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Applied Nonlinear Control.pdf

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    Applied Nonlinear Control JEAN-JACQUES E. SLOTINE Massachusetts Institute of Technology WEIPING LI Massachusetts Institute of Technology' Prentice Hall Englewood Cliffs, New Jersey 07632
    Library of Congress Cataloging-in-Publication Data Slotine, J.-J. E. (Jean-Jacques E.) Applied nonlinear control / Jean-Jacques E. Slotine, Weiping Li p. cm. Includes bibliographical references. ISBN 0-13-040890-5 1, Nonlinear control theory. I. Li, Weiping. QA402.35.S56 1991 629.8'312-dc20 II. Title. 90-33365 C1P Editorial/production supervision and interior design: JENNIFER WENZEL Cover design: KAREN STEPHENS Manufacturing Buyer: LORI BULWIN = ^= © 1991 by Prentice-Hall, Inc. ^=&= A Division of Simon & Schuster T k Englewood Cliffs, New Jersey 07632 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Printed in the United States of America 20 19 18 17 16 15 14 13 12 1] ISBN D-13-DHDfiTa-S Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prentice-Hail Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
    Contents Preface 1. Introduction 1.1 Why Nonlinear Control ? 1.2 Nonlinear System Behavior 1.3 An Overview of the Book 1.4 Notes and References Part I: Nonlinear Systems Analysis Introduction to Part I 14 2. Phase Plane Analysis 2.1 Concepts of Phase Plane Analysis 2.1.1 Phase Portraits 2.1.2 Singular Points 2.1.3 Symmetry in Phase Plane Portraits 18 20 22 2.2 Constructing Phase Portraits 2.3 Determining Time from Phase Portraits 2.4 Phase Plane Analysis of Linear Systems 2.5 Phase Plane Analysis of Nonlinear Systems 2.6 Existence of Limit Cycles 2.7 Summary 2.8 Notes and References 2.9 Exercises xi 1 1 4 12 13 14 17 18 23 29 30 32 36 38 38 38
    VI11 3. Fundamentals of Lyapunov Theory 3.1 Nonlinear Systems and Equilibrium Points 3.2 Concepts of Stability 3.3 Linearization and Local Stability 3.4 Lyapunov's Direct Method 3.4.1 Positive Definite Functions and Lyapunov Functions 3.4.2 Equilibrium Point Theorems 3.4.3 Invariant Set Theorems 61 68 3.5 System Analysis Based on Lyapunov's Direct Method 3.5.1 Lyapunov Analysis of Linear Time-Invariant Systems 3.5.2 Krasovskii's Method 3.5.3 The Variable Gradient Method 3.5.4 Physically Motivated Lyapunov Functions 3.5.5 Performance Analysis 83 86 88 91 58 77 3.6 Control Design Based on Lyapunov's Direct Method 3.7 Summary 3.8 Notes and References 3.9 Exercises 4. Advanced Stability Theory 4.1 Concepts of Stability for Non-Autonomous Systems 4.2 Lyapunov Analysis of Non-Autonomous Systems 4.2.1 Lyapunov's Direct Method for Non-Autonomous Systems 4.2.2 Lyapunov Analysis of Linear Time-Varying Systems 4.2.3 The Linearization Method for Non-Autonomous Systems 114 105 116 4.3 * Instability Theorems 4.4 * Existence of Lyapunov Functions 4.5 Lyapunov-Like Analysis Using Barbalat's Lemma 4.5.1 Asymptotic Properties of Functions and Their Derivatives 4.5.2 Barbalat's Lemma 123 122 4.6 Positive Linear Systems 4.6.1 PR and SPR Transfer Functions 4.6.2 The Kalman-Yakubovich Lemma 4.6.3 Positive Real Transfer Matrices 4.7 The Passivity Formalism 4.7.1 Block Combinations 132 4.7.2 Passivity in Linear Systems 137 126 130 131 40 41 47 53 57 76 94 95 96 97 100 101 105 117 120 122 126 132
    4.8 * Absolute Stability 4.9 * Establishing Boundedness of Signals 4.10 * Existence and Unicity of Solutions 4.11 Summary 4.12 Notes and References 4.13 Exercises 5. Describing Function Analysis 5.1 Describing Function Fundamentals 5.1.1 An Example of Describing Function Analysis 5.1.2 Applications Domain 5.1.3 Basic Assumptions 5.1.4 Basic Definitions 5.1.5 Computing Describing Functions 165 167 162 164 158 5.2 Common Nonlinearities In Control Systems 5.3 Describing Functions of Common Nonlinearities 5.4 Describing Function Analysis of Nonlinear Systems 5.4.1 The Nyquist Criterion and Its Extension 5.4.2 Existence of Limit Cycles 5.4.3 Stability of Limit Cycles 5.4.4 Reliability of Describing Function Analysis 182 184 180 186 5.5 Summary 5.6 Notes and References 5.7 Exercises Part II: Nonlinear Control Systems Design Introduction to Part II 191 6. Feedback Linearization 6.1 Intuitive Concepts 6.1.1 Feedback Linearization And The Canonical Form 6.1.2 Input-State Linearization 6.1.3 Input-Output Linearization 213 216 6.2 Mathematical Tools 6.3 Input-State Linearization of SISO Systems IX 142 147 151 153 153 154 157 158 169 172 179 187 188 188 191 207 208 229 236 208
