Robust Control

Preface xvii List of Abbreviations xix Notations xxi 1 Introduction 1 1.1 Engineering Background of Robust Control 1 1.2 Methodologies of Robust Control 4 1.3 A Brief History of Robust Control 8 2 Basics of Linear Algebra and Function Analysis 10 2.1 Trace, Determinant, Matrix Inverse, and Block Matrix 10 2.2 Elementary Linear Transformation of Matrix and Its Matrix Description 12 2.3 Linear Vector Space 14 2.4 Norm and Inner Product of Vector 18 2.5 Linear Subspace 22 2.6 Matrix and Linear Mapping 23 2.7 Eigenvalue and Eigenvector 28 2.8 Invariant Subspace 30 2.9 Pseudo-Inverse and Linear Matrix Equation 34 2.10 Quadratic Form and Positive Definite Matrix 35 2.11 Norm and Inner Product of Matrix 37 2.12 Singular Value and Singular Value Decomposition 40 2.13 Calculus of Vector and Matrix 43 2.14 Kronecker Product 44 2.15 Norm and Inner Product of Function 45 3 Basics of Convex Analysis and LMI 57 3.1 Convex Set and Convex Function 57 3.2 Introduction to LMI 72 3.3 Interior Point Method* 81 4 Fundamentals of Linear System 85 4.1 Structural Properties of Dynamic System 85 4.2 Stability 100 4.3 Lyapunov Equation 108 4.4 Linear Fractional Transformation 114 5 System Performance 119 5.1 Test Signal 120 5.2 Steady-State Response 122 5.3 Transient Response 130 5.4 Comparison of Open-Loop and Closed-Loop Controls 140 6 Stabilization of Linear Systems 148 6.1 State Feedback 148 6.2 Observer 160 6.3 Combined System and Separation Principle 167 7 Parametrization of Stabilizing Controllers 173 7.1 Generalized Feedback Control System 174 7.2 Parametrization of Controllers 178 7.3 Youla Parametrization 184 7.4 Structure of Closed-Loop System 186 7.5 2-Degree-of-Freedom System 188 8 Relation between Time Domain and Frequency Domain Properties 197 8.1 Parseval's Theorem 197 8.2 KYP Lemma 200 9 Algebraic Riccati Equation 215 9.1 Algorithm for Riccati Equation 215 9.2 Stabilizing Solution 218 9.3 Inner Function 223 10 Performance Limitation of Feedback Control 225 10.1 Preliminaries 226 10.2 Limitation on Achievable Closed-loop Transfer Function 228 10.3 Integral Relation 231 10.4 Limitation of Reference Tracking 237 11 Model Uncertainty 245 11.1 Model Uncertainty: Examples 245 11.2 Plant Set with Dynamic Uncertainty 248 11.3 Parametric System 253 11.4 Plant Set with Phase Information of Uncertainty 264 11.5 LPV Model and Nonlinear Systems 266 11.6 Robust Stability and Robust Performance 269 12 Robustness Analysis 1: Small-Gain Principle 272 12.1 Small-Gain Theorem 272 12.2 Robust Stability Criteria 276 12.3 Equivalence between H infinity Performance and Robust Stability 277 12.4 Analysis of Robust Performance 279 12.5 Stability Radius of Norm-Bounded Parametric Systems 282 13 Robustness Analysis 2: Lyapunov Method 288 13.1 Overview of Lyapunov Stability Theory 288 13.2 Quadratic Stability 290 13.3 Lur'e System 296 13.4 Passive Systems 307 14 Robustness Analysis 3: IQC Approach 312 14.1 Concept of IQC 312 14.2 IQC Theorem 314 14.3 Applications of IQC 316 14.4 Proof of IQC Theorem* 319 15 H2 Control 322 15.1 H2 Norm of Transfer Function 322 15.2 H2 Control Problem 329 15.3 Solution to Nonsingular H2 Control Problem 331 15.4 Proof of Nonsingular Solution 332 15.5 Singular H2 Control 335 15.6 Case Study: H2 Control of an RTP System 337 16 H infinity Control 346 16.1 Control Problem and H infinity Norm 346 16.2 H infinity Control Problem 348 16.3 LMI Solution 1: Variable Elimination 349 16.4 LMI Solution 2: Variable Change 351 16.5 Design of Generalized Plant and Weighting Function 352 16.6 Case Study 354 16.7 Scaled H infinity Control 355 17 mu Synthesis 360 17.1 Introduction to mu 360 17.2 Definition of mu and Its Implication 364 17.3 Properties of mu 365 17.4 Condition for Robust H infinity Performance 368 17.5 D-K Iteration Design 369 17.6 Case Study 371 18 Robust Control of Parametric Systems 375 18.1 Quadratic Stabilization of Polytopic Systems 375 18.2 Quadratic Stabilization of Norm-Bounded Parametric Systems 379 18.3 Robust H infinity Control Design of Polytopic Systems 379 18.4 Robust H infinity Control Design of Norm-Bounded Parametric Systems 382 19 Regional Pole Placement 384 19.1 Convex Region and Its Characterization 384 19.2 Condition for Regional Pole Placement 387 19.3 Composite LMI Region 392 19.4 Feedback Controller Design 394 19.5 Analysis of Robust Pole Placement 396 19.6 Robust Design of Regional Pole Placement 402 20 Gain-Scheduled Control 407 20.1 General Structure 407 20.2 LFT-Type Parametric Model 408 20.3 Case Study: Stabilization of a Unicycle Robot 414 20.4 Affine LPV Model 422 20.5 Case Study: Transient Stabilization of a Power System 428 21 Positive Real Method 436 21.1 Structure of Uncertain Closed-Loop System 436 21.2 Robust Stabilization Based on Strongly Positive Realness 438 21.3 Robust Stabilization Based on Strictly Positive Realness 441 21.4 Robust Performance Design for Systems with Positive Real Uncertainty 442 21.5 Case Study 445 Exercises 448 Notes and References 449 References 450 Index 455