Backstepping Control of Nonlinear Dynamical Systems 0128175826, 9780128175828

Backstepping Control of Nonlinear Dynamical Systems addresses both the fundamentals of backstepping control as well as t

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Table of contents :
Contents
Contributors
Preface
About the book
Objectives of the book
Organization of the book
Book features
Audience
Acknowledgments
1 An introduction to backstepping control
1.1 Introduction
1.2 Backstepping design for a 2-D linear system
1.3 Backstepping design for a 2-D nonlinear system
1.4 Backstepping design for a 3-D linear system
1.5 Backstepping design for the 3-D Vaidyanathan jerk chaotic system
1.6 Backstepping control method
1.7 Examples of backstepping control design
1.8 Conclusions
References
2 A new chaotic system without linear term, its backstepping control, and circuit design
2.1 Introduction
2.2 Properties of the system
2.3 Dynamics of the system
2.4 Backstepping control for the global stabilization of the new chaos system
2.5 Backstepping control for the synchronization of the new chaos systems
2.6 Circuit design
2.7 Conclusions
Acknowledgment
References
3 A new chaotic jerk system with egg-shaped strange attractor, its dynamical analysis, backstepping control, and circuit simulation
3.1 Introduction
3.2 System details
3.3 Backstepping control of the jerk system
3.4 Backstepping synchronization of the jerk system
3.5 Circuit design
3.6 Conclusions
References
4 A new 4-D chaotic hyperjerk system with coexisting attractors, its active backstepping control, and circuit realization
4.1 Introduction
4.2 System model
4.3 Dynamic analysis of the new hyperjerk system
4.4 Active backstepping stabilization of the new hyperjerk system
4.5 Active backstepping synchronization of the new hyperjerk system
4.6 Circuit simulation of the new hyperjerk system
4.7 Conclusions
Acknowledgment
References
5 A new 3-D chaotic jerk system with a saddle-focus rest point at the origin, its active backstepping control, and circuit realization
5.1 Introduction
5.2 System model
5.3 Dynamic analysis of the new jerk system
5.4 Backstepping control of the jerk system
5.5 Backstepping synchronization of the jerk system
5.6 Electronic circuit simulation of the chaotic jerk system
5.7 Conclusions
Acknowledgments
References
6 A new 5-D hyperchaotic four-wing system with multistability and hidden attractor, its backstepping control, and circuit simulation
6.1 Introduction
6.2 System model
6.3 Dynamic analysis of the 5-D hyperchaotic four-wing model
6.3.1 Rest points
6.3.2 Multistability
6.4 Active backstepping control for the global stabilization design of the new 5-D hyperchaotic four-wing system
6.5 Active backstepping control for the global synchronization design of the new 5-D hyperchaotic four-wing systems
6.6 Circuit simulation of the new 5D hyperchaotic four-wing system
6.7 Conclusions
References
7 A new 4-D hyperchaotic temperature variations system with multistability and strange attractor, bifurcation analysis, its active backstepping control, and circuit realization
7.1 Introduction
7.2 System model
7.3 Dynamic analysis of the hyperchaotic temperature variations model
7.3.1 Bifurcation analysis
7.3.2 Rest points
7.3.3 Multistability
7.4 Active backstepping control for the global stabilization design of the new hyperchaotic temperature variations system
7.5 Active backstepping control for the global synchronization design of the new hyperchaos temperature variation systems
7.6 Circuit simulation of the new 4D hyperchaotic temperature variation system
7.7 Conclusions
References
8 A new thermally excited chaotic jerk system, its dynamical analysis, adaptive backstepping control, and circuit simulation
8.1 Introduction
8.2 A new jerk system with two nonlinearities
8.3 Dynamic analysis of the new thermo-mechanical jerk model
8.3.1 Rest points of the new jerk model
8.3.2 Bifurcation analysis
8.3.3 Multistability and coexisting attractors
8.4 Adaptive backstepping control of the new thermo-mechanical jerk system
8.5 Adaptive backstepping synchronization of the new thermo-mechanical jerk systems
8.6 Electronic circuit simulation of the new thermo-mechanical chaotic jerk system
8.7 Conclusions
References
9 A new multistable plasma torch chaotic jerk system, its dynamical analysis, active backstepping control, and circuit design
9.1 Introduction
9.2 A new plasma torch chaotic jerk system with two nonlinearities
9.3 Dynamic analysis of the new plasma torch chaotic jerk model
9.3.1 Rest points of the new chaotic jerk model
9.3.2 Bifurcation analysis
9.3.3 Multistability and coexisting attractors
9.4 Active backstepping control for the global stabilization of the new plasma torch chaotic jerk system
9.5 Active backstepping control for the global synchronization of the new plasma torch chaotic jerk systems
9.6 Electronic circuit simulation of the new plasma torch chaotic jerk system
9.7 Conclusions
References
10 Direct power control of three-phase PWM-rectifier with backstepping control
10.1 Introduction
10.2 Mathematical model of PWM-rectifier
10.2.1 Vector representation
10.2.2 A brief review of direct power control
10.3 Principle and definitions of backstepping control
10.4 Control of DC-voltage by backstepping
10.5 Simulation results
10.6 Conclusion
References
11 Adaptive backstepping controller for DFIG-based wind energy conversion system
11.1 Introduction
11.2 Wind sensor-less rotor speed reference optimization
11.3 Modeling `AC/DC/AC converter-DFIG' association
11.3.1 DFIG-AC/DC modeling
11.3.2 AC/DC rectifier modeling
11.4 Controller design
11.4.1 Control objectives
11.4.2 Speed and stator flux norm regulator design
11.4.3 PFC and DC voltage controller
11.4.3.1 Controlling rectifier output current to meet PFC
11.4.3.2 DC voltage loop
11.5 Simulation results and discussions
11.6 Conclusion
References
12 Dynamic modeling, identification, and a comparative experimental study on position control of a pneumatic actuator based on Soft Switching and Backstepping-Sliding Mode controllers
12.1 Introduction
12.2 Related works
12.3 Experimental setup of the PneuSys
12.4 Dynamic modeling of the pneumatic system
12.4.1 Cylinder dynamics
12.4.2 Pressure dynamics
12.4.3 State space representation of the PneuSys
12.5 GA-based identification of the PneuSys and validation
12.5.1 Identification of the unknown parameters
12.5.2 Validation of the identified dynamic model
12.6 Proposed controllers; Model-free and Model-based controllers
12.6.1 Model-free; Soft Switching controller
12.6.2 Model-based; Backstepping-Sliding Mode controller
12.7 Experimental results
12.8 Discussion
12.9 Conclusion
References
13 Optimal adaptive backstepping control for chaos synchronization of nonlinear dynamical systems
13.1 Introduction
13.2 Chaos detection and chaos synchronization
13.2.1 Chaos detection
13.2.1.1 Lyapunov exponent
13.2.1.2 0-1 test for chaos in dynamical system
13.2.2 Chaos synchronization and recurrence
13.3 Problem statement and preliminaries
13.4 Stability analysis of adaptive backstepping control systems
13.4.1 Lyapunov stability theory and the invariance principle
13.4.2 Adaptive backstepping controller design
13.4.2.1 Principle of backstepping control method
13.4.2.2 Adaptive backstepping control process
13.4.2.3 Optimal backstepping controller based on genetic algorithms
13.5 The PID controller based on genetic algorithms
13.6 Simulation examples and discussion
13.6.1 Lorenz system description
13.6.2 Optimal adaptive backstepping control and genetically optimized PID control for chaos synchronization of Lorenz systems
13.6.2.1 Adaptive backstepping stabilization control for Lorenz system
13.6.2.2 Optimal adaptive backstepping control for Lorenz system synchronization
13.6.2.3 Genetically optimized PID control for chaos synchronization of Lorenz systems
13.6.2.4 Discussion
13.7 Conclusion
References
14 Backstepping controller for nonlinear active suspension system
14.1 Introduction
14.2 Plant model and problem statement
14.2.1 Nonlinear active suspension system
14.2.2 Problem statement
14.3 Backstepping controller synthesis
14.3.1 Backstepping controller
14.3.2 Fuzzy PD controller
14.3.3 Conventional PD controller
14.3.4 Tuning of gains of controllers
14.3.4.1 Grey Wolf optimizing algorithm
14.3.4.2 Cost function
14.4 Results and discussions
14.4.1 Bump road surface
14.4.1.1 Sprung mass uncertainty
14.4.1.2 Uncertainty in height of bump
14.4.2 Multiple bumps road profile
14.5 Conclusions
References
15 Single-link flexible joint manipulator control using backstepping technique
15.1 Introduction
15.2 Single-link flexible joint manipulator model
15.3 Controller design using backstepping technique
15.4 Optimization algorithms
15.4.1 Jaya algorithm
15.4.2 Teaching learning based optimization algorithm
15.4.3 Genetic algorithm
15.5 Results and discussions
15.6 Conclusion
References
16 Backstepping control and synchronization of chaotic time delayed systems
16.1 Introduction
16.2 Related work
16.3 Backstepping stabilization of time delayed systems
16.4 Backstepping synchronization of time delayed chaotic systems
16.5 Numerical examples
16.5.1 Example 1: Stabilization of the time delayed Lorenz chaotic system
16.5.2 Example 2: Synchronization of the time delayed Rössler chaotic system
16.6 Discussion
16.7 Conclusion
References
17 Multi-switching synchronization of nonlinear hyperchaotic systems via backstepping control
17.1 Introduction
17.2 Problem formulation
17.3 System description
17.3.1 Chaotic attractor of the system
17.3.2 Dissipation and existence of chaotic attractor
17.3.3 Symmetry and invariance
17.3.4 Poincaré map
17.4 Simulation results and discussions
17.4.1 Switch 1
17.4.1.1 Design of η1 and η2
17.4.1.2 Design of η3 and η4
17.4.1.3 Numerical simulations
17.4.2 Switch 2
17.4.2.1 Design of η2 and η3
17.4.2.2 Design of η1 and η4
17.4.2.3 Numerical simulations
17.4.3 Switch 3
17.4.3.1 Design of η1 and η2
17.4.3.2 Design of η3 and η4
17.4.3.3 Numerical simulations
17.5 Conclusion
References
18 A 5-D hyperchaotic dynamo system with multistability, its dynamical analysis, active backstepping control, and circuit simulation
18.1 Introduction
18.2 System model
18.3 Dynamic analysis of the 5-D hyperchaotic dynamo model
18.3.1 Rest points
18.3.2 Multistability
18.4 Active backstepping control for the global stabilization design of the new 5-D hyperchaotic dynamo system
18.5 Active backstepping control for the global synchronization design of the new 5-D hyperchaotic dynamo systems
18.6 Circuit simulation of the new 5D hyperchaotic system
18.7 Conclusions
References
19 Design and implementation of a backstepping controller for nonholonomic two-wheeled inverted pendulum mobile robots
19.1 Introduction
19.2 Distributed controller design based on backstepping approach
19.3 Discrete event modeling and control net representation
19.4 Implementation issues on a multi-task processing architecture
19.5 Conclusion
References
20 A novel chaotic system with a closed curve of four quarter-circles of equilibrium points: dynamics, active backstepping control, and electronic circuit implementation
20.1 Introduction
20.2 A new chaotic system with closed-curve equilibrium
20.3 Dynamic analysis of the new chaotic system with a closed-curve equilibrium
20.3.1 Lyapunov exponents analysis
20.3.2 Multistability and coexisting attractors
20.4 Active backstepping control for the global stabilization of the new chaos system with a closed-curve equilibrium
20.5 Active backstepping control for the synchronization of the new chaos systems
20.6 Circuit design for the new chaotic system with a closed-curve equilibrium
20.7 Conclusions
Acknowledgment
References
Index
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Backstepping Control of Nonlinear Dynamical Systems

A volume in the Advances in Nonlinear Dynamics and Chaos series

Series editors Ahmad Taher Azar Sundarapandian Vaidyanathan

Visit the Series webpage at https://www.elsevier.com/books/book-series/ advances-in-nonlinear-dynamics-and-chaos-andc

Backstepping Control of Nonlinear Dynamical Systems

Edited by

Sundarapandian Vaidyanathan Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

Ahmad Taher Azar Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2021 Elsevier Inc. All rights reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-817582-8 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisitions Editor: Sonnini R. Yura Editorial Project Manager: Rachel Pomery Production Project Manager: Prem Kumar Kaliamoorthi Designer: Matthew Limbert Typeset by VTeX

Contents

Contributors Preface

1.

xv xix

An introduction to backstepping control Sundarapandian Vaidyanathan and Ahmad Taher Azar 1.1 1.2 1.3 1.4 1.5

Introduction Backstepping design for a 2-D linear system Backstepping design for a 2-D nonlinear system Backstepping design for a 3-D linear system Backstepping design for the 3-D Vaidyanathan jerk chaotic system 1.6 Backstepping control method 1.7 Examples of backstepping control design 1.8 Conclusions References

2.

1 2 4 7 11 15 21 27 28

A new chaotic system without linear term, its backstepping control, and circuit design Viet-Thanh Pham, Sundarapandian Vaidyanathan, Ahmad Taher Azar, and Vo Hoang Duy 2.1 2.2 2.3 2.4

Introduction Properties of the system Dynamics of the system Backstepping control for the global stabilization of the new chaos system 2.5 Backstepping control for the synchronization of the new chaos systems 2.6 Circuit design 2.7 Conclusions Acknowledgment References

33 34 35 35 40 45 48 48 48 v

vi Contents

3.

A new chaotic jerk system with egg-shaped strange attractor, its dynamical analysis, backstepping control, and circuit simulation Sundarapandian Vaidyanathan, Viet-Thanh Pham, and Ahmad Taher Azar 3.1 Introduction 3.2 System details 3.3 Backstepping control of the jerk system 3.4 Backstepping synchronization of the jerk system 3.5 Circuit design 3.6 Conclusions References

4.

53 55 57 61 65 67 69

A new 4-D chaotic hyperjerk system with coexisting attractors, its active backstepping control, and circuit realization Aceng Sambas, Sundarapandian Vaidyanathan, Sen Zhang, Mohamad Afendee Mohamed, Yicheng Zeng, and Ahmad Taher Azar 4.1 4.2 4.3 4.4 4.5

Introduction System model Dynamic analysis of the new hyperjerk system Active backstepping stabilization of the new hyperjerk system Active backstepping synchronization of the new hyperjerk system 4.6 Circuit simulation of the new hyperjerk system 4.7 Conclusions Acknowledgment References

5.

73 75 78 79 82 88 90 91 91

A new 3-D chaotic jerk system with a saddle-focus rest point at the origin, its active backstepping control, and circuit realization Aceng Sambas, Sundarapandian Vaidyanathan, Sen Zhang, Mohamad Afendee Mohamed, Yicheng Zeng, and Ahmad Taher Azar 5.1 5.2 5.3 5.4 5.5

Introduction System model Dynamic analysis of the new jerk system Backstepping control of the jerk system Backstepping synchronization of the jerk system

95 96 99 100 103

Contents vii

5.6 Electronic circuit simulation of the chaotic jerk system 5.7 Conclusions Acknowledgments References

6.

108 110 111 111

A new 5-D hyperchaotic four-wing system with multistability and hidden attractor, its backstepping control, and circuit simulation Sundarapandian Vaidyanathan, Aceng Sambas, Ahmad Taher Azar, K.P.S. Rana, and Vineet Kumar 6.1 Introduction 6.2 System model 6.3 Dynamic analysis of the 5-D hyperchaotic four-wing model 6.3.1 Rest points 6.3.2 Multistability 6.4 Active backstepping control for the global stabilization design of the new 5-D hyperchaotic four-wing system 6.5 Active backstepping control for the global synchronization design of the new 5-D hyperchaotic four-wing systems 6.6 Circuit simulation of the new 5D hyperchaotic four-wing system 6.7 Conclusions References

7.

115 116 119 119 119 119 124 130 132 134

A new 4-D hyperchaotic temperature variations system with multistability and strange attractor, bifurcation analysis, its active backstepping control, and circuit realization Sundarapandian Vaidyanathan, Aceng Sambas, Ahmad Taher Azar, K.P.S. Rana, and Vineet Kumar 7.1 Introduction 7.2 System model 7.3 Dynamic analysis of the hyperchaotic temperature variations model 7.3.1 Bifurcation analysis 7.3.2 Rest points 7.3.3 Multistability 7.4 Active backstepping control for the global stabilization design of the new hyperchaotic temperature variations system 7.5 Active backstepping control for the global synchronization design of the new hyperchaos temperature variation systems 7.6 Circuit simulation of the new 4D hyperchaotic temperature variation system

139 140 144 144 145 146 148 150 155

viii Contents

7.7 Conclusions References

8.

159 160

A new thermally excited chaotic jerk system, its dynamical analysis, adaptive backstepping control, and circuit simulation Sundarapandian Vaidyanathan, Aceng Sambas, Ahmad Taher Azar, Fernando E. Serrano, and Arezki Fekik 8.1 Introduction 8.2 A new jerk system with two nonlinearities 8.3 Dynamic analysis of the new thermo-mechanical jerk model 8.3.1 Rest points of the new jerk model 8.3.2 Bifurcation analysis 8.3.3 Multistability and coexisting attractors 8.4 Adaptive backstepping control of the new thermo-mechanical jerk system 8.5 Adaptive backstepping synchronization of the new thermo-mechanical jerk systems 8.6 Electronic circuit simulation of the new thermo-mechanical chaotic jerk system 8.7 Conclusions References

9.

165 167 171 171 171 172 173 177 181 184 185

A new multistable plasma torch chaotic jerk system, its dynamical analysis, active backstepping control, and circuit design Sundarapandian Vaidyanathan, Aceng Sambas, Ahmad Taher Azar, and Shikha Singh 9.1 Introduction 9.2 A new plasma torch chaotic jerk system with two nonlinearities 9.3 Dynamic analysis of the new plasma torch chaotic jerk model 9.3.1 Rest points of the new chaotic jerk model 9.3.2 Bifurcation analysis 9.3.3 Multistability and coexisting attractors 9.4 Active backstepping control for the global stabilization of the new plasma torch chaotic jerk system 9.5 Active backstepping control for the global synchronization of the new plasma torch chaotic jerk systems 9.6 Electronic circuit simulation of the new plasma torch chaotic jerk system 9.7 Conclusions References

191 193 196 196 198 198 200 202 205 208 210

Contents ix

10. Direct power control of three-phase PWM-rectifier with backstepping control Arezki Fekik, Hakim Denoun, Ahmad Taher Azar, Nashwa Ahmad Kamal, Mustapha Zaouia, Nabil Benyahia, Mohamed Lamine Hamida, Nacereddine Benamrouche, and Sundarapandian Vaidyanathan 10.1 Introduction 10.2 Mathematical model of PWM-rectifier 10.2.1 Vector representation 10.2.2 A brief review of direct power control 10.3 Principle and definitions of backstepping control 10.4 Control of DC-voltage by backstepping 10.5 Simulation results 10.6 Conclusion References

215 216 218 219 220 225 225 230 232

11. Adaptive backstepping controller for DFIG-based wind energy conversion system Ismail Drhorhi, Abderrahim El Fadili, Chaker Berrahal, Rachid Lajouad, Abdelmounime El Magri, Fouad Giri, Ahmad Taher Azar, and Sundarapandian Vaidyanathan 11.1 Introduction 11.2 Wind sensor-less rotor speed reference optimization 11.3 Modeling ‘AC/DC/AC converter-DFIG’ association 11.3.1 DFIG-AC/DC modeling 11.3.2 AC/DC rectifier modeling 11.4 Controller design 11.4.1 Control objectives 11.4.2 Speed and stator flux norm regulator design 11.4.3 PFC and DC voltage controller 11.5 Simulation results and discussions 11.6 Conclusion References

235 237 238 239 242 244 244 244 250 252 256 258

12. Dynamic modeling, identification, and a comparative experimental study on position control of a pneumatic actuator based on Soft Switching and Backstepping–Sliding Mode controllers Amir Salimi Lafmejani, Mehdi Tale Masouleh, and Ahmad Kalhor 12.1 Introduction 12.2 Related works 12.3 Experimental setup of the PneuSys

261 264 266

x Contents

12.4 Dynamic modeling of the pneumatic system 12.4.1 Cylinder dynamics 12.4.2 Pressure dynamics 12.4.3 State space representation of the PneuSys 12.5 GA-based identification of the PneuSys and validation 12.5.1 Identification of the unknown parameters 12.5.2 Validation of the identified dynamic model 12.6 Proposed controllers; Model-free and Model-based controllers 12.6.1 Model-free; Soft Switching controller 12.6.2 Model-based; Backstepping–Sliding Mode controller 12.7 Experimental results 12.8 Discussion 12.9 Conclusion References

267 267 268 270 270 271 272 275 275 275 279 280 286 287

13. Optimal adaptive backstepping control for chaos synchronization of nonlinear dynamical systems Mohsen Alimi, Ahmed Rhif, Abdelwaheb Rebai, Sundarapandian Vaidyanathan, and Ahmad Taher Azar 13.1 Introduction 13.2 Chaos detection and chaos synchronization 13.2.1 Chaos detection 13.2.2 Chaos synchronization and recurrence 13.3 Problem statement and preliminaries 13.4 Stability analysis of adaptive backstepping control systems 13.4.1 Lyapunov stability theory and the invariance principle 13.4.2 Adaptive backstepping controller design 13.5 The PID controller based on genetic algorithms 13.6 Simulation examples and discussion 13.6.1 Lorenz system description 13.6.2 Optimal adaptive backstepping control and genetically optimized PID control for chaos synchronization of Lorenz systems 13.7 Conclusion References

291 297 297 300 302 303 303 306 314 316 316

325 339 340

14. Backstepping controller for nonlinear active suspension system Vineet Kumar, K.P.S. Rana, Ahmad Taher Azar, and Sundarapandian Vaidyanathan 14.1 Introduction 14.2 Plant model and problem statement 14.2.1 Nonlinear active suspension system 14.2.2 Problem statement

347 350 350 352

Contents xi

14.3 Backstepping controller synthesis 14.3.1 Backstepping controller 14.3.2 Fuzzy PD controller 14.3.3 Conventional PD controller 14.3.4 Tuning of gains of controllers 14.4 Results and discussions 14.4.1 Bump road surface 14.4.2 Multiple bumps road profile 14.5 Conclusions References

353 353 357 359 359 361 361 367 369 370

15. Single-link flexible joint manipulator control using backstepping technique Nishtha Bansal, Aman Bisht, Sruti Paluri, Vineet Kumar, K.P.S. Rana, Ahmad Taher Azar, and Sundarapandian Vaidyanathan 15.1 15.2 15.3 15.4

Introduction Single-link flexible joint manipulator model Controller design using backstepping technique Optimization algorithms 15.4.1 Jaya algorithm 15.4.2 Teaching learning based optimization algorithm 15.4.3 Genetic algorithm 15.5 Results and discussions 15.6 Conclusion References

375 379 381 389 389 391 392 395 400 402

16. Backstepping control and synchronization of chaotic time delayed systems Ahmad Taher Azar, Fernando E. Serrano, Sundarapandian Vaidyanathan, and Nashwa Ahmad Kamal 16.1 16.2 16.3 16.4 16.5

Introduction Related work Backstepping stabilization of time delayed systems Backstepping synchronization of time delayed chaotic systems Numerical examples 16.5.1 Example 1: Stabilization of the time delayed Lorenz chaotic system 16.5.2 Example 2: Synchronization of the time delayed Rössler chaotic system 16.6 Discussion 16.7 Conclusion References

407 409 409 411 413 413 417 418 419 421

xii Contents

17. Multi-switching synchronization of nonlinear hyperchaotic systems via backstepping control Shikha Singh, Sandhya Mathpal, Ahmad Taher Azar, Sundarapandian Vaidyanathan, and Nashwa Ahmad Kamal 17.1 Introduction 17.2 Problem formulation 17.3 System description 17.3.1 Chaotic attractor of the system 17.3.2 Dissipation and existence of chaotic attractor 17.3.3 Symmetry and invariance 17.3.4 Poincaré map 17.4 Simulation results and discussions 17.4.1 Switch 1 17.4.2 Switch 2 17.4.3 Switch 3 17.5 Conclusion References

425 428 429 429 432 432 432 432 434 436 439 442 443

18. A 5-D hyperchaotic dynamo system with multistability, its dynamical analysis, active backstepping control, and circuit simulation Sundarapandian Vaidyanathan, Aceng Sambas, and Ahmad Taher Azar 18.1 Introduction 18.2 System model 18.3 Dynamic analysis of the 5-D hyperchaotic dynamo model 18.3.1 Rest points 18.3.2 Multistability 18.4 Active backstepping control for the global stabilization design of the new 5-D hyperchaotic dynamo system 18.5 Active backstepping control for the global synchronization design of the new 5-D hyperchaotic dynamo systems 18.6 Circuit simulation of the new 5D hyperchaotic system 18.7 Conclusions References

449 450 453 453 454 455 458 465 467 468

19. Design and implementation of a backstepping controller for nonholonomic two-wheeled inverted pendulum mobile robots Gen’ichi Yasuda 19.1 Introduction 473 19.2 Distributed controller design based on backstepping approach 474 19.3 Discrete event modeling and control net representation 477

Contents xiii

19.4 Implementation issues on a multi-task processing architecture 19.5 Conclusion References

480 483 483

20. A novel chaotic system with a closed curve of four quarter-circles of equilibrium points: dynamics, active backstepping control, and electronic circuit implementation Aceng Sambas, Sundarapandian Vaidyanathan, Sukono, Ahmad Taher Azar, Yuyun Hidayat, Gugun Gundara, and Mohamad Afendee Mohamed 20.1 Introduction 20.2 A new chaotic system with closed-curve equilibrium 20.3 Dynamic analysis of the new chaotic system with a closed-curve equilibrium 20.3.1 Lyapunov exponents analysis 20.3.2 Multistability and coexisting attractors 20.4 Active backstepping control for the global stabilization of the new chaos system with a closed-curve equilibrium 20.5 Active backstepping control for the synchronization of the new chaos systems 20.6 Circuit design for the new chaotic system with a closed-curve equilibrium 20.7 Conclusions Acknowledgment References Index

485 487 489 489 489 492 495 499 501 503 503 509

Contributors

Mohsen Alimi, University of Kairouan, Kairouan, Tunisia Ahmad Taher Azar, Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt Nishtha Bansal, Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India Nacereddine Benamrouche, Akli Mohand Oulhadj University, Bouira, Algeria Nabil Benyahia, Electrical Engineering Advanced Technology Laboratory (LATAGE), Mouloud Mammeri University, Tizi-Ouzou, Algeria Chaker Berrahal, Faculté des Sciences et Techniques, Hassan II University of Casablanca, Mohammedia, Morocco University of Caen Basse Normandie, Caen, France Aman Bisht, Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India Hakim Denoun, Electrical Engineering Advanced Technology Laboratory (LATAGE), Mouloud Mammeri University, Tizi-Ouzou, Algeria Ismail Drhorhi, Faculté des Sciences et Techniques, Hassan II University of Casablanca, Mohammedia, Morocco Vo Hoang Duy, Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam Abderrahim El Fadili, Faculté des Sciences et Techniques, Hassan II University of Casablanca, Mohammedia, Morocco Abdelmounime El Magri, Ecole Normale Supérieure d’Enseignement Technique (ENSET), Hassan II University of Casablanca, Mohammedia, Morocco Arezki Fekik, Akli Mohand Oulhadj University, Bouira, Algeria Laboratory of Advanced Technologies of Electrical Engineering (LATAGE), Faculty of Electrical and Computer Engineering, Mouloud Mammeri University (UMMTO), Tizi-Ouzou, Algeria xv

xvi Contributors

Electrical Engineering Advanced Technology Laboratory (LATAGE), Mouloud Mammeri University, Tizi-Ouzou, Algeria Fouad Giri, University of Caen Basse Normandie, Caen, France Gugun Gundara, Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, Indonesia Mohamed Lamine Hamida, Electrical Engineering Advanced Technology Laboratory (LATAGE), Mouloud Mammeri University, Tizi-Ouzou, Algeria Yuyun Hidayat, Department of Statistics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Bandung, Indonesia Ahmad Kalhor, Human and Robot Interaction Laboratory, School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran Nashwa Ahmad Kamal, Cairo University, Giza, Egypt Vineet Kumar, Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India Rachid Lajouad, Ecole Normale Supérieure d’Enseignement Technique (ENSET), Hassan II University of Casablanca, Mohammedia, Morocco Sandhya Mathpal, Department of Mathematics, Faculty of Engineering & Technology, MRIU, Faridabad, Haryana, India Mohamad Afendee Mohamed, Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia Sruti Paluri, Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India Viet-Thanh Pham, Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Hanoi, Vietnam Phenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, Hanoi, Vietnam K.P.S. Rana, Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India Abdelwaheb Rebai, University of Sfax, Sfax, Tunisia Ahmed Rhif, University of Carthage, La Marsa, Tunisia Amir Salimi Lafmejani, School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ, United States Aceng Sambas, Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, Indonesia Fernando E. Serrano, Universidad Tecnológica Centroamericana (UNITEC), Zona Jacaleapa, Tegucigalpa, Honduras Shikha Singh, Department of Mathematics, Jesus and Mary College, University of Delhi, New Delhi, India

Contributors xvii

Sukono, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Bandung, Indonesia Mehdi Tale Masouleh, Human and Robot Interaction Laboratory, School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran Sundarapandian Vaidyanathan, Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India Gen’ichi Yasuda, Nagasaki Institute of Applied Science, Nagasaki, Japan Mustapha Zaouia, Electrical Engineering Advanced Technology Laboratory (LATAGE), Mouloud Mammeri University, Tizi-Ouzou, Algeria Yicheng Zeng, School of Physics and Optoelectric Engineering, Xiangtan University, Xiangtan, China Sen Zhang, School of Physics and Optoelectric Engineering, Xiangtan University, Xiangtan, China

Preface

In control systems engineering, backstepping control is a very useful technique developed around 1990 by Peter V. Kokotovic and others for designing stabilizing controls for a special class of nonlinear dynamical systems. The backstepping approach provides a recursive method for stabilizing the origin of a system in strict-feedback form. These systems are built from subsystems that radiate out from an irreducible subsystem that can be stabilized using the Lyapunov stability method. Because of this recursive structure, the designer can start the design process at the system that is known to be stable and ”back out” new controllers that progressively stabilize each outer subsystem. The process terminates when the final external feedback control is reached. Hence, this control method is known as backstepping control. The backstepping control technique is very resourceful constructing globally stabilizing control laws for a certain class of nonlinear dynamic systems. Backstepping control has several applications in science and engineering such as robot manipulators, aircraft flight control systems, power systems, mechanical systems, biological systems, and chaotic systems.

About the book The new Elsevier book, Backstepping Control of Nonlinear Dynamical Systems, consists of 20 contributed chapters by subject experts who are specialized in the various topics addressed in this book. The special chapters have been brought together in this book after a rigorous review process in the broad areas of backstepping control of nonlinear systems. Special importance was given to chapters offering practical solutions and novel methods for the recent research problems in the mathematical modeling and applications of backstepping control. This book discusses trends and applications of control systems via the backstepping approach.

Objectives of the book This new book focuses upon the latest techniques of backstepping control such as active backstepping control, adaptive backstepping control, fuzzy backstepxix

xx Preface

ping control, and adaptive fuzzy backstepping control. This book also gives modeling and control of many control systems arising in science and engineering via backstepping control. Numerous simulations using MATLAB and circuit design are given to illustrate the main results of theory and applications of backstepping control of nonlinear control systems.

Organization of the book This book consists of 20 chapters.

Book features • The book chapters deal with the recent research problems in the areas of backstepping control. • The book chapters present various techniques of backstepping control such as active backstepping control, adaptive backstepping control, fuzzy backstepping control, and adaptive fuzzy backstepping control. • The book chapters contain a literature survey with a long list of references. • The book chapters present an exposition of the research problem, methodology, block diagrams, and mathematical techniques. • The book chapters are judiciously illustrated with numerical examples and simulations. • The book chapters discuss details of engineering applications and future research areas.

Audience The book is primarily meant for researchers from academia and industry, who are working on nonlinear systems in the research areas - electrical engineering, control engineering, mechanical engineering, and computer science. The book can also be used at the graduate or advanced undergraduate level as a text-book or major reference for courses such as power systems, control systems, electrical devices, scientific modeling, and computational science.

Acknowledgments As the editors, we hope that the chapters in this book will stimulate further research in the mathematical techniques of backstepping control, and utilize them in real-world applications. We hope sincerely that this book, covering so many different topics, will be very useful for all readers. We would like to thank all the reviewers for their diligence in reviewing the chapters. Special thanks go to Elsevier, especially the book Editorial team.

Preface

xxi

Prof. Sundarapandian Vaidyanathan, D.Sc. Professor and Dean, Research and Development Centre, Vel Tech University, Chennai, India Prof. Ahmad Taher Azar, PhD, IEEE senior Member, ISA Member Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

Chapter 1

An introduction to backstepping control Sundarapandian Vaidyanathana and Ahmad Taher Azarb,c a Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, b Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, c Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

1.1

Introduction

Scientific modeling of dynamical systems with state space models and effective control designs for the system models are important areas of research in science and engineering (Levine, 2010). Dynamical systems with exotic properties such as oscillations, bifurcations, chaos, and hyperchaos arise in various scientific and engineering fields such as oscillations (Mir and Tahani, 2020; Stender et al., 2020; Alain et al., 2020; Jafari et al., 2019), circuits (Pitchaimuthu and Kathamuthu, 2020; Bogris et al., 2007; Wang et al., 2019; Wilson, 2019), neural networks (Lin and Wang, 2020; Vaidyanathan et al., 2020; Liu et al., 2013), chemical reactors (Nieto-Villar and Velarde, 2000; Awal et al., 2019), memristive systems (Dong et al., 2020; Sadecki and Marszalek, 2019; Chen et al., 2019; Yuan and Li, 2019), plasma models (Mandi et al., 2019; Koh, 2019; Yang and Qi, 2019), mechanical systems (He et al., 2020; Tusset et al., 2020), lasers (Malica et al., 2020; Chao et al., 2020; Guo et al., 2020; Bonatto, 2018), ecology models (Panday et al., 2020; Samanta et al., 2020). In the last few decades, various control methods have been developed for linear and nonlinear dynamical systems with specific control design goals such as local or global stabilization, output regulation, and synchronization in the control literature (Zhao et al., 2020; Lu et al., 2020; Niu et al., 2020; Li and Long, 2020; Braverman and Rodkina, 2020; Liu and Huang, 2020; Bin and Marconi, 2020; Gu et al., 2019b,a; Zhou et al., 2019; Ouannas et al., 2017). Active state feedback control or output feedback control is a common method for the local or global stabilization of nonlinear control systems (Yuan et al., 2020, 2019; Pisarki and My´sli´nski, 2019; Scaramal et al., 2019; Adib Yaghmaie et al., 2019; Singh et al., 2018a,b). Adaptive feedback control is applied for the stabilization or regulation of control systems when the system parameters are not available for measurement (Hajiloo et al., 2019; Gong et al., 2019; Arif, 2019; Wu et al., 2019; Wang et al., 2015; Vaidyanathan and Azar, 2016d,f,a,c,e). Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00008-8 Copyright © 2021 Elsevier Inc. All rights reserved.

1

2 Backstepping Control of Nonlinear Dynamical Systems

Sliding mode control is a popular variable structure control method that modifies the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to slide along a surface called the sliding manifold, which is a cross-section of the system’s normal behavior (Solis et al., 2020; Armghan et al., 2020; Gang, 2020; Sivaperumal, 2016). In the control design of large and complex systems, passive control is often used to stabilize the control systems. Passive control has many applications in the stabilization and regulation of control systems (Sambas et al., 2019; Sangpet and Kuntanapreeda, 2020; Zhou et al., 2020; Tian et al., 2019). The design of feedback controls via control-Lyapunov functions is a useful technique in the control literature (Khalil, 2002). However, it is a difficult exercise to determine a suitable control-Lyapunov function for a general nonlinear control system. In such cases, one can try to find a local control or global control and establish the stability of the closed-loop system after the implementation of feedback control with the help of Lyapunov stability theory (Khalil, 2002) or center manifold theory (Carr, 1981). The backstepping control method is a recursive design procedure that links the choice of a control Lyapunov function with the design of a feedback controller and guarantees global asymptotic stability of strict feedback systems (Kokotovic, 1992; Kokotovic and Arcak, 2001; Krsti´c et al., 1995; Vaidyanathan et al., 2018b, 2015; Shukla et al., 2018; Vaidyanathan and Azar, 2016b). The integrator backstepping control method (Kokotovic and Arcak, 2001) helps to overcome the limitations of the feedback linearization approach in the control literature. The block backstepping control method (Krsti´c et al., 1995) is a general backstepping control method with more applicability in the control literature. The adaptive backstepping control method is a modified form of backstepping control method that uses estimates for unknown parameters in the control system for general applications (Tran et al., 2020; Yang and Zheng, 2020; Roy et al., 2019). The robust backstepping control method is an effective backstepping technique for control systems with uncertainties (Zhang and Li, 2019; Zheng and Yao, 2019; Yan et al., 2019; Liu et al., 2019). In this chapter, we describe an introduction to backstepping control with many basic examples and simulations. The backstepping control examples are established with Lyapunov stability theory for linear and nonlinear systems (Khalil, 2002). We give basic examples for active backstepping control method for linear and nonlinear control examples. We also state some basic results for active backstepping control method (Khalil, 2002; Krsti´c et al., 1995).

1.2 Backstepping design for a 2-D linear system In this section, we consider the linear system given by ξ˙1 = aξ1 + ξ2 ,

(1.1a)

ξ˙2 = v,

(1.1b)

An introduction to backstepping control Chapter | 1

3

where ξ1 , ξ2 are the states, a is a positive constant, and v is a backstepping control to be designed. In Eq. (1.1a), ξ2 is considered as a virtual controller. Thus, we can rewrite Eq. (1.1a) as ξ˙1 = aξ1 + w,

(1.2)

which is a first order linear system with w as a control input function. For the stability analysis of (1.2), we consider the candidate Lyapunov function 1 V1 (ξ1 ) = ξ12 . (1.3) 2 Clearly, V1 is quadratic and positive definite on R. We find that V˙1 = ξ1 ξ˙1 = ξ1 (aξ1 + w).

(1.4)

We choose the virtual controller w as w = −(a + α)ξ1 (α > 0).

(1.5)

V˙1 = −αξ12 ,

(1.6)

Then (1.4) becomes

which is quadratic and negative definite on R. Thus, by Lyapunov stability theory (Khalil, 2002), the closed-loop system ξ˙1 = aξ1 + w = −αξ1

(1.7)

is globally asymptotically stable. In order for the state ξ1 to be globally asymptotically stable with the desired roots as in Eq. (1.7), the state ξ2 must be equal to the virtual controller w. Since ξ2 and w start from different initial values, we must direct our control design to force ξ2 (t) to track the virtual controller w(t). Therefore, the integrator backstepping control v is designed to regulate the following output: z = ξ2 − w = ξ2 + (a + α)ξ1 .

(1.8)

ξ2 = z − (a + α)ξ1 .

(1.9)

We note that

Thus, we can write Eq. (1.1a) as ξ˙1 = aξ1 + ξ2 = aξ1 + z − (a + α)ξ1 = z − αξ1 .

(1.10)

Furthermore, we note that z˙ = ξ˙2 + (a + α)ξ˙1 = v + (a + α)(z − αξ1 ) = (a + α)z − α(a + α)ξ1 + v. (1.11)

4 Backstepping Control of Nonlinear Dynamical Systems

In the (ξ1 , z) coordinates, the linear system (1.1) can be represented as ξ˙1 = −αξ1 + z, z˙ = (a + α)z − α(a + α)ξ1 + v.

(1.12a) (1.12b)

Finally, we consider the total Lyapunov function given by  1 1 2 ξ1 + z 2 . V (ξ1 , z) = V1 (ξ1 ) + z2 = 2 2

(1.13)

We find that V˙ = ξ1 ξ˙1 + z˙z = ξ1 (−αξ1 + z) + z[(a + α)z − α(a + α)ξ1 + v],

(1.14)

which can be simplified as V˙ = −αξ12 + z[ξ1 + (a + α)z − α(a + α)ξ1 + v].

(1.15)

We consider the integrator backstepping control law given by v = −ξ1 − (a + α + β)z + α(a + α)ξ1 (β > 0).

(1.16)

Substituting (1.16) into (1.15), we obtain V˙ = −αξ12 − βz2 ,

(1.17)

which is globally negative definite on R2 . Hence, by Lyapunov stability theory (Khalil, 2002), we conclude that the linear system (1.12) is globally asymptotically stable. Hence, (ξ1 (t), z(t)) → 0 asymptotically as t → ∞ for all initial conditions ξ1 (0), z(0) ∈ R. From (1.9), it is immediate that ξ2 (t) also converges to the origin for all ξ2 (0) ∈ R. For numerical simulations, we take a = 3 and the gain constants as α = β = 6. We take the initial state of the linear system (1.1) as (ξ1 (0), ξ2 (0)) = (5.2, 3.4). Fig. 1.1 shows the time-history of the states ξ1 (t) and ξ2 (t) when the backstepping control law (1.16) is implemented. From Fig. 1.1, it is clear that both states ξ1 (t) and ξ2 (t) of the linear system (1.1) converge to zero as t → ∞.

1.3 Backstepping design for a 2-D nonlinear system In this section, we consider the nonlinear system given by ξ˙1 = aξ12 + ξ13 + ξ2 ,

(1.18a)

ξ˙2 = v,

(1.18b)

An introduction to backstepping control Chapter | 1

5

FIGURE 1.1 Time-history of the backstepping controlled states ξ1 (t), ξ2 (t) of the linear system (1.1) for a = 3, α = β = 6, and (ξ1 (0), ξ2 (0)) = (5.2, 3.4).

where ξ1 , ξ2 are the states, a is a positive constant, and v is a backstepping control to be designed. In Eq. (1.18a), ξ2 is considered as a virtual controller. Thus, we can rewrite Eq. (1.18a) as (1.19) ξ˙1 = aξ12 + ξ13 + w, which is a first order nonlinear system with w as a control input function. For the stability analysis of (1.19), we consider the candidate Lyapunov function 1 V1 (ξ1 ) = ξ12 . (1.20) 2 Clearly, V1 is quadratic and positive definite on R. We find that V˙1 = ξ1 ξ˙1 = ξ1 (aξ12 + ξ13 + w). (1.21) We choose the virtual controller w as w = −αξ1 − aξ12 − ξ13 (α > 0).

(1.22)

Then (1.21) becomes V˙1 = −αξ12 ,

(1.23)

which is quadratic and negative definite on R. Thus, by Lyapunov stability theory (Khalil, 2002), the closed-loop system ξ˙1 = aξ12 + ξ13 + w = −αξ1 is globally asymptotically stable.

(1.24)

6 Backstepping Control of Nonlinear Dynamical Systems

In order for the state ξ1 to be globally asymptotically stable with the desired roots as in Eq. (1.7), the state ξ2 must be equal to the virtual controller w. Since ξ2 and w start from different initial values, we must direct our control design to force ξ2 (t) to track the virtual controller w(t). Therefore, the integrator backstepping control v is designed to regulate the following output: z = ξ2 − w = ξ2 + αξ1 + aξ12 + ξ13 .

(1.25)

ξ2 = z − αξ1 − aξ12 − ξ13 .

(1.26)

We note that

Thus, we can write Eq. (1.18a) as ξ˙1 = −αξ1 + z.

(1.27)

Furthermore, we note that z˙ = ξ˙2 + (α + 2aξ1 + 3ξ12 )ξ˙1 = (α + 2aξ1 + 3ξ12 )(−αξ1 + z) + v.

(1.28)

In the (ξ1 , z) coordinates, the nonlinear plant (1.18) can be represented as ξ˙1 = −αξ1 + z, z˙ = (α + 2aξ1 + 3ξ12 )(−αξ1 + z) + v,

(1.29a) (1.29b)

Finally, we consider the total Lyapunov function given by  1 1 2 ξ1 + z 2 . V (ξ1 , z) = V1 (ξ1 ) + z2 = 2 2

(1.30)

We find that V˙ = ξ1 ξ˙1 + z˙z = ξ1 (−αξ1 + z) + z[(α + 2aξ1 + 3ξ12 )(−αξ1 + z) + v], (1.31) which can be simplified as V˙ = −αξ12 + z[ξ1 + (α + 2aξ1 + 3ξ12 )(−αξ1 + z) + v].

(1.32)

We consider the integrator backstepping control law given by v = −ξ1 − (α + 2aξ1 + 3ξ12 )(−αξ1 + z) − βz, (β > 0).

(1.33)

Substituting (1.33) into (1.32), we obtain V˙ = −αξ12 − βz2 . It is clear that V˙ is globally negative definite on R2 .

(1.34)

An introduction to backstepping control Chapter | 1

7

Hence, by Lyapunov stability theory (Khalil, 2002), we conclude that the nonlinear system (1.29) is globally asymptotically stable. Hence, (ξ1 (t), z(t)) → 0 asymptotically as t → ∞ for all initial conditions ξ1 (0), z(0) ∈ R. From (1.26), it is immediate that ξ2 (t) also converges to the origin for all ξ2 (0) ∈ R. For numerical simulations, we take a = 4 and the gain constants as α = β = 6. We take the initial state of the nonlinear system (1.18) as (ξ1 (0), ξ2 (0)) = (2.8, 5.3). Fig. 1.2 shows the time-history of the states ξ1 (t) and ξ2 (t) when the backstepping control law (1.33) is implemented. From Fig. 1.2, it is clear that both states ξ1 (t) and ξ2 (t) of the nonlinear system (1.18) converge to zero as t → ∞.

FIGURE 1.2 Time-history of the backstepping controlled states ξ1 (t), ξ2 (t) of the nonlinear system (1.18) for a = 4, α = β = 6, and (ξ1 (0), ξ2 (0)) = (2.8, 5.3).

1.4 Backstepping design for a 3-D linear system In this section, we consider the linear system given by ξ˙1 = aξ1 + ξ2 ,

(1.35a)

ξ˙2 = ξ3 ,

(1.35b)

ξ˙3 = v,

(1.35c)

where ξ1 , ξ2 , ξ3 are the states, a is a positive constant, and v is a backstepping control to be designed.

8 Backstepping Control of Nonlinear Dynamical Systems

In Eq. (1.35a), ξ2 is considered as a virtual controller. Thus, we can rewrite Eq. (1.35a) as ξ˙1 = aξ1 + w1 ,

(1.36)

which is a first order linear system with w1 as a control input function. For the stability analysis of (1.36), we consider the candidate Lyapunov function 1 V1 (ξ1 ) = ξ12 . (1.37) 2 Clearly, V1 is quadratic and positive definite on R. We find that V˙1 = ξ1 ξ˙1 = ξ1 (aξ1 + w1 ).

(1.38)

We choose the virtual controller w1 as w1 = −(a + α)ξ1 (α > 0).

(1.39)

V˙1 = −αξ12 .

(1.40)

Then (1.38) becomes Clearly, V˙1 is quadratic and negative definite on R. Thus, by Lyapunov stability theory (Khalil, 2002), the closed-loop system ξ˙1 = aξ1 + w1 = −αξ1

(1.41)

is globally asymptotically stable. In order for the state ξ1 to be globally asymptotically stable with the desired roots as in Eq. (1.41), the state ξ2 must be equal to the virtual controller w1 . Since ξ2 and w1 start from different initial values, we must direct our control design to force ξ2 (t) to track the virtual controller w1 (t). Therefore, the integrator backstepping control is designed to regulate the following output: z1 = ξ2 − w1 = ξ2 + (a + α)ξ1 .

(1.42)

ξ2 = z1 − (a + α)ξ1 .

(1.43)

We note that

Thus, we can write Eq. (1.35a) as ξ˙1 = aξ1 + ξ2 = aξ1 + z1 − (a + α)ξ1 = z1 − αξ1 .

(1.44)

Furthermore, we note that z˙ 1 = ξ˙2 + (a + α)ξ˙1 = ξ3 + (a + α)(z1 − αξ1 ) = (a + α)z1 − α(a + α)ξ1 + ξ3 . (1.45)

An introduction to backstepping control Chapter | 1

9

In the (ξ1 , z1 , ξ3 ) coordinates, the linear system (1.35) can be represented as follows: ξ˙1 = −αξ1 + z1 ,

(1.46a)

z˙ 1 = −α(a + α)ξ1 + (a + α)z1 + ξ3 ,

(1.46b)

ξ˙3 = v.

(1.46c)

In (1.46b), we consider ξ3 as a virtual controller. Thus, we consider the 2-D linear system given by ξ˙1 = −αξ1 + z1 ,

(1.47a)

z˙ 1 = −α(a + α)ξ1 + (a + α)z1 + w2 .

(1.47b)

Next, we consider the Lyapunov function given by  1 1 2 ξ1 + z12 . V2 (ξ1 , z1 ) = V1 (ξ1 ) + z12 = 2 2

(1.48)

We find that V˙2 = ξ1 ξ˙1 + z1 z˙ 1 = ξ1 (−αξ1 + z1 ) + z1 [−α(a + α)ξ1 + (a + α)z1 + w2 ]. (1.49) Simplifying, we get V˙2 = −αξ12 + z1 [ξ1 − α(a + α)ξ1 + (a + α)z1 + w2 ].

(1.50)

We define the virtual control w2 as w2 = −ξ1 + α(a + α)ξ1 − (a + α + β)z1 (β > 0).

(1.51)

Substituting (1.51) into (1.50), we get V˙2 = −αξ12 − βz12 .

(1.52)

Clearly, V˙2 is globally negative definite on R2 . This shows that the linear system (1.47) is globally asymptotically stable by the action of the control law (1.51). Since ξ3 and w2 start from different initial values, we must direct our control design to force ξ3 (t) to track the virtual controller w2 (t). Therefore, the integrator backstepping control is designed to regulate the following output: z2 = ξ3 − w2 = ξ3 + ξ1 − α(a + α)ξ1 + (a + α + β)z1 .

(1.53)

We note that ξ3 = z2 − ξ1 + α(a + α)ξ1 − (a + α + β)z1 .

(1.54)

10 Backstepping Control of Nonlinear Dynamical Systems

In the (ξ1 , z1 , z2 ) coordinates, the linear system (1.46) can be represented as follows: ξ˙1 = −αξ1 + z1 ,

(1.55a)

z˙ 1 = −ξ1 − βz1 + z2 ,

(1.55b)

z˙ 2 = [1 − α(a + α)](−αξ1 + z1 ) + (a + α + β)(−ξ1 − βz1 + z2 ) + v. (1.55c) Finally, we consider the total Lyapunov function given by  1 1 2 V (ξ1 , z1 , z2 ) = V2 (ξ1 , z1 ) + z22 = ξ1 + z12 + z22 . 2 2

(1.56)

We find that V˙ = ξ1 ξ˙1 + z1 z˙1 + z2 z˙2 .

(1.57)

A simple calculation gives V˙ = −αξ12 − βz12 + z2 S,

(1.58)

where S = z1 + [1 − α(a + α)](−αξ1 + z1 ) + (a + α + β)(−ξ1 − βz1 + z2 ) + v. (1.59) We consider the integrator backstepping control law given by v = −z1 − [1 − α(a + α)](−αξ1 + z1 ) − (a + α + β)(−ξ1 − βz1 + z2 ) − γ z2 (γ > 0).

(1.60)

Substituting (1.60) into (1.58), we obtain V˙ = −αξ12 − βz12 − γ z22 ,

(1.61)

which is globally negative definite on R3 . Hence, by Lyapunov stability theory (Khalil, 2002), we conclude that the linear system (1.55) is globally asymptotically stable. Hence, (ξ1 (t), z1 (t), z2 (t)) → 0 asymptotically as t → ∞ for all initial conditions ξ1 (0), z1 (0), z2 (0) ∈ R. From (1.43) and (1.54), it is immediate that ξ2 (t) and ξ3 (t) also converge to the origin for all ξ2 (0), ξ3 (0) ∈ R. For numerical simulations, we take a = 3 and the gain constants as α = β = γ = 6. We take the initial state of the linear system (1.35) as (ξ1 (0), ξ2 (0), ξ3 (0)) = (12.4, 7.5, 10.9).

An introduction to backstepping control Chapter | 1

11

Fig. 1.3 shows the time-history of the states ξ1 (t), ξ2 (t), and ξ3 (t) when the backstepping control law (1.60) is implemented. From Fig. 1.3, it is clear that all the three states ξ1 (t), ξ2 (t), and ξ3 (t) of the linear system (1.35) converge to zero as t → ∞.

FIGURE 1.3 Time-history of the backstepping controlled states ξ1 (t), ξ2 (t), ξ3 (t) of the linear system (1.35) for a = 3, α = β = γ = 6, and (ξ1 (0), ξ2 (0), ξ3 (0)) = (12.4, 7.5, 10.9).

1.5 Backstepping design for the 3-D Vaidyanathan jerk chaotic system In this section, we first study the properties of the 3-D Vaidyanathan jerk chaotic system (Vaidyanathan et al., 2018a) given by ξ˙1 = ξ2 ,

(1.62a)

ξ˙2 = ξ3 ,

(1.62b)

ξ˙3 = aξ1 − bξ2 − cξ3 − ξ12 − ξ22 .

(1.62c)

In the jerk system (1.62), ξ1 , ξ2 , ξ3 are the states and a, b, c are positive parameters. The 3-D system (1.62) is a nonlinear system with two quadratic nonlinearities in (1.62c). Vaidyanathan et al. (2018a,b) showed that the jerk system (1.62) is chaotic when the system parameters take the values a = 7.3, b = 4, and c = 0.9. Indeed, the Lyapunov exponents of the jerk system (1.62) for (a, b, c) = (7.3, 4, 0.9) and initial state X(0) = (0.3, 0.2, 0.3) are found as ψ1 = 0.1810, ψ2 = 0, and ψ3 = −1.1810. Thus, the Vaidyanathan jerk system (1.62) is a chaotic and dissipative system with a strange attractor. It can be easily verified using Lyapunov stability

12 Backstepping Control of Nonlinear Dynamical Systems

theory (Khalil, 2002) that the Vaidyanathan jerk system (1.62) has two unstable equilibrium points. The phase plots of the Vaidyanathan jerk chaotic system (1.62) are shown in Fig. 1.4.

FIGURE 1.4 Phase plots of the Vaidyanathan jerk chaotic system (1.62) for (a, b, c) = (7.3, 4, 0.9) and initial state X0 = (0.3, 0.2, 0.3). (A) (ξ1 , ξ2 )-plane; (B) (ξ2 , ξ3 )-plane.

Next, we consider the backstepping control design problem for the controlled Vaidyanathan jerk system given by the following dynamics: ξ˙1 = ξ2 ,

(1.63a)

ξ˙2 = ξ3 ,

(1.63b)

ξ˙3 = aξ1 − bξ2 − cξ3 − ξ12 − ξ22 + v.

(1.63c)

In (1.63), ξ1 , ξ2 , ξ3 are the states, a, b, c are positive constants, and v is a backstepping control to be designed. In Eq. (1.63a), ξ2 is considered as a virtual controller. Thus, we can rewrite Eq. (1.63a) as follows: ξ˙1 = w1 ,

(1.64)

which is a first order system with w1 as a control input function. For the stability analysis of (1.64), we consider the candidate Lyapunov function 1 (1.65) V1 (ξ1 ) = ξ12 . 2 Clearly, V1 is quadratic and positive definite on R. We find that V˙1 = ξ1 ξ˙1 = ξ1 w1 .

(1.66)

We choose the virtual controller w1 as w1 = −ξ1 .

(1.67)

An introduction to backstepping control Chapter | 1

13

Then (1.66) becomes V˙1 = −ξ12 .

(1.68)

Clearly, V˙1 is quadratic and negative definite on R. Thus, by Lyapunov stability theory (Khalil, 2002), the closed-loop system ξ˙1 = w1 = −ξ1

(1.69)

is globally asymptotically stable. In order for the state ξ1 to be globally asymptotically stable with the desired roots as in Eq. (1.69), the state ξ2 must be equal to the virtual controller w1 . Since ξ2 and w1 start from different initial values, we must direct our control design to force ξ2 (t) to track the virtual controller w1 (t). Therefore, the integrator backstepping control is designed to regulate the following output: z1 = ξ2 − w1 = ξ2 + ξ1 .

(1.70)

ξ2 = z1 − ξ1 .

(1.71)

We note that

Thus, we can write Eq. (1.63a) as ξ˙1 = ξ2 = z1 − ξ1 .

(1.72)

z˙ 1 = ξ˙2 + ξ˙1 = ξ3 + z1 − ξ1 .

(1.73)

Furthermore, we note that

In the (ξ1 , z1 , ξ3 ) coordinates, the chaotic system (1.63) can be represented as follows: ξ˙1 = z1 − ξ1 ,

(1.74a)

z˙ 1 = z1 − ξ1 + ξ3 ,

(1.74b)

ξ˙3 = aξ1 − ξ12 − b(z1 − ξ1 ) − (z1 − ξ1 )2 − cξ3 + v.

(1.74c)

In (1.74b), we consider ξ3 as a virtual controller. Thus, we consider the 2-D nonlinear system given by ξ˙1 = z1 − ξ1 ,

(1.75a)

z˙ 1 = z1 − ξ1 + w2 .

(1.75b)

Next, we consider the Lyapunov function given by  1 1 2 V2 (ξ1 , z1 ) = V1 (ξ1 ) + z12 = ξ1 + z12 . 2 2

(1.76)

14 Backstepping Control of Nonlinear Dynamical Systems

We find that V˙2 = ξ1 ξ˙1 + z1 z˙ 1 = ξ1 (z1 − ξ1 ) + z1 (z1 − ξ1 + w2 ).

(1.77)

Simplifying, we get V˙2 = −ξ12 + z1 (z1 + w2 ).

(1.78)

We define the virtual control w2 as w2 = −2z1 .

(1.79)

Substituting (1.79) into (1.78), we get V˙2 = −ξ12 − z12 .

(1.80)

Clearly, V˙2 is globally negative definite on R2 . This shows that the nonlinear system (1.75) is globally asymptotically stable by the action of the control law (1.79). Since ξ3 and w2 start from different initial values, we must direct our control design to force ξ3 (t) to track the virtual controller w2 (t). Therefore, the integrator backstepping control is designed to regulate the following output: z2 = ξ3 − w2 = ξ3 + 2z1 = ξ3 + 2(ξ2 + ξ1 ) = 2ξ1 + 2ξ2 + ξ3 .

(1.81)

We note that ξ3 = z2 − 2ξ1 − 2ξ2 .

(1.82)

In the (ξ1 , z1 , z2 ) coordinates, the nonlinear system (1.74) can be represented as follows: ξ˙1 = −ξ1 + z1 ,

(1.83a)

z˙ 1 = −ξ1 − z1 + z2 ,

(1.83b)

z˙ 2 = (a + b − 2)ξ1 + (2c − b − 2)z1 + (2 − c)z2 − ξ12 − (z1 − ξ1 )2 + v. (1.83c) Finally, we consider the total Lyapunov function given by  1 1 2 ξ1 + z12 + z22 . V (ξ1 , z1 , z2 ) = V2 (ξ1 , z1 ) + z22 = 2 2

(1.84)

We find that V˙ = ξ1 ξ˙1 + z1 z˙1 + z2 z˙2 .

(1.85)

An introduction to backstepping control Chapter | 1

15

A simple calculation gives V˙ = −ξ12 − z12 + z2 S,

(1.86)

where S = (a + b − 2)ξ1 + (2c − b − 1)z1 + (2 − c)z2 − ξ12 − (z1 − ξ1 )2 + v. (1.87) We consider the integrator backstepping control law given by v = −(a + b − 2)ξ1 − (2c − b − 1)z1 − (2 − c + k)z2 + ξ12 + (z1 − ξ1 )2 (k > 0).

(1.88)

Substituting (1.88) into (1.86), we obtain V˙ = −ξ12 − z12 − kz22 ,

(1.89)

which is globally negative definite on R3 . Hence, by Lyapunov stability theory (Khalil, 2002), we conclude that the linear system (1.83) is globally asymptotically stable. Hence, (ξ1 (t), z1 (t), z2 (t))→ 0 asymptotically as t → ∞ for all initial conditions ξ1 (0), z1 (0), z2 (0) ∈ R. From (1.71) and (1.82), it is immediate that ξ2 (t) and ξ3 (t) also converge to the origin for all ξ2 (0), ξ3 (0) ∈ R. For numerical simulations, we take the parameter values of the Vaidyanathan jerk system (1.63) as in the chaotic case, viz. (a, b, c) = (7.3, 4, 0.9). Also, we take the gain constant as k = 8. We take the initial state of the jerk chaotic system (1.35) as (ξ1 (0), ξ2 (0), ξ3 (0)) = (2.3, 8.1, 4.9). Fig. 1.5 shows the time-history of the states ξ1 (t), ξ2 (t), and ξ3 (t) when the backstepping control law (1.88) is implemented. From Fig. 1.5, it is clear that all the three states ξ1 (t), ξ2 (t), and ξ3 (t) of the jerk system (1.63) converge to zero as t → ∞. Fig. 1.6 shows the time-history of the outputs z1 (t) and z2 (t) of the jerk system (1.63). It is easy to see that both outputs z1 (t) and z2 (t) are regulated by the backstepping control v and that z1 (t) and z2 (t) converge to zero as t → ∞.

1.6 Backstepping control method In this section, we give a brief summary of the famous backstepping control method in the control literature (Khalil, 2002; Krsti´c et al., 1995). Backstepping control method is a recursive design procedure that links the choice of a control Lyapunov function with the design of a feedback controller and guarantees global asymptotic stability of strict feedback systems (Kokotovic, 1992; Kokotovic and Arcak, 2001; Krsti´c et al., 1995).

16 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 1.5 Time-history of the backstepping controlled states ξ1 (t), ξ2 (t), ξ3 (t) of the Vaidyanathan jerk chaotic system (1.63) for (a, b, c) = (7.3, 4, 0.9), k = 8, and (ξ1 (0), ξ2 (0), ξ3 (0)) = (2.3, 8.1, 4.9).

FIGURE 1.6 Time-history of the outputs z1 and z2 of the Vaidyanathan jerk chaotic system (1.63) for (a, b, c) = (7.3, 4, 0.9), k = 8, and (ξ1 (0), ξ2 (0), ξ3 (0)) = (2.3, 8.1, 4.9).

For the integrating backstepping control design, we consider the control system: ξ˙ = F (ξ ) + G(ξ )η,

(1.90a)

η˙ = v.

(1.90b)

In the system (1.90), X = (ξ, η) ∈ Rn+1 is the state vector and v ∈ R is the control input. The design goal is to find a backstepping control law v such that

An introduction to backstepping control Chapter | 1

17

X(t) → 0 as t → ∞. We assume that both F and G are known and they are continuously differentiable on Rn . We assume that F (0) = 0 and G(0) = 0. Fig. 1.7 shows the block diagram of the control system (1.90). We can view the control system (1.90) as a cascade connection of two components of which the first is an integrator.

FIGURE 1.7 Backstepping control design for the control system (1.90).

In the subsystem (1.90a), we consider η as a virtual controller. We assume that there exists a smooth feedback control law η = ϕ(ξ ) with ϕ(0) = 0 such that ξ = 0 is an asymptotically stable equilibrium of the first system ξ˙ = F (ξ ) + G(ξ )ϕ(ξ ).

(1.91)

We assume also that we know a Lyapunov function V1 (ξ ) that satisfies the inequality ∂V1 [F (ξ ) + G(ξ )ϕ(ξ )] ≤ −W (ξ ), ∂ξ

(1.92)

where W (ξ ) is positive definite on Rn . By adding and subtracting G(ξ )ϕ(ξ ) on the right hand side of (1.90a), we can rewrite the control system (1.90) as follows: ξ˙ = [F (ξ ) + G(ξ )ϕ(ξ )] + G(ξ )[η − ϕ(ξ )],

(1.93a)

η˙ = v.

(1.93b)

We now introduce the change of variables y = η − ϕ(ξ ).

(1.94)

The output y can be also viewed as error between the state η and the pseudocontrol ϕ(ξ ). Thus, a design goal in the backstepping control procedure is to find v so that y(t) → 0 as t → ∞. If we write the initial system (1.93) in the (ξ, y) coordinates, we obtain ξ˙ = [F (ξ ) + G(ξ )ϕ(ξ )] + G(ξ )y,

(1.95a)

y˙ = v − ϕ(ξ ˙ ).

(1.95b)

18 Backstepping Control of Nonlinear Dynamical Systems

Since F , G, and ϕ are known, we can express ϕ(ξ ˙ ) as follows: ∂ϕ [F (ξ ) + G(ξ )η]. ∂ξ

ϕ(ξ ˙ )=

(1.96)

We set u = v − ϕ(ξ ˙ ).

(1.97)

Then the transformed system (1.95) can be represented as follows: ξ˙ = [F (ξ ) + G(ξ )ϕ(ξ )] + G(ξ )y,

(1.98a)

y˙ = u.

(1.98b)

The transformed system (1.98) has the same form as the original control system (1.90). The advantage of the transformed system (1.98) is the following important observation: When the input is zero, the first subsystem (1.98a) is asymptotically stable at ξ = 0. In the backstepping control design, the knowledge of the control Lyapunov function V1 (ξ ) is utilized for the stabilization of the overall control system (1.98). Next, we consider the total Lyapunov function for the original system (1.90) given by V (ξ, η) = V1 (ξ ) +

1 2 1 y = V1 (ξ ) + [η − ϕ(ξ )]2 . 2 2

(1.99)

Next, we establish that V (ξ, η) is a positive definite function on Rn+1 . Since V1 (ξ ) is a positive definite function, it is immediate that V1 (ξ ) ≥ 0 for all ξ ∈ Rn and V1 (ξ ) = 0 ⇐⇒ ξ = 0.

(1.100)

From (1.99) and (1.100), it is immediate that V (ξ, η) ≥ 0 for all (ξ, η) ∈ Rn+1 . Next, we shall show that V (ξ, η) = 0 ⇐⇒ (ξ, η) = (0, 0).

(1.101)

Let (ξ, η) = (0, 0). Then ξ = 0 and η = 0. This implies that V1 (0) = 0 and ϕ(0) = 0. Hence, V (ξ, η) = 0. Next, suppose that V (ξ, η) = 0. Since V is a sum of two nonnegative numbers, it is immediate that V1 (ξ ) = 0 and η − ϕ(ξ ) = 0.

(1.102)

Since V1 is a positive definite function, ξ = 0. Since ϕ(0) = 0, we must have η = 0. Therefore, (ξ, η) = (0, 0).

An introduction to backstepping control Chapter | 1

19

Hence, we have shown that V is a positive definite function on Rn+1 . Next, we calculate the derivative of the candidate Lyapunov function V as follows: V˙

=

∂V1 ∂ξ



−W (ξ ) +

[F (ξ ) + G(ξ )ϕ(ξ )] + ∂V1 ∂z

∂V1 ∂z

G(ξ )y + yv

G(ξ )y + yu.

(1.103)

We choose the backstepping control law u as u=−

∂V1 G(ξ ) − ky (k > 0). ∂z

(1.104)

Substituting (1.104) into the inequality in (1.103), we obtain V˙ ≤ −W (ξ ) − ky 2 .

(1.105)

Hence, by Lyapunov stability theory (Khalil, 2002), we deduce that (ξ, η) = (0, 0) is an asymptotically stable equilibrium for the original system (1.90). From (1.97), we know that u = v − ϕ(ξ ˙ ). Hence, the required backstepping control law is given by v = u + ϕ(ξ ˙ )=

∂ϕ ∂V1 [F (ξ ) + G(ξ )η] − G(ξ ) − k[ξ − ϕ(ξ )] (k > 0). ∂ξ ∂z (1.106)

We summarize the above backstepping calculations in the following result. Theorem 1.1. (Backstepping theorem) (Krsti´c et al., 1995; Khalil, 2002) Consider the control system (1.90) defined on Rn+1 with smooth vector fields F and G with F (0) = 0 and G(0) = 0. Let η = ϕ(ξ ) be a stabilizing state feedback law for the subsystem (1.90a), where ϕ(0) = 0. Suppose that V1 (ξ ) is a Lyapunov function such that ∂V1 [F (ξ ) + G(ξ )ϕ(ξ )] ≤ −W (ξ ), ∂ξ

(1.107)

where W (ξ ) is positive definite on Rn . Then the backstepping control law v = u + ϕ(ξ ˙ )=

∂ϕ ∂V1 [F (ξ ) + G(ξ )η] − G(ξ ) − k[ξ − ϕ(ξ )] (k > 0) ∂ξ ∂z (1.108)

stabilizes the equilibrium (ξ, η) = (0, 0) of the system (1.90) with the total Lyapunov function V (ξ, η) = V1 (ξ ) +

1 [η − ϕ(ξ )]2 . 2

(1.109)

20 Backstepping Control of Nonlinear Dynamical Systems

Next, we consider the backstepping control design for a general system of the form ξ˙ = F (ξ ) + G(ξ )η,

(1.110a)

η˙ = α(ξ, η) + β(ξ, η)u,

(1.110b)

We take the control input u as u=

1 [v − α(ξ, η)]. β(ξ, η)

(1.111)

Substituting (1.111) into (1.110), we get ξ˙ = F (ξ ) + G(ξ )η,

(1.112a)

η˙ = v.

(1.112b)

We can apply the standard backstepping design to the control system (1.112). We can use recursive application of backstepping control design to stabilize systems that are in strict feedback form given as follows: ξ˙

=

F0 (ξ ) + G0 (ξ )η1 ,

η˙ 1

=

F1 (ξ, η1 ) + G1 (ξ, η1 )η2 ,

η˙ 2

=

F2 (ξ, η1 , η2 ) + G2 (ξ, η1 , η2 )η3 ,

.. .

.. .

η˙ k−1

=

Fk−1 (ξ, η1 , . . . , ηk−1 ) + Gk−1 (ξ, η1 , . . . , ηk−1 )ηk ,

η˙ k

=

Fk (ξ, η1 , . . . , ηk ) + Gk (ξ, η1 , . . . , ηk )u,

.. .

(1.113)

where ξ, η1 , η2 , . . . , ηk ∈ R are the states and u ∈ R is the control input. In (1.113), F0 , F1 , . . . , Fk , G0 , G1 , . . . , Gk are known smooth functions. It is noted that in the strict feedback form (1.113), ξ˙i depends only on the states ξ, η1 , . . . , ηi . In most cases, the feedback linearization approach to stabilizing the strict feedback control system (1.113) may lead to cancellation of useful nonlinear terms. However, the standard backstepping control design exhibits more flexibility than the feedback linearization approach since the backstepping design does not require that the final input–output dynamics is a linear system. The backstepping control design is a recursive procedure and using Lyapunov stability theory, a Lyapunov function is designed for the entire system (1.113).

An introduction to backstepping control Chapter | 1

21

1.7 Examples of backstepping control design In this section, we shall illustrate the standard backstepping control design (Theorem 1.1) with some simple examples. Example 1.1. We consider a 2-D nonlinear system given by ξ˙ = ξ 4 + sin ξ + η,

(1.114a)

η˙ = v,

(1.114b)

where ξ, η are the states and v is a backstepping control to be designed. In (1.114a), η is considered as a virtual control. First, we find a stabilizing control law η = ϕ(ξ ) for the subsystem (1.114a). A simple choice is ϕ(ξ ) = −ξ 4 − sin ξ − k1 ξ (k1 > 0).

(1.115)

Indeed, the implementation of η = ϕ(ξ ) in (1.114a) results in the closed-loop system ξ˙ = −k1 ξ,

(1.116)

which is globally asymptotically stable. This can be easily verified by considering the candidate Lyapunov function V1 (ξ ) =

1 2 ξ , 2

(1.117)

which is globally positive definite on R2 . It is easy to verify that V˙1 (ξ ) = ξ ξ˙ = −k1 ξ 2 ,

(1.118)

which is globally negative definite on R2 . Now, we introduce the change of variables z = η − ϕ(ξ ) = η + ξ 4 + sin ξ + k1 ξ.

(1.119)

Then we find that ξ˙ = ξ 4 + sin ξ + η = z − k1 ξ.

(1.120)

We also find that z˙ = η˙ + (4ξ 3 + cos ξ + k1 )ξ˙ = (4ξ 3 + cos ξ + k1 )(z − k1 ξ ) + v.

(1.121)

22 Backstepping Control of Nonlinear Dynamical Systems

In the (ξ, z) coordinates, we can express the 2-D nonlinear plant (1.114) as follows: ξ˙ = z − k1 ξ,

(1.122a)

z˙ = (4ξ 3 + cos ξ + k1 )(z − k1 ξ ) + v.

(1.122b)

In the standard backstepping control design, we consider the total Lyapunov function for the system (1.122) as follows: 1 1 V (ξ, z) = V1 (ξ ) + z2 = (ξ 2 + z2 ). 2 2

(1.123)

Then we find that V˙ = ξ ξ˙ + z˙z = ξ(z − k1 ξ ) + z[(4ξ 3 + cos ξ + k1 )(z − k1 ξ ) + v].

(1.124)

A simple calculation shows that V˙ = −k1 ξ 2 + z[ξ + (4ξ 3 + cos ξ + k1 )(z − k1 ξ ) + v].

(1.125)

Hence, we choose the backstepping control v as v = −ξ − (4ξ 3 + cos ξ + k1 )(z − k1 ξ ) − k2 z (k1 > 0, k2 > 0).

(1.126)

Substituting (1.126) into (1.125), we get V˙ = −k1 ξ 2 − k2 z2 ,

(1.127)

which is negative definite on R2 . Hence, by Lyapunov stability theory, the system (1.122) is globally asymptotically stable. Consequently, the original system (1.114) is also globally asymptotically stable. For numerical simulations, we take k1 = k2 = 5. We take the initial state as (ξ(0), η(0)) = (1.8, 7.4). Fig. 1.8 shows the time-history of the states ξ1 (t) and ξ2 (t) when the backstepping control law (1.126) is implemented. From Fig. 1.8, it is clear that both states ξ1 (t) and ξ2 (t) of the nonlinear system (1.114) converge to zero as t → ∞. Example 1.2. We consider a 2-D nonlinear system given by ξ˙ = ξ 2 + sinh ξ + η,

(1.128a)

η˙ = v,

(1.128b)

where ξ, η are the states and v is a backstepping control to be designed.

An introduction to backstepping control Chapter | 1

23

FIGURE 1.8 Time-history of the backstepping controlled states ξ1 (t), ξ2 (t) of the nonlinear system (1.114) for k1 = k2 = 5 and (ξ(0), η(0)) = (1.8, 7.4).

In (1.128a), η is considered as a virtual control. First, we find a stabilizing control law η = ϕ(ξ ) for the subsystem (1.128a). A simple choice is ϕ(ξ ) = −ξ 2 − sinh ξ − k1 ξ (k1 > 0).

(1.129)

Indeed, the implementation of η = ϕ(ξ ) in (1.128a) results in the closed-loop system ξ˙ = −k1 ξ,

(1.130)

which is globally asymptotically stable. This can be easily verified by considering the candidate Lyapunov function V1 (ξ ) =

1 2 ξ , 2

(1.131)

which is globally positive definite on R2 . It is easy to verify that V˙1 (ξ ) = ξ ξ˙ = −k1 ξ 2 ,

(1.132)

which is globally negative definite on R2 . Now, we introduce the change of variables z = η − ϕ(ξ ) = η + ξ 2 + sinh ξ + k1 ξ.

(1.133)

Then we find that ξ˙ = ξ 2 + sinh ξ + η = z − k1 ξ.

(1.134)

24 Backstepping Control of Nonlinear Dynamical Systems

We also find that z˙ = η˙ + (2ξ + cosh ξ + k1 )ξ˙ = (2ξ + cosh ξ + k1 )(z − k1 ξ ) + v.

(1.135)

In the (ξ, z) coordinates, we can express the 2-D nonlinear plant (1.128) as follows: ξ˙ = z − k1 ξ,

(1.136a)

z˙ = (2ξ + cosh ξ + k1 )(z − k1 ξ ) + v.

(1.136b)

In the standard backstepping control design, we consider the total Lyapunov function for the system (1.136) as follows: 1 1 V (ξ, z) = V1 (ξ ) + z2 = (ξ 2 + z2 ). 2 2

(1.137)

Then we find that V˙ = ξ ξ˙ + z˙z = ξ(z − k1 ξ ) + z[(2ξ + cosh ξ + k1 )(z − k1 ξ ) + v].

(1.138)

A simple calculation shows that V˙ = −k1 ξ 2 + z[ξ + (2ξ + cosh ξ + k1 )(z − k1 ξ ) + v].

(1.139)

Hence, we choose the backstepping control v as v = −ξ − (2ξ + cosh ξ + k1 )(z − k1 ξ ) − k2 z (k1 > 0, k2 > 0).

(1.140)

Substituting (1.140) into (1.139), we get V˙ = −k1 ξ 2 − k2 z2 ,

(1.141)

which is negative definite on R2 . Hence, by Lyapunov stability theory, the system (1.136) is globally asymptotically stable. Consequently, the original system (1.128) is also globally asymptotically stable. For numerical simulations, we take k1 = k2 = 6. We take the initial state as (ξ(0), η(0)) = (3.9, 8.2). Fig. 1.9 shows the time-history of the states ξ1 (t) and ξ2 (t) when the backstepping control law (1.140) is implemented. From Fig. 1.9, it is clear that both states ξ1 (t) and ξ2 (t) of the nonlinear system (1.128) converge to zero as t → ∞.

An introduction to backstepping control Chapter | 1

25

FIGURE 1.9 Time-history of the backstepping controlled states ξ1 (t), ξ2 (t) of the nonlinear system (1.128) for k1 = k2 = 6 and (ξ(0), η(0)) = (3.9, 8.2).

Example 1.3. We consider a 2-D nonlinear system given by ξ˙ = ξ 2 + ξ e−ξ + η,

(1.142a)

η˙ = v,

(1.142b)

where ξ, η are the states and v is a backstepping control to be designed. In (1.142a), η is considered as a virtual control. First, we find a stabilizing control law η = ϕ(ξ ) for the subsystem (1.142a). A simple choice is ϕ(ξ ) = −ξ 2 − ξ e−ξ − k1 ξ (k1 > 0).

(1.143)

Indeed, the implementation of η = ϕ(ξ ) in (1.142a) results in the closed-loop system ξ˙ = −k1 ξ,

(1.144)

which is globally asymptotically stable. This can be easily verified by considering the candidate Lyapunov function V1 (ξ ) =

1 2 ξ , 2

(1.145)

which is globally positive definite on R2 . It is easy to verify that V˙1 (ξ ) = ξ ξ˙ = −k1 ξ 2 , which is globally negative definite on R2 .

(1.146)

26 Backstepping Control of Nonlinear Dynamical Systems

Now, we introduce the change of variables z = η − ϕ(ξ ) = η + ξ 2 + ξ e−ξ + k1 ξ.

(1.147)

Then we find that ξ˙ = ξ 2 + ξ e−ξ + η = z − k1 ξ.

(1.148)

We also find that z˙ = η˙ + (2ξ + (1 − ξ )e−ξ + k1 )ξ˙ = (2ξ + (1 − ξ )e−ξ + k1 )(z − k1 ξ ) + v. (1.149) In the (ξ, z) coordinates, we can express the 2-D nonlinear plant (1.142) as follows: ξ˙ = z − k1 ξ,

(1.150a)

z˙ = (2ξ + (1 − ξ )e−ξ + k1 )(z − k1 ξ ) + v.

(1.150b)

In the standard backstepping control design, we consider the total Lyapunov function for the system (1.150) as follows: 1 1 V (ξ, z) = V1 (ξ ) + z2 = (ξ 2 + z2 ). 2 2

(1.151)

Then we find that V˙ = ξ ξ˙ + z˙z = ξ(z − k1 ξ ) + z[(2ξ + (1 − ξ )e−ξ + k1 )(z − k1 ξ ) + v]. (1.152) A simple calculation shows that V˙ = −k1 ξ 2 + z[ξ + (2ξ + cosh ξ + k1 )(z − k1 ξ ) + v].

(1.153)

Hence, we choose the backstepping control v as v = −ξ − (2ξ + (1 − ξ )e−ξ + k1 )(z − k1 ξ ) − k2 z (k1 > 0, k2 > 0). (1.154) Substituting (1.154) into (1.153), we get V˙ = −k1 ξ 2 − k2 z2 ,

(1.155)

which is negative definite on R2 . Hence, by Lyapunov stability theory, the system (1.150) is globally asymptotically stable. Consequently, the original system (1.142) is also globally asymptotically stable.

An introduction to backstepping control Chapter | 1

27

For numerical simulations, we take k1 = k2 = 6. We take the initial state as (ξ(0), η(0)) = (6.4, 1.9). Fig. 1.10 shows the time-history of the states ξ1 (t) and ξ2 (t) when the backstepping control law (1.154) is implemented. From Fig. 1.10, it is clear that both states ξ1 (t) and ξ2 (t) of the nonlinear system (1.142) converge to zero as t → ∞.

FIGURE 1.10 Time-history of the backstepping controlled states ξ1 (t), ξ2 (t) of the nonlinear system (1.142) for k1 = k2 = 6 and (ξ(0), η(0)) = (6.4, 1.9).

1.8 Conclusions Backstepping control method is a recursive design procedure that links the choice of a control Lyapunov function with the design of a feedback controller and guarantees global asymptotic stability of strict feedback systems. Backstepping control is a popular method and widely used in fields such as mechanical engineering, robotics, electrical engineering, power systems, thermomechanical systems, neural networks, and chaotic systems. Active backstepping control helps to overcome the limitations of the feedback linearization approach in the control literature (Khalil, 2002; Krsti´c et al., 1995). Block backstepping control is a general backstepping control method for nonlinear dynamical systems having a general form (Krsti´c et al., 1995). Adaptive backstepping control is a modified form of backstepping control that uses estimates for unknown parameters in the systems (Krsti´c et al., 1995). Robust backstepping control is an useful backstepping control method for linear and nonlinear control systems with uncertainties. We gave a basic introduction of the backstepping control with many basic examples and simulations. We also described some classical results for active backstepping control method (Khalil, 2002; Krsti´c et al., 1995).

28 Backstepping Control of Nonlinear Dynamical Systems

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Chapter 2

A new chaotic system without linear term, its backstepping control, and circuit design Viet-Thanh Phama,b , Sundarapandian Vaidyanathanc , Ahmad Taher Azard,e , and Vo Hoang Duyf a Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS),

Phenikaa University, Hanoi, Vietnam, b Phenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, Hanoi, Vietnam, c Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, d Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, e Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, f Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2.1

Introduction

For many years, chaos and its roles have been addressed in numerous studies (Lorenz, 1963; Strogatz, 1994; Sprott, 2003; Chen and Yu, 2003). The literature on chaos has highlighted several chaotic systems such as Lorenz’s system (Lorenz, 1963), Rössler’s system (Rössler, 1976), Chen’s system (Chen and Ueta, 1999), Sprott’s systems (Sprott, 1994, 2010), and so on (Wu et al., 2016; Wu and Wang, 2016; Radwan et al., 2017; Yu et al., 2010; Lin et al., 2015; Buscarino et al., 2016). Simple system with chaos has been an object of research because of its elegance (Sprott, 1997, 2000; Munmuangsaen et al., 2011; Minati et al., 2017). Previous research has established that five-term chaotic systems are the most elegant ones (Sprott, 2010). A five-term chaotic attractor was found by Munmuangsaen and Srisuchinwong (2009). Wei and Wang (2012) investigated chaotic attractors in a five-term system including an exponential quadratic term. Chaos synchronization of a five-term system was discovered by Chang and Kim (2013). Sun and Sprott (2009) studied bifurcations of fractional-order Lorenz system having five terms. It is interesting that Wang et al. (2019) introduced the circuit of a fractional five-term system. Moreover, a line-equilibrium system with five terms was reported (Pham et al., 2019). In this chapter, we study a new chaotic system without linear term. Properties and dynamics of the system are presented in Sections 2.2 and 2.3. We investiBackstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00009-X Copyright © 2021 Elsevier Inc. All rights reserved.

33

34 Backstepping Control of Nonlinear Dynamical Systems

gate backstepping control and synchronization for the system in Sections 2.4 and 2.5. The control method via backstepping approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of chaotic systems (Vaidyanathan et al., 2015; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015). A circuit for implementing the system is proposed in Section 2.6. Conclusions are drawn in Section 2.7.

2.2

Properties of the system

We consider the system including six terms, ⎧ ⎪ ⎨ x˙ y˙ ⎪ ⎩ z˙

=

ayz,

=

b − z2 ,

=

cx 3

(2.1)

+ yz,

where a, b, and c are three parameters. System (2.1) only has nonlinear terms. It is worth noting that there has been an increasing interest in a class of systems without linear terms (Xu and Wang, 2014; Mobayen et al., 2018; Zhang et al., 2018). By applying the coordinate transformation (x, y, z) → (−x, y, −z) ,

(2.2)

the invariance of the system (2.1) is confirmed. This also implies rotation symmetry for the system (2.1) about the y-axis. By setting ⎧ ⎪ ⎨ ayz = 0, (2.3) b − z2 = 0, ⎪ ⎩ 3 cx + yz = 0, we obtain the equilibrium points of the system (2.1) √ E1 (0, 0, b), √ E2 (0, 0, − b).

(2.4) (2.5)

The Jacobian matrix of the system (2.1) is given by ⎡ ⎢ J =⎣

0 0

az 0

3cx 2

z

⎤ ay ⎥ −2z ⎦ . y

(2.6)

A new chaotic system without linear term Chapter | 2

35

FIGURE 2.1 Bifurcation diagram of the system (2.1) for a = 1, c = 0.05, and b ∈ [1, 5].

Because of symmetry, by considering the Jacobian matrix at the equilibrium E1 , we get the characteristic equation λ3 + 2bλ = 0

(2.7)

λ1 = 0,

(2.8)

and the eigenvalues √ λ2,3 = ±j 2b.

(2.9)

This calculation shows that the system (2.1) is at a critical case at the rest points E1 and E2 .

2.3 Dynamics of the system We have found that the system displays different behaviors when varying the parameter b. We have displayed the bifurcation diagram and Lyapunov exponents of the system in the range of b from 1 to 5 while keeping a = 1, c = 0.05, and (x(0), y(0), z(0)) = (−1, 1, −1) (see Fig. 2.1 and Fig. 2.2). System generates chaos for b = 1 as illustrated in Figs. 2.3 and 2.4. Limit cycles of system (2.1) are presented in Fig. 2.5.

2.4 Backstepping control for the global stabilization of the new chaos system Here, we adopt backstepping control for globally stabilizing the trajectories of the new chaos system (2.1) for all initial conditions. The controlled chaos system is described by the 3D dynamics ⎧ ⎪ ⎨ x˙ = ayz + ux , (2.10) y˙ = b − z2 + uy , ⎪ ⎩ 3 z˙ = cx + yz + uz .

36 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 2.2 Maximum Lyapunov exponents of the system (2.1) for a = 1, c = 0.05 when varying the value of the parameter b from 1 to 5.

As a first step, we use active control to transform the system (2.10) to a system with triangular structure that aids backstepping control design. We consider the control law ⎧ ⎪ ⎨ ux = −ayz + y, (2.11) uy = −b + z2 + z, ⎪ ⎩ 3 uz = −cx − yz + vz , where vz is a backstepping control to be determined. Substituting (2.11) into (2.10), we get the new system in triangular form as ⎧ ⎨ x˙ y˙ ⎩ z˙

= y, = z, = vz .

(2.12)

We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(2.13)

where ηx = x.

(2.14)

Differentiating W1 with respect to t along the dynamics (2.12), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (x + y).

(2.15)

ηy = x + y.

(2.16)

We define

A new chaotic system without linear term Chapter | 2

37

FIGURE 2.3 Chaotic attractors of the system (2.1) in (A) x–y plane, (B) x–z plane, and (C) y–z plane for a = 1, b = 1, c = 0.05, and (x(0), y(0), z(0)) = (−1, 1, −1).

With the help of Eq. (2.16), we can express (2.15) as W˙ 1 = −ηx2 + ηx ηy . We proceed next with defining the Lyapunov function

1 1 2 ηx + ηy2 . W2 (ηx , ηy ) = W1 (ηx ) + ηy2 = 2 2

(2.17)

(2.18)

38 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 2.4 Chaotic attractor of the system (2.1) in x–y–z space for a = 1, b = 1, c = 0.05, and (x(0), y(0), z(0)) = (−1, 1, −1).

FIGURE 2.5 Representations of limit cycles in the system (2.1): (A) b = 3.5, and (B) b = 4.5.

Differentiating W2 with respect to t along the dynamics (2.12), we get W˙ 2 = −ηx2 − ηy2 + ηy (2x + 2y + z).

(2.19)

We define ηz as follows: ηz = 2x + 2y + z.

(2.20)

With the help of Eq. (2.20), we can express Eq. (2.19) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(2.21)

A new chaotic system without linear term Chapter | 2

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As a final step of the backstepping control design, we set the quadratic Lyapunov function W (ηx , ηy , ηz ) = W2 (ηx , ηy ) +

1 2 1 2 ηz = ηx + ηy2 + ηz2 . 2 2

(2.22)

It is clear that W is a positive define function on R 3 . Differentiating W with respect to t along the dynamics (2.12), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηz + ηy + η˙ z ).

(2.23)

A simple calculation yields the result W˙ = ηx2 − ηy2 − ηz2 + ηz (3x + 5y + 3z + vz ).

(2.24)

We define the control law vz as vz = −3x − 5y − 3z − Kηz ,

(2.25)

where we take K as a positive constant. Substituting (2.25) into (2.24), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(2.26)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that x = ηx , y = ηy − ηx , z = ηz − 2ηy .

(2.27)

As a consequence, it follows that (x(t), y(t), z(t)) → 0 exponentially as t → ∞. Substituting (2.25) into (2.11), the required backstepping control law is given by ⎧ ⎪ ⎨ ux = −ayz + y, (2.28) uy = −b + z2 + z, ⎪ ⎩ 3 uz = −cx − yz − (3 + 2K)x − (5 + 2K)y − (3 + K)z. Thus, we have proved the following result. Theorem 2.1. The backstepping control law defined via (2.28) with gain K > 0 globally and exponentially stabilizes the trajectories of the 3D chaos plant (2.10) for all initial states (x(0), y(0), z(0)) ∈ R 3 .

40 Backstepping Control of Nonlinear Dynamical Systems

For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b, c) = (1, 1, 0.05). We choose K = 6 and the initial state of the system (2.10) as x(0) = 6.2, y(0) = −3.4, and z(0) = 8.7. Fig. 2.6 shows the time-history of the backstepping controlled states x(t), y(t), and z(t). It is easy to note that the controlled states converge to zero exponentially by the action of the backstepping control law (2.28).

FIGURE 2.6 Time-history of the backstepping controlled states x(t), y(t), and z(t) for (a, b, c) = (1, 1, 0.05), K = 6, and (x(0), y(0), z(0)) = (6.2, −3.4, 8.7).

