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