Algorithms for Linear-quadratic Optimization
1996
Contents
Preface iii
1 Linear-Quadratic Optimization Problems 1
1.1 Standard Linear-Quadratic Optimization Problems 2
1.1.1 Basic theory 3
1.1.2 Standard control problems 17
1.2 Problems Reducible to Standard Linear-Quadratic Optimiza-
tion Problems 20
1.2.1 Related problems and control schemes 20
1.2.2 Numerical example 27
1.3 Robust Control 36
1.3.1 Performance specification and the Ято optimization
problem " 37
1.3.2 Standard robust control problem 42
1.3.3 Hi optimal control 49
1.4 Basic Numerical Linear Algebra Algorithms and Software . 53
1.4.1 Algorithms and programs 53
1.4.2 Numerical issues 55
1.4.3 Basic algorithms 58
1.5 Overview of Algorithms 68
1.5.1 Direct iteration algorithms 70
1.5.2 Doubling algorithms 74
1.5.3 Algorithms for sequential state estimation 77
vi Contents
1.5.4 Defect correction 85
References 87
2 Newton Algorithms 97
2.1 Basic Theory 98
2.1.1 Newton's method 98
2.1.2 Stabilization methods 101
2.2 Computation of Real Schur Form and Invariant Subspaces . 105
2.2.1 Basic definitions and properties 105
2.2.2 Preprocessing algorithms 114
2.2.3 The QR algorithm 118
2.2.4 Real Schur form computation and ordering 135
2.3 Solving Sylvester and Lyapunov Equations 143
2.3.1 Solving Sylvester equations 144
2.3.2 Solving Lyapunov equations 159
2.3.3 Solving stable non-negative definite Lyapunov equa-
tions 162
2.4 Stabilization Algorithms 174
2.4.1 Full stabilization algorithms 174
2.4.2 Partial stabilization algorithms 177
2.5 Newton-Based Riccati Solvers 179
2.5.1 Algorithmic templates 180
2.5.2 Computational issues 183
2.5.3 Applicability and limitations 186
References 191
3 Schur and Generalized Schur Algorithms 197
3.1 Basic Theory 198
3.1.1 Schur vectors method 199
3.1.2 Generalized Schur vectors method 202
3.2 Computation of Generalized Real Schur Form and Deflating
Subspaces 211
3.2.1 Basic definitions and properties 212
3.2.2 Computation of generalized real Schur form 217
3.2.3 Ordering generalized real Schur form . , 233
3.3 Schur-Based Riccati Solvers 243
3.3.1 Algorithmic templates 243
3.3.2 Computational issues 250
3.3.3 Applicability and limitations 252
3.4 Generalized Schur-Based Riccati Solvers 260
3.4.1 Algorithmic templates 260
3.4.2 Computational issues 266
Contents vii
3.4.3 Applicability and limitations 267
References 275
4 Structure-Preserving Algorithms 281
4.1 Basic Theory 282
4.1.1 Matrix sign function method 283
4.1.2 Structure-preserving QR-type methods 287
4.1.3 Multishift method 295
4.2 Matrix Sign Function Algorithm 298
4.2.1 Algorithmic templates 298
4.2.2 Computational issues 300
4.2.3 Applicability and limitations 303
4.3 Structure-Preserving QR-Type Algorithms . 306
4.3.1 Algorithmic templates 306
4.3.2 Computational issues 322
4.3.3 Applicability and limitations 325
4.4 Multishift Algorithm 330
4.4.1 Algorithmic templates . . 330
4.4.2 Computational issues 336
4.4.3 Applicability and limitations 336
References 339
Appendixes
A Comparison of Riccati Solvers 345
В Notation and Abbreviations 353
B.I Notation 353
B.2 Abbreviations 356
Index of Algorithms Definitions 357
Index 359