Mohammed Chadli has been Assoc. Professor at the University of Picardy Jules Verne (UPJV) in Amiens, France since 2004 as well as being a researcher in the "Modélisation, Information & Systèmes" (MIS) Laboratory. His research interests 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 ixIntroduction xiiiChapter 1. Multiple Model Representation 11.1. Introduction 11.2. Techniques for obtaining multiple models 21.2.1. Construction of multiple models by identification 31.2.2. Multiple model construction by linearization 81.2.3. Multiple model construction by mathematical transformation 141.2.4. Multiple model representation using the neural approach 221.3. Analysis and synthesis tools 291.3.1. Lyapunov approach 291.3.2. Numeric tools: linear matrix inequalities 311.3.3. Multiple model control techniques 38Chapter 2. Stability of Continuous Multiple Models 412.1. Introduction 412.2. Stability analysis 422.2.1. Exponential stability 482.3. Relaxed stability 492.4. Example 522.5. Robust stability 542.5.1. Norm-bounded uncertainties 562.5.2. Structured parametric uncertainties 572.5.3. Analysis of nominal stability 602.5.4. Analysis of robust stability 622.6. Conclusion 63Chapter 3. Multiple Model State Estimation 653.1. Introduction 653.2. Synthesis of multiple observers 673.2.1. Linearization 683.2.2. Pole placement 703.2.3. Application: asynchronous machine 723.2.4. Synthesis of multiple observers 753.3. Multiple observer for an uncertain multiple model 773.4. Synthesis of unknown input observers 823.4.1. Unknown inputs affecting system state 833.4.2. Unknown inputs affecting system state and output 873.4.3. Estimation of unknown inputs 883.5. Synthesis of unknown input observers: another approach 933.5.1. Principle 933.5.2. Multiple observers subject to unknown inputs and uncertainties 963.6. Conclusion 97Chapter 4. Stabilization of Multiple Models 994.1. Introduction 994.2. Full state feedback control 994.2.1. Linearization 1014.2.2. Specific case 1034.2.3. Stability: decay rate 1064.3. Observer-based controller 1134.3.1. Unmeasurable decision variables 1154.4. Static output feedback control 1194.4.1. Pole placement 1224.5. Conclusion 126Chapter 5. Robust Stabilization of Multiple Models 1275.1. Introduction 1275.2. State feedback control 1295.2.1. Norm-bounded uncertainties 1295.2.2. Interval uncertainties 1315.3. Output feedback control 1375.3.1. Norm-bounded uncertainties 1375.3.2. Interval uncertainties 1475.4. Observer-based control 1505.5. Conclusion 156Conclusion 157APPENDICES 159Appendix 1: LMI Regions 161A1.1. Definition of an LMI region 161A1.2. Interesting LMI region examples 162A1.2.1. Open left half-plane 163A1.2.2. Stability 163A1.2.3. Vertical band 163A1.2.4. Horizontal band 164A1.2.5. Disk of radius R, centered at (q,0) 164A1.2.6. Conical sector 165Appendix 2: Properties of M-Matrices 167Appendix 3: Stability and Comparison Systems 169A3.1. Vector norms and overvaluing systems 169A3.1.1. Definition of a vector norm 169A3.1.2. Definition of a system overvalued from a continuous process 170A3.1.3. Application 172A3.2. Vector norms and the principle of comparison 173A3.3. Application to stability analysis 174Bibliography 175Index 185

Mohammed Chadli has been Assoc. Professor at the University of Picardy Jules Verne (UPJV) in Amiens, France since 2004 as well as being a researcher in the "Modélisation, Information & Systèmes" (MIS) Laboratory. His research interests 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 ixIntroduction xiiiChapter 1. Multiple Model Representation 11.1. Introduction 11.2. Techniques for obtaining multiple models 21.2.1. Construction of multiple models by identification 31.2.2. Multiple model construction by linearization 81.2.3. Multiple model construction by mathematical transformation 141.2.4. Multiple model representation using the neural approach 221.3. Analysis and synthesis tools 291.3.1. Lyapunov approach 291.3.2. Numeric tools: linear matrix inequalities 311.3.3. Multiple model control techniques 38Chapter 2. Stability of Continuous Multiple Models 412.1. Introduction 412.2. Stability analysis 422.2.1. Exponential stability 482.3. Relaxed stability 492.4. Example 522.5. Robust stability 542.5.1. Norm-bounded uncertainties 562.5.2. Structured parametric uncertainties 572.5.3. Analysis of nominal stability 602.5.4. Analysis of robust stability 622.6. Conclusion 63Chapter 3. Multiple Model State Estimation 653.1. Introduction 653.2. Synthesis of multiple observers 673.2.1. Linearization 683.2.2. Pole placement 703.2.3. Application: asynchronous machine 723.2.4. Synthesis of multiple observers 753.3. Multiple observer for an uncertain multiple model 773.4. Synthesis of unknown input observers 823.4.1. Unknown inputs affecting system state 833.4.2. Unknown inputs affecting