2.5 Backstepping control for the synchronization of the new chaos systems The term chaos synchronization has been used mostly in recent years as an indication of the field of studies at the interfaces between the control theory and the theory of dynamic systems which studies methods of controlling non-regular chaotic systems (Alain et al., 2020; Khan et al., 2020b,a; Ouannas et al., 2020, 2019b; Volos et al., 2018; Singh et al., 2017). In the first place Yamada and Fujisaka (Yamada and Fujisaka, 1983) studied the synchronization of the chaotic system with subsequent work by Pecora and Carroll (Pecora and Carroll, 1990; Carroll and Pecora, 1991). One way to explain the vulnerable dependency on initial conditions is the synchronization of chaos (Ouannas et al., 2019a; Alain et al., 2019; Vaidyanathan et al., 2018a). The synchronization of two chaotic systems that recognize the tendency for two or more systems to undergo closely related movements was shown to be coupled. Chaos synchronization is a problem of designing a link between both systems to idealize the chaotic time assessment (Pham et al., 2018; Khettab et

A new chaotic system without linear term Chapter | 2

41

al., 2018; Singh et al., 2018; Alain et al., 2018; Vaidyanathan et al., 2018b; Singh et al., 2018a). The response system output tracks the drive system output asymptotically, i.e. the master system output controls the slave system (Singh et al., 2018b; Grassi et al., 2017; Pham et al., 2017). Backstepping architecture is a sequential control technique characterized by step-by-step interlacing. Each step involves the transformation of the coordinates and the design of a virtual control based on the Lyapunov technique. Finally, under the global stability, the true controller is achieved. Strict feedback systems, blocking systems with strict feedback, and parametric-strict feedback systems can be implemented with a backstepping design (Azar et al., 2020; Shukla et al., 2018; Vaidyanathan et al., 2018c). Here, we adopt backstepping control for globally synchronizing the trajectories of a pair of new chaos systems considered as leader–follower systems. The leader chaos system is depicted by the 3D dynamics ⎧ ⎪ ⎨ x˙1 = ay1 z1 , (2.29) y˙1 = b − z12 , ⎪ ⎩ 3 z˙ 1 = cx1 + y1 z1 . The follower chaos system with controls is described by the 3D dynamics ⎧ ⎪ ⎨ x˙2 = ay2 z2 + ux , (2.30) y˙2 = b − z22 + uy , ⎪ ⎩ 3 z˙ 2 = cx2 + y2 z2 + uz , where ux , uy , uz are feedback controls to be determined. The synchronization chaos error is defined by means of the equations ex = x2 − x1 , ey = y2 − y1 , ez = z2 − z1 . Upon calculating the error dynamics, we obtain the following: ⎧ ⎪ ⎨ e˙x = a(y2 z2 − y1 z1 ) + ux , e˙y = −z22 + z12 + uy , ⎪ ⎩ e˙z = c(x23 − x13 ) + y2 z2 − y1 z1 + uz .

(2.31)

(2.32)

As a first step, we use active control to transform the error system (2.32) to an error system with triangular structure that aids backstepping control design. We consider the control law ⎧ ⎪ ⎨ ux = −a(y2 z2 − y1 z1 ) + ey , (2.33) uy = z22 − z12 + ez , ⎪ ⎩ 3 3 uz = −c(x2 − x1 ) − y2 z2 + y1 z1 + vz , where vz is a backstepping control to be determined.

42 Backstepping Control of Nonlinear Dynamical Systems

Substituting (2.33) into (2.32), we get the new error system in triangular form as ⎧ ⎨ e˙x = ey , (2.34) e˙ = ez , ⎩ y e˙z = vz . We start with the Lyapunov function W1 (ηx ) =

1 2 η , 2 x

(2.35)

where ηx = ex .

(2.36)

Differentiating W1 with respect to t along the error dynamics (2.34), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (ex + ey ).

(2.37)

η y = e x + ey .

(2.38)

We define

With the help of Eq. (2.16), we can express (2.37) as W˙ 1 = −ηx2 + ηx ηy .

(2.39)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

1 2 1 2 ηy = ηx + ηy2 . 2 2

(2.40)

Differentiating W2 with respect to t along the error dynamics (2.34), we get W˙ 2 = −ηx2 − ηy2 + ηy (2ex + 2ey + ez ).

(2.41)

We define ηz as follows: ηz = 2ex + 2ey + ez .

(2.42)

With the help of Eq. (2.42), we can express Eq. (2.41) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(2.43)

As a final step of the backstepping control design, we set the quadratic Lyapunov function W (ηx , ηy , ηz ) = W2 (ηx , ηy ) +

1 2 1 2 ηz = ηx + ηy2 + ηz2 . 2 2

(2.44)

A new chaotic system without linear term Chapter | 2

43

It is clear that W is a positive define function on R 3 . Differentiating W with respect to t along the error dynamics (2.34), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηz + ηy + η˙ z ).

(2.45)

A simple calculation yields the result W˙ = ηx2 − ηy2 − ηz2 + ηz (3ex + 5ey + 3ez + vz ).

(2.46)

We define the control law vz as vz = −3ex − 5ey − 3ez − Kηz ,

(2.47)

where we take the gain K as a positive constant. Substituting (2.47) into (2.46), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(2.48)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that ex = ηx , ey = ηy − ηx , ez = ηz − 2ηy .

(2.49)

As a consequence, it follows that (ex (t), ey (t), ez (t)) → 0 exponentially as t → ∞. Substituting (2.47) into (2.33), the required backstepping control law is given by ⎧ ⎪ ⎪ ux = −a(y2 z2 − y1 z1 ) + ey , ⎪ ⎨ uy = z22 − z12 + ez , (2.50) ⎪ uz = −c(x 3 − x 3 ) − y2 z2 + y1 z1 − (3 + 2K)ex ⎪ ⎪ 2 1 ⎩ − (5 + 2K)ey − (3 + K)ez . Thus, we have proved the following result. Theorem 2.2. The backstepping control law defined via (2.50) with positive gain K globally and exponentially synchronizes the trajectories of the 3D chaos plants (2.29) and (2.30) for all initial states. For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b, c) = (1, 1, 0.05). We choose K = 6. We take the initial state of the leader system (2.29) as x1 (0) = 5.8, y1 (0) = 2.6, and z(0) = −4.1. We also consider the initial state of the follower system (2.30) as x2 (0) = 1.7, y(0) = −5.9, and z(0) = 3.4.

44 Backstepping Control of Nonlinear Dynamical Systems

Figs. 2.7–2.9 display the exponential synchronization between the states of the leader system (2.29) and follower system (2.30). Fig. 2.10 displays the timehistory of the error variables ex (t), ey (t), ez (t) depicting the synchronization error between the chaos systems (2.29) and (2.30).

FIGURE 2.7 Synchronization between the states x1 and x2 of the leader–follower systems (2.29) and (2.30).

FIGURE 2.8 Synchronization between the states y1 and y2 of the leader–follower systems (2.29) and (2.30).

A new chaotic system without linear term Chapter | 2

45

FIGURE 2.9 Synchronization between the states z1 and z2 of the leader–follower systems (2.29) and (2.30).

FIGURE 2.10 Time-history of the chaos synchronization error between the leader–follower systems (2.29) and (2.30).

2.6 Circuit design Implementation of systems with chaos is important for developing chaos-based applications (Yalcin et al., 2004; Wang et al., 2008; Volos et al., 2012, 2013; Valli et al., 2014; Akgul et al., 2016; Vaidyanathan et al., 2019; Tolba et al., 2017a,b). In this section, we design an electronic circuit of system (2.1) by us-

46 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 2.11 Schematic of the circuit designed with common components.

ing electronic components. The schematic of the designed circuit is shown in Fig. 2.11. The circuit includes three operational amplifiers (U1 –U3 ), four multipliers, five resistors, three capacitors, and a DC voltage Vb . Three operational amplifiers (U1 –U3 ) are configured as three integrators. We have the circuital equation by applying Kirchhoff’s circuit laws: ⎧ dvC ⎪ ⎪ dt 1 = R1 C11 10V vC2 vC3 , ⎪ ⎨ dvC2 1 1 2 (2.51) ⎪ dt = − R2 C2 Vb − R3 C2 10V vC3 , ⎪ ⎪ ⎩ dvC3 1 1 3 dt = R C 100V 2 vC1 + R5 C3 10V vC2 vC3 , 4 3

where vC1 , vC2 , and vC3 are the voltages at the three integrators (U1 –U3 ). We t and select normalize Eq. (2.51) with τ = RC 

R2 = R, C1 = C2 = C3 = C.

Thus, the dimensionless system is described by ⎧ ⎪ ⎨ X˙ = aY Z, Y˙ = b − Z 2 , ⎪ ⎩ Z˙ = cX 3 + Y Z. It is noted that

⎧ ⎪ ⎪ ⎨ a= b= ⎪ ⎪ ⎩ c=

R R1 10 , −Vb 1V , R R4 100 .

(2.52)

(2.53)

(2.54)

A new chaotic system without linear term Chapter | 2

47

FIGURE 2.12 PSpice chaotic attractors of the designed circuit in (A) X–Y plane, (B) X–Z plane, and (C) Y –Z plane.

48 Backstepping Control of Nonlinear Dynamical Systems

In (2.53), the three variables X, Y , and Z correspond to voltages vC1 , vC2 , and vC3 , respectively. It is trivial to confirm that system (2.53) is equivalent to the theoretical system (2.1). For implementing system (2.1) for a = 1, b = 1, and c = 0.05, the values of electronic components are ⎧ ⎪ R1 = R3 = R5 = 1 k, ⎪ ⎪ ⎪ ⎪ ⎨ R2 = R = 10 k, (2.55) R4 = 2 k, ⎪ ⎪ ⎪ V = −1V , b DC ⎪ ⎪ ⎩ C1 = C2 = C3 = C = 10 nF. Circuital attractors in OrCAD are presented in Fig. 2.12. Also, refer to Fig. 2.3.

2.7 Conclusions A chaotic system without linear term is proposed in our work. We have investigated the system through its properties and dynamics. Moreover, backstepping control approach is developed for the global stabilization and synchronization of the new system. We also build a circuit to confirm the system. As future research, the system can be used in engineering applications such as secure communication systems.

Acknowledgment The authors acknowledge Prof. GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong for suggesting many helpful references.

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Wu, J., Wang, L., Chen, G., Duan, S., 2016. A memristive chaotic system with heart-shaped attractors and its implementation. Chaos, Solitons and Fractals 92, 20–29. Xu, Y., Wang, Y., 2014. A new chaotic system without linear term and its impulsive synchronization. Optik 125, 2526–2530. Yalcin, M.E., Suykens, J.A.K., Vandewalle, J., 2004. True random bit generation from a doublescroll attractor. IEEE Transactions on Circuits and Systems. I, Regular Papers 51, 1395–1404. Yamada, T., Fujisaka, H., 1983. Stability theory of synchronized motion in coupled-oscillator systems. II: The mapping approach. Progress of Theoretical Physics 70 (5), 1240–1248. Yu, S., Tang, W.K.S., Lu, J., Chen, G., 2010. Design and implementation of multi-wing butterfly chaotic attractors via Lorenz-type systems. International Journal of Bifurcation and Chaos 20, 29–41. Zhang, S., Zeng, Y., Li, Z., 2018. Chaos in a novel fractional order system without a linear term. International Journal of Non-Linear Mechanics 106, 1–12.

Chapter 3

A new chaotic jerk system with egg-shaped strange attractor, its dynamical analysis, backstepping control, and circuit simulation Sundarapandian Vaidyanathana , Viet-Thanh Phamb,c , and Ahmad Taher Azard,e a Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, b Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Hanoi, Vietnam, c Phenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, Hanoi, Vietnam, d Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, e Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

3.1

Introduction

In the recent decades, significant research attention has been devoted to the modeling and applications of dynamical systems exhibiting chaos (Xu et al., 2019; Gusso et al., 2019; Gatabazi et al., 2019; Singh and Roy, 2019; Cabanas et al., 2019; Ginoux et al., 2019; Jahanshahi et al., 2019; Daumann and Rech, 2019). Xu et al. (2019) proposed a chaotic system based on a circuit design involving a memristor model and a meminductor model. Gusso et al. (2019) analyzed the nonlinear dynamical model and the existence of chaos in suspended beam MEMS/NEMS resonators that are actuated by two-sided electrodes. Gatabazi et al. (2019) analyzed 2-D and 3-D Grey Lotka–Volterra Models (GLVM) and explored their application in cryptocurrencies such as Bitcoin, Litecoin, and Ripple. Singh and Roy (2019) studied microscopic chaos control of a chemical reactor system via nonlinear active plus proportional integral sliding mode control. Cabanas et al. (2019) discovered chaos in driven nano-magnets such as spin valves by using the magnetic energy and the magnetoresistance. Ginoux et al. (2019) discovered chaos in a dynamical system modeling the illicit drug consumption in a population comprising drug users and non-users. Jahanshahi et al. (2019) discussed a finance hyperchaos system via entropy analysis and Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00010-6 Copyright © 2021 Elsevier Inc. All rights reserved.

53

54 Backstepping Control of Nonlinear Dynamical Systems

control methods. Daumann and Rech (2019) reported hyperchaos in a heat-flux convection model. Jerk dynamical systems are important classes of mechanical systems. If y(t) denotes the displacement of a moving object, then Dy(t) = dy dt represents its 2

3

velocity, D 2 y(t) = ddt y2 its acceleration, and D 3 y(t) = ddt y3 its jerk. An autonomous jerk differential equation has the general representation given by D 3 x = F (x, Dx, D 2 x).

(3.1)

The jerk differential equation (3.1) can be displayed in a system form as ⎧ ⎪ x˙ ⎪ ⎨ y˙ ⎪ ⎪ ⎩ z˙

=

y,

=

z,

=

F (x, y, z).

(3.2)

Recently, much interest has been given to the finding of both jerk systems in the chaos literature (Vaidyanathan et al., 2018a; Vaidyanathan, 2017; Vaidyanathan et al., 2017; Vaidyanathan, 2016, 2015; El-Nabulsi, 2018). Vaidyanathan et al. (2018a) reported a new chaotic jerk system with two quadratic nonlinearities and discussed its applications to electronic circuit implementation and image encryption. Vaidyanathan (2017) reported a new 3-D chaotic jerk system with two cubic nonlinear terms and discussed its adaptive synchronization using backstepping control. Vaidyanathan et al. (2017) analyzed a new chaotic jerk system with its applications for circuit simulation and voice encryption. Vaidyanathan (2016) announced a new chaotic jerk system and discussed its adaptive synchronization using backstepping control. Vaidyanathan (2015) proposed a new chaotic jerk system with two quadratic nonlinearities. El-Nabulsi (2018) reported jerk and hyperjerk systems in the study of nonlocal effects in fluids, plasmas, and solar physics. This work reports a new 3-D dynamical jerk system with chaos. The proposed nonlinear jerk mechanical system with chaos has two nonlinear terms. This chapter aims to control chaotic behavior of a new jerk mechanical chaotic system with an egg-shaped strange attractor. By exploiting dynamics and properties of the jerk system, we show that the system displays an egg-shaped strange attractor and complex, dissipative, properties. Backstepping control is applied to control and synchronize the chaos in the proposed jerk system. The control method via the backstepping approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of chaotic systems (Vaidyanathan et al., 2015b; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015).

A new chaotic jerk system with egg-shaped strange attractor Chapter | 3

55

Finally, a circuit model using SPICE of the new jerk system with chaos is designed for practical implementation. We show that the SPICE outputs of the jerk system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019; Azar et al., 2020; Vaidyanathan et al., 2018b; Shukla et al., 2018; Vaidyanathan and Azar, 2016b; Vaidyanathan et al., 2015a).

3.2 System details In this chapter, we propose a new jerk mechanical system, which is described as the 3-D model ⎧ ⎪ ⎨ x˙ = y, y˙ = z, (3.3) ⎪ ⎩ 2 2 z˙ = −ax − by − cz − x − y . It is easy to see that the jerk system (3.3) can be also expressed as a jerk differential equation (D 3 + cD 2 + bD + a)x = −x 2 − (Dx)2 ,

(3.4)

d . where D = dt In (3.3) and (3.4), a, b, c are system constants. In this research work, it is established that the jerk plant (3.3) has chaos behavior for the choice of constants as

a = 7.3, b = 4, c = 0.9.

(3.5)

For the initial state X(0) = (0.2, 0.1, 0.3) and (a, b, c) = (7.3, 4, 0.9), the Lyapunov characteristic exponents of the jerk plant (3.3) are estimated as ψ1 = 0.1806, ψ2 = 0, ψ3 = −1.1806,

(3.6)

which confirms that the jerk plant (3.3) is a dissipative chaos system. It is also an easy calculation to determine the Kaplan–Yorke dimension of the jerk plant (3.3) as DKY = 2 +

ψ 1 + ψ2 = 2.1671. |ψ3 |

(3.7)

The Kaplan–Yorke dimension of the jerk plant (3.3) shows the complex characteristic of the dissipative jerk plant (3.3). The system (3.3) has self-excited strange attractor only. It is easy to see that the jerk plant (3.3) has two rest points at E0 = (0, 0, 0) and E1 = (−a, 0, 0). For positive values of a, both rest points are saddle-foci and unstable.

56 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 3.1 Bifurcation diagram of the jerk system for a = 7.3, c = 0.9 while changing b from 4 to 5.

FIGURE 3.2 Maximum Lyapunov exponents of the jerk system for a = 7.3, c = 0.9 when varying b from 4 to 5.

The jerk plant (3.3) is invariant when we change coordinates as (x, y, z) → (−x, −y, z), which shows that the jerk plant (3.3) has rotation symmetry with respect to the z-axis. We have changed the values of parameters to discover the dynamics of the proposed system (3.3). It is worth noting that the system (3.3) displays attractive behaviors when varying the parameter b. Figs. 3.1 and 3.2 present the bifurcation diagram and Lyapunov exponents of the system (3.3) for a = 7.3, c = 0.9, and (x(0), y(0), z(0)) = (0.2, 0.1, 0.3). It is easy to see that the system (3.3) displays periodical and chaotic behavior when varying the parameter b (b ∈ [4, 5]). For example, chaos is observed for b = 4 as shown in Figs. 3.3 and 3.4. Based on the chaotic phase portraits in Figs. 3.3 and 3.4, it is noted that the jerk plant has an egg-shaped strange attractor. Limit cycles in system (3.3) are observed for different values of b as illustrated in Fig. 3.5.

A new chaotic jerk system with egg-shaped strange attractor Chapter | 3

57

FIGURE 3.3 Chaotic attractor of the jerk system in x–y–z space.

3.3 Backstepping control of the jerk system Here, we adopt backstepping control for globally stabilizing the trajectories of the new jerk chaotic system (3.3) for all initial conditions. The controlled jerk system is described by the 3D dynamics ⎧ ⎪ ⎨ x˙ = y, y˙ = z, (3.8) ⎪ ⎩ 2 2 z˙ = −ax − by − cz − x − y + v, where v is a backstepping control to be determined. We start with the Lyapunov function W1 (ηx ) =

1 2 η , 2 x

(3.9)

where ηx = x.

(3.10)

Differentiating W1 with respect to t along the dynamics (3.8), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (x + y).

(3.11)

ηy = x + y.

(3.12)

We define

With the help of Eq. (3.12), we can express (3.11) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(3.13)

58 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 3.4 Chaotic attractors of the jerk system in (A) x–y plane, (B) x–z plane, and (C) y–z plane. It is noted that the strange attractor of the jerk system is egg-shaped.

A new chaotic jerk system with egg-shaped strange attractor Chapter | 3

59

FIGURE 3.5 Representations of the limit cycles in the jerk system: (A) b = 5.0, (B) b = 4.5.

We proceed next with defining the Lyapunov function  1 1 2 ηx + ηy2 . W2 (ηx , ηy ) = W1 (ηx ) + ηy2 = 2 2

(3.14)

Differentiating W2 with respect to t along the dynamics (3.8), we get W˙ 2 = −ηx2 − ηy2 + ηy (2x + 2y + z).

(3.15)

We define ηz as follows: ηz = 2x + 2y + z.

(3.16)

With the help of Eq. (3.16), we can express Eq. (3.15) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(3.17)

60 Backstepping Control of Nonlinear Dynamical Systems

As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 1 W (ηx , ηy , ηz ) = W2 (ηx , ηy ) + ηx2 = (ηx2 + ηy2 + ηz2 ). 2 2

(3.18)

It is clear that W is a positive definite function on R3 . Differentiating W with respect to t along the jerk dynamics (3.8), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ).

(3.19)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 − a)x + (3 − b)y + (2 − c)z − x 2 − y 2 + v]. (3.20) We define the control law v as v = −(2 − a)x − (3 − b)y − (2 − c)z + x 2 + y 2 − Kηz ,

(3.21)

where K is taken as a positive constant. Substituting (3.21) into (3.20), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(3.22)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that x = ηx , y = ηy − ηx , z = ηx − 2ηy .

(3.23)

As a consequence, it follows that (x(t), y(t), z(t)) → (0, 0, 0) exponentially as t → ∞. Thus, we have proved the following result. Theorem 3.1. The backstepping control law defined via (3.21) with gain K > 0 globally and exponentially stabilizes all the trajectories of the 3D jerk chaos plant (3.8) for all initial states (x(0), y(0), z(0)) ∈ R3 . For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b, c) = (7.3, 4, 0.9). We choose K = 6 and the initial state of the jerk chaos system (3.8) as x(0) = 8.4, y(0) = 2.7, and z(0) = 5.3. Fig. 3.6 shows the time-history of the backstepping controlled states x(t), y(t), and z(t). It is easy to see that the controlled states converge to zero exponentially by the action of the backstepping control law (3.21).

A new chaotic jerk system with egg-shaped strange attractor Chapter | 3

61

FIGURE 3.6 Time-history of the backstepping controlled states x(t), y(t), z(t) for (a, b, c) = (7.3, 4, 0.9), K = 6, and (x(0), y(0), z(0)) = (8.4, 2.7, 5.3).

3.4 Backstepping synchronization of the jerk system Recently, there have been published significant results in the literature for chaotic synchronization of complex systems (Vaidyanathan and Azar, 2016e,f; Ouannas et al., 2017b,c,a; Vaidyanathan and Azar, 2016d,c,a; Khan et al., 2020b,a; Alain et al., 2019; Singh et al., 2018; Khettab et al., 2018). A pair of systems called master and slave systems are considered for the synchronization process and the design goal of hybrid synchronization is to build a feedback mechanism so that complete synchronization and anti-synchronization co-exist in the synchronization process. Here, we adopt backstepping control for globally synchronizing the state trajectories of a pair of new jerk chaos systems considered as leader–follower systems. The leader chaos system is depicted by the 3D jerk dynamics ⎧ ⎪ ⎨ x˙1 = y1 , y˙1 = z1 , (3.24) ⎪ ⎩ z˙ 1 = −ax1 − by1 − cz1 − x12 − y12 . The follower chaos system is depicted by the 3D jerk dynamics ⎧ ⎪ ⎨ x˙2 = y2 , y˙2 = z2 , ⎪ ⎩ z˙ 2 = −ax2 − by2 − cz2 − x22 − y22 + v, where v is a backstepping control to be designed.

(3.25)

62 Backstepping Control of Nonlinear Dynamical Systems

The synchronization chaos error is defined by means of the equations ex = x2 − x1 , ey = y2 − y1 , ez = z2 − z1 . Upon calculating the error dynamics, we obtain the following: ⎧ ⎪ ⎨ e˙x = ey , e˙y = ez , ⎪ ⎩ e˙z = −aex − bey − cez − x22 − y22 + x12 + y12 + v.

(3.26)

(3.27)

We start with the Lyapunov function W1 (ηx ) =

1 2 η , 2 x

(3.28)

where ηx = ex .

(3.29)

Differentiating W1 with respect to t along the dynamics (3.27), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (ex + ey ).

(3.30)

η y = e x + ey .

(3.31)

We define

With the help of Eq. (3.31), we can express (3.30) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(3.32)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

 1 2 1 2 ηy = ηx + ηy2 . 2 2

(3.33)

Differentiating W2 with respect to t along the dynamics (3.27), we get W˙ 2 = −ηx2 − ηy2 + ηy (2ex + 2ey + ez ).

(3.34)

We define ηz as follows: ηz = 2ex + 2ey + ez .

(3.35)

With the help of Eq. (3.35), we can express Eq. (3.34) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(3.36)

A new chaotic jerk system with egg-shaped strange attractor Chapter | 3

63

As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 1 W (ηx , ηy , ηz ) = W2 (ηx , ηy ) + ηx2 = (ηx2 + ηy2 + ηz2 ). 2 2

(3.37)

It is clear that W is a positive definite function on R3 . Differentiating W with respect to t along the error dynamics (3.27), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ).

(3.38)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 − a)ex + (3 − b)ey + (2 − c)ez − x22 − y22 + x12 + y12 + v].

(3.39)

We define the control law v as v = −(2 − a)ex − (3 − b)ey − (2 − c)ez + x22 + y22 − x12 − y12 − Kηz , (3.40) where K is taken as a positive constant. Substituting (3.40) into (3.39), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(3.41)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that ex = ηx , ey = ηy − ηx , ez = ηx − 2ηy .

(3.42)

As a consequence, it follows that (ex (t), ey (t), ez (t)) → (0, 0, 0) exponentially as t → ∞. Thus, we have proved the following result. Theorem 3.2. The backstepping control law defined via (3.40) with gain K > 0 globally and exponentially synchronizes the 3D jerk chaos plants (3.24) and (3.25) for all initial states in R3 . For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b, c) = (7.3, 4, 0.9). We choose K = 6. We take the initial state of the leader system (3.24) as x1 (0) = 1.5, y1 (0) = −2.6, and z1 (0) = 7.4. We also consider the initial state of the follower system (3.25) as x2 (0) = 4.8, y2 (0) = 7.1, and z2 (0) = 2.9.

64 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 3.7 Synchronization between the states x1 and x2 of the leader–follower systems (3.24) and (3.25).

FIGURE 3.8 Synchronization between the states y1 and y2 of the leader–follower systems (3.24) and (3.25).

Figs. 3.7–3.9 display the synchronization between the states of the leader system and follower system. Fig. 3.10 shows the exponential convergence of the synchronization error ex (t), ey (t), ez (t) between the jerk systems (3.24) and (3.25).

A new chaotic jerk system with egg-shaped strange attractor Chapter | 3

65

FIGURE 3.9 Synchronization between the states z1 and z2 of the leader–follower systems (3.24) and (3.25).

FIGURE 3.10 Time-history of the synchronization error between the leader–follower systems (3.24) and (3.25).

3.5 Circuit design Implementation of systems with chaos by using physical elements is a vital topic because of its role in chaos-based applications (Yalcin et al., 2004; Bouali et al.,

66 Backstepping Control of Nonlinear Dynamical Systems

2012; Li et al., 2013; Lin et al., 2015; Zhou et al., 2015; Lai and Yang, 2016; Yu et al., 2016; Akgul et al., 2016b,a; Kacar, 2016; Liu et al., 2016b,a; Cicek et al., 2016; Azar et al., 2017; Alain et al., 2020). For illustrating the feasibility of the proposed jerk system (3.3), we design an electronic circuit for the jerk system as shown in Fig. 3.11. The schematic of the designed circuit includes five operational amplifiers (U1 –U5 ), two multipliers, eleven resistors, and three capacitors. Here three integrators are configured by using three operational amplifiers (U1 –U3 ). By applying Kirchhoff’s circuit laws, the circuital equation is described by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dvC1 dt

=

1 R1 C1 vC2 ,

dvC2 dt

=

1 R2 C2 vC3 ,

dvC3 dt

= − R31C3 vC1 −

(3.43) −

1 R4 C3 vC2

1 2 R6 C3 10V vC1





1 R5 C3 vC3

1 2 R7 C3 10V vC2 ,

where vC1 , vC2 , and vC3 are the voltages at three integrators (U1 –U3 ). We nort malize Eq. (3.43) with τ = RC and choose 

R1 = R2 = R, C1 = C2 = C3 = C.

(3.44)

As a result, we obtain the dimensionless system as follows: ⎧ ⎪ X˙ = Y, ⎪ ⎨ Y˙ = Z, ⎪ ⎪ ⎩ Z˙ = −aX − bY − cZ − X 2 − Y 2 .

(3.45)

It is worth noting that ⎧ ⎪ a = RR3 , ⎪ ⎪ ⎨ b = RR4 , ⎪ ⎪ ⎪ ⎩ c = RR5 .

(3.46)

In the dimensionless system (3.45), three variables X, Y , and Z correspond to voltages vC1 , vC2 , and vC3 , respectively. It is simple to confirm that system (3.45) is equivalent to the proposed jerk system.

A new chaotic jerk system with egg-shaped strange attractor Chapter | 3

67

FIGURE 3.11 Schematic of the jerk system designed with electronic components.

For implementing the jerk system with a = 7.3, b = 4, and c = 0.9, the electronic components are selected as ⎧ R1 = R2 = R = 400 k, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R3 = 54.79 k, ⎪ ⎪ ⎪ ⎪ ⎨ R4 = 100 k, ⎪ R5 = 444.44 k, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R6 = R7 = 40 k, ⎪ ⎪ ⎪ ⎩ C1 = C2 = C3 = C = 1 nF.

(3.47)

The circuit was implemented with a SPICE simulation program, in which the power supplies are ±15 VDC . Circuital attractors are reported in Fig. 3.12. From Figs. 3.4 and 3.12, it is simple to verify that the designed circuit works well and generates chaotic signals.

3.6 Conclusions This chapter introduces a 3-D jerk system, which generated chaotic behaviors. By studying its dynamics, we find that the system generates chaotic attractors. In addition, backstepping control of the jerk system is investigated and backstepping synchronization for such a jerk system is reported. We also present results of an electronic circuit, which implemented to confirm the feasibility of the jerk system. In our upcoming works, practical applications will be developed by using complex behaviors of this jerk system.

68 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 3.12 Chaotic attractors in SPICE simulation; (A) X–Y plane, (B) X–Z plane, and (C) Y –Z plane.

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References Akgul, A., Hussain, S., Pehlivan, I., 2016a. A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications. Optik 127, 7062–7071. Akgul, A., Moroz, I., Pehlivan, I., Vaidyanathan, S., 2016b. A new four-scroll chaotic attractor and its engineering applications. Optik 127, 5491–5499. Alain, K.S.T., Azar, A.T., Bertrand, F.H., Romanic, K., 2019. Robust observer-based synchronisation of chaotic oscillators with structural perturbations and input nonlinearity. International Journal of Automation and Control 13 (4), 387–412. Alain, K.S.T., Azar, A.T., Kengne, R., Bertrand, F.H., 2020. Stability analysis and robust synchronisation of fractional-order modified Colpitts oscillators. International Journal of Automation and Control 14 (1), 52–79. Azar, A.T., Serrano, F.E., Flores, M.A., Vaidyanathan, S., Zhu, Q., 2020. Adaptive neural-fuzzy and backstepping controller for port-Hamiltonian systems. International Journal of Computer Applications in Technology 62 (1), 1–12. Azar, A.T., Volos, C., Gerodimos, N.A., Tombras, G.S., Pham, V.-T., Radwan, A.G., Vaidyanathan, S., Ouannas, A., Munoz-Pacheco, J.M., 2017. A novel chaotic system without equilibrium: dynamics, synchronization, and circuit realization. Complexity 2017, 7871467 (1–11). Bouali, S., Buscarino, A., Fortuna, L., Frasca, M., Gambuzza, L.V., 2012. Emulating complex business cycles by using an electronic analogue. Nonlinear Analysis: Real World Applications 13, 2459–2465. Cabanas, A.M., Clerc, M.G., Laroze, D., Leon, A.O., 2019. Chaotic patterns and localized states in spin valves. Journal of Magnetism and Magnetic Materials 476, 589–596. Cicek, S., Ferikoglu, A., Pehlivan, I., 2016. A new 3D chaotic system: dynamical analysis, electronic circuit design, active control synchronization and chaotic masking communication application. Optik 127, 4024–4030. Daumann, C.C., Rech, P.C., 2019. Hyperchaos in convection with the Cattaneo–Christov heat-flux model. The European Physical Journal B 92 (5), 110. El-Nabulsi, R.A., 2018. Time-nonlocal kinetic equations, jerk and hyperjerk in plasmas and solar physics. Advances in Space Research 61 (12), 2914–2931. Gatabazi, P., Mba, J.C., Pindza, E., Labuschagne, C., 2019. Grey Lotka–Volterra models with application to cryptocurrencies adoption. Chaos, Solitons and Fractals 122, 47–57. Ginoux, J.-M., Naeck, R., Ruhomally, Y.B., Dahuhoo, M.Z., Perc, M., 2019. Chaos in a predator– prey-based mathematical model for illicit drug consumption. Applied Mathematics and Computation 347, 502–513. Gusso, A., Viana, R.L., Mathias, A.C., Caldas, I., 2019. Nonlinear dynamics and chaos in micro/nanoelectromechanical beam resonators actuated by two-sided electrodes. Chaos, Solitons and Fractals 122, 6–16. Jahanshahi, H., Yousefpour, A., Wei, Z., Alcaraz, R., Bekiros, S., 2019. A financial hyperchaotic system with coexisting attractors: dynamic investigation, entropy analysis, control and synchronization. Chaos, Solitons and Fractals 126, 66–77. Kacar, S., 2016. Analog circuit and microcontroller based RNG application of a new easy realization 4D chaotic system. Optik 127, 9551–9561. Khan, A., Singh, S., Azar, A.T., 2020a. Combination–combination anti-synchronization of four fractional order identical hyperchaotic systems. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). In: Advances in Intelligent Systems and Computing, vol. 921. Springer International Publishing, Cham, pp. 406–414. Khan, A., Singh, S., Azar, A.T., 2020b. Synchronization between a novel integer-order hyperchaotic system and a fractional-order hyperchaotic system using tracking control. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). In: Advances in Intelligent Systems and Computing, vol. 921. Springer International Publishing, Cham, pp. 382–391.

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Khettab, K., Bensafia, Y., Bourouba, B., Azar, A.T., 2018. Enhanced fractional order indirect fuzzy adaptive synchronization of uncertain fractional chaotic systems based on the variable structure control: robust H∞ design approach. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 597–624. Lai, Q., Yang, L., 2016. Chaos, bifurcation, coexisting attractors and circuit design of a threedimensional continuous autonomous system. Optik 127, 5400–5406. Li, C., Li, H., Tong, Y., 2013. Analysis of a novel three-dimensional chaotic system. Optik 124, 1516–1522. Lin, Y., Wang, C., He, H., 2015. A simple multi-scroll chaotic oscillator employing CCIIs. Optik 126, 824–827. Liu, H., Kadir, A., Li, Y., 2016a. Asymmetric color pathological image encryption scheme based on complex hyper chaotic system. Optik 127, 5812–5819. Liu, H., Kadir, A., Li, Y., 2016b. Audio encryption scheme by confusion and diffusion based on multi-scroll chaotic system and one-time keys. Optik 127, 7431–7438. Nwachioma, C., Humberto Perez-Cruz, J., Jimenez, A., Ezuma, M., Rivera-Blas, R., 2019. A new chaotic oscillator – properties, analog implementation, and secure communication application. IEEE Access 28 (7), 7510–7521. Ouannas, A., Azar, A.T., Abu-Saris, R., 2017a. A new type of hybrid synchronization between arbitrary hyperchaotic maps. International Journal of Machine Learning and Cybernetics 8 (6), 1887–1894. Ouannas, A., Azar, A.T., Vaidyanathan, S., 2017b. A new fractional hybrid chaos synchronisation. International Journal of Modelling, Identification and Control 27 (4), 314–322. http://www. inderscienceonline.com/doi/abs/10.1504/IJMIC.2017.084719. Ouannas, A., Azar, A.T., Vaidyanathan, S., 2017c. New hybrid synchronisation schemes based on coexistence of various types of synchronisation between master–slave hyperchaotic systems. International Journal of Computer Applications in Technology 55 (2), 112–120. https://www. inderscienceonline.com/doi/abs/10.1504/IJCAT.2017.082868. Rajagopal, K., Akgul, A., Jafari, S., Karthikeyan, A., Cavusoglu, U., Kacar, S., 2019. An exponential jerk system: circuit realization, fractional order and time delayed form with dynamical analysis and its engineering application. Journal of Circuits, Systems, and Computers 28 (5), 1950087. Rasappan, S., Vaidyanathan, S., 2012. Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East Journal of Mathematical Sciences 67 (2), 265–287. Shukla, M.K., Sharma, B.B., Azar, A.T., 2018. Control and synchronization of a fractional order hyperchaotic system via backstepping and active backstepping approach. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 559–595. Singh, P.P., Roy, B.K., 2019. Microscopic chaos control of chemical reactor system using nonlinear active plus proportional integral sliding mode control technique. The European Physical Journal Special Topics 228 (1), 169–184. Singh, S., Azar, A.T., Bhat, M.A., Vaidyanathan, S., Ouannas, A., 2018. Active control for multi-switching combination synchronization of non-identical chaotic systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in System Dynamics and Control. In: Advances in Systems Analysis, Software Engineering, and High Performance Computing (ASASEHPC). IGI Global, pp. 129–162. Vaidyanathan, S., 2015. Analysis, control, and synchronization of a 3-D novel jerk chaotic system with two quadratic nonlinearities. Kyungpook Mathematical Journal 55 (3), 563–586. Vaidyanathan, S., 2016. A novel 3-D jerk chaotic system with two quadratic nonlinearities and its adaptive backstepping control. International Journal of Control Theory and Applications 9 (1), 199–219. Vaidyanathan, S., 2017. A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control. Archives of Control Sciences 27 (3), 409–439.

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Vaidyanathan, S., Abba, O.A., Betchewe, G., Alidou, M., 2019. A new three-dimensional chaotic system: its adaptive control and circuit design. International Journal of Automation and Control 13 (1), 101–121. Vaidyanathan, S., Akgul, A., Kacar, S., 2018a. A new chaotic jerk system with two quadratic nonlinearities and its applications to electronic circuit implementation and image encryption. International Journal of Computer Applications in Technology 58 (2), 89–101. Vaidyanathan, S., Azar, A.T., 2016a. A novel 4-D four-wing chaotic system with four quadratic nonlinearities and its synchronization via adaptive control method. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 203–224. Vaidyanathan, S., Azar, A.T., 2016b. Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 249–274. Vaidyanathan, S., Azar, A.T., 2016c. Adaptive control and synchronization of Halvorsen circulant chaotic systems. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 225–247. Vaidyanathan, S., Azar, A.T., 2016d. Dynamic analysis, adaptive feedback control and synchronization of an eight-term 3-D novel chaotic system with three quadratic nonlinearities. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 155–178. Vaidyanathan, S., Azar, A.T., 2016e. Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 275–290. Vaidyanathan, S., Azar, A.T., 2016f. Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in Chaos Theory and Intelligent Control. Springer International Publishing, Cham, pp. 179–202. Vaidyanathan, S., Idowu, B.A., Azar, A.T., 2015a. Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Chaos Modeling and Control Systems Design. In: Studies in Computational Intelligence, vol. 581. Springer, Berlin, Germany, pp. 39–58. Vaidyanathan, S., Jafari, S., Pham, V.-T., Azar, A.T., Alsaadi, F.E., 2018b. A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design. Archives of Control Sciences 28 (2), 239–254. Vaidyanathan, S., Sambas, A., Mamat, M., Sanjaya, W.S.M., 2017. Analysis, synchronisation and circuit implementation of a novel jerk chaotic system and its application for voice encryption. International Journal of Modelling, Identification and Control 28 (2), 153–166. Vaidyanathan, S., Volos, C., Rajagopal, K., Kyprianidis, I., Stouboulos, I., 2015b. Adaptive backstepping controller design for the anti-synchronization of identical WINDMI chaotic systems with unknown parameters and its SPICE implementation. Journal of Engineering Science and Technology Review 8 (2), 74–82. Xu, B., Wang, G., Lu, H.H.-C., Yu, S., Yuan, F., 2019. A memristor–meminductor-based chaotic system with abundant dynamical behaviors. Nonlinear Dynamics 96 (1), 765–788. Yalcin, M.E., Suykens, J.A.K., Vandewalle, J., 2004. True random bit generation from a doublescroll attractor. IEEE Transactions on Circuits and Systems. I, Regular Papers 51, 1395–1404. Yu, F., Li, P., Bo, K., Yin, B., 2016. Research progress of multi-scroll chaotic oscillators based on current-mode devices. Optik 127, 5486–5490. Zhou, W., Wang, Z., Wu, M., Zheng, W., Weng, J., 2015. Dynamics analysis and circuit implementation of a new three-dimensional chaotic system. Optik 126, 765–768.

Chapter 4

A new 4-D chaotic hyperjerk system with coexisting attractors, its active backstepping control, and circuit realization Aceng Sambasa , Sundarapandian Vaidyanathanb , Sen Zhangc , Mohamad Afendee Mohamedd , Yicheng Zengc , and Ahmad Taher Azare,f a Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya,

Indonesia, b Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, c School of Physics and Optoelectric Engineering, Xiangtan University, Xiangtan, China, d Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu,

Malaysia, e Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, f Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

4.1

Introduction

In the recent decades, significant research attention has been devoted to the modeling and applications of dynamical systems exhibiting chaos (Xu et al., 2019; Gusso et al., 2019; Gatabazi et al., 2019; Singh and Roy, 2019; Cabanas et al., 2019; Ginoux et al., 2019; Jahanshahi et al., 2019; Daumann and Rech, 2019). Xu et al. (2019) proposed a chaotic system based on a circuit design involving a memristor model and a meminductor model. Gusso et al. (2019) analyzed the nonlinear dynamical model and the existence of chaos in suspended beam MEMS/NEMS resonators that are actuated by two-sided electrodes. Gatabazi et al. (2019) analyzed 2-D and 3-D Grey Lotka–Volterra Models (GLVM) and explored their application in cryptocurrencies such as Bitcoin, Litecoin, and Ripple. Singh and Roy (2019) studied microscopic chaos control of a chemical reactor system via nonlinear active plus proportional integral sliding mode control. Cabanas et al. (2019) discovered chaos in driven nano-magnets such as spin valves by using the magnetic energy and the magnetoresistance. Ginoux et al. (2019) discovered chaos in a dynamical system modeling the illicit drug consumption in a population comprising drug users and non-users. Jahanshahi et al. (2019) discussed a finance hyperchaos system via entropy analysis and Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00011-8 Copyright © 2021 Elsevier Inc. All rights reserved.

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74 Backstepping Control of Nonlinear Dynamical Systems

control methods. Daumann and Rech (2019) reported hyperchaos in a heat-flux convection model. Jerk and hyperjerk dynamical systems are important classes of mechanical systems. If y(t) denotes the displacement of a moving object, then Dy(t) = dy dt represents its velocity, D 2 y(t) = 4 D 4 y(t) = ddt y4

d2y dt 2

its acceleration, D 3 y(t) =

d3y dt 3

its jerk,

its hyperjerk. and An autonomous hyperjerk differential equation has the general representation given by D 4 x = F (x, Dx, D 2 x, D 3 x).

(4.1)

The hyperjerk differential equation (4.1) can be displayed in a system form as

⎧ x˙1 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 ⎪ x˙3 ⎪ ⎪ ⎪ ⎩ x˙4

=

x2 ,

=

x3 ,

=

x4 ,

=

F (x1 , x2 , x3 , x4 ).

(4.2)

Recently, much interest has been given to the finding of both jerk and hyperjerk systems in the chaos literature (Vaidyanathan et al., 2018a; Vaidyanathan, 2017; Vaidyanathan et al., 2017; Vaidyanathan, 2016, 2015a; El-Nabulsi, 2018; Prousalis et al., 2018; Ahmad and Srisuchinwong, 2018; Tsafack and Kengne, 2018; Daltzis et al., 2018; Vaidyanathan et al., 2018b; Wang et al., 2017). Vaidyanathan et al. (2018a) reported a new chaotic jerk system with two quadratic nonlinearities and discussed its applications to electronic circuit implementation and image encryption. Vaidyanathan (2017) reported a new 3-D chaotic jerk system with two cubic nonlinear terms and discussed its adaptive synchronization using backstepping control. Vaidyanathan et al. (2017) analyzed a new chaotic jerk system with its applications for circuit simulation and voice encryption. Vaidyanathan (2016) announced a new chaotic jerk system and discussed its adaptive synchronization using backstepping control. Vaidyanathan (2015a) proposed a new chaotic jerk system with two quadratic nonlinearities. El-Nabulsi (2018) reported jerk and hyperjerk systems in the study of nonlocal effects in fluids, plasmas, and solar physics. Prousalis et al. (2018) observed extreme multi-stability in a 4-D hyperjerk memristive system with infinite number of rest points. Ahmad and Srisuchinwong (2018) reported a 4-D hyperjerk system with hyperchaos and having no rest point. Tsafack and Kengne (2018) reported a 5-D hyperjerk system with circuit design. Daltzis et al. (2018) reported a 4-D hyperjerk system with hyperchaos and built a real circuit design for the hyperjerk system. Vaidyanathan et al. (2018b) reported a new 4-D chaotic hyperjerk system and discussed its applications in RNG, image encryption, and steganography. Wang et al. (2017) found a new 4-D hyperchaotic hyperjerk system with coexisting attractors.

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75

This work reports a new 4-D hyperjerk system with chaos. The proposed nonlinear hyperjerk mechanical system has a hyperbolic sinusoidal nonlinearity. This chapter aims to analyze the dynamic behavior and control chaotic behavior of the new chaotic hyperjerk system. Using dynamic analysis, we show that the new hyperjerk system exhibits multi-stability and coexisting attractors. Multi-stability is a complex feature for a chaotic system where coexisting attractors are obtained for the same values of parameters but different initial states (Zhang et al., 2018a,b; Wang et al., 2018). Active backstepping control is applied to control and synchronize the chaos in the hyperjerk system. Active control method via backstepping approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015b; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015b; Vaidyanathan et al., 2018; Shukla et al., 2018; Vaidyanathan and Azar, 2016b; Vaidyanathan et al., 2015a). Finally, a circuit model using Multisim of the new hyperjerk system with chaos is designed for practical implementation. We show that the Multisim outputs of the hyperjerk system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019a; Sambas et al., 2019a,b; Vaidyanathan et al., 2019b; Azar et al., 2017).

4.2 System model In this chapter, we propose a new hyperjerk mechanical system, which is described as the 4-D model ⎧ x˙1 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 ⎪ x ˙ ⎪ 3 ⎪ ⎪ ⎩ x˙4

=

x2 ,

=

x3 ,

=

x4 ,

=

−x1 − ax2 − bx4 − x1 sinh x3 .

(4.3)

In this research work, it is proved that the hyperjerk plant (4.3) shows chaos when the parameters take the values a = 3.5, b = 2.3.

(4.4)

For the initial state X(0) = (0.4, 0.2, 0.4, 0.2) and (a, b) = (3.5, 2.3), the Lyapunov characteristic exponents of the hyperjerk plant (4.3) are esti-

76 Backstepping Control of Nonlinear Dynamical Systems

mated as ψ1 = 0.1118, ψ2 = 0, ψ3 = −0.5366, ψ4 = −1.8752,

(4.5)

which confirms that the hyperjerk system (4.3) is a dissipative chaos system. It is also an easy calculation to determine the Kaplan–Yorke dimension of the hyperjerk plant (4.3) as DKY = 2 +

ψ1 + ψ2 = 2.2083. |ψ3 |

(4.6)

The Kaplan–Yorke dimension of the hyperjerk plant (4.3) shows the complex characteristic of the dissipative hyperjerk plant (4.3). The rest points of the hyperjerk plant (4.3) are obtained by solving the following system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

x2

=

0,

x3

=

0,

⎪ ⎪ ⎪ ⎪ ⎩

x4

=

0,

−x1 − ax2 − bx4 − x1 sinh x3

=

0.

(4.7)

A simple calculation shows that the hyperjerk plant (4.3) has the unique rest point x1 = 0, x2 = 0, x3 = 0, x4 = 0,

(4.8)

which is the origin of R4 . The system matrix of the hyperjerk plant (4.3) at O = (0, 0, 0, 0) is given by ⎡ ⎤ 0 1 0 1 ⎢ ⎥ 0 1 0 ⎥ ⎢ 0 A=⎢ (4.9) ⎥. ⎣ 0 0 0 1 ⎦ −1 −a 0 −b For the chaos case, (a, b) = (3.5, 2.3), the matrix A has the eigenvalues λ1 = −0.2738, λ2 = −2.7226, λ3,4 = 0.3482 ± 1.1046i.

(4.10)

Hence, O = (0, 0, 0, 0) is a saddle-focus and unstable rest point for the hyperjerk plant (4.3). Figs. 4.1 and 4.2 are MATLAB simulations of the chaotic hyperjerk plant for the parameter values (a, b) = (3.5, 2.3) and x(0) = (0.4, 0.2, 0.4, 0.2). Fig. 4.1 shows the Lyapunov characteristic exponents and Fig. 4.2 shows the 2-D plots of the chaotic hyperjerk plant (4.3).

A new 4-D chaotic hyperjerk system with coexisting attractors Chapter | 4

77

FIGURE 4.1 Lyapunov characteristic exponents of the hyperjerk plant (4.3).

FIGURE 4.2 Chaotic attractors of the hyperjerk system (4.3) in (A) (x1 , x2 ) plane, (B) (x2 , x3 ) plane, (C) (x3 , x4 ) plane, and (D) (x1 , x4 ) plane.

78 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 4.3 Dynamic analysis of the hyperjerk system (4.3) when we fix b = 2.3, X(0) = (0.4, 0.2, 0.4, 0.2), and vary a in the region of [2.4, 4]: (A) bifurcation diagram and (B) Lyapunov exponents of the hyperjerk system.

4.3 Dynamic analysis of the new hyperjerk system Here, we discuss the dynamic analysis of the hyperjerk system (4.3). First, we fix b = 2.3 and initial state X(0) = (0.4, 0.2, 0.4, 0.2). We vary the parameter a in the region of [2.4, 4]. From the bifurcation diagram shown in Fig. 4.3, we can see that the hyperjerk system (4.3) exhibits chaos, periodic, forward period-doubling route, reverse period-doubling route as well as several periodic windows. Some sample results are plotted in Fig. 4.4 to verify the forward perioddoubling route to chaos. Next, we fix a = 3.5 and initial state X(0) = (0.4, 0.2, 0.4, 0.2). We vary the parameter b in the region of [2.3, 12]. From the bifurcation diagram shown in Fig. 4.5, we can see that the hyperjerk system (4.3) exhibits chaos at first, and then goes out of chaos to periodic behavior via reverse period-doubling route. Next, we discuss coexisting attractors of the hyperjerk system (4.3). We denote the state orbit of the hyperjerk system (4.3) starting with the initial state X0 = (0.4, 0.2, 0.4, 0.2) in blue color (dark gray in print). Also, we denote the state orbit of the hyperjerk system (4.3) starting with the initial state Y0 = (−0.4, −0.2, −0.4, 0.2) in red color (light gray in print). We fix a = 3.5 and vary b in the region of [2.3, 5]. Fig. 4.6 describes the bifurcation diagram. From this figure, we can see that there exist coexisting attractors in the narrow region of b ∈ [2.5, 3.6]. Fig. 4.7 describes some sample results of coexisting attractors as follows. When a = 3.5 and b = 2.7, the hyperjerk system (4.3) exhibits coexisting period-8 and period-2 attractors.

A new 4-D chaotic hyperjerk system with coexisting attractors Chapter | 4

79

FIGURE 4.4 Plots of the hyperjerk system (4.3) when we fix b = 2.3, X(0) = (0.4, 0.2, 0.4, 0.2), and vary a in the region of [2.4, 4]: (A) when a = 3, period-2; (B) when a = 2.8, period-4, and (C) when a = 2.65, chaos.

FIGURE 4.5 Dynamic analysis of the hyperjerk system (4.3) when we fix a = 3.5, X(0) = (0.4, 0.2, 0.4, 0.2), and vary b in the region of [2.3, 12]: (A) bifurcation diagram and (B) Lyapunov exponents of the hyperjerk system.

4.4 Active backstepping stabilization of the new hyperjerk system In this section, we employ active backstepping technique for globally stabilizing the new chaos hyperjerk system.

80 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 4.6 Bifurcation diagram showing coexisting attractors of the hyperjerk plant (4.3).

FIGURE 4.7 Coexisting attractors of the hyperjerk system (4.3) when we fix a = 3.5 and b = 2.7.

The controlled hyperjerk system is depicted by the 4D dynamics ⎧ x˙1 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 ⎪ x ˙ ⎪ 3 ⎪ ⎪ ⎩ x˙4

=

x2 ,

=

x3 ,

=

x4 ,

(4.11)

= −x1 − ax2 − bx4 − x1 sinh x3 + v.

Here, v is an active backstepping control law to be designed. We begin with the quadratic Lyapunov function W1 (η1 ) =

1 2 η 2 1

(4.12)

where η1 = x1 .

(4.13)

Differentiating W1 with respect to t along the dynamics (4.11), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (x1 + x2 ).

(4.14)

A new 4-D chaotic hyperjerk system with coexisting attractors Chapter | 4

81

We define η2 = x 1 + x 2 .

(4.15)

With the help of Eq. (4.15), we can express Eq. (4.14) as W˙ 1 = η1 η˙ 1 = −η12 + η1 η2 .

(4.16)

We proceed next with defining the quadratic Lyapunov function W2 (η1 , η2 ) = W1 (η1 ) +

1 2 1 2 η2 = η1 + η22 . 2 2

(4.17)

Differentiating W2 with respect to t along the dynamics (4.11), we get W˙ 2 = −η12 − η22 + η2 (2x1 + 2x2 + x3 ).

(4.18)

η3 = 2x1 + 2x2 + x3 .

(4.19)

We define

With the help of Eq. (4.19), we can express Eq. (4.18) as W˙ 2 = −η12 − η22 + η2 η3 .

(4.20)

We proceed next with defining the quadratic Lyapunov function W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) +

1 2 1 2 η3 = η1 + η22 + η32 . 2 2

(4.21)

Differentiating W3 with respect to t along the dynamics (4.11), we get W˙ 3 = −η12 − η22 − η32 + η3 (3x1 + 5x2 + 3x3 + x4 ).

(4.22)

η4 = 3x1 + 5x2 + 3x3 + x4 .

(4.23)

We define

With the help of Eq. (4.23), we can express Eq. (4.22) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(4.24)

As a final step of the active backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 ) =

1 2 η1 + η22 + η32 + η42 . 2

(4.25)

Differentiating W with respect to t along the dynamics (4.11), we get W˙ = −η12 − η22 − η32 − η42 + η4 T

(4.26)

82 Backstepping Control of Nonlinear Dynamical Systems

where T = η3 + η4 + η˙ 4 = 4x1 + (10 − a)x2 + 9x3 + (4 − b)x4 − x1 sinh x3 + v. (4.27) The main result of this section is described below. Theorem 4.1. The active backstepping control law defined by v = −4x1 − (10 − a)x2 − 9x3 − (4 − b)x4 + x1 sinh x3 − Kη4

(4.28)

where K > 0 and η4 = 3x1 + 5x2 + 3x3 + x4 , globally and exponentially stabilizes the 4D chaotic hyperjerk system (4.11) for all values of x(0) ∈ R4 . Proof. This result is an application of Lyapunov stability theory. First, we remark that the quadratic function W defined via Eq. (4.25) is positive definite. Substituting the backstepping law (4.28) into Eq. (4.27), we get T = −Kη4 . Substituting T = −Kη4 in (4.26), we get W˙ = −η12 − η22 − η32 − (1 + K)η42 .

(4.29)

This shows that W˙ is quadratic and negative definite on R4 . As a consequence, we deduce that x1 (t), x2 (t), x3 (t), and x4 (t) converge to zero exponentially as t → ∞ for all initial conditions xi (0) ∈ R, (i = 1, 2, 3, 4). This completes the proof. For computer simulations, we take the chaos case for the parameter values, i.e. we take a = 3.5 and b = 2.3. We take the control gain as K = 6 and x(0) = (7.5, 3.1, 2.9, 6.8). Fig. 4.8 shows that the backstep-controlled state x(t) converges to 0 ∈ R4 exponentially with time.

4.5 Active backstepping synchronization of the new hyperjerk system Since the pioneering work by Pecora and Carroll (1990), chaos synchronization problem has been studied extensively in the literature (Vaidyanathan and Azar, 2016f,a,c,d,e; Ouannas et al., 2017a). In most of the synchronization approaches, the master–slave or drive–response formalism is used. If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of synchronization is to use the output of the master system to control the slave system so that the output of the response system tracks the output of the master system asymptotically (Ouannas et al., 2020, 2019, 2017c,b,a,e,d; Pham et al., 2018; Singh et al., 2018a,b,c; Volos et al., 2018).

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FIGURE 4.8 Time-history of the backstep-controlled state x(t) for the chaotic hyperjerk plant (4.11).

Here, we employ active backstepping control method for globally synchronizing the trajectories of a pair of new hyperjerk systems considered as leader– follower systems. The leader hyperjerk system is depicted by the 4D dynamics ⎧ x˙1 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 ⎪ x ˙ ⎪ 3 ⎪ ⎪ ⎩ x˙4

=

x2 ,

=

x3 ,

=

x4 ,

=

−x1 − ax2 − bx4 − x1 sinh x3 .

(4.30)

The follower hyperjerk system is equipped with backstepping control and depicted by the 4D dynamics ⎧ y˙1 ⎪ ⎪ ⎪ ⎪ ⎨ y˙ 2 ⎪ y˙3 ⎪ ⎪ ⎪ ⎩ y˙4

=

y2 ,

=

y3 ,

=

y4 ,

(4.31)

= −y1 − ay2 − by4 − y1 sinh y3 + v.

In (4.31), v is an active backstepping control to be designed. The synchronization error is defined by means of the equations ei = yi − xi , i = 1, 2, 3, 4.

(4.32)

84 Backstepping Control of Nonlinear Dynamical Systems

The synchronization error dynamics is calculated as follows: ⎧ e˙1 = e2 , ⎪ ⎪ ⎪ ⎪ ⎨ e˙ = e , 2

⎪ e˙3 ⎪ ⎪ ⎪ ⎩ e˙4

3

=

e4 ,

(4.33)

= −e1 − ae2 − be4 − y1 sinh y3 + x1 sinh x3 + v.

We begin with the quadratic Lyapunov function W1 (η1 ) =

1 2 η 2 1

(4.34)

where η1 = e1 .

(4.35)

Differentiating W1 with respect to t along the dynamics (4.33), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (e1 + e2 ).

(4.36)

η 2 = e 1 + e2 .

(4.37)

We define

With the help of Eq. (4.37), we can express Eq. (4.36) as W˙ 1 = η1 η˙ 1 = −η12 + η1 η2 .

(4.38)

We proceed next with defining the quadratic Lyapunov function W2 (η1 , η2 ) = W1 (η1 ) +

1 2 1 2 η2 = η1 + η22 . 2 2

(4.39)

Differentiating W2 with respect to t along the dynamics (4.33), we get W˙ 2 = −η12 − η22 + η2 (2e1 + 2e2 + e3 ).

(4.40)

η3 = 2e1 + 2e2 + e3 .

(4.41)

We define

With the help of Eq. (4.41), we can express Eq. (4.40) as W˙ 2 = −η12 − η22 + η2 η3 .

(4.42)

We proceed next with defining the quadratic Lyapunov function W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) +

1 2 1 2 η3 = η1 + η22 + η32 . 2 2

(4.43)

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Differentiating W3 with respect to t along the dynamics (4.33), we get W˙ 3 = −η12 − η22 − η32 + η3 (3e1 + 5e2 + 3e3 + e4 ).

(4.44)

η4 = 3e1 + 5e2 + 3e3 + e4 .

(4.45)

We define

With the help of Eq. (4.45), we can express Eq. (4.44) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(4.46)

As a final step of the active backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 ) =

1 2 η1 + η22 + η32 + η42 . 2

(4.47)

Differentiating W with respect to t along the dynamics (4.33), we get W˙ = −η12 − η22 − η32 − η42 + η4 T

(4.48)

where T = η3 + η4 + η˙ 4 = 4e1 + (10 − a)e2 + 9e3 + (4 − b)e4 − y1 sinh y3 + x1 sinh x3 + v. (4.49) The main result of this section is described below. Theorem 4.2. The active backstepping control law defined by v = −4e1 − (10 − a)e2 − 9e3 − (4 − b)e4 + y1 sinh y3 − x1 sinh x3 − Kη4 (4.50) where K > 0 and η4 = 3e1 + 5e2 + 3e3 + e4 , globally and exponentially synchronizes the 4D chaotic hyperjerk systems (4.30) and (4.31) for all values of x(0), y(0) ∈ R4 . Proof. This result is an application of Lyapunov stability theory. First, we remark that the quadratic function W defined via Eq. (4.47) is positive definite. Substituting the backstepping law (4.50) into Eq. (4.49), we get T = −Kη4 . Substituting T = −Kη4 in (4.48), we get W˙ = −η12 − η22 − η32 − (1 + K)η42 . This shows that W˙ is quadratic and negative definite on R4 .

(4.51)

86 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 4.9 Synchronization of the states x1 and y1 of the new hyperjerk plants (4.30) and (4.31).

FIGURE 4.10 Synchronization of the states x2 and y2 of the new hyperjerk plants (4.30) and (4.31).

As a consequence, we deduce that e1 (t), e2 (t), e3 (t), and e4 (t) converge to zero exponentially as t → ∞ for all initial conditions ei (0) ∈ R, (i = 1, 2, 3, 4). This completes the proof. For computer simulations, we take the chaos case for the parameter values, i.e. we take a = 3.5 and b = 2.3.

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FIGURE 4.11 Synchronization of the states x3 and y3 of the new hyperjerk plants (4.30) and (4.31).

FIGURE 4.12 Synchronization of the states x4 and y4 of the new hyperjerk plants (4.30) and (4.31).

We take the control gain as K = 6. We take the initial states of the systems (4.30) and (4.31) as x(0) = (2.1, 6.7, 4.9, 3.8) and y(0) = (5.3, 1.6, 3.4, 7.2), respectively. Figs. 4.9–4.13 shows the exponential synchronization of the new hyperjerk systems (4.30) and (4.31).

88 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 4.13 Time-plot of the synchronization error between the new hyperjerk plants (4.30) and (4.31).

4.6

Circuit simulation of the new hyperjerk system

In this section, we design an electronic circuit using Multisim software to realize the new hyperjerk chaotic system. TL082CD Op-amp, AD633JN multiplier, resistor, and capacitor are combined on Multisim software to simulate the chaotic attractor. The schematic diagram of the electronic circuit is shown in Fig. 4.14, where x1 , x2 , x3 , and x4 represent the voltages across capacitors C1 , C2 , C3 , and C4 , respectively. To get high-quality validation results, the state variables of the new hyperjerk system model must be transformed. We can choose X1 = 14 x1 , X2 = 12 x2 , X3 = 12 x3 , X4 = 14 x4 . Thus, the new hyperjerk system is rewritten as ⎧ ⎪ X˙ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ X˙ 2 ⎪ ⎪ X˙ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙ X4

=

1 2 X2 ,

=

X3 ,

=

2X4 ,

= −X1 − a2 X2 − bX4 −

(4.52) X1 2

sinh(X3 ).

The implementation electronic circuit of the new hyperjerk system is synthesized as shown in Fig. 4.14. The circuit equations are formulated as follows:

A new 4-D chaotic hyperjerk system with coexisting attractors Chapter | 4

89

FIGURE 4.14 Schematic diagram of the hyperjerk system (4.53).

C1 X˙ 1

=

C2 X˙ 2

=

C3 X˙ 3

=

C4 X˙ 4

= − R14 X1 −

1 R1 X 2 , 1 R2 X 3 , 1 R3 X 4 ,

(4.53) 1 R5 X 2



1 R6 X 4



1 10R7 X1 sinh(X3 ).

Compared with (4.52) and (4.53), the parameters are taken as follows: R1 = 800 k, R2 = R4 = 400 k, R3 = 200 k, R5 = 228.57 k, R6 = 173.913 k, R7 = 80 k, R8 = R9 = R10 = R11 = R12 = R13 = 100 k, and C1 = C2 = C3 = C4 = 1 nF.

90 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 4.15 Chaotic attractors of system (4.53) using Multisim circuit simulation: (A) X1 –X2 plane, (B) X2 –X3 plane, (C) X3 –X4 plane, and (D) X1 –X4 plane.

The oscilloscope graphics represented in the phase portraits of the system (4.53) are shown in Fig. 4.15. We can see that the circuit simulation results of the oscilloscope graphics of Fig. 4.15 agree with the MATLAB simulation results given in Fig. 4.2 (Section 4.2).

4.7 Conclusions In this chapter, we proposed a new 4-D chaotic hyperjerk system with a hyperbolic sinusoidal nonlinearity. Jerk and hyperjerk systems are important classes of mechanical dynamical systems in the literature. We gave an expository discussion on the qualitative behavior of the new hyperjerk system and showed that the new hyperjerk system has chaos and multi-stability with coexisting attractors. We used backstepping control theory to derive new results for the chaos stabilization and synchronization for the new hyperjerk system with hyperbolic sinusoidal nonlinearity. A circuit design of the new chaotic hyperjerk dynamical system was derived using Multisim to aid the practical implementation of the proposed chaotic hyperjerk system.

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Acknowledgment The authors thank the Government of Malaysia for funding this research under the Fundamental Research Grant Scheme (FRGS/1/2018/ICT03/UNISZA/02/2) and also Universiti Sultan Zainal Abidin, Terengganu, Malaysia.

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Chapter 5

A new 3-D chaotic jerk system with a saddle-focus rest point at the origin, its active backstepping control, and circuit realization Aceng Sambasa , Sundarapandian Vaidyanathanb , Sen Zhangc , Mohamad Afendee Mohamedd , Yicheng Zengc , and Ahmad Taher Azare,f a Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya,

Indonesia, b Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, c School of Physics and Optoelectric Engineering, Xiangtan University, Xiangtan, China, d Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu,

Malaysia, e Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, f Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

5.1

Introduction

Jerk dynamical systems are important classes of mechanical systems. If y(t) denotes the displacement of a moving object, then Dy(t) = dy dt represents its 2

3

velocity, D 2 y(t) = ddt y2 its acceleration, and D 3 y(t) = ddt y3 its jerk. An autonomous jerk differential equation has the general representation given by D 3 x = F (x, Dx, D 2 x).

(5.1)

By defining new state variables y = Dx and z = D 2 x, the jerk differential equation (5.1) can be displayed in a system form as ⎧ ⎪ ⎨ x˙ = y, (5.2) y˙ = z, ⎪ ⎩ z˙ = F (x, y, z). Recently, much interest has been given to the finding of jerk systems in the chaos literature (Vaidyanathan et al., 2018; Vaidyanathan, 2017; Vaidyanathan Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00012-X Copyright © 2021 Elsevier Inc. All rights reserved.

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96 Backstepping Control of Nonlinear Dynamical Systems

et al., 2017; Vaidyanathan, 2016, 2015; El-Nabulsi, 2018). Vaidyanathan et al. (2018a) reported a new chaotic jerk system with two quadratic nonlinearities and discussed its applications to electronic circuit implementation and image encryption. Vaidyanathan (2017) reported a new 3-D chaotic jerk system with two cubic nonlinear terms and discussed its adaptive synchronization using backstepping control. Vaidyanathan et al. (2017) analyzed a new chaotic jerk system with its applications for circuit simulation and voice encryption. Vaidyanathan (2016) announced a new chaotic jerk system and discussed its adaptive synchronization using backstepping control. Vaidyanathan (2015) proposed a new chaotic jerk system with two quadratic nonlinearities. El-Nabulsi (2018) reported jerk systems in the study of nonlocal effects in fluids, plasmas, and solar physics. This work reports a new 3-D jerk system with chaos. The proposed nonlinear jerk mechanical system has two quadratic nonlinear terms. We show that the new jerk system has the origin as the unique rest point, which is an unstable saddle-focus. In this chapter, we shall analyze the dynamic behavior and control the chaotic behavior of the new chaotic jerk system. Using dynamic analysis, we show that the new jerk system exhibits multi-stability and coexisting attractors. Multi-stability is a complex feature for a chaotic system where coexisting attractors are obtained for the same values of parameters but different initial states (Zhang et al., 2018a,b; Wang et al., 2018; Azar et al., 2018; Wang et al., 2017). Active backstepping control is applied to controlling and synchronizing the chaos in the jerk system. Active control method via the backstepping approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015b; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015; Vaidyanathan et al., 2015a). Finally, a circuit model using Multisim of the new jerk system with chaos is designed for practical implementation. We show that the Multisim outputs of the jerk system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019a; Sambas et al., 2019a,b; Vaidyanathan et al., 2019b; Azar et al., 2017; Vaidyanathan and Azar, 2016b; Vaidyanathan et al., 2018a).

5.2 System model In this chapter, we propose a new jerk mechanical system, which is described as the 4-D model ⎧ ⎪ x˙ = y, ⎪ ⎨ y˙ = z, (5.3) ⎪ ⎪ ⎩ 2 z˙ = −ax − z − xy + by .

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The jerk system (5.3) is equivalent to the jerk differential equation expressed by ... x + x¨ + ax + x x˙ − b(x) ˙ 2 = 0.

(5.4)

In this research work, it is proved that the jerk plant (5.3) shows chaos when the parameters take the values a = 1 and b = 1. For the initial state X(0) = (0.4, 0.2, 0.4) and (a, b) = (1, 1), the Lyapunov characteristic exponents of the jerk plant (5.3) are estimated as ψ1 = 0.11238, ψ2 = 0, ψ3 = −1.11238,

(5.5)

which confirms that the jerk system (5.3) is a dissipative chaos system. It is also an easy calculation to determine the Kaplan–Yorke dimension of the jerk plant (5.3) as DKY = 2 +

ψ 1 + ψ2 = 2.1010. |ψ3 |

(5.6)

The Kaplan–Yorke dimension of the jerk plant (5.3) shows the complex characteristic of the dissipative jerk plant (5.3). The rest points of the hyperjerk plant (5.3) are obtained by solving the following system ⎧ y = 0, ⎪ ⎨ z = 0, (5.7) ⎪ ⎩ 2 −ax − z − xy + by = 0. A simple calculation shows that the jerk plant (5.3) has the unique rest point x = 0, y = 0, z = 0,

(5.8)

which is the origin of R3 . The system matrix of the hyperjerk plant (5.3) at O = (0, 0, 0, 0) is given by ⎡ ⎤ 0 1 0 ⎢ ⎥ (5.9) A = ⎣ 0 0 1 ⎦. −a 0 −1 For the chaos case, (a, b) = (1, 1), the matrix A has the eigenvalues λ1 = −1.4656, λ2,3 = 0.2328 ± 0.7926i.

(5.10)

Hence, O = (0, 0, 0) is a saddle-focus and unstable rest point for the jerk plant (5.3). Figs. 5.1 and 5.2 are MATLAB simulations of the chaotic jerk plant for the parameter values (a, b) = (1, 1) and x(0) = (0.4, 0.2, 0.4). Fig. 5.1 shows the Lyapunov characteristic exponents and Fig. 5.2 shows the phase plots of the chaotic jerk plant (5.3).

98 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 5.1 Lyapunov characteristic exponents of the jerk plant (5.3).

FIGURE 5.2 Chaotic attractors of the jerk system (5.3) in (A) (x, y) plane, (B) (y, z) plane, (C) (x, z) plane, and (D) R3 .

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FIGURE 5.3 Dynamic analysis of the jerk system (5.3) when we fix b = 1, X(0) = (0.4, 0.2, 0.4), and vary a in the region of [0, 1.5]: (A) bifurcation diagram and (B) Lyapunov exponents of the jerk system.

5.3 Dynamic analysis of the new jerk system Here, we discuss the dynamic analysis of the jerk system (5.3). First, we fix b = 1 and initial state X(0) = (0.4, 0.2, 0.4). We vary the parameter a in the region of [0, 1.5]. From the bifurcation diagram shown in Fig. 5.3, we can see that the jerk system (5.3) exhibits chaos, periodic, forward period-doubling route as well as several periodic windows. Some sample results are plotted in Fig. 5.4 to verify the forward perioddoubling route to chaos. Next, we fix a = 1 and initial state X(0) = (0.4, 0.2, 0.4). We vary the parameter b in the region of [0.5, 1.5]. From the bifurcation diagram shown in Fig. 5.5, we can see that the jerk system (5.3) is in chaotic state in the region of [0.5, 1.5] except for a few periodic windows. Next, we discuss coexisting attractors of the jerk system (5.3). We denote the state orbit of the jerk system (5.3) starting with the initial state X0 = (0.4, 0.2, 0.4) in blue color (dark gray in print). Also, we denote the state orbit of the hyperjerk system (5.3) starting with the initial state Y0 = (−0.4, −0.2, 0.4) in red color (light gray in print). We fix b = 0.8 and vary a in the region of [0.7, 0.9]. Fig. 5.6 describes the bifurcation diagram for the jerk system (5.3). From this figure, we can see that there exist coexisting attractors in the very narrow region of b ∈ [0.76, 0.87]. Fig. 5.7 describes some sample results of coexisting attractors as follows. We fix b = 0.8. The jerk system (5.3) exhibits coexisting chaotic attractors when a = 0.86, and coexisting chaotic and periodic attractors when a = 0.8. Also, the jerk system (5.3) exhibits coexisting periodic attractors when a = 0.77.

100 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 5.4 Plots of the jerk system (5.3) when we fix b = 1, X(0) = (0.4, 0.2, 0.4), and vary a in the region of [0, 1.5]: (A) when a = 0.1, period-1, (B) when a = 0.22, period-2, (C) when a = 0.27, period-4, and (D) when a = 0.3, chaos.

FIGURE 5.5 Dynamic analysis of the jerk system (5.3) when we fix a = 1, X(0) = (0.4, 0.2, 0.4), and vary b in the region of [0.5, 1.5]: (A) bifurcation diagram and (B) Lyapunov exponents of the jerk system.

5.4 Backstepping control of the jerk system Backstepping architecture is a sequential control technique characterized by step-by-step interlacing. Each step involves the transformation of the coordinates and the design of a virtual control based on the Lyapunov technique. Finally, under the global stability, the true controller is achieved. Strict feedback systems, blocking systems with strict feedback, and parametric-strict feedback

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FIGURE 5.6 Bifurcation diagram showing coexisting attractors of the jerk plant (5.3).

FIGURE 5.7 Coexisting attractors of the jerk system (5.3) when we fix b = 0.8: (A) when a = 0.86, coexisting chaotic attractors, (B) when a = 0.8, coexisting chaotic and periodic attractors, and (C) when a = 0.77, coexisting periodic attractors.

systems can be implemented with a backstepping design (Azar et al., 2020; Shukla et al., 2018; Vaidyanathan et al., 2018c). Here, we adopt backstepping control for globally stabilizing the trajectories of the new jerk chaotic system (5.3) for all initial conditions.

102 Backstepping Control of Nonlinear Dynamical Systems

The controlled jerk system is described by the 3D dynamics ⎧ ⎪ ⎨ x˙ = y y˙ = z ⎪ ⎩ z˙ = −ax − z − xy + by 2 + v,

(5.11)

where v is a backstepping control to be determined. We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(5.12)

where ηx = x.

(5.13)

Differentiating W1 with respect to t along the dynamics (5.11), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (x + y).

(5.14)

ηy = x + y.

(5.15)

We define

With the help of Eq. (5.15), we can express (5.14) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(5.16)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

1 2 1 2 ηy = ηx + ηy2 . 2 2

(5.17)

Differentiating W2 with respect to t along the dynamics (5.11), we get W˙ 2 = −ηx2 − ηy2 + ηy (2x + 2y + z).

(5.18)

We define ηz as follows: ηz = 2x + 2y + z.

(5.19)

With the help of Eq. (5.19), we can express Eq. (5.18) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(5.20)

As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 1 W (ηx , ηy , ηz ) = W2 (ηx , ηy ) + ηx2 = (ηx2 + ηy2 + ηz2 ). 2 2

(5.21)

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It is clear that W is a positive definite function on R3 . Differentiating W with respect to t along the jerk dynamics (5.11), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ).

(5.22)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 − a)x + 3y + z − xy + by 2 + v].

(5.23)

We define the control law v as v = −(2 − a)x − 3y − z + xy − by 2 − Kηz

(5.24)

where K is taken as a positive constant. Substituting (5.24) into (5.23), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(5.25)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that x = ηx , y = ηy − ηx , z = ηx − 2ηy .

(5.26)

As a consequence, it follows that (x(t), y(t), z(t)) → (0, 0, 0) exponentially as t → ∞. Thus, we have proved the following result. Theorem 5.1. The backstepping control law defined via (5.24) with gain K > 0 globally and exponentially stabilizes all the trajectories of the 3D jerk chaos plant (5.11) for all initial states (x(0), y(0), z(0)) ∈ R3 . For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b) = (1, 1). We choose K = 5 and the initial state of the jerk chaos system (5.11) as x(0) = 7.2, y(0) = 3.1, and z(0) = 6.8. Fig. 5.8 shows the time-history of the backstepping controlled states x(t), y(t), and z(t). It is easy to see that the controlled states converge to zero exponentially by the action of the backstepping control law (5.24).

5.5 Backstepping synchronization of the jerk system The synchronization of two chaotic systems that recognize the tendency for two or more systems to undergo closely related movements was shown to be cou-

104 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 5.8 Time-history of the backstepping controlled states x(t), y(t), z(t) for (a, b, c) = (1, 1), K = 5, and (x(0), y(0), z(0)) = (7.2, 3.1, 6.8).

pled (Ouannas et al., 2020a; Khan et al., 2020b,a; Ouannas et al., 2020b, 2019; Vaidyanathan and Azar, 2016c,a,d,e; Ouannas et al., 2017b,a). Chaos synchronization is a problem of designing a link between both systems to idealize the chaotic time assessment (Pham et al., 2018; Khettab et al., 2018; Singh et al., 2018a; Alain et al., 2018; Vaidyanathan et al., 2018b; Singh et al., 2018b). The response system output tracks the drive system output asymptotically, i.e. the master system output controls the slave system (Singh et al., 2018c; Grassi et al., 2017; Pham et al., 2017; Ouannas et al., 2017d,c). Here, we adopt backstepping control for globally synchronizing the state trajectories of a pair of new jerk chaos systems considered as leader–follower systems. The leader chaos system is depicted by the 3D jerk dynamics ⎧ ⎪ ⎨ x˙1 y˙1 ⎪ ⎩ z˙ 1

=

y1 ,

=

z1 ,

=

−ax1 − z1 − x1 y1 + by12 .

(5.27)

The follower chaos system is depicted by the 3D jerk dynamics ⎧ ⎪ ⎨ x˙2 y˙2 ⎪ ⎩ z˙ 2

=

y2 ,

=

z2 ,

=

−ax2 − z2 − x2 y2 + by22

where v is a backstepping control to be designed.

(5.28) + v,

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The synchronization chaos error is defined by means of the equations ex = x2 − x1 , ey = y2 − y1 , ez = z2 − z1 . Upon calculating the error dynamics, we obtain the following: ⎧ ⎪ ⎨ e˙x = ey , e˙y = ez , ⎪ ⎩ e˙z = −aex − ez − x2 y2 + x1 y1 + b(y22 − y12 ) + v.

(5.29)

(5.30)

We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(5.31)

where ηx = ex .

(5.32)

Differentiating W1 with respect to t along the dynamics (5.30), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (ex + ey ).

(5.33)

η y = e x + ey .

(5.34)

We define

With the help of Eq. (5.34), we can express (5.33) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(5.35)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

1 2 1 2 ηy = ηx + ηy2 . 2 2

(5.36)

Differentiating W2 with respect to t along the dynamics (5.30), we get W˙ 2 = −ηx2 − ηy2 + ηy (2ex + 2ey + ez ).

(5.37)

We define ηz as follows: ηz = 2ex + 2ey + ez .

(5.38)

With the help of Eq. (5.38), we can express Eq. (5.37) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(5.39)

106 Backstepping Control of Nonlinear Dynamical Systems

As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 1 W (ηx , ηy , ηz ) = W2 (ηx , ηy ) + ηx2 = (ηx2 + ηy2 + ηz2 ). 2 2

(5.40)

It is clear that W is a positive definite function on R3 . Differentiating W with respect to t along the error dynamics (5.30), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ).

(5.41)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 − a)ex + 3ey + ez − x2 y2 + x1 y1 + b(x22 − x12 ) + v].

(5.42)

We define the control law v as v = −(2 − a)ex − 3ey − ez + x2 y2 − x1 y1 − b(x22 − x12 ) − Kηz

(5.43)

where K is taken as a positive constant. Substituting (5.43) into (5.42), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(5.44)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that ex = ηx , ey = ηy − ηx , ez = ηx − 2ηy .

(5.45)

As a consequence, it follows that (ex (t), ey (t), ez (t)) → (0, 0, 0) exponentially as t → ∞. Thus, we have proved the following result. Theorem 5.2. The backstepping control law defined via (5.43) with gain K > 0 globally and exponentially synchronizes the 3D jerk chaos plants (5.27) and (5.28) for all initial states in R3 . For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b) = (1, 1). We choose K = 5.

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FIGURE 5.9 Synchronization between the states x1 and x2 of the leader–follower systems (5.27) and (5.28).

FIGURE 5.10 Synchronization between the states y1 and y2 of the leader–follower systems (5.27) and (5.28).

We take the initial state of the leader system (5.27) as x1 (0) = 4.2, y1 (0) = 3.4, and z1 (0) = −2.8. We also consider the initial state of the follower system (5.28) as x2 (0) = 1.9, y2 (0) = 2.1, and z2 (0) = 5.7. Figs. 5.9–5.11 display the synchronization between the states of the leader system and follower system. Fig. 5.12 shows the exponential convergence of the synchronization error ex (t), ey (t), ez (t) between the jerk systems (5.27) and (5.28).

108 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 5.11 Synchronization between the states z1 and z2 of the leader–follower systems (5.27) and (5.28).

FIGURE 5.12 Time-history of the synchronization error between the leader–follower systems (5.27) and (5.28).

5.6 Electronic circuit simulation of the chaotic jerk system The implementation of electronic circuits in chaotic systems is very important for engineering applications such as robotics, secure communication, higher frequency, random bit generator, etc. A circuit for implementation the chaotic jerk system is designed by using Multisim software. The circuit includes ten resistors, three capacitors, five operational amplifiers, and two analog multipliers.

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FIGURE 5.13 Schematic diagram of the jerk system (5.46).

The circuit equation given in (5.46) can be validated in a straightforward manner from the circuit in Fig. 5.13. We have C1 x˙

=

1 R1 y,

C2 y˙

=

1 R2 z,

C3 z˙

= − R13 x −

(5.46) 1 1 R4 z − 10R5 xy

+

1 2 10R6 y .

Here, x, y, and z are the output voltages of the operational amplifiers U1A, U2A, and U3A, respectively. The values of circuit components are selected as

110 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 5.14 Phase projections of the chaotic jerk system (5.46) using Multisim circuit simulation: (A) x–y plane, (B) y–z plane, and (C) x–z plane.

follows: R5 = R6 = 10 k, R1 = R2 = R3 = R4 = R7 = R8 = R9 = R10 = 100 k, and C1 = C2 = C3 = 1 nF. Multisim results of the circuit are shown in Fig. 5.14, which shows its chaotic behavior.

5.7 Conclusions In this chapter, we announced a new jerk system with two quadratic nonlinear terms exhibiting chaotic behavior. We demonstrated that the jerk system has a unique unstable rest point at the origin. We also showed that the new jerk system has multi-stability and coexisting attractors. We discussed the qualitative behavior of the new jerk system in detail with phase plots, Lyapunov exponents, and bifurcation diagrams. Backstepping control was successfully applied to controlling and synchronizing the chaos in the proposed jerk chaotic system. Finally, a circuit design of the new chaotic jerk dynamical system was designed using Multisim to enable the practical implementation of the proposed jerk system.

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Acknowledgments The authors thank the Government of Malaysia for funding this research under the Fundamental Research Grant Scheme (FRGS/1/2018/ICT03/UNISZA/02/2) and also Universiti Sultan Zainal Abidin, Terengganu, Malaysia.

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Ouannas, A., Azar, A.T., Ziar, T., Radwan, A.G., 2017c. Generalized synchronization of different dimensional integer-order and fractional order chaotic systems. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 671–697. Ouannas, A., Azar, A.T., Ziar, T., Radwan, A.G., 2017d. A study on coexistence of different types of synchronization between different dimensional fractional chaotic systems. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 637–669. Ouannas, A., Grassi, G., Azar, A.T., 2020a. Fractional-order control scheme for Q-S chaos synchronization. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). Springer International Publishing, Cham, pp. 434–441. Ouannas, A., Grassi, G., Azar, A.T., 2020b. A new generalized synchronization scheme to control fractional chaotic systems with non-identical dimensions and different orders. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). In: Advances in Intelligent Systems and Computing, vol. 921. Springer International Publishing, Cham, pp. 415–424. Ouannas, A., Grassi, G., Azar, A.T., Singh, S., 2019. New control schemes for fractional chaos synchronization. In: Hassanien, A.E., Tolba, M.F., Shaalan, K., Azar, A.T. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2018. In: Advances in Intelligent Systems and Computing, vol. 845. Springer International Publishing, Cham, pp. 52–63. Pham, V.-T., Gokul P.M., Kapitaniak, T., Volos, C., Azar, A.T., 2018. Dynamics, synchronization and fractional order form of a chaotic system with infinite equilibria. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 475–502. Pham, V.-T., Vaidyanathan, S., Volos, C.K., Azar, A.T., Hoang, T.M., Van Yem, V., 2017. A threedimensional no-equilibrium chaotic system: analysis, synchronization and its fractional order form. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 449–470. Rajagopal, K., Akgul, A., Jafari, S., Karthikeyan, A., Cavusoglu, U., Kacar, S., 2019. An exponential jerk system: circuit realization, fractional order and time delayed form with dynamical analysis and its engineering application. Journal of Circuits, Systems and Computers 28 (5), 1950087. Rasappan, S., Vaidyanathan, S., 2012. Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East Journal of Mathematical Sciences 67 (2), 265–287. Sambas, A., Vaidyanathan, S., Tlelo-Cuautle, E., Zhang, S., Guillen-Fernandez, O., Sukono, Hidayat, Y., Gundara, G., 2019a. A novel chaotic system with two circles of equilibrium points: multistability, electronic circuit and FPGA realization. Electronics 8 (11), 1211. Sambas, A., Vaidyanathan, S., Zhang, S., Zeng, Y., Mohamed, M., 2019b. A new double-wing chaotic system with coexisting attractors and line equilibrium: bifurcation analysis and electronic circuit simulation. IEEE Access 7, 115454–115462. Shukla, M.K., Sharma, B.B., Azar, A.T., 2018. Control and synchronization of a fractional order hyperchaotic system via backstepping and active backstepping approach. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 559–595. Singh, S., Azar, A.T., Bhat, M.A., Vaidyanathan, S., Ouannas, A., 2018a. Active control for multi-switching combination synchronization of non-identical chaotic systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in System Dynamics and Control. In: Advances in Systems

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Analysis, Software Engineering, and High Performance Computing (ASASEHPC). IGI Global, pp. 129–162. Singh, S., Azar, A.T., Vaidyanathan, S., Ouannas, A., Bhat, M.A., 2018b. Multiswitching synchronization of commensurate fractional order hyperchaotic systems via active control. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 319–345. Singh, S., Azar, A.T., Zhu, Q., 2018c. Multi-switching master–slave synchronization of nonidentical chaotic systems. In: Zhu, Q., Na, J., Wu, X. (Eds.), Innovative Techniques and Applications of Modelling, Identification and Control: Selected and Expanded Reports from ICMIC’17. In: Lecture Notes in Electrical Engineering, vol. 467. Springer Singapore, Singapore, pp. 321–330. Vaidyanathan, S., 2015. Analysis, control, and synchronization of a 3-D novel jerk chaotic system with two quadratic nonlinearities. Kyungpook Mathematical Journal 55 (3), 563–586. Vaidyanathan, S., 2016. A novel 3-D jerk chaotic system with two quadratic nonlinearities and its adaptive backstepping control. International Journal of Control Theory and Applications 9 (1), 199–219. Vaidyanathan, S., 2017. A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control. Archives of Control Sciences 27 (3), 409–439. Vaidyanathan, S., Abba, O.A., Betchewe, G., Alidou, M., 2019a. A new three-dimensional chaotic system: its adaptive control and circuit design. International Journal of Automation and Control 13 (1), 101–121. Vaidyanathan, S., Akgul, A., Kacar, S., 2018. A new chaotic jerk system with two quadratic nonlinearities and its applications to electronic circuit implementation and image encryption. International Journal of Computer Applications in Technology 58 (2), 89–101. Vaidyanathan, S., Azar, A.T., 2016a. A novel 4-D four-wing chaotic system with four quadratic nonlinearities and its synchronization via adaptive control method. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 203–224. Vaidyanathan, S., Azar, A.T., 2016b. Adaptive control and synchronization of Halvorsen circulant chaotic systems. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 225–247. Vaidyanathan, S., Azar, A.T., 2016c. Dynamic analysis, adaptive feedback control and synchronization of an eight-term 3-D novel chaotic system with three quadratic nonlinearities. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 155–178. Vaidyanathan, S., Azar, A.T., 2016d. Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 275–290. Vaidyanathan, S., Azar, A.T., 2016e. Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in Chaos Theory and Intelligent Control. Springer International Publishing, Cham, pp. 179–202. Vaidyanathan, S., Azar, A.T., Rajagopal, K., Alexander, P., 2015a. Design and spice implementation of a 12-term novel hyperchaotic system and its synchronisation via active control. International Journal of Modelling, Identification and Control 23 (3), 267–277. Vaidyanathan, S., Azar, A.T., Rajagopal, K., Sambas, A., Kacar, S., Cavusoglu, U., 2018a. A new hyperchaotic temperature fluctuations model, its circuit simulation, FPGA implementation and an application to image encryption. International Journal of Simulation and Process Modelling 13 (3), 281–296. Vaidyanathan, S., Azar, A.T., Sambas, A., Singh, S., Alain, K.S.T., Serrano, F.E., 2018b. A novel hyperchaotic system with adaptive control, synchronization, and circuit simulation. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in System Dynamics and Control. In: Advances in Systems Analysis, Software Engineering, and High Performance Computing (ASASEHPC). IGI Global, pp. 382–419.

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Vaidyanathan, S., Jafari, S., Pham, V.-T., Azar, A.T., Alsaadi, F.E., 2018c. A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design. Archives of Control Sciences 28 (2), 239–254. Vaidyanathan, S., Sambas, A., Mamat, M., Sanjaya, W.S.M., 2017. Analysis, synchronisation and circuit implementation of a novel jerk chaotic system and its application for voice encryption. International Journal of Modelling, Identification and Control 28 (2), 153–166. Vaidyanathan, S., Sambas, A., Zhang, S., 2019b. A new 4-D dynamical system exhibiting chaos with a line of rest points, its synchronization and circuit model. Archives of Control Sciences 29 (3), 485–506. Vaidyanathan, S., Volos, C., Rajagopal, K., Kyprianidis, I., Stouboulos, I., 2015b. Adaptive backstepping controller design for the anti-synchronization of identical WINDMI chaotic systems with unknown parameters and its SPICE implementation. Journal of Engineering Science and Technology Review 8 (2), 74–82. Wang, Z., Volos, C., Kingni, S.T., Azar, A.T., Pham, V.-T., 2017. Four-wing attractors in a novel chaotic system with hyperbolic sine nonlinearity. Optik - International Journal for Light and Electron Optics 131, 1071–1078. Wang, L., Zhang, S., Zeng, Y.-C., Li, Z.-J., 2018. Generating hidden extreme multistability in memristive chaotic oscillator via micro-perturbation. Electronics Letters 54 (13), 808–810. Zhang, S., Zeng, Y., Li, Z., 2018a. A novel 4D no-equilibrium hyper-chaotic system with grid multiwing hyper-chaotic hidden attractors. Journal of Computational and Nonlinear Dynamics 13 (9), 090908. Zhang, S., Zeng, Y., Li, Z., 2018b. One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics. Chinese Journal of Physics 56 (3), 793–806.

