Mohammed Chadli & Pierre Borne 
Multiple Models Approach in Automation [PDF ebook] 
Takagi-Sugeno Fuzzy Systems

Much work on analysis and synthesis problems relating to the
multiple model approach has already been undertaken. This has been
motivated by the desire to establish the problems of control law
synthesis and full state estimation in numerical terms.

In recent years, a general approach based on multiple LTI models
(linear or affine) around various function points has been
proposed. This so-called multiple model approach is a convex
polytopic representation, which can be obtained either directly
from a nonlinear mathematical model, through mathematical
transformation or through linearization around various function
points.

This book concentrates on the analysis of the stability and
synthesis of control laws and observations for multiple models. The
authors’ approach is essentially based on Lyapunov’s
second method and LMI formulation. Uncertain multiple models with
unknown inputs are studied and quadratic and non-quadratic Lyapunov
functions are also considered.

Table of Content

Notations ix

Introduction xiii

Chapter 1. Multiple Model Representation 1

1.1. Introduction 1

1.2. Techniques for obtaining multiple models 2

1.2.1. Construction of multiple models byidentification 3

1.2.2. Multiple model construction by linearization 8

1.2.3. Multiple model construction by mathematicaltransformation 14

1.2.4. Multiple model representation using the neuralapproach 22

1.3. Analysis and synthesis tools 29

1.3.1. Lyapunov approach 29

1.3.2. Numeric tools: linear matrix inequalities 31

1.3.3. Multiple model control techniques 38

Chapter 2. Stability of Continuous Multiple Models 41

2.1. Introduction 41

2.2. Stability analysis 42

2.2.1. Exponential stability 48

2.3. Relaxed stability 49

2.4. Example 52

2.5. Robust stability 54

2.5.1. Norm-bounded uncertainties 56

2.5.2. Structured parametric uncertainties 57

2.5.3. Analysis of nominal stability 60

2.5.4. Analysis of robust stability 62

2.6. Conclusion 63

Chapter 3. Multiple Model State Estimation 65

3.1. Introduction 65

3.2. Synthesis of multiple observers 67

3.2.1. Linearization 68

3.2.2. Pole placement 70

3.2.3. Application: asynchronous machine 72

3.2.4. Synthesis of multiple observers 75

3.3. Multiple observer for an uncertain multiplemodel 77

3.4. Synthesis of unknown input observers 82

3.4.1. Unknown inputs affecting system state 83

3.4.2. Unknown inputs affecting system state andoutput 87

3.4.3. Estimation of unknown inputs 88

3.5. Synthesis of unknown input observers: anotherapproach 93

3.5.1. Principle 93

3.5.2. Multiple observers subject to unknown inputs anduncertainties 96

3.6. Conclusion 97

Chapter 4. Stabilization of Multiple Models 99

4.1. Introduction 99

4.2. Full state feedback control 99

4.2.1. Linearization 101

4.2.2. Specific case 103

4.2.3. Stability: decay rate 106

4.3. Observer-based controller 113

4.3.1. Unmeasurable decision variables 115

4.4. Static output feedback control 119

4.4.1. Pole placement 122

4.5. Conclusion 126

Chapter 5. Robust Stabilization of Multiple Models 127

5.1. Introduction 127

5.2. State feedback control 129

5.2.1. Norm-bounded uncertainties 129

5.2.2. Interval uncertainties 131

5.3. Output feedback control 137

5.3.1. Norm-bounded uncertainties 137

5.3.2. Interval uncertainties 147

5.4. Observer-based control 150

5.5. Conclusion 156

Conclusion 157

APPENDICES 159

Appendix 1: LMI Regions 161

A1.1. Definition of an LMI region 161

A1.2. Interesting LMI region examples 162

A1.2.1. Open left half-plane 163

A1.2.2. Stability 163

A1.2.3. Vertical band 163

A1.2.4. Horizontal band 164

A1.2.5. Disk of radius R, centered at (q, 0) 164

A1.2.6. Conical sector 165

Appendix 2: Properties of M-Matrices 167

Appendix 3: Stability and Comparison Systems 169

A3.1. Vector norms and overvaluing systems 169

A3.1.1. Definition of a vector norm 169

A3.1.2. Definition of a system overvalued from a continuousprocess 170

A3.1.3. Application 172

A3.2. Vector norms and the principle of comparison 173

A3.3. Application to stability analysis 174

Bibliography 175

Index 185

About the author

Mohammed Chadli is Associate Professor at HDR (Habilitation), University de Picardie Jules Verne, UPJV-Amiens, France.

Professor Pierre Borne, Ecole Centrale de Lille, France. He is President of the IEEE-SMC society and is presently President of the IEEE France Section.