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
