Generalized Riccati theory and robust control: a Popov function approach
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester [u.a.]
Wiley
1999
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 380 S. |
| ISBN: | 0471971472 |
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| 245 | 1 | 0 | |a Generalized Riccati theory and robust control |b a Popov function approach |c Vlad Ionescu ; Cristian Oară ; Martin Weiss |
| 264 | 1 | |a Chichester [u.a.] |b Wiley |c 1999 | |
| 300 | |a XXII, 380 S. | ||
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| 700 | 1 | |a Oară, Cristian |e Verfasser |4 aut | |
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Record in the Search Index
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| adam_text | Contents
Preface xiii
Structure of the book and main contributions xv
Acknowledgments xviii
Acronyms, Notation, and Symbols xix
I General Matrix Theory and Linear Dynamical Systems 1
1 Matrices, Matrix Pencils, and Rational Matrix Functions 3
1.1 The eigenvalue problem 3
1.2 The Jordan canonical form of a square matrix 4
1.3 Invariant subspace of a square matrix 5
1.4 Controllability and observability 7
1.5 Linear matrix equations 9
1.6 The generalized eigenvalue problem: the regular case 12
1.7 The Weierstrass canonical form of a regular matrix pencil 13
1.8 Deflating subspace of a regular matrix pencil 14
1.9 The generalized eigenvalue problem: the general case 16
1.10 The Kronecker canonical form of a general matrix pencil 17
1.11 Proper deflating subspace of a general matrix pencil 18
1.12 Structural elements of a rational matrix 22
1.13 Realization theory for rational matrices 24
1.13.1 The proper case 24
1.13.2 The general case 26
1.14 Structural elements in terms of realizations for a rational matrix ... 28
Notes and references 30
2 Linear Dynamical Systems 31
A. The continuous—time case 31
2.1 Linear systems 31
2.1.1 Stable, antistable, and dichotomic systems 32
2.1.2 Controllability, stabilizability, and duals 33
2.1.3 Spaces of rational matrices, norms, and continuity 33
2.1.4 Gramians and Hankel singular values 36
viii CONTENTS
2.1.5 Connections of systems 37
2.2 State evolutions corresponding to L2 inputs and associated operators . 42
2.2.1 The case: A stable 42
2.2.2 The case: A antistable 44
2.2.3 The case: A dichotomic 45
2.2.4 The case: A arbitrary 45
2.3 The L2 input output operator 46
2.4 Hankel and Toeplitz operators 47
B. The discrete—time case 52
2.5 Linear discrete time systems 52
2.5.1 Direct and inverse time systems 52
2.5.2 Stable, antistable, and dichotomic systems 54
2.5.3 Controllability, stabilizability, and duals 55
2.5.4 Spaces of rational matrices, norms, and continuity 55
2.5.5 Gramians and Hankel singular values 57
2.5.6 Bilinear transformation between a continuous and a discrete
time system 58
2.5.7 Continuity of the ff°° norm 59
2.6 State evolutions corresponding to I2 inputs and associated operators . 62
2.6.1 The direct time case: A stable 63
2.6.2 The inverse time case: A stable 65
2.6.3 The direct time case: A antistable 65
2.6.4 The direct time case: A dichotomic 66
2.6.5 The inverse time case: A dichotomic 67
2.6.6 The case: A arbitrary 68
2.7 I2 input output operators 68
2.7.1 The direct time case 68
2.7.2 The inverse time case 69
2.8 Hankel and Toeplitz operators 70
Notes and references 74
II Generalized Riccati Theory 75
3 Popov Triplets 77
A. The continuous—time case 77
3.1 Popov triplets: definition, significance, and equivalence 77
3.2 Objects associated with a Popov triplet 79
3.2.1 Quadratic indices 79
3.2.2 The continuous time algebraic Riccati system (CTARS) .... 80
3.2.3 The continuous time algebraic Riccati equation (CTARE) ... 82
3.2.4 The Kalman Popov Yakubovich system (KPYS) 83
3.2.5 The continuous time Hamiltonian system (CTHS) 85
3.2.6 The extended Hamiltonian pencil (EHP) 86
3.2.7 The continuous time Popov function 86
3.3 Associated operators 88
3.3.1 The case: A dichotomic 88
CONTENTS ix
3.3.2 The case: A stable 89
3.3.3 The case: A antistable 91
3.3.4 Reduction of the case A antistable to the case A stable .... 92
3.3.5 Tle as the input output operator of the CTHS 93
3.4 Particular relevant cases 94
B. The discrete—time case 95
3.5 Popov triplets: the double interpretation, significance, and equivalence 95
3.6 Objects associated with a Popov triplet in discrete time 97
3.6.1 Quadratic indices 97
3.6.2 The discrete time algebraic Riccati system (DTARS) 98
3.6.3 The discrete time algebraic Riccati equation (DTARE) .... 99