Chapter 6

A new 5-D hyperchaotic four-wing system with multistability and hidden attractor, its backstepping control, and circuit simulation Sundarapandian Vaidyanathana , Aceng Sambasb , Ahmad Taher Azarc,d , K.P.S. Ranae , and Vineet Kumare a Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, b Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya,

Indonesia, c Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, d Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, e Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India

6.1

Introduction

The occurrence of two or more positive Lyapunov exponents is the characterizing property of a dynamical system to be a hyperchaotic system (Pham et al., 2018b; Khan et al., 2020b,a; Ouannas et al., 2019a, 2017e,b,a). Hyperchaotic systems exhibit complex properties and characteristics which make them suitable for several applications such as lasers (Bonatto, 2018; Mahmoud and Al-Harthi, 2020; Yan, 2013), chemical systems (Nieto-Villar and Velarde, 2000), finance (Jahanshahi et al., 2019; Cao, 2018), memristors (Rajagopal et al., 2018a,b; Vaidyanathan et al., 2019b; Rajagopal et al., 2018c), electromechanical systems (Wang and Wu, 2018), hyperjerk systems (Ahmad et al., 2018; Vaidyanathan, 2016; Vaidyanathan et al., 2015b; Vaidyanathan and Azar, 2016a; Vaidyanathan et al., 2018b), neural networks (Vaidyanathan et al., 2020), and encryption (Yang, 2017; Bouslehi and Seddik, 2018b,a; Liu et al., 2019; Vaidyanathan et al., 2018a). In the chaos literature, many new mathematical models of hyperchaotic models have been constructed from popular 3D chaos models such as hyperchaos Lorenz system (Chen, 2018), hyperchaos Chen system (Liu et al., 2011), hyperchaos Lü system (Xiao-Hong and Dong, 2009), hyperchaos Vaidyanathan system (Vaidyanathan et al., 2016), etc. Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00013-1 Copyright © 2021 Elsevier Inc. All rights reserved.

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116 Backstepping Control of Nonlinear Dynamical Systems

Recently, there is good interest in building 5D hyperchaos models in the literature. Using active state feedback or other methods, 5D hyperchaos with interesting qualitative properties have been reported in the chaos literature (Singh et al., 2018d; Zhang et al., 2018; Wang et al., 2018; Koyuncu et al., 2019; Singh et al., 2018a,b; Vaidyanathan et al., 2015a). Singh et al. (2018d) designed a new 5D hyperchaos system with stable equilibrium, which belongs to the category of chaotic systems with hidden attractors. Zhang et al. (2018) developed a new 5D hyperchaos system of Lorenz type with unstable equilibrium points. Wang et al. (2018) proposed a new 5D hyperchaos system with a flux-controlled memristor. Koyuncu et al. (2019) implemented a 5D hyperchaos system in real-time high-speed FPGA. In this research work, we build a 5D hyperchaos four-wing with four quadratic nonlinear terms. It is interesting to note that the new 5D hyperchaos system has no rest point. Hence, it belongs to the category of chaotic systems with hidden attractors (Pham et al., 2018b). Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set. It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems (Naimzada and Pireddu, 2014; Hizanidis et al., 2018; Kengne and Mogue, 2019; Tamba and Fotsin, 2017; Lai and Grebogi, 2017). It is known that multistability can lead to very complex behaviors in a dynamical system. In this work, it is shown that the new 5D hyperchaos four-wing system has multistability and coexisting attractors. Active backstepping control is applied to control and synchronize the chaos in the 5D hyperchaos four-wing system. Active control method via backstepping approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015c; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015; Shukla et al., 2018). Finally, a circuit model using Multisim of the new 5D hyperchaos four-wing system is designed for practical implementation. We show that the Multisim outputs of the 5D hyperchaos four-wing system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019a; Sambas et al., 2019a,b; Vaidyanathan et al., 2019c).

6.2 System model In this chapter, we announce our finding of a new 5-D hyperchaotic four-wing model with multistability and hidden attractor.

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Our new 5-D dynamical model is described by the following autonomous system of differential equations: ⎧ ⎪ x˙1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙2 ⎪ ⎪ ⎨ x˙3 ⎪ ⎪ ⎪ ⎪ ⎪ x˙4 ⎪ ⎪ ⎪ ⎪ ⎩ x˙5

=

x1 + x2 + x2 x3 − ax4 + bx5 ,

=

(x2 − x1 )x3 ,

=

b − x3 − x1 x2 ,

=

x1 ,

=

cx3 .

(6.1)

We use X to denote all the states of the 5-D model (6.1), i.e. X = (x1 , x2 , x3 , x4 , x5 ). It is noted that the system (6.1) has three quadratic nonlinear terms x1 x3 , x1 x2 , and x2 x3 in its dynamics. In (6.1), a, b, and c are positive constant parameters. In this work, we establish that the 5-D model (6.1) is a hyperchaotic fourwing system for the parameter values taken as a = 0.8, b = 0.4, c = 0.2.

(6.2)

For numerical calculations, we fix the parameter vector as (a, b, c) = (0.8, 0.4, 0.2). Also, we fix the initial state as X(0) = (0.3, 0.2, 0.1, 0.2, 0.3). Using MATLAB, the Lyapunov characteristic exponents of the 5-D model (6.1) are calculated and shown in Fig. 6.1. We find the Lyapunov exponents as ψ1 = 0.2271, ψ2 = 0.0483, ψ3 = 0, ψ4 = −0.0014, ψ5 = −1.9998. (6.3) The Kaplan–Yorke dimension of the 5-D model (6.1) is found as DKY = 4 +

ψ 1 + ψ 2 + ψ3 + ψ 4 = 4.1370. |ψ5 |

(6.4)

From Eq. (6.3), it can be deduced that the 5-D model (6.1) is hyperchaotic as there are two positive Lyapunov exponents. The 5-D hyperchaotic model (6.1) is also dissipative since the sum of the Lyapunov characteristic exponents in (6.3) is negative. The 2-D phase plots of the 5-D hyperchaotic model (6.1) are given in Fig. 6.2. Thus, it is clear that the 5-D hyperchaotic model (6.1) has a four-wing hyperchaotic attractor.

118 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 6.1 Lyapunov exponents of the 5-D hyperchaotic four-wing model (6.1) for the initial state (0.3, 0.2, 0.1, 0.2, 0.3) and parameter values (a, b, c) = (0.8, 0.4, 0.2).

FIGURE 6.2 MATLAB plots of the 5-D hyperchaotic four-wing model (6.1) for the initial state (0.3, 0.2, 0.1, 0.2, 0.3) and parameter values (a, b, c) = (0.8, 0.4, 0.2): (A) (x1 , x2 )-plane, (B) (x2 , x3 )-plane, (C) (x3 , x4 )-plane, and (D) (x1 , x5 )-plane.

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6.3 Dynamic analysis of the 5-D hyperchaotic four-wing model 6.3.1 Rest points The rest points or equilibrium points of the 5-D model (6.1) are obtained by solving the following equations: x1 + x2 + x2 x3 − ax4 + bx3 = 0,

(6.5a)

(x2 − x1 )x3 = 0,

(6.5b)

b − x3 − x1 x2 = 0,

(6.5c)

x1 = 0,

(6.5d)

cx3 = 0.

(6.5e)

We take the parameter values a, b, c as all positive. From (6.5d), we must have x1 = 0. From (6.5e), we must have x3 = 0. Substituting x1 = x3 = 0 in Eq. (6.5c), we get b = 0, which is a contradiction. This shows that the 5-D hyperchaotic four-wing system (6.1) has no equilibrium point and that the system (6.1) has a hidden attractor.

6.3.2 Multistability Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set (Azar et al., 2018; Wang et al., 2017). It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems. It is known that multistability can lead to very complex behaviors in a dynamical system. It is interesting that the new hyperchaotic four-wing system (6.1) can exhibit coexisting attractors when choosing different initial conditions. We take parameter values as in the hyperchaotic case, viz. (a, b, c) = (0.8, 0.4, 0.2). We select two initial conditions as X0 = (0.3, 0.2, 0.1, 0.2, 0.3), Y0 = (−0.3, −0.2, 0.1, −0.2, −0.3), and the corresponding state orbits of the system (6.1) are plotted in the colors, blue (dark gray in print) and red (light gray in print), respectively. From Fig. 6.3, it can be observed that the new 5-D hyperchaotic four-wing system (6.1) exhibits multistability with two coexisting hyperchaotic attractors.

6.4 Active backstepping control for the global stabilization design of the new 5-D hyperchaotic four-wing system Here, we employ active backstepping technique for globally stabilizing the new 5-D hyperchaotic four-wing system.

120 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 6.3 Multistability of the 5-D hyperchaotic four-wing system (6.1): Coexisting hyperchaotic attractors for (a, b, c) = (0.8, 0.4, 0.2) and initial conditions X0 = (0.3, 0.2, 0.1, 0.2, 0.3) (blue, dark gray in print) and Y0 = (−0.3, −0.2, 0.1, −0.2, −0.3) (red, light gray in print). (A) (x1 , x2 )-plane; (B) (x2 , x4 )-plane.

The controlled hyperchaos system is depicted by the 5D dynamics ⎧ x˙1 = x1 + x2 + x2 x3 − ax4 + bx5 + u1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 = (x2 − x1 )x3 + u2 , x˙3 = b − x3 − x1 x2 + u3 , ⎪ ⎪ ⎪ ⎪ x˙4 = x1 + u4 , ⎪ ⎪ ⎩ x˙5 = cx3 + u5 ,

(6.6)

where u1 , u2 , u3 , u4 , u5 are the active backstepping controls to be determined. As a first step, we use feedback control to transform the system (6.6) to a system with triangular structure that aids backstepping control design. First, we consider the feedback control law ⎧ ⎪ ⎪ u1 = −x1 − x2 x3 + ax4 − bx5 , ⎪ ⎪ ⎪ ⎪ ⎨ u2 = x3 − (x2 − x1 )x3 , (6.7) u3 = −b + x3 + x4 + x1 x2 , ⎪ ⎪ ⎪ ⎪ u4 = −x1 + x5 , ⎪ ⎪ ⎩ u5 = −cx3 + v, where v is a control. Substituting (6.7) into (6.6), we get the new system in triangular form as ⎧ ⎪ ⎪ x˙1 = x2 , ⎪ ⎪ ⎪ ⎨ x˙2 = x3 , (6.8) x˙3 = x4 , ⎪ ⎪ ⎪ x˙4 = x5 , ⎪ ⎪ ⎩ x˙5 = v.

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We begin with the Lyapunov function W1 (η1 ) =

1 2 η 2 1

(6.9)

where η1 = x 1 .

(6.10)

Differentiating W1 with respect to t along the dynamics (6.8), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (x1 + x2 ).

(6.11)

η2 = x1 + x2 .

(6.12)

We define

With the help of (6.12), we can express (6.11) as W˙ 1 = −η12 + η1 η2 .

(6.13)

We proceed next with defining the Lyapunov function 1 1 W2 (η1 , η2 ) = W1 (η1 ) + η22 = (η12 + η22 ). 2 2

(6.14)

Differentiating W2 with respect to t along the dynamics (6.8), we get W˙ 2 = −η12 − η22 + η2 (2x1 + 2x2 + x3 ).

(6.15)

η3 = 2x1 + 2x2 + x3 .

(6.16)

We define

With the help of (6.16), we can express (6.15) as W˙ 2 = −η12 − η22 + η2 η3 .

(6.17)

We proceed next with defining the Lyapunov function 1 1 W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) + η32 = (η12 + η22 + η32 ). 2 2

(6.18)

Differentiating W3 with respect to t along the dynamics (6.8), we get W˙ 3 = −η12 − η22 − η32 + η3 (3x1 + 5x2 + 3x3 + x4 ).

(6.19)

η4 = 3x1 + 5x2 + 3x3 + x4 .

(6.20)

We define

122 Backstepping Control of Nonlinear Dynamical Systems

With the help of (6.20), we can express (6.19) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(6.21)

We proceed next with defining the Lyapunov function 1 1 W4 (η1 , η2 , η3 , η4 ) = W3 (η1 , η2 , η3 ) + η42 = (η12 + η22 + η32 + η42 ). (6.22) 2 2 Differentiating W4 with respect to t along the dynamics (6.8), we get W˙ 4 = −η12 − η22 − η32 − η42 + η4 (5x1 + 10x2 + 9x3 + 4x4 + x5 ).

(6.23)

We define η5 = 5x1 + 10x2 + 9x3 + 4x4 + x5 .

(6.24)

With the help of (6.24), we can express (6.23) as W˙ 4 = −η12 − η22 − η32 − η42 + η4 η5 .

(6.25)

As a final step of the adaptive backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 , η5 ) =

1 2 (η + η22 + η32 + η42 + η52 ). 2 1

(6.26)

Clearly, W is a positive definite function on R5 . Differentiating W with respect to t along the dynamics (6.8), we get W˙ = −η12 − η22 − η32 − η42 − η52 + η5 T

(6.27)

where T = η4 + η5 + η˙ 5 = 8x1 + 20x2 + 22x3 + 14x4 + 5x5 + v.

(6.28)

We define the control v as v = −8x1 − 20x2 − 22x3 − 14x4 − 5x5 − Kη5

(6.29)

where K > 0 is a positive constant. Substituting (6.29) into (6.28), we get T = −Kη4 . Thus, Eq. (6.27) can be simplified as W˙ = −η12 − η22 − η32 − η42 − (1 + K)η52 , which is quadratic and negative definite.

(6.30)

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123

Hence, by Lyapunov stability theory, it is immediate that ηi (t) → 0, (i = 1, . . . , 5) exponentially as t → ∞ for all values of ηi (0) ∈ R, (i = 1, . . . , 5). As a consequence, it follows that xi (t) → 0, (i = 1, 2, 3, 4) exponentially as t → ∞ for all values of xi (0) ∈ R, (i = 1, . . . , 5). Thus, we have established the following main result of this section. Theorem 6.1. The active backstepping control law defined by ⎧ ⎪ u1 = −x1 − x2 x3 + ax4 − bx5 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 = x3 − (x2 − x1 )x3 , u3 ⎪ ⎪ ⎪ ⎪ u4 ⎪ ⎪ ⎪ ⎩ u5

= −b + x3 + x4 + x1 x2 ,

(6.31)

= −x1 + x5 , = −8x1 − 20x2 − (c + 22)x3 − 14x4 − 5x5 − Kη5 ,

where K > 0 and η5 = 5x1 + 10x2 + 9x3 + 4x4 + x5 , globally and exponentially stabilizes the 5D hyperchaotic four-wing system (6.6) for all values of X(0) ∈ R5 . For computer simulations, we take the hyperchaos case for the parameter values a, b, and c, viz. (a, b, c) = (0.8, 0.4, 0.2). We choose K = 6 for the controller gain. Also, we choose X(0) = (1.4, 2.3, 5.9, 4.7, 1.8). Fig. 6.4 shows the exponential convergence of the backstep-controlled state x(t).

FIGURE 6.4 Time-history of the backstep-controlled state X(t) of the 5-D hyperchaotic four-wing system (6.6).

124 Backstepping Control of Nonlinear Dynamical Systems

6.5 Active backstepping control for the global synchronization design of the new 5-D hyperchaotic four-wing systems The synchronization of two or more interacting dynamic systems can occur when these systems are autonomous oscillators (Alain et al., 2020, 2019; Khettab et al., 2018; Singh et al., 2018c; Volos et al., 2018; Vaidyanathan and Azar, 2016c,b; Pham et al., 2018a). Currently, due to its high potential for applications, the analysis of synchronization phenomena in the evolution of chaotic dynamical systems occupies a large part of many particularly interesting study of chaos control (Ouannas et al., 2020, 2019c,a). Since its advent in the 1990s, the study of chaos synchronization problem has grown considerably in many real applications (Ouannas et al., 2019b; Vaidyanathan and Azar, 2016e,d; Tolba et al., 2017; Ouannas et al., 2017d,c,b,j,h,i,g,f,k). Here, we employ active backstepping control method for globally synchronizing the trajectories of a pair of new 5-D hyperchaos four-wing systems considered as leader–follower systems. The leader hyperchaos four-wing system is depicted by the 5D dynamics ⎧ ⎪ x˙1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 x˙3 ⎪ ⎪ ⎪ ⎪ x˙4 ⎪ ⎪ ⎪ ⎩ x˙5

=

x1 + x2 + x2 x3 − ax4 + bx5 ,

=

(x2 − x1 )x3 ,

=

b − x3 − x1 x2 ,

=

x1 ,

=

cx3 .

(6.32)

The follower hyperchaos four-wing system is equipped with backstepping controls and described by the 4D dynamics ⎧ ⎪ y˙1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y˙2 y˙3 ⎪ ⎪ ⎪ ⎪ y˙4 ⎪ ⎪ ⎪ ⎩ y˙5

=

y1 + y2 + y2 y3 − ay4 + by5 + u1 ,

=

(y2 − y1 )y3 + u2 ,

=

b − y3 − y1 y2 + u3 ,

=

y1 + u4 ,

=

cy3 + u5 ,

(6.33)

where u1 , u2 , u3 , u4 , u5 are feedback backstepping controls to be determined. The synchronization hyperchaos error is defined by means of the equations ei = yi − xi , i = 1, 2, 3, 4, 5.

(6.34)

A new 5-D hyperchaotic four-wing system with multistability Chapter | 6

The synchronization error dynamics is calculated as follows: ⎧ ⎪ e˙1 = e1 + e2 − ae4 + be5 + y2 y3 − x2 x3 + u1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙2 = y2 y3 − x2 x3 − y1 y3 + x1 x3 + u2 , e˙3 = −e3 − y1 y2 + x1 x2 + u3 , ⎪ ⎪ ⎪ ⎪ e˙4 = e1 + u4 , ⎪ ⎪ ⎪ ⎩ e˙5 = ce3 + u5 .

125

(6.35)

As a first step, we use feedback control to transform the system (6.35) to a system with triangular structure that aids backstepping control design. First, consider the feedback control law ⎧ u1 = −e1 + ae4 − be5 − y2 y3 + x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 = e3 − y2 y3 + x2 x3 + y1 y3 − x1 x3 , (6.36) u3 = e3 + e4 + y1 y2 − x1 x2 , ⎪ ⎪ ⎪ ⎪ u4 = −e1 + e5 , ⎪ ⎪ ⎩ u5 = −ce3 + v, where v is a feedback control to be determined. Substituting (6.36) into (6.35), we get the new system in triangular form as ⎧ ⎪ e˙1 = e2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙2 = e3 , e˙3 ⎪ ⎪ ⎪ ⎪ e˙4 ⎪ ⎪ ⎪ ⎩ e˙5

=

e4 ,

=

e5 ,

=

v.

(6.37)

We begin with the Lyapunov function W1 (η1 ) =

1 2 η 2 1

(6.38)

where η1 = e1 .

(6.39)

Differentiating W1 with respect to t along the dynamics (6.37), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (e1 + e2 ).

(6.40)

η2 = e1 + e2 .

(6.41)

We define

126 Backstepping Control of Nonlinear Dynamical Systems

With the help of (6.41), we can express (6.40) as W˙ 1 = −η12 + η1 η2 .

(6.42)

We proceed next with defining the Lyapunov function 1 1 W2 (η1 , η2 ) = W1 (η1 ) + η22 = (η12 + η22 ). 2 2

(6.43)

Differentiating W2 with respect to t along the dynamics (6.37), we get W˙ 2 = −η12 − η22 + η2 (2e1 + 2e2 + e3 ).

(6.44)

η3 = 2e1 + 2e2 + e3 .

(6.45)

We define

With the help of (6.45), we can express (6.44) as W˙ 2 = −η12 − η22 + η2 η3 .

(6.46)

We proceed next with defining the Lyapunov function 1 1 W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) + η32 = (η12 + η22 + η32 ). 2 2

(6.47)

Differentiating W3 with respect to t along the dynamics (6.37), we get W˙ 3 = −η12 − η22 − η32 + η3 (3e1 + 5e2 + 3e3 + e4 ).

(6.48)

η4 = 3e1 + 5e2 + 3e3 + e4 .

(6.49)

We define

With the help of (6.49), we can express (6.48) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(6.50)

We proceed next with defining the Lyapunov function 1 1 W4 (η1 , η2 , η3 , η4 ) = W3 (η1 , η2 , η3 ) + η42 = (η12 + η22 + η32 + η42 ). (6.51) 2 2 Differentiating W4 with respect to t along the dynamics (6.37), we get W˙ 4 = −η12 − η22 − η32 − η42 + η4 (5e1 + 10e2 + 9e3 + 4e4 + e5 ).

(6.52)

We define η5 = 5e1 + 10e2 + 9e3 + 4e4 + e5 .

(6.53)

A new 5-D hyperchaotic four-wing system with multistability Chapter | 6

127

With the help of (6.53), we can express (6.52) as W˙ 4 = −η12 − η22 − η32 − η42 + η4 η5 .

(6.54)

As a final step of the adaptive backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 , η5 ) =

1 2 (η + η22 + η32 + η42 + η52 ). 2 1

(6.55)

Clearly, W is a positive definite function on R5 . Differentiating W with respect to t along the dynamics (6.37), we get W˙ = −η12 − η22 − η32 − η42 − η52 + η5 T

(6.56)

where T = η4 + η5 + η˙ 5 = 8e1 + 20e2 + 22e3 + 14e4 + 5e5 + v.

(6.57)

We define the control v as v = −8e1 − 20e2 − 22e3 − 14e4 − 5e5 − Kη5

(6.58)

where K > 0 is a positive constant. Substituting (6.58) into (6.57), we get T = −Kη4 . Thus, Eq. (6.56) can be simplified as W˙ = −η12 − η22 − η32 − η42 − (1 + K)η52 ,

(6.59)

which is quadratic and negative definite. Hence, by Lyapunov stability theory, it is immediate that ηi (t) → 0, (i = 1, . . . , 5) exponentially as t → ∞ for all values of ηi (0) ∈ R, (i = 1, . . . , 5). As a consequence, it follows that ei (t) → 0, (i = 1, . . . , 5) exponentially as t → ∞ for all values of ei (0) ∈ R, (i = 1, . . . , 5). Thus, we have established the following main result of this section. Theorem 6.2. The adaptive backstepping control law defined by ⎧ u1 = −e1 + ae4 − be5 − y2 y3 + x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 = e3 − y2 y3 + x2 x3 + y1 y3 − x1 x3 , u3 = e3 + e4 + y1 y2 − x1 x2 , ⎪ ⎪ ⎪ ⎪ u4 = −e1 + e5 , ⎪ ⎪ ⎩ u5 = −8e1 − 20e2 − (c + 22)e3 − 14e4 − 5e5 − Kη5 ,

(6.60)

where K > 0 and the parameter estimation law and η5 = 5e1 + 10e2 + 9e3 + 4e4 + e5 , globally and exponentially synchronizes the 5D hyperchaotic fourwing systems (6.32) and (6.33) for all values of X(0), Y (0) ∈ R5 .

128 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 6.5 Synchronization between the states x1 and y1 of the 5-D hyperchaotic four-wing systems (6.32) and (6.33).

FIGURE 6.6 Synchronization between the states x2 and y2 of the 5-D hyperchaotic four-wing systems (6.32) and (6.33).

For computer simulations, we take the hyperchaos case for the parameter values a, b, and c, viz. (a, b, c) = (0.8, 0.4, 0.2). We choose K = 6 for the controller gain. We choose X(0) = (3.1, 0.7, 6.5, 4.2, 1.8) and Y (0) = (5.7, 4.9, 2.1, 3.4, 7.2).

A new 5-D hyperchaotic four-wing system with multistability Chapter | 6

129

FIGURE 6.7 Synchronization between the states x3 and y3 of the 5-D hyperchaotic four-wing systems (6.32) and (6.33).

FIGURE 6.8 Synchronization between the states x4 and y4 of the 5-D hyperchaotic four-wing systems (6.32) and (6.33).

Figs. 6.5–6.9 display the synchronization between the states of the 5-D hyperchaotic four-wing systems (6.32) and (6.33). Fig. 6.10 shows the exponential convergence of the synchronization error ei (t), (i = 1, . . . , 5) between the 5-D hyperchaotic four-wing systems (6.32) and (6.33).

130 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 6.9 Synchronization between the states x4 and y4 of the 5-D hyperchaotic four-wing systems (6.32) and (6.33).

FIGURE 6.10 Time-history of the synchronization error between the 5-D hyperchaotic four-wing systems (6.32) and (6.33).

6.6 Circuit simulation of the new 5D hyperchaotic four-wing system In order to illustrate the feasibility of new 5D hyperchaotic system (6.1), an electronic circuit modeling of the new system is constructed. The voltages across the

A new 5-D hyperchaotic four-wing system with multistability Chapter | 6

131

five capacitances represent the five state variables x1 , x2 , x3 , x4 , and x5 , respectively. The operations of additions, subtraction, and integration are implemented by operational amplifiers AD633JN, and the multipliers AD633JN realize the operation of multiplication in the system. As a result, the new 5D hyperchaotic system has to be rescaled by using the transformation 1 1 1 1 1 X 1 = x 1 , X 2 = x2 , X 3 = x3 , X 4 = x4 , X 5 = x5 . 2 2 2 2 4

(6.61)

After the rescaling, we obtain the following new 5D hyperchaos four-wing system: ⎧ ⎪ ⎪ X˙1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X˙ ⎪ ⎨ 2 X˙3 ⎪ ⎪ ⎪ ⎪ ⎪ X˙4 ⎪ ⎪ ⎪ ⎪ ⎩ ˙ X5

=

X1 + X2 + 2X2 X3 − aX4 + 2bX5 ,

=

(2X2 − 2X1 )X3 ,

=

b 2

=

X1 ,

=

cX3 2 .

− X3 − 2X1 X2 ,

(6.62)

Upon applying Kirchhoff’s electrical circuit laws to the circuit of Fig. 6.11, the following set of the new 5D hyperchaotic system can be derived: ⎧ ⎪ X˙1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ X˙2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

=

1 C1 R1 X1

+

1 C1 R2 X2

− C11R4 X4 + =

1 C2 R6 X3 X2

X˙3

=

1 C3 R8 V1

X˙4

=

1 C4 R11 X1 ,

X˙5

=

1 C5 R12 X3 .



+

1 C1 R3 X2 X3

1 C1 R5 X5 ,



1 C2 R7 X3 X1 ,

1 C3 R9 X3



(6.63)

1 C3 R10 X1 X2 ,

Here X1 , X2 , X3 , X4 , X5 are voltages across the capacitors C1 , C2 , C3 , C4 , C4 , C5 , respectively. The parameters are taken as follows: R1 = R2 = R9 = R11 = 400 k, R3 = R6 = R7 = R10 = 20 k, R4 = R5 = 500 k, R8 = 2 M, R12 = 4 M, R13 = R14 = R15 = R15 = R16 = R17 = R18 = R19 = R20 = 100 k, and C1 = C2 = C3 = C4 = C5 = 1 nF. The power supplies of all active devices are ±15 Volts. In Fig. 6.12, we have reported the Multisim results,

132 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 6.11 Schematic diagram of the 5D hyperchaotic four-wing system (6.63).

which show the chaotic attractors of the circuit. A very good similarity between numerical phase portraits in Fig. 6.2 and Multisim simulation results in Fig. 6.12 can be observed.

6.7 Conclusions In this research work, a new 5D hyperchaos four-wing system with three quadratic nonlinear terms has been proposed and its qualitative properties have

A new 5-D hyperchaotic four-wing system with multistability Chapter | 6

133

FIGURE 6.12 Hyperchaotic attractors of the 4D hyperchaotic four-wing system (6.63) using Multisim circuit simulation: (A) X1 –X2 plane, (B) X2 –X3 plane, (C) X3 –X4 plane, and (D) X1 –X5 plane.

been discussed in detail. To begin with, we showed that the new 5D hyperchaos four-wing system has no rest point. Thus, we deduced that the new 5D hyperchaos four-wing system has hidden attractor. It was further shown that the new 5D hyperchaos four-wing system has multistability with coexisting hyperchaos attractors. Using the backstepping control method, we derived new results for the global hyperchaos stabilization and synchronization for the new 5D hyperchaos four-wing system. Finally, using Multisim, we designed an electronic circuit model realizing the 5D hyperchaos four-wing system. Electronic circuit design of the new 5D hyperchaos four-wing system is very useful for the practical implementation and engineering applications of the proposed 5D hyperchaos four-wing system.

134 Backstepping Control of Nonlinear Dynamical Systems

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Chapter 7

A new 4-D hyperchaotic temperature variations system with multistability and strange attractor, bifurcation analysis, its active backstepping control, and circuit realization Sundarapandian Vaidyanathana , Aceng Sambasb , Ahmad Taher Azarc,d , K.P.S. Ranae , and Vineet Kumare a Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, b Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, Indonesia, c Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, d Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, e Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India

7.1

Introduction

The existence of two or more positive Lyapunov exponents is the characterizing property of a dynamical system rendering it a hyperchaotic system (Pham et al., 2018b). Hyperchaotic systems exhibit complex properties and characteristics which make them suitable for several applications such as lasers (Bonatto, 2018; Mahmoud and Al-Harthi, 2020; Yan, 2013), chemical systems (Nieto-Villar and Velarde, 2000), finance (Jahanshahi et al., 2019; Cao, 2018), memristors (Rajagopal et al., 2018a,b; Vaidyanathan et al., 2019b; Rajagopal et al., 2018c), electromechanical systems (Wang and Wu, 2018), hyperjerk systems (Ahmad et al., 2018; Vaidyanathan, 2016; Vaidyanathan et al., 2015a; Vaidyanathan and Azar, 2016b; Vaidyanathan et al., 2018b), neural networks (Vaidyanathan et al., 2020), and encryption (Yang, 2017; Bouslehi and Seddik, 2018b,a; Liu et al., 2019; Vaidyanathan et al., 2018a). Recently, there has been much interest in building new mathematical hyperchaotic models such as the hyperchaos Lorenz system (Chen, 2018), the hyperchaos Chen system (Liu et al., 2011), the hyperchaos Lü system (XiaoBackstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00014-3 Copyright © 2021 Elsevier Inc. All rights reserved.

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140 Backstepping Control of Nonlinear Dynamical Systems

Hong and Dong, 2009), and the hyperchaos Vaidyanathan system (Vaidyanathan et al., 2016). A famous 3D system for temperature variations in the eastern and western parts of the equatorial ocean was modeled by Vallis (1986). The Vallis model is known to have a significant influence on the global weather climate patterns. In this research work, we build a 4D hyperchaos temperature variation system by means of introducing a state feedback in the 3D Vallis system. In this work, we explore the properties of the new hyperchaos temperature variation system with a detailed bifurcation analysis, Lyapunov exponents, equilibrium analysis, etc. It is established that the new hyperchaos temperature variation system has a two-wing strange hyperchaos attractor with a unique saddle-point equilibrium, which is unstable. Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set. It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems (Naimzada and Pireddu, 2014; Hizanidis et al., 2018; Kengne and Mogue, 2019; Tamba and Fotsin, 2017; Lai and Grebogi, 2017). It is known that multistability can lead to very complex behaviors in a dynamical system. In this work, it is shown that the new hyperchaos temperature variation system has multistability and coexisting attractors. Active backstepping control is applied to control and synchronize the chaos in the hyperchaos temperature variation system. Active control method via backstepping approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015b; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015; Shukla et al., 2018). Finally, a circuit model using Multisim of the new 4D hyperchaos temperature variation system is designed for practical implementation. We show that the Multisim outputs of the hyperchaos temperature variation system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019a; Sambas et al., 2019a,b; Vaidyanathan et al., 2019c).

7.2 System model Many systems in nature can exhibit chaos for certain parameter values. A miscellaneous model for temperature variations in the eastern and western parts of the equatorial ocean was modeled by Vallis (1986) as a 3-D system. The Vallis model is known to have a significant influence on the global weather climate patterns. After a change of coordinates, Vallis system can be expressed as the

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141

3-D autonomous system of differential equations given below. ⎧ ⎪ ⎨ x˙1 x˙2 ⎪ ⎩ x˙3

= −ax1 + bx2 , =

x1 − x 2 + x1 x 3 ,

(7.1)

= −x3 − x1 x2 .

In (7.1), x1 is the water velocity on the surface of the ocean, x2 = 0.5(βw − βe ), x3 = 0.5(βw + βe ) − 1, where βe and βw are the temperatures in the eastern and western part of the ocean respectively. It is also noted that a and b are positive constants. Vallis (1986) observed the chaotic property of the temperature model (7.1) when the parameters take the values a = 5, b = 122.

(7.2)

For numerical simulations, we take the initial state as x1 (0) = 0.3, x2 (0) = 0.2, and x3 (0) = 0.3. We also consider the parameter values as (a, b) = (5, 122). For this case, the Lyapunov characteristic exponents of the Vallis model (7.1) can be calculated using MATLAB as ψ1 = 0.5215, ψ2 = 0, ψ3 = −7.5215.

(7.3)

From (7.3), we deduce that the Vallis model (7.1) is chaotic and dissipative with Kaplan–Yorke dimension calculated as DKY = 2 +

ψ 1 + ψ2 = 2.0693. |ψ3 |

(7.4)

Fig. 7.1 shows the 3-D plot of the strange chaotic attractor of the Vallis temperature model (7.1) for the initial state (0.3, 0.2, 0.3) and parameter values (a, b) = (5, 122). From Fig. 7.1, we find that the Vallis temperature model (7.1) exhibits a two-wing chaotic attractor. In this chapter, we announce our finding of a new 4-D hyperchaotic temperature variation system by the introduction of a state feedback in the 3-D Vallis temperature model (7.1) (Vallis, 1986). Our new 4-D dynamical model is described by the following autonomous system of differential equations: ⎧ x˙1 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 ⎪ x˙3 ⎪ ⎪ ⎪ ⎩ x˙4

=

−ax1 + bx2 + x4 ,

=

x1 − x2 + x1 x3 ,

=

−x3 − x1 x2 ,

=

−cx1 − x1 x2 .

(7.5)

142 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 7.1 3-D plot of the two-wing chaotic attractor of the Vallis temperature model (7.1) for the initial state (0.3, 0.2, 0.3) and parameter values (a, b) = (5, 122).

FIGURE 7.2 Lyapunov exponents of the hyperchaotic temperature variations model (7.5) for the initial state (0.3, 0.2, 0.3, 0.2) and parameter values (a, b, c) = (5, 122, 1).

We use X to denote all the states of the 4-D model (7.5), i.e. X = (x1 , x2 , x3 , x4 ). In this work, we establish that the 4-D temperature model (7.5) is a hyperchaotic system for the parameter values taken as a = 10, b = 150, c = 1.

(7.6)

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FIGURE 7.3 MATLAB plots of the 4-D hyperchaotic temperature variation system (7.5) for (a, b, c) = (10, 150, 1) and X(0) = (0.3, 0.2, 0.3, 0.3): (A) (x1 , x2 )-plane, (B) (x2 , x3 )-plane, (C) (x3 , x4 )-plane, and (D) (x1 , x4 )-plane.

For numerical calculations, we fix the parameter vector as (a, b, c) = (10, 150, 1). Also, we fix the initial state as X(0) = (0.3, 0.2, 0.3, 0.2). Using MATLAB, the Lyapunov characteristic exponents of the 4-D model (7.5) are calculated and shown in Fig. 7.2. We find the Lyapunov exponents as ψ1 = 0.2569, ψ2 = 0.0284, ψ3 = 0, ψ4 = −12.2853.

(7.7)

From Eq. (7.7), it can be deduced that the 4-D temperature model (7.5) is hyperchaotic as there are two positive Lyapunov exponents. The 4-D temperature model (7.5) is also dissipative since the sum of the Lyapunov characteristic exponents in (7.7) is negative. Thus, the 4-D hyperchaotic temperature variations model (7.5) has a strange hyperchaotic attractor. The 2-D phase plots of the 4-D hyperchaotic temperature variations model (7.5) are given in Fig. 7.3. Thus, it is clear that the 4-D hyperchaotic temperature variations model (7.5) has a 2-wing hyperchaotic attractor.

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7.3 Dynamic analysis of the hyperchaotic temperature variations model 7.3.1 Bifurcation analysis Bifurcation diagram is classical tool to investigate the dynamics of nonlinear system. We select the parameter a as the control parameter, while the other parameters are kept as (b = 150, c = 1) and initial conditions X(0) = (0.3, 0.2, 0.3, 0.2). As can be seen from Fig. 7.4A, the system (7.5) shows

FIGURE 7.4 (A) Bifurcation diagram of system (7.5) for b = 150, c = 1, and a ∈ [0, 20], (B) bifurcation diagram of system (7.5) for a = 10, c = 1, and b ∈ [100, 150], and (C) bifurcation diagram of system (7.5) for a = 10, b = 150, and c ∈ [0, 10].

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FIGURE 7.4 (continued)

chaotic behavior and hyperchaotic behavior in the parameter a < 11.6. Also, for 11.6 ≤ a ≤ 20, the system (7.5) shows period limit cycles. Here, we fix a = 10, c = 1, and vary b. We can observe the apparent perioddoubling route to chaos or hyperchaos in Fig. 7.4B. Finally, if we take a = 10, b = 150, and vary c. It is clear to see that the proposed system can generate limits cycles and chaos or hyperchaos (see Fig. 7.4C). A period-doubling route to chaos is observed when decreasing the parameter c from 10 to 0.

7.3.2 Rest points The rest points or equilibrium points of the hyperchaotic temperature variations model (7.5) are obtained by solving the following equations: −ax1 + bx2 + x4 = 0,

(7.8a)

x1 − x2 + x1 x3 = 0,

(7.8b)

−x3 − x1 x2 = 0,

(7.8c)

−x1 (x2 + c) = 0.

(7.8d)

We consider the parameter values as in the hyperchaos case, i.e. (a, b, c) = (5, 122, 1). We have two cases to explore: (A) x1 = 0 and (B) x1 = 0. In Case (A), x1 = 0. Substituting this into Eq. (7.8), we get 122x2 + x4 = 0, x2 = 0, and x3 = 0. Obviously, a simplification yields x2 = x3 = x4 = 0. Thus, the origin, O = (0, 0, 0, 0) is the only equilibrium point of the 4-D model (7.5) in this case.

146 Backstepping Control of Nonlinear Dynamical Systems

In Case (B), x1 = 0. From Eq. (7.8d), we get x2 = −1. Substituting this into Eq. (7.8), we get the following system of three equations: −5x1 + x4 − 122 = 0,

(7.9a)

x1 + x1 x3 + 1 = 0,

(7.9b)

x1 − x3 = 0.

(7.9c)

From (7.9c), x1 = x3 . Substituting this into Eq. (7.9), we get −5x1 + x4 − 122 = 0,

(7.10a)

x12 + x1 + 1 = 0.

(7.10b)

The quadratic equation (7.10b) has only complex roots. Thus, we conclude that the hyperchaotic temperature model (7.5) does not have any real equilibrium point in Case (B). Combining Cases (A) and (B), we conclude that the hyperchaotic temperature model (7.5) has a unique rest point at the origin, O = (0, 0, 0, 0). The Jacobian matrix J of the hyperchaotic temperature model (7.5) at O = (0, 0, 0, 0) for the hyperchaos case (a, b, c) = (5, 122, 1) is obtained as follows: ⎡ ⎤ −5 122 0 1 ⎢ ⎥ ⎢ 1 −1 0 0 ⎥ ⎥. J =⎢ (7.11) ⎢ ⎥ 0 −1 0 ⎦ ⎣ 0 −1 0 0 0 The spectral values are found using MATLAB as 1 = 0.0086, 2 = 8.1749, 3 = −1, 4 = −14.1835.

(7.12)

Thus O is a saddle-point and unstable rest point for the hyperchaotic temperature model (7.5).

7.3.3 Multistability Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set (Azar et al., 2018; Wang et al., 2017). It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems. It is known that multistability can lead to very complex behaviors in a dynamical system.

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It is interesting that the new hyperchaotic system (7.5) can exhibit coexisting attractors when choosing different initial conditions. We take parameter values as in the hyperchaotic case, viz. (a, b, c) = (5, 122, 1). We select two initial conditions as X0 = (0.3, 0.2, 0.3, 0.2), Y0 = (0.3, −0.2, 0.3 − 0.2), and the corresponding state orbits of the system (7.5) are plotted in the colors, blue (dark gray in print) and red (light gray in print), respectively. From Fig. 7.5, it can be observed that the new hyperchaotic temperature variation system (7.5) exhibits multistability with two coexisting hyperchaotic attractors.

FIGURE 7.5 Multistability of the hyperchaotic temperature variations system (7.5): Coexisting hyperchaotic attractors for (a, b, c) = (5, 122, 1) and initial conditions X0 = (0.3, 0.2, 0.3, 0.2) (blue, dark gray in print) and Y0 = (0.3, −0.2, 0.3, −0.2) (red, light gray in print).

148 Backstepping Control of Nonlinear Dynamical Systems

7.4 Active backstepping control for the global stabilization design of the new hyperchaotic temperature variations system Here, we employ active backstepping technique for globally stabilizing the new hyperchaotic temperature variation system. The controlled hyperchaos system is depicted by the 4D dynamics ⎧ x˙1 = −ax1 + bx2 + x4 + u1 , ⎪ ⎪ ⎪ ⎪ ⎨ x˙ = x − x + x x + u , 2 1 2 1 3 2 (7.13) ⎪ ⎪ ⎪ x˙3 = −x3 − x1 x2 + u3 , ⎪ ⎩ x˙4 = −cx1 − x1 x2 + u4 , where u1 , u2 , u3 , u4 are the active backstepping controls to be determined. As a first step, we use feedback control to transform the system (7.13) to a system with triangular structure that aids backstepping control design. First, we consider the feedback control law ⎧ u1 = ax1 − (b − 1)x2 − x4 , ⎪ ⎪ ⎪ ⎨ u = −x + x + x − x x , 2 1 2 3 1 3 (7.14) ⎪ = x + x + x x , u 3 3 4 1 2 ⎪ ⎪ ⎩ u4 = cx1 + x1 x2 + v, where v is a control. Substituting (7.14) into (7.13), we get the new system in triangular form as ⎧ ⎪ x˙1 = x2 , ⎪ ⎪ ⎪ ⎨ x˙ = x , 2 3 (7.15) ⎪ = x x ˙ ⎪ 3 4, ⎪ ⎪ ⎩ x˙ = v. 4 We begin with the Lyapunov function W1 (η1 ) =

1 2 η 2 1

(7.16)

where η1 = x1 .

(7.17)

Differentiating W1 with respect to t along the dynamics (7.15), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (x1 + x2 ).

(7.18)

η2 = x1 + x2 .

(7.19)

We define

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With the help of (7.19), we can express (7.18) as W˙ 1 = −η12 + η1 η2 .

(7.20)

We proceed next with defining the Lyapunov function 1 1 W2 (η1 , η2 ) = W1 (η1 ) + η22 = (η12 + η22 ). 2 2

(7.21)

Differentiating W2 with respect to t along the dynamics (7.15), we get W˙ 2 = −η12 − η22 + η2 (2x1 + 2x2 + x3 ).

(7.22)

η3 = 2x1 + 2x2 + x3 .

(7.23)

We define

With the help of (7.23), we can express (7.22) as W˙ 2 = −η12 − η22 + η2 η3 .

(7.24)

We proceed next with defining the Lyapunov function 1 1 W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) + η32 = (η12 + η22 + η32 ). 2 2

(7.25)

Differentiating W3 with respect to t along the dynamics (7.15), we get W˙ 3 = −η12 − η22 − η32 + η3 (3x1 + 5x2 + 3x3 + x4 ).

(7.26)

η4 = 3x1 + 5x2 + 3x3 + x4 .

(7.27)

We define

With the help of (7.27), we can express (7.26) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(7.28)

As a final step of the adaptive backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 ) =

1 2 (η + η22 + η32 + η42 ). 2 1

(7.29)

Clearly, W is a positive definite function on R4 . Differentiating W with respect to t along the dynamics (7.15), we get W˙ = −η12 − η22 − η32 − η42 + η4 T

(7.30)

T = η3 + η4 + η˙ 4 = 5x1 + 10x2 + 9x3 + 4x4 + v.

(7.31)

where

150 Backstepping Control of Nonlinear Dynamical Systems

We define the control v as v = −5x1 − 10x2 − 9x3 − 4x4 − Kη4

(7.32)

where K > 0 is a positive constant. Substituting (7.32) into (7.31), we get T = −Kη4 . Thus, Eq. (7.30) can be simplified as W˙ = −η12 − η22 − η32 − (1 + K)η42 ,

(7.33)

which is quadratic and negative definite. Hence, by Lyapunov stability theory, it is immediate that ηi (t) → 0, (i = 1, 2, 3, 4) exponentially as t → ∞ for all values of ηi (0) ∈ R (i = 1, 2, 3, 4). As a consequence, it follows that xi (t) → 0, (i = 1, 2, 3, 4) exponentially as t → ∞ for all values of xi (0) ∈ R (i = 1, 2, 3, 4). Thus, we have established the following main result of this section. Theorem 7.1. The active backstepping control law defined by ⎧ u1 ⎪ ⎪ ⎪ ⎨ u 2 ⎪ u3 ⎪ ⎪ ⎩ u4

= ax1 − (b − 1)x2 − x4 , = −x1 + x2 + x3 − x1 x3 , = x3 + x 4 + x1 x 2 , =

(7.34)

(c − 5)x1 − 10x2 − 9x3 − 4x4 + x1 x2 − Kη4 ,

where K > 0 and η4 = 3x1 + 5x2 + 3x3 + x4 , globally and exponentially stabilizes the 4D hyperchaotic temperature variation system (7.13) for all values of x(0) ∈ R4 . For computer simulations, we take the hyperchaos case for the parameter values a, b, and c, viz. (a, b, c) = (5, 122, 1). We choose K = 6 for the controller gain. Also, we choose, X(0) = (3.9, 8.2, 11.5, 4.7). Fig. 7.6 shows the exponential convergence of the backstep-controlled state x(t).

7.5 Active backstepping control for the global synchronization design of the new hyperchaos temperature variation systems The chaos synchronization problem has been studied extensively in the literature (Vaidyanathan and Azar, 2016f,d,a,c,e; Ouannas et al., 2017a; Alain et al., 2019, 2020; Khan et al., 2020b,a). In most of the synchronization approaches, the master–slave or drive–response formalism is used. If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of synchronization is to use the output of the master system to control the slave system so that the output of the response

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FIGURE 7.6 Time-history of the backstep-controlled state X(t) of the hyperchaotic temperature variation system (7.13).

system tracks the output of the master system asymptotically (Ouannas et al., 2020, 2019a, 2017c,b,a,e,d, 2019b; Pham et al., 2018a; Singh et al., 2018a,b,c; Volos et al., 2018). Here, we employ active backstepping control method for globally synchronizing the trajectories of a pair of new hyperchaos temperature variation systems considered as leader–follower systems. The leader hyperchaos temperature variation system is depicted by the 4D dynamics ⎧ x˙1 = −ax1 + bx2 + x4 , ⎪ ⎪ ⎪ ⎪ ⎨ x˙ = x − x + x x , 2 1 2 1 3 (7.35) ⎪ x˙3 = −x3 − x1 x2 , ⎪ ⎪ ⎪ ⎩ x˙4 = −cx1 − x1 x2 . The follower hyperchaos temperature variation system is equipped with backstepping controls and depicted by the 4D dynamics ⎧ y˙1 ⎪ ⎪ ⎪ ⎪ ⎨ y˙ 2 ⎪ y˙3 ⎪ ⎪ ⎪ ⎩ y˙4

= −ay1 + by2 + y4 + u1 , =

y1 − y2 + y1 y3 + u2 ,

= −y3 − y1 y2 + u3 ,

(7.36)

= −cy1 − y1 y2 + u4 ,

where u1 , u2 , u3 , u4 are feedback backstepping controls to be determined.

152 Backstepping Control of Nonlinear Dynamical Systems

The synchronization hyperchaos error is defined by means of the equations ei = yi − xi , i = 1, 2, 3, 4.

(7.37)

The synchronization error dynamics is calculated as follows: ⎧ ⎪ ⎪ ⎪ e˙1 = −ae1 + be2 + e4 + u1 , ⎪ ⎨ e˙ = e − e + y y − x x + u , 2 1 2 1 3 1 3 2 ⎪ = −e − y y + x x + u , e ˙ ⎪ 3 3 1 2 1 2 3 ⎪ ⎪ ⎩ e˙ = −ce − y y + x x + u . 4

1

1 2

1 2

(7.38)

4

As a first step, we use feedback control to transform the system (7.38) to a system with triangular structure that aids backstepping control design. First, consider the feedback control law ⎧ u1 = ae1 − (b − 1)e2 − e4 , ⎪ ⎪ ⎪ ⎨ u = −e + e + e − y y + x x , 2 1 2 3 1 3 1 3 (7.39) ⎪ ⎪ ⎪ u 3 = e3 + e4 + y 1 y 2 − x 1 x 2 , ⎩ u4 = ce1 + y1 y2 − x1 x2 + v, where v is a feedback control to be determined. Substituting (7.39) into (7.38), we get the new system in triangular form as ⎧ ⎪ e˙1 = e2 , ⎪ ⎪ ⎪ ⎨ e˙ = e , 2 3 (7.40) ⎪ = e e ˙ ⎪ 3 4, ⎪ ⎪ ⎩ e˙ = v. 4 We begin with the Lyapunov function W1 (η1 ) =

1 2 η 2 1

(7.41)

where η1 = e1 .

(7.42)

Differentiating W1 with respect to t along the dynamics (7.40), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (e1 + e2 ).

(7.43)

η 2 = e 1 + e2 .

(7.44)

We define

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With the help of (7.44), we can express (7.43) as W˙ 1 = −η12 + η1 η2 .

(7.45)

We proceed next with defining the Lyapunov function 1 1 W2 (η1 , η2 ) = W1 (η1 ) + η22 = (η12 + η22 ). 2 2

(7.46)

Differentiating W2 with respect to t along the dynamics (7.40), we get W˙ 2 = −η12 − η22 + η2 (2e1 + 2e2 + e3 ).

(7.47)

η3 = 2e1 + 2e2 + e3 .

(7.48)

We define

With the help of (7.48), we can express (7.47) as W˙ 2 = −η12 − η22 + η2 η3 .

(7.49)

We proceed next with defining the Lyapunov function 1 1 W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) + η32 = (η12 + η22 + η32 ). 2 2

(7.50)

Differentiating W3 with respect to t along the dynamics (7.40), we get W˙ 3 = −η12 − η22 − η32 + η3 (3e1 + 5e2 + 3e3 + e4 ).

(7.51)

η4 = 3e1 + 5e2 + 3e3 + e4 .

(7.52)

We define

With the help of (7.52), we can express (7.51) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(7.53)

As a final step of the adaptive backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 ) =

1 2 (η + η22 + η32 + η42 ). 2 1

Clearly, W is a positive definite function on R4 .

(7.54)

154 Backstepping Control of Nonlinear Dynamical Systems

Differentiating W with respect to t along the dynamics (7.40), we get W˙ = −η12 − η22 − η32 − η42 + η4 T

(7.55)

T = η3 + η4 + η˙ 4 = 5e1 + 10e2 + 9e3 + 4e4 + v.

(7.56)

where

We define the control v as v = −5e1 − 10e2 − 9e3 − 4e4 − Kη4

(7.57)

where K > 0 is a positive constant. Substituting (7.57) into (7.56), we get T = −Kη4 . Thus, Eq. (7.55) can be simplified as W˙ = −η12 − η22 − η32 − (1 + K)η42 ,

(7.58)

which is quadratic and negative definite. Hence, by Lyapunov stability theory, it is immediate that ηi (t) → 0, (i = 1, 2, 3, 4) exponentially as t → ∞ for all values of ηi (0) ∈ R, (i = 1, 2, 3, 4). As a consequence, it follows that xi (t) → 0, (i = 1, 2, 3, 4) exponentially as t → ∞ for all values of ei (0) ∈ R, (i = 1, 2, 3, 4). Thus, we have established the following main result of this section. Theorem 7.2. The adaptive backstepping control law defined by ⎧ ⎪ u1 ⎪ ⎪ ⎪ ⎨ u 2 ⎪ u 3 ⎪ ⎪ ⎪ ⎩ u4

=

ae1 − (b − 1)e2 − e4 ,

= −e1 + e2 + e3 − y1 y3 + x1 x3 , =

e3 + e4 + y 1 y 2 − x1 x 2 ,

=

(c − 5)e1 − 10e2 − 9e3 − 4e4 + y1 y2 − x1 x2 − Kη4 ,

(7.59)

where K > 0 and the parameter estimation law and η4 = 3e1 + 5e2 + 3e3 + e4 , globally and exponentially synchronize the 4D hyperchaos temperature variations systems (7.35) and (7.36) for all values of X(0), Y (0) ∈ R4 . For computer simulations, we take the hyperchaos case for the parameter values a, b, and c, viz. (a; b; c) = (5; 122; 1). We choose K = 6 for the controller gain. We choose X(0) = (3.2, 8.1, 4.9, 5.3) and Y (0) = (7.5, 3.8, 2.6, 9.1). Figs. 7.7–7.10 display the synchronization between the states of the hyperchaotic temperature variation systems (7.35) and (7.36). Fig. 7.11 shows the exponential convergence of the synchronization error ei (t), (i = 1, 2, 3, 4) between the hyperchaotic temperature variation systems (7.35) and (7.36).

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FIGURE 7.7 Synchronization between the states x1 and y1 of the hyperchaotic temperature variations systems (7.35) and (7.36).

FIGURE 7.8 Synchronization between the states x2 and y2 of the hyperchaotic temperature variations systems (7.35) and (7.36).

7.6 Circuit simulation of the new 4D hyperchaotic temperature variation system In this study, we have verified of the new 4D hyperchaotic system model, and the validation approach uses Multisim software. The circuit diagram of Fig. 7.12

156 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 7.9 Synchronization between the states x3 and y3 of the hyperchaotic temperature variations systems (7.35) and (7.36).

FIGURE 7.10 Synchronization between the states x4 and y4 of the hyperchaotic temperature variations systems (7.35) and (7.36).

consists of operational amplifiers associated with resistors and capacitors exploited to implement the basic operations such as integration, inverting, and subtraction.

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FIGURE 7.11 Time-history of the synchronization error between the hyperchaotic temperature variations systems (7.35) and (7.36).

The four state variables (x1 , x2 , x3 , x4 ) of the new hyperchaotic system have been rescaled as follows: 1 1 1 1 X 1 = x1 , X 2 = x 2 , X 3 = x3 , X 4 = x4 . 4 2 2 4

(7.60)

After re-scaling the state variables, the new 4D hyperchaotic system is given by ⎧ ⎪ X˙1 ⎪ ⎪ ⎪ ⎪ ⎨ X˙ 2 ⎪ ⎪ X˙3 ⎪ ⎪ ⎪ ⎩ X˙4

=

−aX1 + b2 X2 + X4 ,

=

2X1 − X2 + 4X1 X3 ,

=

−X3 − 4X1 X2 ,

=

−cX1 − 2X1 X2 .

(7.61)

The implementation electronic circuit of the new 4D hyperchaotic system is synthesized as shown in Fig. 7.12. The circuit equations are formulated as C1 X˙ 1

= − R11 X1 +

C2 X˙ 2

=

C3 X˙ 3

= − R17 X3 −

1 R8 X 1 X 2 ,

C4 X˙ 4

= − R19 X1 −

1 R10 X1 X2.

1 R4 X 1



1 R2 X 2

1 R5 X 2

+

+

1 R3 X 4 ,

1 R6 X 1 X 3 ,

(7.62)

158 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 7.12 Schematic diagram of the 4-D hyperchaotic temperature variation system (7.62).

Compared with (7.61) and (7.62), the parameters are taken as follows: R1 = 40 k, R2 = 5.33 k, R3 = R5 = R7 = R9 = 400 k, R4 = R10 = 200 k, R6 = R8 = R10 = R11 = R12 = R13 = R14 = R15 = R15 = 100 k, and C1 = C2 = C3 = C4 = 1 nF. The power supplies of all active devices are ±15 Volts. The oscilloscope graphics represented the phase portraits of system (7.62) are

A new 4-D hyperchaotic temperature variations system Chapter | 7

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FIGURE 7.13 Chaotic attractors of system (7.62) using Multisim circuit simulation: (A) X1 –X2 plane, (B) X2 –X3 plane, (C) X3 –X4 plane, and (D) X1 –X4 plane.

shown in Fig. 7.13. We can see that the circuit simulation results of Fig. 7.13 agree well with the numerical simulation results of Fig. 7.3.

7.7 Conclusions The Vallis model is a famous 3D system for temperature variations in the eastern and western parts of the equatorial ocean, which was modeled by Vallis (1986). The Vallis model has a significant influence on the global weather climate patterns. In this research work, we developed a 4D hyperchaos temperature variation system by means of introducing a state feedback in the 3D Vallis system. The hyperchaos temperature variation system developed in this work has a two-wing strange hyperchaos attractor with a unique saddle point equilibrium at the origin. We showed that the new hyperchaos temperature variation system has multistability with coexisting hyperchaos attractors. Using backstepping control method, we derived global hyperchaos stabilization and synchronization successfully for the new hyperchaos temperature variation system. Finally, using Multisim, we developed an electronic circuit model realizing the 4D hyperchaos

160 Backstepping Control of Nonlinear Dynamical Systems

temperature variation system. Electronic circuit design of the new hyperchaos temperature variation system aids practical implementation of the proposed hyperchaos system. As future work, the hyperchaos temperature variation system can be implemented in engineering applications such as encryption, communication systems, and steganography.

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Chapter 8

A new thermally excited chaotic jerk system, its dynamical analysis, adaptive backstepping control, and circuit simulation Sundarapandian Vaidyanathana , Aceng Sambasb , Ahmad Taher Azarc,d , Fernando E. Serranoe , and Arezki Fekikf,g a Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, b Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, Indonesia, c Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, d Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, e Universidad Tecnológica Centroamericana (UNITEC), Zona Jacaleapa, Tegucigalpa, Honduras, f Akli Mohand Oulhadj University, Bouira, Algeria, g Laboratory of Advanced Technologies of Electrical Engineering (LATAGE), Faculty of Electrical and Computer Engineering, Mouloud Mammeri University (UMMTO), Tizi-Ouzou, Algeria

8.1

Introduction

In the recent decades, significant research attention has been given to the modeling and applications of dynamical systems exhibiting chaos (Xu et al., 2019; Gusso et al., 2019; Gatabazi et al., 2019; Singh and Roy, 2019; Cabanas et al., 2019; Ginoux et al., 2019; Jahanshahi et al., 2019; Daumann and Rech, 2019). Xu et al. (2019) proposed a chaotic system based on a circuit design involving a memristor model and a meminductor model. Gusso et al. (2019) analyzed the nonlinear dynamical model and the existence of chaos in suspended beam MEMS/NEMS resonators that are actuated by two-sided electrodes. Gatabazi et al. (2019) analyzed 2-D and 3-D Grey Lotka–Volterra Models (GLVMs) and explored their application in cryptocurrencies such as Bitcoin, Litecoin, and Ripple. Singh and Roy (2019) studied microscopic chaos control of a chemical reactor system via nonlinear active plus proportional integral sliding mode control. Cabanas et al. (2019) discovered chaos in driven nano-magnets such as spin valves by using the magnetic energy and the magnetoresistance. Ginoux et al. (2019) discovered chaos in a dynamical system modeling the illicit drug consumption in a population comprising drug users and non-users. Jahanshahi Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00015-5 Copyright © 2021 Elsevier Inc. All rights reserved.

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166 Backstepping Control of Nonlinear Dynamical Systems

et al. (2019) discussed a finance hyperchaos system via entropy analysis and control methods. Daumann and Rech (2019) reported hyperchaos in a heat-flux convection model. Jerk dynamical systems are special classes of mechanical systems. If y(t) describes the displacement of a moving object, then Dy(t) = dy dt represents its 2

3

velocity, D 2 y(t) = ddt y2 its acceleration, and D 3 y(t) = ddt y3 its jerk. An autonomous jerk differential equation has the general representation given by D 3 x = F (x, Dx, D 2 x).

(8.1)

By defining new state variables y = Dx and z = D 2 x, the jerk differential equation (8.1) can be displayed in a system form as ⎧ ⎪ ⎨ x˙ y˙ ⎪ ⎩ z˙

=

y,

=

z,

=

F (x, y, z).

(8.2)

Recently, much interest has been given to the finding of jerk systems in the chaos literature (Vaidyanathan et al., 2018a; Vaidyanathan, 2017; Vaidyanathan et al., 2017c; Vaidyanathan, 2016, 2015; El-Nabulsi, 2018; Vaidyanathan et al., 2015a; Vaidyanathan and Azar, 2016b; Vaidyanathan et al., 2018b). Vaidyanathan et al. (2018a) reported a new chaotic jerk system with two quadratic nonlinearities and discussed its applications to electronic circuit implementation and image encryption. Vaidyanathan (2017) reported a new 3-D chaotic jerk system with two cubic nonlinear terms and discussed its adaptive synchronization using backstepping control. Vaidyanathan et al. (2017c) analyzed a new chaotic jerk system with its applications for circuit simulation and voice encryption. Vaidyanathan (2016) announced a new chaotic jerk system and discussed its adaptive synchronization using backstepping control. Vaidyanathan (2015) proposed a new chaotic jerk system with two quadratic nonlinearities. El-Nabulsi (2018) reported jerk systems in the study of nonlocal effects in fluids, plasmas, and solar physics. In 1966, Moore and Spiegel introduced a classical chaotic system which models the nonperiodic oscillations and irregular variability of the luminosity of the stars (Moore and Spiegel, 1966). In fact, the Moore–Spiegel dynamical system is a classical example of a 3-D jerk chaotic system with a single cubic nonlinearity. In this work, we announce a new 3-D jerk system with chaos which is obtained by adding a quadratic nonlinearity to the famous Moore–Spiegel jerk system. The proposed nonlinear jerk mechanical system has two nonlinear terms—a quadratic term and a cubic term. We show that the new jerk system has the origin as the unique rest point, which is a saddle-focus and unstable.

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In this chapter, we shall analyze the dynamic behavior and control the chaotic behavior of the new chaotic jerk system. Using dynamic analysis, we show that the new jerk system exhibits multistability and coexisting attractors. Multistability is a complex feature for a chaotic system where coexisting attractors are obtained for the same values of parameters but different initial states (Zhang et al., 2018a,b; Wang et al., 2018; Azar et al., 2018). The backstepping control approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015b; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015; Azar et al., 2020; Shukla et al., 2018). The adaptive control method is used for the control design of nonlinear systems with unknown system parameters (Medhaffar et al., 2019; Liu et al., 2019; Gong et al., 2019; Hajiloo et al., 2019; Li and Sun, 2019; Vaidyanathan et al., 2017a,d,b; Vaidyanathan and Azar, 2016d,a,c,e). Medhaffar et al. (2019) described an adaptive fuzzy sliding mode controller for stabilizing the periodic orbits of Chua’s circuit and their controller has good robustness properties. Liu et al. (2019) proposed an adaptive controller that realizes the synchronization of two different fractional-order dissipative chaotic systems using fractional-order Mittag-Leffler stability under the conditions of determined parameters and uncertain parameters. Gong et al. (2019) discussed control results for the adaptive synchronization for a class of fractional-order financial systems. Hajiloo et al. (2019) proposed an adaptive nonlinear delayed feedback controller for stabilizing an unstable periodic orbit of a class of uncertain chaotic systems via the sliding mode control approach. Li and Sun (2019) proposed an adaptive neural network backstepping controller for the control of uncertain fractional-order Chua–Hartley chaotic system. In this work, we use the adaptive backstepping control technique for the global stabilization and synchronization of the new jerk system with unknown parameters. Finally, a circuit model using Multisim of the new chaotic jerk system is designed for practical implementation. We show that the Multisim outputs of the jerk system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019a; Sambas et al., 2019a,b; Vaidyanathan et al., 2019b).

8.2 A new jerk system with two nonlinearities We start this section with a classical thermo-mechanical jerk system with a single cubic nonlinearity, viz. the Moore–Spiegel jerk system, which models the nonperiodic oscillations and irregular variability of the luminosity of the stars (Moore and Spiegel, 1966).

168 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 8.1 MATLAB 3-D plot of the Moore–Spiegel system (8.3) for the initial state X(0) = (0.3, 0.2, 0.3) and (a, b) = (5, 9).

The Moore–Spiegel thermo-mechanical jerk system is described by the 3-D model ⎧ ⎪ x˙ = y, ⎪ ⎨ y˙ = z, (8.3) ⎪ ⎪ ⎩ 2 z˙ = −ax + by − z − x y. Moore and Spiegel showed that the dynamics (8.3) is chaotic, when we take the parameters as a = 5, b = 9.

(8.4)

In fact, for the initial state X(0) = (0.3, 0.2, 0.3) and (a, b) = (5, 9), the Lyapunov characteristic exponents of the Moore–Spiegel jerk system (8.3) can be calculated in MATLAB as ψ1 = 0.0725, ψ2 = 0, ψ3 = −1.0725.

(8.5)

Accordingly, the Kaplan–Yorke dimension for the Moore–Spiegel jerk system (8.3) is found as DKY = 2 +

ψ1 + ψ2 = 2.0725. |ψ3 |

(8.6)

Fig. 8.1 shows the chaotic attractor of the Moore–Spiegel jerk system (8.3) for the initial state X(0) = (0.3, 0.2, 0.3) and (a, b) = (5, 9).

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FIGURE 8.2 Lyapunov exponents of the new jerk system (8.7) for the initial state (0.3, 0.2, 0.3) and parameter values (a, b, c) = (6, 12, 0.1).

In this work, we derive a new thermo-mechanical jerk system by adding a quadratic nonlinearity to the Moore–Spiegel jerk system (8.3) and choosing different values for the system parameters. Our new jerk system is deployed by the 3-D dynamics ⎧ ⎪ x˙ = y, ⎪ ⎨ y˙ = z, (8.7) ⎪ ⎪ ⎩ z˙ = −ax + by − z − x 2 y − cxy, where X = (x, y, z) is the state and (a, b, c) is the parameter set. In this work, we show that the new thermo-mechanical jerk dynamics (8.7) is chaotic when we take the parameters as a = 6, b = 12, c = 0.1.

(8.8)

The Lyapunov characteristic exponents of the new thermo-mechanical jerk dynamics (8.7) for X(0) = (0.3, 0.2, 0.3) and (a, b, c) = (6, 12, 0.1) are estimated using MATLAB as ψ1 = 0.09128, ψ2 = 0, ψ3 = −1.09128.

(8.9)

The time-series estimation of the Lyapunov exponents of the new thermomechanical jerk dynamics (8.7) is shown in Fig. 8.2. From Eq. (8.9), it can be deduced that the 3-D thermo-mechanical jerk model (8.7) is chaotic as there is a positive Lyapunov exponent, ψ1 . The new thermomechanical jerk model (8.7) is also dissipative since the sum of the Lyapunov

170 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 8.3 MATLAB plots of the new thermo-mechanical jerk model (8.7) for (a, b, c) = (6, 12, 0.1) and X(0) = (0.3, 0.2, 0.3): (A) (x, y)-plane, (B) (y, z)-plane, (C) (x, z)-plane, and (D) R3 .

characteristic exponents in (8.9) is negative. Thus, the 3-D chaotic thermomechanical jerk model (8.7) has a strange chaotic attractor. It is also noted that the maximal Lyapunov characteristic exponent of the new thermo-mechanical jerk model (8.7) is found as ψ1 = 0.09128, which is greater than the maximal Lyapunov characteristic exponent of the Moore–Spiegel jerk model (8.3), viz. ψ1 = 0.0725. Furthermore, the Kaplan–Yorke dimension of the new thermo-mechanical jerk model (8.7) is determined as DKY = 2 +

ψ1 + ψ2 = 2.0836. |ψ3 |

(8.10)

Comparing the Kaplan–Yorke dimensions (8.6) and (8.10), we conclude that the new thermo-mechanical jerk model (8.7) has more complexity than the Moore–Spiegel jerk model (8.3). The MATLAB phase plots of the 3-D thermo-mechanical jerk model (8.7) are given in Fig. 8.3.

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8.3 Dynamic analysis of the new thermo-mechanical jerk model 8.3.1 Rest points of the new jerk model The rest points of the new thermo-mechanical jerk model (8.7) are calculated by means of solving the following system of equations: y = 0,

(8.11a)

z = 0,

(8.11b)

−ax + by − z − x 2 y − cxy = 0.

(8.11c)

From (8.11a) and (8.11b), we get y = 0 and z = 0. Substituting these values in (8.11c), we get x = 0. Thus, we conclude that 0 = (0, 0, 0) is the unique rest point of the new thermo-mechanical jerk model (8.7). For the stability analysis of the rest point 0 = (0, 0, 0), we take the parameters as in the chaotic case, viz. (a, b, c) = (6, 12, 0.1). The Jacobian matrix of the new thermo-mechanical jerk model (8.7) at 0 is determined as follows: ⎡

⎤ 0 1 0 ⎢ ⎥ J0 = ⎣ 0 0 1 ⎦. −6 12 −1

(8.12)

The matrix J0 has the spectral values λ1 = 0.5369, λ2 = 2.6616, and λ3 = −4.1985. This shows that 0 is a saddle-point rest point, which is unstable.

8.3.2 Bifurcation analysis When the system parameters a = 6, b = 12, c = 0.1 and the initial conditions (0.3, 0.2, 0.3) are fixed, while the control parameter a varies in the range 0 ≤ a ≤ 8, the bifurcation diagrams of the state variable Zmax obtained from the thermally excited chaotic oscillator are illustrated in Fig. 8.4A. The corresponding Lyapunov exponents are illustrated in Fig. 8.4B, which shows a good coincidence with the bifurcation diagrams. So, the new thermo-mechanical jerk system (8.7) displays limit cycles and chaotic behavior. When the control parameter b varies in the range 0 ≤ b ≤ 15, Figs. 8.5A and 8.5B present the bifurcation diagram and Lyapunov exponent spectrum of system. As in the previous case, the new thermo-mechanical jerk system (8.7) displays limit cycles and chaotic behavior.

172 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 8.4 Dynamical behavior of the thermo-mechanical jerk system (8.7) depending on control parameter a: (A) bifurcation diagrams of the state variable Zmax and (B) Lyapunov exponents.

FIGURE 8.5 Dynamical behavior of the thermo-mechanical jerk system (8.7) depending on b: (A) bifurcation diagrams of the state variable Zmax and (B) Lyapunov exponents.

8.3.3 Multistability and coexisting attractors Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set. It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems. It is known that multistability can lead to very complex behaviors in a dynamical system.

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Our results show that the new thermo-mechanical jerk model (8.7) exhibits coexisting attractors when choosing different initial conditions. We take parameter values as in the chaotic case, viz. (a, b, c) = (6, 12, 0.1). We select two initial conditions as X0 = (0.3, 0.2, 0.3), Y0 = (−0.8, −0.2, −0.6), and the corresponding state orbits of the system (8.7) are plotted in the colors, blue (dark gray in print) and red (light gray in print), respectively. From Fig. 8.6, it can be observed that the new thermo-mechanical jerk model (8.7) exhibits multistability with two coexisting chaotic attractors.

FIGURE 8.6 Multistability of the new thermo-mechanical jerk model (8.7): Coexisting chaotic attractors for (a, b, c) = (6, 12, 0.1) and initial conditions X0 = (0.3, 0.2, 0.3) (blue, dark gray in print) and Y0 = (−0.8, −0.2, −0.6) (red, light gray in print).

8.4 Adaptive backstepping control of the new thermo-mechanical jerk system Here, we deploy adaptive backstepping control for globally stabilizing the trajectories of the new thermo-mechanical jerk chaotic system with unknown system constants for all initial conditions. We consider the controlled jerk system given by the 3D dynamics ⎧ ⎪ x˙ ⎪ ⎨ y˙ ⎪ ⎪ ⎩ z˙

=

y,

=

z,

(8.13)

= −ax + by − z − x 2 y − cxy + v,

where a, b, c are unknown system constants. In (8.13), v is a backstepping control to be determined using estimates A(t), B(t), and C(t) for the unknown system constants a, b, and c, respectively.

174 Backstepping Control of Nonlinear Dynamical Systems

The parameter estimation errors are defined as follows: ⎧ ⎪ ⎨ ea (t) = a − A(t), eb (t) = b − B(t), ⎪ ⎩ ec (t) = c − C(t). A simple calculation shows that ⎧ ⎪ e˙ ⎪ ⎨ a e˙b ⎪ ⎪ ⎩ e˙c

(8.14)

˙ = −A, ˙ = −B,

(8.15)

˙ = −C.

We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(8.16)

where ηx = x.

(8.17)

Differentiating W1 with respect to t along the dynamics (8.13), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (x + y).

(8.18)

ηy = x + y.

(8.19)

We define

With the help of Eq. (8.19), we can express (8.18) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(8.20)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

1 2 1 2 ηy = ηx + ηy2 . 2 2

(8.21)

Differentiating W2 with respect to t along the dynamics (8.13), we get W˙ 2 = −ηx2 − ηy2 + ηy (2x + 2y + z).

(8.22)

We define ηz as follows: ηz = 2x + 2y + z.

(8.23)

With the help of Eq. (8.23), we can express Eq. (8.22) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(8.24)

A new thermally excited chaotic jerk system Chapter | 8

175

As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 W (ηx , ηy , ηz , ea , eb , ec ) = W2 (ηx , ηy ) + ηx2 2 1 2 1 = (ηx + ηy2 + ηz2 ) + (ea2 + eb2 + ec2 ) 2 2

(8.25)

where ea , eb , ec are parameter estimation errors defined in Eq. (8.14). It is clear that W is a positive definite function on R6 . Differentiating W with respect to t along the jerk dynamics (8.13) and (8.15), we get ˙ W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ) − ea A˙ − eb B˙ − ec C.

(8.26)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 − a)x + (3 + b)y + z − x 2 y − cxy + v] ˙ − ea A˙ − eb B˙ − ec C. (8.27) We define the control law v as v = −[2 − A(t)]x − [3 + B(t)]y − z + x 2 y + C(t)xy − Kηz

(8.28)

where K > 0 is a control gain. Substituting (8.28) into (8.27), we get   W˙ = −ηx2 − ηy2 − (1 + K)ηz2 + ea −xηz − A˙     + eb yηz − B˙ + ec −xyηz − C˙ .

(8.29)

From Eq. (8.29), we can choose the parameter update law as ⎧ ⎪ A˙ ⎪ ⎨ B˙ ⎪ ⎪ ⎩ C˙

= −x ηz , =

y ηz ,

(8.30)

= −xy ηz .

Next, we prove the main result of this section. Theorem 8.1. The backstepping control law defined via (8.28) with gain K > 0 and the parameter update law (8.30) globally and exponentially stabilizes all the trajectories of the new thermo-mechanical jerk chaos plant (8.13) for all initial states (x(0), y(0), z(0)) ∈ R3 . Proof. We prove this result by adaptive control theory and Lyapunov stability theory.

176 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 8.7 Time-history of the backstepping controlled states x(t), y(t), z(t) of the new thermomechanical jerk chaotic system (8.28).

First, we note that W defined by Eq. (8.25) is a quadratic and positive definite function on R6 . Substituting the parameter update law (8.30) into Eq. (8.29) simplifies the time-derivative of W as W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(8.31)

which is quadratic and negative semi-definite on R3 . Hence, it follows by Barbalat’s lemma (Khalil, 2002) that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that x = ηx , y = ηy − ηx , z = ηx − 2ηy .

(8.32)

As a consequence, it follows that (x(t), y(t), z(t)) → (0, 0, 0) exponentially as t → ∞. For simulations, we pick the values of the parameters of the new thermomechanical jerk model (8.13) as in the chaos case, viz. (a, b, c) = (6, 12, 0.1). We choose K = 8 and the initial state of the jerk system (8.13) as x(0) = 2.4, y(0) = 5.8, and z(0) = 3.7. We also take A(0) = 2.9, B(0) = 3.1, and C(0) = 6.4. Fig. 8.7 shows the time-history of the backstepping controlled states x(t), y(t), and z(t). It is easy to note that the controlled states converge to zero exponentially by the action of the backstepping control law (8.28) and parameter update law (8.30).

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8.5 Adaptive backstepping synchronization of the new thermo-mechanical jerk systems In most of the chaos synchronization approaches, the master–slave or drive– response formalism is used (Ouannas et al., 2017c,a,d,j,h,i,g,f,k,e,b). If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of synchronization is to use the output of the master system to control the slave system so that the output of the slave system tracks the output of the master system asymptotically. A variety of schemes for ensuring the control and synchronization of such systems have been demonstrated based on their potential applications in various fields (Grassi et al., 2017; Ouannas et al., 2019; Pham et al., 2017; Singh et al., 2017; Alain et al., 2020, 2019; Khan et al., 2020b,a) Here, we deploy adaptive backstepping control for globally synchronizing the state trajectories of a pair of new thermo-mechanical jerk chaos systems considered as leader–follower systems. The leader chaos system is depicted by the 3D thermo-mechanical jerk dynamics ⎧ ⎪ ⎨ x˙1 = y1 , y˙1 = z1 , (8.33) ⎪ ⎩ z˙ 1 = −ax1 + by1 − z1 − x12 y1 − cx1 y1 . The follower chaos system is depicted by the 3D thermo-mechanical jerk dynamics ⎧ ⎪ ⎨ x˙2 = y2 , y˙2 = z2 , (8.34) ⎪ ⎩ z˙ 2 = −ax2 + by2 − z2 − x22 y2 − cx2 y2 + v, where v is a backstepping control to be designed. In the general case, when the parameters a, b, and c in the jerk systems (8.33) and (8.34) are unavailable for measurement, we use adaptive backstepping control for globally synchronizing their respective state trajectories. The synchronization chaos error is defined by means of the equations ex = x2 − x1 , ey = y2 − y1 , ez = z2 − z1 . Upon calculating the error dynamics, we obtain the following: ⎧ ⎪ ⎨ e˙x = ey , e˙y = ez , ⎪ ⎩ e˙z = −aex + bey − ez − x22 y2 + x12 y1 − c(x2 y2 − x1 y1 ) + v.

(8.35)

(8.36)

In (8.36), v is a backstepping control to be determined using estimates A(t), B(t), and C(t) for the unknown system constants a, b, and c, respectively.

178 Backstepping Control of Nonlinear Dynamical Systems

The parameter estimation errors are defined as follows: ⎧ ⎪ ⎨ ea (t) = a − A(t), eb (t) = b − B(t), ⎪ ⎩ ec (t) = c − C(t). A simple calculation shows that ⎧ ⎪ e˙ ⎪ ⎨ a e˙b ⎪ ⎪ ⎩ e˙c

(8.37)

˙ = −A, ˙ = −B,

(8.38)

˙ = −C.

We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(8.39)

where ηx = ex .

(8.40)

Differentiating W1 with respect to t along the dynamics (8.36), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (ex + ey ).

(8.41)

η y = e x + ey .

(8.42)

We define

With the help of Eq. (8.42), we can express (8.41) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(8.43)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

1 2 1 2 ηy = ηx + ηy2 . 2 2

(8.44)

Differentiating W2 with respect to t along the dynamics (8.36), we get W˙ 2 = −ηx2 − ηy2 + ηy (2ex + 2ey + ez ).

(8.45)

We define ηz as follows: ηz = 2ex + 2ey + ez .

(8.46)

With the help of Eq. (8.46), we can express Eq. (8.45) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(8.47)

A new thermally excited chaotic jerk system Chapter | 8

179

As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 W (ηx , ηy , ηz , ea , eb , ec ) = W2 (ηx , ηy ) + ηx2 2 1 2 1 = (ηx + ηy2 + ηz2 ) + (ea2 + eb2 + ec2 ) 2 2

(8.48)

where ea , eb , ec are parameter estimation errors defined in Eq. (8.37). It is clear that W is a positive definite function on R6 . Differentiating W with respect to t along the error dynamics (8.36), we get ˙ W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ) − ea A˙ − eb B˙ − ec C.

(8.49)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 − a)ex + (3 + b)ey + ez − x22 y2 ˙ − c(x2 y2 − x1 y1 ) + v] − ea A˙ − eb B˙ − ec C.

(8.50)

We define the control law v as v = −[2 − A(t)]ex − [3 + B(t)]ey − ez + x22 y2 − x12 y1 + C(t)(x2 y2 − x1 y1 ) − Kηz

(8.51)

where K > 0 is a control gain. Substituting (8.51) into (8.50), we get     W˙ = −ηx2 − ηy2 − (1 + K)ηz2 + ea −ex ηz − A˙ + eb ey ηz − B˙   + ec (−x2 y2 + x1 y1 )ηz − C˙ . (8.52) From Eq. (8.52), we can choose the parameter update law as ⎧ ⎪ A˙ = −ex ηz , ⎪ ⎨ B˙ = ey ηz , ⎪ ⎪ ⎩ C˙ = (−x2 y2 + x1 y1 ) ηz .

(8.53)

Next, we prove the main result of this section. Theorem 8.2. The backstepping control law defined via (8.51) with gain K > 0 and the parameter update law (8.53) globally and exponentially synchronizes the new thermo-mechanical jerk chaos plants (8.33) and (8.34) for all initial states in R3 . Proof. We prove this result by adaptive control theory and Lyapunov stability theory.

180 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 8.8 Synchronization between the states x1 and x2 of the leader–follower systems (8.33) and (8.34).

First, we note that W defined by Eq. (8.48) is a quadratic and positive definite function on R6 . Substituting the parameter update law (8.53) into Eq. (8.52) simplifies the time-derivative of W as W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(8.54)

which is quadratic and negative semi-definite on R3 . Hence, it follows by Barbalat’s lemma (Khalil, 2002) that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that ex = ηx , ey = ηy − ηx , ez = ηx − 2ηy .

(8.55)

As a consequence, it follows that (ex (t), ey (t), ez (t)) → (0, 0, 0) exponentially as t → ∞ for all values of ex (0), ey (0), and ez (0). This completes the proof. We take the initial state of the leader system (8.33) as x1 (0) = 1.8, y1 (0) = 7.2, and z1 (0) = 3.4. We also consider the initial state of the follower system (8.34) as x2 (0) = 4.9, y2 (0) = 3.6, and z2 (0) = 0.7. We take K = 8. We also take A(0) = 2.7, B(0) = 5.8, and C(0) = 6.1. Figs. 8.8–8.10 display the synchronization between the states of the leader system and follower system. Fig. 8.11 shows the exponential convergence of the synchronization error ex (t), ey (t), ez (t) between the new thermo-mechanical chaotic jerk systems (8.33) and (8.34).

A new thermally excited chaotic jerk system Chapter | 8

181

FIGURE 8.9 Synchronization between the states y1 and y2 of the leader–follower systems (8.33) and (8.34).

FIGURE 8.10 Synchronization between the states z1 and z2 of the leader–follower systems (8.33) and (8.34).

8.6 Electronic circuit simulation of the new thermo-mechanical chaotic jerk system According to Fig. 8.12, the integrator, and phase inverter can be realized using operational amplifiers, and the analog multipliers included into this circuit

182 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 8.11 Time-history of the synchronization error between the leader–follower systems (8.33) and (8.34).

is realized by the model AD633JN, which is packaged unit using Multisim. The analog circuit components including resistors, capacitors, operational amplifiers, and multipliers is designed to implement system (8.57) is shown in Fig. 8.12. For the scaling process, let X = 12 x, Y = 12 y, Z = 14 z. Then setting the original state variables x, y, z, instead of the variables X, Y, Z, the scaled system transforms into the system described below: X˙

=

Y,



=

2Z,



= − a2 X + b2 Y − Z − 12 X 2 Y − cXY.

(8.56)

According to Kirchhoff circuit laws, the electric equations can be modeled as follows: C1 X˙

=

1 R1 Y,

C2 Y˙

=

1 R2 Z,

C3 Z˙

= − R13 X +

(8.57) 1 R4 Y



1 R5 Z



1 2 100R6 X Y



1 10R7 XY.

Here, X, Y , and Z are the output voltages of the operational amplifiers U1A, U2A, and U3A, respectively. The values of circuit components are se-

A new thermally excited chaotic jerk system Chapter | 8

183

FIGURE 8.12 Schematic diagram of system (8.57).

lected as follows: R1 = R5 = R7 = 400 k, R2 = 200 k, R3 = 133.33 k, R4 = 66.67 k, R6 = 8 k, R8 = R9 = R10 = R11 = 100 k, and C1 = C2 = C3 = 1 nF. Multisim results of the circuit (8.57) are described in Fig. 8.13, which shows its chaotic behavior. The analog results using Multisim are consistent with the MATLAB results described in Fig. 8.3.

184 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 8.13 Phase projections in Multisim displaying in (A) x–y plane, (B) y–z plane, and (C) x–z plane.

8.7 Conclusions Moore and Spiegel (1966) developed a thermally excited chaotic attractor, which consists of just a cubic nonlinearity in its dynamics. By adding a quadratic

A new thermally excited chaotic jerk system Chapter | 8

185

FIGURE 8.13 (continued)

nonlinearity and taking different parameter values, this work announces a new thermo-mechanical chaotic jerk system having two nonlinearities and complex chaotic behavior. We showed that the new thermo-mechanical chaotic jerk system has a self-excited chaotic attractor with a unique saddle-point equilibrium at the origin. We presented a detailed analysis with bifurcation diagrams and Lyapunov exponents of the new thermo-mechanical chaotic jerk system. We also reported that the new thermal chaotic jerk system has multistability and coexisting chaotic attractors. As control applications, we designed adaptive backstepping controllers for the global chaos stabilization and global chaos synchronization for the new thermo-mechanical chaotic jerk system with unknown parameters. Finally, we design an electronic circuit of the new thermo-mechanical chaotic jerk system using Multisim for practical implementation in engineering.

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form. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 449–470. Rajagopal, K., Akgul, A., Jafari, S., Karthikeyan, A., Cavusoglu, U., Kacar, S., 2019. An exponential jerk system: circuit realization, fractional order and time delayed form with dynamical analysis and its engineering application. Journal of Circuits, Systems, and Computers 28 (5), 1950087. Rasappan, S., Vaidyanathan, S., 2012. Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East Journal of Mathematical Sciences 67 (2), 265–287. Sambas, A., Vaidyanathan, S., Tlelo-Cuautle, E., Zhang, S., Guillen-Fernandez, O., Sukono, Hidayat, Y., Gundara, G., 2019a. A novel chaotic system with two circles of equilibrium points: multistability, electronic circuit and FPGA realization. Electronics 8 (11), 1211. Sambas, A., Vaidyanathan, S., Zhang, S., Zeng, Y., Mohamed, M., 2019b. A new double-wing chaotic system with coexisting attractors and line equilibrium: bifurcation analysis and electronic circuit simulation. IEEE Access 7, 115454–115462. Shukla, M.K., Sharma, B.B., Azar, A.T., 2018. Control and synchronization of a fractional order hyperchaotic system via backstepping and active backstepping approach. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 559–595. Singh, S., Azar, A.T., Ouannas, A., Zhu, Q., Zhang, W., Na, J., 2017. Sliding mode control technique for multi-switching synchronization of chaotic systems. In: 9th International Conference on Modelling, Identification and Control (ICMIC 2017). July 10–12, 2017, Kunming, China. IEEE, pp. 1–6. Singh, P.P., Roy, B.K., 2019. Microscopic chaos control of chemical reactor system using nonlinear active plus proportional integral sliding mode control technique. The European Physical Journal Special Topics 228 (1), 169–184. Vaidyanathan, S., 2015. Analysis, control, and synchronization of a 3-D novel jerk chaotic system with two quadratic nonlinearities. Kyungpook Mathematical Journal 55 (3), 563–586. Vaidyanathan, S., 2016. A novel 3-D jerk chaotic system with two quadratic nonlinearities and its adaptive backstepping control. International Journal of Control Theory and Applications 9 (1), 199–219. Vaidyanathan, S., 2017. A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control. Archives of Control Sciences 27 (3), 409–439. Vaidyanathan, S., Abba, O.A., Betchewe, G., Alidou, M., 2019a. A new three-dimensional chaotic system: its adaptive control and circuit design. International Journal of Automation and Control 13 (1), 101–121. Vaidyanathan, S., Akgul, A., Kacar, S., 2018a. A new chaotic jerk system with two quadratic nonlinearities and its applications to electronic circuit implementation and image encryption. International Journal of Computer Applications in Technology 58 (2), 89–101. Vaidyanathan, S., Azar, A.T., 2016a. A novel 4-D four-wing chaotic system with four quadratic nonlinearities and its synchronization via adaptive control method. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 203–224. Vaidyanathan, S., Azar, A.T., 2016b. Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 249–274. Vaidyanathan, S., Azar, A.T., 2016c. Adaptive control and synchronization of Halvorsen circulant chaotic systems. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 225–247. Vaidyanathan, S., Azar, A.T., 2016d. Dynamic analysis, adaptive feedback control and synchronization of an eight-term 3-D novel chaotic system with three quadratic nonlinearities. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 155–178. Vaidyanathan, S., Azar, A.T., 2016e. Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 275–290.

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Vaidyanathan, S., Azar, A.T., Ouannas, A., 2017a. An eight-term 3-D novel chaotic system with three quadratic nonlinearities, its adaptive feedback control and synchronization. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 719–746. Vaidyanathan, S., Azar, A.T., Ouannas, A., 2017b. Hyperchaos and adaptive control of a novel hyperchaotic system with two quadratic nonlinearities. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 773–803. Vaidyanathan, S., Idowu, B.A., Azar, A.T., 2015a. Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Chaos Modeling and Control Systems Design. In: Studies in Computational Intelligence, vol. 581. Springer, Berlin, Germany, pp. 39–58. Vaidyanathan, S., Jafari, S., Pham, V.-T., Azar, A.T., Alsaadi, F.E., 2018b. A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design. Archives of Control Sciences 28 (2), 239–254. Vaidyanathan, S., Sambas, A., Mamat, M., Sanjaya, W.S.M., 2017c. Analysis, synchronisation and circuit implementation of a novel jerk chaotic system and its application for voice encryption. International Journal of Modelling, Identification and Control 28 (2), 153–166. Vaidyanathan, S., Sambas, A., Zhang, S., 2019b. A new 4-D dynamical system exhibiting chaos with a line of rest points, its synchronization and circuit model. Archives of Control Sciences 29 (3), 485–506. Vaidyanathan, S., Volos, C., Rajagopal, K., Kyprianidis, I., Stouboulos, I., 2015b. Adaptive backstepping controller design for the anti-synchronization of identical WINDMI chaotic systems with unknown parameters and its SPICE implementation. Journal of Engineering Science and Technology Review 8 (2), 74–82. Vaidyanathan, S., Zhu, Q., Azar, A.T., 2017d. Adaptive control of a novel nonlinear double convection chaotic system. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 357–385. Wang, L., Zhang, S., Zeng, Y.-C., Li, Z.-J., 2018. Generating hidden extreme multistability in memristive chaotic oscillator via micro-perturbation. Electronics Letters 54 (13), 808–810. Xu, B., Wang, G., Lu, H.H.-C., Yu, S., Yuan, F., 2019. A memristor-meminductor-based chaotic system with abundant dynamical behaviors. Nonlinear Dynamics 96 (1), 765–788. Zhang, S., Zeng, Y., Li, Z., 2018a. A novel 4D no-equilibrium hyper-chaotic system with grid multiwing hyper-chaotic hidden attractors. Journal of Computational and Nonlinear Dynamics 13 (9), 090908. Zhang, S., Zeng, Y., Li, Z., 2018b. One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics. Chinese Journal of Physics 56 (3), 793–806.

