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1 Introduction 1

1.1 What Is This Book About？ 1

1.2 Highlights of This Book 3

1.3 Notes and References 9

1.4 Problems 10

2 Linear Algebra 11

2.1 Linear Subspaces 11

2.2 Eigenvalues and Eigenvectors 12

2.3 Matrix Inversion Formulas 13

2.4 Invariant Subspaces 15

2.5 Vector Norms and Matrix Norms 16

2.6 Singular Value Decomposition 19

2.7 Semidefinite Matrices 23

2.8 Notes and References 24

2.9 Problems 24

3 Linear Systems 27

3.1 Descriptions of Linear Dynamical Systems 27

3.2 Controllability and Observability 28

3.3 Observers and Observer-Based Controllers 31

3.4 Operations on Systems 34

3.5 State-Space Realizations for Transfer Matrices 35

3.6 Multivariable System Poles and Zeros 38

3.7 Notes and References 41

3.8 Problems 42

4 H2 and H∞ Spaces 45

4.1 Hilbert Spaces 45

4.2 H2 and H∞ Spaces 47

4.3 Computing L2 and H2 Norms 53

4.4 Computing L∞ and H∞ Norms 55

4.5 Notes and References 61

4.6 Problems 62

5 Internal Stability 65

5.1 Feedback Structure 65

5.2 Well-Posedness of Feedback Loop 66

5.3 Internal Stability 68

5.4 Coprime Factorization over RH∞ 71

5.5 Notes and References 77

5.6 Problems 77

6 Performance Specifications and Limitations 81

6.1 Feedback Properties 81

6.2 Weighted H2 and H∞ Performance 85

6.3 Selection of Weighting Functions 89

6.4 Bode’s Gain and Phase Relation 94

6.5 Bode’s Sensitivity Integral 98

6.6 Analyticity Constraints 100

6.7 Notes and References 102

6.8 Problems 102

7 Balanced Model Reduction 105

7.1 Lyapunov Equations 106

7.2 Balanced Realizations 107

7.3 Model Reduction by Balanced Truncation 117

7.4 Frequency-Weighted Balanced Model Reduction 124

7.5 Notes and References 126

7.6 Problems 127

8 Uncertainty and Robustness 129

8.1 Model Uncertainty 129

8.2 Small Gain Theorem 137

8.3 Stability under Unstructured Uncertainties 141

8.4 Robust Performance 147

8.5 Skewed Specifications 150

8.6 Classical Control for MIMO Systems 154

8.7 Notes and References 157

8.8 Problems 158

9 Linear Fractional Transformation 165

9.1 Linear Fractional Transformations 165

9.2 Basic Principle 173

9.3 Redheffer Star Products 178

9.4 Notes and References 180

9.5 Problems 181

10 μ and μ Synthesis 183

10.1 General Framework for System Robustness 184

10.2 Structured Singular Value 187

10.3 Structured Robust Stability and Performance 200

10.4 Overview of μ Synthesis 213

10.5 Notes and References 216

10.6 Problems 217

11 Controller Parameterization 221

11.1 Existence of Stabilizing Controllers 222

11.2 Parameterization of All Stabilizing Controllers 224

11.3 Coprime Factorization Approach 228

11.4 Notes and References 231

11.5 Problems 231

12 Algebraic Riccati Equations 233

12.1 Stabilizing Solution and Riccati Operator 234

12.2 Inner Functions 245

12.3 Notes and References 246

12.4 Problems 246

13 H2 Optimal Control 253

13.1 Introduction to Regulator Problem 253

13.2 Standard LQR Problem 255

13.3 Extended LQR Problem 258

13.4 Guaranteed Stability Margins of LQR 259

13.5 Standard H2 Problem 261

13.6 Stability Margins of H2 Controllers 265

13.7 Notes and References 267

13.8 Problems 267

14 H∞ Control 269

14.1 Problem Formulation 269

14.2 A Simplified H∞ Control Problem 270

14.3 Optimality and Limiting Behavior 282

14.4 Minimum Entropy Controller 286

14.5 An Optimal Controller 286

14.6 General H∞ Solutions 288

14.7 Relaxing Assumptions 291

14.8 H2 and H∞ Integral Control 294

14.9 H∞ Filtering 297

14.10Notes and References 299

14.11 Problems 300

15 Controller Reduction 305

15.1 H∞ Controller Reductions 306

15.2 Notes and References 312

15.3 Problems 313

16 H∞ Loop Shaping 315

16.1 Robust Stabilization of Coprime Factors 315

16.2.Loop-Shaping Design 325

16.3 Justification for H∞ Loop Shaping 328

16.4 Further Guidelines for Loop Shaping 334

16.5 Notes and References 341

16.6 Problems 342

17 Gap Metric and v-Gap Metric 349

17.1 Gap Metric 350

17.2 v-Gap Metric 357

17.3 Geometric Interpretation of v-Gap Metric 370

17.4 Extended Loop-Shaping Design 373

17.5 Controller Order Reduction 375

17.6 Notes and References 375

17.7 Problems 375

18 Miscellaneous Topics 377

18.1 Model Validation 377

18.2 Mixed μ Analysis and Synthesis 381

18.3 Notes and References 389

18.4 Problems 390

Bibliography 391

Index 407