Attractive Ellipsoids in Robust Control (Systems & Control: Foundations & Applications)

This monograph introduces a newly developed robust-control design technique for a wide class of continuous-time dynamical systems called the “attractive ellipsoid method.” Along with a coherent introduction to the proposed control design and related topics, the monograph studies nonlinear affine control systems in the presence of uncertainty and presents a constructive and easily implementable control strategy that guarantees certain stability properties. The authors discuss linear-style feedback control synthesis in the context of the above-mentioned systems. The development and physical implementation of high-performance robust-feedback controllers that work in the absence of complete information is addressed, with numerous examples to illustrate how to apply the attractive ellipsoid method to mechanical and electromechanical systems. While theorems are proved systematically, the emphasis is on understanding and applying the theory to real-world situations. Attractive Ellipsoids in Robust Control will appeal to undergraduate and graduate students with a background in modern systems theory as well as researchers in the fields of control engineering and applied mathematics. Preface 8 Contents 14 List of Figures 20 1 Introduction 23 1.1 Complete Information Case: Classical Control Approaches 23 1.1.1 System Description 24 1.1.2 Feasible and Admissible Control 26 1.1.3 Problem Setting in the General Bolza Form 26 1.1.4 Specific Features of Classical Optimal Control 27 1.2 Case of Incomplete Information 27 1.2.1 Robust Tracking Problem Formulation 27 1.2.2 What Is the Effectiveness of a Designed Control in the Case of Incomplete Information? 29 1.3 Ellipsoid-Based Feedback Control Design 30 1.4 Overview of the Book 31 2 Mathematical Background 33 2.1 The Class of Nonlinear Uncertain Models 33 2.1.1 Quasi-Lipschitz Dynamical Systems 33 2.1.2 Examples of Quasi-Lipschitz Systems 36 2.1.3 Differential Inclusions and General Solution Concept 38 2.1.4 The Filippov Regularization Procedure 41 2.2 The Lyapunov Approach to Quasi-Lipschitz DynamicalSystems 44 2.3 Elements of LMIs 48 2.3.1 Main Concepts 48 2.3.2 Existence of Solutions of LMIs 53 2.3.3 Numerical Approaches to LMIs 60 2.4 S-Lemma and Some Useful Mathematical Facts 63 3 Robust State Feedback Control 68 3.1 Introduction 69 3.2 Proportional Feedback Design 70 3.2.1 Model Description 70 3.2.2 Problem Formulation 71 3.3 S-Procedure-Based Approach 71 3.4 Storage Function Method 74 3.5 Minimization of the Attractive Ellipsoid 75 3.6 Practical Stabilization 77 3.7 Other Restrictions on Control and Uncertainties 79 3.8 Illustrative Example 81 Dynamic Model 81 Numerical Simulations Results 83 3.9 What to Do If We Don't Know the Matrix A? 84 3.9.1 Description of the Dynamic Model in This Case 84 3.9.2 Sufficient Conditions of Attractiveness 86 3.9.3 Optimal Robust Linear Feedback as a Solution of an Optimization Problem with LMI Constraints 89 3.10 Conclusions 89 4 Robust Output Feedback Control 91 4.1 Static Feedback Control 92 4.1.1 System Description and Problem Statement 92 4.1.2 Application of the Attractive Ellipsoids Method 93 4.1.3 Example: Stabilization of a Discontinuous System 96 4.2 Observer-Based Feedback Design 98 4.2.1 State Observer and the Extended Dynamic Model 98 4.2.2 Stabilizing Feedback Gains K and F 99 4.2.3 Numerical Aspects 105 4.2.4 Example: Robust Stabilization of a Spacecraft 107 4.3 Dynamic Regulator 112 4.3.1 Full-Order Linear Dynamic Controllers 112 4.3.2 Main Result on the Attractive Ellipsoidfor a Dynamic Controller 113 4.4 Conclusions 116 5 Control with Sample-Data Measurements 117 5.1 Introduction and Motivation 118 5.2 Problem Formulation and Some Preliminaries 119 5.3 Linear Feedback Proportional to a State Estimate Vector 121 5.3.1 Description in Extended Form 121 5.3.2 Lyapunov-Like Analysis 123 5.3.3 Numerical Aspects 130 5.4 Full-Order Robust Linear Dynamic Controller 133 5.4.1 The Structure of a Dynamic Controller 133 Stability Analysis of the Invariant Set 134 The Lie Derivative Estimation 134 5.4.2 The ``Minimal-Size'' Attractive Ellipsoid and LMI Constrained Optimization 138 5.4.3 On Numerical Realization 140 5.5 Conclusion 141 6 Sample Data and Quantifying Output Control 143 6.1 Introduction 143 6.2 Problem Formulation 145 6.3 A Lyapunov–Krasovskii Functional 148 6.3.1 Main Result 153 6.4 Numerical Aspects 154 6.5 Numerical Examples 160 6.5.1 Example 1 160 6.5.2 Example 2 162 6.6 Conclusions 164 7 Robust Control of Implicit Systems 167 7.1 Introduction 167 7.2 Some Preliminaries 169 7.2.1 Model Description 169 7.2.2 Useful Concepts and Facts 170 Regular Matrix Pairs and Their Properties 170 7.2.3 Transformation to