Systems with Hysteresis Systems with Hysteresis Analysis, Identification and Control using the BoucWen Model Fayỗal Ikhouane Department of Applied Mathematics III School of Technical Industrial Engineering Technical University of Catalunya Barcelona, Spain José Rodellar Department of Applied Mathematics III School of Civil Engineering Technical University of Catalunya Barcelona, Spain Copyright © 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone +44 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved 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 under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 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(Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Anniversary Logo Design: Richard J Pacifico Library of Congress Cataloging in Publication Data Ikhouane, Fayỗal Systems with hysteresis : analysis, identification and control using the Bouc-Wen model / Fayỗal Ikhouane, Josộ Rodellar p cm Includes bibliographical references and index ISBN 978-0-470-03236-7 (cloth) Hysteresis—Mathematical models I Rodellar, José II Title QC754.2.H9I34 2007 621—dc22 2007019894 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13 978-0-470-03236-7 Typeset in 11/13pt Sabon by Integra Software Services Pvt Ltd, Pondicherry, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production To my mother and brothers, Imad and Hicham To Anna, Laura and Silvia J Rodellar F Ikhouane Contents Preface xi List of Figures xv List of Tables xix Introduction 1.1 Objective and Contents of the Book 1.2 The Bouc–Wen Model: Origin and Literature Review Physical Consistency of the Bouc–Wen Model 2.1 Introduction 2.2 BIBO Stability of the Bouc–Wen Model 2.2.1 The Model 2.2.2 Problem Statement 2.2.3 Classification of the BIBO-Stable Bouc–Wen Models 2.2.4 Practical Remarks 2.3 Free Motion of a Hysteretic Structural System 2.3.1 Problem Statement 2.3.2 Asymptotic Trajectories 2.3.3 Practical Remarks 2.4 Passivity of the Bouc–Wen model 2.5 Limit Cases 2.5.1 The Limit Case n = 2.5.2 The Limit Case = 2.5.3 The Limit Case = 2.5.4 The Limit Case + = 2.6 Conclusion 13 13 16 16 16 17 23 24 24 25 31 32 33 33 34 34 34 34 viii CONTENTS Forced Limit Cycle Characterization of the Bouc–Wen Model 37 3.1 Introduction 37 3.2 Problem Statement 38 3.2.1 The Class of Inputs 38 3.2.2 Problem Statement 39 3.3 The Normalized Bouc–Wen Model 39 3.4 Instrumental Functions 42 3.5 Characterization of the Asymptotic Behaviour of the Hysteretic Output 46 3.5.1 Technical Lemmas 49 3.5.2 Analytic Description of the Forced Limit Cycles for the Bouc–Wen Model 56 3.6 Simulation Example 59 3.7 Conclusion 61 Variation of the Hysteresis Loop with the Bouc–Wen Model Parameters 4.1 Introduction 4.2 Background Results and Methodology of the Analysis 4.2.1 Background Results 4.2.2 Methodology of the Analysis 4.3 Maximal Value of the Hysteretic Output 4.3.1 Variation with Respect to 4.3.2 Variation with Respect to 4.3.3 Variation with Respect to n 4.3.4 Summary of the Obtained Results 4.4 Variation of the Zero of the Hysteretic Output 4.4.1 Variation with Respect to 4.4.2 Variation with Respect to 4.4.3 Variation with Respect to n 4.4.4 Summary of the Obtained Results 4.5 Variation of the Hysteretic Output with the Bouc–Wen Model Parameters 4.5.1 Variation with Respect to 4.5.2 Variation with Respect to 4.5.3 Variation with Respect to n 4.5.4 Summary of the Obtained Results 4.6 The Four Regions of the Bouc–Wen Model 4.6.1 The Linear Region Rl 63 63 64 64 67 69 69 71 74 77 79 79 81 82 84 85 87 89 94 94 96 97 CONTENTS 4.6.2 The Plastic Region Rp 4.6.3 The Transition Regions Rt and Rs 4.7 Interpretation of the Normalized Bouc–Wen Model Parameters 4.7.1 The Parameters and 4.7.2 The Parameter 4.7.3 The Parameter n 4.8 Conclusion ix 105 107 107 107 109 110 110 Robust Identification of the Bouc–Wen Model Parameters 5.1 Introduction 5.2 Parameter Identification of the Bouc–Wen Model 5.2.1 Class of Inputs 5.2.2 Identification Methodology 5.2.3 Robustness of the Identification Method 5.2.4 Numerical Simulation Example 5.3 Modelling and Identification of a Magnetorheological Damper 5.3.1 Some Insights into the Viscous + Bouc–Wen Model for Shear Mode MR Dampers 5.3.2 Alternatives to the Viscous + Bouc–Wen Model for Shear Mode MR Dampers 5.3.3 Identification Methodology for the Viscous + Dahl Model 5.3.4 Numerical Simulations 5.4 Conclusion 113 113 Control of a System with a Bouc–Wen Hysteresis 6.1 Introduction and Problem Statement 6.2 Control Design and Stability Analysis 6.3 Numerical Simulation 6.4 Conclusion 165 165 167 175 177 Appendix Mathematical Background A.1 Existence and Uniqueness of Solutions A.2 Concepts of Stability A.3 Passivity and Absolute Stability A.3.1 Passivity in Mechanical Systems A.3.2 Positive Realness 179 179 181 182 182 184 115 115 116 119 137 142 142 147 154 156 164 ... x¯ with the parameter with the values = and n = 81 Figure 4.8 Variation of x¯ with the parameter with the values = and n = 83 Figure 4.9 Variation of x¯ with the parameter n with the. .. some physical properties of the model, and the parameteric identification and control of systems that include the Bouc–Wen model The book has been written to compile the results of this research... of the Book 1.2 The Bouc–Wen Model: Origin and Literature Review Physical Consistency of the Bouc–Wen Model 2.1 Introduction 2.2 BIBO Stability of the Bouc–Wen Model 2.2.1 The Model 2.2.2 Problem