A State Space Approach to Canonical Factorization with Applications (eBook)

Table of Contents 6
Preface 11
Chapter 0 Introduction 13
Part I Convolution equations, canonical factorization and the state space method 19
Chapter 1 The role of canonical factorization in solving convolution equations 20
1.1 Wiener-Hopf integral equations and factorization 20
1.2 Block Toeplitz equations and factorization 24
1.3 Singular integral equations and factorization 26
Notes 28
Chapter 2 The state space method and factorization 29
2.1 Preliminaries on realization 29
2.2 Realization of rational matrix functions 32
2.3 Realization of analytic operator functions 33
2.4 Inversion 36
2.5 Products 37
2.6 Factorization 40
Notes 43
Part II Convolution equations with rational matrix symbols 44
Chapter 3 Explicit solutions using realizations 45
3.1 Canonical factorization of rational matrix functions in state space form 45
3.2 Wiener-Hopf integral operators 50
3.3 Block Toeplitz operators 54
3.4 Singular integral equations 58
3.5 The Riemann-Hilbert boundary value problem 59
Notes 64
Chapter 4 Factorization of non-proper rational matrix functions 65
4.1 Preliminaries about matrix pencils 65
4.2 Realization of a non-proper rational matrix function 67
4.3 Explicit canonical factorization 69
4.4 Inversion of singular operators with a rational matrix symbol 76
4.5 The Riemann-Hilbert boundary value problem revisited (1) 79
Notes 82
Part III Equations with non-rational symbols 83
Chapter 5 Factorization of matrix functions analytic in a strip 84
5.1 Exponentially dichotomous operators and bisemigroups 85
5.2 Spectral splitting and proof of Theorem 5.2 88
5.3 Realization triples 94
5.4 Construction of realization triples 98
5.5 Inverting matrix functions analytic in a strip 100
5.6 Inverting full line convolution operators 105
5.7 Inverting Wiener-Hopf integral operators 107
5.8 Explicit canonical factorization 113
5.9 The Riemann-Hilbert boundary value problem revisited (2) 118
Notes 120
Chapter 6 Convolution equations and the transport equation 121
6.1 The transport equation 122
6.2 The case of a finite number of scattering directions 124
6.3 Wiener-Hopf equations with operator-valued kernel functions 128
6.4 Construction of a canonical factorization 130
6.5 The matching of the subspaces 141
6.6 Formulas for solutions 144
Notes 148
Chapter 7 Wiener-Hopf factorization and factorization indices 149
7.1 Canonical factorization of operator functions 149
7.2 Proof of Theorem 7.2 153
7.3 Wiener-Hopf factorization and spectral invariants 163
Notes 173
Part IV Factorization of selfadjoint rational matrix functions 174
Chapter 8 Preliminaries concerning minimal factorization 175
8.1 Minimal realizations 175
8.2 Minimal factorization 178
8.3 Pseudo-canonical factorization 180
Notes 182
Chapter 9 Factorization of positive definite rational matrix functions 184
9.1 Preliminaries on selfadjoint rational matrix functions 184
9.2 Spectral factorization 188
9.3 Positive definite functions on the unit circle 192
Notes 198
Chapter 10 Pseudo-spectral factorizations of selfadjoint rational matrix functions 200
10.1 Nonnegative rational matrix functions 200
10.2 Selfadjoint rational matrix functions and further generalizations 208
Notes 212
Chapter 11 Review of the theory of matrices in indefinite inner product spaces 213
11.1 Subspaces of indefinite inner product spaces 213
11.2 H-selfadjoint matrices 214
11.3 H-dissipative matrices 217
Notes 218
Part V Riccati equations and factorization 219
Chapter 12 Canonical factorization and Riccati equations 220
12.1 Preliminaries on spectral angular subspaces 220
12.2 Angular operators and factorization 222
12.3 Riccati equations and canonical factorization 228
12.4 Left versus right canonical factorization 230
Notes 232
Chapter 13The symmetric algebraic Riccatiequation 233
13.1 Spectral factorization and Riccati equations 233
13.2 Stabilizing solutions 238
13.3 Symmetric Riccati equations and pseudo-spectral factorization 242
Notes 247
Chapter 14 J-spectral factorization 248
14.1 Definition of J-spectral factorization 248
14.2 J-spectral factorizations and invariant subspaces 250
14.3 J-spectral factorizations and Riccati equations 255
14.4 Two special cases of J-spectral factorization 258
14.5 J-spectral factorization with respect to other contours 261
14.6 Left versus right J-spectral factorization 272
14.7 J-spectral factorization relative to the unit circle revisited 275
Notes 287
Part VI Factorizations and symmetries 288
Chapter 15 Factorization of positive realrational matrix functions 289
15.1 Rational matrix functions with a positive definite real part 289
15.2 Canonical factorization of functions with a positive definite real part 292
15.3 Generalization to pseudo-canonical factorization 295
Notes 298
Chapter 16 Contractive rational matrix functions 299
16.1 State space analysis of contractive rational matrix functions 299
16.2 Strictly contractive rational matrix functions 302
16.3 An application to spectral factorization 303
16.4 An application to canonical factorization 305
16.5 A generalization to pseudo-canonical factorization 307
Notes 310
Chapter 17 J-unitary rational matrix functions 311
17.1 Realizations of J-unitary rational matrix functions 311
17.2 Factorization of J-unitary rational matrix functions 319
17.3 Factorization of unitary rational matrix functions 322
17.4 Intermezzo on the Redheffer transformation 326
17.5 J-inner rational matrix functions 331
17.6 Inner-outer factorization 334
17.7 Unitary completions of minimal degree 337
17.8 Bi-inner completions of inner functions 339
Notes 343
Part VII Applications of J-spectral factorizations 344
Chapter 18 Application to the rational Nehari problem 345
18.1 Problem statement and main result 345
18.2 Intermezzo about linear fractional maps 348
18.3 The J-spectral factorization approach 355
18.4 Proof of the main result 357
18.5 The case of a non-stable given function 362
18.6 The Nehari-Takagi problem 364
Notes 366
Chapter 19 Review of some control theory for linear systems 367
19.1 Stability and feedback 367
19.2 Parametrization of internally stabilizing compensators 370
Notes 374
Chapter 20 H-infinity control applications 375
20.1 The standard problem and model matching 376
20.2 The one-sided model matching problem 378
20.3 The two-sided model matching problem 382
20.4 State space solution of the standard problem 388
Notes 400
Bibliography 401
List of symbols 411
Index 415