Differential-Algebraic Form 171 7.2.4 Problem Formulation 173 7.3 Attractive Ellipsoid for Implicit Systems 174 7.3.1 Descriptive Method Application 174 7.3.2 Reduction of Nonlinear Matrix Inequalities to LMIs 176 7.4 Concluding Remarks 180 8 Attractive Ellipsoids in Sliding Mode Control 182 8.1 Minimization of Unmatched Uncertainties Effect in Sliding Mode Control 183 8.1.1 Problem Statement 183 8.1.2 LMI-Based Sliding Mode Control Design 185 8.1.3 Optimal Sliding Surface 187 8.1.4 Numerical Aspects of Sliding Surface Design 191 8.1.5 Numerical Example 193 8.2 Gain Matrix Tuning in Dynamic Actuators 195 8.2.1 Problem Statement 195 8.2.2 Controller Design 197 8.2.3 Example 201 8.3 Conclusion 204 9 Robust Stabilization of Time-Delay Systems 205 9.1 Time-Delay Systems with Known Input Delay 205 9.1.1 Brief Historical Remark 205 9.1.2 System Description and Problem Statement 206 9.1.3 Unavoidable Stabilization Error 208 9.1.4 Minimal Invariant Ellipsoid for the Prediction System 209 9.1.5 Minimal Attractive Ellipsoid of the Original System 215 9.1.6 Computational Aspects 220 9.1.7 Numerical Example 223 9.2 Control of Systems with Unknown Input Delay 225 9.2.1 Introduction 225 9.2.2 Problem Statement 226 9.2.3 Attractive Ellipsoid Method for Time-Delay Systems 228 9.2.4 Predictor-Based Output Feedback Design 228 9.2.5 Adjustment of Control Parameters: Computational Aspects 234 9.2.6 Numerical Example 238 9.3 Conclusion 239 10 Robust Control of Switched Systems 242 10.1 Introduction 243 10.1.1 Some Preliminaries 244 10.1.2 Problem Formulation 245 10.2 Application of the Attractive Ellipsoid Method 249 10.2.1 Practical Stability 250 10.2.2 Intersection of Ellipsoids 255 10.2.3 Bilinear Matrix Inequality Representation 261 10.2.4 Simulation Results 264 10.3 Switched Systems with Quantized and Sampled OutputFeedback 268 10.3.1 System Description 268 10.3.2 Lyapunov–Krasovskii-Like Functional 271 10.3.3 On Practical Stability 274 Intersection of Ellipsoids 278 Main Result 278 Illustrative Example 279 10.4 Conclusions 282 11 Bounded Robust Control 284 11.1 Introduction 285 11.2 The Class of Uncertain Nonlinear Systemsand Problem Formulation 285 11.2.1 System Description 285 11.2.2 Basic Assumptions 287 11.2.3 Extended Dynamic Form 289 11.2.4 Problem Formulation 290 11.3 Robust Bounded Output Control Synthesis 291 11.3.1 Storage Function 291 11.3.2 Zone-Convergence Analysis 294 11.3.3 The Attractive Ellipsoid of ``Minimal Size'' 300 11.4 Numerical Aspects 303 11.4.1 Transformation of BMI Constraints into LMI Constraints 303 11.4.2 Computational Aspects 305 11.5 Illustrative Example 306 11.5.1 Dynamic Model 306 11.5.2 Numerical Results 308 11.5.3 Simulation Results 309 11.6 Conclusion 310 12 Attractive Ellipsoid Method with Adaptation 312 12.1 Introduction 313 12.2 Attractive Ellipsoid Method with KL-Adaptation 314 12.2.1 Basic Assumptions and Constraints 315 12.2.2 System Description and Problem Formulation 315 12.2.3 Main Assumptions 316 12.2.4 Extended Quasilinear Format 317 12.2.5 Problem Formulation 318 12.2.6 Learning Laws, Storage Function Properties, and the ``Minimal Size'' Ellipsoid 318 12.2.7 Attractive Ellipsoid for Robust Control with KL-Adaptation 322 12.2.8 On the Attractive Ellipsoid in the State Space 325 12.2.9 On the Effectiveness of the Adaptation Process 327 Specific Persistent Excitation Condition 327 12.2.10 On Transformation BMI Constraints into LMI Constraints 330 12.2.11 Numerical Aspects 333 12.2.12 Illustrative Example 333 12.3 A-Adaptation in the Attractive Ellipsoid Method 335 12.3.1 Quasilinear Model with Adjusted Feedback and Problem Formulation 337 12.3.2 ``A''-Adaptation 337 The Extended System with ``A''-Adaptation 338 12.3.3 Closed-Loop Representation and Storage Function 341 12.3.4 Stability Analysis 344 Lyapunov Function and the Convergence Zone 345 12.3.5 On the ``Minimal Size'' of the Attractive Ellipsoid 349 Attractive Ellipsoid in the Extended z-Space 349 Attractive Ellipsoid in the State Space 349 12.3.6 Numerical Aspects 350 Numerical Example 351 Numerical Data 352 Simulation Results 352 12.4 Conclusion 353 Bibliography 356 Index 364 Front Matter....Pages i-xxi Introduction....Pages 1-10 Mathematical Background....Pages 11-45 Robust State Feedback Control....Pages 47-69 Robust Output Feedback Control....Pages 71-96 Control with Sample-Data Measurements....Pages 97-122 Sample Data and Quantifying Output Control....Pages 123-146 Robust Control of Implicit Systems....Pages 147-161 Attractive Ellipsoids in Sliding Mode Control....Pages 163-185 Robust Stabilization of Time-Delay Systems....Pages 187-223 Robust Control of Switched Systems....Pages 225-266 Bounded Robust Control....Pages 267-294 Attractive Ellipsoid Method with Adaptation....Pages 295-338 Back Matter....Pages 339-348