Strategies to be fair, it must be established that the performance must take into account the spatial position of each cell. For example, based on Figs. 2, 3 and 4, it is possible to compare the performance of each strategy by observing the behavior in the exploration of C25 (in Strategy 1), with the exploration of C23 (in Strategy 2) and the research during C24 (in Strategy 3). These three cells correspond to the same spatial point. In this case, considering this spatial point, it is possible to notice that in Strategies 1 and 2, UAVs make fewer jumps and thus carry out the mapping in a uniform way in relation to C1 . In other words, UAVs take longer to distance themselves from C1 . It is possible to observe that in these two strategies each UAV needs more interactions to reach cells with higher values. It is enough to observe the behavior at the same spatial point to verify the number of necessary interactions that each UAV took to arrive at a coordinate. The number of cells found should be the same for all strategies, since the space is the same size for all tests. A possible explanation for the number of cells found not being equal during the three tests is the tolerance that the simulator can deliver in terms of sensing. When comparing the three tests, it is observed that the identification of cells in the corners of the environment walls can cause disturbances in the laser sensor used in each UAV.

5

Conclusions and Future Work

This paper presented a multi-UAVs task planning system for mapping unknown indoor environments. The proposed approach was inspired on how bees build their hive. The multi-UAVs explored the indoor environment in order to map it in a efficient way. The proposed strategies were tested and validated in simulation environment and with case studies that simulates the rescue of people in difficult access places. Two cases were tested, with one UAV and with three UAVs to

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map the unstructured environment. The obtained results allow to conclude that the task planning should combine the distance from initial Cell, to ensure a uniform mapping from the initial place, and also information about the cell location of the UAV. As future work, the strategy III should be combined with a populational optimization algorithm to minimize the time and the travelled distance. According to the final task manager method, an implementation with a real scenario and UAVs should be arranged. Acknowledgements. This work is supported by Grant #337/2014 (Funda¸ca ˜o Arauc´ aria - Brazil), the grant from the bi-national cooperation scheme of UTFPR - IPB and by FCT – Funda¸ca ˜o para a Ciˆencia e Tecnologia within the Projects Scope UIDB/05757/2020.

References 1. Oliveira, D., Wehrmeister, M.A.: Using deep learning and low-cost RGB and thermal cameras to detect pedestrians in aerial images captured by multirotor UAV. Sensors 18(7), 2244, 1–33 (2018) 2. Otto, A., et al.: Optimization approaches for civil applications of unmanned aerial vehicles (UAVs) or aerial drones: a survey. Networks 72(4), 411–458 (2018) 3. Shakhatreh, H., et al.: Unmanned aerial vehicles (UAVs): a survey on civil applications and key research challenges. IEEE Access 7, 48572–48634 (2019) 4. Jordan, S., et al.: State-of-the-art technologies for UAV inspections. IET Radar Sonar Navig. 12(2), 151–164 (2017) 5. Burgard, W., Moors, M., Stachniss, C., Schneider, F.E.: Coordinated multi-robot exploration. IEEE Trans. Robot. 21(3), 376–386 (2005) 6. Chandler, P.R., Pachter, M., Rasmussen, S.: UAV cooperative control. In: Proceedings of the 2001 American Control Conference, vol. 1, pp. 50–55. IEEE (2001) 7. Rathbun, D., et al.: An evolution based path planning algorithm for autonomous motion of a UAV through uncertain environments. In: Proceedings of the 21st Digital Avionics Systems Conference, vol. 2. IEEE (2002) 8. Vincent, P., Rubin, I.: A framework and analysis for cooperative search using UAV swarms. In: Proceedings of the 2004 ACM Symposium on Applied Computing, pp. 79–86. ACM (2004) 9. Duan, H., Luo, Q., Shi, Y., Ma, G.: Hybrid particle swarm optimization and genetic algorithm for multi-UAV formation reconfiguration. IEEE Comput. Intell. Mag. 8(3), 16–27 (2013) 10. Yin, C., et al.: Offline and online search: UAV multiobjective path planning under dynamic urban environment. IEEE Internet Things J. 5(2), 546–558 (2017) 11. Deng, L., et al.: Post-earthquake search via an autonomous UAV: hybrid algorithm and 3D path planning. In: 2018 14th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD), pp. 1329–1334. IEEE (2018) 12. Chen, H.X., Nan, Y., Yang, Y.: Multi-UAV reconnaissance task assignment for heterogeneous targets based on modified symbiotic organisms search algorithm. Multidisc. Digit. Publishing Inst. 19(3), p734 (2019) 13. Cabreira, T.M., Brisolara, L.B., Ferreira Jr., P.R.: Survey on coverage path planning with unmanned aerial vehicles. Drones 3, 4 (2019)

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14. Goerzen, C., et al.: A survey of motion planning algorithms from the perspective of autonomous UAV guidance. J. Intell. Robot. Syst. 57, 65 (2010) 15. Galceran, E., Carreras, M.: A survey on coverage path planning for robotics. Robot. Auton. Syst. 61(12), 1258–1276 (2013) 16. da Rosa, R., Wehrmeister, M.A., Brito, T., Lima, J., Pereira, A.: Honeycomb Map: a bioinspired topological map for indoor search and rescue unmanned aerial vehicles. Sensors 20(3), 907 (2020). https://doi.org/10.3390/s20030907 17. Hornung, A., et al.: OctoMap: an efficient probabilistic 3D mapping framework based on octrees. Auton. Robots 34(3), 189–206 (2013) 18. Rohmer, E. Singh, S.P.N., Freese, M.: V-REP: a versatile and scalable robot simulation framework. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (2013) 19. Sede˜ no-Noda, A., Colebrook, M.: A biobjective Dijkstra algorithm. Eur. J. Oper. Res. 276(1), 106–118 (2019)

Robustness Issues in Event-Based PI Control Systems: Internal Model Control Tuning R. Vilanova1(B) , C. Pedret1 , M. Barbu2 , M. Beschi3 , and A. Visioli3 1 2

Universitat Aut` onoma de Barcelona, Barcelona, Spain {Ramon.Vilanova,Carles.Pedret}@uab.cat “Dunarea de Jos” University of Galati, Galat¸i, Romania [email protected] 3 University of Brescia, Brescia, Italy {manuel.beschi,antonio.visioli}@unibs.it

Abstract. Event-based solutions for process control applications are of clear increasing interest. The approach is really appealing because of clear advantages for what matters to minimal use of communication bandwidth, energy consumption, etc. Therefore, completely appropriate for the wireless sensor-actuator networks. One of the solutions that, at least within the process control domain, has reached more acceptance is that of event-based control on the basis of a Symmetric-Send-OnDelta (SSOD) event detector. In this work, a specific implementation of this smart sensor is proposed with clear advantages with respect to the usual ones. The major advantage is that the resulting event-based control system has better robustness properties than the usual implementation that is found in the literature.

Keywords: Event-based control

1

· Robustness margins · PI control

Introduction

The more and more frequent use of wireless systems in process control in the last years has motivated a large effort in the development of event-based control systems. In fact, in this context it is essential to minimize the transmissions between the agents of the control systems in order to reduce their power consumption (and therefore to reduce the costs of the maintenance of the batteries). The rationale of event-based control systems is indeed to send a transmission only when it is really necessary, on the contrary to time-triggered control systems where the transmissions are performed at a periodic rate [6]. In other words, signals are sampled and transmitted only when an event occurs and for this reason an event generator has to be selected when an event-based control system is designed. Among those proposed in the literature [10], the most common one is c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 518–527, 2021. https://doi.org/10.1007/978-3-030-58653-9_50

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the so-called Send-On-Delta (SOD) sampling [5] (also called deadband sampling, or level-triggered sampling, or Lebesgue sampling), which consists in generating an event when a given signal (typically the control error or the control variable) changes more than a given threshold. Among the different event-based control strategies proposed in the literature, the most significant ones from an industrial viewpoint are surely those based on proportional-integral-derivative (PID) controllers [4], as (time-domain) PID controllers are surely the most employed controllers in the process industry. For this reason, many event-based PID control methods have been proposed [1,9,11] and some of them have used a modification of the SOD method, called Symmetric-Send-on-Delta (SSOD) [2,8]. Another common issue in all control problems is that of ensuring a certain degree of robustness. This is of special concern with event-based control systems. Whereas for conventional control systems, the definition of robustness margins and associated controller design approaches has reached a certain degree of maturity, this is not the case of event-based industrial control systems. In such cases, PI/PID controllers are the ones with most success, even more if we concentrate at the field loop level. In such cases it is common practice the use of simple tuning approaches as well as robustness measures such as gain and phase margins. On that respect, this paper proposes the implementation of PI controllers with the use of the Internal Model Control (IMC) equivalent. If the IMC tuning is applied, that implementation will ensure large gain and phase robustness margins. As the IMC tuning is one of the most widely used, the fact of obtaining larger robustness margins will help in the spreading of using such event-based approaches. As an additional advantage, the event-based control system, being more robust, will provide better performance in the sense of generating a lower number of events and also better resemblance with the continuous time solution. This implementation is based on the IMC event-based control configuration presented in [13] without any formal analysis regarding advantages such as the ones reported in this work. As some of the elements this work is based on are well known within the event-based literature, they are presented here in a very succinct way, by referring the reader to the appropriate literature.

2

Materials and Methods

This sections presents the different elements needed to conduct the robustness analysis of the event-based IMC control system. 2.1

Feedback and Equivalent PI IMC Controllers

Even quite well known, for easy reference, we introduce first the PI controller formulated as the usual feedback controller form as well as its implementation in terms of the IMC configuration. As long as we are dealing with stable processes, the IMC framework constitutes a bloc diagram implementation of the parameterisation of stabilising controllers in terms of Q(s). This means that for every

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feedback controller C(s) one can find the corresponding Q(s) and viceversa. Both controllers are related by the following relations: C(s) =

C(s) Q(s) ⇐⇒ Q(s) = 1 − Q(s)Pm (s) 1 + C(s)Pm (s)

Therefore, given a feedback PI controller, C(s) = ing Q(s), is computed as: Q(s) =

Kp (Ti s+1) Ti s

(1)

the correspond-

Kp (Ti s + 1)(Tm s + 1) Ti s(Tm s + 1) + Km Kp (Ti s + 1)

(2)

Note that in the special case where the integral time is chosen equal to the process time constant Ti = Tm , then C(s) =

2.2

(Tm s + 1) Kp (Tm s + 1) ⇒ Q(s) = Tm s Km [(Tm /Km Kp )s + 1]

(3)

Feedback and Equivalent PI IMC Controllers - Event-Based Implementations

Assume we have a feedback controller C(s) and its corresponding IMC controller Q(s). The proposed implementation of the event-based control system in terms of Q(s) instead of C(s), is according to the schema shown in Fig. (1). This event-based configuration was first presented in [13] for a design based on IMC principles. In this paper, a more practical approach is addressed where the IMC is presented just as an option for implementation that can be completely transparent to the control operator.

Fig. 1. Event-based control system implemented in terms of the IMC controller

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Describing Function Analysis

The approach to deal with the SSOD event-generator as a non-linear element, the Describing Function (DF) analysis will be used. The DF method allows for the characterisation of the potential existence of limit cycles experimented in the presence of nonlinear elements. The relevance and practical use of the method relies on the adaptation of the Nyquist stability test to detect the presence of limit cycles in a nonlinear control system [3]. In order to apply the DF analysis the control system is first converted to the equivalent form where the nonlinear and linear parts are identified in separate blocks and connected through the feedback signal. In the particular case of our event-based control system, the nonlinear block corresponds to the SSOD sampling unit whereas the GL (s) representes the resulting linear dynamics part by taking into account all the implied blocks. The DF of a non linear element is defined as the ratio of the fundamental component of the output with respect to the magnitude of a sinusoidal input signal, x(t) = Asin(ωt), applied to the nonlinear element. If we denote by N (A, w) such ratio, the characteristic equation of the system is expressed as: 1 + N (A, w)GL (s) = 0 −→ GL (jw) = −

1 N (A, w)

and provides a characterization of potential limit cycles. From this equation we see that the existence of limit cycles is translated into a Nyquist stability problem where the critical point (−1,0) has been replaced by a locus of critical points in the complex plane given by −1/N (A, w).

Fig. 2. Describing function analysis casuistic. Possibilities for the system’s response: Stable no-oscillating (solid), limit situation for oscillation (dash), oscillation with stable limit cycle (dash-dot), critical stability (dot)

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In the particular case of our EB control system, the nonlinear block corresponds to the SSOD sampling unit whereas the GL (s) representes the resulting linear dynamics part by taking into account all the implied blocks. See for example Fig. (2) for different possibilities regarding the intersection of GL (jω) and N (A). If we are to apply the describing function analysis in order to elaborate on the robustness margins defined above, the basic entity for such analysis is the aggregated linear part of the control system GL (s). This transfer function takes quite different form for the feedback and the IMC implementations. Specifically: Kp (Ti s + 1) Km e−Lm s Ti s Tm s + 1

(4)

Kp (Ti s + 1)Km e−Lm s Ti s(Tm s + 1) + Km Kp (Ti s + 1)

(5)

B GF L (s) = K(s)Pm (s) =

C GIM (s) = Q(s)Pm (s) = L

where the main difference is the presence of an integrator in K(s) and not in Q(s). This pole at the origin will be the main fact in characterising the phase B IM C (s). Therefore, the characterization of of GF L (s) with respect to that of GL the robustness margins. Just to show the different behaviour of the event-based control system, depending on the sensor unit implementation, will consider the −6.15s following process as an example P (s) = e2s+1 . This is a lag dominated process that is to be controlled by a PI controller tuned by applying the Internal Model T ; Ti = T . In this case, the corresponding Control (IMC) tuning Kp = K(λ+L) IMC controller Q(s), according to (3) results as Q(s) =

(Tm s + 1) Km [(λ + L)s + 1]

(6)

In both cases, the SSOD unit is selected with exactly the same parameters, say δ = 0.05 and sampling time Ts = 0.1s. Figure (3) shows the time responses corresponding to both implementations. Also, for the sake of completeness, the corresponding continuous time implementations are also shown. Therefore it is possible to compare between each one of the corresponding time responses, by looking at the time-based (in this case continuous time) implementation and its event-based counterpart. For what regards to the event-based implementations, event generation is also shown at the bottom part of the control action plot. Events corresponding to the IMC implementation are shown half size of those corresponding to the feedback based implementation. In the case shown, the total number of events to take care of the set-point change and the load disturbance is 63 events for the feedback and 21 for the IMC. It is worth to notice the event that takes place at tt ≈ 200 that allows to remove the steady state error. This event is driven by an internal IAE element that is reset at every new event and allows for accumulating steady state error as explained in [13]. Its behaviour is determined by an IAE threshold level.

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Fig. 3. Closed-loop time responses of the event based PI control system on the basis of a feedback and IMC implementations. Number of events generated: Feedback (63) and IMC(21).

2.4

Robusntess Margins

According to the definitions introduced in [7], the idea will be to establish the new critical point as P2 . The idea is to avoid any crossing with the N (A) locus in order to minimise undesired system output oscillations. As dealing with process control applications, the usual dynamics can be represented by a first order plus −Ls time delay (FOPTD) model of the form P (s) = Ke T s+1 the critical point to be avoided is point P2 that has coordinates −π/4 − jπ/4, that corresponds to a radius with angle −3π/4 and a radius of 1.11. This is the reason of the outer circle in Fig. (2). Point P2 will be the reference from now on for establishing the robustness properties of the event-based control system. In an analogous way as for conventional control systems, we can define the event-based gain and phase margins as – Event-based gain margin (gmEB ): If ωg,EB is the frequency where arg{GL (jωg,EB )} = 3π/4 then the gain margin of the event based control system is computed as: 1.1 (7) gmEB = |GL ωg,EB | – Event-based phase margin (pmEB ): If ωc,EB is the frequency there |GL (jωc,EB ) = 1.1 then the phase margin of the event based control system is computed as: π pmEB = 3 arg{GL (jωc,EB )} (8) 4

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Robustness Margins Analysis

In this section we analyse the robustness margins of the two considered eventbased PI control systems implementations: the one defined in terms of a conventional feedback configuration, where the sensor essentially incorporates a SSOD unit, and the implementation defined in terms of the IMC configuration where the sensor implements some intelligence in terms of the IMC elements controller. Notice first that the PI controller can be obtained by applying any of the existing tuning approaches, even the manual or empirical one. It is important to distinguish between the approach to tune the PI controller parameters and the approach used to implement it. Even in both cases we have referred to IMC, we can tune the controller by using the IMC but implement the controller as a feedback PI controller. However, as the IMC tuning is one of the most extensively used, it deserves special attention and in what follows we concentrate the analysis on the PI-IMC tuning. When using the IMC approach, the different options for the resulting tuning formulae, depends on the choice for the delay information that is incorporated into the controller. There do exist in the literature, quite a few proposals for tuning of PI (and PID, of course) controllers on the basis of the IMC framework. It is not the purpose to review here all of them. Here we are interested in the IMC tuning as per construction, the tuning of the integral time constant is based on Ti = Tm , the process model time constant. This generates a cancelling controller. Any other option, even IMC based, that proposes modifications of the integral term, see the S-IMC [12] for example, should be analysed within the framework of the generic PI tuning. A more general approach is needed in order to extrapolate the results to a generic PI tuning and this will be presented in a subsequent work. Considering the FOPTD process model Pm (s) =

Km e−Lm s Tm s + 1

(9)

we do have the following two tuning options: {Kp1 =

T T , Ti1 = Tm } {Kp2 = , Ti2 = Tm } Km λ Km (λ + Lm )

(10)

B,1 B,2 (s) and GF (s) to denote to the aggregated linear part We refer to GF L L of the event-based control system when applying each one of the two tuning options when implemented on a feedback configuration. We get: B,1 (s) = GF L

e−Lm s λs

B,2 GF (s) = L

e−Lm s (λ + Lm )s

(11)

We can now compute the associated frequencies ωg,EB and ωc,EB F B,1 F B,2 = ωg,EB = ωg,EB

π 4Lm

F B,1 ωc,EB =

1 1 ω F B,2 = 1.1λ c,EB 1.1(λ + Lm )

(12)

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and the associated gain and phase margins. The gain and phase margins are obtained as B,1 gmF EB = 1.1 B,1 pmF EB =

πλ 4Lm

Lm π − 1.1 4 λ

B,2 gmF EB = 1.1 B,2 pmF EB =

π λ + Lm B,1 > gmF EB 4 Lm

Lm π B,1 − 1.1 > pmF EB 4 (λ + Lm )

(13) (14)

C,1 (s) If now we turn attention to the IMC implementation, denote by GIM L IM C,2 (s) the corresponding aggregated linear parts of the resulting eventand GL based control system. It turns out that C,1 (s) = GIM L

e−Lm s λs + 1

C,2 GIM (s) = L

e−Lm s (λ + Lm )s + 1

(15)

C,x C,1 Therefore, as in both cases, |GIM (jw)| ≤ 1 we do have that pmIM = L EB = ∞. Regarding the gain margins:

C,2 pmIM EB

C,1 arg{GIM (s)(jw)} = −Lm w − arctan(λw) L π C,1 F B,1 (s)(jωc,EB )} > 3 −→ arg{GIM L 4 C,2 arg{GIM (s)(jw)} = −Lm w − arctan((λ + Lm )w) L π IM C,2 F B,2 (s)(jωc,EB )} > 3 −→ arg{GL 4

(16) (17)

(18) (19)

F B,x IM C,x which implies that ωc,EB = π/(4Lm ) < ωc,EB . Therefore

 C,1 IM C,1 2 gmIM = 1.1 1 + (λωc,EB ) EB  B,1 > 1.1 1 + (λπ/4Lm )2 > gmF EB

(20)

 C,2 IM C,2 2 gmIM = 1.1 1 + ((λ + Lm )ωc,EB ) EB  B,2 > 1.1 1 + ((λ + Lm )π/4Lm )2 > gmF EB

(22)

(21)

(23)

Let us continue with the motivational example from the previous sections. Table (1) shows the gain and phase margins for both IMC-PI tunings under both implementations, feedback and IMC. As it can be seen, the IMC implementation provides better robustness characteristics as predicted by the previous developments. For the special case of tuning 1, it shows a negative phase margin when implemented as a feedback controller, whereas when implemented as an eventbased IMC controller, provides infinite phase margin and considerably larger gain margin. It is worth to remember that this negative phase margin does not imply instability but the presence of oscillations. Figure (4) shows the frequency

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Fig. 4. Robustness margins and closed-loop time responses for tuning 1 under a feedback and an IMC event-based implementations

responses corresponding to the describing function analysis as well as the time responses for tuning 1. The deterioration in performance because of the oscillations is clear. Also it generates a quite more communication activity. Whereas the IMC implementation just generates 23 events, the feedback one do generate 153 events. Therefore, the performance implications of the better robustness margins are clear. Table 1. Gain and Phase margins for feedback and IMC implementations F B 1 IM C 1 F B 2 IM C 2

4

gm 0.86 1.9

1.73 2.84

pm −18 ∞

13.5 ∞

Conclusions

This work has analysed a PI event-based control system. The analysis has been focused on the robustness margins of the event-based control system when implementing the PI controller by using its IMC equivalent. When comparing the IMC with respect to the conventional implementation the IMC one provides larger robustness margins. Specifically, the advantage is analytically proved when the PI controller is tuned by using the IMC tuning: The Integral time constant is chosen as the process model time constant (therefore generating a cancellation of the process model mode). This advantage is added to the fact that the tuning that can be used is the usual one as per the feedback controller. Therefore, no

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additional, new, different tuning approaches from that of the feedback implemented controller is needed.

Acknowledgements. This research is supported by the Catalan Government under Project 2017 SGR 1202 and also by the Spanish Government under Project DPI201677271-R co-funded with the European Regional Development Funds of the European Union.

References 1. Arzen, K.: A simple event-based PID controller. In: Proceedings of 14th World Congress of IFAC, Beijing, China (1999) 2. Beschi, M., Dormido, S., Sanchez, J., Visioli, A.: Characterization of symmetric send-on-delta PI controllers. J. Process Control 22(10), 1930–1945 (2012). https:// doi.org/10.1016/j.jprocont.2012.09.005 3. Gelb, A., Vander Velde, W.E.: Multiple-Input Describing Functions and Nonlinear System Design. McGraw-Hill, New York (1968) 4. S´ anchez, J., Visioli, A., Dormido, S.: Event-based PID control. In: Vilanova, R., Visioli, A. (eds.) PID Control in the Third Millennium. Springer (2008) 5. Miskowicz, M.: Send-on-delta: an event-based data reporting strategy. Sensors 6, 49–63 (2009) 6. Miskowicz, M.: Reducting communication by event-triggered sampling. In: EventBased Control and Signal Processing. CRC Press (2015) 7. Perez, J.A.R., Llopis, R.S.: Tuning and robustness analysis of event-based pid controllers under different event-generation strategies. Int. J. Control 91(7), 1567– 1587 (2018). https://doi.org/10.1080/00207179.2017.1322716 8. P´erez, J.A.R., Llopis, R.S.: A new method for tuning PI controllers with symmetric send-on-delta sampling strategy. ISA Trans. 64, 161–173 (2016). https://doi.org/ 10.1016/j.isatra.2016.05.011 ´ Beschi, M., Visioli, A., Dormido, S., Jim´enez, J.E.: A unified event-based 9. Ruiz, A., control approach for FOPTD and IPTD processes based on the filtered smith predictor. J. Franklin Inst. 354(2), 1239–1264 (2017). https://doi.org/10.1016/j. jfranklin.2016.11.017 10. Sanchez, J., Guarnes, M.A., Dormido, S.: On the application of different eventbased sampling strategies to the control of a simple industrial process. Sensors 9(9), 6795–6818 (2009) 11. S´ anchez, J., Visioli, A., Dormido, S.: A two-degree-of-freedom PI controller based on events. J. Process Control 21(4), 639–651 (2011). https://doi.org/10.1016/j. jprocont.2010.12.001 12. Skogestad, S.: Simple analytic rules for model reduction and PID controller tuning. J. Process Control 13, 291–309 (2003) 13. Vilanova, R.: An internal model control approach to event-based control. In: 2017 3rd International Conference on Event-Based Control, Communication and Signal Processing (EBCCSP), pp 1–6 (2017). https://doi.org/10.1109/EBCCSP.2017. 8022819

A Fractional Order Predictive Control for Trajectory Tracking of the AR.Drone Quadrotor Ricardo Cajo1,2,3(B) , Shiquan Zhao1,4 , Douglas Plaza3 , Robain De Keyser1,2 , and Clara Ionescu1,2 1

Department of Electromechanical, Systems and Metal Engineering, Ghent University, Tech Lane Science Park 125, 9052 Ghent, Belgium {ricardoalfredo.cajodiaz,shiquan.zhao,robain.dekeyser, claramihaela.ionescu}@ugent.be 2 Flanders Make, EEDT Decision & Control, 9052 Ghent, Belgium 3 Escuela Superior Polit´ecnica del Litoral ESPOL, Campus Gustavo Galindo km 30.5 Via Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador {rcajo,douplaza}@espol.edu.ec 4 College of Automation, Harbin Engineering University, Harbin 150001, China

Abstract. A fractional-order model predictive control with extended prediction self-adaptive control (FOMPC-EPSAC) strategy is proposed for the AR.Drone quadrotor system. The objective is to achieve an optimal trajectory tracking control for an AR.Drone quadrotor by using a fractional order integral cost function in the conventional MPC-EPSAC algorithm. In addition, a particle swarm optimization (PSO) algorithm is applied to find the optimal weighting matrices, which depend on the terms (α, β) of the fractional order cost function. Some simulation results show the superiority of FOMPC-EPSAC over conventional MPC-EPSAC with respect to trajectory tracking and robustness under wind disturbances. Keywords: Fractional calculus · Fractional order control · Model predictive control (MPC) · Unmanned aerial vehicles (UAVs) · Particle swarm optimization (PSO) algorithm

1

Introduction

Lately, unmanned aerial vehicles (UAVs) have aroused the interest of the research community, because of its potential use in different applications such as reconnaissance, search and rescue, agricultural imaging, surveillance, among others [1]. This is because drones can perform dangerous tasks and access remote locations and dangerous environments. In addition, the drones are fast dynamic

c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 528–537, 2021. https://doi.org/10.1007/978-3-030-58653-9_51

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systems, which allows them to perform rapid flight maneuvers and coordination with other UAVs (e.g. swarm, flocking, etc. [2]). Given the large number of applications involving drones, different types of drones in shape and size have been developed to be used in various tasks [3]. The main types of UAVs are the following: helicopters, quadrotors, fixed-wing and combinations of them [4]. Each platform has its own advantages and disadvantages depending on the task to be performed. Model Predictive Control (MPC) and PID control are two of the most widely applied control strategies in the industry [5]. In the literature, several authors have presented works related to trajectory tracking control based on PID control [6], sliding mode control [7], L1 , MPC [8], etc. Fractional calculus allows to represent derivatives and integrals to non-integer orders, with which it is possible to obtain the function derived to a real or complex order [9]. Thanks to these advantages of fractional calculus, many researchers are developing and applying fractional methods to control systems such as servo-mechanisms, UAVs, water tank system, unmanned ground vehicles (UGVs) and other industrial applications. Better results are obtained in terms of robustness, payload variations, modeling uncertainties, etc. [10]. In addition, some applications have been developed with fractional-order MPC (FOMPC) strategies. However, they are related with islanded microgrid, industrial heating furnaces, lateral and longitudinal speed control of a rail-vehicle and a commercial vehicle [11]. According to the state of the art and a recent review on fractional order controllers applied to unmanned aerial vehicle (UAV) and unmanned ground vehicles, there is no application reported with fractional order MPC with these platforms [12]. This work proposes a FOMPC with extended prediction self-adaptive control (EPSAC) framework, termed as the FOMPC-EPSAC, to reach optimal trajectory tracking for the AR.Drone quadrotor, in which a fractional-order cost function is used into the MPC-EPSAC algorithm. Besides, a particle swarm optimization (PSO) algorithm is employed to find the optimal weighting matrices, which depend on the fractional-integral terms(α, β) in the cost function. These fractional terms are calculated with the respective optimal closed-loop response, while minimizing another objective function composed of performance indices. This comprises of the Integral of the Absolute Error (IAE) and the integral of the squared control signal (ISU) based on weighted-sum method [13]. The paper is systematized as follows. A brief description of the AR.Drone quadrotor and its dynamic model are presented in Sect. 2. Section 3 describes the theory of the FOMPC-EPSAC with the generalized fractional-order cost function based on Gr¨ unwald-Letnikov (G-L) scheme. Section 4 proposes the PSO algorithm-based optimal weighting methodology for calculating of matrices associated to the fractional-integral terms (α, β) of the FOMPC-EPSAC controller. Finally, the results and conclusions are summarized in Sect. 5 and Sect. 6, respectively.

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System Description

AR.Drone is based on a classic quadrotor design composed by four brushless motors that transfer power to the four fixed propellers, making flight and maneuverability possible. The four basic movements of the UAV (pitch, roll, throttle and yaw) are illustrated in Fig. 1. The drone has different motion sensors such as an ultrasonic sensor, a pressure sensor for altitude estimation, a 3-axis magnetometer sensor for the orientation of the quadrotor. More details about this flight platform can be found in [14].

Fig. 1. Movements of the AR.Drone

The quadrotors are multivariable, highly non-linear and unstable systems. Nevertheless, the AR.Drone has an embedded inner control system by which it can be considered as a Linear Time Invariant (LTI) system. By using parametric identification techniques, the linear models for each movement has been obtained as reported in [15]. The dynamic models of the quadrotor are: Gx,y (s) = Gz (s) =

0.72 z(s) = e−0.1s z Uin (s) s(0.23s + 1)

Gyaw (s) =

3

7.27 x(s) −0.1s x (s) = s(1.05s + 1) e Uin

ψout (s) ψ Uin (s)

=

(1)

2.94 e−0.1s s(0.031s + 1)

Control Formulation

The predictive control algorithm employed hereafter is the EPSAC formulation [16]. This section does not describe in detail the algorithm as it has been previously extensively described in several works [17]. The MPC-EPSAC law is

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obtained by minimizing a cost function subject to a set of constraints as follows: JM P C =

N2  k=N1

γ[w(t + k|t) − y(t + k|t)]2 +

Subject to :

umin ≤ u(t) ≤ umax ymin ≤ y(t) ≤ ymax

N u −1

λ[u(t + k|t)]2

k=0

(2)

where, w is the trajectory to follow; y is the process output; u is the control effort; N1 is the minimum prediction horizon; N2 is the maximum prediction horizon; Nu is the control horizon and γ and λ are nonnegative weighting elements, which have been assumed constants in this case. The cost function (2) is expressed in matrix form as follow: JM P C (U) = [(W − Y) − GU]T Q[(W − Y) − GU] + UT PU

(3)

where, W = [w(t + N1 |t). . .w(t + N2 |t)]T is the reference trajectory vector; Y = [ybase (t + N1 |t). . .ybase (t + N2 |t)]T is the effect of base future control; U = [u(t|t). . .u(t + Nu − 1|t)]T is the future control action optimized; Q = γI(N2 −N1 )×(N2 −N1 ) and P = λINu ×Nu are weighting vectors used as design parameters. A standard quadratic cost function representation of the previous expression is given as follow: min JM P C (U) = UT HU + 2f T U + c such that A · U ≤ b U

(4)

with, H = GT QG + P, f = −GT Q(W − Y) and c = (W − Y)T Q(W − Y) where, A is a matrix and b is a vector with the process constraints. FOMPC-EPSAC formulation results of the generalization of cost function (2) based on the fractional-order operator α Iab (·) [18]. N2 JF OM P C = α IN γ[w(t) − y(t)]2 + β I0Nu −1 λ[u(t)]2 1

Subject to :

umin ≤ u(t) ≤ umax ymin ≤ y(t) ≤ ymax

(5)

The cost function can be discretized with a sampling period Ts and evaluated [18]. Hence, the FOMPC cost function in matrix form is defined as follow: JF OM P C (U) = [(W − Y) − GU]T Q(Υ(α, Ts ))[(W − Y) − GU]+ UT P(Ω(β, Ts ))U

(6)

where Υ and Ω are the weighting matrices, which depend on the fractional terms (α, β) ∈ , respectively: Υ(α, Ts ) = Tsα diag(mn mn−1 . . . m0 )

(7)

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with mj = vj − vj−n , n = N2 − N1 , vk = (−1)k

−α k

and vk = 0, ∀k < 0.

Ω(β, Ts ) = Tsβ diag(mn mn−1 . . . m0 )   and vk = 0, ∀k < 0 with mj = vj − vj−n , n = Nu − 1, vk = (−1)k −β k

4

(8)

Predictive Controllers Tuning

The predictive controllers tuning involve setting N1 , N2 , Nu , together with (γ, λ) for MPC-EPSAC and (α, β) for FOMPC-EPSAC, respectively. However, N1 , N2 and Nu can be chosen by using the thumb-rules given in [19] (for instance, N1 = 1 + system time-delay, N2 ≈ system rise-time and Nu = number of badlydamped poles). It allows us to reduce the optimization problem to two unknown parameters for each controller. The weights of the controller’s objective functions are selected based on the optimal closed-loop response, while minimizing another objective function composed of IAE and ISU performance indices (This method has been used to tune a linear quadratic regulator (LQR) [20]). Therefore, a PSO algorithm presented in [21] is applied to find the optimal weighting elements and fractional-integral terms associated to each controller, respectively. A composite objective function is formulated based on weighted-sum method [13] as follows: J(X) =

N s−1

s1 |w(k) − y(k)| + s2 (u(k))2

(9)

k=0

where, X = [γ λ] are the weights to be optimized for MPC-EPSAC and X = [α β] are the fractional-terms to be optimized for FOMPC-EPSAC. Ns is the number of samples, s1 , s2 are the corresponding weights of IAE and ISU indices, respectively. For this case study presented in this paper, the parameters N1 , N2 and Nu are chosen following the thumb-rules previously announced and a default disturbance model has been selected as an integrator to guarantee zero steady-state error. It is important to mention that both controllers will be tuned using the same horizons and disturbance model for comparison purposes. The parameters for MPC strategies are provided in Table 1. The parameters of PSO algorithm to tune the controller of each movement of the UAV are: number of particles (Np = 50), number of interactions (Tmax = 20), γ, λ ∈ [0 10], α, β ∈ [−5 5], social factor (c2 = 0.5), cognitive factor (c1 ) defined by the randomness parameter (c0 = 0.5) and control parameter (ζ = 0.95). Cognitive and social factors affect the size of the step the particle takes towards its best individual and global solution, respectively. These values are selected based on the ranges specified in [21]. Finally, the optimal parameters for each movement of the drone are obtained by PSO algorithm using the objective function defined by (9) with weights set as s1 = 0.6 and s2 = 0.4 according to the priority of the objectives. The parameters obtained are summarized in Table 2.

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Table 1. MPC parameters MPC-EPSAC/FOMPC-EPSAC Parameter x, y control z control yaw control N1

3

3

3

N2

17

18

6

Nu

2

2

2

Ts (s)

0.066

0.066

0.066

Table 2. Optimal parameters for the controllers MPC-EPSAC PSO x, y control z control yaw control γ

1.784

4.535

3.628

λ

0.319

0.5590

0.652

J

0.5003

1.1793

0.3316 5.0

FOMPC-EPSAC

5

α

4.759

4.559

β

1.019

1.096

4.277

J

0.4994

1.1461

0.3255

Results

In this section, the performance and effectiveness of MPC-EPSAC and FOMPCEPSAC are analyzed with the AR.Drone quadrotor by simulation. As mentioned before both controllers are tuned based on closed-loop response, while minimizing another objective function composed of the IAE and ISU performance indices. The controller parameters for each movement of the UAV are summarized in Table 2, while the tracking performance for each position, altitude and orientation are shown in Fig. 2. It is important to notice that the FOMPC-EPSAC exhibits better behavior in the steady state without overshoot and a faster response than the MPC-EPSAC. Additionally, the stabilization under wind gust condition is essential for drones. And a poor disturbance rejection of the controller can cause a deterioration in UAV performance and it will even induce crash. The disturbance rejection is studied for each controller, so wind gusts between [0.5–1.5] m/s with duration of 0.5 s are added to each movement of the UAV. The results indicate that the fractional controller has the shorter settling time under wind disturbance with respect to the classical MPC (see Fig. 3). Finally, a quantitative analysis is used to evaluate the trajectory tracking and disturbance rejection based on performance indices as IAE, integral of time multiply absolute error (ITAE), integral squared error (ISE), integral of time

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Fig. 2. Step response and circular trajectory with AR.drone quadrotor

Fig. 3. Wind disturbance rejection

multiply squared error (ITSE) and ISU. IAE =

N s−1

|w(k) − y(k)|;

IT AE =

N s−1

k=0

ISE =

N s−1

k|w(k) − y(k)|

(10)

k(w(k) − y(k))2

(11)

k=0

(w(k) − y(k))2 ;

IT SE =

N s−1

k=0

k=0

ISU =

N s−1 k=0

(u(k))2

(12)

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The performance indices values for each controller of the UAV are presented in Table 3. Table 3. Performance indices Control Index MPC-EPSAC FOMPC-EPSAC x

IAE ITAE ISE ITSE ISU

0.8016 1.6046 0.4947 0.4197 0.5514

0.7162 1.1746 0.4726 0.2955 0.6068

y

IAE ITAE ISE ITSE ISU

0.9410 2.4953 0.5786 0.9505 0.8097

0.8284 1.8789 0.5307 0.6583 0.8806

z

IAE ITAE ISE ITSE ISU

1.8374 4.9272 0.9768 1.3026 1.2254

1.4503 2.9431 0.8819 0.8630 1.5725

yaw

IAE ITAE ISE ITSE ISU

0.6203 1.5330 0.3568 0.4783 0.3226

0.5537 1.2581 0.3357 0.3822 0.3622

The IAE and ITAE indices allow to penalize the initial and later values of cumulative errors during trajectory tracking. Moreover, ISE and ITSE penalize the convergence time to a steady state, which will be used to evaluate the performance of disturbance rejection of each controller. Finally, ISU index is a measure related with the energy employed by the controller (control effort). According to the values of IAE and ITAE, the FOMPC-EPSAC has a better trajectory tracking than MPC-EPSAC. In addition, FOMC also achieves better disturbance rejection for each movements of the UAV, which is demonstrated by ISE and ITAE indices. Regarding the control effort, the ISU index shows that both controllers use almost the same energy to achieve their goal.

6

Conclusions

In this article, FOMPC and MPC approaches for the AR.drone quadrotor are studied. Both controllers are tuned with PSO, while minimizing another objec-

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tive function composed of the IAE and ISU performance indices. In MPC the weighting sequences are always nonnegative (which can be a time variant weights, but it produces an optimization problem of high dimension). For this reason in practice these sequences are commonly chosen constants. On the other hand, there are only two parameters in FOMPC required to be optimized (α and β), and this automatically generates non-constant weighting sequences. Therefore, these terms are considered as high level parameters to construct non-constant weighting sequences Υ and Ω, which can have negative element (It is an unusual case that deserves more research). The results demonstrate a best behavior of the fractional controller in trajectory tracking and wind disturbance rejection for translational, altitude and orientation control thanks to the characteristics inherent before mentioned. Acknowledgment. This work was supported in part funded by National Secretariat of Higher Education, Science, Technology and Innovation of Ecuador (SENESCYT), Ghent University Special Research Fund, project MIMOPREC-Starting grant 2018 and Flanders Make-CONACON project nr HBC.2018.0235.

References 1. Mellinger, D., Kumar, V.: Minimum snap trajectory generation and control for quadrotors. In: International Conference on Robotics and Automation, pp. 2520– 2525 (2011). https://doi.org/10.1109/ICRA.2011.5980409 2. Zhao, W., Chu, H., Zhang, M., Sun, T., Guo, L.: Flocking control of fixed-wing UAVs with cooperative obstacle avoidance capability. IEEE Access 7, 17798–17808 (2019). https://doi.org/10.1109/ACCESS.2019.2895643 3. Valavanis, K.P.: Advances in unmanned aerial vehicles: state of the art and the road to autonomy, vol. 33. Springer (2008). https://doi.org/10.1007/978-1-40206114-1 4. Kanellakis, C., Nikolakopoulos, G.: Survey on computer vision for UAVs: current developments and trends. J. Intell. Robot. Syst., 141–168 (2017). https://doi.org/ 10.1007/s10846-017-0483-z 5. Zhang, R.D., Xue, A.K., Gao, F.R.: Model Predictive Control. Springer, Singapore (2019). https://doi.org/10.1007/978-981-13-0083-7 6. Mac, T.T., Copot, C., Duc, T.T., De Keyser, R.: AR.Drone UAV control parameters tuning based on particle swarm optimization algorithm. In: 2016 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR), pp. 1–6 (2016). https://doi.org/10.1109/AQTR.2016.7501380 7. Shah, M.Z., Samar, R., Bhatti, A.I.: Guidance of air vehicles: a sliding mode approach. IEEE Trans. Control Syst. Technol. 23(1), 231–244 (2015). https://doi.org/ 10.1109/TCST.2014.2322773 8. Hernandez, A., Murcia, H., Copot, C., De Keyser, R.: Model predictive pathfollowing control of an AR.Drone quadrotor. In: Memorias del XVI Congreso Latinoamericano de Control Automatico, Proceedings, pp. 618–23 (2014) 9. Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu, V.: Fractional Order Systems and Controls: Fundamentals and Applications. Advances in Industrial Control, 1st edn. Springer, London (2010). https://doi.org/10.1007/978-1-84996-335-0

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10. Cajo, R., Copot, C., Ionescu, C.M., De Keyser, R., Plaza, D.: Fractional order PD path-following control of an AR.Drone quadrotor. In: 2018 IEEE 12th International Symposium on Applied Computational Intelligence and Informatics (SACI), Timisoara, pp. 291–296 (2018). https://doi.org/10.1109/SACI.2018.8440944 11. Romero, M., de Madrid, A.P., Ma˜ noso, C., Vinagre, B.M.: A survey of fractionalorder generalized predictive control. In: 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), Maui, HI, pp. 6867–6872 (2012). https://doi.org/10.1109/ CDC.2012.6426244 12. Cajo, R., Mac, T.T., Plaza, D., Copot, C., De Keyser, R., Ionescu, C.: A survey on fractional order control techniques for unmanned aerial and ground vehicles. IEEE Access 7, 66864–66878 (2019). https://doi.org/10.1109/ACCESS.2019.2918578 13. Marler, R.T., Arora, J.S.: The weighted sum method for multiobjective optimization: new insights. Struct. Multidisc. Optim. 41(6), 853–862 (2010). https://doi. org/10.1007/s00158-009-0460-7 14. Bristeau, P., Callou, F., Vissiere, D., Petit, N.: The navigation and control technology inside the AR.Drone micro UAV. In: Proceedings of the World Congress, vol. 18, no. 1, pp. 1477–1484 (2011). https://doi.org/10.3182/20110828-6-IT-1002. 02327 15. Mac, T.T., Copot, C., Hernandez, A., De Keyser, R.: Improved potential field method for unknown obstacle avoidance using UAV in indoor environment. In: 2016 IEEE 14th International Symposium on Applied Machine Intelligence and Informatics (SAMI), Herlany, pp. 345–350 (2016). https://doi.org/10.1109/SAMI. 2016.7423032 16. De Keyser, R.: Model Based Predictive Control. Invited Chapter in UNESCO Encyclopedia of Life Support Systems (EoLSS), vol. 83. Eolss Publishers, Oxford (2003). Article 6.43.16.1 17. Muresan, C.I., Ionescu, C.M., Dulf, E.H., Rusu-Both, R., Folea, S.: Advantage of low-cost predictive control: study case on a train of distillation columns. Chem. Eng. Technol. 41(10), 1936–1948 (2018). https://doi.org/10.1002/ceat.201700529 18. Romero, M., de Madrid, A.P., Vinagre, B.M.: Arbitrary real-order cost functions for signals and systems. Sig. Process. 91(3), 372–378 (2011). https://doi.org/10. 1016/j.sigpro.2010.03.018 19. Clarke, D.W., Mohtadi, C., Tuffs, P.S.: Generalized predictive control. Part I. The basic algorithm. Automatica 23(2), 137–148 (1987). https://doi.org/10.1016/00051098(87)90087-2 20. Das, S., Pan, I., Halder, K., Das, S., Gupta, A.: LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index. Appl. Math. Model. 37, 4253–4268 (2013). https://doi. org/10.1016/j.apm.2012.09.022 21. Mac, T.T., Copot, C., De Keyser, R., Ionescu, C.M.: The development of an autonomous navigation system with optimal control of an UAV in partly unknown indoor environment. Mechatronics 49, 187–196 (2018). https://doi.org/10.1016/j. mechatronics.2017.11.014

Practical Validation of a Dual Mode Feedforward-Feedback Control Scheme in an Arduino Kit P. B. de Moura Oliveira1,2(B) and Damir Vranˇci´c3 1 INESC-TEC Technology and Science, Campus da FEUP, Porto, Portugal 2 Department of Engineering, University of Trás-os-Montes and Alto Douro (UTAD),

Vila Real, Portugal [email protected] 3 Department of Systems and Control, Jožef Stefan Institute, Jamova Cesta 39, Ljubljana, Slovenia [email protected]

Abstract. Two major control design objectives are set-point tracking and disturbance rejection. How to design a control system to achieve the best possible performance for both objectives is a classical research issue. For most systems these design objectives are conflicting meaning that a single controller cannot cope in providing overall good performance. In this paper, a dual mode control system is reported using a feedforward controller to achieve optimum set-point tracking and PID control to deal with disturbance rejection. A particle swarm optimization algorithm is deployed to design the feedforward controller and the magnitude optimum multiple integration method applied to design the PI/PID controllers. The proposed control system is tested on a custom-made laboratory control temperature kit based on Arduino system. Preliminary results are presented showing the dual-mode control potential merits.

1 Introduction Two of the main control objectives are set-point tracking (SPT) and disturbance rejection (DR). For most systems dynamics, by using a single controller, it is necessary to establish a trade-off between the SPT and the DR performance. A possibility to overcome this drawback is to use a control configuration with more than one controller (often termed two-degrees of freedom) (Araki and Taguchi 2003). For the second order underdamped systems and some oscillatory systems, a dual mode control configuration was proposed in (Moura Oliveira et al. 2012) which can provide optimum performance for both SPT and DR. The objective of this dual mode configuration is to use input shaping technique to achieve a fast SPT response with no overshoot, in open-loop, and then switch to feedback control to deal with disturbance rejection. The input shaping technique adopted is based in a Posicast control technique originally proposed by Smith (1957) which originated most of the current input command shaping techniques (e.g. Huey et al. (2008); Singhose (2009)). The Posicast amplitudes and the time instants of the steps © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 538–547, 2021. https://doi.org/10.1007/978-3-030-58653-9_52

Practical Validation of a Dual Mode Feedforward-Feedback Control

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applied to the system, were optimized by a particle swarm optimization (PSO) algorithm. The feedback controller used is a proportional, integrative and derivative (PID) controller. Another dual mode approach inspired in a concept described in (Shynskey et al. 1988) was proposed by Jones and Moura Oliveira (1996) in which the feedforward controller is based on an on-off profile and PID controller both optimized with a genetic algorithm. A similar approach is explored in this paper, using a PSO as optimizer and the magnitude optimum multiple integration method (MOMI) proposed by Vranˇci´c et al. (1999, 2001) to design PI and PID controllers. Besides blending an artificial intelligence technique (PSO) with a classical PID design methodology (MOMI), this paper contribution lies in the control technique practical test. For this purpose, a simple custom-made kit using an Arduino board was build which is inspired by the Temperature Control Laboratory (TCLab) proposed by Hedengren (2020). The rest of the paper is organized as follows: Sect. 2 presents open-loop Posicast feedforward input shaping concepts; Sect. 3 presets the proposed dual mode feedforwardfeedback configuration; Sect. 4 presents Feedforward Control with TCLab; The basic concepts regarding the design of PID controller with the MOMI technique are presented in Sect. 5; Sect. 6 presents tests results and Sect. 7 presents some conclusions and outlines further work.

2 Feedforward Control In this section a similar approach to the reference Posicast Control input shaping approach is reported. As it was shown in (Moura Oliveira et al. 2012), by using a feedforward Posicast input shaper governed by Eqs. (1–2), it is possible to achieve fast SPT responses without overshoot. Gfts (s) = A1 + A2 e−t1 s + A3 e−t2 s 0 < t1 < t2

(1)

A1 + A2 + A3 = 1

(2)

subjected to:

where A1 , A2 and A3 represent the three step amplitudes; t 1 and t 2 represent the time instant of the second and the third step applied to the system reference input. However, this type of input shaping is mostly adequate for certain systems (e.g. underdamped and oscillatory systems). Illustrative examples of possible deadbeat responses for a double pole with time delay (3) and a non-minimum phase system (4): Gp1 (s) = Gp2 (s) =

1

e−s

(3)

α = 0.5,

(4)

(1 + s)

1 − αs (s+1)3

2

obtained with the input shapers (5) and (6) are represented in Fig. 1 a) and b), respectively. Gfts1 (s) = 2.0 − 2.00e−1.481s + 1.0e−1.916s

(5)

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a) For system Gp1.

b) For system Gp2.

Fig. 1. Input shaping feedforward control illustrative examples.

Gfts2 (s) = 2.0 − 2.0e−1.87s + 1.0e−2.648s

(6)

The design of these type of input feedforward input shapers requires determining optimal values for the step amplitude: A1 , A2 and A3 and step times: t 1 and t 2 . In this case the values were optimized using a PSO algorithm. The basic idea reported in (Moura Oliveira et al. 2012), consisted in using an input shaping reference input shaper to bring the system output toward the set-point and then switch to the feedback control by using a PID controller. A similar approach is proposed here, using an optimized on-off feedforward control instead of the input shaper as it will be described in the following section.

3 Dual Mode FeedForward-Feedback Control The proposed dual model feedforward-feedback control scheme is represented in Fig. 2 where Gc is a PID controller transfer function, Gff the feedforward controller transfer function, Gp is the process transfer function, r is the setpoint, uPID is the PI(D) controller output and ur is the process input. FF is the feedforward signal and d is the process input disturbance. Open-loop feedforward (FF) control mode is applied first to obtain a fast set-point tracking response. When the transient response reaches the steady-state the control is switch to feedback (FB) mode. The switch selects which signal (FF or u-PID ) is connected to the process input. After setpoint change, the switch is in “M” position (FF control). The switching time between FF and FB is represented in Fig. 2 as t s . After t = t s , the switch changes to “A” position allowing the PI(D) controller to take over. The FF controller is governed by (7): ⎧ ⎨ umax ⇐ t < t1 uff = umin ⇐ t1 ≤ t < t2 (7) ⎩ uss ⇐ t ≥ t2   s2 Kd + sKp + Ki 1 Gc (s) = (8) s 1 + sTf

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with uff representing the controller output, umax and umin the actuator maximum and minimum values, respectively, uss the steady state control value, t 1 and t 2 the switching times. The PID controllers is governed by the following function:

ds 1 Tf s

+

e

+

K

d +

Kp

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1 Tf s

+ -

+

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+

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1/Kp -

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FF

Fig. 2. Dual model feedforward-feedback control configuration.

with K p , K i , K d representing respectively, the proportional, integrative and derivative gains and T f the filter time constant. The controller design follows a hybrid methodology, using a PSO algorithmn to optimize values for the feedforward controller (FF), namely: [t 1 , t 2 , uss ] and Magnitude Optimum Multiple Integration (MOMI) tuning method (Vranˇci´c et al. 2001) to determine the PID gains, namely: [K p , K i , K d ]. The filter time constant, T f , is also selected according to the process dynamics. Both these methodologies are briefly described in the following sections.

4 Particle Swarm Optimization Feedforward Design The particle swarm optimization (PSO) algorithm originally proposed by (Kennedy and Eberhart 1995) was selected to be used in this study to perform the FF controller optimization. This optimizer selection is based on several reasons, namely the PSO: i) is by now one of the most well-established nature inspired search and optimization algorithm with a wide range of successful engineering applications; ii) has proven quite robust to design PID controllers, ranging from uni-objective to many-objective optimization (Freire et al. 2017); iii) it is quite suitable for the time-steps instant optimization, as it will be detailed hereafter. As PSO is by now a reference metaheuristic, only the main mechanisms are presented here. PSO is inspired in the animal swarm behavior. An analogy is established between the swarm animal movement and a particle, which is characterized by two variables: position, x, and velocity, v. Every iteration, t, a d-dimensional i-particle position is updated using (9). The position updating term, the velocity, can be evaluated using (10)

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where: ω represents a ponderation factor of the previous iteration velocity value, known as inertia weight; bi is the best position reached by the i-particle until iteration t; g is a neighborhood global-best particle. In this case, as the neighborhood is the entire swarm, this is known as the fully-connected model, with particles knowing the swarm best position; c1 and c2 are know as the cognitive and social error constants. Following the classical PSO formulation, here c1 = c2 = 2, providing the same relevance to the cognitive and social knowledge terms; Finally, φ1 and φ2 are random numbers uniformly generated in the interval [0, 1], which act as search disturbers to the cognitive and social errors, respectively. The inertia weight, ω, is linearly decayed between a maximum value, ωmax = 0.9 to ωmin = 0.4 along the search. xid (t + 1) = xid (t) + vid (t + 1)

(9)

  vid (t + 1) = ωvid (t) + c1 φ1 [bid (t) − xid (t)] + c2 φ2 gd (t) − xid (t)

(10)

To design the FF controller, each particle represents a 3-dimensional vector representing [t 1 , t 2 , uss ]. The swarm is randomly initialized within the search space defined by the decision variables, represented as [t 1min , t 1max ], [t 2min , t 2max ] and [ussmin , ussmax ]. As it can be observed in (7) t1 ≤ t t 2min . Moreover, for certain types of processes, this search interval overlap, may prove to be better. In Fig. 3 two types of feedforward control signal profiles are presented. In Fig. 3a) the PSO converged to an up-down-up profile with t 1 < t 2 . In Fig. 3b) the PSO converged to an up-down profile with t 1 > t 2 .

a) Second-order system.

b) First order plus time delay system.

Fig. 3. Feedforward controller examples.

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5 PID Design with Magnitude Optimum Multiple Integration The magnitude optimum (MO) method is known for many decades (Whiteley 1946) and aims at finding the closed-loop amplitude frequency response flat in wide frequency region. The closed-loop time responses are, therefore, relatively fast and stable. In the last two decades the calculation of controller parameters became even simpler by using the magnitude optimum multiple integration (MOMI) method (Vranˇci´c et al. 1999; 2001). With MOMI, the controller parameters can be calculated either from the process transfer function or directly from the process open-loop time response. Besides using MOMI tuning method, one of the key points in integrating switching (FF) control with feedback (FB) PI(D) control is selection of anti-windup protection. Namely, when the actual control signal output is not the same to the PI(D) controller output signal, the integrating term of the controller might reach undesired values. Here we chose the conditioning technique (Hanus et al. 1987; Peng et al. 1998), since it appropriately solves windup and bump transfer problems. Consider the FB-FF control configuration presented in Fig. 2. Note that during the FF control, the controller windup is prevented by the anti-windup protection.

6 Practical Results The proposed solution was tested on a Arduino based kit prototype. The “process” part of the kit (see Fig. 4) is similar to the Temperature Control Laboratory (TCLab), proposed by (Hedengren 2020). However, due to some difficulties in obtaining faster sampling time under Phyton environment and lower-frequency disturbances, we have rather used Octave-based Arduino environment (Octave 2020). Moreover, by making our own “process”, we added some dynamics by appropriately placing heaters (transistors) and temperature sensors on the same aluminum bar (see Fig. 4). As can be seen the kit is still under development, but the first results are very promising. The kit consists of two heaters (transistors) and two temperature sensors. The kit can use any combination of heaters and sensors including for controlling multivariable process (two-input-twooutput process). A block diagram of the prototype is given in Fig. 5.

Fig. 4. Photo of the Arduino based temperature control kit prototype.

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S1 signal amplificaƟon and filtering

to Arduino analog inputs

from Arduino digital outputs

duty-cycle filtering and power control

T1 T2

S2 temperature sensors

power transistors

Fig. 5. Block diagram of temperature control kit prototype.

The Arduino board is used as A/D converter, where D/A function is realized by changing duty-cycle of a digital signal (between 0% and 100%). The installation of the Arduino firmware is automatic by executing appropriate commands in Octave (Octave 2020). Note that all the calculations of controller output during the experiment are carried on in Octave program package. Besides being a multivariable process and having better control of dissipated power by the transistors, one of the main differences compared to the original TCLab kit (Hedengren 2020) is the sensors position, with one of the sensors clearly separated from the transistor heater. In our test we used the first heater (T 1 ) and the second sensor (S 2 ). As will be shown later, due to the temperature transfer, the actual process transfer function will be of the second-order. The disturbance on the process was achieved by adding the step-like signal (10% change in duty cycle) to the process input (signal d in Fig. 2). A second order model for the kit temperature was obtained by applying an open-loop step change of power duty cycle to transistor T 1 and measuring the temperature variation using sensor S 2 . The model is represented by transfer function (11). The step response and model data mismatch can be observed in Fig. 6. Gp (s) =

0.49 2119s2 + 182.5s + 1

(11)

Fig. 6. The process input (upper figure) in % of duty cycle and the process and the model output (lower figure) in °C.

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The optimization results obtained with the PSO algorithm for the FF controller are represented by (12) considering that the step reference input was applied at t = 20 s. The PI tuning parameters obtained with the MOMI method are represented by (13). The sampling time was Ts = 0.2 s and a measurement filter time constant was set to T f = 1 s. The response obtained with the Arduino kit, using these dual mode control setting and performing the switching between feedforward to feedback mode at t = 86.93 s, is presented in Fig. 7. From this response it is clearly noticeable the set-point tracking performance improvement obtained with the dual mode control configuration over the PI control.

Fig. 7. Responses obtained with the Arduino kit using a dual mode and conventional PI control.

t1 = 64.60 s, t2 = 86.93 s, uss = 20.28

(12)

Kp = 10, Ki = 0.06

(13)

The same experiment was replicated for a PID controller using the same feedforward gains and the PID gains (14). As it can be observed in Fig. 8, the differences between the dual-mode and the PID controller, with control settings (14), are not significant. The PID controller overshoots slightly. Kp = 20.26, Ki = 0.113, Kd = 164 Tf = 5s

(14)

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Fig. 8. Responses obtained with the Arduino kit using a dual mode and conventional PID control.

7 Conclusion and Further Work A two-degrees of freedom control configuration called dual mode control was readdressed and tested in a custom-made Arduino based temperature control kit. It consists of using a feedforward controller to deal with set-point tracking and a PID controller for disturbance rejection. The feedforward control signal is an on-off profile like a three-step Posicast, which is designed using a particle swarm optimization algorithm. The PI(D) controller is designed using the magnitude optimum multiple integration method. The tests results obtained by the custom-made Arduino Kit, indicate that when using a PI controller, there is a clear performance improvement by using the feedforward control in the tracking mode. On the other hand, the performance improvement was not significant with the PID controller. The actual obtained in practice is naturally constrained by the actuator limitations. Further work will be pursuit in using the feedforward on-off control profile to find optimal PID controller gains for the reference following stage which may replace the FF signal best match the optimal on-off signal. Thus two PID controllers may be used one for the tracking phase and the second one for disturbance rejection phase. Acknowledgments. This work is financed by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project UIDB/50014/2020.

References Araki, M., Taguchi, H.: Two-degree-of-freedom PID controllers. Int. J. Control Autom. Syst. 1(4), 401–411 (2003) Moura Oliveira, P.B., Vranˇci´c, D., Boaventura Cunha, J.: 10th Portuguese Conference on Automatic Control, July, Madeira, Portugal, pp. 27– 32 (2012) Smith, O.J.M.: Posicast control of damped oscillatory systems. Proc. IRE 45(9), 1249–1255 (1957) Huey, J.R., Sorensen, K.L., Singhose, W.P.: Useful applications of closed-loop signal shaping controllers. Control Eng. Pract. 16, 836–846 (2008)

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Singhose, W.: Command shaping for flexible systems: a review of the first 50 years. Int. J. Precis. Eng. Manuf. 10(4), 153–168 (2009) Shynskey, F.G.: Process Control Systems: Application, Design and Tuning, 3rd edn. McGraw-Hill, New York (1988) Jones, A.H., Moura Oliveira, P.B.: Genetic Design of dual mode controllers through a process of co-evolution. In: Proceedings of the 4th IEEE Mediterranean Symposium on New Directions in Control and Automation, pp. 794–798 (1996) Vranˇci´c, D., Peng, Y., Strmˇcnik, S.: A new PID controller tuning method based on multiple integrations. Control Eng. Pract. 7, 623–633 (1999). ISSN 0967–0661 Vranˇci´c, D., Strmˇcnik, S., Juriˇci´c, D.: A magnitude optimum multiple integration tuning method for filtered PID controller. Automatica 37, 1473–1479 (2001) Freire, H., Oliveira, P.B.M., Pires, E.J.S.: From single to many-objective PID controller design using particle swarm optimization. Int. J. Control Autom. Syst. 15(2), 918–932 (2017) Hedengren, J.D.: Temperature Control Lab Kit (2020). https://apmonitor.com/heat.htm. Accessed 25 Feb 2020 Octave: Arduino package (2020). https://wiki.octave.org/Arduino_package. Accessed 14 May 2020 Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, pp. 1942–1948 (1995) Whiteley, A.L.: Theory of servo systems, with particular reference to stabilization. J. Inst. Electr. Eng.-Part II: Power Eng. 93(34), 353–367 (1946) Hanus, K.M., Henrotte, J.L.: Conditioning technique, a general anti-windup and bumpless transfer method. Automatica 23(6), 729–739 (1987) Peng, Y., Vranˇci´c, D., Hanus, R., Weller, S.R.: Anti-windup designs for multivariable controllers. Automatica 34, 1559–1565 (1998)

On the Use of a Maximum Correntropy Criterion in Kalman Filtering Based Strategies for Robot Localization and Mapping Matheus F. Reis(B) , Hamed Moayyed, and A. Pedro Aguiar Faculty of Engineering, Department of Electrical and Computer Engineering, University of Porto, Porto, Portugal {matheusreis,hamed.moayed,pedro.aguiar}@fe.up.pt

Abstract. One of the applications of the Kalman filter in the field of robotics is to solve the problem of Simultaneous Localization and Mapping (SLAM). The main drawback of the Kalman filter is that its performance can degrade in the presence of non-Gaussian measurement noise. In robotic systems using laser range finders such as the LiDAR, often optical properties of the beam-environment interaction introduce nonGaussian noise into the system, which can significantly affect performance. In this paper, we investigate this problem and propose a SLAM algorithm similar to the Extended Kalman filter but based on the Maximum Correntropy Criteria (MCC), which aims to exhibit better performance than the classical Extended Kalman filter for some types of nonGaussian noises. The performance of the proposed MCC-EKF SLAM and the classical EKF SLAM are compared by means of numerical simulations.

1

Introduction

Nowadays, mobile robots encounter a wide number of applications in many different areas, such as industry, security, surveillance and environmental exploration [14]. In particular, the Simultaneous Localization and Mapping (SLAM) problem is of particular importance for autonomous operation, and consists of estimating a structure for the unknown environment surrounding the robot (a map) using data from its exteroceptive sensors, while at the same time localizing the robot on the environment. This work was supported in part by R&D Unit UIDB+P/00147/2020 funded FCT/MCTES (PIDDAC) and projects: STRIDE – NORTE-01-0145-FEDER-000033, funded by N2020, ERDF; IMPROVE - POCI-01-0145-FEDER-031823, MAGIC PTDC/EEI-AUT/32485/2017 and HARMONY - POCI-01-0145-FEDER-031411 funded by FEDER funds through COMPETE2020 – POCI and by national funds (PIDDAC). c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 548–558, 2021. https://doi.org/10.1007/978-3-030-58653-9_53

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Achieving successful localization and mapping in an unknown environment is a challenging task, since it depends on a very intricate interaction between specialized sub-tasks, such as feature detection and matching. Feature detection consists of extracting representative classes of features from sensory input. Extraction of high level geometric primitives from raw sensor data, such as corners, edges, lines or planes is a particular requirement in geometric approaches for SLAM [14]. Feature matching is essentially a data classification problem, and consists of finding patterns between previously mapped and new observed features. In the context of SLAM, observed geometric primitives must be matched to previously mapped ones, such that the robot can correctly update the belief about its state. Recently, machine learning techniques have been proposed to solve the feature detection and matching problems, such as Deep Neural Networks (DNNs), Convolutional Neural Networks (CNNs) and Variation Autoencoders (VAEs) [4,6]. In [16], the Extended Kalman filter (EKF) was proposed as a central estimator for the robot pose and map states. The prediction step of the EKF is interpreted as the robot future prediction about its own pose and map states using known robot and map models, respectively, while the update step updates these states using new obtained information from observed features in the environment and a known observation model. This solution has been extensively studied in the SLAM literature [1,2,7,12,17]. Recently, some works have addressed the use of different state estimators to tackle particular limitations of the EKF method in the SLAM context. For example, [5] proposed a SLAM algorithm using an Unscented Kalman filter (UKF) to compensate the loss of performance of the EKF due to the linearization of the process model. However, the UKF can be significantly more expensive than the classical EKF from a computational point of view. In [13], the authors proposed a Constrained Kalman Filter (CKF) based on the Maximum Correntropy Criterion (MCC) ([3,10,11]) to achieve high-accuracy Time-of-Arrival (ToA) ranging. This ranging process is affected by multipath interference and non-line-of-sight (NLOS) factors, which introduces colored Gaussian noise into the measurements. The motivation for the use of the MCC-based CKF lies in the fact that it was shown to exhibit better performance than the Kalman filter for some types of non-Gaussian noises [8]. In this work, we propose a SLAM algorithm based on the Extended Kalman filter approach using the Maximum Correntropy Criterion. The proposed method is called MCC-EKF SLAM and uses 2D points and lines as features, but can be readily generalized to general geometric primitives, such as circles and planes. The MCC-EKF is very similar to the classical EKF, but has a small modification in the update step that allows better performance under the presence of nonGaussian measurement and process noises. Therefore, it is a computationally inexpensive method when compared to other filtering approaches for SLAM. From a practical point of view, another important advantage is that any EKFbased SLAM algorithm can be easily modified to the MCC-EKF SLAM with little implementation effort, since only the update equations need to be modified.

