systems with hysteresis. analysis_ identification and control using the bouc–wen model

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):
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 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of
the Publisher. Requests to the Publisher should be addressed to the Permissions Department,
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ,
England, or emailed to , or faxed to (+44) 1243 770620.
This publication is designed to provide accurate and authoritative information in regard to
the subject matter covered. It is sold on the understanding that the Publisher is not engaged
in rendering professional services. If professional advice or other expert assistance is
required, the services of a competent professional should be sought.
Other Wiley Editorial Offices
John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA
Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA
Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany
John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia
John Wiley & Sons (Asia) Pte Ltd, 2 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)
1. 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

1 Introduction
1
1.1 Objective and Contents of the Book
1
1.2 The Bouc–Wen Model: Origin and Literature Review 5
2 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 = 1
2.5.2 The Limit Case  = 1
2.5.3 The Limit Case  = 0
2.5.4 The Limit Case  +  = 0
2.6 Conclusion

13
13
16
16
16
17
23
24
24
25
31
32
33
33
34
34
34
34


viii

CONTENTS

3 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
4 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

5 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

6 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

CONTENTS

A.3.3 Sector Functions
A.3.4 Absolute Stability
A.4 Input–Output Properties

185
186
188

References

189

Index

199


Preface

This book deals with the analysis, the identification and the control of
a special class of systems with hysteresis. This nonlinear behaviour is

encountered in a wide variety of processes in which the input–output
dynamic relations between variables involve memory effects. Examples
are found in biology, optics, electronics, ferroelectricity, magnetism,
mechanics and structures, among other areas. In mechanical and structural systems, hysteresis appears as a natural mechanism of materials
to supply restoring forces against movements and dissipate energy.
In these systems, hysteresis refers to the memory nature of inelastic
behaviour where the restoring force depends not only on the instantaneous deformation but also on the history of the deformation.
The detailed modelling of these systems using the laws of physics is
an arduous task, and the obtained models are often too complex to be
used in practical applications involving characterization of systems,
identification or control. For this reason, alternative models of these
complex systems have been proposed. These models do not come, in
general, from the detailed analysis of the physical behaviour of the
systems with hysteresis. Instead, they combine some physical understanding of the hysteretic system along with some kind of black-box
modelling. For this reason, some authors have called these models
‘semi-physical’.
Within this context, a hysteretic semi-physical model was proposed
initially by Bouc early in 1971 and subsequently generalized by Wen
in 1976. Since then, it is known as the Bouc–Wen model and has
been extensively used in the current literature to describe mathematical components and devices with hysteretic behaviours, particularly
within the areas of civil and mechanical engineering. The model
essentially consists of a first-order nonlinear differential equation


xii

PREFACE

that relates the input displacement to the output restoring force in
a hysteretic way. By choosing a set of parameters appropriately, it

is possible to accommodate the response of the model to the real
hysteresis loops. This is why the main efforts reported in the literature have been devoted to the tuning of the parameters for specific
applications.
This book is the result of a research effort that was initiated by
the first author (Prof. Fayỗal Ikhouane) in 2002 when he joined the
research group on Control, Dynamics and Applications (CoDAlab)
in the Department of Applied Mathematics III at the Technical
University of Catalonia in Barcelona (Spain). During the last five
years, the authors have explored various issues related to this model
as an analysis of 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
effort in a comprehensive and self-contained organized body. Part of
these results have been published in scientific journals and presented
in international conferences within the last three years as well as in
lectures and seminars for graduate students. The contents cover four
topics:
1. Analysis of the compatibility of the model with some laws of
physics.
2. Relationship between the model parameters and the
hysteresis loop.
3. Identification of the model parameters.
4. Control of systems that include a Bouc–Wen hysteresis.
Although mathematical rigour has been the main pursued feature,
the authors have also tried to make the book attractive for, say,
end users of the model. Thus, the mathematical developments are
completed with practical remarks and illustrated with examples.
Their final goal is that the analytical studies and results give a solid
framework for a systematic and well-supported practical use of the

Bouc–Wen model. It is their hope that this has been achieved and that
the book might be of interest to researchers, engineers, professors
and students involved in the design and development of smart structures and materials, vibration control, mechatronics, smart actuators
and related issues in engineering areas such as civil, mechanical,
automotive, aerospace and aeronautics.


