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OPTIMAL CONTROL SYSTEMS ~,.~t~.~ -: .u&:iuut ~ ".. ....aIMlJ f~t jS ~ J '1 6 .3 z. : ~ OJ"':' lrAr If I 1 a : ~ i!J1i

Electrical Engineering Textbook Series Richard C. Dorf, Series Editor University of California, Davis Forthcoming and Published Titles Applied Vector Analysis Matiur Rahman and Isaac Mulolani Continuous Signals and Systems with MATLAB Taan EIAli and Mohammad A. Karim Discrete Signals and Systems with MATLAB Taan EIAIi Electromagnetics Edward J. Rothwell and Michael J. Cloud Optimal Control Systems Desineni Subbaram Naidu

OPTIMAL CONTROL SYSTEMS Desineni Subbaram Naidu Idaho State Universitv . Pocatello. Idaho. USA o CRC PRESS Boca Raton London New York Washington, D.C.

Cover photo: Terminal phase (using fuel-optimal control) of the lunar landing of the Apollo 11 mission. Courtesy of NASA. TJ "l13 N1. b'~ <'l ~ot Library of Congress Cataloging-in-Publication Data Naidu, D. s. (Desineni S.), 1940- Optimal control systems I by Desineni Subbaram N aidu. p. cm.- (Electrical engineering textbook series) Includes bibliographical references and index. ISBN 0-8493-0892-5 (alk. paper) 1. Automatic control. 2. Control theory. 3. Mathematical optimization. I. Title II. Series. 2002067415 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2003 by CRC Press LLC No claim to original u.S. Government works International Standard Book Number 0-8493-0892-5 Library of Congress Card Number 2002067415 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

v "Because the shape of the whole universe is most per fect and, in fact, designed by the wisest Creator, nothing in all of the world will occur in which no maximum or minimum rule is somehow shining forth. " Leohard Euler, 1144

vi Dedication My deceased parents who shaped my life Desineni Rama Naidu Desineni Subbamma and My teacher who shaped my education Buggapati A udi Chetty

Preface vii Many systems, physical, chemical, and economical, can be modeled by mathematical relations, such as deterministic and/or stochastic dif ferential and/or difference equations. These systems then change with time or any other independent variable according to the dynamical re lations. It is possible to steer these systems from one state to another state by the application of some type of external inputs or controls. If this can be done at all, there may be different ways of doing the same task. If there are different ways of doing the same task, then there may be one way of doing it in the "best" way. This best way can be minimum time to go from one state to another state, or maximum thrust developed by a rocket engine. The input given to the system corresponding to this best situation is called "optimal" control. The measure of "best" way or performance is called "performance index" or "cost function." Thus, we have an "optimal control system," when a system is controlled in an optimum way satisfying a given performance index. The theory of optimal control systems has enjoyed a flourishing period for nearly two decades after the dawn of the so-called "modern" control theory around the 1960s. The interest in theoretical and prac tical aspects of the subject has sustained due to its applications to such diverse fields as electrical power, aerospace, chemical plants, economics, medicine, biology, and ecology. Aim and Scope In this book we are concerned with essentially the control of physical systems which are "dynamic" and hence described by ordinary differ ential or difference equations in contrast to "static" systems, which are characterized by algebraic equations. Further, our focus is on "deter ministic" systems only. The development of optimal control theory in the sixties revolved around the "maximum principle" proposed by the Soviet mathemati cian L. S. Pontryagin and his colleagues whose work was published in English in 1962. Further contributions are due to R. E. Kalman of the United States. Since then, many excellent books on optimal control theory of varying levels of sophistication have been published. This book is written keeping the "student in mind" and intended to provide the student a simplified treatment of the subject, with an

viii appropriate dose of mathematics. Another feature of this book is to assemble all the topics which can be covered in a one-semester class. A special feature of this book is the presentation of the procedures in the form of a summary table designed in terms of statement of the prob lem and a step-by-step solution of the problem. Further, MATLAB© and SIMULINK© 1 , including Control System and Symbolic Math Toolboxes, have been incorporated into the book. The book is ideally suited for a one-semester, second level, graduate course in control sys tems and optimization. Background and Audience This is a second level graduate text book and as such the background material required for using this book is a first course on control sys tems, state space analysis, or linear systems theory. It is suggested that the student review the material in Appendices A and B given at the end of the book. This book is aimed at graduate students in Electrical, Mechanical, Chemical, and Aerospace Engineering and Applied Math ematics. It can also be used by professional scientists and engineers working in a variety of industries and research organizations. Acknowledgments This book has grown out of my lecture notes prepared over many years of teaching at the Indian Institute of Technology (IIT), Kharagpur, and Idaho State University (ISU), Pocatello, Idaho. As such, I am indebted to many of my teachers and students. In recent years at ISU, there are many people whom I would like to thank for their encouragement and cooperation. First of all, I would like to thank the late Dean Hary Charyulu for his encouragement to graduate work and research which kept me "live" in the area optimal control. Also, I would like to mention a special person, Kevin Moore, whose encouragement and cooperation made my stay at ISU a very pleasant and scholarly productive one for many years during 1990-98. During the last few years, Dean Kunze and Associate Dean Stuffie have been of great help in providing the right atmosphere for teaching and research work. IMATLAB and SIMULINK are registered trademarks of The Mathworks, Inc., Natick, MA, USA.

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