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Numerical simulations illustrate further differences between the two methods under different types of non-Gaussian noises.

2 2.1

Simultaneous Localization and Mapping Robot and Map Modeling

As illustrated in Fig. 1a, the robot pose on the plane is completely characterized by its planar position with respect to an inertial reference frame {I}, denoted T  by p = xR yR ∈ R2 and its planar angle φR ∈ R with the x-axis. Then,  T the robot pose is defined by the vector R = xR yR φR ∈ R3 . The robot nonlinear continuous dynamics is given by ˙ = f (R, u) + wR , R

(1)

where function f : R3 × Rm → R3 maps the robot state velocities, u ∈ Rm is the robot control input and wR ∈ R3 is the process noise. In many cases (with particular emphasis in Kalman filtering frameworks), this signal is assumed to be a white Gaussian noise with known covariance matrix QR ∈ R3×3 .

3

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(a) Robot localization with points and lines. (b) Simulated room with 5 straight walls.

Fig. 1. Robot and map models.

A simple model widely used to describe ground mobile robots is the unicycle T  model, described by f (R, u) = v cos(φR ) v sin(φR ) ω with control input  T given by u = v ω ∈ R2 , where v, ω are the robot forward and angular velocities, respectively. In this work, we assume that the vehicle is described by the unicycle model and has an inner-loop autopilot controller that is responsible for tracking linear and angular velocity commands. After the discretization of the continuous model, we get the following discrete-time model Rk+1 = fk (Rk , uk ) + wk ,

(2)

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where fk (Rk , uk ) ∈ R3 denotes the transition function for the robot state and the subscripts denote the samples at the k-th time step. The equivalent discrete covariance matrix associated to wk is QRk ∈ R3×3 , and the Jacobian of the ∂f k ∈ R3×3 . transition function is expressed by FRk = ∂R k The map state of the surrounding environment at time k is represented by  T T T N 1 k a vector composed of Nk planar landmarks Mk = Lk . . . Lk of different types, where Lik ∈ Rl denotes the corresponding l parameters associated to the i-th landmark Lik , expressed in the inertial reference frame {I}. The landmarks are assumed to be stationary, thus the discrete kinematic model of the map is Mk+1 = Mk

(3)

Note that no state transition error is assumed to affect the map state transition model. The state of the system of robot pose and map configurations   Tcomposed T at time k is given by xT k = Rk Mk , and evolves according to (2) and (3). 2.2

Feature Observation Model

Usually, measurements of observed geometric landmarks in the environment are expressed in the sensor coordinate frame. We assume that these measurements are obtained from a particular observation model that depends both on the robot state and on the corresponding landmark state, and can be affected by some type of measurement noise. This model can be described by F ik = hi (Rk , Lik ) + rk ,

(4)

where F ik ∈ Rl denotes the sensor coordinates associated to the i-th feature detected at time k, and hi : R3 × Rl → Rl is the observation function. The vector rk ∈ Rli is assumed to be drawn from a white Gaussian distribution with known covariance matrix Rk ∈ Rl×l . The observation function can be written with respect to the complete system state xk ∈ R3+2Nk as hi (xk ). The Jacobian of this vector function with respect to xk has a sparse structure due to independence of observations:   ∂h ∂h Hki = ∂Rik 0 · · · ∂Lii · · · 0 . (5) k

If the measurement noise rk can be properly filtered and if the observation : R3 × Rl → Rl can be mapping hi is bijective, then the inverse mapping h−1 i used to retrieve the landmark state coordinates from the corresponding landmark measurements and from the robot state, with Lik = gi (Rk , F ik ),

(6)

where gi = h−1 i is the inverse observation model associated to the i-th landmark. The Jacobian matrices associated to the inverse observation model are GiRk = ∂g i ∂g i l×3 l×l and GiFk = ∂F . i ∈ R ∂Rk ∈ R k

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Feature Detection

Line Detection. Lines are a simple geometric feature that can be used for robust localization and mapping. They can be accurately detected using raw sensory data from a 2D point cloud by using a well known feature extraction technique called the Hough transform [15], which returns the polar coordinates ρ, θ of observed lines expressed in the robot frame {R}. T  Let the local coordinates of the i-th detected line be F ik = ρ¯ik θ¯ki ∈ R2 . The observation function for line-based features is simply given by the rigid body transformation of a line expressed in the inertial frame to the robot frame {R}:   cos(θki ) sin(θki ) 0 i i i i (7) hi (Rk , Lk ) = Lk − A(θk ) Rk , A(θk ) = 0 0 1 T  where Lik = ρik θki are the coordinates of the line in the inertial frame {I}, which is simply the corresponding line state in the map. Therefore, (7) is the expression for the observation function in (4) using line-based features. Point Detection. Corners can be detected from raw data using intersections of observed lines. Therefore, an algorithm was designed, which consists of finding the endpoints of a detected line segment, followed by detecting overlap between endpoints from two different lines and their corresponding intersection point. The Cartesian coordinates of the i-th observed corner at instant k with respect to the robot frame {R} are F ik ∈ R2 . Since the Cartesian coordinates of the corner point can be expressed with respect to the inertial frame {I} as Lik ∈ R2 , the observation function is also a rigid body transformation of a point from the inertial frame {I} to the robot frame {R}, given by  (8) hi (Rk , Lik ) = R(φRk ) Lik − pk , T  where pk = xRk yRk and φRk are the robot position and orientation states at instant k, while R(·) ∈ SO(2) denotes the planar rotation matrix. 2.4

Feature Matching

Feature matching algorithms are responsible for associating new feature observations to corresponding landmarks on the current map. This is essentially a data classification problem where observations of environmental features must be compared to the current features already added to the map. In this work, we used a classification method based on a generalized notion of distance between possibly √ matching features. Define the weighted norm of a vector x ∈ Rn as xA = xT Ax, where A ∈ Rn×n is a positive definite matrix. Assuming the observed 2D features are drawn from a distribution with covariance matrix Σ ∈ Rn×n , the distance between the i-th observed feature and the j-th mapped feature is Dij = F ik − hj (Rk , Ljk )Σ −1

(9)

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which is simply the Mahalanobis distance between the feature coordinates. The matching algorithm consists of computing all (i, j) pairs of observed and mapped features such that Dij ≤ δ, where δ is a fixed matching threshold. Then, the best match is the one with the lowest value for Dij .

3

Extended Kalman Filter with Maximum Correntropy Criterion

An interesting descriptor that can be estimated directly from data, with intention to substitute the conventional statistical descriptors used in nonlinear adaptive filters, machine learning and signal processing, is the correntropy descriptor [9]. Correntropy is extended to a so-called general case of similarity measure between two arbitrary random variables. Given the two random variables X, Y ∈ R with joint probability distribution FXY (x, y), correntropy is a generalized similarity measure defined by [9]:

V (X, Y ) = E[κ(X, Y )] = κ(x, y)dFXY (x, y), (10) where E is the expectation operator and  κ(·, ·) is a shift invariant Mercer kernel, e2 , with e = x−y. This definition such as the Gaussian kernel Gσ (e) = exp − 2σ 2 expresses correntropy as a measure of similarity that directly indicates the probability density of the similarity between two random variables are in a specific window controlled by the kernel size σ. In other words, the kernel bandwidth acts as a zoom lens which controls the observation window in which similarity is assessed. For linear systems, the traditional Kalman filter is optimal in the sense that it minimizes the mean square estimation error in the presence of Gaussian noises. However, its performance may deteriorate significantly under non-Gaussian scenarios. The main reason for this is that the Kalman filter is developed based on the MMSE criterion, which captures only the second order statistics of the error signal and is sensitive to large outliers. It is possible to demonstrate that correntropy on (10) is dependent on second and higher order moments of the random variable X − Y [8,9]. Therefore, as shown in [8], a Maximum Correntropy Criterion (MCC) can be used to derive a new type of KF-like estimator, which may perform much better than the traditional KF in non-Gaussian noise environments. Actually, in [8], the MCCbased KF was shown to exhibit better performance than the traditional KF for linear systems and some types of non-Gaussian noises. A complete derivation of the MCC-based KF can be found in [8]. The MCC-based Extended Kalman filter (EKF) is very similar to the traditional EKF, with a modification on the update equations [11]. In the proposed MCC-EKF SLAM, we apply this modification in the update step in an incremental fashion by updating the system state using each observation at a time. This is possible due to the independence property among different feature observations,

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k ∈ R3 , M  k ∈ R2Nk and P k ∈ R3×2Nk as the estimated as seen in (5). Define R robot pose, map state and covariance matrix at instant k. The covariance of the system estimation error is given by  

RR P RM P k , P k = k PMR PMM k

k

where P RRk ∈ R3×3 and P MMk ∈ R2Nk ×2Nk are the robot pose and map T ∈ R3×2Nk is the cross covaricovariances, respectively, while P RMk = P MR k ance between the robot and map estimated states. With these definitions, the proposed MCC-EKF SLAM method can be described by Algorithm 1. Algorithm 1. MCC-EKF SLAM Algorithm  0 and process covariance P0 Require: Initial robot state R 0 as an empty list Initialize map state M for each time step k = 0, 1, . . . do Get odometry data u k ∈ R2  k , u k ), k+1 = M k  k+1 = f (R M R Prediction step:    T    0 FR FR 0 QR k k k Pk+1 = Pk + 0 IN 0 0 IN k k

M

2 k Update step: get feature observations F 1 k, Fk, · · · , Fk for each observed feature F ik , i = 1, 2, Mk do Match observed feature F ik to the current features on the map if F ik is a new feature then    i = g i (R k , F i ) Initialize new feature state L k k  k ← Mk Append new feature state to the current map: M i   L  k T Pk+1 PLx Append initial feature covariances: Pk+1 ← , where: PLx PLL i PLx = G R

k+1

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k+1

i , PLL = G R

k+1

PRR

k+1

iT i iT +G F R G k+1 k Fk+1 k+1

GR

else i k , L  ji ) = F ik − h ji (R Using F ik , compute the measurement innovation: z k k Compute the MCC Kalman gain K i : −1 iT −1 i −1 iT −1 Li H k R k K i = (Pk+1 + Li H k R k H k )

Gσ Li =

⎛

i  −1 z k R k

⎞ ,

 k , u k ) −1  k+1 − f (R Gσ ⎝R P

⎠

σ=

1 i z k −1 R k

RRk+1

Update system state and error covariance: i  k+1 ← x  k+1 + K i z k x i i T T Pk+1 ← (I − K i H k ) Pk+1 (I − K i H k ) + K i R k K i

end if end for end for

4

Experiments

Comparisons between the proposed MCC EKF and the nominal EKF SLAM were made from simulation results using a Matlab simulator and different types

On the Use of a Maximum Correntropy Criterion in Kalman Filtering

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of noises affecting the simulated data. The simulator generates the LiDAR point cloud for the range sensor, and the two SLAM algorithms are tested under different types of noises affecting the range measurements. The simulated map consists of a convex polygon with five sides of 2 m each, resulting in 5 different lines and corners to be detected, show in Fig. 1b. The lines are shown in red, while their endpoints are shown as green crosses. Figure 1b shows the simulated unicycle robot performing a circular maneuver centered on the origin with 1 m of radius and in the clockwise direction, for about 80 s. The SLAM algorithm uses the simulated point cloud to extract the line and point features without any prior knowledge about the map, similarly to a real experimental scenario. Three different types of simulations where performed for both types of SLAM algorithms, each one with a different type of noise affecting the range measurements: (i) white Gaussian noise, (ii) mixture of Gaussians noise and (iii) white Gaussian with spike-like noise. We have performed 100 Monte Carlo (MC) runs for each simulation, and computed the RMS errors for the estimated robot pose, line coordinates and point coordinates. The mapping estimation error is given in terms of the detected lines and points separately. The estimation error for the

i Σ −1 , i-th map feature is computed using the Mahalanobis distance Lik − L k where Σ = diag(0.005, 0.01) (units in meters) and Σ = I2 are the measure matrices used for the matching of lines and points, respectively. In (9), we have used the matching threshold δ = 10 for the line features and δ = 0.5 for the point features. Figure 2 illustrates the RMS errors for the robot estimated pose and for all 10 map features (5 lines and 5 corners) when the range measurements are affected by white Gaussian noise with variance 2 cm. The EKF SLAM seems to perform slightly better than the MCC EKF SLAM in terms of mapping performance for almost all map features, specially for the lines. This is expected since the traditional Kalman filter is optimal in terms of the mean squared estimation error in the presence of Gaussian noises, although here we are using the EKF estimator, which is sub-optimal. However, for systems with mild nonlinearities, the performance of the EKF is very similar to the KF. Figure 3 illustrates the results obtained with noise drawn from a mixture of two Gaussians with means μ1 = 2 cm, μ2 = 4 cm and variances σ1 = 2 cm, σ2 = 4 cm affecting the range measurements. In this situation, the mapping performance of the MCC-EKF SLAM is slightly superior to the EKF SLAM for the mapping error, with smaller RMS values for almost all line features. Figure 4 illustrates the results obtained with noise drawn from a white Gaussian distribution with spike-like noise to the range measurements. Every timestep, there is roughly 10% of change for a spike to occur, and its magnitude follows a normal Gaussian distribution. Here, the mapping performance of the MCC-EKF SLAM is clearly superior than the EKF SLAM, specially in terms of line parameter errors. Note that, although the difference may seem small, the line estimation error was computed using polar line parameters, which include the line angles θ. That means that the small numerical differences between the two

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algorithms have an impact on the estimated line orientation in radians, which can be crucial for the construction of an accurate map. Table 1 shows the line RMS errors for all simulations. Regarding the line features, the EKF SLAM clearly presents better performance under Gaussian noise, while the MCC-EKF SLAM performs better in the simulated non-Gaussian scenarios. Regarding the point features, we observed no significant difference between the two methods.

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Conclusions

In this work, we have proposed a SLAM algorithm using an Extended Kalman filter derived with a Maximum Correntropy Criterion, known as the MCC-EKF SLAM, for the localization of a mobile robot equipped with a range sensor. We have tested the proposed SLAM method using 100 Monte Carlo simulations, and concluded that it has better mapping performance than the traditional EKF SLAM in the presence of non-Gaussian measurement noises affecting the range measurements, specially regarding the line features.

References 1. Bailey, T., Nieto, J., Guivant, J., Stevens, M., Nebot, E.: Consistency of the EKF-SLAM algorithm. In: 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3562–3568. IEEE (2006) 2. Castellanos, J.A., Martinez-Cantin, R., Tard´ os, J.D., Neira, J.: Robocentric map joining: improving the consistency of EKF-SLAM. Robot. Auton. Syst. 55(1), 21– 29 (2007)

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3. Chen, B., Liu, X., Zhao, H., Principe, J.C.: Maximum correntropy Kalman filter. Automatica 76, 70–77 (2017) 4. Chen, Z., Lam, O., Jacobson, A., Milford, M.: Convolutional neural network-based place recognition. arXiv preprint arXiv:1411.1509 (2014) 5. Cheng, J., Kim, J., Jiang, Z., Yang, X.: Compressed unscented Kalman filter-based SLAM. In: 2014 IEEE International Conference on Robotics and Biomimetics, ROBIO 2014, pp. 1602–1607 (December 2014) 6. Fischer, P., Dosovitskiy, A., Brox, T.: Descriptor matching with convolutional neural networks: a comparison to sift (2014) 7. Huang, G.P., Mourikis, A.I., Roumeliotis, S.I.: Observability-based rules for designing consistent EKF SLAM estimators. Int. J. Robot. Res. 29(5), 502–528 (2010) 8. Izanloo, R., Fakoorian, S.A., Yazdi, H.S., Simon, D.: Kalman filtering based on the maximum correntropy criterion in the presence of non-gaussian noise. In: 2016 Annual Conference on Information Science and Systems (CISS), pp. 500– 505 (March 2016) 9. Liu, W., Pokharel, P.P., Principe, J.C.: Correntropy: properties and applications in non-gaussian signal processing. IEEE Trans. Sig. Process. 55(11), 5286–5298 (2007) 10. Liu, X., Chen, B., Zhao, H., Qin, J., Cao, J.: Maximum correntropy Kalman filter with state constraints. IEEE Access 5, 25846–25853 (2017) 11. Liu, X., Qu, H., Zhao, J., Chen, B.: Extended Kalman filter under maximum correntropy criterion. In: 2016 International Joint Conference on Neural Networks (IJCNN), pp. 1733–1737 (July 2016) 12. Paz, L.M., Tard´ os, J.D., Neira, J.: Divide and conquer: EKF SLAM in O(n). IEEE Trans. Robot. 24(5), 1107–1120 (2008) 13. Qi, Y., Ji, M., Xu, C., Wan, J., He, J.: An improved Kalman filter for TOA localization using maximum correntropy criterion. In: 2019 28th Wireless and Optical Communications Conference (WOCC), pp. 1–4 (May 2019) 14. Siciliano, B., Khatib, O.: Springer Handbook of Robotics. Springer (2016) 15. Siegwart, R., Nourbakhsh, I.R., Scaramuzza, D.: Introduction to Autonomous Mobile Robots. MIT press (2011) 16. Smith, R., Cheeseman, P.: On the representation and estimation of spatial uncertainty. Int. J. Robot. Res. 5, 56–68 (1986) 17. Sola, J., Vidal-Calleja, T., Civera, J., Montiel, J.M.M.: Impact of landmark parametrization on monocular EKF-SLAM with points and lines. Int. J. Comput. Vis. 97(3), 339–368 (2012)

Extrinsic Sensor Calibration Methods for Mobile Robots: A Short Review Ricardo B. Sousa1(B) , Marcelo R. Petry2 , and Ant´ onio Paulo Moreira1,2 1

2

Faculdade de Engenharia da Universidade do Porto, Porto, Portugal [email protected] INESC TEC - Instituto de Engenharia de Sistemas e Computadores, Tecnologia e Ciˆencia, Porto, Portugal

Abstract. Data acquisition is a critical task for localisation and perception of mobile robots. It is necessary to compute the relative pose between onboard sensors to process the data in a common frame. Thus, extrinsic calibration computes the sensor’s relative pose improving data consistency between them. This paper performs a literature review on extrinsic sensor calibration methods prioritising the most recent ones. The sensors types considered were laser scanners, cameras and IMUs. It was found methods for robot–laser, laser–laser, laser–camera, robot– camera, camera–camera, camera–IMU, IMU–IMU and laser–IMU calibration. The analysed methods allow the full calibration of a sensory system composed of lasers, cameras and IMUs. Keywords: Mobile robots · Calibration · Sensor · Extrinsic parameters · Laser scanners · LiDAR · Inertial Measurement Unit IMU · Cameras

1

·

Introduction

A critical task in autonomous navigation is acquiring information about the environment for, e.g., localisation or perception. It can be accomplished using onboard sensors. However, it is necessary to know the relative poses between them and to the robot. Indeed, the so-called extrinsic parameters transform all the sensors measurements into a common frame. The estimation of these parameters is known as the extrinsic sensor calibration [11]. Thus, accurate extrinsic calibration can lead to improvements in the robot’s perception and localisation. This paper is a literature review on extrinsic sensor calibration methods. The sensors types considered are laser scanners, cameras and Inertial Measurement Units (IMUs). Also, the review analyses the most recent works developed on extrinsic calibration covering the most setups of sensors possible between the three types considered. The manuscript is organised as follows. Section 2 presents the search considerations taken into account and the methods found. Section 3 analyses them by the pair of sensor types that the method intends to compute the extrinsic parameters. Section 4 presents the conclusions from this literature review. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 559–569, 2021. https://doi.org/10.1007/978-3-030-58653-9_54

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Method

This paper analyses the research about extrinsic sensor calibration methods, and covering the most sensors setups possible between laser scanners, IMUs and cameras for ground mobile robots. Different setups of these sensors allow the implementation of multi-sensor fusion algorithms for mapping or localisation. So, the extrinsic sensor calibration can lead to improvements in these areas. An in-depth search was performed on Scopus, Inspec, IEEE Xplore, and Google Scholar databases only considering full-text articles published in English. Given the different sensors considered and the focus on ground robots, it was used the logic operator AND to combine the following keywords: mobile robots, extrinsic calibration, extrinsic parameters, and sensor. First, the works experimented on mobile robots were prioritised. Second, the most recent ones were selected. Table 1 presents the obtained 19 methods categorised by the intended setup to be calibrated. Table 1. Synthesis of the search results Sensors setup

References

Robot–Laser

Gao and Spletzer [6] (2010), Underwood et al. [18] (2010)

Laser–Laser

Almeida et al. [1] (2012), Pereira et al. [17] (2016), K¨ uhner and K¨ ummerle [11] (2019), Oliveira et al. [16] (2020)

Laser–Camera

Gomez-Ojeda et al. [7] (2015), Pereira et al. [17] (2016), Guindel et al. [8] (2017), K¨ uhner and K¨ ummerle [11] (2019), Oliveira et al. [16] (2020)

Robot–Camera

Mueller and Wuensche [15] (2017)

Camera–Camera Carrera et al. [3] (2011), Warren et al. [19] (2013), Ling and Shen [13] (2016), K¨ uhner and K¨ ummerle [11] (2019), Oliveira et al. [16] (2020) Camera–IMU

Yang and Shen [20] (2017), Huang and Liu [9] (2018), Kim et al. [10] (2018), Arbabmir and Ebrahimi [2] (2019), Eckenhoff et al. [4] (2019), Liu et al. [14] (2019)

Robot–IMU

Liu et al. [14] (2019)

IMU–IMU

Kim et al. [10] (2018)

Laser–IMU

Le Gentil et al. [12] (2018), Kim et al. [10] (2018)

3

Discussion

The analysis categorises the methods by the intended setup aimed for calibration. It is described briefly each method, its requirements, and the experiments made. If there are more than 2 methods for a setup, a comparison table is also presented. 3.1

Robot–Laser Scanner

Gao and Spletzer [6] relaxed the non-linear calibration problem to a SecondOrder Cone Program (SOCP). The SOCP minimises the distance error of known

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features (retro-reflective tape on vertical poles). Due to the type of features needed, [6] requires that the lasers provide distance and remission values. Underwood et al. [18] used the MatLab function fmincon to solve the non-linear calibration problem by minimising the distance error of known features (vertical pole). [6] and [18] have two common requirements: the robot 6 Degrees of Freedom (DoFs) relative to the world and an extrinsic parameter initial estimation. As for experiments made, [6] used a vehicle platform with two 2D laser scanners and the OxTS RT-3050 localisation system. Given that the SOCP is a relaxation of the original non-linear problem, [6] evaluated its computational efficiency compared with a fmincon implementation, and it was two orders of magnitude faster achieving similar results for the sensors extrinsic parameters. The approach [18] used a vehicle platform with four 2D laser scanners and a Novatel commercial unit for localisation. Works [6] and [18] compared their methods’ accuracy to measured extrinsic parameters, in terms of mapping inconsistencies. Both methods obtained improvements over manual measurements. 3.2

Laser Scanner–Laser Scanner

Almeida et al. [1] used the 3D laser’s point cloud as a reference to fit the data from a 2D laser and extract common points. The user sets the calibration object (conic-based) centre position in the point cloud readings. Then, the Visualization Toolkit computes the rigid transformation. Pereira et al. [17] implemented an Iterative Closest Point (ICP) algorithm with closed-form equations minimising the point-to-point error. This error uses the centre position of the ball calibration object. K¨ uhner and K¨ ummerle [11] used a weighted least-squares algorithm to minimise the point-to-point distance error. It needs a sphere calibration object to compute the point-to-point error. Oliveira et al. [16] used the least-squares algorithm to minimise the point-to-plane distance error. The calibration object used to compute this error was a checkerboard. [16] also needs user interactions to label checkerboard readings in the sensors data, and to set initial estimations (to address the local minima problem) using the rviz tool (ROS framework). Furthermore, [1] considered only the setup 2D–3D laser and [16] only 2D–2D. Even tough [11] proposed an error function for 2D lasers, [11] was only tested with 3D laser scanners. In contrast, [17] considers any combination between 2D and 3D lasers. The four methods need an overlapping field between the sensors. In terms of experiments made, [1] used a vehicle platform with two 2D and one 3D laser scanner. [1] noted that its accuracy is highly dependent on the number of 2D laser measures that intersect the calibration object. It also evaluated data inconsistencies between the sensors. The approach [17] used a vehicle platform with a camera, a 3D and two 2D laser scanners, and a ToF sensor. It obtained an experimental standard deviation below 10 cm for the laser–laser setup. Work [11] used a setup with two 3D laser scanners. It made simulations using 3D laser scanners of low and high precisions showing good noise robustness for distance and angles errors. As for [16], it used a vehicle platform with two 2D laser scanners. The calibration of a laser–laser setup resulted in distance errors relative to the checkerboard plane in the order of centimeters. Table 2 summarises the comparisons made between the methods.

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3.3

[1] [17] [11] [16] Iterative Closest Point (ICP) Least-squares Visualization Toolkit (VKT) x

Requirements Calibration object Overlapping field of view User interactions

x x x

Experiments

x x

Simulations Real data

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Laser Scanner–Camera

Gomez-Ojeda et al. [7] used Maximum Likelihood Estimation (MLE) processes to minimise line-to-plane (rotation) and point-to-plane (translation) errors using scene corners. Pereira et al. [17] used the Levenberg-Marquardt optimisation algorithm to minimise the reprojection error (Perspective-n-Point problem) using a ball. Guindel et al. [8] used the ICP algorithm with closed-form equations to minimise the point-to-point error. It has a limitation: needs at least two cameras to compute a 3D point cloud by stereo matching. The points correspondence between sensors used the holes’ centres of the calibration object (board with four circular holes). K¨ uhner and K¨ ummerle [11] used a weighted least-squares algorithm to minimise the point-to-ray distance error and, similar to the laser-laser calibration, it needs a sphere object. Oliveira et al. [16] used the least-squares algorithm to minimise the error distance point-to-plane (laser) and reprojection error (camera). It has the same requirements as described in Subsect. 3.2. All five methods need an overlapping field of view between sensors. Moreover, [7] and [16] only considered 2D laser scanners. [8] needs a stereopair system and a 3D laser to produce two point clouds. As for [17], it can calibrate both 2D and 3D lasers with a camera. The 3D laser–camera calibration was not tested. It was only tested the 2D–2D/3D laser and 2D laser–camera setups, although the authors justify that the 2D laser was the reference sensor because of its accuracy. Work [11] experimented only with 3D laser scanners and cameras. Still, the 2D laser was considered in the development of the method. In terms of experiments made, [7] used a rig with a 2D laser scanner and a camera. It performed Monte Carlo simulations to analyse the method’s accuracy using a different number of features correspondence. A higher number of correspondences showed better results, analysing the extrinsic parameters rotational and translational error. Also, the accuracy of [7] using a low number of correspondences outperformed other state-of-art methods referred to in the article. With real data, analysing the reprojection error by increasing the camera’s exposure time, the method showed good results in evaluating these visually. Approach [17] used a vehicle platform with a camera, a 3D and two 2D laser

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scanners, and a ToF sensor. It evaluated its performance analysing the mean reprojection error. This error was less than 4 pixels error in the experiments made. Work [8] used a vehicle platform with a 3D laser scanner and a trinocular camera. It only performed simulations in order to have ground-truth data for the extrinsic sensors parameters. It was compared with other state-of-art methods referred to in the article obtaining lower rotational and translational error. Although real experiments are mentioned in [8], no results were presented. The approach [11] used a setup with two cameras and two 3D laser scanners. It simulated the sphere detection error and evaluated the impact of the sensors noise model. The results were compared to an euclidean approach and the [11] showed better results. With real data, it obtained standard deviations below 0.1 cm and 0.01◦ for translation and rotational parameters, respectively. As for [16], it used a vehicle platform with two cameras and two 2D laser scanners. Work [16] compared the initial estimation with the extrinsic calibration, and the distance errors to the checkerboard and the reprojection errors decreased in most of the experiments. Table 3 summarises the comparisons made between the methods. Table 3. Synthesis of the methods for laser–camera setups Description Algorithm

[7] [17] [8] [11] [16] Iterative Closest Point (ICP) Levenberg-Marquardt optimisation Maximum Likelihood Estimation (MLE) x Least-squares

Requirements Calibration object Overlapping field of view User interactions Experiments

3.4

Simulations Real data

x x x

x x x

x

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x x

x x

x x x

x

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x x

x

Robot–Camera

Mueller and Wuensche [15] was the only method found for the robot–camera setup. By using an Extended Kalman Filter (EKF), it estimates the twelve extrinsic parameters of robot’s left and right cameras, and the position of the tracked Scale-Invariant Feature Transform (SIFT) features. The Inertial Navigation System (INS) used allowed to estimate the vehicle’s motion, and, so, predict the position of the tracked features (static relative to the world frame). In order to [15] be executed in dynamic environments, a convolutional neural network was used to label the features as dynamic or static rejecting the dynamic ones. As for experiments made, [15] used a vehicle platform with a high-end INS and two cameras. Work [15] evaluated its performance in urban environments, and it

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did converge (corrected the false initial estimative for the extrinsic parameters) demonstrating that it could run online. It was noted that the extrinsic parameters could be computed in straight-line motion because these were sensitive to the road small imperfections. Lastly, [15] is not analysed in Subsect. 3.5 because [15] did not performed specific tests for the camera–camera setup. Indeed, the EKF considers always the robot–camera extrinsic parameters in its state. 3.5

Camera–Camera

Carrera et al. [3] used a bundle adjustment to minimise the weighted reprojection error. The MonoSLAM was used to construct a Speeded Up Robust Features (SURF) 3D map and to match the features between cameras. The cameras need to be synchronised, and it was noted that high-texture scenes helped generate more features. [3] was the only method of the selected that did not require an overlapping field of views for the camera–camera calibration. Warren et al. [19] is similar to [3], but it did not use MonoSLAM. The calibration algorithm is executed at any time if a specific heuristic defined in [19] is not met. After the calibration, the method continues to perform a visual odometry also proposed in [19]. It also needs an initial estimation for the extrinsic parameters. Although [19] did not specify if the method needs synchronised cameras or high-texture scenes, the experimental setup used synched cameras, and the tests were performed in an outdoor environment (high-texture scenes). Ling and Shen [13] proposed an optimisation-based algorithm that minimises the overall epipolar error using Binary Robust Independent Elementary Features (BRIEF) matching. However, it only computes 5 DoFs between the cameras. [13] considers only a stereo configuration for the cameras where the distance between them is known. K¨ uhner and K¨ ummerle [11] used a weighted least-squares algorithm to minimise the rayto-ray distance error. It requires a sphere object for the calibration procedure. Oliveira et al. [16] used the least-squares algorithm to minimise the reprojection error. It has the same requirements as described in Subsect. 3.2. As for the experiments made, [3] used a robot with two and four cameras. Also, it was used the commercial photogrammetry solution Photomodeler to model the calibration room and provide ground-truth data. Work [3] accomplished good results in terms of reprojection error. Also, it was noted that adding more cameras improved the accuracy because it added more constraints to the optimisation process. Approach [19] used a vehicle platform wit two cameras. It performed simulations for different stereo-pair cameras poses always converging. Also, [19] experimented in outdoors and evaluated its online capability by comparing the visual odometry with an INS. It had similar localisation results in comparison to the INS. Approach [13] used a setup with two cameras on a frontparallel stereo configuration. [13] compared its accuracy using a checkerboard or BRIEF features to an offline-checkerboard-based calibration method available in OpenCV. The proposed method using a calibration object or BRIEF features obtained similar performance to the OpenCV method in terms of the experimental mean and standard deviation. Approach [11] used a setup with two cameras. It compared low-resolution to high-resolution cameras (the last

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as ground-truth). The method showed noise robustness in the simulations by estimating the extrinsic parameters accurately when observation noise was associated with the cameras. Work [16] used a vehicle platform with two cameras. It evaluated the reprojection error difference between an initial estimative and a calibrated setup. Although in one of the cameras was reduced the reprojection error, for the other camera, it increased. So, [16] could not be suited for camera–camera calibration. Table 4 summarises the comparisons made between the methods. Table 4. Synthesis of the methods for camera–camera setups Description Algorithm

[3] [19] [13] [11] [16] Bundle adjustment Least-squares Optimisation

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Requirements Calibration object High texture environment x Overlapping field of view Synched cameras x User interactions Experiments

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Camera–IMU

Yang and Shen [20] used a probabilistic optimisation-based approach to minimise the Mahalanobis norm. Huang and Liu [9] used an optimisation-based algorithm integrated with ORB SLAM. Kim et al. [10] used an optimisation-based algorithm to minimise residual functions defined in [10]. It requires 3D landmarks with known localisation. Arbabmir and Ebrahimi [2] opted for a hybrid optimisation algorithm based on particle swarm optimisation and genetic algorithm. [9] and [2] require a synched camera–IMU setup. Eckenhoff et al. [4] used a MultiState Constraint Kalman Filter that also computes the time offsets between camera and IMU readings, and could be executed online. Liu et al. [14] proposed a Visual-Inertial System (VINS) coupled with encoder performing a online extrinsic calibration based on optimisation. It requires the encoder’s data and landmarks (not their localisation). [10,20] and [14] did not specify if synched data is a requirement. Only [14] requires initial estimations. In terms of experiments made, [20] used a sensor suite with a camera and an IMU. [20] compared its performance with the Kalibr [5] method. It achieved an experimental standard deviation lower than 1 cm (3 cm for [5]). A calibrated VINS was also compared to a ground-truth provided by the OptiTrack system achieving similar localisation results. [9] used the EuRoC dataset acquired from a

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micro aerial vehicle with two cameras and an IMU. The extrinsic parameters had rotational errors lower than 1◦ and translation errors lower than 15 cm. [10] used a sensor suite with one camera and two IMUs. [10] compared its performance to [5] by performing two experiments: 45◦ rotation of part of the sensor suite, and 5 cm change in translation. It had lower translation and rotational error than [5] comparing the second experiment with the first. [2] performed simulations using the KIITI dataset and compared the results with a state-of-art method. It obtained similar results in terms of the parameters accuracy, even though it has fewer requirements (e.g., no calibration object, no calibration scenarios). Also, executing [2] on a VINS and comparing its accuracy to a GPS/INS system, it showed good navigation results in outdoors. [4] used a Snapdragon Flight with onboard IMU and two/three cameras. [4] evaluated its online convergence and obtained similar accuracy relative to [5]. [14] used a car with a camera, an IMU, and a wheel encoder on the left rear wheel. It compared its VINS with a similar one that does not perform the online calibration. [14] improved the positioning and reconstruction accuracy of the system. Table 5 summarises the comparisons made between the methods. Table 5. Synthesis of the methods for camera–IMU setups Description Algorithm

[20] [9] [10] [2] [4] [14] Multi-state constraint Kalman filter Optimisation x

Requirements Landmarks Synched camera–IMU Experiments

3.7

Real data

x x

x

x

x

x x x

x

x x

x

x

x

x

Robot–IMU

Liu et al. [14] proposed a VINS coupled with wheel encoder that performs an online extrinsic calibration based on optimisation. It requires the use of landmarks and the encoder’s data. The experimental setup is the same as described in Subsect. 3.7. [14] showed that the online calibration of robot–IMU–camera has better reconstruction and positioning accuracy than only camera–IMU. 3.8

IMU–IMU

Kim et al. [10] defined residual functions minimised by nonlinear least-squares solvers. It estimated the extrinsic parameters, and the initial gravity and velocity vectors of the IMUs. [10] was tested with two IMUs performing the same experiments as for the camera–IMU setup. The results compared to the expected ones were 1.03◦ for rotational and 5.88 cm for translational errors.

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3.9

567

IMU–Laser Scanner

Le Gentil et al. [12] is an optimisation-based algorithm that minimises the point-to-plane distance associated with the 3D laser and the residuals associated with the IMU. Kim et al. [10] defined residual functions to be minimised by an optimisation-algorithm. Both methods require a scene corner and only [12] addressed the synched data problem by modelling the inertial data continuously. Approach [12] used a 3D laser–IMU setup. It performed Monte Carlo simulations to analyse the rotation and translational errors. Athough ideal IMUs resulted in accurate extrinsic parameters, the translation and rotational errors increased (0.57 to 34 cm and 0.02 to 0.96◦ , respectively) when the IMU had perturbations. [12] showed accurate results in terms of reprojection error when compared to a chained calibration method (Kalibr [5] for camera–IMU, and point-to-plane optimisation for laser–camera). Even though it is referred in [12] experiments with real data, no conclusions were taken. Work [10] used a 2D laser–IMU setup, and evaluated the relative errors between two experiments. It achieved a 0.32 cm translational and a 0.94◦ rotation errors.

4

Conclusions

This paper analysed the most recent works on extrinsic sensor calibration. Its accuracy is determinant for transforming the sensor readings into a common frame and it can improve localisation, mapping performances and robot perception. Thus, it was analysed 19 methods. In the in-depth search, it was noted that the laser-camera and the camera-IMU were the setups most well studied, and the robot–camera and robot–IMU are the less studied. In terms of the most general methods, [11,17] and [16] calibrate two or more setups of lasers and cameras, and [10] setups with IMUs except for robot–IMUs. For robot–laser, [6] seems to be accurate but also computationally efficient. [15] is the only method found for the robot–camera setup. Lastly, [14] was the only method found for robot-IMU. This literature review presents some methods that allow the calibration of a sensory system composed by lasers, IMUs and cameras. So, we aim that this paper, apart from analysing some of the most recent work on extrinsic calibration, helps the scientific community to propose new methods to improve this field. Acknowledgement. This work is financed by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme, and by National Funds through the Portuguese funding agency, FCT - Funda¸ca ˜o para a Ciˆencia e a Tecnologia, within project SAICTPAC/0034/2015- POCI-01-0145-FEDER-016418.

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References 1. Almeida, M., Dias, P., Oliveira, M., Santos, V.: 3D-2D laser range finder calibration using a conic based geometry shape. In: Image Analysis and Recognition, pp. 312– 319. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31295-3 37 2. Arbabmir, M., Ebrahimi, M.: Visual-inertial state estimation with camera and camera-IMU calibration. Robot. Auton. Syst. 120, 103–249 (2019). https://doi. org/10.1016/j.robot.2019.103249 3. Carrera, G., Angeli, A., Davison, A.J.: SLAM-based automatic extrinsic calibration of a multi-camera rig. In: IEEE International Conference on Robotics and Automation (ICRA), pp. 2652–2659. IEEE, Shanghai (2011). https://doi.org/10. 1109/ICRA.2011.5980294 4. Eckenhoff, K., Geneva, P., Bloecker, J., Huang, G.: Multi-camera visual-inertial navigation with online intrinsic and extrinsic calibration. In: 2019 IEEE International Conference on Robotics and Automation (ICRA), pp. 3158–3164. IEEE, Montreal (2019). https://doi.org/10.1109/ICRA.2019.8793886 5. Furgale, P., Rehder, J., Siegwart, R.: Unified temporal and spatial calibration for multi-sensor systems. In: 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1280–1286. IEEE, Tokyo (2013). https://doi.org/ 10.1109/IROS.2013.6696514 6. Gao, C., Spletzer, J.R.: On-line calibration of multiple LIDARs on a mobile vehicle platform. In: 2010 IEEE International Conference on Robotics and Automation (ICRA), pp. 279–284. IEEE, Anchorage (2010). https://doi.org/10.1109/ROBOT. 2010.5509880 7. Gomez-Ojeda, R., Briales, J., Fernandez-Moral, E., Gonzalez-Jimenez, J.: Extrinsic calibration of a 2D laser-rangefinder and a camera based on scene corners. In: 2015 IEEE International Conference on Robotics and Automation (ICRA), pp. 3611– 3616. IEEE, Seattle (2015). https://doi.org/10.1109/ICRA.2015.7139700 8. Guindel, C., Beltr´ an, J., Mart´ın, D., Garc´ıa, F.: Automatic extrinsic calibration for lidar-stereo vehicle sensor setups. In: 2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), pp. 1–6. IEEE, Yokohama (2017). https://doi.org/10.1109/ITSC.2017.8317829 9. Huang, W., Liu, H.: Online initialization and automatic camera-IMU extrinsic calibration for monocular visual-inertial SLAM. In: 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 5182–5189. IEEE, Brisbane (2018). https://doi.org/10.1109/ICRA.2018.8460206 10. Kim, D., Shin, S., Kweon, I.S.: On-line initialization and extrinsic calibration of an inertial navigation system with a relative preintegration method on manifold. IEEE Trans. Autom. Sci. Eng. 15(3), 1272–1285 (2018). https://doi.org/10.1109/ TASE.2017.2773515 11. K¨ uhner, T., K¨ ummerle, J.: Extrinsic multi sensor calibration under uncertainties. In: 2019 IEEE International Conference on Intelligent Transportation Systems (ITSC), pp. 3921–3927. IEEE, Auckland (2019). https://doi.org/10.1109/ITSC. 2019.8917319 12. Le Gentil, C., Vidal-Calleja, T., Huang, S.: 3D lidar-IMU calibration based on upsampled preintegrated measurements for motion distortion correction. In: 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 2149– 2155. IEEE, Brisbane (2018). https://doi.org/10.1109/ICRA.2018.8460179

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CDM Controller Design of a Grid Connected Photovoltaic System Jo˜ao Paulo Coelho1(B) , Wojciech Giernacki2 , Jos´e Gon¸calves1 , and Jos´e Boaventura-Cunha3,4 1

Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal {jpcoelho,goncalves}@ipb.pt 2 Institute of Control and Information Engineering, Poznan University of Technology, ul. Piotrowo 3A, 60-965 Poznan, Poland [email protected] 3 Escola de Ciˆencias e Tecnologia, Universidade de Tr´ as-os-Montes e Alto Douro, Vila Real, Portugal [email protected] 4 INESC TEC Technology and Science, Campus da FEUP, 4200-465 Porto, Portugal

Abstract. Distributed power sources will become increasingly ubiquitous in the near future. In this power production paradigm, photovoltaic conversion systems will play a fundamental role due to the growing tendency of energy price, and an opposed trend for the photovoltaic panels. This will lead to increased pressure for the installation of this particular renewable energy source in home buildings. In particular, on-grid photovoltaic systems where the generated power can be injected directly to the main power grid. This strategy requires the use of DC-AC inverters whose output is synchronized, in phase, with the main grid voltage. In order to provide steady output in the presence of load disturbances, the inverter must work in closed-loop. This work presents a new way to design an inverter controller by resorting to the CDM design technique. The obtained results suggest that the controller achieved with this method, although simpler than other methods, leads to an acceptable and robust closed-loop response. Keywords: CDM

1

· Power converter · Renewable energy

Introduction

At the present time, there is a strong tendency toward a decentralised electric energy production paradigm. That is, an electrical energy consumer can be, at the same time, also an electric energy producer. This paradigm implies that the classical electrical grid topology will evolve in the future to a micro-grid network mesh. This trend results in several phenomena that can be identified at different social levels. Among them, is the increasing value paid by each kilowatt-hour c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 570–581, 2021. https://doi.org/10.1007/978-3-030-58653-9_55

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(kWh) of delivered energy from the main power grid. In particular, for Portugal, this statement is evident by analysing the graph depicted in Fig. 1. The plotted curve represents the price, per kWh, registered from the last twenty years for domestic type consumer [1].

Fig. 1. Price tendency, per kilowatt-hour of electric energy, in Portugal from 1999 to 2019 (for typical domestic consumers).

On the other hand, the price per watt-peak (Wp) for photovoltaic has drooped from around 6 e/Wp, at the end of 2001, to a value lower than 2 e/Wp in 2019. For these reasons, photovoltaic solar energy solutions are reaching a point of both technical maturity and economic convergence which causes the electric energy auto-production to be economically attractive. Domestic photovoltaic power generation can be grouped into two main different categories: off-grid and on-grid installations. The former, as the name suggests, represents a broad family of technological solutions where the generated solar energy is used for self-consumption only and no power grid interface is considered. This setup may require batteries if power delivery capability is necessary during the periods of solar radiation absence. On the other hand, grid-tied systems are able to use the main power grid as a virtual power reservoir: during hours where the production is higher than the consumption, the user sells surplus energy to the grid and, when the energy consumption is higher than the energy production, the consumer is able to buy it again from the same place. Whenever possible, this last photovoltaic power management scenario is preferred since it prevents the use of batteries which are expensive, require maintenance, and have limited charge capacity and life cycle. Moreover, in the off-grid solution, significant energy loss is wasted in battery charging and discharging cycles. Hence, due to economic and robustness conditions, the on-grid photovoltaic power generation system is a better solution leaving the off-grid system to rural places that are too far away from the power grid.

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A simplified schematic diagram of a grid-tie photovoltaic system is presented in figure Fig. 2. An array of electrically connected solar panels feeds a DC-DC boost converter that increases the photovoltaic voltage vpv to an intermediate, and regulated DC, voltage Vb . Additionally, a single-phase inverter converts the boosted voltage to the sinusoidal voltage pattern required by the power grid.

Fig. 2. Diagram of a string inverter composed of a DC-DC boost converter and a single-phase inverter.

Besides the voltage level translation, the DC-DC converter is responsible to operate the photovoltaic panels on its maximum power-point. That is, the boost converter modifies the electrical operating point of the solar panels to ensure they deliver the maximum amount of power. This involves knowing the electrical current or voltage of the solar panel at which maximum power can be generated. The DC to AC power inverter takes the DC power and converts it to AC so it can be injected into the electric utility company grid. One major requirement of this system is its ability to synchronize the generated power signal frequency with that of the power grid. This device must have a controller that locks into the current AC grid waveform and outputs a voltage synchronised with the one provided by the grid. Usually, proportional integral derivative (PID) controllers are employed to control both the DC-DC boost converters and the power inverter. Frequently they are tuned based on certain defined operating ranges using averaged mathematical models. As a matter of fact, common mathematical models for switchmode power converters rely on state variables averaging and linearization around some quiescent point. This approximation can lead to a very different dynamic behaviour between the true power system and its mathematical model. Hence any controller tuned to operate adequately within the linearized operating point can perform poorly when subjected to true operating conditions. In particular when severe load changes take place. For this reason, a more robust approach must be addressed in the controller design process. Bearing this in mind, this paper presents an alternative inverter design controller strategy based on the coefficient diagram method (CDM).

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The controller design by the coefficient diagram method appears in the late nineteen nineties as a sequence of the seminal work carried out by Shunji Manabe in the search for an easy to apply control analysis and design method [2,3]. Since then, many articles have been published in both CDM theoretical extensions [4,5] and practical applications [6–8]. From the user point-of-view, the main feature of CDM is its simplicity since the design process only requires the designer to define a single parameter: the equivalent time-constant. Then the controller transfer function is automatically obtained via an algebraic method similar to pole placement. However, unlike the latter, the characteristic polynomial in CDM is defined in a straightforward manner. In particular, an improved version of Kessler’s standard form [9], commonly called Manabe’s standard form, is selected as the target polynomial. This choice will lead to a zero overshoot closed-loop step response and a settling time within 2.5 to 3 time-constants. Besides the CDM algebraic nature, this method also includes a diagram that can be used to anticipate the system behaviour. In fact, the precise name of this method derives exactly from this diagram. The plotted curves in this diagram can be used to analyse the system dynamics, its robustness regarding modelling errors and stability limits. The latter curves are added in the diagram by resorting to the Lypatov-Sokolov sufficient stability conditions [10]. The CDM design procedure can be summarized as follows: first, a plant mathematical model, in polynomial format, is obtained. Then, the characteristic equation is established regarding the desired dynamic performance. The next step concerns the definition of the controller order and its mathematical formulation in polynomial format. The controller coefficients are then obtained by solving a design equation similar to the Diophantine equation. The last step is to analyse the coefficient diagram and interpret it bearing in mind both the desired and obtained system characteristics. Computer simulation of the overall system, taking into consideration disturbances and measurement noise or sensor faults should also be carried out. This paper begins by presenting a short overview of the CDM controller paradigm where the controller state equations are described. Section 3 applies this method to the design of a power inverter current controller. Simulation results will be provided and compared to the ones obtained by the PID controller. This article ends after a brief conclusion section where final considerations regarding this method will be presented.

2

CDM Controller Structure

Figure 3 presents the block diagram for a closed-loop feedback system within a CDM control strategy. Notice that the variables are expressed in the time-domain and that, in this context, μ refers to the differential operator. The target system is described by a differential equation expressed by polynomials C(μ) and D(μ) and the controller behaviour depends on A(μ), B(μ) and E(μ). The variable r(t) concerns the reference signal, u(t) the control signal and y(t) the controlled variable. In addition δ(t) denotes a generic system disturbance and

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n(t) represents the measurement noise. Note that in this case the disturbance δ(t) is placed at the system input. So, all disturbances applied in other system points must be referred to this one. The same applies to the measurement noise which, in this case, is placed at the plant output.

Fig. 3. Overall CDM closed-loop architecture.

For the sake of simplicity, let the order of both polynomials A(μ) and B(μ) be equal to n and the orders of polynomials C(μ) and D(μ) be equal to m. Then, the m + n order characteristic polynomial P (μ) is: P (μ) = A(μ) · D(μ) + C(μ) · B(μ)

(1)

which can be generically defined as: P (μ) =

nP 

pi · μi

(2)

i=0

where pi , for i = 0, · · · , nP , are the polynomial coefficients and nP = m + n is its order. From the polynomial coefficients, the predominant time-constant and the stability index are then defined. The former is defined by τ = pp10 and the latter by : p2i (3) γi = pi+1 pi−1 for i = 1, · · · , nP − 1. Each one of the characteristic polynomial coefficients pi can be written as a function of both stability indexes and predominant time-constant. Hence, Eq. (2) can be expressed alternatively as: ⎧ ⎛ ⎞⎫ nP ⎨ i−1 ⎬  1 P (s)  ⎠ + τμ + 1 (4) = (τ μ)i ⎝ ⎩ p0 γji−j ⎭ i=2

j=1

The characteristic polynomial coefficients are then defined, after the pioneering work of Manabe, by forcing the stability index values to be [3]:  2.5 if i = 1 (5) γi = 2 if i = 2, · · · , nP which leads to a transient response that subsides within 2.5 to 3 equivalent time-constants with minimum overshoot.

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575

The Coefficient Diagram

The name of the described controller design technique derives from the fact that, in addition to the algebraic component, it involves a diagram that can be used to predict the performance of the closed-loop system. This behavior can be analyzed following three distinct vectors: dynamic response, stability and robustness. Conceptually, the coefficient diagram consists of a set of plots represented over a logarithmic ordinate axis system. The abscissa is defined by a dimensionless variable that represents the coefficient indexes. For example, considering a nP order characteristic polynomial, the abscissa will contain nP + 1 integer values from 0 to nP . Within this pair of axes, several plots are sketched. Each plot is obtained by a first-order interpolation between a set of points derived from the values of the polynomial coefficients of P (μ), B(μ) and the stability measures γi . From the polynomial P (μ), a plot denoted by the coefficients curve is obtained. In the same way, the stability curve concerns the curve obtained by linear interpolation trough all the γi points for i = 1, · · · , nP − 1. Finally, the partial coefficient curve regards the curve obtained by plotting the coefficients of the controller polynomial B(μ) [11]. Intuition on system stability can be obtained by observing the characteristic polynomial convexity curve since a larger curvature implies a stabler system. Moreover, system stability margin increases when the vertical distance between the stability curve and the stability limit curve increases. The system dynamic response can be estimated by the overall inclination of the coefficients curve. A measure of the system speed is the equivalent time constant represented as the characteristic polynomial slope between i = 0 and i = 1. The coefficient curve shape sensibility, due to plant parameter variation, is a measure of system robustness. Robustness assessment is based on the closeness of both coefficient and partial coefficient curves. Immunity to parameters change is higher if the coefficient curve is above the partial coefficient curve. Additionally, this robustness increase as both curves get closer.

3

CDM Power Inverter Design

Commonly, on a grid-connected PV system, a set of eight to ten solar panels are wired in series leading to a net increase of the photovoltaic voltage output to values that can reach 800 V. This string of panels is electrically connected to a combiner box that merges all the multiple wires that come from the solar panel array into the electrical cable that leaves for the inverter. The main function of the grid-tied inverter is to provide grid synchronization and to control active and reactive power flow or, seen differently, to control the grid currents and the DC-link capacitor voltage. Most grid-tie inverters are controlled using voltage-oriented control (VOC). Which uses a rotational direct quadrature (d-q) reference frame transformation oriented with the grid voltage vector to convert all AC quantities into DC. This will simplify the control system design and will enable the use of PI controllers. VOC is based on a cascaded

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voltage and current control loop where the outer voltage loop controls the DClink voltage using a PI controller whose output is proportional to the active power. Current control is also regulated using PI controllers, whose outputs are the d-q reference voltages. The fundamental components of a string photovoltaic on-grid system can be broken down into the basic constituent elements represented in Fig. 4. Notice that this schematic is not meant to be an exhaustive representation of such systems. For example, the maximum power point tracking block, surge protection, and fuses are not shown.

Fig. 4. Block diagram of a typical string inverter highlighting the closed-loop feedback loop associated to the DC to AC converter.

From the above figure, one can observe that besides the string of solar panels, the system is composed by a DC-DC boost converter, that increase the string photovoltaic voltage vpv to an intermediate DC voltage Vb , and a single-phase inverter, that converts the boosted voltage to the sinusoidal voltage pattern required by the power grid. The amount of electrical power generated by the photovoltaic string panels depends on several factors such as solar irradiance and temperature. Moreover, the relationship between the delivered power and the panel’s output voltage is non-linear. Hence, the point at which the solar modules generate the maximum power is not static and not known in advance. This value is commonly tracked by electronic circuits by measuring several variables such as the panels output electric voltage and current and the ambient temperature. The output of the maximum power-point tracker circuit will be used as a set-point for the boost converter duty-cycle controller in order to regulate the converter input voltage and current.

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The bulk capacitance C is used to damp the imbalance between the DC photovoltaic power, available at the DC-DC output, and the AC power absorbed. Without it, the AC disturbances at the inverting bridge will propagate up to the boost converter input. Finally, the high voltage bus from the DC-DC converter is fed to the DC-AC power inverter which will be responsible for supplying the load and make the connection to the grid. At the heart of the DC to AC converter is the electronic H-bridge circuit. This setup involves four electronic solid-state switches which can be either bipolar transistors, MOSFET, or IGBT. The four bridge transistors are driven by PWM signal leading the transistors to operate away from the linear region. Therefore, the power dissipated in the switches is very low contributing to the overall converter efficiency, which typically attains a value close to 90%. In order to regulate the load current, the inverter must operate in a closedloop. The output current is measured by a Hall-effect sensor with negligible dynamics and gain ks . This measurement is compared to a given current setpoint provided by a current reference generator block (for example a phase-locked loop (PLL) based circuit). This device generates the sinusoidal signal vref with the same frequency and in phase with the AC grid voltage. Ignoring higher-order dynamics, its transfer function can be approximated to K(s) = VVmp where Vm is the PWM gain stage and Vp is the vac grid peak voltage. The error signal, computed by the difference between the actual current and the reference, is delivered to a compensator with transfer function Gc (s) Then the control signal is applied to a pulse-width modulator (PWM) responsible to control, through a driver circuit, the semiconductors state. The purpose of this work is to test the CDM design technique within this framework. In order to do this, a mathematical model for the DC-AC converter must be obtained. This will be addressed in the following subsection. 3.1

The Inverter Open-Loop Transfer Function

In this section, the open-loop transfer function of the system represented in Fig. 4 will be addressed. First, we assume that the voltage provided, upstream, by the DC-DC boost converter is equal to 400 V and that the inverter should supply a sinusoidal alternating voltage with 240 V root-mean-square value at a frequency of 50 Hz. Notice that, due to the inherent step-down characteristic of the Hbridge, the DC bus voltage must be higher than that of the peak AC voltage. The inverter filter inductance is established as 1 mH and the PWM modulation frequency is set to 40 kHz. Moreover, the PWM gain module Vm is equal to 5 V and the Hall-effect sensor sensitivity ks is 0.2 V/A. Since the switching frequency is much higher than that of the grid, during transistors switching cycle is considered to be constant. Moreover, the voltage vb generated by the DC-DC converter is considered fixed with a negligible ripple. Now, assume that during some fraction of the switching cycle, the solid-state switches S2 and S4 are on and the remainder is off. Then, the voltage across the inductor L is equal to vL = Vb − Vac . During the complementary cycle,

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vL = −Vb −Vac . For this reason, the average voltage over one complete switching cycle, denoted by < vL >, is equal to: < vL >= Vb (2d − 1) − Vac

(6)

where d ∈ [0, 1] regards the duty-cycle value. From the relationship between the average voltage and current in an inductor, 2L

d < iac >= Vb (2d − 1) − Vac dt

(7)

By applying a very small AC disturbance dˆ and ˆiac to the states variable iac and d around a quiescent point (D, Iac ) leads to: 2L

d (Iac + ˆiac ) = 2Vb D + 2Vb dˆ − Vb − Vac dt

(8)

Now, after canceling out the DC terms, the following small-signal model is obtained: Vb ˆ dˆ iac = d (9) dt L which, after applying the Laplace transform, and taking into consideration zero initial conditions, the inverter current to duty-cycle transfer function is: Gid (s) =

Vb s·L

(10)

Finally, the system open-loop transfer function Gol (s) is equal to the product of the sensor transfer function H(s), the pulse-width modulator M (s) and the duty-cycle to current transfer function Gid (s). That is, Gol (s) = Gid (s) · H(s) · M (s) Vb · ks = Vm · L · s 3.2

(11)

Controller Design

Having the open-loop transfer function, the controller design by the CDM technique is straightforward. First, the desired closed-loop bandwidth is selected by providing a suitable value for τ . In the current problem, one is able to achieve a settling time around 1 ms. Hence, a value of τ = 4 × 10−4 is selected. Since it is desirable to suppress step type disturbances, the controller order will be set to 2 which translates to a third-order characteristic polynomial. Taking into consideration Manabe’s standard values for γ, the characteristic polynomial P (μ) was found, after solving the Diophantine equation, to be: P (μ) = 5.12 × 10−12 μ3 + 6.4 × 10−8 μ2 + 4 × 10−4 μ + 1

(12)

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leading to the following controller equations: E(μ) = 0.0125 A(μ) = 1.02 × 10−9 μ2 −10 2

B(μ) = 8 × 10

(13) −6

μ + 5 × 10

−2

μ + 1.25 × 10

The coefficient diagram associated with this controller design is presented in Fig. 5. Insights on the closed-loop system behaviour can be obtained by analysing the shape of the curves. The system is stable since both stability and stability limit curves do not intersect themselves and the stability curve is always above the stability limit. Moreover, since the coefficients curve is above the partial coefficient curve and both are very close, the closed-loop system immunity to eventual model mismatches is high.

Fig. 5. The coefficient diagram for the closed-loop inverter control system.

Using the above design controller, a Matlab simulation of the closed-loop system was carried out. A Simulink model of the system described in Fig. 4 was implemented. Its appearance is presented in figure Fig. 6. The simulation results have shown that the closed-loop system exhibits, besides zero steady-state error, a settling time lower than 0.8 ms and an overshoot below 1%. In addition, several simulations were performed by changing the values of the components within a range of ± 20%. The results obtained show that the overall closed-loop system performance is almost unchanged and is able to systematically track the set-point.

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Fig. 6. Simulink model for the inverter closed-loop control system.

4

Conclusion

This work presents the preliminary results regarding the controller design of a DC-AC switched converter for grid-tied photovoltaic power systems. Usually, the controller is a PD or PID but, in this paper, an alternative algebraic approach was tested. In particular the use of the CDM design method. When comparing to other controller techniques, this design strategy is friendlier and does not require a strong knowledge of control systems theory. From the obtained results, it is possible to conclude that, although simpler, the closed-loop performance regarding stability, robustness, and dynamic behaviour was attained. In the future, this design technique will be applied also to the DC-DC boost converter which, besides being a non-minimum phase system, usually requires double-loop control. Acknowledgment. This work has been supported by FCT – Funda¸ca ˜o para a Ciˆencia e Tecnologia within the Project Scope: UIDB/05757/2020.

References 1. Data taken from PORDATA, February 2020 2. Manabe, S.: Coefficient diagram method as applied to the attitude control of controlled-bias-momentum satellite. In: 13th IFAC Symposium on Automatic Control in Aerospace, pp. 322–327, September 1994 3. Manabe, S.: The coefficient diagram method. In: 14th IFAC Symposium on Automatic Control in Aerospace, pp. 199–210, August 1998 4. Manabe, S.: Coefficient diagram method in mimo application: an aerospace case study. In: Proceedings of the 16th IFAC World Congress, pp. 1961–1966 (2005) 5. Ocal, O., Bir, A., Tibken, B.: Digital design of coefficient diagram method. In: 2009 American Control Conference, pp. 2849–2854 (2009)

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6. Cahyadi, A., Isarakorn, D., Benjanarasuth, T., Ngamwiwit, J., Komine, N.: Application of coefficient diagram method for rotational inverted pendulum control. In: Control, Automation, Robotics and Vision Conference, vol. 3, pp. 1769–1773 (2004) 7. Budiyono, A., Sudiyanto, T.: An algebraic approach for the mimo control of small scale helicopter. In: Proceedings of ICIUS 2007, pp. 64–72 (2007) 8. Kongratana, V., Numsomran, A., Roengruen, P., Suksri, T., Thuengsripan, S.: Smith predictor design by cdm for temperature control system. In: Proceedings ICCAS 2007, pp. 1472–1477 (2007) 9. Kessler, C.: Ein beitrag zur theorie mehrschleifiger regulungen. Regelungst 8, 261– 266 (1960) 10. Lipatov, A.V., Sokolov, N.: Some sufficient conditions for stability and instability of continuous linear stationary systems. Autom. Remote Control 39, 1285–1291 (1979) 11. Koksal, M., Hamamci, S.E.: A program for the design of linear time invariant control systems: CDMCAD. Comput. Appl. Eng. Educ. 12–3, 165–174 (2004)

Classification of Car Parts Using Deep Neural Network Salik Ram Khanal1(B) , Eurico Vasco Amorim1,2 , and Vitor Filipe1,2 1 School of Science and Technology, Universidade de Trás-os-Montes e Alto Douro,

5000-801 Vila Real, Portugal 2 INESC TEC – Institute for Systems and Computer Engineering, Technology and Science,

4200-465 Porto, Portugal

Abstract. Quality automobile inspection is one of the critical application areas to achieve better quality at low cost and can be obtained with the advance computer vision technology. Whether for the quality inspection or the automatic assembly of automobile parts, automatic recognition of automobile parts plays an important role. In this article, vehicle parts are classified using deep neural network architecture designed based on ConvNet. The public dataset available in CompCars [1] were used to train and test a VGG16 deep learning architecture with a fully connected output layer of 8 neurons. The dataset has 20,439 RGB images of eight interior and exterior car parts taken from the front view. The dataset was first separated for training and testing purpose, and again training dataset was divided into training and validation purpose. The average accuracy of 93.75% and highest accuracy of 97.2% of individual parts recognition were obtained. The classification of car parts contributes to various applications, including car manufacturing, model verification, car inspection system, among others. Keywords: Machine vision · Quality control inspection · Automobile inspection · Deep learning · Convolutional neural network · VGG16

1 Introduction Computer vision, machine learning, and deep learning have many potential applications in various areas, including the automotive domain during manufacturing, and aftersales processes. Those include quality inspection of the vehicle, advanced driving assistance systems, autonomous driving, automatic vehicle parts detection either inside the automobile or outside [2–4]. Advance system of automatic quality inspection is fully automated, implemented modern technology to save time, money and labour, and to obtain good quality and better reliability. In general quality inspection is performed in three classes; manual or by hand, semi-automated, and fully automated [5]. One of the emerging technologies to automatize quality inspection is computer vision which can be applied to recognise various components, defects, position, and others of vehicle parts and location. With an introduction to deep learning technology, the accuracy on identification, © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 582–591, 2021. https://doi.org/10.1007/978-3-030-58653-9_56

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classification, and estimation of objects are improved dramatically. Therefore, the quality inspection process might be more accurate using deep learning which can identify objects in various stages of quality inspection during manufacturing. One of the most important tasks in the assembly process and quality inspection is to detect various automobile parts. The car parts recognition is performed at early stages of the automatic assembly process, and quality inspection so that parts can be positioned accurately, and location of defects can be detected and visualised. However, classification of car parts contributes to various applications including, car model verification, car attribute prediction, defect detection, among others [1]. The automatic vehicle parts recognition in an automobile is different from standard object detection. The location of defects in manufacturing can be identified only after identifying the parts or location. In general, manual defect detection by human inspectors is impractical, slow, expensive, and less accurate [6]. These issues can be overcome by implementing a computer vision-based non-contact inspection technique, whether for machine parts detection and defect detection [7, 8]. The presented study may help the human inspector or inspector robot used either in the assembly process or quality inspection process. Deep learning is the emerging technology with high accuracy for object identification and classification. However, only a limited study was carried out to classify vehicle parts using static images. Many studies applied deep learning to identify car parts and detection of location from whole-body images of vehicles [9–11]. Chavez-Aragon [9] proposed a vision-based detection and labelling model for multiple vehicles, but it only detects parts of the car such as front, rear, and others. Before fault detection in automatic quality inspection of a vehicle, the mandatory early stage is a vehicle part detection. For instance, to detect painting defects on the side mirror, we need to detect a mirror first. It is hypothesised that the machine parts can be classified using VGG16 with high accuracy, which is acceptable in the further application, including an automatic quality inspection.