Chapter 9

A new multistable plasma torch chaotic jerk system, its dynamical analysis, active backstepping control, and circuit design Sundarapandian Vaidyanathana , Aceng Sambasb , Ahmad Taher Azarc,d , and Shikha Singhe a Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, b Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya, Indonesia, c Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, d Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, e Department of Mathematics, Jesus and Mary College, University of Delhi, New Delhi, India

9.1

Introduction

Jerk chaotic systems have many applications in science and engineering. In recent decades, significant attention has been paid to the modeling of new jerk differential equations arising in engineering (Xu et al., 2019; Gusso et al., 2019; Gatabazi et al., 2019; Singh and Roy, 2019; Cabanas et al., 2019; Ginoux et al., 2019; Jahanshahi et al., 2019; Daumann and Rech, 2019; Vaidyanathan et al., 2018c; Vaidyanathan and Azar, 2016; Vaidyanathan et al., 2015a). Jerk dynamical systems are special classes of mechanical systems. If y(t) describes the displacement of a moving object, then Dy(t) = dy dt rep2

3

resents its velocity, D 2 y(t) = ddt y2 its acceleration, and D 3 y(t) = ddt y3 its jerk. An autonomous jerk differential equation has the general representation given by D 3 x = F (x, Dx, D 2 x). Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00016-7 Copyright © 2021 Elsevier Inc. All rights reserved.

(9.1) 191

192 Backstepping Control of Nonlinear Dynamical Systems

By defining new state variables y = Dx and z = D 2 x, the jerk differential equation (9.1) can be displayed in a system form as ⎧ ⎪ ⎨ x˙ y˙ ⎪ ⎩ z˙

=

y,

=

z,

=

F (x, y, z).

(9.2)

Recently, much interest has been given to the finding of jerk systems in the chaos literature (Vaidyanathan et al., 2018a; Vaidyanathan, 2017; Vaidyanathan et al., 2017; Vaidyanathan, 2016, 2015; El-Nabulsi, 2018). Vaidyanathan et al. (2018a) reported a new chaotic jerk system with two quadratic nonlinearities and discussed its applications to electronic circuit implementation and image encryption. Vaidyanathan (2017) reported a new 3-D chaotic jerk system with two cubic nonlinear terms and discussed its adaptive synchronization using backstepping control. Vaidyanathan et al. (2017) analyzed a new chaotic jerk system with its applications for circuit simulation and voice encryption. Vaidyanathan et al. (2017) announced a new chaotic jerk system and discussed its adaptive synchronization using backstepping control. Vaidyanathan (2016) proposed a new chaotic jerk system with two quadratic nonlinearities. Vaidyanathan (2015) reported jerk systems in the study of nonlocal effects in fluids, plasmas, and solar physics. El-Nabulsi (2018) introduced a classical chaotic jerk system which models the nonperiodic oscillations and irregular variability of the luminosity of the stars (Ghorui et al., 2000). In fact, the Ghorui dynamical system is a recent example of a 3-D jerk chaotic system with just a single cubic nonlinearity and linear terms. In this work, we announce a new 3-D jerk system with chaos which is obtained by adding a quadratic nonlinearity to the recent Ghorui plasma torch jerk system (Ghorui et al., 2000). The proposed nonlinear jerk mechanical system has two nonlinear terms—a quadratic term and a cubic term. We show that the new jerk system has the origin as the unique rest point, which is a saddle-focus and unstable. In this chapter, we shall analyze the dynamic behavior and control the chaotic behavior of the new plasma torch chaotic jerk system. Using dynamic analysis, we show that the new plasma torch jerk system exhibits multi-stability and coexisting attractors. Multistability is a complex feature for a chaotic system where coexisting attractors are obtained for the same values of parameters but different initial states (Zhang et al., 2018a,b; Wang et al., 2018). Backstepping control approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015b; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015). In this

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work, we use active backstepping control technique for the global stabilization and synchronization of the new plasma torch chaotic jerk system. Finally, a circuit model using Multisim of the new plasma torch chaotic jerk system is designed for practical implementation. We show that the Multisim outputs of the plasma torch jerk system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019a; Sambas et al., 2019a,b; Vaidyanathan et al., 2019c,b, 2018b; Tolba et al., 2017a,b).

9.2 A new plasma torch chaotic jerk system with two nonlinearities We start this section with a recent plasma torch chaotic jerk system with a single cubic nonlinearity, viz. the Ghorui jerk system, which models the chaotic oscillations and vibrations of a rod-type plasma torch jerk system (Ghorui et al., 2000). The Ghorui plasma torch jerk system is described by the 3-D model ⎧ ⎪ x˙ ⎪ ⎨ y˙ ⎪ ⎪ ⎩ z˙

=

y,

=

z,

= ax − by

(9.3) − z − x3.

It is noted that the dynamics (9.3) has a total of six terms on the right-handside with five linear terms and a cubic nonlinearity. Ghorui et al. (2000) showed that the dynamics (9.3) is chaotic, when we take the parameters as a = 130, b = 50.

(9.4)

In fact, for the initial state X(0) = (0.4, 0.2, 0.4) and (a, b) = (130, 5), the Lyapunov characteristic exponents of the Ghorui jerk system (9.3) can be calculated in MATLAB as ψ1 = 0.3702, ψ2 = 0, ψ3 = −1.3702.

(9.5)

Thus, the Ghorui jerk system (9.3) is dissipative and chaotic with the maximal Lyapunov exponent given by ψ1 = 0.3702. Fig. 9.1 shows the chaotic attractor of the Ghorui jerk system (9.3) for the initial state X(0) = (0.3, 0.2, 0.3) and (a, b) = (5, 9).

194 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 9.1 MATLAB 3-D plot of the Ghorui jerk system (9.3) for the initial state X(0) = (0.4, 0.2, 0.4) and (a, b) = (130, 5).

The Kaplan–Yorke dimension for the Ghorui jerk system (9.3) is determined as DKY = 2 +

ψ1 + ψ2 = 2.2702. |ψ3 |

(9.6)

The value of DKY gives a measure of the chaotic complexity of the Ghorui jerk system (9.3). In this work, we derive a new plasma torch chaotic jerk system by adding a quadratic nonlinearity to the Ghorui jerk system (9.3). Our new jerk system is deployed by the 3-D dynamics ⎧ ⎪ x˙ = y, ⎪ ⎨ y˙ = z, (9.7) ⎪ ⎪ ⎩ z˙ = ax − by − z − cxy − x 3 , where X = (x, y, z) is the state and (a, b, c) is the parameter set. In this work, we show that the new plasma torch jerk dynamics (9.7) is chaotic when we take the parameters as a = 130, b = 5, c = 0.1.

(9.8)

The Lyapunov characteristic exponents of the new plasma torch jerk dynamics (9.7) for X(0) = (0.4, 0.2, 0.4) and (a, b, c) = (130, 5, 0.1) are estimated using MATLAB as ψ1 = 0.4755, ψ2 = 0, ψ3 = −1.4755.

(9.9)

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FIGURE 9.2 Lyapunov exponents of the new plasma torch jerk system (9.7) for the initial state (0.4, 0.2, 0.4) and parameter values (a, b, c) = (130, 5, 0.1).

The time-series estimation of the Lyapunov exponents of the new plasma torch jerk dynamics (9.7) is shown in Fig. 9.2. From Eq. (9.9), it can be deduced that the 3-D plasma torch jerk model (9.7) is chaotic as there is a positive Lyapunov exponent, ψ1 . The new plasma torch jerk model (9.7) is also dissipative since the sum of the Lyapunov characteristic exponents in (9.9) is negative. Thus, the 3-D chaotic plasma torch jerk model (9.7) has a strange chaotic attractor. It is also noted that the maximal Lyapunov characteristic exponent of the new plasma torch jerk model (9.7) is found as ψ1 = 0.4755, which is greater than the maximal Lyapunov characteristic exponent of the Ghorui jerk model (9.3), viz. ψ1 = 0.3702. Furthermore, the Kaplan–Yorke dimension of the new plasma torch jerk model (9.7) is determined as DKY = 2 +

ψ 1 + ψ2 = 2.3223. |ψ3 |

(9.10)

Comparing the Kaplan–Yorke dimensions (9.6) and (9.10), we conclude that the new thermo-mechanical jerk model (9.7) has more complexity than the Ghorui plasma torch jerk system (9.3). The MATLAB phase plots of the 3-D plasma torch jerk model (9.7) are given in Fig. 9.3.

196 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 9.3 MATLAB plots of the new plasma torch jerk model (9.7) for (a, b, c) = (130, 5, 0.1) and X(0) = (0.4, 0.2, 0.4): (A) (x, y)-plane, (B) (y, z)-plane, (C) (x, z)-plane, and (D) R3 .

9.3 Dynamic analysis of the new plasma torch chaotic jerk model 9.3.1 Rest points of the new chaotic jerk model The rest points of the new plasma torch chaotic jerk model (9.7) are calculated by means of solving the following system of equations: y = 0,

(9.11a)

z = 0,

(9.11b)

ax − by − z − cxy − x 3 = 0.

(9.11c)

From (9.11a) and (9.11b), we get y = 0 and z = 0. Substituting these values in (9.11a), we get ax − x 3 = x(a − x 2 ) = 0.

(9.12)

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Thus, the new plasma-torch chaotic jerk model (9.7) has three rest points given by √ √ (9.13) E0 = (0, 0, 0), E1 = ( a, 0, 0), E2 = (− a, 0, 0). For the chaotic case (a, b, c) = (130, 5, 0.1), the rest points of the system (9.7) are obtained as E0 = (0, 0, 0), E1 = (11.4018, 0, 0), E2 = (−11.4018, 0, 0).

(9.14)

For the stability analysis of the rest points E0 , E1 , and E2 , we take the parameters as in the chaotic case, viz. (a, b, c) = (130, 5, 0.1). The Jacobian matrix of the new plasma torch jerk model (9.7) at E0 is determined as follows: ⎡ ⎤ 0 1 0 ⎢ ⎥ J0 = J (E0 ) = ⎣ 0 (9.15) 0 1 ⎦. 130 −5 −1 The matrix J0 has the spectral values λ1 = 4.4476 and λ2,3 = −2.7238 ± 4.6701i. This shows that E0 is a saddle-focus rest point, which is unstable. The Jacobian matrix of the new plasma torch jerk model (9.7) at E1 is determined as follows: ⎡ ⎤ 0 1 0 ⎢ ⎥ (9.16) J1 = J (E1 ) = ⎣ 0 0 1 ⎦. −260 −6.1402 −1 The matrix J1 has the spectral values λ1 = −6.3959 and λ2,3 = 2.6979 ± 5.7769i. This shows that E1 is a saddle-focus rest point, which is unstable. The Jacobian matrix of the new plasma torch jerk model (9.7) at E2 is determined as follows: ⎡ ⎤ 0 1 0 ⎢ ⎥ (9.17) J2 = J (E2 ) = ⎣ 0 0 1 ⎦. −260 −3.8598 −1 The matrix J2 has the spectral values λ1 = −6.5215 and λ2,3 = 2.7607 ± 5.6786i. This shows that E2 is a saddle-focus rest point, which is unstable. Thus, all the three rest points of the new plasma torch jerk model (9.7) are saddle-foci rest points and unstable. Hence, the new plasma torch jerk model (9.7) has a self-excited attractor.

198 Backstepping Control of Nonlinear Dynamical Systems

9.3.2 Bifurcation analysis First, we fix the parameters b = 50, c = 0.1, and the initial state as X(0) = (0.4, 0.2, 0.4). We let the system parameter a to vary in the range [130, 140]. The corresponding bifurcation diagram and Lyapunov exponent spectrum for the new plasma torch jerk system (9.7) are plotted in Figs. 9.4A and 9.4B, respectively. From Fig. 9.4, we see that the new plasma torch jerk system (9.7) shows chaotic behavior and periodic behavior. The jerk system (9.7) is periodic for a ≥ 138. Next, we fix the parameters a = 130, c = 0.1, and the initial state as X(0) = (0.4, 0.2, 0.4). We let the system parameter b to vary in the range [47, 57]. The corresponding bifurcation diagram and Lyapunov exponent spectrum for the new plasma torch jerk system (9.7) are plotted in Figs. 9.5A and 9.5B, respectively. It can be seen from Fig. 9.5 that periodic behavior occurs for 47 ≤ b ≤ 47.6 and that the new jerk system (9.7) generates chaotic oscillations for 47.6 ≤ b ≤ 51.4. When b ≥ 51.4, the jerk system (9.7) displays a periodic state with the Lyapunov exponents having the signs (0, −, −).

FIGURE 9.4 Dynamical behavior of the new plasma torch jerk system (9.7) depending on the control parameter a: (A) bifurcation diagrams of the state variable Zmax and (B) Lyapunov exponents.

9.3.3 Multistability and coexisting attractors Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set (Azar et al., 2018). Multistability is an interesting phenomenon and can usually be found in many nonlinear dynamical systems. It is known that multistability can lead to very complex behaviors in a dynamical system.

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FIGURE 9.5 Dynamical behavior of the new plasma torch jerk system (9.7) depending on the control parameter b: (A) bifurcation diagrams of the state variable Zmax and (B) Lyapunov exponents.

Our results show that the new plasma torch jerk model (9.7) exhibits coexisting attractors when choosing different initial conditions. We take parameter values as in the chaotic case, viz. (a, b, c) = (130, 5, 0.1). We select two initial conditions as X0 = (0.4, 0.2, 0.4, Y0 = (−0.4, 0.4, −0.4), and the corresponding state orbits of the system (9.7) are plotted in the colors, blue (dark gray in print) and red (light gray in print), respectively. From Fig. 9.6, it can be observed that the new plasma torch chaotic jerk model (9.7) exhibits multistability with two coexisting chaotic attractors.

FIGURE 9.6 Multistability of the new plasma torch chaotic jerk model (9.7): Coexisting chaotic attractors for (a, b, c) = (130, 5, 0.1) and initial conditions X0 = (0.4, 0.2, 0.4) (blue, dark gray in print) and Y0 = (−0.4, 0.4, −0.4) (red, light gray in print).

200 Backstepping Control of Nonlinear Dynamical Systems

9.4 Active backstepping control for the global stabilization of the new plasma torch chaotic jerk system In this section, we deploy active backstepping control for globally stabilizing the trajectories of the new plasma torch chaotic jerk chaotic system for all initial conditions. The controlled plasma torch chaotic jerk system is described by the 3D dynamics ⎧ ⎪ ⎨ x˙ = y, y˙ = z, (9.18) ⎪ ⎩ 3 z˙ = ax − by − z − cxy − x + v, where v is a backstepping control to be determined. We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(9.19)

where ηx = x.

(9.20)

Differentiating W1 with respect to t along the dynamics (9.18), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (x + y).

(9.21)

ηy = x + y.

(9.22)

We define

With the help of Eq. (9.22), we can express (9.21) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(9.23)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

1 2 1 2 ηy = ηx + ηy2 . 2 2

(9.24)

Differentiating W2 with respect to t along the dynamics (9.18), we get W˙ 2 = −ηx2 − ηy2 + ηy (2x + 2y + z).

(9.25)

We define ηz as follows: ηz = 2x + 2y + z.

(9.26)

With the help of Eq. (9.26), we can express Eq. (9.25) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(9.27)

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As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 1 W (ηx , ηy , ηz ) = W2 (ηx , ηy ) + ηx2 = (ηx2 + ηy2 + ηz2 ). 2 2

(9.28)

It is clear that W is a positive definite function on R3 . Differentiating W with respect to t along the jerk dynamics (9.18), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ).

(9.29)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 + a)x + (3 − b)y + z − cxy − x 3 + v]. (9.30) We define the control law v as v = −(2 + a)x − (3 − b)y − z + cxy + x 3 − Kηz

(9.31)

where K is taken as a positive constant. Substituting (9.31) into (9.30), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(9.32)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that x = ηx , y = ηy − ηx , z = ηx − 2ηy .

(9.33)

As a consequence, it follows that (x(t), y(t), z(t)) → (0, 0, 0) exponentially as t → ∞. Thus, we have proved the following result. Theorem 9.1. The backstepping control law defined via (9.31) with gain K > 0 globally and exponentially stabilizes all the trajectories of the new plasma torch chaotic jerk chaos plant (9.18) for all initial states (x(0), y(0), z(0)) ∈ R3 . For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b, c) = (130, 5, 0.1). We choose K = 8 and the initial state of the new plasma torch jerk chaos system (9.18) as x(0) = 6.5, y(0) = 4.9, and z(0) = 2.7. Fig. 9.7 shows the time-history of the backstepping controlled states x(t), y(t), and z(t). It is easy to note that the controlled states converge to zero exponentially by the action of the backstepping control law (9.31).

202 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 9.7 Time-history of the backstepping controlled states x(t), y(t), z(t) of the plasma torch jerk system (9.18) for (a, b, c) = (130, 5, 0.1), K = 8, and (x(0), y(0), z(0)) = (6.5, 4.9, 2.7).

9.5 Active backstepping control for the global synchronization of the new plasma torch chaotic jerk systems Synchronization of chaotic systems is a phenomenon that may occur when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator (Ouannas et al., 2017c,a,d,j,h,i,g,f,k,e,b). Because of the butterfly effect which causes the exponential divergence of the trajectories of two identical chaotic systems started with nearly the same initial conditions, synchronizing two chaotic systems is seemingly a very challenging problem. A variety of schemes for ensuring the control and synchronization of such systems have been demonstrated based on their potential applications in various fields (Grassi et al., 2017; Ouannas et al., 2019b; Pham et al., 2017; Singh et al., 2017; Singh and Azar, 2020; Ouannas et al., 2020; Alain et al., 2020; Ouannas et al., 2019a; Alain et al., 2019; Khan et al., 2020b,a). Here, we deploy active backstepping control for globally synchronizing the state trajectories of a pair of new plasma torch jerk chaos systems considered as leader-follower systems. The leader chaos system is depicted by the 3D plasma torch jerk dynamics ⎧ ⎪ ⎨ x˙1 y˙1 ⎪ ⎩ z˙ 1

= =

y1 , z1 ,

=

ax1 − by1 − z1 − cx1 y1 − x13 .

(9.34)

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The follower chaos system is depicted by the 3D plasma torch jerk dynamics ⎧ ⎪ ⎨ x˙2 = y2 , y˙2 = z2 , (9.35) ⎪ ⎩ z˙ 2 = ax2 − by2 − z2 − cx2 y2 − x23 + v, where v is a backstepping control to be designed. The synchronization chaos error is defined by means of the equations ex = x2 − x1 , ey = y2 − y1 , ez = z2 − z1 . Upon calculating the error dynamics, we obtain the following: ⎧ ⎪ ⎨ e˙x = ey , e˙y = ez , ⎪ ⎩ e˙z = aex − bey − ez − c(x2 y2 − x1 y1 ) − x23 + x13 + v.

(9.36)

(9.37)

We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(9.38)

where ηx = ex .

(9.39)

Differentiating W1 with respect to t along the dynamics (9.37), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (ex + ey ).

(9.40)

η y = e x + ey .

(9.41)

We define

With the help of Eq. (9.41), we can express (9.40) as W˙ 1 = ηx η˙ x = −ηx2 + ηx ηy .

(9.42)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

1 2 1 2 ηy = ηx + ηy2 . 2 2

(9.43)

Differentiating W2 with respect to t along the dynamics (9.37), we get W˙ 2 = −ηx2 − ηy2 + ηy (2ex + 2ey + ez ).

(9.44)

We define ηz as follows: ηz = 2ex + 2ey + ez .

(9.45)

204 Backstepping Control of Nonlinear Dynamical Systems

With the help of Eq. (9.45), we can express Eq. (9.44) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(9.46)

As a final step of the backstepping control design, we set the quadratic Lyapunov function 1 1 W (ηx , ηy , ηz ) = W2 (ηx , ηy ) + ηx2 = (ηx2 + ηy2 + ηz2 ). 2 2

(9.47)

It is clear that W is a positive definite function on R3 . Differentiating W with respect to t along the error dynamics (9.37), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηx + ηy + η˙ z ).

(9.48)

A simple calculation yields the result W˙ = −ηx2 − ηy2 − ηz2 + ηz [(2 + a)ex + (3 − b)ey + ez − c(x2 y2 − x1 y1 ) − x23 + x13 + v].

(9.49)

We define the control law v as v = −(2 + a)ex − (3 − b)ey − ez + c(x2 y2 − x1 y1 ) + x23 − x13 − Kηz (9.50) where K is taken as a positive constant. Substituting (9.50) into (9.49), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(9.51)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that ex = ηx , ey = ηy − ηx , ez = ηx − 2ηy .

(9.52)

As a consequence, it follows that (ex (t), ey (t), ez (t)) → (0, 0, 0) exponentially as t → ∞.

A new multistable plasma torch chaotic jerk system Chapter | 9

205

FIGURE 9.8 Synchronization between the states x1 and x2 of the plasma torch jerk systems (9.34) and (9.35).

Thus, we have proved the following result. Theorem 9.2. The backstepping control law defined via (9.50) with gain K > 0 globally and exponentially synchronizes the 3D plasma torch jerk chaos plants (9.34) and (9.35) for all initial states in R3 . For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b, c) = (130, 5, 0.1). We choose K = 8. We take the initial state of the leader system (9.34) as x1 (0) = 2.4, y1 (0) = 3.8, and z1 (0) = −1.4. We also consider the initial state of the follower system (9.35) as x2 (0) = 0.8, y2 (0) = −0.7, and z2 (0) = 2.9. Figs. 9.8–9.10 display the synchronization between the states of the leader system and follower system. Fig. 9.11 shows the exponential convergence of the synchronization error ex (t), ey (t), ez (t) between the jerk systems (9.34) and (9.35).

9.6 Electronic circuit simulation of the new plasma torch chaotic jerk system This section is presented to the design of electronic circuits in order to validate the theoretical analysis and confirm the applicability of the mathematical model of the plasma torch chaotic jerk system (9.7).

206 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 9.9 Synchronization between the states y1 and y2 of the plasma torch jerk systems (9.34) and (9.35).

FIGURE 9.10 Synchronization between the states z1 and z2 of the plasma torch jerk systems (9.34) and (9.35).

In order to generate enough larger signals in an electronic circuit, the three state variables x, y, z of the jerk system (9.7) are rescaled as follows: X=

1 1 1 x, Y = y, Z = z. 10 10 100

(9.53)

A new multistable plasma torch chaotic jerk system Chapter | 9

207

FIGURE 9.11 Time-history of the synchronization error between the plasma torch jerk systems (9.34) and (9.35).

Thus, we transform the plasma torch jerk system (9.7) into the following dynamical system in the new coordinates: X˙

=

Y,



=

10Z,



=

a b 10 X − 10 Y

(9.54) − Z − cXY − 10X 3 .

The basic elements of the circuit are the resistors, capacitors, operational amplifiers (TL082CD) and multipliers (AD633JN), as shown in Fig. 9.12. According to Kirchhoff circuit laws, the electric equations can be written as follows: C1 X˙

=

1 R1 Y,

C2 Y˙

=

1 R2 Z,

C3 Z˙

=

1 1 R3 X − R4 Y

(9.55) −

1 R5 Z



1 R6 XY



1 3 R7 X .

Here X, Y, Z are the output voltages of the operational amplifiers U1A, U2A, and U3A, respectively. The values of circuit components have been chosen as R1 = R5 = 400 k, R2 = R7 = 40 k, R3 = 30.769 k, R4 = 80 k, R6 = 4 M, R8 = R9 = R10 = R11 = R12 = R13 = 100 k, and C1 = C2 = C3 = 1 nF. Fig. 9.13 represents Multisim outputs in different planes. Chaotic

208 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 9.12 Schematic diagram of the plasma torch system (9.55).

behavior showed in Fig. 9.3 is also captured in Multisim with good agreement as Fig. 9.13 illustrates.

9.7 Conclusions There is good interest in the modeling, simulation and control of thermomechanical systems in the control literature. An important model of a plasma

A new multistable plasma torch chaotic jerk system Chapter | 9

209

FIGURE 9.13 Phase projections of the plasma torch system (9.55) using Multisim circuit simulation: (A) x–y plane, (B) y–z plane, and (C) x–z plane.

210 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 9.13 (continued)

torch jerk differential equation exhibiting chaos was reported by Ghorui et al. (2000). In this work, we modeled a new plasma torch jerk chaotic system by adding a quadratic nonlinearity to the Ghorui jerk chaotic system with just a cubic nonlinearity. We showed that the new plasma torch jerk system has a self-excited chaotic attractor with three unstable saddle-foci rest points. We described a detailed bifurcation analysis of the new plasma torch chaotic jerk system. We also exhibited that the new plasma torch chaotic jerk system has multistability and coexisting chaotic attractors. As control applications, we designed active backstepping control laws for the global chaos stabilization and global chaos synchronization for the new plasma torch chaotic jerk system. We illustrated the backstepping control results with MATLAB plots. Using Multisim, we designed an electronic circuit of the new plasma torch chaotic jerk system for practical implementation in engineering.

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Chapter 10

Direct power control of three-phase PWM-rectifier with backstepping control Arezki Fekika,b , Hakim Denounb , Ahmad Taher Azarc,d , Nashwa Ahmad Kamale , Mustapha Zaouiab , Nabil Benyahiab , Mohamed Lamine Hamidab , Nacereddine Benamrouchea , and Sundarapandian Vaidyanathanf a Akli Mohand Oulhadj University, Bouira, Algeria, b Electrical Engineering Advanced Technology

Laboratory (LATAGE), Mouloud Mammeri University, Tizi-Ouzou, Algeria, c Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, d Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, e Cairo University, Giza, Egypt, f Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

10.1

Introduction

In recent years, static converters of power electronics are the most exploited in various sectors momentarily related to human activities, alike as the electrical network, industrial, and domestic areas. These converters include diode or thyristor rectifiers that are used in diverse devices, alike as asynchronous variable speed drives, computers, and their peripherals, as well as a new generation of lighting devices. In addition, there are some increasing exploitation of these converters in electrical installations has largely participated to the progress of their performances. However, they are one of the fundamental generators of harmonics in power distribution networks. Thus, to overcome the difficulties of this type of rectifier, a different power converter is deployed, named the Pulse Width Modulator (PWM) which is similarly considered as a nonlinear system. Due to rapid advancement in the digital signal processing system and power semiconductor technology, the implementation of three-phase PWM rectifiers is feasible (Hamida et al., 2017, 2018; Cao et al., 2016; Yan et al., 2012). A much more advanced control system for this type of power converter is now available (Bouafia et al., 2010; Fekik et al., 2018a,c; Wai and Yang, 2019). Strategies for three-phase PWM rectifiers have been enhanced through advanced progress in the use of semiconductor components and numerical methods (Malinowski et al., 2003; Fnaiech et al., 2018). For the regulation of the three-phased PWM rectifiers, various control methods were suggested. In recent decades, the control Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00017-9 Copyright © 2021 Elsevier Inc. All rights reserved.

215

216 Backstepping Control of Nonlinear Dynamical Systems

strategy has employed the voltage oriented control (VOC), which can indirectly regulate input power through the d-q co-ordinates regulation of the input current vector (Tahiri et al., 2019; Malinowski et al., 2003; Fekik et al., 2019; Zarif and Monfared, 2015). The main characteristics of VOC are fixed switching rate, good steady state stability, and high performance dynamics. The efficiency of this control program depends greatly, given its merits, on the modification of parameters for the proportional-integrate function. Another popular three-phase PWM rectifiers control technique has recently been direct power control (DPC) (Kahia et al., 2018; Fekik et al., 2017; Fischer et al., 2014). This control strategy allows for instant control, high robustness to parameter changes, and fast dynamic response to be decoupled (Zhang et al., 2019; Fekik et al., 2018b). DPC is the same as the DTC induction motor strategy (Gadoue et al., 2009). Consequently, a co-ordinate, independent blocks of PWM, and internal regulations are not needed for the DPC. Important studies have thus been carried out to improve system performance, improving the conventional DPC technique. In addition, other control algorithms, including the Fuzzy Logic (FL) approach and artificial neural networks (ANN) method, based on artificial intelligence techniques were developed recently (Coteli et al., 2017; Fekik et al., 2018b; Jamma et al., 2018). In this chapter, an improvement DPC technique based on backstepping controller for three-phase PWM rectifiers is proposed to satisfy the unity-powerfactor (UPF) condition. The results of the simulation show the practicability, adaptation of the backstepping controller, and transient or stationary voltage regulation with an almost sinusoidal current source (a low total harmonic distortion) which meets the IEEE 519 standard for harmonic distortion. This book chapter is organized into six sections. Section 10.1 is introductory. In Section 10.2, mathematical model of the PWM-rectifier is given. In Section 10.3, principle and definition of backstepping control are described. In Section 10.4, DC-voltage adjustment established in backstepping control is presented. Simulations results and discussions are presented in Section 10.5. Finally in Section 10.6, concluding remarks are given.

10.2 Mathematical model of PWM-rectifier The system of PWM rectifiers is shown in Fig. 10.1. This converter consists of three arms. Each one is composed of two IGBTs with antiparallel diode to ensure the continuity of the current. Each phase consists of a sinusoidal electromotive force with a resistance R and an inductance L. In this manner, the load block is collected of a capacitor C position in parallel with a resistance Rd , instantaneous line currents (ia , ib , ic ), and the simple voltages at the converter input (uea , ueb , uec ). A three-phase PWM converter is able to perform eight voltage vectors with six active vectors (V1 to V6 ) and two inactive vectors (V0 , V7 ). The representation of voltage vectors various of zero in the (α–β) coordinate frame forms a hexagon as shown in Fig. 10.2.

Direct power control of three-phase PWM-rectifier Chapter | 10 217

FIGURE 10.1 Diagram of the PWM rectifier.

FIGURE 10.2 Representation of the switching polygon.

The mathematical model of three-phase PWM rectifier in the (a, b, c) coordinates is given by: dia = ea − Ria − uea , dt dib = eb − Rib − ueb , dt dic = ec − Ric − uec . dt

(10.1)

The rectified current is given by the following equation: is = Sa ia + Sb ib + Sc ic .

(10.2)

218 Backstepping Control of Nonlinear Dynamical Systems

Knowing the state of each switch, we can define the conversion matrix of the rectifier. Simple voltages are expressed by the following relation: ⎛ ⎞⎛ ⎞ ⎛ ⎞ uea 2 −1 −1 Sa ⎟⎝ ⎠ ⎜ ⎟ Vdc ⎜ u = (10.3) −1 2 −1 ⎝ ⎠ Sb . ⎝ eb ⎠ 3 Sc −1 −1 2 uec By replacing Eq. (10.3) by (10.1), we find 2Sa − Sb − Sc dia = ea − Ria − Vdc , dt 3 2Sb − Sa − Sc dib = eb − Rib − Vdc , dt 3 dic 2Sc − Sa − Sb = ec − Ric − Vdc . dt 3

(10.4)

10.2.1 Vector representation The vector representation consists of placing the control vector in the two-phase reference obtained by the Clark transformation. The possible commutations of the switches can be carried out in three states (Sa , Sb , Sc ). The commands are given by Table 10.1, which makes it possible to find any combination of the given switches. The vector is obtained in the coordinate system (α–β). This gives the switching polygon. TABLE 10.1 Order states. Sa

Sb

Sc

Uea

Ueb

Uec

Vk



0

0

0

0

0

0

V0

0

0

1

V − 3dc

V − 3dc

V 2 3dc

V5

V 2 3dc

V − 3dc

V3

−V √dc 6 −V √dc 6

−V √dc 2 V√dc 2

0

0

0

1

0

V − 3dc

0

1

1

−2 3dc

− 3dc

V

− 3dc

V

V4

V − 3dc V −2 3dc

V − 3dc V − 3dc

V1

V

1

0

0

1

0

1

V 2 3dc V − 3dc

1

1

0

− 3dc

V

− 3dc

V

−2 3dc

V

1

1

1

0

0

0

 − 23 Vdc  2V 3 dc



0 0

V2

V√dc 6 −V √dc 6

−V √dc 2 −V √dc 2

V7

0

0

V6

The eight possible voltage vectors are expressed by the following equation: 2 π Vdc ej k (10.5) Vk = 3 3 with k = 1, 2, ..., 6; V0 = V7 = 0.

Direct power control of three-phase PWM-rectifier Chapter | 10 219

10.2.2 A brief review of direct power control The DPC is a control strategy that applies the instantaneous active and reactive power as command variables. This strategy works by variation the current variables in the systems via a switching table whose inputs are the errors (Sp ; Sq ) between the reference and captured values of the instantaneous powers, as well as the voltage vector position (θn ). It is similar to the direct torque control (DTC) for the electrical machines, in that the stator flux and the electromagnetic torque are the quantities commanded. The complete design of DPC for a PWM rectifier is shown in Fig. 10.3.

FIGURE 10.3 General configuration of DPC.

The reference of the active power Pref is obtained by controlling the DC-bus voltage Vdc . The reference value of active power is the output of voltage regulation while the reference value of reactive power is set to zero to achieve unity power factor condition. We have dib dic a Pest = L(ia di dt + ib dt + ic dt ) + Vdc (Sa ia + Sb ib + Sc ic ),

Qest =

√1 [L(ic dia dt 3

c − ia di dt ) + Vdc (Sa (ib − ic ) + Sb (ic − ia ) + Sc (ia − ib ))].

(10.6) The DPC technique makes the position of the voltage vector of the grid, for this, the plane (α–β) is divided inside twelve sectors, as shown in Fig. 10.4. The

220 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 10.4 Sectors and vectors of the rectifier voltages.

sectors can be made explicit numerically by this relation: (n − 2)

π π ≤ θn ≤ (n − 1) . 6 6

(10.7)

Here n is the sector number. It is instantly set by the position of the grid voltage vector and is calculated by θn = arctan

(vβ ) . (vα )

(10.8)

The errors between the calculated and the reference powers are input into two levels hysteresis controllers. The output of these controllers given by the digitized signals Sp and Sq by the two following equations:



1 if Pref − P ≥ Hp , 1 if Qref − Q ≥ Hq , Sq = Sp = 0 if Pref − P ≤ Hp , 0 if Qref − Q ≤ Hq , where Hp and Hq perform the deviations of hysteresis comparators. The switching states of the three-phase PWM rectifier for every sector are fixed by the table of state vectors as shown in Table 10.2.

10.3 Principle and definitions of backstepping control The backstepping control method is relatively recent in the control theory of nonlinear systems. It is a control method for nonlinear systems that permits one sequentially and systematically to build stabilizing Lyapunov functions for backstepping. It can be applied less restrictive and does not require the system to be linear (Hou and Fei, 2015; Dehkordi et al., 2017; Azar et al., 2020). Backstepping architecture is a sequential control technique characterized by stepby-step interlacing. Each step involves the transformation of the coordinates

Sp

Sq

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

θ9

θ10

θ11

θ12

0

0

101

100

100

110

110

010

010

011

011

001

001

101

0

1

100

110

110

010

010

011

011

001

001

101

101

100

1

0

001

001

101

101

100

100

110

110

010

010

011

011

1

1

010

010

011

011

001

001

101

101

100

100

110

110

Direct power control of three-phase PWM-rectifier Chapter | 10 221

TABLE 10.2 Table of states vectors.

222 Backstepping Control of Nonlinear Dynamical Systems

and the design of a virtual control based on the Lyapunov technique (Shukla et al., 2018; Vaidyanathan and Azar, 2016; Vaidyanathan et al., 2018, 2015). Finally, under the global stability, the true controller is achieved. Strict feedback systems, blocking systems with strict feedback, and parametric-strict feedback systems can be implemented with a backstepping design. Backstepping applies to standard triangular nonlinear systems (strict feedback systems). Consider the following nonlinear system: x˙ = f (x) + g(x)u; f (0) = 0, y = f (x),

(10.9)

where x = [x1 , x2 , ...xn ]; u: the command or system input; h(x): analytical function of x; y: the output of the system; f, g: fields of the infinitely differentiable vectors. To be able to write the system in the form of “strict feedback”, we apply a change of variable, the system (10.9) becomes ⎧ ϕ˙1 = ϕ2 , ⎪ ⎪ ⎪ ⎪ ⎪ ϕ˙2 = ϕ3 , ⎪ ⎪ ⎪ ⎪ ⎨ .. . (10.10) ⎪ ⎪ϕn−1 ˙ = ϕ , ⎪ n ⎪ ⎪ ⎪ ⎪ = u, ϕ ˙ n ⎪ ⎪ ⎩ y = ϕ1 , where ϕ = [ϕ1 , ϕ2 , ..., ϕn ] is the new state vector. The purpose of this change of variable is to find the first equation of system (10.10), a command called virtual via the variable ϕ2 , that one is controlled by ϕ3 , until the last equation, the global system is controlled by the command u, this procedure is explained step by step thereafter. Step 1: The system must be able to follow a given trajectory. This corresponds to doing the design of a tracking controller. The error between the output and its reference is Z1 = y ∗ − y = y ∗ − ϕ1 .

(10.11)

The instantaneous variation of this error is: Z˙1 = y˙∗ − y˙ = y˙∗ − ϕ2 .

(10.12)

The initial function of Lyapunov is selected as 1 v1 = Z 2 . 2

(10.13)

Direct power control of three-phase PWM-rectifier Chapter | 10 223

The instantaneous variation of this function is v˙1 = Z1 Z˙1 = Z1 (y˙∗ − ϕ2 ).

(10.14)

For the first variable to converge to its reference, the instantaneous variation of the Lyapunov function must be negative, for which the choice is y˙∗ − ϕ2 = −K1 Z1

(10.15)

where k1 > 0 is a positive coefficient. From relation (10.15), ϕ2 = y˙∗ + K1 Z1 .

(10.16)

The above equation defines the value that ϕ2 will take in order to stabilize Lyapunov. However, it is impossible to act directly on the ϕ2 state. The notation ϕ2∗ will be used to indicate the desired value (reference) of the state. The desired value obtained from the state is given by ϕ2∗ = y˙∗ + K1 Z1 .

(10.17)

Step 2: It is therefore unlikely that this state follows exactly its trajectory, this is why another error term is introduced: Z2 = ϕ˙2∗ − ϕ2 = y˙∗ + K1 Z1 − ϕ2 .

(10.18)

Z˙2 = y¨∗ + K1 Z˙1 − ϕ˙2 .

(10.19)

Its derivative is then

From (10.12) and (10.18), we find Z˙1 = y˙∗ − ϕ˙2 = Z2 − K1 Z1 .

(10.20)

Substituting (10.19) into (10.20), we find Z˙2 = y¨∗ + K1 (Z2 − K1 Z2 ) − ϕ˙2 .

(10.21)

Lyapunov’s function is augmented by another term to take into consideration the possible error on the ϕ2 state. The new candidate function is given by 1 v2 = (Z12 + Z22 ). 2

(10.22)

The derivative of this function is v˙2 = Z1 Z˙1 + Z2 Z˙2 = −K1 Z12 + Z2 [Z1 − ϕ˙2 + ϕ˙2∗ ].

(10.23)

For the Lyapunov criterion to be met, the expression in parentheses must be equal to (−K2 Z2 ) as illustrated by the following equation: Z1 (1 − K12 ) − K1 Z2 − ϕ3 + y¨∗ = −K2 Z2 .

(10.24)

224 Backstepping Control of Nonlinear Dynamical Systems

From this we can choose the second ϕ3 virtual command like ϕ3∗ = (1 − K12 )Z1 + (K1 + K2 )Z2 + y¨∗ .

(10.25)

Where k2 is a non-zero positive parameter like k1 , this would bring the function of Lyapunov into the following form: v˙2 = −K1 Z12 − K2 Z22 .

(10.26)

In this way, the V2 function meets Lyapunov’s criteria. The chosen control law ensures that the function V2 is ever positive, and its instantaneous variation v˙2 , is ever negative, the function of the error is then converged to zero at all times. Step i: For step i, the equation is of the form Zi = ϕi∗ − ϕi .

(10.27)

The function of Lyapunov is defined by 1 2 Zj 2 i

Vi =

(10.28)

j =1

and Z˙ i=1 = Zi − Ki=1 Zi=1 − Zi=2 , v˙i = −

i=1 

Kj Zj2 + Zi (Zi=1 − ϕ˙i + ϕ˙i∗ ).

(10.29)

j =1

The virtual command in this case represents the actual command u: u = ϕ˙n∗ , u = Kn Zn − Zn−1 + ϕ˙n∗ . Here Kn > 0. (See Fig. 10.5.)

FIGURE 10.5 Schematic diagram of control by backstepping.

(10.30)

Direct power control of three-phase PWM-rectifier Chapter | 10 225

10.4

Control of DC-voltage by backstepping

The error variable e1 is defined by e1 = Vdcref − Vdc .

(10.31)

The first subsystem allows the design of the DC voltage regulator. The variation of the error e1 is defined by e˙1 = V˙dcref − V˙dc , e˙1 = V˙dcref −

Pref . Vdc C

(10.32)

The initial function of Lyapunov is selected as follows: 1 v1 = e12 . 2

(10.33)

The variation of this function is

  Pref . v˙1 = e1 e˙1 = e1 V˙dcref − Vdc C

(10.34)

In order for the variation of the Lyapunov function to be negative, −K1 e1 = V˙dcref −

Pref . Vdc C

(10.35)

Hence the command may be defined as follows: Pref = Vdc C(V˙dcref + K1 e1 ).

(10.36)

In the case where the reference voltage is selected as constant, its derivative will be null, and the command will be Pref = Vdc CK1 e1 .

10.5

(10.37)

Simulation results

To approve the performance of the proposed controller, numerical simulations were carried out in the MATLAB® / SIMULINK environment. The circuits parameters discussed in this section are specific below: f = 50 Hz, Vdcref = 300 V, C = 4700 µF, R = 0.25 , L = 16 mH, eabc = 120 V, Rd = 100 . In this part, two tests will be considered. A. Test with the variation of the reference voltage In this test a step of the reference DC voltage is applied at t = 0.5 s from 300 V to 350 V.

226 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 10.6 DC-voltage (Vdc = 300 to 350 V).

FIGURE 10.7 Active power (Vdc = 300 to 350 V).

Figs. 10.6, 10.7, and 10.9, respectively, exhibit the responses of the DC side voltage, active instantaneous power, and absorbed currents during a transient regime. Fig. 10.6 illustrates that the system response has become very fast (t < 0.09 s) and follows its reference without overshoot with the backstepping controller. Indeed, an augmentation in the reference of the DC bus voltage causes an increase in the active power.

Direct power control of three-phase PWM-rectifier Chapter | 10 227

FIGURE 10.8 AC-voltage and current line (Vdc = 300 to 350 V).

FIGURE 10.9 Line currents (Vdc = 300 to 350 V).

In this state, the increase of power is limited, which is safe for the execution of the system. The direct control of the power based on backstepping controller ensures a good follow-up of the active power reference and the absorption of the sinusoidal currents is also ensured during this transient. In Fig. 10.8, the current has a sinusoidal form, and in phase with the line voltage, so operation under a unit power factor is ensured.

228 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 10.10 Line voltage and voltage estimation (Vdc = 300 to 350 V).

FIGURE 10.11 Harmonic spectrum (Vdc = 300 to 350 V).

Fig. 10.10 indicates the estimated voltage eaest and the measured voltage ea . It is to be noted that the estimated voltage follows the measured value on the alternative side. This is useful for accurately estimating the P and Q. The signal from the grid current is locked to the sine signal form and thus THD has been reduced to 3.09% (Fig. 10.11) which ensures a non-contaminated circuit and thus a good electrical energy.

Direct power control of three-phase PWM-rectifier Chapter | 10 229

B. Test with the variation of the load In this test, a scale of load resistance variation Rd is applied at t = 0.5 s from 70  to 110 .

FIGURE 10.12 DC-voltage (Rd = 70 to 110 ).

Figs. 10.12, 10.13, and 10.15 show that a decrease in load power causes an over-voltage at the DC bus. After a short transient period, the DC bus voltage remains constant near its reference and the active power follows its novel reference with good precision and stability. This time, the backstepping controller compensates for this surge by reducing the reference of the active power. In this case, the absorption of sinusoidal currents is observed. Fig. 10.14 shows that the line current has a sinusoidal form and is in phase with the voltage of the source, which implies that the reactive power remains constant near its reference zero, which confirms that the DPC developed from the controller guarantees a perfect decoupling of instantaneous active and reactive power regulation. Fig. 10.16 shows the estimated voltage eaest and the line voltage with the load change Rd . It can be noted that the estimated line voltage is indeed the sensed voltage even with the change of the load. This controller also guarantees a weak THD current (T H Di = 3.09%), which is therefore a high energy output for the electrical network if the charge is adjusted as shown in Fig. 10.17.

230 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 10.13 Active power (Rd = 70 to 110 ).

FIGURE 10.14 AC-voltage and current line (Rd = 70 to 110 ).

10.6 Conclusion This chapter introduces a study and analysis of the three-phase PWM rectifier DPC based on backstepping controller, which aims to show the contribution of these proposed controllers. The testing of the simulation results demonstrated the robustness, efficiency, and suitable performance of the backstepping controller founded on the DPC scheme. The benefits of support by this controller

Direct power control of three-phase PWM-rectifier Chapter | 10 231

FIGURE 10.15 Line currents (Rd = 70 to 110 ).

FIGURE 10.16 Line voltage and voltage estimation (Rd = 70 to 110 ).

are very significant, in particular on the harmonic compensation side, the response time, and the monitoring of the reference values. It is to be noted that the nonlinear controller has a high performance for decoupled regulation of active and reactive power as well as better precision in DC bus voltage regulation and small current THD. For future studies, the regulation of the DC voltage will be ensured by adaptive controllers.

232 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 10.17 Harmonic spectrum (Rd = 70 to 110 ).

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Yan, S., Wei, F., Du, H., Liu, X., 2012. Fast tracking control of three-phase PWM rectifier for microturbine. In: Wang, J., Yen, G.G., Polycarpou, M.M. (Eds.), Advances in Neural Networks – ISNN 2012. Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 525–533. Zarif, M., Monfared, M., 2015. Step-by-step design and tuning of VOC control loops for grid connected rectifiers. International Journal of Electrical Power & Energy Systems 64, 708–713. Zhang, Y., Jiao, J., Liu, J., Gao, J., 2019. Direct power control of PWM rectifier with feedforward compensation of DC-bus voltage ripple under unbalanced grid conditions. IEEE Transactions on Industry Applications 55 (3), 2890–2901.

Chapter 11

Adaptive backstepping controller for DFIG-based wind energy conversion system Ismail Drhorhia , Abderrahim El Fadilia , Chaker Berrahala,c , Rachid Lajouadb , Abdelmounime El Magrib , Fouad Giric , Ahmad Taher Azard,e , and Sundarapandian Vaidyanathanf a Faculté des Sciences et Techniques, Hassan II University of Casablanca, Mohammedia, Morocco, b Ecole Normale Supérieure d’Enseignement Technique (ENSET), Hassan II University of

Casablanca, Mohammedia, Morocco, c University of Caen Basse Normandie, Caen, France, d Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, e Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, f Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

11.1

Introduction

Due to its high production capability and its lower cost installation, the technology of wind energy extraction made considerable progress over the last few decades. It is estimated that the part of wind energy in total electricity production worldwide by 2050 will be 15 ∼ 18% (Ni et al., 2017). One major component of any wind energy conversion system (WECS) is the generator that converts the wind energy into electricity. In this respect, the Doubly Fed Induction Generator (DFIG) is widely used in wind power generation technology today. (See Fig. 11.1.) Indeed, this machine proves to be cost effective, efficient, and reliable, (Abad et al., 2011). Furthermore, DFIG gives the possibility to separate control of reactive and active power, generation of electrical power at lower wind speeds or the control of the wind turbines power factor. Moreover, DFIG machines are more energy efficient, since there is little power dissipated in the converter and the good performances in grid compatibility. For these reasons, DFIGs have received recently a great deal of interest (Rekioua, 2014; Abad et al., 2011; Abdelmalek et al., 2018a, 2017, 2018b). Although the stator is directly connected to the power grid, the back-to-back converters, which are controlled with Pulse Width Modulation (PWM), separate the rotor machine-side from grid-side portions, which are connected to each other via a DC-link capacitor. This topology makes it possible to regulate the turbine torque and thus the rotor speed. In addition, the grid-side converter Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00018-0 Copyright © 2021 Elsevier Inc. All rights reserved.

235

236 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 11.1 The wind energy conversion system based on doubly fed induction generator.

also regulates the reactive power injection (Rekioua, 2014; Meghni et al., 2018; Ben Smida et al., 2018; Meghni et al., 2017). On the other hand, it is seen from the turbine characteristic (wind speed vs extracted power), the wind turbine model contains a nonlinear term depending on the wind speed, the pitch angle, and the turbine speed. In the last decade, a lot of research devoted to the problem of maximizing the extracted wind power for each wind speed, called “Maximum Power Point Extraction” (MPPT) (Rekioua, 2014; Lajouad et al., 2019; Soetedjo et al., 2011). It resulted a number of techniques to optimize the transfer of wind energy to the grid have been designed and reported on in the literature. The hill-climb searching (HCS) method, also called perturbation and observation (P&O), has been proposed by several authors i.e. (Koutroulis and Kalaitzakis, 2006; Kazmi et al., 2011; Kamal et al., 2020). However, despite its good results in photovoltaic systems (fast dynamics), this technique shows its limits in large-inertia wind turbine systems. Another approach to achieving the optimal tip-speed ratio (turbine velocity / wind speed) has been proposed in Cardenas and Pena (2004); Datta and Ranganathan (2003), the rotor speed must be variable and proportional to the wind speed value. Despite the simplicity and flexibility of this method, an accurate on-line measurement of wind speed and rotor speed is required. This is practically a delicate task. In this chapter, the Reference Speed Optimizer (RSO), developed in Lajouad et al. (2019), is used to ensure the MPPT whatever the wind speed. The RSO method performs an online search of the optimal value of the turbine reference

Adaptive backstepping controller for DFIG Chapter | 11 237

speed ωmopt . If the speed of the rotor ωm reaches ωmopt , the maximum wind energy is captured and transmitted to the electric grid. Compared to the other techniques, the RSO search technique enjoys, in addition to the sensorless feature, a faster adaptation of wind changes and soft convergence (non-oscillating) towards the optimum point. The main purpose of this chapter, is to provide the modeling and control of wind energy conversion system constituted of DFIG and AC/DC/AC converters connected to the grid. The adaptive backstepping control technique, Lyapunov stability, and the RSO MPPT technique without using sensors of wind velocity and generator torque are adopted. To reduce the cost of implementation and maintenance, only the currents and voltages (electrical) variable measurements are needed. There are two operational control objectives: (i) speed regulation, to ensure that the system extracts the maximum power possible at variable wind speed, (ii) power factor correction, to guarantee better quality for the current injected into the power grid and avoid harmonic pollution. In addition to this, two additional control objectives are ensured: stator flux norm and the DC-Link voltage. This book chapter is organized into six sections. Section 11.1 is introductory. Section 11.2 is devoted to the online generation of optimal rotor speed reference to extract maximum power regardless the wind speed. Section 11.3 presents the modeling wind energy system conversion, constituted of doubly fed induction generator and AC/DC/AC converters. In Section 11.4, a multi-loop nonlinear controller is synthesized using the backstepping adaptive technique and Lyapunov stability. In Section 11.5, the simulation results are reported that illustrate the control performances. Finally, in Section 11.6, concluding remarks are given.

11.2

Wind sensor-less rotor speed reference optimization

The objective is to construct a speed-reference optimizer in order to meet the MPPT requirement. The speed reference optimizer design is based on the turbine power characteristic (Fig. 11.2) without the requirement of the wind velocity measurement. The optimizer is expected to compute on-line the optimal speed value ωmopt so that, if the turbine rotor speed ωm is made equal to ωmopt then maximal wind energy is captured, and transmitted to the grid through the aerogenerator (Lajouad et al., 2019, 2013, 2015; El Fadili et al., 2018). For a given wind speed, each power curve has a maximum power point (Fig. 11.2), where speed of the wind turbine must be adjusted to ensure the best extraction. The peaks of these curves give the maximum ‘extractable’ power and thus the optimal point. This point is characterized by the optimal speed. These curves give the maximum power Popt according to ωmopt , in other words for i , there is a unique couple (ω , P ) that presents the MPP. The set each vwin mi i of all optimal couples (ωmi , Pi ) has thus been obtained in Fig. 11.2 and interpolated to get a polynomial function, which demonstrates the speed-reference

238 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 11.2 Turbine Power Characteristics (Pitch angle β = 0◦ ).

FIGURE 11.3 Optimal Power-Speed (OPS) function.

optimizer ωmopt = F (Popt ) given by F (P ) = hn P n + hn−1 P n−1 + . . . + h1 P + h0

(11.1)

where the coefficients hi have numerical values corresponding to mechanical characteristic of the wind turbine. Such a polynomial function is referred to optimal power-speed (OPS) characteristic and is illustrated in Fig. 11.3. Remark 11.1. The polynomial interpolation yielding the function has been obtained the MATLAB® functions MAX, POLYVAL, SPLINE, and POLYFIT.

11.3 Modeling ‘AC/DC/AC converter-DFIG’ association A wind energy conversion system (WECS) transforms wind kinetic energy to mechanical energy by using rotor blades. This energy is then transformed into

Adaptive backstepping controller for DFIG Chapter | 11 239

electric energy by a generator (Lang et al., 2011; Jadhav and Roy, 2013; Kazmi et al., 2011).

11.3.1 DFIG-AC/DC modeling The controlled system, illustrated in Fig. 11.4, includes a combination of doubly fed induction generator-inverter, on the one hand, and a three-phase DC/AC rectifier, on the other hand. The inverter is an DC/AC converter operating, like the AC/DC rectifier, according to the well-known Pulse Wide Modulation (PWM) principle.

FIGURE 11.4 Doubly fed induction generator connected to the AC/DC/AC converters.

Considering the flux components, φsd , φsq , and the current components, ird , irq , as state variables and assuming that magnetic circuit to be linear, the two-phase model of the doubly fed induction generator, represented in a rotating reference frame (d, q), can be given by the following differential equations (Leonard, 2001; El Fadili et al., 2013b,c,d; Giri, 2013; El Magri et al., 2013b): dωm dt dφsd dt dφsq dt dird dt

Tg Msr F (φsq ird − φsd irq ) − , ωm + p (11.2) J J Ls J Msr 1 ird + vsd , (11.3) = − φsd + ωs φsq + τs τs Msr 1 irq + vsq , (11.4) = − φsq − ωs φsd + τs τs γ2 = −γ1 ird + (ωs − pωm )irq + φsd − pωm γ2 φsq − γ2 vsq + γ3 vrd , τs (11.5) =−

240 Backstepping Control of Nonlinear Dynamical Systems

dirq γ2 = −γ1 irq − (ωs − pωm )ird + φsq + pωm γ2 φsd − γ2 vsd + γ3 vrq , dt τs (11.6) where the following notations are used: • • • • • • • • • • • •

φsd , φsq , stator flux dq-components. ird , irq , rotor current dq-components. vrd , vrq , rotor voltage dq-components. ωm , rotor speed. ωs , dq-frame speed. Rs , Rr , stator and rotor resistances. Ls , Lr , stator and rotor self-inductances. Msr , mutual inductance between the stator and the rotor. F , friction coefficient. J , rotor inertia. Tg , generator torque. p, number of pole pairs. The remaining parameters are defined as follows: 2 Rr L2s + Rs Msr , 2 σ Lr Ls Msr γ2 = , σ Ls Lr

γ1 =

σ =1−

2 Msr , Ls Lr

Ls , Rs 1 γ3 = . σ Lr τs =

When the stator voltage is linked to the d-axis of the frame, vsd = Vg and vsq = 0, the stator and grid currents will be related directly to the active and reactive power. An adapted control of these currents will thus permit to control the power exchanged between the generator and the grid. The inverter is characterized by the fact that the rotor d- and q-voltages can be controlled independently. To this end, these voltages are expressed in function of the corresponding control action (El Fadili et al., 2011; El Magri et al., 2009b,a; El Fadili et al., 2010, 2013c,a, 2012a,b,c, 2014b,c; Fekik et al., 2020): vrd = vdc u1 ,

vrq = vdc u2 ,

ire = u1 ird + u2 irq ,

(11.7)

where u1 , u2 represent the average d-axis and q-axis (Park’s transformation) of the three-phase duty ratio system (s1 , s2 , s3 ), ire designates the input current inverter, and vdc is DC-link voltage (see Fig. 11.5).  si =

1 if Si 0 if Si

On and Si Off and Si

Off On

i = 1, 2, 3.

(11.8)

Adaptive backstepping controller for DFIG Chapter | 11 241

Now, let us introduce the state variables: x 1 = ωm ;

x2 = φ sd ;

x3 = φ sq ;

x4 = i rd ;

x5 = i rq ;

x6 = v dc ;

u1 x6 = v rd ;

u2 x6 = v rq , (11.9)

where (•) denote the average value on the modulation (PWM) period of (•).

FIGURE 11.5 DC/AC inverter.

Then substituting (11.7)–(11.9) in (11.2)–(11.6) yields the following statespace representation of the association of doubly fed induction generatorinverter: dx1 dt dx2 dt dx3 dt dx4 dt dx5 dt

Tg Msr F (x3 x4 − x2 x5 ) − , x1 + p J J Ls J Msr 1 x4 + Vg , = − x2 + ω s x3 + τs τs Msr 1 x5 , = − x3 − ω s x2 + τs τs γ2 = −γ1 x4 + (ωs − px1 )x5 + x2 − pγ2 x1 x3 − γ2 Vg + γ3 x6 u1 , τs γ2 = −γ1 x5 − (ωs − px1 )x4 + x3 + pγ2 x1 x2 + γ3 x6 u2 . τs =−

(11.10) (11.11) (11.12) (11.13) (11.14)

242 Backstepping Control of Nonlinear Dynamical Systems

11.3.2 AC/DC rectifier modeling The power grid is connected to the rectifier which consists of six semiconductors IGBTs supposed ideal switches, the electrical equations are given by Lo

d[iin ]123 = vdc [k]123 − [vg ]123 , dt dvdc 1 = (iot − ire ), dt C iot = [k]T123 [iin ]123 ,

(11.15) (11.16) (11.17)

T  is the input currents rectifier, [vg ]123 = where [iin ]123 = iin1 iin2 iin3 T  vg1 vg2 vg3 is the sinusoidal three-phase net voltages, iot is the output current rectifier; Lo is the grid inductance; C is the capacitor DC-link bus, and ki is the switch position function taking values in the discrete set {0, 1} (see Fig. 11.6):  ki =

1 if Ki 0 if Ki

On and Ki Off and Ki

Off On

i = 1, 2, 3.

(11.18)

To simplify the three-phase representation (11.15)–(11.16) for the synthesis of control laws, the Park transformation, where the d-axis of the frame is linked to the stator voltage, is used: Vg vdc u3 diind + , = ωs iinq + dt Lo Lo diinq vdc u4 , = −ωs iind + dt Lo 1 dvdc = (iot − ire ), dt C

(11.19) (11.20) (11.21)

where (iind , iinq ) denotes the rectifier side network current in dq-coordinates and u3 , u4 represent the average d- and q-axis components of the three-phase duty ratio system (k1 , k2 , k3 ), and ωs is grid pulsation. Using the average value of vdc ; iind and iinq x6 = v dc ;

x7 = i ind ;

x8 = i inq ,

(11.22)

and replace iot by iot = u3 x7 + u4 x8 .

(11.23)

Adaptive backstepping controller for DFIG Chapter | 11 243

FIGURE 11.6 AC/DC rectifier connected to the power grid.

Eqs. (11.19)–(11.21) become 1 dx6 = (u3 x7 + u4 x8 − ire ), dt C Vg x6 u3 dx7 + , = ω s x8 + dt Lo Lo dx8 x6 u4 . = −ωs x7 + dt Lo

(11.24) (11.25) (11.26)

The state-space equations obtained up to now are put together to get a statespace model of the whole system including the AC/DC/AC converters combined with the doubly fed induction generator. For convenience, the whole model is rewritten here for future reference: dx1 dt dx2 dt dx3 dt dx4 dt

Tg Msr F (x3 x4 − x2 x5 ) − , x1 + p (11.27) J J Ls J Msr 1 x4 + Vg , (11.28) = − x2 + ω s x3 + τs τs Msr 1 x5 , (11.29) = − x3 − ω s x2 + τs τs γ2 = −γ1 x4 + (ωs − px1 )x5 + x2 − pγ2 x1 x3 − γ2 Vg + γ3 x6 u1 , τs (11.30) =−

244 Backstepping Control of Nonlinear Dynamical Systems

dx5 dt dx6 dt dx7 dt dx8 dt

11.4

= −γ1 x5 − (ωs − px1 )x4 +

γ2 x3 + pγ2 x1 x2 + γ3 x6 u2 , τs

1 (u3 x7 + u4 x8 − ire ), C Vg x6 u3 + , = ω s x8 − Lo Lo x6 u4 . = −ωs x7 + Lo =

(11.31) (11.32) (11.33) (11.34)

Controller design

11.4.1 Control objectives The controller is designed to verify four objectives. CO1: Optimization of the wind energy extraction in order to extract the maximum power possible. CO2: Speed regulation: the machine speed ωm must track, as closely as possible, a given reference signal ωmref , despite the generator torque Tg and the rotor inertia J uncertainty. The reference signal ωmref has been obtained from the MPPT strategy as in Lajouad et al. (2019, 2013, 2015). CO3: The rectifier input currents (iin1 , iin2 , iin3 ) must be sinusoidal with the same phase as the power grid. The reactive power must be null. CO4: Controlling the continuous voltage vdc must track a given reference signal vdcref . This generally is set to a constant value equal to the converter nominal voltage.

11.4.2 Speed and stator flux norm regulator design The problem of controlling the rotor speed and stator flux norm is presently addressed for the doubly fed induction generator described by (11.27)–(11.31). The speed reference x1∗ = ωmref = F (P ) is a bounded, the stator flux reference ∗s is fixed to its nominal value. The controller design will now be performed in two steps using the tuning-functions adaptive backstepping technique (Krsti´c et al., 1995). The tracking errors are defined as follows: z1 = x1∗ − x1 , z2 = ∗s 2

(11.35)

− (x2 + x3 ). 2

2

(11.36)

Step 1. Using the state equations (11.27)–(11.31), the errors z1 and z2 obey the differential equations: z˙ 1 = x˙1∗ +

Tg F Msr (x3 x4 − x2 x5 ) + , x1 − p J J Ls J

(11.37)

Adaptive backstepping controller for DFIG Chapter | 11 245

˙ ∗s − 2(x˙2 x2 + x˙3 x3 ) z˙ 2 = 2∗s  2 2Msr ˙ ∗s + (x2 2 + x3 2 ) − = 2∗s  (x2 x4 + x3 x5 ) − 2x2 Vg . τs τs

(11.38)

sr In (11.37) and (11.38), the quantities p MLsrs (x3 x4 − x2 x5 ) and 2M τs (x2 x4 + x3 x5 ) stand up as virtual control signals. If these were the actual control signals, the error system (11.37)–(11.38) could be globally asymptotically stabilized letsr ting p MLsrs (x3 x4 − x2 x5 ) = μ1 and 2M τs (x2 x4 + x3 x5 ) = ν1 with

def

μ1 = J (c1 z1 + x˙1∗ ) + F x1 + Tg , 2 def ˙ ∗s + (x22 + x32 ) − 2x2 Vg . ν1 = c2 z2 + 2∗s  τs

(11.39) (11.40)

Presently, the load torque Tg and the rotor inertia J are not assumed to be known. This suggests that one must consider the certainty equivalence form of Eq. (11.39). That is, one has def μ1 = Jˆ(c1 z1 + x˙1∗ ) + F (x1∗ − z1 ) + Tˆg

(11.41)

where c1 and c2 are any positive design parameters, and Tˆg and Jˆ are the estimates of Tg and J , respectively (yet to be determined). sr As the quantities p MLsrs (x3 x4 − x2 x5 ) = μ1 and 2M τs (x2 x4 + x3 x5 ) = ν1 are not the actual control signals, they cannot be set equal to μ1 and ν1 , respectively. Nevertheless, we retain the expressions of μ1 and ν1 as first stabilizing functions and introduce the new errors: Msr (x3 x4 − x2 x5 ), Ls 2Msr (x2 x4 + x3 x5 ). z4 = ν1 − τs

z3 = μ1 − p

(11.42) (11.43)

Then, using the notations (11.42)–(11.43), the dynamics of the errors z1 and z2 , can be rewritten as follows: Tg F 1 + x1 , (μ1 − z3 ) + J J J    T  1 F g Jˆ c1 z1 + x˙1∗ + Tˆg + F x1 − z3 + + x1 , z˙ 1 = x˙1∗ − J J J ˜   Tg J 1 c1 z1 + x˙1∗ + + z3 , z˙ 1 = −c1 z1 + J J J z˙ 2 = −c2 z2 + z4 , z˙ 1 = x˙1∗ −

(11.44) (11.45) (11.46) (11.47)

where T˜g = Tg − Tˆg ,

J˜ = J − Jˆ.

(11.48)

246 Backstepping Control of Nonlinear Dynamical Systems

Step 2. The second design step consists in choosing the actual control signals, u1 and u2 , so that all errors (z1 , z2 , z3 , z4 ) converge to zero. To this end, we should express these errors depend on the actual control signals (u1 , u2 ). We start focusing on z3 ; it follows from (11.42) that z˙ 3 = μ˙ 1 − p

Msr (x˙3 x4 + x3 x˙4 − x˙2 x5 − x2 x˙5 ). Ls

(11.49)

Assume that the load torque Tg and the rotor inertia J are constants or slowly time-varying and using (11.27)–(11.31), (11.41), and (11.48), one gets from (11.49):



g ˜  T J 1 c1 z1 + x˙1∗ + + z3 z˙ 3 = c1 Jˆ − F −c1 z1 + J J J   ˙ Msr γ3 x6 (x3 u1 − x2 u2 ) + Jˆx¨1∗ + F x˙1∗ − J˙˜ c1 z1 + x˙1∗ − T g −p Ls  Msr px1 (x3 x5 + x2 x4 ) + pγ2 x1 2s + (γ2 x3 + x5 )Vg +p Ls  1 + (γ1 + ) (x3 x4 − x2 x5 ) , (11.50) τs

 T g J˜  c1 z1 + x˙1∗ + z˙ 3 = μ2 + (c1 Jˆ − F ) J J

− c1 z3

J˜ F − z3 J J

 ˙  Msr γ3 x6 (x3 u1 − x2 u2 ), − J˙˜ c1 z1 + x˙1∗ − T g −p Ls

(11.51)

with μ2 = −c1 z1 (Jˆc1 − F ) + c1 z3 + Jˆx¨1∗ + F x˙1∗  Msr p px1 (x3 x5 + x2 x4 ) + pγ2 x1 2s + (γ2 x3 + x5 )Vg Ls  1 + (γ1 + ) (x3 x4 − x2 x5 ) . τs

(11.52)

Similarly, it follows from (11.43) that z4 becomes the following differential equation: z˙ 4 = ν˙ 1 −

2Msr (x˙2 x4 + x2 x˙4 + x˙3 x5 + x3 x˙5 ). τs

(11.53)

Using (11.27)–(11.31) and (11.40), it follows from (11.53): z˙ 4 = ν2 −

2Msr γ3 x6 (x2 u1 + x3 u2 ) τs

(11.54)

Adaptive backstepping controller for DFIG Chapter | 11 247

with ˙ ∗s )2 + 2∗s  ¨ ∗s ν2 = c2 (−c2 z2 + z4 ) + 2( Msr 3 4 1 +2 ( + γ1 )(x2 x4 + x3 x5 ) + (− 2s + Vg x2 ) τs τs τs τs 1 Msr Msr 2 2 x4 + Vg ) − 2( ) (x4 + x52 ) − 2Vg (− x2 + ωs x3 + τs τs τs   Msr γ2 2  + px1 (x3 x4 − x2 x5 ) + x4 Vg − γ2 x2 Vg . −2 τs τs s

(11.55)

Remark 11.2. The derivatives x˙1∗ and x¨1∗ are obtained using Eq. (11.1). To analyze the error system, composed of Eqs. (11.46), (11.47), (11.51), and (11.54), let us consider the following Lyapunov function candidate: 1 1 1 1 T˜g2 1 J˜2 1 + . V = z12 + z22 + z32 + z42 + 2 2 2 2 2 J 2 J

(11.56)

Its time derivative along the trajectory of the state vector (z1 , z2 , z3 , z4 ) is V˙ = z1 z˙ 1 + z2 z˙ 2 + z3 z˙ 3 + z4 z˙ 4 +

T˙˜g T˜g J˙˜J˜ + . J J

(11.57)

Placing (11.46), (11.47), (11.51), and (11.54), into (11.57) leads to   ˜ T˜g 1 J ∗ V˙ = z1 −c1 z1 + z3 + (c1 z1 + x˙1 ) + + z2 (−c2 z2 + z4 ) J J J

   T g  ˙  J˜  ˙ ∗ ∗ ˆ ˜ + z3 μ2 + (c1 J − F ) c1 z1 + x˙1 + − J c1 z1 + x˙1 − T g J J   J˜ F Msr + z3 −p γ3 x6 (x3 u1 − x2 u2 ) − c1 z3 − z3 Ls J J   T˜g 2Msr J ˙ γ3 x6 (x2 u1 + x3 u2 ) + T˙˜g + J + z4 ν2 − . (11.58) τs J J Adding c3 z32 − c3 z32 + c4 z42 − c4 z42 to the right side of (11.58) and rearranging terms yield F 1 V˙ = −c1 z12 − c2 z22 − c3 z32 − c4 z42 − z32 + z1 z3 J J     Msr ˙ ∗ ˙ ˜ + z3 μ2 + c3 z3 − J c1 z1 + x˙1 − T g − p γ3 x6 (x3 u1 − x2 u2 ) Ls   2Msr + z4 ν2 + c4 z4 + z2 − γ3 x6 (x2 u1 + x3 u2 ) τs

248 Backstepping Control of Nonlinear Dynamical Systems

 T˜g  (c1 Jˆ − F )z3 + z1 + T˙˜g J

    J˜  ˙˜  + J + c1 z1 + x˙1∗ z1 + z3 c1 Jˆ − F c1 z1 + x˙1∗ − c1 z32 , J +

(11.59)

which suggests the following parameter adaptation laws: ˙ = −λ , T g Tg with

J˙˜ = −λJ ,

    λJ = −c1 z32 + z1 c1 z1 + x˙1∗ + z3 c1 Jˆ − F c1 z1 + x˙1∗ ,

λTg = z1 + c1 Jˆ − F z3 .

(11.60)

(11.61) (11.62)

Substituting the parameter adaptation laws (11.60) to T˙˜g and J˙˜ in the right side of (11.59) yields F 1 V˙ = −c1 z12 − c2 z22 − c3 z32 − c4 z42 − z32 + z1 z3 J J     Msr ∗ + z3 μ2 + c3 z3 + λJ c1 z1 + x˙1 + λTg − p γ3 x6 (x3 u1 − x2 u2 ) Ls   2Msr + z4 ν2 + c4 z4 + z2 − γ3 x6 (x2 u1 + x3 u2 ) (11.63) τs where c3 and c4 are two new random positive real design parameters. Eq. (11.63) suggests that the control signals u1 , u2 must set to zero the two quantities between curly brackets (on the right side of (11.63)). Letting these quantities equal to zero and solving the resulting second-order linear equation system with respect to (u1 , u2 ), gives the following control law:

    ∗ +λ u1 c + c z + λ z + x ˙ μ −1 2 3 3 J 1 1 Tg 1 = (11.64) u2 ν2 + c4 z4 + z2 with  =

λ2 =

λ0 λ2

λ1 λ3

 ;

2Msr γ3 x 6 x 2 , τs

Msr γ3 x 6 x 3 , Ls Msr λ1 = p γ3 x 6 x 2 , Ls 2Msr λ3 = γ3 x 6 x 3 . τs λ0 = −p

(11.65)

Adaptive backstepping controller for DFIG Chapter | 11 249

It is worth noting that the matrix is non-singular. Indeed, it is easily checked M2

that its determinant is D = λ0 λ3 − λ2 λ4 = −2p Ls srτs γ32 x62 (x22 + x32 ) and s =  (x22 + x32 ) never vanish in practice because of the machine nonzero remnant flux and the small variations of x6 with respect to its high nominal. Finally, with supposing the unknown parameters (Tg , J ) be constants, one gets from (11.60) and (11.48), the following parameter adaptive laws:   J˙ˆ = −c1 z32 + z1 (c1 z1 + x˙1∗ ) + z3 (c1 Jˆ − F ) c1 z1 + x˙1∗ , ˙ Tˆg = z1 + (c1 Jˆ − F )z3 . (11.66) The properties of the speed/flux regulator thus designed are described in the following proposition. Proposition 11.1 (Speed regulation). Consider the closed-loop system composed of the doubly fed induction generator (DFIG)-DC/AC inverter association, described by model (11.27)–(11.31), the adaptive backstepping controller defined by the control law (11.64), and the parameter update laws (11.66). Then one has the following properties: • The closed-loop error system undergoes, in the (z1 , z2 , z3 , z4 ) coordinates, the following equations: 1 ), z3 + ζ1 (z1 , z3 , J z˙ 2 = −c2 z2 + z4 , F ), z˙ 3 = −(c3 + )z3 + ζ3 (z1 , z3 , J z˙ 4 = −c4 z4 − z2 , z˙ 1 = −c1 z1 +

(11.67) (11.68) (11.69) (11.70)

which can be given the more compact form ) Z˙ 1 = A1 Z1 + χ(Z1 ,

(11.71)

with Z1 =



z1

z2 ⎡

⎢ ⎢ A1 = ⎢ ⎣

z3 −c1 0 0 0

z4

T

0 −c2 0 −1

,

=



J T g

1/J 0 − (c3 + F /J ) 0

0 1 0 −c4

T ,

(11.72)

⎥ ⎥ ⎥, ⎦

(11.73)



250 Backstepping Control of Nonlinear Dynamical Systems

    = ζ1 Z1 , 0 χ Z1 ,

T   0 , ζ3 Z1 ,

(11.74)

  T g  J˜  = ζ1 Z1 , c1 z1 + x˙1∗ + , (11.75) J J   J˜

T˜g = (c1 Jˆ − F )(c1 z1 + x˙1∗ ) − c1 z3 + (c1 Jˆ − F ) . (11.76) ζ2 Z1 , J J • Let the design parameters c1 and c3 be sufficiently large in the sense that 1 1 F c1 > 2J and c3 > 2J − 2J . Then the above system is stable with respect to T˜ 2

˜2

the Lyapunov function V = 12 z12 + 12 z22 + 12 z32 + 12 z42 + 12 Jg + 12 JJ , and the errors (z1 , z2 , z3 , z4 ) are vanishing, whatever their initial values. Proof. Eqs. (11.67)–(11.68) are immediately obtained from (11.46)–(11.47). Eq. (11.69) is obtained substituting the control law (11.64) and the parameter update law (11.60) to (u1 , u2 ) on the right side of (11.51). Eq. (11.70) is obtained by substituting the control law (11.64) to (u1 , u2 ) on the right side of (11.54). Part 1 is proved. On the other hand, substituting the control law (11.64) to (u1 , u2 ) on the right side of (11.63) yields F 1 V˙ = −c1 z12 − c2 z22 − c3 z32 − c4 z42 − z32 + z1 z3 . J J

(11.77)

Using the inequality |z1 z3 | ≤ 12 z12 + 12 z32 one obtains from (11.77) 1 2 F 1 2 V˙ ≤ −(c1 − )z1 − c2 z22 − (c3 + − )z − c4 z42 . 2J J 2J 3

(11.78)

1 1 F and c3 > 2J − 2J , it follows Using the assumption in Part 2, c1 > 2J from (11.78) that V˙ ≤ 0 whenever the state vector (z1 , z2 , z3 , z4 ) is nonzero. Then, applying Lasalle’s invariant set principle, it follows that V is bounded and (z1 , z2 , z3 , z4 ) converges to zero whatever the initial conditions. Proposition 11.1 is established.

11.4.3 PFC and DC voltage controller The PFC requirement amounts to ensure a sinusoidal output current and the injection of a desired reactive power into the electric network. DC voltage regulation entails the control of the continuous voltage vdc so that it takes a given reference value vdcref . The achievement of these objectives necessitates two control loops. The first one ensures the regulation of the squared DC voltage x6 , and the second ensures the injection of the desired reactive power.

Adaptive backstepping controller for DFIG Chapter | 11 251

11.4.3.1 Controlling rectifier output current to meet PFC The PFC objective means that the reactive power Q = 0 and by consequence the output current of the overall system should be sinusoidal and in phase with the AC supply voltage. The reactive power is given by Q = vsq igd − vsd igq .

(11.79)

As mentioned above, the stator voltage is linked to the d-axis of the frame, vsd = Vg and vsq = 0, then the reactive power becomes Q = −Vg igq . Therefore, one seeks a regulator that enforces the current i gq = i sq + i inq to track a reference signal equal to zero to impose Q = 0 and ig in phase with the voltage supply vg . ∗ As the reference signal i gq is null, it follows that the tracking error z5 = ∗ i gq − i gq undergoes z5 = −x3 − x8 , as x3 = Ls isq + Msr x5 , z5 becomes z5 = −

x3 Msr + x 5 − x8 . Ls Ls

(11.80)

In view of (11.29), (11.31), and (11.34), the above error undergoes the following equation: 1 1 Msr x6 u4 (− x3 − ωs x2 + x5 ) + ω s x7 − Ls τ s τs Lo Msr γ2 + (−γ1 x5 − (ωs − px1 )x4 + x3 + pγ2 x1 x2 + γ3 x6 u2 ). Ls τs

z˙ 5 = −

(11.81)

To get a stabilizing control law for this first-order system, consider the quadratic Lyapunov function V5 = 0.5z52 . It can be easily checked that the time derivative V˙5 is a negative definite function of z5 if the control input u4 x6 is chosen as follows: u4 x6 = c5 Lo z5 + Lo h1 (x),

(11.82)

with c5 > 0 as a design parameter and 1 1 Msr (− x3 − ωs x2 + x5 ) + ω s x7 Ls τ s τs Msr γ2 + (−γ1 x5 − (ωs − px1 )x4 + x3 + pγ2 x1 x2 + γ3 x6 u2 ). Ls τs

h1 (x) = −

(11.83)

11.4.3.2 DC voltage loop The objective of this subsection is to design a control law u3 in order that the ∗ . rectifier output voltage x6 = v dc is steered to a given reference value x6∗ = vdc

252 Backstepping Control of Nonlinear Dynamical Systems ∗ is chosen (not mandatory) set to the nominal value of As mentioned above, vdc the rotor voltage amplitude. Multiply both sides of Eq. (11.32) by 2x6 . The squared voltage (y = x62 ) varies according to the equation:

y˙ =

2 (u3 x7 x6 + u4 x6 x8 − x6 ire ); C

(11.84)

replace the quantities x6 u4 by their value given by Eq. (11.82), (11.84) becomes y˙ =

2 2 u3 x7 x6 + (c5 Lo z5 x8 + Lo h1 (x)x8 − x6 ire ). C C

(11.85)

∗ 2 (squared DC-link As previously mentioned, the reference signal y ∗ = vdc ∗ voltage x6 = vdc ) is chosen to be constant (i.e. y˙ = 0), it is given the nominal value of rotor voltage amplitude. Then it follows from (11.84) that the tracking error z6 = y ∗ − y undergoes the following equation:

z˙ 6 = y˙ ∗ −

2 2 2 2 u3 x7 x6 − c5 Lo z5 x8 + Lo h1 (x)x8 − x6 ire . C C C C

(11.86)

To get a stabilizing control law for the system (11.86), consider the quadratic Lyapunov function V6 = 12 z62 . Deriving V6 along the trajectory of (11.86) yields V˙6 = z˙ 6 z6 .

(11.87)

This suggests that the control input u3 x6 can be chosen as follows: u3 x 6 = c 6 C

z5 + h2 (x) 2x7

(11.88)

with c6 > 0 as a design parameter and h2 (x) =

C ∗ 1 y˙ − (c5 Lo z5 x8 + Lo h1 (x)x8 − x6 ire ). 2x7 x7

(11.89)

11.5 Simulation results and discussions The adaptive backstepping controller, designed in Section 11.4, was evaluated by simulation, within MATLAB/Simulink environment. The simulated system has considering the experimental setup described by Fig. 11.7, using the electromechanical characteristics summarized in Table 11.1, and including the control laws (11.64), (11.82), (11.88), and the parameter adaptive laws (11.66) and (11.60). The experimental protocol is described by Figs. 11.8 to 11.13. The experimental setup conditions are imposed by wind speed and moment of inertia J variations. Fig. 11.8A shows the proposed change from moment of inertia J ,

Adaptive backstepping controller for DFIG Chapter | 11 253

FIGURE 11.7 Control system including AC/DC/AC converters and a doubly fed induction generator.

254 Backstepping Control of Nonlinear Dynamical Systems

TABLE 11.1 System characteristics. Characteristics

Symbol

Value

Unity

Nominal power Stator resistor

Pn

1.5

kW

Rs

1.75



Stator cyclic inductor

Ls

0.295

H

Rotor resistor

Rr

1.68



Ms r

0.195

H

Rotor inertia

J

0.35

Nm/rd/s2

Viscous friction

F

0.026

Nm/rd/s

Number of pole pairs

p

2

Grid inductance

L0

0.01

H

Capacitor

C

47.00

mF

Rotor resistor

Rr

0.40



Grid voltage

Vg

220/380

V

Grid frequency

fn

50.00

Hz

Mutual inductor

FIGURE 11.8 The experimental setup conditions. (A) Rotor inertia variation [kg.m2 ]; (B) Wind speed variation [m/s].

its variation is 25% percent of nominal. The wind speed (Fig. 11.8B) is chosen for the wind turbine being operated in two zones of the level, low and high wind speed. The closed-loop inputs are kept constants sref = 0.56 wb, and vdcref = 200 V. The following values of the controller design parameters have been selected using a ‘try-and-error’ search method and proved to be suitable: c1 = 10,

c2 = 50,

c3 = 100,

c4 = 200,

c5 = 40,

c6 = 15.

The simulation results are depicted in Figs. 11.9 to 11.13. For different values of the wind speed over the time, 6 m/s in [0, 20 s], and 9 m/s in [20, 40 s],

Adaptive backstepping controller for DFIG Chapter | 11 255

FIGURE 11.9 MPPT checking P [W].

FIGURE 11.10 Rotor speed of DFIG ωm [rd/s].

Fig. 11.9 and Fig. 11.10 show that the active power P and rotor speed reference ωmref changes with wind speed variations to meet MPPT requirement. Fig. 11.9 illustrated the performance of MPPT described in Section 11.2; it can be seen that the extracted active power is equal to the maximum power corresponding to wind speed as shown by Fig. 11.2. Figs. 11.10 and 11.11 show that the rotor speed and the stator flux norm, converge perfectly to their respective references, after some variations of mechanical parameters (generator torque Tg and moment of inertia J ) as confirmed by Proposition 11.1. Figs. 11.10 and 11.11 show that the tracking quality is quite satisfactory for both controlled variables (ωm ; s ) as the response time (after each change in speed reference or mechanical parameters) is less than 0.3 s. Fig. 11.12 shows that the DC-link voltage x6 = vdc is well regulated and quickly settles down after each change in the speed reference or mechanical parameters. Fig. 11.13 shows that the grid current in the q-axis and reactive

256 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 11.11 Stator flux norm s and ∗s [wb].

FIGURE 11.12 DC-link voltage vdc [V].

power are null, and then vg1 and ig1 are in the same phase, which guarantees a high quality of the current grid injected into the power grid. Fig. 11.14 shows the generator torque and moment of inertia adaptive law variations corresponding to wind speed and rotor speed changes. The generator torque negative value is explained by the receiver convention used when modeling the doubly fed induction machine. Fig. 11.14B shows, as indicated by Eqs. (11.75)–(11.76), that Tˆg tends to Tg only if the tracking errors tend to zero and the reference rotor speed is constant. However, the adaptive law Jˆ varies as a function of the control errors and does not necessarily converge to J .

11.6 Conclusion In this chapter, the modeling and adaptive backstepping technique control of the wind energy conversion system, based on DFIG and the AC/DC/AC converters

Adaptive backstepping controller for DFIG Chapter | 11 257

FIGURE 11.13 Unity power factor checking. (A) Reactive power Q [Var]; (B) Current iqn [A].

FIGURE 11.14 Mechanical parameter adaptation. (A) Rotor inertia Jˆ [kg.m2 ]; (B) Generator torque Tg (N.m) and Tˆg [N.m].

association, despite mechanical parameter uncertainty or variation, is presented. The system control is described by the nonlinear model (11.27)–(11.34) and by the multi-loop nonlinear controller defined by the control laws (11.64), (11.66), (11.82), and (11.88). A formal analysis using Lyapunov stability is carried out to design the control system performances. In addition to closed-loop global asymptotic stability, it is proven, by simulation, that all control objectives have been successfully achieved. In order to avoid mechanical sensors of speed and torque and to reduce costs and improve the reliability of DFIG control, sensorless control (also called output feedback) can be one of the perspectives of this work. The output feedback control (as El Fadili et al. (2012b, 2014c,a); El Magri et al. (2013a)) even in the presence of mechanical sensors, can implement fault-tolerant control strategies.

258 Backstepping Control of Nonlinear Dynamical Systems

References Abad, G., Lopez, J., Rodriguez, M.A., Marroyo, L., Iwanski, G., 2011. Doubly Fed Induction Machine. IEEE, WILEY. Abdelmalek, S., Azar, A.T., Dib, D., 2018a. A novel actuator fault-tolerant control strategy of DFIGbased wind turbines using Takagi–Sugeno multiple models. International Journal of Control, Automation, and Systems 16 (3), 1415–1424. Abdelmalek, S., Azar, A.T., Rezazi, S., 2018b. An improved robust fault-tolerant trajectory tracking controller (FTTTC). In: 2018 International Conference on Control, Automation and Diagnosis (ICCAD). Marrakech, Morocco. IEEE, pp. 1–6. Abdelmalek, S., Rezazi, S., Azar, A.T., 2017. Sensor faults detection and estimation for a DFIG equipped wind turbine. In: Materials & Energy I (2015). Energy Procedia 139, 3–9. Ben Smida, M., Sakly, A., Vaidyanathan, S., Azar, A.T., 2018. Control-based maximum power point tracking for a grid-connected hybrid renewable energy system optimized by particle swarm optimization. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in System Dynamics and Control. In: Advances in Systems Analysis, Software Engineering, and High Performance Computing (ASASEHPC). IGI Global, pp. 58–89. Cardenas, R., Pena, R., 2004. Sensorless vector control of induction machines for variable-speed wind energy applications. IEEE Transactions on Energy Conversion 19 (1), 196–205. Datta, R., Ranganathan, V., 2003. A method of tracking the peak power points for a variable speed wind energy conversion system. IEEE Transactions on Energy Conversion 18 (1), 163–168. El Fadili, A., Boutahar, S., Dhorhi, I., Stitou, M., Lajouad, R., El Magri, A., Kheddioui, E., 2018. Reference speed optimizer controller for maximum power tracking in wind energy conversion system involving DFIG. In: 2018 Renewable Energies, Power Systems Green Inclusive Economy (REPS-GIE), pp. 1–6. El Fadili, A., Giri, F., El Magri, A., Besançon, G., 2014a. Sensorless induction machine observation with nonlinear magnetic characteristic. International Journal of Adaptive Control and Signal Processing 28 (2), 149–168. El Fadili, A., Giri, F., El Magri, A., Dugard, L., Chaoui, F., 2012a. Adaptive nonlinear control of induction motors through AC/DC/AC converters. Asian Journal of Control 14 (6), 1470–1483. El Fadili, A., Giri, F., El Magri, A., 2013a. Backstepping control for maximum power tracking in single- phase grid-connected photovoltaic systems. IFAC Proceedings Volumes 46 (11), 659–664. El Fadili, A., Giri, F., El Magri, A., 2013b. Control Models for Induction Motors. John Wiley & Sons Ltd, pp. 15–40. El Fadili, A., Giri, F., El Magri, A., 2014b. Reference voltage optimizer for maximum power point tracking in triphase grid-connected photovoltaic systems. International Journal of Electrical Power & Energy Systems 60, 293–301. El Fadili, A., Giri, F., El Magri, A., Dugard, L., 2013c. Nonlinear controller for doubly fed induction motor with bi-directional AC/DC/AC converter. IFAC Proceedings Volumes 46 (11), 134–139. El Fadili, A., Giri, F., El Magri, A., Dugard, L., Ouadi, H., 2011. Induction motor control in presence of magnetic saturation: speed regulation and power factor correction. In: Proceedings of the 2011 American Control Conference, pp. 5406–5411. El Fadili, A., Giri, F., El Magri, A., Lajouad, R., Chaoui, F., 2012b. Output feedback control for induction machine in presence of nonlinear magnetic characteristic. IFAC Proceedings Volumes 45 (21), 600–605. El Fadili, A., Giri, F., El Magri, A., Lajouad, R., Chaoui, F., 2012c. Towards a global control strategy for induction motor: speed regulation, flux optimization and power factor correction. International Journal of Electrical Power & Energy Systems 43 (1), 230–244. El Fadili, A., Giri, F., El Magri, A., Lajouad, R., Chaoui, F.Z., 2014c. Adaptive control strategy with flux reference optimization for sensorless induction motors. Control Engineering Practice 26, 91–106.

