Skip to main content

Systems with Hysteresis: Analysis, Identification and Control Using the Bouc-Wen Model

Systems with Hysteresis: Analysis, Identification and Control Using the Bouc-Wen Model

Fayçal Ikhouane, José Rodellar

ISBN: 978-0-470-51319-4

Sep 2007

222 pages



Hysterisis is a system property that is fundamental to a range of engineering applications as the components of systems with hysterisis are able to react differently to different forces applied to them. Control theory is used to model these complex systems and cause them to behave in the desired manner; the Bouc-Wen model is a well-known semi-physical model that is used extensively to describe the hysterisis of systems in the areas of smart structures and civil engineering.

The Bouc-Wen model for system hysterisis has increased in popularity due to its capability of capturing in an analytical form a range of shapes of hysteretic cycles that match the behaviour of a wide class of hysteretic systems. “Systems with Hysterisis: Analysis, Identification and Control using the Bouc-Wen Model” deals with the analysis, identification and control of these systems, and offers a comprehensive and self-contained framework for the study of the Bouc-Wen model.

  • Includes the latest techniques for modelling smart structures and materials
  • Provides a rigorous mathematical treatment of the subject along with practical comments, numerical solutions and a case study of magentorheological (MR) dampers.
  • Begins by analysing the compatibility of the Bouc-Wen model with the laws of physics, and continues to cover the relationship between the model parameters and hysterisis loop, identification of the model parameters and control of systems that include a hysteretic part described by the Bouc-Wen model.
  • Includes case studies covering the identification and control of smart material transducers for use in automotive, aerospace and structural control

Systems with Hysterisis: Analysis, Identification and Control using the Bouc-Wen Model offers an invaluable source of ideas, concepts and insights for engineers, researchers, lecturers and senior/ postgraduate students involved in the research, design and development of smart structures and related areas within civil and mechanical engineering. It will also be of interest to readers involved in the wider disciplines of electrical & control engineering, applied mathematics, applied physics and material science.


List of Figures.

List of Tables.                                                                                           

1. Introduction                                                                      

1.1 Objective and contents of the book

1.2 The Bouc-Wen model: origin and literature review

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 Classi¯cation 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

3 Forced limit cycle characterization of the Bouc-Wen model            

3.1 Introduction 

3.2 Problem statement

3.2.1 The class of inputs

3.2.2 Problem statement

3.3 The normalized Bouc-Wen model

3.4 Instrumental functions

3.5 Characterization of the asymptotic behavior of the hysteretic


3.5.1 Technical Lemmas

3.5.2 Analytic description of the forced limit cycles for the

Bouc-Wen model

3.6 Simulation example

3.7 Conclusion 

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 ¾

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 ¾

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


4.5.1 Variation with respect to ±

4.5.2 Variation with respect to ¾

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

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 ½ and ±

4.7.2 The parameter ¾

4.7.3 The parameter n

4.8 Conclusion

5 Robust identification of the Bouc-Wen model parameters          

5.1 Introduction

5.2 Parameter identi¯cation for the Bouc-Wen


5.2.1 Class of inputs

5.2.2 Identi¯cation methodology

5.2.3 Robustness of the identi¯cation method 

5.2.4 Numerical simulation example

5.3 Modeling and identi¯cation 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

5.4 Identi¯cation methodology for the viscous + Dahl model . .          

5.4.1 Numerical simulations 

5.5 Conclusion

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 

A 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

A.3.3 Sector functions

A.3.4 Absolute stability

A.4 Input-output properties