2 Related Works Usually, the computer vision or machine vision system uses a real image to feed into the decision-making system. Zhou [12] used a computer vision system that does not use data collected from real images to teach decision-making algorithms applied in the quality assurance system. Computer-Aided Design (CAD) models were used to detect a variation of assembly lines and misplacement of components. Especially, in case of quality control or inspection system, car parts detection task is a more critical task to avoid the error [13]. Automatic defect detection often followed by vehicle part detection or parts segmentation. In literature, various approaches were presented in automatic quality inspection of an automobile using computer vision techniques. Chung [13] proposed an inspection system to automatically verify whether all the doors and windows are closed in real-time with high accuracy. The doors and windows inspection were performed separately. The algorithm consists the steps: detecting a sensor signal, identifying automobile model, selecting proper sensor to the model, detecting sensor signal, capturing four-door images, computing average value of the background, setting

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search area for each door images. Even if the study was not only focused on parts identification, identifying an automobile model is a more critical task to be done with less error. Efficient 2D computer vision-based approach to recognise machine parts and detect assembly parts in an assembly line was proposed by [14]. Images captured from moving car in assembly line and the camera fitted in various location to view the car. As an image processing procedure, they used background subtraction, binarization, scaling, noise and hole removal, and contour detection. A 1D array consists of the distance between centroid to subsequent normalised contour points. They have used a traditional approach in automatic detection of parts in the assembly line. The performance of classified was evaluated using a confusion matrix. NEC Visual Inspection System [8] enables manufacturers to accurately identify defects and dramatically reduce costs with an efficient solution that can be easily and affordably deployed. The system utilizes a deep learning one-class classification architecture that allows it to be trained using only images of non-defective components, as opposed to architectures that require images of both good and defective parts to learn the classifiers. It was already tested in more than 80 manufacturing facilities. In the case of automobile parts, it can classify to detect the defects on material parts, resin parts and fabrics and can detect the scratch, crack, dirt, dent, and burr. Chavez [9] presented a method for the visual detection of parts of interest on the outer surface of vehicles. Authors applied image processing for feature extraction and applied machine learning algorithms to detect the parts. The parts were detected from an image of the whole car. Fourteen parts (Real wheel, front wheel, mirror, rear handle, front handle, headlight, rear lamp, roof etc.) and its location were identified. A rectangle of Region of Interest represented identified parts and location. Wheels detection has 100% accuracy, whereas the mirror has only 53% of accuracy. Zelener [11] proposed a different approach to classify the vehicle parts. The vehicle parts were segmented and then classified jointly and separately. The steps carried out are local feature extraction from 3D images, Random sample consensus (RANSAC) parts segmentation, part-level segmentation, a structured model for parts in objects, and classification using general machine learning algorithms including Hidden Markova model (HMM) and Support Vector Machine (SVM). In the case of car parts identification, some literature presented a new concept of segmentation of car parts. Lu [10] presented an approach using semantic segmentation, where each pixel of an image is assigned a part. Authors segmented body, window, lights, license plates, wheels etc. Initially, landmarks are detected and apply the hierarchy of segments. One of the latest techniques in object classification, where an object is classified into subcategory based on specific features, is called fine-grained classification. It is wellapplicable in various areas, including automobile parts identification. Dia [15] proposed a fine-grained classification technique using a bilinear convolutional neural network where an object is classified based on the bilinear combination on Convolutional Neural Network (CNN). The interclass similarities were measured and analysed. Standford cars-196 car databased and Caltech-UCSD Birds-200-2011 to classify the parts of cars. Various classifies were used including SoftMax, CNN, k- nearest neighbour (KNN), etc. and obtained up to 91.92% of accuracy in car classification.

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3 Materials and Methods 3.1 Dataset Description The experiments were carried out using CompCars database [1]. The dataset contains 20,439 images of eight car parts taken from 163 car makers with 1,716 different car models, covering most of the commercial car models. All the images have an identical size of 908 × 926 pixels. The images were captured from various distances and the front direction. The images collected from various car models include four exterior parts (headlight, taillight, fog light, and air intake) and four interior parts (console, steering wheel, dashboard, and gear lever). The distribution of labelled car parts images is given in Table 1. Sample images of four exteriors and four interior parts are shown in Fig. 1. Table 1. Distribution of the label’s car part images [1] Parts

# Total

Average # per model

Headlight

3705

2.2

Taillight

3563

2.1

Fog light

3177

1.9

Air intake

3407

2.0

Console

3350

2.0

Steering wheel

3503

2.1

Dashboard

3478

2.1

Gear lever

3435

2.0

3.2 Proposed CNN Architecture A deep neural network was designed to classify the image car parts based on the wellfamous deep learning architecture VGG16 proposed by [16]. The VGG16 architecture was based on ConvNet contains nineteen deep layers with 16 convolutional layers and three fully connected layers including output layer. Input Layer [224 × 224 × 3]: holds the raw pixel values of RGB images of the car parts. This architecture contains various layers, including convolutional layer, max-pooling, fully connected, and SoftMax. A stack of convolutional layers is followed by three fully connected layers or dense layers among them first two layers are dense layer with 4096 neurons and one output layer with 8 neurons respectively. The SoftMax is also used as out loss function. The convolutional layer consists small filter of size 3 × 3. The overall architecture is shown in Fig. 2. The first convolutional layer consists of 64 3 × 3 filters, second one has 128 3 × 3 filters, third one has 256 3 × 3, forth one has 512 3 × 3 filters, and last one also has 512 3 × 3 filters. In all the hidden layers, a stride size of 1, batch normalization, max-pooling of size 2 × 2, and drop out of 0.6 and ReLU as the activation function were used.

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a. Exterior parts

b. Interior parts Fig. 1. Car parts from the database. a. Exterior parts contains headlight, taillight, fog light, and aircondition (from left) and b. Interior parts contains four internal parts console, steering, dashboard, and gear lever (from left) [1].

Fig. 2. Proposed Convolutional Neural Network Architecture based on VGG16.

We implemented this architecture in a well-known python library Keras. The architecture was implemented in GeForce GTX 1080 GPU with dedicated video memory of 8192 MB GDDR5X in windows 10 Enterprise, 64-bit, with the quart core CPU with Intel (R) Core™ i7-6700 K CPU @ 4.00 GHz. The training was performed with 50 epochs with the batch size of 32. We have an image set of 20,439 of eight car parts. Top label classification was carried out even if the database was proposed for fine-grained classification with a higher value of classification accuracy. The training and testing dataset were split before training and store in a separate folder. Based on the file provided, the complete dataset was spitted into training and testing dataset. The partition results in a training set containing 11,104 images and testing set contains 9,335 images. From training dataset, 20% of images were used for validation purpose during training the architecture. The number of images in each class may be slightly different.

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3.3 Experimental Setup The experimental setup of car part classification using deep learning architecture is shown in Fig. 3. In the training phase, the weights of proposed deep learning architecture are calculated and saved, so that it can be used during the testing phase. The proposed system can be applied in real-time where images can be acquired using a fixed camera, mobile camera, or head-mounted camera, which depends on the purpose of classification. The results of classification can be interpreted in various ways such as object (parts) detection, car parts validation, model verification.

Fig. 3. Flowchart of car part classification using deep learning architecture.

4 Experimental Results Before feeding neural network architecture, the images were resized to 224 × 224 × 3 to prepare the input parameter for the deep network architecture. The training and validation loss history, and training and validation accuracy during training 50 epoch are shown in Fig. 4(a) and 4(b) respectively. The green-blue curve indicates training and red curve indicates validation. The performance of the classifier is visualised using a confusion matrix. The accuracy of each category of car parts is shown using a confusion matrix, as shown in Fig. 5. The average accuracy on the testing dataset was 95.75 with the loss 0.22. The best accuracy was on steering wheels, that means this part is easier to classify. It is shown that the miss-classification of parts is with a similar category that means light is miss-classified to other types of light. Let us say, headlights are more miss-classified to taillight not to other parts like a dashboard, handle etc. The least accurate classification was on fog lights. From the result, it is shown that 7.3 per cent of fog lights are miss-classified to a headlight. Console has the most accurate classification result with accuracy of 97.2% of correct prediction.

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Fig. 4. Training and validation loss (a) and accuracy (b) vs. epoch.

5 Discussion Identification of parts of automobiles reveals to be a key idea to detect them with better accuracy, but also to automatic vehicle assembly and for quality inspection. In automatic quality inspection and fault detection, identification of vehicle parts plays a crucial role. It is an era of automation, including industry application. For the better accuracy with a smaller number of employees, camera-based automatic assembly and automatic quality inspection, as well as automatic fault detection, is implemented with an application of computer vision, machine learning, and deep learning techniques [4, 6, 17, 18]. Using deep learning to detect the parts of vehicles will be the open-door technology in these

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Fig. 5. Confusion matrix for four class car parts classification (0-Headline, 1-Traillight, 2Foglight, 3-Air intake, 4-Console, 5-Steering Wheel, 6-Dashboard, 7-Gear lever).

areas. Using the proposed architecture to implement in a real-time application, the accuracy of classification is acceptable. This approach can be applied in different ways, including mobile tab, fixed camera, attached camera, among others [19]. In this paper, the classification of vehicle parts was carried out with the accuracy better than state of the art studies. In the dataset verification, authors exhibit car model identification using car parts and got very poor results in top-1 accuracy [1]. However, the purpose of the study was not similar. The purpose of classification was to identify the car model from car parts identification and the purpose of presented study is only to classify the parts. We obtain high accuracy up to 97.2% of accuracy in car part identification which indicates that it is reliable to apply this technique for various purposes including quality inspection, automatic car assembly and so forth. Various vehicle company used camera-based automatic fault detection, whether internal or external car parts [12]. Let us suppose that we can inspect the mirror whether painted or not, in this case, before detecting the faulty, we must detect the mirror from the background image (which includes car itself). So, detection or car part identification will further be implemented in fault detection. Likewise, in the automatic assembly of car parts using a robot, before locating the part to the baseline, the part and its location

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must be identified; therefore, accurate vehicle part identification plays a vital role in robotics. Study has a few limitations. The dataset has only around 11,000 images which are not well enough for training the architecture. A few higher numbers of images may result in better accuracy. The study can only classify eight parts. The images used for the experiments were capture in good illumination condition; the result may vary with illumination condition. The deep neural network architecture was designed only using VGG16. Other deep learning architecture might exhibit better result.

6 Conclusion and Future Work Using a deep neural network to classify the car parts reveals to be a key technology to achieve more accurate fault detection and quality inspection. The average accuracy of 93.96% and highest accuracy of 97.2% of individual part recognition were obtained. The results obtained from the experiments indicate that parts classification using VGG16 has acceptable results and the techniques have a vast area of application including automatic assembly, fault detection, and automatic quality inspection before lunch in the market. The dataset covers a wide range of car models; therefore, the result can be generalising for all the car models. This work can be extended to identify the parts from the video so that the operator can easily identify the parts and location whether the operator is human or robot. The experiments were performed only using car parts so that this work can be expended to experiments with various types of vehicles such as ambulance, truck, busses, and others. The deep learning architecture was designed only with VGG16, but other architectures can be adopted and compared to find better accuracy. Acknowledgement. This work was funded by Project “INDTECH 4.0 – New Technologies for smart manufacturing”, no. POCI- 01-0247-FEDER-026653, financed by the European Regional Development Fund (ERDF), through the COMPETE 2020 - Competitiveness and Internationalization Operational Program (POCI).

References 1. Yang, L., Luo, P., Loy, C.C., Tang, X.: A large-scale car dataset for fine-grained categorization and verification. cs.CV, pp. 1–11 (2015) 2. Manual, Semi-Automated or Automated: What Type of Assembly System is Right for You? 29 April 2019. http://www.invotec.com/news/manual-semi-automated-or-automated-whattype-of-assembly-system-is-right-for-you/ 3. Simonyan, K., Zisserman, A.: Very deep convolutionl networks for large-scale image recognition. cs.CV, pp. 1–14 (2015) 4. Chauhan, V., Surgenor, B.: Fault detection and classification in automated assembly machines using machine vision. Int. J. Adv. Manuf. Technol. 90(9–12), 2491–2512 (2017) 5. Zhou, H., Xu, H., He, P., Song, Z., Geng, C.: Automatic inspection of LED indicators on automobile meters based on a seeded region growing algorithm. J. Zhejiang Univ.-SCIENCE C (Comput. Electron.) 11(3), 199–205 (2010)

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6. Artificial Intelligence (AI), deep learning that automates factory defect inspection, NEC, 29 April 2019. https://www.necam.com/AI/DefectInspection/. Accessed 28 Feb 2019 7. Chavez-Aragon, A., Laganiere, R., Payeur, P.: Vision-based detection and labelling of multiple vehicle parts. In: 14th International IEEE Conference on Intelligent Transportant Systems, Washington, DC, USA (2011) 8. Lu, W., Lian, X., Yuille, A.: Parsing Semantic Parts of Cars Using Graphical Models and Segment Approach Consistency, Center for Brains, Minds and Machines, pp. 1–12 (2018) 9. Zelener, A., Mordohai, P., Stamos, I.: Classification of vehicle parts in unstructured 3D point clouds. In: 2014 2nd International Conference on 3D Vision, Tokyo, Japan (2014) 10. Chung, Y.K.K., Kim, H.: Automated visual inspection system of automobile doors and windows using the adaptive feature extraction. In: Second International Conference. KnowledgeBased Intelligent Electronic Systems. Proceedings (KES 1998) (Cat. No. 98EX111), Adelaide, SA, Australia (1998) 11. Campos, M., Martins, T., Ferreira, M., Santos, C.: Detection of defects in automotive metal components through computer vision. In: IEEE International Symposium on Industrial Electronics, Cambridge, UK (2008) 12. Zhou, Q., Chen, R., Huang, B., Liu, C., Yu, J., Yu, X.: An automatic surface defect inspection system for automobiles using machine vision methods. Sensor (Basel) 19(3), E644 (2019) 13. Prasad, P.B., Radhakrishnan, N., Bharathi, S.S.: Machine vision solutions in automotive industry. Soft Comput. Tech. Eng. Appl. 543, 1–14 (2014) 14. Arjun, P., Mirnalinee, T.T.: Machine parts recognition and defect detection in automated assembly systems using computer vision techniques. Rev. Téc. Ing. Univ. Zulia 39(1), 71–80 (2016) 15. Piero, N., Schmitt, M.: Virtual commissioning of camera-based quality assurance systems for mixed model assembly lines. Procedia Manuf. 11, 914–921 (2017) 16. Pei, Z., Chen, L.: Welding Component Identification and Solder Joint Inspection of Automobile Door Panel Based on Machine Vision. In: The 30th Chinese Control and Decision Conference, Shenyang, China (2018) 17. Essid, O., Laga, H., Samir, C.: Automatic detection and classification of manufacturing defects in metal boxes using deep neural networks. PLoS ONE 13(11), e0203192 (2018) 18. Dai, X., Gong, S., Zhong, S., Bao, Z.: Bilinear CNN model for fine-grained classification based on subcategory-similarity measurements. Appl. Sci. 301(9), 1–16 (2019) 19. Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. cs.CV, pp. 1–14 (2014)

Soiling Monitoring Modelling for Photovoltaic System Vitor H. Pagani1,2(B) , Nelson A. Los1,2 , Wellington Maidana1 , Paulo Leit˜ ao1 , 2 2 Marcio M. Casaro , and Claudinor B. Nascimento 1

Research Center in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal {vitorpagani,nelsonlos}@alunos.utfpr.edu.br, {maidana,pleitao}@ipb.pt 2 Universidade Tecnol´ ogica Federal do Paran´ a, Campus de Ponta Grossa, Ponta Grossa, PR, Brazil {casaro,claudionor}@utfpr.edu.br

Abstract. Soiling on photovoltaic panels is a factor that has a significant impact on the photovoltaic production. The monitoring of the soiling index appears as a relevant alternative for the maintenance of solar systems. This work proposes a soiling index modelling for photovoltaic systems based on two input variables, namely the solar radiation and the generated current, providing a simple, programmable and reliable way to check the efficiency and be able to establish the parameters for cleaning the system. The study was based on the adaptation of an existing mathematical modelling, that besides to estimate the soiling index also allows to establish an optimal point for cleaning. The proposed model is compared with the results provided by the PVSyst software aiming its validation. The achieved results show that, despite the developed system only consider two input variables, it presents a low relative error, i.e. 2.07%, when compared to the PVSyst software, allowing to conclude that the proposed modelling system is valid and presents excellent reliability, having a vast applicability in the monitoring of solar producers of any model or size. Keywords: Soiling · Photovoltaic systems Maintenance management.

1

· Solar panels · Modeling ·

Introduction

The growing global energy demand has been a significant challenge for authorities today. Environmental commitments to reduce greenhouse gases are being made, leading many countries to invest in other energy alternatives [1], mainly from sources that aim to reduce and prevent the impacts caused by conventional energy sources, e.g., coal, natural gas and fossil co-generation, as reported by the International Energy Agency (IEA) [2]. In this scenario, the implementation of c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 592–601, 2021. https://doi.org/10.1007/978-3-030-58653-9_57

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photovoltaic (PV) systems appears as a promising alternative that aims to meet this growing need, since it derives a renewable source and its form of adaptation, with the implementation of a solar tracker, increases the generation of energy. According to the International Renewable Energy Agency (IRENA) [3], the PV power in the world was 22,816.0 MW in 2009, rising to 480,619.0 MW in 2018. According to the Portuguese Renewable Energy Association (APREN), Portugal data for these periods were 110 MW in 2009, rising to 673 MW in 2018. However, similarly to other energy sources, PV has its limitations and problems, even with the implementation of a solar tracker, which significantly increases the system energy generation [4]. The accumulation of dust on the surface of the panels shows a decrease in the power generation of the modules, and this problem may vary in different regions [5]. The determination of a module soiling is a very relevant problem to this source of energy. According to [6], the current can suffer a reduction of 2.78% per day due to the dust accumulation, where it represents the impact caused by the accumulation of debris in the generation of PV energy [7]. A simplified single-axis solar tracker prototype was developed, which has a soiling monitoring system that only considers the solar irradiation and the output current as data source, which may contribute to the increase of the power generation by PV modules. This paper describes the development of a soiling index modelling that also only uses the solar irradiation and the output current as input data, but take into account the cost/benefit ratio. From the application of a soiling index calculation model, it is possible to improve the cleaning cycles, avoiding the late cleaning and the loss of electricity production, since in different regions the dirt indices vary, which makes difficult to stabilise a cleaning schedule. The application of the soiling index modelling was based on the meteorological data from Bragan¸ca, Portugal, where the study was conducted. The rest of this paper is organised as follows: Sect. 2 presents the related work and Sect. 3 presents the modelling methodology for the soiling index. Section 4 discusses the obtained results, and finally, Sect. 5 rounds up the work with conclusions, contributions and limitations, and points out the future work.

2

Related Work

This section analyses the different solar tracker models, as well as the reasons for selecting the target model used in this work. Additionally, it also addresses the impact of dirt on the electrical generation of the PV system. 2.1

Solar Tracker

The solar tracker is a technology that allows to monitor the path of the sun during the period of exposure of the PV plates to the solar irradiation, in order to increase the absorption of light irradiation, consequently increasing the electrical generation. The cost of implementing this technology provides an increase of

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approximately 20% in the total cost of the project and may generate an increase in energy production of 40% [4]. Solar trackers can be classified into two broad groups, those with one axis and two axes. The trackers with double axes represent 42.57% and single-axis 41.58% of searches [8], showing a profound difference in the interest between the two models. Among the two large groups, the literature still divides the models according to the form of screening, namely azimuth/elevation for the double axis, and azimuth, horizontal and polar for the single-axis. The differences regarding the presented models are related to the number of actuators that move their structure and the shape of the structure. For the single-axis trackers, the azimuth model presents the pile structure. In contrast, the polar and horizontal models present a roll-tilt structure, differentiating with each other due to the existence of an inclination of the axis in the polar model, which makes it more efficient in the generation in places with more considerable latitude. The determining factors in choosing the tracker model are the geographic location, the cost of implementation and the increase in the generation. Comparing the different types of solar tracking, the North-South inclined single-axis tracker had an absorption of 93.2% in relation to the dual-axis tracker [9]. A study carried out in Changdu / China, that is located at approximately 31◦ of latitude, demonstrated a close performance, with a more simplified structure and with less investment, justifying the decision for this model, i.e. a single polar axis, as the target of the application of this study. 2.2

Soiling Impact

Among the external factors that impact the generation of the PV energy, the accumulation of debris is one of the most important to be considered [5,10], as shown in Fig. 1. The impact caused by the accumulation of debris can be observed as a progressive loss of power, due to the shading caused by the dirt [5], presenting values of loss by soiling between 40–70% for an exposure of the system for 6 months, in environments with a very dry climate and with low incidence of rain. However, in contrast, it was observed that more humid climates could end up increasing the adhesion of dirt on the surface of the modules, which must be taken into account when cleaning the modules, in order to clean more carefully and not end up damaging the system [11]. The difference between the types of debris that affect the plates surface is another relevant factor analysed in the literature. The influence of the sizes and composition of the dirt can cause different impacts on the modules [11]. Among the impacts treated in this paper is the loss of light transmittance to the cells depending on the size of the debris, as the slope of the system can help to reduce the dirt, making possible to escape the dirt without the cleaning intervention. Modelling the impact of the dirt is a way to identify this drop in the PV generation and to improve the system’s operating conditions. According to the analysed literature, the models where the level of dirt was modelled based on climatic factors [13,14] presented the best results [5], but other models use the behaviour of the system as the base [15].

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Fig. 1. Soiling impact on energy generation [12].

The analysed models have very complex monitoring forms, which may end up making them challenging to implement, possibly because they have a high implementation cost. Moreover, the number of sensors required and the need for big data processing are other factors, as they require to use artificial neural networks (ANN) and very complex calculations, making these methods economically unviable [5]. The objective of this paper is to develop a low-cost monitoring system for the soiling index on the PV system, avoiding the use of sophisticated analysis tools, such as neural networks, and avoiding the use of many input variables as presented in [13–15]. In this way, allowing the application of the system with fewer sensors, using a microcontroller system with less processing power, therefore with a less cost, and presenting output current values close to the simulated database, with a low error level. Such solution is useful, reliable and low-cost, making this technology easier to be introduced in the market.

3

Methodology

The experimentation model is based on the values obtained from the modelling presented in [16,17] and applied to the PVSyst software. In order to calculate the generated current (Iout ) and to estimate the level of soiling existing on the PV system, the methodology follows the diagram shown in Fig. 2. 3.1

Model Calibration

The soiling monitoring uses as input parameters in the system modelling, the specifications provided by the manufacturer of the modules, to obtain the best result for each specific case. The necessary constants for the system calibration are: Voc - module open circuit voltage (V), Vmp - maximum module power voltage (V), Isc - Module

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Fig. 2. Methodology flowchart of modelling’s system.

short-circuit current (A), Imp - maximum module power current (A), α - temperature coefficient of the current change in the insolation reference (A/◦ C ), β - temperature coefficient of voltage change in the insolation reference (V/◦ C), Rs - module series resistance (Ω), Sref - insolation reference (1000 W/m2) and Tref - reference temperature (25 ◦ C). These data are provided by the tests carried out by the manufacturers and are available in the datasheet of the photovoltaic plates and must be updated for each case of application of the study according to the module used. 3.2

Real-Time Data Acquisition

Considering the system modelling developed by [16], it was found that the following equations can define the behaviour of the PVS: Vout −ΔV   Iout = Isc ∗ 1 − C1 ∗ (e C2∗Voc − 1) + ΔI

(1)

V

C2 =

( Vmp )−1 oc ln(1 −

C1 = (1 −

Imp Isc )

−Vmp Imp ) ∗ e C2∗Voc Isc

(2)

(3)

The Eqs. 1, 2 and 3 show the behaviour of the generated current (Iout ) before the circuit modelling system. This behaviour is related to the specific constants of the module model, other three system variables, namely temperature (T), irradiation (S) and output voltage (Vout ), and also with ΔI and ΔV explained in Eqs. 4 and 5.

Soiling Monitoring Modelling for Photovoltaic System

ΔI = α ∗

S Sref

∗ ΔT + [

S Sref

− 1] ∗ Isc

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(4)

By ΔI, we observe the influence suffered by the Iout caused by the variation of the temperature and the solar irradiation, and if the α value is minimal, the temperature has a reduced impact in the variation of the current. In addition, the variation of the voltage in relation to the temperature and ΔI is observed by the Eq. 5. However, contrary to the current behaviour, the Vout suffers a small decrease with temperature. ΔV = −β ∗ ΔT − Rs ∗ ΔI

(5)

However, this system focuses on a simplified and less costly implementation, being decided to decrease the number of necessary sensors. After analyzing the effect of dirt on some systems, demonstrated in their results that the most significant impact of the accumulation of debris is on the generated current, as the debris limits the amount of irradiation absorbed by photocells [18,19]. Based on these studies and our goal of reducing the price and complexity, it was decided to decrease the number of input variables, considering the output voltage (Vout ) at Eq. 1 constant and equal to the Vmp value provided by the manufacturer. To simplify the analysis, the impact of the photovoltaic cell temperature on the output current was observed by the ratio of coefficients presented by the manufacturer, where α presents the effect of temperature on the current and β the impact of temperature on the voltage, in the model of the module chosen for this study: α = 5.0 mA/C and β = −163 mV/C. Observing the low proportional impact of the temperature in Iout , it was decided to work only with the irradiation relation with the current to determine the soiling index present in the system. Hence, discarding the need for a thermal sensor in the application, it was decided that T in Eq. 6 is equal to Tref . ΔT = T − Tref → T = Tref → ΔT = 0 3.3

(6)

Calculus of the Soiling Index

After the validation, made by comparing the Iout determined by the Eq. 1 applied to the programming, with the current from the PVSyst software [20], it was possible to calculate the soiling index of the module. For such, the Iout was used for modelling the relation to the radiation measured by the sensor, and the current generated by the system (Iact ), measured by an ammeter sensor. From a direct relation expressed in Eq. 7, it is possible to check the value of the soiling index present in the system by comparing the current value that should be generated if the system is clean (Iout ), with the current value measured in the actual state (Iact ). Soiling% =

Iout − Iact ∗ 100 Iout

(7)

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The value obtained when implemented in the real system, can be used to monitor the level of accumulated soiling index of the modules.

4

Analysis of Results

The proposed methodology was applied to a case study, and the achieved results compared with the ones generated by the PVSyst software [20], which served as a database for validating the methodology. 4.1

Soiling Index Modelling Application

The designed PV system considers the existing models in the literature [13–15] and the mathematical formulations for the electronic model of a PV module [16,17], allowing to obtain the real values of PV modules to be used in the methodology. The PV case study contains six PV modules with a total generation capacity of 1.5kW. After collecting the data from the PV system, it was simulated, obtaining the values that made up the database. The entire data collected was used to apply the proposed methodology. According to the proposed model, the data obtained from the simulation allowed to perform a comparison with the generated values, as shown in Fig. 3, noticing a similarity between the database and the modelled values.

Fig. 3. Comparison between some simulated and modelled values

Figure 4 represents the relative error between the database values and the data generated by the modelling approach.

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Fig. 4. Relative error between simulated and modeled values

From the behaviour observed in Fig. 3, it was possible to identify the current decay related to the increased impact of dirt on the PV modules, demonstrating a direct relationship between the current and the soiling index, due to the drop absorption of solar radiation. The decay analysed earlier shows a pattern in the current drop, which can also be analysed in the graph presented in Fig. 4, showing how this pattern is maintained for the most usual tilt angle. According to a comparative analysis between the achieved results, it is possible to identify that the error remained constant for the most of analysis, presenting a variation only in the values of greater inclination, and due to this variation the errors range between 0.33% to 2.48%. These values have a consistency to be taken into account when compared to existing studies used as a basis for carrying out this research [13–15]. With all the obtained results, it is possible to add the weather information and create a database, which may come to build a new model in order to predict the dirtiness index for the studied region. From the changes of the climate with the variation of the soiling index, it is possible to discover the pattern of the incidence of dirt on the modules, making the creation of cleaning cycles even easier. As can be seen in Fig. 4, for an angle of 90◦ inclination and with a soiling index of 40%, it was possible to notice an unusual variation compared with the others, an error that may come from the analysis at very low values, where a small difference can raise this error unexpectedly as seen, but remaining within the values presented by the others.

5

Conclusions

This paper proposes a soiling index modelling for photovoltaic systems, that only consider two input variables, the solar radiation and the generated current. The

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PVsyst software was used to generate the database, and these data were used to test the model, obtaining an average relative error of 2.04%, a good result when compared with the other empirical studies found in the existing literature. As it was a modelling based on simulated data, an actual application was not performed, being the most significant limitation of the work. Another limitation is that the results and what was observed from them was all obtained from computational simulation. Thus, not being from a real application, it limits the data and the interpretations. For future work, the practical application of this modelling to a real system is desirable in order to make a comparison between the values obtained in theory and to be able to validate them in practice. Additionally, along with the experimental validation, it is recommended to study the system’s behaviour in comparison with the climate in which the system is exposed. This will allow a database to be provided so that the system is only a real-time index model, and can perform the soiling index forecast according to the climate changes around it. Acknowledgment. This work has been supported by FCT- Funda¸ca ˜o para a Ciˆencia e Tecnologia within the Project Scope: UIDB/05757/2020.

References 1. European Commission, “Novel carbon capture and utilisation technologies.” https://ec.europa.eu/research/sam/pdf/sam ccu report.pdf#view=fit&pagemode =none. Accessed June 2019 2. IEA, “World energy outlook 2019.” https://www.iea.org/reports/world-energyoutlook-2019 3. IRENA, “Renewable energy statistics 2019.” https://www.irena.org/publications/ 2019/Jul/Renewable-energy-statistics-2019 4. Gil, F.G., Martin, M., Vara, J.P., Calvo, J.R., Perlovsky, L., Dionysiou, D., Zadeh, L., Kostic, M., Gonzales-Concepcion, C., Jaberg, H., et al.: A review of solar tracker patents in Spain. In: Proceedings of the Energy Problems and Environmental Engineering, pp. 292–297 (2009) 5. Said, S.A., Hassan, G., Walwil, H.M., Al-Aqeeli, N.: The effect of environmental factors and dust accumulation on photovoltaic modules and dust-accumulation mitigation strategies. Renew. Sustain. Energy Rev. 82, 743–760 (2018) 6. Ibrahim, A., et al.: Effect of shadow and dust on the performance of silicon solar cell. J. Basic Appl. Sci. Res. 1(3), 222–230 (2011) 7. Mani, M., Pillai, R.: Impact of dust on solar photovoltaic (PV) performance: research status, challenges and recommendations. Renew. Sustain. Energy Rev. 14(9), 3124–3131 (2010) 8. Hafez, A., Yousef, A., Harag, N.: Solar tracking systems: technologies and trackers drive types-a review. Renew. Sustain. Energy Rev. 91, 754–782 (2018) 9. Wu, J., Hou, H., Yang, Y.: Annual economic performance of a solar-aided 600 MW coal-fired power generation system under different tracking modes, aperture areas, and storage capacities. Appl. Therm. Eng. 104, 319–332 (2016)

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10. Kishor, N., Villalva, M.G., Mohanty, S.R., Ruppert, E.: Modeling of PV module with consideration of environmental factors. In: IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT Europe), pp. 1–5. IEEE (2010) 11. Adinoyi, M.J., Said, S.A.: Effect of dust accumulation on the power outputs of solar photovoltaic modules. Renew. Energy 60, 633–636 (2013) 12. Vaishak, S., Bhale, P.V.: Effect of dust deposition on performance characteristics of a refrigerant based photovoltaic/thermal system. Sustain. Energy Technol. Assess. 36, 100548 (2019) 13. Javed, W., Guo, B., Figgis, B.: Modeling of photovoltaic soiling loss as a function of environmental variables. Solar Energy 157, 397–407 (2017) 14. Laarabi, B., Tzuc, O.M., Dahlioui, D., Bassam, A., Flota-Ba˜ nuelos, M., Barhdadi, A.: Artificial neural network modeling and sensitivity analysis for soiling effects on photovoltaic panels in morocco. Superlattices Microstruct. 127, 139–150 (2019) 15. Hammad, B., Al-Abed, M., Al-Ghandoor, A., Al-Sardeah, A., Al-Bashir, A.: Modeling and analysis of dust and temperature effects on photovoltaic systems’ performance and optimal cleaning frequency: Jordan case study. Renew. Sustain. Energy Rev. 82, 2218–2234 (2018) 16. Rauschenbach, H.S.: Solar Cell Array Design Handbook: The Principles and Technology of Photovoltaic Energy Conversion. Springer Science & Business Media, New York (2012) 17. Salameh, Z.M., Borowy, B.S., Amin, A.R.: Photovoltaic module-site matching based on the capacity factors. IEEE Trans. Energy Convers. 10(2), 326–332 (1995) 18. Kumar, E.S., Sarkar, B., Behera, D.: Soiling and dust impact on the efficiency and the maximum power point in the photovoltaic modules. Int. J. Eng. Res. Technol. (IJERT) 2(2), 1–8 (2013) 19. Tripathi, A.K., Aruna, M., Murthy, C.S.: Output power loss of photovoltaic panel due to dust and temperature. Int. J. Renew. Energy Res. 7(1), 439–442 (2017) 20. V.M. Mermoud, A.: “Pvsyst.” https://www.pvsyst.com/help/

Vision-Based Object Detection and Localization for Autonomous Airborne Payload Delivery James Sewell1 , Theo van Niekerk1(B) , Russell Phillips1 , Paul Mooney1 , and Riaan Stopforth2 1 Nelson Mandela University, Port Elizabeth, Eastern Cape, South Africa

[email protected] 2 University of KwaZulu–Natal, Durban, KwaZulu–Natal, South Africa

Abstract. This paper follows the development of a vision–based object detection and localization system for implementation in an autonomous aircraft for payload delivery. Application of such a system could see use in the delivery of packages to offshore freighter ships and inaccessible inland areas. This system was developed in a modular fashion, such that it could be interchanged and adapted between various airframes. This system comprised of three core elements, namely autonomous flight control, vision–based object detection and localization and, payload release and delivery modelling. The final integrated system was tested and able to achieve fully autonomous flight, and simultaneously model a payload release trajectory from an altitude of 75 m to deliver the given payload with an average displacement of 1.8 m of the designated drop-zone. The drop-zone location was determined via the onboard vision system through the implementation of an object detection and localization algorithm. Keywords: MAVROS · ROS · Pixhawk · Autonomous · Computer vision

1 Introduction Package delivery forms an essential part of everyday life, from the delivery of medical supplies to the delivery of goods and services. With today’s ever evolving technology, the efficiency and execution of a delivery system has become the focus of optimization with competing markets, with rapid transportation of goods and services being the status quo. The introduction of modern technologies has also brought about a largely competitive market for innovative ways to transport goods and services effectively. Although this market place is no stranger to new forms of innovation, it has not necessarily been exposed to all viable solutions. As such, the potential to introduce a cost-effective mode of delivery that meets all other offering criteria is always an option worth exploring. This paper follows the development of a vision-based object detection and localization system for implementation in an autonomous aircraft payload delivery system. Where, this integrated system looked at bridging the cost-effective delivery system gap in the delivery market. The system utilized an onboard companion computer Pixhawk © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 602–615, 2021. https://doi.org/10.1007/978-3-030-58653-9_58

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flight controller and a webcam-based computer vision system to detect, authenticate and localize a desired drop-zone location for a given payload. The aircraft was automated from takeoff to mission to landing and was only provided with an approximate location of the desired drop-zone. This uncertainty in the desired drop-zone’s location defined the role of the vision-based object detection system, where implementations of such a system in such a scenario could be viable in search and rescue missions. With the approximate drop-zone location, the onboard computer was able to plot a mission which would include an autonomous survey of the area surrounding the given object’s approximate location. Upon identification of the desired drop-zone location, the onboard computer would update the current mission waypoints to include the, now precise, drop-zone location. With the precise location of the drop-zone now known, the aircraft could calculate an appropriate flightpath in order to achieve successful payload delivery. Through the implementation of this vision-based object detection and identification system, payload delivery was achieved within an average of 1.82 m of the desired dropzone location from an altitude of 75 m. This result was further complimented by the performance of the computer vision system, as object detection and localization was achieved within an average accuracy of 1.36 m of the true drop-zone location from an altitude of 75 m.

2 Background Autonomous package delivery has made the headlines in recent years, with Amazon Prime Air being one of many systems which exhibit these features and hopes to see full deployment in years to come. Another company which has introduced autonomous package delivery, but in the medical field, is Zipline. Zipline is an American-based company which operates in several countries, including Tanzania and Rwanda. Zipline specializes in the delivery of medical supplies to both remote and urban areas where access to the necessary medical supplies is either several hours away or completely inaccessible. Zipline’s system implements a custom fixed wing aircraft which flies to the designated delivery coordinates and releases the payload package to land within 10 m of the designated drop-zone [1]. The difference between Zipline and this research is that this research utilizes active drop-zone tracking and localization via the onboard vision system, whereas Zipline’s payload delivery is done based on the GPS coordinates provided by the recipient. This use of active tracking means that this research’s system is able to implement precise drop-zone localization and recipient authentication. Figure 1 illustrates the Zipline aircraft during takeoff [2].

3 System Architecture To achieve the desired testing and results, a mechatronic system was developed in order to integrate the various subsystems into a functional test platform. 3.1 Hardware Architecture As this system was designed for aerial use, a remote control (RC) fixed wing aircraft, the Skywalker 2013, was selected. Due to the use of nonconventional hardware for the

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Fig. 1. Zipline aircraft during takeoff

onboard computing and vision processing, several structural alterations were made to the aircraft. In order to introduce autonomous flight control into the aircraft, the Pixhawk flight control unit (FCU) was utilized. The significance of the Pixhawk was its functionality to enable onboard companion computer use. This onboard companion computer, the ODroid-XU4, was responsible for all onboard processing and computation, and due to its compatibility with the Pixhawk, flight path alterations. Other than additional flight diagnostic monitoring sensors, the final components introduced into this mechatronic system were the onboard camera, an ELP-USB500W04AF-A60, and an image stabilization camera gimbal [3]. Figure 2 illustrates the integrated hardware architecture and Fig. 3 depicts the aircraft after mechatronic integration and alterations.

Fig. 2. Integrated hardware architecture

3.2 Hardware Architecture In order to achieve sufficient computation times and to enable near real-time multiprocessing without the use of accelerated hardware, the Robotic Operating System (ROS)

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Fig. 3. Aircraft used in the development of this research

and the Micro Air Vehicle ROS package (MAVROS) were utilized onboard the ODroid XU4 [4]. These middleware packages were defining elements within this system as they allowed for near simultaneous processing to occur and enabled the use of the MAVLink protocol to communicate with the Pixhawk in order to receive and update mission specific parameters. The ROS network developed for this system consisted of five nodes, four of which were slave nodes to the master MAVROS node. The master MAVROS node was responsible for ensuring sustained communication between the ODroid and the Pixhawk, in addition to hosting the four remaining nodes. The four remaining nodes include the image capture node, the image processing node, the autonomous flight node and the payload release node. The nodes were written in the C++ coding language and the Open Source Computer Vision (OpenCV) library was used in order to achieve the necessary image processing capabilities. Figure 4 illustrates the interconnectivity of these nodes.

Fig. 4. ROS node network

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4 Vision System As mentioned in the system architecture, the ELP USB500W04AF A60 camera was utilized for image capture onboard the aircraft. In order to reduce the risk of hardware damage, the camera and gimbal were mounted inside the aircraft’s fuselage in a downward facing orientation and a viewing window was cut into the underside of the aircraft’s fuselage. 4.1 Ground Plane to Image Plane Relationship Due to the downward facing orientation of the vision system, image capture onboard the aircraft could be taken in two-dimensions and the displacement between the image plane and ground plane was given by the altitude of the aircraft. For reference, the image plane refers to the image displayed by the camera and the ground plane represents the ground or surface the aircraft is flying above and visualizing through its vision system. The fixed camera orientation and known altitude meant that ground plane displacements between the image plane origin and the detected object’s location within the image plane could be translated from pixel coordinates to ground plane displacements based on camera resolution and field of view (FOV). Figure 5 illustrates the various reference frames used in the implementation of this vision system.

Fig. 5. Vision system reference frames

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4.2 Drop-Zone Identifier With the relationship between the ground plane and the image plane established, image processing could be undertaken without concern for image orientation offsets. As the function of this vision system was to identify a designated drop-zone location, a specific drop-zone identifier was defined in order to demarcate and differentiate these drop-zone locations from their surroundings. As this research was undertaken to provide a proof of concept for future development, the drop-zone identifier was defined to be a red circle, 0.5 m in diameter. The colour and shape of this drop-zone identifier was critical to the function of the image processing algorithm. As such, image processing was undertaken in two distinct phases, namely image colour thresholding and geometry detection. 4.3 Image Processing – Colour Thresholding With the colour of the drop-zone identifier known, colour thresholding could be implemented. The image stream from the onboard camera was converted from the BGR (Blue, Green, Red) colourspace to the HSV (Hue, Saturation, Value) colourspace. The distinct variation in these colourspaces is due to the three-dimensional representation of each colourspace, where the BGR colourspace is defined to be a cube and the HSV colourspace is defined to be a cone or in some cases a cylinder. As such, converting the image stream to the HSV colourspace broadens the range within which colours can be filtered as only two values, an upper and lower hue range, are necessary to encapsulate variations in the appearance of the drop-zone identifier during different testing environments. Figure 6 illustrates these colourspace geometries [5].

Fig. 6. Colourspace representation for BGR (left) and HSV (right)

From converting the image stream into the HSV colourspace, the upper and lower hue images were then combined into a single equally weighted binary image. Elements within the original image which met the hue range filtering criteria were displayed as white pixels and all other colours would be displayed as black pixels. In this case, red objects were converted into white pixels. This isolation of the colour range of the drop-zone identifier into a binary image allowed for optimized object detection, as the algorithm would not waste time processing elements within the binary image that were not white pixels. In addition to colour thresholding, a Gaussian blur was introduced to the binary image in order to reduce noise. This blur reduced the number of false positives for the drop-zone identifier detected by the object detection algorithm by forcing the algorithm to refine its search for the correct object geometry [6].

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4.4 Object Detection – Shape Geometry With the streamed image now represented as a binary image, object detection was undertaken through the implementation of OpenCV’s Hough Circle Transform algorithm [7]. As such, all circles fitting the colour threshold and circle parameter specifications within the binary image were detected. This raised the issue of multiple circles with the same colour threshold being detected. The solution to this issue was the designation of the drop-zone indicator’s size, where knowing the size of the indicator and the given altitude meant a direct correlation could be made between the physical size of the drop-zone indicator and the pixel size measured by Circle Hough Transform. Further development of a unique drop-zone indicator would mitigate this issue, but for the purpose of this research, this method of elimination was sufficient [8]. 4.5 Localization From the object detection, pixel coordinates of the detected drop-zone identifier could be translated to ground plane displacements due to the relationship established between the ground plane and image plane. Pixel coordinate offsets were necessary due to OpenCV’s origin reference point being situated in the top left-hand side of the image plane. As such, a reference point was established in the centre of the image plane as this allowed for the drop-zone identifier’s position relative to the aircraft’s position to be established. To maximize the localization of the relative position of the drop-zone identifier relative to the aircraft, the image plane was divided into four quadrants. From these quadrants, the relative location of the drop-zone indicator could be defined in terms of in front or behind the aircraft and to the left or right of the aircraft. In addition to these quadrants, the heading of the aircraft was defined to be the direction the camera’s image plane negative y-axis (−yi ). Knowing these details about the drop-zone identifier’s relative position to the aircraft allowed for GPS localization of the precise coordinates of the drop-zone identifier through image frame to global frame transformations. To assist in these transformations, the GPS was located directly above the onboard camera and as such, GPS coordinates of the drop-zone indicator could be directly derived from the aircraft’s current GPS location and the determined ground plane displacements to the indicator. This relationship between the global frame and the image frame can be seen in Fig. 5. A notable result obtained with the use of the GPS sensor onboard the aircraft was that due to the resolution of the GPS sensor, GPS coordinates obtained from the device were offset from the true GPS location of the aircraft. This being said, due to the fact that both the camera and GPS were onboard the aircraft and communicated in a closedloop system, the determined location of the drop-zone indicator would possess the same offset due to its reference being the GPS. As such, relative to the physical location of the drop-zone indicator, the determined location would not necessarily be correct but due to the closed loop system and consistent reference, the drop-zone indicator was defined correctly with respect to the aircraft’s global frame of reference. Figure 7 illustrates the four quadrants in the image plane.

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Fig. 7. Image plane quadrant designation

5 Waypoint Allocation To achieve seamless missions in which the aircraft would be able to function whether detecting the drop-zone indicator or not, missions were defined in a generic manner as a fixed number of waypoints. These generic missions were defined as the following sequence of waypoints; takeoff, survey, environmental parameter determination, payload release approach, payload release, confirmation flyover, return to launch and land. With these generic waypoints defined for each mission, new waypoint initialization and declaration was not necessary during missions and the given waypoints could be updated during missions instead. As such, in mission scenarios where the drop-zone indicator was not detected, the aircraft would transition from the survey waypoints to the return to launch waypoint, as the intermittent waypoints would remain in their default state and were deemed unachievable by the FCU. However, with the precise GPS location of the drop-zone indicator known, the respective waypoint updates could be communicated to the Pixhawk and the aircraft would transition from survey to the environmental parameter determination waypoints. The use of the environmental parameter determination waypoints was crucial in the achievement of simplified payload trajectory modelling. Environmental parameter determination was used to determine the headwind velocity and direction. Knowing these parameters meant that the payload release approach could be defined in such a manner as to ensure the aircraft would fly directly into the headwind. Flying directly into the headwind allowed for the payload release to be modelled in two dimensions as opposed to three dimensions, as the headwind velocity could be modelled as a coplanar force imparted on the payload during its release trajectory, provided the aircraft flew directly into the headwind.

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6 Payload Release Payload release was modelled as a two-dimensional projectile motion, where headwind was introduced as an external force imparted on the payload during its release. The method for modelling the payloads trajectory was derived from first principles and was defined as two distinct phases, namely Phase A and Phase B. Phase A represented the period when the payload was released from the aircraft and would begin to decelerate horizontally due to the force imparted upon it by the headwind and drag due to air resistance. Phase A concluded when the payload’s relative horizontal velocity was zero, implying that the forward momentum imparted on the payload from the aircraft during flight had been cancelled out by the headwind and drag. Phase B proceeded Phase A and represented the period in which the payload would begin to accelerate horizontally in the same direction as the headwind until it reached the ground. These two phases were used to derive the nature in which the payload reacted during its trajectory, but these phases were not always guaranteed. Instances where there was little to no headwind, the payload could reach the ground prior to ever achieving a zero velocity in the horizontal plane. Similarly, if a payload with a significant mass was modelled, the headwind would not necessarily be capable of overcoming the payload’s inertia prior to impact with the ground. As such, these phases were heavily time dependent, which in the case of this system was constant as vertical acceleration of the payload was taken to be a constant 9.81 m/s2 due to gravity and with the altitude of the aircraft known, the time taken for the payload to fall the given distance could be approximated. It should be noted that this research assumed the headwind velocity vectors to follow a horizontal trajectory, parallel to the ground and as such, vertical gusts were not considered. Knowing the time constant for the payload release to fall to the ground allowed for the model to determine whether the payload would remain in Phase A for the duration of the trajectory. In this case, if the time necessary for the payload’s horizontal velocity to reach zero exceeded that of the fall duration time, the payload would remain in Phase A, otherwise the payload would enter Phase B until this fall time elapsed. Knowing which phase the payload would transition to allowed for the determination of the horizontal displacement of the payload relative to the GPS location of the point of release. Phase A displacements would provide a positive displacement from the point of release, where a positive displacement implied a displacement in the direction the aircraft was flying at the moment of payload release. Phase B displacements would result in negative displacements relative to the final displacement achieved in Phase A. As such, for greater headwinds, the payload would ultimately land at a negative displacement relative to the point of release. Knowing this final relative displacement of the payload meant that the GPS location at which the payload needed to be released in order to achieve payload delivery at the determined drop-zone location could be defined and updated on the Pixhawk’s waypoint list. The equations used to define the Phase A and Phase B displacements were as follows.  tEq  1 a dt t (1)  xA = vi t + 2 to

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And, ⎛   xB =

tfall

tEq

⎜ ⎜ vwalt ⎜ ⎝

VT2 2g

ln

 −VT tan

V2  T 2

halt

gt VT

⎞ a1 +VT2

⎟ ⎟ ⎟ dt ⎠

(2)

Where,  xA and  xB represented the Phase A and Phase B displacements respectively. vi represented the initial velocity of the payload, VT represented terminal velocity, halt represented the altitude of release, vwalt represented the headwind velocity at the altitude of release and t represented time.

7 Testing and Results As this system comprised of several subsystems, testing of each subsystem was necessary in order to optimize the final integrated mechatronic system. Within this mechatronic system, three subsystems formed the core elements for the success of this research, namely autonomous flight control, the vision system and payload release trajectory plotting. Each of these core elements were tested separately and their individual performances were combined to define the overall system performance. The system was tested both in simulation via Software in the loop (SITL) and Hardware in the loop (HITL) and physically onboard the testbench aircraft. From these simulations and physical tests, comparisons were drawn as to the performance of the system. 7.1 Autonomous Flight Control Of the three core elements, autonomous flight control represented the most challenging in terms of success quantification as success of the mission could simply be deemed as the ability of the aircraft to complete the mission. In addition to this ambiguity of success classification, elements such as waypoint accuracy was handled by the Pixhawk and once user-defined waypoint parameters were defined, the FCU would ensure said goals were met and provided minimal data to truly gauge the system’s performance. With these drawbacks to the physical system performance classification highlighted, a system efficiency analysis approach was taken in order to quantify the success of the autonomous flight control. This efficiency analysis compared the simulated power consumption of the aircraft with the power consumption of the aircraft during autonomous missions and during manual control. This comparison of power consumption was used to justify the implementation of an autonomous flight controller over the use of manual flight control of the aircraft. An example of the comparative power consumption of the aircraft during takeoff can be seen in Fig. 8. From Fig. 8, it can be noted that both the simulation’s (blue) and the autonomous flight control’s (red) instantaneous power consumption increased rapidly until a consistent output power was attained for takeoff. Comparing this instantaneous power consumption with that of the manual flight control (green), it can be noted that the instantaneous power consumption increased slowly overtime until the desired takeoff power was achieved.

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Fig. 8. Instantaneous power consumption during takeoff

From these trends, it was concluded that not only did the autonomous flight control follow the simulation within a respectable tolerance, but also provided the necessary evidence to justify the use of an autonomous flight controller over manual flight control due to the reduction in overall power consumed during takeoff, derived from the graph area below each plot. Although the autonomous flight controller implemented a more aggressive throttle ramp in order to achieve takeoff, the manual control of the aircraft resulted in a longer duration of instantaneous power consumption prior to achieving flight. Consumption comparisons were further assessed in various elements of a standardized mission where maneuvers, which were possible to execute during manual control, were compared to the simulation and autonomous flight control. From the results gathered, it was found that the simulation and autonomous flight control were more consistent in their application of throttle control during mission execution and were able to maintain a consistent altitude when compared to that of the manual flight control. However, a variation in the trajectory planning of the simulation and autonomous flight control was noted and the simulation was able to attain superior flight paths. This variation in flight path planning was believed to be due to the variations in the simulated aircraft and the physical aircraft’s structure, in addition to a variation in flight controller firmware releases as compared to the simulated version. This variation being noted, the simulation could still be taken as an almost direct comparison as to how the autonomous flight control would perform and mission optimization could be performed iteratively through the simulation. 7.2 The Vision System Much like the autonomous flight control, testing of the vision system could be done through simulation where image streams were feed into the image processing algorithm. During these simulations, flight parameters could also be simulated, such as

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altitude, heading and GPS location, to test the performance of the image processing algorithm in determining the true location of the drop-zone identifier. Through simulation, a greater understanding of the OpenCV Circle Hough Transform was gained, as the basic requirements for object, in this case a circle, detection could be determined. Once the vision system had been refined through simulation, physical testing was implemented. In these physical tests, the aircraft was flown through a standard mission and the drop-zone identifier was placed at a random location within the survey area. As the aircraft flew the survey, the mission waypoints were checked to see if waypoint allocation for the drop-zone had been achieved. During these tests, it was found that the vision system was able to detect and define the drop-zone indicator’s location within an average of 1.36 m from an altitude of 75 m. This result was deemed acceptable as the area defined for a successful payload delivery was within a 5 m radius of the drop-zone indicator. Figure 9 illustrates the testing of the vision system of footage captured onboard a DJI Phantom and fed into the image processing algorithm. In Fig. 9, the drop-zone indicator can be seen to be detected and encircled in blue.

Fig. 9. Testing of the image processing algorithm

7.3 Payload Delivery In terms of simulation, payload release trajectory was modelled with the assistance of MATLAB, where Phase A and Phase B release trajectories were modelled. From these simulations, lookup tables were developed for implementation onboard the aircraft. The significance of these lookup tables was the reduction in the required amount of onboard processing the ODroid would need to undertake in order to determine the appropriate release coordinates. The lookup tables consisted of Phase A completion times for various altitudes, various aircraft airspeeds and headwind velocities. With these tables implemented on the ODroid, a less computation intensive interpellation between lookup table data and current flight data could be done to determine the Phase

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A completion time. From this time and after calculating the time for the payload to fall from the given altitude to the ground, the ODroid could determine whether the payload would enter Phase B. From these trajectory phase predictions, the ODroid could calculate the displacement from the drop-zone indicator in order to achieve successful payload delivery. Through implementing the aforementioned lookup tables and working from the location of the drop-zone indicator ascertained by the vision system, payload delivery was achieved successfully within an average displacement of 1.82 m of the drop-zone location from a release altitude of 75 m. This result was well within the defined performance requirement of the system. Figure 10 illustrates a plot of the transition from payload release to Phase A to Phase B and finally, to delivery. The data in Fig. 10 was captured during physical testing of the payload release system.

Fig. 10. Payload transition during payload release

8 Conclusion This integrated system was developed to provide the ability to achieve accurate payload delivery to inaccessible areas. From the results gathered, this system illustrates its capability in achieving this goal by successfully implementing fully autonomous flight control, active object detection and localization and payload delivery within an average

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displacement of 1.82 m of the desired delivery location. Further development of this system would see improvements made to the object detection system to include more complex drop-zone indicators which could provide greater payload delivery security. Acknowledgements. We would like to show our gratitude towards the South African Department of Science and Technology’s ROSSA programme, the Eskom TESP programme, Nation Research Foundation and MerSETA, for providing the necessary funding, which enabled the acquisition of the necessary tools and equipment required for this research.

References 1. Zipline (2019). http://www.flyzipline.com 2. Shankland, S.: Zipline drone launch. [Internet]: CNET, April 2018. https://www.cnet.com/pic tures/take-a-look-at-ziplines-new-drone-delivery-system 3. Elpcctv.com. MINI 5MP AF USB camera module OV5640 COLOR CMOS sensor. [Internet]: Ailipu Technology Co., Ltd. (2019). http://www.elpcctv.com/mini-5mp-af-usb-camera-mod ule-ov5640-color-cmos-sensor-p-234.html 4. Vooon. MAVROS. [Internet]: ROS.org; 201. http://wiki.ros.org/mavros 5. OpenCV dev team. Thresholding Operations using inRange. [Internet]: OpenCV; 2014. https:// docs.opencv.org/3.4/da/d97/tutorial_threshold_inRange.html 6. OpenCV dev team. Image Filtering. [Internet]: OpenCV (2014). https://docs.opencv.org/2.4/ modules/imgproc/doc/filtering.html#image-filtering 7. Opencv dev team. Hough Circle Transform. [Internet]: OpenCV (2014).: https://docs.ope ncv.org/2.4/doc/tutorials/imgproc/imgtrans/hough_circle/hough_circle.html#hough-circle-tra nsform 8. Sol. Detect red circles in an image using OpenCV. [Internet]: Solarian Programmer; May 2015 https://solarianprogrammer.com/2015/05/08/detect-red-circles-image-using-opencv

Stabilization Using In-domain Actuator: A Numerical Method for a Non Linear Parabolic Partial Differential Equation Th´er`ese Azar1 , Laetitia Perez1 , Christophe Prieur2 , Emmanuel Moulay3 , and Laurent Autrique1(B) 1

LARIS-Polytech, Angers, France {therese.azar,laetitia.perez,laurent.autrique}@univ-angers.fr 2 Gipsa-Lab, Grenoble, France [email protected] 3 XLIM, Poitiers, France [email protected]

Abstract. This paper deals with the problem of null controllability for an unstable nonlinear parabolic partial differential equation (PDE) system considering in-domain actuator. The main objective of this communication is to provide an efficient control law in order to stabilize the system state close to zero in a desired time whatever the initial state is. A numerical approach is developed and in order to highlight the relevance of the proposed control strategy, a realistic physical problem is investigated. Thermal evolution of a thin rod with homogeneous Dirichlet boundaries conditions is considered. Thermal state is described by the heat equation and assuming that thermal conductivity is temperature dependent, a nonlinear mathematical model has to be taken into account. Considering that all the model inputs are known, a direct problem is numerically solved (regarding a finite element method) in order to estimate the temperature at each point of the 1D geometry and at each instant. Then an inverse problem is formulated in such a way as to determine the in-domain control which ensures a final temperature close to zero. An iterative regularization method based on the conjugate gradient method (CGM) is developed for the minimization of a quadratic cost function (output error). Several numerical experimentations are provided in order to discuss the numerical approach attractiveness.

Keywords: Parabolic partial differential equation Conjugate gradient method.

1

· Inverse problem ·

Introduction

Evolution of numerous physical phenomena could be described by a mathematical model based on a set of PDE: wave equation, Navier-Stokes equations that c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 616–627, 2021. https://doi.org/10.1007/978-3-030-58653-9_59

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describe the motion of viscous fluid substances, Saint-Venant equations that describe the flow below a pressure surface in a fluid in one direction, heat equation that describes the temperature evolution ... The last one is issued from energy balance (Fourier’s Law) and leads to a set of a parabolic PDE with initial and boundaries conditions. Control and stabilization of physical processes described by PDE systems require multidisciplinary study and are one of the most extensive research of the past decades [1]. Recent developments could be mentioned in [2–10] and from the theoretical point of view the investigated mathematical models are usually linear. However, in numerous realistic problems non linearities could not be neglected and development of numerical approaches in order to build relevant control law is a crucial requirement. In the specific case of nuclear fusion for example, the control of the spatial distribution of the safety profile in tokamak plasma is highly complex [11,12] and should be investigated carefully by taken into account non linearities and coupled phenomena (magnetohydrodynamics equation and heat equations). In the following, our purpose is to stabilize in finite time the spatial distribution of the solution of a parabolic PDE system near to the stable desired state. The main challenge is to identify the in-domain flux (heating or cooling) which ensures the null controllability. It is well known that such inverse heat conduction problem (IHCP) is ill posed in Hadamard sense [13]. In the framework of thermal properties identification, the iterative regularization method based on the CGM has been developed in [14]. In recent works, authors have proposed new developments for mobile heating source tracking in 2D and 3D geometries [15– 17] and for quasi online identification of a temperature dependent characteristics in [18]. The paper is organized as follows. Next section is devoted to the mathematical model of the physical system: direct problem is defined in order to describe the heat conduction in the investigated domain (thermal characteristics of the heated material are also given). A finite element method implemented with Comsol Multiphysics Solver and Matlab software is used for numerical simulation [19–21]. Several numerical configurations are tested in order to highlight the instability of the PDE system. In the third section, identification of the control law is formulated as a minimization problem and the optimization method is based on iterative descent scheme. This approach is based on iterative numerical resolution of three well-posed problems: direct problem, sensitivity problem and adjoint problem. Considering the constructive method of control proposed in the third section, several numerical configurations are tested in the fourth section. Finally, concluding remarks and outlooks are discussed in the last section.

2 2.1

Studied Configuration Problem Statement

In this section, the investigated thermal system is briefly presented. Let us study a thin plate. Considering that the plate thickness is neglected (temperatures are

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equal on the upper and on the lower face of the plate), each point of the domain is defined by its space variables (x, y) ∈ [0, L] × R. The time variable is t ∈ T = [0, tf ] where tf is the final time. The in-domain actuator location is centered on 0 < a < L and the control ua (x, t) acts on ω = [a − ε, a + ε] ⊂ Ω = [0, L] (see Fig. 1). Such localized internal control can be encountered in the context of nuclear fusion for a current controlled by microwaves, for example [22,23].

Fig. 1. Studied geometry.

The initial temperature distribution at t = 0 is denoted by f (x). Let us denote by θ(x, t) the temperature in K at point x and time t. On the both boundaries {x = 0; x = L}, homogeneous Dirichlet condition is considered. Then, heat exchanges on the two faces of the plate could be described by the following system: ⎧   ∂θ(.) ∂ ∂θ(.) ⎪ ⎪ ρC − ∀(x, t) ∈ Ω × T λ(θ) = ξθ(.) + ua (.) ⎨ ∂t ∂x ∂x θ(0, t) = θ(L, t) = 0 ∀t ∈ T ⎪ ⎪ ⎩ θ(x, 0) = f (x) ∀x ∈ Ω (1) The first equation of system (1) is the parabolic PDE describing the evolution of the temperature distribution where: ρC is the volumic heat in J.m−3 . K−1 and λ is the thermal conductivity in W.m−1 .K−1 . The non linearity of this heat equation is due to the thermal dependence of the thermal conductivity λ(θ). The coefficient ξ in W.m−3 .K−1 can be negative or positive and ξθ(x, t) describes the heat transfer with the surrounding and affects the system stability. Control ua (x, t) in W.m−3 acts on ω and aims to stabilize the temperature such as θ(x, tf ) = 0, ∀x ∈ Ω. If all the input parameters of (1) are known, this system defines a well posed direct problem and its resolution leads to the determination of the temperature distribution evolution θ(x, t), ∀(x, t) ∈ Ω × T . 2.2

Direct Problem Resolution

In the following example, realistic input parameters are taken into account: L = 0.1 m, ρC = 106 J.m−3 . K−1 and initial temperature is defined as follows:      −(x − 0.02)2 −(x − 0.05)2 1 f (x) = 25 exp + exp . 2 5 × 10−5 10−4

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 θ Thermal conductivity is assumed to be: λ(θ) = 33 + exp . Without any 40 internal control (ua (x, t) = 0), if ξ = 0 the temperature naturally converges towards 0 and if ξ < 0, the term ξθ(x, t) increases the convergence speed. It is well known that there exists a threshold K such that if ξ > K > 0, the system

L 2 2 is not stable and lim θ(t)L2 (Ω) = lim [θ(x, t)] dx = +∞. t→∞

t→∞

0

Effect of different values of ξ are taken into account and direct problem (1) is solved until tf = 100 s. Temperature are determined using a software based on a finite element method (Comsol-Multiphysics solver interfaced with Matlab program). Temperature norms θ(t)2L2 (Ω) for several values of parameter ξ are shown in Fig. 2. 20 18

Temperature norm

16 14 12 5

ξ=−10 ξ=0

10 8

ξ=2× 104

6

4

ξ=4× 10

4 2 0 0

20

40

60

80

100

t

Fig. 2. Evolution of temperature norm.

Considering Fig. 2, it is obvious that according to the first three small values of ξ, system (1) is stable: without control, temperature norm decreases and temperature converges towards zero in the domain. For ξ = 4 × 104 , system (1) is unstable; ξθ brings a non-stability that tends to move the system away from 0. Such behaviour is shown in Fig. 3. In the following section, unstable configuration where ξ = 4 × 104 is considered and an inverse problem is proposed in order to identify the unknown control law ua (x, t) which will be able to stabilize system (1).

3

Inverse Problem

The aim of the following inverse problem is to identify an in-domain control ua (.) such that θ(x, tf )2L2 (Ω) ≈ 0. An iterative minimization method is developed in

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Fig. 3. Temperature evolution - ξ = 4 × 104 .

order to minimize at each iteration k the quadratic cost function: J(θk ) =

1 2



L

 k 2 θ (x, tf ) dx

(2)

0

A discrete formulation could be considered and leads to a similar minimization of θ(x, tf )2L2 (Ω) : Jdiscrete =

N  2

θˆi (tf ) , ∀i = 1, . . . , N i=1

where N is the number of sensors and θˆi is the measured temperature at the sensor location xi . An iterative algorithm based on CGM is implemented in order to solve this inverse problem written as a minimization one. CGM is a descent method that solves the problem of parametric identification by stopping the minimization when a relevant threshold Jstop is obtained. Such method has been developed for thermal applications in [14–18]. At each iteration k of the algorithm, three well-posed problems have to be solved: 1. The direct problem in order to determine the temperature distribution θk (x, tf ), and then to estimate the criterion J(θk ), 2. The adjoint problem to determine the gradient of the cost function J(θk ) and thus to define the next descent direction dk , 3. The sensitivity problem to estimate the descent depth γ k (in the descent direction).

