Attractive Ellipsoids in Robust Control

Preface......Page 8
Contents......Page 14
List of Figures......Page 20
 1.1 Complete Information Case: Classical Control Approaches......Page 23
  1.1.1 System Description......Page 24
  1.1.3 Problem Setting in the General Bolza Form......Page 26
  1.2.1 Robust Tracking Problem Formulation......Page 27
  1.2.2 What Is the Effectiveness of a Designed Control in the Case of Incomplete Information?......Page 29
 1.3 Ellipsoid-Based Feedback Control Design......Page 30
 1.4 Overview of the Book......Page 31
  2.1.1 Quasi-Lipschitz Dynamical Systems......Page 33
  2.1.2 Examples of Quasi-Lipschitz Systems......Page 36
  2.1.3 Differential Inclusions and General Solution Concept......Page 38
  2.1.4 The Filippov Regularization Procedure......Page 41
 2.2 The Lyapunov Approach to Quasi-Lipschitz DynamicalSystems......Page 44
  2.3.1 Main Concepts......Page 48
  2.3.2 Existence of Solutions of LMIs......Page 53
  2.3.3 Numerical Approaches to LMIs......Page 60
 2.4 S-Lemma and Some Useful Mathematical Facts......Page 63
3 Robust State Feedback Control......Page 68
 3.1 Introduction......Page 69
  3.2.1 Model Description......Page 70
 3.3 S-Procedure-Based Approach......Page 71
 3.4 Storage Function Method......Page 74
 3.5 Minimization of the Attractive Ellipsoid......Page 75
 3.6 Practical Stabilization......Page 77
 3.7 Other Restrictions on Control and Uncertainties......Page 79
  Dynamic Model......Page 81
  Numerical Simulations Results......Page 83
  3.9.1 Description of the Dynamic Model in This Case......Page 84
  3.9.2 Sufficient Conditions of Attractiveness......Page 86
 3.10 Conclusions......Page 89
4 Robust Output Feedback Control......Page 91
  4.1.1 System Description and Problem Statement......Page 92
  4.1.2 Application of the Attractive Ellipsoids Method......Page 93
  4.1.3 Example: Stabilization of a Discontinuous System......Page 96
  4.2.1 State Observer and the Extended Dynamic Model......Page 98
  4.2.2 Stabilizing Feedback Gains K and F......Page 99
  4.2.3 Numerical Aspects......Page 105
  4.2.4 Example: Robust Stabilization of a Spacecraft......Page 107
  4.3.1 Full-Order Linear Dynamic Controllers......Page 112
  4.3.2 Main Result on the Attractive Ellipsoidfor a Dynamic Controller......Page 113
 4.4 Conclusions......Page 116
5 Control with Sample-Data Measurements......Page 117
 5.1 Introduction and Motivation......Page 118
 5.2 Problem Formulation and Some Preliminaries......Page 119
  5.3.1 Description in Extended Form......Page 121
  5.3.2 Lyapunov-Like Analysis......Page 123
  5.3.3 Numerical Aspects......Page 130
  5.4.1 The Structure of a Dynamic Controller......Page 133
   The Lie Derivative Estimation......Page 134
  5.4.2 The ``Minimal-Size\'\' Attractive Ellipsoid and LMI Constrained Optimization......Page 138
  5.4.3 On Numerical Realization......Page 140
 5.5 Conclusion......Page 141
 6.1 Introduction......Page 143
 6.2 Problem Formulation......Page 145
 6.3 A Lyapunov–Krasovskii Functional......Page 148
  6.3.1 Main Result......Page 153
 6.4 Numerical Aspects......Page 154
  6.5.1 Example 1......Page 160
  6.5.2 Example 2......Page 162
 6.6 Conclusions......Page 164
 7.1 Introduction......Page 167
  7.2.1 Model Description......Page 169
   Regular Matrix Pairs and Their Properties......Page 170
  7.2.3 Transformation to Differential-Algebraic Form......Page 171
  7.2.4 Problem Formulation......Page 173
  7.3.1 Descriptive Method Application......Page 174
  7.3.2 Reduction of Nonlinear Matrix Inequalities to LMIs......Page 176
 7.4 Concluding Remarks......Page 180
8 Attractive Ellipsoids in Sliding Mode Control......Page 182
  8.1.1 Problem Statement......Page 183
