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1 Introduction1

1.1 What Is This Book About？1

1.2 Highlights of This Book3

1.3 Notes and References9

1.4 Problems10

2 Linear Algebra11

2.1 Linear Subspaces11

2.2 Eigenvalues and Eigenvectors12

2.3 Matrix Inversion Formulas13

2.4 Invariant Subspaces15

2.5 Vector Norms and Matrix Norms16

2.6 Singular Value Decomposition19

2.7 Semidefinite Matrices23

2.8 Notes and References24

2.9 Problems24

3 Linear Systems27

3.1 Descriptions of Linear Dynamical Systems27

3.2 Controllability and Observability28

3.3 Observers and Observer-Based Controllers31

3.4 Operations on Systems34

3.5 State-Space Realizations for Transfer Matrices35

3.6 Multivariable System Poles and Zeros38

3.7 Notes and References41

3.8 Problems42

4 H2 and H∞ Spaces45

4.1 Hilbert Spaces45

4.2 H2 and H∞ Spaces47

4.3 Computing L2 and H2 Norms53

4.4 Computing L∞ and H∞ Norms55

4.5 Notes and References61

4.6 Problems62

5 Internal Stability65

5.1 Feedback Structure65

5.2 Well-Posedness of Feedback Loop66

5.3 Internal Stability68

5.4 Coprime Factorization over RH∞71

5.5 Notes and References77

5.6 Problems77

6 Performance Specifications and Limitations81

6.1 Feedback Properties81

6.2 Weighted H2 and H∞ Performance85

6.3 Selection of Weighting Functions89

6.4 Bode’s Gain and Phase Relation94

6.5 Bode’s Sensitivity Integral98

6.6 Analyticity Constraints100

6.7 Notes and References102

6.8 Problems102

7 Balanced Model Reduction105

7.1 Lyapunov Equations106

7.2 Balanced Realizations107

7.3 Model Reduction by Balanced Truncation117

7.4 Frequency-Weighted Balanced Model Reduction124

7.5 Notes and References126

7.6 Problems127

8 Uncertainty and Robustness129

8.1 Model Uncertainty129

8.2 Small Gain Theorem137

8.3 Stability under Unstructured Uncertainties141

8.4 Robust Performance147

8.5 Skewed Specifications150

8.6 Classical Control for MIMO Systems154

8.7 Notes and References157

8.8 Problems158

9 Linear Fractional Transformation165

9.1 Linear Fractional Transformations165

9.2 Basic Principle173

9.3 Redheffer Star Products178

9.4 Notes and References180

9.5 Problems181

10 μ and μ Synthesis183

10.1 General Framework for System Robustness184

10.2 Structured Singular Value187

10.3 Structured Robust Stability and Performance200

10.4 Overview of μ Synthesis213

10.5 Notes and References216

10.6 Problems217

11 Controller Parameterization221

11.1 Existence of Stabilizing Controllers222

11.2 Parameterization of All Stabilizing Controllers224

11.3 Coprime Factorization Approach228

11.4 Notes and References231

11.5 Problems231

12 Algebraic Riccati Equations233

12.1 Stabilizing Solution and Riccati Operator234

12.2 Inner Functions245

12.3 Notes and References246

12.4 Problems246

13 H2 Optimal Control253

13.1 Introduction to Regulator Problem253

13.2 Standard LQR Problem255

13.3 Extended LQR Problem258

13.4 Guaranteed Stability Margins of LQR259

13.5 Standard H2 Problem261

13.6 Stability Margins of H2 Controllers265

13.7 Notes and References267

13.8 Problems267

14 H∞ Control269

14.1 Problem Formulation269

14.2 A Simplified H∞ Control Problem270

14.3 Optimality and Limiting Behavior282

14.4 Minimum Entropy Controller286

14.5 An Optimal Controller286

14.6 General H∞ Solutions288

14.7 Relaxing Assumptions291

14.8 H2 and H∞ Integral Control294

14.9 H∞ Filtering297

14.10Notes and References299

14.11 Problems300

15 Controller Reduction305

15.1 H∞ Controller Reductions306

15.2 Notes and References312

15.3 Problems313

16 H∞ Loop Shaping315

16.1 Robust Stabilization of Coprime Factors315

16.2.Loop-Shaping Design325

16.3 Justification for H∞ Loop Shaping328

16.4 Further Guidelines for Loop Shaping334

16.5 Notes and References341

16.6 Problems342

17 Gap Metric and v-Gap Metric349

17.1 Gap Metric350

17.2 v-Gap Metric357

17.3 Geometric Interpretation of v-Gap Metric370

17.4 Extended Loop-Shaping Design373

17.5 Controller Order Reduction375

17.6 Notes and References375

17.7 Problems375

18 Miscellaneous Topics377

18.1 Model Validation377

18.2 Mixed μ Analysis and Synthesis381

18.3 Notes and References389

18.4 Problems390

Bibliography391

Index407