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Let us consider that the control uka (x, t) is a piecewise linear continuous function depending on time and space and can be formulated as follows: ua (x, t) =

M

uaj (x)sj (t)

j=1

where

⎧ x−(a−ε) uaj ⎨ ε uaj (x) = (a+ε)−x uaj ε ⎩ 0

if x ∈ [a − ε, a] if x ∈ [a, a + ε] if not

and

sj (t) =

⎧ (M +1)t−(j−1)tf ⎪ ⎪ tf ⎨

if t ∈

⎪ ⎪ ⎩

if t ∈ if not

(j+1)tf −(M +1)t tf

0





j−1 tf , Mj+1 tf   M +1 j j+1 t , t f f M +1 M +1

Functions sj (t) are the “hat functions”, basis for piecewise linear functions. According to this notation uka (x, 0) = uka (x, tf ) = 0, a pointwise actuator could be investigated if ε is close to zero. In the following three subsections, sensitivity problem, adjoint problem and a conjugate gradient algorithm are presented. 3.1

Sensitivity Problem

In order to calculate at iteration k the descent depth γ k in the descent direction dk , the sensitivity problem has to be solved. Let us consider temperature variation θ(x, t) + ε0 δθ(x, t) induced by a variation of the thermal flux ua (x, t) + ε0 δua (x, t). Formulation of system (1) satisfied by the temperature θ+ = θ + ε0 δθ and considering the control u+ a = ua + ε0 δua leads to the following system while ε0 → 0. Sensitivity function δθ is solution of this so-called sensitivity problem: ⎧   ⎪ ∂δθk (.) ∂ 2 λ(θ)δθk (.) ⎪ ⎨ ρC − = ξδθk (.) + δuka (.) ∀(x, t) ∈ Ω × T 2 ∂t ∂x k k δθ (0, t) = δθ (L, t) = 0 ∀t ∈ T ⎪ ⎪ ⎩ k ∀x ∈ Ω δθ (x, 0) = 0 (3) In this study, the variation of thermal flux provided by the actuator is: δua (x, t) =

M

δuaj (x)sj (t).

j=1

At each iteration k, the identified control uk+1 aj , ∀j = 1, . . . , M is given by: k k k uk+1 aj = uaj − γ dj

(4)

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and γ k is the optimal depth determined as follow:

L − θk (x, tf )δθk (x, tf )dx k k k 0 γ = arg minJ(ua − γd ) =

L  k 2 γ∈R δθ (x, tf ) dx

(5)

0 k

where θ (x, tf ) is the solution of the direct problem (1) and δθk (x, tf ) is the solution of the sensitivity problem (3) (solved in descent direction dk ). In the following section adjoint problem is developed in order to determine the descent direction dk . 3.2

Adjoint Problem

∂J(θk ) and the descent ∂ukaj direction dk , an adjoint problem is formulated. Let us denote by a Lagrangian formulation which is a function of uka (.), θk (.) and ψ k (.) where ψ k (.) is the ∂ k ∂ adjoint function, fixed such as: δθ = 0 and δψ k = 0. Furthermore, if ∂θk ∂ψ k θ(.) is solution of system (1) then: In order to calculate at each iteration k, the gradient

(uka , θk , ψ k ) = J(θk ) ⇒ δ (uka , θk , ψ k ) = δJ(θk ) and variation of Lagrangian

L δ (uka , θk , ψ k ) = θk (x, tf )δθk (x, tf )dx 

tf 0 L   ∂2  ∂δθk k k k − λ(θ)δθ − ξδθ − δua ψ k dxdt. ρC + ∂t ∂x2 0 0

(6)

After several integrations by parts with respect to x and t, ψ k (x, t) is solution of the following adjoint problem: ⎧ ∂ψ k (.) ∂ 2 ψ k (.) ⎪ ⎪ −ρC − λ(θ) = ξψ k (.) ∀(x, t) ∈ Ω × T ⎪ 2 ⎨ ∂t k ∂x k (7) ψ (0, t) = ψ (L, t) = 0 ∀t ∈ T ⎪ ⎪ 1 ⎪ ⎩ θk (x, tf ) ∀x ∈ Ω ψ k (x, tf ) = − ρC This system is numerically solved using a retrograde scheme in time. For ψ k (x, t) solution of system (7), variation of Lagragian is: δ (uka , θk , ψ k ) =

∂ k δu = δJ(uka ). ∂uk

According to the previous equations, cost function gradient is defined as follows:

tf a+ε ∂J = − ψ k (x, t)sj (t)dxdt (8) ∂ukaj 0 a−ε

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The descent direction can be estimated at each new iteration k from the previous gradient (at iteration k − 1), as follows:   ∂J k + βk dk−1 (9) d =− ∂ukaj j=1,...,M

2       ∂J     ∂uk  aj j=1,...,M  with βk =  2 and . is the Euclidean norm.      ∂J    ∂uk−1    aj j=1,...,M 3.3

Conjugate Gradient Algorithm

In order to solve the previous inverse problem, an offline algorithm is implemented as follows: 1. Initialization  of the unknown parameter (control flux) at the first iteration k = 0: ukaj j=1,...,M = 0,

2. Resolution of the direct problem (1) in order to determine θk (x, t), 3. Determination of the criterion J(θk ) according to (2): – If the criterion is below the stopping threshold Jstop , then the algorithm is stopped and uka is considered as the estimate of the unknown parameter (internal control law ua (x, t) is then identified). – Otherwise, continue to step 4, 4. Solve the adjoint problem (7) in order to determine the Lagrange multiplier ψ k (x, t) and the cost function gradient according to (8). Determination of the descent direction dk according to (9), 5. Resolution of the sensitivity problem (3) in the descent direction dk to calculate the sensitivity function δθk (x, t) and determination of the descent depth γ k according to (5), 6. Update new estimations of the control according to (4), 7. Increment of the iteration k = k + 1 and back to step 2.

In the next section, several numerical configurations are investigated in order to highlight the method relevance.

4

Numerical Results

In this paragraph, specific cases are proposed considering thermal properties given in Sect. 2.2. Non stability is induced by ξ = 4 × 104 and without control system evolution is shown in Fig. 3. The previous numerical method is used for the determination of a relevant control law and the following values presented in Table 1 are taken into account with Jstop = 10−3 :

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100

100

100

100

40

40

40

40

a × 10

3

3

9

9

3

3

9

9

ε × 10−3

3

1

3

1

3

1

3

1

−2

Furthermore, in order to compare the previous cases together, the energy

L tf 2 required by each control law is calculated as follows: E = [ua (x, t)] dxdt 0

and is given in the following Table 2.

0

Table 2. The cost of each cases. Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 12

E × 10

 2.8

8.5

20

63

9.3

28

99

290

In Fig. 4. several results are shown: the evolution of identified control uaj for cases 1, 2, 3 and 4 on the left and the evolution of identified control uaj for cases 5, 6, 7 and 8 on the right. 7

2

8

x 10

1 case 1 case 2 case 3 case 4

1

0.5

0 a

control u (t)

0 control u (t) a

x 10

−1

−0.5

−2

−1

−3

−1.5

−4 0

20

40

60 Time in s

80

100

−2 0

case 5 case 6 case 7 case 8

5

10

15

20 25 Time in s

30

35

40

Fig. 4. Evolution of identified control laws: cases 1, 2, 3 and 4 (left), cases 5, 6, 7 and 8 (right).

In Fig. 5 temperature evolution without control on the left and with control on the right for the case 8 are shown.

Stabilization Using In-domain Actuator 30

30 t=0 t=10 t=20 t=30 t=40

20

10 θ in K

θ in K

t=0 t=10 t=20 t=30 t=40

20

10

0

0

−10

−10

−20

−20

−30 0

625

0.02

0.04

0.06 x in m

0.08

0.1

−30 0

0.02

0.04

0.06

0.08

0.1

x in m

Fig. 5. Temperature evolution without control (left) and with control (right) - case 8.

According to the previous results, several remarks could be proposed: – If the spatial support ω of the actuator is small, then the control flux has to be big. Comparing cases 1&2, 3&4, 5&6 and 7&8, it is shown that if ω is divided by 3, then the energy required by the control is multiplied by 3. – If the final time tf of the process is small, then the control flux has to be big. Moreover, due to the small time horizon and to the wide control flux, evolution of the control law is complex (with positive and negative commutation). This behaviour is meaningful and could be compare to a bang-bang point wise controller.

5

Conclusions

In this paper, the stabilization of a non linear parabolic PDE system in 1D geometry has been investigated. Considering a thermal realistic configuration, non stability has been numerically put in evidence. In order to ensure the convergence of the system state toward zero, an in-domain actuator has been implemented. An iterative regularization method for identification of such unknown control is presented. Numerical results of in-domain control determination for several studied configuration are provided and show that the suggested method is relevant in order to identify an in-domain control law in such non linear system. Effect of actuator spatial support (size, location) has been studied and it has been shown how the desired final time (for stabilization purposes) affect the determination of the control. Future works will consist in adaptation of online implementation in order to ensure disturbances rejection. Moreover, the effect of several non collocated actuators and sensors could be easily investigated considering the proposed method. The control of few moving actuators seems to be quite attractive and could ensure null controllability in an elegant way. Acknowledgments. This work was supported by the regional programme Atlanstic 2020, funded by the French Region Pays de la Loire and the European Regional Development Fund.

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18. Vergnaud, A., Beaugrand, G., Gaye, O., Perez, L., Lucidarme, P., Autrique, L.: Online identification of temperature-dependent thermal conductivity. In: European Control Conference, Strasbourg, France (2014) 19. Pepper, D.W., Heinrich, J.C.: The Finite Element Method - Basic Concepts and Applications, p. 240. Taylor & Francis, Group, Boca Raton (1992) 20. Zimmerman, W.B.J.: Multiphysics Modeling with Finite Element Methods, p. 432. World Scientific Publishing, Singapore (2006) 21. Baker, A.J.: Finite Elements: Computational Engineering Sciences, p. 288. Wiley, New York (2012) 22. Garrido, A.J., et al.: Nuclear fusion control-oriented plasma physics. In: Proceedings of the 13th WSEAS International Conference on systems, WSEAS CSCC Multiconference, Crete Island, Greece (2009) 23. Biel, W., et al.: Diagnostics for plasma control – from ITER to DEMO. Fusion Eng. Des. 146(A), 465–472 (2019)

Direct Power Control of a Doubly Fed Induction Generator Using a Lyapunov Based State Space Approach Yassine Boukili(B) , A. Pedro Aguiar, and Adriano Carvalho Department of Electrical and Computer Engineering, Faculty of Engineering, University of Porto (FEUP), Porto, Portugal {yassine,pedro.aguiar,asc}@fe.up.pt

Abstract. This paper addresses the modelling and control design of a DoublyFed Induction Generator (DFIG) when connected to a grid subject to unbalanced grid voltage conditions. Using Lyapunov theory, we derive a robust direct power control law that tracks the active and reactive power references that are adequately manipulated accordingly with the presence of unbalanced grid voltage dips. Simulation results illustrate the robustness and effectiveness of the proposed approach. Keywords: DFIG · Direct power control · Symmetrical voltage dips · Lyapunov

1 Introduction Contrary to conventional power generation using fossil resources, Renewable Energy Resources (RES) can have a significant impact in reducing the pollution in the world [1]. Among them, Wind Energy Source (WES) has been largely exploited through the use of Doubly-Fed Induction Generators (DFIG) and Permanent Magnet Synchronous generators (PMSG). Nowadays, the most widely used are WES generators above 1 MW in the DFIG type because of its reliability and cost advantages [2–4]. A consequence of the increasing penetration of RER in the electrical network is that the quality of power generation becomes more challenging to keep it, especially during unbalanced grid. Additionally, the grid code requires that the Low-Voltage Ride Through (LVRT) and the reactive power injected by the DFIG in the electrical network must obey some limits [5, 6]. In this regard, WES based on the DFIG generator require robust control strategies because the DFIG is very sensitive to the electrical network This work was supported in part by R&D Unit UIDB+P/00147/2020 funded FCT/MCTES (PIDDAC) and projects: STRIDE – NORTE-01-0145-FEDER-000033, funded by N2020, ERDF; IMPROVE - POCI-01-0145-FEDER-031823, MAGIC PTDC/EEI-AUT/32485/2017 and HARMONY - POCI-01-0145-FEDER-031411 funded by FEDER funds through COMPETE2020 – POCI and by national funds (PIDDAC). c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gonc¸alves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 628–637, 2021. https://doi.org/10.1007/978-3-030-58653-9_60

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Fig. 1. DFIG configuration.

variation. Moreover, unbalanced grid may cause the increasing of the stator and rotor currents and the DC voltage between the back-to-back converters [7]. In the literature one can found several works regarding the control of the DFIG. The works in [8, 9], present the vectorial control by using the stator voltage or flux vectors; however, it does not work well during unbalanced grid. In [4] the vectorial controller is improved to balance the system during symmetrical voltage deeps and achieve the Low Voltage Ride through. In [10], the authors presents two controllers by employing the positive and negative sequences in case of asymmetrical voltage deeps, but the separation of the current loops increases the control delay. Despite the mentioned work, the DFIG generator operations are still limited during grid faults using vectorial controllers [11]. In [12], the authors propose to reduce the harmonics in the Grid Side Converter (GSC), presented in Fig. 1, during the grid faults. A control strategy for the GSC with the purpose of stabilising the DC voltage is described in [13]. The most used control methods for the DFIG are the Direct Torque Control (DTC) and the Direct Power Control (DPC) [14]. This paper contributes with the modelling of DFIG based on state space equations, and proposes a DPC law using Lyapunov theory to stabilise and protect the system during symmetrical voltage dips. This method is simpler than using the fluxes of the stator and rotor. Several simulation experiments done in Matlab/Simulink are presented to illustrate the performance and effectiveness of the described control strategies. This paper is organized as follows: Sect. 2 presents the modelling of the wind turbine and the DFIG dynamics. Section 3 addresses the DFIG state equations modelling and the proposed DPC controller for the DFIG connected into an unbalanced grid. In particular, symmetrical voltage dips is studied. Finally, the results obtained are discussed in Sect. 4.

2 State Equation Based Modelling of the DFIG 2.1 DFIG Modelling The DFIG configuration is presented in Fig. 1, where the wind turbine transfers the wind power to the asynchronous generator through the gearbox. The stator is connected into the Grid directly while the rotor through the back-to-back converters. The wind power that is extractable by the turbine can be computed from   (1) Pm = 0.5ρ ACp λ , β Vω3 ,

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Value

f

Frequency

50 Hz

Ps

Power delivered by the Stator

10 kW

ω

Synchronous speed

1500 tr/min

Vns

Stator voltage

380 V

Ins

Stator current

5.2 A

p

Number of pole

2

Rs

Stator resistance

0.22 Ω

Ls

Stator inductance

4.25×10−2 H

Lm

Mutual inductance

4×10−2 H

Rr

Rotor resistance

0.209 Ω

Lr

Rotor inductance

4.3×10−2 H

J

Inertia moment of the rotating parts 0.04 Kg.m2

D

Coefficient of viscous friction

2×10−3 n.s/rad

N

Multiplier coefficient

2

R

Radius of the turbine

5m

C pmax

Coefficient of maximum power

0.44

λopt

Specific speed

7.2

ρ

Air density

1.22 Kg/m3

where Vω is the wind speed (m/s), ρ is the density of the air (kg/m3 ), A is the rotor swept area (m2 ) and Cp is the power coefficient in function of the tip speed ratio λ and the pitch angle β [15]. The power coefficient depends on the specific wind turbine system. In this paper, we follow the reference [15] and approximate it by the function   24 116 − 0.4β − 5 e −λi + 0.0068λ , Cp λ , β = 0.5176 λi 1 0.035 λi = − λ + 0.008β β 3 + 1 



(2) (3)

Applying the Newton’s second law for a rotational mass, we obtain the dynamics associated with the rotor angular velocity ωr (rad/s) that satisfies [16] 1 d ωr = (Tem − F ωr − Tm ), dt J

(4)

where J is the inertia moment (kg.m2 ), Tem is the electromagnetic torque (N.m), F is the viscous friction coefficient, Tm is the shaft mechanical torque. The torques can be computed from Tem =

3 Lm p (ψqs idr − ψds iqr ), 2 Ls

Tm =

Pm , ωr

(5)

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Fig. 2. Equivalent single phase electrical scheme of the DFIG.

where p is the number of pole pairs of the machine, and ψ is the flux (W b). The subindices r and s denote the rotor and the stator while the sub-indices d and q indicate the components of the Park transformation axis frame. The equivalent electrical circuit of the DFIG is presented in Fig. 2, where ω is the synchronous reference frame angular speed (rad/s). By applying the mesh law the following equations are deduced Vs = Rs is + jωψs +

d ψs , dt

Vr = Rr ir + j(ω − ωr )ψr +

d ψr , dt

(6)

where Vs is stator voltage (V ) and Vr is the rotor voltage (V ). In the d/q frame, the dynamic equations can be written as [17] d ψqs d ψds , Vqs = Rs iqs + ωψds + , dt dt d ψqr d ψdr , Vqr = Rr iqr + (ω − ωr )ψdr + , Vdr = Rr idr − (ω − ωr )ψqr + dt dt Vds = Rs ids − ωψqs +

(7) (8)

where the stator and rotor flux equations in Park transformation frame are presented as

ψds = Ls ids + Lm idr , ψqs = Ls iqs + Lm iqr , ψdr = Lr idr + Lm ids . ψqr = Lr iqr + Lm iqs .

(9) (10)

From the flux equations, the following current equations can be deduced as Is,dq = where Cr =

Lm Lr , Cs

1 Cr ψs,dq − ψr,dq , σ Ls σ Ls =

Lm Ls

Ir,dq =

1 Cs ψr,dq − ψs,dq , σ Lr σ Lr

(11)

2

and σ = 1 − LLsmLr .

The grid-side-converter presented in Fig. 1 is connected into the grid through a passive filter shown in Fig. 3, where R f and L f are the resistance and inductance of the filter. The sub-indices ac, bc and dc are the three phases of the grid-side-converter, while the sub-indices ag, bg and cg present the three phases of the grid.

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Fig. 3. Electrical circuit of the filter between the grid-side-converter and the electrical network.

In Fig. 3, the potential differences between the converter and the grid can be written in d/q frame as didg dt diqg vqc − vqg = R f iqg + 2π f L f idg + L f dt

vdc − vdg = R f idg − 2π f L f iqg + L f

(12) (13)

Finally, the active and reactive power can be obtained as 3 Ps = (Vds ids +Vqs iqs ), 2

3 Qs = (Vqs ids −Vds iqs ). 2

(14)

3 Lyapunov Based State Space Approach 3.1

DFIG State Equations Modelling

Collecting all the Eq. (8)–(11) and denoting by x = [ids iqs idr iqr ωr ]T , the state equation of the DFIG can be described as x˙ = f (x) + B1 u1 + B2 u2 , where

⎡

(15)

⎤



1 2 + ω (L L − L2 )]i + R L i + ω L L i −Rs Lr ids + [ωr Lm s s r r m dr r r m qr m qs L L −L2

⎥ ⎢ s r1 m 2 ) + ω L2 ]i + R L i − ω L L i ⎢ Ls Lr −L2 −Rs Lr iqs − [ωs (Ls Lr − Lm r m ds r m qr r r m dr ⎥ ⎢ 1 m

⎥ 2 ) − ω L L ]i − R L i − ω L L i ⎥ ⎢ Rs Lm ids + [ωs (Ls Lr − Lm r s r qr r s dr r s m qs ⎥ 2 Ls Lr −Lm f (x) = ⎢

⎥ ⎢ 1 2 ) − ω L L ]i − R L i + ω L L i ⎢ Ls Lr −L2 Rs Lm iqs − [ωs (Ls Lr − Lm r s r dr r s qr r s m ds ⎥ m ⎥ ⎢    1 C λ ,β ρπ R2 v3 ⎦ ⎣ 2 p 1 J 3Lm (iqs idr − ids iqr ) − F ωr − ωr

⎡

⎤ Lr 0 ⎢ 0 Lr ⎥ ⎥ 1 ⎢ ⎢−Lm 0 ⎥ , B1 = ⎢ σ Ls Lr ⎣ 0 −L ⎥ m⎦ 0 0

⎡ ⎤ −Lm 0 ⎢ 0 −Lm ⎥ ⎥ 1 ⎢ ⎢ Ls 0 ⎥ B2 = , ⎢ σ Ls Lr ⎣ 0 L ⎥ s ⎦ 0 0

(16)

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u1 = [uds uqs ]T , u2 = [udr uqr ]T

633

(17)

Note that if ωr is assumed to be constant, the model simplifies to x˙ = Ax + B1 u1 + B2 u1 , where1 ⎡

⎤ 2 +ω σL L −Rs Lr ωr Lm Rr Lm ωr Lr Lm s s r 2 ⎥ 1 ⎢ −Rs Lr −ωr Lr Lm Rr Lm ⎢−ωr Lm − ωs σ Ls Lr ⎥ A= Rs Lm −ωr Ls Lm −Rr Ls −ωr Ls Lr − ωs σ Ls Lr ⎦ σ Ls Lr ⎣ ωr Ls Lm Rs Lm ωr Ls Lr − ωs σ Ls Lr −Rr Ls

It is interesting to note that in our specific case, the matrix A is not stable since not all eigenvalues have negative real part. 3.2 Direct Power Control In this section the direct power control based on the Lyapunov technique is presented. By substituting the Eq. (9) and Eq. (10) in Eq. (14), and by setting the q component as the reference, the equations of the active and reactive power can be rewritten as Ps = Vs

Cr ψds ( − ψdr ), σ Ls Cr

Qs = −Vs

Cr ψqs ( − ψqr ) σ Ls Cr

(18)

By substituting Eq. (9) and Eq. (10) in the active and reactive power equations, the following equations can be deduced: Ps = Cids , where

Qs = Kiqs ,

  2 m C = Vs LsσLLr +L 0 0 0 0 s Lr   2 −L L Lm K = 0 Vs σ L Ls r 0 0 0 s r

(19)

(20) (21)

The derivative of the active and reactive powers can be written as dPs = C ( f (x) + B1 u1 + B2 u2 ) dt dQs = K ( f (x) + B1 u1 + B2 u2 ) dt

(22) (23)

To control the power, we now introduce the errors of the active and reactive powers defined as eP = Ps∗ − Ps ,

1

eQ = Q∗s − Qs ,

(24)

Here, obviously the vector x = [ids iqs idr iqr ωr ]T and matrices B1 and B2 do not have the last row.

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where Ps∗ and Q∗s are the desired references of the active and reactive powers that are considered to be constant. Consequently, the errors satisfy the following dynamics e˙P = −C f (x) − α uds − β udr ,

(25)

e˙Q = −K f (x) − γ uqs − λ uqr ,

(26)

where we have used the following notation CLr uds , σ Ls Lr −CLm β udr = CB2 u2 = udr , σ Ls Lr KLr γ uqs = KB1 u1 = uqs , σ Ls Lr −KLm λ uqr = KB2 u2 = uqr . σ Ls Lr

α uds = CB1 u1 =

(27) (28) (29) (30)

To control the active and reactive powers of the DFIG, we start with the control Lyapunov functions 1 v p = e2P , 2

1 vq = e2Q , 2

whose time derivative yield v˙ p = −e p [C f (x) + α uds + β udr ] v˙q = −eq [K f (x) + γ uqs + λ uqr ]

(31) (32)

Therefore, by selecting the control laws 1 [− (C f (x) + α uds ) − k p e p ] , β 1 uqr = [− (K f (x) + γ uqs ) − kq eq ] . λ

udr =

(33) (34)

with k p and kq positive constants, it follows that v˙ p = −k p e2p < 0,

(35)

v˙q = −kq e2q

(36)

< 0,

and hence the origin of the errors are asymptotically stable. When the symmetrical voltage dips happens in the terminals of the stator, the direct power controller must compute the references of the active and reactive powers. Consequently, the maximum apparent power Smax can be computed as [18] Smax = InsVns ,

(37)

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where Ins and Vns are the stator nominal current and voltage, respectively. Following the grid code [19], the references of the active and reactive powers for a symmetrical voltage dips for a duration of 1000 s maximum are computed as follow Ps∗ = 0,   Vqs . Q∗s = −InsVqs 1 − Vns

(38) (39)

4 Results and Discussion This section illustrates the behavior of the proposed control strategy thought computer simulations that were performed Matlab/Simulink software, considering a constant wind speed of 12 m/s. The power maximal of the DFIG used in the simulation is 10 kW and all the parameters used in the simulation are presented in Table 1.

3000

6000 5000

2000

4000 1000 3000 0

eq (VAR)

ep (W)

2000 1000 0

-1000 -2000

-1000 -3000 -2000 -4000

-3000 -4000

-5000 0.5

1

1.5

Time (s)

2

2.5

0.5

1

1.5

2

2.5

Time (s)

Fig. 4. e p : Error of the stator active power (Watt), eq : error of the stator reactive power (VAR).

For testing the robustness of the proposed controller, 60% of the voltage dips is applied in the stator terminals between the seconds 1 and 1.5. Figure 4 presents the errors of the active and reactive powers, where the errors are converging to zero quickly, with some picks at the grid fault moment and recovery, which validated the proposed controller. In Fig. 5, the evolutions of the stator active power (Ps ) and the stator reactive power (Qs ) are presented. Ps is generating 5 kW in the stable period, while Qs is 0 VAR. In the first second the grid fault occurred, the controller computes the new references. Consequently, the active power decrease to 0 Watt to restrain the high current caused by the voltage dips, and the reactive power is increased to take the value 2500 VAR, that is injected into the grid for recovering the voltage.

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Y. Boukili et al. 1000

500 0

0 -500 -1000

Q s (VAR)

Ps (W)

-1000 -2000 -3000

-1500 -2000 -2500 -3000

-4000

-3500 -5000 -4000 -6000

-4500 0.5

1

1.5

Time (s)

2

2.5

0.5

1

1.5

2

2.5

Time (s)

Fig. 5. Ps : Stator active power (Watt), Qs : stator reactive power (VAR).

Fig. 6. Vs : stator volatge (V), Vr : rotor voltage (V).

The stator voltage (Vs ) and rotor voltage (Vr ) are presented in the Fig. 6, Vs has a voltage dips between the seconds 1 and 1.5, during this period Vr is also sinusoidal with some perturbation.

5 Conclusion We presented a control design strategy using a generalized model of the DFIG and Lyapunov theory to address the direct power control when the DFIG is connected to a grid subject to unbalanced grid voltage conditions. The proposed was tested and validated in simulation by applying a symmetrical voltage dips using Matlab/Simulink software. The controller used in this system will be improved and applied in practical studies and renewable hybrid systems. Future work include the experimental validation of the proposed controller, that has the advantaging of being simpler in terms of implementation. Experimental results will be compared with the obtained results. Additionally, an adaptive controller for the DFIG connected through an unbalanced grid will be studied.

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References 1. Borenstein, S.: The private and public economics of renewable electricity generation. J. Econ. Perspect. 26(1), 67 (2012) 2. Zin, A.A.B.M., Ha, M.P., Khairuddin, A.B., Jahanshaloo, L., Shariati, O.: An overview on doubly fed induction generators’ controls and contributions to wind based electricity generation. Renew. Sustain. Energy Rev. 27, 692–708 (2013) 3. El Ouanjli, N., Motahhir, S., Derouich, A., El Ghzizal, A., Chebabhi, A., Taoussi, M.: Improved DTC strategy of doubly fed induction motor using fuzzy logic controller. Energy Rep. 5, 271 (2019) 4. L´opez, J., Gub´ıa, E., Olea, E., Ruiz, J., Marroyo, L.: Ride through of wind turbines with doubly fed induction generator under symmetrical voltage dips. IEEE Trans. Ind. Electron. 56(10), 4246 (2009) 5. Massmann, J., Erlinghagen, P., Schnettler, A.: 2018 Power Systems Computation Conference (PSCC), pp. 1–8. IEEE (2018) 6. Piya, P., Ebrahimi, M., Karimi-Ghartemani, M., Khajehoddin, S.A.: Fault ride-through capability of voltage-controlled inverters. IEEE Trans. Ind. Electron. 65(10), 7933 (2018) 7. Firouzi, M., Gharehpetian, G.B.: LVRT performance enhancement of DFIG-based wind farms by capacitive bridge-type fault current limiter. IEEE Trans. Sustain. Energy 9(3), 1118 (2017) 8. Li, S., Haskew, T.A., Williams, K.A., Swatloski, R.P.: Control of DFIG wind turbine with direct-current vector control configuration. IEEE Trans. Sustain. Energy 3(1), 1 (2011) 9. Li, S., Haskew, T.A.: 2007 IEEE Power Engineering Society General Meeting, pp. 1–7. IEEE (2007) 10. Chen, L., Hui, Z.B., Fan, X.: IEEE Trans. Energy Conv. (2019) 11. Li, S.Y., Sun, Y., Wu, T., Liang, Y.Z., Yu, X., Zhang, J.M.: 2010 International Conference on Electrical and Control Engineering, pp. 3331–3334. IEEE (2010) 12. Etxeberria-Otadui, I., De Heredia, A.L., Gazta˜naga, H., Bacha, S., Reyero, M.R.: A single synchronous frame hybrid (SSFH) multifrequency controller for power active filters. IEEE Trans. Ind. Electron. 53(5), 1640 (2006) 13. Tremblay, E., Atayde, S., Chandra, A.: Comparative study of control strategies for the doubly fed induction generator in wind energy conversion systems: a DSP-based implementation approach. IEEE Trans. Sustain. Energy 2(3), 288 (2011) 14. Lucas, H., Pinnington, S., Cabeza, L.F.: Education and training gaps in the renewable energy sector. Solar Energy 173, 449 (2018) 15. Robinett III, R.D., Wilson, D.G.: Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov Analysis. Springer Science & Business Media, London (2011) 16. Mensou, S., Essadki, A., Nasser, T., Idrissi, B.B.: An efficient nonlinear backstepping controller approach of a wind power generation system based on a DFIG. Int. J. Renew. Energy Res. 7(4), 1520 (2017) 17. Novotny, D.W., Lipo, T.A.: Vector Control and Dynamics of AC Drives, vol. 1. Oxford University Press, New York (1996) 18. Franco, R., Capovilla, C.E., Jacomini, R., Altana, J., Sguarezi Filho, A.J.: IECON 2014-40th Annual Conference of the IEEE Industrial Electronics Society, pp. 1906–1911. IEEE (2014) 19. Mensou, S., Essadki, A., Nasser, T., Idrissi, B.B.: A direct power control of a DFIG basedWECS during symmetrical voltage dips. Prot. Control Mod. Power Syst. 5(1), 5 (2020)

Model of a DC Motor with Worm Gearbox V´ıtor H. Pinto1,3(B) , Jos´e Gon¸calves2 , and Paulo Costa1,3 1

Faculty of Engineering, FEUP - University of Porto, Porto, Portugal {vitorpinto,paco}@fe.up.pt 2 IPB - Polytechnic Institute of Bragan¸ca, Bragan¸ca, Portugal [email protected] 3 INESC TEC - Institute for Systems and Computer Engineering, Technology and Science, CRIIS - Centre for Robotics in Industry and Intelligent Systems, Porto, Portugal

Abstract. In this paper, the modeling of a system based on a DC Motor with Worm Gearbox is presented. Worm gearboxes are typically applied when its compactness is an important factor, as well as an orthogonal redirectioning is required. One of the greatest advantages of worm gears is its unique self-locking characteristic. This means that the gear can only rotate by its input side, and cannot be actuated through the load side. Using a DC motor with a worm gearbox is a solution that guarantees that, for instance, in a robotic manipulator, when the arm’s joint reaches a desired angle, it does not move until a next required setpoint. Modeling accurately this system is crucial in order to develop its control in a more efficient way.

Keywords: Modeling system

1

· DC motor · Worm gearbox · Motor gearbox

Introduction

DC (Direct-Current) motors are important components on electrical machines such as wind turbines, domestic devices, computer fans and even the vibration system of current days mobile phones. Also, a field of study where DC motors are widely used is robotics. Whether in the use of these to actuate the traction system of a vehicle, to change the angle of a joint of a robotic manipulator, or even to rotate the sensor of a LiDAR (Light Detection And Ranging), there are countless uses for this kind of motors in robotics. However, these motors have a non-linear behaviour which comes from the fact that it cannot rotate at any powering voltage, having always a minimum rotation torque (starting torque) only overtaken when a certain voltage is reached, commonly called the motor dead zone. To compensate for these, several control methods can be applied, such as [1–3], just to state a few. Also, as Mahajan and c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 638–647, 2021. https://doi.org/10.1007/978-3-030-58653-9_61

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Deshpande [4] present, when considering the full DC motor model, friction can also introduce some non-linearities. In case the use of a DC motor is not very influenced in this dead zone, or if the control can also be linearized outside it, the most applied model is the steady state one [5]. Nevertheless, if it is necessary to control the system in all its operating regimes, namely when starting and stopping it, as an instance for speed control, it is necessary to consider the adequate model and find compatible control solutions. Worm gearboxes are typically applied when its compactness is an important factor, as well as an orthogonal redirectioning is required. One of the greatest advantages of worm gears is its unique self-locking characteristic. This means that the gear can only rotate by its input side, and cannot be actuated through the load side. As Ye and Wu [6] state, these features make worm gearboxes useful for multiple types of robotic vehicles and manipulators. However, any gearbox has a main non-linearity, which is the movement of its output shaft without rotating its input, called backlash. The vibratory behaviour is also associated with worm gears when systems are affected by gravity loading, as is the case with robotic manipulators, as mentioned by May et al. [7]. Using a DC motor with a worm gearbox is a solution that guarantees that, for instance, in a robotic manipulator, when the arm’s joint reaches the desired angle it does not move until the next required change. However, the friction from the two subsystems, and also from the coupling between them can surely be a source of motion disturbance and can cause a performance degradation, if not considered. Modeling the system is crucial to understand its behaviour under multiple conditions and also to develop its control. Having the model, one can put a signal in the process input and analyse how the system reacts, obtaining the system output. The system’s transfer function can be interpreted as the process model. Robotic systems often include DC motors, being this crucial to improve the control performance. Thus, obtaining a realistic model of the system to be controlled is an advantage when defining and tuning the controller. In addition to being more intuitive to decide which kind of controller is more appropriate, obtaining the system response to various inputs is also more reliable. Obtaining a more realistic model allows to know, as close to reality as possible, how the system will behave after designing and tuning the controllers [8]. Given this, estimation techniques must be applied to estimate the unknown or difficult to quantify parameters with accuracy. These techniques can be applied online or offline, as mentioned by Dupuis et al. [9], but are a very complex problem, and can result in an inaccurate estimation of the parameters. Examples of online parameter identification are present in [10,11], while for offline methods, examples can be [12–14]. Considering that online estimation methods are relatively more complex, as well as that the conditions associated with the motor presumably will not change when it is actuated, an offline identification method will be applied.

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The modeling of the motor + worm gearbox will be performed through real hardware tests. Throughout this paper, the process of modeling the motor-gearbox system will be presented. To present these, an introduction to the topic was developed in Sect. 1, and the proposed models will be described across Sect. 2. Section 3 will present the results obtained by the performed experiments, and in Sect. 4 there are presented the conclusions and future work.

2

Proposed Model

First, it is important to mention that, in order to understand the impact that the gearbox had on the obtained data, it was decoupled from the motor shaft, and the same tests were carried out. The conclusion is that the obtained data are practically the same, only affected by the correction factors related to speed and torque, due to the reduction of the gearbox. Along this section, the generic model for a DC motor will be presented, which is equivalent to the motor-gearbox system. Also, taking into account that some hardware tests must be performed to estimate some parameters, they will also be detailed. Despite what was mentioned in previous paragraph, all the tests were performed with the gearbox coupled to the motor, in order to obtain a reliable model of the motor+gearbox system. Taking into account the decision of using a motor with a worm gearbox, it was decided to use an off-the-shelf solution, specifically the JGY-371, which is a low cost popular actuator, being a 12 VDC motor with built-in encoders, attached to a 340:1 reduction worm gearbox. It was chosen that the motor speed would be 20 rpm, which is the best solution for the purpose defined for the project. With this choice, despite the low maximum angular speed of the motor, the joints can benefit from more torque. In addition to the fact that the incorporated encoder can be applied to collect data in the modeling process, it will also be extremely useful during the motor movement, making it possible to control in closed-loop with access to speed information in real time. Taking into account the above mentioned considerations, the motor model is further described using the equations associated with it. These are present all over the literature associated with DC motors, so no references will be presented to specific publications. Thus, the general model of a DC motor is as follows: di + Ev dt Ev = k · ω(t)

vi (t) = R · i(t) + L ·

dω 1 = · (Tm − Bv · ω(t) − Tq ) dt J Tm = k · i(t)

(1) (2) (3) (4)

where vi is the voltage applied to the system (V), i is the electrical current (A), R the resistance (Ω), L the impedance (H), and Ev the Back Electromotive Force. This force is obtained through the second presented equation, where k

DC Motor with Worm Gearbox

641

is the motor constant, and ω the angular speed (rad/s). In the last equation J represents the moment of inertia (kg · m2 ), Tm the motor torque (N·m), obtained by the fourth equation, Bv the viscous friction coefficient and Tq the static friction coefficient. In the Laplace domain, the system can be represented as: vi (s) = R · i(s) + s · L · i(s) + k · ω(s) J · s · ω(s) = k · i(s) − Bv · ω(s)

(5) (6)

The transfer function of the system will then be: ω(s) k = vi (s) (s · J + Bv ) · (R + s · L) + k 2

(7)

Adjusting some variables, to facilitate its later use in the estimation of some parameters, it becomes: s2 + s( R L +

k J·L Bv J )+

2

v +k ( R·BJ·L )

(8)

There are some hardware tests that allow to estimate some unknown motor constants, like R, k, Bv and Tq . These tests are simple to perform and consist on: – Powering the motor with multiple pre-known values of supply voltage, allowing it to reach the steady-state; – Input a voltage step to the motor terminals; From these experiments, some important variables must be obtained, such as motor speed, using the built-in encoder, motor withdraw current, using a current sensor, and applied motor voltage, with a voltage sensor connected directly to the motor terminals. The referred acquisition system is shown in Fig. 1. How these values are applied in practice will be detailed in Sect. 3.

Fig. 1. Acquisition system diagram

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Results

Both tests mentioned in the previous section were performed in the motorgearbox system. For the first set of tests, an average of the measurements was obtained, namely motor speed, voltage applied to the motor, absorbed current and power. The voltage applied to the motor terminals started at 0.5VDC and was increased by steps of 0.5 VDC until it reached the 12VDC. Finally, the values were processed when required and compiled in the graphs from Figs. 2, 4 and 3.

Fig. 2. Plot of angular speed vs. voltage

By slightly changing Eqs. 1–4, it is possible to obtain first-order functions, which can be compared with the trendlines obtained in the graphs and consequently obtained the values of the mentioned constants. Note that, for these di changes it was considered the steady state where, for instance, the value of L · dt dω can be unvalued, as well as dt , becoming: i(t) vi (t) =R +k ω(t) ω(t) k · i(t) = Bv · ω(t) + Tq

(9) (10)

Analyzing the trendlines, and since Eq. 9 corresponds to Fig. 3, it is possible to directly obtain that the motor resistance R and constant k are equivalent to 8,6538 and 0,0174, respectively. Also, considering that Eq. 10 is depicted in Fig. 4, the values of Tq and Bv are 0.608 × 10−3 and 5.975 × 10−7 , respectively.

DC Motor with Worm Gearbox

Fig. 3. Plot of

V oltage AngularSpeed

vs.

643

Current AngularSpeed

Fig. 4. Plot of k · current vs. angular speed ω

At this point, it is possible to estimate the parameters that are missing from the model, using the already calculated parameters. Thus, considering the values of R, k, Bv and Tq , it is possible to estimate the values of L and J associated with the system. For this estimation, the second test was performed, in which the system’s response to a voltage step in the motor supply is analyzed over time. To analyze the adequacy of the model to the real system, first and second

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Fig. 5. Plot of angular speed vs. time

order models were used, as described in Eqs. 11 and 13, respectively. Then, the following equations refer to that same system in the time-domain. k1 (s + a1 )

(11)

k1 · (1 − e−a1 ·t )

(12)

Using a1 as the average of the angular speeds after the system stabilizes, reaching the steady-state, and solving the equation to obtain the minimum error between the real and the model’s angular speed, the values of k1 and a1 are 0.3684 and 42.4062, respectively. The data from the optimization can be found in Table 1. k2 · a2 · b2 (13) (s + a2 )(s + b2 ) k2 s2 + s(a2 + b2 ) + (a2 · b2 ) a2 =

R L

b2 =

Bv J

k2 k2 k2 · e−a2 ·t − · e−b2 ·t − a2 · b2 a2 (b2 − a2 ) b2 · (a2 − b2 )

(14) (15) (16)

The second-order model considers constants k2 , a2 and b2 , and the optimization cycle was also performed to reach the minimum error between the real and modeled angular speed. The data obtained from this optimization is also presented in Table 1, and the parameters k2 , a2 and b2 reached values

DC Motor with Worm Gearbox Table 1. First and second order models results Time (s) ω (rad/s) ωest 1st order (rad/s) error ωest 2nd order (rad/s) error 0.000

0.000

0.000

0.000 0.000

0.000

0.005

0.039

0.024

0.015 0.040

0.001

0.010

0.100

0.074

0.026 0.102

0.002

0.015

0.157

0.130

0.027 0.157

0.000

0.020

0.205

0.181

0.023 0.202

0.003

0.025

0.239

0.226

0.013 0.237

0.002

0.030

0.269

0.261

0.007 0.265

0.004

0.035

0.285

0.289

0.004 0.287

0.002

0.040

0.305

0.310

0.005 0.304

0.001

0.045

0.316

0.326

0.011 0.318

0.002

0.050

0.325

0.338

0.014 0.329

0.004

0.055

0.335

0.347

0.012 0.337

0.002

0.060

0.344

0.353

0.009 0.344

0.000

0.065

0.347

0.358

0.011 0.349

0.002

0.070

0.351

0.361

0.010 0.353

0.002

0.075

0.358

0.363

0.005 0.356

0.002

0.080

0.362

0.365

0.003 0.359

0.003

0.085

0.362

0.366

0.004 0.361

0.001

0.090

0.362

0.367

0.005 0.362

0.001

0.095

0.366

0.367

0.001 0.364

0.002

0.100

0.362

0.367

0.006 0.365

0.003

0.105

0.369

0.368

0.001 0.365

0.003

0.110

0.363

0.368

0.005 0.366

0.003

0.115

0.366

0.368

0.002 0.367

0.000

0.120

0.369

0.368

0.001 0.367

0.002

0.125

0.363

0.368

0.005 0.367

0.004

0.130

0.370

0.368

0.002 0.367

0.002

0.135

0.369

0.368

0.001 0.368

0.001

0.140

0.368

0.368

0.001 0.368

0.000

0.145

0.370

0.368

0.002 0.368

0.002

0.150

0.369

0.368

0.000 0.368

0.001

0.155

0.370

0.368

0.001 0.368

0.002

0.160

0.370

0.368

0.001 0.368

0.002

0.165

0.369

0.368

0.000 0.368

0.001

645

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V. H. Pinto et al.

0.3684, 315.9816 and 47.7916, respectively. Then, these functions were graphically designed and compared with the obtained data to understand which is the one that fits the best. The representation of both functions can be found in Fig. 5, where the blue graphic represents the obtained data, the green one is the first-order system and the orange one represents the second-order system. As can be seen, the second order model is undoubtedly the most suitable model, so its values will be used for the estimation of the engine parameters. Thus, it is obtained that J and L take values of 8.5075 and 0.0238, respectively.

4

Conclusions and Future Work

Using a simple acquisition system, with low-cost sensors, it was possible to obtain reliable data on several interesting variables about a motor-gearbox system. The monitored variables were the angular speed of the motor, as well as the current consumed by it, and the supply voltage. Based on these data, it was possible to build graphs that allowed, using equations from the physical model of the DC motor, to estimate simpler parameters related to the system, namely its resistance and constants. Then, two models, first and second order, were considered for comparison with the data obtained in reality of angular velocity in order of time. Using previously obtained data and optimizing to minimize the angular velocity error, it was possible to realize that the second-order model is the one that best reflects the physical behavior of the motor and to estimate its impedance and moment of inertia parameters. In terms of future work, it will be interesting to design a controller suitable for the system, considering its specificities, namely the gearbox unique feature of self-locking. Acknowledgements. This work is financed by National Funds through the Portuguese funding agency, FCT - Funda¸ca ˜o para a Ciˆencia e a Tecnologia, within project UIDB/50014/2020.

References 1. Zhou, Z., Tan, Y., Xie, Y., Dong, R.: State estimation of a compound non-smooth sandwich system with backlash and dead zone. Mech. Syst. Signal Process. 83, 439–449 (2017) 2. Galuppini, G., Magni, L., Raimondo, D.M.: Model predictive control of systems with deadzone and saturation. Control Eng. Pract. 78, 56–64 (2018) 3. Castillo, O., Aguilar, L.T.: Fuzzy control for systems with dead-zone and backlash. In: Type-2 Fuzzy Logic in Control of Nonsmooth Systems, pp. 55–71. Springer (2019) 4. Mahajan, N.P., Deshpande, S.: Study of nonlinear behavior of DC motor using modeling and simulation. Int. J. Sci. Res. Publ. 3(3), 576–580 (2013) 5. Pinto, V.H., Gon¸calves, J.A., Costa, P.: Introduction to DC motors for engineering students based on laboratory experiments. In: 2019 6th International Conference on Control, Decision and Information Technologies (CoDIT), pp. 1522–1527. IEEE (2019)

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6. Yeh, T.J., Wu, F.K.: Modeling and robust control of worm-gear driven systems. Simul. Model. Pract. Theory 17(5), 767 – 777 (2009). http://www.sciencedirect. com/science/article/pii/S1569190X09000045 7. May, D., Jayasuriya, S., Mooring, B.: Modeling and control of a manipulator joint driven through a worm gear transmission. J. Vib. Control 6(1), 85–111 (2000) 8. Brynjarsd´ ottir, J., OHagan, A.: Learning about physical parameters: the importance of model discrepancy. Inverse Prob. 30(11), 114007 (2014) 9. Dupuis, A., Ghribi, M., Kaddouri, A.: Multiobjective genetic estimation of dc motor parameters and load torque. In: 2004 IEEE International Conference on Industrial Technology, IEEE ICIT 2004, vol. 3, pp. 1511–1514. IEEE (2004) 10. Liu, G., Hu, C.: Adaptive modeling of brushless DC motor drive system. In: 2018 IEEE 4th International Conference on Control Science and Systems Engineering (ICCSSE), pp. 173–177. IEEE (2018) 11. Campos, R.D.F., Couto, E., de Oliveira, J., Nied, A.: On-line parameter identification of an induction motor with closed-loop speed control using the least square method. J. Dyn. Syst. Meas. Control. 139(7), 071010 (2017) 12. Saab, S.S., Kaed-Bey, R.A.: Parameter identification of a dc motor: an experimental approach. In: ICECS 2001. 8th IEEE International Conference on Electronics, Circuits and Systems (Cat. No. 01EX483), vol. 2, pp. 981–984. IEEE (2001) 13. Gon¸calves, J., Lima, J., Costa, P.G.: Dc motors modeling resorting to a simple setup and estimation procedure. In: CONTROLO 2014–Proceedings of the 11th Portuguese Conference on Automatic Control, pp. 441–447. Springer (2015) 14. Nayak, B., Sahu, S.: Parameter estimation of DC motor through whale optimization algorithm. Int. J. Power Electron. Drive Syst. 10(1), 83 (2019)

Trajectory Planning for Landing with a Direct Optimal Control Algorithm Bertinho A. Costa(B)

and Jo˜ ao M. Lemos

INESC-ID/IST/ULisbon, Rua Alves Redol, 9, 1000-029 Lisboa, Portugal {bac,jlml}@inesc-id.pt

Abstract. This article describes a methodology to introduce graduated students to the topic of optimal trajectory planning, where the initial and final states are known but the trajectory between these states is not known. To motivate the problem, a spacecraft landing manoeuvre is considered. An analysis of the dynamic model is performed to simplify the application of the optimization method that uses a direct method. Unconstrained optimization is used iteratively with adjustable weighting factors to compute the control actions and to progressively fulfil the problem constraints.

Keywords: Trajectory optimization optimization

1

· State predictors · Constrained

Introduction

Trajectory planning and optimization play an important role in the design of automatic control systems. It can be seen as a first stage in tracking control problems where trajectory reference signals are not known and must be designed. It is interesting to observe that in control systems courses it is usually assumed that references are provided. But there are cases where no explicit information is available to connect the initial state and final target state. The references must be designed from several constraints on manipulated variables such as saturation values or the maximum quantity of some kind that may be used, such as fuel or the nutrients. Depending on the problem to be addressed, a simplification of the dynamics may help trajectory design. The design of optimal trajectories is not a new problem and has been tackled by applying optimization principles and techniques such as optimal control [3], dynamic programming [1,2], and techniques that mimic phenomena in nature such as the behaviour of insects (ants, ...) and particle swarm optimization [4]. In the case of optimal control, using the “classic” approach (indirect methods), the initial problem that has a This work was supported by Funda¸ca ˜o para a Ciˆencia e Tecnologia under the research project HARMONY AAC n.2/SAICT/2017 - 031411 the pluriannual funding program UIDB/50021/2020. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 648–657, 2021. https://doi.org/10.1007/978-3-030-58653-9_62

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state dimension d is transformed in a problem where the state dimension is 2 ∗ d because it requires the solution of the state and costate equations with boundary values. This problem may be very difficult to solve. An alternative approach is the application of direct methods, where the system dynamics is discretized and the problem is transformed in a minimization of a cost function with a set of linear or non-linear algebraic constraints. The objective of this paper is to contribute to address the design of trajectories, by presenting a methodology that involves learning objectives: 1) Formulation of a continuous-time statespace model; 2) Transformation of the continuous-time model into a discretetime model in order to explore the direct method; 3) To understand the relation, expressed as a mathematical expression, that relates the discrete-time initial state vector to the state vector in a future time instant; 4) To formulate and to solve the unconstraint minimization problem; 5) To develop the code/software to solve the unconstraint minimization problem; 6) And finally, to be able to evaluate the results and to apply the methodology to other similar problems. To motivate the problem, the landing on the moon of the Apollo Lunar Module spacecraft is considered, where this vehicle starts from a given altitude and attitude, near the surface of the moon, and must land on a hypothetical location. This article is organized as follows. Section 2 describes the model of spacecraft. Section 3 addresses the analysis of the model and describes a transformation that allows to treat the model as being linear. The fuel consumption minimization and the minimum time problem are considered briefly. Section 4 describes the approach used to discretize the continuous-time dynamics into the discrete-time framework to explore the direct method approach. A normalization is applied to improve/avoid numerical errors. Section 5 presents the steps to obtain the set of algebraic equations that relate the initial state to the final target state and the unconstrained optimization. The optimization algorithm is described in Sect. 6. Computer numerical simulation results are presented in Sect. 7. The conclusions are presented in the last section where additional problems to the application of the methodology are proposed.

2

Spacecraft Dynnamic Model

A simplified representation of a spacecraft, and references (body and local) systems used, are presented in Fig. 1 where the movement is in the Y Z plane and a flat planet model is used. Lateral rockets are employed to generate (thrust) forces, F1 (t) ≥ 0 and F2 (t) ≥ 0, that are used to move the spacecraft during the landing manoeuvre and to change its attitude that is described by the angle θ(t). The mass of the spacecraft, without fuel, is represented by mS . The fuel mass, meaning the fuel and oxidizer, is represented by mF (t) and is consumed at a rate that depends on the thrust (forces) F1 (t) and F2 (t) that are the manipulated variables. In this problem, the equations that describe the dynamics of the spacecraft are dm(t)vy (t) = −(F1 (t) + F2 (t)) sin(θ(t)) , dt

dsy (t) = vy (t) , dt

(1)

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Fig. 1. Description of the reference systems used to describe the dynamics of the spacecraft.

dm(t)vz (t) dsz (t) = (F1 (t) + F2 (t)) cos(θ(t)) − gp m(t) , = vz (t) , dt dt dmF (t) = −ξ(F1 (t) + F2 (t))) , m(t) = mS + mF (t) , dt where vy (t) and vz (t) represent the horizontal and vertical speeds (centre of mass) at the continuous time t, and sy (t) and sz (t) represent the position of the spacecraft. The dynamics of the attitude is described by dI(t)ω(t) = RI (F1 (t) − F2 (t)) , dt

dθ(t) = ω(t) , dt

(2)

where the angular rate of the attitude ω(t) dy depends on the moment of inertia I(t) and on the difference between the forces F1 (t) and F2 (t) and the normalized factor RI . As a simplification, the shape of the spacecraft is considered spherical, where the mass is uniformly distributed. As an example to choose the parameters for the model, the Apollo Lunar Module data was used, mS = 7000 kg, the initial mass of the fuel (at time t0 ) is mF (t0 ) = 8000 kg, maximum value for F1 (and F2 ) is 22500 N, and thrust-tofuel mass ratio ξ = 4.0 × 10−4 kg/(N.s). The initial position of the spacecraft is sy (t0 ) = 10000 m , sz (t0 ) = 5000 m, with initial velocity vy (t0 ) = 100 m/s and vz (t0 ) = 0 m/s. The local gravity value gp = 1.63 m/s2 . The objective is to find a trajectory, an optimal trajectory that minimizes fuel consumption, to land the spacecraft at position (0,0), with zero terminal velocity and zero attitude angle of the spacecraft.

3

Model Analysis and Reduction

Given that the fuel mass changes over time, it is important to evaluate its impact on the dynamic model. Considering the thrust-to-mass coefficient ξ = 4.0 × 10−4 kg/(N.s) and the maximum thrust 45000 N, the fuel consumption is 18 kg/s. For a flight time of 100 s the fuel consumption will be 1800 kg which is 12% of

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the initial total mass. Based on this result the mass change can be considered a slow variation. It will be assumed that attitude dynamics is much faster than the translation dynamics and thus, in a first approach it will not be included in the design of the translation trajectory. A key point on the translation dynamics is the presence of the non-linear terms (F1 (t) + F2 (t)) sin(θ(t)) and (F1 (t) + F2 (t)) cos(θ(t)) that may cause difficulties. However, it is possible to define the transformation uy (t) =

(F1 (t) + F2 (t)) sin(θ(t)) , m(t)

uz (t) =

(F1 (t) + F2 (t)) cos(θ(t)) , m(t)

(3)

that represent accelerations, and transforms the non-linear translation dynamics into a linear dynamics, where the fuel consumption is described by  dmF (t) = −ξmF (t)( (u2y (t) + u2z (t)) , m(t) = mS + mF (t) , dt where uy (t) and uz (t) are the new (virtual) manipulated variables. Using the new manipulated variables, the time evolution of the fuel mass is described by  t  ( (u2y (t) + u2z (t))dt) , (4) mF (t) = mF (t0 )exp(−ξ t0

from it can be concluded that the minimization of the fuel consumption is equivt  alent to the minimization of time integral t0 ( (u2y (t) + u2z (t))dt), that is the integral of the thrust over time. At this point it is interesting to consider a vertical landing manoeuvre such that y = 0, with the zero horizontal velocity (vy (t) = 0) and vz (t0 ) = 0. The objective is that at the terminal time tf , vz (tf ) = 0 and z(tf ) = 0. For this case the fuel mass consumption is given by  t uz (t)dt) , (5) mF (t) = mF (t0 )exp(−ξ t0

being remarked that uz (t) ≥ 0. Integrating the equation that defines  tf uz (t)dt = gp (tf − t0 ) .

dvz (t) dt

yields (6)

t0

This equation shows that the fuel consumption minimization problem is related to the minimum flight time problem. Extending the flight time implies an additional fuel consumption to compensate the local gravity gp . But, if tf is too small, then the spacecraft may not have enough time to reach the target position. Considering now the vertical position, it is related to uz (.) and gp by the expression  tf  τ (tf − t0 )2 . (7) uz (μ)dμdτ = −sz (t0 ) + gp 2 t0 t0  From   this example occurs the question how to select uz (t) such that uz (t)dt and uz (t)dμτ fulfil the constraint described by Eq. (6) and (7)? This problem will be addressed using optimization based on a direct method.

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Discretisation and Normalization of the Dynamics

The discrete-time model can be obtained from the continuous-time model using several methods, such as the Euler method. But in this work, the Zero-orderHold (ZoH) method is applied. The continuous-time t is discretized such that the discrete-time k is defined by t = kh + t0 , being h the sampling time. The continuous-time model is integrated during the sampling time interval with the applied forces F1 (.) and F2 (t) being constant. An important aspect that must be considered is the normalization of the vector state (model) to avoid computational numerical errors that may occurs for large prediction horizons. The normalization applied uses the horizontal distance Ly = sy (tf ) − sy (t0 ) and the vertical distance Lz = sz (tf ) − sz (t0 ) from which a new a new state is defined in discrete-time domain by s[k] = [vy [k]/Ly , sy [k]/Ly , vz [k]/Lz , sy [k]/Lz ]. Using this normalization, the initial state is selected as s[0] = [vy [0]/Ly , −1, vz [0]/Lz , 1] and the final state is selected as s[kf ] = [0, 0, 0, 0] . This terminal state describes the coordinates of the landing site and the velocity. The following discrete-time model is obtained by integrating the continuous-time translation dynamics using the ZoH strategy, s[k + 1] = As[k] + B1 u[k] + B2 gp ,

(8)

where the input is

u[k] = [uy [k], uz [k]] ⎡ h ⎤ ⎡ ⎡ ⎤ − Ly 0 0 1000 ⎢ − h2 0 ⎥ 0 ⎢h 1 0 0⎥ ⎢ ⎢ 2Ly ⎥ ⎢ ⎥ and A = ⎢ h ⎥ ; B2 = ⎣ − h ⎣ 0 0 1 0 ⎦ ; B1 = ⎢ ⎣ 0 ⎦ Ly Lz h2 00h1 h2 − 2L 0 z

⎤ ⎥ ⎥. ⎦

2Lz

5

State Predictors and Unconstrained Optimization

The objective is to find a trajectory that connects the initial state s[0] to the final state s[n], that minimizes the fuel consumption and fulfil the constraints on uy [.] and uz [.]. The predictor for s[n], with tf = t0 + nh (n defines the discrete-time prediction horizon), can be obtained from s[0] by iteratively applying Eq. (8), s[k + 1] = As[k] + B1 u[k] + B2 gp s[k + 2] = As[k + 1] + B1 u[k + 1] + B2 gp = A2 s[k] + AB1 u[k] + B1 u[k + 1] + (AB1 + B1 )g ,

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that yielding s[k + n] = S(n)s[k] + W (n)U [k] + G(n)gp , where

(9)

U [k] = [u[k] , u[k + 1] , . . . , u[k + (n − 1)] ] ]

S(n) = An ,

 W (n) = An−1 B1 , . . . , AB1 , B1 ,

G(n) =

n−1

Ai B2 ,

(10)

i=0

with adequate dimensions. In order to compute U [k] the following cost function is defined J[n] = (sT [n] − s[n]) Q(sT [n] − s[n]) + U  [k]F U [k] ,

(11)

where the vector sT [n] represents the target state. The matrices Q and F are symmetric and positive definite and are used as adjustable knobs to penalize undesired values of the components of the final state s[n], or high values of the control actions u[.]. Matrices Q and F are adjusted iteratively such that the sequence of manipulated variables are kept in the adequate range and the terminal error is acceptable. The minimization of the cost function is subjected to (9). Inserting (9) in Eq. (11) and computing the gradient of J[n] in respect to vector U [t], and equating it to zero yields the solution U [k] = (F + W (n) QW (n))−1 W  (n)Q)(sT [n] − S(n)s[k] − G(n)) .

6

Minimization Procedure

Constrained optimization is a difficult problem to solve and there is no closed form available. Eq. (5) has 3 degrees of freedom, the discrete-time horizon n, the matrix Q and the matrix F. For small values for n it is plausible that the system state has no time to evolve to be near the target state, or if so the computed manipulated variables values u[.] could be above the saturation values, thus not fulfilling the constraints. To address this problem a heuristic procedure can be devised. This procedure is summarized by the following steps: 1. Find a time horizon value n such that the state final error is acceptable. Knowing the time scale of the dynamics helps to select an initial value for n. 2. Evaluate the state terminal error. Increase the time horizon to obtain a terminal state with lower error. Additionally, the corresponding components on the matrix Q can be increased. 3. Evaluate the new terminal state error. 4. Verification if u(.) fulfil the constraints. Increase the time horizon to lower the levels of u(.). 5. Penalize high levels of u(.) by increasing the corresponding components on the matrix F. 6. Repeat the previous points until a satisfactory solution is obtained, or just quit without a satisfactory solution after x iterations!

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7

Numerical Results

In order to demonstrate the application of the algorithm, the following scenario is considered. The initial position is defined by sy (0) = −10000 m (t0 = 0s), sz (0) = 5000 m, initial velocity vy (0) = 100 m/s, vz (0) = 0 m/s. With these values, the spacecraft takes, in free-flight, 100 s to go from sy (0) to the target 0 m (horizontal displacement), and it takes 78.8 s to go from sy (t0 ) to target 0 m (vertical displacement). A sampling time of h = 1 s is chosen. From the above figures a reasonable initial value for n = tf /h is n = 78. This fact means that the spacecraft has time to approach the surface. The thrust much be applied to change the velocity of the spacecraft such that it will reach the target position (0, 0). For an initial computer numerical simulation, the matrix Q is set to the identity matrix Q = I, and the matrix F is set to F = 1 × 10−4 I. yz trajectory

1

v y [t]/L y and vz [t]/L z (.-)

0.02

0.9

0.8

0.7

0.01 0 -0.01 -0.02

s z [t]/L z [m]

0.6 -0.03 0

10

20

30

0.5

40

50

60

70

80

50

60

70

80

t [s] 1

sy [t]/L y and sz [t]/L z (.-)

0.4

0.3

0.2

0.1

0 -1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

s y [t]/L y [m]

(a) Trajectory yz.

-0.2

-0.1

0

0.5

0

-0.5

-1 0

10

20

30

40

t [s]

(b) Top, evolution of the normalized velocities. Bottom evolution of the normalized position.

Fig. 2. Sim1: in this simulation the spacecraft terminal position is near the target position but the terminal velocity has large error.

The results for this configuration are presented respectively in Figs. 2(a), 2(b), 3(a) and 3(b). From Fig. 2(a) it can be concluded that the spacecraft terminal position is near the target position, but the horizontal terminal error can be considered large. From Fig. 2(b) the terminal speed is not adequate. From Fig. 3(a) it can be concluded that the saturation values on the thrust are not reached. From the plot of the attitude it can be concluded that the spacecraft is accelerated during the initial phase of the flight and at a later phase the attitude changes, reducing the horizontal velocity. Figure 3(b) justifies the hypothesis that the fuel mass changes slowly. From the results obtained with the initial simulation, it was decided to decrease the terminal velocity error by increasing the components of Q associated with speed, such that diag(Q) = [10000, 1, 10000, 1]. The implication of this change was an increase on the manipulated variables values that became

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higher that than the constraints. This motivates the change on the value of the prediction horizon from n = 80 to an n = 150, to have more time to perform the manoeuver and by that to keep the thrust below the constraint. Additionally, it was decided to penalize the control action at the terminal time tf = 151 s because it is assumed that corresponds to the touch-down phase where engines must stop. The results are shown in Figs. 4(a), 4(b), 5(a) and 5(b). yz trajectory

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trajectory is changed to cope with the initial velocity, the yz trajectories become larger, and the terminal position error increases, and the thrust is higher than the maximum allowed value. Other problems can be explored with this methodology such as “rendezvous” spacecraft problems. The interesting aspect of this approach is that it allows to explore and learn the implications of using the parameters of the algorithm, n, Q and F .

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This article describes a methodology to introduce control students the topic of optimal trajectory planning, where the trajectory is defined by the initial and

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final trajectory state but there is no explicit information on how to connect these two states. To motivate the problem, a spacecraft landing manoeuvre is considered to present the steps of the methodology. It uses a direct method with state predictors to describe the relation between the final state to the initial and the manipulated variables. A function cost is minimized using state predictors. Numerical computer simulations are presented to describe the selection of the time horizon, and the weighting factors of the cost function (n, Q and F ). In order to assess student’s knowledge, similar problems can be used, for example, 1) Vertical landing and vertical departure imply a change in the number of manipulated variables; 2) To control the attitude angle at the terminal state, this implies an expansion of the model and 3) A more complex problem to explore is the “rendezvous” manoeuvre that can be divided into two phases, approach phase and docking phase.

References 1. Chai, R., Savvaris, A., Tsourdos, A., Chai, S., Xia, Y.: A review of optimization techniques in spacecraft flight trajectory design. Prog. Aerosp. Sci. 109, 100543 (2019) 2. Chu, H., Ma, L., Wang, K., Shao, Z., Song, Z.: Trajectory optimization for lunar soft landing with complex constraints. Adv. Space Res. 60(9), 2060–2076 (2017) 3. Conway, B.: Spacecraft Trajectory Optimization. Cambridge University Press, New York (2010) 4. Zheng, J., Lu, C., Gao, L.: Multi-objective cellular particle swarm optimization for wellbore trajectory design. Appl. Soft Comput. 77, 106–117 (2019)

Accelerated Generalized Correntropy Interior Point Method in Power System State Estimation Hamed Moayyed(B) , Diyako Ghaderyan, Yassine Boukili, and A. Pedro Aguiar Faculty of Engineering, University of Porto, Porto, Portugal {hmoayyed,dghaderyan,yassine,pedro.aguiar}@fe.up.pt

Abstract. Classical Weighted Least Squares (WLS) is a well-known and broadly applicable method in many state estimation problems. In power system networks, WLS is particularly used because of its stability and reliability in the cases that measurement noise are Gaussian. Nowadays, with the use of renewable energy sources and the migration to smart grids WLS is no more appropriate because the noises are far from being Gaussian. Recently, a novel state estimation algorithm denoted Generalized Correntropy Interior-Point method (GCIP) was presented that can deal with measurements contaminated by gross errors. Under that conditions, the superiority of GCIP is confirmed in a variety of tests. This paper presents an improved GCIP in terms of computational efficiency. The main computational burden of GCIP arises from a large dimension matrix of the correction equation. By looking into the structure of the data, a new arrangement for this matrix with lower order is presented that helps to reduce computational time remarkably. The efficiency of new method was tested with different IEEE benchmark systems. Keywords: Power system · State estimation · Weighted Least Squares · Generalized Correntropy · Interior Point Method

1 1.1

Introduction Motivation and Relevant Literature

State Estimation (SE) has been widely used in the power control centers, in order to compensate the errors in the measurements. During the last decade, different SE strategies have been suggested, being the Weighted Least Square (WLS) approach the most common method [1,2]. The conventional WLS is considered to be one of the best approaches when measurement noises are Gaussian. However, when this is not the case, which happens in recent smart grids networks, standard WLS is not a good method particularly when dealing with Gross Errors (GEs). To overcome this problem, many improvements have been presented based on WLS method, like snapshot WLS [3], and Kalman filter for the transmission lines c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 658–667, 2021. https://doi.org/10.1007/978-3-030-58653-9_63

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[4]. Moreover, several SE approaches have been suggested to deal with different kinds of GEs. M-estimators are among these typical robust state estimation methods [5–7]. Other few recent SE works claimed their ability in dealing with GEs [8–10]. The important point which has not been adequately considered in the abovementioned works is the type of induced error. The fact is that in practice the measurement errors do not follow the assumed kind of errors [11]. Therefore, designing an SE methodology that could be adapted with different types of noises poses still some challenges. In order to deal with different type of GEs, recent literature suggests applying Information Theoretic Learning (ITL) concepts. One of these approaches with interesting properties is Correntropy loss function [12], which is based on Gaussian kernel. However, applying a fixed kernel causes unsatisfactory effects. In the case of flexibility and robustness, the Generalized Correntropy (GC) [13] is one step further, when uses generalized Gaussian density (GGD) function as the kernel. This property makes GC worthy, which helps GC to show considerably better performance than Correntropy. 1.2

Contributions and Organization

A novel SE approach was recently reported by author, based on Generalized Correntropy Interior Point (GCIP) method [14–16]. This method is based on two features: i) Generalized Correntropy as the cost function, and ii) Primal Dual Interior Point method as the solution methodology. This combination makes GCIP prominent in comparing other SE models, with various interesting properties and 99% confidence level in a variety of tests. In this paper, we present an Accelareted GCIP (A-GCIP) in terms of computation efficiency. The main contributions of this paper are as follows: 1. The description of a new arrangement for the correction equation of GCIP that helps to reduce computational time remarkably and leading to A-GCIP. 2. The development of a technique that improves the computation efficiency of GCIP leading to the A-GCIP. 3. The demonstration of the efficiency and robustness of the A-GCIP on different IEEE benchmark systems. This paper is organized as follows. An introduction to the Generalized Correntropy and Interior Point method is presented in the following sections. The theoretical basis and the mathematical properties of Improved-GCIP are addressed in Sect. 4. Case studies on standard IEEE test systems are given in Sect. 5. Finally, the conclusions are drawn in Sect. 6.