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El Fadili, A., Giri, F., Ouadi, H., El Magri, A., Dugard, L., Abouloifa, A., 2010. Induction motor control through AC/DC/AC converters. In: Proceedings of the 2010 American Control Conference, pp. 1755–1760. El Fadili, A., Van Assche, V., El Magri, A., Fouad, G., 2013d. Backstepping controller for DFIM with bidirectional AC/DC/AC converter. In: Giri, F. (Ed.), AC Electric Motors Control. John Wiley & Sons Ltd, pp. 253–274. El Magri, A., Giri, F., Abouloifa, A., El Fadili, A., 2009a. Nonlinear control of associations including wind turbine, PMSG and AC/DC/AC converters. In: 2009 European Control Conference (ECC), pp. 4338–4343. El Magri, A., Giri, F., Abouloifa, A., Lachkar, I., Chaoui, F.Z., 2009b. Nonlinear control of associations including synchronous motors and AC/DC/AC converters: a formal analysis of speed regulation and power factor correction. In: 2009 American Control Conference, pp. 3470–3475. El Magri, A., Giri, F., Besancon, G., El Fadili, A., Dugard, L., Chaoui, F.Z., 2013a. Sensorless adaptive output feedback control of wind energy systems with PMS generators. Control Engineering Practice 21 (4), 530–543. El Magri, A., Giri, F., El Fadili, A., 2013b. Control Models for Synchronous Machines. John Wiley & Sons Ltd, pp. 41–56. Fekik, A., Denoun, H., Azar, A.T., Kamal, N.A., Zaouia, M., Yassa, N., Hamida, M.L., 2020. Direct torque control of three phase asynchronous motor with sensorless speed estimator. In: Hassanien, A.E., Shaalan, K., Tolba, M.F. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019. In: Advances in Intelligent Systems and Computing, vol. 1058. Springer International Publishing, Cham, pp. 243–253. Giri, F., 2013. AC Electric Motors Control: Advanced Design Techniques and Applications. Wiley & sons. Jadhav, H., Roy, R., 2013. A comprehensive review on the grid integration of doubly fed induction generator. International Journal of Electrical Power & Energy Systems 49 (1), 8–18. Kamal, N.A., Azar, A.T., Elbasuony, G.S., Almustafa, K.M., Almakhles, D., 2020. PSO-based adaptive perturb and observe MPPT technique for photovoltaic systems. In: Hassanien, A.E., Shaalan, K., Tolba, M.F. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019. In: Advances in Intelligent Systems and Computing, vol. 1058. Springer International Publishing, Cham, pp. 125–135. Kazmi, R., Goto, H., Hai-Jiao, G., Ichinokura, O., 2011. A novel algorithm for fast and efficient speed-sensorless maximum power point tracking in wind energy conversion systems. IEEE Transactions on Industrial Electronics 58 (1), 29–36. Koutroulis, E., Kalaitzakis, K., 2006. Design of a maximum power tracking system for wind-energyconversion applications. IEEE Transactions on Industrial Electronics 53 (2), 486–494. Krsti´c, M., Kanellakopoulos, I., Kokotovi´c, P., 1995. Nonlinear and Adaptive Control Design, 1st Edition. Wiley & Sons. Lajouad, R., El Magri, A., El Fadili, A., Chaoui, F., Giri, F., Besançon, G., 2013. State feedback control of wind energy conversion system involving squirrel cage induction generator. IFAC Proceedings Volumes 46 (11), 299–304. Lajouad, R., El Magri, A., El Fadili, A., Chaoui, F., Giri, F., 2015. Adaptive nonlinear control of wind energy conversion system involving induction generator. Asian Journal of Control 17 (4), 1934–6093. Lajouad, R., Giri, F., Chaoui, F.Z., El Fadili, A., El Magri, A., 2019. Output feedback control of wind energy conversion system involving a doubly fed induction generator. Asian Journal of Control, 2027–2037. https://doi.org/10.1002/asjc.2116. Lang, Y., Zargari, N., Kouro, S., 2011. Power Conversion and Control of Wind Energy Systems. IEEE, WILEY. Leonard, W., 2001. Control of Electrical Drives, 1st Edition. Springer. Meghni, B., Dib, D., Azar, A.T., 2017. A second-order sliding mode and fuzzy logic control to optimal energy management in wind turbine with battery storage. Neural Computing and Applications 28 (6), 1417–1434.

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Meghni, B., Dib, D., Azar, A.T., Saadoun, A., 2018. Effective supervisory controller to extend optimal energy management in hybrid wind turbine under energy and reliability constraints. International Journal of Dynamics and Control 6 (1), 369–383. Ni, K., Hu, Y., Liu, Y., Gan, C., 2017. Performance analysis of a four-switch three-phase grid-side converter with modulation simplification in a doubly-fed induction generator-based wind turbine (DFIG-WT) with different external disturbances. Energies 10 (6), 706. Rekioua, D., 2014. Wind Power Electric Systems: Modeling, Simulation and Control. Springer. Soetedjo, A., Lomi, A., Mulayanto, W.P., 2011. Modeling of wind energy system with MPPT control. In: Electrical Engineering and Informatics (ICEEI), 2011 International Conference on. IEEE, pp. 1–6.

Chapter 12

Dynamic modeling, identification, and a comparative experimental study on position control of a pneumatic actuator based on Soft Switching and Backstepping–Sliding Mode controllers Amir Salimi Lafmejania , Mehdi Tale Masoulehb , and Ahmad Kalhorb a School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ,

United States, b Human and Robot Interaction Laboratory, School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran

12.1

Introduction

Pneumatic systems play an important role in the industrial and mechatronic applications. In the past two decades, Pneumatic Systems (PneuSys) have been extensively used in many applications such as aerospace (Grewal et al., 2012), automotive and locomotion systems (Liu et al., 2018; Yoshimura and Takagi, 2004), and especially in robotics (Fan et al., 2017; Machiani et al., 2014; Salimi Lafmejani et al., 2017a). In spite of some drawbacks of PneuSys, which mainly originates from nonlinear behavior of the air compressibility, they entail many advantages compared to their counterpart, hydraulic systems, such as cleanness for the working environment, easy maintenance, and rapid movement and reactions (Fu et al., 2018). The nonlinear behavior of a pneumatic system falls into three areas: (1) compressibility of the air, (2) different regimes of air flow rate through the valve, and (3) friction force of the pneumatic actuator as a function of pneumatic actuator’s velocity. Thus, some nonlinear parameters emerge from the aforementioned features in the dynamic model of the system. The nonlinear parameters in a dynamic model of a PneuSys should be identified in order to have an applicable model of the pneumatic system (Salimi LafmeBackstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00019-2 Copyright © 2021 Elsevier Inc. All rights reserved.

261

262 Backstepping Control of Nonlinear Dynamical Systems

jani et al., 2016). Additionally, a highly accurate control of PneuSys would not be possible without having a relatively precise dynamic model of the system. There are a lot of challenges in dynamic modeling and position control of a pneumatic actuator in order to achieve a higher precision in trajectory tracking. One of the challenging issues is modeling friction force of a pneumatic actuator due to its nonlinear behavior. The friction force of a pneumatic actuator shows a dynamic behavior which depends on the velocity of the actuator’s cylinder. Different models have been proposed for modeling friction force in the previous researches. Using simple models for friction force results in more uncertainties in the model of the pneumatic system. On the other hand, more complicated models, e.g. the LuGre model, have unknown parameters which should be identified by performing experimental tests. As another challenge, there is an important issue in control of a pneumatic system consisting a proportional valve without spool feedback and a pneumatic actuator with nonlinear behavior of friction force. Spool position feedback is necessary for measuring the mass flow rate through the electrical valve. Although a servo-valve is equipped with a spool position feedback, it is expensive compared to proportional valves. Thus, by using a proportional valve, the cost of the pneumatic components is reduced, while, the aforementioned challenges are risen during identification of model of the pneumatic system. Moreover, the nonlinear behavior of a pneumatic actuator which has been controlled by nonlinear electrical valve, contributes to big challenges that are unavoidable in order to achieve a high accuracy in trajectory tracking based on uncertainties in the identified model of the pneumatic system. Furthermore, trajectory tracking has been investigated for a certain frequency of sinusoidal desired trajectory in different researches. There is a challenging problem in trajectory tracking of different sinusoidal trajectories by a pneumatic actuator in order to evaluate performance of the employed controllers, which has not been addressed in previous research. The main contribution of this chapter consists of designing a hybrid controller including the BS-SMC for trajectory tracking of a pneumatic actuator which is supplied just by one proportional valve. The LuGre model is taken into account for modeling the friction force behavior of the pneumatic actuator. Moreover, mass flow rate model is derived according to the International Standard Atmosphere (ISA) model of compressible air through chambers of the cylinder which flows from electrical proportional valve. Unknown parameters of the friction force and mass flow rate models are identified by GA. Furthermore, this research provides a comparative study on the experimental application of the two controllers: (1) SSC and (2) BS-SMC. Trajectory tracking control is investigated based on employing the proposed controllers, which aims at obtaining a high accuracy performance in tracking and position control of the pneumatic actuator. The novelties of this study lie in the following: (1) using only a 5/3 proportional directional control valve instead of using two 4/2 or four 2/2 pneumatic valves like similar previous research studies, (2) deriving a precise dynamic

Dynamic modeling, identification, and experimental study Chapter | 12 263

model of the pneumatic system instead of using simple friction force or air flow models which contribute to more uncertainties, (3) comparing two controllers including a simple Soft-Switching Model-free and a nonlinear Backstepping– Sliding Model-based controller for position control of a pneumatic cylinder, (4) presenting two different control strategies in position control of a pneumatic actuator for various requirements such as accuracy of tracking, frequency of motions, velocity of tracking, and the boundary of stability. With respect to the design of a pneumatic system, a 5/3 proportional valve does not occupy a large space compared to two or four pneumatic valves, which costs more if several pneumatic valves would be employed for a specific application. For instance, a 6-DoF pneumatically-actuated Gough–Stewart parallel robot needs six pneumatic systems in order to generate the six degrees of freedom in end-effector of the robot (Salimi Lafmejani et al., 2017b). Assume that each link of the robot should have 4 pneumatic valves, results in a robot with 24 pneumatic valves. Not only would it not be an intelligent and practical design for a pneumatic system, but it also costs more compared to using a proportional 5/3 valve for each of links (Salimi Lafmejani et al., 2018). In addition, the more uncertainties in the derived dynamic model of the pneumatic system, the more complex will be position control of the pneumatic actuator. In previous studies, simple linear models have been suggested for the friction force model. In this chapter, the LuGre is employed for modeling friction force of the pneumatic cylinder in order to derive a more accurate model of the pneumatic system. Sliding Mode controller guarantees robustness of the proposed controller, and stability of the controller is proved by the Backstepping method. There is a big challenge in control of a pneumatic system consisting a proportional valve without spool feedback and a pneumatic actuator with nonlinear behavior of friction force. Thus, the two SSC (Model-free) and BS-SMC (Model-based) controllers are proposed for trajectory tracking control of the pneumatic actuator. The SSC controller is very simple in terms of implementation but does not result in an accurate tracking. On the other hand, the BS-SMC is more complicated but is beneficial for high accuracy applications of trajectory tracking for pneumatic actuators. The remainder of this chapter is organized as follows. The studies in the literature which are related to the dynamic modeling, identification, and control of PneuSys are summarized in Section 12.2. The experimental setup of the PneuSys including electronic and pneumatic equipment are introduced in Section 12.3. In Section 12.4, load dynamic equation and dynamic equations of the proportional valve are expressed which leads to a mathematical representation of the nonlinear system. Moreover, the obtained dynamic equations are verified based on the comparison of the experimental step response test and result from the identified model in Section 12.5. Section 12.6 is devoted to introducing two control strategies from Model-free, i.e., the SSC to the nonlinear Model-based control method, i.e., the BS-SMC. The position control and trajectory tracking are experimentally implemented using the identified model and the proposed control strategies in Section 12.7. In Section 12.8, the results are discussed and

264 Backstepping Control of Nonlinear Dynamical Systems

interpreted for different experimental tests and the efficiency of the two controllers are put into contrast. Finally, the chapter ends with some conclusions and future work.

12.2 Related works There are several studies in the literature which have been conducted in modeling, identification, and control of a pneumatic system. In Sobczyk et al. (2016), the LuGre model has been utilized for modeling of friction force in order to overcome its nonlinear behavior. The GA is among different methods for identification of the unknown parameters of a nonlinear system. In Sharifzadeh et al. (2017), nonlinear parameters of dynamic model of a 3-DoF decoupled parallel robot have been identified based on GA. In Sorli et al. (2010), static and dynamic model of a proportional valve has been derived in which the dynamic model was validated by experimental and simulation results in terms of spool position and flow rate. In Righettini et al. (2013), an alternative and practical approach method has been introduced for characterizing a pneumatic servo valve. They considered the steady state of the cylinder’s motion to measure the sonic regime conductance without using any pressure sensor. Although the nonlinear behavior of PneuSys lead to the aforementioned challenges in modeling and control of such systems, this feature has paved the way for many innovative studies in this field. Various controllers have been proposed for tracking and position control of a pneumatic actuator, including, among others, Fuzzy logic (Nazari and Surgenor, 2016), Nonlinear Proportional-Integral-Derivative (NPID) (Syed Salim et al., 2014), Backstepping–Sliding (Taheri et al., 2014; Salimi Lafmejani et al., 2018), Feedback Linearization (Wang and Gordon, 2012), Robust control (Krupke and Wang, 2015), adaptive approaches (Shang et al., 2016; Li et al., 2015), and Model Predictive Control (MPC) (Bone et al., 2015), have been investigated by researchers in this realm. On the other hand, some hybrid methods, e.g. Fuzzy-PID (Yang et al., 2017), have been employed for tracking trajectory of a pneumatic actuator (Meng et al., 2013). In Asaeikheybari et al. (2017), a method has been proposed based on combination of Neural Networks and evolutionary algorithms for tracking of predetermined trajectories. In Wang et al. (1999), a modified PID controller associated with an acceleration feedback and two nonlinear compensations, namely time-delay minimization and null offset compensation, have been employed in order to solve the time-delay and dead-zone problems in a pneumatic actuator. In Yang et al. (2017), based on dynamic model of a pneumatic system, a fuzzy proportional integral derivative controller has been developed with an asymmetric fuzzy compensator in which the chamber pressure and charge and discharge states have been chosen as inputs of the compensator. In Takosoglu et al. (2009), a fuzzy ProportionalDerivative (PD) controller based on trapezoid type adjusted by the Mamdani model was proposed. The results of the foregoing study revealed that the system is robust to disturbance. A Neural Network (NN) has been used for tuning the

Dynamic modeling, identification, and experimental study Chapter | 12 265

gain of Proportional-Integral (PI) controller as feedback control corresponding to position error and changes in external forces, (Kothapalli and Hassan, 2008). In Arteaga-Pérez et al. (2015), a linear observer has been implemented for a pneumatic system. Besides, they utilized generalized proportional integral approach in order to estimate unmeasured parameters of the system. In Barth et al. (2003), a controller has been proposed based on transforming discontinuous nonlinear switching model of a pneumatically-actuated Pulse Width Modulator (PWM) controlled servo system. Moreover, loop-shaping method was used to increase robustness of their proposed controller, which demonstrates a good performance in tracking. In Pandian et al. (1997), a Sliding Mode observer has been designed to estimate unknown time-variant parameters like temperature and discharge coefficient end to control of a pneumatic actuator. They concluded that combination of Sliding Mode with Feedback Linearization controller shows a better performance than a standard Sliding Mode controller. In Song and Ishida (1997), a robust Sliding Mode controller has been proposed in such a way that the dynamic error converges to zero at infinity. From experimental tests, it has been concluded that the proposed controller, which was applied to a pneumatic system, guarantees the stability of tracking in which the error converges to zero. In Richer and Hurmuzlu (2000), a model-based controller was studied by using Sliding Mode controller. The Sliding Mode controller was designed based on a mathematical model of the pneumatic system which includes electrical valve dynamic where the time-delay due to the tubes was neglected. In Bone and Ning (2007), a Sliding Mode controller was studied for linear and nonlinear plants for a pneumatic system. A Linearized Sliding Mode (LSMC) controller is more robust than a nonlinear Sliding Mode controller (NSMC) if the external load is higher than the nominal value of load. Conversely, the NSMC has more robustness than LSMC if the payload is smaller than its nominal value. Hybrid methods such as SMC with Fuzzy controller have been investigated based on Mamdani type in order to tune the controller gains, (Papoutsidakis et al., 2009). In Tsai and Huang (2008), Multiple Surface Sliding Mode controller (MS-SMC) was designed according to Lyaponov’s stability. By using the latter controller, position tracking by using this controller has some beneficial, such as robustness to uncertainty originates from payload changes. Several researches in the context of nonlinear control of a pneumatic system have been conducted by Karpenko and Sepehri (2006). The authors carried out experimental studies on the proposed dynamical adaptive Backstepping–Sliding mode control for position tracking of a pneumatic cylinder (Rahman et al., 2016). In Estrada and Plestan (2014), a second order sliding mode controller has been designed based on switching gains output feedback for position control of a pneumatic actuator. By employing this method, chattering effect has been reduced in their study. Moreover, the BS-SMC strategy has been employed in order to control and trajectory tracking of a pneumatically-actuated 6-DoF Gough–Stewart parallel robot in Salimi Lafmejani et al. (2018).

266 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 12.1 A general overview of the experimental setup of the under study pneumatic system, electrical and pneumatic equipment, and connections.

12.3

Experimental setup of the PneuSys

Fig. 12.1 demonstrates the experimental electro-pneumatic setup of the pneumatic system, including pneumatic cylinder, directional proportional electrical valve, which is equipped with three pressure sensors, potentiometer as position sensor. The regulator adjusts supply pressure of the pneumatic system measuring by a pressure sensor connected to the input port of the electrical valve. As it shown in Fig. 12.1, two pressure sensors are employed in order to measure the air pressure of both chambers of the pneumatic cylinder. In experimental tests, dP2 1 the time derivative of the air pressure, dP dt and dt , are obtained according to an online derivative calculator. It computes derivative of the air pressure simultaneously over each loop of the program written down in Arduino board. A Kalman filter is implemented in the program aims to obtain smooth and accurate values as far as possible. The proportional valve guides the air flow through chambers of the pneumatic actuator according to the spool position which depends on the input voltage of the valve. Air pressure of the actuator’s chambers are obtained by pressure sensors which are placed in the way of air flow. On the other hand, the pneumatic force enables the actuator to have upward and downward strokes with different velocities. Position of the piston is calculated by a variable resistor in the potentiometer sensor and the derivative of the cylinder’s displacement results in the velocity. The detailed specification of all components are listed in Table 12.1.

Dynamic modeling, identification, and experimental study Chapter | 12 267

TABLE 12.1 Specifications of the under study electro-pneumatic experimental setup components. Component

Model

Manufac.

Actuator

DSNU-20-100-PPV-A

Festo

Electrical valve

MPYE-5-1/8-HF-010-B

Festo

Position sensor

MLO-POT-100-LWG

Festo

Pressure sensor

SED1-D10-G2-H18

Festo

Pressure regulator

LFR-D-MINI

Festo

Accelerometer

IMU GY-80

Sparkfun

12.4

Dynamic modeling of the pneumatic system

Referring to Salimi Lafmejani et al. (2016), the mathematical model of the PneuSys including pneumatic actuator and proportional control valve has been investigated in this section.

12.4.1 Cylinder dynamics Fig. 12.2, demonstrates the free body diagram for the pneumatic actuator’s cylinder in which external forces depicted. According to Fig. 12.2, dynamic equations of the actuator are obtained from the second Newton’s law for upward and downward strokes in Eq. (12.1) as follows: Ma = P1 A1 − P2 A2 − Pa Ar − Mg ∓ Ff

(12.1)

where Ff is modeled based on a prominent friction model, the so-called LuGre model. The LuGre model of friction force was introduced in Fung et al. (2008), as a function of cylinder velocity, Stribeck velocity, static and coulomb friction forces. The LuGre model came into being as a result of the Dahl model modification consisting in adoption of the hysteresis shape coefficient, the Stribeck curve s(v), and viscous damping coefficients δ1 and δ2 : F = δ0 z + δ1 z˙ + δ2 v, δ0|v| z, z˙ = v − s(v) s(v) = Fc + (Fs − Fc ) exp(−

(12.2) v δvs ), vs

where δ0 , δvs indicate asperity stiffness and shape factor of Stribeck curve, respectively. The z, F , Fc , Fs , v, vs stand for state variable interpreted as elastic deformation of surface asperities of adjacent bodies, friction force, static friction, Coulomb friction, sliding velocity, and Stribeck velocity.

268 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 12.2 Free body diagram of the forces applied to the pneumatic actuator’s cylinder.

In this chapter, a simplified LuGre model has been suggested for modeling friction force of the pneumatic cylinder without taking into account the effects of hysteresis, contact pre-sliding, elastic deformation of surface asperities of adjacent bodies and with δ2 = 1. Based on the latter reference, the friction model can be expressed mathematically as Ff = Fc sgn(v) + (Fs − Fc ) exp(−(

v 2 ) )sgn(v) + Bv. vs

(12.3)

Moreover, the Stribeck velocity, vs , viscous coefficient B, static friction, Fs , and coulomb friction, Fc , are unknown parameters which should be identified and then substituted into the dynamic model based on Eq. (12.1).

12.4.2 Pressure dynamics Referring to Najafi et al. (2009), a model for the mass flow rate of a compressible air through chambers of the cylinder was proposed as follows: m(P ˙ u , Pd ) = ⎧ 1 k−1 1 ⎪ 2 12 √ ⎪ ) Pu ( PPdu ) k (1 − ( PPdu ) k ) 2 , ⎨Cf Av ( Rk k−1

Pd Pu

> Pc ,

⎪ k+1 ⎪ ⎩Cf Av √Pu ( k ( 2 ) k−1 ) 12 , R k+1

Pd Pu

< Pc .

T

T

(12.4)

Furthermore, the critical pressure, Pc is determined by Pc = (

k 2 ) k−1 . k+1

(12.5)

Dynamic modeling, identification, and experimental study Chapter | 12 269

TABLE 12.2 Constant parameters in the dynamic model corresponding to the environment conditions of the experiment location. Symbol

Value

Dimension

k

1.4



T

294

K

R

287

J/K.kg

Pa

105

N/m2

According to the standard ISO 6358, specific heat ratio, gas constant, temperature, and atmosphere pressure have been determined based on the environment conditions of the experiment which are listed in Table 12.2. Thus, by substituting value of k in Eq. (12.5), Pc will be 0.528. The relationship between spool displacement and input command voltage described in Kothapalli and Hassan (2008), is calculated as follows: Av = π

Xs2 4



Xs = Cv u

(12.6)

where effective area of the proportional valve, Av is 15 × 10−6 (m2 ). The upstream and downstream is different for upward and downward strokes. Tables 12.3 and 12.4 represent the upstream and downstream pressures at each condition for the upward and downward strokes, respectively. However, temperature of the air through the proportional valve is one of the most crucial parameters in dynamic behavior of the air, the air temperature has been assumed unchanged. In other words, the temperature has a constant value; nevertheless, the uncertainties resulted from the air temperature inaccuracy are compensated in the Sliding Mode controller. Consequently, the derivative of the air temperature is equal to zero. Finally, the dynamic equations of the PneuSys are obtained by Eq. (12.1) and considering the changes of heat ratio in the cylinder, one has 1 (P1 A1 − P2 A2 − Pa Ar − Mg ∓ Ff ), M RT P1 (γi m ˙ i,1 − γo m ˙ o,1 ) − γk ( )V˙1 , P˙1 = V1 V1 RT P 2 P˙2 = (γi m ˙ i,2 − γo m ˙ o,2 ) + γk ( )V˙2 , V2 V2 T = cte.  T˙ = 0, a=

(12.7)

where P˙1 and P˙2 stand for pressure dynamics in the chambers of the cylinder.

270 Backstepping Control of Nonlinear Dynamical Systems

TABLE 12.3 Relative pressure for the upward strokes. Chamber

Upstream

Downstream

Chamber 1

Ps

P1

Chamber 2

P2

Pa

TABLE 12.4 Relative pressure for the downward strokes. Chamber

Upstream

Downstream

Chamber 1

P1

Pa

Chamber 2

Ps

P2

12.4.3 State space representation of the PneuSys Assume that x1 = x, x2 = v, x3 = P1 , and x4 = P2 . By considering the aforementioned equations, and the load and pressure dynamics, and upon rewriting Eq. (12.7), the state space representation of the PneuSys consists in dynamic equations of pneumatic actuator and proportional directional control valve as follows: x˙1 = x2 , 1 x˙2 = (x3 A1 − x4 A2 − Pa Ar − Mg ∓ Ff ), M RT x3 (γi m ˙ i,1 − γo m ˙ o,1 ) − γk ( )V˙1 , x˙3 = V1 V1 RT x4 (γi m ˙ i,2 − γo m ˙ o,2 ) + γk ( )V˙2 , x˙4 = V2 V2

(12.8)

where the sign of double-signed Ff is determined based on the stroke whether velocity of the cylinder is positive or negative. The upper sign is related to the positive velocity or upward strokes and the lower sign refers to the negative velocity or downward strokes. In practice, position of the piston, x1 , gathering real-time, is observed by potentiometer sensor, and x2 stands for velocity which is the derivative of the displacement of the piston. A gaussian filter is employed to remove noise on velocity, which is originated from derivation procedure, and leads to have a smooth velocity data. Furthermore, P1 and P2 are observed by pressure sensors.

12.5 GA-based identification of the PneuSys and validation This section is devoted to describe the GA-based identification of the unknown parameters using and the validation of the identified parameters.

Dynamic modeling, identification, and experimental study Chapter | 12 271

TABLE 12.5 GA parameters for identification of the unknown parameters. Parameter

Value

Description

nVar

7

number of variables

VarMin

0

minimum of variables

1

maximum of variables

VarMax MaxIt

1000

maximum iteration

nPop

100

number of populations

pc

0.8

crossover percentage

pm

0.3

mutation percentage

mu

0.02

mutation rate

12.5.1 Identification of the unknown parameters For the purpose of this study, GA is applied in order to obtain the unknown parameters by minimizing the Normalized Root Mean Square Error (NRMSE) criterion as the objective function. The GA parameters for identification of the unknown variables are included in Table 12.5. The number of variables is determined according to the number of unknown parameters. All the solutions are proposed by GA in the range of [0 1], or between the minimum of variables (VarMin) and maximum of variables (VarMax) for each unknown variable. According to different range of variations of the variables, the proposed solutions between 0 and 1 are mapped to the corresponding feasible range of them, which are predefined in the MATLAB® program. Furthermore, stop condition of the GA is based on the value of maximum iteration (MaxIt). Number of population (nPop) is adjusted to the value by which the GA achieve to optimal solutions in a short time. Moreover, the proposed solutions are normalized according to their feasible ranges and the cost function is defined based on minimizing the NRMSE criterion. In applying the evolutionary-based identification approach, without losing generality, it is assumed that the viscous coefficient has a constant value. The friction force is modeled according to the positive and negative velocities which are related to the upward and downward strokes, respectively. Furthermore, three parameters γ0 , γi , and γk for pressure dynamic equations are identified based on the evolutionary method. Eventually, the unknown parameters are calculated according to the obtained experimental data, gathered from the potentiometer and pressure sensors, so that the dynamic equations of the PneuSys should be satisfied. The identification of unknown parameters of dynamic equations of the PneuSys, which is performed based on the results of different step response test, are listed in Table 12.6.

272 Backstepping Control of Nonlinear Dynamical Systems

TABLE 12.6 The identified parameters of the dynamic equations of the PneuSys using GA method. Parameter

Dimension

Up. stroke

Down. stroke

vs

m/s

0.5641

-0.2894

B

N.s/m

34.4445

-3.2342

Fs

N

86.6540

-91.9421

Fc

N

49.1371

-78.8793 10.0013

γi



1.78 × 10−14

γo



4.6263

4.2568

γk



0.1440

0.1965

12.5.2 Validation of the identified dynamic model In this section, the identified model is verified by considering a specific step response of the PneuSys. The verification procedure is performed to the end of showing the accuracy of the achieved model by comparing the results of the step response for identified model and experimental test. Acceleration is obtained by substituting friction force which has been identified by applying the so-called LuGre model in Eq. (12.3) into the dynamic model of the cylinder. Figs. 12.3, 12.4, and 12.5 depict the comparison between acceleration values obtained from the identified model and the experimental results for the upward and downward strokes, respectively. The velocity and position of the actuator can be calculated by integration of the obtained acceleration. On the other hand, dynamics of the pressures 1 and 2 resulted from the identified model comparing to experimental results are demonstrated in Figs. 12.6, 12.7, and 12.8. The sign of dP1 is related to the motion of the pneumatic cylinder during the downward strokes in which the inlet air pressure is P2 and the outlet pressure is P1 . During the downward motion, the pressure sensor measure P1 as the outlet pressure to the exhaust of the proportional valve. The pneumatic cylinder stops in 0.2 (sec). After it stops by the mechanical constraint at the end of the stroke, the transition to the complete stop takes place; thus, the derivative of the air pressure reaches to zero and due to the increasing force behind the cylinder from pressure P2 , the air pressure P1 is trapped in the tube which leads to increase the air pressure through 1 the pressure sensor in a reverse direction. Thus, the sign of dP dt is negative when the pneumatic cylinder is moving and the positive sign shows the transition to a complete stop. The latter arguments can be provided for upward motions. According to nonlinear behaviors generated from the friction force nonlinearity, and also compressibility of the air, deriving an accurate model of the pneumatic system is a big challenge. The LuGre model for the friction force and the ISA model of compressible air through the chambers of the pneumatic cylinder are employed in order to derive the dynamic model more precise. Nevertheless, the large differences between the derived model and experimental tests which are reported in Figs. 12.3, 12.4, 12.5, 12.6, 12.7, 12.8 are linked

Dynamic modeling, identification, and experimental study Chapter | 12 273

FIGURE 12.3 Comparison of acceleration obtained by the identified model and the experimental test for the upward stroke.

FIGURE 12.4 Comparison of pressure 1 variations obtained by the identified model and the experimental test for the upward stroke.

FIGURE 12.5 Comparison of pressure 2 variations obtained by the identified model and the experimental test for the upward stroke.

to (1) measurement errors of the pressure and potentiometer sensors, (2) uncertainties of constant parameters in dynamic model of the pneumatic system, (3) simplifications in dynamic behavior of the pneumatic system, e.g. temperature has been assumed constant during the experimental tests. However, the authors have not claimed that the derived model is an exact model, the uncertainties of the dynamic behavior of the pneumatic system have been compensated by the proposed nonlinear Backstepping–Sliding Mode controller. The difference

274 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 12.6 Comparison of acceleration obtained by the identified model and the experimental test for the downward stroke.

FIGURE 12.7 Comparison of pressure 1 variations obtained by the identified model and the experimental test for the downward stroke.

FIGURE 12.8 Comparison of pressure 2 variations obtained by the identified model and the experimental test for the downward stroke.

between the values of γ for upward and downward strokes is due to direction of the pneumatic cylinder during the experimental tests. Since, the setup is the pneumatic system of each link of a 6-DoF Gough–Stewart robot, the pneumatic cylinder has been installed and fixed to a table vertically, (Salimi Lafmejani et al., 2016, 2017b). However, it has been tested horizontally in all similar previous researches, which eases identification, dynamic modeling, and control of motions based on the symmetrical motion of the pneumatic cylinder rather than an asymmetric position control. As shown in Fig. 12.2 and Eq. (12.1), the verti-

Dynamic modeling, identification, and experimental study Chapter | 12 275

cal force of weight has been considered in dynamic equations of the pneumatic cylinder. The asymmetric motion of the cylinder leads one to measure different γ s in the identification procedure.

12.6

Proposed controllers; Model-free and Model-based controllers

In this section, the SSC and the BS-SMC controllers are designed, respectively, as a Model-free and a Model-based control strategies.

12.6.1 Model-free; Soft Switching controller Tangent hyperbolic function, based on its smooth behavior, is a function which is used to design the SSC. Moreover, relation between input control signal, u, and error, e, is defined by u = τ0 + τ1 tanh(τ2 e)

(12.9)

where τ1 and τ2 are adjusting parameters of the SSC. By adjusting τ1 , the range of input control signal can be controllable. On the other hand, by adjusting τ2 , sensitivity of the input control signal to variation of error will be differed. Thus, adjusting the two parameters are very critical in decreasing the tracking error. Regarding the model-free controller, adjusting the parameters should be performed by trial and error tests. Thus, after some practical experiments, τ1 and τ2 are adjusted according to Table 12.7. TABLE 12.7 Adjusted parameter of the SSC controller. Parameter

Adjusted value

τ0

5

τ1

5

τ2

40

12.6.2 Model-based; Backstepping–Sliding Mode controller Referring to Eq. (12.8), the state space representation of the BS-SMC is proposed for the trajectory tracking purpose of the pneumatic actuator. By taking into consideration the state space of the PneuSys according to Eq. (12.8), the state space equations include four state variables, X = [x v P1 P2 ] = [x1 x2 x3 x4 ], which stand for the position, velocity, pressure of chambers 1 and 2, respectively. By rewriting Eq. (12.8) and defining C as follows: x2 2

C = −Pa Ar − Mg − [Fc sgn(x2 ) + (Fs − Fc )e−( vs ) sgn(x2 )]

(12.10)

276 Backstepping Control of Nonlinear Dynamical Systems

where C indicates constant terms of the load dynamic equation. Thus, the state space representation of the pneumatic system Eq. (12.8) can be written down as follows: x˙1 = x2 , x˙2 = x3 A1 − x4 A2 − Bx2 + C, x˙3 = F1 (X, u), x˙4 = F2 (X, u),

(12.11)

y = x1 , in which F1 and F2 are the right side of the third and fourth equations of Eq. (12.8), respectively. Assume that φ consists of state variables corresponding to dynamic equations of proportional valve, x3 and x4 . Thus, φ can be obtained by φ = x3 A1 − x4 A2 .

(12.12)

Eventually, the load dynamic equations are as follows: x˙1 = x2 , x˙2 = φ − Bx2 + C.

(12.13)

x¨1 = φ − B x˙1 + C.

(12.14)

Thus one has

For the sake of applying the concept of SMC, the sliding surface is defined as follows: S = (x˙1 − x˙d,1 ) + λ(x1 − xd,1 ) = e˙x + λex

(12.15)

where ex and e˙x indicate the position error and variation of position error, respectively. The derivative of sliding surface with respect to the time can be calculated as follows: S˙ = e¨x + λe˙x = x¨1 − x¨d,1 + λe˙x = φ − B x˙1 + C − x¨d,1 + λe˙x .

(12.16)

Assuming that C and B are unknown parameters which are estimated by Cˆ ˆ respectively. Thus, it can be assumed that and B, ˆ < C, ¯ |C − C| ˆ < B. ¯ |B − B|

(12.17)

Dynamic modeling, identification, and experimental study Chapter | 12 277

By considering φ, φ = φˆ − Ksgn(S)

(12.18)

φˆ = argφ (S˙ = 0|B=B,C= ˆ Cˆ )

(12.19)

where K > 0. Also,

By rewriting Eq. (12.18), one has φˆ − Bˆ x˙1 + Cˆ − x¨1,d + λe˙x = 0.

(12.20)

φˆ = Bˆ x˙1 − Cˆ + x¨1,d − λe˙x .

(12.21)

Finally,

By substituting φˆ into Eq. (12.18), φ = Bˆ x˙1 − Cˆ + x¨d − λe˙x − Ksgn(S).

(12.22)

The Lyaponov function is defined as follows: 1 Vs = S 2 . 2

(12.23)

By calculating the derivative of Vs with respect to the time, one has V˙s = S S˙ = S(φ − B x˙1 + C − x¨1,d + λe˙x ) = S(Bˆ x˙1 − Cˆ − B x˙1 − Ksgn(S) + C)

(12.24)

ˆ − Ksgn(S)). = S((Bˆ − B)x˙1 + (C − C) Finally, V˙ can be summarized by ˆ − K|S|. V˙ = S(Bˆ − B)x˙1 + (C − C)S

(12.25)

According to Eq. (12.17), K is defined as follows: ¯ ¯ x˙ + C, K = η + Bγ

η > 0,

(12.26)

where K is the gain of sliding surface which should be designed in order that V˙ become negative in Eq. (12.30). According to the so-called Schwartz inequality and regarding Eq. (12.17), it can be concluded that ˆ x˙1 | ≤ |S|Bγ ¯ x˙ , S(Bˆ − B)x˙1 ≤ |S||B − B|| ˆ ≤ |S||C − C| ˆ ≤ |S|C, ¯ S(C − C)

(12.27)

278 Backstepping Control of Nonlinear Dynamical Systems

where γx˙ is the upper bound of the following term: A1 F1 (X, u) − A2 F2 (X, u) = −γx˙ |x|. ˙

(12.28)

In other words, the parameter γx stands for the value for which the following inequality reaches its maximum value: ˆ x˙1 | ≤ |S|Bγ ¯ x˙ . S(Bˆ − B)x˙1 ≤ |S||B − B||

(12.29)

Thus, V˙ can be expressed as follows: ˆ x˙ + C¯ − K)|S|. V˙ ≤ (Bγ

(12.30)

V˙s ≤ −η|S|.

(12.31)

Thus, As a result, S will be converged to zero in a limited time T where T ≤ η1 . By defining z according to ˆ z = x3 A1 − x4 A2 − φ.

(12.32)

The Lyaponov function for the main state space system is defined by 1 W = Vs + z2 . 2

(12.33)

By taking the derivative of the Lyaponov function with respect to the time one has W˙ = S S˙ + z˙z ˆ − K × sgn(S)) = S((Bˆ − B)x˙1 + (C − C)

(12.34)

ˆ˙ + z(A1 F1 (X, u) − A2 F2 (X, u) − φ). Finally, it leads to the following equation in which the control strategy can be computed once it is solved for u: ˆ˙ = −βz (A1 F1 (X, u) − A2 F2 (X, u) − φ)

(12.35)

where β is a positive constant parameter. Thus, W˙ is negative semi-definite (W ≤ 0), and W converges to zero. Consequently, S→0 z→0

⇒ ⇒

x1 → x1,d , x3 , x4 → bounded.

(12.36)

In other words, by converging S to zero, it can be concluded that x1 converges to its desired value. On the other hand, when z converges to zero, internal states of the system, x3 and x4 will be bounded.

Dynamic modeling, identification, and experimental study Chapter | 12 279

By solving Eq. (12.35), the control signal will be obtained. According to Meng et al. (2011), the relationship between input voltage, u, and spool displacement, Xspool of the 5/3-way MPYE5-1/8-HF-010B Festo proportional control valve, can be expressed by a linear relation as follows: Xspool = au + b

(12.37)

where a = −0.3 × 10−3 and b = 1.5 × 10−3 . By considering Eqs. (12.37), it can be concluded that the mass flow rate, m, is related to the input signal u by a second order polynomial function: π m(P ˙ u , Pd ) = Hi (Pu , Pd ) (a × u + b)2 , 4

i = 1, 2,

(12.38)

where Hi (Pu , Pd ) indicates the terms of mass flow rate equation which is related to upstream and downstream pressures for chambers 1 and 2. Finally, by substituting H and Xspool and rewriting Backstepping–Sliding Mode controller strategy in Eq. (12.35), leads to the control strategy for trajectory tracking of the PneuSys as follows:   A1 γk Vx31 V˙1 − A2 γk Vx42 V˙2 − β + φ ∗ b 1 u=  (12.39) − π A1 RT γi H1 a a ( − A2 RT γo H2 ) 4

V1

V2

where u is the control signal. It should be noted that the denominator of the control signal will never be equal to zero because H1 = H2 for all the values of upstream and downstream pressures. In summary, the Model-free controller, SSC, acts as a P controller with a dynamic gain and shows a tangent hyperbolic behavior. The parameters of the SSC are adjusted empirically. On the other hand, the Model-based controller, BS-SMC, guarantees robustness of the proposed controller, and stability of the controller have been proved by Backstepping method.

12.7

Experimental results

In this section, the BS-SMC and the SSC controllers which are described and designed in the latter sections, are implemented practically for tracking the desired trajectories defined in Table 12.8. Sinusoidal trajectories with different frequencies are experimented in order to examine the accuracy of the controllers. A sinusoidal trajectory is defined by x = xref + sin(wt)

(12.40)

where xref , w, and t indicate reference position of the sinusoidal trajectory, frequency of the sinus function, and time, respectively. The experimental tests include different frequencies from 0.1 Hz to 0.5 Hz of a sinusoidal trajectory

280 Backstepping Control of Nonlinear Dynamical Systems

TABLE 12.8 Desired sinusoidal trajectories for the experimental test. Number

Frequency (Hz)

Trajectory (m)

1

0.1

0.05 + 0.03 sin(0.1 ∗ 2π t)

2

0.2

0.05 + 0.03 sin(0.2 ∗ 2π t)

3

0.3

0.05 + 0.03 sin(0.3 ∗ 2π t)

4

0.4

0.05 + 0.03 sin(0.4 ∗ 2π t)

5

0.5

0.05 + 0.03 sin(0.5 ∗ 2π t)

for the pneumatic actuator which is more applicable for industrial applications. The xref is determined in the middle of the pneumatic stroke length. Since, the pneumatic actuator’s stroke is 10 cm, xref would be 5 cm. The prescribed sinusoidal trajectories are listed in Table 12.8. The maximum amplitude of the sinusoidal trajectories is 3 cm and the bias value of the sinusoidal trajectory is 5 cm. In other words, the pneumatic actuator fluctuate around the height of 5 cm with the maximum positive and negative displacement of 3 cm. According to the prescribed trajectories given in Table 12.8, different frequencies of the desired sinusoidal trajectory are experimented aiming at determining the performance of the two proposed controllers. The sinusoidal trajectories given in Table 12.8, are considered as desired trajectories for tracking procedures based on implementation of the two proposed control strategies. Fig. 12.9 depicts the results of applying the Model-free controller for the desired sinusoidal trajectories with different frequencies. In turn, Fig. 12.10 represents the results of applying the nonlinear Backstepping–Sliding Mode controller for the desired sinusoidal trajectories with different frequencies.

12.8 Discussion In this section, the experimental results are interpreted for the implementation of the two proposed controllers for tracking of different desired sinusoidal trajectories. The bigger distortion in the first cycle is based on the fact that the pneumatic actuator is tried to conquer static friction at the zero velocity. According the bigger amount of static friction as compared to dynamic frictions, a bigger force is necessary for the initial motion. Hence, based on the input control signal, the electrical control valve allows a larger amount of the air to flow toward the pneumatic actuator. Consequently, the actuator starts to move with a higher velocity which is difficult to control and the bigger distortions are happened. The asymmetrical trajectory tracking is due to the Soft Switching controller’s symmetric acting which is based on tangent hyperbolic function Tanh(.), whereas the pneumatic actuator is not symmetric in terms of its mechanical structure and direction of motion in a sinusoidal trajectory. In other words, however, the desired trajectories are symmetric, the asymmetric pneumatic actuator needs asymmetric control strategies for upward and downward strokes

Dynamic modeling, identification, and experimental study Chapter | 12 281

FIGURE 12.9 Comparison of the desired trajectory for different frequencies and trajectory tracking of the SSC. (A) 0.1 (Hz); (B) 0.2 (Hz); (C) 0.3 (Hz); (D) 0.4 (Hz); (E) 0.5 (Hz).

282 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 12.9 (continued)

TABLE 12.9 Comparing Backstepping–Sliding Mode and Soft Switching controllers for trajectory tracking. Controller Soft Switching Backstepping–Sliding Mode

Type

RMSE

BW

MAE

Model-free

0.534

0.3 (Hz)

8 ∼ 30 (mm)

Model-based

0.126

1 (Hz)

2 ∼ 6 (mm)

based on different area of the two sides of the cylinder, A1 and A2 , and the different effects of gravity force in upward and downward strokes. Efficiencies of the designed SSC and BS-SM controllers for trajectory tracking are compared based on three main criteria: (1) Root Mean Square Error (RMSE) (2) Bandwidth (BW) (3) Maximum Absolute Error (MAE) The first criterion, RMSE, is very crucial for trajectory tracking purposes, by which accuracy of tracking procedure and performance of implemented control strategies can be determined. The second criterion is based on maximum frequency of motions which can be handled by the designed controller and the system remains in stability circumstance. The third criterion determines how a designed controller can compensate uncertainty of the system by avoiding large errors from desired trajectory and ability of the controller to remove spikes from the generated motion. Furthermore, RMSE, BW, and MAE values for each controller are reported in Table 12.9. The value of MAE is between 0.8 cm to 3 cm for Soft-Switching controller and between 0.2 cm to 0.6 cm for Backstepping–Sliding Mode controller. The amount of error would be acceptable by taking into account the nonlinear behavior of this kind of pneumatic setup. The pneumatic system setup consists of only a 5/3 proportional valve in order to control a pneumatic actuator, which resulted in challenges in modeling and control of this system. On the other hand, two 4/2 or four 2/2 pneumatic valves have been utilized to control of the air through a proportional valve for position control of a pneu-

Dynamic modeling, identification, and experimental study Chapter | 12 283

FIGURE 12.10 Comparison of the desired trajectory for different frequencies and trajectory tracking of the BS-SMC. (A) 0.1 (Hz); (B) 0.2 (Hz); (C) 0.3 (Hz); (D) 0.4 (Hz); (E) 0.5 (Hz).

284 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 12.10 (continued)

matic actuator in similar research studies. In this way, position control of a pneumatic actuator would not be a difficult problem because both chambers of the pneumatic cylinder behave dependently. Using a 5/3 proportional valve has some advantages such as decreasing costs of the pneumatic system, and lessening occupied space of the system. But it leads to the above-mentioned challenges in modeling and control due to the dependency of the air pressure, P1 and P2 through the proportional valve. As a result, the value of the reported maximum absolute error will be satisfying. The maximum error is, approximately, 0.8 cm to 3 cm for Soft-Switching controller; however, by implementing the Backstepping–Sliding Mode controller the error is diminished to the range of 0.2 to 0.6 cm. In Fawaz and Aziz (2016), the pneumatic setup consists of four 2/2 digital pneumatic valves. The trajectory tracking has been performed for a 0.1 Hz sinusoidal trajectories in which tracking error is 1.5 mm. While in this chapter, the experimental tests are performed for different sinusoidal trajectories between 0.1 Hz and 0.5 Hz. As compared to the aforementioned reference, for instance, RMSE of tracking is 0.089 cm for a sinusoidal trajectory of 0.1 Hz. Furthermore, only one 5/3 proportional valve is employed for the experimental tests. This difference contributes to serious challenges in the control procedure of the pneumatic actuator. Since using a proportional valve the dynamic behavior of air through the two chambers of a proportional valve affect each other, the dynamic behaviors through the chambers are not independent. Thus, the equations related to the air pressure dynamics, P˙1 and P˙2 , are mutually dependent. Figs. 12.11A and 12.11B demonstrate the trajectory tracking error for the 0.5 Hz desired sinusoidal trajectory which are represented for comparing performance of the Soft Switching and Backstepping–Sliding Mode controllers. Moreover, the input control signal of 0.5 Hz sinusoidal tracking for Soft Switching and Backstepping–Sliding Mode controllers are shown in Figs. 12.12A and 12.12B, respectively. According to the aforementioned criteria, the trajectory tracking problem by applying the two controllers are compared based on their performance in Table 12.9. The MAE of the position control of the pneu-

Dynamic modeling, identification, and experimental study Chapter | 12 285

FIGURE 12.11 Tracking error of the 0.5 (Hz) sinusoidal desired trajectory for (A) Soft Switching and (B) Backstepping–Sliding Mode controllers.

FIGURE 12.12 Control signal in trajectory tracking of 0.5 (Hz) sinusoidal desired trajectory for (A) Soft Switching and (B) Backstepping–Sliding Mode controllers.

286 Backstepping Control of Nonlinear Dynamical Systems

matic actuator is 2 to 6 mm for different frequencies. As shown in Fig. 12.11, however, it is so difficult for a pneumatic system; the tracking error by implementing the Backstepping–Sliding Mode controller has not exceeded 7 mm for a sinusoidal desired trajectory with frequency of 0.5 Hz. As a result, by comparing performance of Backstepping–Sliding Mode and Soft Switching controllers, nonlinear model-based controller has a better performance in comparison to the Soft Switching controller. Tracking error, from 0.8 to 3 cm for different desired sinusoidal trajectory of 0.1 to 1 Hz, has been reported by applying the Model-free controller is larger than, 0.2 to 0.6 cm for different desired sinusoidal trajectory of 0.1 to 1 Hz, which has been measured by applying Backstepping– Sliding Mode controller. The BS-SMC benefits from Sliding Mode controller which guarantees robustness of the closed-loop system in such a way that it could conquer the uncertainty of the dynamic model of the system.

12.9 Conclusion In this chapter, dynamic equations of a pneumatic system comprising a pneumatic actuator and a proportional electrical valve were obtained to the end of deriving the state space representation of the system. Friction forces of the pneumatic cylinder were modeled based on the LuGre model instead of using a simple model which led to a more accurate model. The unknown parameters of the extracted model were identified by using GA. The identified model was validated by step response experimental tests which revealed the accuracy of the identified model. A Backstepping–Sliding Mode (Model-based) and a Soft Switching (Model-free) controller were designed for trajectory tracking purpose of the pneumatic actuator. Moreover, sinusoidal trajectories with different frequencies from 0.1 to 1 Hz with 5 cm amplitude were considered as desired trajectories. Three different criteria, Root Mean Square Error, Maximum Absolute Error and frequency Bandwidth, were considered to evaluate the performance of the two proposed controllers for trajectory tracking purposes. From the experimental results, it can be inferred that in spite of easy and fast implementation of the Soft Switching Control strategy, the proposed Backstepping–Sliding Mode control strategy led to a better performance in point of view of all the aforementioned criteria in trajectory tracking for different frequencies. As a result, the Backstepping–Sliding Mode controller can be employed for trajectory tracking applications up to 1 Hz frequencies with maximum absolute error lesser than 0.8 cm and high-precision tracking. As future work, the proposed BS-SMC strategy can be used for trajectory tracking control of a pneumatically-actuated Gough–Stewart parallel robot. The accuracy the proposed dynamic model of the pneumatic system and performance of the proposed controller can be evaluated in different experimental tests.

Dynamic modeling, identification, and experimental study Chapter | 12 287

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Chapter 13

Optimal adaptive backstepping control for chaos synchronization of nonlinear dynamical systems Mohsen Alimia , Ahmed Rhifb , Abdelwaheb Rebaic , Sundarapandian Vaidyanathand , and Ahmad Taher Azare,f a University of Kairouan, Kairouan, Tunisia, b University of Carthage, La Marsa, Tunisia, c University of Sfax, Sfax, Tunisia, d Research and Development Centre, Vel Tech University,

Chennai, Tamil Nadu, India, e Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, f Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

13.1

Introduction

Since the appearance of the deterministic chaos concept proposed by Poincaré (1890), Ruelle and Takens (1971), Devaney (1987) as a complex behavior that causes many patterns with strange vicissitudes to come from the internal structure of the dynamical system, the term chaos still remains difficult to define precisely in a scientific and rigorous way, despite the practice of the modern scientific method. Loosely, some people think that chaos, is just a fancy word for instability or degradation in performance system, and it should be eliminated in many cases. While in the absence of a single, definitive and universally accepted definition, almost everyone would agree, more formally, on the chaos definition proposed by Strogatz (1994) as an aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions, which has been discovered to be satisfying two other basic properties: boundedness and infinite recurrence. After that, chaos has been developed over time as a descriptive theory of qualitative study for unstable aperiodic behavior in deterministic nonlinear dynamical systems, which was mentioned in Vaidyanathan and Volos (2016b). But, studying chaotic behavior was considered, for a long time, as a very difficult subject, and generally more complex in that it is effectively impossible to control, not only because of the presence of a virtual strange phenomenon produced with many dynamic evolving endogenous structures, but Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00020-9 Copyright © 2021 Elsevier Inc. All rights reserved.

291

292 Backstepping Control of Nonlinear Dynamical Systems

also because of the uncertainty arising mainly from its unpredictability, see Takens (1981), Grassberger and Procaccia (1983). In the last decades, after the emergence of this concept and its fully diffusion into the scientific literature, that perception has been completely changed and clearly the chaotic behavior has become completely controllable as indicated by Yagasaki and Uozumi (1998) and recently confirmed by Vaidyanathan (2013a), Vaidyanathan et al. (2014). Therefore, Boccaletti et al. (2002), Andrievsky and Fradkov (2003) have estimated that the control of a nonlinear chaotic system will be possible, using many approaches and techniques via various analytical and numerical investigations. Besides, the main controllability conditions for the control of chaotic dynamics system are discussed by Chen (1997). This was confirmed currently as well in Andrievsky and Fradkov (2004), either by reducing or, on the contrary, by increasing its degree of chaoticity. This development has changed the scientific examination of chaos theory in profound ways, more specifically about the synchronization theory of nonlinear dynamical systems. When one chaotic system drives the other, both chaotic systems are synchronized, and must be controlled. Because of high sensitivity to initial condition, they may have exponentially diverging state trajectories. So that, due to the intense interdependency among control and synchronization, the integration of the chaos synchronization problem in the framework of control theory has become, currently, the most important open research problem in the chaos theory study. In addition, the most current research on synchronization focuses on how to study the consensual issue of chaotic synchronization in many coupling or forcing schemes for two identical or non-identical chaotic systems, which were originally founding paper of Pecora and Carroll (1990). In fact, these authors have defended the idea of controlling the response signal by using the control system output signal until both signals types be synchronized. Then, this was followed by the discovery of many different new types of complex systems synchronization, such as chaotic and hyperchaotic systems. Hence, the field of chaotic synchronization has grown significantly since its appearance in 1990s, to currently cover a lot of the other popular areas of highly sophisticated chaotic complex system modeling, such as fractional order chaotic systems. In this context, Alimi et al. (2018) stated that synchronization of two fractional order chaotic systems in the master–slave topology may be performed using different analysis methods such as recurrence plot and cross recurrence plot. Furthermore, the synchronization problems inherent in coupled chaotic dynamical systems have been widely gaining increasing attention in control domain, by reference to Kurths et al. (2003). Since then, the synchronization of dynamic systems with complex behaviors such as hidden oscillators and chaotic and hyperchaotic systems has attracted increasing interest by the emergence of numerous fields of applications or potential investigations in different scientific and engineering areas such as physics (Li et al., 2007), chemistry (Vaidyanathan, 2015a), electronics (Vaidyanathan et al., 2015b), information processing and secure communication (Kocarev and Parlitz, 1995; Andrievsky and Fradkov, 2000;

Optimal adaptive backstepping control Chapter | 13 293

Feki, 2003), biology (Kyriazis, 1991; Vaidyanathan, 2015b), ecology (Sprott et al., 2005), and economics (Guégan, 2009; Caraiani, 2013; Idowu et al., 2018). Added to the practical importance, Lee and Markus (1976), Piccardi and Ghezzi (1997) have mentioned that in theory this control scheme aims to develop applied models that achieve these theoretical dimensions. It is, therefore, attractive for two reasons. On the one hand, as nonlinear chaotic systems have an unlimited number of unstable periodic orbits, one dynamical system can thus include a wide range of controlled behaviors. On the other hand, since these schemes are based primarily on very general characteristics of chaotic dynamics, they are, consequently, applicable to a wide range of seemingly unrelated dynamical systems. According to Liao and Yu (2006), the tasks of control of chaotic systems can be divided into two steps. First, it is to eliminate the chaotic behavior after its properly detection. Second, it is necessary to stabilize the system at one of its equilibrium points. In this interesting framework of analysis, the control performance improvement of chaos synchronization has become a very important goal. As a result, many effective and intelligent methods have been developed in order to solve the problem of how to ameliorate the control of chaos synchronization of nonlinear dynamical systems. In the last few years, specially, some numerically efficient, active, robust, and accurate solutions are given to verify the control and synchronization performance of nonlinear dynamical systems with or without external excitation. But, to achieve this strategy, these basic instruments make the controllers very complex for many application cases. Moreover, due to the presence of explosive instability that can affect the required performance of these control systems during the transitions of the estimated parameters, it is essential to proceed by the adaptive control. Many recent analyzes show that the adaptive control method has been regarded as the most exciting and potentially ground-breaking research topics. According to Bernardo (1996), Liao and Lin (1999), Ge et al. (2000), Wang and Fan (2015), Hua and Guan (2004), Yu and Zhang (2004), Yassen (2006), Wu and Zhang (2009), Vaidyanathan (2012a), Vaidyanathan (2013b), Vaidyanathan (2013c), Vaidyanathan and Madhavan (2013), Vaidyanathan (2016a), because it does not need fixed parameters in advance, but modified online. In this context, to improve the performance of chaos control, the backstepping technique was systematically developed in 1990s firstly by Peter V. Kokotovi´c and others. This methodology consists of organizing a global system into several subsystems, provided that the degree of each subsystem does not exceed that of the whole system, and recursively use certain states as virtual controls to establish the intermediate control laws using the Control-Lyapunov Function (CLF), proposed by Artstein (1983) and then generalized by Sontag (1989). It was extremely used in control theory, as powerful tools for adaptive control for different classes of nonlinear systems with a triangular structure which only have state equation as strict feedback form, as indicated in Krasovskii (1963), Mascolo (1997). Obviously, it is used for designing stabilizing controls for a special class of nonlinear chaotic sys-

294 Backstepping Control of Nonlinear Dynamical Systems

tems which have synchronization error dynamics as the strict feedback form, specifically, both for tracking and regulation purposes, as explained in Kokotovi´c et al. (1991). Because of its ability to globally stabilize a large class of nonlinear systems without any cancellation at each system level, even in the presence of unknown parameters and perturbations. For this reason, this popular method can be creative in comparison with some earlier control theories such as the feedback linearization method or the Linear-Quadratic Regulator (LQR) method (Freeman and Kokotovi´c, 1993, 1995). Because it can contribute considerably to reduce the controller complexity. However, the traditional backstepping method suffers from solving the implicit nonlinear algebraic equation and has the disadvantage of too much complexity that is revealed in the different forms of nonlinearity treated where the state can diverge infinitely for a finite period. This, therefore, suggests that optimal control may be advantageous in these situations, if the cost of control is to be taken into account. Particularly, the optimal control is defined as a method for obtaining the performance index, when seeking to satisfy a desired control law for a given system, if a certain optimality criterion is reached. As a consequence, many researchers have thought of studying the optimal control problem for adaptive backstepping chaotic synchronization of nonlinear dynamical systems, see, for example Guessas and Benmahammed (2011), Lei and Wang (2016), Yu et al. (2017), and the references therein. Since the pioneering work of Ott et al. (1990) in controlling chaotic systems, several other improved methods based on the Lyapunov exponent are used for controlling after detecting the presence of any potentially chaotic behavior in these systems. In particular, the Lyapunov exponent which is initially developed by Wolf et al. (1985), is based on the principle that the stabilization of unstable periodic orbits of a nonlinear system can be simplified to the relatively small perturbations to a chaotic system. So, this effective tool is considered as a practical instrument for controlling the presence of chaos by quantifying the sensitive dependence on initial states, often referred to as «butterfly effect». We deduce from this useful tool that if the Lyapunov exponent is positive, then the dynamical system behavior is chaotic. In fact, this implies the existence of an exponential divergence of nearby chaotic trajectories. On the contrary, the presence of a negative Lyapunov exponent ensures absence of chaos in favor of the determinism. Thus, Freeman and Kokotovi´c (1992a) have proved that chaos synchronization is one of the most important phenomena based on the CLF used in backstepping feedback designs. Next, the backstepping controller design is extended to the problem of global chaos synchronization. As a result, many effective backstepping controller methods have been proposed (Vaidyanathan et al., 2015a). Recently, this leads us to think that the backstepping procedure is among the most important nonlinear controllers for multiple chaos synchronization schemes of nonlinear dynamical systems. Until now, many advanced theories and backstepping methods have been proposed for controlling chaos in nonlinear dynamic systems, such as state

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feedback backstepping control (Cheng et al., 2011; Vaidyanathan et al., 2015b), robust backstepping control (Freeman and Kokotovi´c, 1992b; Ezal et al., 2000), bounded backstepping control (Freeman and Praly, 1998; Bowong, 2007), active backstepping control (Vincent, 2008; Feng et al., 2017; Shukla et al., 2018), adaptive backstepping control (Vaidyanathan, 2012b; Vaidyanathan and Volos, 2016a; Vaidyanathan, 2016b; Vaidyanathan et al., 2018b,a; Azar et al., 2020; Vaidyanathan and Azar, 2016b), sliding mode backstepping control (Koshkouei and Zinober, 2000; Koshkouei et al., 2004), optimal backstepping control (Freeman and Kokotovi´c, 1994; Fouladi and Mojallal, 2018), and fuzzy backstepping control (Wang and Ge, 2001; Azar and Vaidyanathan, 2017). Recently, several researchers have tried to study how to establish adaptive backstepping control based on the Lyapunov theory, to ensure the stability of a nonlinear chaotic system. On the contrary, few are the researchers who tried to widen this field of investigation in a guaranteed framework of optimization, in order to establish an optimal control guarantee for chaos synchronization of nonlinear dynamical systems. Indeed, the adaptive backstepping technique proposed by Tan et al. (2003), remains until now as one of the most popular nonlinear techniques for nonlinear adaptive control design, which is treated first as an effective methodology capable of solving the problem being addressed. According to Laoye et al. (2009), this recursive procedure consists in finding a stabilizing function which is a virtual command for each subsystem, which is based on the stability of Lyapunov, until it is possible to determine the stable overall command to the system. In particular, they have shown that these nonlinear dynamical systems which can be put under the model of strict return loop form a triangular shape, transformations of the state variables and translations to equilibrium points are introduced to represent them under these forms. It is a necessary condition for the application of the backstepping method. They have also tried to show that it is possible to achieve global asymptotic stabilization at the origin using such a method. As a result, the backstepping method ensures overall stability, tracking, and transient performance for a broad class of strict feedback systems. In this light, the adaptive backstepping method has received considerable attention and it also approached as a robust tool for achieving the control of chaos synchronization for various complex systems. Due to its simplicity and robustness, this control method requires less effort in comparison with other control methods and has the particularity of having some or all of its parameters are unknown, as well as the design and application of control laws. Updating laws with a gain suitable for chaotic nonlinear systems, are basic for modeling systems and validating algorithms and approaches, such as second-order nonautonomous systems including the Duffing and Van der Pool chaos oscillators, and third-order autonomous systems including the Lorenz systems of Chua and Rössler. For some chaotic systems extremely sensitive to small perturbations, the stabilization and the continuation tracking are done by an arbitrary choice of design constants, but for others, the task is done only through an adequate

296 Backstepping Control of Nonlinear Dynamical Systems

optimized choice of these constants. For instance, by using the Genetic Algorithms (GAs), an improvement of the time of convergence and a total tracking of the reference signal can be noticed clearly. To see the effectiveness of proposed technique, a comparison based on another classical control method must be used. In practice, synchronization of the chaos makes it possible to mask a useful signal I (t) by adding it to a chaotic signal n(t) and transmitting the superposition of these two signals. The chaotic signals make it possible to use a broad frequency spectrum and, therefore, to increase the quantities of information transmitted, as well as to make the security of the information conveyed more efficiently. The information can be retrieved after comparing the received signal I (t) + n(t) with the original n(t) chaotic signal. With this method, the chaotic signals in the transmitter and receiver systems must be synchronized. This method of securing communications is difficult to unmask. Thus, due to the recent development of control communication complexity problem, control laws making via chaotic synchronization based on the method of backstepping and adaptive backstepping, is one of the new applications most currently used in the secure transmission of data discussed in Feki (2003). In some applications the information can only be reached through several systems. We will exploit the approached procedure to solve the problem of coordination of a group of chaotic nonlinear dynamical systems. Thus, system-independent dynamics are coordinated to have a unique overall structure. Although the issue of the complexity of communication and control appears as a standard part of the optimization problem, we have no effective control over what an optimal deterministic protocol might offer. So, in the absence of definitive solutions, we propose this new method which can be very advantageous in this situation. Therefore, the important interest brought to the procedure of backstepping is that all nonlinearities encountered can be treated in several ways. Useful nonlinearities that act for stabilization can be retained, and the rest of other nonlinearities can be treated with a linear control. Retaining nonlinearities instead of eliminating them requires less accurate models and also minimal control effort. We expect our control law simulation results to be optimal with respect to the performance index that guarantees certain robustness properties. In this chapter, the focus is meant to propose a new design methodology for optimal adaptive backstepping control for chaos synchronization of nonlinear dynamical systems. This proposed new nonlinear controller is an intelligent backstepping controller which is obtained by a combination of standard backstepping method and GAs. In order to clearly assimilate the efficiency of the proposed method, we make a comparison of the convergence in time and the level of the command with another known method. For example, we propose to use the Proportional-Integral-Derivative (PID) feedback controller, whose gains are optimized by GAs. The methodology is based on the comparison of two main tools: the PID controller based on GAs and

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the recursive backstepping theory for the model stability evaluation and control structuring based on Lyapunov’s stability theory. The originality of this research work is to contribute to the following three pathways. First, after a sophisticated diagnosis for the detection of chaos, optimal chaos control is implemented by applying the intelligent command by backstepping approach with integration of the GAs, which can optimize the controller to gain optimal and proper values for the parameters. Second, synchronization of chaos by applying adaptive backstepping technique. Third, generation of new independent chaotic attractors using anti-chaos control in some nonlinear dynamic systems. The results show the effectiveness and benefits of the proposed approach which presents a better performance, as well as a good robustness. The rest of this chapter is organized in the following way. The first section, introduces a brief description of 0 − 1 test for chaos detection in dynamical systems and the synchronization and recurrence of chaotic systems. The second section presents in brief the general problem statement and preliminaries. The third section focuses on the basic notion of the recursive backstepping theory, and describes the procedure steps of the adaptive backstepping controller design for suppressing chaotic motion as well as the synchronization of chaos and the stability condition analysis. The fourth section exposes the PID controller based on GAs. The fifth section illustrates the content of the work, which consists of a new scheme on the optimal adaptive backstepping control for chaos synchronization of nonlinear dynamical systems. Numerical simulation is given for illustration and verification of the effectiveness and feasibility of the given optimal control technique. At the end, the work will be summarized by a general conclusion followed by a bibliography.

13.2

Chaos detection and chaos synchronization

In chaos theory literature, there are many approaches and techniques to detect chaos. Several graphical and analytical methods are proposed for the detection of chaos in deterministic discrete or continuous dynamical systems (Vaidyanathan and Azar, 2016e,f,d,c,a; Ouannas et al., 2017a,b). As part of this research, we are mainly limited to the study of two tools, such as the Lyapunov exponent and the 0 − 1 test for chaos.

13.2.1 Chaos detection 13.2.1.1 Lyapunov exponent The Lyapunov’s exponent of a dynamical system can detect the presence of chaos and quantify the stability or instability of the system. It is defined as follows.

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Definition 13.1. Let E be a finite real dimensional vector space and FE the vector space of the functions of the real variable, with values in E and defined on an interval I ⊂ R with form [t0 , +∞[, t0 > 0. If f does not cancel on I, the Lyapunov exponent of f ∈ FE is defined by, ln f (t) , and λ(0) = −∞ t t→+∞, t∈I

λ(f ) = lim . sup

(13.1)

where . is the chosen Euclidean norm in E. If the quantity of the right-hand side is a limit (instead of being an upper limit), then f admits for exact Lyapunov exponent λ(f ) . Definition 13.2. Let σi (t, x0 ) (i = 1, ..., n) be the singular values of the Jacobian matrix J t (x0 ) (i.e., the square roots of the eigenvalues of J t (x0 )T J t (x0 )). The exact Lyapunov’s exponents values λ(σi (.,x0 )) of dynamical system are the logarithms of the eigenvalues of the limit (x0 ) , given by (x0 ) = lim

t→+∞

 1/(2t) . J t (x0 )T J t (x0 )

(13.2)

If Lyapunov’s exponent is positive, then the behavior of the dynamical system is chaotic. If, Lyapunov’s exponent is negative, then the behavior of the dynamical system is non-chaotic.

13.2.1.2 0 − 1 test for chaos in dynamical system The 0−1 test proposed by Gottwald and Melbourne (2009) is one of the most effective analytical tools for distinguishing chaos dynamics from regular behavior. According to Gottwald and Melbourne (2004), due to its simplicity, robustness and its ability to be used successfully to test a wide variety of dynamical systems, this test has become very popular. In this context, we first introduce the detailed algorithm steps of 0 − 1 test. Second, we will apply this test in order to detect the presence of chaos dynamics in deterministic continuous dynamical systems by distinguishing chaos from regular behavior. The 0 − 1 test algorithm is based on the following six steps. Step 1: Compute the average mutual information among φ0 (i) and φ0 (i + ω) defined by I (ω) =

 φ0 (i), φ0 (i+ω)

P (φ0 (i), φ0 (i + ω)) × log2 [

P (φ0 (i), φ0 (i + ω)) ] P (φ0 (i)) P (φ0 (i + ω)) (13.3)

where ω is the time delay, φ0 (i) is the considered set of measurement data generated from a continuous time system. The value ω associated with the first local minimum of the mutual information I (ω), is the optimal sampling period.

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The initial measurement data φ0 (i) are rearranged to form the finely sampled measurement data φ(k), using the following formula: φ(k) = φ0 (i + kω), ∀ i = 1, 2, . . . , T , ∀ k = 1, 2, . . . , N,

(13.4)

in which T is the amount of the data. Step 2: For a random number c ∈ [0; π], evaluate the new coordinates (Pc (n); Qc (n)) of translation variables as Pc (n) =

n 

φ(k) cos(α(k)), and Qc (n) =

k=1

n 

φ(k) sin(α(k))

(13.5)

k=1

where αk = kc +

k 

φ(k), ∀ k = 1, 2, . . . , n.

(13.6)

i=1

Step 3: Apply the mean square displacement Mc (n) defined as N 1  Mc (n) = lim (Pc (k + n) − Pc (k))2 + (Qc (k + n) − Qc (k))2 , N →∞ N k=1

N ] ∀ n ∈ [1; 10

(13.7)

where Mj (n) is able to analyze the diffusive (or non-diffusive) behavior in the (Pc (n); Qc (n)) plane of translation variables. According to this test, it must be noted that if the underlying dynamics is regular, the mean square displacement is a bounded function in time, whereas if the underlying dynamics is chaotic, the mean square displacement is proportional to the time (i.e., unbounded). Step 4: Formulate the modified mean square displacement Dc (n) given by N 1  1 − cos nc . φ(k))2 N →∞ N 1 − cos c

Dc (n) = Mc (n) − ( lim

(13.8)

k=1

Step 5: Calculate the median value of correlation coefficient K defined as K = median(Kc )

(13.9)

cov(μ; δ) Kc = √ ∈ [−1; 1] var(μ)var(δ)

(13.10)

where

in which μ = (1, 2, . . . , ncut ), δ = (Dc (1), Dc (2), . . . , Dc (ncut )), ncut = round(N/10), and the covariance and variance are formulated for vectors v

300 Backstepping Control of Nonlinear Dynamical Systems

and w of length p as 1 cov(v; w) = [(v(k) − v)(w(k) ¯ − w)], ¯ p p

k=1

p 1 v¯ = v(k), p

and

var(v) = cov(v; v).

(13.11)

k=1

Step 6: Interpret the results as follows. The decision rule implies that: – If K tends to zero, then this indicates that the underlying dynamics is regular (i.e., periodic or quasi-periodic). – If K tends to 1, then this indicates that the underlying dynamics is chaotic.

13.2.2 Chaos synchronization and recurrence Historically, the word synchronization was first proposed by Huygens (1673). According to him, the synchronization of two or more interacting dynamic systems can occur when these systems are autonomous oscillators (Alain et al., 2020; Khettab et al., 2018; Singh et al., 2018a,b,c; Volos et al., 2018). Since then, this phenomenon immediately emerged as a fundamental property of the phase transition, characterizing the mostly complex evolution of a wide variety of coupled or forced, identical or non-identical nonlinear dynamical systems. In particular, all forms of identical synchronization occur when two or more dynamic systems perform the same behavior at the same time. Currently, due to its high potential for applications, the analysis of synchronization phenomena in the evolution of chaotic dynamical systems occupies a large part of many particularly interesting study of chaos control (Ouannas et al., 2020a,b, 2019a,b). Since its advent in 1990s, the study of chaos synchronization problem has grown considerably in many real applications. In this context, to use a chaotic signal in transmission of communications, we require that the receiver receives a copy of the chaotic signal from the transmitter or, better, be synchronized with it. In fact, chaotic synchronization is a requirement of many types of communication systems. Hence, Boccaletti et al. (2000) defined chaos synchronization as the phenomenon that refers to a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy). In order to define this concept in this framework, let consider two identical chaotic systems. The first is the slave system which can be synchronized with the second which is the master system. Both systems are, respectively, represented as X˙ 1 = f1 (X1 , X10 , β1 ), X˙ 2 = f2 (X2 , X20 , β2 ) + U,

(13.12) (13.13)

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where X1 ∈ Rn and X2 ∈ Rm are the multidimensional states vectors of the two considered chaotic systems respectively; X10 and X20 are, respectively, the initial states of the two systems, f1 : Rn → Rn and f2 : Rm → Rm are, respectively, two nonlinear functions, β1 and β2 are the two multidimensional vectors value of respectively the two original systems parameters, and U ∈ Rm is a control vector to be determined. We assume that these two systems can manifest some traditional types of synchronization for a specific time, independent of initial conditions X10 and X20 in a large zone of Rn+m . Different notions of chaotic systems synchronization are possible. The most useful traditional types of synchronization which can be satisfied by chaotic systems are complete synchronization, antisynchronization, projective synchronization, generalized synchronization, Q-S synchronization and topological synchronization. In Alimi et al. (2018), we give a definition of each possible synchronization type and we explore the fundamental property of its characteristics for invariant trajectories associated to the synchronous chaotic systems. In the general case, complete synchronization condition of two chaotic systems is defined, for the error synchronization (t) = X2 (t) − X1 (t), as lim  (t) = 0

(13.14)

t→∞

where . is the Euclidean norm. If f1 is equal to f2 , the relation becomes an identical complete synchronization. If f1 is not equal to f2 , then the complete synchronization is not identical. It should be noted that, due to the presence of the recurrence phenomenon, several diverse interactions will appear in phase space because of the extreme sensitivity of the initial conditions and the divergence in the trajectories of the synchronous chaotic dynamical systems. Thus, their dynamics are not longer independent and the correlation effects can appear. This is the reason why synchronization of chaos becomes observed. In order to justify this statistically, Eckmann et al. (1987) have defined the recurrence plot (RP) indicator based on the following assumption: if each point zi of the phase portrait {zi }i=1,2,...,N is or is not close to another point zj , then a recurrence between point’s zi and zj of the trajectory will take place. The RP indicator denoted Ri,j is considered as an associated diagram to a square matrix of recurrence constructed around the indicated previous assumption, is formulated as follows: Ri,j = (˜ − zi − zj )i, j =1,2,...,N ,

zi , zj ∈ R,

(13.15)

where N is the total number of considered dynamical system states, (.) is the Heaviside function, ˜ is a threshold distance, . is the Euclidean norm, and zi (i = 1, 2, . . . , N) and zj (j = 1, 2, . . . , N) are two points of trajectory of dynamical system states. According to Alimi et al. (2018), the principle of this technique consists in the visual analysis and inspection of the graph matrix of recurrence constructed with vector distances when each portion of the curve is compared to all the others and represented on a recurrence map, as well as

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the quantitative characteristics are necessary for the evaluation of the distances, between the points in the temporal space and in the phase space. On another side, if to be synchronized, two chaotic systems have the original states equation are strict feedback form, then the adaptive backstepping design can be used directly to control chaos synchronization of these systems. As already mentioned, the adaptive backstepping design technique has been widely recognized as the powerful method for control the chaos synchronization. Until now, various methods based on the adaptive backstepping design technique have been proposed to achieve this interesting and challenging issue of control and synchronization of chaotic systems. But, the treatment of this subject in an optimization framework, remains again as an open question. In order to contribute for solve this optimization problem of chaos control and synchronization of chaotic systems, in this chapter, a new optimal adaptive backstepping control method is proposed to control the chaos synchronization of two identical systems.

13.3 Problem statement and preliminaries In order to test whether a nonlinear dynamical system is feedback stabilizable using the backstepping method which based on the CLF, we will try to know, for any state X, if there exists a control u such that a system can be brought to the zero state by applying the control u. The backstepping is a recursive, Lyapunov-based procedure, which allows deducing the control law of a nonlinear dynamical system, in order to guarantee its global stability. The principle of adaptive backstepping control method consists in choosing a Lyapunov candidate function to ensure the stability of the first subsystem, then increasing the system by adding the second subsystem and selecting the most appropriate Lyapunov candidate function for the increased system stability, until reaching a global Lyapunov function that ensures overall system stability. To illustrate this principle, let us consider the following nth-order nonlinear system with strictfeedback form: ⎧ ⎪ x˙i (t) = gi (xi , t)xi+1 + fi (xi , t) + βiT Fi (xi , t) ⎪ ⎪ ⎪ ⎪ ⎪ (i = 1, . . . , n − 1), ⎨ (13.16) x˙n (t) = gn (xn , t)u + fn (xn , t) + βnT Fn (xn , t), ⎪ ⎪ ⎪ ⎪ y = x1 , ⎪ ⎪ ⎩ x t0 = x 0 , where xi = [x1 , x2 , . . . , xi ] ∈ Ri (i = 1, . . . , n) is the unmeasured state vector of the ith subsystem; u ∈ R and y ∈ R are the input, and the measured output of the overall system, respectively; β ∈ Rp is the vector of unknown constant parameters, which must be estimated; gi (.) = 0, fi (.), Fi (.) (i = 1, . . . , n) are the known, differentiable and smooth nonlinear functions. While gn (.) = 0,

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fn (.), Fn (.) are known continuous nonlinear functions, and a known reference model, derivable and bounded as follows: ⎧ ⎪ ⎨ x˙si (t) = fsi (xsi , t) (i = 1, . . . , m); n ≤ m, ⎪ ⎩ ys = xs1 ,

(13.17)

where, xi = [x1 , x2 , . . . , xsi ] ∈ Rm , ys ∈ R are the states and output, respectively; and fsi (.) (i = 1, . . . , m) are known, smooth nonlinear functions. The problem dealt with here, therefore, consists in designing an adaptive controller in the feedback for the system that guarantees overall stability, and forces the system output y = x1 (t) to asymptotically follows the output ys = xs1 (t) of the reference model, i.e. lim y(t) − ys (t) = 0

t→∞

(13.18)

where . is the Euclidean norm. The purpose of this chapter is to design a control u that ensures perfect compliance between the tracking output y and a specified trajectory ys so that all system variables are bounded. It should be noted that the resolution of this problem can be solved by using stability analysis. In the following section, we propose a scheme based on Lyapunov stability theory for nonlinear dynamical system.

13.4

Stability analysis of adaptive backstepping control systems

In this section, we begin by presenting the Lyapunov stability theory on which the backstepping control technique is based. Then, we use it to explain the fundamental principle and the steps of the backstepping controller design methodology.

13.4.1 Lyapunov stability theory and the invariance principle The stability of Lyapunov characterizes the non-divergence or convergence of system trajectories. Lyapunov’s theory is based on the characterization of this stability by demonstrating that a system is bounded and possibly converges to a minimum. The Lyapunov stability theory has the advantage to demonstrate stability without solving differential equations. Hence, the stability control design can be easily expanded in particular for nonlinear dynamical systems. In this sense, the object of this subsection is to present the additional conditions on the CLF and its derivative in time to guarantee the asymptotic stability.

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Definition 13.3. Let us consider the following nonlinear dynamical system: X˙ = f (X, X0 , β)

(13.19)

where X ∈ Rn is the state vector of a given system of Eq. (13.19); X0 is the initial state of the system, f : Rn → Rn is a nonlinear continuous function, and β is the multidimensional vector value of the original system parameters. If there exists a positive definite function V (X) : Rn → R that is continuously differentiable, and if its time derivative V˙ (X) = ∇V (X).f (X, X0 , β), all along the trajectory of the system of Eq. (13.19), is semi-negative: ∀ X = 0, ∃ u, V˙ (X) ≤ 0.

(13.20)

In Lyapunov’s theory, this function V (X) is called CLF for a given system. It was used to test whether system of Eq. (13.19) under given initial conditions is feedback stabilizable. By studying this function V (X), one of the important criteria presented shows that if it fulfills certain conditions, then the equilibrium point X ∗ of the system of Eq. (13.19) is qualified as stable. Nevertheless, the theorem will not give a proof of strict stability in the sense of Lyapunov, but rather it provides an asymptotic convergence criterion, this being valid for both the equilibrium points and the limit cycles. In addition, under certain conditions, the tool that will be presented also makes it possible to treat the asymptotically stable convergence towards an equilibrium point or a limit cycle. Theorem 13.1. If the CLF V (X) of a system is positive definite in Rn and V˙ (X) is semi-negative definite in Rn , then the equilibrium point X ∗ is stable in the sense of Lyapunov. If V˙ (X) is strictly negative definite in Rn , then the equilibrium point X ∗ is asymptotically stable in the sense of Lyapunov. In order to prove the Lyapunov global asymptotic stability of a given system of Eq. (13.19), it suffices to add to the above theoretical conditions due to Lyapunov, the condition that the CLF V (X) related to the considered system is radially unlimited in Rn . The last condition is crucial; since it implies that, for each state X, we can find a control u which will reduce V to zero. As a result, the system loses its energy, which means it stops immediately. The existence of a regular stabilizing feedback u is made rigorous by the following result, proposed by Artstein (1983). Theorem 13.2. The dynamical system of Eq. (13.19) has a differentiable CLF, if and only if, there exists a regular stabilizing feedback u. If there is a solution u∗ , it is in fact to solve, for each state X, the following nonlinear optimization problem: u∗ (X) = arg minu V˙ (X).

(13.21)

In order to guarantee convergence towards the local minimum of the CLF V at the fixed point X ∗ for autonomous systems, let us now recall the LaSalle’s

Optimal adaptive backstepping control Chapter | 13 305

invariance principle in LaSalle (1968), which can satisfy the two conditions of invariance at the equilibrium point only for autonomous systems. If both conditions are satisfied, we can conclude asymptotic stability of fixed point even when V˙ (X) is not locally positive definite. While, for non-autonomous systems, Barbalat’s lemma is used to prove asymptotic stability. Theorem 13.3. Consider a fixed point X ∗ of system of Eq. (13.19) and an associated Lyapunov function V (X) in a region D ⊂ Rn around X ∗ . Suppose that V˙ (X) ≤ 0, ∀ X ∈ D; and the only whole solution for which V˙ (X) = 0 is X = X ∗ . Then X ∗ is asymptotically stable. In fact, a formal proof of the theorem reveals that all the trajectories of the system that start in a particular bounded region of the state space are approaching a positive limit set. These may contain asymptotically stable equilibria and stable limit cycles. After presenting the essentials of Lyapunov’s stability theory, the LaSalle’s invariance principle and Barbalat’s lemma, in the rest of this section, we present now some interesting tools for adaptive control, including in the order Barbalat’s lemma, the LaSalle–Yoshizawa Theorem, and the Krazovskii–LaSalle theorem. As already mentioned, the previous LaSalle invariance theorem is applied to prove the asymptotic stability of autonomous systems, whereas for non-autonomous systems, Barbalat’s lemma is used. Theorem 13.4. Let ϕ : R → R be a uniformly continuous function on [0, + ∞[. t Suppose that limt→∞ 0 ϕ(τ )dτ exists and is finite. Then, ϕ → 0 when t → ∞. For the original proof of Barbalat’s lemma see Barbalat (1959). By combining Lyapunov’s direct method and Barbalat’s lemma, we obtain the LaSalle– Yoshizawa theorem. Theorem 13.5. Consider the system of Eq. (13.19) where f is locally Lipschitz in X uniformly in t. If there exists a continuously differentiable function V (X, t) : Rn × [t0 , ∞) → R+ such that W1 (X) ≤ V (X, t) ≤ W2 (X) and ∀t ≥ t0 , ∀X ∈ Rn , V˙ (X, t) ≤ −ψ(X) ≤ 0 where W1 (X) and W2 (X) are continuous positive definite and radially unbounded functions and ψ(X) is a continuous positive semi-definite function, then the state is bounded and satisfies limt→∞ ψ(X(t)) = 0. Moreover, if ψ(X) is positive definite, then the equilibrium point X = 0 is uniformly globally asymptotically stable. For the proof, see Yoshizawa (1968), Yoshizawa (1966), LaSalle (1966), Krasovskii (1963). So far the stability of the systems has been defined from the stability of the state of equilibrium around a point in a fixed domain. It is also possible to consider the input / output stability using the Krazovskii–LaSalle theorem. Theorem 13.6. Let V (X) : Rn → R+ a continuously differentiable positive definite function such that V (X) → ∞ as X → ∞ and V˙ (X) ≤ 0, ∀ X. Let 

306 Backstepping Control of Nonlinear Dynamical Systems

be the set of all points where V˙ (X) = 0, that is  = {X ∈ Rn | V (X) = 0}, and K be the largest invariant set in . Then all solutions X(t) converge to K. If K = X ∗ then the equilibrium point X ∗ of system of Eq. (13.19) is globally asymptotically stable. For the proof, see Krstic et al. (1995), LaSalle (1968). Then, we present in the following subsection, the adaptive controller design based on the backstepping command.

13.4.2 Adaptive backstepping controller design 13.4.2.1 Principle of backstepping control method The backstepping methodology is a systemic and recursive design method for nonlinear feedback control. As mentioned previously, the backstepping control fundamental principle is to organize the global given system of Eq. (13.16) into several subsystems in cascade where the degree of each subsystem does not exceed the initial system. A Lyapunov candidate function is chosen to ensure the stability of the first subsystem. Then the system is augmented by adding the second subsystem and the new Lyapunov candidate function is chosen for the stability of the augmented system, and so on. We repeat this process recursively, until we find an universal Lyapunov function, that ensures the global system stability. The advantages of this technique are numerous, including its recursion, minimum control effort, and cascading structure that divides a high-order system into several smaller, lower-order systems. In the following subsection we illustrate the procedure of adaptive backstepping control. 13.4.2.2 Adaptive backstepping control process In order to illustrate the backstepping principle, let’s consider a previous nonlinear system presented by Eq. (13.16), where the control u is determined using the backstepping method. The backstepping controller u must be determined using the states xi (i = 1, 2, . . . , n) and estimates βˆ of the unknown parameters β of the system. The design procedure based on the backstepping theory is recursive, which involves n steps. During the ith stage, the ith subsystem is stabilized under an appropriate CLF Vi by the development of a stabilizing function θi and ˆ and the a setting function πi . The update law for the estimated parameter β(t) current control u is only determined in the last step n. The detailed procedure of the backstepping design which was developed in Montaseri and Yazdanpanah (2012) is described in the following steps. The first step: Concerning the first subsystem x1 from Eq. (13.16), the first variable of the backstepping introduced as the tracking error 1 = x1 − xs1 and the error of the unknown parameters β˜ = βˆ − β; where β is the value of the adaptive parameter run, βˆ is the value of the estimated adaptive parameter and β˜ is the value of

Optimal adaptive backstepping control Chapter | 13 307

the error on the adaptive parameter. In addition, the second variable 2 = x2 − xs2 − θ1 ; where 1 and 2 are virtual variables of the procedure, x2 is taken as a virtual control input, and θ1 is used as a control law to stabilize the first system equation that will be defined later. The time derivative ˙1 of the indicated error variable 1 of the first equation x1 of the system of Eq. (13.16) is written as ˙1 = x˙1 − x˙s1

(13.22)

where ˙1 = g1 (x2 − xs2 ) + f1 + βˆ T F1 − β˜ T F1 − fs1 + g1 xs2 . For uniformity and simplification of scoring for all stages, the following approximation is considered F1p = F1 and f1p = f1 − fs1 + g1 xs2 , hence ˙1 = g1 2 + g1 θ1 + f1p + βˆ T F1p − β˜ T F1p . To find this control law, a quadratic partial Lyapunov function is constructed as follows: 1 1 V1 ( 1 ) = 12 + β˜ T −1 β˜ 2 2

(13.23)

where  = T is the adaptation gain matrix. Hence, its derivative in time is ˙ˆ After substituting ˙ by its value, this derivative takes V˙1 = 1 ˙1 + β˜ T −1 β. 1 the expression V˙1 = 1 (g1 2 + g1 θ1 + f1p + βˆ T F1p ) + β˜ T −1 (β˙ˆ − 1 F1p ). If x2 was the current control law, then 2 would be null, which implies that x2 = θ1 . So β˜ and V˙1 will be eliminated by an update law β˙ˆ = π1 , where π1 = 1 F1p is the adjustment function. Thus, to make V˙1 = −ψ1 12 negative, θ1 must be chosen as follows: 1 θ1 = (−ψ1 1 − f1p − βˆ T F1p ) (13.24) g1 where ψ1 > 0 is a positive design constant. As long as x2 will not consider the current control law, then the virtual variable 2 will no longer be zero and β˙ˆ = π1 is no longer used as an update law. π1 must, however, be retained as a first adjustment function and θ1 as a first stabilizing function. Therefore, the decision made approximately on β˙ˆ by tolerating the presence of β˜ in V˙1 , implies that V˙ = −ψ 2 + g + β˜ T −1 (β˙ˆ − π ) (13.25) 1

1 1

1 1 2

1

knowing that, for global stability, the second term g1 1 2 will be eliminated in the next step. Thus, the closed-loop equation takes the following form: ˙1 = −ψ1 1 + g1 2 + β˜ T −1 (β˙ˆ − π1 ).

(13.26)

The second step: In the same way, the time derivative ˙2 of the indicated error variable 2 of the second equation x2 of the system of Eq. (13.16) is written as ˙2 = x˙2 − x˙s2 − θ˙1

(13.27)

308 Backstepping Control of Nonlinear Dynamical Systems

knowing that θ˙1 =

∂θ1 ∂θ1 ∂θ1 ˙ T x˙1 + x˙s1 + βˆ ∂x1 ∂xs1 ∂ βˆ T

after substitution, this implies that θ˙1 = ∂θ1 ∂xs1 (fs1 )

∂θ1 ∂x1 (g1

(13.28)

+ f1 + βˆ T F1 − β˜ˆ T F1 ) +

+ ∂θˆ 1T  1 F1p . After having produced the third variable of the pro∂β cedure 3 = x3 − xs3 − θ2 , where x3 is taken as a virtual control input and θ2 is used as a control law to stabilize the second equation of the system ∂θ1 ˙2 = g2 ( 3 + θ2 ) + f2 + βˆ T F2 − β˜ T F2 − fs2 + g2 xs3 − [ ∂x (g1 + f1 + βˆ T F1 − 1 β˜ˆ T F ) + ∂θ1 (f ) + ∂θ1  F ]. For uniformity and simplification of scoring 1

∂xs1

s1

1 1p

∂ βˆ T

∂θ1 for all stages, it is posited that F2p = F2 − ∂x F1 and f2p = f2 − fs2 + g2 xs3 − 1 ∂θ1 ∂θ1 ∂θ1 ∂x1 (g1 x2 + f1 ) − ∂xs1 (fs1 ) − ∂ βˆ T

 1 F1p . Thus, the closed loop equation of the

system becomes ˙2 = g2 3 + g2 θ2 + f2p + βˆ T F2p − β˜ T F2p .

