STABLE ADAPTIVE NEURAL NETWORK CONTROL

The Kluwer International Series on ASIAN STUDIES IN COMPUTER AND INFORMATION SCIENCE

Series Editor Kai-Yuan Cai

Beijing University of Aeronautics and Astronautics, Beijing, CHINA

Editorial Advisory Board Han-Fu Chen, Institute of System Science, Chinese Academy of Sciences Jun-Liang Chen, Beijing University of Post and Telecommunication Lin Huang, Peking University Wei Li, Beijing University of Aeronautics and Astronautics Hui-Min Lin, Institute of Software Technology, Chinese Academy of Sciences Zhi-Yong Liu, Institute of Computing Technology, Chinese Academy of Sciences Ru-Qian Lu, Institute of Mathematics, Chinese Academy of Sciences Shi-Tuan Shen, Beijing University of Aeronautics and Astronautics Qing-Yun Shi, Peking University You-Xian Sun, Zhejiang University Lian-Hua Xiao, National Natural Science Foundation of China Xiao-Hu You, Southeast University Bo Zhang, Tsinghua University Da-Zhong Zheng, Tsinghua University Bing-Kun Zhou, Tsinghua University Xing-Ming Zhou, Changsha University of Technology Also in the Series:

FULLY TUNED RADIAL BASIS FUNCTION NEURAL NETWORKS FOR FLIGHT CONTROL by N. Sundararajan, P. Saratchandran and Yan Li; ISBN: 0-7923-7518-1

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS by Qiang Lu, Yuanzhang Sun, Shengwei Mei; ISBN: 0-7923-7312-X

DATA MANAGEMENT AND INTERNET COMPUTING FOR IMAGE/PATTERN ANALYSIS David Zhang, Xiobo Li and Zhiyong Liu; ISBN: 0-7923-7456-8

COMMON WAVEFORM ANALYSIS: A New and Practical Generalization of Fourier Analysis, by Yuchuan Wei and Qishan Zhang; ISBN: 0-7923-7905-5

DOMAIN MODELING-BASED SOFTWARE ENGINEERING: A Formal Approach, by Ruqian Lu and Zhi Jin; ISBN: 0-7923-7889-X

AUTOMATED BIOMETRICS: Technologies and Systems, by David D. Zhang; ISBN: 0-7923-7856-3

FUZZY LOGIC AND SOFT COMPUTING, by Guoqing Chen, Mingsheng Ying Kai-Yuan Cai; ISBN: 0-7923-8650-7

INTELLIGENT BUILDING SYSTEMS, by Albert Ting-pat So, Wai Lok Chan; ISBN: 0-7923-8491-1

PERFORMANCE EV ALUA TION, PREDICTION AND VISUALIZATION OF PARALLEL SYSTEMS by Xingfu Wu; ISBN: 0-7923-8462-8

ROBUST MODEL-BASED FAULT DIAGNOSIS FOR DYNAMIC SYSTEMS by Jie Chen and Ron J. Patton; ISBN: 0-7923-8411-3

STABLE ADAPTIVE NEURAL NETWORK CONTROL

by

S. S. Ge

C. C. Hang

T. H. Lee

T. Zhang

Department 0/ Electrical Engineering National University a/Singapore

SPRINGER SCIENCE+BUSINESS MEDIA, LLC

Library of Congress Cataloging-in-Publication Data

Stable adaptive neural network control/by S. S. Ge ... ret al.J. p. cm.-- (The Kluwer international series on Asian studies in computer and

information science; 13) Includes bibliographical references and index. ISBN 978-1-4419-4932-5 ISBN 978-1-4757-6577-9 (eBook) DOI 10.1007/978-1-4757-6577-9

1. Adaptive control systems. 2. Neural networks (Computer science) 1. Ge, S.S. (Shuzhi S.) II. Series.

TJ217.S7362001 629.8'3--dc21

Copyright © 2002 by Springer Science+Business· Media New York Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1 st edition 2002

2001050337

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC.

Printed on acidjree paper.

To

Jinlan, Sao Chin, Ruth, Yuan

and

our loved ones

SERIES EDITOR'S ACKNOWLEDGMENTS

I am pleased to acknowledge the assistance to the editorial work by Beijing University of Aeronautics and Astronautics and the National Natural Science Foundation of China

Kai-Yuan Cai Series Editor Department of Automatic Control Beijing University of Aeronautics and Astronautics Beijing 100083 China

I do not know what I may appear to the world, bu.t to myself I seem to have been only like a boy playing on the seashore, diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all u.ndiscovered before me.

