Linear Parameter-Varying Control

About the Author xv Preface xvii Acronyms xxi About the Companion Website xxiii Introduction xxv Part I Some Theoretical Aspects on LPV Systems: From Modeling to Control 1 1 Some Modeling Approaches for LPV and qLPV Systems 3 1.1 Introduction 3 1.2 Dynamical Systems 4 1.2.1 Nonlinear Dynamical Systems 4 1.2.2 Linear Time-Invariant (LTI) Dynamical Systems 5 1.3 An Introduction to LPV Models 5 1.3.1 Definition 6 1.3.2 About the Time-Varying Parameters 7 1.3.3 Is an LPV System Linear or Nonlinear? 9 1.3.4 Discrete-Time LPV Systems 9 1.4 Specific Classes of LPV Systems 10 1.4.1 Class 1: Affine Parameter Dependence 10 1.4.2 Class 2: Polytopic Representations 11 1.4.3 Class 3: Polynomial Parameter Dependence 14 1.4.4 Class 4: Rational Parameter Dependence 14 1.4.5 Class 5: LFT Representations 15 1.4.6 Class 6: Takagi–Sugeno (TS) Representations 18 1.5 From a Nonlinear Model to an LPV Representation 19 1.5.1 Jacobian Linearization-Based Method 19 1.5.2 Linear Differential Inclusion-Based Method 21 1.5.3 Other Modeling Approaches 23 1.6 An Introduction to Identification of LPV Systems 23 1.7 The Nonuniqueness Issue: A Control-Oriented LPV Modeling Perspective 25 1.8 Illustrative Example 1: A Single Tank System 26 1.8.1 A Jacobian Linearization-Based Model 27 1.8.2 An LDI-Based Model 28 1.8.3 A Polytopic Model 28 1.8.4 LFT Models 29 1.8.4.1 LFT from the Jacobian-Linearized Model 29 1.8.4.2 LFT from the Nonlinear Model 29 1.9 Illustrative Example 2: qLPV Modeling and Time-Varying Characteristics 30 1.9.1 A Polytopic Model 30 1.9.2 An LFT Model 31 1.9.3 Analysis 31 1.9.4 Considerations for the Simulation and Implementation of Polytopic Models 32 1.10 Conclusion 34 Bibliography 34 2 Properties of LPV Systems 41 2.1 Introduction 41 2.2 Controllability 42 2.2.1 Definition 43 2.2.2 A First Analysis Following Odd Characterizations 43 2.2.3 The Correct Time-Varying Characterization 45 2.3 Observability 46 2.3.1 Definition 47 2.3.2 Characterizations 47 2.4 Comments on State-Space Realizations of LPV Systems 49 2.5 Stability 50 2.5.1 A Few Background 50 2.5.1.1 Stability of Nonlinear Systems 50 2.5.1.2 Stability of LTI Systems 51 2.5.2 Problem Statement and Facts 52 2.5.3 Quadratic Stability of LTI Uncertain Systems 53 2.5.4 Quadratic Stability of LPV Systems 53 2.5.5 Robust Stability or Parameter-Dependent Lyapunov Stability 54 2.6 Performance Criteria:  , and Pole Placement 55 2.6.1 LTI Continuous-Time Systems 56 2.6.1.1 Performance 56 2.6.1.2 Generalized Performance 57 2.6.1.3 Pole Placement 58 2.6.2 LPV Continuous-Time Systems 59 2.6.2.1 Stability (Performance) of LPV Systems 60 2.6.2.2 Generalized Performance of LPV Systems 61 2.7 About Stabilizability and Detectability 62 2.7.1 Quadratic Stabilizability and Detectability 62 2.7.2 Parameter-Dependent Stabilizability and Detectability 63 2.8 The Case of Discrete-Time LPV Systems 63 2.8.1 Stability of Discrete-Time LPV Systems 64 2.8.1.1 Quadratic Stability 64 2.8.1.2 Parameter-Dependent Stability 65 2.8.2 Performance Criteria for Discrete-Time LPV Systems 66 2.9 Conclusion 68 Bibliography 68 3 Control of LPV Systems 75 3.1 Introduction 75 3.2 LPV State-Feedback Control 77 3.2.1 Some Facts and Preliminary Results 79 3.2.1.1 Case 1: Robust State-Feedback Control 79 3.2.1.2 Case 2: LPV State-Feedback Control with Fixed Performances 81 3.2.1.3 Case 3: LPV State-Feedback Control with Varying (Adaptive) Performances 