(13.29)

To find this control law, a partial quadratic Lyapunov function is introduced, as follows: 1 1 ˜ (13.30) V2 ( 1 , 2 ) = V1 + 22 + β˜ T −1 β. 2 2 Its derivative in time gives V˙2 = V˙1 + 2 ˙2 + β˜ T −1 β˙ˆ

(13.31)

where V˙2 = −ψ1 12 + g1 1 2 + β˜ T −1 (β˙ˆ − F1p ) + 2 (g2 3 + g2 θ2 + f2p + ˙ˆ After development, this gives the expression V˙ = βˆ T F2p − β˜ T F1p )β˜ T −1 β. 2 −ψ 2 + g + (g + g θ + f + βˆ T F ) − β˜ T −1 (β˙ˆ −  F − 1 1

2 2 3

2

1 1

2 2

2p

2p

1 1p

 2 F2p ). Hence, if x3 is chosen as the current control law, then 3 will be zero, which implies that x3 = θ2 . So, β˜ will be eliminated from V˙2 with an update law β˙ˆ = π2 , where π2 = π1 + 2 F2p is the adjustment function. In addition, to make V˙1 = −ψ1 12 − ψ2 22 negative, θ2 will be chosen as follows: 1 θ2 = (−ψ2 2 − g1 1 − f2p − βˆ T F2p ) (13.32) g2 where ψ2 > 0 is a positive design constant and  = T is the adaptation gain matrix. Hence, as long as x3 is not the current control law then 3 will not be null and thereafter β˙ˆ = π2 will not be used as an updated law. However, π2 will be retained in this case as a second adjustment function and θ2 will be considered

Optimal adaptive backstepping control Chapter | 13 309

as the second stabilizing function. So, a decision will be made approximately ˙ˆ which is to tolerate the presence of β˜ in V˙ , given by on β, 2 V˙2 = −ψ1 12 − ψ2 22 + g2 2 3 + β˜ T −1 (β˙ˆ − π2 ).

(13.33)

So, for overall stability, the last term g2 2 3 will be eliminated in the next step. The closed loop equation is then defined as follows: ˙2 = −ψ2 2 + g2 3 + β˜ T −1 (β˙ˆ − π2 ).

(13.34)

The ith step (3 ≤ i ≤ n − 1): After some algebraic manipulation, the virtual controller will be chosen to make the ith error variable i = xi − xis − θi−1 for all i = 3, 4, . . . , n − 1 of the ith equation xi of the system of Eq. (13.16) as small as possible. The time derivative ˙i of the error variable i is expressed as ˙i = gi i+1 + fip + β T Fip

(13.35)

where i  ∂θi−1 (−1) (gk xk+1 + fk ) fip = fi − fsi + gi xs(i+1) − ∂xk k=1

i i   ∂θi−1 ∂θi−1 (−1) (fsk ) − (−1) ( k Fkp ) − ∂xsk ∂ βˆ T k=1

k=1

Fk . Hence, the Lyapunov function candidate of and Fip = Fi − ik=1 (−1) ∂θ∂xi−1 k quadratic type is defined as follows: 1 ˜ Vi ( 2 , 4 , . . . , i ) = Vi−1 + i2 + β˜ T −1 β. 2

(13.36)

Its time derivative is ˙ˆ V˙i = V˙i−1 + i ˙i + β˜ T −1 β.

(13.37)

The control law u is designed to make the derivative of the global Lyapunov function V˙i defined negative, is given by V˙i = i (gi−1 i−1 + gi i+1 + gi θi +

fip + βˆ T Fip ) − ik=1 (−1)ψk k + β˜ T −1 β˙ˆith . The unknown β is replaced by its estimator βˆ in the θi expression, the equation for θi gives θi =

1 (−ψi i − gi−1 i−1 − fip − βˆ T Fip ) gi

(13.38)

310 Backstepping Control of Nonlinear Dynamical Systems

where ψi > 0 is a positive design constant. After substituting θi in ˙i and V˙i , the obtained result is V˙i = −

i−1  (ψk k2 ) + gi i i+1 + fip + β˜ T −1 (β˙ˆith −  i Fip ).

(13.39)

k=1

So, the error term on the parameter β˜ T will be eliminated by using the updated law β˙ˆith =  i Fip . For overall stability, the last error term gi i i+1 will be eliminated in the next step. The nth step: The last step of the design procedure, starts as soon as the current control u focuses on the time derivative ˙n of the nth error variable ˙n of the last equation xn of the system of Eq. (13.16), given by ˙n = xn − xsn − θn−1 .

(13.40)

That is to say, ˙n = gn u + fnp + β T Fnp , where fnp = fn − fsn −

n−1 ∂θn−1

n−1 ∂θn−1

n−1 ∂θn−1 k=1 ∂xk (gk xk+1 + fk ) − k=1 ∂xsk (fsk ) − k=1 ∂ βˆ T ( k Fkp ) and

n−1 ∂θn−1 Fnp = Fn − k=1 ∂xk Fk . Hence, the global Lyapunov function of quadratic type is constructed as follows: 1 ˜ Vn ( 1 , 2 , . . . , n ) = Vn−1 + n2 + β˜ T −1 β. 2

(13.41)

The control law u is designed to make the derivative of the global Lyapunov 2 function V˙n defined negative, with V˙n = − n−1 k=1 ψk k + n (gn−1 n−1 + gn u + ˙ fnp + βˆ T Fnp ) + β˜ T −1 βˆnth , where the choice of the current control is u=

1 (−ψn n − gn−1 n−1 − βˆ T Fnp ). gn

(13.42)

After substituting u in n and V˙ , the nth system equation gives ˙n = −ψn n − gn−1 n−1 + β˜ T Fnp .

(13.43)

The time derivative of the global Lyapunov function is given by V˙n = −

n  [ψk k2 + β˜ T −1 (β˙ˆnth −  n Fnp )].

(13.44)

k=1

So, the term of the error β˜ in V˙n will be eliminated by final update law β˙ˆnth =  n Fnp . The result is that the derivative of the global Lyapunov functions are

Optimal adaptive backstepping control Chapter | 13 311

defined negative: V˙n = −

n  (ψk k2 )

(13.45)

k=1

where ψk > 0 is a positive design constant. V˙n becomes a negative definite and according to LaSalle–Yoshizawa theorem the equilibrium (0; 0; . . . ; 0) of the system ( 1 ; 2 ; . . . ; n ) at the origin point, is globally asymptotically stable. It is, therefore, obvious that the controller u is not able to modify the system equilibrium. In this respect, the new chaotic system is originally stabilized under the controller of Eq. (13.42). So, the used backstepping method is able to set the states (x1 ; x2 ; . . . ; xn ) to the origin point (0; 0; . . . ; 0) via the controller u, which is calculated with the previous n steps of the given design procedure. In addition, this control law is used for asymptotically stabilizing the system ( 1 ; 2 ; . . . ; n ) = (0; 0; . . . ; 0). Hence, the closed loop equation of the overall system in the coordinates 1 , 2 , . . . , n is given by ⎧ ⎪ ˙1 = −ψ1 1 − g1 2 + β˜ T F1 , ⎪ ⎪ ⎪ ⎨ ˙i = −ψi i − gi−1 i−1 + gi i+1 + β˜ T Fip (13.46) ⎪ (i = 2, . . . , n − 1), ⎪ ⎪ ⎪ ⎩ ˙n = −ψn n − gn−1 n−1 + β˜ T Fnp , where i = xi − xsi − θi−1 for all i = 1, 2, . . . , n and θ0 = 0. So, for all i = 1, 2, . . . , n, the used update law is defined by β˙ˆith =  i Fip .

(13.47)

Consequently, when t tends to infinity and simultaneously β˜ tends to zero, then the global Lyapunov function V˙n will be defined negative and therefore, the system of error equation in i (i = 1, 2, . . . , n) is corresponding to the final system in closed loop. The system of updating laws βˆi (i = 1, 2, . . . , n) admits then an equilibrium point eq = [ 1 , 2 , . . . , n ]T converge to zero and is globally uniformly asymptotically stable. However, since the reference model xs1 is bounded and knowing that 1 = x1 − xs1 , this guarantees that the state x(t), the current control law u, the vector of the estimated parameters βˆ are all globally bounded, consequently limt→∞ (t) = 0 and so limt→∞ y(t) − ys (t) = 0. In the following subsection we illustrate the procedure of the proposed optimal backstepping controller, based on GAs.

13.4.2.3 Optimal backstepping controller based on genetic algorithms Until now for controlling chaos synchronization of nonlinear dynamical systems, many various methods have been used. For several cases of these methods,

312 Backstepping Control of Nonlinear Dynamical Systems

the designed controller suffers a significant overshoot and the stability of the system is reached slowly after a long time. For other controller types, the stability of the system is reached in the exact time, but the system error persists for some time. In order to solve this problem, several optimization algorithms are proposed. Most of these algorithms are based on the gradient of the cost function and they are very sensitive to the choice of initial conditions. So, for the wrong choice of initial point or interval search, these algorithms can easily be erroneous and misleading on the locally optimum, and subsequently, they cannot reach the globally optimum. In order to avoid this constraint, we propose to use a very specific computational class of evolutionary optimization algorithms, such as GAs. According to Michalewicz (1994), the principle of GAs is first to start from a very large number of initial points in order to cover all the search intervals, then they encode a potential solution to a specific problem on a single chromosome, such as a data structure. Then they apply recombination operators to these structures to preserve the critical information. In the end, the execution of GA requires a large population of chromosomes, which are implemented randomly. Each chromosome has an objective function called a fitness function that must be evaluated correctly. It must be noted that in order to apply both genetic reproductive operations which are called crossover and mutation, two individuals called parents must be selected randomly. The first crossover operation is applied if its probability is reached, between parents by exchanging a part of their bits to begets two children. The second mutation operation is applied to each single child by inverting its bit if the probability is reached. After this step, two populations are obtained: a parental population and an infant population, only the individual which has a goodness solution must be preserved. Once a string is selected for mutation, a randomly chosen element of the string is changed or mutated. Fig. 13.1 illustrates the diagram of GA process. The different steps of the previous diagram of the GA process are ordered as follows: Step 1: Generate a randomly initial individuals population for a fixed size. Step 2: Evaluate their fitness. Step 3: Select the fittest members of the population. Step 4: Apply the mutation operation using a probabilistic method for chromosomes reproduce. Step 5: Implement crossover operation on the reproduced chromosomes. Step 6: Execute mutation operation with low probability. Step 7: Repeat Step 2, until a predefined convergence criterion is met. It must be noted that the convergence criterion of a GA is a condition that must be specified by the user, that is, the maximum number of generations or when the string value exceeds a certain threshold. The advantage of integrating GAs is to minimize a fitness function in order to determine the minimum current value. Then, the fitness function is used to quantify the minimum value that can minimize the least squares errors. The used fitness function h given

Optimal adaptive backstepping control Chapter | 13 313

FIGURE 13.1 Diagram of the genetic algorithm process.

by Michalewicz (1994) is n  1 h(xi , xdi ) = (xi − xdi )2 n

(13.48)

i=1

where xi is the state of system and xdi is a favorite mood for xi . Based on the purpose of the system to set the states to zero value; xdi is equal with zero. The AGs are used to search the optimal positive unknown parameters (ci , i = 1, 2, . . . , n) able to ensure system stability by guaranteeing negativity of the Lyapunov function and having an appropriate time response. The structure of the proposed optimal backstepping controller based on GAs is shown in Fig. 13.2. It used for optimizing the key settings of the backstepping controller. The utility of combining GAs with the backstepping controller is not only to select the appropriate fitness function Eq. (13.48) by using, but also to rationally determine the optimal values for the parameters. The adequate selected fitness function Eq. (13.48) is used, in this hybrid forms, for minimizing the least square error. Indeed, the fitness function of Eq. (13.48) is able to force the error of the controller system to converge to zero quickly. Consequently, the response of the controller system will be optimized with a short adjustment time. Thus, the fit-

314 Backstepping Control of Nonlinear Dynamical Systems

ness function is based on optimal controller and intervenes effectively to exceed to reach its minimum value. As a result, this proposed hybrid form given in Fig. 13.2 is an optimal backstepping controller.

FIGURE 13.2 The structure of the proposed optimal backstepping controller with GAs.

Fortunately, we can evaluate the efficiency of the proposed optimal control method. We prefer to make a comparison of the time of convergence and the level of the command with another known method based on the PID controller. In the following section we illustrate the procedure of the PID controller based on GAs.

13.5

The PID controller based on genetic algorithms

In general, the control of nonlinear systems is difficult in the absence of an efficient systematic procedure such as that available for linear systems. Several design techniques and approaches have been developed for controlling this kind of non-linear systems. These methods can be classified into two broad categories. The first category, known as feedback linearization, is to linearize the system to use one of the linear control techniques. The second category, on the contrary, uses nonlinear controllers without linearization, such as backstepping. Over the past decades, control theory based on adaptive backstepping has recognized major developments, and many various intelligent control algorithms have been developed to combine with this popular method, such as fuzzy logic, neural network, GA, and particle swarm optimization. In particular, the technique of GAs developed by Michalewicz (1994), as a heuristic technique is effective in solving problems that are difficult to formalize mathematically. In particular, Fleming and Purshouse (2002) have been successful in proving the utility of GA in optimized adaptive nonlinear control with a discontinuous, nondifferentiable, nonconvex, and/or multimodal search space. According to these researchers, GAs can reduce, to a large extent, the arbitrary design of a controller. The best known controller form of feedback, widely used in requiring continuously modulated control, is the PID algorithm. Thanks to its simple structure and its robust performance over a wide range of operational conditions,

Optimal adaptive backstepping control Chapter | 13 315

this controller is considered in adaptive control processes as a control loop feedback mechanism. In fact, this type of PID controller remains, until now, the most popular in the industry. This is due namely to: the simplicity of the PID control law and the minimum number of adjustment parameters it requires. In practice, the principle of the PID controller consists of continuously calculating an error value et as being the difference between a desired setpoint and a measured process variable and then automatically applying an accurate and responsive correction to a control function. The applied correction is on the three basic types control values of the Proportional (P), Integral (I) and Derivative (D). The simplest diagram that illustrates the PID controller scheme is given by Guessas and Benmahammed (2011) as follows:  ui = KP ei + KI 0

t

ei dπ + KD

∂ei , i = 1, 2, . . . , n, ∂t

(13.49)

where ui is the PID action control for i = 1, 2, . . . , n, ei is a proportion of the system error used for system control, and KP , KI , KD are the gains respectively for the proportional, integral, and derivative controllers to synchronize chaotic and hyperchaotic systems. On the one hand, the use of a part of the error of the system to control the system itself, admits a greater limit. Indeed, this makes it possible to introduce a static compensation of the error in the considered system. Thus, since the output of the integrator controller is proportional to the time-increasing summation of the errors present in the system, then the integral action removes the compensation introduced by the proportional controller. On the other hand, it introduces a phase shift in the system. In addition, since the output of the drift controller is proportional to the rate of change of the error, then the drift controller is used to reduce or eliminate the overshoot and introduce an angular main action that removes the phase shift introduced by the integral action. We will use GA to optimize the controller gains such that the error ei (i = 1, 2, . . . , n) is minimized. In other words, as the design of an objective function is the most difficult part to create a GAs, an objective function could be created to find a controller that gives the smallest offset, the fastest rise time or the fastest fall time. To combine all these objectives, an objective function that will minimize the error of the controlled system is chosen. Each chromosome of the population is evaluated in both objective functions. As a result, the largest chromosome rated is the best. The GA uses the value of the chromosome’s ability to create a new population consisting of members of greater cost. To evaluate the values of the PID controller chosen by the GAs, an objective function must be correctly written based on the sum of the criterion of the absolute error performance. Practically, this kind of performance criterion of the error is good for the simulation. In the following sections we illustrate all the previously procedures in a practical application.

316 Backstepping Control of Nonlinear Dynamical Systems

13.6 Simulation examples and discussion It must be noted that the implementation of the proposed optimal backstepping controller in this practical application, can be divided in two parts. First, the model specification consists of four steps, namely: Steps 1: Defining all parameters of the nonlinear system and identify their values. Steps 2: Identify all the state variables of the nonlinear system and specify their initial conditions. Steps 3: The simulation of system equations that describe and characterize the dynamics of trajectories variations states in both time and phase space. Steps 4: Detecting the presence of chaos. Second, the chaotic nonlinear system optimal control application includes three steps, namely: Steps 1: Implementing the optimal adaptive backstepping control for chaos synchronization systems and plotting the controller results. Steps 2: Implementing the genetically optimized PID control for chaos synchronization systems and plotting the controller results. Steps 3: Comparing the efficiency of the two controllers and decide what should be chosen for a desired optimized performance.

13.6.1 Lorenz system description The Lorenz’s differential dynamic system is developed by Lorenz (1963) in studies of convection and instability in planetary atmospheres, following this model: ⎧ ⎪ ⎨ x˙1 = β11 (x2 − x1 ), (13.50) x˙2 = −x1 x3 + β12 x1 − x2 , ⎪ ⎩ x˙ = x x − β x , 3 1 2 13 3 where β11 , β12 , and β13 are three positive real parameters and x˙1 , x˙2 , and x˙3 are the three dynamic variables specifying the system status over time. It must be noted that the previous procedure implementation is based on Runge–Kutta methods and all simulations are given with computational time 100 s for time step equal to 0.01. For the initial condition (x10 ; x20 ; x30 ) = (0; −4; −1) and the given values of the control parameters β11 = 10, β12 = 28, and β13 = 83 , the Lorenz system of Eq. (13.50) in its phase portrait, without any controller and AG, exhibits a good chaotic behavior. Therefore, for the started initial condition (x10 ; x20 ; x30 ) = (0; −4; −1) and given control parameters, the Lorenz system of Eq. (13.50) in state space has a strange attractor with shaped like butterfly wings, as is illustrated in the phase space in Fig. 13.3. The simulation result of the Lorenz system of Eq. (13.50) in x1 –x2 plane for β11 = 10, β12 = 28, β13 = 83 , and initial conditions (x10 ; x20 ; x30 ) = (0; −4; −1) is shown in Fig. 13.4.

Optimal adaptive backstepping control Chapter | 13 317

FIGURE 13.3 Chaotic attractor of Lorenz system of Eq. (13.50).

FIGURE 13.4 Lorenz system of Eq. (13.50) in x1 –x2 plane.

The simulation result of the Lorenz system of Eq. (13.50) in x1 –x3 plane for β11 = 10, β12 = 28, β13 = 83 , and initial conditions (x10 ; x20 ; x30 ) = (0; −4; −1) is shown in Fig. 13.5. The simulation result of the Lorenz system of Eq. (13.50) in x2 –x3 plane for β11 = 10, β12 = 28, β13 = 83 , and initial conditions (x10 ; x20 ; x30 ) = (0; −4; −1) is shown in Fig. 13.6. The regrouped chaotic motion of the system of Eq. (13.50) states with started initial conditions (x10 ; x20 ; x30 ) = (0; −4; −1) is shown in Fig. 13.7.

318 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 13.5 Lorenz system of Eq. (13.50) in x1 –x3 plane.

FIGURE 13.6 Lorenz system of Eq. (13.50) in x2 –x3 plane.

The separated chaotic motion of the system of Eq. (13.50) states with started initial conditions (x10 ; x20 ; x30 ) = (0; −4; −1) is shown in Fig. 13.8. First, as illustrated in Fig. 13.9, the estimation of the embedding dimension of the Lorenz system of Eq. (13.50) by using both techniques of the autocorrelation function ACF expressed in terms of lag and the average mutual information AMI expressed in terms of time lag show that the best computed time-lag param-

Optimal adaptive backstepping control Chapter | 13 319

FIGURE 13.7 Regrouped states trajectories variation of Lorenz system of Eq. (13.50).

FIGURE 13.8 Separated states trajectories variation of Lorenz system of Eq. (13.50).

320 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 13.9 Simulation result of the autocorrelation function and the average mutual information of Lorenz system of Eq. (13.50).

eter is τ = 37. Hence, an embedding dimension can be determined by using the well-known Cao algorithm; see Cao (1997). The embedding dimension value can also be used for the estimation of the correlation dimension. To measure the fractal dimension of the phase space of Lorenz system of Eq. (13.50), we can use the correlation dimension technique. This function can be used to specify the range in which the linear behavior appears. First, we verify that the estimated correlation dimension value does not depend on the computed embedding dimension, by using the correlation sums technique. Thus, as is shown in Fig. 13.10, the selected embedding dimension of Lorenz system of Eq. (13.50) is m = 4. According to Takens’ embedding theorem, the embedding dimension can be empirically detected by the plots of two mean quantities E1(d) and E2(d) which are expressed as a function of the dimension d as defined in Takens (1981). As illustrated in Fig. 13.11, the obtained optimal embedding dimension of Lorenz system of Eq. (13.50) is m = 4. Then, we noted that all these computations are usually used also for estimating several other nonlinear statistics, such as the generalized correlation dimensions, the sample entropy, and the proper Lyapunov exponent estimation. As indicated previously, the important characteristic of a chaotic system is its sensitivity to initial conditions. As a result, the close trajectories diverge exponentially fast. In order de verify the presence of this sensitivity, we can measure the divergence through time between system states. So, we use the maximum Lyapunov exponent that can compute the average rate of divergence of close trajectories in the system. Thus, graphically, the detection of chaos can be further verified by a simple examination of the curve of the maximal Lyapunov exponent. The maximum Lyapunov exponent for the considered system equal to

Optimal adaptive backstepping control Chapter | 13 321

FIGURE 13.10 Simulation result of the correlation sum and the correlation dimension of Lorenz system of Eq. (13.50) as a function of the radius r, with different embedding dimension values of m (m = 4, 5, 6, 7).

0.00006258, which is positive, thereby indicating the chaotic dynamics. Indeed, as illustrated in Fig. 13.12 the maximal Lyapunov exponents versus embedding dimension values of the studied Lorenz system of Eq. (13.50) is represented by nonlinear curves all evolve positively in increasing ways. This is an indicator that graphically justifies the presence of close trajectories diverging over time. As a consequence of this sensitivity to initial conditions we conclude to the existence of a chaotic attractor for the considered Lorenz system of Eq. (13.50). In addition to the detection of chaos, synchronization of chaos can also occur during the evolution of the Lorenz system, where the synchronous system becomes perfectly periodic. In order to verify this, the RP for the Lorenz system of Eq. (13.50) is shown in Fig. 13.13. The visual analysis and inspection of the system’s RP shows that its dynamics are no longer independent and correlation effects due to the endogenously structural change by recomposition of invariances can therefore appear. This is the reason why chaos synchronization is taken into consideration. Thus, the chaos synchronization is evident in the recurrence graph of Fig. 13.13. Besides graphical methods, the detection of chaos can be validated with the help of several various tests, including surrogate data testing and the 0 − 1 test

322 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 13.11 Simulation result of the embedding dimension of Lorenz system of Eq. (13.50).

FIGURE 13.12 Simulation result of the Lyapunov exponent of Lorenz system of Eq. (13.50) with different embedding dimension values of m (m = 4, 5, 6, 7).

for chaos. For instance, we apply here the 0 − 1 test for chaos, that detailed previously in Section 13.2.1. The graphical representation of both functions of the mean square displacement Mc (n), given in Step 3 and the modified mean square displacement Dc (n) given in Step 4 is illustrated in Fig. 13.14. These two functions select together the best value of the random number c = 1.9566 ∈ [0; π], indicated in Step 2.

Optimal adaptive backstepping control Chapter | 13 323

FIGURE 13.13 Simulation result of the recurrence plot of Lorenz system of Eq. (13.50).

FIGURE 13.14 Simulation result of both regions of the mean square displacement Mc (blue, dark gray in print) and the modified mean square displacement Dc (green, mid gray in print) for optimal determined c = 1.9566 value of Lorenz system of Eq. (13.50).

The correlation coefficient function Kc , given in Step 5 is expressed as a function of the random number c and represented in Fig. 13.15. By applying this statistical test indicates K = 0.5. As K tends to 1, then the underlying dynamics is chaotic. It should be noted that the chaotic dynamics of the Lorenz system of Eq. (13.50) is characterized by some statistical indicators that determine dynamics of the system states xi (t) (i = 1, 2, 3). The main statistical characteristics of each of these three states have been calculated and are shown in Table 13.1.

324 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 13.15 Simulation result of correlation coefficient Kc versus various values of random number c ∈ [0; 2.5] of Lorenz system of Eq. (13.50).

TABLE 13.1 The statistical characteristics of states. x1

x2

x3

Size

10001

10001

10001

Minimum

−20.200

−28.367

−1.000

1st quartile

−4.484

−4.076

18.183

Median

1.788

1.379

24.682

Mean

1.201

1.207

24.190

3rd quartile

7.991

7.897

29.492

Maximum

17.976

24.140

49.070

Standard deviation

7.961

8.887

7.972

The system of Eq. (13.50) has three equilibrium points: E1∗ = (0; 0; 0) that exist regardless of the values of the actual parameters β11 , β12 , and β13 , and ∗ = (±√β β − β ; ±√β β − β ; other two symmetrical fixed points E2,3 12 13 13 12 13 13 β12 − 1), that only exist when β12 > 1. The equilibrium points of the system with the given start point and parameters values are E1∗ = (0; 0; 0), E2∗ = (8.4853; 8.4853; 27), and E3∗ = (−8.4853; −8.4853; 27). The Jacobian matrix related to the Lorenz system at the critical point E ∗ = (x1∗ ; x2∗ ; x3∗ ) is written as ⎛ ⎞ −β11 β11 0 ⎜ ⎟ J ∗ (x1∗ ; x2∗ ; x3∗ ) = ⎝(β12 − x3∗ ) −1 −x1∗ ⎠ . x2∗ x1∗ −β13

Optimal adaptive backstepping control Chapter | 13 325

The eigenvalues λi (i = 1, 2, 3) are determined by solving the following matrix determinant:   ∗ ∗ ∗ ∗ J (x ; x ; x ) − λi .I3  = 0. 1 2 3 It should be noted that if we change the settings of the given Lorenz system of Eq. (13.50) as the parameter values are β11 = 16, β12 = 45.92, β13 = 4, and the initial conditions are (0; −4; −1). In that case, the resulting Lyapunov exponents are λ1 = 2.16, λ2 = 0.00, and λ3 = −32.4. Thus, according to Kaplan and Yorke (1979), the Kaplan–Yorke dimension of the novel chaotic system of Eq. (13.50) is calculated as follows:

j λi λ 1 + λ2 = 2.07. DKY = j +  i=1  = 2 + λj +1  |λ3 |

13.6.2 Optimal adaptive backstepping control and genetically optimized PID control for chaos synchronization of Lorenz systems 13.6.2.1 Adaptive backstepping stabilization control for Lorenz system According to Figs. 13.3 and 13.4, it is easy to see that even if there is no control input, the Lorenz system has a chaotic behavior. Backstepping design is proposed by Mascolo and Grassi (1997) to control a Lorenz chaotic system. In this section, we design an adaptive backstepping controller for globally stabilizing the Lorenz system with unknown parameters. For this purpose, a control signal u is added to the third equation system from Eq. (13.50). Let us consider the following parametric form Eq. (13.51) of Lorenz’s system, with unknown adaptive parameters: ⎧ ⎪ ⎪ x˙1 = β1T (x1 − x2 ), ⎪ ⎪ ⎨ x˙2 = −x1 x3 + β2T x1 − x2 , (13.51) ⎪ ⎪ x˙3 = x1 x2 − β3T x3 + u, ⎪ ⎪ ⎩ y=x , 1 where u is a backstepping controller to be designed using the states x1 , x2 , x3 and estimates βˆ1T , βˆ2T , βˆ3T of respectively the unknown parameters β1T , β2T , β3T of the system. It must be noted that the parameters estimation errors can be determined by presuming that β˜jT = βˆjT − βjT , ∀ j = 1, 2, 3, where β˜jT is the value of the error on adaptive parameter, βˆjT is the value of the estimated adaptive parameter, and βjT is the value of the current adaptive parameter. The backstepping method is applied to set states x10 , x20 , x30 to the origin

326 Backstepping Control of Nonlinear Dynamical Systems

point (0; 0; 0) by using the proposed control signal u of Eq. (13.52), which can be estimated by three steps of the procedure detailed in Section 13.4.2.2. On the other hand, the main result for the adaptive backstepping controller design for the Lorenz system of Eq. (13.50) is described by the following theorem. Theorem 13.7. The Lorenz chaotic system of Eq. (13.50) with unknown parameters β1 , β2 , and β3 is globally and exponentially stabilized for all values of (x10 ; x20 ; x30 ) ∈ R3 by applying the proposed adaptive backstepping controller of Eq. (13.52), given as follows: u = −ψ3 3 + 2 x1 − x1 x2 + βˆ3T x3 − [1 − τ12 ]β˙ˆ1T − β˙ˆ2T

(13.52)

where βˆ1T , βˆ2T , βˆ3T are estimates of the unknown parameters β1T , β2T , β3T , and β˙ˆ T , β˙ˆ T , β˙ˆ T are their parameters update law, respectively. 1

2

3

Proof. Let us start with the subsystem x1 , and introduce the variables of the backstepping for the different subsystems: 1 = x1 , 2 = x2 − θ 1 , 3 = x3 − θ 2 .

(13.53)

It must be noted that we establish the main result using Lyapunov stability theory developed by Hahn (1967). The adaptive backstepping for the stabilization control of the Lorenz’s system requires the application of the relevant steps of the procedure mentioned above. If we pose ψ1 = τ1 βˆ1T with τ1 > 0, βˆ1T > 0, and ψ1 > 0, then, in the new basis set ( 1 , 2 , 3 ), the three equations of total closed-loop system will be rewritten as follows: ˙1 = βˆ1T 2 − ψ1 1 − β˜1T [ 2 − 1 (1 + τ1 )], ˙1 = −x1 3 − x1 βˆ1T − 2 (1 − τ1 βˆ1T ) − β˜1T τ1 [ 2 − 1 (1 + τ1 )] − β˜2T x1 , ˙3 = −ψ3 3 + 2 x1 + β˜3T x3 . (13.54) The equations of stabilizing functions of the subsystems and the final control law are θ1 = −τ1 1 , θ2 = βˆ1T [1 − τ12 ] + βˆ2T + τ1 ,

u = −ψ3 3 + 2 x1 − x1 x2 + βˆ3T x3 − [1 − τ12 ]β˙ˆ1T − β˙ˆ2T .

(13.55)

Optimal adaptive backstepping control Chapter | 13 327

The final update rules are π1 =  1 [ 2 − 1 (1 + τ1 )], π2 =  1 [ 2 − 1 (1 + τ1 )] +  2 [ 1 − τ1 ( 2 − 1 (1 + τ1 ))], π3 =  1 [ 2 − 1 (1 + τ1 )] + 2 [x1 + τ1 ( 2 − 1 (1 + τ1 ))] −  3 x3 . (13.56) The Lyapunov functions system of the three equations are V˙1 = −ψ1 12 + βˆ1T 1 2 , V˙2 = −ψ1 2 − ψ2 2 − 2 3 x1 , V˙3 =

1 −ψ1 12

2 − ψ2 22

(13.57)

− ψ3 32 .

For all positive constants ψi (i = 1, 2, 3), the update law V˙3 from Eq. (13.57) is negative definite and thus satisfies the LaSalle–Yoshizawa theorem. The controller given by Eq. (13.57) does not change the equilibrium of the error dynamics. So, the Lorenz chaotic system of Eq. (13.50) with unknown parameters β1 , β1 , and β1 is globally and exponentially stabilized for all values of (x10 ; x20 ; x30 ) ∈ R3 by using the proposed backstepping controller of Eq. (13.52). This proposition is proved.

13.6.2.2 Optimal adaptive backstepping control for Lorenz system synchronization In this study, the essential feature of chaos synchronization is that two identical dynamical systems in master–slave configuration are synchronized such that all the states of slave system are synchronized with those of the master. Let us consider two Lorenz chaotic systems in parametric forms. The first is the slave system which is defined as follows: ⎧ ⎪ ⎨ x˙11 = β11 (x12 − x11 ), (13.58) x˙12 = −x11 x13 + β12 x11 − x12 , ⎪ ⎩ x˙13 = x11 x12 − β13 x13 , where x1i (i = 1, 2, 3) are the state variables and β1i (i = 1, 2, 3) are unknown system parameters. As seen the slave system of Eq. (13.58) is chaotic when the parameter values are taken as β11 = 10, β12 = 28, β13 = 3/8, and the initial states are (0; −4; −1). While, the second is the master system, which is defined with the unknown control parameters values U , as follows: ⎧ ⎪ ⎨ x˙21 = β21 (x22 − x21 ), (13.59) x˙22 = −x21 x23 + β22 x21 − x22 , ⎪ ⎩ x˙23 = x21 x22 − β23 x23 + U,

328 Backstepping Control of Nonlinear Dynamical Systems

where x2i (i = 1, 2, 3) are the state variables, β2i (i = 1, 2, 3) are unknown system parameters, and U is a control function to be determined later. It should be noted that the synchronization error states for each state variable are given by 1 = x21 − x11 , 2 = x22 − x12 , 3 = x23 − x13 .

(13.60)

Theorem 13.8. The identical Lorenz chaotic systems of Eq. (13.58) and Eq. (13.59) with unknown system parameters are globally and exponentially synchronized for all initial conditions by the adaptive backstepping control law of Eq. (13.82) and the parameter update law of Eq. (13.83). Proof. The synchronization problem is thus transformed into a problem of stabilization of the error model around zero. This involves establishing the relationship ( 1 , 2 , 3 ) = (0; 0; 0) which means to (x11 ; x12 ; x13 ) = (x21 ; x22 ; x23 ). If we drift over time the error states, then after substituting Eq. (13.58) and Eq. (13.59) into Eq. (13.60), we obtain ˙1 = β21 (x22 − x21 ) − β11 (x12 − x11 ), ˙2 = −x21 x23 + β22 x21 − x22 + x11 x13 − β12 x11 + x12 , ˙3 = x21 x22 − β23 x23 + U − x11 x12 + β13 x13 .

(13.61)

We treat this system as three cascade subsystems, each with one input and one output. We start the design with the first equation of the system of Eq. (13.61) and we contend the procedure until the last subsystem. It must be noted that during the design process, we make a change of coordinates w = φ(x1 , x2 , x3 ). First step: Let us consider the stability of the following first subsystem: ˙1 = β21 (x22 − x21 ) − β11 (x12 − x11 ) = β21 (x22 − x12 ) + β21 x12 − β21 (x21 − x11 ) − β21 x11 − β11 (x12 − x11 ) = β21 ( 2 − 1 + x12 − x11 ) − β11 (x12 − x11 ), (13.62) where βj 1 (j = 1, 2) is the value of the current adaptive parameter, βˆj 1 (j = 1, 2) is the value of the estimated adaptive parameter, and β˜j 1 = βˆj 1 − βj 1 (j = 1, 2) is the value of the error on adaptive parameter. If we pose

Optimal adaptive backstepping control Chapter | 13 329

f1 (x11 , x12 ) = −β11 (x12 − x11 ), then after substitution in Eq. (13.62) this gives ˙1 = β21 ( 2 − 1 + x12 − x11 ) + f1 (x11 , x12 ) = βˆ21 ( 2 − 1 + x12 − x11 ) − β˜21 ( 2 − 1 + x12 − x11 ) + f1 (x11 , x12 ). (13.63) For this first subsystem, we choose 2 as a virtual command. We then define the first virtual variable of the backstepping w1 = 1 and the virtual command w2 = 2 − θ1 . θ1 is the linearization function of the first subsystem and w2 is a new variable. It must be noted that w2 will not be used in the first step, but its presence is necessary to link the first subsystem of Eq. (13.63) which is expressed in function of w1 to the second subsystem expressed in terms of w2 , which will be used in the next step. Hence, the first subsystem becomes w˙ 1 = ˙1 = βˆ21 (w2 + θ1 − w1 + x12 − x11 ) − β˜21 (w2 + θ1 − w1 + x12 − x11 )+ f1 (x11 , x12 ). (13.64) In order to find the stabilizing function θ1 in Eq. (13.64), we choose an appropriate Lyapunov function, as follows: 1 1 T −1  β˜21 . V1 (w1 , β21 ) = w12 + β˜21 2 2

(13.65)

The time derivative of Eq. (13.64) is written as V˙1 = V˙1 (w1 , β21 ) T −1 βˆ˙ = w1 w˙ 1 + β˜21 21 ˆ = w1 [β21 (w2 + θ1 − w1 + x12 − x11 ) − β˜21 (w2 + θ1 − w1 + x12 − x11 )+ T −1 β˙ˆ f1 (x11 , x12 )] + β˜21 21 = βˆ21 w12 + βˆ21 w1 w2 + w1 [βˆ21 (θ1 + x12 − x11 ) + f1 (x11 , x12 )]+ β˜ T −1 [βˆ˙ − w (w + θ − w + x − x )]. 21

21

1

2

1

1

12

11

(13.66) In order to make V˙1 (w1 , β21 ) negative, just choose the stabilizing function as θ1 = −x12 + x11 −

f1 (x11 , x12 ) βˆ21

(13.67)

with the updated law β˙ˆ21 = π1 , where π1 = w1 [w2 + θ1 − w1 + x12 − x11 ] is the adjustment function of the first to be estimated parameter. Then, the time

330 Backstepping Control of Nonlinear Dynamical Systems

derivative V˙1 (w1 , β21 ) becomes T −1 ˙ˆ  (β21 − π1 ). V˙1 (w1 , β21 ) = −βˆ21 w12 + βˆ21 w1 w2 + β˜21

(13.68)

T −1 (β˙ˆ − π ) in V˙ (w , β ) will be elimiWe note that the second term β˜21 21 1 1 1 21 nated in the second step. Hence, the equation of the first subsystem becomes T w˙ 1 = βˆ21 (−w1 + w2 ) + β˜21 [w1 − (w2 + θ1 ) + x12 − x11 ].

(13.69)

So, the design coordinates have changed from ( 1 , 2 ) to (w1 , w2 ). Second step: Let us now consider the stability of the following second subsystem: ˙2 = −x21 3 − 2 − x13 1 + βˆ22 x21 − β˜22 x21 − β12 x11 .

(13.70)

Since w1 = 1 and w2 = 2 − θ1 , the second subsystem of Eq. (13.70) is expressed in function of (w1 , w2 ), as w˙ 2 = −x21 3 − w2 − θ1 − x13 w1 + βˆ22 x21 − β˜22 x21 − β12 x11 .

(13.71)

Now we introduce the new virtual variable of the backstepping w3 = 3 − θ2 , where θ2 is the stabilizing function for the transformed second subsystem of Eq. (13.71). We obtain w˙ 2 = −x21 w3 − x21 θ2 − w2 − θ1 − x13 w1 + βˆ22 x21 − β˜22 x21 − β12 x11 − θ˙1 . (13.72) If we pose f2 (x11 ) = −β12 x11 − θ˙1 , then the chosen appropriate Lyapunov function for the second subsystem of Eq. (13.70) is as follows: 1 1 T −1  β˜22 . V2 (w1 , w2 , β21 , β22 ) = V1 (w1 ) + w12 + β˜22 2 2

(13.73)

The time derivative of Eq. (13.73) is written as V˙2 = V˙2 (w1 , w2 , β21 , β22 ) T −1 β˙ˆ = V˙1 (w1 ) + w2 w˙ 2 + β˜22 22 2 2 ˆ = −β21 w − w − x21 w2 w3 1

(13.74)

2

+w2 [βˆ21 w1 − x21 θ2 − θ1 − x13 w1 + θ2 x21 + f2 (x11 )] +β˜ T −1 (β˙ˆ − π ) + β˜ T −1 (β˙ˆ T − w x ). 22

22

1

22

22

2 21

In order to make V˙2 (w1 , w2 , β21 , β22 ) negative, just choose the stabilizing function as βˆ21 w1 − θ1 − x13 w1 + βˆ22 x21 + f2 (x11 ) θ2 = (13.75) x21

Optimal adaptive backstepping control Chapter | 13 331

with the updated law β˙ˆ22 = π2 , where π2 = w2 x21 is the adjustment function of the second to be estimated parameter control. Then, the time derivative V˙2 (w1 , w2 , β21 , β22 ) becomes T −1 ˙ˆ V˙2 (w1 , w2 , β21 , β22 ) = −βˆ21 w12 − w22 − x21 w2 w3 + β˜21  (β21 − π1 ) + β˜ T −1 (βˆ˙ − π ). (13.76) 21

22

2

T −1 (β˙ˆ − π ) in V˙ (w , w , β , β ) will be We note that the third term β˜21 22 2 1 1 2 21 22 eliminated in the next step. Hence, the equation of the second subsystem of Eq. (13.66) becomes

w˙ 2 = −x21 w3 − w2 − βˆ21 w1 − β˜22 x21 .

(13.77)

Third step: Now, we consider the stability of the following third subsystem of Eq. (13.78): ˙3 = x21 x22 − βˆ32 x23 + U − x11 x12 + β13 x13 .

(13.78)

Since the virtual variable of the backstepping w3 = 3 − θ2 , where θ2 is the stabilizing function for the second subsystem, then w˙ 3 = ˙3 − θ˙2 . The transformed third subsystem of Eq. (13.79) is expressed in function of (w1 , w2 , w3 ), as w˙ 3 = −βˆ32 x23 − β˜32 x23 + U + x21 x22 + β13 x13 − x11 x12 − θ˙2 .

(13.79)

If we pose f3 (x11 , x12 , x13 , x21 , x22 ) = x21 x22 + β13 x13 − x11 x12 − θ˙2 , then the chosen appropriate Lyapunov function for the third subsystem of Eq. (13.78), is as follows: 1 1 T −1 V3 (w1 , w2 , w3 , β21 , β22 , β23 ) = V1 + V2 + w32 + β˜23  β˜23 . 2 2

(13.80)

The time derivative of Eq. (13.80) is written as V˙3 = V˙3 (w1 , w2 , w3 , β21 , β22 , β23 ) = V˙ + V˙ + w w˙ + β˜ T −1 β˙ˆ 1

=

2 −βˆ21 w12

3 3 23 − w22 − βˆ23 w32

23

+ w3 [−βˆ23 x23 + U + T −1 (β˙ˆ − π )+ f3 (x11 , x12 , x13 , x21 , x22 )] + β˜21 21 1 ˙ T −1 β˜  (βˆ − π )+ 22

22

(13.81)

2

T −1 (β˙ˆ T − w x ). β˜23 3 23 23

In order to make V˙3 (w1 , w2 , w3 , β21 , β22 , β23 ) negative, just choose the control function U as U = βˆ23 x23 − f3 (x11 , x12 , x13 , x21 , x22 )

(13.82)

332 Backstepping Control of Nonlinear Dynamical Systems

with the updated law β˙ˆ23 = π3 , where π3 = w3 x23 is the adjustment function of the third control parameter that to be estimated. Then, the time derivative V˙3 (w1 , w2 , w3 , β21 , β22 , β23 ) becomes V˙3 (w1 , w2 , w3 , β21 , β22 , β23 ) = −βˆ21 w12 − w22 − w32 + β˜ T −1 (β˙ˆ − π )+ 21 21 ˙ T −1 (βˆ β˜21 22 ˙ T −1 β˜23  (βˆ23

1

− π2 )+

(13.83)

− π3 ).

Theorem 13.8 is proved. It should be noted that the control signal of Eq. (13.83) can have some positive parameters. These parameters must be chosen correctly and an inappropriate choice of these parameters can skew the performance of the system as well as its instability. However, the determination of these parameters values by trial and error process is an expensive task and can sometimes lead to very flawed results. In the proposed controller of Eq. (13.83), let us take U extracted from Eq. (13.82) as an actual control input. In this context, the GAs allow one to find the appropriate values of these unknown parameters by minimizing the fitness function of Eq. (13.48). Thus, the given parameters values of the proposed GA, that is used for estimating the optimal both controllers parameters are defined in Table 13.2. TABLE 13.2 The genetic algorithms parameters. Parameters

Values

Variable

[−100; 100]

Population size

100

Maximum of generation

300

The selection function

Uniform stochastic

The probability of the crossover

0.75

Mutation rate

0.001

ci search interval

[0.1; 10]

The stopping criterion

The error performance criterion

In the simulation, the sampling time is 0.001. The search ranges for the backstepping parameters ki , i = 1, 2, 3 are bounded in [0, 10]. The parameters values are estimated for 20 iterations. In order to obtain the desired output synchronization signal of the master–slave Lorenz systems, we based on two assumptions. First the considered parameters values and the related initial state of the master system are taken as in the chaotic case, i.e. β11 = 10, β12 = 28, β13 = 3/8, and (0; −4; −1). Also, suppose that the considered parameters values and the related

Optimal adaptive backstepping control Chapter | 13 333

initial state of the slave system are taken as β21 = 16, β22 = 45.92, β23 = 4, and (0; −4; −1). Second, the selected design positive parameters values of both controllers are taken as k1 = k2 = 3.5. By applying the GA summarized in the previous Table 13.2, for the backstepping controller of Eq. (13.83), the estimated optimal parameters values of backstepping controller are given as c1 = 5.254, c2 = 3.753, and c3 = 4.672; see Fig. 13.16. The setting time of optimal backstepping controller obtain at 4.362 s, after 13 iterations. The obtained results are shown in Fig. 13.16, Fig. 13.17, Fig. 13.18, and Fig. 13.19. Besides, the fitness value obtained by the GA is 1.5010. Fig. 13.16 shows that trajectories variations of all system parameters in time ci (t), i = 1, 2, 3 obtained by using the optimal adaptive backstepping controller are different and fluctuating, but does not converge towards zero.

FIGURE 13.16 The values and trajectories of optimal adaptive backstepping controller of master– slave Lorenz systems parameters.

The response of master–slave Lorenz systems with backstepping controller and used GA is given in Fig. 13.17, which shows all trajectories’ variation xi (t), i = 1, 2, 3 for master–slave Lorenz systems states. In particular, all these generated states trajectories variation on starting with initial condition (0; −4; −1) may converge to zero after a shorter finite length of time. Fig. 13.18 shows that variations of error states i (t), i = 1, 2, 3 of the considered master–slave Lorenz systems may converge to zero too after a shorter finite length of time. Thus, the two chaotic master–slave Lorenz systems have been synchronized. Fig. 13.19 shows the x2 –x3 plane output of the synchronized master–slave Lorenz systems. After applying the optimal adaptive backstepping controller

334 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 13.17 Controlled synchronization states trajectories in time of master–slave Lorenz systems via optimal adaptive backstepping.

FIGURE 13.18 Controlled error states trajectories in time of master–slave Lorenz systems via optimal adaptive backstepping.

which combined with GA, we can see that good tracking performance is obtained. Thus, the synchronized master–slave Lorenz systems output converge asymptotically and globally towards a limit cycle. Fig. 13.20 shows the 3D result of synchronized of master–slave Lorenz systems after using the optimal adaptive backstepping controller with GA.

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FIGURE 13.19 The x2 –x3 plane of chaos synchronization master–slave Lorenz systems response by optimal adaptive backstepping controller combined with GA.

FIGURE 13.20 Phase portrait of synchronized master–slave Lorenz systems by optimal adaptive backstepping controller combined with GA.

13.6.2.3 Genetically optimized PID control for chaos synchronization of Lorenz systems In this subsection, we are interested in optimizing the gains of an efficient PID controller by using GAs. For this reason, we will need three channels assigned

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to each member of the population. These members will consist of a chain P, a chain I, and a chain D which will be evaluated everywhere or course of the GA, as true (floating point), where the numbers will be used to encrypt the population. The three terms are introduced into the GA by the declaration of a third line. Between −100 and 100, the controlled system makes it possible to give a step input and the error is distributed using a criterion of the performance of the appropriate error. This implies that a value of the total aptitude according to the magnitude of the error is assigned to the chromosome, the smallest error corresponds to the greatest value of the aptitude. Based on the given parameters values of the GA which are summarized in the previous Table 13.2. The obtained PID gain values of the best solution tracked over generations, for the Lorenz chaotic system are cP = 3.9201, cI = 1.3579, and cD = 4.9805; see Fig. 13.21. Fig. 13.21 shows that trajectories variations of all system parameters in time ci (t), i = P , I, D, obtained by using the genetically optimized PID controller are different and fluctuating, but do not converge towards zero.

FIGURE 13.21 The values and trajectories of genetically optimized PID controllers of master– slave Lorenz systems parameters.

Fig. 13.22 shows that trajectories variations of all system states in time xi (t), i = 1, 2, 3 using the genetically optimized PID controller may converge to zero after a shorter finite length of time. Fig. 13.23 shows that trajectories variations of all system error states in time i (t), i = 1, 2, 3 using the genetically optimized PID controller may converge to zero too after a shorter finite length of time.

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FIGURE 13.22 Controlled synchronization states trajectories in time of master–slave Lorenz systems via optimal adaptive backstepping via genetically optimized PID.

FIGURE 13.23 Average error dynamics in time of genetically optimized PID controllers synchronization of master–slave Lorenz systems.

13.6.2.4 Discussion In order to examine and compare the synchronization performance of two studied methods, we apply the synchronization quantity indicator proposed by Baker et al. (1998). This indicator is simply the average error propagation e on the sys-

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tem state variables, defined as e=



12 + 22 + 32 .

(13.84)

Fig. 13.24 compares the synchronization performance in time of both used controllers via the synchronization quantity indicator.

FIGURE 13.24 Comparison of controllers synchronization performances of master–slave Lorenz systems.

As illustrated in Fig. 13.24, the synchronization error e for both studied methods when controls are started at t = 0 are converging towards zero, but differently with a slight shift of time. It must be noted that we observe at t = 1 s, the synchronization using the backstepping control was already reached. While for the PID control, the synchronization was reached later at t = 3 s. So, the delay offset is 2 s. Although it is clear that the backstepping control works faster and is much easier to design, it requires only a single controller in the design, while three controllers are required for the PID design case. Despite the fact that the backstepping command has received considerable attention because of its many advantages, including simplicity and robustness, the controller structure in this scheme remains more complex, especially for optimized implementation. By comparing Eq. (13.1) defining the chaotic system with Eq. (13.4) defining the active controllers, it can be easily deduced that the controller is more complex than the system itself. In order to achieve the goal of optimal control of chaos synchronization, the proposed intelligent controller proves its efficiency and its rational ability to reduce considerably the controller complexity. Indeed, according to Corron et al. (2000), the problem of controller complexity always remain a crucial problem in many practical implementations. For instance, in communication and other interesting technical applications. In this study, the proposed intelligent backstepping controller can effectively contribute to solving the control communication complexity problem, which can be approximated to a standard part

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of optimization problem. However, according to Wong and Baillieut (2009), in this case, two main problems can be raised. First, the cost and skills needed to design intelligent control units. Second, the key is to better improve the complexity of this controller and to make it easily comparable, or even inferior, to that of the controlled device, if the intelligent controller optimization technique is desired to achieve a useful goal and not just a simple scientific curiosity.

13.7

Conclusion

In this chapter, we have studied the optimal adaptive backstepping control for chaos synchronization of nonlinear dynamical systems. As a result, a new nonlinear controller is presented via the rational determination of backstepping technique parameters by combining it with GAs. This proposed a new optimal adaptive backstepping controller which is used to control rationally the chaos synchronization of two identical Lorenz systems in master–slave configuration. In this optimized controller, parameters are obtained automatically and intelligently using GAs without trial and error by minimizing the fitness function, which is based on the squared control signal. This was established through virtual control laws, final control laws, and negative quadratic Lyapunov functions for chaos control to a stable equilibrium point for the used third-order autonomous chaotic systems (the Lorenz systems). In the proposed controller, the parameters of the backstep method are determined so that the system chaos is controlled in a minimal time. Uniformly globally asymptotic stabilization is obtained according to LaSalle’s theorem and LaSalle–Yoshizawa’s theorem. At first, we put these systems in the strict parametric form, necessary condition for the application of the method of backstepping, laws of command and control in pursuit were then developed by adopting a formalism based on the backstepping. However, since third-order autonomous chaotic systems may have some singularities, then an algorithm for optimizing design constants and even gains for PID correctors was essential for improving convergence to the stable equilibrium point. The global pursuit for a reference trajectory. In addition, by applying the backstepping technique, the nonlinearities can be processed by two different means: the useful nonlinearities which contribute in the stabilization were retained in the Lyapunov function, while the nonlinear non-usefulnesses were replaced by a linear control which requires the establishment of optimal resulting control laws ensuring a certain property of robustness. To obtain good control results, the most common method is to possibly choose a smaller optimization algorithm. The GA uses more memory space and therefore takes longer to find the best solution; optimality is the major disadvantage of GA. For this reason, we believe that it is possible to further improve the resulting solution. It suffices to simply replace the GAs combined with the backstepping method by other artificial intelligence techniques such as the particle swarm optimization technique.

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Chapter 14

Backstepping controller for nonlinear active suspension system Vineet Kumara , K.P.S. Ranaa , Ahmad Taher Azarb,c , and Sundarapandian Vaidyanathand a Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India, b Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University,

Riyadh, Saudi Arabia, c Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, d Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

14.1

Introduction

Generally, passive suspension system (PSS) in a car is responsible for ride comfort and road holding. PSS holds the car body and transfer all forces between car body and the road surface. In any classical car, PSS comprises a spring and a damping component. Usually, the spring and damping coefficients of PSS are selected according to the requirement of ride comfort, road holding, and handling specifications. On the other hand, PSS has its own limitation, it can only provide comfort ride up to an extent and generally it fails to offer the desired ride comfort to the occupants of vehicle (Rajamani, 2011; Liu et al., 2014; Kashem et al., 2018). Therefore, natural limitations of conventional PSS have motivated the scientist and researchers to investigate and explore the active control of suspension systems and this scheme has become famous as active suspension system (ASS). Generally, the active control schemes require a significant amount of energy to generate the essential control actions. A fully ASS can possibly offer better performance than PSS (Guglielmino et al., 2008; Savaresi et al., 2010; Rajamani, 2011; Liu et al., 2014; Kashem et al., 2018). Also, the dynamical model of suspension system comprises nonlinear differential equations and therefore, as per the literature, it is not recommended to employ conventional linear controllers. So, engineers are trying out nonlinear controller along with intelligent controller for such kind of complex and nonlinear system (Aström, 1983; 1989; Aström and McAvoy, 1992; Aström and Hägglund, 1995; 2006; Johnson and Moradi, 2005; Kler et al., 2018a; Ouannas et al., 2017; Kumar et al., 2011a; 2011b; 2012; Ghoudelbourk et al., 2016; Kumar and Rana, 2016; Vaidyanathan and Azar, 2016a,b,c; Kumar and Mittal, 2010; Passino and Yurkovich, 1988). Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00021-0 Copyright © 2021 Elsevier Inc. All rights reserved.

347

348 Backstepping Control of Nonlinear Dynamical Systems

For the past two decades, intelligent controllers such as fuzzy logic controller, neuro-fuzzy controller, etc., have become quite popular among the researchers, especially in the process and manufacturing industries due to their better performances in comparison with conventional controller (Ammar et al., 2018; Pilla et al., 2019; Aström, 1989; Aström and McAvoy, 1992; Kumar and Rana, 2016; Kumar et al., 2011a; 2011b; 2012; 2018a; Abdelmalek et al., 2018; Kumar and Mittal, 2010; Khettab et al., 2018; Djeddi et al., 2019; Passino and Yurkovich, 1988; Vaidyanathan and Azar, 2016e). Many potential works have been reported along this line for the following fields: robotic manipulator (Sharma et al., 2014a; 2014b; Kumar and Rana, 2017a; 2017b; Kumar et al., 2018b; 2018c; 2019a; 2019b; Azar et al., 2020a,b), hybrid electric vehicle (Kumar et al., 2016), integrated power system (Nithilasaravanan et al., 2018; Gorripotu et al., 2019), binary distillation column (Mishra et al., 2015), sensor and controller noise suppression in closed loop (Kumar et al., 2017; Kumar and Rana, 2017a; 2017b), electrodynamic shaker for automotive and aerospace vibration testing (Rana, 2011), vibration actuator (Rana et al., 2010), etc. Further, in order to improve the performance of ASS, lots of studies on efficient control of ASS have been carried out using intelligent controllers such as type 1 and 2 fuzzy logic controllers, adaptive neuro-fuzzy control, adaptive genetic-based optimal fuzzy control, etc., and have been reported in the literature (Liu et al., 2014; Guglielmino et al., 2008; Savaresi et al., 2010). Some of the significant work in this domain are cited here. Lin and Lian (2011) proposed an intelligent controller namely hybrid self-organizing fuzzy and radial basis function neural network controller (HSFRBNC) and successfully tested on ASS. Simulation investigation revealed that HSFRBNC out-perform conventional self-organizing fuzzy controller (SOFC) by enriching the service life of the PSS and the ride quality of a vehicle (Lin and Lian, 2011). Chiou et al. (2012) presented a novel design technique for tuning the scaling factors of an optimal fuzzy PID controller for ASS using particle swarm optimization and considerably improved the ride comfort in the car (Chiou et al., 2012). Further, Lin and Lian (2013) developed a gray prediction SOFC (GPSOFC) for an ASS. Experimental outcomes clearly showed the supremacy of GPSOFC over SOFC and PSS by improving the ride comfort of driver in a car (Lin and Lian, 2013). Wang et al. (2015) presented a fuzzy gain scheduling of conventional parallel PID controller for an ASS. The gains of fuzzy controller were tuned using improved cultural optimization by minimizing the vertical acceleration of a quarter car model. Presented simulation studies clearly showed that fuzzy PID control scheme successfully eliminate the sprung mass vertical acceleration and provide enhance ride quality as compared to PSS (Wang et al., 2015). Bououden et al. (2016) proposed a robust nonlinear predictive control technique by utilizing the Takagi–Sugeno fuzzy method for nonlinear ASS. It has been observed that proposed control scheme out performed model predictive control method and provide improved ride comfort (Bououden et al., 2016). Recently, Kumar et al. developed a self-tuned fractional-order fuzzy PD (FO-FPD) controller for an un-

Backstepping controller for nonlinear active suspension system Chapter | 14 349

certain and nonlinear ASS wherein simulation results clearly revealed that the developed FO-FPD controller offered much better comfort drive with respect to its counterpart (Kumar et al., 2018a,b,c,d). It has been revealed from the various reported work that in case of FLC, one has to establish stability of overall closed loop control system separately (Kumar et al., 2011a,b; Passino and Yurkovich, 1988). Generally, for a nonlinear process, one would require a nonlinear controller which ensure the stability of overall closed loop control system. In this regard, backstepping control scheme is one of the best possible control methods which utilize Lyapunov function to provide control solution which guarantees stability of overall closed loop control system (Vaidyanathan et al., 2018; 2015; Shukla et al., 2018; Vaidyanathan and Azar, 2016d). One of the limitations of this control scheme is that it is totally based on the mathematical model. If there is a mismatch in the mathematical model it will affect the performance of nonlinear control scheme (Khalil, 1991; Krstic et al., 1995; Zhou and Wen, 2008; Wang et al., 2017; Rudra et al., 2017; Yagiz and Hacioglu, 2008). Further, many works have utilized backstepping control techniques for effectively control the ASS and improving the ride quality and road holding, some of the current work is as follows. Yagiz and Hacioglu (2008) presented a backstepping control scheme to regulate an ASS of full vehicle. The performance of control scheme was assessed both in time and frequency domain under different road conditions, and it was found that the proposed control method performed effectively by considerably improving the ride quality of passengers in the vehicle (Yagiz and Hacioglu, 2008). Sun et al. (2013) developed an adaptive control scheme based on the backstepping control strategy for half car ASS with rigid constraints. The efficacy of the developed method has been demonstrated by a design example (Sun et al., 2013). Goyal et al. (2015) performed a comparative investigation of Barrier Lyapunov Function (BLF) and Quadratic Lyapunov Function (QLF) based backstepping controllers for quarter car nonlinear ASS. The gains of both controllers were tuned using a genetic algorithm. Interestingly, it has been noted that an optimized QLF-based controller demonstrated much superior performance as compared to an optimized BLF based controller (Goyal et al., 2015). Basturk (2016) design an adaptive control scheme based on backstepping approach for maintaining the ride quality and safety of the car body. The worthiness of control scheme is demonstrated with the help of simulated examples and by performing the different road test (Basturk, 2016). Nkomo et al. (2017) conducted a comparative study of backstepping and sliding mode control (SMC) approaches for a half car ASS. The experimental investigations demonstrated that backstepping based control scheme out-performed SMC and PID in terms of ride quality improvement (Nkomo et al., 2017). Pang et al. (2018) presented a constraint adaptive control technique based on backstepping approach for nonlinear ASS with parameter ambiguities and safety restrictions. The effectiveness of proposed control technique is demonstrated by performing numerical simu-

350 Backstepping Control of Nonlinear Dynamical Systems

lations (Pang et al., 2018). Recently, Zheng et al. (2019) developed an adaptive control method by utilizing backstepping approach for a cloud-aided nonlinear ASS for full vehicle with 7-degree of freedom (DOF) model. Simulation outcomes demonstrated the efficacy of the developed control technique over the PSS (Zheng et al., 2019). From the literature survey, it is clear that the performance of any controller will totally depend on its scaling factors. For, intelligent and nonlinear controllers there is no specific method is available in the literature to tune the parameters of controllers in contrast to linear conventional PID controllers the available tuning methods are Ziegler–Nichols (Ziegler and Nichols, 1942) and Cohen–Coon (Cohen and Coon, 1953). These days, due to the development of many metaheuristic optimization techniques, one can tune the controllers for customized performance indices for any complex process or plant. Many such applications are reported in the literature were parameters of fuzzy and nonlinear controller are optimized by various optimizing techniques (Bhatnagar et al., 2016; Kathuria et al., 2018; Kler et al., 2018b; Sharma et al., 2014a; 2014b). In the preset of this chapter, a comparative study is performed among the considered controllers i.e. backstepping based controller, fuzzy PD controller with 7 triangular memberships for input and output variables, and conventional PD controller for quarter car nonlinear ASS model for bumpy road surface. Scaling factors of considered controllers were optimized using the Grey Wolf Optimizing (GWO) (Mirjalili et al., 2014; Muro et al., 2011) technique by minimizing the root mean square of the vertical acceleration (RMSVA). Simulation results revealed that nonlinear backstepping controller out-performed the rest of the control schemes and provide superior ride quality under varying bumpy road profile. The chapter is organized as follows: after a brief introduction and literature survey in Section 14.1, the dynamic model of nonlinear ASS and problem statement is presented in Section 14.2. Backstepping, FPD, and conventional PD controllers are synthesized in Section 14.3. Also, a brief introduction of the optimizing method i.e. GWO and corresponding performance index, RMSVA are introduced in this section. Section 14.4 presents a comparative performance assessment of backstepping, fuzzy PD, and conventional PID controllers for nonlinear ASS for bumpy road disturbance in closed loop. Finally, conclusions are drawn in Section 14.5.

14.2 Plant model and problem statement In this section dynamic model of nonlinear ASS is presented. Also, the problem statement is stated with clearly specifying all hard constraints.

14.2.1 Nonlinear active suspension system A 2 DOF, nonlinear active suspension system of a quarter car, as shown in Fig. 14.1, has been considered for study. The two DOF comprises the verti-

Backstepping controller for nonlinear active suspension system Chapter | 14 351

cal motions of the sprung mass ‘zs ’ and unsprung mass ‘zus ’. They are linked by a passive suspension system having spring-damper system along with an actuator to transform this combine solution to an active suspension system. Also, the unsprung mass is connected to the road surface through tire which can be further modeled as an additional spring-damper system.

FIGURE 14.1 Schematic diagram of quarter car.

The dynamic mathematical model of quarter car is as follows: ms z¨s + Fs + Fd = uf

(14.1)

mus z¨us − Fs − Fd + Fts + Ftd = −uf

(14.2)

where ms and mus are the sprung and unsprung mass, respectively. Fs and Fd are the forces due to spring and damper, respectively. Fts and Ftd are the elastic and damping forces of the tire, respectively. z¨s and z¨us are the vertical acceleration of sprung and unsprung mass, respectively. uf is the required control action for the suspension system. These forces are defined as Fs = Ks (zs − zus ) + Ksn (zs − zus )3  Bd1 (˙zs − z˙ us ) , if (˙zs − z˙ us ) ≥ 0 Fd = Bd2 (˙zs − z˙ us ) , if (˙zs − z˙ us ) < 0

(14.3) (14.4)

Fts = Kt (zus − zr )

(14.5)

Ftd = Bt (˙zus − z˙ r )

(14.6)

where Ks and Ksn are the stiffness coefficients of linear and nonlinear terms of the spring, respectively. z˙ s and z˙ us are the vertical speed of sprung and unsprung mass, respectively. Bd1 and Bd2 are damping coefficients for extension and compression, respectively. Kt and Bt are spring and damping constants

352 Backstepping Control of Nonlinear Dynamical Systems

for tire, respectively. zr is the road surface input disturbance and z˙ r is its first derivative. (zs − zus ) and (zus − zr ) are the suspension and tire deflection, respectively. Normally, the sprung mass, i.e. the mass of car body ms fluctuates as the number of travelers varies and introducing the uncertainty in the system. Now, describing the state variables as x1 = zs , x2 = z˙ s , x3 = zus , and x4 = z˙ us , the system can be defined as x˙1 = x2  1  −Fs − Fd + uf x˙2 = ms x˙3 = x4 1 x˙4 = (Fs + Fd − Fts − Ftd − uf ) mus

(14.7) (14.8) (14.9) (14.10)

14.2.2 Problem statement The complete performance of an ASS can be evaluated based on the following aspects: A. Ride comfort In any vehicle, ride comfort of the occupants is one of the main parameters to judge its performance. Essentially what it means is that one has to design a compensator which can suppress the vertical acceleration i.e. z¨s of the sprung mass. B. Road holding Safety of the vehicle is the main concern of the automobile engineer. While designing this feature in any vehicle, one has to ensure that there should be a strong and never-ending contact between tire and road surface. This term is satisfied if the dynamic load on wheels is less than the static load. Fts + Ftd < (ms + mus ) g

(14.11)

Fts + Ftd 0 is a constant. Substituting the value of φ0 from Eq. (14.27) in Eq. (14.26), yields  ∂V0  f1 + g1 φ0 = −K1 e12 ≤ −W (e1 ) ∂e1

(14.28)

W (e1 ) = K1 e12 ≥ 0

(14.29)

Therefore,

is positive definite and e1 is guaranteed to converge to zero asymptotically. Step – II: Re-writing sprung mass system defined by Eq. (14.17) in the form x˙2 = f2 + g2 uf x˙2 = −MFf + M ∗ uf

(14.30) (14.31)

where f2 = −MFf and g2 = M. Let uf be uf =

 1  ua − f2 g2

(14.32)

Substituting uf from Eq. (14.32) in Eq. (14.30), one gets x˙2 = f2 + g2 ∗

 1  ua − f2 g2

x˙2 = ua

(14.33) (14.34)

Now, adding and subtracting g1 φ0 in Eq. (14.30) yield e˙1 = f1 + g1 x2 + g1 φ0 − g1 φ0

(14.35)

Rearranging Eq. (14.35), e˙1 = f1 + g1 φ0 + g1 (x2 − φ0 )

(14.36)

Substituting e2 = x2 − φ0 in Eq. (14.36) e˙1 = f1 + g1 φ0 + g1 e2

(14.37)

356 Backstepping Control of Nonlinear Dynamical Systems

Now, differentiating Eq. (14.20), e˙2 = x˙2 − φ˙ 0

(14.38)

Substituting x˙2 = ua from Eq. (14.34) in Eq. (14.38), e˙2 = ua − φ˙ 0

(14.39)

e˙2 = v2

(14.40)

Let

where v2 is now being a control signal. Consider a control Lyapunov function 1 V1 = V0 + e22 2

(14.41)

Now, differentiating Eq. (14.41), V˙1 = V˙0 + e2 e˙2

(14.42)

Substituting e˙2 = v2 in Eq. (14.42), or ∂V0 e˙1 + e2 v2 V˙1 = ∂e1

(14.43)

Substituting e˙1 from Eq. (14.37) in Eq. (14.43),  ∂V0  f1 + g1 φ0 + g1 e2 + e2 v2 V˙1 = ∂e1

(14.44)

 ∂V0 ∂V0  f1 + g1 φ0 + g1 e 2 + e2 v 2 V˙1 = ∂e1 ∂e1

(14.45)

∂V0 V˙1 = −W (e1 ) + g1 e 2 + e2 v 2 ∂e1

(14.46)

or

Using Eq. (14.28),

Therefore, according to the Lyapunov stability criterion, subsystem – II will be asymptotically stable if V˙1 is negative definite. Thus, in order to fulfill this criterion, we choose v2 as v2 = −K2 e2 − where K2 > 0 is a constant.

∂V0 g1 ∂e1

(14.47)

Backstepping controller for nonlinear active suspension system Chapter | 14 357

Substituting, v2 from Eq. (14.47) in Eq. (14.46),

∂V0 ∂V0 ˙ V1 = −W (e1 ) + g1 e2 + e2 −K2 e2 − g1 ∂e1 ∂e1 V˙1 = −W (e1 ) − K2 e22 ≤ 0

(14.48) (14.49)

Now, rearranging Eq. (14.39) and putting e˙2 = v2 , ua = v2 + φ˙ 0

(14.50)

Substituting v2 from Eq. (14.47) in Eq. (14.50) and rearranging it, ua = −K2 e2 −

∂V0 ∂φ0 g1 + e˙1 ∂e1 ∂e1

Further, substituting ua from Eq. (14.51) in Eq. (14.32),

1 ∂V0 ∂φ0 uf = g1 + e˙1 − f2 −K2 e2 − g2 ∂e1 ∂e1

(14.51)

(14.52)

Furthermore, substituting f2 = −MFf and g2 = M in Eq. (14.52), uf =

 1  −K2 e2 − e1 − K1 e˙1 − (−MFf ) M

(14.53)

Now, putting e˙1 = x˙1 = x2 and e2 = x2 − φ0 = x2 + K1 x1 in Eq. (14.53), uf = Ff +

1 [−K2 (x2 + K1 x1 ) − x1 − K1 x2 ] M

(14.54)

Therefore, aggregated control action is uf = Ff +

1 [−K2 x2 − K1 K2 x1 − x1 − K1 x2 ] M

(14.55)

14.3.2 Fuzzy PD controller The classical PD controller in time domain and in position form is defined as uPD (t) = KP e(t) + KD

de(t) dt

(14.56)

where KP and KD are the proportional and derivative gains, respectively. e(t) and uPD (t) are the error (between reference setpoint and process variable) and control signals, respectively. The FPD controller is implemented based upon the position form of conventional PD controller defined in Eq. (14.56). Re-writing Eq. (14.56) with new

358 Backstepping Control of Nonlinear Dynamical Systems

controller gains K1FPD and K2FPD for error and rate of change of error signals, respectively. de(t) (14.57) = uPD (t) dt Therefore, the FPD controller is realized based on position form of classical PD controller. Input to the FPD controller are error signal ‘K1FPD e(t)’ and rate of change of error signal ‘K2FPD de(t) dt ’, respectively, while output signal is uPD (t) having absolute value. Further, in order to increase the overall DOF of current control solution and another gain KUFPD is introduced in the right-hand side of Eq. (14.57). We have K1FPD e(t) + K2FPD

K1FPD e(t) + K2FPD

de(t) = KUFPD uPD (t) dt

(14.58)

de(t) = uFPD (t) dt

(14.59)

Re-writing Eq. (14.58), K1FPD e(t) + K2FPD where uFPD (t) = KUFPD uPD (t)

(14.60)

is the output of FPD controller. FPD controller was realized using seven triangular membership functions (MFs) fully overlapped for the input and output variables. The abbreviations of considered MFs are as follows: “negative big (NB), negative medium (NM), negative small (NS), zero (Z), positive small (PS), positive medium (PM) and positive big (PB)”, and the range of input and output MFs are [−1, 1]. Also, the control engineer knowledge is utilized to form the rule base for FPD controller (Passino and Yurkovich, 1988); see Table 14.2. For inference mechanism, max-min inference is used and for defuzzification center of the area method is considered. TABLE 14.2 Rule base (Passino and Yurkovich, 1988). r

NB

NM

NS

NB

NB

NS

PS

PB

PB

PB

PB

NM

NB

NM

ZE

PM

PM

PB

PB PB

e

ZE

PS

PM

PB

NS

NB

NM

NS

PS

PM

PB

ZE

NB

NM

NS

ZE

PS

PM

PB

PS

NB

NB

NM

NS

PS

PM

PB

PM

NB

NB

NM

NM

ZE

PM

PB

PB

NB

NB

NB

NB

NS

PS

PB

Backstepping controller for nonlinear active suspension system Chapter | 14 359

14.3.3 Conventional PD controller The classical PD controller is considered for the comparative investigation and is mathematically defined in Eq. (14.56).

14.3.4 Tuning of gains of controllers The scaling factors of any controller plays a vital role in the overall performance of that controller in closed loop as per the customize performance indices. As such there is no particular tuning method is, reported in the literature, for the nonlinear and intelligent controllers. Consequently, various optimization algorithms have been used to tune the scaling factors of the controllers with specify performance indices. In the present study, GWO has been considered to optimize the parameters of backstepping, FPD, and PD controllers.

14.3.4.1 Grey Wolf optimizing algorithm GWO algorithm is based upon the concept of swarm intelligence and proposed by Mirjalili et al. (2014). It is influenced by the behavior of gray wolfs (Canis Lupus). In other words, it copies the leadership hierarchy and hunting method of gray wolfs. There are 4 types of gray wolves namely, alpha, beta, delta, and omega that belong to the Canidae family. Due to this hierarchy structure it shows another exciting social feature known as group hunting. The main stages of group hunting according to Muro et al. (2011) are tracking, chasing, and approaching the prey; pursuing, encircling, and harassing it until it stops moving; attack the prey. For more details of the GWO algorithm, the reader is advised to refer to Muro et al. (2011) and the references therein. Furthermore, one can consult Gupta et al. (2015) for a step by step development of the GWO algorithm toolkit in LabVIEW environment. For tuning of controllers, population size was selected as 30 and the maximum number of iterations was retained as 100. 14.3.4.2 Cost function In the present study, performances of the controllers are evaluated by a cost function f which comprises RMS value of the vertical acceleration i.e., RMSVA (m/s2 ). We have N 1   2 Z¨ s  (14.61) f =J = N t=1

where N is the number of samples. In the present study, for tuning of controllers, a very common bumpy road profile is considered and is defined as follows: ⎧     ⎨ A 1 − cos 2πv t , 0 ≤ t ≤ Lv , 2 L (14.62) Zr (t) = ⎩ 0, t > Lv ,

360 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 14.2 Cost function v/s generation curve.

FIGURE 14.3 Assessment of RMS of the vertical acceleration of controllers.

TABLE 14.3 Tuned gains of backstepping, FPD, and PD controllers. Controllers

Backstepping

FPD

Gains

K1

K2

K1FPD

K2FPD

PD KUFPD

KP

KD

Value

5.0342

5.0066

−8490.32

9.99315

7637.59

9.41546

9997.88

where L and A are the length and the height of the bump, respectively, and v is the car forward velocity. In the current work, A = 0.1 m, L = 5 m, and v = 20 m/s are considered for tuning the controllers. The cost function versus iteration curves for backstepping, FPD, and PD controllers for bumpy road surface is presented in Fig. 14.2 and their cost function (RMSVA) values are compared using the bar chart in Fig. 14.3. Also, the tuned gains of all the three controllers are tabulated in Table 14.3. Further, in the subsequent section, the performances of backstepping, FPD, and PD controllers are evaluated for different type of bumpy road profiles. Also, the tuned controllers have been examined under model uncertainty.

Backstepping controller for nonlinear active suspension system Chapter | 14 361

14.4

Results and discussions

As one knows that ride quality can be significantly enhanced by suppressing the vertical acceleration. This objective can be accomplished by tracking a zero setpoint for vertical displacement of the sprung mass while maintaining all hard constraints within their given range. In the present work, exhaustive simulations were carried out to evaluate the performances of ASS having backstepping, FPD, and PD controllers, and PSS under bumpy road profiles. The tuned parameters of all the controllers under study were retained unaffected throughout the simulation investigations. The various hard constraints and simulation parameters are listed in Table 14.1. Simulations are performed using National Instrument software, LabVIEW, and its add-ons “Simulation and Control Design Toolkit”.

14.4.1 Bump road surface Single bump road profile is mathematically defined by Eq. (14.62). It is very common and usually required to check the speed of vehicle near school, hospital, villages, etc. Therefore, to assess the ride quality of any vehicle it is a perfect road input disturbance. For single bump surface, the variation in sprung mass vertical displacement, vertical acceleration, and tire deflection for backstepping, FPD, and PD controllers in closed loop are shown in Fig. 14.4A–C.

FIGURE 14.4 Performance comparison of controllers for single bump road surface of height 0.1 m and length 5 m: (A) sprung mass vertical displacement; (B) sprung mass vertical acceleration; (C) tire deflection; (D) tracking force; (E) ratio between dynamic and static load of tire; (F) suspension movement.

362 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 14.4 (continued)

It has been clearly observed that ASS with backstepping, FPD, and PD controllers demonstrate excellent ride comfort in comparison of PSS. Among the three considered controller’s backstepping and FPD conventional PD controllers out-performed the conventional PD controller for single bumpy input. Both the backstepping and FPD controllers significantly reduced the vertical displace-

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ment of the sprung mass and are sufficiently capable to suppress the vertical acceleration and considerably improve the ride quality of vehicle. Also, RMSVA of ASS with backstepping, FPD, and PD controllers in closed loop are 0.00708363, 0.0360061, and 0.540906, respectively. It is clear from the cost function (RMSVA) that backstepping controller give much superior performance in contrast to FPD controller. Further, it may be noted that all the controllers are able to hold the three hard constraints namely force, road holding, and suspension movement within limits, as shown in Fig. 14.4D–F.

14.4.1.1 Sprung mass uncertainty Generally, number of passengers vary in any vehicle. Due to the uncertainty in sprung mass of vehicle, the dynamic model of vehicle become highly uncertain. Therefore, robustness of controllers can be assessed by handling this model uncertainty in terms of cost function i.e. RMSVA.

FIGURE 14.5 RMSVA for variation in sprung mass of vehicle from 300 kg to 900 kg for single bump road surface.

The sprung mass of vehicle was varied from 300 kg to 900 kg and the corresponding variation in RMSVA is shown in Fig. 14.5. It has been observed that when passenger occupancy of vehicle is lowest i.e. sprung mass of vehicle is 300 kg the value of RMSVA is higher for conventional PD controller as compared to backstepping and FPD controllers. But the backstepping controller demonstrates much better performance as compared to FPD controller. As the number of passengers increases in the vehicle, RMSVA of conventional PD controller decreases. On the other hand, in the case of backstepping and FPD controllers it almost remained same and when the sprung mass is 900 kg the backstepping controller retains minimum RMSVA and stood first among three controllers under study. A typical variation of the vertical acceleration, tire deflection, road holding, and suspension movement for sprung mass of 300 kg are depicted in Fig. 14.6 and the resulting cost functions i.e. RMSVA for controllers under study are compared with the help of the bar chart in Fig. 14.7. From Fig. 14.6C, it is clear that conventional PD controller is not able to hold the road holding parameter below unity and thus there may be a chance that the vehicle having ASS with conventional PD may topple and a serious accident may occur.

364 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 14.6 Performance comparison of controllers for single bump road surface for sprung mass 300 kg: (A) sprung mass vertical acceleration; (B) tire deflection; (C) ratio between dynamic and static load of wheel; (D) suspension travel.

14.4.1.2 Uncertainty in height of bump Normally, the height of the bump may not be constant. It generally may vary place to place. Therefore, the controller must be capable to handle such uncer-

Backstepping controller for nonlinear active suspension system Chapter | 14 365

FIGURE 14.7 Comparison of RMS of the vertical acceleration of controllers for sprung mass 300 kg.

FIGURE 14.8 RMSVA for variation in height of bump from 0.1 m to 0.2 m with a constant bump length of 5 m.

tain variations in the bump height and provide a comfort ride to the passengers seating in the vehicle. Also, one has to be very careful while designing the controller i.e. the control action should be such that it strictly maintains all considered hard constraints, particularly the “ratio of dynamic and static load of vehicle tire” as it is required to hold the vehicle on the ground i.e. to maintain firm contact between vehicle tire and road. In the current work, height of the bump was varied from 0.1 m to 0.2 m, while the length of a bump remained fixed to 5 m. The variation of bump’s height and associated change in RMSVA is plotted in Fig. 14.8. It has been observed that as height of bump increased gradually the cost function i.e. RMSVA for all the considered controllers also increased steadily but in the case of FPD and conventional PD controller the changes in RMSVA is higher than the backstepping controller. As far as hard constraints are concerned, once the bump height crosses 0.14 m, conventional PD and FPD controllers failed to keep the dynamic load on tire less than the static load and thus it is very difficult for both these controllers to hold the vehicle in contact with road surface. In such case it is very difficult to avoid any accident. On the other hand, backstepping controller comfortably kept the ratio of dynamic and static load of wheel less than one up to the height of bump 0.19 m. The typical variation of the vertical acceleration, tire deflection, road holding, and suspension movement for the bump height 0.19 m and length 5 m is depicted in Fig. 14.9. The RMSVA value for

366 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 14.9 Performance for single bump road surface of height 0.19 m and length 5 m: (A) sprung mass vertical acceleration; (B) tire deflection; (C) ratio between dynamic and static load of wheel; (D) suspension travel.

backstepping, FPD, and conventional PD controllers are 0.399064, 0.746168, and 1.02778, respectively. Again, it has been established that the backstepping controller out-performed the rest of the controllers under study and give a much superior comfort ride.

Backstepping controller for nonlinear active suspension system Chapter | 14 367

14.4.2 Multiple bumps road profile Generally, it has been observed near the toll plaza on the highway, successive bumps are provided to check the speed of vehicles. Also, it has been noted that while passing such a road profile usually the vehicle drivers lower their speed but still passengers can feel significant vertical movement and sometimes it can hurt the occupants of the vehicles. In other words, the ride comfort during such kind of road surface is significantly reduced in the vehicle having PSS. Therefore, in this study, such a kind of road profile is considered to critically evaluate all controllers in ASS. In the present investigation, to assess the controllers, a road surface having multiple bumps is considered and is given by ⎧       ⎪ L3 A ⎪ 1 − cos 2πv t , 0.75 ≤ t ≤ 0.75 + , ⎪ ⎪ 4 L v 3 ⎪ ⎪   ⎪ ⎪ ⎪ 0, 0.75 + Lv3 < t < 3.0, ⎪ ⎪ ⎪ ⎪       ⎪ ⎪ L2 A 2πv ⎪ 1 − cos t , 3.0 ≤ t ≤ 3.0 + ⎪ ⎪ 3 L2 v , ⎪ ⎪   ⎪ ⎪ ⎪ 0, 3.0 + Lv3 < t < 5.0, ⎪ ⎪ ⎪  ⎪      ⎪ ⎪ ⎨ A 1 − cos 2πv t , 5.0 ≤ t ≤ 5.0 + L1 , 2 L1 v (14.63) Zr (t) =   ⎪ ⎪ 5.0 + Lv1 < t < 7.0, ⎪ 0, ⎪ ⎪ ⎪       ⎪ ⎪ L2 A 2πv ⎪ 1 − cos t , 7.0 ≤ t ≤ 7.0 + ⎪ ⎪ 3 L2 v , ⎪ ⎪   ⎪ ⎪ ⎪ 0, 7.0 + Lv2 < t < 9.0, ⎪ ⎪ ⎪ ⎪       ⎪ ⎪ A ⎪ 1 − cos 2πv t , 9.0 ≤ t ≤ 9.0 + Lv3 , ⎪ 4 L ⎪ 3 ⎪ ⎪   ⎪ ⎪ ⎩ 0, 9.0 + L3 < t < 12, v

where L1 , L2 , and L3 are the lengths of different bumps in the considered road profile, and A is amplitude. v is the car forward velocity. In the current investigation, A = 0.19 m, L1 = 5 m, L2 = 10 m, L3 = 15 m, and v = 20 m/s are considered. For a road surface containing multiple bumps, the deviation in sprung mass vertical displacement, vertical acceleration, and bending of tire for backstepping, FPD, and PD controllers in closed loop are depicted in Fig. 14.10A–C. From Fig. 14.10A–D, it is absolutely clear that ASS with backstepping and FPD controllers demonstrated excellent performance in comparison to ASS with conventional PD controller. The cost function i.e. RMSVA for multiple bumps road profile for backstepping, FPD, and conventional PD controllers are 0.364294, 0.681166, and 1.05429, respectively. It has been revealed from the performance of controllers under study that ASS with backstepping controller again provide outstanding ride comfort for passengers seating in the vehicle as compared to the FPD and conventional controllers with ASS. Further, only backstepping

368 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 14.10 Performance comparison of controllers for multiple bumps road surface: (A) sprung mass vertical displacement; (B) sprung mass vertical acceleration; (C) tire deflection; (D) tracking force; (E) ratio between dynamic and static load of tire; (F) suspension movement.

controller fulfill all the hard constraints i.e. tracking force, ratio of dynamic and static load of tire for road holding, and suspension movement as shown in Fig. 14.10D–F. While FPD and conventional PD controllers are not able to keep

Backstepping controller for nonlinear active suspension system Chapter | 14 369

FIGURE 14.10 (continued)

the hard constraint related to road holding less than 1 as depicted in Fig. 14.10E and thus vehicle having ASS with these controllers are prone to overturn and cause a life risk of passengers. Therefore, the backstepping controller clearly out-performed the rest of the two controllers namely FPD and conventional PD for such a tough road profile.

14.5

Conclusions

In the present work, comfort ride of passengers in the vehicle is enhanced by using a backstepping controller for a nonlinear active suspension system (ASS) of a quarter car. Generally, a vehicle is a highly uncertain system due to variation in the sprung mass as the number of passengers in the vehicle varies from time to time. Also, the passive suspension system fails to provide a comfort ride for varying bumpy road surface. The level of the ride comfort of the occupants in the car can be improved significantly by reducing the root mean square of the vertical acceleration (RMSVA) of the sprung mass and simultaneously preserving all the hard constraints namely force, road holding in terms of static and dynamic load of wheel, and suspension alteration. A comparative study is performed to assess the ride comfort performance of ASS with backstepping, Fuzzy PD (FPD), and conventional PD controllers. The FPD controller was designed by seven overlapped membership function of triangular shape for inputs and output variables. The gains of controllers were tuned by Grey Wolf Optimization for a single bump road profile. Further, the ride comfort of ASS with

370 Backstepping Control of Nonlinear Dynamical Systems

all the considered controllers were evaluated for sprung mass and bump height uncertainties and for road surface having multiple bumps. Extensive simulation investigations were carried out and it has been clearly revealed that ASS with backstepping controller provides excellent comfort ride in contrast to FPD and conventional PD controllers and comfortably meet all hard constraints within specified limits.

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Chapter 15

Single-link flexible joint manipulator control using backstepping technique Nishtha Bansala , Aman Bishta , Sruti Paluria , Vineet Kumara , K.P.S. Ranaa , Ahmad Taher Azarb,c , and Sundarapandian Vaidyanathand a Department of Instrumentation and Control Engineering, Netaji Subhas University of Technology, New Delhi, India, b Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University,

Riyadh, Saudi Arabia, c Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, d Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India

15.1

Introduction

Nowadays manipulators are being used extensively in several engineering domains (Azar et al., 2020b). From their conception as systems that perform long repetitive cycles of work, to systems used for high accuracy tasks, their application domain is expanding. While the rigid joint manipulators have been studied for several years and a high tracking precision has been obtained for them, the need for flexibility in joints was realized when these systems were found to be not perfect enough for certain applications, even though they were working properly. Moreover, the design of the controllers for manipulators with slightly lesser stiff joints is a challenging task. This is due to the fact that the elasticity in the joint introduces nonlinearities in the system. The controllers that offer high precision tracking results for rigid joints are found to be inefficient (Akyuz et al., 2011). A possible way for obtaining better results was to modify these linear controllers in order to deal with the flexibility introduced in the joint. So, the researchers began to identify the effect of this flexibility on the basis of its source. A general case was taken into account, where the source was the elasticity of gears, belts, tendons, bearings, hydraulic lines, etc. For such models, two of the nonlinear control techniques were compared, namely, feedback linearization, and integral manifold and composite control (Marino and Spong, 1986). While feedback linearization method is based on the diffeomorphic coordinate transformations and the nonlinear static state feedback, it requires the measurement of dynamic parameters such as, position and velocity of both the link and the motor shaft. It also needs the link acceleration and jerk measurements in order Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00022-2 Copyright © 2021 Elsevier Inc. All rights reserved.