-Isaac Newton

Contents

Preface xiii

1 Introduction 1 1.1 Introduction . 1 1.2 Adaptive Control . 2 1.3 Neural Network Control 3 1.4 Instability Mechanisms in Adaptive Neural Control Systems .5 1.5 Outline of the Book 9 1.6 Conclusion 10

2 Mathematical Preliminaries 11 2.1 Introduction . 11 2.2 Mathematical Preliminaries 11

2.2.1 Norms for Vectors and Signals 12 2.2.2 Properties of Matrix 15

2.3 Concepts of Stability . 16 2.4 Lyapunov Stability Theorem 17 2 .. 5 Useful Theorems and Formula. 19

2.5.1 Sliding Surface 19 2 .. 5.2 Mean Value Theorem 20 2.5.3 Integral Formula 20 2.5.4 Implicit Function Theorem 23 2.5.5 Input-Output Stability . 25

2.6 Conclusion 26

3 Neural Networks and Function Approximation 27 3.1 Introduction. 27 3.2 Function Approximation . 27 3.3 Linearly Parametrized Neural Networks 29 3.4 Non-linearly Parametrized Networks 35 3.5 Neural Networks for Control Applications 44 3.6 Conclusion 46

x Contents

4 SISO Nonlinear Systems 47 4.1 Introduction.............. 47 4.2 NN Control with Regional Stability. 49

4.2.1 Desired Feedback Control . . 49 4.2.2 HONN Controller Design Based on (4.7) . 51 4.2.3 MNN Control Based on (4.10) ...... 59

4.3 VSC - Semi-Global Stability .......... 70 4.3.1 VSC-based Adaptive NN Control Design. 73 4.3.2 Elimination for Controller Chattering 77 4.3.3 Simulation Study. 79

4.4 Conclusion ....... 79

5 ILF for Adaptive Control 81 5.1 Introduction............. 81 5.2 Matching SISO Nonlinear Systems 82

5.2.1 Integral Lyapunov Function 83 5.2.2 Choice of Weighting Function Q(x) 83 5.2.3 Adaptive NN Control Based on DFCs 92

.5.3 Backstepping Adaptive NN Design . . . . . . 10.5 .5.3.1 Adaptive Design for a First-order System 108 .5.3.2 Design for nth-order Systems . . . . . . . 112 .5.3.3 Controller Design with Reduced Knowledge 121 5.3.4 Simulation Studies . . . . . . . . . 123

.5.4 NN Control for MIMO Nonlinear Systems . . . . . 127 5.4.1 System Description. . . . . . . . . . . . . . 128 5.4.2 Lyapunov Function Design and Control Structure 130 .5.4.3 Adaptive MIMO Control Using MNNs 132

.5.5 Conclusion .......................... 138

6 Non-affine Nonlinear Systems 139 6.1 Introduction................. 139 6.2 System Description and Properties . . . . 140

6.2.1 Implicit Desired Feedback Control 142 6.2.2 High-gain Observer. . . . . 146

6.3 Controller Design Based on LPNN 147 6.3.1 State Feedback Control . 149 6.3.2 Output Feedback Control . 153 6.3.3 Simulation Study. . . . . . 159

6.4 Controller Design Based on MNN . 160 6.4.1 State Feedback Control . 163 6.4.2 Output Feedback Control 6.4.3 Application to CSTR ..

168 176

Contents

6.5 Conclusion ......... .

7 Triangular Nonlinear Systems 7.1 Introduction................. 7.2 Special Systems in Strict-Feedback Form.

7.2.1 Direct Adaptive NN Control 7.2.2 Simulation studies ....... .

7.3 Partially Known Nonlinear Systems .. 7.3.1 Adaptive Neural Control Design 7.3.2 Numerical Simulation ..... .

7.4 Pure-feedback Nonlinear Systems ... . 7.4.1 Direct Adaptive NN Control for El . 7.4.2 Direct Adaptive NN Control for E2 . 7.4.3 Simulation studies

