Multiple models approach in automation : 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.

Author Notes

Mohammed Chadli has been Assoc. Professor at the University of Picardy Jules Verne (UPJV) in Amiens, France since 2004 as well as being a researching in the "Modlisation, Information Systmes" (MIS) Laboratory. His research interest include, on the theoretical side, analysis and control of singular (switched) systems, analysis and control of fuzzy/LPV polytopic models, the multiple model approach, robust control, fault detection and isolation (FDI), fault tolerant control (FTC), analysis and control via LMI optimization techniques and Lyapunov methods. On the application side he is mainly interested in automotive control.
Pierre Borne is Professor at Ecole Centrale de Lille in France. He has received honorary degrees from the University of Moscow, Russia, the Politehnica University of Bucharest, Romania and the University of Waterloo, Canada. He is a Fellow of the IEEE, and currently President of the IEEE France section.

Notations	p. ix
Introduction	p. xiii
Chapter 1 Multiple Model Representation	p. 1
1.1 Introduction	p. 1
1.2 Techniques for obtaining multiple models	p. 2
1.2.1 Construction of multiple models by identification	p. 3
1.2.2 Multiple model construction by linearization	p. 8
1.2.3 Multiple model construction by mathematical transformation	p. 14
1.2.4 Multiple model representation using the neural approach	p. 22
1.3 Analysis and synthesis tools	p. 29
1.3.1 Lyapunov approach	p. 29
1.3.2 Numeric tools: linear matrix inequalities	p. 31
1.3.3 Multiple model control techniques	p. 38
Chapter 2 Stability of Continuous Multiple Models	p. 41
2.1 Introduction	p. 41
2.2 Stability analysis	p. 42
2.2.1 Exponential stability	p. 48
2.3 Relaxed stability	p. 49
2.4 Example	p. 52
2.5 Robust stability	p. 54
2.5.1 Norm-bounded uncertainties	p. 56
2.5.2 Structured parametric uncertainties	p. 57
2.5.3 Analysis of nominal stability	p. 60
2.5.4 Analysis of robust stability	p. 62
2.6 Conclusion	p. 63
Chapter 3 Multiple Model State Estimation	p. 65
3.1 Introduction	p. 65
3.2 Synthesis of multiple observers	p. 67
3.2.1 Linearization	p. 68
3.2.2 Pole placement	p. 70
3.2.3 Application: asynchronous machine	p. 72
3.2.4 Synthesis of multiple observers	p. 75
3.3 Multiple observer for an uncertain multiple model	p. 77
3.4 Synthesis of unknown input observers	p. 82
3.4.1 Unknown inputs affecting system state	p. 83
3.4.2 Unknown inputs affecting system state and output	p. 87
3.4.3 Estimation of unknown inputs	p. 88
3.5 Synthesis of unknown input observers: another approach	p. 93
3.5.1 Principle	p. 93
3.5.2 Multiple observers subject to unknown inputs and uncertainties	p. 96
3.6 Conclusion	p. 97
Chapter 4 Stabilization of Multiple Models	p. 99
4.1 Introduction	p. 99
4.2 Full state feedback control	p. 99
4.2.1 Linearization	p. 101
4.2.2 Specific case	p. 103
4.2.3 ¿-stability: decay rate	p. 106
4.3 Observer-based controller	p. 113
4.3.1 Unmeasurable decision variables	p. 115
4.4 Static output feedback control	p. 119
4.4.1 Pole placement	p. 122
4.5 Conclusion	p. 126
Chapter 5 Robust Stabilization of Multiple Models	p. 127
5.1 Introduction	p. 127
5.2 State feedback control	p. 129
5.2.1 Norm-bounded uncertainties	p. 129
5.2.2 Interval uncertainties	p. 131
5.3 Output feedback control	p. 137
5.3.1 Norm-bounded uncertainties	p. 137
5.3.2 Interval uncertainties	p. 147
5.4 Observer-based control	p. 150
5.5 Conclusion	p. 156
Conclusion	p. 157
Appendices	p. 159
Appendix 1 LMI Regions	p. 161
A1.1 Definition of an LMI region	p. 161
A1.2 Interesting LMI region examples	p. 162
A1.2.1 Open left half-plane.	p. 163
A1.2.2 ¿-stability	p. 163
A1.2.3 Vertical band	p. 163
A1.2.4 Horizontal band	p. 164
A1.2.5 Disk of radius R, centered at (q,0)	p. 164
A1.2.6 Conical sector	p. 165
Appendix 2 Properties of M-Matrices	p. 167
Appendix 3 Stability and Comparison Systems	p. 169
A3.1. Vector norms and overvaluing systems p. 169
A3.1.1 Definition of a vector norm	p. 169
A3.1.2 Definition of a system overvalued from a continuous process	p. 170
A3.1.3 Application	p. 172
A3.2 Vector norms and the principle of comparison	p. 173
A3.3 Application to stability analysis	p. 174
Bibliography	p. 175
Index	p. 185

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