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Generalized Correntropy

Correntropy [12] measures the similarity between two ransom variable and has a strong relationship with Entropy [17,18]. In the framework of adaptive system’s parameters optimization, Correntropy maximization has impacts similar

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to those of Entropy minimization. Given two scalar random variables X and Y , Correntropy can be defined as   υσ (X, Y ) = E[κσ (X − Y )] = κσ (X − Y ) p(x, y) dxdy (1) xy

where κσ is a kernel operator with parameter (width) σ. Correntrpoy is symmetric, positive, bounded, and based on Gaussian kernel. If the Gaussian kernel is replaced by a GGD, it is denoted as the Generalized Correntropy [13]. The GGD is a family of distributions which its expression is given by r c exp (−| |c ) (2) Gc,σ (r) = 2σΓ (1 /c ) σ where c > 0 is a shape parameter, Γ (.) is the gamma function, and σ is the bandwidth parameter. The GGD is the kernel function of GC, which is defined as N 1  υˆc,σ (X, Y ) = Gc,σ (xi − yi ) (3) N i=1 where X and Y are two scalar random variables. The flexibility of GC is apparent when modifying the shape parameter of c produces different distributions functions, e.g., c = 1 – Laplace density function, c = 2 – Gaussian function and c → ∞ – uniform distribution.

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Interior Point Method

The SE problems in power systems usually involves an optimization formulation with equality and inequality constraints. Interior Point Methods (IPM) are suitable to address that type of constrained optimization problems. The first suggestions for using IPM in SE problems related with power systems can be traced to the 80’s [19–22]. In general, SE resumes to a nonlinear programming problem that can be formulated in Primal-Dual IPM. In brief, the solution methodology is as follows: 1. Transform the inequalities constraints into equalities by adjoining nonnegative slack variables. 2. State the Lagrange optimality conditions. 3. Derive the Karush-Kuhn-Tucker (KKT) conditions. 4. Apply Newton’s method to the optimality conditions. In the following section, the theoretical basis and the mathematical formulation of Improved-GCIP are discussed in detail.

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In the following, a formulation of GCIP solver will be detailed. The optimization problem is set up as follows max J(x) = x

m   r c  c 1   i exp −  m i=1 2σΓ (1 /c ) σ

(4)

s.t. g(x) = 0, r = z − h(x) where r is denoted as the residual vector; z is the measurement vector (m × 1); x is the state vector (n×1); h(x) is function of measurement and g(x) : (Rn → Ro ) are equality constraints, where o is the number of equality constraints. The main objective function of GCIP is non-differentiable that poses difficulties in the implementation of classical optimisation methods. The key idea is to make it differentiable by introducing two slack variables p, q (see details in [14]), which are assumed to be non-negative, and rewrite (4) as max J(x) = x

  c  M pi + qi c 1  exp − M i=1 2σΓ (1 /c ) σ

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s.t. f (x) = 0, g(x) = 0, p, q ≥ 0 where f (x) = z − h(x) − p + q. The new function (5) is continuous and differentiable, therefore we can proceed with conventional optimization methods. 4.1

Solution Methodology

The solution methodology of GCIP is based on primal-dual IPM, as it was mentioned above. By introducing the Lagrange multipliers β ∈ Ro ; λ, γ, α ∈ Rm , we obtain the Lagrangian function L := J(x) − αT f (x) − β T g(x) − γ T q + λT p

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Applying the Karush-Khun-Tucker (KKT) optimization conditions to (6), we obtain a set of equations, being the following ones a subset of them (see datails in [14]) Lγi := γi qi = 0 Lλi := λi pi = 0

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A discussed in [23], the above complementary conditions need to be modified since it is not possible to directly solve them by Newton’s method. To this end, we introduce a parameter μ > 0 that relaxes (7) and (8) as Lγi μ := γi qi − μ = 0 Lλi μ := λi pi − μ = 0

(9) (10)

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The parameter μ takes the value μ = ρGap/2m where Gap = γ T q + λT p and ρ ∈ (0, 1) is the centering parameter. After the above considerations, we can now apply the Newton’s method to the resulting optimality conditions. To this end, we obtain the following set of equations (see details in [14]): γi dqi + qi dγi = −Lγi μ

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ei dpi + ei dqi − dαi − dγi = −Lqi

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where ∇2 h(x) and ∇2 g(x) are Hessian matrices of h(x) and g(x), d(.) denote the increments from iteration (k) to (k + 1), that is, d(.) = (.)ki+1 − (.)ki , and Eqs. (11)–(12) arise from (9)–(10), and the others from the rest of the KKT conditions not displayed in this paper. After some simplifications and ignoring the second derivative terms we reach to the following correction equation to obtain dα, dx and dβ ⎤⎡ ⎤ ⎡ ⎤ ⎡ dα ν A H 0 ⎣H T 0 −GT ⎦ ⎣ dx ⎦ = ⎣Lx ⎦ (18) Lβ dβ 0 −G 0 where G = ∇g(x); and H = ∇h(x). To obtain dγ and dλ we use (13)–(14), but first we need to compute dp and dq. To this end, denote

 −1 ai bi qi ei + γi qi ei = ci di pi ei pi ei + λi

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The following expressions are thus obtained dqi = n1i dαi + t1i dpi = n2i dαi + t2i

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where n1i = ai qi −bi pi ; n2i = ci qi −di pi , t1i = −ai (qi Lqi +Lμγi )−bi (pi Lpi +Lμλi ) and t2i = −ci (qi Lqi + Lμγi ) − di (pi Lpi + Lμλi ). Substituting (20) and (21) in (16) one obtains ∇h(x) dx + A dα = ν (22) where A ∈ Rm×m is a diagonal matrix, with the elements Ai = −n1i + n2i ; ν = z − h(x) − p + q + t1 + t2 and t1 , t2 ∈ Rm as defined in (20) and (21).

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To compute the primal and dual step sizes denoted by ΔP and ΔD we follow [24] , by setting ΔP = 0.9995 min{min(−qi /dqi : dqi < 0; −pi /dpi : dpi < 0), 1}

(23)

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The main computational burden of GCIP arises from the large dimension of the correction matrix in (18). The order of the correction Eq. (18) can be reduced by eliminating the variables corresponding to voltage magnitude measurements and line power flow measurements from (18) [25], taking into account that the number of voltage magnitude measurements and line power flow measurements is much larger than that of injection power measurements. The specific implementation method is given as follows. 4.2

Reducing the Order of the Correction Equation

Looking into the structure of the data, it turns out that the measurements in the SE problem contain voltage magnitude, power flow, and injection power. In power networks, as suggested in [25], we can make use of the fact that the number of injection power measurements is much less than the two other measurements. Thus, the main idea is to take out the voltage magnitude and power flow measurements in the correction matrix. More precisely, let the measurement vector z be split into two components T ]T = [hTi hTv+p ]T + [eTi eTv+p ]T z = [ziT zv+p

(25)

where the components with the subscript “i” are associated with injection power measurements and components with “v+p” correspond to voltage magnitude and power flow measurements. This classification is also implemented in other related variables. Therefore, T T T ]T , dα = [αiT αv+p ]T , and v = [viT vv+p ]T , and the correction H = [HiT Hv+p Eq. (18) can be rewritten as ⎡ ⎤ ⎤⎡ ⎤ ⎡ Ai 0 Hi 0 vi dαi ⎢ 0 Av+p Hv+p 0 ⎥ ⎢dαv+p ⎥ ⎢vv+p ⎥ ⎢ T T ⎥ ⎥⎢ ⎥ ⎢ (26) ⎣Hi Hv+p 0 −GT ⎦ ⎣ dx ⎦ = ⎣ Lx ⎦ Lβ dβ 0 0 −G 0 where Ai and Av+p are the corresponding to matrix A. Eliminating dαv+p from (26), the new correction equation with sufficient lower order can be achieved ⎡ ⎤⎡ ⎤ ⎡ ⎤ Qin Hin 0 dαi vi T ⎣Hin 0 −GT ⎦ ⎣ dx ⎦ = ⎣ φx ⎦ (27) Lβ dβ 0 −G 0 T −1 where φ(x) = Lx − Hvl Qvl vvl .

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The advantage of this dimension reduction is notable. As abovementioned, in comparison, the total number of measurements, the number of injection power measurements, is much less than others. Therefore, this modification improves considerable the GCIP estimator. The overview of the A-GCIP algorithm is described in Table 1. Table 1. The overall algorithm of A-GCIP Step 1: (Initialization) Choose σ ≥ 0, λ(0) = π (0) = 0 and c, u(0) , v (0) , α(0) , β (0) ≥ 0, ρ ∈ (0, 1) and  = 10−3 ; Step 2: Compute the Gap, If Gap <  then go to Step 7; else, compute the perturbed parameter μ; Step 3: Solve the new reduced correction equation (27) for [dx, dβ, dαi ]T and then compute [dp, dq, dγ, dλ]T from (20), (21), (13) and (14); Step 4: Compute the primal and dual step sizes denoted by ΔP and ΔD from (23) and (24); Step 5: Update the primal and dual variables as  [x(k) , p(k) , q (k) ]T + ΔP [dx, dp, dq]T (28) [β (k) , α(k) , γ (k) , λ(k) ]T + ΔD [dβ, dα, dγ, dλ]T Step 6: Set k = k + 1 and go back to step 2; Step 7: Output the optimal solution and END

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Simulations Results

In this section we illustrate the performance of the A-GCIP in a variety of IEEE power systems networks. All the tests were performed on a desktop device with processor of COREi7-Inetel and 16,0 GB installed RAM. The algorithm run in Matlab R2018b using the Matpower package to access the robust power system calculation tools [26]. For the validation of A-GCIP, a fist test was done on a IEEE-14 bus system contaminated by GEs. Table 2 shows the robustness of the A-GCIP estimator. As expected the estimated state is perfectly robust although there were multiple GEs in measurements. In this case study, the measurement vector is (52×1). The number of measurements included voltage magnitude, power flow and injection power was 38 measurements, when the number of injection power measurement was 14. Therefore, we implemented around 63% of dimension reduction for each element of the correction matrix. Figure 1 illustrates the effects of the dimension reduction of the correction equation, when compared with the GCIP in a variety of IEEE power systems networks. Furthermore, one can see that the A-GCIP rises linearly with expanding the system network.

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Table 2. Confirming the Robustness of A-GCIP Measurements True value Measured value Estimated value Residual Error P1

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6

Conclusions

WLS is a widely used method for SE in power systems. The drawback of this method is its sensitivity to GRs. A recent reported SE model, GCIP can address this problem, but in some cases when applied to power system control centers may not be feasible given the computational/memory burden. In this paper, we describe an improved GCIP. The main computational burden of GCIP arises from a large dimension matrix of the correction equation. A new arrangement for this matrix with lower order is presented that reduces computational time remarkably. The performance of A-GCIP was confirmed in variety of IEEE benchmark systems.

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Acknowledgements. This work was supported in part by R&D Unit UIDB+P/ 00147/2020 funded FCT/MCTES (PIDDAC) and by projects: STRIDE – NORTE-010145-FEDER-000033, funded by N2020, ERDF; IMPROVE - POCI-01-0145-FEDER031823, MAGIC PTDC/EEI-AUT/32485/2017 and HARMONY - POCI-01-0145FEDER-031411 funded by FEDER funds through COMPETE2020 – POCI and by national funds (PIDDAC).

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16. Pesteh, S., Moayyed, H., Miranda, V.: Favorable properties of interior point method and generalized correntropy in power system state estimation. Electr. Power Syst. Res. 178, 106035 (2020) 17. Wu, W., Guo, Y., Zhang, B., Bose, A., Hongbin, S.: Robust state estimation method based on maximum exponential square. IET Gener. Transm. Distrib. 5(11), 1165–1172 (2011). https://doi.org/10.1049/iet-gtd.2011.0100 18. Renyi, A.: Some fundamental questions of information theory. In: Selected Papers of Alfred Renyi, vol. 2, Akademia Kiado, pp. 526–552 (1976) 19. Ponnambalam, K., Quintana, V., Vannelli, A.: A fast algorithm for power system optimization problems using an interior point method. In: IEEE Power Industry Computer Application Conference (1991) 20. Momoh, J.A., Guo, S.X., Ogbuobiri, E.C., Adapa, R.: The quadratic interior point method solving power system optimization problems. IEEE Trans. Power Syst. 9(3), 1327–1336 (1994) 21. Wei, H., Sasaki, H., Nagata, T.: An application of interior point quadratic programming algorithm to power system optimization problems. IEEE Trans. Power Energy 116(2), 174–180 (1996) 22. Clements, K.A., Davis, P.W., Frey, K.D.: An interior point algorithm for weighted least absolute value power, system state estimation. In: IEEE Winter Power Meeting (1991) 23. Wei, H., et al.: An interior point method for power system weighted nonlinear L1 norm static state estimation. IEEE Trans. Power Syst. 13(2), 617–623 (1998) 24. Wei, H., Sasaki, H., Kubokawa, J., Yokoyama, R.: An interior point method for power system weighted nonlinear L/sub 1/norm static state estimation. IEEE Trans. Power Syst. 13(2), 617–623 (1998) 25. Wei, H., et al.: An interior point method for power system weighted nonlinear L/sub 1/norm static state estimation. IEEE Trans. Power Syst. 13(2), 617–623 (1998) 26. Zimmerman, R.D., Murillo-S´ anchez, C.E., Thomas, R.J.: MATPOWER: steadystate operations, planning and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19 (2011)

Your Turn to Learn – Flipped Classroom in Automation Courses Filomena Soares1(B)

, P. B. de Moura Oliveira2,3

, and Celina P. Leão1

1 Centro Algoritmi, Engineering School, University of Minho, 4800-058 Guimarães, Portugal

[email protected], [email protected] 2 ECT, UTAD-University of Trás-os-Montes and Alto Douro, 5000-801 Vila Real, Portugal

[email protected] 3 INESC TEC Technology and Science, Campus da FEUP, 4200-465 Porto, Portugal

Abstract. Flipped Classroom approach was implemented in an Automation course with around 100 students. Videos focused on GRAFCET topics were given to students prior to class and problem-based challenges were solved in class by the students in a collaborative way. The teacher’s role was to guide students in their learning process. The goal was to identify students’ behavior regarding this learning approach, and the videos in particular, by using questionnaires. Result analysis shows a positive feedback from students motivating teachers to enlarge this learning approach to other courses. Keywords: Control engineering education · Flipped Classroom · Videos · Automation · GRAFCET

1 Introduction Students are different from they used to be fifteen, even ten years ago. We are assisting an evolution, or revolution as some argue, in Education. Technology is closely following healthcare, industry, daily live and education is not an exception. Education 4.0 is here [1–3]. The current scenario is to promote collaborative learning where students learn by themselves and with their peers and where teachers are the facilitators in the learning process [1]. Fisk [1] considers nine trends related to Education 4.0, namely: • • • • •

students can learn anytime and anywhere; students can personalize their learning; students can choose how they want to learn; students will have more project-based learning; students will have more hands-on learning (internships, mentoring projects and collaborative projects); • students will be exposed to critical thinking and data interpretation; • students will be assessed differently; • students’ feedback will be considered in the curriculum design; © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 668–675, 2021. https://doi.org/10.1007/978-3-030-58653-9_64

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• students will become more independent in their own learning, being guided by teachers. ‘Students’ is the word listed in the nine trends. In fact, Education 4.0 puts students in the center of the learning process. Students are responsible for their learning. Nonetheless, teachers do have a relevant role. They should guide students in this learning path, being their support and counselors. We are social beings and face-to-face communication keeps its relevance. Students need their peers and their teachers to effectively learn. The only difference is that each actor should retune his/her role. There are different tools, frameworks, methodologies to put in practice the referred trends. Among them is Flipped Classroom (FC) (or Inverted Classroom) approach. Bergmann and Sams [4] refer to FC approach as a pedagogical model where the teacher gives to students digital resources to be studied outside the classroom. Then, in class, problem-solving activities are carried out in a collaborative and active way [5, 6]. So, the main issue is that the theoretical parts of the topic are learned outside the class time and interactive and collaborative problem-solving learning is carried out in class. Literature points to videos as the main digital resources given to students [7, 8]. Cabi [9] presents a study on the impacts of the FC approach on students’ academic achievements. In order to compare FC learning to traditional learning, two groups were formed: a control group with participants taught through traditional blended learning and an experimental group with students learning through the FC approach. The results indicated that there were no statistically significant differences between the scores of the pre- and post-test scores of each group (two-way ANOVA for mixed measures). In fact, the results obtained show that the FC approach did not increase students’ achievement. There is not a consensus in the literature regarding the FC impact on students´ success in the course (no impact [10]), (with impact [11]). The goal of this paper was to perceive if students positively react to FC approach to learn GRAFCET topics and if the videos delivered are a suitable tool for learning. It was not the goal to infer the impact on students’ final grades as it was a one run experiment and with no control group. Also, it was to be explored if FC approach can be easily applied in big classes (100 students) carried out in auditorium. The paper is organized in 4 sections: Sect. 2 details the learning experience in an Automation Course; Sect. 3 presents the students’ feedback towards FC approach by using questionnaires; and Sect. 4 concludes with the final remarks of the study.

2 The Learning Experience in Automation Course This section presents the learning experiment objective, the course characterization and the target group, as well as the methodology followed in the experiment and its assessment. 2.1 Learning Experiment Goal The goal of this teaching/learning experiment was to assess how students react to a Flipped Classroom configuration, where they should watch the videos prior to classes

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and in class they should solve problems and exercises in group. The authors wanted to perceive if students were actively engaged in the activity, namely: • Do they prepare themselves to classes? • Do they feel more motivated with these activities? • Do they consider this learning methodology effective? 2.2 Course and Target Group This experiment took place in the University of Minho, north Portugal, in the compulsory Course Unit (CU) Process Control and Automation (PCA), running in the 1st semester 3rd year of the Integrated Master of Engineering and Industrial Management (IMEIM). PCA runs in for 15 weeks with a time slot of 3 h theoretical class run in an auditorium. One of PCA main goals is to design, evaluate and solve automation problems, in particular, to design combinatorial and sequential controllers using GRAFCET and ladder diagrams. One hundred students were enrolled in the course in 2019/20 curricular year. Although class attendance was not mandatory, around 80 (sometimes more) regularly attended classes. Classes were usually conducted by the teacher as lectures and problemsolving exercises regarding combinational, sequential controllers, and ladder diagrams. This year a new framework was introduced and tested. 2.3 Methodology Implemented - Flipped Classroom A set of three videos [13] were prepared by one of this paper authors and are focused on GRAFCET rules and formalism illustrated by practical examples. The videos included PowerPoint presentation and audio explanation of the topic (in Portuguese). The videos last from 10 to 15 min. The three topics covered in the videos were: a) Fundamental notions of Grafcet; b) Exclusive sequences; c) Parallel sequences. One week prior to the Flipped Classroom (FC) activity, the teacher posted in the Blackboard platform the task guideline. The task included the learning goal, the topics to be studied, the topics to be reviewed in class and the link to the video. Students were then able to prepare the activity prior to the scheduled class by analysing the videos. In class, after a short presentation of the topic by the teacher, students solve in group a set of questions and exercises related to the subject he/she studied. A group resolution was presented to the teacher. The first class was carried out in 3 h by 88 students divided in 11 groups. The teacher went around the room to feel the students’ doubts, helping in the resolution, if needed, and correcting the exercises. At the end of the class the student took the resolution to assure they were correct and brought them back in the next class. As 11 groups are too many for a single teacher to provide suitable supervision, the next two classes were divided in two: half of the groups came in the first one hour and a half and the others came in the second hour. By doing that the teacher had more time dedicated to each group. In the second FC activity, 86 students attended the class and in the third and last activity, 83 participated in the task.

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2.4 Assessment The evaluation of this component corresponds to 10% of the overall evaluation (maximum 2 out of 20 points) and takes into account the student’s participation in the three inverted classroom activities. This motivates the students to attend classes In fact, the participation in the activity is the key point in the evaluation rather than the correctness of the problem resolution. The goal was to engage students in the FC activities in order to promote GRAFCET learning.

3 Analyzing Students’ Feedback In order to obtain feedback from students an anonymous questionnaire was applied. This survey aims to assess students’ perceptions about the use of new teaching/learning tools and strategies, in particular the use of videos in Flipped Classrooms and in Team Based Learning. 3.1 Material and Method Students were asked to fulfil a questionnaire, after the five FC, on a voluntary basis, in order to obtain their perceptions regarding this experiment. The questionnaire fulfilment took around 5 min. The questionnaire was divided into five main parts: (1) student identification, SI (course, sex, age, curricular year, first time attending the course), (2) video analysis, VA (how much in advance, at what time, local and if the task was performed in group or individually), (3) students’ perceptions regarding the activity, SPA (two Y/N questions and two open questions), (4) motivation, Mo (set of 14 items with answers given in a five-point Likert scale) and (5) two open questions highlighting the most positive aspects, PA, and the points that could be improved, PI. 3.2 Questionnaire Results To fulfil the prime objective of this paper, some items of the questionnaire were used, as identified in Table 1, where each research question is matched against some of the items of the questionnaire. In the following subsections it will be presented the students’ sample characterization and the general experience evaluation. Students’ Characterization A total of 89 students accepted the challenge to respond to the questionnaire. The majority of the students are female (59.6%) and their average age is 20.59 years old (SD = 1.96; interval range 19–36 years old, and the 72th percentile was 20 years old). The oldest student (with 36 years old) already holds a university degree. 96.5% of the students are from the 3rd year and attend for the first time the Automation course. In total, 92.1% refers that they do not have performed experiments of this kind, or similar. The data collected from the students’ questionnaire do not follow normality (based on the Shapiro-Wilk test for normality), therefore non-parametric tests were considered in the analysis: Mann-Whitney, U, for the comparison of two independent

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F. Soares et al. Table 1. Research questions by topic and questionnaire items.

Topic

Research question

Questionnaire items

Learning experiment goal

Do students prepare themselves to classes?

Mo12, Mo13

Do students feel more motivated with these activities?

Mo1, Mo7, Mo10

Do students consider this learning methodology effective?

Mo4, Mo11

Do students positively react to FC approach?

Mo5, Mo6 (reverse), Mo14

General

samples means (alternative to the independent sample t-test. A significance level of 5% was considered, that is, statistically significant differences for p < .05, and for the analysis it was used the statistical software SPSS 26.0 [12]. Mo1: Lessons become more interesting with this activity. Mo4: This activity prepare me better for evaluation. Mo5: I become satisfied with classes. Mo6: This activity wasted a lot of time in class. Mo7: It improved my attention and participation in class. Mo10: It stimulated my intellectual activity. Mo11: It provided study tools. Mo12: I always analyzed the videos before class. Mo13: All my team members always analyzed the videos before class. M014: Videos and Flipped Classroom are a suitable learning tool.

General Experience Evaluation To evaluate the general experience, the three research questions are answered based on the identified 10 items, see Table 1. Table 2 presents more details on the obtained descriptive statistics relative to the students’ perceptions about the use of new teaching/learning tools and strategies, in particular the use of videos in Flipped Classrooms. Figure 1 illustrates the distribution of the students’ rating for those 10 items, in descendent order of the statistics median. As expected the item Mo6: This activity wasted a lot of time in class, obtained the lowest value for the statistics and central tendency measures (median and mean), since it has been presented negatively. Three items obtained mean values higher than 4 reflecting agreement and a positive opinion regarding M014: Videos and Flipped Classroom are a suitable learning tool; Mo1: Lessons become more interesting with this activity, and Mo11: It provided study tools. These results suggest the effectiveness and the positive reaction towards FC approach.

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Table 2. Statistics U regarding the questionnaire items under analysis. Item

Min Max Median Mean Std. deviation

Mo12 2

5

4

3.64

1.05

Mo13 1

5

3

3.19

1.10

Mo1

2

5

4

4.21

.57

Mo7

2

5

4

3.82

.67

Mo10 1

5

4

3.70

.70

Mo4

1

5

4

3.98

.74

Mo11 3

5

4

4.21

.61

Mo5

1

5

4

3.98

.77

M06

1

4

2

2.40

.90

M014 2

5

4

4.25

.67

Fig. 1. Students’ rating for the 10 items under study in descendent order of the statistics median.

It is interesting to observe that the two items with the highest variability (higher standard deviation) and the lowest mean are Mo12: I always analyzed the videos before class, and Mo13: All my team members always analyzed the videos before class (3.64 and 3.19, respectively). These two items reflect how students prepare themselves to classes, answering the first research question: not all students prepare in advance for classes. Moreover, the analysis of the obtained data was done based on gender in order to perceive how and if students’ level of rating differ according to gender. Table 3, summarizes the statistics, regarding the 10 items under analysis, using the non-parametric Mann-Whitney U test to compare the medians between these two groups (difference between level of agreement for female and male students). It can be seen that in one case the statistical differences obtained are significant between gender (with p < .05).

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F. Soares et al. Table 3. Statistics Mann-Whitney U, for the questionnaire items under analysis. Item Mo12

Mo13

Mo1

Mo7

Mo10

U

874.5

898.5

738.0

899.5

p

822.5 .42

.64

.58

.04*

.61

Item Mo4

Mo11

Mo5

Mo6

Mo14

U

770.0

814.0

904.0

859.5

p

805.0 .58

.08

.23

.77

.52

* significant at the .05 level

Based on the collected data, for Mo7: It improved my attention and participation in class, it can be concluded that the level of agreement for female students is statistically significantly higher than the male students (U = 738.0, p = .04), meaning that female students considered that the use of these tools raise attention and motivation.

4 Final Remarks In this paper a learning experience using Flipped Classroom approach in an Automation Course was presented. The goal was to perceive students perceptions towards this learning method using videos as the main tool. Three FC classes were carried out with around 85 students in an auditorium. In order to gather students´ feedback regarding FC approach and video learning materials a questionnaire was given to students. The results from the questionnaires pointed out that students feel motivated with the activities in the FC approach, considering this learning methodology effective. The students’ preparation before class is reflected in two items that received the highest variability and the lowest mean (Mo12: I always analyzed the videos before class, and Mo13: All my team members always analyzed the videos before class). So, not all students prepare themselves prior to class. Students are not accustomed to this teaching/learning methodology. Some are still a little passive regarding learning. This is a point to improve in the next edition of the course as the work developed prior to classes is a key issue in FC methodology. An extra analysis was carried out: do gender influence students’ perception towards FC approach? In the 10 items under analysis, only one points to significant statistical differences between gender (Mo7: It improved my attention and participation in class). Female students consider that the use of these tools raise attention and motivation in class. As limitations of the approach we point out the large number of students and the university facilities. We believe that FC approach works better with small number of students and run in a room with several tables for group work. Nevertheless, we do believe that these limitations cannot prevent following these new ways of teaching/learning. It may be harder for the teacher but for students it is still an appropriate and efficient tool for learning. As future work, we are considering producing more videos in other topics and spreading this learning approach to other courses in engineering.

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Acknowledgements. The authors would like to express their acknowledgments to all students for their voluntary co-operation. This work has been supported by COMPETE: POCI-01-0145FEDER-007043 and by FCT – Fundação para a Ciência e Tecnologia within the R&D Units Project Scope: UIDB/00319/2020.

References 1. Fisk, P.: Education 4.0 … the future of learning will be dramatically different, in school and throughout life (2017). http://www.thegeniusworks.com/2017/01/future-education-youngeveryone-taught-together. Accessed 27 Feb 2020 2. Salmon, G.: May the fourth be with you: creating Education 4.0. J. Learn. Dev. 6(2), 95–115 (2019) 3. Hussin, A.A.: Education 4.0 made simple: ideas for teaching. Int. J. Educ. Lit. Stud. 6(3), 92–98 (2018) 4. Bergmann, J., Sams, A.: Flip your classroom: reach every student in every class every day. International Society for Technology in Education, Washington, D.C. (2012). http://i-lib.imu.edu.my/NewPortal/images/NewPortal/CompE-Books/Flip-Your-Cla ssroom.pdf. Accessed 27 Feb 2020 5. Toto, R., Nguyen, H.: Flipping the work design in an industrial engineering course. In: ASEE/IEEE Frontiers in Education Conference, San Antonio, TX (2009). https://ieeexplore. ieee.org/stamp/stamp.jsp?tp=&arnumber=5350529. Accessed 27 Feb 2020 6. Suwapaet, N.: Introducing a flipped classroom to engineering students: a case study in mechanics of materials course. In: AIP Conference Proceedings, vol. 1941 (2018). https://doi.org/10. 1063/1.5028099 7. Graziano, K.: Peer teaching in a flipped teacher education classroom. TechTrends 61, 121–129 (2017). https://doi.org/10.1007/s11528-016-0077 8. Hsu, T.: Behavioural sequential analysis of using an instant response application to enhance peer interactions in a flipped classroom. Interact. Learn. Environ. 26, 1–15 (2017). https:// doi.org/10.1080/10494820.2017.1283332 9. Cabi, E.: The impact of the flipped classroom model on students’ academic achievement. Int. Rev. Res. Open Distrib. Learn. 19(3) (2018). https://doi.org/10.19173/irrodl.v19i3.3482 10. Smallhorn, M.: The flipped classroom: a learning model to increase student engagement not academic achievement. Student Success 8(2) (2017). https://doi.org/10.5204/ssj.v8i2.381 11. El-Sheikh, A.A., El-Sayad, H.E.: Impact of a flipped classroom on academic achievement and perception among first year nursing students. Clin. Nurs. Stud. 7(3) (2019) 12. Field, A.: Discovering Statistics Using SPSS. SAGE Publications Ltd., London (2009) 13. Moura Oliveira, P.B.: GRAFCET Videos (2020). https://www.youtube.com/playlist?list= PLHlMBjz_1_oSYzzMgfGf7gKWQWZMjaHMs. Accessed 27 Feb 2020

Modeling of an Elastic Joint: An Experimental Setup Approach V´ıtor H. Pinto1,3(B) , Jos´e Lima2 , Jos´e Gon¸calves2 , and Paulo Costa1,3 1

Faculty of Engineering, FEUP - University of Porto, Porto, Portugal {vitorpinto,paco}@fe.up.pt 2 IPB - Polytechnic Institute of Bragan¸ca, Bragan¸ca, Portugal {jllima,goncalves}@ipb.pt 3 CRIIS - Centre for Robotics in Industry and Intelligent Systems, INESC TEC - Institute for Systems and Computer Engineering, Technology and Science, Porto, Portugal

Abstract. Throughout this paper it is presented a novel elastic joint configuration, being compared with other similar joints found in recent literature. It is presented its modeling, being its estimation process developed offline, based on a proposed experimental setup. This setup enables to monitor and collect data from an absolute encoder and a load cell. Some data obtained from these sensors is then graphically represented, like angle and torque, obtaining some parameters. Finally, through an optimization process, where the error of the angle is minimized, the remaining parameters of the joint are estimated, thus obtaining a realistic model of the system. Keywords: Elastic joint · Modeling estimation · Optimization

1

· Experimental setup · Parameter

Introduction

To increase safety and robustness in the performance of industrial robots, which otherwise could easily break their joints, or hurt someone, the use of elastic joints is increasingly found in the literature and in the industry. These mechanisms contain elastic components (such as springs), which introduce a mechanical energy storage in the system, which can increase the maximum output power [1]. In [2], Giusti et al. have designed and implemented successfully a controller based on the analysis of the inverse dynamics of the elastic joint, and applied a structured controller. The testbed mixes rigid and elastic joints on an interesting configuration. The usage of elastic joints for exoskeletons is also expanding, possibly due to the decrease in the size of the actuators. An example is the work is presented in [3] for a variable viscoelastic joint system. Another interesting application example for elastic joints are deployable mechanisms, as the one presented by c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 676–685, 2021. https://doi.org/10.1007/978-3-030-58653-9_65

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Zhang et al. [4], where the singularities of a system like the presented are analyzed and successfully avoided. The Quasi-Direct Drive is also an example of the technology applied to actuate an elastic joint, as presented in [5] for a quadruped. As it can be easily understood, all the mentioned research has a same primary step the modeling phase, applied to translate mathematically the behavior of the system. This is required to be able to design suitable controllers, test them in simulation and obtain relevant results. Thus, the closer to reality the developed model is, the better designed the controller can be in theory, and be applied in practice. However, all the robotic systems mentioned above include components that adds uncertainties and non-linearities to it, making it more complex to model. Estimation techniques can be applied to help modeling the system, since these can be more suited to the obtained solution. The techniques applied can be diverse, from evolutionary algorithms, such as Montazeri et al. use in [6], recursive methods as Gonzalez et al. [7] use for their third-order modeling, through the use of neural networks, as explained in [8] or even experimentally, as the authors of [9] and [10] present. Considering that the essential characteristics of the elastic joint developed for this work are not expected to suffer structural changes, the estimation of its dynamic parameters will be carried out previously, with offline techniques, and using experiments that allow to obtain them. This paper presents a novel approach to an elastic joint and the process of modeling it is described. An introduction to the topic was performed in Sect. 1, and the proposed model will be introduced throughout Sect. 2. Section 3 will state the obtained results by the carried out experiments, and in Sect. 4 there are presented the conclusions and future work.

2

Proposed Model

Throughout this section, the model for the elastic joint will be presented. Some hardware tests were performed to estimate its parameters, and these will be further detailed. For the elastic joint construction, a passive system was developed. Applying 4 springs, attached to a circular support part as represented in Fig. 1, a rotational force is applied in the support’s centre of rotation, is passed to the next link through the springs, providing elasticity and cushioning to the rotational movement. To connect all the parts, as can be noticeable by the mentioned figure, bearings are used to decrease the rotational friction, as well as to allow a tighter connection between all the parts. The mathematical/physical representation of the system, with its respective constants, velocities and torques is present in Fig. 2. Given the proposed elastic joint, its model will become as follow: τK = K · (θm − θj )

(1)

τB = B · (θ˙m − θ˙j )

(2)

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Fig. 1. Simulated representation of the elastic joint

Fig. 2. Motor joint representation

J · θ¨ =



τ = τK + τ B

1 dω = (−Kθ − B ω) ˙ dt J

(3) (4)

d2 θ(t) dθ(t) +B· + K · θ(t) = 0 (5) dt dt dθ J · (s2 θ(s) − sθ(t = 0) − (t = 0)) + B · (sθ(s) − θ(t = 0)) + K · θ(s) = 0 (6) dt J·

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Considering that the initial conditions can be not null, the following variables take values: (7) θ(t = 0) = θ0 dθ(t = 0) = ω(t = 0) = 0 dt

(8)

Js2 θ(s) − Jsθ0 + Bsθ(s) − Bθ0 + Kθ(s) = 0

(9)

θ(s) = θ(s) = θ0 ·

θ0 · (Js + B) Js2 + Bs + K (s + B J) s2 + ( B )s + (K J J )

(10) (11)

Similar to what was described in the above subsection, for modeling the elastic joint, as well as for estimating the parameters associated with it, some hardware tests will also be performed, in order to obtain parameters J, K and B, consisting of: – Pushing/Pulling the link attached to the elastic joint until a certain angle; – Pushing/Pulling the link attached to the elastic joint and releasing, allowing it to stabilize; To conduct these tests, it is needed to know the joint’s angular position, as well as the applied torque. So, to obtain the exact angular displacement, an absolute encoder was placed exactly concentrically with the axis of rotation of the joint. Also, to obtain the torque, a load cell was placed in the center of the link to evaluate the exerted force which can, using the distance to the rotational axis, be traduced to the torque. Figure 3 presents the implemented solution, and the interaction between the subsystems.

Fig. 3. Hardware diagram

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3

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Results

In order to perform the hardware tests, a system with the components presented in Fig. 3 was built. This is presented in Fig. 4, and it was designed by the authors, being posteriorly 3D printed. The parts already provide space specially allocated for the load cell, as well as for its amplifier, and also for the absolute encoder. The most complicated task was to physically align the rotation axis of the joint, which has the magnet placed in its center, and the center of the Hall sensor. Taking into account the collected data, this task was considered to be performed successfully, since the angular displacement read by the sensor and by a physical angle meter were very similar.

Fig. 4. Real joint

For the first set of tests, the system was acquiring data continuously, in order to recognize the variations in force exerted on the cell, with information of the angle. This test allows to know what is the behavior of the joint when its center of rotation is locked, and the posterior link is applying a force on it. This force is traduced to a torque, since it was put with some distance from the center. The results from this test are present graphically in Fig. 5. Since the joint is already coupled to a motor attached to a gearbox system, and since the gearbox present some backlash, the joint presents a dead zone. This behaviour is clearly visible in the figure, in the red line of the graph where, although the exerted force is practically zero, the angle has a significant variation. Considering that the motor is stopped, the difference between the angles of the motor shaft and the joint is obtained by the angle in the absolute encoder. So, using Eq. 1, and calculating the average of the straight slopes of the values from the positive and the negative torques, the value of K is obtained and is equal to 7.3035.

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Fig. 5. Plot of Torque vs Angle

Then, to find the value of J and B, it was necessary to proceed to a second set of tests. This consists of placing the system to acquire data, rotating the posterior link to a certain angle and releasing it, allowing its free oscillation until reaching the equilibrium point again. The results obtained are compiled in Table 2. Using Eq. 11, and the inverse Laplace transform, the following Equations could be used to represent the system in the time domain.   ( ωαn − ξ · ωn )2 + 1 · e−ξωn t sin(ωn · 1 − ξ 2 · t + φ) (12) θ(t) = θ0 · 2 1−ξ −1

φ = tan

 ωn 1 − ξ 2 ( ) α − ξωn

(13)

Having defined the approximation function, and since the value of K was already estimated, the optimization process that minimizes the absolute error to the obtained angular position was put to run and the values of the variables for Eq. 12 were found. Establishing the parallelism between the function in the Laplace domain used and Eq. 11, after the optimization process, the values found for B and J were 0.0416 and 0.0085.

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Considering a simple pendulum configuration, J = m·r2 . If a weight of 269 gr. is put on a known point of the link, there is a known moment of inertia. Taking the value obtained for J = 0.0085, and placing the known mass approximately 18 cm from the axis of rotation, one obtain an estimated value for the distance r = 17.78 cm, an error of less than 1%, which indicates that the estimation of the parameters associated with the joint has been carried out successfully. All the obtained estimated parameters are compiled in Table 1. Table 1. Estimated joint parameters Parameters K 7.3035 N/m B 0.0416 N·s/m J 0.0085 Kg·m2

In Fig. 6, in blue there is the real measured behaviour of the joint, while in red there is the estimated model of the joint. Actually, approximately from t = 1s, it begins to be noticed that the model differs somewhat from the performed measurements. However, this difference can be explained by the non-linear behavior presented by the dead zone of the joint, which will have to be compensated by a suitable controller.

Fig. 6. Plot of Angle vs Time

Elastic Joint Model

Table 2. Second-order model Time (s) θ(◦ )

θest (◦ ) Error

0

38.14

34.28

3.85

0.02

23.77

28.75

4.98

0.04

4.42

14.61

10.19

0.06

−13.36 −2.86

10.50

0.08

−26.17 −17.79 8.38

0.1

−30.19 −25.69 4.50

0.12

−24.04 −24.70 0.66

0.14

−11.84 −15.96 4.12

0.16

2.37

−2.97

5.34

0.18

14.85

9.76

5.09

0.2

21.6

18.21

3.39

0.22

22.24

20.09

2.15

0.24

15.42

15.42

1.24E-06

0.26

4.9

6.29

1.39

0.28

−4.94

−3.99

0.95

0.3

−12.13 −12.05 0.08

0.32

−15.36 −15.53 0.17

0.34

−12.85 −13.76 0.91

0.36

−6.61

−7.79

1.18

0.38

0.88

0.11

0.77

0.4

7.23

7.23

1.18E-08

0.42

11.1

11.40

0.30

0.44

11.56

11.56

1.46E-07

0.46

7.65

8.04

0.39

0.48

2.37

2.29

0.08

0.5

−3.19

−3.65

0.46

0.52

−7.23

−7.88

0.65

0.54

−8.48

−9.21

0.73

0.56

−6.79

−7.50

0.71

0.58

−2.59

−3.57

0.98

0.6

1.65

1.14

0.51

0.62

4.92

5.04

0.12

0.64

7.12

6.98

0.14

0.66

6.64

6.53

0.11

0.68

4.28

4.04

0.24

683

684

4

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Conclusions and Future Work

As a first conclusion, the physical implementation of the idealized and simulated prototype of a novel elastic joint have met the expectations, adding a damping component to the joint. The addition of the 4 springs allowed the joint to have an elastic behavior with only passive components, and consequently without energy consumption. Then, it can be concluded that, based on a purely low-cost sensor acquisition system, it is possible to carry out tests and consider measurements that allow to estimate parameters of a complex model. Regarding the model, the estimated parameters were used to compare measurements with real values and have a very low error, so this process can be considered as successfully achieved. Finally, in the overlap between the function in order of time of the model developed and the measures that were obtained from the real system, differences are effectively noticed, which come from the joint’s dead zone. These differences will therefore have to be compensated in the design of a suitable controller. Therefore, the future work is to design a suitable closed-loop controller that allows to control the joint in position, and that tries to reduce the non-linear effects that the dead zone adds to the model. Acknowledgements. This work is financed by National Funds through the Portuguese funding agency, FCT - Funda¸ca ˜o para a Ciˆencia e a Tecnologia, within project UIDB/50014/2020.

References 1. Liu, H., Cui, S., Liu, Y., Ren, Y., Sun, Y.: Design and vibration suppression control of a modular elastic joint. Sensors 18(6), 1869 (2018) 2. Giusti, A., Malzahn, J., Tsagarakis, N.G., Althoff, M.: On the combined inversedynamics/passivity-based control of elastic-joint robots. IEEE Trans. Rob. 34(6), 1461–1471 (2018) 3. Okui, M., Iikawa, S., Yamada, Y., Nakamura, T.: Variable viscoelastic joint system and its application to exoskeleton. In: 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3897–3902. IEEE (2017) 4. Zhang, T., Huang, H., Guo, H., Li, B.: Singularity avoidance for a deployable mechanism using elastic joints. J. Mech. Des. 141(9), 094501 (2019) 5. Kau, N., Schultz, A., Ferrante, N., Slade, P.: Stanford doggo: an open-source, quasi-direct-drive quadruped. In: 2019 International Conference on Robotics and Automation (ICRA), pp. 6309–6315. IEEE (2019) 6. Montazeri, A., West, C., Monk, S.D., Taylor, C.J.: Dynamic modelling and parameter estimation of a hydraulic robot manipulator using a multi-objective genetic algorithm. Int. J. Control 90(4), 661–683 (2017) 7. Gonz´ alez, A., Cerda-Lugo, A., Cardenas, A., Maya, M., Piovesan, D.: A third-order model of hip and ankle joints during balance recovery: modeling and parameter estimation. J. Comput. Nonlinear Dyn. 14(10), 101001 (2019) 8. Ge, W., Wang, B., Mu, H.: Dynamic parameter identification for reconfigurable robot using adaline neural network. In: 2019 IEEE International Conference on Mechatronics and Automation (ICMA), pp. 319–324. IEEE (2019)

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9. Miranda-Colorado, R., Moreno-Valenzuela, J.: Experimental parameter identification of flexible joint robot manipulators. Robotica 36(3), 313–332 (2018) 10. Ni, H., Zhang, C., Hu, T., Wang, T., Chen, Q., Chen, C.: A dynamic parameter identification method of industrial robots considering joint elasticity. Int. J. Adv. Rob. Syst. 16(1), 1729881418825217 (2019)

On the Control Models in the Trajectory Tracking Problem of a Holonomic Mechanical System Aleksandr Andreev and Olga Peregudova(B) Ulyanovsk State University, 42, Leo Tostoi str., Ulyanovsk 432017, Russia [email protected], [email protected]

Abstract. Studies on the trajectory tracking and stabilization problems of a holonomic mechanical system have numerous applications in the design of the control structure of manipulators, wheeled robots, aircraft, and other technical objects. For all its great importance and relevance, the problem of substantiating a simpler control structure remains the subject of many works. The paper suggests a new control model that solves the trajectory tracking problem. The results are achieved on the basis of the approach and methods previously used by the authors in solving the stabilization problem of a given position and stationary motion of a holonomic mechanical system. The performance of the proposed controller is demonstrated on a five joints robot manipulator. Keywords: Holonomic mechanical system · Trajectory tracking Lyapunov functional · 5-DOF robot manipulator

1

·

Introduction

The basis of numerous studies on the trajectory tracking of controlled mechanical systems was the solution to the problem of the forces structure influence on the equilibrium position and stationary motion stability of a holonomic mechanical system [1,2]. Indeed, the construction of a control law in the form of a proportional-derivative (PD) controller is not difficult to imagine it in the framework of this problem as the action of dissipative and potential forces. Adding an integral component transforms the PD controller into a proportional-integroderivative (PID) controller. The use of a PID controller to stabilize the desired position of a holonomic mechanical system made it possible to qualitatively improve the control process [3–6]. A detailed analysis of the relevant works is presented in the papers [7–10]. Great attention has been paid to solving the trajectory tracking control problem of a mechanical system without velocity measurements. One approach to solving this problem is to add a position filter to obtain an estimate of the velocities and their integral solutions [11–14]. The application of the stability theorems for Volterra integro-differential equations made it possible to substantiate c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 686–695, 2021. https://doi.org/10.1007/978-3-030-58653-9_66

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another approach in the form of a nonlinear proportional-integral (PI) controller [15–17]. The presented application of PD, PID, and PI controllers has been largely developed for the position stabilization problem of a holonomic mechanical system. Despite a number of significant results on the application of these controllers in the position stabilization problem [18–20], its study is far from complete. The aim of this paper is to solve the trajectory tracking problem of a holonomic mechanical system by a nonlinear PID controller of a fairly general form.

2

Problem Formulation

The motion of controlled mechanical systems with holonomic constraints is described by the Lagrange equation   d ∂T ∂T =Q+U , (1) − dt ∂ q˙ ∂q where q ∈ Rn , q = (q1 , q2 , . . . , qn ) is the vector of generalized coordinates that determines the position of the system, (·) is the transpose operation, ˙ is the kinetic energy with a positive definite matrix A ∈ Rn×n , T = (q˙ A(q)q)/2 Q = (Q1 , Q2 , . . . , Qn ) is the vector of uncontrolled generalized forces, U = U (t, q, q) ˙ is the vector of control forces. The paper considers the type of control models such that U1 , U2 , . . ., Un are direct control forces. The control forces are defined such as U = U (t, W ), where W = (W1 , W2 , . . . , Wn ) is the vector of control signals to the actuators. Another type of the control models for the system (1) arises when it is necessary to take into account the functioning of the actuators as well as the processes in the control structure and the operation of measuring devices. For convenience, we restrict ourselves to the first type of the control models assuming that U1 , U2 , . . ., Un are the direct control forces. Let q (0) (t) be some motion from the set Fq of mechanical system motions. Assume that due to the use of sensors, all or the part of the phase variables q1 (t), q2 (t), . . ., qn (t), q˙1 (t), q˙2 (t), . . ., q˙n (t) are available to measurement. The feedback tracking control problem of the robot trajectory q (0) (t) is to find the controller U = U (t, q(t), q(t)) ˙ such that the motion q (0) (t) is asymptotically stable. Moreover, the property of uniform asymptotic stability is more effective, since it provides the stability of the robot motion q (0) (t) under constantly acting perturbations [2]. If the property of attraction for the motion q (0) (t) is achieved for any initial perturbations (q0 , q˙0 ), then the global asymptotic stability property of q (0) (t) holds. Accordingly, one can introduce the global uniform asymptotic stability property [2]. Introduce the tracking errors x = q − q (0) (t) that can be considered as new variables. The kinetic energy of the system in the variables x can be represented

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as T = T2 + T1 + T0 ,

T2 =

1  (1) x˙ A (t, x)x˙ , 2

1 (0) (q˙ (t)) A(q (0) (t) + x)q˙(0) (t) , 2 A(1) (t, x) = A(q (0) (t) + x), B(t, x) = A(q (0) (t) + x)q˙(0) (t) . ˙ T1 = B  (t, x)x,

T0 (t, x) =

The motion equations of the mechanical system in the variables x can be written as follows   ∂T0 ∂B d ∂T2 ∂T2 = X + U + Gx˙ + − , (2) − dt ∂ x˙ ∂x ∂x ∂t where X = X(t, x, x) ˙ = Q(t, q (0) (t) + x, q˙(0) (t) + x), ˙ ∂B  ∂B = −G , − ∂x ∂x ˙ U (1) (t, 0, 0) ≡ 0 , U = U (0) (t) + U (1) (t, x, x), ∂B ∂T U (0) (t) = −X(t, 0, 0) − (t, 0) + (t, 0) . ∂q ∂t G=

The trajectory tracking control problem is reduced to find a controller U (t, x, x) ˙ such that the zero solution x˙ = x = 0 of the system (2) is uniformly asymptotically stable (globally asymptotically stable).

3

Problem Solution

From the generalized force X one can find the dissipative-accelerating and potential forces ∂Π1 (t, x) + X (2) (t, x) , ∂x ˙ = X(t, x, x) ˙ − X(t, x, 0) , X (1) (t, x, x) ∂Π1 (t, x) + X (2) (t, x) , X(t, x, 0) = − ∂x

X = X (1) (t, x, x) ˙ −

where Π1 = Π1 (t, x) is the scalar function expressing the potential forces effect on the mechanical system. Assume that the deviations x1 , x2 , . . . , xn and their velocities x˙ 1 , x˙ 2 , . . . , x˙ n are continuously measured. Consider the feedback controller U = U (t, x, x) ˙ =  +

∂f (x) ∂x

  t

∂B(t, x) ∂Π2 (t, x) − X (2) (t, x) − + ∂t ∂x

P1 (τ − t)f (x

(1)

0

t

(τ ))dτ − F (t, x)x˙ − P2 (0)

P2 (τ − t)x(τ ˙ )dτ 0

(3)

On the Control Models

689

where Π2 (t, x) is some scalar potential function, vector-function f : Rm → Rm , f = f (x(1) ) (x(1) ∈ Rm ), f (0) = 0 and matrix P1 : (−∞, 0] → Rm×m determine the control component which is integral one with respect to the coordinates x1 , x2 , . . . , xm , matrices F : R × Rn → Rn×n and P2 : (−∞, 0] → Rn×m determine the components of the controller which are linear and integral linear respectively in x. ˙ In this construction, the control feedback satisfies the following conditions 1) P1 = P1 , α1 (s)||x(1) ||2 ≤ (x(1) ) P1 (s)x(1) ≤ α2 (s)||x(1) ||2 , (x(1) )

dP1 (s) (1) x ≥ 2α3 (s)||x(1) ||2 , αk (s) > 0, ds

0 αk (s)ds < +∞, k = 1, 2, 3 ; −∞

(4) 2) the set {∀c ∈ Rn : f (x(1) ) = c, ||x(1) || ≤ H = constant, 0 < H < +∞} is finite; 3)   (1) ∂A (t, x)  (1)  − F (t, x) x˙ ≤ −2μ0 ||x˙ (2) || , ˙ + x˙ (5) x˙ X (t, x, x) ∂t where μ0 > 0, x(2) = (xm+1 , . . . , xm+l ) ; 4) dP2 (s) = μ1 P2 (s), μ1 > 0 , ds ||P2 (s)x|| ≥ α5 (s)||x(3) ||, x(3) = (xm+l+1 , . . . , xn ) , 0 αk (s) > 0, αk (s)ds < +∞, k = 4, 5 ; ||P2 (s)|| ≤ α4 (s),

−∞

5) 1 a1 (||x||) ≤ Π(t, x) − Π(t, 0) − 2

t

f  (x(1) (τ ))P (τ − t)f (x(1) (τ ))dτ ≤ a2 (||x||) ,

0

   ∂Π(t, x)  ∂  (Π(t, x) − Π(t, 0)) ≤ 0, a3 (||x||) ≤   ∂x  ≤ a4 (||x||) , ∂t Π(t, x) = Π1 (t, x) + Π2 (t, x) − T0 (t, x) ,

where ak (s) are Hahn functions [2], k = 1, 2, 3, 4.

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Choose the Lyapunov functional candidate as follows 1  x˙ A(t, x)x˙ + Π(t, x) − Π(t, 0) 2 t 1 (f (x(1) (t)) − f (x(1) (τ ))) P1 (τ − t)(f (x(1) (t)) − f (x(1) (τ )))dτ + 2

V =

0

(6)

2  t     1  . P (τ − t) x(τ ˙ )dτ +  2   2  0

For the time derivative of the functional (6), by virtue of the motion equations (2) with the controller (3) according to the conditions 1–5, one can find   (1) ∂A (t, x) 1 dV = x˙  − F (t, x) x˙ dt 2 ∂t t dP1 (τ − t) 1 (f (x(1) (t)) − f (x(1) (τ )))dτ − (f (x(1) (t)) − f (x(1) (τ ))) 2 dτ 0

2  t     1  − ∂ (Π(t, x) − Π(t, 0)) P (τ − t) x(τ ˙ )dτ − μ1  2   2  ∂t  0

t ≤ −μ0 ||x˙

|| −

α3 (s)||f (x(1) (t) − f (x(1) (τ ))||2 dτ

(2) 2

0

2  t     1 (3)  ≤0. P (τ − t) x ˙ (τ )dτ − μ1  2   2   0

Further, applying the asymptotic stability theorem from [17], one can get the following. Theorem 1. A feedback controller (3) constructed in accordance with the conditions 1–5 ensures the uniform asymptotic stability property of the desired motion q (0) (t) of the system (1). Remark 1. The advantage of the control model (3) in comparison with the models substantiated in [5,6] consists in a simpler structure and better representability of the conditions with respect to control parameters. The disadvantage of this model is the necessary compensation for the part of inertial forces.

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Fig. 1. Model of 5DOF robot manipulator

4

Motion Control Problem of a 5-DOF Serial Robot Manipulator with Revolute and Prismatic Joints

The dynamics of a 5-DOF serial robot manipulator with revolute and prismatic joints (see, Fig. 1) is defined by the following equations A(q)¨ q + C(q, q) ˙ q˙ + g(q) = U ,

(7)

where q ∈ R represents the vector of joint positions, A(q) ∈ R is the inertia matrix, the Coriolis and centrifugal forces are described by the vector C(q, q) ˙ q, ˙ the function g(q) represents the gravitational forces or torques, and U ∈ R5 is the vector of control forces. Assume that the generalized coordinates q1 = ϕ1 , q2 = ϕ2 , q3 = ϕ3 , and q4 = ϕ4 are the angular displacements of the revolute joints O1 , O2 , O3 , and O4 , respectively, and the other q5 = x is the linear displacement of the prismatic joint. The robot parameters are given as 5

5×5

J1 = 0.01 kg · m2 , m2 = 2 kg, m3 = 2 kg, m4 = 1 kg, m5 = 3.3 kg , l2 = 0.5 m, lC2 = 0.2 m, lC3 = 0.2 m, lC4 = 0.2 m, lC5 = 0.1 m .

(8)

The desired trajectory is chosen as (0)

(0)

(0)

q1 (t) = 0.5t rad, q2 (t) = cos(0.5t) rad, q3 (t) = sin(0.5t) rad , (0) (0) q4 (t) = cos(0.5t) rad, q5 (t) = 0.5 + 0.1 sin(0.5t) m .

(9)

The controller (3) is given by (0)

(0)

U1 = U1 (t) + b1 μ cos((q1 − q1 (t))/2)

t 0

Ui =

(0) Ui (t)

− b1 (q˙i −

(0) q˙i (t))

eα(τ −t) sin((q1 (τ ) − q1 (τ ))/2)dτ (0)

(0)

− b2 sin((q1 − q1 (t))/2) , (0) − b2 sin((qi − qi (t))/2), i = 2, 3, 4, 5 . (10)

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angular position (rad)

10 5 0 −5 −10 −15 −20 −25 −30 0

5

10

15

20

25

30

time (sec)

Fig. 2. The time response of angular position and reference of the manipulator first link. 1

angular position (rad)

0.5 0 −0.5 −1 −1.5 −2 −2.5 0

5

10

15

20

25

30

time (sec)

Fig. 3. The time response of angular position and reference of the manipulator second link.

The control gain parameters are chosen such as b1 = 100,

b2 = 100,

μ = 1,

α=3.

(11)

We consider the simulations results using the initial conditions for the robot manipulator such as (0) (0) q1 (0) = −3.1 + q1 (0) rad, q2 (0) = −2.8 + q2 (0) rad (0) (0) (0) q3 (0) rad, q4 (0) = −3.1 + q4 (0) rad, q5 (0) = 0.05 + q5 (0) m

, , q3 (0) = 2.5 + q˙1 (0) = −10 rad/sec, q˙2 (0) = −20 rad/sec, q˙3 (0) = −15 rad/sec , q˙4 (0) = −20 rad/sec, q˙5 (0) = 0.3 m/sec . (12)

On the Control Models

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3.5

angular position (rad)

3 2.5 2 1.5 1 0.5 0 −0.5 −1 0

5

10

15

20

25

30

time (sec)

Fig. 4. The time response of angular position and reference of the manipulator fourth link. 1.5

angular position (rad)

1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0

5

10

15

20

25

30

time (sec)

Fig. 5. The time response of angular position and reference of the manipulator fifth link.

In Figs. 2, 3, 4, 5 and 6 we show the link trajectories as well as the references for the robot (7). From these results, it can be seen that controller (10) provides asymptotic convergence to the reference trajectory plus 2πk, where k = (k1 , k2 , k3 , k4 , 0) , ki ∈ Z, i = 1, 2, 3, 4.

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linear displacement (m)

2.5 2 1.5 1 0.5 0

0

5

10

15

20

25

30

time (sec)

Fig. 6. The time response of linear position and reference of the manipulator third link.

5

Conclusions

In this paper, we proposed a nonlinear PID controller to solve the trajectory tracking problem of a holonomic mechanical system. A proposed control scheme is the development of the previously obtained control forms for the stationary motion stabilization problem of holonomic mechanical systems. The advantage of the control structure is that it is robust with respect to the mass-inertial parameters of the system. It allows us to take into account the action of potential and viscous friction forces on the system, and it is also convenient in choosing the control type by combining PD, PID, and PI controllers depending on the completeness of the generalized coordinates measurement. Note that the proposed control structure for the serial robot manipulator does not require the angular velocity measurement of the base vertical link. Notes and Comments. This work is financially supported by Russian Foundation for Basic Research (projects no. 19-01-00791, 18-41-730022).

References 1. Merkin, D.R.: Introduction to the Theory of Stability. Springer, New York (1996) 2. Rouche, N., Habets, P., Laloy, M.: Stability Theory by Lyapunov’s Direct Method. Springer, New York (1977) 3. Arimoto, S., Miyazaki, F.: Stability and robustness of PID feedback control for robot manipulators of sensory capability. In: Brady, M., Paul, R.P. (eds.) Robotics Researches: First International Symposium, pp. 783–799. MIT Press, Cambridge (1984) 4. Arimoto, S.: Control Theory of Non-Linear Mechanical Systems: A Passivity-Based and Circuit-Theoretic Approach. Clarendon Press, Oxford (1996)

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5. Orrante, J., Santibanez, V., Campa, R.: On saturated PID controllers for industrial robots: the PA10 robot arm as case of study. In: Ehsan Shafiei, S. (ed.) Advanced Strategies for Robot Manipulators. InTech (2010) 6. Santibanez, V., Camarillo, K., Moreno-Valenzuela, J., Campa, R.: A practical PID regulator with bounded torques for robot manipulators. Int. J. Control Autom. Syst. 8(3), 544–555 (2010) 7. Kelly, R., Santibanez, V., Loria, A.: Control of Robot Manipulators in Joint Space. Springer, Berlin (2005) 8. Andreev, A.S., Peregudova, O.A.: Nonlinear regulators in position stabilization problem of holonomic mechanical system. Mech. Solids 3, S22–S38 (2018) 9. Andreev, A.S.: On motion stabilization of a mechanical system with cyclic coordinates. In: Proceedings of 2018 14th International Conference “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference) (STAB). IEEE Xlpore (2018). https://ieeexplore.ieee.org/abstract/document/8408342 10. Andreev, A., Peregudova, O., Sutyrkina, K., Filatkina, E.: On global output feedback trajectory tracking control of a wheeled mobile robot. In: 23rd International Conference on Mechatronics Technology (ICMT). IEEE Xplore (2019). https:// doi.org/10.1109/ICMECT.2019.8932122 11. Ortega, R., Loria, A., Kelly, R.: A semiglobally stable output feedback PI2D regulator for robot manipulators. IEEE Trans. Autom. Control 40(8), 1432–1436 (1995) 12. Berghuis, H., Nijmeijer, H.: A passivity approach to controller-observer design for robots. IEEE Trans. Robot. Autom. 9(6), 740–754 (1993) 13. Loria, A., Lefeber, E., Nijmeijer, H.: Global asymptotic stability of robot manipulators with linear PID and PI2D control. Stab. Control Theory Appl. 3(2), 138–149 (2000) 14. Burkov, I.V.: Stabilization of a natural mechanical system without measuring its velocities with application to the control of a rigid body. J. Appl. Math. Mech. 62(6), 853–862 (1998) 15. Andreev, A.S., Peregudova, O.A.: Stabilization of the preset motions of a holonomic mechanical system without velocity measurement. J. Appl. Math. Mech. 81(2), 95– 105 (2017) 16. Andreev, A., Peregudova, O.: Non-linear PI regulators in control problems for holonomic mechanical systems. Syst. Sci. Control Eng. 6(1), 12–19 (2018). https:// doi.org/10.1080/21642583.2017.1413437 17. Andreev, A.S., Peregudova, O.A.: On the stability and stabilization problems of Volterra integral-differential equations. Russ. J. Nonlinear Dyn. 14(3), 387–407 (2018). http://ndtest1.ics.org.ru/nd180309/ 18. Yarza, A., Santibanez, V., Moreno-Valenzuela, J.: An adaptive output feedback motion tracking controller for robot manipulators: uniform global asymptotic stability and experimentation. Int. J. Appl. Math. Comput. Sci. 23(3), 599–611 (2013). https://doi.org/10.2478/amcs-2013-0045 19. Salinas, A., Moreno-Valenzuela, J., Kelly, R.: A family of nonlinear PID-like regulators for a class of torque-driven robot manipulators equipped with torqueconstrained actuators. Adv. Mech. Eng. 8, 1–14 (2016) 20. Andreev, A.S., Peregudova, O.A.: On global trajectory tracking control of robot manipulators in cylindrical phase space. Int. J. Control (2019). https://www. tandfonline.com/doi/full/10.1080/00207179.2019.1575526

LMI-Based Sliding Mode Controller Design for an Uncertain Single-Link Flexible Robot Manipulator Jos´e Manuel Andrade1(B) and Christopher Edwards2 1

2

Department of Electronics, Computing and Mathematics, College of Engineering and Technology, University of Derby, Markeaton St., Derby DE22 3AW, UK [email protected] College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter EX4 4QF, UK [email protected]

Abstract. A conventional (first order) sliding mode controller (SMC) is applied to an uncertain planar single-link flexible robot arm in the context of the regulator problem (disturbance rejection). The planar single-link flexible robot arm exhibits nonlinearities and uncertainties associated with Coulomb’s friction, the payload, and viscous friction. A SMC design framework that takes into account matched and mismatched uncertainties is proposed in this paper. The design methodology involves linear matrix inequality (LMI) methods and polytopic models of the uncertain planar single-link flexible robot arm. Computer simulation results demonstrate the effectiveness of the proposed sliding mode approach. Keywords: Sliding mode control · Linear matrix inequalities (LMIs) · Polytopic models · Flexible robot manipulator · Flexible link manipulator

1 Introduction Control of flexible robot arms has been extensively studied and can be classified according to the different numbers of flexible links (i.e. single, double and multi-link flexible manipulators [1]) and the different control objectives (end-effector regulation, endeffector to rest in a desired fixed time, joint trajectory tracking, and end-effector trajectory tracking [2]). Control techniques including feed-forward (for vibration suppression) and PID control, adaptive control (model reference and self-tuning control), inversion based control, robust control, optimal control, Lyapunov based control, and sliding mode control (SMC) have all been applied to regulate the motion of flexible manipulators [1, 2]. Feedback linearization combined with SMC has been applied to a flexible joint robot in [3] under the assumption that the state variables are directly measurable and hence the inclusion of a robust state estimator is suggested to overcome the difficulty of measuring some variables. Yeung and Chen [4] consider an arbitrary number of flexible modes c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gonc¸alves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 696–706, 2021. https://doi.org/10.1007/978-3-030-58653-9_67

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when solving the regulation problem and attenuating vibrations of a single-link flexible manipulator using SMC. Again, in this work all the state variables are assumed to be available for control. Qian and Ma in [5] consider the control of the end-point position of a robot arm in a non-collated manner by using a sliding surface which is constructed considering the end-point position and its derivative. Although Qian and Ma assume that the position variables are directly measured, velocities are estimated by means of first order estimators [5]. Choi et al. [6] applied sliding mode theory to control the tip (end-point) position of a flexible manipulator subject to uncertain parameters, and in this work, a decoupled reduced order observer is employed to estimate velocity state variables which are not directly measurable. Chen and Hsu [7] tackled modelling issues related to assumed-mode and finite-element methods. Then, a sliding mode controller involving a finite-element based model of a single-link flexible manipulator, is applied for regulation control and vibration suppression considering payload changes. Jalili [8] applied an exponentially stable variable structure controller for angular velocity regulation considering the angular positions of the robot arm base and the tip as the only measurable signals. In addition, on-line estimation of perturbations, associated with the unmodeled dynamics and parameter uncertainties, are considered. Bazzi and Chalhoud [9] present conventional and fuzzy sliding mode controllers in which the fuzzy sliding mode controller design involves Lyapunov functions and a boundary layer. In this paper, a first-order (conventional) sliding mode controller, whose design accounts for both matched and mismatched uncertainties, is proposed. The synthesis methodology involves polytopic models of an uncertain planar single-link flexible robot arm and linear matrix inequality (LMI) methods. Both the sliding surface and control law are designed using LMIs considering the mismatched uncertainties present in the planar single-link flexible robot manipulator. To the best knowledge of the authors, the LMI and polytopic-based sliding mode control approach, presented in this paper, has not been applied to the regulation control problem of single-link flexible manipulators before. The control approach proposed in this paper allows for parametric uncertainties to be explicitly considered when designing a sliding mode controller using LMI techniques. Furthermore, specifications such as fast decay, appropriate damping, and reasonable controller dynamics can be imposed via placement of poles in an LMI region defined by the designer. Another important feature to point out is that LMIs can be solved numerically using efficient optimization algorithms available in software tools such as MATLAB. For all these reasons, the control design methodology presented in this paper offers a new attractive control tool to the designer.

