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.

Table of Contents:
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 by identification 3 1.2.2. Multiple model construction by linearization 8 1.2.3. Multiple model construction by mathematical transformation 14 1.2.4. Multiple model representation using the neural approach 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 multiple model 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 and output 87 3.4.3. Estimation of unknown inputs 88 3.5. Synthesis of unknown input observers: another approach 93 3.5.1. Principle 93 3.5.2. Multiple observers subject to unknown inputs and uncertainties 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 continuous process 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