82 3.2.2 Static State-Feedback Control: A Polytopic Approach 84 3.2.3 Static State-Feedback Control: A Grid-Based Approach 86 3.3 The LPV Dynamic Output Feedback Control 88 3.3.1 LPV Control Problems 91 3.3.2 LPV Dynamic Output Feedback Control – A Polytopic Approach 92 3.3.2.1 Requirements and Definitions 92 3.3.2.2 Solution of the LPV Polytopic Control Problems 94 3.3.2.3 About the Conservatism of the Polytopic Approach 96 3.3.2.4 Computation and Implementation of the Polytopic Controller 97 3.3.3 LPV Dynamic Output Feedback Control – A Grid-Based Approach 100 3.3.3.1 The LPV Grid-Based Design Solution 101 3.3.3.2 Comments on the Grid-Based Approach 103 3.4 LPV Observer Design 104 3.4.1 Introduction and Problem Statement 104 3.4.2 LPV Polytopic Observer with H ∞ Performance Method 106 3.4.3 LPV Polytopic Observer with Pole Placement Method 107 3.4.4 Concluding Remarks 108 3.5 About Control of Discrete-Time LPV Systems 109 3.6 Conclusion 111 Bibliography 111 Part II LPV Methods for Nonlinear Systems 121 4 Control and Observer Design for Nonlinear Systems Using Quasi-LPV Models: An Illustration Through Examples 123 4.1 Introduction 123 4.2 LPV Control of a Nonlinear System 124 4.2.1 qLPV and Polytopic Models 125 4.2.2 LTI/LPV Control: Problem Formulation 127 4.2.3 Performance Specification Using Weighting Functions 129 4.2.4 Problem Solution and Analysis 129 4.2.5 Polytopic Controller and Scheduling Parameters 131 4.2.6 Simulation Results 133 4.3 An LPV Observer of a Three-Tank Nonlinear System 134 4.3.1 Physical Model 135 4.3.2 LPV Modeling and qLPV State Space Model 136 4.3.3 LPV Observer Design: Problem Formulation 137 4.3.4 LPV Observer Design: Problem Solution 137 4.3.5 Simulation Results 138 4.4 Conclusion 140 Bibliography 140 5 Observer Design for Semi-active Suspension Systems: qLPV Approaches 143 5.1 Introduction 143 5.2 Illustrative Case Study: The INOVE Testbench, a Semi-active Suspension System 145 5.3 Electro-Rheological Dampers: Modeling Approaches 147 5.3.1 Static Models of Semi-active Dampers 148 5.3.2 Dynamical Model of Semi-active Dampers 149 5.3.3 Observer-Design Oriented ER Damper Models 150 5.4 qLPV Quarter Car Semi-active Suspension Models 152 5.4.1 qLPV Model: Method 1 155 5.4.2 qLPV Model: Method 2 156 5.4.3 qLPV Model: Method 3 157 5.5 Method 1: An Observer for Suspension State Estimation 158 5.5.1 Problem Formulation 158 5.5.2  Observer Polytopic Design Method 159 5.5.3 Problem Solution and Simulation Results 160 5.6 Method 2: A Filtering Approach for Damper Force Estimation 163 5.6.1 Problem Formulation 164 5.6.2 Filter Design: A Polytopic Approach 164 5.6.3 Problem Solution and Simulation Results 166 5.7 Method 3: A Nonlinear Parameter Varying Approach for State Estimation 168 5.7.1 NLPV Observer Definition 169 5.7.2 Problem Formulation 169 5.7.3 NLPV Observer Polytopic Design 170 5.7.4 NLPV Observer Grid-Based Design 171 5.7.5 Problem Solution and Simulation Results 173 5.8 Concluding Remarks 175 Bibliography 176 6 Lateral Control of Autonomous Vehicle 181 6.1 Introduction 181 6.2 Modeling 182 6.2.1 Dynamical Bicycle Models 183 6.2.1.1 Grid-Based Bicycle Model 184 6.2.1.2 Polytopic Bicycle Model 185 6.2.2 Augmented Model with Actuator Dynamics 186 6.3 LPV Control Design 187 6.3.1 Problem Formulation 187 6.3.2 Performance Specifications Using Weighting Functions 188 6.3.3 Polytopic Approach 189 6.3.4 Grid-Based Approach 190 6.4 Analysis of