2 Mathematical Model and Problem Formulation The mathematical model of the single-link flexible robot arm, considered in this paper, is based on the following assumptions: (i) the mass of the flexible link is negligible with respect to the load mass concentrated at the tip, (ii) there is no torque at the extreme of the arm, (iii) the motion of the robot arm is confined to the horizontal plane (x, y) and hence the effect of gravity is assumed negligible, and (iv) the actuator involves a DC motor, a servo-amplifier and a reduction gear. Thus, the dynamic behaviour of the plant is described by the following differential equations [10–12]:

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knv(t) = Jn2 θ¨m (t) + νm n2 θ˙m (t) + Γcl (t) + Γcp (t)   ml 2 θ¨t (t) = c θm (t) − θt (t)

(1) (2)

where k is a constant parameter associated with the actuator (i.e., the DC motor and servo-amplifier) [Nm/V], v is the input voltage of the motor [V] (i.e. the control signal) and is assumed to be within the range [−10, +10] V, J is the inertia of the DC motor [kgm2 ], n is the motor-gear reduction ratio [non-dimensional], νm is the viscous friction coefficient [Nms], Γcl is the unknown Coulomb friction [Nm], Γcp is the motor-arm coupling torque [Nm], m is the mass of the load [kg], l is the length of the arm [m], c is the stiffness of the arm [Nm]. The motor-gear angular position [rad], speed [rad/s] and acceleration [rad/s2 ] are denoted by θm , θ˙m , and θ¨m respectively. With regard to the arm tip, θt and θ¨t denote angular position [rad] and acceleration [rad/s2 ] respectively. Notice that Newton’s notation for differentiation, which is also known as the dot notation for differentiation, (e.g. θ˙m (t) = d θm (t)/dt) is used in this paper. Higher derivatives are represented using multiple dots (e.g. θ¨m (t) = d 2 θm (t)/dt 2 ). A schematic diagram of the planar single-link flexible manipulator, considered in this paper, is shown in Fig. 1 whereas its nominal parameter values are presented in Table 1. Table 1. Nominal parameters for the singlelink flexible manipulator [10].

Parameter Units

Fig. 1. Planar coordinate system for the singlelink flexible robot arm.

Value

k

[N · m/V] 0.21

J

[kg · m2 ]

6.87 ×10−5

νm0 n c l m0

[N · m/s]

1.041 ×10−3 50

[N · m]

1.584

[m]

0.5

[kg]

0.03

The motor-arm coupling torque Γcp (t) in (1) is related to the motor and tip angular positions by the following algebraic equation [10, 11]:   Γcp (t) = c θm (t) − θt (t) (3) In this paper, it is assumed that all state variables are available for the implementation of a state-feedback control law. Sensors for measuring both the angular position of the motor θm (t) and the coupling torque Γcp (t) may be employed. More specifically, an incremental encoder may be used to measure the angular position whereas a couple of strain gauges, installed at the base of the robot arm, may be used to measure the coupling torque [10]. On the other hand, the angular speed θ˙m (t) and the time derivative of the coupling torque Γ˙cp (t) may be estimated using, for example, the robust exact differentiator proposed in [13], which is based on sliding mode concepts.

LMI-Based Sliding Mode Controller Design

699

Without loss of generality, the viscous friction coefficient ν and load mass m are assumed to be the uncertain parameters in the flexible link manipulator system. In the particular case of a robot arm, the uncertainty associated with the load mass stems from payload changes that affect the manipulator during different operation conditions. The uncertainty in the viscous friction coefficient is related to the use of different types of lubricants and/or a deterioration in the properties of the fluid. In this paper, it is assumed a variation of the viscous friction coefficient up to three times the nominal value, and that the payload varies between 0.7100m0 and 6.6633m0 where m0 is the nominal load mass. Therefore, the following bounds are established: 0

N·m N·m ≤ νm ≤ 3νm0 = 0.0031 s s

and

0.0213 kg ≤ m ≤ 0.1999 kg

(4)

The problem to be addressed in this paper is stated as follows: consider the uncertain nonlinear single-link flexible robot manipulator described in (1)–(3), which involves parameter uncertainties (payload m and viscous friction νm ) and the nonlinear effect associated with Coulomb’s friction Γcl (t) (which is assumed to be a bounded disturbance), then assuming that all state variables are available for control, design a sliding mode controller that regulates the angular position of the tip θt (t).

3 Sliding Mode Control Using Polytopic Models The controller design methodology proposed in this section involves a polytopic description of the parametric mismatched uncertainties affecting the state matrix. Both the sliding surface and control law are designed using linear matrix inequality methods considering the mismatched uncertainties present in the dynamical system. Consider an uncertain dynamical system described in state-space form ∀ t ≥ 0 by       Δ A11 (t) Δ A12 (t) 0 A11 A12 u(t) + ξ (t, x, u) (5) x(t) + + x˙ (t) = A21 A22 0 0 B2 where x ∈ ℜn is the state vector and u ∈ ℜm is the control input vector. Furthermore, matrices A11 ∈ℜ(n−m)×(n−m) , A12 ∈ℜ(n−m)×m , A21 ∈ℜm×(n−m) , A22 ∈ℜm×m and B2 ∈ ℜm×m are known constant matrices. The function ξ : ℜ+ ×ℜn ×ℜm → ℜm , which represents the lumped sum of matched nonlinearities and/or uncertainties, is unknown but bounded. The input matrix sub-block B2 is non-singular. Without loss of generality, it has been assumed that the nominal pair (A, B) is already in regular form [14]. Matrices Δ A11 ∈ ℜ(n−m)×((n−m)) and Δ A12 ∈ ℜ(n−m)×(m) are unknown matrices that are assumed to be affine with respect to the uncertain parameters written in vector form as ϑ (t) = [ϑ1 (t) ϑ2 (t) · · · ϑr (t)]T . These uncertain parameters satisfy

ϑ i ≤ ϑi (t) ≤ ϑ i for i ∈ I(1, r)

(6)

The matched uncertainty ξ (t, x, u) is assumed to be bounded by ξ (t, x, u) ≤ k1 u(t) + ϕ (t, x(t)) + k2

(7)

where ϕ : ℜ+ × ℜn → ℜ+ is a known function, and k1 , k2 ∈ ℜ+ are known positive scalars.

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Sliding Surface Design: A sliding surface S is a subspace of the state space X given by

 S = x ∈ X ⊆ ℜn : σ (t) = Γ x(t) = Γ 1 Γ 2 x(t) = 0 , σ (t) ∈ ℜm (8) where the design matrix Γ ∈ ℜm×n can be parameterised, as shown in [15], as  Γ = Γ 2 K Im

(9)

with the matrix Γ 2 ∈ ℜm×m is a scaling factor and the gain matrix K ∈ ℜm×(n−m) is to be designed. Since the switching function σ (t) and its time derivative are equal to zero when in the sliding mode [14], it can be demonstrated that the sliding mode dynamics are governed by   (10) x˙ 1 (t) = AΔ 11 (t) − AΔ 12 (t)K x1 (t)     where AΔ 11 (t) = A11 + Δ A11 (t) and AΔ 12 (t) = A12 + Δ A12 (t) . The reduced-order system defined in (10) reveals that the invariance property [16] is not attained because of the unmatched uncertainties. By assumptionthe uncertain matrix  Δ A(t) is affine in ϑ (t), and hence a polytopic model for the pair AΔ 11 (t), AΔ 12 (t) can be constructed as   N N  Pσ = ∑ μ j AΔ 11 j AΔ 12 j : ∑ μ j = 1, μ j ≥ 0 (11) j=1

j=1

  where N is the number of vertices of Pσ and the pairs AΔ 11 j , AΔ 12 j are assumed to be stabilisable. Let Q1 ∈ ℜ(n−m)×(n−m) be a symmetric positive definite (s.p.d.) matrix and L1 be defined such that K = L1 Q−1 1 . Then, it can be proved that the reduced-order system (10) is quadratically stable if and only if the following LMIs are satisfied: Q1 ATΔ 11 j + AΔ 11 j Q1 − LT1 ATΔ 12 j − AΔ 12 j L1 ≺ 0 for j = 1, 2, · · · , N .

(12)

Quadratic stability is guaranteed when (12) is feasible, and hence a gain matrix K can be computed for designing a sliding surface. Nevertheless, the designer does not have freedom regarding the placement of poles in a designated area of the complex left half-plane. This lack of freedom is overcome with a proposition, which allows the designer to define a particular convex region in the complex plane for pole placement, that will be stated in the sequel. A convex set D ⊂ C is said to be an LMI region [17] if   (13) D = z ∈ C : fD (z) = Ξ + z Φ + z Φ T ≺ 0 where z¯ is the conjugate of the complex variable z, the complex function fD is the characteristic function of D, where Ξ = Ξ T and Φ are known constant matrices of appropriate dimensions. Theorem 1 [17]. A matrix M ∈ ℜn×n has all its eigenvalues in the LMI region D, and is called D-stable, if and only if there exists an s.p.d. matrix X ∈ ℜn×n such that MD (M, X) = Ξ ⊗ X + Φ ⊗ (MX) + Φ T ⊗ (MX)T ≺ 0

(14)

LMI-Based Sliding Mode Controller Design

701

The symbol ⊗ in (14) denotes the Kronecker product. The synthesis of the gain matrix K, involved in the parameterisation of the switching gain matrix Γ defined in (9), is formulated as an LMI problem. If a solution is found, then all the reduced-order closed-loop eigenvalues will lie within the convex region D(h, cn , rd , α ). Let Q1 ∈ ℜ(n−m)×(n−m) be an s.p.d. matrix and define L1 such that K = L1 Q−1 1 . It can be proved that the reduced-order system (10) is quadratically stable and λ (AΔ 11 j − AΔ 12 j K) ⊆ D(h, cn , rd , α ) for j ∈ I(1, N), if and only if the following LMIs are satisfied Ψ + 2hQ1 ≺ 0 (15)     −rd Q1 AΔ 11 j Q1 − AΔ 12 j L1 + cn Q1 sin(α )Ψ cos(α )ϒ ≺ 0 and ≺ 0 (16) ∗ −rd Q1 ∗ sin(α )Ψ where and

Ψ = Q1 ATΔ 11 j + AΔ 11 j Q1 − LT1 ATΔ 12 j − AΔ 12 j L1

(17)

ϒ = AΔ 11 j Q1 − Q1 ATΔ 11 j − AΔ 12 j L1 + LT1 ATΔ 12 j

(18)

Control Law Synthesis: The next step, after synthesising the sliding surface defined in (8), is the design of a control law to induce and maintain sliding on S . To this end, T  consider the following change of coordinates x(t) → Tu x(t) = x(t) = xT1 σ T that produces       ˙x(t) = A11 (t) A12 (t) x(t) + 0 u(t) + ξ (t, x, u) (19) B2 A21 (t) A22 (t)  where A11 (t) = AΔ11 (t) − AΔ12 (t)K, A12 (t) = AΔ12 (t)Γ −1 , A21 (t) = Γ 2 KAΔ11 (t) + 2    A21 − A22 K , A22 (t) = Γ 2 KAΔ12 (t) + A22 , and B2 = Γ 2 B2 . Let P be an s.p.d. matrix partitioned compatibly with (19) as follows   P 0 P= 1 (20) 0 P2 where P1 ∈ ℜ(n−m)×(n−m) and P2 ∈ ℜm×m . The diagonal form of P ∈ ℜn×n results naturally from the interrelation between the stability of the reduced-order system and the form of the closed-loop system due to the regular form (19). The proposed control has the form u(t) = −Fx(t) + uNL (t) where the nonlinear component is given by ⎧ ⎨−ρ (t, x, u)B−1 P2σ (t) 2 P σ (t) 2 uNL (t)= ⎩ 0

(21)

if σ (t) = 0 otherwise

(22)

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The scalar function ρ (t, x, u) satisfies   B2  k1 uL (t) + ϕ (t, x) + k2 + η ρ (t, x, u) ≥ 1 − k1 κ (B2 )

(23)

where κ (B2 ) denotes the condition number of B2 , and η ∈ ℜ+ is a design parameter. An LMI formulation for designing the gain matrix F ∈ ℜm×n is described in what follows. The starting point corresponds to the Lyapunov inequality  T   A(t) − BF P + P A(t) − BF ≺ 0 .

(24)

By using similar arguments as those considered in the previous section, concerned with the  of a sliding surface, a polytope Pu can be constructed for the system pair  design A(t), B as follows   N N  (25) Pu = ∑ μ j A j B : ∑ μ j = 1, μ j ≥ 0 . j=1

j=1

Thus, the matrix inequality defined in (24) can be written as N

∑ μj



T   A j − BF P + P A j − BF ≺ 0 .

(26)

j=1

Since μ j ≥ 0 for j = 1, 2, · · · , N a necessary and sufficient condition for (26) to be satisfied is that  T   A j − BF P + P A j − BF ≺ 0 for j = 1, 2, · · · , N (27) ¯ ¯ where P has been defined in (20). Considering the structure of the system pair (A(t), B) given in (19), and the partition of the gain matrix  F = F1 F2 (28) where F1 ∈ ℜm×(n−m) and F2 ∈ ℜm×m , it follows that   A11 j A12 j A j − BF = . A21 j − B2 F1 A22 j − B2 F2

(29)

By applying the change of variable Q  P−1 and a congruence transformation to the matrix inequalities given in (27), and then by defining L  FQ the following LMIs are obtained: T T (30) QA j + A j Q − LT B − BL ≺ 0 for j = 1, 2, · · · , N . An LMI problem is formulated in the sequel to design the gain matrix F defined in (28) so that the eigenvalues of A j − BF for j = 1, 2, · · · , N, lie in a convex region of the complex left half-plane C− specified by the designer.

LMI-Based Sliding Mode Controller Design

Define    −1  Q1 0 P1 0 Q = = P−1 0 Q2 0 P−1 2

,

L 1  F1Q 1

703

and L 2  F2Q 2 (31)

It can be demonstrated that the bilinear matrix inequalities (BMIs) in (27) can be written as LMIs after applying a congruence transformation. Thus, the following LMI feasibility problem can be formulated L 2 and Q find L 1 ,L s.t.   T T ¯ T + A11 Q 1 L T1 B2 + A12 j Q 2 Q 1 A21 j −L Q 1A j 11 j ≺ 0 for j ∈ I(1, N) (32) T T L T2 B2 + A22 j Q 2 − B2L 2 ∗ Q 2 A22 j −L Q 0

(33)

If there exists a feasible solution to the problem above, the gain matrix F is computed as follows  −1 (34) F = L 1 Q −1 1 L 2 Q2 Consequently, the linear part of the control law in the original coordinates is given by uL (t) = −FTu x(t)

(35)

It can be proved by using the Lyapunov theory that the control law (21), with nonlinear component (22), guarantees that a sliding motion takes  place in finite time on the sliding surface S defined in (8) inside the sliding patch Ω = x1 ∈ ℜ(n−m) , x2 ∈ ℜm :    x1  < ηγ −1 where γ = max A21 j − B2 F1  . j=1,2,···n

4 Design and Simulation Results The planar single-link flexible manipulator described in (1)–(3) can be written in statespace form as follows   (36) x˙ (t) = A0 + Δ A x + Bu(t) + DΓcl (t) where ⎤ ⎡ 0 1 0 0 0 0 0 νm 0 δ νm 1 ⎥ ⎢ ⎢0 − − Jn2 0⎥ ⎢0 − J 0 J + A0 + Δ A = ⎢ ⎣0 0 0 0 1⎦ ⎣0 0   c νm 0 − cδJνm − cδlm2 Δ 0 − J 0 − Jnc2 + cml 2Δ 0 ⎡

 B= 0

k Jn

0

ck T Jn

and

 D=− 0

1 Jn2

0

c T Jn2

.

⎤ 0 0⎥ ⎥ 0⎦ 0

(37)

(38)

The nominal and uncertain viscous friction terms are denoted by νm0 and δ νm respectively. The uncertain payload m is not an affine parameter and hence the parameter transformation mΔ = m1 has been applied. The nominal value of the mass is denoted by mΔ0 whilst the mass uncertainty is given by δ mΔ .

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By applying a similarity transformation, the original system (36) can be written in the so-called regular form given in (19). Then, a polytope with four vertices is constructed and a robust sliding surface is designed considering h = 6, cn = 0, rd = 35, and α = 0.9273 rad. The following switching gain vector is obtained  Γ = −22.296357 −4.198279 −38.843124 1.467823 . (39) The linear component of the control law has been designed as  uL (t) = − 1.693008 0.266858 9.648876 0.194599 x(t) whereas the nonlinear component has been designed as ⎧ 0.011400 σ (t) ⎨ρ (t, x, u) 1.305586 σ (t) + 0.01 uNL (t) = ⎩ 0

(40)

if σ (t) = 0

(41)

otherwise

The discontinuous term of the nonlinear control component has been implemented as a sigmoid-like function in order to avoid the high-frequency switching nature of this part of the control law. Furthermore, the scalar function ρ (·) is straightforwardly calculated using the matched uncertainty ξ (t, x, u) involving the Coulomb’s friction term in (36) and the matched uncertainty associated with the uncertain state matrix given in (37). Computer simulations were carried out using different values for load mass m and viscous friction vm , and an artificial disturbance d(t) injected onto the angular position θm (t). The unmatched disturbance d(t) is introduced into the system in order to prove the effectiveness of the sliding mode controller given by the switching gain matrix (39) and the control law (21) with linear component (40) and nonlinear component (41) to reject the disturbances and hence regulate the state variables to the steady state which in this case corresponds to the origin of the state space. The initial condition used in  T all simulations is given by x0 = 0.2618 0 0 0 . Effective regulation of the angular position of the tip θt (t) can be seen in Figs. 2 and 3 for different values of load mass m and viscous friction vm , and disturbance d(t) simulated as a step signal (black dashed signal) of amplitude 0.1 [rad] and a step time of 10 s. 0.3

0.3 (t) using m and v 0

t

disturbance (t) using m & lower bound of v

0.2

t

0.15

t

(t) using m and v 0

0

0

m

(t) using vm & lower bound of m 0

0.1 0.05 0

m

0

disturbance (t) using m 0 & upper bound of vm t

0.2

t

t

(t) and d(t) [rad]

0.25

m

(t) and d(t) [rad]

t

t

(t) using v

m

& upper bound of m 0

0.1

0

-0.1

-0.05 0

5

10

15

20

0

5

10

time [s]

time [s]

(a)

(b)

15

20

Fig. 2. Regulation of angular position of the tip θt (t) considering (a) nominal values m0 and vm0 , and lower bounds of m and vm ; (b) nominal values m0 and vm0 , and upper bounds of m and vm .

LMI-Based Sliding Mode Controller Design 0.3

0.3

t

t

m

0.25 m

0.1

0

t

(t) and d(t) [rad]

0.2

t

(t) using upper bound of m & lower bound of v

(t) and d(t) [rad]

t

disturbance (t) using upper bound of m & upper bound of v

(t) using lower bound of m & upper bound of v

disturbance (t) using lower bound of m & lower bound of v t

0.2

705

m

m

0.15 0.1 0.05 0

-0.1

-0.05 0

5

10

15

20

0

5

10

time [s]

time [s]

(a)

(b)

15

20

Fig. 3. Regulation of angular position of the tip θt (t) considering (a) upper bound of m and lower bound of vm , and upper bounds of m and vm , and (b) lower bound of m and upper bound of vm , and lower bounds of m and vm .

5 Conclusions A conventional (first-order) sliding mode control scheme based on linear matrix inequalities, and a polytopic representation of a single-link flexible manipulator, has been proposed in this paper. The sliding mode existence and reachability problems have been tackled using polytopic models. The design of a sliding surface, which accounts for mismatched uncertainties, was carried out based on robust pole clustering in LMI regions and a feasibility LMI problem. The control law consists of two components and its design has also taken into account both matched and mismatched parametric uncertainties. The results obtained through computer simulations have demonstrated the effectiveness of the proposed sliding mode control scheme despite the parameter uncertainties and the injection of an external disturbance directly affecting the motorgear angular position. For future work, the authors will consider only a subset of the state variables involving physically measurable signals (i.e. the angular position of the motor θm (t) and the coupling torque Γcp (t)) and will explore the application of robust exact differentiators, based on sliding mode concepts, for obtaining the first derivatives of the measurable state variables (i.e. θ˙m (t) and Γ˙cp (t)) required for the control law. Furthermore, the application of higher-order sliding mode control algorithms will also be investigated considering uncertain parameters, disturbances and measurement noise.

References 1. Dwivedy, S.K., Eberhard, P.: Dynamic analysis of flexible manipulators, a literature review. Mech. Mach. Theory 41, 749–777 (2006) 2. Benosman, M., Vey, G.L.: Control of flexible manipulators: a survey. Robotica 22, 533–545 (2004) 3. Sira-Ram´ırez, H., Spong, M.W.: Variable structure control of flexible joint manipulators. Int. J. Robot. Autom. 3(2), 57–64 (1988) 4. Yeung, K.S., Chen, Y.P.: Regulation of a one-link flexible robot arm using sliding mode techique. Int. J. Control 49(6), 1965–1978 (1989)

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5. Qian, W.T., Ma, C.C.H.: A new controller design for a flexible one-link manipulator. IEEE Trans. Autom. Control 37, 133–137 (1992) 6. Choi, S.-B., Cheong, C.-C., Shin, H.-C.: Sliding mode control of vibration in a single-link flexible arm with parameter variations. J. Sound Vib. 5, 737–748 (1995) 7. Chen, Y.P., Hsu, H.T.: Regulation and vibration control of an FEM-based single-link flexible arm using sliding mode theory. J. Vib. Control 7(5), 741–752 (2001) 8. Jalili, N.: Regulation of a lightweight one-link flexible robot arm using an exponentially stable variable structure controller. In: Proceedings of the American Control Conference, Arlington, VA, USA, vol. 2, pp. 1569–1574, June 25–27, 2001 9. Bazzi, B.A., Chalhoub, N.G.: Fuzzy sliding mode controller for a flexible single-link robotic manipulator. J. Vib. Control 11, 295–314 (2005) 10. Becedas, J., Trapero, J., Feliu, V., Sira-Ram´ırez, H.: Adaptive controller for single-link flexible manipulators based on algebraic identification and generalized proportional integral control. IEEE Trans. Syst. Man Cybern. Part B Cybern. 39, 735–751 (2009) 11. Feliu, V., Ramos, F.: Strain gauge based control of single-link flexible very lightweight robots robust to payload changes. Mechatronics 15, 547–571 (2005) 12. Feliu, V., Rattan, K.S., Brown Jr., H.B.: Control of flexible arms with friction in the joints. IEEE Trans. Robot. Autom. 9, 467–475 (1993) 13. Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379– 384 (1998) 14. Utkin, V.I.: Sliding Modes in Control Optimization. Springer, Heidelberg (1992) 15. Edwards, C., Spurgeon, S.K.: Sliding Mode Control - Theory and Applications. Systems and Control Series. Taylor & Francis, London (1998) 16. Dra˘zenovi´c, B.: The invariance conditions in variable structure systems. Automatica 5, 287– 295 (1969) 17. Chilali, M., Gahinet, P.: H∞ design with pole placement constraints: an LMI approach. IEEE Trans. Autom. Control 41(3), 358–367 (1996)

Control of Bio-Inspired Multi-robots Through Gestures Using Convolutional Neural Networks in Simulated Environment A. A. Saraiva2,6 , D. B. S. Santos1 , Nuno M. Fonseca Ferreira3,4,5(B) , and Jos´e Boaventura-Cunha3,4,5 1

4

UESPI-University of State Piaui, Piripiri, Brazil [email protected] 2 University of Tr´ as-os-Montes and Alto Douro, Vila Real, Portugal [email protected] 3 Coimbra Polytechnic - ISEC, Coimbra, Portugal [email protected], [email protected] Knowledge Engineering and Decision-Support Research Center (GECAD) of the Institute of Engineering, Polytechnic Institute of Porto, Porto, Portugal 5 INESC-TEC Technology and Science, Porto, Portugal 6 University of Sao Paulo Sao Carlos, S˜ ao Carlos, Brazil

Abstract. In this paper the comparison between three convolutional neural networks, used for the control of bio-inspired multi-robots in a simulated environment, is performed through manual gestures captured in real time by a webcam. The neural networks are: VGG19, GoogLeNet and Alexnet. For the training of networks and control of robots, six gestures were used, each gesture corresponding to one action, collective and individual actions were defined, the simulation contains four bioinspired robots. In this work the performance of the networks in the classification of gestures to control robots is compared. They proved to be efficient in the classification and control of agents, with Alexnet achieving an accuracy of 98.33%, VGG19 98.06% e Googlelenet 96.94%. Keywords: Bio-inspired Alexnet

1

· Gestures · Robotic · VGG19 · Googlenet ·

Introduction

With increasing interest, research was developed around the use of Artificial Intelligence for the development of pattern recognition systems [1]. Such systems, incorporated into manipulator robots, were used in automated assembly operations, resulting in systems that can operate in structured and unstructured environments, through the use of advanced sensory feedback mechanisms, making decisions using learning algorithms and reasoning [1]. c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 707–718, 2021. https://doi.org/10.1007/978-3-030-58653-9_68

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In modern times, it is also widely diffusing robotics controlled by various methods. It can be commanded by gestures, voice or even programmed to work autonomously, executing routines and following commands pre-programmed for the necessary purposes of the application [2]. One line of research and development in robotics that has received attention in recent years is the development of bio-inspired walking robots [3]. Based on this inspired the design of manipulation of multi-robot spiders with 8 simulated feet, by means of manual gestures. The objects of interest that must be detected and analyzed are the manual coordinates that will be processed in real time, and the control of the spider robots. The method chosen and implemented is in the capture and detection of gestures by a webcam, then the real-time classification is performed by an artificial intelligence, consisting of a deep learning known as Convolutional Neural Network (CNN). Three different CNNs were used, these being pre-trained neural networks known as AlexNent, VGG19 and GoogLeNet, in this way an analysis is made of which network has a better performance for the proposed system. The last step is to perform the control of the robots in a simulated environment containing obstacles. To make the locomotion of the robots in an efficient way is implemented the algorithm A* Pathfinding. According to the classified gesture the robots will perform an action being collective or individual. The method approached ensures a robust coverage in image recognition, under certain assumptions that will be clarified throughout the text. The developed structure allows the robots to maintain or alter the formation of the specified trajectories. The document is divided into 5 sections, in which Sect. 2 deals with the state of the art, since Sect. 3 deals with the methodology, containing the system structure, the robots and the simulation environment used, follows the formulation of the applied central algorithm and the statistical method to verify the reliability of the system used. The results after application of the proposal are presented in Sect. 4 and the conclusion in Sect. 5.

2

Related Works

In [4], it is one that addresses the problem of deploying a network of robots to gather information in a risky environment. This may mean that there are adversary agents in the environment trying to disable robots. A probabilistic model of the environment is formulated, in which recursive Bayesian filters are used to estimate environmental events and hazards. Robots should control their positions both to avoid sensor failures and to provide useful information to the sensor. A method for gesture recognition is discussed in [5], where temporal information is more significant compared to general video classification tasks. A deep architectures for video gesture recognition is explored and a new end-to-end trainable neural network architecture is proposed, incorporating temporal convolutions and bidirectional recurrence.

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A general gesture recognition model for human-robot collaboration is also proposed in [6]. Four essential technical components are used for gesture recognition model for human-robot collaboration: sensor technologies, identification of gestures, tracking of gestures and classification of gestures. The proposed approaches are classified according to the four essential technical components. Statistical analysis is also presented in this paper. In the work [7], it presents the control of an omnidirectional wheelchair based on manual gestures using inertial measurement unit and myoelectric units as wearable sensors. Seven gestures are recognized and graded using formbased resource extraction and the Dendogram Support Vector Machine classifier. Dynamic gestures are mapped to the omni-directional motion commands to control the wheelchair. In [8], a bio-inspired robot with waving fins and its methods of control is proposed. The control consists of some basic movements, swimming back and forth, diving movement and climb, these movements are implemented and evaluated by experiments. Then, a hybrid control is presented that combines the control of rejection of active perturbations with a fuzzy strategy to obtain the depth of the environment.

3

Methodology

In this section will be discussed the structure of the adopted system, the manual gestures chosen, the structure of the control algorithm, the robots and the simulated environment, the neural networks used, the validation metrics of the neural networks and finally the control method used for robots. 3.1

Structure of the System

The structure of the system corresponds to a sequence of steps that will allow the tracking of gestures in real time Fig. 1. The first step consists in the training

Fig. 1. Structure of the system.

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of the neural networks with the dataset, after the training a webcam is used to capture the manual gestures, then the gesture is classified in real time by a CNN, after that the classification of the gesture is sent via Socket communication [9], for Game Engine Unity3D where robots were built and the simulated environment. For the locomotion of the robots in the environment the algorithm A* Pathfinding is used. According to the classified gesture the robots performed an action being individual or collective. 3.2

Bio-inspired Model and Simulation Environment

For this work is used 4 bio-inspired robots, in which the same consist of an 8legged spider, the model of the used robots can be seen in Fig. 2. This model was developed and applied in the Unity 3D game development environment, where the simulation and visualization of robots was controlled, and certain instructions are performed when a gesture is recognized.

Fig. 2. Bio-inspired robot.

The method chosen for simulation consists of an environment containing obstacles Fig. 3, thus hindering the locomotion of the robots. To solve the robot locomotion problem, the A* Pathfinding algorithm is used and implemented, which consists of an informed search algorithm, or a search for the first best point, which means that it solves the search problem between possible paths, in order to reduce the cost, ie lower distance traveled and less time. The objective of choosing this type of environment is to perform the simulation of a forest Fig. 3, which is a difficult-to-access environment, with the purpose of proposing a multi-robot control system using CNN, which can be an aid tool in various tasks such as rescue, surveillance, exploration and others. 3.3

A* Pathfinding

In the field of Artificial Intelligence, A* algorithm is a computer algorithm that is often used in pathfinding and crossing chartt. [10]. In 1964 Nils Nilsson created

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Fig. 3. Bio-inspired simulation environment.

a heuristic-based approach to speed up the Dijkstra algorithm. This algorithm was called A1 [11]. A * is an informed search algorithm, or a search of the first best, it means that it solves the search problem between possible paths, in order to reduce the cost, ie less distance traveled and less time [12]. Such an algorithm is based on graphs, from a specific node of the graph, it builds a tree of paths from that node, expanding the paths one step at a time, until one of its paths ends in the target node [13]. At each iteration, A* needs to determine which of its partial paths will be expanded in one or more paths, formulation of the algorithm can be seen in Eq. 1. f (n) = g(n) + h(n)

(1)

where n is the last node in the path, g(n) follows the cost of the initial node path to n and h(n) is the best path cost estimate of n to the goal. 3.4

Classification of Gestures

In practice, very few people train an entire CNN from scratch because it is relatively rare to have a dataset of sufficient size. To perform the classification of the gestures was used Transfer of learning. Transfer learning is commonly used in deep learning applications. Where it is possible to use a pre-trained network and use it as a starting point to learn a new task [14]. Fine-tuning a network with Learning Transfer is often much faster and easier than training a network with weights randomly initialized from scratch. With this you can quickly transfer the resources learned to a new task using a smaller number of training images [15]. In order to classify the database was used fine-tuning with Learning Transfer, three pre-trained networks known as AlexNet, VGG19 and GoogLeNet were used, thus analyzing which network has a better performance for the proposed system. Three layers of neurons of each network were replaced by beds with the

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number of neurons equal to the number of classes in the database, so that it is possible to make the correct prediction of each class. The data set was used NUS hand posture dataset-II of the work of [16]. This set of images contains 10 classes of gestures, where each class has 200 images, all of them 160 × 120 and containing variable background. For the tests of this method was chosen the 6 classes of gestures Fig. 4, totaling 1200 images, 70% (840 images) for training and 30% (360 images) for validation were used. AlexNet. It is a pre-trained network of approximately 1.3 million highresolution images of the ImageNet training set in its pre-training dataset, containing 1000 different classes images [17]. The network contains 8 layers, 5 convolutionals, some followed by grouping layers, two layers connected and a final softmax of 1000 outputs. It has 60 million parameters and 500,000 neurons optimized to reduce overfitting. It is worth noting that AlexNet was the first network to use dropout to assist in fully connected layer training. VGG19. VGG-19 is a convolutional neural network that is trained in more than one million images of the ImageNet database [18]. The network has 19 layers of depth and can classify images into 1000 categories of objects, such as keyboard, mouse, pencil and many animals. As a result, the network learned rich resource representations for a wide range of images. GoogLeNet. GoogLeNet is a pre-trained CNN, winner of ImageNet 2014, GoogLeNet contains 22 layers. The main feature of this architecture is the best use of computing resources within the network. With a carefully crafted design, we increase the depth and width of the net, keeping the computational costs constant. To optimize quality, architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing [19]. 3.5

Metrics of the Evaluation

As a statistical tool is used the confusion matrix that provides the basis for describe the accuracy of the classification and characterize the errors, helping refine the ranking [20]. The confusion matrix is formed by an arrangement of squares of numbers arranged in rows and columns expressing the number of sample units of a particular relative category inferred by a decision rule compared to the current category found in the field. Typically below the columns is the set of reference data that is compared to the product data of the classification that are represented along the lines. The measures derived from the confusion matrix are: total accuracy being chosen by the present work, individual class accuracy, producer precision, user precision and Kappa index, among others. The total accuracy is calculated by dividing the sum of the principal diagonal of the error matrix xii , by the total number of samples collected n. According to the Eq. 2.

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a

xii (2) n Accuracy is also used as a statistical measure of how well the classification validation test is. That is, to estimate the accuracy of a test, one must calculate the proportion of true and negative true in all evaluated cases. T =

Accuracy =

i= 1

tp + tn tp + tn + f p + f n

(3)

Where tp is the true positives, tn is true negatives, f p is the false positives and f n false negatives. 3.6

Control of Robots

Six gestures were used to control the robots Fig. 4, each gesture transmits an action to the robots being able to be of two categories, collective or individual. The “G1” gesture conveys a collective action to the robots that consists of the same ones executing a movement of walk forward, “G2” just as the previous gesture transmits a collective action causing the robots to return to the initial position,“G3” in turn consists of an individual action that causes the robot 3 to move to the center, “G4” transmits an individual action that makes the robot 2 go to the scepter. The “G5” gesture makes the second robot 4 go to the center, and finally the “G6” gesture causes robot 1 to go to the center. The functioning of the system control algorithm adopted can be seen in Fig. 5.

Fig. 4. Gestures used.

In Fig. 5 it is possible to visualize the operation of the system. The first step consists of dividing the dataset into training and testing, where 70% for training and 30% for test. In the second step, already with the trained model is realized

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the capture of the frames of the webcam and submitted to the model already trained, the same classifies the gesture in one of the 6 trained, it is also checked if the robot is already performing some movement [21], because if the action is not executed and the process is restarted, then if it is not performing the movement the command is sent to the robot in which the gesture corresponds.

Fig. 5. System operation.

4

Results

In this section, the accuracy of the proposed method is evaluated using a standard validation technique, where the accuracy of the CNNs is measured by the method presented in Eq. 3, as well as the results of the confusion matrices of each network. The performance of proposed CNNs is shown in Table 1, according to their matrices of confusion, as well as the training time for each of the networks.

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The processing was performed using a Quadro P6000 video card, which has 3840 CUDA cores (processors), 24 GB dedicated memory, 12 GB RAM and a fourth generation Core i7 processor. The training time for the networks was as follows: AlexNet took 2 min and 57 s to be trained, VGG consumed 8 min and 40 s, GoogleNet in turn took 9 min and 48 s and can be checked in the Table 1. With this it can be observed that alexNet was the network that took less time to carry out the training, in Fig. 7 it is possible to analyze the performance of training and validation of GoogLeNet, being the network that took more time to carry out the training Table 1.

Fig. 6. Example of the final result with robot control.

Fig. 7. Training and validation Googlelanet

The final result can be seen in Fig. 6, already in the 3D simulation environment, where the “G6” gesture is executed, transmitting an individual action, thus causing robot 1 to move to the center. The classification of gestures in real time is made by a webcam using code developed in MATLAB in communication with Unity 3D. How image capturing and processing are done in real time, the answer is automatic. Table 1 shows the final performance of each CNN, with

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AlexNet delivering superior performance over other CNNs, with an accuracy of 98.33%, but the other networks also perform well, achieving an accuracy of over 95%. Table 1. Accuracy of the interactions the model learning. Network

G1 G2 G3 G4 G5 G6 Acurracy Training time

AlexnetG1 1.0 0 G2 0 0.96 G3 0 0 G4 0.03 0 G5 0 0 G6 0 0

0 0 0 0 98.33 % 0 0 0 0.03 1.0 0 0 0 2 min 57 sec 0 0.96 0 0 0 0 0.96 0.03 0 0 0 1.0

VGG19G1 1.0 G2 0 G3 0 G4 0.01 G5 0 G6 0

0 0 0 0 98.06 % 0 0 0 0 1.0 0 0 0 8 min 40 sec 0 0.98 0 0 0 0 0.93 0.05 0 0 0.03 0.96

0 1.0 0 0 0.01 0

GooogLeNetG10.98 0.01 0 0 0 0 96.94 % G2 0 0.98 0 0 0 0.01 G3 0 0 1.00 0 0 0 9 min 48 sec G4 0.01 0 0 0.98 0 0 G5 0 0 0 0 0.980.01 G6 0 0.08 0 0.01 0.01 0.88

5

Conclusion

This article presents a real-time multi-robot control system from robust and reliable hand signals. Six gestures were used, and for each gesture there is an associated action, whether individual or collective. The agents were arranged in an environment built in Unity 3D, being that they used the algorithm A * to perform the locomotion within the environment. In addition, the neural networks used to classify the gestures performed well, with Alexnet having 98.33 % accuracy, VGG19 98.06 % and GoogLeNet had 96.94%. The system showed satisfactory results in terms of real-time video processing, obtaining analysis and classification of the gestures, to transmit information to the robots and with this actions are performed instantaneously, with this the proposed method proved to be a potential tool for aiding in various tasks, such as rescue, surveillance, exploration and other. Future work intends to carry out the construction of a real controlled environment, for the evaluation of the performance of, taking into account the latency

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and speed of responses of robots, as well as the limitations and capabilities of bio-inspired robots. Acknowledgment. The elaboration of this work would not have been possible without the collaboration of the INESC-TEC Technology and Science, Porto, Portugal. This work is financed by National Funds through the Portuguese funding agency, FCT - Funda¸ca ˜o para a Ciˆencia e a Tecnologia, within project UIDB/50014/2020.

References 1. Melo, R., de Ara´ ujo, T.P., Saraiva, A.A., Sousa, J.V.M., Ferreira, N.M.F.: Computer vision system with deep learning for robotic arm control. In: 2018 Latin American Robotic Symposium, 2018 Brazilian Symposium on Robotics (SBR) and 2018 Workshop on Robotics in Education (WRE), pp. 357–362. IEEE (2018) 2. Junior, F.D.C.F.M., Saraiva, A.A., Sousa, J.V.M., Ferreira, N.M.F., Valente, A.: Manipulation of bio-inspired robot with gesture recognition through fractional calculus. In: 2018 Latin American Robotic Symposium, 2018 Brazilian Symposium on Robotics (SBR) and 2018 Workshop on Robotics in Education (WRE), pp. 230–235. IEEE (2018) 3. Saraiva, A.A., Santos, D.S., Junior, F.M., Sousa, J.V.M., Ferreira, N.F., Valente, A.: Navigation of quadruped multirobots by gesture recognition using restricted Boltzmann machines. In: Memorias de Congresos UTP, vol. 1, pp. 431–438 (2018) 4. Schwager, M., Dames, P., Rus, D., Kumar, V.: A multi-robot control policy for information gathering in the presence of unknown hazards. In: Robotics Research, pp. 455–472. Springer (2017) 5. Pigou, L., Van Den Oord, A., Dieleman, S., Van Herreweghe, M., Dambre, J.: Beyond temporal pooling: recurrence and temporal convolutions for gesture recognition in video. Int. J. Comput. Vision 126(2–4), 430–439 (2018) 6. Liu, H., Wang, L.: Gesture recognition for human-robot collaboration: a review. Int. J. Ind. Ergon. 68, 355–367 (2018) 7. Kundu, A.S., Mazumder, O., Lenka, P.K., Bhaumik, S.: Hand gesture recognition based omnidirectional wheelchair control using IMU and EMG sensors. J. Intell. Robot. Syst. 91(3–4), 529–541 (2018) 8. Wang, S., Wang, Y., Wei, Q., Tan, M., Yu, J.: A bio-inspired robot with undulatory fins and its control methods. IEEE/ASME Trans. Mechatron. 22(1), 206–216 (2017) 9. Brandstatter, K.: Computer communication system for communication via public networks. US Patent 9,288,188, March 15, 2016 10. Chaudhari, A.M., Apsangi, M.R., Kudale, A.B.: Improved a-star algorithm with least turn for robotic rescue operations. In: International Conference on Computational Intelligence, Communications, and Business Analytics, pp. 614–627. Springer (2017) 11. Ji, X., Liu, L., Zhao, P., Wang, D.: A-star algorithm based on-demand routing protocol for hierarchical LEO/MEO satellite networks. In: 2015 IEEE International Conference on Big Data (Big Data), pp. 1545–1549. IEEE (2015) 12. Duchoˇ n, F., et al.: Path planning with modified a star algorithm for a mobile robot. Procedia Eng. 96, 59–69 (2014) 13. Nanda, A., Rath, A.K.: Fuzzy a-star based cost effective routing (facer) in WSNs. In: Progress in Advanced Computing and Intelligent Engineering, pp. 557–563. Springer (2018)

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14. Nguyen, L.D., Lin, D., Lin, Z., Cao, J.: Deep CNNs for microscopic image classification by exploiting transfer learning and feature concatenation. In: 2018 IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1–5. IEEE (2018) 15. Hurtado, D.M., Uziela, K., Elofsson, A.: Deep transfer learning in the assessment of the quality of protein models. arXiv preprint arXiv:1804.06281 (2018) 16. Pisharady, P.K., Vadakkepat, P., Loh, A.P.: Attention based detection and recognition of hand postures against complex backgrounds. Int. J. Comput. Vision 101(3), 403–419 (2013) 17. Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. In: Advances in Neural Information Processing Systems, pp. 1097–1105 (2012) 18. Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556 (2014) 19. Szegedy, C., et al.: Going deeper with convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–9 (2015) 20. Saraiva, A., Melo, R., Filipe, V., Sousa, J., Ferreira, N.F., Valente, A.: Mobile multirobot manipulation by image recognition (2018) 21. Saraiva, A.A., Costa, N., Sousa, J.V.M., De Araujo, T.P., Ferreira, N.F., Valente, A.: Scalable task cleanup assignment for multi-agents. In: Memorias de Congresos UTP, vol. 1, pp. 439–446 (2018)

Addendum to “Seeking a Unique View to Control of Simple Systems” ˇ ak1 , and A. Serbezov2 Mikulas Huba1(B) , M. Hypiusov´ a1 , P. Tap´ 1

Slovak University of Technology in Bratislava, 812 19 Bratislava, Slovakia [email protected] 2 Rose-Hulman Institute of Technology, Terre Haute, IN 47803, USA [email protected]

Abstract. This article extends the spectrum of basic structures with disturbance reconstruction and compensation considered in the paper from IFAC ACE’19 symposium and discusses problems met in their application, interpretation, teaching and learning. The objective is to foster a comprehensive understanding of basic control problems wrapped around the simplest first order plant models, which may be useful in dealing with their generalization to more complex tasks. Also included are remarks to some of the results and comments in the IFAC pilot survey on the introductory control course content.

Keywords: PI control

1

· Disturbance observer · ESO · IMC · ADRC

Introduction

In the paper [12], an introduction to a learning object focussed on controllers with integral (I) action for compensating input and output disturbances related to the simplest first order plant models has been given. It started with discussing pole assignment 2DOF proportional (P) control of first order plants with possible feedforwards from measurable input and output disturbances. This core structure has then been used in more advanced controllers with I action, as: – Input disturbance di reconstruction by FIR filters yielding the simplest structure of the Model Free Control (MFC) denoted as Intelligent P control [2]; – State-space reconstruction of di by extended state observer ESO; Its application to first order plant yields identical (Luenberger) observer; In combination with integrative first order models it yields structure typically used in active disturbance rejection control (ADRC) [3]. – ESO based approaches may be shown as special case of transfer functions based disturbance observer (DOB) for di reconstruction with inverse plant model considering loop stabilization by the setpoint tracking channel [18,19]. – State-space reconstruction of the output disturbance do shows to lead for integral plant models to unobservable situation which requires special elimination c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 719–728, 2021. https://doi.org/10.1007/978-3-030-58653-9_69

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of the unobservability impact [6]. In a structure considered typically within internal model control (IMC) with a parallel plant model based reconstruction of do for time delayed integrative plants it may lead to modification of the Smith Predictor [27] named as filtered Smith Predictor (FSP) considering stabilizing controller in the disturbance compensation channel [16]. – Finally, different tuning approaches to traditional PI control have been treated to illustrate differences in dealing with stable, integrative, or unstable plants. To explain success of the simplified plant modeling and control considered firstly in [33] and exploited today by ADRC and MFC, all structures have been applied twice by using two types of linear models - the “ultra local” integrative models (a = 0) and the “usual - local” first order models including an internal feedback characterized by a coefficient a = 0 of the transfer function S (s) =

Ks Y (s) = U (s) s+a

(1)

With respect to limited space offered by standard conference publication it was not possible to include also the structure [15,24,28,30–32] with “decoupled” setpoint and disturbance responses based on stabilizing DOB (SDOB) with inverse plant model. Therefore, it was left to next contributions. Of course, as several comments on our article have recalled, there are numerous other interesting approaches to control based on simple models - let’s just mention unstable systems control, for which many traditional approaches are inappropriate, and therefore cascading solutions are used. Or a dynamic setpoint feedforward which requires application of reference model control [29]. But, from the point of view of the introduction to the most common structures and the development of block diagram manipulation skills, we consider the introduction to be sufficient. As already mentioned above, besides of adding SDOB control structure, this paper continues in the unified introduction to the most common controllers with I action used for regulation and tracking of simple first order plants (equivalents of PI control) by extending discussion of the educational and the scientific framework. In this context, it also notes, at least in part, the problems raised within the framework of IFAC pilot survey devoted to the needs of a single subject on automatic control included in the bachelor study (see e.g. [22,23], or http:// iolab.sk/ifac/results.php).

2

Decoupled Setpoint and Disturbance Response

The DOB based control with inverse plant model [18,19] and its modification to integrator plus dead time (IPDT) plant [5] have been reported for situations with the stabilizing controller located in the setpoint tracking channel. The subsequent decoupled setpoint and disturbance responses for IPDT systems have been proposed in [15,30–32] without explaining shift of the stabilizing controller to the disturbance compensation channel and with a baffling comment [32] “The

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original structure is not causal and sometimes is not internally stable”. Since it required implementation by a modified equivalent scheme, we decided to analyze firstly the situation without a dead time. 2.1

Model Uncertainties and Constrained Setpoint Feedforward

For a piece-wise constant reference setpoint w filtered with a first order low-pass filter Qw (s), simple feedforward control may be accomplished according to Ff f (s) =

s+a Uf f (s) 1 = Qw (s); Qw (s) = W (s) 1 + Tc s Ks

(2)

Obviously, for a model uncertainty expressed as a = a + Δa, K s = Ks Y (s) =

s + a + Δa Δa Qw (s)W (s) = (1 + )Qw (s)W (s) s+a s+a

(3)

The model uncertainty is equivalent to an equivalent external input disturbance die =

ΔaQw (s)W (s) Ks

(4)

which, in the case of unstable systems, leads to an unrestricted output increase and prevents the concept from being usable.

Fig. 1. Constrained setpoing and di feedforward

Furthermore, already for a = a = 0, constraints put on the applied control uf f lead to a permanent control error also in nominal systems without disturbances. For stable plants, effect of control constraints may be eliminated by implementing the feedforward (instead of using just the transfer function Ff f (s)), by a primary loop (Fig. 1) with KP = (1/Tc − a)/K s

(5)

Of course, in situations with acting disturbances and a ≤ 0, such a control is not able to guarantee stable responses with zero permanent error. A feedback stabilization needs to be used. In [7–10] it has been located into the setpoint tracking channel. Later appearing papers on FSP, or SDOB introduced the disturbance response stabilization via the disturbance rejection channels.

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Stabilizing Disturbance Feedforward by SDOB

In the setpoint tracking channel, unstable plants may be stabilized by simple P control. However, stabilization via the DOB channel requires a proportionalderivative (PD) controller. Qd (s) (Fig. 2) has to be composed (as in [15]) as Qd (s) =

1 + βs (1 + Tf s)2

(6)

Fig. 2. Decoupled setpoing and disturbance feedforward

Its time constant β has to guarantee stable disturbance response Fiy (s) =

s(2Tf − β + Tf2 s)Ks Y (s) = Di (s) s(2Tf − β + Tf2 s)(s + a) + (s + a)(1 + βs)Ks /K s

(7)

For a = a = 0, Ks = K s and Tc = Tf , under requirement of zero steady-state error Fiy (0) = 0, the plant pole s = 0 may be cancelled by a numerator zero for β = 2Tf Fiy (s) =

sKs Tf2 Ks Y (s) = ; IEw = Tc ; IAEi = Ks Tc2 ; Fuy = 2 (1 + Tf s) U (s) s

(8) (9)

It means that with respect to u the plant block with SDOB behaves as a single integrator. On the other hand, with respect to disturbances, it keeps zero total disturbance di − dif whereby it guarantees stable di responses. Simple feedforward control might thus be expected to guarantee precise setpoint tracking also in presence of disturbances and uncertainties. For a = 0, a = 0, Tf = Tc with (8) sKs Tf2 (1 + Tf s)2 Ks ; F (s) = uy (1 + Tf s)2 + saTf2 s (1 + Tf s)2 + saTf2 2 = Tc (1 + aTc ); IAEi = Ks Tc ; Tc = Tf

Fiy (s) = IEw0

(10)

These results are the same as for DOB with stabilizing P controller in the setpoint tracking channel [12] with the first order filter. The disturbance response remains

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stable for 2Tf + aTf2 > 0, i.e. 2 + aTf > 0. For a < 0 Tf may not be chosen arbitrarily large. For a = a = 0, from zero steady-state error Fiy (0) = 0 and from cancellation of the unstable pole s = −a expressed by Fiy (−a) = 0 follows sKs Tf2 Ks ; Fuy (s) = 2 (1 + Tf s) s+a IEw = Tc ; IAEi = Ks Tc2 ; Tc = Tf β = Tf (2 − aTf ); Fiy (s) =

(11)

From comparison of these results with [12] follows that no new performance appeared. When simulating transients for various model and plant parameters, students may encounter situations where the proposed structures work perfectly. However, when attempting longer transients with unstable systems, they discover collapsing responses with signals growing above all limits. The moment of collapse can be influenced by choice of the simulation parameters and the numerical integration method, but only partially. After trying all possible solutions to eliminate the problem, students are more receptive to following its explanation - stabilizing the disturbance responses does not mean stabilizing the state of the system. The aim of SDOB, by introducing feedback to follow dynamics given by the parameter a < 0 even in the presence of uncertainties and disturbances [25], may not guarantee permanently stable loop behavior for unstable plants. Thus, achieving an ideally matching model and system dynamics leads here to a conflicting requirement in terms of overall stability. The task formulated in this way is meaningless and it is always necessary either to worsen the dynamics of transients by choosing a stable model with a > 0, or to add a regulator ensuring stabilization of the plant state. Due to the decoupled setpoint tracking and disturbance compensation, this moment may here be easier understood than for the stabilizing disturbance feedforward of the IMC like structures (as e.g. in [12]). The final lesson of the experiment for students is to critically examine all claims, even if they are published in the top journals and books. The different effects of the experts’ opinions on the solution of unstable process control can also be nicely illustrated by the results of the corona virus pandemic control in different countries, which is an example of unstable time-delayed dynamics. Since such untreated problems, together with limitations of the control action, significantly influence usability of the SDOB concept, they could explain reasons for the above-mentioned unintelligible remark “The original structure is not causal and sometimes is not internally stable”.

3

Educational Framework of the Complete Batch

The impulses for writing this article come from several sources. They include curiosity that forces us to ask what, how, whom and why we teach, whether students are interested and what it gives them, etc. Since such questions have been addressed since the 1980s, the answers to them include much of the development of the theory of automatic control. Recently, numerous similar questions appeared also in preparing and conducting the IFAC pilot survey devoted to the

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needs of a single subject on automatic control included in the bachelor study. Although education in areas dominating our courses is certainly not enough with a single semester, the essence of the questions to be answered remains the same. And we do not avoid situations when, forced by the environment changes, longterm stabilized blocks of education and knowledge need to be broken first and then picked up from them isolated stones that fit into a newly created mosaic. 3.1

Plant Modeling, Block Diagrams and Content Flipping

Most traditional control curricula start from rigorous mathematical models, then go through simplifications (e.g., linearization), then Laplace and transfer functions, to arrive at the simple models. It typically takes a few weeks and all that time students are asking “Where is the control?” Students lose motivation and fail to see the connection to real world practice. In their minds control is just another applied math class. Project-based learning [14,20] allows us to flip this progression backwards. Start with the simplified model (do not derive, just make them axioms) and build control concepts from there. Let students do trial and error tuning and controller design. Let them experience firsthand what setpoint tracking and disturbance rejection is. Let them get frustrated and fail a few times... Then, show how mathematical analysis can streamline the process. Thus, as the first priority of our course, the ability to work with block diagrams in Matlab/Simulink has been defined. This is crucial for both control analysis and synthesis and also for system modeling. While in the distant past students tested their skills while working with some “artificial” schemes, by considering all the basic approaches to designing integrating controllers, we gained a wide range of schemes interesting both from the point of their functionality and for developing the simulation skills. Work with them includes experimenting, watching lectures, attentive listening and team work including writing and presenting reports. Considered problems also enable to illustrate historical development of control technology, including simple application examples and relevant terminology (aspects neglected in the running survey). Lecture, or classroom flipping represent an instructional strategy used also in the control area [11,21]. In principle, it shifts instructions from a teachercentered to a learner-centered model. It may also be extended to content flipping, when some its items are treated in an inverse order. Due to the recently decreased ability of students to study mathematically oriented courses (including low motivation for such a study), we may use control experiments to encourage their motivation and to show the practical needs of math study. From the Kolb’s learning cycle [13] follows that it may favor different type of student than the traditional study. As e.g. reported in [11], such control course may start with much lower student knowledge from the math area covering just work with exponential functions. By assuming solutions of differential equations in form y(t) = cest it is possible to interpret “s” as an operator of differentiation, or “z −1 ” as a shift operator, which opens the way to using the Simulink blocks, to introduce the closed loop pole as quotient of signal increase/decrease, block algebra, etc. A detailed interpretation of some steps relevant for the Laplace, or Z transforms may be

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shifted to later stages, when students have already advanced in a parallel running math course and due to increased motivation they better understand the overall context. One comment received to the paper was: “Is not the 2-parameter model too big simplification of reality?” Another note to this point claims that “even the system has 1 dominant time constant and exhibits open loop time domain behavior close to first order model, the frequency response could be significantly different and closing loop may cause instability when higher order are neglected. Then, from controller perspective the system is of higher order, although has a dominant time constant. Also small dead-time caused by sampling and signal processing could cause problems when moving from virtual space to real plant.” These comments are appreciated, because they bring up highly important and for a long time neglected questions. As shown by our analysis, all disturbance responses do not depend on the plant parameter a. This allows for the possibility of a simplified plant identification, which has already been used by Ziegler and Nichols way back in 1942. Also MFC and ADRC use this simplification which is significant especially in dealing with nonlinear systems, where the linear plant approximations change with the operating point and require application of gainscheduling. By experimenting on real plant with different values of Tc students see when the used plant model becomes inadequate (evidenced by increasingly oscillatory transients with growing shape-related performance measures). With respect to the level of their preparation, detailed traditional analysis and design in the frequency domain is out of the possibilities. However, several structures from the batch have been included especially as a preparation to design of deadtime compensators (DTCs) for the time delayed systems. Above approach shows to be consistent with the results from the survey question [22,23]: “A first course should focus more on concepts, philosophy and motivation-reasons to use control, illustrating principles such as uncertainty handling with case studies but not get drawn into mathematics too quickly”. But, without showing, how, such statements turn to be more or less just political slogans. A bit more specific is the sentence: “A first course should focus on classical tools such as Laplace, closed-loop transferences and lead/lag/PID design.” Basically, it is to agree also with other similar statement “PID analysis and tuning is essential for all students.” However, without a rigorous definition it is not clear, what should be understood under the notion PID (see e.g. different types of PIDs considered in standard textbooks [1,29]). And numerous authors extend this notion, under denotations as IMC-PID, DOB-PID, or iPID, to a much broader context including several structures of our test set. As noted by [17], design of PID controllers is most frequently accomplished using FOTD and IPDT models. Most of the existing DTCs follow from generalization of the approaches discussed in our case study, based on the same plant models as PID control. Furthermore, also PID has been shown as a special case of a model based approach with deadtime approximated by Pad´e or Taylor series expansions [26]. From this point of view, better performance could be expected from more precise methods not requiring such approximations. Of course, historically, the use of transport delays was a problem for analogue-based controllers, but this is no longer the case for

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modern digital controllers. Thus, when seeing doubts regarding use of DTCs [4], something must be wrong in the approaches used. And when the preliminary survey shows relatively low interest regarding development of the last 60 years, this suspicion is yet increasing. 3.2

Non Multa Sed Multum Versus Non Multum Sed Multa

Some comments expressed doubts, whether the chosen design methods fit well to the introductory control course and if they can be considered as the most common approaches to control system design. Traditional control curricula usually begin with classical methods such as root-locus or loop-shaping which require iterative “hand-tuning” but allow the students to develop deeper understanding of how the feedback loop affects the overall system behavior... Exactly these aims may be achieved by individual experimenting on laboratory plant models, produced in large numbers, given to every student and adjusted the course topics, enabling students to play, iterate and to develop deeper understanding. The presented structures may seem too complex for bachelor students and cause confusion and misunderstanding. But, at the same time, other comments appeared that “the complexity of choices the designer must make grows considerably for higher order models and the results achieved for the first order plant cannot be extrapolated simply. Also the interpretation of robustness becomes much more involved than presented.” When discussing inflation of different approaches, it is just to remind that it started long time ago - see e.g. the arguments given by introducing the Modern Control Theory, IMC, ADRC, MFC, etc. Which of these approaches should be omitted? Should we deal just with the “traditional” PID control (do we know its definition?) and let all the alternatives untouched? The result would be that many students, upon discovering some of these “revolutionary” alternatives, would start complaining about the obsolescence of our education and full of enthusiasm continue disseminating unfounded optimistic information on their use. However, as shown by the analysis, nominally all discussed structures may be included into few equivalent classes. No matter how “attractive” and “trendy” titles they use.

4

Conclusions and Future Work

As concluded already in the source paper [12], mastering of the presented structures and tuning procedures opens the door to deeper and rigorous study of more complex tasks as, for example: control of first order time delayed systems, or control of (time delayed) systems with higher (2nd) order dominant dynamics with possibly constrained control and the discrete time implementation. The analysis also pointed to the need for a fundamental revision of several conclusions regarding the control of unstable systems. In view of the ongoing survey, it can be stated that one cannot expect from it straightforward and unambiguous instructions for optimal design of his course. Therefore, it will be appropriate to discuss a number of possible approaches, including their detailed syllabi.

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Acknowledgement. Supported by the grants APVV SK-IL-RD-18-0008 and VEGA 1/0745/19.

References 1. ˚ Astr¨ om, K.J., H¨ agglund, T.: Advanced PID Control. ISA, Research Triangle Park, NC (2006) 2. Fliess, M., Join, C.: Model-free control. Int. J. Control 86(12), 2228–2252 (2013) 3. Gao, Z.: On the centrality of disturbance rejection in automatic control. ISA Trans. 53, 850–857 (2014) 4. Grimholt, C., Skogestad, S.: Should we forget the smith predictor? IFACPapersOnLine. In: 3rd IFAC Conference on Advances in Proportional-IntegralDerivative Control PID 2018, vol. 51(4), pp. 769–774 (2018) 5. Hong, K., Nam, K.: A load torque compensation scheme under the speed measurement delay. IEEE Trans. Ind. Electron. 45, 283–290 (1998) 6. Huba, M., B´elai, I.: Limits of a simplified controller design based on IPDT models. Proc. Inst. Mech. Eng. Part I: J Syst. Control Eng. 232(6), 728–741 (2018) 7. Huba, M., Kuˇlha, P.: Saturating control for the dominant first order plants. In: Proceedings of the IFAC Workshop Motion Control, pp. 197–204 (1995) ˇ akova, K.: Observer-based control of unstable process with 8. Huba, M., Kuˇlha, P., Z´ dead time. In: Preprints 10th Conference Process Control, vol. 1, pp. 35–38 (1995) ˇ akov´ 9. Huba, M., Bist´ ak, P., Skachov´ a, Z., Z´ a, K.: P- and PD-controllers for I1 and I2 models with dead time. In: 6th IEEE Mediterranean Conference on Control and Automation, vol. 11, pp. 514–519 (1998) ˇ akov´ 10. Huba, M., Bist´ ak, P., Skachov´ a, Z., Z´ a, K.: Predictive anti windup PI and PIDcontrollers based on I1 and I2 models with dead time. In: 6th IEEE Mediterranean Conference, vol. 11, pp. 532–535 (1998) ˇ akov´ 11. Huba, M., Z´ a, K., Bist´ ak, P.: Flipping the courses on automatic control: why and how. In: CONTROLO 2018, Ponta Delgada, Portugal, pp. 236–241 (2018) ˇ akov´ 12. Huba, M., Z´ a, K., Bist´ ak, P., Hypiusov´ a, M., Tap´ ak, P.: Seeking a unique view to control of simple models. IFAC-PapersOnLine. In: 12th IFAC Symposium on Advances in Control Education ACE 2019, vol. 52(9), pp. 91–96 (2019) 13. Kolb, D.A.: The Learning Style Inventory: Technical Manual. McBer, Boston (1976) 14. Mills, J., Treagust, D.: Engineering education, is problem-based or project-based learning the answer. Aust. J. Eng. Educ. 3(2), 2–16 (2003) 15. Normey-Rico, J., Camacho, E.: Dead-time compensators: a survey. Control Eng. Pract. 16(4), 407–428 (2008) 16. Normey-Rico, J., Camacho, E.: Unified approach for robust dead-time compensator design. J. Process Control 19(1), 38–47 (2009) 17. O’Dwyer, A.: Handbook of PI and PID Controller Tuning Rules, 3rd edn. Imperial College Press, London (2009) 18. Ohishi, K.: A new servo method in mechatronics. Trans. Jpn. Soc. Elect. Eng. 107–D, 83–86 (1987) 19. Ohishi, K., Nakao, M., Ohnishi, K., Miyachi, K.: Microprocessor-controlled DC motor for load-insensitive position servo system. IEEE Trans. Ind. Electron. IE– 34(1), 44–49 (1987) 20. O’Mahony, T.: Project-based learning in control engineering. In: IET Irish Signals and Systems Conference (ISSC 2008), pp. 72–77 (2008)

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Open Hardware and Software Robotics Competition for Additional Engagement in ECE Students - The Robot@Factory Lite Case Study V´ıtor H. Pinto1,3(B) , Armando Sousa1,3 , Jos´e Lima2 , Jos´e Gonc¸alves2 , and Paulo Costa1,3 1

3

Faculty of Engineering, FEUP - University of Porto, Porto, Portugal {vitorpinto,asousa,paco}@fe.up.pt 2 IPB - Polytechnic Institute of Braganc ¸ a , Braganc¸a, Portugal {jllima,goncalves}@ipb.pt CRIIS - Centre for Robotics in Industry and Intelligent Systems, INESC TEC - Institute for Systems and Computer Engineering, Technology and Science, Porto, Portugal Abstract. Throughout this paper, a competition created to enable an interconnection between the academic and industrial paradigms is presented, using Open Hardware and Software. This competition is called Robot at Factory Lite and serves as a case study as an additional enrollment for students to apply knowledge in the fields of programming, perception, motion planning, task planning, autonomous robotic, among others. Keywords: Education · Robotic competitions · Mobile robotics · Portuguese Robotics Open · Robot@Factory

1 Introduction Robotics has more and more impact on daily tasks and that their development allows an increase in the performance of tasks that, until now, were costly and practically impossible for humans. Furthermore, industry is more and more engaged on improving flexibility and adaptability, being an example the Industry 4.0 movement, enabling a global management of material and human resources, optimizing these two aspects. Professors in several Engineering fields, which typically are researchers in that area, begin to realize the advantages that robotics can bring to their specific field of interest. Specifically, teachers from Electrical and Computers Engineering (ECE) should promote students’ interest in areas they like and where they are most likely to find a job. Project Based Learning (PBL) is proven to be a very effective method in several areas, like Science, Technology, Engineering and Mathematics (STEM) [1], Medicine [2], Biology [3], Economics [4], being these just examples of the multiplicity of areas in which PBL is effective. Typically, students in the ECE field of study are attracted for practical activities but also for activities framed in areas that can provide useful learning for career opportunities and curriculum. Throughout this paper, a competition created to enable an interconnection between the academic and industrial paradigms is presented, using Open Hardware c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gonc¸alves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 729–739, 2021. https://doi.org/10.1007/978-3-030-58653-9_70

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and Software. This competition is called Robot at Factory Lite and serves as a case study as an additional enrollment for students to apply knowledge in the fields of programming, perception, motion planning, task planning, autonomous robotic, among others. This paper is organized with the Introduction in Sect. 1, followed by the Related Work in Sect. 2. In Sections 3 and 4 the open-source Hardware and Software modules of the robotic platform are presented, respectively. Section 5 presents the obtained results and in Sect. 6 the conclusions withdrawn from this work, as well as possible future work.