system state and output 873.4.3. Estimation of unknown inputs 883.5. Synthesis of unknown input observers: another approach 933.5.1. Principle 933.5.2. Multiple observers subject to unknown inputs and uncertainties 963.6. Conclusion 97Chapter 4. Stabilization of Multiple Models 994.1. Introduction 994.2. Full state feedback control 994.2.1. Linearization 1014.2.2. Specific case 1034.2.3. Stability: decay rate 1064.3. Observer-based controller 1134.3.1. Unmeasurable decision variables 1154.4. Static output feedback control 1194.4.1. Pole placement 1224.5. Conclusion 126Chapter 5. Robust Stabilization of Multiple Models 1275.1. Introduction 1275.2. State feedback control 1295.2.1. Norm-bounded uncertainties 1295.2.2. Interval uncertainties 1315.3. Output feedback control 1375.3.1. Norm-bounded uncertainties 1375.3.2. Interval uncertainties 1475.4. Observer-based control 1505.5. Conclusion 156Conclusion 157APPENDICES 159Appendix 1: LMI Regions 161A1.1. Definition of an LMI region 161A1.2. Interesting LMI region examples 162A1.2.1. Open left half-plane 163A1.2.2. Stability 163A1.2.3. Vertical band 163A1.2.4. Horizontal band 164A1.2.5. Disk of radius R, centered at (q,0) 164A1.2.6. Conical sector 165Appendix 2: Properties of M-Matrices 167Appendix 3: Stability and Comparison Systems 169A3.1. Vector norms and overvaluing systems 169A3.1.1. Definition of a vector norm 169A3.1.2. Definition of a system overvalued from a continuous process 170A3.1.3. Application 172A3.2. Vector norms and the principle of comparison 173A3.3. Application to stability analysis 174Bibliography 175Index 185

Mohammed Chadli has been Assoc. Professor at the University of Picardy Jules Verne (UPJV) in Amiens, France since 2004 as well as being a researcher in the "Modélisation, Information & Systèmes" (MIS) Laboratory. His research interests 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 ixIntroduction xiiiChapter 1. Multiple Model Representation 11.1. Introduction 11.2. Techniques for obtaining multiple models 21.2.1. Construction of multiple models by identification 31.2.2. Multiple model construction by linearization 81.2.3. Multiple model construction by mathematical transformation 141.2.4. Multiple model representation using the neural approach 221.3. Analysis and synthesis tools 291.3.1. Lyapunov approach 291.3.2. Numeric tools: linear matrix inequalities 311.3.3. Multiple model control techniques 38Chapter 2. Stability of Continuous Multiple Models 412.1. Introduction 412.2. Stability analysis 422.2.1. Exponential stability 482.3. Relaxed stability 492.4. Example 522.5. Robust stability 542.5.1. Norm-bounded uncertainties 562.5.2. Structured parametric uncertainties 572.5.3. Analysis of nominal stability 602.5.4. Analysis of robust stability 622.6. Conclusion 63Chapter 3. Multiple Model State Estimation 653.1. Introduction 653.2. Synthesis of multiple observers 673.2.1. Linearization 683.2.2. Pole placement 703.2.3. Application: asynchronous machine 723.2.4. Synthesis of multiple observers 753.3. Multiple observer for an uncertain multiple model 773.4. Synthesis of unknown input observers 823.4.1. Unknown inputs affecting system state 833.4.2. Unknown inputs affecting system state and output 873.4.3. Estimation of unknown inputs 883.5. Synthesis of unknown input observers: another approach 933.5.1. Principle 933.5.2. Multiple observers subject to unknown inputs and uncertainties 963.6. Conclusion 97Chapter 4. Stabilization of Multiple Models 994.1. Introduction 994.2. Full state feedback control 994.2.1. Linearization 1014.2.2. Specific case 1034.2.3. Stability: decay rate 1064.3. Observer-based controller 1134.3.1. Unmeasurable decision variables 1154.4. Static output feedback control 1194.4.1. Pole placement 1224.5. Conclusion 126Chapter 5. Robust Stabilization of Multiple Models 1275.1. Introduction 1275.2. State feedback control 1295.2.1. Norm-bounded uncertainties 1295.2.2. Interval uncertainties 1315.3. Output feedback control 1375.3.1. Norm-bounded uncertainties 1375.3.2. Interval uncertainties 1475.4. Observer-based control 1505.5. Conclusion 156Conclusion 157APPENDICES 159Appendix 1: LMI Regions 161A1.1. Definition of an LMI region 161A1.2. Interesting LMI region examples 162A1.2.1. Open left half-plane 163A1.2.2. Stability 163A1.2.3. Vertical band 163A1.2.4. Horizontal band 164A1.2.5. Disk of radius R, centered at (q,0) 164A1.2.6. Conical sector 165Appendix 2: Properties of M-Matrices 167Appendix 3: Stability and Comparison Systems 169A3.1. Vector norms and overvaluing systems 169A3.1.1. Definition of a vector norm 169A3.1.2. Definition of a system overvalued from a continuous process 170A3.1.3. Application 172A3.2. Vector norms and the principle of comparison 173A3.3. Application to stability analysis 174Bibliography 175Index 185