7.5 MIMO Nonlinear Systems 7.6 Conclusion ....... .

8 Conclusion 8.1 Conclusion 8.2 Design Flexibility . 8.3 Further Research

References

Index

xi

182

183 183 185 188 198 201 203 215 217 220 235 240 242 260

261 261 262 263

265

281

Preface

Recent years have seen a rapid development of neural network control tech­niques and their successful applications. Numerous simulation studies and actual industrial implementations show that artificial neural network is a good candidate for function approximation and control system design in solving the control problems of complex nonlinear systems in the presence of different kinds of uncertainties. Many control approaches/methods, reporting inventions and control applications within the fields of adaptive control, neural control and fuzzy systems, have been published in various books, journals and conference proceedings. In spite of these remarkable advances in neural control field, due to the complexity of nonlinear systems, the present research on adaptive neural control is still focused on the development of fundamental methodologies. From a theoretical viewpoint, there is, in general, lack of a firmly mathematical basis in stability, robustness, and performance analysis of neural network adaptive control systems.

This book is motivated by the need for systematic design approaches for stable adaptive control using approximation-based techniques. The main objec­tives of the book are to develop stable adaptive neural control strategies, and to perform transient performance analysis of the resulted neural control systems analytically. Other linear-in-the-parameter function approximators can replace the linear-in-the-parameter neural networks in the controllers presented in the book without any difficulty, which include polynomials, splines, fuzzy systems, wavelet networks, among others.

Stability is one of the most important issues being concerned if an adaptive neural network controller is to be used in practical applications. In this book, Lyapunov stability techniques playa critical role in the design and stability analysis of the adaptive systems. Under different operating conditions and a priori knowledge, stable neural controller designs are presented for several classes of nonlinear systems, including (i) single-input single-output (SISO) nonlinear systems, (ii) nonlinear systems in strict-feedback form, (iii) non affine nonlinear systems, and (iv) multi-input and multi-output (MIMO) nonlinear in triangular form. Stability of the overall neural network systems is rigorously

xiii

xiv Preface

proven through Lyapunov stability analysis. Transient performance of adaptive neural control systems is also essential for the control applications. It has been shown that poor initial conditions may result in unacceptably poor transient behaviour in adaptive systems. It is highly desirable for a control engineer to have an estimate of the transient performance before a neural network con­troller is put into practice. In this book, for different neural control designs, the effects of controller parameters, initial conditions and reference signals on sys­tem stability and control performance are investigated for providing valuable insights into performance improvement and design trade-offs.

The main special features of this book are as follows: (i) singularity-free controllers are presented for a class of nonlinear SISO systems by exploiting its property of 8b(x}j8xn = 0, (ii) through the introduction of integral Lya­punov function (ILF) candidates, novel design methodologies are introduced to solve the control problems for a wide class of nonlinear adaptive control problems without encountering the controller singularity problem, (iii) besides affine nonlinear systems, controller design for nonaffine nonlinear systems are also be treated using implicit function theorem, neural network approxima­tion and adaptive control techniques, and (iv) most of the results presented are analytical with repeatable design algorithms because closed-loop stability is proven mathematically, and detailed performance analysis of the proposed adaptive neural controllers is performed rigorously.

The book starts with a brief introduction of adaptive control, neural network control, and the possible instability mechanisms in adaptive neural control systems in Chapter 1.

For completeness, Chapter 2 gives a brief summary of the basic mathe­matical tools of norms, stability theorems, implicit function and mean value theorems and properties of integrations, which are used for controller design, stability and performance analysis in the subsequent chapters of the book.

Chapter 3 presents two classes of function approximators, namely, linearly parameterized neural networks (LPNN) and non-linearly parameterized (multi­layer) neural networks (MNN) for function approximation. Main properties of such two kinds of networks are discussed for control applications. In addition, their advantages and shortcomings are studied when they are used in system identification and adaptive control design.

In Chapter 4, a regionally stable NN design is firstly proposed for nonlin­ear systems in a Brunovsky form. The control performance of the systems is analytically quantified by the mean square criterion and Loo criterion. Then, a semi-global NN controller is provided by using variable structure control tech­nique. Furthermore, the transient behaviour of the adaptive neural system has been investigated, and several methods are provided for improving the system response.