2 Related Work Robotic competitions are a proven way to find “out of the box” solutions for difficult problems, that otherwise would be extremely hard. These events bring together a teams of people focused on the same objective, interested in a common set of thematic and working to build a solution to solve a problem. This competitive mindset thrills the participants and encourage them to overcome complex problems in different areas of competition [5]. Thanks to the participants’ spirit, this kind of competitions lead to innovative solutions for societal problems and great technological advances [6]. There are numerous robotic competitions in all robotics fields, such as SAUC [7] for Marine Robotic Vehicles, the DARPA Robotics Challenge for Ground Robots [8] and even Multi-Type Competitions, as euRathlon, that combines all the mentioned types of vehicles to achieve a certain goal [9]. Using robotic competitions to improve the field’s technology is a very powerful tool that allows a more motivated learning, with a marked interpersonal component that allows at the same time a sharing of knowledge that may even surpass that of a conference. Portuguese Robotics National Competition have a competition called Robot@Factory [10] providing challenges that can be found in real industrial factories, using Autonomous Guided Vehicles (AGVs) to transport boxes from an entrance warehouse to the exit one, passing through some production machinery. Robot@Factory Lite (R@F Lite) competition was inspired in this major league competition, adding new components, such as RFID identification of the boxes, the introduction of an electromagnet to transport the boxes, among other changes [11], and served as a case study. As the authors of [12] clearly support, educational robots can be used as programming project, learning focus and collaborator. Robotic Competitions for teaching graduate students are a way to increase knowledge in the areas of programming languages, mechanical construction, sensors and their respective data acquisition, actuators and the respective drivers, etc. To do this, it is necessary to create a way for students to use this knowledge, creating platforms that allow this development. There are several open hardware and software platforms used to teach students in broad areas, such as [13], [14] or [15]. These are good examples of how open-source platforms are a great asset for STEM education.

3 Open Hardware Description In order to motivate students to the proposed challenge, a hardware and software prototypes of a robot are provided at: https://github.com/P33a/SimTwo/tree/ master/RobotFactoryLite. The hardware module is composed by several components,

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as described. Table 1, with a Bill of Materials, is presented in the end of this section, containing links to a sample provider’s page, as well as indicative prices. – Arduino based microcontroller: Arduino UNO or arduino nano with a easy connections shield is suggested to facilitate connections between peripherals and the microcontroller and also to use the arduino IDE, a well-known programming environment. – RFID: A SPI protocol radio-frequency identification reader to identify the part type. – Part switch detector: a switch to be assembled on the front of the robot to detect the presence of the part. – Batteries: two 18650 lithium batteries to supply the robot – Step Down converter: A switching converter to supply the 5 V components in an efficient way. – Motor driver: Receives the signals (PWM and direction) from the microcontroller and actuates the motors. – Electro-Magnet: A magnet controller by a bit is used to hold the part while moving on the floor. – Motors: Two Geared motors (left and right) are used to move the robot in a differential architecture. – Floor line detector: A PCB composed by 5 infra-red emitters and receivers is used to detect the white line on the floor. It is used to follow the line during the competition. – Reverse voltage protection circuit: based on a N channel mosfet with a low resistance Ron DS protects the electronic components against wrong batteries connections (see Fig. 1 for the circuitry). – Power button: an electronic switch (Fig. 2) is used to control the supply of the robot. The main function is to power off the robot when the batteries voltage is below a threshold to protect them against low state of charge. – stl files: All 3D parts that compose the robot are available in stl format and ready to be printed in a 3D printer.

Fig. 1. Reverse voltage protection circuit.

Fig. 2. Power switch.

It is also provided a schematic that helps students and supervisors to build, in a easy way, the mobile robot. The schematic is presented in Fig. 3. The complete assembly of the robot can be seen in Fig. 4. Due to the restricted number of I/O pins the end switch sensor shares the same pin as the MOSI line the is connected to the RFID reader. The end switch can pull the

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Fig. 3. Robot schematic.

Fig. 4. Assembled robot

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MOSI pin to the ground through a 1 kΩ resistor. That way, when the MOSI pin is being used to communicate with the RFID reader it acts as an output and the state of the end switch can’t affect the signal level. When the communication is paused, the MOSI pin can be toggled to act as an input with the internal pull-up on. That way, if the end switch connects the 1 kΩ resistor to the ground the input level will be near 0 V and the micro-controller will read the input as a zero. byte readTouchSwicth(void) { byte ret; pinMode(TOUCHSW, INPUT_PULLUP); // Change to input with pullup ret = !digitalRead(TOUCHSW); // Read zero means pressed pinMode(TOUCHSW, OUTPUT); // Change back to output return ret; }

Table 1. Bill of materials Robot@Factory Lite Component

Link

Unit price Quantity Total

Motor + wheel (2x) https://www.botnroll.com/en/dc-motor/2975-hobbygearmotor-200rpm-65mm-wheel.html

e2.90

2

e5.80

RFID reader

https://www.botnroll.com/en/rf-lora/2580-rfid-modulerc522-kit-13-56mhz-6cm-with-tags-s50.html

e4.90

1

e4.90

Line sensor

https://www.botnroll.com/en/infrared/2586-trackersensor-infrared-line-tracking.html

e6.50

1

e6.50

Magnet

https://www.botnroll.com/en/solenoides/976-groveelectromagnet.html

e12.60

1

e12.60

Castor

https://www.botnroll.com/en/wheel/604-rodiziominiatura.html

e2.90

1

e2.90

Battery support (2x) https://www.botnroll.com/en/accessories/1299-supportfor-1-battery-mr18650-w-wires.html

e1.00

2

e2.00

Motors driver

https://www.botnroll.com/en/controllers/1957-adafruittb6612-12a-dcstepper-motor-driver-breakout-board-. html

e5.90

1

e5.90

Arduino nano

https://www.botnroll.com/en/arduino-boards/934arduino-nano-30-compativel.html

e12.90

1

e12.90

Arduino nano shield https://www.botnroll.com/en/shield-prototype/2373nano-io-shield-for-arduino-nano.html

e9.60

1

e9.60

Lithium battery (2x) https://www.botnroll.com/en/batteries/2300-re-batteryli-ion-mr18650-37v-2550mah.html

e4.90

2

e9.80

Step down converter https://www.botnroll.com/en/dcdc-converters/937lm2596-step-down.html

e3.90

1

e3.90

e0.90

1

e0.90

Total

e77.70

Micro switch

https://www.botnroll.com/en/switchs-buttons/2026microswitch-with-roller-.html

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4 Software Architecture There are two ways of developing the software solution. The first one is to program the robot directly on the Arduino IDE and test it on the maze. On the other way, it can be used the Hardware-in-the-loop approach, where a simulator (SimTwo provided environment - see Fig. 5) is applied and the Arduino communicates with the simulator thought USB port. This way, solution can be easily validated and adjusted with a simulator with the real limitations of the microcontroller. The same code can be used both for real and simulated robot.

Fig. 5. SimTwo simulator

The robot can be controlled by a high level state machine that decides the current action. Possible current actions, already implemented, are provided by the functions: void void void void

moveRobot ( f l o a t Vnom , f l o a t Wnom ) ; f o l l o w L i n e ( f l o a t Vnom , f l o a t K ) ; f o l l o w L i n e L e f t ( f l o a t Vnom , f l o a t K ) ; f o l l o w L i n e R i g h t ( f l o a t Vnom , f l o a t K ) ;

The action moveRobot sets the linear speed v and the angular speed w. It can be used to blindly, go straight ahead, turn, perform a curve with a certain radius or simply stop (when v = 0 and w = 0). The units for v and w are not calibrated as the absence o motor encoders makes the speed dependent on the battery voltage level and the friction from the transported payload. In the future, a version with motors fitted with encoders can overcome this problem at a slightly higher cost. The other actions make the robot follow a black line using an reference, respectively the center of the line, the left edge or the right edge. The parameter Vnom sets the desired linear speed and K is the feedback gain that controls how much correction effort is applied on the angular speed. A simple state machine that makes the robot travel the first part to be moved a bring that part to its destination was made available as an example. It could be used as a starting point and as a way to verify if the hardware was working properly. The state machine is presented in next diagram.

Open Hardware and Software Robotics Competition

go

Touch

Stopped start

moveRobot v = 0, w = 0

735

Travel followLineLeft

Approach followLineLeft solenoid on after 0.4 s

Travel followLineRight reset Crosses

Turn 180◦

Back

moveRobot moveRobot v = 0, w = 50 v = -60, w = 0 Line after 1.2 s after 1.2 s

after 5 Crosses Line after 1.2 s Maneuver moveRobot v = 40, w = -40

after 2 s

BackOut

Go In

followLineRight

moveRobot v = -40, w = 0 solenoid off

5 Findings As mentioned before, The R@FLite competition targets mainly late secondary education and early higher education (HE) students and is meant to be an entry level competition for busy students in a challenging academic environment. This article will focus on HE students in the 2019 Portuguese Robotics Open (PRO), the first year that this competition occurred. The current section will address findings. The data gathered comes from 3 data sources: (i) official enrollment data from the R@FLite competition; (ii) a survey to participants and (iii) interviews with mentors of the teams and organizers. From official enrollment data it is known that a total of 20 students entered the competition on 5 teams, all teams with 4 students. One team came from the Instituto Politecnino de Braganc¸a (IPB) and 4 from the Faculdade Engenharia da Universidade do Porto (FEUP) both from Portugal. 5.1 Survey to Participants To assess the general and educational interest of this competition the participants were asked to answer a quiz. The quiz includes answerer characterization, time invested in the competition and learning/interest in the competition. The mentioned quiz also has open responses regarding learning, strong points and points to improve. The students targeted in this quiz are 18 year old or older and were contacted by email roughly one

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month after the competition. The email directed to an anonymous quiz implemented with Google Forms. The 20 emails were sent by each team mentor twice, one week apart. 5.2

Responses and Analysis

The quiz totaled 15 answers (4 from IPB and 11 from FEUP). From IPB, 3 students were enrolled in the 5th final year and one in the 1st initial year of the Electrical and Computer Engineering (ECE). Regarding quiz responders from the FEUP’s ECE course, 1 student came from the first year, 4 from the 3rd year and 6 from the 4th year. Globally, less students (6) had this participation articulated with an academic degree that the opposite (6).

Fig. 6. Histogram of quiz responses

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The answers are summarized in Fig. 6 show that most common working time is form one week to 4 weeks full time equivalent (FTE, weeks of 35 h of work). This seems articulated with most common answer of total work time for the team ranging from 4 person week to 16 person week. Not surprisingly, a manual detailed analysis reveals that individual students that have academic coordination have spent, in average, more time with the competition. The next section of the quiz asked for Lickert scale agreement to some questions using numeric encoding as 1 = No or Disagree, 3 = Average or Neutral, 5 = Yes or Agree. Further inquiries from the same quiz reveal that the vast majority had not participated in previous robotics competitions (average 2.3) and that the participation was interesting (average 4.6) and engaging (4.5). Interestingly, participants seem eager to repeat a future participation, likely due to the competitive mindset. The same quiz (Fig. 6) reveals some dispersion in learning issues, 4 responders declaring that learning with the participation was “3 Neutral” and average of answers is 4.0. A most interesting question is if the robotics issues that students “learned” had been addressed in the program. Answers reveal some dispersion, likely related to student’s academic year - students state “Learned things not in the Program” average score of 3.8 and standard deviation of 0.9 (which is larger than for other questions). In the open text, 2 students identified State Machines as a main topic of interest and another student pointed out the importance of building a robot mostly from elementary components. These two ideas were the only two ideas all mentors mentioned. This hints the importance of open software (inducing healthy programming habits) and open hardware further reinforces deep learning as the robot is all done by the team and the team has mechanic, electronic and programming layouts readily accessible. Mentors also state the importance of open hardware and open software to keep complexity at an interesting level and promote problem solving capabilities.

6 Conclusions and Future Work The shown results hint that the Robot at Factory Lite competition is interesting to students that state high levels of interest and engagement even if not a huge amount of time is dedicated. This is interesting to allow for easy coordination with other academic duties such as classes and intermediate tests that keep their normal pace during competition time. The students affirm some learning and mentors and organizers find this knowledge to be of practical nature, crossing all of the hardware and software levels. Such knowledge type is not easy to address in ECE regular courses, especially in massified faculties. Admittedly, the same kind of knowledge might be attained by Project Based Learning or in capstone courses dedicated to those outcomes. Most students state that they worked less for this competition than they would have for a 6 ECTS course (162 h). Many of these students did this work extra class for no academic merit (and this further hints attractiveness). Open hardware and software are also a part of this attractiveness as otherwise the challenge would not be solvable in such a low amount of time. It makes sense that such competitions be articulated into academic curricula in HE in order to achieve practical skills and “learn by doing” and “problem solving” knowledge. Participants recognize that some learning is not addressed in normal curricula.

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Mentors state the importance of open hardware and software to complete full knowledge of a fully functional system robotic made by the students themselves. Many of the involved also state transversal engineering skills, soft skills and global engineering skills to be improved by participation in the mentioned competition. Further inquiries are necessary to establish statistical relevance but answers hint that the Robot at Factory Lite competition motivates and complements a common ECE curricula. Admittedly, such benefits are likely to be common to other educational practices. Longer and more detailed surveys are necessary to have solid hints as to what is being learned in this competition that is not addressed in the curricula or if it is only a matter of something that has not yet been addressed at the year the student is presently at. Acknowledgements. This work is financed by National Funds through the Portuguese funding agency, FCT - Fundac¸a˜ o para a Ciˆencia e a Tecnologia, within project UIDB/50014/2020.

References 1. Han, S., Rosli, R., Capraro, M., Capraro, R.: The effect of science, technology, engineering and mathematics (STEM) project based learning (PBL) on students’ achievement in four mathematics topics. J. Turk. Sci. Educ. 13(Specialissue), 3–30 (2016). Accessed by 8. https:// www.scopus.com/inward/record.uri?eid=2-s2.0-84999098096&doi=10.12973 2. Alrahlah, A.: How effective the problem-based learning (PBL) in dental education. A critical review. Saudi Dent. J. 28(4), 155–161 (2016). http://www.sciencedirect.com/science/article/ pii/S1013905216300396 3. Thakur, P., Dutt, S.: Problem based learning in biology: its effect on achievement motivation of students of 9th standard. Int. J. Multi. Educ. Res. 2(2), 99–104 (2017) 4. Madsen, M.O., Olesen, F.: Teaching economics at Aalborg University using the PBL approach. Department of Business and Management Working Paper Series, No. 5, 2016. Department of Business and Management (2016). http://www2.business.aau.dk/digitalAssets/226/ 226176 wp-pbl-16.pdf 5. Costa, V., Rossetti, R., Sousa, A.: Simulator for teaching robotics, ROS and autonomous driving in a competitive mindset. Int. J. Technol. Hum. Interact. 13(4), 19–32 (2017). https:// doi.org/10.4018/IJTHI.2017100102 6. Dias, J., Althoefer, K., Lima, P.U.: Robot competitions: what did we learn? [competitions]. IEEE Rob. Autom. Mag. 23(1), 16–18 (2016) 7. Ferri, G., Ferreira, F., Djapic, V.: Fostering marine robotics through competitions: from SAUC-E to ERL emergency 2018. In: OCEANS 2018 MTS/IEEE Charleston, pp. 1–7, October 2018 8. Krotkov, E., Hackett, D., Jackel, L., Perschbacher, M., Pippine, J., Strauss, J., Pratt, G., Orlowski, C.: The DARPA robotics challenge finals: results and perspectives, pp. 1–26. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-74666-1 1 9. Sousa, P., Ferreira, A., Moreira, M., Santos, T., Martins, A., Dias, A., Almeida, J., Silva, E.: ISEP/INESC TEC aerial robotics team for search and rescue operations at the euRathlon 2015. J. Intell. Rob. Syst. 93(3–4), 447–460 (2019) 10. Costa, P.J., Moreira, N., Campos, D., Gonc¸alves, J., Lima, J., Costa, P.L.: Localization and navigation of an omnidirectional mobile robot: the robot@factory case study. IEEE Revista Iberoamericana de Tecnologias del Aprendizaje 11(1), 1–9 (2016) 11. Lima, J., Costa, P., Brito, T., Piardi, L.: Hardware-in-the-loop simulation approach for the robot at factory lite competition proposal. In: 2019 IEEE International Conference on Autonomous Robot Systems and Competitions (ICARSC), pp. 1–6, April 2019

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12. Miller, D.P., Nourbakhsh, I.R., Siegwart, R.: Robots for Education, pp. 1283–1301. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-30301-5 56 13. Mej´ıas, A., Herrera, R., M´arquez, M., Calder´on, A., Gonz´alez, I., And´ujar, J.: Easy handling of sensors and actuators over TCP/IP networks by open source hardware/software. Sensors 17(1), 94 (2017) 14. Arvin, F., Espinosa, J., Bird, B., West, A., Watson, S., Lennox, B.: Mona: an affordable opensource mobile robot for education and research. J. Intell. Rob. Syst. 94(3), 761–775 (2019). https://doi.org/10.1007/s10846-018-0866-9 15. Pinto, V.H., Monteiro, J.M., Gonc¸alves, J., Costa, P.: Prototyping and programming a multipurpose educational mobile robot - NaSSIE. In: Lepuschitz, W., Merdan, M., Koppensteiner, G., Balogh, R., Obdrˇza´ lek, D. (eds.) Robotics in Education, pp. 199–206. Springer, Cham (2019)

Engine Labels Detection for Vehicle Quality Verification in the Assembly Line: A Machine Vision Approach Sílvio Capela1 , Rita Silva1 , Salik Ram Khanal1(B) , Ana Teresa Campaniço1 , João Barroso1,2 , and Vítor Filipe1,2 1 School of Science and Technology, Universidade de Trás-os-Montes e Alto Douro,

5000-801 Vila Real, Portugal [email protected] 2 INESC TEC–Institute for Systems and Computer Engineering, Technology and Science, 4200-465 Porto, Portugal

Abstract. The automotive industry has an extremely high-quality product standard, not just for the security risks each faulty component can present, but the very brand image it must uphold at all times to stay competitive. In this paper, a prototype model is proposed for smart quality inspection using machine vision. The engine labels are detected using Faster-RCNN and YOLOv3 object detection algorithms. All the experiments were carried out using a custom dataset collected at an automotive assembly plant. Eight engine labels of two brands (Citroën and Peugeot) and more than ten models were detected. The results were evaluated using the metrics Intersection of Union (IoU), mean of Average Precision (mAP), Confusion Matrix, Precision and Recall. The results were validated in three folds. The models were trained using a custom dataset containing images and annotation files collected and prepared manually. Data Augmentation techniques were applied to increase the image diversity. The result without data augmentation was 92.5%, and with it the value was up-to 100%. Faster-RCNN has more accurate results compared to YOLOv3. Keywords: Assembly verification · Machine vision · Automotive industry · Industry 4.0 · Deep learning · Object detection

1 Introduction The automotive industry has an extremely high-quality product standard, not just for the security risks each faulty component can present, but the very brand image it must uphold at all times to stay competitive [1, 2]. A situation that can get complicated in the current market, where vehicle customization is an important factor for many customers, and where factories can assemble cars for multiple brands at a time. Even with the tight quality control that occurs at multiple points of the assembly line, this increases © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 740–751, 2021. https://doi.org/10.1007/978-3-030-58653-9_71

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the likelihood of non-conformity errors to occur [3–5]. Which, when undetected, can negatively impact the brand’s reputation [1]. While the use of automated processes is commonplace in this industry, much the of quality inspection still relies on the manual approach. Especially in situations where Computer Vision struggles to identify in real-time what is observing, such as detecting defects on highly reflective or transparent surfaces [6, 7]. Or, in our specific case, accurately classifying the labels placed inside the much dimmer, and visually noisy, area of the vehicle engine. However, thanks to the increases in computational power and the growing adoption of Industry 4.0 methodologies, new machine vision approaches are overcoming these past obstacles [1, 8]. This not only results in a more cost-efficient automated solution, but it also aids the workers by decreasing subjectiveness and ocular and mental fatigue [4, 7, 9]. And it frees them to perform the more flexible, abstract problem-solving tasks their automated partners cannot [10, 11]. When discussing vehicle quality inspection via sensors, machine vision or other automated processes [12] their study usually focuses on two broad groups: fault detection [2, 12–15] and surface defects [6, 7, 16–18]. Each require specialized approaches to tackle the specific problems each present, which in turn led to a multitude of solutions, such as binarization [13, 19], edge detection [14, 20], template matching [4, 14, 15], fuzzy decision trees [17] and a wide variety of machine learning algorithms [1, 2, 7, 8, 12, 16, 18, 21, 22]. However, this branch of Artificial Intelligence also covers many different methods, such as k-nearest-neighbors, random forest [2, 8], support vector machine [7, 8], l1-regularized logistic regression [1], decision trees, naive Bayes [2], neural networks and its sub-branch deep learning [16, 18, 21, 22]. This is particularly true when it comes to the common problem of precisely identifying and verifying small parts and labels in increasingly more complex assembly processes. In the last decades, many computer vision studies have been proposed to ameliorate the manual assessment but, to our knowledge, none of them focused on the final assembly verification. Which is a key point of occurring errors, such as parts misplacement, missing labels, etc. [5, 23]. To overcome this limitation, we proposed a model for components assembly verification in car parts assembly line using machine vision. The experiments focuses on engine labels detection, which were detected using deep learning object detection algorithms (Faster R-CNN [24] and YOLOv3 [25]) and demonstrated the effectiveness of their application in assembly verification. The comparison of the scores produced by both prediction models also help us evaluate the potential such solutions have in reducing the repetitiveness of everyday work to the human operators and the level of mitigation of human error. Section 2 describes the methods used to collect and process the data, while Sect. 3 describes the experiment itself. Section 4 presents the results of the object detection algorithms, as well as their comparison. Section 5 discusses the results and Sect. 6 presents us the conclusions and future works.

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2 Materials and Methods 2.1 Data Collection and Preparation All the dataset used in the experiments was collected prepared manually. Images were collected at an automotive assembly plant. They were taken at the factory’s assembly line using a video camera, lighted by florescent lights located inside the assembly line room in order to maintain the Realtime scenario, as exemplified by Fig. 1. The images were taken from three different angles (straight, 45° left, and 45° right), and various orientations. The object and camera distances were 1 m. The featured vehicles are the Citroën Berlingo and Peugeot Partner/Rifter, which came is various colors and configurations. The number of images and objects for both brands were equally distributed. An image can contain one or more objects depending on the camera’s orientation and distance.

Fig. 1. A sample image collected for the experiment containing the oil label.

Table 1. Distribution of vehicle engine labels.

Quantity

Red battery

Green battery

Smoke

Start & stop

Peugeot oil

Citroen oil

Gas big

Gas small

Total

83

23

49

54

29

48

15

20

321

As shown in Table 1, a total of 321 objects were annotated from 250 images to be used for training and testing purposes. As we can see the eight different objects are not equally distributed. After collecting the image dataset all the objects from the images were manually annotated using LabelImg image annotator [26]. It produced an XML file containing all the information about the image and the location of the objects. During the experiments both image file and corresponding XML file are loaded, as exemplified in Fig. 2. 2.2 Prototype Model In order to achieve a less error-prone, fast, and reliable vehicle assembly verification the during overall assembly procedure, we propose a model where a camera captures images

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Fig. 2. The object manual selection of the vehicle’s oil label (left) and the corresponding XML annotation information (right).

in real time, detects the particular object, compares it with the database containing parts information, and displays the validation result (see Fig. 3).

Fig. 3. Block diagram of the smart inspection system using machine vision in the assembly line.

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In the proposed model, continuous monitoring of the vehicle’s interior and exterior parts can be captured either by fixed cameras, mobile phone, head mounted displays (HMD), such as smart glasses, or other similar devices. After receiving the input image/video, the object detection algorithms will detect car parts based on their extracted features that had been previously trained with similar data. In this study, the chosen object detection algorithms were YOLOv3 and Faster-RCNN. After predicting the engine label object(s), the model’s results are compared with a database containing the parts checklist of that specific vehicle. The results of the non-conformity comparison are sent to the display interface, be it a fixed screen placed at the control station, a tablet PC/mobile phone, smart glasses or other similar device. 2.3 Proposed Object Detection Algorithms With the advancements of computer vision technology, there has been a lot of innovation and creative methodologies which enable researchers to use it in various domains of object detection. Before deep learning emerged in the 2010s the object detection tasks were performed manually, which usually were not very accurate and required a high amount of processing time. After the amazing results obtained by the Convolutional Neural Networks (CNN) in image classification problems, researchers started seeing these networks as a new solution for object detection. In the proposed model, vehicle’s engine labels were detection using two object detection algorithms called Faster-RCNN [24] and YOLOv3 [25] which are improved versions of RCNN and YOLO respectively. The RCNN object detection algorithm divides an image into multiple regions and processes each for several iterations. In the first version of the RCNN model, the image was divided into 2000 regions according to a selective search algorithm, which were then feed to the CNN. This method achieved great results, but the amount of time needed to train a network was still a major problem, as the it had to classify 2000 regions per image, a process that took approximately 47 s per individual analysis. Meaning this solution is not viable for real time problems. The same authors later proposed another version named Fast RCNN [27] to fix the major problems of the previous system. Instead of feeding the network with 2000 separate images, it receives the initial input image to compute its feature map, which the model can then use to select the regions of interest. And in 2017 another version was developed, named Faster R-CNN, which uses a separated CNN to predict and reshape the regions of interest that are fed to the network [24]. An extension of the Faster-RCNN that locates the exact pixels of each object instead of just the bounding boxes is known as Mask-RCNN [28]. Another notable object detection algorithm, outside of RCNN family, is the You Only Look Once, or YOLO [29]. It makes predictions after a single evaluation of the input image by dividing it into an S x S grid of cells and predicting the bounding boxes and class probabilities for each. The confidence value of an object existing in a given cell is calculated by the probability of that object class multiplied by the Intersection of Union (IoU), which will be zero if no object is present. However, a limitation that results from this grid subdivision is when predicting the bounding boxes cells, it can fail in detecting objects that are too close to each other or that have unusual aspect ratio. Like the RCNN it received improvements as new versions were developed. The one we implemented was YOLOv3 [25].

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3 Experiments The algorithms chosen to support this proposal were the latest versions of RCNN and YOLO, which were trained and tested with our custom dataset. In both cases, the training was performed with transfer learning which states that the initial weights were taken from another pre-trained CNN architecture. The Faster-RCNN training process took about 1.5 h to complete, while the YOLOv3 took 2.45 h. The dataset provided to both algorithms was composed of different images, 960 * 540 px in dimension, with a total of 321 objects. 281 objects, or 87% of the dataset, was used as the training data, while the remainder 13% was used as the test data. Also due to the limited number of objects and the imbalance of the number of ground truths for each, 5 were taken from each and placed in the training data to better balance the test data. Data augmentation was also performed to the increase the training data to 100 objects for each of the 8 classes, to improve the networks’ performance. This augmentation consisted of the application of random rotations between −180º and 180º and preserving the objects’ scale and proportions.

Fig. 4. A formula illustration to calculate intersection of union.

For the evaluation of the object detection performance we applied the IoU, which measures the ratio of the Area of Overlap and the Area of Union between the ground truth box and the detection box (Fig. 4). It’s a particularly important metric for object detection assessment, as its score determines if a detection can be considered a True Positive (TP) prediction or not. The threshold value for the IoU was 0.5, meaning that if the overlapped area is less than that value then the object was not detected. Therefore, considered as a True Negative (TN). Another performance metric used was the confusion matrix, which allows us to measure the amount of correct and incorrect positive and negative predictions. It is a matrix containing the amount for each of the TP, TN, False Positive (FP), and False Negative (FN) results. And, from those values, we can calculate precision and Recall as illustrated in the expressions 1 and 2. Recall =

TP TP + FP

(1)

Recall =

TP TP + FN

(2)

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Besides Precision and Recall, Prediction Confidence and Mean Average Precision (mAP) were also used to evaluate the predictions. The mAP is the most common object detection and multiclassification metric [30], as it calculates the mean value of the average precision (AP) of all classes. Due to the number of objects in the dataset, for validation proposes a K-fold cross validation with 3 folds was used.

4 Results As we can see in Fig. 5, the object detection performed by the algorithm displays the identification name of the detected object, the location of its bounding box and the probability of occurrence as the output.

Fig. 5. Detection of the oil label using faster-RCNN.

The system also calculates a detailed evaluation of the object detection, which allows us to determine the performance metrics of the proposed model. Specifically, the Confusion Matrix, Precision, Recall, Intersect Over Union (IoU), Mean Average Precision (mAP) and Mean Prediction Confidence, which were calculated in a three-fold validation for both the Faster R-CNN and YOLOv3 (Tables 2 and 3). Table 2. Confusion matrix using faster-RCNN and YOLOv3 Faster-RCNN Fold

TP

FN FP

1st

92.5 2.5 10

2nd 3rd

YOLOv3 TN TP

FN

FP TN

–

57.5 40

2.5 –

90

2.5 12.5 –

62.5 35

2.5 –

85

10

5

–

62.5 37.5 0

5

9

–

60.8 37.5 1.7 –

Average 89

–

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Table 3. Precision, recall, IoU, mAP, and confidence using faster RCNN and YOLOv3. Faster-RCNN Fold P

RC

YOLOv3

IoU mAP C

P

RC

IoU mAP C

1st

90 97.4 0.85 91.7

89.9

96 47.9 0.81 56.67 86

2nd

88 97.3 0.85 87.2

92.3

96 64.1 0.77 55.38 91

3rd

94 89.5 0.85 82

93.7 100 62.5 0.76 62.5

A

91 94.7 0.85 85.0

91.9

89

97 58.2 0.78 58.18 89

*P-Precision, RC-Re-Call, IoU-mean Intersection of Union, C-confidence, A-Average. Each value is represented in %.

As seen in Table 3, there’s not a relevant difference between each fold results for both networks. According to the overall metrics, the Faster RCNN achieved better results when considering the number of objects that were correctly detected and classified, with an 89% of True Positives results, compared to YOLOv3’s 60.8%. However, YOLOv3 predicted less False Positives, meaning 97% of all the objects were detected correctly, against the 91% of the Faster R-CNN. Also, when comparing both Confusion Matrixes, YOLOv3 achieved a 37.5% of False Negatives in the test set, instead of Faster R-CNN’s 5%, which displays a failure at detecting a large number of objects. In general, the application of the Data Augmentation increased the performance of both networks. The similar results represented in Tables 2 and 3 are also presented in Tables 4 and 5 using Data Augmentation applied in all the image dataset. The Faster RCNN achieved a 97.5% of True Positives and 0% of False Negatives, meaning it correctly detected all test objects. The YOLOv3 suffered a smaller increase, only achieving 71.6% of True Positives and 25% of False Negatives while retaining its inability of detecting several objects at once. The Data Augmentation approach also increased the Precision and Recall in both models but, compared to the previous results, the Faster R-CNN achieved the highest Precision and a 100% Recall. Table 4. Data augmentation confusion matrix for faster-RCNN YOLOv3. Each value is represented in %. Faster-RCNN

YOLOv3

Fold

TP

FN FP

TN TP

FN FP TN

1st

95

0

–

25

2nd

97.5 0

2.5 –

72.5 25

2.5 –

3rd

100

0

–

72.5 25

2.5 –

2.5 –

71.6 25

3.3 –

0

Average 97.5 0

5

70

5

–

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Table 5. Precision, recall, IoU, mAP and mean prediction confidence using data augmentation for faster-RCNN and YOLOv3. Faster-RCNN

YOLOv3

Fold P

RC IoU mAP C

1st

95

100 0.91 96.9

98.1 94 73.7 0.86 79.8

P

RC

IoU mAP C 92

2nd

97.5 100 0.94 100

99.6 97 74.3 0.85 70.4

93

3rd

100 100 0.91 96.5

99.0 97 74.3 0.87 71.5

94

A

97.5 100 0.92 97.8

98.9 96 74.1 0.86 73.9

93

5 Discussions In this study we proposed the creation of a prototype model for vehicle assembly verification using machine vision, with specific focus on engine labels detection through the use of object detection algorithms. We also had the secondary objective to see which of the proposed algorithms, Faster-RCNN and YOLOv3, had the highest accuracy and reliability for this specific setting. And how much impact Data Augmentation would have on their predictive performance, compared to the original image dataset. According to the results both networks achieved similar Prediction Confidence and IoU scores, with the Recall and mAP scores being what set them the most apart. For the original custom dataset, the YOLOv3 achieved a Precision value of 97% than FasterRCNN’s 91%, but when it came to Recall it only achieved a value of 58.2% compared to the other model’s amazing 94.7%. Meaning that while YOLOv3 has a higher proportion of positive results in the correctly predicted values, the Faster-RCNN vastly outperforms it in its ability to correctly predict the positive results. The Faster-RCNN also achieved a higher IoU, mAP and Confidence scores with an average of 85%, 85% and 91.9% respectively, compared to YOLOv3’s 0.78%, 58.18% and 89% respectively. Proving it possesses a much greater precision and level of accuracy in multi-object detection and classification. This difference between both algorithms is similar to the study presented by Alganci [31]. As hypothesized the data augmentation helped increase the object detection performance of both algorithms [32, 33]. Here the Faster-RCNN outperformed YOLOv3 in Precision, with an average of 97.5% rather than 96%, and achieved an outstanding 100% Recall rate, compared to YOLOv3’s 74.1%. Meaning that the Faster-RCNN, in average, made zero incorrect detections. The IoU, mAP and Confidence results also improved significantly for both algorithms, with the Faster-RCNN still outperforming YOLOv3’s in all metrics to the point of near perfect scores. Most of misclassifications that did occur were associated with the level of similarity between objects. To exemplify, the “Oil Peugeot” and “Oil Citroen” are two classes that share the exact same design, apart from the brand logo, which lead to a high change of misclassification between these two items. However, a major limitation of this study was number of objects being restricted to engine labels, not being possible to generalize the results to other vehicle components.

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Another important limitation is the limited number of images in the dataset. With only 321 images used for training and testing the model, we intend to improve the results by increasing the size of the input data. Also, only two object detection algorithms were tested.

6 Conclusion When it comes to the detection of non-conformities in a field traditionally performed by highly trained human operators, the necessity of the object detection model to perform the quality inspection accurately, reliably and efficiently is paramount. The results achieved by the Faster R-CNN, especially when combined with the data augmentation, not only achieved the best performance in all the metrics, but it was also better at detecting all the objects in the test set. It also proved to possess enough high levels of performance and reliability to justify its implementation in the quality inspection of vehicle engine labels. Further addition of other detection algorithms is important to determine the most adequate solution to this given context. Especially when it comes to reducing the occurrence of misclassification in highly similar classes, as it was the case in this current study. Therefore, the study can be extended with higher number of images/objects including various internal and external car parts so that the proposed prototype model can be completed in industry application. Acknowledgement. This work was funded by Project “INDTECH 4.0 – New Technologies for smart manufacturing”, No. POCI- 01-0247-FEDER-026653, financed by the European Regional Development Fund (ERDF), through the COMPETE 2020 - Competitiveness and Internationalization Operational Program (POCI).

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8. Peres, R.S., et al.: Multistage quality control using machine learning in the automotive industry. IEEE Access 7, 79908–79916 (2019) 9. Abdulla, W.: Mask R-CNN for object detection and instance segmentation on Keras and TensorFlow. GitHub repository (2017) 10. Wang, Q., Sowden, M., Mileham, A.R.: Modelling human performance within an automotive engine assembly line. Int. J. Adv. Manuf. Technol. 68(1), 141–148 (2013) 11. Krugh, M., Mears, L.: A complementary cyber-human systems framework for industry 4.0 cyber-physical systems. Manuf. Lett. 15, 89–92 (2018) 12. Chauhan, V., Surgenor, B.: Fault detection and classification in automated assembly machines using machine vision. Int. J. Adv. Manuf. Technol. 90(9), 2491–2512 (2017). https://doi.org/ 10.1007/s00170-016-9581-5 13. Campos, M., et al.: Detection of defects in automotive metal components through computer vision. In: 2008 IEEE International Symposium on Industrial Electronics (2008) 14. Pei, Z., Chen, L.: Welding component identification and solder joint inspection of automobile door panel based on machine vision. In: 2018 Chinese Control And Decision Conference (CCDC) (2018) 15. Zhou, H., et al.: Automatic inspection of LED indicators on automobile meters based on a seeded region growing algorithm. J. Zhejiang Univ. Sci. C 11(3), 199–205 (2010) 16. Chang, F., et al.: A mobile vision inspection system for tiny defect detection on smooth car-body surfaces based on deep ensemble learning. Meas. Sci. Technol. 30(12), 125905 (2019) 17. Doring, C., et al.: Improved classification of surface defects for quality control of car body panels. In: 2006 IEEE International Conference on Fuzzy Systems (2006) 18. Sun, X., et al.: Surface defects recognition of wheel hub based on improved faster R-CNN. Electronics 8(5), 481 (2019) 19. Arjun, P., Mirnalinee, T.T.: Machine parts recognition and defect detection in automated assembly systems using computer vision techniques. Revista Tecnica De La Facultad De Ingenieria Universidad Del Zulia 39(1), 71–80 (2016) 20. Yu, B.J., Yang, M., Zhao, G.Q.: Method of automobile-sensitive component of safety belt’s dimensional inspection based on machine vision. Appl. Mech. Mater. 442, 397–404 (2014) 21. Iqbal, R., et al.: Fault detection and isolation in industrial processes using deep learning approaches. IEEE Trans. Industr. Inf. 15(5), 3077–3084 (2019) 22. Luckow, A., et al.: Deep learning in the automotive industry: applications and tools. In: 2016 IEEE International Conference on Big Data (Big Data). IEEE (2016) 23. Baraldi, E.C., Kaminski, P.C.: Reference model for the implementation of new assembly processes in the automotive sector. Cogent Eng. 5(1), 1482984 (2018) 24. Ren, S., et al.: Faster R-CNN: towards real-time object detection with region proposal networks. IEEE Trans. Pattern Anal. Mach. Intell. 39(6), 1137–1149 (2017) 25. Redmon, J., Farhadi, A.: Yolov3: an incremental improvement. arXiv 2018 arXiv preprint arXiv:1804.02767 (2019) 26. Tzutalin, LabelImg. Git code (2015) 27. Girshick, R.: Fast R-CNN. In: 2015 IEEE International Conference on Computer Vision (ICCV) (2015) 28. He, K., et al.: Mask R-CNN. In: 2017 IEEE International Conference on Computer Vision (ICCV) (2017) 29. Redmon, J., et al.: You only look once: unified, real-time object detection. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2016) 30. Everingham, M., Winn, J.: The PASCAL visual object classes challenge 2012 (VOC 2012) development kit (2012). http://host.robots.ox.ac.uk/pascal/VOC/voc2012/htmldoc/dev kit_doc.html

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31. Alganci, U., Soydas, M., Sertel, E.: Comparative research on deep learning approaches for airplane detection from very high-resolution satellite images. Remote Sens. 12(3), 458 (2020) 32. Shorten, C., Khoshgoftaar, T.M.: A survey on image data augmentation for deep learning. J. Big Data 6(1), 60 (2019). https://doi.org/10.1186/s40537-019-0197-0 33. Mikołajczyk, A., Grochowski, M.: Data augmentation for improving deep learning in image classification problem. In: 2018 International Interdisciplinary Ph.D. Workshop (IIPhDW) (2018)

Mitigation of Earthquake-Induced Structural Pounding Between Adjoining Buildings – State-of-the-Art Pedro Folhento1(B)

, Rui Barros1

, and Manuel Braz-César2

1 Faculty of Engineering, University of Porto, Porto, Portugal

{up201811645,rcb}@fe.up.pt 2 Polytechnic Institute of Bragança, Bragança, Portugal

[email protected]

Abstract. The investigation of collisions between contiguous building structures due to severe earthquakes is of great importance, particularly in large cities where there is a high population density. These collisions will produce strong impact forces that will significantly influence the dynamic behavior of building structures. Moreover, these impacts may provoke serious structural damage, that can lead to local collapse, or in the worst-case scenario to complete structural collapse. Different measures and techniques in mitigating pounding effects between adjacent buildings during seismic hazard events were extensively developed and studied by several researchers in recent years. This study presents an overview of these different pounding mitigation solutions, namely regarding the required separation seismic gap, link elements, shock absorber devices, structure stiffening, and supplemental energy and control devices. The main conclusions of several researches, from pioneer to state-of-the-art studies, concerning code provisions for minimum gap sizes and solutions for pounding mitigation between buildings, are compiled and presented in this study to demonstrate its diversity, effectiveness, and practical applications. Keywords: Earthquake-induced pounding · Building structures · Mitigation

1 Introduction Several pounding mitigation methods have been proposed over the years. According to Valles and Reinhorn [1], these methods can be classified according to the problem of pounding in question: methods to prevent pounding, methods to strengthen structures to withstand pounding, and techniques to reduce pounding effects in structures. Methods to prevent pounding contemplate the establishment of minimum gap size between buildings and the reduction of lateral deformation of the structures. The use of link elements, dampers, bumpers, shock absorbers, crushable devices, or collision shear walls placed in specific locations can help to avoid sudden shocks due to collisions of structures [2], and can also be seen as techniques to reduce the negative effects of © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 752–761, 2021. https://doi.org/10.1007/978-3-030-58653-9_72

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pounding between structures. Methods to strengthen the structure depend on increasing the lateral stiffness of buildings [3] to reduce the lateral displacements of structures. Pounding effects can also be reduced by the use of techniques that enhance the seismic performance of contiguous structures with small gaps. Active, passive, and semi-active vibration control systems can add supplemental energy dissipation and thus constitutes one of the techniques to mitigate pounding effects [2]. The need to mitigate the negative effects arising from pounding between adjacent buildings is very important. This survey intends to briefly outline some mitigation solutions proposed over the years presenting its main results and practical application. Further investigation will be suggested based on some problems and drawbacks verified in the solutions presented.

2 Critical or Minimum Gap Size A spectral difference method (SPD) named double difference combination rule (DDC) or complete quadratic combination method (CQC), based on the vibration phase and damping, gives the minimum gap among structures to avoid pounding [4]  2 2 Gapmin = xmax (1) ,L + xmax,R − 2 ρLR xmax,L xmax,R , in which x max,L , and x max,R are, respectively, the peak displacements of the left and right structures represented in Fig. 1 and ρLR is the correlation factor [5],

ρLR

   3 √ 2 8 ξL ξR ξR + ξL TTRL TTRL = ,   2 2  2    2  TR 2 TR 2 1 − TTRL + 4 ξ + 4ξL ξR 1 + TTRL + ξ L R TL TL

(2)

where T L and T R are the natural periods and ξL and ξR are the damping ratios, respectively of the left and right structures. When the two structures are uncorrelated (ρLR = 0), the expression for the minimum gap becomes as follows [1]  2 2 Gapmin = xmax (3) ,L (t) + xmax,R (t), that is equivalent to the square root of the sum of the squares method (SRSS) [6]. For negatively correlated responses of the two structures (ρLR = −1), the minimum gap is equivalent to the absolute sum method (ABS) Gapmin = xmax,L (t) + xmax,R (t).

(4)

ABS method gives the most conservative approach for the establishment of the minimum gap since it considers that both structures attain the peak displacement at the same time [2]. The SRSS method is the most used in seismic codes in defining the critical gap (e.g., [7, 8]). Equation 3 and the correlation factor in Eq. 2 were studied by many authors to extend the applicability of DDC rule to non-linear hysteretic systems, whose assumption is not so accurate as for linear systems [1, 9].

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Lopez-Garcia and Soong [10] examined the accuracy of the DDC rule’s expression considering pounding among linear structural systems. It was found that the accuracy of DDC’s rule depends on the ratio of natural periods of the structures and the relationship between natural periods and the period of the main frequency of excitation. To compute the minimum separation distance between structures to avoid collisions, Naderpour et al. [11] created a new formula. The accuracy of this formula in computing the optimum gap size between two adjacent structures was guaranteed by using an artificial neural network tool and a program (CRVK) that was specifically developed to solve a large number of dynamic simulations of lateral displacement. The numerical analysis confirmed the accurate minimum gap sizes obtained. Abdel Raheem et al. [12] determined the relative displacement time history response of three adjacent buildings in series with unequal heights, for several input earthquake excitations. All possible alignment configurations were considered. This structural analysis was performed using the software ETABS and the minimum separation distance was evaluated according to the ABS, SRSS, and DDC methods. A new equation that computes the effective periods of inelastic buildings, based on ductility demands, was developed by Khatami et al. [13], to accurately assess the minimum separation gap to avoid earthquake-induced pounding between structures. Numerical analyses were performed by comparing other similar formulations with the proposed formula, presenting better results in ensuring the appropriate seismic gap. The recent seismic design codes for new constructions recommend the critical distance between adjacent structures to avoid pounding (e.g., [7, 8]). However, due to certain facts, these provisions are not always effective or applicable, e.g., the inconsistencies with modern codes that imply large deformations due to inelastic responses during seismic events, small lot sizes and high costs of land in populated cities making the seismic gap difficult to apply, inadequate seismic gaps due to limitations were not provided among adjacent buildings designed with old earthquake codes, etc. [14].

3 Pounding Mitigation Techniques 3.1 Bumpers Shock absorber devices constitute a technique in mitigating negative effects of pounding between structures, e.g., the interposition of a shock-absorbing material element in gapped structures, or the use of collision walls at boundary lines of structures [15]. Bumpers are energy dissipation devices that are triggered once the gap between the structures closes, reducing in this way the impulsive forces that are transmitted between structures [1]. Collision walls act as bumpers providing seismic resistance to the structure and minimize the risk of collapse due to pounding by filling the seismic separation gap [16]. A schematic representation of a bumper damper element and collision shear walls are presented in Fig. 1 (respectively, on the middle and on right). An impact model based on non-linear inelastic forces was proposed and investigated by Polycarpou et al. [17] to mitigate earthquake-induced pounding in structures. The behavior of rubber under impact loading is then evaluated to implement such material in collision bumpers. The authors considered based on experimental data that a non-linear

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impact model is the most suitable approach for emulating the response of rubber during a collision. The authors then validated the impact model according to experimental results from the literature, showing a very good correlation. Jankowski and Mahmoud [2] used this impact model to simulate the application of rubber bumpers in building pounding, verifying that this technique is less effective than the coupling elements since in the latter the buildings vibrate together and with bumpers, it may lead to a significant rebound of one of the structures after pounding. To control interstory deflections and avoid pounding between two buildings, Khatami et al. [18] performed comparative analyses considering some mitigation techniques: modification of structural properties, application of TMD, base isolation, and base isolation with rubber bumpers. The best solution to avert earthquake-induced pounding was verified to be the application of base isolation with rubber bumpers.

Fig. 1. Representation of two adjacent buildings as lumped masses under a ground motion (left), shock absorber devices: bumper damper (middle), and collisions walls (right).

3.2 Link Elements The connection of adjacent buildings by the use of link elements can be considered as a technique to mitigate pounding. This connection results in combining two structural systems into one, effectively coupling the motion of structures [1]. Link elements can be linear elastic [19], i.e., a linear spring, can have energy dissipation properties when the structures are linked by dashpot elements [20], and can have both spring and dashpot elements together, named viscoelastic elements [2]. Considering Fig. 2, the equations of motion can be written, respectively, as [2]   (5) M −x¨ (t) + C −x˙ (t) + K + K s x−(t) = −M −i −x¨ (t), g

  M −x¨ (t) + C + C D −x˙ (t) + K −x(t) = −M −i −x¨ (t),

(6)

    M −x¨ (t) + C + C D −x˙ (t) + K + K s x−(t) = −M −i −x¨ (t),

(7)

g

g

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where M is the mass matrix of the system, C and C D are, respectively, the damping matrices of the system and the dashpot link element, K and K S are, respectively, the stiffness matrices of the system and the spring element, i is the vector of influence coefficients, and x¨ g (t) is the ground acceleration. Jankowski and Mahmoud [2, 21] studied the effectiveness of link elements (spring, dashpot, viscoelastic) in mitigating earthquake-induced pounding by considering different values of the stiffness and damping, computing the optimum values that lead to the largest reduction of the structural response. Licari et al. [22] developed a multi-link viscoelastic (MLV) finite element model, that reproduces Jankowski’s non-linear viscoelastic relationship. This model comprises an in-series assemblage of linear dampers associated with linear springs in parallel. This link element was then applied to a real case-study, connecting two adjacent multi-story buildings by FVDs. Reductions of the story displacements and story shear were verified by the implementation of this mitigation strategy. The effect of the localized interconnection of adjacent buildings for pounding mitigation was studied by Mooty et al. [3]. This research was carried using the software ETABS and a set of five buildings with different heights and configurations. The proposed connections of the adjacent buildings were achieved by connecting slabs with a reinforced concrete patched or steel plates, or connecting columns at the joint by a steel belt. The equation of motion used in the analysis is identical to Eq. 6. Results were found to be effective in mitigating earthquake-induced pounding. Viscous and viscoelastic dampers were investigated by Roshan et al. [23] as two types of link elements between two buildings, to mitigate pounding. SAP2000 was used to analyze several frames with unequal heights, configurations, and structural systems. The effectiveness of the link elements was verified, significant reductions of the building’s seismic response, mitigating in this way the effects of pounding.

Fig. 2. Lumped mass model of two-story buildings with link elements: spring (left), dashpot (middle), and viscoelastic (right).

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3.3 Structure Stiffening The reduction of excessive lateral displacements by stiffening structures likely to collide can be seen as a solution to avoid pounding [1]. This solution may also reduce the acceleration responses of the structure since increased stiffness leads to the increased natural frequency, avoiding the range of the dominant frequencies of excitations [2]. Many ways to add stiffness to a structure can be found in the literature, e.g., the addition of shear walls, bracing systems, and implementation of active stiffeners [2]. Jankowski and Mahmoud [2] studied the effect of adding stiffness to a structure of a three-story building under earthquake excitations by modifying the stiffness matrix of the building in the elastic dynamic equation of motion. The authors validated the solution of enhancing stiffness as being effective in reducing the story deformations and thus avoiding pounding. However, it was pointed out that the implementation of this solution may be costly and technically difficult. To prevent impacts between adjacent structures with unequal heights under different seismic excitations, Roy and Das [24] proposed some mitigation measures based on stiffening the structures, viz., RC shear walls, and reinforced cement concrete (RCC) cross bracings. Performing time history analyses, the authors verified that these pounding mitigation measures proved to be effective. Additionally, the best location of these elements was also investigated, highlighting the fact that a proper arrangement of these elements leads to a safe and economical structural design. 3.4 Supplemental Energy Dissipation and Control Devices The installation of devices in structures capable of providing supplemental energy dissipation, improving its overall behavior to seismic events, and thus reducing structural displacements, can be seen as a technique that substantially reduces the pounding forces between buildings and in the best-case scenario, it can prevent from it [2]. Abdullah et al. [25] proposed a solution to reduce structural vibrations and pounding probability between two adjacent eight-story buildings, subjected to earthquake excitations. This solution sketched in Fig. 3 (middle), consists of connecting the adjacent buildings using a shared tuned mass damper (STMD). Its design is based on a performance function, used to obtain the optimum parameters of the STMD, which will lead to the best overall system response. The authors compared this innovative solution with the usual scenario of each building possessing an individual tuned mass damper (TMD) (Fig. 3 on the left). Results of the numerical analysis using the software MATLAB and concerning the STMD solution, proved to be more effective than the TMD solution in reducing structural vibrations and mitigating pounding. Other investigations concerning the application of TMDs, to effectively prevent pounding were carried out. Bekdas and Nigdeli [26] studied the application of an optimum TMD in a slender structure surrounded by two adjacent stiffer structures. Mate et al. [27] considered the application of a TMD in an elastic or inelastic system with a singledegree-of-freedom (SDOF) subjected to seismic excitations, considering or ignoring pounding. Ismail et al. [28, 29] developed and further investigated an innovative seismic isolation device called “roll-n-cage” (RNC) isolator, which essentially consists of a rolling

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object with a special quasi-elliptical form that is enclosed between two plates and an elastomeric cylinder. This mechanism is installed at the base isolation level and in a single unit is capable of providing rigid support, horizontal flexibility, and energy dissipation features. Through numerical analyses, this isolation device with hybrid control schemes showed effective results controlling and preventing pounding between adjacent structures under severe near-fault ground motions. Magneto-rheological (MR) dampers for vibration control of buildings, extensively studied by Braz-César [30], can be used as a technique to mitigate pounding effects in adjacent buildings. Abdeddaim et al. [31], investigated the use of the coupling strategy (link elements) by connecting two ten-story adjacent buildings with an MR damper for different control strategies (depending on the voltage induced in the damper), viz., passive-off, passive-on, semi-active on-off, and fuzzy logic controller, to mitigate the effects of earthquake-induced pounding. The authors [31] then verified the effectiveness of this technique using a single damper, reducing the system response and consequently the minimum gap required to avoid pounding. Additionally, the fuzzy logic controller led to better control results. The authors later extended this research to the application of this pounding mitigation technique for different configurations of the adjacent buildings and locations of the MR damper [32], and different floor levels and structural parameters [33], verifying through numerical analyses effective pounding mitigation between the structures.

Fig. 3. Adjacent structures as lumped mass models with different vibration control strategies: TMDs (left), STMD (middle), and SSTMD (right).

Kim [34] proposed a semi-active STMD (SSTMD – Fig. 3 on the right) that consists of one TMD connected to two adjacent 8-story buildings by MR dampers. The MR damper was effectively controlled by a multi-input multi-output (MIMO) fuzzy logic algorithm, developed by the author. Numerical analyses were performed considering the application of the SSTMD and the conventional TMD solutions to the adjacent buildings under an artificially generated ground motion. These solutions presented considerable reductions

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concerning the uncontrolled situation. Moreover, the solution SSTMD presented greater results even though it only uses half of the mass of the conventional TMD solution. Another recent version of a TMD application in reducing structural response and consequently mitigating pounding effects between structures with insufficient separation distance is the pounding TMD (PTMD). This solution was developed by Song et al. [35] and has been studied by many researchers (e.g., [36, 37]), providing better results in reducing structural responses than the conventional TMD. Nishath and Abhilash [38] investigated the implementation of passive vibration control systems, friction, and non-linear fluid viscous dampers in chevron form, in two adjacent buildings with unequal heights subjected to a seismic acceleration. The dampers were considered in different positions in the buildings: at the mid bays, end bays, or all bays. A non-linear dynamic time history analysis was carried out and the effectiveness of the dampers in mitigating pounding between buildings was proved. Regarding serviceability and economic reasons, the best options in mitigating pounding were found to be the one that considered viscous dampers at mid bays of the buildings and friction dampers at end bays of the buildings.

4 Conclusions It was found in this literature review that establishing a minimum seismic gap size between structures constitutes the best way to mitigate and prevent the negative effects of earthquake-induced pounding. This solution is widely recommended by various building and seismic codes but presents some difficulties in its applicability. Link elements are an accurate solution in mitigating earthquake-induced pounding and is of easy application, although it may have some disadvantages depending on the dynamic response features of the adjacent structures (high forces in the link elements, redistribution of forces in the structure, promotion of torsional effects, etc.), design considerations (different design criteria of the buildings, etc.), political problems (buildings with different owners) and the projected use of the buildings. Thus, the application of this solution may not be appropriate in some situations, since it will alter the dynamic response of the structural system. Structure stiffening is a strategy of mitigating earthquake-induced pounding, that to be effective requires a prior and thorough planning and design of adjacent structures. Shock absorber devices, viz., bumpers have found to be one solution in mitigating earthquake-induced pounding between structures, that generally provides greater results in comparison with other solutions. However, some innovative solutions regarding the implementation of supplemental energy devices or control mechanisms, or even the combination of these with other pounding mitigation strategies, revealed even greater results in avoiding or reducing pounding effects. Further investigation is essential to provide more information concerning the effectiveness and practical applicability of the existing techniques in mitigating earthquakeinduced pounding between contiguous buildings. Acknowledgments. This paper is within the scope of the first author’s project thesis investigation (PTI), part of his Ph.D. degree in progress. The first author gratefully acknowledges the funding

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by Fundação para a Ciência e a Tecnologia (FCT), Portugal, through the funding of the Ph.D. studentship with reference SFRH/BD/139570/2018.

References 1. Valles, R., Reinhorn, A.: Evaluation, prevention and mitigation of pounding effects in building structures. Technical report NCEER-97-0001, National Center for Earthquake Engineering Research, State University of New York, Buffalo, USA (1997) 2. Jankowski, R., Mahmoud, S.: Earthquake-Induced Structural Pounding. Springer, Cham (2015) 3. Abdel-Mooty, M., Ahmed, N.: Pounding mitigation in buildings using localized interconnections. In: The 2017 World Congress on Advances in Structural Engineering and Mechanics (ASEM17), 28 August–1 September, Ilsan (Seoul), Korea (2017) 4. Jeng, V., Kasai, K., Maison, B.: A spectral difference method to estimate building separations to avoid pounding. Earthq. Spectra 9(2), 201–223 (1992) 5. Kiureghian, A.: On response of structures to stationary excitation - Report No. UCB/EERC - 79/32. Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley, California, December 1979 6. Anagnostopoulos, S.: Pounding of buildings in series during earthquakes. Earthq. Eng. Struct. Dyn. 16, 443–456 (1988) 7. European Committee for Standardization, EC8: Design of structures for earthquake resistance: General rules seismic actions and rules for buildings (EN1998-1), Belgium (2004) 8. American Society of Civil Engineers (ASCE), Minimum design loads for buildings and other structures, ASCE/SEI Standard 7–10, Reston, VA, USA (2010) 9. Lopez Garcia, D.: Separation between adjacent nonlinear structures for prevention of seismic pounding. In: Thirteenth World Conference on Earthquake Engineering, Paper No. 478, 1–6 August 2004, Vancouver, BC, Canada (2004) 10. Lopez-Garcia, D., Soong, T.: Assessment of the separation necessary to prevent seismic pounding between linear structural systems. Probab. Eng. Mech. 24, 210–233 (2009) 11. Naderpour, H., Khatami, S., Barros, R.: Prediction of critical distance between two MDOF systems subjected to seismic excitation in terms of artificial neural networks. Period. Polytech. Civil Eng. 61, 516–529 (2017) 12. Abdel Raheem, S., Fooly, M., Abdel Shafy, A., Taha, A., Abbas, Y., Abdel Latif, M.: Numerical simulation of potential seismic pounding among adjacent buildings in series. Bull. Earthq. Eng. 17, 439–471 (2019) 13. Khatami, S., Naderpour, H., Barros, R., Jankowski, R.: Verification of formulas for periods of adjacent buildings used to assess minimum separation gap preventing structural pounding during earthquakes. Adv. Civil Eng. 2019, 1–8 (2019) 14. Favvata, M.: Minimum required separation gap for adjacent RC frames with potential interstory seismic pounding. Eng. Struct. 152, 643–659 (2017) 15. Anagnostopoulos, S., Spiliopoulos, K.: An investigation of earthquake induced pounding between adjacent buildings. Earthq. Eng. Struct. Dyn. 21, 289–302 (1992) 16. Anagnostopoulos, S., Karamaneas, C.: Use of collision shear walls to minimize seismic separation and to protect adjacent buildings from collapse due to earthquake-induced pounding. Earthq. Eng. Struct. Dyn. 37, 1371–1388 (2008) 17. Polycarpou, P., Komodromos, P., Polycarpou, A.: A nonlinear impact model for simulating the use of rubber shock absorbers for mitigating the effects of structural pounding during. Earthq. Eng. Struct. Dyn. 42, 81–100 (2013)

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18. Khatami, S., Naderpour, H., Razavi, S., Barros, R., Jakubczyk-Gałczy´nska, A., Jankowski, R.: Study on methods to control interstory deflections. Geosciences 10(2), 75 (2020) 19. Westermo, B.: The dynamics of interstructural connection to prevent pounding. Earthq. Eng. Struct. Dyn. 18, 687–699 (1989) 20. Kobori, T., Yamada, T., Takenaka, Y., Maeda, Y., Nishimura, I.: Effect of dynamic tuned connector on reduction of seismic response -application to adjacent office buildings-. In: Proceedings of 9th World Conference on Earthquake Engineering, Tokyo-Kyoto, Japan, 2–9 August, vol. 5, pp. 773–778 (1988) 21. Jankowski, R., Mahmoud, S.: Linking of adjacent three-storey buildings for mitigation of structural pounding during earthquakes. Bull. Earthq. Eng. 14, 3075–3097 (2016) 22. Licari, M., Sorace, S., Terenzi, G.: Nonlinear modeling and mitigation of seismic pounding between R/C frame buildings. J. Earthq. Eng. 19, 431–460 (2015) 23. Roshan, A., Taleshian, H., Eliasi, A.: Seismic pounding mitigation by using viscous and viscoelastic dampers. J. Fundam. Appl. Sci. 9(7S), 377–390 (2017) 24. Roy, G., Das, P.: Noble methods to prevent pounding between adjacent buildings. J. Mech. Continua Math. Sci. 13(4), 134–146 (2018) 25. Abdullah, M., Hanif, J., Richardson, A., Sobanjo, J.: Use of a shared tuned mass damper (STMD) to reduce vibration and pounding in adjacent structures. Earthq. Eng. Struct. Dyn. 30, 1185–1201 (2001) 26. Bekdas, G., Nigdeli, S.: Preventing the pounding of adjacent buildings with harmony search optimized tuned mass damper. In: Recent Advances in Engineering: Proceedings of the 3rd European Conference of Chemical Engineering, France, pp. 283–288. WSEAS Press (2012) 27. Mate, N., Bakre, S., Jaiswal, O.: Seismic pounding response of singled-degree-of-freedom elastic and inelastic structures using passive tuned mass damper. Int. J. Civil Eng. 15, 991–1005 (2017) 28. Ismail, M., Rodellar, J., Ikhouane, F.: A seismic isolation system for supported objects. Spanish Office of Patents and Marks Patente P200802043 (2008) 29. Ismail, M.: Elimination of torsion and pounding of isolated asymmetric structures under near-fault ground motions. Struc. Contr. Health Monit. 22, 1295–1324 (2015) 30. Braz-César, M.: Vibration control of building structures using MagnetoRheological dampers. Ph.D. thesis, FEUP, Porto, Portugal (2015) 31. Abdeddaim, M., Ounis, A., Djedoui, N., Shrimali, M.: Pounding hazard mitigation between adjacent planar buildings using coupling strategy. J. Civil Struct. Health Monit. 6, 603–617 (2016) 32. Abdeddaim, M., Ounis, A., Djedoui, N., Shrimali, M.: Reduction of pounding between buildings using fuzzy controller. As. J. Civil Eng. (BHRC) 17(7), 985–1005 (2016) 33. Abdeddaim, M., Ounis, A., Shrimali, M.: Pounding hazard reduction using a coupling strategy for adjacent buildings. In: 16th World Conference on Earthquake, 9–13 January 2017, Chile (2017) 34. Kim, H.-S.: Seismic response control of adjacent buildings coupled by semi-active shared TMD. Int. J. Steel Struct. 16(2), 647–656 (2016) 35. Song, G., Li, L., Singla, M., Mo, Y.-L.: Pounding tuned mass damper with viscoelastic material. US Patent 9,500,247 B2 (2016) 36. Song, G., Zhang, P., Li, L., Singla, M., Patil, D., Li, H., Mo, Y.: Vibration control of a pipeline structure using pounding tuned mass damper. J. Eng. Mech. 142(6), 1–10 (2016) 37. Xue, Q., Zhang, J., He, J., Zhang, C.: Control performance and robustness of pounding tuned mass damper for vibration reduction in SDOF structure. Shock Vib. 2016, 1–15 (2016) 38. Nishath, P., Abhilash, P.: Mid-column pounding effects on adjacent tall buildings and its mitigation using viscous dampers and friction dampers. Int. J. Sci. Eng. Res. 8(11), 168–173 (2017)

Prototyping of a Low-Cost Stroboscope to Be Applied in Condition Maintenance: An Open Hardware and Software Approach Laiany Brancali˜ ao1(B) , Caio Camargo1 , Jos´e Gon¸calves2 , and Jos´e Lima2 1

Polytechnic Institute of Bragan¸ca, Bragan¸ca, Portugal [email protected], [email protected] 2 Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Polit´ecnico de Bragan¸ca, Campus de Santa Apol´ onia, 5300-253 Bragan¸ca, Portugal {goncalves,jllima}@ipb.pt https://portal3.ipb.pt/index.php/pt/ipb

Abstract. This paper aims to develop a low-cost stroboscope, which consists of an optical equipment capable of generating flashes of light at different frequencies, allowing to measure the rotation velocity of machines and contributing to maintenance processes in the industry. This device is based on the stroboscopic effect, a visual event that occurs when a continuous movement is presented by a series of samples, generated by flashes of light. When the frequency of the rotation movement is the same frequency of light pulses, the process will appear stationary. Based on the high cost of the commercial stroboscopes, it was developed a stroboscope prototype based on the Arduino platform, LED technology and 3D printing with an open hardware and software. The final prototype went through calibration and validation processes, achieving a performance very similar to a commercial instrument. Keywords: Stroboscope

1

· Maintenance · Prototyping · Open source

Introduction

In the industry is very common the applying of resources and machines for implementation of the processes and creation of the products. For this reason, the measurement and control of such processes are essential to achieve better performance, efficiency, and consequently best quality, as well as, monitor the operation of the equipment, identify failures and ensure the security for the industry and the employees [1]. The measure instruments are the set of tools used with the objective of obtaining data about the particular processes. In general, these devices measure features like pressure, temperature, velocity, humidity, vibration, and are widely applied in machines such as motors, heaters, reactors, refrigerators, air conditioners, compressors, ovens, and other equipment. It is important to obtain the c The Editor(s) (if applicable) and The Author(s), under exclusive license  to Springer Nature Switzerland AG 2021 J. A. Gon¸ calves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 762–772, 2021. https://doi.org/10.1007/978-3-030-58653-9_73

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periodic calibration of these instruments so that the results obtained through them be reliable and accurate [1]. The inspection and evaluation of equipment that work through continuous or periodic movements need to be done while it is in operation. Some devices such as automobiles, motors, propellers, textile looms, electric razors, blenders, and blowers [2], when they exhibit some defect, the best way to identify the problem is with the device turn on. However, this task becomes difficult when the movement is in high frequency. The solution for this problem is the use of a stroboscope, an optical physical device that consists of the emission of blinking light in different frequencies desired, allowing the study of the velocity of an object and visualizing details of the movement that the human eyes fail to see [3]. Besides that, there is the possibility to do measurements without mechanical contact with the moving object, representing an advantage regarding traditional measurement methods [4]. Section 1.1 describes the stroboscopic effect, on which the device is based, the Sect. 1.2 presents the analysis about the commercial stroboscopes available on the market, while the Sect. 1.3 is intended to the advantages of prototyping in 3D printing. The prototype description is presented in Sect. 2, including firmware and hardware. The stroboscope calibration is shown in Sect. 3, followed by the validation and results discuss in Sect. 4. Lastly, the conclusions and future work are presented in Sect. 5. 1.1

The Stroboscope and the Stroboscopic Effect

This instrument is based on the stroboscopic effect, a visual phenomenon that happens when a blinking light source glows an object in movement, generating a sequence of samples. Depending on the blink light frequency, the process can seem forward or backward, so it is important to coincide the frequencies to see the stationary image, which enables the study and the maintenance of moving parts with high frequency [5]. A periodic movement repeats at equal time intervals, that is, it returns to the same position after complete a specific period. If a pulse of light illuminates a periodically moving object always when it is in a single position, the object appears stopped and the movement frozen [6]. Figure 1 represents an illustrative example of the stroboscopic effect through the use of a fan with a mark put on one of its propellers and a blink light source illuminating it. If the illumination frequency f1 is bigger than the frequency of movement f2, the body will appear to move backward, as presented in Fig. 1 a), if f2 is bigger than f1, the body will appear to move forward, as in Fig. 1 b), and if the frequencies coincide, the process will appear to be stationary, as in Fig. 1 c). The stationary visualization of a rotate movement can also be observed if the flashes velocity are multiples or submultiples of the rotation velocity. If the flashing rate is twice the fundamental velocity of the machine, or also called second harmonic, two marks will be seen stopped in a distance of 180◦ . If the flashing

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Fig. 1. Stroboscopic effect

rate is three times the fundamental velocity of the machine, third harmonic, three marks will be seen stopped in a distance of 120◦ and so on [2]. When the light flashes velocity is half the rotation velocity, it will also seen one mark stopped, because the machine completes two rotations every flash. For this reason, to avoid errors measurements, it is recommended to start the velocity measurements at a high flash rate or the instrument’s maximum flash rate, and then slow down until only one mark is seen [2]. For the visualization of the phenomenon, the light pulse length must be short enough compared with the periodic movement which intends to analyze, otherwise, image freezing may not be detected [6]. According to [6], the duty cycle must be below 10% of the total period [7], for the better visualization of frozen motion by the human eyes without blur. Therefore, this exhibition of slow-motion provided by stroboscopic effect is useful to determine the angular velocity and analyze irregularities in machines with high velocity, because many times the problems can’t be seen by the human eyes or the contact with the machine is very difficult and dangerous [2]. It’s important to highlight the care that should be taken with this phenomenon, because depending on the situation, some moving parts can be seen as being stopped, of an undesirable way, causing accidents [7]. 1.2

Analysis of Existing Commercial Solutions

Performing research about the stroboscopes commercially available nowadays, it could be found diverse industries that manufacture stroboscopes of many types. Basically, there are the stroboscope hand-held, the more compact called pocketstroboscope, the fixed and the high intensity ones, which have more than 1000 LEDs. The velocities ranges from 30 RPM to 500000 RPM, the light intensities varies from 3000 to 27000 lx, and the prices range from e200 to e3000, depending on the model [8–10].

