Bert Van Keulen 1Coo-Control for Distributed Parameter

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Wonham, University of Toronto Bert van Keulen 1Coo-Control for Distributed Parameter Systems: A State-Space Approach Springer Science+Business Media, LLC Bett van Keulen Department of Mathematics University of Groningen The Netherlands Library of Congress Cataloging In-Publication Data Keulen, Bert van. H [infinity]-control for distributed parameter systems / astate space approach / Bert van Keulen. p. cm. -- (Systems & control) On t.p. "[infinity]" appears as the infinity symbol. Includes bibliographical references. ISBN 978-1-4612-6718-8 ISBN 978-1-4612-0347-6 (eBook) DOI 10.1007/978-1-4612-0347-6 1. H [infinity symbol] control. I. Distributed paramter systems. 3. State-space methods. I. Title. 11. Series. QA402.3.K42 1993 93-24351 003'.78--dc20 CIP Printed on acid-free paper © Springer Science+Business Media New York 1993 Originally published by Birkhäuser Boston in 1993 Softcover reprint of the hardcover 1st edition 1993 Copyright is not c1aimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrievaI system, or transmitted, in any form or by any means, electronic, mechanical, photocopy ing, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific c1ients is granted by Springer Science+Business Media, LLC for Iibraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC. ISBN 978-1-4612-6718-8 Typeset by the Author in LATEX. 987654321 Contents Preface 1 Introduction 1 1.1 'Heo-control ..... ... 1 1.2 Unbounded inputs and outputs 9 1.3 Organization of the book . 15 2 Pritchard-Salamon systems 17 2.1 Notation. ...... 17 2.2 Definitions. .... .. 19 2.3 Frequency domain results 28 2.4 Perturbation results . 31 2.5 Duality theory. 37 2.6 Stability theory . 48 2.7 Dynamic output-feedback 56 2.8 Riccati equations . 69 3 Linear quadratic control and frequency domain inequalities 75 3.1 Introduction....... ...... 75 3.2 Preliminary results . 76 3.3 Problem formulation and main result 88 3.4 Proof of the main result .. 92 4 'Heo-control with state-feedback 101 4.1 Problem formulation and main result 101 4.2 Proof of the state-feedback result . 106 4.3 Relaxation of the a priori assumptions 124 4.3.1 Feedthrough from disturbance to output 124 4.3.2 How to 'remove' the regularity assumptions 128 5 'Heo-control with measurement-feedback 130 5.1 Problem formulation and main result 131 5.2 Redheffer's Lemma . 139 Preface VI 5.3 Proof of the measurement-feedback result. 144 5.4 Relaxation of the a priori assumptions .. 165 5.4.1 Including the feedthroughs . 165 5.4.2 How to 'remove' the regularity assumptions 174 6 Examples and conclusions 177 6.1 Delay systems in state-space . 177 6.1.1 Dynamic controllers for delay systems. 180 6.1.2 A linear quadratic control problem . 184 6.1.3 Duality ........ .. ..... 189 6.2 The mixed-sensitivity problem for delay systems 192 6.2.1 Introduction and statement of the problem. 192 6.2.2 Main result .............. 194 6.3 Conclusions and directions for future research. 200 A Stability theory 205 A.1 205 A.2 206 B Differentiability and some convergence results 207 B.l 207 B.2 208 B.3 209 B.4 209 B.5 209 B.6 211 B.7 213 C The invariant zeros condition 214 C.1 214 D The relation between P, Q and P 221 D.1 .. .......... ... 221 Bibliography 230 Index 239 Preface Control of distributed parameter systems is a fascinating and challenging top ic, from both a mathematical and an applications point of view. The same can be said about Hoc-control theory, which has become very popular lately. I am therefore pleased to present in this book a complete treatment of the state-space solution to the Hoo-control problem for a large class of distributed parameter systems. The class of distributed parameter systems considered in this book allows for a certain degree of unboundedness of the input and output operators and is usually referred to as the Pritchard-Salamon class. In addition to deriving the state-space solution to the Hoc-control problem for this class of systems, it is'my aim to show the very nice system theoretic properties of the class. Some basic knowledge about infinite-dimensional systems theory will be assumed: the book is intended for graduate students and other researchers interested in infinite-dimensional systems theory and Hoc-control. Most of the research that lead to this book was performed when I was a PhD student in the Systems & Control group of the Department of Math ematics of the University of Groningen in The Netherlands. My supervisor was Ruth Curtain and, since this book would not have appeared without her advice and support, lowe her many thanks. It is my pleasure to acknowledge here the help of all the others who in some way or another contributed to this book. In particular, I would like to thank the members of the Systems & Control group in Groningen for the pleasant (working) atmosphere. I thank Professors Bensoussan, Kaashoek and Pritchard for their comments on the thesis that forms the core of this book, the Systems and Control Theory Network in The Netherlands for the high level courses and the extra funding, and the staff of Birkhauser Boston for excellent cooperation. Finally, and most importantly of all, I thank my family and my friends for their interest and their support over the last few years. Bert van [(eulen June 1993 .

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