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Real-Time Optimization by Extremum-Seeking Control

Real-Time Optimization by Extremum-Seeking Control KARTIK B. ARIYUR MIROSLAV KRSTIC´ A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2003 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, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, 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, e-mail: permreq@wiley.com. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation 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 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 on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Ariyur, Kartik B. Real-time optimization by extremum-seeking control / Kartic B. Ariyur, Miroslav Kristic´. p. cm. “A Wiley-Interscience publication.” Includes bibliographical references and index. ISBN 0-471-46859-2 (cloth) 1. Adaptive control systems. I. Kristic´, Miroslav. II.Title. TJ217.A665 2003 629.8⬘36—dc21 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 2003053460

Contents Preface I THEORY 1 SISO Scheme and Linear Analysis 1.1 Extremum Seeking for a Static Map 1.2 Single Parameter Extremum Seeking for Plants with Dynamics 1.2.1 Single Parameter LTV Stability Test 1.2.2 Single Parameter LTI Stability Test 1.2.3 Single Parameter Compensator Design 1.3 Single Parameter Example Notes and References 2 Multiparameter Extremum Seeking 2.1 Output Extremization in Multiparameter Extremum Seeking 2.1.1 Multivariable LTV Stability Test 2.1.2 Multivariable LTI Stability Test 2.2 Multiparameter Design 2.3 Multiparameter Simulation Study 2.3.1 Step Variations in B*(t) and f*(t) 2.3.2 General Variations in B*(t) and f*(t) Notes and References 3 Slope Seeking 3.1 Slope Seeking on a Static Map 3.2 General Single Parameter Slope Seeking 3.3 Compensator Design 3.4 Multiparameter Gradient Seeking Notes and References v ix 1 3 4 6 8 11 17 18 19 21 21 23 26 34 39 40 41 43 4 7 48 50 57 58 60

vi CONTENTS 4 Discrete Time Extremum Seeking 4.1 Discrete-Time Extremum Seeking Control 4.2 Closed-Loop System 4.3 Stability Analysis 4.4 Example Notes and References 5 Nonlinear Analysis 5.1 Extremum Seeking: Problem Statement 5.2 Extremum Seeking Scheme 5.3 Averaging Analysis 5.4 Singular Perturbation Analysis Notes and References 6 Limit Cycle Minimization 6.1 Scheme for Limit Cycle Minimization 6.2 Vander Pol Example 6.3 Analysis Notes and References II APPLICATIONS 7 Antilock Braking 7.1 Model of a Slipping Wheel 7.2 ABS via Extremum Seeking Notes and References 8 Bioreactors 8.1 Dynamic Model of a Continuous Stirred Tank Reactor 8.2 Optimization Objective 8.3 Bifurcation Analysis of the Open-Loop System 8.3.1 Monad Model 8.3.2 Haldane Model 8.4 Extremum Seeking via the Dilution Rate 8.4.1 Monod Model 8.4.2 Haldane Model 8.5 Feedback with Washout Filters for the Haldane Model 8.5.1 Control Design 8.5.2 Simulation Results Notes and References 61 61 62 65 68 69 71 71 72 74 77 79 81 81 82 84 89 91 93 93 95 96 99 100 102 102 102 104 106 107 108 109 110 111 112

CONTENTS Vll 9 Formation Flight 9.1 Wingman Dynamics in Close Formation Flight 9.1.1 Wake Model 9.1.2 Forces and Moments on the Wingman in the Wake 9.1.3 Wingman Equilibrium in the Wake 9.1.4 Wingman Dynamics in the Wake 119 122 122 124 126 127 129 133 9.3.1 Formulation as a Standard Extremum Seeking Problem 133 137 9.3.2 Extremum Seeking Design for Formation Flight 138 141 9.2 Formation-Hold Autopilot 9.3 Extremum Seeking Control of Formation Flight 9.4 Simulation Study Notes and References 10 Combustion Instabilities 10.1 Identification of Averaged Pressure Magnitude Dynamics 10.2 Controller Phase Tuning via Extremum-Seeking 10.3 Experiments with the Adaptive Algorithm 10.4 Instability Suppression during Engine Transient Notes and References 11 Compressor Instabilities: Part I 11.1 Model Derivation 11.2 Equilibria in the E-MG3 Model 11.3 Skewness 11.4 Open-Loop Bifurcation Diagrams 11.5 Pre-Control Analysis: Critical Slopes 11.6 Control Design 11.6.1 Enforcing a Supercritical Bifurcation 11.6.2 Linearization at a Stall Equilibrium 11.6.3 Stability at the Bifurcation Point 11.7 Pressure Peak Seeking for the Surge Model 11.8 Peak Seeking for the Full Moore-Greitzer Model 11.9 Simulations for the Full MG Model Notes and References 12 Compressor Instabilities: Part II 12.1 Extremum Seeking on the Caltech Rig 12.1.1 Actuation for Stall Stabilization 12.1.2 Filter Design 12.2 Experimental Results 12.2.1 Initial Point on the Axisymmetric Characteristic 12.2.2 Initial Point on the Nonaxisymmetric Characteristic 12.3 Near Optimal Compressor Operation via Slope Seeking Notes and References 143 144 146 149 152 154 157 160 163 164 165 167 169 170 172 176 176 178 181 184 187 187 189 190 191 191 191 194 196

viii Appendices A Continuous Time Lemmas B Discrete Time Lemmas C Aircraft Dynamics in Close Formation Flight C.l C-5 and Flight Condition Data C.2 Wake-Induced Velocity Field C.3 Free Flight Model Data C.4 Formation Flight Model: Influence Matrices C.5 Formation-Hold Autopilot Parameters D Derivation of Equations {11.8) and (11.10) E Derivation of the Critical Slopes F Proof of Lemma 11.1 Bibliography Index CONTENTS 199 201 203 207 207 207 208 210 210 213 217 219 223 235

Preface Extremum seeking, a popular tool in control applications in the 1940s-1960s, has seen a return as an exciting research topic and industrial real-time opti mization tool in the 1990's. Extremum seeking is also a method of adaptive control but it does not fit into the classical paradigm or model reference and related schemes, which deal with the problem of stabilization of a known ref erence trajectory or set point. A second distinction between classical adaptive control and extremum seek ing is that the latter is not model based. As such, it provides a rigorous, high performance alternative to control methods involving neural networks. Its non-model based character explains the resurgence in popularity of extremum seeking in the last half a decade: the recent applications in fluid flow, com bustion, and biomedical systems are all characterized by complex, unreliable models. Extremum seeking is applicable in situations where there is a nonlinear ity in the control problem, and the nonlinearity has a local minimum or a maximum. The nonlinearity may be in the plant, as a physical nonlinearity, possibly manifesting itself through an equilibrium map, or it may be in the control objective, added to the system through a cost functional of an opti mization problem. Hence, one can use extremum seeking both for tuning a set point to achieve an optimal value of the output, or for tuning parameters of a feedback law. The parameter space can be multivariable, a case we cover extensively in this book. This book overviews the efforts made over the last seven years to put extremum seeking on a rigorous analytical footing and to make improvement of performance in extremum seeking schemes systematic. Stability guidelines that have been developed are applicable not only to static maps but also to systems that combine static maps with dynamics in virtually any form, with the single restriction that the dynamics be open loop stable. 1The main accomplishment during the recent period, to which this book is dedicated, is achieving convergence to the optimum on a time scale comparable to the ix

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