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The modern electronic stroboscopes includes a light source and a voltage pulse generator [11], responsible for controlling the frequency of light flashes. The frequencies can be adjusted by the user through buttons until the stroboscope effect is observed, that is, until the object appears stopped. The current stroboscopes also include a display, where the information is shown, usually this information is the angular velocity in RPM. Currently, most of the stroboscopes are digital and are manufactured using LED technology, but one still can be found the analog models. There are also found stroboscopes with xenon lamps, with built-in tachometers and those with high-intensity. In some device catalogs [9,10] there is a recommended distance to the object of 20–50 cm, the most stroboscopes found are powered by AA, Li-Ion or rechargeable batteries and all of them have a cold color temperature, around 6000K [8]. 1.3

Prototyping in 3D Printing

According to [13] 3D printing can be considered a technological revolution, it has gained the attention and interest of industries and research laboratories because this tool enables infinite possibilities [14] to design objects quickly, easily and creating physical prototypes with accuracy and precision [15]. 3D printing is a technology applied to create three-dimensional objects through the successive deposition of layers of material and this is controlled digitally. This process is also called additive manufacturing, because objects are created adding, rather than removing material, like happened in subtractive manufacturing. Besides that, this kind of process reduces energy costs and material waste [14]. Nowadays, there are many kinds of materials that can be used for 3D printing, but the most common is plastic, being ABS, PLA, and Nylon. However, metal, ceramics, wood particles, and even chocolate have been used for it. The popularization and the constant cost savings of 3D printers has allowed the fabrication of several kinds of objects, such as, prototypes, tools, molds, prosthetics, toys, and among others [13].

2

Prototype Description

For the stroboscope prototyping, it was developed an electronics part, which included hardware and firmware, and a mechanical design. Figure 2 presents the systems diagram and the electronics components. The LCD Keypad Shield performed as the interface, its buttons are the inputs for the Arduino Uno, which is responsible for processing all the information and generating two outputs. One of them is the velocity presentation on the display and the other is a digital modulated signal, which is sent to the driver circuit that switches the LEDs. The power supply for the stroboscope is a power bank, which one of its outputs power the Arduino board and the other power a boost converter that increases the voltage to power the driver circuit.

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Power source Power source InformaƟon presentaƟon

BuƩons data

Actuator Power source

Processor

Digital modulated signal

Driver circuit

LEDs Switching

Fig. 2. Prototype components

The 3D model of the stroboscope design, presented in Fig. 3, was drawn using the SolidWorks, a 3D CAD software. The structure was elaborated to be as compact as possible and considering all the electronics components that would be assembled inside it.

Fig. 3. Prototype 3D model a) front view b) back view c) bottom view d) top view

Figure 3 a) represents the front view of the stroboscope, intended to fix the LCD display and the buttons. Figure 3 b) shows the back view, where the LEDs are exposed. Figure 3 c) corresponds to a bottom view, there is an extension where the power bank is dovetail with the stroboscope structure. A top view is

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presented in Fig. 3 d), without the lid, where it is possible to view the space for all the components. 2.1

Firmware

The firmware was developed using the Arduino Uno platform and based on three main tasks, buttons reading, angular velocity presentation on the display and generation of a digital modulated signal, as presented in Fig. 4. This signal must have a variable period and a fixed duty cycle, which represents the period that the signal keeps active. The duty cycle must be below 10% of the total period of the signal for the stroboscopic effect visualization without drag or blur [6,7].

Fig. 4. Digital modulated signal

According to the frequency increase, the period tends to decrease and therefore, the value set for the duty cycle can become greater than 10% of the signal period. This way, when the duty cycle is close to overcoming 10% of the signal period, it is changed to a value below of that percentage. Therefore, the duty cycle is constant but not for all velocity values because it is necessary an adjustable duty cycle. According to the frequency increase, the digital modulated signal generated by Arduino began to present a relative error significant. For this reason, it was established a velocity limit for the stroboscope prototype that ranges from 60 RPM to 100000 RPM with a duty cycle period in microseconds, which is adjusted according to high frequency values. 2.2

Hardware

For the hardware development, it was created a driver circuit, which included ten LEDs SMD5730 to act as the stroboscopic light, and a MOSFET responsible for switching the LEDs at different frequencies. This electronic component has 3 operation regions, cut, triode, and saturation, to perform as a switch it must

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work within the cut and triode regions [16]. This is possible sending the digital modulated signal from Arduino to the MOSFET gate. The system was tested on the bench and then a PCB was created, which all the components were attached. The last step consisted to print the 3D model of the stroboscope prototype using a 3D printer. The parts went through the sanding and painting process in order to obtain the final result. Figure 5 a) presented the front view, where the LCD display and the buttons are attached. The back view can be seen in Fig. 5 b), where the LEDs are exposed behind a transparent plastic.

Fig. 5. Final prototype a) front view b) back view

The firmware developed in the Arduino IDE, driver circuit schematic and 3D model drawing created to develop the stroboscope prototype were available in a repository [17]. The final stroboscope prototype represents a lower cost compared to commercial stroboscopes presented in Sect. 1.2, due to the use of low-cost electronic components and tools, such as Arduino and prototyping in 3D printing, which has become very popular due to cost and material savings, as presented in Sect. 1.3.

3

Calibration Process

The calibration of the stroboscopic light was based on an LDR sensor, a luminosity sensor that varies its resistance according to the intensity of light focused on it [18]. It was used a circuit composed of the sensor in series with a 10 kΩ resistor, powered by a 5V DC source, and an oscilloscope connected over the resistor. The stroboscopic light was positioned in front of the sensor and then, every light flash, the LDR resistance decrease, and the resistor voltage increase. Allowing to visualize the voltage pulses over the resistor, referring to the light flashes.

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Then, it was possible to measure the period between each flash and compare it with the digital modulated signal period. In Fig. 6, the yellow signal represents the digital modulated signal sent by Arduino at 1000 RPM and the blue signal represents the light pulses captured by the sensor. The periods of both signals coincided with each other, confirming that the LED response was consistent with the signal sent by the microcontroller. It is possible to observe that the voltage level of the blue signal didn’t return to zero, because it wasn’t possible to realize these measurements in a place totally dark, so the LDR always detected a little of luminosity. At high frequencies, above 10000 RPM, the calibration could not be done, because the LDR response isn’t instantaneous, there is a latency time until the sensor answers by the transition from dark to light and vice-versa. Therefore, according to the frequency increase, the period decreases, and the transition is no longer clear, preventing the measurement of the period between each flash.

Fig. 6. Calibration - comparison between digital modulated signal and light pulses response.

4

Validation of the Prototype and Results Discussion

This chapter presents the validation tests to verify if the stroboscope prototype was able to measure the velocity of a rotating machine by visualizing the stroboscopic effect. The velocity of a three-phase motor, controlled in closed-loop and with a constant velocity, was measured using the stroboscope and a commercial tachometer. The measured results of both devices were compared in order to realize the validation of the stroboscope. The results comparison between the stroboscope and the tachometer can be analyzed in Fig. 7, which shows the graphic of the devices’ performance. The motor was subjected to different voltages and the graphic behavior shows the velocity increase according to the voltage increase and its stabilization around

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1500 RPM. Through this graphic, it is possible to notice that the results of both instruments remained aligned and close, with an error practically insignificant. These results are similar to those obtained in [12], in which the performance comparison between a low-cost tachometer and a commercial one is presented.

Fig. 7. Graphic of the performance comparison between stroboscope and tachometer.

The velocities measured by the stroboscope were obtained visually through the stroboscopic effect, that is, at the moment that the machine appears stopped. A white mark was done in the three-phase motor for the visualization of the phenomenon, which could be visualized successfully and without drag or blur. It is important to start the measurement process with the stroboscope in high frequencies or the maximum frequency of the device to have the assurance to find the right velocity. Figure 8 a) was captured at the moment that the motor seemed to be stopped, that is, the moment that the rotation velocity of the motor coincides with the light flashes velocity of the stroboscope. In this situation each flash of light illuminates the mark always at the same position, when the motor completes one rotation, causing a frozen movement. Therefore, the velocity value of 1497.5 RPM, shown on the display, is the rotation velocity of the three-phase motor. The stroboscopic effect can also be seen at velocities multiples of the real velocity. In Fig. 8 b) are seen two marks stopped because of the second harmonic, the flashes velocity is the double of the rotation velocity, that is, 2995 RPM. Then, at each rotation, the machine is illuminated by two flashes and as this process is very fast, in microseconds, the human eye does not notice this difference. The same situation occurs in Fig. 8 c), but in this case, it is the fourth harmonic, that is, the quadruple of the blinking velocity of light, 5990 RPM, so 4 marks are seen stopped.

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Fig. 8. Stroboscopic effect visualization a) fundamental frequency b) second harmonic c) fourth harmonic

5

Conclusions and Future Work

This study contributed to the development of a stroboscope prototype, in which its performance is very similar to the commercial devices, and can be applied in condition maintenance. However, with some differences, such as low-cost, ergonomics, and open hardware and software, allowing any person that desires or needs this kind of equipment can develop their own stroboscopes. The stroboscope prototype was able to generate light blinkings at different frequencies, allowing to measure the velocity of a rotate machine, through the stroboscopic effect visualization. Such an effect could be seen successfully and without drag, proving that the period used for the duty cycle of the modulated signal was good. Three-phase motor velocities measured by the prototype proved to be very close to the velocities obtained by the commercial tachometer. The stroboscope behavior towards the user’s action was shown plausible and the stroboscope worked as expected, providing the tools and techniques used to manufacture the prototype were enough. Through the calibration process with an LDR sensor, it was possible to compare the period of each light flash with the period of the digital modulated signal sent to the LEDs and both were consistent. At high velocities, above 10000 RPM, the calibration process could not be done properly due to LDR sensor limitation, an alternative would be to use a faster light sensor, like a photodiode. As a way to improve the stroboscope, it is suggested as future work, the development of a communication interface, allowing to send the velocity value read on the display to a computer for registration or processing. Acknowledgements. This work has been supported by FCT – Funda¸ca ˜o para a Ciˆencia e Tecnologia within the Project Scope: UIDB/05757/2020.

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References 1. A importˆ ancia da Instrumenta¸ca ˜o. https://br.omega.com/prodinfo/instrumenta cao.html 2. Van Veen, F.: Handbook of Stroboscopy. GenRad Inc., Massachusetts (1977) 3. Mayer, V.V., Varaksina, E.I.: A stroboscopic light source for experiments in mechanics. In: Physics Education, vol. 52, pp. 1–7. IOP Publishing (2017). https:// doi.org/10.1088/1361-6552/aa8836 ˇ Lovrin, N., Gregov, G.: Primjene stroboskopa applications of strobo4. Vrcan, Z., scopes. Eng. Rev. 29(1), 95–106 (2009) 5. Lai, H.W., Chow, M.W.K., Ma, C.K., Yan, A.Y.K.: Calibration of the frame rate of high-speed digital video recorders by stationary counting method: application of the stroboscopic effect. NCSLI Measure, 1–7 (2019). https://doi.org/10.1080/ 19315775.2019.1572480 6. Kassamakov, I., Hanhij¨ arvi, K., Aaltonen, J., Sainiemi, L., Grigoras, K., Franssila, S., Hæggstr¨ om, E.: Stroboscopic white light interferometry for dynamic characterization of capacitive pressure sensors. J. Acoust. Soc. Am. 2523–2527. Proceedings European Conference on Noise Control (2002). https://doi.org/10.1121/1.2933663 7. Guedes, M.V.: Laborat´ orio de M´ aquinas El´etricas - Estroboscopia. Faculdade de Engenharia da Universidade do Porto, Porto (2002) 8. Direct Industry - O sal˜ ao online da ind´ ustria. https://www.directindustry.com/pt/ fabricante-industrial/estroboscopio-66061.html 9. Schmidt Control Instruments - Stroboscopes. https://www.hans-schmidt.com/en/ produkte/stroboscopes/ 10. MonarchInstrument - Stroboscopes. https://monarchinstrument.com/collections/ stroboscopes 11. Mayer, V.V., Varaksina, E.I.: A LED stroboscope with computer control of light flashes. Phys. Educ. 53, 1–6 (2018) 12. Bakibillah, A.S.M., Uddin, M.A., Haque, S.A.: Design, implementation and performance analysis of a low-cost optical tachometer. In: IIUC Stud. 7, 107–116. (2011) 13. Rayna, T., Striukova, L.: From rapid prototyping to home fabrication: how 3D printing is changing business model innovation. In: Technological Forecasting and Social Change, vol. 102, pp. 214–224. Elsevier B.V. (2016). https://doi.org/10. 1016/j.techfore.2015.07.023 14. Ambrosi, A., Pumera, M.: 3D-printing technologies for electrochemical applications. Chem. Soc. Rev. 45, 2740–2755 (2016). https://doi.org/10.1039/c5cs00714c 15. Gross, B.C., Erkal, J.L., Lockwood, S.Y., Chen, C., Spence, D.M.: Evaluation of 3D printing and its potential impact on biotechnology and the chemical sciences. Anal. Chem. 86, 3240–3253 (2014). https://doi.org/10.1021/ac403397r 16. Sedra, A.S., Smith, K.C.: Microeletrˆ onica. Oxford University Press, Pearson (2007) 17. Repository - stroboscope files. https://github.com/Laiany/stroboscopeLaiany 18. Sunrom Technologies: Light Dependent Resistor - LDR, July 2008

Dynamic Survey of a Telecommunication Tower by Interferometric Radar Technique Fábio Paiva1 , Rui Barros2(B) , Jorge Henriques1 , Tiago Cunha3 , and Pierre Feyfant4 1 Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

[email protected] 2 Faculty of Engineering, University of Porto, Porto, Portugal

[email protected] 3 BSSE, Porto, Portugal 4 ITAS Pylônes, Beauzelle, France

Abstract. The dynamic behavior (in terms of resonant frequencies and modal damping ratios) of a Telecommunication Tower located in France is studied using an Interferometric Radar (IR). The tower under investigation suffers from a dynamic instability phenomenon, designated as vortex shedding that induce large lateral vibration on the tower. As a possible solution for vortex shedding phenomena a Tuned Liquid Damper (TLD) device was developed and installed, with the purpose of mitigating the vibrations by adding damping into the system. From the investigation conducted and with the help of modal identification techniques in the frequency and time domain, some conclusions can be drawn in terms of modal estimation parameters for the Telecommunication Tower before and after TLD implementation. In both the TLD stages it was possible to identify the frequencies for the first two global modes of the tower, with a positive degree of certainty. The same statement cannot be asserted for the modal damping ratio. The time domain technique (SSI-UPC method), applied in this work to estimate the damping ratio, has displayed a high variability. Factors like duration of the measurements, number of output sensors considered, spectral range and others related with specific parameters of the method, affected significantly the values achieved. The weak ambient excitation during the dynamic survey also limited the reliability of the modal damping ratio estimates. The TLD implementation affected the structure in two ways: by reducing the first frequency from 0.960 Hz to 0.823 Hz; and through the increased energy dissipation in the system, where a more evident effect in the structure first mode was also detected. Keywords: Structural health monitoring · Interferometric Radar · Telecommunication Tower · Dynamic survey

1 Introduction The present work intends to characterize the dynamic behavior of a Telecommunication Tower located in northern France, in terms of modal frequencies and damping ratios. The © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 773–782, 2021. https://doi.org/10.1007/978-3-030-58653-9_74

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tower under investigation suffers from a dynamic instability phenomenon, designated as vortex shedding, that induce large lateral vibration of the tower [1, 2]. As a possible solution for vortex shedding phenomena, a Tuned Liquid Damper (TLD) device was developed and installed, with the purpose of mitigating the lateral vibrations by adding damping to the system. The tower before the TLD installation has a height of 30.0 m and after the TLD installation has an increment of an additional 1.0 m. A dynamic survey was conducted with the help of an Interferometric Radar (IR) [3–5] (IBIS-FS). The survey was performed in two stages, the first one “before” the complete installation of Tuned Liquid Damper (TLD) and the second one “after” the TLD installation. The data obtained with IR equipment will be analyzed with the help of two software. The Data-Viewer software for selecting the best “bins” (reflecting points), that work as virtual displacement sensors on the tower, and ARTeMIS Modal software [6] for identifying the modal natural frequencies and modal damping ratio factors of critical damping (mainly for the two first global structural tower modes) from the measured displacement time histories. Therefore in the ARTeMIS Modal, Operational Modal Analysis methods [7–9] in the frequency and time domain will be used to estimate the dynamic behavior of the tower before and after the TLD installation. This work begins with a short description of the main technical characteristics of the IR used in this dynamic investigation. It follows by presenting how the dynamic survey was performed (number of radar positions considered, distance to the tower, radar head tilt rotation and survey duration). Then the main modal parameters (natural frequency and critical damping factor) estimated from the surveys before and after the TLD installation are presented and discussed. It finishes with some conclusions about the obtained results and the effect of the TLD; the degree of uncertainty (variability) found in the estimated parameters (specially the modal damping factor) is also commented. 1.1 Technical Characteristics of the Interferometric Radar IBIS-FS IBIS-FS (Image by Interferometric survey) microwave interferometry based system [10] is shown in Fig. 1. IBIS-FS system. The system can measure remotely the displacement of several points along the structure to be monitored in the Line of Sight (LOS), with the help of two radar techniques: interferometry and frequency modulated-continuous wave [11]. The IBIS-FS system performs day/night in all weather conditions and as an autonomy of 8 h with batteries (or indefinitely if a power source exists). The IBIS-FS system is particularly suitable for use in all applications where it is important to measure the displacements of a structure without having access to it, or when short investigations must be performed quickly (installation time of 10–15 min). The main IBIS-FS system specifications [10] are stated in Table 1. By remotely performing the measurements without the need to access the structure being monitored, IBIS-FS permits measurements to be performed even when: • The area of interest is not accessible; • The structure is very tall, such as an antenna or a tower; • The structure is covered with frescoes or sculptures and contact measurements are not feasible.

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Fig. 1. IBIS-FS system

In addition, in emergency situations when monitoring activities are required to guarantee safety, the possibility of performing remote monitoring may be an essential condition towards saving lives. Table 1. IBIS-FS Interferometric Radar System specifications [10] IBIS – FS system specifications

Technical characteristics

Displacement accuracy

0.01 mm ± 0.1 (depending on range)

Maximum range

Up to 1000 m

Spatial resolution in LOS

0.75 m

Acquisition frequency

Up to 200 Hz

Frequency band (ku)

17.1–17.3 GHZ

2 Radar Dynamic Survey As already mentioned the survey was conducted in two stages: “before” TLD installation; and then “after” TLD installation. A total of 3 different position were considered for the radar placement, as represented in Fig. 2a), with one position serving as a reference for comparison purposes (the position 2, shown in Fig. 2b). The selected positions were picked based on factors such as good visibility to the target tower and stability of the ground surface where the radar support tripod rested. The duration of each individual survey was around 15–20 min. Other survey parameters that were considered constant during this investigation were: the range resolution of 0.75 m and the sampling frequency of 98.7 Hz, respectively.

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Fig. 2. a) Identification of the adopted radar positions around the tower; b) Perspective of radar position 2, during the radar survey measurements

2.1 Survey TLD Installation Parameters Two main survey phases were considered in this study. The main survey parameters that characterize the dynamic data acquisition for “before-B” and “after-A” TLD installation are identified in Table 2. Table 2. Survey parameters for the ‘before-B’ and “after-A” TLD installation stage Survey label

Radar position

Distance to the tower (m)

Radar vertical tilt (º)

Radar height to the ground (m)

20/11/19-1 B

2

25.0

45

1.20

20/11/19-2 B

2

25.0

45

1.20

21/11/19-1 B

1

5.1

45

1.20

21/11/19-2 B

1

5.1

70

1.20

21/11/19-1 A

2

25.0

45

1.40

21/11/19-2 A

2

25.0

45

1.40

21/11/19-3 A

3

18.0

30

1.40

21/11/19-4 A

3

18.0

45

1.40

3 Main Results of the Dynamic Survey In general, for all the surveys conducted it was possible to identify 2–4 good reflector points depending on the particular survey. These “reflective points” were then processed and corrected within ARTeMIS Modal software with the purpose of estimating the resonant frequencies and damping ratios (modal properties) of the first two structural modes. For this particular structure, two methods in the frequency domain and one in the time domain were used. The Enhanced Frequency Domain Decomposition (EFDD) method and the Curve-fit Frequency Domain Decomposition (CFDD) method, were the

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selected methods for the frequency domain; and the Stochastic Subspace Identification Unweighted Principal Components (SSI-UPC) method, for the time domain. For more details about these methods, help guide [6] and references [7, 8] provide sufficient background. In terms of signal processing, three different setups (applied to all the surveys) were explored with the purpose of estimating the tower modal properties at different frequency ranges and with variable spectral density resolutions: • Setup A: Frequency range 0–9.870 Hz, Spectral density resolution variable (depends of the survey); • Setup B: Frequency range 0–4.935 Hz, Spectral density resolution variable (depends of the survey); • Setup C: Frequency range 0–2.468 Hz, Spectral density resolution variable (depends of the survey). In addition, the de-trending option (that removes mean and linear trend from the survey measurements) was enabled in all the different setups, and 100 maximum state space dimension was considered for SSI-UPC. For some selected surveys (identified in Sect. 4) a graphical representation of the singular values of spectral densities and the stabilization diagram are presented for the stages ‘before’ and ‘after’ TLD installation. 3.1 Before TLD Installation For the mentioned surveys a summary of the results in table format is presented. The results focus on the estimation of the modal parameters (natural frequency and damping ratio) and Table 3 contain one example of that information. Survey 20/11/19-1 B Table 3. Summary of the modal estimation results for the survey 20/11/19-1 B Setup

Method

Mode

Frequency (Hz)

A

EFDD

1

0.959

0.004

1

1.018

0.007

CFDD

1

0.959

0.004

1

1.018

SSI-UPC

1

0.957

0.852

0.014

1

1.019

0.414

0.005

B

EFDD

Damping (%)

Complexity (%)

0.007

2

2.428

15.167

1

0.959

0.004

1

1.018

0.007 (continued)

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Setup

C

Method

Mode

Frequency (Hz)

CFDD

1

0.959

1

1.018

SSI-UPC

1

0.957

0.477

0.002

1

1.019

0.189

0.003

1

0.959

0.004

1

1.018

0.007

CFDD

1

0.959

0.004

1

1.018

SSI-UPC

1

0.958

0.467

0.002

1

1.018

0.248

0.001

EFDD

Damping (%)

Complexity (%) 0.004 0.007

0.007

In Fig. 3 the setup B was chosen for the graphical representation of the Singular Values of Spectral Density (SVSD), for the survey under analysis.

Fig. 3. Singular values of spectral densities for survey 20/11/19-1 B (setup B)

From the results of all surveys (only 20/11/19-1 B shown here), it is possible to identify two frequency ranges that are related with the first mode shape, with values of 0.960–0.962 Hz and 1.018–1.019 Hz respectively. It is also possible to detect other frequency range at 2.428–2.509 Hz, in this case related with the second mode shape. The damping variability found in these results was quite expressive and showed a strong dependency on the survey under analysis, on the estimation method and also on how the data was prepared (signal processing options).

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In this work, only the SSI method (time-domain) was considered, given its superior reliability for estimating modal damping parameters, in particular in cases where the length of the time series is reduced (operational constraints). For the first natural frequency range of 0.960–0.962 Hz, the first mode critical damping factor using the SSI-UPC method was estimated with values in the range 0.207– 0.852% (with a strong tendency around 0.2–0.3%). For the first natural frequency range of 1.018–1.019 Hz, the first mode critical damping factor using the SSI-UPC method was estimated with values in the range 0.189–0.414%. For the second natural frequency range of 2.428–2.509 Hz (of the second mode), given the low reliability found in the results no value is assumed for the critical damping factor from the data obtained. 3.2 After TLD Installation Table 4 provides an example of the modal parameter’s estimation for the stage where the TLD was already implemented in the telecommunication tower. Survey 21/11/19-1 A Table 4. Summary of the modal estimation results for the survey 21/11/19-2 A Setup

Method

Mode

Frequency (Hz)

A

EFDD

1

0.826

0.002

1

1.063

0.002

2

2.483

0.423

Complexity (%)

2

2.945

0.11

1

0.826

0.002

1

1.061

0.002

2

2.482

0.423

2

2.943

0.11

SSI-UPC

1

0.842

2.541

1

1.064

0.756

EFDD

1

0.834

0.001

1

1.063

0.011

2

2.482

1.0104

CFDD

B

Damping (%)

CFDD

0.008 0.011

2

2.943

0.064

1

0.835

0.001

1

1.061

0.011

2

2.481

1.104 (continued)

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F. Paiva et al. Table 4. (continued)

Setup

Method

Mode

Frequency (Hz)

2

2.946

1

0.844

1.126

0.007

1

1.06

0.623

0.006

EFDD

1

0.838

1

1.065

0.002

CFDD

1

0.835

0.002

1

1.061

0.002

1

0.84

1.904

0.008

1

1.053

0.666

0.002

SSI-UPC C

SSI-UPC

Damping (%)

Complexity (%) 0.064

0.000

Fig. 4. Singular values of spectral densities survey 21/11/19-1 A (setup B)

In Fig. 4 the setup B was again chosen for the graphical representation of the SVSD, for the survey under analysis. Taking into account all the surveys performed for this after stage (the paper only shows in detail survey 21/11/19-1- A) some observations are made. In the first place, two frequency ranges were identified for the first modal frequency corresponding to the values of 0.823–0.860 Hz and 1.052–1.064 Hz, respectively. For this first mode, it is also possible to visualize double peak pattern in the spectral densities graphics. For the second mode, also two frequency ranges were identified for the second modal frequency corresponding to the values of 2.481–2.486 Hz and 2.943–2.955 Hz, respectively.

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For the first natural frequency range of 0.823–0.860 Hz, the first mode critical damping factor using the SSI-UPC method was estimated with values in the range 0.307– 2.541%. For the first natural frequency range of 1.052–1.064 Hz, the first mode critical damping factor using the SSI-UPC method was estimated with values in the range 0.37– 1.63%. In the case of second mode with frequency around 2.48 Hz the SSI-UPC method did not provide any reliable damping ratio. In general, the last two surveys (Survey 21/11/19-3 A and Survey 21/11/19-4 A) did provide low damping ratio estimates; this could be due to the few measurements points (radar virtual sensors) considered for data post-processing and the weak ambient excitation level present during that particular survey of this structural health monitoring.

4 Conclusions From the dynamic surveys and investigations conducted in this work of structural health monitoring, for the mitigation of lateral vibrations of a Telecommunication Tower due to vortex shedding, some conclusions can be drawn in terms of the estimation of modal parameters for the ‘before’ and ‘after’ TLD implementation in the tower. In both the TLD stages of the tower, it was possible to identify the frequencies for the first two global modes of the tower, with a high degree of certainty. The same statement cannot be asserted for the damping ratio. The SSI-UPC method applied in this work to estimate the damping ratio has exhibited a high variability from the results obtained; it was confirmed that some factors – like: survey measurements duration, total number of output sensors in the analysis, spectral range, and particular parameters of the (frequency domain, time domain) method used – affected significantly the values achieved. The weak ambient excitation during this dynamic survey, also limited the reliability of the modal damping ratio estimates. The TLD implementation affected the structure in two ways: by reducing the first frequency from 0.960 Hz to 0.823 Hz; and through the increased energy dissipation in the system, where a more evident effect in the structure first mode is detected. Acknowledgments. This work was financially supported by Base Funding UIDB/04708/2020 of the CONSTRUCT R&D Institute on Structures and Constructions (Instituto de I&D em Estruturas e Construções), through national funds of FCT/MCTES (PIDDAC). The first two authors also acknowledge the possibility granted by BSSE (Best Structural Solutions Engineering) – Porto – Portugal and ITAS Pylônes – France, to publish this research work based on their international external service provided in November 2019.

References 1. Vieira, D., Barros, R.C.: Tubular steel lattice telecommunication tower subjected to wind loading and vortex shedding. In: COMPDYN 2017 - Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes/Greece, 15–17 June 2017 2. Freitas, S., Barros, R.C., Paiva, F.: Telecommunication tubular towers with helicoid straps: design considerations vs CFD modeling. In: Proceedings IRF 2016: 5th International Conference Integrity-Reliability-Failure, Porto/Portugal, 24–28 July 2016

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3. Barros, R.C., Paiva, F.: On the use of radar interferometry for the structural monitoring of bridges. In: Proceedings IRF 2018: 6th International Conference Integrity-Reliability-Failure, Lisbon/Portugal, 22–26 July 2018 4. Silva, L.: Monitorização de Estruturas com recurso a Radar Interferométrico. Master of Science Thesis, Faculdade de Engenharia da Universidade do Porto (2015). (in Portuguese) 5. Silva, L., Barros, R.C., Paiva, F., Henriques, J.: Structural monitoring of a telecommunication mast by radar interferometry. In: Proceedings IRF 2016: 5th International Conference Integrity-Reliability-Failure, Porto/Portugal, 24–28 July 2016 6. SVS. ARTeMIS Modal Pro. http://www.svibs.com 7. Rainieri, C., Fabbrocino, G.: Operational Modal Analysis of Civil Engineering Structures – An Introduction and Guide for Applications. Springer, New York (2014) 8. Brincker, R., Ventura, C.E.: Introduction to Operational Modal Analysis. Wiley, New York (2015) 9. Au, S.K.: Operational Modal Analysis - Modeling, Bayesian Inference, Uncertainty Laws. Springer, Singapore (2017) 10. IDS Georadar homepage. https://idsgeoradar.com/products/interferometric-radar/ibis-fs. Accessed 11 Oct 2019 11. Gentile, C., Bernardini, G.: Application of radar technology to deflection measurement and dynamic testing of bridges. In: IOMAC 2009 – 3rd International Operational Modal Analysis (2009)

Motion-Based Design of Semi-active Tuned Mass Dampers to Control Pedestrian-Induced Vibrations in Footbridges Under Uncertainty Conditions Javier Fernando Jiménez-Alonso1(B) , José Manuel Soria Herrera1 , Carlos Martín de la Concha Renedo1 , and Francisco Guillen-González2 1 Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos,

Universidad Politécnica de Madrid, 28040 Madrid, Spain [email protected] 2 Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain

Abstract. Modern slender footbridges are sensitive to human-induced vibrations together with the uncertainty associated with the variation of the operational and environmental conditions. In order to overcome these limitations, semi-active damping devices have been widely employed due to their adequate balance between their effectiveness and their cost when they are used to control the pedestrian-induced vibrations in footbridges. Different design methods have been proposed to guarantee that the footbridges, controlled by these damping devices, meet the vibration serviceability limit state without compromising their budget. Among these proposals, the motion-based design method has shown a high performance when it has been implemented to design passive damping devices for footbridges. Herein, the motion-based design method under uncertainty conditions has been adapted and further implemented for the robust optimum design of semi-active tuned mass dampers when they are employed to control the pedestrianinduced vibrations in slender footbridges. According to this method, the design problem can be transformed into two sub-problems: (i) a multi-objective optimization sub-problem; and (ii) a reliability analysis sub-problem. Thus, its main objective is to find the parameters of the semi-active damping device which guarantee an adequate comfort level without compromising its cost. In order to take into account the effect of the modification of the structural modal properties associated with the variation of the operational and environmental conditions, the compliance of the design requirements has been formulated via a reliability index. Therefore, a reliability analysis must be performed to assess the probability of failure associated with the abovementioned serviceability limit state. Keywords: Motion-based design · Structural control · Semi-active tuned mass damper · Pedestrian-induced vibrations · Footbridges · Uncertainty conditions

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 783–793, 2021. https://doi.org/10.1007/978-3-030-58653-9_75

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1 Introduction The improvement of the strength of the construction materials together with the current aesthetic requirements of the modern societies have increased the slenderness of footbridges [1]. Accordingly, this higher slenderness has increased the sensitivity of footbridges to two phenomena: (i) the human-structure interaction [2] and (ii) the variability of their modal properties associated with the modification of their operational and environmental conditions [3]. In order to overcome these limitations, external damping devices have been widely installed on these structures to guarantee their robust structural behavior during their overall life cycle [4]. Among these damping devices, semi-active damping systems have shown an adequate balance between their effectiveness and cost when they are implemented under stochastic conditions [5]. Different algorithms and control laws have been proposed to address successfully the design process of these damping devices [6]. Among these proposals, the motion-based design method under a stochastic approach is presented and further implemented herein to design semi-active tuned mass dampers (STMD) when they are employed to control the pedestrian-induced vibrations in slender footbridges under uncertainty conditions [7]. Thus, the main contribution of this study is to adapt the abovementioned method, which has been previously implemented for the design of different passive damping systems [8], for the design of STMDs. According to this method, the design problem may be transformed into two coupled sub-problems: (i) a multiple-objective optimization sub-problem [9]; and (ii) a reliability analysis sub-problem [10]. The objective function of the first sub-problem is defined in terms of two different elements: (i) the parameters of the STMD which are needed to be determined; and (ii) the design requirements of the structure which are needed to be met [2]. As the main objective of this design process is to control the pedestrian-induced vibrations in footbridges, the design requirements may be defined in terms of the comfort level of the footbridge. Thus, these design requirements are met if the maximum accelerations of the structure are lower than an allowable acceleration established by the designer [2]. Additionally, due to the sensitivity of the modal properties of the structure to the variation of the operational and environmental conditions, the abovementioned design requirements have been re-formulated following a probabilistic approach. According to this approach, the design requirements are met if a reliability index, β, which reflects the probability of compliance of the abovementioned vibration serviceability limit state (VSLS), is greater than an allowable reliability index, βlim [10]. Sampling techniques are usually considered to compute this reliability index [10]. Herein a conventional Monte Carlo method has been regarded [10]. Finally, the performance of the proposed algorithm has been validated via the analysis of a numerical case-study [2]. Concretely, the VSLS of a steel footbridge under uncertainty conditions has been met via the installation of a STMD designed according to the proposed algorithm. The paper is organized as follows. In Sect. 2, the mathematical model of the STMDfootbridge interaction system is formulated and its dynamic response under pedestrian action is computed numerically in time domain. Subsequently, in Sect. 3, the motionbased design method under a stochastic approach is described in detail. Later, in Sect. 4, a numerical case-study is presented to assess the performance of the proposed method when it is implemented for the design of a STMD employed to control the dynamic

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response of a steel footbridge under pedestrian action. Finally, some concluding remarks are included in Sect. 5 to finish the paper.

2 Semi-active Tuned Mass Damper-Footbridge Interaction Model Under Pedestrian Load In order to design the semi-active damping device according to the mentioned method, the dynamic response of a STMD-footbridge interaction system under pedestrian action must be assessed. For this purpose, the following steps must be given: (i) the formulation of a STMD-footbridge interaction model; (ii) the definition of the pedestrian load; (iii) the definition of the driving force (a control law); and (iv) the numerical integration of the equations of motion which governs the STMD-footbridge interaction model. For the formulation of the STMD-footbridge interaction model, the following assumptions have been considered herein: (i) the behavior of the structure (the footbridge) is simulated via a single vibration mode (modal coordinates) since it is assumed that only a vibration mode is prone to suffer from pedestrian-induced vibrations [11]; (ii) the STMD is modelled via a single degree of freedom system (physical coordinates); (iii) the pedestrian load is simulated by an equivalent harmonic load; and (iv) the STMD is located at the point with the maximum modal displacement. Figure 1 shows a scheme of the STMD-footbridge interaction model.

̈ Inertial Mass

( ( )

−

) ̈

Footbridge

∗(

) ̇

Fig. 1. STMD-footbridge interaction model.

Thus, the equations of motion of the STMD-footbridge interaction model can be obtained via the implementation of the second Newton’s law to the two masses (STMD and equivalent modal mass). These equations may be expressed as follows: mf x¨ f (t) + cf x˙ f (t) + kf xf (t) = p∗ (t) + fd (t)

(1)

  ma x¨ a (t) + ka xa (t) − xf (t) = fa (t)

(2)

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  fd (t) = ka xa (t) − xf (t) − fa (t) = −ma x¨ a (t)

(3)

where mf [kg], cf [sN/m] and kf [N/m] are respectively the mass, damping and stiffness of the considered vibration mode of the footbridge; p∗ (t) = φ T p(t) [N] is the projection of the pedestrian load on the considered vibration (being p(t) the pedestrian load [N], φ the considered vibration mode and the transpose function); x¨ f (t) [m/s2 ], x˙ f (t) [m/s] and xf (t) [m] are respectively the acceleration, velocity and displacement of the footbridge; ma [kg] and ka [N/m] are respectively the mass and stiffness of the STMD, x¨ a (t) [m/s2 ] and xa (t) [m] are respectively the acceleration and displacement of the STMD; fa (t) [N] is the driving force generated by the semi-active damper and fd (t) [N] is the control force generated by the STMD. These equations can be re-organized in matrix form as follows:          x¨ f (t) xf (t) mf 0 cf 0 x˙ f (t) k + ka −ka + + f x¨ a (t) xa (t) −ka ka 0 ma 0 0 x˙ a (t)     −1 1 ∗ (4) = fa (t) p (t) + 1 0 [M ]{¨x(t)} + [C]{˙x(t)} + [K]{x(t)} = {B0 }p∗ (t) + {Bc }fa (t)

(5)

    c 0 mf 0 is the mass matrix; [C] = f is the damping matrix; where [M ] = 0 ma 0 0   k + ka −ka is the stiffness matrix; {B0 } is the input vector associated with the [K] = f −ka ka pedestrian load; {Bc } is the input vector associated with the driving force; {¨x(t)} is the acceleration vector; {˙x(t)} is the velocity vector and {x(t)} is the displacement vector. In order to integrate this equation system and make easier the implementation of a control law for the determination of the driving force, fa (t), the above-mentioned equation system has been transformed into a state space formulation [11]. According to this formulation, the dynamic behavior of a linear invariant system may be expressed by the following differential equation system with initial conditions ({z(0)} = z0 ):  {˙z (t)} = [A]{z(t)} + [B] p∗ (t) + [B]{fa (t)} (6) {y(t)} = [E]{z(t)} + [D]{fa (t)}

(7)

where {z(t)} is the state vector; {y(t)} is the output vector; [A] is the system matrix; [B] is the input matrix; [E] is the output matrix; and [D] is the feedthrough matrix [11]. It is possible via matrix transformations to convert the abovementioned equation system into its state space formulation [11]. The state space matrices can be defined as:   0 I (8) [A] = −[M ]−1 [K] −[M ]−1 [C]

  {B0 } for p∗ (t) 0 where {Bi } = (9) [B] = −1 −[M ] {Bi } {Bc } for fa (t)

Motion-Based Design of Semi-active Tuned Mass Dampers

 [E] = −[Ea ] [M ]−1 [K] [M ]−1 [C]

787

(10)

(11) [D] = [Ea ][M ]−1 [B]

 where the state vector {z(t)} = xf (t) xa (t) x˙ f (t) x˙ a (t) is defined in terms of the displacements and velocities of both the footbridge and the STMD; and the output vector, {y(t)}, is defined in terms of the accelerations experienced by the footbridge (being [Ea ] the acceleration matrix which indicates the elements in which the acceleration is computed). In order to obtain the response of the above state space equation system both the pedestrian force, p(t), and the driving force, fa (t), which simulates the behavior of the semi-active damper, must be defined. The pedestrian force, p(t), has been simulated according to the recommendations of the French guidelines [2]. Only the vertical contribution of the walking pedestrian action has been considered herein. According to these guidelines, the walking pedestrian action, p(t), can be determined via an equivalent harmonic force defined as follows:   (12) p(t) = 280 · cos 2π · ff · t · neq · ψ where ff [Hz] is the natural frequency of the considered vibration mode of the footbridge; neq [-] is the equivalent number of pedestrians; and ψ is a reduction factor which takes into account the probability that the natural frequency is within the range which characterizes the pedestrian-structure interaction in vertical direction (1.25 ≤ ff ≤ 2.3 Hz) [2]. The driving force, fa (t), is determined via the implementation of a feedback controller to the above mentioned system in the state space [11]. Figure 2 shows the general layout of the feedback controller considered.

Fig. 2. Design of a feedback controller in a state space formulation [11].

The implementation of this feedback controller allows modifying the system equation as follows:  {˙z (t)} = [A]{z(t)} + [B0 ] p∗ (t) + [Bc ]{fa (t)}] (13)

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where [B0 ] and [Bc ] are obtained from Eq. (9) considering as pattern load vectors {B0 } and {Bc } respectively. Thus, according to the feedback controller (Fig. 2), the driving force, fa (t), may be determined in terms of a gain matrix, −[G], and the state vector, {z(t)}. Thus, the system equation may be expressed as:  {˙z (t)} = [A]{z(t)} + [B0 ] p∗ (t) − [Bc ][G]{z(t)} (14)  {˙z (t)} = ([A] − [Bc ][G]){z(t)} + [B0 ] p∗ (t)

(15)

There are several algorithms [11] to determine the value of the gain matrix, −[G]. Among these algorithms, the linear quadratic regular (LQR) method [11] has been considered herein due to its extensive use for practical engineering applications. According to this method, the value of the gain matrix, −[G], is obtained via the minimization of the following performance-index function, J .  ∞   {z(t)}T Q {z(t)} + {[G]{z(t)}}T [R]{[G]{z(t)}} dt (16) J = 0

 where Q and [R] are two weighting matrices which may be computed in terms of the mass, [M ], and stiffness, [K], matrices of the interaction model [11]. According to the proposals of several authors [11], the weighting matrices may be determined as follows:    [K] 0 Q = αd (17) 0 [M ] [R] = βd [I ]

(18)

where αd [-] and βd [-] are the weighting factors. In this manner, once the value of these factors has been fixed, the value of the different element of the gain matrix, −[G], which minimizes the value of the performance-index function, J , is obtained. Additionally, in each time step, the driving force, fa (t), determined by the LQR controller, must be modified by a clipped force algorithm [11]. This algorithm adapts the behavior of a general actuator to the particular limitations of a semi-active damper. Thus, the driving force, fa (t), is bounded between a minimum, fmin [N], and a maximum, fmax [N], force which reflect the extreme values of the constitutive model of a semi-active damper (for instance a magneto-rheological damper). For this purpose, the following relationship has been considered herein:     ⎧ flqr (t) · u˙ r (t) < 0 & flqr(t) >fmax ⎨ sgn flqr (t) · fmax if flqr (t) · u˙ r (t) < 0 & fmin < flqr(t) < fmax (19) fa (t) =  flqr  ⎩   sgn flqr(t) · fmin flqr (t) · u˙ r (t) > 0 & flqr (t) < fmin where u˙ r (t) [m/s] is the relative velocity of the semi-active damper; sgn() is the sign function; and flqr (t) [N] is the driving force computed by the LQR controller. Subsequently, the response of the STMD-footbridge interaction model has been assessed via the integration of the abovementioned state space system using a RungeKutta method, as it is implemented in the Matlab software [12].

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To sum up, the adjustment of the driving force, fa (t), is addressed via the compliance of the following steps: (i) to develop a linear model of the STMD-footbridge interaction model in the state space domain (where the value of the mass and of the STMD  stiffness are fixed); (ii) to adjust the value of the weighting matrices, Q and [R] in terms of the weighting factors, αd and βd ; (iii) to determine the optimum value of the driving force, fa (t), via the minimization of the performance-index function, J , and the clipped force algorithm; (iv) to simulate numerically the response of the linear system; and (v) to modify the values of the weighting factors and repeat the steps (ii) to (iv) until the dynamic response of the footbridge meets the design requirements [2] (established by the designer).

3 Motion-Based Design Under Uncertainty Conditions The abovementioned control algorithm has been hybridized with the motion-based design method under uncertain conditions [8] to improve the performance of the design process of semi-active damping devices. The performance-based design method [13] transforms the design problem into two coupled sub-problems: (i) a multi-objective optimization sub-problem; and (ii) a reliability analysis sub-problem. Thus, the main objective of this design problem is to determine the parameters of the STMD that, minimizing its cost, ensures the compliance of the design requirements [2]. As the design requirements, which need to be accomplished, are defined in terms of the accelerations of the structure, x¨ f (t), this design process may be understood as a motion-based design optimization method [7]. Therefore, the general formulation of the proposed method may be expressed as follows [13]: Find θi i = 1, .., nd  Minimizing f(θi ) = f1 (θi ) . . . fj (θi ) . . . fnf (θi )

(20)

Subjected to θil ≤ θi ≤ θiu where f(θi ) is the multi-objective function; fj (θi ) is the jth terms of the multi-objective function; nf is the number of terms of multi-objective function; θi is the ith design variable; θil is the lower bound of the ith design variable; θiu is the upper bound of the ith design variables, and nd is the total number of design variables. For this case, the design of a STMD, the above multi-objective function may be formulated in terms of two aspects: (i) the different parameters of the STMD; and (ii) the compliance of the VSLS of the footbridge. According to the most advances guidelines [2], the VSLS is met if the maximum acceleration of the structure, x¨ f (t), is lower than an allowable value, x¨ f ,lim , established by the designer [2]. Additionally, in order to take into account the random character of the modal properties of the footbridge, which are sensitive to the variation of the operational and environmental conditions [3], the VSLS is re-formulated via a probabilistic approach [10]. According to this approach, the modal properties of the footbridge are assumed to be random variables which follow normal probability distribution function. Hence, the

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response of the system is also a random variable, and it is possible to determine the probability of failure associated with the compliance of the different design requirements (in this case, the VSLS). According to this, the VSLS is defined in terms of a reliability index, β, which establishes certain probability of failure of its compliance. Thus, this limit state is met if this reliability index, β, is greater than an allowable value, βlim , established by the design guidelines [14]. In order to compute the reliability index, β, analytical and numerical method can be used [10]. Among these methods, a Monte Carlo simulation has been considered herein [10]. In order to establish the formulation of this design problem, according to the motionbased design, the design parameters must be determined. For this particular case, four are the design variables: (i) the mass ratio, μ = ma /mf [-]; (ii) the frequency ratio, δ = fa /ff [-] (where fa [Hz] is the natural frequency of the STMD); and (iii) the weighting factors, αd and βd (the driving force). In order to reduce the number of design variables, simplifying the problem, a hybrid strategy has been considered herein. Thus, the frequency ratio, δ, of the STMD has been selected following a conventional criterion for the adjustment of tuned mass dampers [2]. According to this criterion, the frequency ratio of the STMD is set as, δ = 1/(1 + μ). In this manner, the formulation of the motion-based design method under uncertainty conditions for this particular problem may be formulated as follows: Find μ, αd , βd   Minimizing {f(µ, αd , βd )} = f1 (µ) f2 (µ, αd , βd ) = μ ββlim  Subjected to μ ∈ [0.01 0.10] αd ∈ [10 1000] βd ∈ 10−5 10−8

(21)

Subsequently, in order to solve this optimization problem, global optimization algorithms are normally employed due to their good efficiency to find optimum solutions in nonlinear optimization problems [9]. Among these computational algorithms, a natureinspired computational algorithm, genetic algorithms, has been considered herein [9]. As result of this optimization process, a set of non-dominated solutions is obtained [9]. This set of solutions may be represented in a functional space, generating the so-called Pareto front [9]. Finally, a subsequent decision making problem must be addressed, the selection of the best solution among the different elements of the Pareto front. Therefore, an additional condition, ββlim , has been included herein; to select the best element of the Pareto front.

4 Application Example: Motion-Based Design of a STMD for a Vibrating Footbridge Under Uncertainty Conditions In order to illustrate the performance of the motion-based design method, when it is implemented to design STMDs for vibrating footbridges under uncertainty conditions, the following numerical case-study is presented. Thus, the compliance of the VSLS of a numerical footbridge is guaranteed via the installation of a STMD. A detailed description of this numerical footbridge can be found in the French guidelines [2]. The structural behavior of the structure has been simulated via the finite element (FE) method. The FE package Ansys [15] has been used for this purpose. A numerical model using 646

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beam elements and 540 shell elements (BEAM188 and SHELL181 respectively) has been built (Fig. 3). A structural damping ratio of ζf = 0.6% has been considered [2]. The numerical modal parameters of the footbridge have been obtained via a numerical modal analysis. As result of this analysis, it was checked that the first vertical vibration mode (ff = 2.14 Hz) is prone to vibrate under pedestrian action (Fig. 3). The modal mass, mf , of this vertical vibration mode is about 34706 kg.

Fig. 3. Finite element model of the benchmark footbridge and first vertical vibration mode [2].

In order to check the VSLS of this footbridge, the recommendations of the French guidelines [2] have been considered herein. Thus, one design scenario has been taken into account with the following design parameters: (i) a pedestrian traffic of 1 P/m2 [2]; (ii) an allowable vertical acceleration, x¨ f ,lim , of 1.00 m/s2 [2]; and (iii) an allowable reliability index, βlim , of 1.5 [14]. If this requirement is not met, a STMD must be installed to reduce the amplitude of the pedestrian-induced vibrations according to the mentioned thresholds. The driving force, fa , has been limited between the range 10–50 N (fmin − fmax ) in order to characterize adequately the behavior of the semi-active damper. According to the results provided by several researchers [3] a range of variation of ±10% has been considered for both the first vertical natural frequency of the footbridge and its associated damping ratio. Consequently, it has been checked that the VSLS term of the multi-objective function is a random variable which follows a log-normal probability distribution. In order to obtain the reliability index, β, associated with the VSLS of the footbridge, a Monte Carlo simulation has been performed [10]. For the selection of the sample size (50000 simulations), a convergence analysis has been carried out. The mathematical package Matlab [12] has been employed for this study. As result of this study, Table 1 shows the reliability index, β, associated with the VSLS without and with the STMD. As Table 1 shows, the motion-based design method allows control the dynamic response of the footbridge under uncertainty conditions without compromising the cost of the control system. Additionally, the parameters of the STMD designed according to this method have been included in Table 1.

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J. F. Jiménez-Alonso et al. Table 1. Reliability index, β, of the VSLS of the footbridge (without and with STMD).

STMD

x¨ f [m/s2 ]

β [-]

αd [-]

βd [-]

ma [kg]

ka [N/m]

|fa | [N]

No

–

−0.53

–

–

–

–

–

224.19

771.64 10−8

625

1.08 105

50

Yes

1.00

1.50

5 Conclusions In this manuscript, the motion-based design method has been presented and further implemented for the design of STMDs when they are employed to control the pedestrian-induced vibrations in footbridges under uncertainty conditions. According to this method, the design problem may be formulated via two coupled sub-problems: (i) a multi-objective optimization sub-problem; and (ii) a reliability analysis sub-problem. Thus, the multi-objective function of the problem is defined in terms of the parameters of the STMD which are needed to be determined and a reliability index which establishes the probability of compliance of the VSLS of the footbridge. Sampling techniques, as the Monte Carlos simulation method, are usually used to estimate numerically this reliability index. As application example, a footbridge, which is prone to vibrate due to walking pedestrian action, has been regarded. A STMD has been installed at its mid-span to reduce the pedestrian-induced vibrations. The STMD parameters have been obtained via the implementation of the mentioned proposal. As result of this study, the performance of the proposed method has been shown up. In spite of its good performance, further studies are needed, both to better determine the probabilistic distribution function of the different modal properties and to assess experimentally the performance of the structure designed according to this proposal. Acknowledgements. This work was partially funded by two research projects: (i) research project RTI2018-094945-B-C21 (Ministerio de Economía y Competitividad of Spain and the European Regional Development Fund) and (ii) research project SEED-SD RTI2018-099639-B-I00 (Ministerio de Ciencia, Innovación y Universidades of Spain).

References 1. Van Nimmen, K., Lombaert, G., De Roeck, G., Van den Broeck, P.: Vibration serviceability of footbridges: evaluation of the current codes of practice. Eng. Struct. 59, 448–461 (2014) 2. Setra/AFGC Guide méthodologique passerelles piétonnes (Technical Guide Footbridges: Assessment of vibration behaviour of footbridge under pedestrian loading) (2006) 3. Hu, W.H., Caetano, E., Cunha, A.: Structural health monitoring of a stress-ribbon footbridge. Eng. Struct. 57, 578–593 (2013) 4. Weber, F., Feltrin, G., Huth, O.: Guidelines for Structural Control. Structural Engineering Research Laboratory. Swiss Federal Laboratories for Materials Testing Research. Dübendort, Switzerland (2006) 5. Moutinho, C.: Testing a simple control law to reduce broadband frequency harmonic vibrations using semi-active tuned mass dampers. Smart Mater. Struct. 24, 055007 (2015)

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6. Preumont, A., Seto, K.: Active Control of Structures, 1st edn. Wiley, Cornwall (2008) 7. Connor, J.: Introduction to Structural Motion Control, 1st edn. Prentice Hall, Upper Saddle River (2003) 8. Jimenez-Alonso, J.F., Saez, A.: Motion-based design of TMD for vibrating footbridges under uncertainty conditions. Smart Struct. Syst. 21(6), 727–740 (2018) 9. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. Evol. Comput. 6(2), 182–197 (2002) 10. Holický, M.: Reliability Analysis for Structural Design, 1st edn. Sun Media, Stellenbosch (2009) 11. Xu, Z.D., Guo, Y.-Q., Zhu, J.-T., Xu, F.-H.: Intelligent Vibration Control in Civil Engineering Structures, 1st edn. Zhejiang University Press Co., Ltd., Hangzhou (2017) 12. Matlab (2020) R2020a. http://www.mathworks.com/ 13. Liang, Q.Q.: Performance-based optimization: a review. Adv. Struct. Eng. 10(6), 739–753 (2007) 14. EN 1990 Eurocode 0: Basis of Structural Design; European Committee for Standardization, Brussels (2002) 15. Ansys (2020) Mechanical Release. http://www.ansys.com/

Author Index

A Abreu, Antonio, 383 Acién, Francisco Gabriel, 190 Aftab, Muhammad Saleheen, 12 Aguiar, A. Pedro, 466, 548, 628, 658 Amorim, Eurico Vasco, 582 Andrade, José Manuel, 696 Andreev, Aleksandr, 686 Anes, Vitor, 383 Araújo, Eduardo, 487 Astolfi, Giacomo, 263 Autrique, Laurent, 616 Azar, Thérèse, 616 B Balsa, Carlos, 148, 232 Barbosa, Ramiro S., 415 Barbu, M., 518 Barros, Rui, 752, 773 Barroso, João, 740 Batzies, Ekkehard, 180 Belfo, João P., 466 Berenguel, Manuel, 190 Beschi, M., 518 Boaventura-Cunha, José, 23, 114, 570, 707 Bot, K., 332 Botto, Miguel Ayala, 487 Boukili, Yassine, 628, 658 Brancalião, Laiany, 762 Braun, João, 303 Braz-César, Manuel, 49, 752 Brito, Thadeu, 303, 507

C Cajo, Ricardo, 528 Calado, João M. F., 383 Camargo, Caio, 762 Camarinha, Margarida, 322 Campaniço, Ana Teresa, 740 Cano Marchal, Pablo, 284 Capela, Sílvio, 740 Carvalho, Adriano, 628 Casaro, Marcio M., 592 Castaño, Fernando, 170 Chertovskih, Roman, 373 Cocchioni, Francesco, 263, 455 Coelho, João Paulo, 49, 570 Colombo, Leonardo J., 363 Costa, Bertinho A., 648 Costa, Paulo, 49, 63, 303, 638, 676, 729 Costa, S. J., 293 Cots, Olivier, 232 Crouch, Peter, 322 Cunha, Bruno, 313 Cunha, Tiago, 773 D De Keyser, Robain, 528 de la Concha Renedo, Carlos Martín, 783 de Moura Oliveira, P. B., 23, 538, 668 Díaz, Iván M., 221, 394, 477 dos Santos, P. Lopes, 159 E Edwards, Christopher, 696

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. A. Gonçalves et al. (Eds.): CONTROLO 2020, LNEE 695, pp. 795–797, 2021. https://doi.org/10.1007/978-3-030-58653-9

796 F Ferreira, Filipe, 313 Ferreira, R., 293 Feyfant, Pierre, 773 Filipe, Vítor, 582, 740 Findeisen, Rolf, 404 Folhento, Pedro, 752 Fonseca Ferreira, Nuno M., 114, 707 Fonseca, Joana, 33 Fornés, José M., 170 G Gallegos, Christian, 477 Gama, Sílvio, 148, 352 Gámez García, Javier, 284 Garcia, Claudio, 104 García-Palacios, Jaime H., 221 Garrido, Paulo, 425 Gergaud, Joseph, 232 Ghaderyan, Diyako, 658 Giernacki, Wojciech, 570 Godoy Molina, Mauricio, 82 Gómez Ortega, Javier, 284 Gonçalves, José, 49, 63, 72, 570, 638, 676, 729, 762 Goodman, Jacob R., 363 Guillen-González, Francisco, 783 Guzmán, José Luis, 190, 200 H Hägglund, Tore, 200 Hedengren, John D., 23 Henriques, Jorge, 773 Hernández, Elder, 313 Herrera, José Manuel Soria, 783 Hoyo, Ángeles, 200 Huba, Mikulas, 719 Hüper, Knut, 274, 435 Hypiusová, M., 719 I Ibrahim, Mohamed, 404 Igreja, J. M., 293 Illana Rico, Sergio, 284 Imsirovic, Edin, 126 Ionescu, Clara, 528 Ishizone, Tsuyoshi, 342 J Jiménez-Alonso, Javier Fernando, 783 Johansen, Tor Arne, 33

Author Index Johansson, Karl Henrik, 33 Jurdjevic, Verlimir, 136 K Kallies, Christian, 404 Kern, Moritz, 93 Khanal, Salik Ram, 582, 740 L Leal, Marta, 200 Leão, Celina P., 668 Leitão, Paulo, 592 Leite, Fátima Silva, 180 Lemos, João M., 253, 466, 648 Lima, José, 303, 507, 676, 729, 762 Los, Nelson A., 592 M Machado, Luís, 180 Maidana, Wellington, 592 Markina, Irina, 82, 274 Marques, Gil, 352 Martínez Gila, Diego, 284 Martins, José Duarte Moleiro, 383 Mendonça, Teresa, 497 Mineiro, Nuno, 72 Moayyed, Hamed, 548, 658 Mooney, Paul, 602 Moreira, António Paulo, 559 Moulay, Emmanuel, 616 N Nabais, João Lemos, 487 Nabais, Margarida, 253 Nakamura, Kazuyuki, 342 Nascimento, Claudinor B., 592 Ntogramatzidis, Lorenzo, 210 O Omerdic, Edin, 126 Omerdic, Emir, 126 Orlietti, Lorenzo, 263 Ortega, Manuel G., 170 Osmic, Jakub, 126 Otálora, Pablo, 190 P Pagani, Vitor H., 592 Paiva, Fábio, 773 Pedret, C., 518 Pepe, Crescenzo, 263, 455

Author Index

797

Perdicoúlis, T.-P. Azevedo, 159 Peregudova, Olga, 686 Pereira, Ana I., 507 Pereira, E., 394 Pereira, Fernando Lobo, 373 Pereira, Ricardo, 210 Pérez Serrano, Manuel A., 104 Perez, Laetitia, 616 Pérez-Aracil, J., 394 Petry, Marcelo R., 559 Phillips, Russell, 602 Piardi, Luis, 303 Pina, Fátima, 243 Pinto, Vítor H., 638, 676, 729 Plaza, Douglas, 528 Prieur, Christophe, 616

Sewell, James, 602 Silva Leite, Fátima, 243, 274, 322 Silva, Jorge, 497 Silva, Rita, 740 Silvestre, Carlos, 445 Silvestre, Daniel, 445 Smajlovic, Tima O., 126 Soares, Filomena, 668 Soares, José Miguel, 383 Soria, José Manuel, 477 Sousa, Armando, 729 Sousa, Cristóvão, 313 Sousa, Ricardo B., 559 Staritsyn, Maxim, 373 Stegemeyer, Maximilian, 435 Stopforth, Riaan, 602

R Ramírez-Senent, José, 221 Rebelo, Rui, 313 Reis, Matheus F., 548 Renedo, Carlos Martín Concha, 477 Reyes Dreke, Victor D., 104 Reynolds, P., 394 Ribeiro, João, 63, 72 Ribeiro, Rafael, 445 Rocha, Paula, 210, 497 Rodrigues, Maria João, 352 Rosa, Ricardo, 507 Rossiter, John Anthony, 1, 12 Ruano, A., 332 Ruano, M. Graça, 332 Rubio, Francisco R., 170

T Ťapák, P., 719

S Santos, D. B. S., 707 Santos, Ricardo, 383 Saraiva, A. A., 707 Satué, Manuel G., 170 Serbezov, A., 719

U Uhl, Christian, 93 V Valzecchi, Chiara, 455 van Niekerk, Theo, 602 Vilanova, R., 518 Visioli, A., 518 Vrančić, Damir, 538 W Warmuth, Monika, 93 Wehrmeister, Marco A., 507 Wei, Jieqiang, 33 Wembe, Boris, 232 Z Zanoli, Silvia Maria, 263, 455 Zhang, Zhiming, 12 Zhao, Shiquan, 528