In Chapter 5, by introducing an integral Lyapunov function, adaptive NN

Preface xv

controller is firstly developed for a class of SISO nonlinear systems. The con­trol singularity problem, which usually met in feedback linearization adaptive control, is completely solved using the newly proposed control method. The developed control schemes ensure global stability of the adaptive systems and asymptotic convergence of output tracking error. Then, adaptive control design is developed for strict-feedback nonlinear systems through combining multilayer NNs with backstepping technique. It is proven that under certain conditions, the semi-globally uniformly ultimate boundedness is achievable for the closed­loop adaptive systems. The relationship between the transient performance and the design parameters is also investigated to guide the tuning of the neural controller. Finally, adaptive NN Control is presented for a class of MIMO non­linear systems having triangular structure in control inputs using multi-layer neural networks. Without imposing any constraints on the system interconnec­tions, the developed controller guarantees the stability of the adaptive neural system and the convergence of the mean square tracking errors to small residual sets.

In Chapter 6, adaptive NN control is investigated for a class of non affine nonlinear systems. Both state and output (using a high-gain observer for state estimation) feedback controllers are given for linearly parameterized and mul­tilayer neural networks, and their effectiveness are verified by numerical simu­lation.

In Chapter 7, controller design is investigated for several classes of trian­gular nonlinear systems using quadratic Lyapunov function for its convenience of analysis and simplicity of the resulting controllers. Firstly, we investigate a class of systems in strict-feedback form with gn(Xn-l) which is indepen­dent of X n . This nice properties can be exploited for better controller design. Secondly, we study the nonlinear strict-feedback systems which include both parametric uncertainty and unknown nonlinear functions, and constant gi so that the parametric certainties can be solved using model based adaptive con­trol techniques and the unknown nonlinear functions be approximated using NN approximation. Thirdly, we investigate the control problem a class of non­linear pure-feedback systems with unknown nonlinear functions. This problem is considered difficult to be dealt with in the control literature, mainly because that the triangular structure of pure-feedback systems has no affine appear­ance of the variables to be used as virtual controls. Finally, the extension from SISO nonlinear systems in triangular forms to MIMO nonlinear systems in block-triangular forms have also been considered in this chapter.

In summary, this book covers the analysis and design of neural network based adaptive controller for different classes of nonlinear systems, which in­clude SISO nonlinear systems, nonlinear systems in strict-feedback and pure­feedback forms, MIMO nonlinear systems in triangular form, and nonaffine nonlinear systems. Numerical simulation studies are used to verify the effec-

xvi Preface

tiveness and the performance of the control schemes. This book is aimed at a wide readership and is a convenient and useful reference for research stu­dents, academics and practicing engineers in the areas of adaptive control, neural/fuzzy modelling and control.

For the creation of the book, we are very fortunate to have the appropri­ate suggestions from and helpful discussions with our colleagues, friends and co-workers. In particular, we would like to express our sincere gratitude to C. Canudas de Wit of Laboratoire d'Automatique de Grenoble, K Y. Cai of Bei­jing University of Aeronautics and Astronautics, C. J. Harris of University of Southampton, F. L. Lewis of University of Texas at Arlington, 1. M. Y. Mareels of University of Melbourne, Y. Miyasato of Institute of Statistical Mathematics, T. Parisini of Politecnico di Milano, M. Polycarpou of University of Cincinnati, 1. Postlethwaite of University of Leicester, J. Si of Arizona State University, M. W. Spong of University of Illinois at Urbana-Champaign, G. Tao of University of Virginia, C. W. de Silva of University of British Columbia, H. Wang of the University of Manchester Institute of Science and Technology, B. Yao, Purdue University, Y. H. Tan, Guilin Institute of Electronic Technology, A. P. Loh and J. X. Xu of the National University of Singapore for their constructive and helpful suggestions and comments.

The first author owes a big thank you to his parents and his wife for their love, supports and sacrifice throughout the years, to his children, Yaowei Jas­mine, Yaolong George and Yaohong Lorraine, for their understanding of not being able to play with them for numerous weekends and evenings.

Last but not the least, the first author would like to thank his current and former postdoctoral fellows and graduate students, especially Z. Sun, J. Wang, and G.Y. Li, C. Wang, Z.P. Wang, J. Zhang, and J.Q. Gong for their help in technical analysis, and many critical discussions. In particular, he is in great debt to J. Wang, C. Wang and J. Zhang for their unconditional help in formatting the book without any hesitation. Special thanks go to M. Fearon for her assistance and help in the process of publishing the book.

Shuzhi S. Ge, Chang C. Hang, Tong H. Lee and Tao Zhang Department of Electrical fj Computer Engineering

The National University of Singapore