    6.4 Input-Output Linearization of SISO Systems 6.5 * Multi-Input Systems 6.6 Summary 6.7 Notes and References 6.8 Exercises 7. Sliding Control 7.1 Sliding Surfaces 7.1.1 A Notational Simplification 7.1.2 * Filippov's Construction of the Equivalent Dynamics 7.1.3 Perfect Performance - At a Price 7.1.4 Direct Implementations of Switching Control Laws 7.2 Continuous Approximations of Switching Control Laws 7.3 The Modeling/Performance Trade-Offs 278 285 283 289 7.4 * Multi-Input Systems 7.5 Summary 7.6 Notes and References 7.7 Exercises 8. Adaptive Control 8.1 Basic Concepts in Adaptive Control 8.1.1 Why Adaptive Control ? 8.1.2 What Is Adaptive Control ? 8.1.3 How To Design Adaptive Controllers ? 312 315 323 8.2 Adaptive Control of First-Order Systems 8.3 Adaptive Control of Linear Systems With Full State Feedback 8.4 Adaptive Control of Linear Systems With Output Feedback 8.4.1 Linear Systems With Relative Degree One 8.4.2 Linear Systems With Higher Relative Degree 340 346 8.5 Adaptive Control of Nonlinear Systems 8.6 Robustness of Adaptive Control Systems 8.7 * On-Line Parameter Estimation 246 266 270 271 271 276 277 290 301 303 306 307 307 311 312 326 335 339 350 353 358 8.7.1 Linear Parametrization Model 8.7.2 Prediction-Error-Based Estimation Methods 8.7.3 The Gradient Estimator 8.7.4 The Standard Least-Squares Estimator 364 359 370 364
    8.7.5 Least-Squares With Exponential Forgetting 8.7.6 Bounded-Gain Forgetting 8.7.7 Concluding Remarks and Implementation Issues 376 374 381 1.8 Composite Adaptation 1.9 Summary 1.10 Notes and References 1.11 Exercises 9. Control of Multi-Input Physical Systems 9.1 Robotics as a Prototype 9.1.1 Position Control 9.1.2 Trajectory Control 397 9.2 Adaptive Robot Trajectory Control 404 394 9.2.1 The Basic Algorithm 9.2.2 * Composite Adaptive Trajectory Control 9.3 Putting Physics in Control 9.3.1 High-Frequency Unmodeled Dynamics 9.3.2 Conservative and Dissipative Dynamics 9.3.3 Robotics as a Metaphor 419 9.4 Spacecraft Control 9.4.1 The Spacecraft Model 9.4.2 Attitude Control 425 422 9.5 Summary 9.6 Notes and References 9.7 Exercises BIBLIOGRAPHY INDEX 411 416 418 382 388 389 389 392 393 403 416 422 432 433 433 437 459
    Preface In recent years, the availability of powerful low-cost microprocessors has spurred great advances in the theory and applications of nonlinear control. In terms of theory, major strides have been made in the areas of feedback linearization, sliding control, and nonlinear adaptation techniques. In terms of applications, many practical nonlinear control systems have been developed, ranging from digital "fly-by-wire" flight control systems for aircraft, to "drive-by-wire" automobiles, to advanced robotic and space systems. As a result, the subject of nonlinear control is occupying an increasingly important place in automatic control engineering, and has become a necessary part of the fundamental background of control engineers. This book, based on a course developed at MIT, is intended as a textbook for senior and graduate students, and as a self-study book for practicing engineers. Its objective is to present the fundamental results of modern nonlinear control while keeping the mathematical complexity to a minimum, and to demonstrate their use and implications in the design of practical nonlinear control systems. Although a major motivation of this book is to detail the many recent developments in nonlinear control, classical techniques such as phase plane analysis and the describing function method are also treated, because of their continued practical importance. In order to achieve our fundamental objective, we have tried to bring the following features to this book: • Readability: Particular attention is paid to the readability of the book by carefully organizing the concepts, intuitively interpreting the major results, and selectively using the mathematical tools. The readers are only assumed to have had one introductory control course. No mathematical background beyond ordinary differential equations and elementary matrix algebra is required. For each new result, interpretation is emphasized rather than mathematics. For each major result, we try to ask and answer the following key questions: What does the result intuitively and physically mean? How can it be applied to practical problems? What is its relationship to other theorems? All major concepts and results are demonstrated by examples. We believe that learning and generalization from examples are crucial for proficiency in applying any theoretical result. • Practicality: The choice and emphasis of materials is guided by the basic xiii
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