  8.1.2 LMI-Based Sliding Mode Control Design......Page 185
  8.1.3 Optimal Sliding Surface......Page 187
  8.1.4 Numerical Aspects of Sliding Surface Design......Page 191
  8.1.5 Numerical Example......Page 193
  8.2.1 Problem Statement......Page 195
  8.2.2 Controller Design......Page 197
  8.2.3 Example......Page 201
 8.3 Conclusion......Page 204
  9.1.1 Brief Historical Remark......Page 205
  9.1.2 System Description and Problem Statement......Page 206
  9.1.3 Unavoidable Stabilization Error......Page 208
  9.1.4 Minimal Invariant Ellipsoid for the Prediction System......Page 209
  9.1.5 Minimal Attractive Ellipsoid of the Original System......Page 215
  9.1.6 Computational Aspects......Page 220
  9.1.7 Numerical Example......Page 223
  9.2.1 Introduction......Page 225
  9.2.2 Problem Statement......Page 226
  9.2.4 Predictor-Based Output Feedback Design......Page 228
  9.2.5 Adjustment of Control Parameters: Computational Aspects......Page 234
  9.2.6 Numerical Example......Page 238
 9.3 Conclusion......Page 239
10 Robust Control of Switched Systems......Page 242
 10.1 Introduction......Page 243
  10.1.1 Some Preliminaries......Page 244
  10.1.2 Problem Formulation......Page 245
 10.2 Application of the Attractive Ellipsoid Method......Page 249
  10.2.1 Practical Stability......Page 250
  10.2.2 Intersection of Ellipsoids......Page 255
  10.2.3 Bilinear Matrix Inequality Representation......Page 261
  10.2.4 Simulation Results......Page 264
  10.3.1 System Description......Page 268
  10.3.2 Lyapunov–Krasovskii-Like Functional......Page 271
  10.3.3 On Practical Stability......Page 274
   Main Result......Page 278
   Illustrative Example......Page 279
 10.4 Conclusions......Page 282
11 Bounded Robust Control......Page 284
  11.2.1 System Description......Page 285
  11.2.2 Basic Assumptions......Page 287
  11.2.3 Extended Dynamic Form......Page 289
  11.2.4 Problem Formulation......Page 290
  11.3.1 Storage Function......Page 291
  11.3.2 Zone-Convergence Analysis......Page 294
  11.3.3 The Attractive Ellipsoid of ``Minimal Size\'\'......Page 300
  11.4.1 Transformation of BMI Constraints into LMI Constraints......Page 303
  11.4.2 Computational Aspects......Page 305
  11.5.1 Dynamic Model......Page 306
  11.5.2 Numerical Results......Page 308
  11.5.3 Simulation Results......Page 309
 11.6 Conclusion......Page 310
12 Attractive Ellipsoid Method with Adaptation......Page 312
 12.1 Introduction......Page 313
 12.2 Attractive Ellipsoid Method with KL-Adaptation......Page 314
  12.2.2 System Description and Problem Formulation......Page 315
  12.2.3 Main Assumptions......Page 316
  12.2.4 Extended Quasilinear Format......Page 317
  12.2.6 Learning Laws, Storage Function Properties, and the ``Minimal Size\'\' Ellipsoid......Page 318
  12.2.7 Attractive Ellipsoid for Robust Control with KL-Adaptation......Page 322
  12.2.8 On the Attractive Ellipsoid in the State Space......Page 325
   Specific Persistent Excitation Condition......Page 327
  12.2.10 On Transformation BMI Constraints into LMI Constraints......Page 330
  12.2.12 Illustrative Example......Page 333
 12.3 A-Adaptation in the Attractive Ellipsoid Method......Page 335
  12.3.2 ``A\'\'-Adaptation......Page 337
   The Extended System with ``A\'\'-Adaptation......Page 338
  12.3.3 Closed-Loop Representation and Storage Function......Page 341
  12.3.4 Stability Analysis......Page 344
   Lyapunov Function and the Convergence Zone......Page 345
   Attractive Ellipsoid in the State Space......Page 349
  12.3.6 Numerical Aspects......Page 350
   Numerical Example......Page 351
   Simulation Results......Page 352
 12.4 Conclusion......Page 353
Bibliography......Page 356
Index......Page 364