《Applied Nonlinear Control》 by Slotine.pdf

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Slotine • Li APPLIED NONLINEAR CONTROL ! i

APPLIED NONLINEAR CONTROL Jean-Jacques E Slotine Weiping Li

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

To Our Parents

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

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