H. K. KHALIL-NonlinearSystems 3rd Ed.pdf

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Nonlinear SysteIlls Third Edition HASSAN K. KHALIL Department of Electrical and Computer Engineering Michigan State University "~ j Pl1~D!1Ji~e Hall ~" PRENTICE HALL Upper Saddle River, New Jersey 07458

Library of Congress Cataloging-in-Publication Data elP data on file. Vice President and Editorial Director, ECS: Marcia Horton Associate Editor: Alice Dworkin Vice President and Director of Production and Manufacturing, ESM: David W. Riccardi Executive Managing Editor: Vince O'Brien Managing Editor: David A. George Production Editor: Tamar Savir Composition: PreTEX, Inc. Director of Creative Services: Paul Belfanti Creative Director: Carole Anson Art Director: Jayne Conte Art Editor: Greg Dulles Cover Designer: Bruce K enselaar Manufacturing Manager: Trudy Pisciotti Manufacturing Buyer: Lisa McDowell Marketing Manager: Holly Stark © 2002, 1996 by Prentice Hall Prentice-Hall, Inc. Upper Saddle River, NJ 07458 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. The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Printed in the United States of America 10 9 8 7 6 5 4 ISBN 0-13-067389-7 Pearson Education Ltd., London Pearson Education Australia Pty. Ltd., Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., TOTOnto Pearson Educaclon de Mexico, S.A. de C.V. Pearson Education-Japan, Tokyo Pearson Education Malaysia, Pte. Ltd. Pearson Education, Upper Saddle River, New Jersey

To my mentor, Petar V. K okotovic A min a, Mohammad, Omar, Yousuf, and Suzanne and my family

Contents Preface 1 Introduction 1.1 Nonlinear Models and Nonlinear Phenomena 1.2 Examples . . . . . . . . . . . 1.2.1 Pendulum Equation . 1.2.2 Tunnel-Diode Circuit 1.2.3 Mass-Spring System 1.2.4 Negative-Resistance Oscillator 1.2.5 Artificial Neural Network 1.2.6 Adaptive Control ... 1.2.7 Common Nonlinearities 1.3 Exercises . . . . . 2 Second-Order Systems 2.1 Qualitative Behavior of Linear Systems 2.2 Multiple Equilibria . . . . . . . . . . . 2.3 Qualitative Behavior Near Equilibrium Points 2.4 Limit Cycles . . . . . . . . . . . . . . . . 2.5 Numerical Construction of Phase Portraits 2.6 Existence of Periodic Orbits. 2.7 Bifurcation 2.8 Exercises.... .. 3 Fundamental Properties 3.1 Existence and Uniqueness . . . . . . . . . . . 3.2 Continuous Dependence on Initial Conditions and Parameters . . . . . . . . . . . . . . . 3.3 Differentiability of Solutions and Sensitivity Equations ., .... 3.4 Comparison Principle 3.5 Exercises....... vii xiii 1 1 5 5 6 8 11 14 16 18 24 35 37 46 51 54 59 61 69 76 87 88 95 99 102 105

viii 4 Lyapunov Stability 4.1 Autonomous Systems 4.2 The Invariance Principle . . . . . 4.3 Linear Systems and Linearization 4.4 Comparison Functions . . . . . . 4.5 Nonautonomous Systems . . . . 4.6 Linear Time-Varying Systems and Linearization 4.7 Converse Theorems . . . . . . . . . . . 4.8 Boundedness and Ultimate Boundedness 4.9 4.10 Exercises . . . . . . Input-to-State Stability 5 .c Stability . . . . . Input-Output Stability 5.1 5.2 £ Stability of State Models 5.3 £2 Gain . . . . . . . . . . 5.4 Feedback Systems: The Small-Gain Theorem 5.5 Exercises 6 Passivity 6.1 Memoryless Functions . . . . . . 6.2 State Models. . . . . . . . . . . 6.3 Positive Real Transfer Functions 6.4 £2 and Lyapunov Stability . , . 6.5 Feedback Systems: Passivity Theorems. 6.6 Exercises................. 7 Frequency Domain Analysis of Feedback Systems 7.1 Absolute Stability . . . 7.1.1 Circle Criterion . . . . . 7.1.2 Popov Criterion . . . . . 7.2 The Describing Function Method 7.3 Exercises . . . . . . . . . 8 Advanced Stability Analysis 8.1 The Center Manifold Theorem 8.2 Region of Attraction . . . . . . 8.3 Invariance-like Theorems .. , 8.4 Stability of Periodic Solutions. 8.5 Exercises............ CONTENTS 111 112 126 133 144 147 156 162 168 174 181 195 195 201 209 217 222 227 228 233 237 241 245 259 263 264 265 275 280 296 303 303 312 322 329 334

CONTENTS 9 Stability of Perturbed Systems 9.1 Vanishing Perturbation ., 9.2 Nonvanishing Perturbation 9.3 Comparison Method . . . . 9.4 Continuity of Solutions on the Infinite Interval 9.5 Interconnected Systems 9.6 Slowly Varying Systems 9.7 Exercises . . . . . . . . 10 Perturbation Theory and Averaging 10.1 The Perturbation Method . . . . . . . . . . . . 10.2 Perturbation on the Infinite Interval . . . . . . 10.3 Periodic Perturbation of Autonomous Systems. 10.4 Averaging . . . . . . . . . . . . . . . . . . 10.5 Weakly Nonlinear Second-Order Oscillators 10.6 General Averaging 10.7 Exercises . . . . . 11 Singular Perturbations 11.1 The Standard Singular Perturbation Model. 11.2 Time-Scale Properties of the Standard Model 11.3 Singular Perturbation on the Infinite Interval . 11.4 Slow and Fast Manifolds 11.5 Stability Analysis 11.6 Exercises ... 12 Feedback Control 12.1 Control Problems 12.2 Stabilization via Linearization 12.3 Integral Control . . . . . . . 12.4 Integral Control via Linearization 12.5 Gain Scheduling 12.6 Exercises . . . . . . 13 Feedback Linearization 13.1 Motivation . . . . . 13.2 Input-Output Linearization 13.3 Full-State Linearization 13.4 State Feedback Control 13.4.1 Stabilization 13.4.2 Tracking. 13.5 Exercises . . . . . . . ix 339 340 346 350 355 358 365 372 381 382 393 397 402 411 413 419 423 424 430 439 443 449 460 469 469 475 478 481 485 499 505 505 509 521 530 530 540 544

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