PREFACE

xiii

The research work leading to this book has been mainly sponsored
by the Ministry of Education and Science of Spain through research
project grants. Particularly significant is the grant ‘Ramón y Cajal’
awarded to Prof. F. Ikhouane for the last five years. Additional
support from the Research Agency of the Government of Catalonia
is appreciated. This work has also benefited from the participation
of the group CoDAlab in the program CONVIB (Innovative Control
Technologies for Vibration Sensitive Civil Engineering Structures)
sponsored by the European Science Foundation (ESF) during the
period 2001–2005.
The School of Industrial Technical Engineering of Barcelona
(EUETIB) and the School of Civil Engineering of Barcelona
(ETSECCPB) have provided a pleasant environment for our work.
There is also appreciation for the support of colleagues and graduate
students at the Technical University of Catalonia.
We are grateful to Prof. Shirley Dyke, Prof. Víctor Mosa and
Prof. Jorge Hurtado for their coauthoring of some of the papers that
have been used in this book, and also to anonymous reviewers for
their valuable comments.

Finally, we owe a special gratitude for the permanent support of
our respective families during the research work and the writing of
this book.
Fayỗal Ikhouane and Josộ Rodellar



List of Figures

Figure 1.1 Graph force versus displacement for a
hysteresis functional

5

Figure 1.2 Evolution of the Bouc–Wen model literature

8

Figure 2.1 Example of a Bouc–Wen model that is unstable

14

Figure 2.2 Example of a Bouc–Wen model that does not
dissipate energy

16

Figure 2.3 Base isolation device (a) and its physical model (b)

24

Figure 2.4 Equivalent description of system (2.30)–(2.32)

29

Figure 2.5 Limit cycles for a class III Bouc–Wen model

32

Figure 3.1 Illustration of the notation related to the input
signal

39

+
w
n

−
n
w

(dash-dot),
(dashed)
Figure 3.2 Functions
and n w (solid), with the values  = 2 and n = 2

45

+

−
 (dash-dot), n
 (dashed)
Figure 3.3 Functions n
and n  (solid), with the values  = 2 and n = 2

46

Figure 3.4 Upper left: input signal x for 0 ≤ ≤ T .
Upper right: dashed, the graph xt

BW t for t ∈ 0 5T
;
¯ BW  for
solid, the graph of the limit cycle x

0 ≤ ≤ T . Lower: dashed, the Bouc–Wen model output
¯ BW t both for t ∈ 0 5T


BW xt; solid, the limit function

60

Figure 4.1 Example of a wave T -periodic signal

65

Figure 4.2 Symmetry property of the hysteresis loop of
the Bouc–Wen model

67


xvi

LIST OF FIGURES

Figure 4.3 Methodologies of the analysis of the variation
of w
¯ x
¯

68

Figure 4.4 Variation of the maximal hysteretic output
n  with the parameter , for the values of  = 2 and n = 2

71

Figure 4.5 Variation of the maximal hysteretic output
n  with the parameter , for the values of  = 1
and n = 2 (semi-logarithmic scale). Observe that for
 = 05 the corresponding value is 052 1 = 04786 and
lim→ 2 1 = 07610 = 2+ 1

74

Figure 4.6 Variation of the maximal value n  with
the parameter n for three values of  and with  = 14. In

this case, we have  ∗  11

78



Figure 4.7 Variation of x¯ with the parameter  with the
values  = 2 and n = 2

81

Figure 4.8 Variation of x¯  with the parameter  with the
values  = 1 and n = 2

83



Figure 4.9 Variation of x¯ with the parameter n with the
values  = 1 and  = 2

84

Figure 4.10 Variation of w
¯ x
¯ with the normalized input
x,
¯ with the values  = 2  = 2 and n = 2

86

Figure 4.11 Variation of w
¯ x
¯ with  for x¯ = 05  = 1
and n = 2

88

Figure 4.12 Variation of w
¯ x
¯ with  for
x¯ = −05  = 100 and n = 2

89

Figure 4.13 Variation of w
¯ x
¯ with  for x¯ = 05  = 1
and n = 2. In this case, 2/1/2n  − 1 = −00429

92

Figure 4.14 Variation of w
¯ x
¯ with  for different values
of x,
¯ with  = 14 and n = 2. Upper curve, x¯ = −08;
middle, x¯ = −087; lower, x¯ = −095. In this case.
2/1/2n  − 1 = −00851 and x¯ ∗ = −087

