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OPTIMAL CONTROL Third Edition FRANK L. LEWIS Department of Electrical Engineering Automation & Robotics Research Institute University of Texas at Arlington Arlington, Texas DRAGUNA L. VRABIE United Technologies Research Center East Hartford, Connecticut VASSILIS L. SYRMOS Department of Electrical Engineering University of Hawaii at Manoa Honolulu, Hawaii JOHN WILEY & SONS, INC.

This book is printed on acid-free paper. Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Cataloging-in-Publication Data: Lewis, Frank L. Optimal control / Frank L. Lewis, Draguna L. Vrabie, Vassilis L. Syrmos.—3rd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-470-63349-6 (cloth); ISBN 978-1-118-12263-1 (ebk); ISBN 978-1-118-12264-8 (ebk); ISBN 978-1-118-12266-2 (ebk); ISBN 978-1-118-12270-9 (ebk); ISBN 978-1-118-12271-6 (ebk); ISBN 978-1-118-12272-3 (ebk) 1. Control theory. 2. Mathematical optimization. I. Vrabie, Draguna L. II. Syrmos, Vassilis L. III. Title. QA402.3.L487 2012 629.8’312–dc23 2011028234 Printed in the United States of America 10 9 8 4 3 7 6 5 2 1

To Galina, Roma, and Chris, who make every day exciting —Frank Lewis To my mother and my grandmother, for teaching me my potential and supporting my every choice —Draguna Vrabie To my father, my ﬁrst teacher —Vassilis Syrmos

CONTENTS PREFACE 1 STATIC OPTIMIZATION 1.1 1.2 1.3 Optimization without Constraints / 1 Optimization with Equality Constraints / 4 Numerical Solution Methods / 15 Problems / 15 xi 1 2 OPTIMAL CONTROL OF DISCRETE-TIME SYSTEMS 19 2.1 2.2 2.3 2.4 2.5 Solution of the General Discrete-Time Optimization Problem / 19 Discrete-Time Linear Quadratic Regulator / 32 Digital Control of Continuous-Time Systems / 53 Steady-State Closed-Loop Control and Suboptimal Feedback / 65 Frequency-Domain Results / 96 Problems / 102 3 OPTIMAL CONTROL OF CONTINUOUS-TIME SYSTEMS 3.1 3.2 3.3 The Calculus of Variations / 110 Solution of the General Continuous-Time Optimization Problem / 112 Continuous-Time Linear Quadratic Regulator / 135 110 vii

viii CONTENTS 3.4 3.5 Steady-State Closed-Loop Control and Suboptimal Feedback / 154 Frequency-Domain Results / 164 Problems / 167 4 THE TRACKING PROBLEM AND OTHER LQR EXTENSIONS 177 4.1 4.2 4.3 4.4 4.5 4.6 The Tracking Problem / 177 Regulator with Function of Final State Fixed / 183 Second-Order Variations in the Performance Index / 185 The Discrete-Time Tracking Problem / 190 Discrete Regulator with Function of Final State Fixed / 199 Discrete Second-Order Variations in the Performance Index / 206 Problems / 211 5 FINAL-TIME-FREE AND CONSTRAINED INPUT CONTROL 5.1 5.2 Final-Time-Free Problems / 213 Constrained Input Problems / 232 Problems / 257 6 DYNAMIC PROGRAMMING 6.1 6.2 6.3 Bellman’s Principle of Optimality / 260 Discrete-Time Systems / 263 Continuous-Time Systems / 271 Problems / 283 213 260 7 OPTIMAL CONTROL FOR POLYNOMIAL SYSTEMS 287 7.1 7.2 Discrete Linear Quadratic Regulator / 287 Digital Control of Continuous-Time Systems / 292 Problems / 295 8 OUTPUT FEEDBACK AND STRUCTURED CONTROL 297 8.1 8.2 8.3 8.4 8.5 8.6 Linear Quadratic Regulator with Output Feedback / 297 Tracking a Reference Input / 313 Tracking by Regulator Redesign / 327 Command-Generator Tracker / 331 Explicit Model-Following Design / 338 Output Feedback in Game Theory and Decentralized Control / 343 Problems / 351

9 ROBUSTNESS AND MULTIVARIABLE FREQUENCY-DOMAIN TECHNIQUES Introduction / 355 9.1 9.2 Multivariable Frequency-Domain Analysis / 357 9.3 9.4 9.5 9.6 H∞ DESIGN / 430 Robust Output-Feedback Design / 380 Observers and the Kalman Filter / 383 LQG/Loop-Transfer Recovery / 408 Problems / 435 10 DIFFERENTIAL GAMES CONTENTS ix 355 438 10.1 Optimal Control Derived Using Pontryagin’s Minimum Principle and the Bellman Equation / 439 10.2 Two-player Zero-sum Games / 444 10.3 Application of Zero-sum Games to H∞ Control / 450 10.4 Multiplayer Non-zero-sum Games / 453 11 REINFORCEMENT LEARNING AND OPTIMAL ADAPTIVE CONTROL 461 11.1 Reinforcement Learning / 462 11.2 Markov Decision Processes / 464 11.3 Policy Evaluation and Policy Improvement / 474 11.4 Temporal Difference Learning and Optimal Adaptive Control / 489 11.5 Optimal Adaptive Control for Discrete-time Systems / 490 11.6 Integral Reinforcement Learning for Optimal Adaptive Control of Continuous-time Systems / 503 11.7 Synchronous Optimal Adaptive Control for Continuous-time Systems / 513 APPENDIX A REVIEW OF MATRIX ALGEBRA 518 Partitioned Matrices / 519 A.1 Basic Deﬁnitions and Facts / 518 A.2 A.3 Quadratic Forms and Deﬁniteness / 521 A.4 Matrix Calculus / 523 A.5 The Generalized Eigenvalue Problem / 525 REFERENCES INDEX 527 535

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