Robust Control Theory and Applications

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

Exercises 53

Notes and References 56

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

Exercises 83

Notes and References 84

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

Exercises 117

Notes and References 118

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

Exercises 146

Notes and References 147

6 Stabilization of Linear Systems 148

6.1 State Feedback 148

6.2 Observer 160

6.3 Combined System and Separation Principle 167

Exercises 170

Notes and References 172

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

Exercises 193

Notes and References 196

8 Relation between Time Domain and Frequency Domain Properties 197

8.1 Parseval’s Theorem 197

8.2 KYP Lemma 200

Exercises 214

Notes and References 214

9 Algebraic Riccati Equation 215

9.1 Algorithm for Riccati Equation 215

9.2 Stabilizing Solution 218

9.3 Inner Function 223

Exercises 224

Notes and References 224

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

Exercises 244

Notes and References 244

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

Exercises 270

Notes and References 271

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_∞ Performance and Robust Stability 277

12.4 Analysis of Robust Performance 279

12.5 Stability Radius of Norm-Bounded Parametric Systems 282

Exercises 283

Notes and References 287

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

Exercises 310

Notes and References 311

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

Notes and References 321

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

Exercises 342

Notes and References 345

16 H_∞ Control 346

16.1 Control Problem and H_∞ Norm 346

16.2 H_∞ 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_∞ Control 355

Exercises 358

Notes and References 359

17 μ Synthesis 360

17.1 Introduction to μ 360

17.2 Definition of μ and Its Implication 364

17.3 Properties of μ 365

17.4 Condition for Robust H_∞ Performance 368

17.5 D–K Iteration Design 369

17.6 Case Study 371

Exercises 373

Notes and References 374

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_∞ Control Design of Polytopic Systems 379

18.4 Robust H_∞ Control Design of Norm-Bounded Parametric Systems 382

Exercises 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 39219.4 Feedback Controller Design 394

19.5 Analysis of Robust Pole Placement 396

19.6 Robust Design of Regional Pole Placement 402

Exercises 405

Notes and References 406

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

Exercises 434

Notes and References 435

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

Professor Kang-Zhi Liu, Dept. of Electrical and Electronic Engineering, Chiba University, Japan. Professor Liu achieved his Ph.D. degree in 1991 from Chiba University, Japan.  His areas of expertise include Control Theory, Control and Operation of Power Systems, and System Integration of Smart-Grid, and he has worked in these related areas for 27 years (4 years as a professor, 13 years as an associate professor, 5 years as an assistant professor, and 5 years as a graduate student). He is currently Associate Editor of both the International Journal of Control Theory and Applications, and the International Journal of Systems Science.  He is the author of 6 books (two in Chinese and four in Japanese).

Dr. Yu Yao is a Cheng Kong Scholar Chair Professor at the Harbin Institute of Technology, China. He also serves as Vice President of Harbin University of Engineering, China. His research interests include nonlinear systems, robust control and flight control. He has published over 100 journal papers.