93

¯ x
¯ for  = 5
Figure 4.15 The limit function limn→ w

96

¯ lt > n 
Figure 4.16 The linear region Rl in the case w

99

¯ sl < −n 
Figure 4.17 The linear region Rl in the case w

100

Figure 4.18 The linear region Rl in the case
w
¯ sl < −n  and w
¯ lt > n 

101

Figure 4.19 The linear region Rl in the case
−n  ≤ w
¯ sl < w
¯ lt ≤ n 

102


LIST OF FIGURES

xvii

Figure 4.20 Interpretation of the parameter 

108

Figure 5.1 Parametric identification algorithm scheme

114

Figure 5.2 Identification in a noisy environment

120

Figure 5.3 Upper left: solid, input signal xt; dashed,
input signal x1 t. Lower left: solid, output
BW xt;
¯ BW 
dashed, output
BW1 xt. Right: limit cycles x

¯
(solid) and x1 
BW1  (dashed) that have been obtained
for the time interval 4T 5T

138

Figure 5.4 Input–output representation of the MR damper

142

Figure 5.5 Mechanical model of the MR damper

143

Figure 5.6 Upper: input signal. Lower: response of
the Bouc–Wen model (5.98)–(5.99) to the two sets of
parameters (5.100) and (5.101)

144

Figure 5.7 Hysteresis loop corresponding to the part
vzt of the Bouc–Wen model (5.98)–(5.99) with the
set of parameters (5.100) to the input displacement
xt = sint

146

Figure 5.8 Response of the normalized Bouc–Wen
model (5.102)–(5.103) to a random input signal with a
frequency content within the interval [0,10HZ ]: solid
set of parameters (5.104); dotted, set of parameters
(5.106)

148

Figure 5.9 Coulomb model for dry friction

149

Figure 5.10 Viscous + Coulomb model for the shear
mode MR damper (also called the Bingham model [136])

150

Figure 5.11 Response to a random input signal with a
frequency content that covers the interval [0,10 Hz]: solid,
standard Bouc–Wen model (5.98)–(5.99) with the set of
parameters (5.100); dotted, viscous + Coulomb model
(5.107) with the set of parameters (5.110)

151

Figure 5.12 Viscous + Dahl model for the MR damper

153

Figure 5.13 Response to a random input signal with a
frequency content that covers the interval [0,10 Hz]: solid,
normalized Bouc–Wen model (5.102)–(5.103) with the set
of parameters (5.104) (or equivalently the standard
Bouc–Wen model (5.98)–(5.99) with the set of parameters
(5.100)); dotted, viscous + Dahl model (5.116)–(5.117)

with the set of parameters (5.118)

153


xviii

LIST OF FIGURES

Figure 5.14 Shear mode MR damper

157

Figure 5.15 Response of the MR damper model

159

Figure 5.16 Function x. The marker corresponds to the
point whose abscissa is x∗1

160

Figure 5.17 Response of the viscous + Dahl model to a
random input signal with a range of frequencies that
covers the interval [0,10 Hz]: solid,  = 600 cm−1 ; dashed,
 = 3204 cm−1 ; dotted  = 1281.7 cm−1

162

Figure 5.18 Force in N versus displacement in cm:

solid, viscous + Bouc–Wen; dotted, viscous + Dahl
with  = 3204 cm−1 ; dashed, viscous + Dahl with
 = 12817 cm−1 ; dotted-dashed, viscous + Dahl with
 = 64918 cm−1

163

Figure 5.19 Force in N versus velocity in cm/s:
solid, viscous + Bouc–Wen; dotted, viscous + Dahl
with  = 3204 cm−1 ; dashed, viscous + Dahl with
 = 12817 cm−1 ; dotted-dashed, viscous + Dahl with
 = 64918 cm−1

164

Figure 6.1 Equivalent description of Equation (6.18)

173

Figure 6.2 Closed-loop signals

176

Figure A.1 Equivalent description of system (A.10)

183

Figure A.2 Example of a function that belongs to the
sector K1  K2

185

Figure A.3 Equivalent representation of the system
(A.19)–(A.21)

186


List of Tables

Table 2.1 Classification of the BIBO-stable Bouc–Wen
models

19

Table 3.1 Classification of the BIBO, passive and
thermodynamically consistent normalized Bouc–Wen models

42

Table 4.1 Variation of the maximal hysteretic output
n  with the Bouc–Wen model parameters   n

78

Table 4.2 Variation of the hysteretic zero x¯  with the
Bouc–Wen model parameters   n

85

Table 4.3 Variation of the hysteretic output with the
Bouc–Wen model parameters   n

95

Table 5.1 Procedure for identification of the Bouc–Wen
model parameters

118

Table 5.2 Procedure for the identification of the viscous
+ Dahl model

156