375

376 Backstepping Control of Nonlinear Dynamical Systems

to maintain robustness to parametric uncertainty. The second method, integral manifold, overcomes the need for external measurements and allows a linearizable reduced order model to be obtained. However, it loses its robustness to parametric uncertainty. Another control scheme included an adaptive controller that was similar to the one used for rigid joints except for the addition of a simple correction term for damping out the elastic oscillations in the joint (Ghorbel et al., 1989). A major drawback of these control schemes was that the abovementioned sources only caused weak elasticity in the joints. The performance was found to deteriorate as the elasticity of the joints increased and there was a possibility of breakdown of system stability, and there was no value defined for the maximum flexibility the system could handle effectively. In order to obtain better control in the presence of non-negligible flexibility, the controller designs started using the full order dynamic model of the system directly. Mrad and Ahmad (1992) presented a controller that involved the use of Lyapunov’s second method. The Lyapunov function was selected on the basis of the analysis of system energy and the control scheme was designed using two sliding surfaces to obtain a switching control law. This design did not use link jerk and acceleration feedback, nor did it require the numerical differentiation of the velocity signal or the inversion of inertial matrices. Though it overcame the restrictions on the joint flexibility, it required the complete knowledge of the inertia parameters of the manipulator. Later, Dawson et al. (1994) designed a controller on the basis of complete knowledge of the model. It used the integrator backstepping technique. The technique can be described as a recursive approach, governed by Lyapunov’s theory for stability, which involved the partitioning of the nth-order dynamic model into series cascaded subsystems. The states of the first subsystem act as the control variables for the next subsystem, and the control input derived for the last subsystem ensures that the states of the first subsystem follow the desired trajectory. This technique gave rise to a new band of nonlinear controllers which were capable of meeting the required control objective in a global fashion, in the presence of parameter uncertainty and only needed true system state measurements. To extend the application of this technique to systems with unknown model parameters and that too without the use of link acceleration and jerk measurements, various other techniques were combined with it. Oh and Lee (1998) presented a controller which required only the joint position and velocity measurements. This was achieved by addition of a robust control law to the backstepping method. Yim (2001) presented an adaptive output-feedback controller in which the backstepping technique was used along with the filter design technique, used for estimating the unknown parameters of the system while only motor and link position measurements were used for the synthesis of the controller. Compared to the same adaptive controller which measured just a link displacement angle, the order of the proposed controller became half and the complexity of the control law was reduced dramatically. Another controller design was based on

Single-link flexible joint manipulator control Chapter | 15 377

the combination of the backstepping technique with the generalized small gain approach (Yang et al., 2004). In addition, the Takagi–Sugeno type fuzzy logic systems were used to approximate the uncertainties (Pilla et al., 2019; Khettab et al., 2018; Djeddi et al., 2019; Vaidyanathan and Azar, 2016a). The algorithm had two advantages – semi-global uniform ultimate boundedness of adaptive control system was achieved in the presence of unstructured uncertainties, and the adaptive mechanism required minimal learning parameterizations. A complete output-feedback controller was designed for two-link flexible joint manipulators by Chatlatanagulchai et al. (2004). The control scheme consisted of a nonlinear Luenberger-type observer, multilayer neural network plant identifier, and controller based on a backstepping framework and variable structure controller. Only link angular positions were measured as outputs. The controller achieved good performance despite the presence of additive external disturbances, unmodeled dynamics, actuator nonlinearities, i.e., deadzone and backlash, and payload changes. Combining the advantages of sliding-mode control (SMC) and backstepping methodology, Ma et al. (2006) presented a controller design that used networks of Gaussian radial basis functions with variable weights to compensate the model uncertainties. While SMC ensures robustness in the presence of model uncertainties, it has many drawbacks such as the classic estimation schemes require the model to be linearly parameterized and the nonlinearities exactly known (Singh et al., 2017; Mekki et al., 2015; Vaidyanathan et al., 2015b, 2019; Meghni et al., 2017; Vaidyanathan and Azar, 2015a,b). Furthermore, there is a matching condition that needs to be satisfied in order to compensate for the uncertainty and disturbances. Also, the switching gain had to be higher than the known norm of uncertainties in order to maintain the sliding mode. By integrating the backstepping algorithm into the design of SMC, these requirements were eliminated and the efficiency of the system was enhanced. Another variation of the backstepping method involved the use of the nonlinear H-infinity method and the saturation-type nonlinear control based on Lyapunov’s second method for designing the virtual and real controls (Lee et al., 2007). The designed robust inputs were made to satisfy the L2 -gain. The need for acceleration and jerk measurements of the link was eliminated completely. All of these previous designs were derived without considering the actuator dynamics. It was later noted that the inclusion of the actuator dynamics further enhanced the performance of the system. Li et al. (2013) presented a controller, which used an adaptive fuzzy output feedback approach, a combination of adaptive backstepping and dynamic surface control, for a single-link flexible joint manipulator (SLFJM) coupled to a brushed direct current motor with a non-rigid joint. It required only measurements of link position and used fuzzy logic system to approximate the unknown nonlinearities, i.e., an adaptive fuzzy filter observer was designed to estimate the immeasurable states. A major advantage was that the problem of “explosion of complexity” existing in the conventional backstepping control methods was avoided. Mbede and

378 Backstepping Control of Nonlinear Dynamical Systems

Ahanda (2014) presented another design for exponential tracking control using backstepping approach for voltage-based control of a SLFJM electrically driven robot. The controller was able to cope with the difficulty introduced by the cascade structure in the dynamic model, while simultaneously dealing with flexibility in joints and ensuring fast tracking performance. It also improved the tracking when the velocity of the link is very high. Another backstepping controller, based on position control, in combination with a hysteresis controller for current control was presented by Zouari et al. (2014). The source of flexibility was considered to be the harmonic drive in the mechanical arm. While including the motor dynamics of a brushless DC motor, the system was able to track the desired trajectory with minimum vibration. Further, this controller was able to give satisfactory performance in the presence of uncertainties and unknown bounded disturbances (Zouari et al., 2015). A new tuning functions-based adaptive backstepping controller using combined direct and indirect adaptation for SLFJM was presented by Soukkou and Labiod (2015). In this approach, the parameter estimation was driven by a weighted combination of tracking and identification errors, and an x-swapping filter identifier with a gradient-type update law. Modified further by Kien and Hanh (2016), the different filter structures (K-filter) and identifiers with these intermediate update laws were instead used to compensate for the effect of parameter estimation transients to obtain the tuning functions to give an adaptive backstepping control law. Only the final tuning function was used as the parameter update law. Pan et al. (2017) presented an adaptive command-filtered backstepping control for series elastic actuator driven robot arms which involved the use of second-order command filters and compensated for the discontinuous friction simultaneously with the help of an adaptive mechanism. While the closed-loop stability was rigorously established by the Lyapunov synthesis and time-scales separation, the control scheme gave an improved performance while alleviating the dependence of the control law on precise dynamic models. Many other controllers were developed in order to control manipulator systems (Sharma et al., 2014b,c; Pranav et al., 2015, 2016; Kumar et al., 2016a,b, 2018a,b,c; Kumar and Rana, 2017a,b; Kumar et al., 2017; Azar et al., 2017; Kathuria et al., 2018). As far as control schemes for tracking problem of SLFJM are concerned, the tuning of controller is the most crucial part of control design. Tuning is needed as when the backstepping technique is applied to the system’s mathematical model and as a result few gain terms get introduced. Further, the system’s performance is highly sensitive to these gain terms, and therefore, to find optimal values for these terms, optimization techniques are applied. In most of the previously reported works, the Genetic Algorithm (GA) was used for tuning. However, in the past decades, several efficient algorithms have been developed and they have proven to give better solutions than GA. Narula et al. (2016) presented a comparative study between three optimization algorithms, namely GA, Cuckoo Search Algorithm (CSA), and Teaching Learning Based Optimization

Single-link flexible joint manipulator control Chapter | 15 379

(TLBO) algorithm (Gorripotu et al., 2019). Integral Absolute Error (IAE) was used as a measure for comparison (Ammar et al., 2018). The simulation results showed that TLBO was able to offer better results as compared to the others. Many other comparative studies were performed (Kathuria et al., 2017; Kumar et al., 2016a,b, 2015; Sharma et al., 2014a,b). In this chapter, the backstepping technique has been applied on the SLFJM system, and in the derivation of the control law, four gains were introduced. These constants were tuned using Jaya algorithm, TLBO algorithm, and GA, leading to the comparison of their performance in terms of algorithm efficiency, and their ability to reach the best solution in the given search space. Rest of the chapter has been organized as follows. Section 15.2 gives the description of the mathematical model of SLFJM. In Section 15.3, the controller design steps using backstepping method have been described. Section 15.4 presents brief of the used optimization algorithms used for determination of optimum values for the gains introduced in the design steps. The trajectory tracking results have been discussed in Section 15.5. Finally, conclusions are presented in Section 15.6.

15.2

Single-link flexible joint manipulator model

In the era of industrial automation, industrial robot manipulator is a major development that has helped in increasing the production rate and quality of products simultaneously. Robot motion control is a problem that has been analyzed for decades now in order to improve the robot performance, reduce the robot cost, improve safety, while introducing new functionalities. Thus, the tracking problem of a SLFJM has been considered.

FIGURE 15.1 Single-link flexible joint manipulator.

Fig. 15.1 represents the model of a typical single-link flexible joint manipulator. The system equations for this model are: Il θ¨ + mgl sin θ + k (θ − θm ) = 0 Im θ¨m − k (θ − θm ) + μθ˙m = u

(15.1) (15.2)

380 Backstepping Control of Nonlinear Dynamical Systems

where θ is the link angle, θm is the angle of rotation of the motor, Il is the inertia of link, Im is the inertia of motor actuator, k is the elastic stiffness of the flexible link, m is the mass of the link, g is the acceleration due to gravity, l is the link length, μ is the viscosity, and u is the control torque generated by the motor. For converting the model into state space form, assume the following: x1 = θ x2 = θ˙

(15.3)

x3 = θ m x4 = θ˙m

(15.5)

(15.4) (15.6)

Thus, x1 is the link angle, x2 is the link angular velocity, x3 is the motor rotor angle, and x4 is the angular velocity of the rotor. The state space model hence obtained is given by x˙1 = x2

(15.7)

mgl k sin x1 − (x1 − x3 ) x˙2 = − Il Il x˙3 = x4 k 1 μ x˙4 = (x1 − x3 ) − x4 + u Im Im Im

(15.8) (15.9) (15.10)

The model above shows that the system is a 4th-order system, i.e., a complex as well as nonlinear system. Even then, a major advantage of the above model is that it is in the Brunovsky canonical form (Nicosia et al., 1986) – a state space description of the system in which the state vector elements are connected to each other through a chained integration procedure and the last state variable is equal to the integral of the control input. The values of the parameters associated with the model are given in Table 15.1. TABLE 15.1 Model parameters. S. No.

Variables

Description

Values

1.

m

Mass of the link

1.2756 kg

2.

g

Acceleration due to gravity

9.8 m/sec2

3.

l

Length of link

0.4 m

4.

Im

Inertia of motor actuator

0.3 kg m2

5.

Il

Inertia of link

1 kg m2

6.

k

Elastic stiffness of the flexible link

100 Nm

7.

μ

Viscosity

0.1 kg m2 /sec

Single-link flexible joint manipulator control Chapter | 15 381

15.3

Controller design using backstepping technique

Backstepping method is a recursive approach governed by Lyapunov theory (Vaidyanathan et al., 2018, 2015a,b; Shukla et al., 2018; Azar et al., 2020a; Vaidyanathan and Azar, 2016b). It involves the extension of the controlled stability of the subsystem to the larger system by partitioning the system into cascaded subsystems. The states of first subsystem are taken as the control variables for the next subsystem and so on. The derived control input obtained for the last subsystem ensures that the first subsystem realizes the desired state. This technique ensures that the system is stable asymptotically. It is easily applicable to dynamic systems while giving flexibility in designing a feedback control law. It completely eliminates the need to linearize the systems. It is also applicable to systems of any order and does not involve the use of derivatives in the control law. This method is applicable to systems that are in strict feedback form, i.e. the following form of equations: x˙ = fx (x) + gx (x) z1 z˙ 1 = f1 (x, z1 ) + g1 (x, z1 ) z2 z˙ 2 = f2 (x, z1 , z2 ) + g2 (x, z1 , z2 ) z3 .. . z˙ i = fi (x, z1 , . . . , zi−1 , zi ) + gi (x, z1 , . . . , zi−1 , zi ) zi+1 .. . z˙ k = fk (x, z1 , . . . , zk−1 , zk ) + gk (x, z1 , . . . , zk−1 , zk ) u

for 1 ≤ i ≤ k − 1

(15.11) The following are the steps for designing a controller using the backstepping method (Khalil, 1991). Here, the objective is to find a control law, u (t) such that the link position of the manipulator tracks the desired or given trajectory. Let x1 track a given trajectory r(t) and x¯1 be the deviation of x1 from r(t). Therefore x¯1 = x1 − r

(15.12)

The SLFJM system equations, Eqs. (15.7)–(15.10), can thus be manipulated to obtain x˙¯1 = x2 − r˙ mgl k x˙2 = − sin (x¯1 + r) − (x¯1 + r − x3 ) Il Il x˙3 = x4 k 1 μ x˙4 = (x¯1 + r − x3 ) − x4 + u(t) Im Im Im

(15.13) (15.14) (15.15) (15.16)

382 Backstepping Control of Nonlinear Dynamical Systems

The errors for each state variable in the manipulated SLFJM system equations are defined as follows: e1 = x¯1

(15.17)

e2 = x2 − φ0

(15.18)

e3 = x3 − φ1

(15.19)

e4 = x4 − φ2

(15.20)

where φ0 , φ1 , and φ2 are the stabilizing functions that are to be designed. Here x2 , x3 , and x4 are taken as virtual controls for x1 , x2 , and x3 , respectively. Step 1: In this step, subsystem 1 (x¯1 ) is considered, which is given by the following equation: x˙¯1 = x2 − r˙

(15.21)

Then, x2 can be viewed as the virtual control input for stabilizing the system at x¯1 = 0. This step involves the design of stabilizing function φ0 . The following Lyapunov function candidate is selected: 1 V0 = e12 2

(15.22)

The derivative of V0 can be obtained by differentiating Eq. (15.22) as follows: V˙0 = e1 e˙1

(15.22a)

Similarly, the derivative of e1 can be obtained by differentiating Eq. (15.17) as follows: e˙1 = x˙¯1

(15.23)

Now, substituting the value of x˙¯1 from Eq. (15.21) in Eq. (15.23), the following expression for e˙1 is obtained: e˙1 = x2 − r˙

(15.23a)

Further substituting the value of x2 from Eq. (15.18) in Eq. (15.23a), the following expression for e˙1 is obtained: e˙1 = e2 + φ0 − r˙

(15.23b)

Using the value of e˙1 from Eq. (15.23b) in Eq. (15.22a), the expression for V˙0 gets modified as follows: V˙0 = e1 (e2 + φ0 − r˙ )

(15.24)

Single-link flexible joint manipulator control Chapter | 15 383

Rearranging Eq. (15.24), the following expression for V˙0 is obtained: V˙0 = e1 (φ0 − r˙ ) + e1 e2

(15.24a)

In order for the subsystem to be asymptotically stable, V˙0 should be negative definite according to Lyapunov stability criteria. Thus, to ensure the negative definiteness of V˙0 i.e., for value of V˙0 in Eq. (15.24a) to be negative, φ0 is selected as follows: φ0 = −λ1 e1 + r˙

(15.25)

where λ1 > 0 is a constant. Substituting the value of e1 from Eq. (15.17) in Eq. (15.25), φ0 is obtained as follows: φ0 = −λ1 x¯1 + r˙

(15.26)

Further substituting the value of φ0 from Eq. (15.25) in Eq. (15.24a), the following expression for V˙0 is obtained: V˙0 = −λ1 e12 + e1 e2

(15.27)

From Eq. (15.27), it can be seen that V˙0 will be negative definite if e2 is zero, i.e. x2 = φ0 (from Eq. (15.18)). Step 2: For this step, subsystem 2 (x1 , x2 ) is considered which is given by the following equations: x˙¯1 = x2 − r˙ mgl k x˙2 = − sin (x¯1 + r) − (x¯1 + r − x3 ) Il Il

(15.28a) (15.28b)

Then, x3 can be viewed as the virtual control input for stabilizing the subsystem at e2 = 0. This step involves the design of stabilizing function φ1 . A Lyapunov function candidate V1 is obtained by augmenting V0 with a quadratic function to obtain the following expression: 1 V1 = V0 + e22 2

(15.29)

The derivative of V1 can be obtained by differentiating Eq. (15.29) as follows: V˙1 = V˙0 + e2 e˙2

(15.29a)

Similarly, the derivative of e2 can be obtained by differentiating Eq. (15.18) as follows: e˙2 = x˙2 − φ˙ 0

(15.30)

384 Backstepping Control of Nonlinear Dynamical Systems

Now, substituting the value of x˙2 from Eq. (15.28b) in Eq. (15.30), the following expression for e˙2 is obtained: e˙2 = −

mgl k sin (x¯1 + r) − (x¯1 + r − x3 ) − φ˙ 0 Il Il

(15.30a)

Rearranging Eq. (15.30a), the following expression is obtained: e˙2 = −

mgl k k sin (x¯1 + r) − (x¯1 + r) + x3 − φ˙ 0 Il Il Il

(15.30b)

Again, substituting the value of x3 from Eq. (15.19) in Eq. (15.30b), the following expression for e˙2 is obtained: e˙2 = −

mgl k k sin (x¯1 + r) − (x¯1 + r) + (e3 + φ1 ) − φ˙ 0 Il Il Il

(15.30c)

Using the value of e˙2 from Eq. (15.30c) and the value of V˙0 from Eq. (15.27) in Eq. (15.29a), the expression for V˙1 gets modified as follows:  k −mgl 2 ˙ sin (x¯1 + r) − (x¯1 + r) V1 = −λ1 e1 + e1 e2 + e2 Il Il  k + (e3 + φ1 ) − φ˙ 0 Il

(15.31)

Rearranging Eq. (15.31), the following expression for V˙0 is obtained:  mgl k 2 ˙ sin (x¯1 + r) − (x¯1 + r) V1 = −λ1 e1 + e2 e1 − Il Il  k k + φ1 − φ˙ 0 + e2 e3 Il Il

(15.31a)

In order for the subsystem to be asymptotically stable, V˙1 should be negative definite. Thus, to ensure the negative definiteness of V˙1 i.e., for value of V˙1 in Eq. (15.31a) to be negative, φ1 is selected as follows: φ1 =

Il k

  mgl k sin (x¯1 + r) + (x¯1 + r) + φ˙ 0 −λ2 e2 − e1 + Il Il

(15.32)

Single-link flexible joint manipulator control Chapter | 15 385

where λ2 > 0 is a constant. Next, φ˙ 0 is obtained by differentiating Eq. (15.26) as follows: φ˙ 0 = −λ1 x˙¯1 + r¨

(15.33)

Substituting the values of e1 , e2 , and φ˙ 0 from Eqs. (15.17), (15.18), and (15.33), respectively, in Eq. (15.32), the expression for φ1 is obtained as follows: Il φ1 = k



 mgl k ˙ sin (x¯1 + r) + (x¯1 + r) − λ1 x¯1 + r¨ −λ2 (x2 − φ0 ) − x¯1 + Il Il (15.34)

Further substituting the value of φ1 from Eq. (15.32) in Eq. (15.31a), the following expression for V˙1 is obtained: k V˙1 = −λ1 e12 − λ2 e22 + e2 e3 Il

(15.35)

From Eq. (15.35), it can be seen that V˙1 will be negative definite if e3 is zero, i.e. x3 = φ1 (from Eq. (15.19)). Step 3: For this step, subsystem 3 (x¯1 , x2 , x3 ) is considered, which is given by the following equations: x˙¯1 = x2 − r˙ −mgl k x˙2 = sin (x¯1 + r) − (x¯1 + r − x3 ) Il Il x˙3 = x4

(15.36a) (15.36b) (15.36c)

Then, x4 can be viewed as the virtual control input for stabilizing the system at e3 = 0. This step involves the design of stabilizing function φ2 . A Lyapunov function candidate V2 is obtained by augmenting V1 with a quadratic function to obtain the following expression: 1 V2 = V1 + e32 2

(15.37)

The derivative of V2 can be obtained by differentiating Eq. (15.37) as follows: V˙2 = V˙1 + e3 e˙3

(15.37a)

Similarly, the derivative of e3 can be obtained by differentiating Eq. (15.19) as follows: e˙3 = x˙3 − φ˙ 1

(15.38)

386 Backstepping Control of Nonlinear Dynamical Systems

Now, substituting the value of x˙3 from Eq. (15.36c) in Eq. (15.38), the following expression for e˙3 is obtained: e˙3 = x4 − φ˙ 1

(15.38a)

Further substituting the value of x4 from Eq. (15.20) in Eq. (15.37a), the following expression for e˙3 is obtained: e˙3 = e4 + φ2 − φ˙ 1

(15.38b)

Using the value of e˙3 from Eq. (15.38b) and the value of V˙1 from Eq. (15.35) in Eq. (15.37a), the expression for V˙2 gets modified as follows:   k V˙2 = −λ1 e12 − λ2 e22 + e2 e3 + e3 e4 + φ2 − φ˙ 1 Il

(15.39)

Rearranging Eq. (15.39), the following expression for V˙2 is obtained:   k 2 2 ˙ ˙ e2 + φ2 − φ1 + e3 e4 (15.39a) V2 = −λ1 e1 − λ2 e2 + e3 Il In order for the subsystem to be asymptotically stable, V˙2 should be negative definite. Thus, to ensure the negative definiteness of V˙2 i.e., for value of V˙2 in Eq. (15.39a) to be negative, φ2 is selected as follows: φ2 = −λ3 e3 −

k e2 + φ˙ 1 Il

(15.40)

where λ3 > 0 is a constant. Next, φ˙ 1 is obtained by differentiating Eq. (15.34) as follows:    Il mgl k ˙ ˙ cos (x¯1 + r) + φ1 = − (λ1 + λ2 ) x˙2 − (λ1 λ2 + 1) x¯1 + x2 k Il Il  ... + (λ2 + λ1 ) r¨ + r (15.41) Substituting the values of e2 , e3 , and φ˙ 0 from Eqs. (15.18), (15.19), and (15.41), respectively, in Eq. (15.40), the expression for φ2 is obtained as follows: k φ2 = −λ3 (x3 − φ1 ) − (x2 − φ0 ) Il    mgl k Il cos (x¯1 + r) + − (λ1 + λ2 ) x˙2 − (λ1 λ2 + 1) x˙¯1 + x2 + k Il Il  ... + (λ2 + λ1 ) r¨ + r (15.42)

Single-link flexible joint manipulator control Chapter | 15 387

Further substituting the value of φ2 from Eq. (15.40) in Eq. (15.39a), the following expression for V˙2 is obtained: V˙2 = −λ1 e12 − λ2 e22 − λ3 e32 + e3 e4

(15.43)

From Eq. (15.43), it can be seen that V˙2 will be negative definite if e4 is zero, i.e. x4 = φ2 (from Eq. (15.20)). Step 4: For this step, the last subsystem (x¯1 , x2 , x3 , x4 ), which is the complete system given by Eqs. (15.13)–(15.16), is considered. Here, u is the actual control input used for stabilizing the system at e4 = 0. A Lyapunov function candidate V3 is obtained by augmenting V2 with a quadratic function to obtain the following expression: 1 V3 = V2 + e42 2

(15.44)

The derivative of V3 can be obtained by differentiating Eq. (15.44) as follows: V˙3 = V˙2 + e4 e˙4

(15.44a)

Similarly, the derivative of e4 can be obtained by differentiating Eq. (15.20) as follows: e˙4 = x˙4 − φ˙ 2

(15.45)

Now, substituting the value of x˙4 from Eq. (15.16) in Eq. (15.45), the following expression for e˙4 is obtained: e˙4 =

k μ 1 (x¯1 + r − x3 ) + u − x4 − φ˙ 2 Im Im Im

(15.45a)

Using the value of e˙4 from Eq. (15.45a) and the value of V˙2 from Eq. (15.43) in Eq. (15.44a), the expression for V˙3 gets modified as follows: V˙3 = −λ1 e12 − λ2 e22 − λ3 e32   k μ 1 + e4 e3 + (x¯1 + r − x3 ) + u − x4 − φ˙ 2 Im Im Im

(15.46)

In order for the subsystem to be asymptotically stable, V˙3 should be negative definite. Thus, to ensure the negative definiteness of V˙3 i.e., for value of V˙2 in Eq. (15.46) to be negative, u is selected as follows:   k μ ˙ u = Im −λ4 e4 − e3 − (x¯1 + r − x3 ) + x4 + φ2 Im Im

(15.47)

388 Backstepping Control of Nonlinear Dynamical Systems

where λ4 > 0 is a constant. Next, φ˙ 2 is obtained by differentiating Eq. (15.42) as follows:   Il k φ˙ 2 = − (λ1 + λ2 + λ3 ) x˙3 + − (λ1 λ2 λ3 + λ3 ) − λ1 x˙¯1 k Il    k Il mgl k + − − cos (x¯1 + r) − λ 1 λ 2 + λ2 λ 3 + λ 1 λ 3 + 1 − x˙2 Il k Il Il   Il k mgl mgl + (λ1 + λ2 + λ3 ) cos (x¯1 + r) + sin (x¯1 + r) x22 x2 − k Il Il k k Il  ... .... + r¨ + (λ1 λ2 + λ2 λ3 + λ1 λ3 + 1) r¨ + (λ1 + λ2 + λ3 ) r + r Il k (15.48) Substituting the values of e3 , e4 , and φ˙ 2 from Eqs. (15.19), (15.20), and (15.48), respectively, in Eq. (15.47), the expression for u is obtained as follows:  k μ u(t) = Im −λ4 (x4 − φ2 ) − (x3 − φ1 ) − (x¯1 + r − x3 ) + x4 Im Im   Il k − (λ1 + λ2 + λ3 ) x˙3 + − (λ1 λ2 λ3 + λ3 ) − λ1 x˙¯1 k Il    k Il mgl k + − − cos (x¯1 + r) − λ 1 λ 2 + λ 2 λ 3 + λ1 λ 3 + 1 − x˙2 Il k Il Il   Il k mgl mgl + (λ1 + λ2 + λ3 ) cos (x¯1 + r) + sin (x¯1 + r) x22 x2 − k Il Il k  k Il  ... .... + r¨ + (λ1 λ2 + λ2 λ3 + λ1 λ3 + 1) r¨ + (λ1 + λ2 + λ3 ) r + r Il k (15.49) Further substituting the value of u from Eq. (15.47) in Eq. (15.46), the following expression for V˙3 is obtained: V˙3 = −λ1 e12 − λ2 e22 − λ3 e32 − λ4 e42

(15.50)

It can be seen that V˙3 is negative definite following the condition that λ1 , λ2 , λ3 , and λ4 are greater than zero which is true. Thus, the designed controller takes care of the asymptotic stability of the system. To obtain the most suitable value of these gains, the optimization algorithms are applied to the complete system and its performance measured with the help of a cost function, which is selected as the Integral Absolute Error (IAE) for the link deviation x¯1 .

Single-link flexible joint manipulator control Chapter | 15 389

15.4

Optimization algorithms

The objective of an optimization algorithm is to find an optimal or near-optimal solution for a given problem. Presently, there exist innumerable optimization algorithms which have been developed either for generic purposes or for solving very specific problems. The algorithms for gain tuning are generally the population-based heuristic algorithms. These algorithms can be grouped into two categories – Evolutionary Algorithms and Swarm Intelligence based algorithms. Besides these, there is a new generation of algorithms which do not require any algorithm-specific parameters, thus do not need any internal tuning, and are much easier to apply. Here, an evolutionary algorithm – GA, and two algorithms without algorithm-specific parameters – TLBO algorithm and Jaya algorithm have been used to tune the controller gains and the results have been compared. A brief description of working of each algorithm is given below.

15.4.1 Jaya algorithm This algorithm was developed by Rao (2015). It is based on the concept that the solution obtained for a given problem should move towards the best solution and should avoid the worst solution. This algorithm requires only the common control parameters and does not require any algorithm-specific control parameters. However, unlike the TLBO algorithm, which has two phases (i.e. teacher phase and the learner phase), the Jaya algorithm has only one phase and it is comparatively simpler to apply and requires less computational power. It is a simple and powerful optimization algorithm for solving the constrained and unconstrained optimization problems. The proposed algorithm secures first rank for the ‘best’ and ‘mean’ solutions in Friedman’s rank test. The algorithm begins with randomly initializing a set of points, i.e. the initial population in the given sample space. The cost function for each individual in the population is computed, assuming f (x) as the required objective function to be minimized (or maximized). Let there be p number of design variables and n number of candidate solutions. The best candidate is the one with the best value of f (x) (i.e., f (x)best ) among the entire set of candidate solutions and the worst candidate is the one with the worst value of f (x) (i.e., f (x)worst ) among the entire set of candidate solutions. If Xj,k,i is the value of the j th variable for the kth candidate during the ith iteration, then this value is modified as follows:



   Xj,k,i = Xj,k,i + r1,j,i Xj,best,i − Xj,k,i − r2,j,i Xj,worst,i − Xj,k,i (15.51) where Xj,best,i is the value of j th variable for the best candidate and Xj,worst,i  is the value of j th variable for the worst candidate. Xj,k,i is the updated value

390 Backstepping Control of Nonlinear Dynamical Systems

of Xj,k,i , and r1,j,i and r2,j,i are the two random numbers

 for the j th variable during the ith iteration in the range [0, 1]. The term “r1,j,i Xj,best,i − Xj,k,i ” indicates the tendency of the solution to move closer to the best solution and the  term “−r2,j,i Xj,worst,i − Xj,k,i ” indicates the tendency of the solution to  avoid the worst solution. Xj,k,i is accepted if it gives a better function value. All the accepted function values at the end of iteration are maintained and these values become the input to the next iteration. The algorithm strives to become victorious by reaching the best solution and hence it is named as Jaya, a Sanskrit word meaning victory. Fig. 15.2 shows a flow chart of the steps involved in this algorithm.

FIGURE 15.2 Flowchart for Jaya algorithm.

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15.4.2 Teaching learning based optimization algorithm TLBO is a population-based algorithm which simulates the teaching learning process of the class room (Rao et al., 2010). This algorithm requires only the common control parameters such as the population size and the number of generations and does not require any algorithm-specific control parameters unlike the GA. The population represents a group of learners or a class of learners. Distinct subjects given to the learners are considered as the different design variables of the optimization problem and a learner’s result is analogous to the fitness value of the optimization problem. The working of TLBO is divided into two parts – teacher phase and learner phase, as explained below: Teacher phase: It is the first part of the algorithm where learners learn through the teacher. During this phase, a teacher tries to increase the mean result of the class in the subject taught by him or her depending on his or her capability. At any iteration i, assume that there are m number of subjects and n number of learners. Let X¯ j,i be the mean result of the learners in a particular subject, where j varies from 1, 2, . . . , m. The best overall result XT otal−kbest ,i considering all the subjects together obtained in the entire population of learners can be considered as the result of the best learner kbest . However, as the teacher is usually considered as a highly learned person who trains learners so that they can have better results, the best learner identified is considered as the teacher by the algorithm. The difference between the existing mean result of each subject and the corresponding result of the teacher for each subject is given by   Difference_Meanj,k,i = ri Xj,kbest ,i − TF X¯ j,i

(15.52)

where Xj,kbest ,i is the result of the best learner in subject j , TF is the teaching factor which decides the value of mean to be changed, and ri is the random number in the range [0, 1]. Value of TF can be either 1 or 2. The value of TF is decided randomly with equal probability as follows: 

TF = round 1 + rand (0, 1) {2 − 1}

(15.53)

It may be noted that TF is not a parameter of the TLBO algorithm and therefore, the value of TF is not given as an input to the algorithm and its value is randomly decided by the algorithm using Eq. (15.53). After conducting a number of experiments on many benchmark functions, it is concluded that the algorithm performs better if the value of TF is between 1 and 2. However, the algorithm is found to perform much better if the value of TF is either 1 or 2.

392 Backstepping Control of Nonlinear Dynamical Systems

Based on the Difference_Meanj,k,i , the existing solution is updated in the teacher phase according to the following expression.  = Xj,k,i + Difference_Meanj,k,i Xj,k,i

(15.54)

 is the upwhere Xj,k,i is the result of kth learner in the subject j , and Xj,k,i  dated value of Xj,k,i . Xj,k,i is accepted if it gives better function value. All the accepted function values at the end of the teacher phase are maintained and these values become the input to the learner phase.

Learner phase: It is the second part of the algorithm where learners increase their knowledge by interacting among themselves. A learner interacts randomly with other learners for enhancing his or her knowledge. A learner learns new things if the other learner has more knowledge than him or her. Considering a population size of n, the learning phenomenon of this phase is explained below.  Two learners P and Q are randomly selected such that Xtotal−P ,i =    Xtotal−Q,i (where Xtotal−P ,i and Xtotal−Q,i are the updated values of Xtotal−P ,i and Xtotal−Q,i , respectively, at the end of teacher phase). The value of P is updated as follows.  Xj,P ,i

⎧     ⎨ X j,P ,i + ri Xj,P ,i − Xj,Q,i , =    ⎩ X  j,P ,i + ri Xj,Q,i − Xj,P ,i ,

  if Xtotal−P ,i < Xtotal−Q,i   < Xtotal−P if Xtotal−Q,i ,i (15.55)

 Xj,P ,i is accepted if it gives a better function value. Eq. (15.55) is for minimization problems. In the case of maximization problems, Eq. (15.56) is used.

 Xj,P ,i

⎧     ⎨ X j,P ,i + ri Xj,P ,i − Xj,Q,i , =    ⎩ X  j,P ,i + ri Xj,Q,i − Xj,P ,i ,

  < Xtotal−P if Xtotal−Q,i ,i   if Xtotal−P ,i < Xtotal−Q,i (15.56)

Fig. 15.3 shows the flowchart for the algorithm.

15.4.3 Genetic algorithm GA is a stochastic global and adaptive heuristic search and optimization algorithm that is similar to the natural biological evolution. It is based on Darwin’s theory of biological evolution, i.e., the principle of survival of the fittest

Single-link flexible joint manipulator control Chapter | 15 393

FIGURE 15.3 Flowchart – TLBO algorithm.

(Bennett and Shapiro, 1994). Thus, it operates on a population of potential solutions and generates better approximations to a solution successively. In each generation of GA, a new set of approximations is generated by the process of selecting individuals according to their level of fitness and reproducing them using operators borrowed from natural genetics. This process leads to the evolution of population of the individuals that are better suited to their environment than the individuals from which they were created, just as in natural adaptation.

394 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 15.4 Flowchart for GA.

The Genetic algorithm is divided into five phases – generation of initial population, computation of the fitness function, selection, crossover, and mutation. The initial population is generated randomly in the given sample space such that each member in the population represents a solution to the given problem. The fitness function is computed for each member, respectively, followed by the selection phase, wherein the fittest individuals are allowed to pass on their genes to the next generation. Next is the Crossover phase, where for each pair of parents to be mated, a crossover point is chosen at random from within the genes. Offspring are created by exchanging the genes of parents among themselves until the crossover point is reached. Finally, in the mutation phase, in certain offspring formed, some of their genes undergo mutation, though with a low random probability. The algorithm terminates if the population has converged, i.e. it does not produce offspring which are significantly different from the previous generation. This process is shown in the flowchart in Fig. 15.4.

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TABLE 15.2 Parameters of optimization algorithms. S. No.

Algorithm-specific parameters

1.

Population Size

2.

No. of Generations

3.

Gain Bounds

4.

Initial Bounds

5.

Creation Function

6.

Jaya

TLBO

GA

50 50 [0.5 250] [0.5 10]

[0.5 250]

[0.5 10]

Scaling Function





Rank

7.

Selection Function





Stochastic Uniform

8.

Mutation Function





Gaussian

9.

Crossover Function





Scattered

Uniform

In this work, the GA toolkit of MATLAB® was used and the corresponding parameters used to instantiate the algorithm are given in Table 15.2.

15.5

Results and discussions

The simulations were performed on Simulink and MATLAB environment, and were carried out using an ode-4 Runge–Kutta solver with a fixed step size of 0.001 s. The controller output was limited by adding a saturation of ±10 N-m. IAE was taken as the cost function, which can be defined as the integral of the absolute error between the link position and the desired trajectory to be tracked. This IAE was minimized in order for the link position to track the desired trajectory efficiently. The effectiveness of the controller designed using the backstepping technique for the given model of SLFJM is verified for three different inputs. (i) Unit step input: Step function was used as the desired trajectory, with the initial sampling time set to zero. Table 15.3 presents the optimized gains for all the inputs. Table 15.4 lists the performances in terms of percent overshoot, the rise time as well as the settling time for each of the sets of values obtained from the tuning of the controller with the help of the three optimization algorithms. Figs. 15.5–15.8 show the convergence plots for various optimization techniques, manipulator response, controller output and error signal for the manipulator considering step input. From the obtained responses it can be clearly inferred that Jaya outperforms TLBO and GA for the case of step input.

Input

Optimization Technique

Value of Gains After Optimization k1

k2

k3

k4

Unit step

Jaya

9.76192

0.50000

250

13.34535

0.38108

TLBO

5.09562

43.46918

220.83534

15.61003

0.41612

GA

7.21033

7.55948

10.27277

5.91089

0.42360

Jaya

15.04069

0.50000

250

9.74645

0.02660

TLBO

2.91901

203.57935

0.50000

19.95135

0.05460

GA

2.99307

19.01074

15.39525

3.31922

0.15067

Jaya

4.54042

0.50000

221.70207

20.91744

0.03614

TLBO

1.71834

21.35815

71.23031

4.15233

0.16414

GA

2.67900

11.79227

11.73334

3.05618

0.15165

x (t) = 1 − e−t

x (t) = sin t + cos t − 1

IAE

396 Backstepping Control of Nonlinear Dynamical Systems

TABLE 15.3 Optimized controller gains.

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TABLE 15.4 Transient performance: Unit step input. S. No.

Optimization Algorithm

Overshoot (%)

Rise Time (s)

Settling Time (s)

1.

Jaya

1.232

0.403

0.674

2.

TLBO

0.000

0.508

1.007

3.

GA

0.000

0.534

1.021

FIGURE 15.5 Convergence plots for various optimization techniques: Unit step input.

FIGURE 15.6 Manipulator response for unit step input.

398 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 15.7 Controller outputs for optimized gains: Unit step input.

FIGURE 15.8 Error signal for unit step input.

(ii) Exponential input: The link position was made to follow an exponential trajectory, given by the following equation: x (t) = 1 − e−t Figs. 15.9–15.12 show the convergence plots for various optimization techniques, manipulator response, controller output and error signal for the manipulator considering exponential input. From the obtained responses it can be clearly inferred that Jaya outperforms TLBO and GA for the case of exponential input.

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FIGURE 15.9 Convergence plots for various optimization techniques: Exponential input.

FIGURE 15.10 Manipulator response for exponential input.

(iii) Sinusoid input: Another case was taken into account where the link position of SLFJM was made to track a sinusoid trajectory. The input equation fed is given by x (t) = sin t + cos t − 1 Figs. 15.13–15.16 show the convergence plots for various optimization techniques, manipulator response, controller output, and error signal for the manipulator considering sinusoid input. From the obtained responses it can be clearly inferred that Jaya outperforms TLBO and GA for the case of sinusoid input.

400 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 15.11 Controller outputs for optimized gains: Exponential input.

FIGURE 15.12 Error signal for exponential input.

15.6 Conclusion The single-link flexible joint manipulator system was efficiently controlled using the backstepping control technique. The three optimization algorithms, namely Jaya algorithm, Teaching Learning Based Optimization (TLBO) algorithm, and Genetic Algorithm (GA) were considered for controller tuning, and the control performances were assessed. The Integral Absolute Error (IAE) for Jaya, TLBO, and GA were found to be 0.3818, 0.41612, and 0.4236, respectively for unit step input. Jaya algorithm

Single-link flexible joint manipulator control Chapter | 15 401

FIGURE 15.13 Convergence plots for various optimization techniques: Sinusoid input.

FIGURE 15.14 Manipulator response for sinusoid input.

offered better results for the step input. Furthermore, two more inputs, namely an exponential input (given by, x (t) = 1 − e−t ) and a trigonometric input (given by, x (t) = sin t + cos t − 1) were also considered. The IAE for Jaya, TLBO, and GA were found to be 0.02660, 0.05460, and 0.15067, respectively, for exponential input and 0.164, 0.036, and 0.152, respectively, for sinusoid input. Based on the detailed simulation studies, it is concluded that the Jaya optimization algorithm is able to obtain the minimum fitness value for all the three reference inputs.

402 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 15.15 Controller outputs for optimized gains: Sinusoid input.

FIGURE 15.16 Error signal for sinusoid input.

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Oh, J.H., Lee, J.S., 1998. Backstepping control design of flexible joint manipulator using only position measurements. In: Proceedings of the 37th IEEE Conference on Decision & Control. 18–18 Dec. 1998, Tampa, Florida, USA, pp. 931–936. https://doi.org/10.1109/CDC.1998.760814. Pan, Y., Wang, H., Li, X., Yu, H., 2017. Adaptive command-filtered backstepping control of robot arms with compliant actuators. IEEE Transactions on Control Systems Technology 26 (3), 1149–1156. Pilla, R., Azar, A.T., Gorripotu, T.S., 2019. Impact of flexible AC transmission system devices on automatic generation control with a metaheuristic based fuzzy PID controller. Energies 12 (21), 4193. https://doi.org/10.3390/en12214193. Pranav, Kumar, J., Kumar, V., Rana, K.P.S., 2016. Efficient reaching law for SMC with PID surface applied to a manipulator. In: Proceedings of 2nd International Conference on Innovative Applications of Computational Intelligence on Power, Energy and Controls with Their Impact on Humanity, KIET. 18–19 Nov. 2016, Ghaziabad, India, pp. 51–56. https:// doi.org/10.1109/CIPECH.2016.7918736. Pranav, Raman, Mehra, K.S., Kumar, J., Rana, K.P.S., Kumar, V., 2015. Some studies on SMC applied to a robotic manipulator. In: Proceedings of IEEE 12th International Conference on Electronics, Energy, Environment, Communication, Computer, Control, IINDICON. 17–20 Dec. 2015, Jamia Millia Islamia, New Delhi, India. https://doi.org/10.1109/INDICON.2015. 7443349. Rao, R.V., 2015. Jaya: a simple and new optimization algorithm for solving constrained and unconstrained optimization problems. International Journal of Industrial Engineering Computations 7, 19–34. Rao, R.V., Savsani, V.J., Vakharia, D.P., 2010. Teaching–learning-based optimization: a novel method for constrained mechanical design and optimization problems. Computer Aided Design 43 (3), 303–315. Sharma, R., Rana, K.P.S., Kumar, V., 2014a. Comparative study of controller optimization techniques for a robotic manipulator. In: Pant, M., Deep, K., Nagar, A., Bansal, J. (Eds.), Proceedings of the Third International Conference on Soft Computing for Problem Solving. In: Advances in Intelligent Systems and Computing, vol. 258. Springer, New Delhi. Sharma, R., Rana, K.P.S., Kumar, V., 2014b. Statistical analysis of GA based PID controller optimization for robotic manipulator. In: Proceedings of the IEEE International Conference on Issues and Challenges in Intelligent Computing Techniques. 7–8 Feb. 2014, Ghaziabad, India, pp. 713–718. https://doi.org/10.1109/ICICICT.2014.6781368. Sharma, R., Rana, K.P.S., Kumar, V., 2014c. Performance analysis of fractional order fuzzy PID controllers applied to a robotic manipulator. Expert Systems with Applications 41 (9), 4274–4289. Elsevier. Shukla, M.K., Sharma, B.B., Azar, A.T., 2018. Control and synchronization of a fractional order hyperchaotic system via backstepping and active backstepping approach. In: Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 597–624. Singh, S., Azar, A.T., Ouannas, A., Zhu, Q., Zhang, W., Na, J., 2017. Sliding Mode Control Technique for Multi-switching Synchronization of Chaotic Systems. In: Proceedings of 9th International Conference on Modelling, Identification and Control (ICMIC 2017). July 10–12, 2017, Kunming, China. IEEE, pp. 880–885. Soukkou, Y., Labiod, S., 2015. Adaptive backstepping control using combined direct and indirect adaptation for a single-link flexible-joint robot. International Journal of Industrial Electronics and Drives 2 (1), 11–19. Vaidyanathan, S., Azar, A.T., 2015a. Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan–Madhavan chaotic systems. In: Studies in Computational Intelligence, vol. 576. Springer-Verlag GmbH, Berlin/Heidelberg, pp. 527–547. https://doi.org/10.1007/978-3-319-11173-5_19.

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Vaidyanathan, S., Azar, A.T., 2015b. Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidyanathan chaotic systems. In: Studies in Computational Intelligence, vol. 576. Springer-Verlag GmbH, Berlin/Heidelberg, pp. 549–569. https:// doi.org/10.1007/978-3-319-11173-5_20. Vaidyanathan, S., Azar, A.T., 2016a. Takagi–Sugeno fuzzy logic controller for Liu–Chen fourscroll chaotic system. International Journal of Intelligent Engineering Informatics (IJIEI) 4 (2), 135–150. Vaidyanathan, S., Azar, A.T., 2016b. Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. In: Advances in Chaos Theory and Intelligent Control. In: Studies in Fuzziness and Soft Computing, vol. 337. Springer-Verlag, Germany, pp. 249–274. Vaidyanathan, S., Azar, A.T., Akgul, A., Lien, C.H., Kacar, S., Cavusoglu, U., 2019. A memristorbased system with hidden hyperchaotic attractors, its circuit design, synchronisation via integral sliding mode control and voice encryption. International Journal of Automation and Control (IJAAC) 13 (6), 644–667. Vaidyanathan, S., Idowu, B.A., Azar, A.T., 2015a. Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. In: Studies in Computational Intelligence, vol. 581. Springer-Verlag GmbH, Berlin/Heidelberg, pp. 39–58. https://doi.org/10.1007/9783-319-13132-0_3. Vaidyanathan, S., Jafari, S., Pham, V.T., Azar, A.T., Alsaadi, F.E., 2018. A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design. Archives of Control Sciences 28 (2), 239–254. Vaidyanathan, S., Sampath, S., Azar, A.T., 2015b. Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system. International Journal of Modelling, Identification and Control (IJMIC) 23 (1), 92–100. Yang, Y., Feng, G., Ren, J., 2004. A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems. IEEE Transactions on Systems, Man and Cybernetics. Part A. Systems and Humans 34 (3), 406–420. Yim, W., 2001. Adaptive control of a flexible joint manipulator. In: Proceedings of the 2001 IEEE International Conference on Robotics & Automation. 21–26 May 2001, Seoul, South Korea, pp. 3441–3446. https://doi.org/10.1109/ROBOT.2001.933150. Zouari, L., Abid, H., Abid, M., 2014. Flexible joint manipulator based on backstepping controller. In: 15th International Conference on Sciences and Techniques of Automatic Control & Computer Engineering – STA’2014. 21–23 Dec. 2014, Hammamet, Tunisia, pp. 330–334. https:// doi.org/10.1109/STA.2014.7086747. Zouari, L., Abid, H., Abid, M., 2015. Backstepping controller for electrically driven flexible joint manipulator under uncertainties. International Journal of Robotics and Automation 4 (2), 156–163.

Chapter 16

Backstepping control and synchronization of chaotic time delayed systems Ahmad Taher Azara,b , Fernando E. Serranoc , Sundarapandian Vaidyanathand , and Nashwa Ahmad Kamale a Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, b Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, c Universidad Tecnológica Centroamericana (UNITEC), Zona Jacaleapa, Tegucigalpa, Honduras, d Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, e Cairo University, Giza, Egypt

16.1

Introduction

The stabilization and control of chaotic systems have been recently studied because this phenomenon is found in many kinds of physical systems such as mechanical, electrical, chemical, biological among others. One of the main purposes of the chaotic system stabilization is to eliminate the chaotic behavior yielded by unstable limit cycles, or oscillations, in order to follow a desired trajectory or to reach the equilibrium points (Azar et al., 2017; Ouannas et al., 2020c, 2019c, 2020a, 2019b). This objective is achieved to avoid this characteristic because some physical systems can be destroyed if a chaotic behavior is found, besides, a desired operating point can be reached (Alain et al., 2019, 2020; Vaidyanathan and Azar, 2016c; Singh et al., 2018a; Singh and Azar, 2020). Nowadays, there are many control and stabilization strategies such as backstepping controller as found in Gholipour et al. (2015) in which an optimal backstepping controller is designed for chaos control of a rod-type plasma torch system in which the optimal parameters of the backstepping controller are found by a bees algorithm. In Shukla and Sharma (2017) the backstepping stabilization of a fractional order chaotic system is evinced, where the backstepping design procedure is done by a Mittag-Leffler and Lyapunov approach and the results are tested in a chaotic Lorenz systems. Another example can be found in Chen et al. (2008) where a chaotic system is stabilized by a backstepping controller without the parameter perturbation, this proposed strategy allows the parameter perturbation systems to be stabilized. Finally, in Laoye et al. (2009); Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00023-4 Copyright © 2021 Elsevier Inc. All rights reserved.

407

408 Backstepping Control of Nonlinear Dynamical Systems

Njah and Sunday (2009) recursive backstepping controllers for the stabilization of 4D Lorenz Stenflo systems are shown. The synchronization of chaotic systems has been studied also in recent years when the synchronization is needed in coupled physical chaotic systems (Khan et al., 2020b,a; Ouannas et al., 2019a, 2017, 2020b; Vaidyanathan et al., 2019; Vaidyanathan and Azar, 2016b,a). For example, in Huang (2008), an adaptive synchronization controller is implemented for the synchronization of two hyper-chaotic systems. In Zheng (2016), the synchronization of two coupled time delayed complex chaotic system is proposed, besides, in Lei and Wang (2016), the synchronization of Genesio and Chen systems is achieved with a reduced number of active inputs methods. Finally, the synchronization of two time delayed fractional order chaotic systems with different structure and order is shown in Song et al. (2016) in which this synchronization technique includes a compensator and an optimal controller. The control of time delayed systems has been studied nowadays in which the delays are considered in the inputs or in the states. Some examples of controllers implemented for the stabilization of time delayed systems are found, for example, in Bresch-Pietri et al. (2018) where the design of a prediction based controller for linear systems with time varying input delays is proposed. Other examples of controllers and stability criteria for time delayed nonlinear systems can be found in Parlakçi and Küçükdemiral (2011); Erol and ˙Iftar (2013); Lee et al. (2014) considering that the results obtained in these studies are very important for the design of the proposed control technique explained in this chapter because the conditions for the stabilization of state delayed chaotic systems are provided. In this study, the stabilization and synchronization of time delayed chaotic system is presented. These objectives are achieved by the design of a backstepping controller and synchronizer for chaotic systems of any dimension, considering that in the synchronization case the chaotic systems must be identical. The backstepping design procedures for the synchronization and stabilization of chaotic systems is done by a recursive method designing appropriate Lyapunov functions in which the delayed states in the chaotic system are considered and so, some conditions must be taken into account to deal with delays. It is important to notice that the results obtained in this study can be extended to hyper-chaotic systems and the obtained results are implemented for the stabilization of a time delayed Lorenz system and the synchronization of a Rössler system. The chapter is divided in the following sections. In Section 16.2, the related work is explained. In Sections 16.3 and 16.4, the backstepping control and synchronization design are depicted. Then in Section 16.5 a numerical example is shown. Finally in Sections 16.6 and 16.7, the discussion and conclusions of this study are shown, respectively.

Backstepping control and synchronization Chapter | 16 409

16.2

Related work

Some related work to the control and stabilization of chaotic systems can be found in Zhang et al. (2005) where a backstepping controller is designed to stabilize a Rössler system to a steady state. Then in Yassen (2006) another backstepping controller is used to stabilize a Lorenz, Chen, and Lü systems. Finally in Gambino et al. (2006), a Chen chaotic system is controlled by a nonlinear stabilizing control law. In Mobayen and Ma (2018), a synchronization of time delayed chaotic systems is presented considering the presence of disturbances. Backstepping synchronization examples can be found in Wang and Wen (2007); Yu and Zhang (2004); Park (2006) in which the system synchronization is designed to obtain the desired system state values and to drive the error variables between the drive–response system to zero as time goes to infinity. Finally in Zhou et al. (2008), an adaptive synchronization approach is used to synchronize two identical hyper-chaotic systems implementing a Lyapunov approach where the parameters of the system are uncertain. In Chen and Li (2006); Azar and Serrano (2015), some conditions for time delayed systems are explained to synchronize and stabilize the time delayed chaotic system when delays are found in the states. The backstepping approach designed and explained in this chapter considers the previous mentioned conditions in the design of the Lyapunov function because it is not necessary to implement Lyapunov–Krasovskii functions.

16.3

Backstepping stabilization of time delayed systems

As previously explained, the design of a backstepping controller to stabilize the variables of a state delayed chaotic system is shown in this section. The backstepping design procedure is done following a recursive procedure by first designing the Lyapunov functional for the first state until the last state (Vaidyanathan et al., 2018; Shukla et al., 2018; Vaidyanathan et al., 2015). Virtual control variables are useful in the design procedure to make the derivative of the positive definite to be negative definite. The results obtained in this section are based on the studies of Jiao et al. (2005); Chen and Li (2006) in which nonlinear systems with time delays are considered. First, consider the following nonlinear time delayed system: x˙1

=

f1 (X) + f1 (Xτ ),

x˙2

= .. .

f2 (X) + f2 (Xτ ),

x˙n

=

fn (X) + fn (Xτ ) + U,

(16.1)

where X ∈ Rn is X = [x1 (t), x2 (t), ..., xn (t)]T and Xτ = [x1 (t − τ1 ), x2 (t − τ2 ), ..., xn (t −τn )]T , and U ∈ R is the control input, and τi ≥ 0 for i = 1, 2, ..., n.

410 Backstepping Control of Nonlinear Dynamical Systems

The backstepping design procedure is done by defining the following auxiliary variables and virtual control inputs: z1

=

x1 ,

z2

= .. .

x2 − α1 (X),

zn

=

xn − αn−1 (X).

(16.2)

Then, by selecting the Lyapunov function V1 = 12 z12 , the derivative of this Lyapunov function is V˙1 = z1 (f1 (X) + f1 (Xτ )).

(16.3)

Then by making α1 = f1 (Xτ ) (16.3) becomes V˙1 = z1 (f1 (X) + f1 (Xτ ) − x2 ) + z1 z2 .

(16.4)

Now selecting the following Lyapunov function: 1 V2 = V1 + z22 . 2

(16.5)

Taking the first derivative of (16.5) yields V˙2 = z1 (f1 (X) + f1 (Xτ ) − x2 ) + z1 z2 + z2 (f2 (X) + f2 (Xτ ) − α˙ 1 ).

(16.6)

Now defining α2 = f2 (Xτ ) − α˙ 1 (16.6) becomes V˙2 = z1 (f1 (X) + f1 (Xτ ) − x2 ) + z1 z2 + z2 (f2 (X) + f2 (Xτ ) − α˙ 1 − x3 ) + z2 z3 .

(16.7)

So, to ensure the stability by using all the Lyapunov functions, the following derivative of Vn is obtained: V˙n = z1 f1 (X) + z2 f2 (X) + ... ... + zn fn (X) + zn fn (Xτ ) + zn U − zn α˙ n−1 .

(16.8)

Now selecting the following control law: U =−

z1 z2 f1 (x) − f2 (x) − ... − fn (x) + α˙ n−1 − 2fn (xτ ), zn zn

(16.9)

Backstepping control and synchronization Chapter | 16 411

Eq. (16.8) becomes V˙n = −zn fn (xτ ).

(16.10)

As shown in Jiao et al. (2005) |fi (Xτ )| ≤

i 

bij |xj t |

(16.11)

j =1

where bij > 0 with i = 1, ..., n and j = 1, ..., i. For this reason (16.10) becomes V˙n ≤ −|zn |

i 

bij |xj t |.

(16.12)

j =1

With this result, the stability of the system is ensured and it will be corroborated in Section 16.5 for the chaotic system where it is stabilized until the equilibrium point is reached. In the next section the synchronization of identical chaotic system by backstepping control is presented.

16.4

Backstepping synchronization of time delayed chaotic systems

For the backstepping synchronization of chaotic system, a similar procedure as used in the previous section is followed. For this purpose consider the following drive system: x˙1

=

f1 (X) + f1 (Xτ ),

x˙2

= .. .

f2 (X) + f2 (Xτ ),

x˙n

=

fn (X) + fn (Xτ ),

(16.13)

where X ∈ Rn is X = [x1 (t), x2 (t), ..., xn (t)]T and Xτ = [x1 (t − τ1 ), x2 (t − τ2 ), ..., xn (t − τn )]T for τi ≥ 0 with i = 1, ..., n. The response system is given by y˙1

=

g1 (Y ) + g1 (Yτ ),

y˙2

= .. .

g2 (Y ) + g2 (Yτ ),

y˙n

=

gn (Y ) + gn (Yτ ) + U,

(16.14)

where Y ∈ Rn with Y = [y1 (t), y2 (t), ..., yn (t)]T and Yτ = [x1 (t − τ1 ), x2 (t − τ2 ), ..., xn (t − τn )]T where τi ≥ 0 for i = 1, ..., n. Defining the error as

412 Backstepping Control of Nonlinear Dynamical Systems

ei = xi − yi , the following error system is obtained: e˙1

=

f1 (X) + f1 (Xτ ) − g1 (Y ) − g1 (Yτ ),

e˙2

= .. .

f2 (X) + f2 (Xτ ) − g2 (Y ) − g2 (Yτ ),

e˙n

=

fn (X) + fn (Xτ ) − gn (Y ) − gn (Yτ ) − U.

(16.15)

Then by making F Gi (X, Y ) = fi (X) − gi (Y ) and F Gi (Xτ , Yτ ) = fi (Xτ ) − gi (Yτ ), the system (16.15) becomes e˙1

=

F G1 (X, Y ) + F G1 (Xτ , Yτ ),

e˙2

= .. .

F G2 (X, Y ) + F G2 (Xτ , Yτ ),

e˙n

=

F Gn (X, Y ) + F Gn (Xτ , Yτ ) − U.

(16.16)

To design the synchronization control law, consider the following change of variable with virtual control laws (Jiao et al., 2005) z1 = e1 , z2 = e2 − α1 , and zn = en − αn−1 . So, by selecting the Lyapunov function V1 = 12 z12 , its derivative is V˙1 = z1 (F G1 (X, Y ) + F G1 (Xτ , Yτ ))

(16.17)

then, by making α1 = F G1 (Xτ , Yτ ), (16.17) becomes V˙1 = z1 (F G1 (X, Y ) + F G1 (Xτ , Yτ ) − e2 ) + z1 z2 .

(16.18)

Now define a new Lyapunov function: 1 V2 = V1 + z22 . 2

(16.19)

So, the derivative of (16.19) is V˙2 = z1 (F G1 (X, Y ) + F G1 (Xτ , Yτ ) − e2 ) + z1 z2 + z2 (F G2 (X, Y ) + F G2 (Xτ , Yτ ) − α˙ 1 ).

(16.20)

Now making α2 = F G2 (Xτ , Yτ ) − α˙ 1 and implementing z3 in (16.20), the following result is obtained: V˙2 = z1 (F G1 (X, Y ) + F G1 (Xτ , Yτ ) − e2 ) + z1 z2 + z2 (F G2 (X, Y ) + F G2 (Xτ , Yτ ) − α˙ 1 − e3 ) + z2 z3 .

(16.21)

Backstepping control and synchronization Chapter | 16 413

So, the derivative of Vn is given by V˙n = z1 F G1 (X, Y ) + z2 F G2 (X, Y ) + ... ... + zn F Gn (X, Y ) + zn F Gn (Xτ , Yτ ) − zn U − zn α˙ n−1 .

(16.22)

So, with the following control law: z2 z1 F G1 (X, Y ) + F G2 (X, Y ) + ... + F Gn (X, Y ) zn zn − α˙ n−1 + 2F Gn (Xτ , Yτ ),

U=

(16.23)

Eq. (16.22) becomes ⎛ V˙n ≤ −|zn | ⎝

i  j =1

aj |xj t | +

i 

⎞ bj |yj t |⎠

(16.24)

j =1

for aj , bj > 0. With these results the time delayed chaotic system is stable and the derivatives of the Lyapunov functionals are solved recursively, so the synchronization of two identical time delay chaotic systems is done successfully. In the next section, two numerical experiments are shown, one to test and validate the controller and the other to validate synchronizer controller.

16.5

Numerical examples

In this section, two numerical examples are shown for the stabilization and synchronization of time delayed chaotic systems. The first example consists in the stabilization of a time delayed Lorenz chaotic systems and the second example consists in the synchronization of two identical time delayed Rössler systems under different initial conditions. The following examples verify and validate the theoretical results obtained in this study, including a comparative analysis to corroborate the superior performance and novelty of the proposed strategy.

16.5.1 Example 1: Stabilization of the time delayed Lorenz chaotic system In this section the stabilization of a Lorenz time delayed chaotic system is shown. The following dynamical system demonstrates the Lorenz time delayed state controlled system: x(t) ˙

=

a(y(t − τ2 ) − x(t − τ1 )),

y(t) ˙

=

cx(t − τ1 ) − x(t − τ1 )z(t − τ3 ) − y(t − τ2 ),

z˙ (t) =

x(t − τ1 )y(t − τ2 ) − bz(t − τ3 ) + U.

(16.25)

414 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 16.1 Phase portrait x1 , x2 of the Lorenz chaotic time delayed system.

FIGURE 16.2 Phase portrait x2 , x3 of the Lorenz chaotic time delayed system.

The simulation parameters were selected as a = 10, b = 8/3, and c = 28 with the initial conditions x(0) = −8, y(0) = 8, and z(0) = 27 and the delay constants τ1 = 0.02 s, τ2 = 0.02 s, and τ3 = 0 s. The phase portrait for x1 and x2 along with the phase portrait of x2 and x3 are shown in Fig. 16.1 and Fig. 16.2 considering an uncontrolled system. In Figs. 16.3, 16.4, and 16.5, the time evolution of these state variables are depicted noticing that in Fig. 16.3 the proposed controller stabilizes this variable with less overshoot than the results obtained by the control strategies shown in Chen and Gao (2018); Singh et al. (2018b), noticing that the results obtained

Backstepping control and synchronization Chapter | 16 415

FIGURE 16.3 Time evolution of the variable x1 .

FIGURE 16.4 Time evolution of the variable x2 .

by the proposed controller yields less oscillations than the strategy shown in Singh et al. (2018b). The same occurs for Fig. 16.4 where less oscillations and smaller overshoot are obtained by the proposed control strategy. It is important to notice that in Fig. 16.5 the equilibrium point is reached efficiently by the proposed control strategy and it is noticeable also that the system response of x3 yielded by Chen and Gao (2018); Singh et al. (2018b) is significantly deviated. Another interesting characteristic can be found in Fig. 16.5, where the results provided by the strategy found in Singh et al. (2018b) stabilizes the system in other equilibrium point and not the equilibrium point reached by the proposed

416 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 16.5 Time evolution of the variable x3 .

FIGURE 16.6 Time evolution of the input variable U .

control strategy. This occurs because the controller performance shown in Singh et al. (2018b) is not the optimal for time delayed chaotic systems. Finally in Fig. 16.6, the control input of the stabilized chaotic time delayed system is shown noticing that the proposed controller strategy yields a smaller control effort and less oscillations in comparison with the control effort yielded by the strategies shown in Chen and Gao (2018); Singh et al. (2018b). The difference between the control efforts shown in Fig. 16.6 is due to the strategy shown in Singh et al. (2018b) stabilize the system in other equilibrium point because its performance is not optimal for this kind of time delayed chaotic systems, something that can be corroborated also in Fig. 16.5.

Backstepping control and synchronization Chapter | 16 417

FIGURE 16.7 Phase portrait x1 , x2 of the Rössler chaotic time delayed system.

16.5.2 Example 2: Synchronization of the time delayed Rössler chaotic system The second example evinces the synchronization of two time delayed identical chaotic systems which is a Rössler system. The drive and response systems are defined as follows (Yu and Zhang, 2004): x˙1 (t)

=

x2 (t − τ2 ) + ax1 (t − τ1 ),

x˙2 (t)

= −x1 (t − τ1 ) − x3 (t − τ3 ),

x˙3 (t)

=

x2 (t − τ2 )x3 (t − τ3 ) − bx3 (t − τ3 ) + c,

y˙1 (t)

=

y˙2 (t)

= −y1 (t − τ1 ) − y3 (t − τ3 ),

y˙3 (t)

=

(16.26)

y2 (t − τ2 ) + ay1 (t − τ1 ), y2 (t − τ2 )y3 (t − τ3 ) − by3 (t − τ3 ) + c + U,

(16.27)

where the simulation constants and initial conditions for the drive and response systems are a = 0.2, b = 5.7, and c = 0.2 with x1 (0) = 10, x2 (0) = 1, x3 (0) = 5, y1 (0) = 12, y2 (0) = 5, y3 (0) = 11, and the time delays are τ1 = 0.02 s, τ2 = 0.02 s, and τ3 = 0 s. In Figs. 16.7 and 16.8, the phase portraits of the chaotic drive system shown in (16.26) are depicted. In Figs. 16.9, 16.10, and 16.11, the synchronization is achieved by the backstepping proposed controller where that the error between the drive and system response is small as corroborated later. Finally, Figs. 16.12, 16.13, and 16.14 corroborate that the error is significantly small proving that this synchronization strategy provides an efficient system performance.

418 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 16.8 Phase portrait x2 , x3 of the Rössler chaotic time delayed system.

FIGURE 16.9 Synchronization of x1 and y1 .

16.6 Discussion The results obtained in this study offers a contribution for the stabilization and synchronization of time delayed chaotic systems. One of the issues that has to do with time delayed systems of any kind is the conditions on which time delays are treated, some conditions such as the Wirtinger based inequality helps to ensure the stability of time delayed systems (Lee et al., 2014). In this study and in order to overcome this problem, the property (16.11) that is found in Jiao et al. (2005) provides sufficient conditions in order to stabilize and synchronize any

Backstepping control and synchronization Chapter | 16 419

FIGURE 16.10 Synchronization of x2 and y2 .

FIGURE 16.11 Synchronization of x3 and y3 .

kind of time delayed systems. The backstepping design procedure used in this study is done recursively where all Lyapunov functional derivatives ensure the stability or convergence for synchronization or stabilization. One final comment that is important to mention, is that the backstepping control strategy shown in this study can be implemented in hyper-chaotic time delayed system for synchronization and stabilization purposes.

16.7

Conclusion

In this study, the design of a backstepping controller for stabilization and synchronization of time delayed chaotic system is shown. The design procedure

420 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 16.12 Error between x1 and y1 .

FIGURE 16.13 Error between x2 and y2 .

for the backstepping controller is done by selecting appropriate Lyapunov functions, and in this case, the use of Lyapunov–Krasovskii function is not necessary. In order to overcome the time delays in the states of chaotic systems, some property was used to assure the stability of these systems and to make the error between the variables of the drive and response systems approach zero as times goes to infinity in the synchronization case. Two illustrative examples corroborates the results obtained in this study, first, the stabilization of a Lorenz time delay system and then the synchronization of two identical Rössler systems.

Backstepping control and synchronization Chapter | 16 421

FIGURE 16.14 Error between x3 and y3 .

References Alain, K.S.T., Azar, A.T., Bertrand, F.H., Romanic, K., 2019. Robust observer-based synchronisation of chaotic oscillators with structural perturbations and input nonlinearity. International Journal of Automation and Control 13 (4), 387–412. Alain, K.S.T., Azar, A.T., Kengne, R., Bertrand, F.H., 2020. Stability analysis and robust synchronisation of fractional-order modified Colpitts oscillators. International Journal of Automation and Control 14 (1), 52–79. Azar, A.T., Serrano, F.E., 2015. Deadbeat control for multivariable discrete time systems with time varying delays. In: Chaos Modeling and Control Systems Design, pp. 97–132. Azar, A.T., Volos, C., Gerodimos, N.A., Tombras, G.S., Pham, V.-T., Radwan, A.G., Vaidyanathan, S., Ouannas, A., Munoz-Pacheco, J.M., 2017. A novel chaotic system without equilibrium: dynamics, synchronization, and circuit realization. Complexity 2017, 7871467 (1–11). Bresch-Pietri, D., Mazenc, F., Petit, N., 2018. Robust compensation of a chattering time-varying input delay with jumps. Automatica 92, 225–234. Chen, F., Chen, L., Zhang, W., 2008. Stabilization of parameters perturbation chaotic system via adaptive backstepping technique. Applied Mathematics and Computation 200 (1), 101–109. Chen, Q., Gao, J., 2018. Delay feedback control of the Lorenz like system. Mathematical Problems in Engineering, 1–13. Chen, W., Li, J., 2006. Backstepping tracking control for nonlinear time-delay systems. Journal of Systems Engineering and Electronics 17 (4), 846–852. Erol, H.E., ˙Iftar, A., 2013. A necessary and sufficient condition for the stabilization of decentralized time-delay systems by time-delay controllers. IFAC Proceedings Volumes 46 (13), 62–67. Gambino, G., Lombardo, M.C., Sammartino, M., 2006. Global linear feedback control for the generalized Lorenz system. Chaos, Solitons and Fractals 29 (4), 829–837. Gholipour, R., Khosravi, A., Mojallali, H., 2015. Multi-objective optimal backstepping controller design for chaos control in a rod-type plasma torch system using bees algorithm. Applied Mathematical Modelling 39 (15), 4432–4444. Huang, J., 2008. Chaos synchronization between two novel different hyperchaotic systems with unknown parameters. Nonlinear Analysis: Theory, Methods and Applications 69 (11), 4174–4181. Jiao, X., Sun, Y., Shen, T., 2005. Backstepping design for robust stabilizing control of nonlinear systems with time-delay. IFAC Proceedings Volumes 38 (1), 375–380.

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Khan, A., Singh, S., Azar, A.T., 2020a. Combination–combination anti-synchronization of four fractional order identical hyperchaotic systems. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). In: Advances in Intelligent Systems and Computing, vol. 921. Springer International Publishing, Cham, pp. 406–414. Khan, A., Singh, S., Azar, A.T., 2020b. Synchronization between a novel integer-order hyperchaotic system and a fractional-order hyperchaotic system using tracking control. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). In: Advances in Intelligent Systems and Computing, vol. 921. Springer International Publishing, Cham, pp. 382–391. Laoye, J., Vincent, U., Kareem, S., 2009. Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller. Chaos, Solitons and Fractals 39 (1), 356–362. Lee, T.H., Park, J.H., Jung, H., Kwon, O., Lee, S., 2014. Improved results on stability of time-delay systems using Wirtinger-based inequality. IFAC Proceedings Volumes 47 (3), 6826–6830. Lei, J., Wang, L., 2016. Backstepping synchronous control of chaotic system with reduced number of active inputs. Optik 127 (23), 11364–11373. Mobayen, S., Ma, J., 2018. Robust finite-time composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and time-delay. Chaos, Solitons and Fractals 114, 46–54. Njah, A., Sunday, O., 2009. Generalization on the chaos control of 4-D chaotic systems using recursive backstepping nonlinear controller. Chaos, Solitons and Fractals 41 (5), 2371–2376. Ouannas, A., Azar, A.T., Vaidyanathan, S., 2017. On a simple approach for Q-S synchronization of chaotic dynamical systems in continuous-time. International Journal of Computing Science and Mathematics 8 (1), 20–27. Ouannas, A., Azar, A.T., Ziar, T., 2019a. Control of continuous-time chaotic (hyperchaotic) systems: F-M synchronisation. International Journal of Automation and Control 13 (2), 226–242. Ouannas, A., Grassi, G., Azar, A.T., 2020a. Fractional-order control scheme for Q-S chaos synchronization. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). Springer International Publishing, Cham, pp. 434–441. Ouannas, A., Grassi, G., Azar, A.T., 2020b. A new generalized synchronization scheme to control fractional chaotic systems with non-identical dimensions and different orders. In: Hassanien, A.E., Azar, A.T., Gaber, T., Bhatnagar, R., Tolba, M.F. (Eds.), The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2019). In: Advances in Intelligent Systems and Computing, vol. 921. Springer International Publishing, Cham, pp. 415–424. Ouannas, A., Grassi, G., Azar, A.T., Gasri, A., 2019b. A new control scheme for hybrid chaos synchronization. In: Hassanien, A.E., Tolba, M.F., Shaalan, K., Azar, A.T. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2018. In: Advances in Intelligent Systems and Computing, vol. 845. Springer International Publishing, Cham, pp. 108–116. Ouannas, A., Grassi, G., Azar, A.T., Khennaouia, A.A., Pham, V.-T., 2020c. Chaotic control in fractional-order discrete-time systems. In: Hassanien, A.E., Shaalan, K., Tolba, M.F. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019. In: Advances in Intelligent Systems and Computing, vol. 1058. Springer International Publishing, Cham, pp. 207–217. Ouannas, A., Grassi, G., Azar, A.T., Singh, S., 2019c. New control schemes for fractional chaos synchronization. In: Hassanien, A.E., Tolba, M.F., Shaalan, K., Azar, A.T. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2018. In: Advances in Intelligent Systems and Computing, vol. 845. Springer International Publishing, Cham, pp. 52–63.

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Park, J.H., 2006. Synchronization of Genesio chaotic system via backstepping approach. Chaos, Solitons and Fractals 27 (5), 1369–1375. Parlakçi, M.A., Küçükdemiral, Ä.B., 2011. Further stability criteria for time-delay systems with interval time-varying delays. IFAC Proceedings Volumes 44 (1), 3867–3872. Shukla, M.K., Sharma, B., 2017. Stabilization of a class of fractional order chaotic systems via backstepping approach. Chaos, Solitons and Fractals 98, 56–62. Shukla, M.K., Sharma, B.B., Azar, A.T., 2018. Control and synchronization of a fractional order hyperchaotic system via backstepping and active backstepping approach. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 559–595. Singh, S., Azar, A.T., 2020. Controlling chaotic system via optimal control. In: Hassanien, A.E., Shaalan, K., Tolba, M.F. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019. In: Advances in Intelligent Systems and Computing, vol. 1058. Springer International Publishing, Cham, pp. 277–287. Singh, S., Azar, A.T., Bhat, M.A., Vaidyanathan, S., Ouannas, A., 2018a. Active control for multi-switching combination synchronization of non-identical chaotic systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in System Dynamics and Control. In: Advances in Systems Analysis, Software Engineering, and High Performance Computing (ASASEHPC). IGI Global, pp. 129–162. Singh, P., Singh, K., Roy, B., 2018b. Chaos control in biological system using recursive backstepping sliding mode control. The European Physical Journal Special Topics, 731–746. Song, X., Song, S., Li, B., 2016. Adaptive synchronization of two time-delayed fractional-order chaotic systems with different structure and different order. Optik 127 (24), 11860–11870. Vaidyanathan, S., Azar, A.T., 2016a. Dynamic analysis, adaptive feedback control and synchronization of an eight-term 3-D novel chaotic system with three quadratic nonlinearities. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 155–178. Vaidyanathan, S., Azar, A.T., 2016b. Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In: Advances in Chaos Theory and Intelligent Control. Springer, Berlin, Germany, pp. 275–290. Vaidyanathan, S., Azar, A.T., 2016c. Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in Chaos Theory and Intelligent Control. Springer International Publishing, Cham, pp. 179–202. Vaidyanathan, S., Azar, A.T., Akgul, A., Lien, C.-H., Kacar, S., Cavusoglu, U., 2019. A memristorbased system with hidden hyperchaotic attractors, its circuit design, synchronisation via integral sliding mode control and an application to voice encryption. International Journal of Automation and Control 13 (6), 644–667. Vaidyanathan, S., Idowu, B.A., Azar, A.T., 2015. Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Chaos Modeling and Control Systems Design. In: Studies in Computational Intelligence, vol. 581. Springer, Berlin, Germany, pp. 39–58. Vaidyanathan, S., Jafari, S., Pham, V.-T., Azar, A.T., Alsaadi, F.E., 2018. A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design. Archives of Control Sciences 28 (2), 239–254. Wang, B., Wen, G., 2007. On the synchronization of a class of chaotic systems based on backstepping method. Physics Letters A 370 (1), 35–39. Yassen, M., 2006. Chaos control of chaotic dynamical systems using backstepping design. Chaos, Solitons and Fractals 27 (2), 537–548. Yu, Y., Zhang, S., 2004. Adaptive backstepping synchronization of uncertain chaotic system. Chaos, Solitons and Fractals 21 (3), 643–649. Zhang, H., Ma, X.-k., Li, M., Zou, J.-l., 2005. Controlling and tracking hyperchaotic Rössler system via active backstepping design. Chaos, Solitons and Fractals 26 (2), 353–361.

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Zheng, S., 2016. Synchronization analysis of time delay complex-variable chaotic systems with discontinuous coupling. Journal of the Franklin Institute 353 (6), 1460–1477. Zhou, X., Wu, Y., Li, Y., Xue, H., 2008. Adaptive control and synchronization of a novel hyperchaotic system with uncertain parameters. Applied Mathematics and Computation 203 (1), 80–85.

Chapter 17

Multi-switching synchronization of nonlinear hyperchaotic systems via backstepping control Shikha Singha , Sandhya Mathpalb , Ahmad Taher Azarc,d , Sundarapandian Vaidyanathane , and Nashwa Ahmad Kamalf a Department of Mathematics, Jesus and Mary College, University of Delhi, New Delhi, India, b Department of Mathematics, Faculty of Engineering & Technology, MRIU, Faridabad, Haryana, India, c Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, d Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, e Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, f Cairo University, Giza, Egypt

17.1

Introduction

Over the last few decades, the random chaotic behavior of deterministic systems has shown great interest. Chaotic attractor can be defined as deterministic random behavior in bounded phase space of the underlying nonlinear dynamical system that has extreme sensitiveness to infinitesimal perturbations in initial conditions (Volos et al., 2018; Wang et al., 2017; Vaidyanathan et al., 2018b). Deterministic because it arises from intrinsic causes and not from some extraneous noise or interference and random due to irregular, unpredictable behavior, which is characterized by exponential divergence of nearby trajectories on average. Chaos theory, once considered to be the third revolution in physics following relativity theory and quantum mechanics, has been studied extensively in the past 30 years. During the last few decades, chaotic dynamics has moved from mystery to familiarity. In the last few decades, a large number of theoretical investigations, numerical simulations, and experimental work have been carried out on various dynamical systems in an effort to understand the different features associated with the occurrence of chaotic behavior (Alain et al., 2020, 2019; Khan et al., 2020b; Ouannas et al., 2019a,b,c,d). A lot of chaotic phenomena have been found and enormous mathematical strides have been taken. Nowadays, it has been agreed by scientists and engineers that chaos is ubiquitous in natural sciences and social sciences, such as in physics, chemistry, mathematics, biology, ecology, physiology, economics Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00024-6 Copyright © 2021 Elsevier Inc. All rights reserved.

425

426 Backstepping Control of Nonlinear Dynamical Systems

(Strogatz, 2001; Kassim et al., 2017; Pano-Azucena et al., 2017). Wherever nonlinearity exists, chaos may be found. For a long time, chaos was thought of as a harmful behavior that could decrease the performance of a system and therefore should be avoided when the system is running. One remarkable feature of a chaotic system distinguishing itself from other non-chaotic systems is that the system is extremely sensitive to initial conditions. Any tiny perturbation of the initial conditions will significantly alter the long-term dynamics of the system. This fact means that when one wants to control a chaotic system one must make sure that the measurement of the needed signals is absolutely precise. Otherwise any attempt of controlling chaos would make the dynamics of the system go to an unexpected state. Chaos control refers to the manipulation of chaotic dynamism in certain complicated nonlinear systems. In fact, chaos control as a new and young discipline played a role in many traditional scientific and technological developments. Examples include data traffic congestion control in the Internet, encryption, and secure communication at different levels of communications, high-performance circuits and devices (e.g., delta-sigma modulators and power converters), liquid mixing, chemical reactions, power systems collapse prediction and protection, oscillators design, biological systems modeling and analysis (e.g., the brain and the heart), crisis management (e.g., jet-engine surge and stall), nonlinear computing and information processing, and critical decisionmaking in political, economic, and military events (Strogatz, 2001; Ouannas et al., 2020c; Singh and Azar, 2020; Ouannas et al., 2019a,b,c). There are many practical reasons for controlling chaos. In a system where chaotic response is undesired or harmful, it should be reduced as much as possible, or totally suppressed. Examples of this include avoiding fatal voltage collapse in power networks, eliminating deadly cardiac arrhythmias, guiding disordered circuit arrays (e.g., multi-coupled oscillators and cellular neural networks) to reach a certain level of desirable pattern formation, regulating dynamical responses of mechanical and electronic devices (e.g., diodes, laser machines, and machine tools), and organizing a multi-agency corporation to achieve optimal performance. It has been shown that the sensitivity of chaotic systems to small perturbations can be used to rapidly direct system trajectories to a desired target using a minimal control effort. A suitable manipulation of chaotic dynamics, such as stability conversion or bifurcation delay can significantly extend the operational range of machine tools and aircraft engines, enhance the artificial intelligence of neural networks, and also increases coding/decoding efficiency in signal and image encryption and communications. It has been demonstrated that data traffic through the Internet is likely to be chaotic. Special chaos control strategies may help network designers in better congestion control, thereby further benefiting the rapidly evolving and expanding Internet, to handle the exponentially increasing demands from the industry and the commercial market.

Multi-switching synchronization Chapter | 17 427

Within the context of biological systems, chaos control seems to be a crucial mechanism employed by the human brain in carrying out many of its tasks such as learning, perception, memorization, and particularly conceptualization. Additionally, some recent laboratory studies reveal that the complex dynamical variability in a variety of normal-functioning physiological systems demonstrates features reminiscent of chaos. Some medical evidence supports the idea that a safe and highly efficient intelligent pacemaker can soon be developed to control certain chaotic cardiac arrhythmias. In fact, chaos has become a public focal point in various research areas of life sciences, medicine research, and biomedical engineering. Chaos synchronization involves the coupling of two chaotic systems so that both systems achieve identical dynamics asymptotically with time (Ouannas et al., 2020a; Vaidyanathan et al., 2018a; Pham et al., 2018; Alain et al., 2018). There are two forms of coupling: mutual (bidirectional) coupling and the drive-response (unidirectional) coupling. In mutual coupling, the two systems influence or alter each other dynamically until both systems achieve identical dynamics. In the unidirectional coupling, control functions are designed to force the dynamics of one system referred to as the response system to track the unaltered dynamics of the other system referred to as the drive system. Chaos synchronization is of great theoretical significance and practical value and has made great contributions due to its potential application in many scientific and engineering fields during the recent years (Pham et al., 2017; Ouannas et al., 2017h,f,j). The history of synchronization goes back to the 17th century when the Dutch physicist Christiaan Huygens reported on his observation of phase synchronization of two pendulum clocks (Huygens, 1986). In 1990, Pecora and Carroll (1990) gave the synchronization of chaotic systems using the concept of master and slave system of integer-order chaotic systems. Different types of methods have been developed for controlling chaos and synchronization of non-identical and identical systems, for instance adaptive control (Khan and Singh, 2017b), active control (Singh et al., 2018a; Khan and Singh, 2017c; Vaidyanathan et al., 2017, 2015a), optimal control (Singh and Azar, 2020), robust control (Khettab et al., 2018), backstepping control (Wang et al., 2007; Vaidyanathan et al., 2018b; Wang, 2013; Shukla et al., 2018), sliding mode control (Azar and Serrano, 2020; Singh et al., 2017; Vaidyanathan et al., 2015b, 2019), and adaptive sliding mode control (Khan and Singh, 2017a). In past few years, active control have been extensively used as vigorous control method for controlling chaos and synchronization. This control method gives the flexibility to establish a control law to be used widely for controlling chaos and synchronization of various chaotic and hyperchaotic nonlinear dynamical system. As a result of fast growing interest in chaos control and synchronization, various synchronization types and schemes have been proposed and reported. For instance, complete synchronization (Azar et al., 2017; Bhat and Shikha, 2019), phase synchronization (Nobukawa et al., 2019), generalized synchronization

428 Backstepping Control of Nonlinear Dynamical Systems

(Zhang et al., 2019b; Ouannas et al., 2020b, 2017e,g), generalized projective synchronization (Zhang et al., 2019a), lag synchronization (Sader et al., 2019), anti-synchronization (Hu et al., 2005; Khan et al., 2020a), projective synchronization (Khan and Singh, 2016; Ouannas et al., 2017i), function projective synchronization (Azar et al., 2018), modified-function projective synchronization (Du et al., 2009), hybrid synchronization (Ouannas et al., 2017a,b,c), hybrid function projective synchronization (Ouannas et al., 2017d; Khan and Singh, 2017b), and different-order synchronization (Ayub and Singh, 2017; Khan and Singh, 2017c). In 2008, Ucar et al. (2008) proposed the multi-switching synchronization of coupled chaotic systems via active controls, and has achieved the multiswitching synchronization of the Lorenz system. Several manuscripts have investigated multi-switching synchronization (Singh et al., 2018b,a). The relevance of such kinds of synchronization studies to information security is evident in the wide range of possible synchronization directions that exist due to multi switching synchronization. Motivated by the above discussions, in this chapter we have described the combination multi-switching synchronization of nonidentical chaotic systems via active control technique. This book chapter is organized into five sections. Section 17.1 is introductory. In Section 17.2, description of problem formulation is given. In Section 17.3, a system description of a hyperchaotic system is presented. In Section 17.4, simulation results with discussions are performed to validate the theoretical results. Finally in Section 17.5, concluding remarks are given.

17.2 Problem formulation In this section, we describe the problem formulation to achieve the multiswitching complete synchronization between hyperchaotic systems via active backstepping method. Consider the following hyperchaotic system as master system: u(t) ˙ = F (u(t))

(17.1)

where u(t) ∈ Rn×1 is the state vector. F (u(t)) ∈ Rn×1 represents the nonlinear terms in the system (17.1). Consider the subsequent hyperchaotic system which acts as a slave system: v(t) ˙ = G(v(t)) + ηij (u(t), v(t))

(17.2)

where v(t) ∈ Rn×1 is the state vector. G(v(t)) ∈ Rn×1 describes the nonlinear terms in the system (17.2) and ηij (u(t), v(t)) ∈ R n×1 is the real feedback controller which is to be designed via a backstepping controller.

Multi-switching synchronization Chapter | 17 429

Definition 17.1. The two systems (17.1) and (17.2) can achieve complete synchronization, if lim eij (t) = lim vj (t) − ui (t) = 0

t→+∞

t→+∞

where the symbol  •  symbolizes the matrix norm. From Eqs. (17.1) and (17.2), the error dynamical system is obtained: eij (t) = G(v(t)) + ηij (u(t), v(t)) − F (u(t)).

(17.3)

The goal is to design suitable controller ηij (u(t), v(t)), such that the system (17.1) and (17.2) achieve multi-switching complete synchronization in accordance with Definition 17.1. In this chapter, we are considering 4-D hyperchaotic system for which the possible switches can be obtained by imposing the conditions on i, j = 1, 2, 3, 4. The list of the possible errors for hyperchaotic system whose possible combinations can be used to form the switches is as follows: For i = j , we have e11 , e22 , e33 , e44 . For i = j , we have e12 , e13 , e14 , e21 , e23 , e24 , e31 , e32 , e34 .

17.3

System description

The 4-D hyperchaotic system is given by ⎧ u˙ 1 = a(u2 − u1 ), ⎪ ⎪ ⎪ ⎪ ⎨ u˙ 2 = −u1 u3 − cu2 + ku4 , ⎪ u˙ 3 = −b + u1 u2 , ⎪ ⎪ ⎪ ⎩ u˙ 4 = −mu2 ,

(17.4)

where (u1 , u2 , u3 , u4 ) ∈ R4 are the state variables and a, b, c, k, m ∈ R are parameters.

17.3.1 Chaotic attractor of the system When the parameters a = 10, c = −2.5, k = 1, b = 25, m = 1, and initial condition (u1 , u2 , u3 , u4 ) = (0.1, 0.2, 0.15, 1) are chosen, then the system displays

430 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 17.1 3D phase portraits of hyperchaotic system.

FIGURE 17.2 2D phase portraits of hyperchaotic system.

chaotic attractor as shown in Fig. 17.1. Also the 2-D phase portrait and time series of this hyperchaotic attractor is shown in Fig. 17.2 and Fig. 17.3. The corresponding Lyapunov exponents of the hyperchaotic attractor are γ1 = 0.93179, γ2 = 0.032026, γ3 = −0.0090145, γ4 = −8.4543 as shown

Multi-switching synchronization Chapter | 17 431

FIGURE 17.3 Time series of hyperchaotic system in u1 , u2 , u3 , and u4 .

FIGURE 17.4 Lyapunov exponents of the 4D hyperchaotic system.

in Fig. 17.4. The Kaplan–Yorke dimension is defined by

D=j +

j  i=1

γi |γj +1 |

= 3.112936

432 Backstepping Control of Nonlinear Dynamical Systems

j j +1 where j is the largest integer satisfying i=1 γj ≥ 0 and i=1 γj < 0. Therefore the Kaplan–Yorke dimension of the chaotic attractor is D = 3.0579, which means that the Lyapunov dimension of the chaotic attractor is fractional.

17.3.2 Dissipation and existence of chaotic attractor The divergence of the novel integer-order 4-D hyperchaotic system (17.4) is ∂ u˙ 1 ∂ u˙ 2 ∂ u˙ 3 ∂ u˙ 4 + + + ∂u1 ∂u2 ∂u3 ∂u4 = −a − c = −7.50 < 0.

∇.V =

So, the system (17.4) is dissipative and it converges by the index rate of e−7.50 . It means that a volume element V0 is contracted into V0 e−7.50t . That is, each volume containing the system orbit shrinks to zero as t → 0, at an exponential rate ∇V , which is independent of u1 , u2 , u3 , u4 . Consequently, all system orbits will ultimately be confined to a specific subset of zero volume and the asymptotic motion settles into an attractor. Then the existence of attractor is proved.

17.3.3 Symmetry and invariance The hyperchaotic system (17.4) is invariant under the transformation (u1 , u2 , u3 , u4 ) → (−u1 , −u2 , u3 , −u4 ). This means that the system (17.4) is symmetric about u3 -axis. Moreover, the u3 -axis is an orbit of the system, and the orbit on the u3 -axis tends to the origin as b > 0 and t → ∞.

17.3.4 Poincaré map As an important analysis technique, the Poincaré map reflects the periodic, quasi-periodic, chaotic, and hyper-chaotic behavior of the system. When a = 10, b = 2, c = 28, and k = 0.1 one may take u1 = 0 as the crossing section which shows the hyperchaotic behavior as shown in Fig. 17.5.

17.4 Simulation results and discussions Consider the master system as follows: ⎧ u˙ 1 = a(u2 − u1 ), ⎪ ⎪ ⎪ ⎨ u˙ = −u u − cu + ku , 2 1 3 2 4 ⎪ u ˙ = −b + u u , 3 1 2 ⎪ ⎪ ⎩ u˙ 4 = −mu2 ,

(17.5)

Multi-switching synchronization Chapter | 17 433

FIGURE 17.5 Poincaré map on the crossing section u2 = 0.

and the response system as follows: ⎧ v˙1 = a(v2 − v1 ) + η1 , ⎪ ⎪ ⎪ ⎨ v˙ = −v v − cv + kv + η , 2 1 3 2 4 2 ⎪ v˙3 = −b + v1 v2 + η3 , ⎪ ⎪ ⎩ v˙4 = −mv2 + η4 , where ηi , i = 1, 2, 3, 4, are the controllers which are to be computed. In this manuscript, we consider the following three switches: ⎧ e11 = v1 − u1 , ⎪ ⎪ ⎪ ⎨ e22 = v2 − u2 , Switch 1 ⎪ e33 = v3 − u3 , ⎪ ⎪ ⎩ e44 = v4 − u4 , ⎧ e12 = v2 − u1 , ⎪ ⎪ ⎪ ⎨ e23 = v3 − u2 , Switch 2 ⎪ e34 = v4 − u3 , ⎪ ⎪ ⎩ e41 = v1 − u4 , ⎧ e12 = v1 − u2 , ⎪ ⎪ ⎪ ⎨e = v − u , 23 2 3 Switch 3 ⎪ e = v − u 34 3 1, ⎪ ⎪ ⎩ e41 = v4 − u4 ,

(17.6)

(17.7)

(17.8)

(17.9)

434 Backstepping Control of Nonlinear Dynamical Systems

where we refer to Eqs. (17.7), (17.8), and (17.9) as switch 1, switch 2, and switch 3, respectively.

17.4.1 Switch 1 Let us consider switch 1 where i = j = 1, 2, 3, 4. Define the error system as follows: ⎧ e11 = v1 − u1 , ⎪ ⎪ ⎪ ⎨e = v − u , 22 2 2 ⎪ = v − u e 33 3 3, ⎪ ⎪ ⎩ e44 = v4 − u4 .

(17.10)

The corresponding error dynamical system is given by ⎧ e˙11 = a(e22 − e11 ) + η1 , ⎪ ⎪ ⎪ ⎨ e˙ = −ce + ke − v v + u u + η , 22 22 44 1 3 1 3 2 ⎪ e˙33 = v1 v2 − u1 u2 + η3 , ⎪ ⎪ ⎩ e˙44 = −me22 + η4 .

(17.11)

17.4.1.1 Design of η1 and η2 Step 1. Define z1 = e11 . Its derivative is z˙ 1 = a(e22 − z1 ) + η1 .

(17.12)

Select e22 = α(z1 ) as a virtual controller and define the following Lyapunov function: 1 (17.13) V1 = z12 . 2 The derivative is V˙1 = z1 z˙ 1 (17.14) = z1 [α1 (z1 ) − az1 + η1 ]. Select α1 (z1 ) = 0 and η1 = 0, we get V˙1 = −az12 < 0.

(17.15)

Step 2. Let z2 = e22 − α1 (z1 ). Its derivative is z˙ 2 = e˙22 = −ce22 + ke44 − v1 v3 + u1 u3 + η2 .

