FUZZY CONTROL SYSTEMS DESIGN AND ANALYSIS

A Linear Matrix Inequality Approach 1 INTRODUCTION 1 1.1 A Control Engineering Approach to Fuzzy Control/1 1.2 Outline of This Book/2 2 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION 5 2.1 Takagi-Sugeno Fuzzy Model/6 2.2 Construction of Fuzzy Model/9 2.2.1 Sector Nonlinearity/10 2.2.2 Local Approximation in Fuzzy Partition Spaces/23 2.3 Parallel Distributed Compensation/25 2.4 A Motivating Example/26 2.5 Origin of the LMI-Based Design Approach /29 2.5.1 Stable Controller Design via Iterative Procedure/30 2.5.2 S table Controller Design via Linear Matrix Inequalities/34 v vi CONTENTS 2.6 Application: Inverted Pendulum on a Cart /38 2.6.1 Two-Rule Modeling and Control/38 2.6.2 Four-Rule Modeling and Control/42 Bibliography/47 3 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS 49 3.1 Stability Conditions/49 3.2 Relaxed Stability Conditions/52 3.3 Stable Controller Design/58 3.4 Decay Rate/62 3.5 Constraints on Control Input and Output /66 3.5.1 Constraint on the Control Input/66 3.5.2 Constraint on the Output/68 3.6 Initial State Independent Condition/68 3.7 Disturbance Rejection/69 3.8 Design Example: A Simple Mechanical System/76 3.8.1 Design Case 1: Decay Rate/78 3.8.2 Design Case 2: Decay RateqConstraint on the Control Input/79 3.8.3 Design Case 3: Stability qConstraint on the Control Input/80 3.8.4 Design Case 4: Stability qConstraint on the Control InputqConstraint on the Output/81 References/81 4 FUZZY OBSERVER DESIGN 83 4.1 Fuzzy Observer/83 4.2 Design of Augmented Systems/84 4.2.1 Case A/85 4.2.2 Case B/90 4.3 Design Example/93 References/96 5 ROBUST FUZZY CONTROL 97 5.1 Fuzzy Model with Uncertainty/98 5.2 Robust Stability Condition/98 5.3 Robust Stabilization/105 References/108 CONTENTS vii 6 OPTIMAL FUZZY CONTROL 109 6.1 Quadratic Performance Function and Stabilization Control/110 6.2 Optimal Fuzzy Controller Design/114 Appendix to Chapter 6/118 References/119 7 ROBUST-OPTIMAL FUZZY CONTROL 121 7.1 Robust-Optimal Fuzzy Control Problem /121 7.2 Design Example: TORA/125 References/130 8 TRAJECTORY CONTROL OF A VEHICLE WITH MULTIPLE TRAILERS 133 8.1 Fuzzy Modeling of a Vehicle with Triple-Trailers/134 8.1.1 Avoidance of Jack-Knife Utilizing Constraint on Output/142 8.2 Simulation Results/144 8.3 Experimental Study/147 8.4 Control of Ten-Trailer Case r 150 References/151 9 FUZZY MODELING AND CONTROL OF CHAOTIC SYSTEMS 153 9.1 Fuzzy Modeling of Chaotic Systems/154 9.2 Stabilization/159 9.2.1 Stabilization via Parallel Distributed Compensation /159 9.2.2 Cancellation Technique/165 9.3 Synchronization/170 9.3.1 Case 1 /170 9.3.2 Case 2 /179 9.4 Chaotic Model Following Control/182 References/192 10 FUZZY DESCRIPTOR SYSTEMS AND CONTROL 195 10.1 Fuzzy Descriptor System/196 10.2 Stability Conditions/197 10.3 Relaxed Stability Conditions/206 10.4 Why Fuzzy Descriptor Systems?/211 References/215 viii CONTENTS 11 NONLINEAR MODEL FOLLOWING CONTROL 217 11.1 Introduction/217 11.2 Design Concept/218 11.2.1 Reference Fuzzy Descriptor System/218 11.2.2 Twin-Parallel Distributed Compensations/219 11.2.3 The Common B Matrix Case/223 11.3 Design Examples/224 References/228 12 NEW STABILITY CONDITIONS AND DYNAMIC FEEDBACK DESIGNS 229 12.1 Quadratic Stabilizability Using State Feedback PDC/230 12.2 Dynamic Feedback Controllers/232 12.2.1 Cubic Parametrization/236 12.2.2 Quadratic Parameterization/243 12.2.3 Linear Parameterization/247 12.3 Example/253 Bibliography/256 13 MULTIOBJECTIVE CONTROL VIA DYNAMIC PARALLEL DISTRIBUTED COMPENSATION 259 13.1 Performance-Oriented Controller Synthesis/260 13.1.1 Starting from Design Specifications/260 13.1.2 Performance-Oriented Controller Synthesis/264 13.2 Example/271 Bibliography/274 14 T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR 277 14.1 Approximation of Nonlinear Functions Using Linear T-S Systems/278 14.1.1 Linear T-S Fuzzy Systems/278 14.1.2 Construction Procedure of T-S Fuzzy Systems/279 14.1.3 Analysis of Approximation/281 14.1.4 Example/286 CONTENTS ix 14.2 Applications to Modeling and Control of Nonlinear Systems/287 14.2.1 Approximation of Nonlinear Dynamic Systems Using Linear Takagi-Sugeno Fuzzy Models/287 14.2.2 Approximation of Nonlinear State Feedback Controller Using PDC Controller/288 Bibliography/289 15 FUZZY CONTROL OF NONLINEAR TIME-DELAY SYSTEMS 291 15.1 T-S Fuzzy Model with Delays and Stability Conditions/292 15.1.1 T-S Fuzzy Model with Delays/292 15.1.2 Stability Analysis via Lyapunov Approach/294 15.1.3 Parallel Distributed Compensation Control/295 15.2 Stability of the Closed-Loop Systems/296 15.3 State Feedback Stabilization Design via LMIs/297 15.4 H Control/299 15.6 Design Example/300