the Polytopic and Grid-Based Design Methods 191 6.5 Simulation Results 192 6.6 Conclusion 197 Bibliography 197 Part III LPV Adaptive-Like Control Methods 203 7 Methods and Tools for LPV Adaptive-Like Control 205 7.1 Introduction 205 7.2 The Framework: A Generic Tool for “Adaptive-Like” Control 206 7.3 LPV Adaptive Control with Varying Closed-Loop Performances (Function of External Parameters) 208 7.3.1 LPV Control Problem Formulation 208 7.3.2 Performance Specification Using Weighting Functions 209 7.3.3 Problem Solution and Analysis 210 7.3.4 Polytopic Controller and Scheduling Strategy 211 7.3.5 Simulation Results 212 7.4 LPV Adaptive Control Function of Varying Endogeneous Parameters 215 7.4.1 A Polytopic Model 216 7.4.2 LPV Control Problem Formulation 217 7.4.3 Performance Specification Using Weighting Functions 218 7.4.4 Problem Solution and Analysis 218 7.4.5 Polytopic Controller and Scheduling Strategy 219 7.4.6 Simulation Results 221 7.5 Concluding Remarks 223 Bibliography 223 8 LPV Road Adaptive Suspension Control 227 8.1 Introduction 227 8.2 The Semi-active Suspension Quarter-Car Model 230 8.2.1 Control Design Damper Model 231 8.2.2 Simulation Damper Model 232 8.3 Road Roughness Estimator 233 8.4 Synthesis of a Semi-active Suspension Control 237 8.4.1 qLPV Quarter-Car Model 238 8.4.2 LPV Road Adaptive Control: Problem Formulation 240 8.4.3 Performance Specification Using Weighting Functions 241 8.4.4 LPV Control Synthesis and Solution 242 8.4.4.1 Polytopic Approach 242 8.4.4.2 Grid-Based Approach 244 8.4.5 Scheduling Strategy in View of Road Adaptive Performances 245 8.5 Simulation Results 246 8.6 Conclusions 249 Bibliography 249 9 LPV Fault-Tolerant Control Strategies for Suspension Systems 257 9.1 Introduction 257 9.2 Related Works 259 9.2.1 About Fault Estimation 259 9.2.2 About FTC 259 9.2.3 About Vehicle Applications 260 9.3 Fault Diagnosis Problem Formulation for Semi-active ER Suspension Systems 261 9.3.1 Approach 1: Static Bingham Damper Model with Multiplicative Fault 264 9.3.2 Approach 2: Dynamical Guo Damper Model with Additive Fault 264 9.4 Fault Estimation Using LPV PI Observers 265 9.4.1 Approach 1: An LPV Observer Design Method for Multiplicative Damper Fault Estimation 265 9.4.2 Approach 2: An NLPV Observer Design Method for Additive Damper Fault Estimation 269 9.5 FTC LPV Control of Semi-active Suspension Systems 276 9.5.1 Quarter-Car Model 277 9.5.2 Reconfigurable Semi-active Suspension Control: Problem Formulation 277 9.5.3 Performance Specifications Using Weighting Functions 279 9.5.4 Problem Solution and Analysis 279 9.5.5 Polytopic Controller and Scheduling Strategy 281 9.5.6 Simulation Results 282 9.6 Conclusion 284 Bibliography 284 10 Lateral LPV Adaptive-Like Control of Automated Vehicles Adapted to Driver Performance 293 10.1 Introduction 293 10.2 LPV Observer-Based Control Structure for ADAS Systems 294 10.3 Driver Fault Estimation Using a Discrete-Time LPV PI Observer 295 10.3.1 Driver Model 295 10.3.2 Fault Model and Problem Formulation 296 10.3.3 Design Method of the Discrete-Time PI Observer for Fault Estimation 298 10.3.4 Problem Solution 300 10.4 Robust ADAS Strategy 301 10.4.1 Modeling Step 301 10.4.2 LPV Steering/Braking Control Problem Formulation 303 10.4.3 Synthesis of LPV State-Feedback Grid-Based Controllers 305 10.4.4 Problem Solution and Analysis 307 10.5 Simulation Results 308 10.6 Conclusion 313 Bibliography 313 Index 317