(17.16)

Multi-switching synchronization Chapter | 17 435

Then, we obtain (z1 , z2 ) subsystem z˙ 1 = a(z2 − z1 ), z˙ 2 = −ce22 + ke44 − v1 v3 + u1 u3 + η2 .

(17.17)

Let us choose η2 = −ke44 + v1 v3 − u1 u3 + cz2 − e22 . Define the Lyapunov function V2 as 1 1 V2 = z12 + z22 . 2 2

(17.18)

The derivative of V2 is V˙2 = z1 z˙ 1 + z2 z˙ 2 = −az12 + z2 (−z2 ) < 0.

(17.19)

This implies that (z1 , z2 ) subsystem is asymptotically stable.

17.4.1.2 Design of η3 and η4 Step 1. Define z3 = e3 . Its derivative is z˙ 3 = e˙33 = v1 v2 − u1 u2 + η1 .

(17.20)

Select e44 = α2 (z3 ) as a virtual controller and define the following Lyapunov function: 1 V3 = z32 . (17.21) 2 The derivative is V˙3 = z3 z˙ 3 (17.22) = z3 [v1 v2 − u1 u2 + η3 ]. Select α2 (z3 ) = 0 and η3 = −v1 v2 + u1 u2 − z3 , we get V˙3 = −z32 < 0.

(17.23)

Step 2. Let z4 = e44 − α2 (z3 ). Its derivative is z˙ 4 = e˙44 = −me22 + η4 .

(17.24)

Then, we obtain the (z3 , z4 ) subsystem, z˙ 3 = −z3 , z˙ 4 = −me22 + η4 .

(17.25)

436 Backstepping Control of Nonlinear Dynamical Systems

Let us choose η4 = −me22 − z4 . Define the Lyapunov function V4 as 1 1 V4 = z32 + z42 . 2 2

(17.26)

The derivative of V4 is V˙4 = z3 z˙ 3 + z4 z˙ 4 = −z32 + z4 [−me22 + η4 ] = −z32 − z42 < 0.

(17.27)

This implies that (z3 , z4 ) subsystem is asymptotically stable.

17.4.1.3 Numerical simulations Numerical simulation is performed to illustrate the validity and feasibility of the presented synchronization technique. The parameters values of the hyperchaotic system are chosen so that system shows chaotic behavior in the absence of controllers as shown in Fig. 17.2. The initial conditions of the master systems and slave system are chosen as (u1 , u2 , u3 , u4 ) = (0.2, 0.1, 0.75, −2) and (v1 , v2 , v3 , v4 ) = (0.3, 0.3, 0.9, −1). The corresponding initial condition for error system is obtained as (e11 , e22 , e33 , e44 ) = (0.1, 0.2, 0.15, 1). The convergence of error state variables in Fig. 17.6 shows that the synchronization between master and slave hyperchaotic systems is achieved when controllers are activated at t > 0. 17.4.2 Switch 2 In this case, we use switch 2 to apply the given methodology. For switch 2, the error system is defined as follows: ⎧ e12 = v2 − u1 , ⎪ ⎪ ⎪ ⎨e = v − u , 23 3 2 (17.28) ⎪ e = v − u , 34 4 3 ⎪ ⎪ ⎩ e41 = v1 − u4 . The corresponding error dynamical system is given by ⎧ e˙12 = −v1 v3 + (a − c)v2 + kv4 − au2 − ae12 + η2 , ⎪ ⎪ ⎪ ⎨ e˙23 = −b + v1 v2 + u1 u3 + ce23 − cv3 + 2cu2 − ku4 + η3 , ⎪ e˙34 = −mv2 + b − u1 u2 + η4 , ⎪ ⎪ ⎩ e˙41 = av2 − ae41 − au4 + mu2 + η1 .

(17.29)

Multi-switching synchronization Chapter | 17 437

FIGURE 17.6 Synchronization error between states of master and slave systems for switch 1.

17.4.2.1 Design of η2 and η3 Step 1. Define z1 = e12 . Its derivative is z˙ 1 = −v1 v3 + (a − c)v2 + kv4 − au2 − ae12 + η2 .

(17.30)

Select e23 = α1 (z1 ) as a virtual controller and define the following Lyapunov function: 1 V1 = z12 . 2

(17.31)

The derivative is V˙1 = z1 z˙ 1 = z1 [−v1 v3 + (a − c)v2 + kv4 − au2 − az1 + η2 ].

(17.32)

Select α1 z1 = 0 and η2 = v1 v3 − (a − c)v2 − kv4 + au2 , we get V˙1 = −az12 < 0.

(17.33)

Step 2. Let z2 = e23 − α1 (z1 ). Its derivative is z˙ 2 = e˙23 = −b + v1 v2 + u1 u3 + ce23 − cv3 + 2cu2 − ku4 + η3 .

(17.34)

438 Backstepping Control of Nonlinear Dynamical Systems

Then, we obtain the (z1 , z2 ) subsystem: z˙ 1 = −az1 , z˙ 2 = −b + v1 v2 + u1 u3 + cz2 − cv3 + 2cu2 − ku4 + η3 .

(17.35)

Let us choose η2 = b − v1 v2 − u1 u3 + cv3 − 2cu2 + ku4 Define the Lyapunov function V2 as 1 1 V2 = z12 + z22 . 2 2

(17.36)

The derivative of V2 is V˙2 = z1 z˙ 1 + z2 z˙ 2 = z1 (−az1 ) + z2 (cz2 ) = −az12 + cz22 < 0 (∵ c < 0).

(17.37)

This implies that the (z1 , z2 ) subsystem is asymptotically stable.

17.4.2.2 Design of η1 and η4 Step 1. Define z3 = e34 . Its derivative is z˙ 3 = e˙33 = −mv2 + b − u1 u2 + η4 .

(17.38)

Select e41 = α2 z3 as a virtual controller and define the following Lyapunov function: 1 V3 = z32 . 2

(17.39)

V˙3 = z3 z˙ 3 = z3 [−mv2 + b − u1 u2 + η4 ].

(17.40)

The derivative is

Select α2 (z3 ) = 0 and η4 = mv2 − b + u1 u2 − z3 + η4 , we get V˙3 = −z32 < 0.

(17.41)

Step 2. Let z4 = e41 − α2 (z3 ). Its derivative is z˙ 4 = e˙41 = av2 − ae41 − au4 + mu2 + η1 .

(17.42)

Multi-switching synchronization Chapter | 17 439

Then, we obtain the (z3 , z4 ) subsystem: z˙ 3 = −z3 , z˙ 4 = av2 − az4 − au4 + mu2 + η1 .

(17.43)

Let us choose η1 = −av2 + au4 − mu2 . Define the Lyapunov function V4 : 1 1 V4 = z32 + z42 . 2 2

(17.44)

The derivative of V4 is V˙4 = z3 z˙ 3 + z4 z˙ 4 = −z32 + z4 [av2 − az4 − au4 + mu2 + η1 ] = −z32 − az42

(17.45)

< 0. This implies that the (z3 , z4 ) subsystem is asymptotically stable.

17.4.2.3 Numerical simulations Numerical simulation is performed to illustrate the validity and feasibility of the presented synchronization technique. The parameters values of the hyperchaotic system are chosen so that system shows chaotic behavior in the absence of controllers as shown in Fig. 17.2. The initial conditions of the master systems and slave system are chosen as (u1 , u2 , u3 , u4 ) = (0.2, 0.1, 0.75, −2) and (v1 , v2 , v3 , v4 ) = (0.3, 0.3, 0.9, −1). The corresponding initial condition for error system is obtained as (e12 , e23 , e34 , e41 ) = (0.1, 0.8, −3.75, 2.3). The convergence of error state variables in Fig. 17.7 shows that the synchronization between master and slave hyperchaotic systems is achieved when controllers are activated at t > 0. 17.4.3 Switch 3 In this case, we use switch 3 to apply the given methodology. F or switch 3, the error system is defined as follows: ⎧ e12 = v1 − u2 , ⎪ ⎪ ⎪ ⎨e = v − u , 23 2 3 ⎪ e = v − u 34 3 1, ⎪ ⎪ ⎩ e41 = v4 − u4 .

(17.46)

440 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 17.7 Synchronization error between states of master and slave systems for switch 2.

The corresponding error dynamical system is given by ⎧ e˙21 = av2 − ae21 − (a − c)u2 + u1 u3 − ku4 + η1 , ⎪ ⎪ ⎪ ⎨ e˙ = −v v − cv + kv + b − u u + η , 32 1 3 2 4 1 2 2 ⎪ e˙13 = −b + v1 v2 − au2 − ae13 + av3 + η3 , ⎪ ⎪ ⎩ e˙44 = −mv2 + mu2 + η4 .

(17.47)

17.4.3.1 Design of η1 and η2 Step 1. Define z1 = e21 . Its derivative is z˙ 1 = av2 − ae21 − (a − c)u2 + u1 u3 − ku4 + η1 .

(17.48)

Select e32 = α1 (z1 ) as a virtual controller and define the following Lyapunov function: 1 (17.49) V1 = z12 . 2 The derivative is V˙1 = z1 z˙ 1 = z1 [av2 − az1 − (a − c)u2 + u1 u3 − ku4 + η1 ].

(17.50)

Select α1 z1 = 0 and η1 = −av2 + (a − c)u2 − u1 u3 + ku4 , we get V˙1 = −az12 < 0.

(17.51)

Multi-switching synchronization Chapter | 17 441

Step 2. Let z2 = e32 − α1 (z1 ). Its derivative is z˙ 2 = e˙32 = −v1 v3 − cv2 + kv4 + b − u1 u3 + η2 .

(17.52)

Then, we obtain the (z1 , z2 ) subsystem z˙ 1 = −az1 , z˙ 2 = −v1 v3 − cv2 + kv4 + b − u1 u3 + η2 .

(17.53)

Let us choose η2 = v1 v3 + cv2 − kv4 − b + u1 u3 − z2 . Define the Lyapunov function V2 : 1 1 V2 = z12 + z22 . 2 2

(17.54)

The derivative of V2 is V˙2 = z1 z˙ 1 + z2 z˙ 2 = z1 (−az1 ) + z2 (−z2 ) = −az12 − z22 < 0.

(17.55)

This implies that the (z1 , z2 ) subsystem is asymptotically stable.

17.4.3.2 Design of η3 and η4 Step 1. Define z3 = e34 . Its derivative is z˙ 3 = e˙13 = −b + v1 v2 − au2 − ae13 + av3 + η3 .

(17.56)

Select e44 = α2 z3 as a virtual controller and define the following Lyapunov function: 1 (17.57) V3 = z32 . 2 The derivative is V˙3 = z3 z˙ 3 = z3 [−b + v1 v2 − au2 − ae13 + av3 + η3 ].

(17.58)

Select α2 (z3 ) = 0 and η4 = b − v1 v2 + au2 − av3 , we get V˙3 = −z32 < 0.

(17.59)

442 Backstepping Control of Nonlinear Dynamical Systems

Step 2. Let z4 = e44 − α2 (z3 ). Its derivative is z˙ 4 = e˙44 = −mv2 + mu2 + η4 .

(17.60)

Then, we obtain the (z3 , z4 ) subsystem z˙ 3 = −az3 , z˙ 4 = −mv2 + mu2 + η4 .

(17.61)

Let us choose η4 = mv2 − mu2 − z4 . Define the Lyapunov function V4 : 1 1 V4 = z32 + z42 . 2 2

(17.62)

The derivative of V4 is V˙4 = z3 z˙ 3 + z4 z˙ 4 = −az32 + z4 [−mv2 + mu2 + η4 ] = −az32 − z42

(17.63)

< 0. This implies that the (z3 , z4 ) subsystem is asymptotically stable.

17.4.3.3 Numerical simulations Numerical simulation is performed to illustrate the validity and feasibility of the presented synchronization technique. The parameters values of the hyperchaotic system are chosen so that system shows chaotic behavior in the absence of controllers as shown in Fig. 17.2. The initial conditions of the master systems and slave system are chosen as (u1 , u2 , u3 , u4 ) = (0.2, 0.1, 0.75, −2) and (v1 , v2 , v3 , v4 ) = (0.3, 0.3, 0.9, −1). The corresponding initial condition for error system is obtained as (e12 , e23 , e34 , e41 ) = (0.1, 0.8, −3.75, 2.3). The convergence of error state variables in Fig. 17.8 shows that the synchronization between master and slave hyperchaotic systems is achieved when controllers are activated at t > 0.

17.5 Conclusion In this chapter, we investigated multi-switching synchronization scheme between hyperchaotic systems via backstepping control by considering three switches. Some fundamental dynamical properties of the integer-order hyperchaotic system such as Lyapunov exponent, Poincaré section, Kaplan–Yorke dimension, and phase portraits are also discussed. The idea of multi-switching

Multi-switching synchronization Chapter | 17 443

FIGURE 17.8 Synchronization error between states of master and slave systems.

complete synchronization is implemented by considering two identical hyperchaotic systems. The controllers are obtained using backstepping control technique. Based on the Lyapunov stability criteria, the stability of the dynamical system is obtained. Lastly, numerical results are given to confirm the efficiency of the proposed synchronization scheme. Theoretical and numerical results are in excellent agreement.

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Pano-Azucena, A.D., de Jesus Rangel-Magdaleno, J., Tlelo-Cuautle, E., de Jesus Quintas-Valles, A., 2017. Arduino-based chaotic secure communication system using multi-directional multi-scroll chaotic oscillators. Nonlinear Dynamics 87, 2203–2217. Pecora, L.M., Carroll, T.L., 1990. Synchronization in chaotic systems. Physical Review Letters 64, 821. Pham, V.-T., Gokul P.M., Kapitaniak, T., Volos, C., Azar, A.T., 2018. Dynamics, synchronization and fractional order form of a chaotic system with infinite equilibria. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 475–502. Pham, V.-T., Vaidyanathan, S., Volos, C.K., Azar, A.T., Hoang, T.M., Van Yem, V., 2017. A threedimensional no-equilibrium chaotic system: analysis, synchronization and its fractional order form. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 449–470. Sader, M., Abdurahman, A., Jiang, H., 2019. General decay lag synchronization for competitive neural networks with constant delays. Neural Processing Letters, 1–13. Shukla, M.K., Sharma, B.B., Azar, A.T., 2018. Control and synchronization of a fractional order hyperchaotic system via backstepping and active backstepping approach. In: Azar, A.T., Radwan, A.G., Vaidyanathan, S. (Eds.), Mathematical Techniques of Fractional Order Systems. In: Advances in Nonlinear Dynamics and Chaos (ANDC). Elsevier, pp. 559–595. Singh, S., Azar, A.T., 2020. Controlling chaotic system via optimal control. In: Hassanien, A.E., Shaalan, K., Tolba, M.F. (Eds.), Proceedings of the International Conference on Advanced Intelligent Systems and Informatics 2019. In: Advances in Intelligent Systems and Computing, vol. 1058. Springer International Publishing, Cham, pp. 277–287. Singh, S., Azar, A.T., Bhat, M.A., Vaidyanathan, S., Ouannas, A., 2018a. Active control for multi-switching combination synchronization of non-identical chaotic systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Advances in System Dynamics and Control. In: Advances in Systems Analysis, Software Engineering, and High Performance Computing (ASASEHPC). IGI Global, pp. 129–162. Singh, S., Azar, A.T., Ouannas, A., Zhu, Q., Zhang, W., Na, J., 2017. Sliding mode control technique for multi-switching synchronization of chaotic systems. In: Modelling, Identification and Control (ICMIC), 2017 9th International Conference on. IEEE, pp. 880–885. Singh, S., Azar, A.T., Zhu, Q., 2018b. Multi-switching master–slave synchronization of nonidentical chaotic systems. In: Innovative Techniques and Applications of Modelling, Identification and Control. Springer, pp. 321–330. Strogatz, S.H., 2001. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder. Ucar, A., Lonngren, K.E., Bai, E.-W., 2008. Multi-switching synchronization of chaotic systems with active controllers. Chaos, Solitons and Fractals 38, 254–262. Vaidyanathan, S., Azar, A.T., Akgul, A., Lien, C.-H., Kacar, S., Cavusoglu, U., 2019. A memristorbased system with hidden hyperchaotic attractors, its circuit design, synchronisation via integral sliding mode control and an application to voice encryption. International Journal of Automation and Control 13, 644–667. Vaidyanathan, S., Azar, A.T., Boulkroune, A., 2018a. A novel 4-D hyperchaotic system with two quadratic nonlinearities and its adaptive synchronisation. International Journal of Automation and Control 12, 5–26. Vaidyanathan, S., Azar, A.T., Ouannas, A., 2017. Hyperchaos and adaptive control of a novel hyperchaotic system with two quadratic nonlinearities. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 773–803. Vaidyanathan, S., Azar, A.T., Rajagopal, K., Alexander, P., 2015a. Design and spice implementation of a 12-term novel hyperchaotic system and its synchronisation via active control. International Journal of Modelling, Identification and Control 23, 267–277.

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Chapter 18

A 5-D hyperchaotic dynamo system with multistability, its dynamical analysis, active backstepping control, and circuit simulation Sundarapandian Vaidyanathana , Aceng Sambasb , and Ahmad Taher Azarc,d a Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, b Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya,

Indonesia, c Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, d Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt

18.1

Introduction

The existence of two or more positive Lyapunov exponents is the defining property of a dynamical system to be a hyperchaotic system (Pham et al., 2018b; Khan et al., 2020a,b; Singh et al., 2018a; Wang et al., 2017; Ouannas et al., 2017; Vaidyanathan et al., 2017; Ouannas et al., 2020, 2019; Azar et al., 2018b). Hyperchaotic dynamical systems possess complex properties and characteristics which make them suitable for several applications such as lasers (Bonatto, 2018; Mahmoud and Al-Harthi, 2020; Yan, 2013), chemical systems (NietoVillar and Velarde, 2000), finance (Jahanshahi et al., 2019; Cao, 2018), memristors (Rajagopal et al., 2018a,b; Vaidyanathan et al., 2019b; Rajagopal et al., 2018c), electromechanical systems (Wang and Wu, 2018), hyperjerk systems (Ahmad et al., 2018; Vaidyanathan, 2016; Vaidyanathan et al., 2015a), neural networks (Vaidyanathan et al., 2020), encryption (Yang, 2017; Bouslehi and Seddik, 2018a,b; Liu et al., 2019), etc. In the chaos literature, many new mathematical models of hyperchaotic models have been constructed from popular 3D chaos models such as the hyperchaos Lorenz system (Chen, 2018), the hyperchaos Chen system (Liu et al., 2011), the hyperchaos Lü system (Xiao-Hong and Dong, 2009), and the hyperchaos Vaidyanathan system (Vaidyanathan et al., 2016). Recently, there is good interest in building 5D hyperchaos models in the literature. Using active state feedback or other methods, 5D hyperchaos with inBackstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00025-8 Copyright © 2021 Elsevier Inc. All rights reserved.

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450 Backstepping Control of Nonlinear Dynamical Systems

teresting qualitative properties have been reported in the chaos literature (Singh et al., 2018b; Zhang et al., 2018; Wang et al., 2018; Koyuncu et al., 2019). Singh et al. (2018b) designed a new 5D hyperchaos system with stable equilibrium, which belongs to the category of chaotic systems with hidden attractors. Zhang et al. (2018) developed a new 5D hyperchaos system of Lorenz type with unstable equilibrium points. Wang et al. (2018) proposed a new 5D hyperchaos system with a flux-controlled memristor. Koyuncu et al. (2019) implemented a 5D hyperchaos system in real-time high-speed FPGA. In this research work, we build a new 5D hyperchaotic dynamo system by introducing two state feedback controls to the classical chaotic dynamo system reported by Rikitake (1958). Rikitake’s two-disk dynamo system is a 3D chaotic dynamo model that describes the reversals of the earth’s magnetic field (Rikitake, 1958). Our new 5D hyperchaos dynamo system has three quadratic nonlinear terms. It is worthwhile to observe that the new 5D hyperchaos dynamo system has no rest point. Hence, it belongs to the category of chaotic systems with hidden attractors (Pham et al., 2018b; Azar and Serrano, 2020; Pham et al., 2018a). Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set. It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems (Naimzada and Pireddu, 2014; Hizanidis et al., 2018; Kengne and Mogue, 2019; Tamba and Fotsin, 2017; Lai and Grebogi, 2017). It is well known that multistability can lead to very complex behaviors in a dynamical system. In this work, it is shown that the new 5D hyperchaos dynamo system has multistability and two coexisting hyperchaos attractors. Active backstepping control is applied to control and synchronize the chaos in the 5D hyperchaos dynamo system. Active control method via backstepping approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015b; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015; Shukla et al., 2018; Vaidyanathan et al., 2018). Finally, a circuit model using Multisim of the new 5D hyperchaos dynamo system is designed for practical implementation. We show that the Multisim outputs of the 5D hyperchaos dynamo system exhibit a good match with the MATLAB® simulations of the same system. Circuit realizations of chaotic dynamical systems are useful for real-world implementations (Rajagopal et al., 2019; Nwachioma et al., 2019; Vaidyanathan et al., 2019a; Sambas et al., 2019a,b; Vaidyanathan et al., 2019c).

18.2

System model

In 1958, Rikitake built a classical 3-D dynamical model that describes the reversals of the earth’s magnetic field (Rikitake, 1958), which is given by the

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 451

following dynamics: ⎧ x˙1 ⎪ ⎪ ⎨ x˙2 ⎪ ⎪ ⎩ x˙3

= −ax1 + x2 x3 , = −ax2 − bx1 + x1 x3 , =

(18.1)

1 − x1 x2 .

It is well-known that the Rikitake dynamo system (18.1) exhibits a 2-scroll chaotic attractor when the parameters a and b take the values a = 1, b = 1.

(18.2)

For numerical simulations, we fix the parameter values at (a, b) = (1, 1) and the initial state at (0.3, 0.2, 0.1). For this data, the Lyapunov exponents of the Rikitake dynamo system (18.1) are numerically estimated as: ψ1 = 0.1252, ψ2 = 0, ψ3 = −1.1252,

(18.3)

which shows that the Rikitake dynamo system (18.1) is chaotic and dissipative. The phase plots in Fig. 18.1 describe the 2-scroll chaotic attractor of the Rikitake dynamo system (18.1). In this chapter, we find a new 5-D hyperchaotic dynamo system by introducing two feedback controls (x4 and x5 ) in the Rikitake 2-scroll chaotic dynamo model (18.1). Our new 5-D dynamo system is described by the following autonomous system of differential equations: ⎧ ⎪ x˙1 = −ax1 + x2 x3 − dx4 + x5 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙2 = −ax2 − bx1 + x1 x3 − dx4 + x5 , ⎪ ⎪ ⎨ (18.4) x˙3 = 1 − x1 x2 , ⎪ ⎪ ⎪ ⎪ ⎪ x˙4 = cx1 , ⎪ ⎪ ⎪ ⎪ ⎩ x˙5 = x1 . We use X to denote all the states of the 5-D model (18.4), i.e. X = (x1 , x2 , x3 , x4 , x5 ). It is noted that the system (18.4) has three quadratic nonlinear terms x2 x3 , x1 x3 , and x1 x2 in its dynamics. In (18.4), a, b, c, and d are positive constant parameters. In this work, we establish that the 5-D dynamo model (18.4) is a hyperchaotic 2-scroll system for the parameter values taken as a = 1, b = 1, c = 6, d = 6.

(18.5)

452 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 18.1 MATLAB phase plots of the Rikitake 2-scroll chaotic dynamo model (18.1) for the initial state (0.3, 0.2, 0.1) and parameter values (a, b) = (1, 1): (A) (x1 , x2 )-plane, (B) (x2 , x3 )-plane, (C) (x1 , x3 )-plane, and (D) R3 .

For numerical calculations, we fix the parameter vector as (a, b, c, d) = (1, 1, 6, 6). Also, we fix the initial state as X(0) = (0.3, 0.2, 0.1, 0.2, 0.3). Using MATLAB, the Lyapunov characteristic exponents of the 5-D model (18.4) are calculated and shown in Fig. 18.2. We find the Lyapunov characteristic exponents of the 5-D model (18.4): ψ1 = 0.1796, ψ2 = 0.0291, ψ3 = 0.0021, ψ4 = 0, ψ5 = −2.2106.

(18.6)

The Kaplan–Yorke dimension of the 5-D hyperchaos dynamo model (18.4) is found to be DKY = 4 +

ψ 1 + ψ2 + ψ 3 + ψ4 = 4.0954. |ψ5 |

(18.7)

From Eq. (18.6), it can be deduced that the 5-D dynamo model (18.4) is hyperchaotic as there are three positive Lyapunov exponents. The 5-D hyper-

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 453

FIGURE 18.2 Lyapunov exponents of the 5-D hyperchaotic dynamo model (18.4) for the initial state (0.3, 0.2, 0.1, 0.2, 0.3) and parameter values (a, b, c, d) = (1, 1, 6, 6).

chaotic dynamo model (18.4) is also dissipative since the sum of the Lyapunov characteristic exponents in (18.6) is negative. The 2-D phase plots of the 5-D hyperchaotic dynamo model (18.4) are given in Fig. 18.3.

18.3

Dynamic analysis of the 5-D hyperchaotic dynamo model

18.3.1 Rest points The rest points or equilibrium points of the 5-D hyperchaotic dynamo model (18.4) are obtained by solving the following equations: −ax1 + x2 x3 − dx4 + x5 = 0,

(18.8a)

−ax2 − bx1 + x1 x3 − dx4 + x5 = 0,

(18.8b)

1 − x1 x2 = 0,

(18.8c)

cx1 = 0,

(18.8d)

x1 = 0.

(18.8e)

We take the parameter values a, b, c, d as all positive. From (18.8d), we have x1 = 0. Substituting x1 = 0 in Eq. (18.8c), we get a contradiction. This shows that the 5-D hyperchaotic dynamo system (18.4) has no equilibrium point and that the system (18.4) has hidden attractor (Pham et al., 2018b).

454 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 18.3 MATLAB plots of the 5-D hyperchaotic dynamo model (18.4) for the initial state (0.3, 0.2, 0.1, 0.2, 0.3) and parameter values (a, b, c, d) = (1, 1, 6, 6): (A) (x1 , x2 )-plane, (B) (x2 , x3 )-plane, (C) (x3 , x4 )-plane, and (D) (x2 , x5 )-plane.

18.3.2 Multistability Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set (Azar et al., 2018a). It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems. It is well known that multistability can lead to very complex behaviors in a dynamical system. It is interesting that the new 5-D hyperchaotic dynamo system (18.4) can exhibit coexisting attractors when choosing different initial conditions. We take parameter values as in the hyperchaotic case, viz. (a, b, c, d) = (1, 1, 6, 6). We select two initial conditions as X0 = (0.3, 0.2, 0.1, 0.2, 0.3), Y0 = (0.3, −0.4, −0.3, −0.4, 0.3), and the corresponding state orbits of the system (18.4) are plotted in the colors, blue (dark gray in print) and red (light gray in print), respectively. From Fig. 18.4, it can be observed that the new 5-D hyperchaotic dynamo system (18.4) exhibits multistability with two coexisting hyperchaotic attractors.

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 455

FIGURE 18.4 Multistability of the 5-D hyperchaotic dynamo system (18.4): Coexisting hyperchaotic attractors for (a, b, c) = (0.8, 0.4, 0.2) and initial conditions X0 = (0.3, 0.2, 0.1, 0.2, 0.3) (blue, dark gray in print) and Y0 = (0.3, −0.4, −0.3, −0.4, 0.3) (red, light gray in print). (A) (x1 ; x3 )-plane; (B) (x2 ; x5 )-plane.

18.4

Active backstepping control for the global stabilization design of the new 5-D hyperchaotic dynamo system

In this section, we employ active backstepping technique for globally stabilizing the new 5-D hyperchaotic dynamo system. The controlled hyperchaos dynamo system is described by the 5D dynamics ⎧ ⎪ x˙1 = −ax1 + x2 x3 − dx4 + x5 + u1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙2 = −ax2 − bx1 + x1 x3 − dx4 + x5 + u2 , ⎪ ⎪ ⎨ (18.9) x˙3 = 1 − x1 x2 + u3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙4 = cx1 + u4 , ⎪ ⎪ ⎩ x˙5 = x1 + u5 , where u1 , u2 , u3 , u4 , u5 are the active backstepping controls to be determined. As a first step, we use feedback control to transform the system (18.9) to a system with triangular structure that aids backstepping control design. First, we consider the feedback control law ⎧ u1 = ax1 + x2 + dx4 − x5 − x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 = bx1 + ax2 + x3 + dx4 − x5 − x1 x3 , (18.10) u3 = x4 − 1 + x1 x2 , ⎪ ⎪ ⎪ ⎪ u4 = −cx1 + x5 , ⎪ ⎪ ⎩ u5 = −x1 + v, where v is a backstepping control to be determined.

456 Backstepping Control of Nonlinear Dynamical Systems

Substituting (18.10) into (18.9), we get the new system in triangular form as ⎧ ⎪ x˙1 = x2 , ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 = x3 , (18.11) x˙3 = x4 , ⎪ ⎪ ⎪ x ˙ = x , 4 5 ⎪ ⎪ ⎩ x˙5 = v. We begin with the Lyapunov function W1 (η1 ) =

1 2 η 2 1

(18.12)

where η1 = x 1 .

(18.13)

Differentiating W1 with respect to t along the dynamics (18.11), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (x1 + x2 ).

(18.14)

η2 = x1 + x2 .

(18.15)

We define

With the help of (18.15), we can express (18.14) as W˙ 1 = −η12 + η1 η2 .

(18.16)

We proceed next with defining the Lyapunov function 1 1 W2 (η1 , η2 ) = W1 (η1 ) + η22 = (η12 + η22 ). 2 2

(18.17)

Differentiating W2 with respect to t along the dynamics (18.11), we get W˙ 2 = −η12 − η22 + η2 (2x1 + 2x2 + x3 ).

(18.18)

η3 = 2x1 + 2x2 + x3 .

(18.19)

We define

With the help of (18.19), we can express (18.18) as W˙ 2 = −η12 − η22 + η2 η3 .

(18.20)

We proceed next with defining the Lyapunov function 1 1 W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) + η32 = (η12 + η22 + η32 ). 2 2

(18.21)

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 457

Differentiating W3 with respect to t along the dynamics (18.11), we get W˙ 3 = −η12 − η22 − η32 + η3 (3x1 + 5x2 + 3x3 + x4 ).

(18.22)

We define η4 = 3x1 + 5x2 + 3x3 + x4 .

(18.23)

With the help of (18.23), we can express (18.22) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(18.24)

We proceed next with defining the Lyapunov function 1 1 W4 (η1 , η2 , η3 , η4 ) = W3 (η1 , η2 , η3 ) + η42 = (η12 + η22 + η32 + η42 ). (18.25) 2 2 Differentiating W4 with respect to t along the dynamics (18.11), we get W˙ 4 = −η12 − η22 − η32 − η42 + η4 (5x1 + 10x2 + 9x3 + 4x4 + x5 ).

(18.26)

We define η5 = 5x1 + 10x2 + 9x3 + 4x4 + x5 .

(18.27)

With the help of (18.27), we can express (18.26) as W˙ 4 = −η12 − η22 − η32 − η42 + η4 η5 .

(18.28)

As a final step of the adaptive backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 , η5 ) =

1 2 (η + η22 + η32 + η42 + η52 ). 2 1

(18.29)

Clearly, W is a positive definite function on R5 . Differentiating W with respect to t along the dynamics (18.11), we get W˙ = −η12 − η22 − η32 − η42 − η52 + η5 T

(18.30)

where T = η4 + η5 + η˙ 5 = 8x1 + 20x2 + 22x3 + 14x4 + 5x5 + v.

(18.31)

We define the control v as v = −8x1 − 20x2 − 22x3 − 14x4 − 5x5 − Kη5 where K > 0 is a positive constant.

(18.32)

458 Backstepping Control of Nonlinear Dynamical Systems

Substituting (18.32) into (18.31), we get T = −Kη4 . Thus, Eq. (18.30) can be simplified to W˙ = −η12 − η22 − η32 − η42 − (1 + K)η52 ,

(18.33)

which is quadratic and negative definite. Hence, by Lyapunov stability theory, it is immediate that ηi (t) → 0 (i = 1, . . . , 5) exponentially as t → ∞ for all values of ηi (0) ∈ R (i = 1, . . . , 5). As a consequence, it follows that xi (t) → 0 (i = 1, 2, 3, 4) exponentially as t → ∞ for all values of xi (0) ∈ R (i = 1, . . . , 5). Thus, we have established the following main control result of this section. Theorem 18.1. The active backstepping control law defined by ⎧ ⎪ u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 u3 ⎪ ⎪ ⎪ ⎪ u4 ⎪ ⎪ ⎪ ⎩ u 5

=

ax1 + x2 + dx4 − x5 − x2 x3 ,

=

bx1 + ax2 + x3 + dx4 − x5 − x1 x3 ,

=

x4 − 1 + x1 x2 ,

=

−cx1 + x5 ,

=

−9x1 − 20x2 − 22x3 − 14x4 − 5x5 − Kη5 ,

(18.34)

where K > 0 and η5 = 5x1 + 10x2 + 9x3 + 4x4 + x5 , globally and exponentially stabilizes the 5D hyperchaotic dynamo system (18.9) for all values of X(0) ∈ R5 . For computer simulations, we take the hyperchaos case for the parameter values a, b, c, and d, viz. (a, b, c, d) = (1, 1, 6, 6). We choose K = 8 for the controller gain. As the initial state for the hyperchaos dynamo system (18.9), we choose, X(0) = (5.2, 1.9, 2.6, 8.3, 6.1). Fig. 18.5 shows the exponential convergence of the backstep-controlled state x(t).

18.5 Active backstepping control for the global synchronization design of the new 5-D hyperchaotic dynamo systems In this section, we employ active backstepping control method for globally synchronizing the trajectories of a pair of new 5-D hyperchaos dynamo systems considered as leader–follower systems.

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 459

FIGURE 18.5 Time-history of the backstep-controlled state X(t) of the 5-D hyperchaotic dynamo system (18.9).

The leader hyperchaos dynamo system is described by the 5D dynamics ⎧ ⎪ x˙1 = −ax1 + x2 x3 − dx4 + x5 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙2 = −ax2 − bx1 + x1 x3 − dx4 + x5 , ⎪ ⎪ ⎨ (18.35) x˙3 = 1 − x1 x2 , ⎪ ⎪ ⎪ ⎪ ⎪ x˙4 = cx1 , ⎪ ⎪ ⎪ ⎪ ⎩ x˙5 = x1 . The follower hyperchaos dynamo system is equipped with backstepping controls and depicted by the 4D dynamics ⎧ ⎪ y˙1 = −ay1 + y2 y3 − dy4 + y5 + u1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y˙2 = −ay2 − by1 + y1 y3 − dy4 + y5 + u2 , ⎪ ⎪ ⎨ (18.36) y˙3 = 1 − y1 y2 + u3 , ⎪ ⎪ ⎪ ⎪ ⎪ y˙4 = cy1 + u4 , ⎪ ⎪ ⎪ ⎪ ⎩ y˙5 = y1 + u5 , where u1 , u2 , u3 , u4 , u5 are feedback backstepping controls to be determined. The synchronization hyperchaos error is defined by means of the equations ei = yi − xi , i = 1, 2, 3, 4, 5.

(18.37)

460 Backstepping Control of Nonlinear Dynamical Systems

The synchronization error dynamics is calculated as follows: ⎧ ⎪ e˙1 = −ae1 − de4 + e5 + y2 y3 − x2 x3 + u1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙2 = −ae2 − be1 − de4 + e5 + y1 y3 − x1 x3 + u2 , e˙3 = −y1 y2 + x1 x2 + u3 , ⎪ ⎪ ⎪ ⎪ e˙4 = ce1 + u4 , ⎪ ⎪ ⎪ ⎩ e˙5 = e1 + u5 .

(18.38)

As a first step, we use feedback control to transform the system (18.38) to a system with triangular structure that aids backstepping control design. First, consider the feedback control law ⎧ u1 = ae1 + e2 + de4 − e5 − y2 y3 + x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 = be1 + ae2 + e3 + de4 − e5 − y1 y3 + x1 x3 , (18.39) u3 = e4 + y1 y2 − x1 x2 , ⎪ ⎪ ⎪ ⎪ u4 = −ce1 + e5 , ⎪ ⎪ ⎩ u5 = −e1 + v, where v is a feedback control to be determined. Substituting (18.39) into (18.38), we get the new system in triangular form as ⎧ ⎪ e˙1 = e2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙2 = e3 , e˙3 ⎪ ⎪ ⎪ ⎪ e˙4 ⎪ ⎪ ⎪ ⎩ e˙5

=

e4 ,

=

e5 ,

=

v.

(18.40)

We begin with the Lyapunov function W1 (η1 ) =

1 2 η 2 1

(18.41)

where η1 = e 1 .

(18.42)

Differentiating W1 with respect to t along the dynamics (18.40), we get W˙ 1 = η1 η˙ 1 = −η12 + η1 (e1 + e2 ).

(18.43)

η 2 = e 1 + e2 .

(18.44)

We define

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 461

With the help of (18.44), we can express (18.43) as W˙ 1 = −η12 + η1 η2 .

(18.45)

We proceed next with defining the Lyapunov function 1 1 W2 (η1 , η2 ) = W1 (η1 ) + η22 = (η12 + η22 ). 2 2

(18.46)

Differentiating W2 with respect to t along the dynamics (18.40), we get W˙ 2 = −η12 − η22 + η2 (2e1 + 2e2 + e3 ).

(18.47)

η3 = 2e1 + 2e2 + e3 .

(18.48)

We define

With the help of (18.48), we can express (18.47) as W˙ 2 = −η12 − η22 + η2 η3 .

(18.49)

We proceed next with defining the Lyapunov function 1 1 W3 (η1 , η2 , η3 ) = W2 (η1 , η2 ) + η32 = (η12 + η22 + η32 ). 2 2

(18.50)

Differentiating W3 with respect to t along the dynamics (18.40), we get W˙ 3 = −η12 − η22 − η32 + η3 (3e1 + 5e2 + 3e3 + e4 ).

(18.51)

We define η4 = 3e1 + 5e2 + 3e3 + e4 .

(18.52)

With the help of (18.52), we can express (18.51) as W˙ 3 = −η12 − η22 − η32 + η3 η4 .

(18.53)

We proceed with defining the Lyapunov function 1 1 W4 (η1 , η2 , η3 , η4 ) = W3 (η1 , η2 , η3 ) + η42 = (η12 + η22 + η32 + η42 ). (18.54) 2 2 Differentiating W4 with respect to t along the dynamics (18.40), we get W˙ 4 = −η12 − η22 − η32 − η42 + η4 (5e1 + 10e2 + 9e3 + 4e4 + e5 ).

(18.55)

We define η5 = 5e1 + 10e2 + 9e3 + 4e4 + e5 .

(18.56)

462 Backstepping Control of Nonlinear Dynamical Systems

With the help of (18.56), we can express (18.55) as W˙ 4 = −η12 − η22 − η32 − η42 + η4 η5 .

(18.57)

As a final step of the adaptive backstepping control design, we set the quadratic Lyapunov function W (η1 , η2 , η3 , η4 , η5 ) =

1 2 (η + η22 + η32 + η42 + η52 ). 2 1

(18.58)

Clearly, W is a positive definite function on R5 . Differentiating W with respect to t along the dynamics (18.40), we get W˙ = −η12 − η22 − η32 − η42 − η52 + η5 T

(18.59)

where T = η4 + η5 + η˙ 5 = 8e1 + 20e2 + 22e3 + 14e4 + 5e5 + v.

(18.60)

We define the control v as v = −8e1 − 20e2 − 22e3 − 14e4 − 5e5 − Kη5

(18.61)

where K > 0 is a positive constant. Substituting (18.61) into (18.60), we get T = −Kη4 . Thus, Eq. (18.59) can be simplified to W˙ = −η12 − η22 − η32 − η42 − (1 + K)η52 ,

(18.62)

which is quadratic and negative definite. Hence, by Lyapunov stability theory, it is immediate that ηi (t) → 0 (i = 1, . . . , 5) exponentially as t → ∞ for all values of ηi (0) ∈ R (i = 1, . . . , 5). As a consequence, it follows that ei (t) → 0 (i = 1, . . . , 5) exponentially as t → ∞ for all values of ei (0) ∈ R (i = 1, . . . , 5). Thus, we have established the following main result of this section. Theorem 18.2. The adaptive backstepping control law defined by ⎧ u1 = ae1 + e2 + de4 − e5 − y2 y3 + x2 x3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u2 = be1 + ae2 + e3 + de4 − e5 − y1 y3 + x1 x3 , u 3 = e4 + y1 y 2 − x 1 x 2 , ⎪ ⎪ ⎪ ⎪ u4 = −ce1 + e5 , ⎪ ⎪ ⎩ u5 = −9e1 − 20e2 − 22e3 − 14e4 − 5e5 − Kη5 ,

(18.63)

where K > 0 and the parameter estimation law and η5 = 5e1 + 10e2 + 9e3 + 4e4 + e5 , globally and exponentially synchronizes the 5D hyperchaotic dynamo systems (18.35) and (18.36) for all values of X(0), Y (0) ∈ R5 .

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 463

FIGURE 18.6 Synchronization between the states x1 and y1 of the 5-D hyperchaotic dynamo systems (18.35) and (18.36).

FIGURE 18.7 Synchronization between the states x2 and y2 of the 5-D hyperchaotic dynamo systems (18.35) and (18.36).

For computer simulations, we take the hyperchaos case for the parameter values a, b, c, and d viz. (a, b, c, d) = (1, 1, 6, 6). We choose K = 8 for the controller gain. We choose the initial states of the 5-D dynamo systems (18.35) and (18.36) as X(0) = (8.4, 2.3, 7.1, 4.9, 3.6) and Y (0) = (6.1, 2.8, 3.9, 1.4, 8.7), respectively.

464 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 18.8 Synchronization between the states x3 and y3 of the 5-D hyperchaotic dynamo systems (18.35) and (18.36).

FIGURE 18.9 Synchronization between the states x4 and y4 of the 5-D hyperchaotic dynamo systems (18.35) and (18.36).

Figs. 18.6–18.10 display the synchronization between the states of the 5-D hyperchaotic dynamo systems (18.35) and (18.36). Fig. 18.11 shows the exponential convergence of the synchronization error ei (t), (i = 1, . . . , 5) between the 5-D hyperchaotic dynamo systems (18.35) and (18.36).

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 465

FIGURE 18.10 Synchronization between the states x5 and y5 of the 5-D hyperchaotic dynamo systems (18.35) and (18.36).

FIGURE 18.11 Time-history of the synchronization error between the 5-D hyperchaotic dynamo systems (18.35) and (18.36).

18.6

Circuit simulation of the new 5D hyperchaotic system

In this section, the 5D hyperchaotic system is realized by Multisim software. In this system, operational amplifiers U1A, U3A, U5A, U7A, and U8A have the voltages x1 , x2 , x3 , x4 , and x5 at the outputs, respectively (see Fig. 18.12). The circuit includes eight operational amplifiers, three multipliers, five capacitors, and 19 resistors. It is noted that five operational amplifiers (U1A, U3A,

466 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 18.12 Schematic of circuit diagram for the 5-D hyperchaotic dynamo system (18.64).

A 5-D hyperchaotic dynamo system with multistability Chapter | 18 467

U5A, U7A, U8A) are configured as integrators while three operational amplifiers (U2A, U6A, U9A) are configured as inverting amplifier. We rescale the state variables of the new 5D hyperchaotic dynamo system (18.4) by using X1 = 2x1 , X2 = 2x2 , X3 = 2x3 , X4 = 2x4 , and X5 = 2x5 . After the rescaling, we obtain the following new 5-D hyperchaotic dynamo system: ⎧ ⎪ ⎪ X˙1 = −aX1 + 12 X2 X3 − dX4 + X5 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X˙2 = −aX2 + 12 X1 X3 − bX1 − dX4 + X5 , ⎨ (18.64) X˙3 = 2 − 12 X1 X2 , ⎪ ⎪ ⎪ ⎪ ⎪ X˙4 = cX1 , ⎪ ⎪ ⎪ ⎩ ˙ X 5 = X1 . Upon applying Kirchhoff’s electrical circuit laws to the circuit of Fig. 18.12, the following set of the new 5D hyperchaotic system can be derived: ⎧ ⎪ X˙1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X˙ ⎪ ⎪ ⎨ 2 X˙3 ⎪ ⎪ ⎪ ⎪ ⎪ X˙4 ⎪ ⎪ ⎪ ⎪ ⎩ ˙ X5

= − C11R1 X1 +

=

1 1 C1 R2 X2 X3 − C1 R3 X4 − C21R5 X2 + C21R6 X1 X3 − C21R7 X1 1 1 C3 R10 V1 − C3 R11 X1 X2 , 1 C4 R12 X1 ,

+

=



=

1 C5 R13 X1 .

=

1 C1 R4 X5 , 1 1 C2 R8 X4 + C2 R9 X5 ,

(18.65) Here, X1 , X2 , X3 , X4 , X5 are voltages across the capacitors C1 , C2 , C3 , C4 , C5 , respectively. The parameters are taken as follows: R1 = R4 = R5 = R7 = R9 = R13 = 400 k, R2 = R6 = R11 = 800 k, R3 = R8 = R12 = 66.67 k, R10 = 200 k, R14 = R15 = R16 = R17 = R18 = R19 = 100 k, and C1 = C2 = C3 = C4 = C5 = 1 nF. The power supplies of all active devices are ±15 Volts. Multisim phase portraits in Fig. 18.13 verify the hyperchaotic behavior of the 5-D dynamo system (18.64).

18.7

Conclusions

In this work, we detailed a new 5-D hyperchaotic dynamo system with three quadratic nonlinearities, which was derived by introducing two feedback controls in the famous 3-D Rikitake dynamo system (1958). We established that the new hyperchaos dynamo system has multistability and coexisting attractors. We also pointed out that there are no rest points for the hyperchaos dynamo system. Hence, the new 5-D dynamo system has hidden hyperchaos attractors.

468 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 18.13 Hyperchaotic attractors of the 5-D hyperchaotic dynamo system (18.64) using Multisim circuit simulation: (A) X1 –X2 plane, (B) X2 –X3 plane, (C) X3 –X4 plane, and (D) X2 –X5 plane.

As control applications, we deployed active backstepping control for the global hyperchaos stabilization and synchronization for the new 5-D hyperchaos dynamo system. Finally, using Multisim, we built an electronic circuit for the new 5-D hyperchaos dynamo system. This circuit realization of the new hyperchaos dynamo system enables practical implementation for engineering applications.

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Vaidyanathan, S., 2015. Analysis, control, and synchronization of a 3-D novel jerk chaotic system with two quadratic nonlinearities. Kyungpook Mathematical Journal 55 (3), 563–586. Vaidyanathan, S., 2016. Active controller design for the output regulation of Vaidyanathan hyperjerk system. In: Vaidyanathan, S., Volos, C. (Eds.), Advances and Applications in Nonlinear Control Systems. Springer, Berlin, Germany, pp. 185–209. Vaidyanathan, S., Abba, O.A., Betchewe, G., Alidou, M., 2019a. A new three-dimensional chaotic system: its adaptive control and circuit design. International Journal of Automation and Control 13 (1), 101–121. Vaidyanathan, S., Azar, A.T., Akgul, A., Lien, C.-H., Kacar, S., Cavusoglu, U., 2019b. A memristorbased system with hidden hyperchaotic attractors, its circuit design, synchronisation via integral sliding mode control and an application to voice encryption. International Journal of Automation and Control 13 (6), 644–667. Vaidyanathan, S., Azar, A.T., Ouannas, A., 2017. Hyperchaos and adaptive control of a novel hyperchaotic system with two quadratic nonlinearities. In: Azar, A.T., Vaidyanathan, S., Ouannas, A. (Eds.), Fractional Order Control and Synchronization of Chaotic Systems. In: Studies in Computational Intelligence, vol. 688. Springer International Publishing, Cham, pp. 773–803. Vaidyanathan, S., Idowu, B.A., Azar, A.T., 2015a. Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. In: Azar, A.T., Vaidyanathan, S. (Eds.), Chaos Modeling and Control Systems Design. In: Studies in Computational Intelligence, vol. 581. Springer, Berlin, Germany, pp. 39–58. Vaidyanathan, S., Jafari, S., Pham, V.-T., Azar, A.T., Alsaadi, F.E., 2018. A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design. Archives of Control Sciences 28 (2), 239–254. Vaidyanathan, S., Pehlivan, I., Dolvis, L.G., Jacques, K., Alcin, M., Tuna, M., Koyuncu, I., 2020. A novel ANN-based four-dimensional two-disk hyperchaotic dynamical system, bifurcation analysis, circuit realisation and FPGA-based TRNG implementation. International Journal of Computer Applications in Technology 62 (1), 20–35. Vaidyanathan, S., Sambas, A., Zhang, S., 2019c. A new 4-D dynamical system exhibiting chaos with a line of rest points, its synchronization and circuit model. Archives of Control Sciences 29 (3), 485–506. Vaidyanathan, S., Volos, C.K., Pham, V.-T., 2016. Adaptive control and circuit simulation of a novel 4-D hyperchaotic system with two quadratic nonlinearities. In: Vaidyanathan, S., Volos, C. (Eds.), Advances and Applications in Chaotic Systems. Springer International Publishing, Berlin, Germany, pp. 163–187. Vaidyanathan, S., Volos, C., Rajagopal, K., Kyprianidis, I., Stouboulos, I., 2015b. Adaptive backstepping controller design for the anti-synchronization of identical WINDMI chaotic systems with unknown parameters and its SPICE implementation. Journal of Engineering Science and Technology Review 8 (2), 74–82. Wang, R., Li, M., Gao, Z., Sun, H., 2018. A new memristor-based 5D chaotic system and circuit implementation. Complexity 2018, 6069401. Wang, Z., Volos, C., Kingni, S.T., Azar, A.T., Pham, V.-T., 2017. Four-wing attractors in a novel chaotic system with hyperbolic sine nonlinearity. Optik - International Journal for Light and Electron Optics 131, 1071–1078. Wang, Z., Wu, H., 2018. Stabilization in finite time for fractional-order hyperchaotic electromechanical gyrostat systems. Mechanical Systems and Signal Processing 111, 628–642. Xiao-Hong, Z., Dong, L., 2009. Fuzzy modelling and impulsive control of the hyperchaotic Lü system. Chinese Physics B 18 (5), 1774–1779. Yan, S.L., 2013. Synchronizations of quasi-period and hyperchaos in injected two-section semiconductor lasers. Journal of Optical Communications 34 (1), 9–14. Yang, N., 2017. Digital image encryption algorithm design based on genetic-hyperchaos. Chinese Journal of Liquid Crystals and Displays 32 (6), 474–481. Zhang, F., Chen, R., Wang, X., Chen, X., Mu, C., Liao, X., 2018. Dynamics of a new 5D hyperchaotic system of Lorenz type. International Journal of Bifurcation and Chaos 28 (3), 1850036.

Chapter 19

Design and implementation of a backstepping controller for nonholonomic two-wheeled inverted pendulum mobile robots Gen’ichi Yasuda Nagasaki Institute of Applied Science, Nagasaki, Japan

19.1

Introduction

In recent years, functionality and complexity of real-time controllers dramatically increased, beyond the complexity of hardware components. Due to the growing complexity of the software components in advanced robot controllers, including timing requirements of run time elaboration of information from external sensors, the need of early validation of the design of control software is increasing (Noda et al., 2010; Wu et al., 2013). Usually, the design of control laws for nonlinear physical systems like robotic vehicles and arms is made in continuous time and allows for checking mainly qualitative properties like stability. At run time, these control laws are implemented as multi-tasks programs controlled by a real-time operating system on a single or multiprocessors target. The performance indices depend on the actual implementation, including sampling rates and computing latencies, besides the algorithm in use. Research on real-time operating systems for nonlinear system control do not provide tools to analyze or synthesize such sampled control laws with respect to performance indices, measuring the impact of the organization of the control system on the controlled process. While traditional techniques for designing the software of complex real-time control systems, such as a finite state automaton, offer limited support for validating design before producing final code, the use of formal methods can increase the possibility of validating design (David and Alla, 1994; Martinez et al., 1987). This chapter presents a formal approach to designing software for sensorbased nonlinear motion control with a backstepping approach, focusing on the Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00026-X Copyright © 2021 Elsevier Inc. All rights reserved.

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design of control software for nonholonomic two-wheeled inverted pendulum mobile robots. Control algorithms are decomposed into groups of cooperating tasks; where these tasks run in sequence, executing with the same period, or in parallel (Wang and Sarides, 1993; Music and Matko, 1999). Multi-task programs with multi-rate controllers permit modular programming and software reusability, which is useful because the hardware of the control system exhibits subsystems with different dynamics, for example, feed-forward paths to update some parameters and measurements to come from sensors of different kind running at different rates. The feature can be useful to optimize computing resources, because both onboard space and energy are strongly limited, especially for autonomous robots working in unstructured or critical environments.

19.2 Distributed controller design based on backstepping approach A wheeled inverted pendulum mobile robot is a self-balancing vehicle with two wheels attached on the sides of its body. Two motors are attached between the wheels and the body. It controls its own posture, traveling speed, and direction. Furthermore, the software generates robot trajectories based on online elaboration of sensors information and manages kinematic inversion. The traveling control or navigation problem may be divided into three basic problems: tracking a reference trajectory, following a path, and point stabilization. The main idea behind nonlinear feedback control for solving these problems is to define velocity control inputs based on the perfect velocity tracking assumption. So, such velocity control inputs should be converted into torques, taking into account the actual vehicle dynamics. The complete dynamics of the mobile robot consists of the kinematic steering system and vehicle dynamics. The motion and orientation are achieved by independent actuators providing the necessary torques to the wheels (Takei et al., 2009; Fierro and Lewis, 1997). The nonholonomic constraint states that the vehicle can only move in the direction normal to the axis of the driving wheels; the mobile base satisfies the conditions of pure rolling and non-slipping. For the balancing control, the pitch angle x1 and its derivative x2 = x˙1 are used as the state variables, where the posture angle is 0 when the center of gravity of the robot is right above the contact point of a wheel to the ground. The reference values of the posture angle and the angular velocity are zero. Then, the state-space equations are generally represented as follows: x˙1 = x2 , 1 x˙2 = (u − h(x1 , x2 )) . g1 (x1 )

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As the first step of backstepping approach, a virtual control law is defined such that lim z1 (t) = 0, as follows: t→∞

 α = k1 z1 + k2

z1 (τ )dτ.

Then, an input torque u is designed such that, for the error z2 = α − x2 , lim z2 (t) = 0. t→∞

The Lyapunov function is defined by 2  1 1 z1 (τ )dτ , V1 = z12 + 2 2 1 2 V2 = V1 + z2 , 2 V˙2 = V˙1 + z2 z˙ 2 = −k1 z12 + z1 z2 + z2 z˙ 2 . For V˙2 < 0, z˙ 2 = −k3 z2 − z1 . Then, an input torque is determined as    u = g z1 , z1 (τ )dτ, z2 . Since V˙1 < 0 is derived, it is ensured that the tracking error converges on zero. Fig. 19.1 shows the structure of software modules for motion control of twowheeled mobile robot. Fig. 19.2 shows a view of the wheeled inverted pendulum mobile robot using LEGO Mindstorms NXT.

FIGURE 19.1 Software structure of motion control of two-wheeled mobile robot.

The motion controller consists of three cooperative modules: motion planning module, balancing control module, and traveling control module. The planning module computes nominal trajectories in terms of positional setpoints in

476 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 19.2 View of experimental mobile robot.

the operational space and passes to the traveling control module, which corrects them according to balancing control. Existing components, such as shared memory, flags, toggles, which satisfy the modeling requirement of the multi-tasking control applications, are adopted. A task is the maximum granularity considered by the control systems engineer at the device control level, characterizing continuous time closed-loop control laws, with their temporal features related to the management of associated events. For example, the balancing control task performs the activation of a control scheme structurally invariant along the task duration, where a logical behavior is associated with a set of signals during the task execution. The continuous time specification is translated into a description by adding temporal properties, such as discretization of time, durations of computations, communication and synchronization between the involved processes. Consequently, each task is defined in terms of communicating real-time computing modules, which each implement an elementary part of the control law. Most of the modules are periodically executed. Some modules perform the calculations involved in the computation of the control algorithm, for example, pitch angular velocity observer, pitch angle estimation, gyro output drift compensation, kinematic inversion, calculation of virtual control, and calculation of control torque. Furthermore, other modules monitor conditions like reaching a task space limit, and may trigger events which are classified into preconditions, post conditions, and exceptions, participating in the reactive management of the task. The non-periodic reactive behavior of the task is handled by a special module which may be awakened by events coming from the task itself or others; these

Design and implementation of a backstepping controller Chapter | 19 477

events represent the external view of the task and are used to compose tasks into more complex tasks. Because, in the case of most robotic systems, periodic and multi-rate modules are distributed over a multiprocessor target architecture, various message passing mechanisms over type ports should be available. The structure of the periodic modules is provided by separating calculations, communications, and calls to the underlying operating system. The temporal attributes of a module for calculation are duration of execution, activation period, input and output ports, and their communication protocols, priority for scheduling at run time, and assigned processor in case of a multiprocessors target.

19.3

Discrete event modeling and control net representation

The Petri net is gaining increasing importance in the robotics research for the discrete event modeling and analysis of control systems, since it offers a convenient way of expressing system behavior which is both parallel and asynchronous, and therefore distributed. In addition to the precedence constraints among computing modules, including sequences of modules and the repetition of certain modules, looser couplings associated with shared resources, such as memory and devices, can also be directly expressed (Fleury et al., 1994). Besides modeling capabilities, the Petri net can be analyzed in a formal way to obtain information about the dynamic behavior of the modeled system (Peterson, 1981; Murata, 1989). Fig. 19.3 shows a Petri-net model of computing module in motion control software. The simple Petri nets associate events with transitions and conditions with places. Places and transitions are linked by directed edges called arcs. Since the presence of a token in a place indicates that the corresponding condition is true, the marking describes the state of the Petri net in terms of conditions which are true and those which are false. When each of the input places becomes marked, i.e. has a token, the transition fires, i.e. the event occurs, and the firing removes a token from each of the input places, making the preconditions false, and deposits a token in each of the output places, making the postconditions true.

FIGURE 19.3 Petri-net model of computing module in motion control software.

478 Backstepping Control of Nonlinear Dynamical Systems

Nowadays, the state diagrams or statecharts are adopted to extract the discrete event system dynamics of the UML framework design. Statecharts are designed as state machines and built by initial and final pseudostates, and transitions having one input state and one output state. Each transition arrow has a label with three optional parts: event, condition, activity; if the current state is the specified source state and the appropriate event occurs with the specified condition, the transition is fired and the specified action is taken. Finite state machines are equivalently represented by a Petri-net subclass, where a place having two or more output transitions is referred to as a conflict, decision or choice and the representation of the synchronization of parallel activities does not be allowed. The translation from a UML statechart to a Petri net is simple such that each statechart start is converted into a Petri-net place and each statechart transition into a Petri-net transition with corresponding labels and tokens. Based on a bottom-up approach, the first terms to be designed are the devicelevel tasks or closed-loop control laws encapsulated in a reactive logical behavior. The main features of the device-level control software are modeled using extended notations based on the ordinary Petri nets. Events are modeled by transitions as instantaneous action occurring in a system. States can either represent a passive condition, such as waiting for input data, or the execution of an active program, such as path planning, object recognition, and control computation. As a simple representation of interaction with the environment, a firing transition may trap sensory data and send output data to actuators. The transitions behave as agents for communication with the environment by use of the preand postprocessors of a firing transition; an enabled transition can call a preprocessor before it starts firing and after firing it can call a postprocessor. The preprocessor also sends signal to the actuator to turn on its effector. The execution of an active program with finite duration can be modeled with two events; start transition that represents the beginning of the computing action, end transition that represents the termination of the action, and place that models the action in progress. A typical robotic task is sequentially composed of the following modules: reading an input port, performing a calculation, and writing to an output port, and may be modeled by a condition/event net with three transitions for execution of the actions and a fourth transition for periodic activation of the task with a real-time clock, which is also modeled by a Petri net. In repetitive execution, control data can be associated as integer iteration number with tokens in order to control transition firing by means of predicates. By associating durations with transitions or places, a timed Petri net can be obtained. Considering the duration of calculation of the transitions, reading and writing can be assumed to be instantaneous events. In modeling repetitive behavior of periodic control laws, a basic property to be ensured is structural deadlock freedom, which means that given any natural initial marking, deadlock cannot occur. This property has corresponding definitions in the Petri net. A Petri net is live with respect to an initial marking if,

Design and implementation of a backstepping controller Chapter | 19 479

from any marking in the set of markings reachable from the initial marking, there exists for each transition a legal firing sequence leading to a marking in which that transition is enabled. Because deadlock arises when no transition can be longer enabled, liveness implies deadlock freedom. Partially synchronizing some computing modules can improve the practical performance of the implementation. However, using synchronizations may lead to deadlocks or temporal inconsistencies. The consequences of introducing such synchronization should be examined in terms of structural and temporal problems using Petri-net modeling and analysis. Some guidelines about how to add such synchronization are given to design deadlock-free and efficient implementations of real-time periodic control laws. In the case each place has just one input transition and one output transition, that is, each associated condition may only become true in one way and can have one consequence, the task behavior is deterministic and the resulting Petri net is called a marked graph. For a marked graph the token count in a directed circuit is invariant under any firing. Using the marked graph, parallel construct of modules and their synchronization can be modeled with two transitions (events): transition that enables two parallel actions and transition that fires when both actions are completed, where both start transitions fire with no delay and end transitions are delayed by the duration of the modeled actions. Generally, shared memories are used to exchanging data among modules, holding data produced by some processes and consumed by other processes. A shared memory is modeled with a place marked with a token that stores the shared data. Shared memories with capacity 1 and no access control, which store only one value that can be written or read without limitations, can be used for storing sampled data to be used for some control functions, because only the most recent value is important regardless of the use of older values (Silva et al., 2014; Mohan et al., 2004). Fig. 19.4 shows a net model of shared memory.

FIGURE 19.4 Petri-net model of shared memory.

The procedure to merge two Petri nets to generate a composed module or task using shared memory is as follows. For corresponding transitions between two Petri nets, a communication place labeled as “Signal sent” or “Buffer” is built, and an output arc from the transition labeled as “Send” to the place and an input arc from the place to the transition labeled as “Wait” are added. If there is a transition that corresponds to a transition already connected to a communication place, then an input or output arc between the transition and the communication pace is added. Because initial marking of the composed Petri net must be referenced to the Petri nets, numerical labels of places and transitions are reorganized

480 Backstepping Control of Nonlinear Dynamical Systems

to create a joined marking and transition enumeration for the composed Petri net. The number of the input and output places of the corresponding transitions is not one. As a more practical method of communication between modules, a flag records a Boolean value with two places, either of which holds a token indicating the current state of the flag, and it can be set to either of its values independently from its previous value. The value of a flag can be tested checking for the presence of a token in the places. Since transitions need to remove a token to test a flag state, they are also required to put the token back so that the state of the flag will be preserved. The device-level tasks are logically and temporally composed to build more complex tasks, then the higher-level software is designed in terms of a set of robotic tasks to be executed under a real-time operating system. The modules communicate synchronously through a set of shared flags to notify the states of execution of control tasks. For example, when the path planning task is completed, the corresponding flag is set. Then the transitions for the sensor module can be enabled and the module will start. Setpoints and corrected setpoints are asynchronously communicated from the planning module to the sensor module and in reverse, respectively, through the shared memory. Trajectory correction is computed and applied to the reference trajectory generated by the planning module. Path profiles are handed over to the planning module. Fig. 19.5 shows a detailed net model of backstepping controller using shared memory and flags.

FIGURE 19.5 Petri-net model of backstepping controller.

19.4 Implementation issues on a multi-task processing architecture The overall control system has been configured to introduce exteroceptive sensor-based control capabilities as an evolution of the current industrial controller. The control unit is based on the VME-bus and includes a multiprocessor board equipped with Motorola 68020/68882 CPU’s for path and motion planning, and a DSP (Texas 320C25) for balancing control and traveling control. The planning module performs trajectory generation in operational space and velocity calculation by kinematic inversion both with a repetition interval of

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10 ms. The control module communicates with the planning module and executes micro-interpolation and position control in joint space of the vehicle at a sampling interval of 1 ms. Additionally, some sensing modules are provided to handle the acquisition of exteroceptive gyro sensor data and encoder pulses from the wheels. During control of physical processes, data always flow from sensors to actuators along several control paths. Since feed-forward paths and parameter adjustments are typically executed more slowly (less frequently) than feedback error computations, multi-rate sampling can optimize computing power. Clocks generate periodic events to be used by processes for activation and synchronization, and they model controller clocks that generate interrupts needed to activate control tasks each sampling time. The transition generates events with periodicity given by its firing time. Events with different periodicity can be generated by different transitions synchronized in the initial marking. The console is modeled with the transition, and its input and output places. The transition represents an instruction loader, by popping a command from a queue of commands according to its time-stamp in the input place and putting it in the output place which is connected to the sensor module for execution of sensor-based control algorithms. The abstract data type of the queue is defined with the usual operations such as popping which extracts the top element from the queue, testing which is true if the queue is empty, etc. The place which represents the interface of the sensor contains the current value of the sensor. The input transition simulates the sensor, by updating the value of the token in the place, popped from the queue in the input place of the transition. The control system development methodology consists of three phases to assist implementation of the control software (Yasuda, 2014). In the first phase, the robotic task and environment requirements are specified to identify functionalities and attributes of the hardware and software architecture needed to complete the task. The net model for system control is designed as follows. First, the net model of the task or recipe specifications is constructed using the task-oriented approach. Second, the net model of resource specifications corresponding to each task specification is provided using the mutual exclusion concept. Then, the task and resource models are composed to yield the basic system control model. By analyzing and refining the basic model, a deadlock-free, safe, and reversible model is obtained. A customized Unified Modeling Language (UML) framework is used to describe every component of the control system, including class attributes, methods, and their relations of inheritance, composition and association providing a structural view of the software with the hardware locations of the components. Based on the above class diagrams, the interaction sequence and the messages passed among classes are described using UML sequence, collaboration, and use-case diagrams. The state charts are generated to provide a global view of the inherent states of the software components behavior. Then, the dynamics of the UML schematics is translated to Petri nets.

482 Backstepping Control of Nonlinear Dynamical Systems

Although designing a task using several modules promote a modular approach (Mohan et al., 2004), mapping each module to a thread at run time increases operating system overhead due to numerous context switches and inter-process communication. Therefore, assigning a group of modules which are sequentially executed to a thread by adding a certain amount of synchronization is desirable, where communication between groups does not add synchronization constraints, so that the synchronization skeleton of the task is not changed. Because, in case that each module is running freely (or independently), the communication does not add synchronization, it can lead to an unexpected ordering of tasks at run time by the operating system. So, to translate control laws into computing tasks, a synchronization scheme should be adopted for ensuring that the robotic task is deadlock free and minimizing computing latencies to improve control performance. Although different communication mechanisms for message passing between the sender and the receiver modules can be available with respect to synchronization to link pairs of modules via their ports, closedloop control software should be concerned with the followings protocols, so that the last or the next available one is the best data for closed-loop control without loss of data. In synchronous communication, each module acts synchronously with the other; the first module to reach the rendez-vous is blocked until the second one is ready. In asynchronous communication, while the sender is running freely and posts messages on its output ports at each period, the receiver either receives the message if a new one is available or is blocked until the next message is posted. Symmetrically, the receiver runs freely, the sender is blocked up to the next request except if a new message was posted since the last receiving. Although communication between planning and control modules can be asynchronous or synchronous depending motion commands. The communication between the modules is specified to be synchronous at the established sampling frequencies. Application programs for Petri-net-based modeling and simulation of discrete event dynamic systems have been developed and used as a real-time control simulator. The Petri-net simulator is realized as a toolbox for the MATLAB® platform, so that the general purpose MATLAB tools for simulation of communication actions such as SimEvents and SIMULINK can be used in the Petri-net models. The MATLAB environment with RWTH-Mindstorms NXT Toolbox allows control of LEGO Mindstorms NXT through Bluetooth wireless connection Data acquisition toolbox is used for receiving and sending input and output signals from/to external sensors and effectors, communicating with USB hardware. The execution of the control system keeps periodically scanning; in each scan a Petri net is executed in the similar manner as a multi-tasking real-time operating system. The simulator checks for any enabled transitions. If there is any enabled transition it is placed into a queue-like data structure, then the respective transition definition files is run to make sure whether it can actually fire, satisfying all the guard conditions defined in the file including reserving resources for use by

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it. If an enabled transition can fire, then it is moved to another data structure. At the same time, input tokens for this transition are consumed from the respective input places. The simulator also checks whether any firing transition is completing. If there is any transition that is completing, then it is removing from the structure and the output tokens are deposited into the output places of the transition. For optimal management of system resources, the following functions are provided: requesting and releasing resources, declaring and changing priorities of different transitions, and reporting resource usage idle time.

19.5

Conclusion

Because backstepping control design can promote distributed implementation of the control software, this chapter has described a unified and formal approach of modular, cooperative control software synthesis for complex mobile robot hardware using extended Petri nets. The modeling and control of robotic motion can be executed in real time using Petri-net simulation tools on a general computing architecture such as a client server system connected on an Ethernet network, where the system controller is the server and the local controllers are the clients. For robust-adaptive controller based on neural networks and more complex command tasks with real-time trajectory tracking including obstacle avoidance, the net model can be easily implemented, but it should be more efficiently executed on a high-performance multiprocessor architecture.

References David, R., Alla, H., 1994. Petri nets for modeling of dynamic systems. Automatica 30 (2), 175–202. Fierro, R., Lewis, F.L., 1997. Control of a nonholonomic mobile robot: backstepping kinematics into dynamics. Journal of Robotic Systems 14 (3), 149–163. Fleury, S., Herbb, M., Chatila, R., 1994. Design of a modular architecture for autonomous robots. In: IEEE International Conference on Robotics and Automation. Martinez, J., Muro, P., Silva, M., 1987. Modeling, validation, and software implementation of production systems using high level Petri nets. In: IEEE International Conference on Robotics and Automation, pp. 1180–1185. Mohan, S., Yalcin, A., Khator, S., 2004. Controller design and performance evaluation for deadlock avoidance in automated flexible manufacturing cells. Robotics and Computer-Integrated Manufacturing 20, 541–551. Murata, T., 1989. Petri nets: properties, analysis and applications. Proceedings of the IEEE 77, 541–580. Music, G., Matko, D., 1999. Petri net based control of a modular production system. In: IEEE International Symposium on Industrial Electronics, vol. 3, pp. 1383–1388. Noda, A., et al., 2010. Intelligent robot technology for cell production systems. In: ISCIE/ASME 2010 International Symposium on Flexible Automation (ISFA2010). Tokyo, Japan. ISFA2010-JPS2562. Peterson, J.L., 1981. Petri Net Theory and the Modeling of Systems. Prentice Hall. Silva, E., Campos-Rebelo, R., Hirashima, T., Moutinho, F., Malo, P., Costa, A., Gomes, L., 2014. Communication support for Petri nets based distributed controllers. In: Proceedings of 2014 IEEE International Symposium on Industrial Electronics, pp. 1111–1116. Takei, T., Imamura, R., Yuta, S., 2009. Baggage transportation and navigation by a wheeled inverted pendulum mobile robot. IEEE Transactions on Industrial Electronics 56 (10), 3985–3994.

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Wang, F.Y., Sarides, G.N., 1993. Task translation and integration specification in intelligent machines. IEEE Transactions on Robotics and Automation RA-9 (3), 257–271. Wu, N.Q., Zhou, M.C., Chu, F., Chu, B.C., 2013. A Petri-net-based scheduling strategy for dualarm cluster tools in wafer fabrication. IEEE Transactions on Systems, Man and Cybernetics: Systems 43 (5), 1182–1194. Yasuda, G., 2014. Distributed coordination architecture for cooperative multi-robot task control using Petri nets. In: ISCIE/ASME 2014 International Symposium on Flexible Automation (ISFA2014). Awaji-Island, Hyogo, Japan, 2014. ISFA2014-131.

Chapter 20

A novel chaotic system with a closed curve of four quarter-circles of equilibrium points: dynamics, active backstepping control, and electronic circuit implementation Aceng Sambasa , Sundarapandian Vaidyanathanb , Sukonoc , Ahmad Taher Azard,e , Yuyun Hidayatf , Gugun Gundaraa , and Mohamad Afendee Mohamedg a Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya,

Indonesia, b Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India, c Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas

Padjadjaran, Bandung, Indonesia, d Robotics and Internet-of-Things Lab (RIOTU), Prince Sultan University, Riyadh, Saudi Arabia, e Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt, f Department of Statistics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Bandung, Indonesia, g Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia

20.1

Introduction

A nonlinear and aperiodic dynamical system is called chaotic if it exhibits sensitive dependence on the initial conditions. Thus, any autonomous continuoustime dynamical system with three or more states is chaotic if there is a positive element in the Lyapunov exponents of the system (Pham et al., 2018b). Investigation of chaotic behavior is one of the main topics in the field of nonlinear dynamics (Azar and Vaidyanathan, 2016; Azar et al., 2017; Pham et al., 2017b). This is evidenced by the identification of dynamic behavior in various scientific and engineering applications (Strogatz, 2015). Dynamical systems with chaotic behavior have various engineering applications such as robotics (Wu et al., 2018; Toz, 2020; Moysis et al., 2020), Backstepping Control of Nonlinear Dynamical Systems. https://doi.org/10.1016/B978-0-12-817582-8.00027-1 Copyright © 2021 Elsevier Inc. All rights reserved.

485

486 Backstepping Control of Nonlinear Dynamical Systems

electronic circuits (Sambas et al., 2019b,a; Vaidyanathan et al., 2019a; Yao et al., 2020; Azar et al., 2018b; Vaidyanathan et al., 2018b; Alain et al., 2020, 2018), secure communication (Liang et al., 2019; Liu et al., 2020), random bits generator (Ye et al., 2020; Yoshiya et al., 2020; Rezk et al., 2020), image encryption (Wang et al., 2020; Cheng et al., 2019; Vaidyanathan et al., 2018a), voice encryption (Vaidyanathan et al., 2019c; Elkouny et al., 2002), DC motors (Liu et al., 2019; Medeiros et al., 2019; Faradja and Qi, 2020), FPGA design (Tuna et al., 2019; Koyuncu et al., 2020; Tolba et al., 2017a,b), memristors (Kengne et al., 2019; Chen et al., 2019; Vaidyanathan et al., 2019b), and control systems (Alain et al., 2019; Singh et al., 2018, 2017; Ouannas et al., 2019). In the recent decades, significant research attention has been devoted to the modeling and applications of dynamical systems exhibiting chaos (Xu et al., 2019; Gusso et al., 2019; Gatabazi et al., 2019; Singh and Roy, 2019; Cabanas et al., 2019; Ginoux et al., 2019; Jahanshahi et al., 2019; Daumann and Rech, 2019; Khan et al., 2020a,b). Xu et al. (2019) proposed a chaotic system based on a circuit design involving a memristor model and a meminductor model. Gusso et al. (2019) analyzed the nonlinear dynamical model and the existence of chaos in suspended beam MEMS/NEMS resonators that are actuated by two-sided electrodes. Gatabazi et al. (2019) analyzed 2-D and 3-D Grey Lotka–Volterra Models (GLVM) and explored their application in cryptocurrencies such as Bitcoin, Litecoin, and Ripple. Singh and Roy (2019) studied microscopic chaos control of a chemical reactor system via nonlinear active plus proportional integral sliding mode control. Cabanas et al. (2019) discovered chaos in driven nano-magnets such as spin valves by using the magnetic energy and the magnetoresistance. Ginoux et al. (2019) discovered chaos in a dynamical system modeling the illicit drug consumption in a population comprising drug users and non-users. Jahanshahi et al. (2019) discussed a finance hyperchaos system via entropy analysis and control methods. Daumann and Rech (2019) reported hyperchaos in a heat-flux convection model. In recent years, studies on finding chaotic systems with closed curves of rest points have received much attention in the chaos literature (Gotthans and Petrzela, 2015; Pham et al., 2016, 2017a; Sambas et al., 2018; Vaidyanathan et al., 2018c). Gotthans and Petrzela (2015) introduced a new class of 3-D system with a circular equilibrium. Pham et al. (2016) proposed a new chaotic system with different shape of equilibria such as circular equilibrium, ellipse equilibrium, square-shaped equilibrium, and rectangle-shaped equilibrium. Pham et al. (2017a) discussed a new chaotic system with heart-shaped equilibrium and detailed its circuit design. Sambas et al. (2018) proposed a new chaotic system with pear-shaped equilibrium and detailed its circuit design. Vaidyanathan et al. (2018c) introduced a new chaotic system with axe-shaped equilibrium and discussed adaptive synchronization results. All these chaotic systems with closed curves of equilibrium points have infinite number of equilibrium points and such systems are said to belong to the class of chaotic systems with hidden attractors (Pham et al., 2018b).

A novel chaotic system with a closed curve Chapter | 20 487

In this paper, a new chaotic system with a closed curve of four quarter circles of rest points is reported. We show that the equilibrium curve is symmetric about the straight lines x = ±y. Using MATLAB® , we give the phase portraits of the new chaotic system and confirm its chaotic behavior with a calculation of Lyapunov exponents of the system. We discuss the dynamic analysis of the new chaotic system with Lyapunov exponents. We show that the new chaotic system shows multistability and has coexisting attractors. Multistability is a complex feature for a chaotic system where coexisting attractors are obtained for the same values of parameters but different initial states (Zhang et al., 2018a,b; Wang et al., 2018). Backstepping control approach is a recursive procedure for the stabilization of a control system about an equilibrium in strict-feedback design form and the backstepping method is popularly used for the control of systems (Vaidyanathan et al., 2015; Rasappan and Vaidyanathan, 2012; Vaidyanathan, 2015). In this work, we use the active backstepping control technique for the global stabilization and synchronization of the new chaotic system with a closed curve of four quarter circles of rest points. This chapter is organized as follows: In Section 20.2, the mathematical model of the new chaotic system and phase plots are presented. In Section 20.3, a dynamical analysis for the new chaotic system is described. In Section 20.4, backstepping control design for the global stabilization of the new chaotic system is presented. In Section 20.5, active backstepping control design for the global synchronization of a pair of new chaotic systems taken as leader–follower systems is presented. In Section 20.6, an electronic circuit implementation of the new chaotic system using Multisim is described. Section 20.7 concludes this work with a summary of the main results.

20.2

A new chaotic system with closed-curve equilibrium

In this section, we propose a new chaotic system modeled by the dynamics ⎧ x˙ = z, ⎪ ⎪ ⎨ (20.1) y˙ = −z(ay + by 2 + cxz), ⎪ ⎪ ⎩ z˙ = x 2 + y 2 − |x| − |y| − 4. For simplicity, we use X = (x, y, z) to denote the state vector of (20.1). In the dynamics, (20.1), a, b, c are positive constants. The rest points of the system (20.1) are derived by solving the following equations: z = 0,

(20.2a)

−z(ay + by 2 + cxz) = 0,

(20.2b)

x 2 + y 2 − |x| − |y| − 4 = 0.

(20.2c)

488 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 20.1 A closed-curve equilibrium of four quarter-circles in the (x, y)-plane of the system (20.1).

Solving (20.2), it is easy to see that the rest points of the system (20.1) are represented by a closed curve of four quarter-circles in the (x, y)-plane given by S = {X ∈ R3 | x 2 + y 2 − |x| − |y| − 4 = 0, z = 0}.

(20.3)

Fig. 20.1 shows the closed curve of rest points of the system (20.1) represented by the set S on the (x, y)-plane. It is also easy to note that the closed equilibrium curve is symmetric about the lines x = ±y. This follows because the closed equilibrium curve remains invariant under the change of coordinates (x, y) → (y, x) and also the change of coordinates (x, y) → (y, −x). In this work, we establish that the 3-D model (20.1) is a chaotic system if the parameter values are taken to be a = 5, b = 2, c = 1.2.

(20.4)

For numerical calculations, we fix the parameter vector as (a, b, c) = (5, 2, 1.2). Also, we fix the initial state as X(0) = (0.01, 0.02, 0.01). Using MATLAB, the Lyapunov characteristic exponents of the 3-D model (20.1) are calculated and shown in Fig. 20.2. We find the Lyapunov characteristic exponents of the 3-D model (20.1) to be ψ1 = 0.2806, ψ2 = 0, ψ3 = −4.9181.

(20.5)

The Kaplan–Yorke dimension of the 3-D novel chaotic system (20.1) is found to be ψ1 + ψ2 DKY = 2 + = 2.0571. (20.6) |ψ3 | From Eq. (20.5), it can be deduced that the 3-D model (20.1) with closedcurve equilibrium is chaotic as there is a positive Lyapunov exponent. The 3-D

A novel chaotic system with a closed curve Chapter | 20 489

FIGURE 20.2 Lyapunov exponents of the 3-D chaotic system (20.1) with a closed-curve equilibrium for the initial state (0.01, 0.02, 0.01) and parameter values (a, b, c) = (5, 2, 1.2).

chaotic model (20.1) is also dissipative since the sum of the Lyapunov characteristic exponents in (20.5) is negative. The phase plots of the 3-D chaotic model (20.1) with a closed-curve equilibrium are given in Fig. 20.3. Since the chaotic model (20.1) has infinitely many rest points, it follows that the system exhibits hidden chaotic attractor (Pham et al., 2018b).

20.3

Dynamic analysis of the new chaotic system with a closed-curve equilibrium

20.3.1 Lyapunov exponents analysis Here, Lyapunov exponents are drawn to validate the chaotic behavior of the novel chaotic system (20.1) with a closed-curve equilibrium. The Lyapunov exponents of the system (20.1) versus a ∈ [0, 10] are shown in Fig. 20.4. When a = 5, the system (20.1) has chaotic behavior. It is similar to Lyapunov exponent in Fig. 20.5 with variations in parameters b ∈ [0, 10]. When b = 2, system (20.1) is chaotic. This is indicated by the positive LE value. For c = 1.2, system (20.1) has a chaotic attractor as shown in Fig. 20.6. When 1.6 ≤ c ≤ 6.7, the system (20.1) has periodic attractor and when 6.8 ≤ c ≤ 12, the system (20.1) chaotic attractor.

20.3.2 Multistability and coexisting attractors Multistability means the coexistence of two or more attractors under different initial conditions but with the same parameter set (Azar et al., 2018a; Pham et al., 2018a; Wang et al., 2017). It is an interesting phenomenon and can usually be found in many nonlinear dynamical systems. It is well known that multistability can lead to very complex behaviors in a dynamical system.

490 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 20.3 MATLAB plots of the 3-D novel chaotic system (20.1) for (a, b, c) = (5, 2, 1.2) and X(0) = (0.01, 0.02, 0.01): (A) (x, y)-plane, (B) (y, z)-plane, (C) (x, z)-plane, and (D) R3 .

FIGURE 20.4 Lyapunov exponents of the system (20.1) with (b, c) = (2, 1.2) and X(0) = (0.01, 0.02, 0.01).

A novel chaotic system with a closed curve Chapter | 20 491

FIGURE 20.5 Lyapunov exponents of the system (20.1) with (a, c) = (5, 1.2) and X(0) = (0.01, 0.02, 0.01).

FIGURE 20.6 Lyapunov exponents of the system (20.1) with (a, b) = (5, 2) and X(0) = (0.01, 0.02, 0.01).

It is interesting that the new chaotic system with a closed-curve equilibrium (20.1) can exhibit coexisting chaotic attractors when choosing different initial conditions. We take parameter values as in the chaotic case, viz. (a, b, c) = (5, 2, 1.2). We select two initial conditions as X0 = (0.01, 0.02, 0.01), Y0 = (0.04, −0.04, −0.04), and the corresponding state orbits of the system (20.1) are plotted in the colors blue (dark gray in print) and red (light gray in print), respectively.

492 Backstepping Control of Nonlinear Dynamical Systems

FIGURE 20.7 Multistability of the 3-D new chaotic system (20.1) with a closed-curve equilibrium: Coexisting chaotic attractors for (a, b, c) = (5, 2, 1.2) and initial conditions X0 = (0.01, 0.02, 0.01) (blue, dark gray in print) and Y0 = (0.04, −0.04, −0.04) (red, light gray in print). (A) (x; y)-plane; (B) (y; z)-plane.

From Fig. 20.7, it can be observed that the new chaotic system with a closedcurve equilibrium (20.1) exhibits multistability with two coexisting chaotic attractors.

20.4 Active backstepping control for the global stabilization of the new chaos system with a closed-curve equilibrium Here, we deploy active backstepping control for globally stabilizing the trajectories of the new chaos system with a closed-curve equilibrium for all initial conditions. The controlled chaos system is described by the 3D dynamics ⎧ ⎪ ⎨ x˙ y˙ ⎪ ⎩ z˙

=

z + ux ,

=

−z(ay + by 2 + cxz) + uy ,

=

x2

+ y2

(20.7)

− |x| − |y| − 4 + uz .

As a first step, we use active control to transform the system (20.7) to a system with triangular structure that aids the backstepping control design. We consider the control law ⎧ ⎪ ⎨ ux uy ⎪ ⎩ uz

=

y − z,

=

z(ay + by 2 + cxz + 1),

=

−x 2 − y 2 + |x| + |y| + 4 + v,

where v is an active backstepping control to be determined.

(20.8)

A novel chaotic system with a closed curve Chapter | 20 493

Substituting (20.8) into (20.7), we get the new system in triangular form as ⎧ ⎨ x˙ y˙ ⎩ z˙

= y, = z, = v.

(20.9)

We start with the Lyapunov function W1 (ηx ) =

1 2 η 2 x

(20.10)

where ηx = x.

(20.11)

Differentiating W1 with respect to t along the dynamics (20.9), we get W˙ 1 = ηx η˙ x = −ηx2 + ηx (x + y).

(20.12)

ηy = x + y.

(20.13)

We define

With the help of Eq. (20.13), we can express (20.12) as W˙ 1 = −ηx2 + ηx ηy .

(20.14)

We proceed next with defining the Lyapunov function W2 (ηx , ηy ) = W1 (ηx ) +

 1 2 1 2 ηy = ηx + ηy2 . 2 2

(20.15)

Differentiating W2 with respect to t along the dynamics (20.9), we get W˙ 2 = −ηx2 − ηy2 + ηy (2x + 2y + z).

(20.16)

We define ηz as follows: ηz = 2x + 2y + z.

(20.17)

With the help of Eq. (20.17), we can express Eq. (20.16) as W˙ 2 = −ηx2 − ηy2 + ηy ηz .

(20.18)

As a final step of the backstepping control design, we set the quadratic Lyapunov function W (ηx , ηy , ηz ) = W2 (ηx , ηy ) +

 1 2 1 2 ηz = ηx + ηy2 + ηz2 . 2 2

(20.19)

494 Backstepping Control of Nonlinear Dynamical Systems

It is clear that W is a positive define function on R 3 . Differentiating W with respect to t along the dynamics (20.9), we get W˙ = −ηx2 − ηy2 − ηz2 + ηz (ηz + ηy + η˙ z ).

(20.20)

A simple calculation yields the result W˙ = ηx2 − ηy2 − ηz2 + ηz (3x + 5y + 3z + v).

(20.21)

We define the control law v as v = −3x − 5y − 3z − Kηz

(20.22)

where we take K to be a positive constant. Substituting (20.22) into (20.21), we get W˙ = −ηx2 − ηy2 − (1 + K)ηz2 ,

(20.23)

which is quadratic and negative definite. By Lyapunov stability theory, it is immediate that (ηx (t), ηy (t), ηz (t)) → 0 exponentially as t → ∞. We know that x = ηx , y = ηy − ηx , z = ηz − 2ηy .

(20.24)

As a consequence, it follows that (x(t), y(t), z(t)) → 0 exponentially as t → ∞. Substituting (20.22) into (20.8), the required backstepping control law is given by ⎧ ⎪ ⎨ ux uy ⎪ ⎩ uz

=

y − z,

=

z(ay + by 2 + cxz + 1),

=

−x 2

− y2

(20.25)

+ |x| + |y| + 4 − 3x − 5y − 3z − Kηz .

Thus, we have proved the following result. Theorem 20.1. The backstepping control law defined via (20.25) with gain K > 0 globally and exponentially stabilizes the trajectories of the 3D chaos plant (20.7) for all initial states (x(0), y(0), z(0)) ∈ R3 . For simulations, we pick the values of the parameters as in the chaos case, viz. (a, b, c) = (5, 2, 1.2). We choose K = 6 and the initial state of the system (20.7) as x(0) = 8.5, y(0) = −4.7, and z(0) = 3.2.

A novel chaotic system with a closed curve Chapter | 20 495

FIGURE 20.8 Time-history of the backstepping controlled states x(t), y(t), and z(t) for (a, b, c) = (5, 2, 1.2), K = 6, and (x(0), y(0), z(0)) = (8.5, −4.7, 3.2).

Fig. 20.8 shows the time-history of the backstepping controlled states x(t), y(t), and z(t). It is easy to note that the controlled states converge to zero exponentially by the action of the backstepping control law (20.25).

20.5

Active backstepping control for the synchronization of the new chaos systems

Here, we deploy active backstepping control for globally synchronizing the trajectories of a pair of new chaos systems considered as leader–follower systems. The leader chaos system is described by the 3D dynamics ⎧ x˙ = z1 , ⎪ ⎪ ⎨ 1 (20.26) y˙1 = −z1 (ay1 + by12 + cx1 z1 ), ⎪ ⎪ ⎩ z˙ 1 = x12 + y12 − |x1 | − |y1 | − 4. The follower chaos system with controls is described by the 3D dynamics ⎧ x˙ = z2 + ux , ⎪ ⎪ ⎨ 2 (20.27) y˙2 = −z2 (ay2 + by22 + cx2 z2 ) + uy , ⎪ ⎪ ⎩ z˙ 2 = x22 + y22 − |x2 | − |y2 | − 4 + uz , where ux , uy , uz are feedback controls to be determined. The synchronization chaos error is defined by means of the equations ex = x2 − x1 , ey = y2 − y1 , ez = z2 − z1 .

(20.28)

496 Backstepping Control of Nonlinear Dynamical Systems

Upon calculating the error dynamics, we obtain the following: ⎧ e˙ = ez + ux , ⎪ ⎪ ⎨ x e˙y = −a(y2 z2 − y1 z1 ) − b(y22 z2 − y12 z1 ) − c(x2 z22 − x1 z12 ) + uy , ⎪ ⎪ ⎩ e˙z = x22 − x12 + y22 − y12 − |x2 | + |x1 | − |y2 | + |y1 | + uz . (20.29) As a first step, we use active control to transform the error system (20.29) to an error system with triangular structure that aids backstepping control design. We consider the control law ⎧ ux = ey − ez , ⎪ ⎪ ⎨ uy = a(y2 z2 − y1 z1 ) + b(y22 z2 − y12 z1 ) + c(x2 z22 − x1 z12 ) + ez , ⎪ ⎪ ⎩ uz =