3.6.4 The Kalman Szego Popov Yakubovich system (KSPYS) ... 100
3.6.5 The discrete time Hamiltonian system (DTHS) 102
3.6.6 The extended symplectic pencils (ESPs) 103
3.6.7 The discrete time Popov functions 104
3.7 The time reversed Popov triplet 105
3.8 Associated operators 108
3.8.1 The direct time case: A dichotomic 108
3.8.2 The direct time case: A stable 109
3.8.3 The inverse time case: A dichotomic 110
3.8.4 The inverse time case: A stable Ill
3.8.5 Reduction of the inverse time case to the direct time case ... 112
3.8.6 The direct time case: A antistable 114
3.8.7 TZe and TZe as input output operators 115
Notes and references 116
4 Riccati Theory: An Operator based Approach 117
A. The continuous—time case 118
4.1 Existence of the stabilizing solution: the stable case 118
4.1.1 The main result 118
4.1.2 Proof of the implication 2. = 1 119
4.1.3 The positiveness case 121
4.2 Removing the stability assumption 123
4.2.1 The positiveness case: the standard Riccati equation for control 124
4.3 Existence of the stabilizing solution: the antistable case 125
4.3.1 The main result 125
4.3.2 The positiveness case 127
4.4 The signature condition 128
4.5 Frequency domain conditions 132
4.5.1 The main result 132
4.5.2 The positiveness case: A stable 134
4.5.3 The positiveness case: A arbitrary 135
4.5.4 The signature condition in the frequency domain 135
4.6 Algebraic Riccati inequalities 144
B. The discrete time case 145
4.7 Existence of the stabilizing solution: the stable case 145
4.7.1 The main result 146
x CONTENTS
4.7.2 Proof of the implication 2. = 1 147
4.7.3 The positiveness case 150
4.8 Removing the stability assumption 152
4.9 Existence of the stabilizing solution: the inverse time case 152
4.10 Existence of the stabilizing solution: the antistable case 152
4.10.1 The main result 153
4.10.2 The positiveness case 155
4.11 The signature condition 156
4.12 Frequency domain conditions 157
4.12.1 The positiveness case: A arbitrary 159
4.12.2 The signature condition in the frequency domain 159
4.13 Algebraic Riccati inequalities 161
Notes and references 162
5 Riccati Equations and Matrix Pencils: the Regular Case 163
A. The continuous—time case 163
5.1 The eigenstructure of a regular EHP 163
5.2 The CTARE and the EHP 166
5.3 The continuous time Bernoulli equation 169
B. The discrete—time case 171
5.4 The eigenstructure of a regular ESP 171
5.5 The DTARE and the ESP 174
5.6 The discrete time Bernoulli equation 177
5.7 The time reversed DTARE 179
Notes and references 185
6 Riccati Systems and Matrix Pencils: the General Case 187
A. The continuous—time case 187
6.1 The eigenstructure of a general EHP 187
6.2 The generalized continuous time Riccati system and the EHP 191
B. The discrete—time case 194
6.3 The eigenstructure of a singular ESP 194
6.4 The generalized discrete time Riccati system and the ESP 201
6.5 Numerical algorithms 204
6.5.1 Basic algorithms 204
6.5.2 Computation of proper deflating subspaces 209
6.5.3 Computation of solutions to Riccati equations and systems . . 212
Notes and references 214
III Applications to Systems Theory and Robust Control 215
7 Applications to Systems Theory 217
A. The continuous—time case 217
7.1 The bounded real lemma 217
7.2 The CTARE associated with the positiveness Popov triplet 219
7.3 Normalized coprime factorizations 224
CONTENTS xi
7.4 The Small Gain Theorem 225
7.5 A quadratic index with constrained dynamics 226
7.6 Spectral and inner outer factorizations: the arbitrary rank case .... 228
B. The discrete—time case 231
7.7 The bounded real lemma 231
7.8 The DTARE associated with the positiveness Popov triplet 232
7.9 Normalized coprime factorizations 233
7.10 The Small Gain Theorem 234
7.11 A quadratic index with constrained dynamics 235
7.12 Spectral and inner outer factorization with respect to the unit circle . 237
Notes and references 240
8 The Four Block Nehari Problem 241
A. The continuous—time case 241
8.1 The Nehari problem and the signature condition 241
8.2 Parrott s problem 245
8.3 Necessary and sufficient conditions 247
8.4 Uncoupled solvability conditions 254
B. The discrete time case 257
8.5 The four block Nehari problem 257
Notes and references 262
9 The Optimal #2 Control Problem 263
A. The continuous—time case 263
9.1 Problem formulation 263
9.1.1 An evaluation of the i?2 norm 265
9.2 Main result 265
B. The discrete—time case 271
9.3 Problem formulation 271
9.3.1 An evaluation of the H2 norm 271
9.4 Main result 272
Notes and references 274
10 The H°° Control Problem 275
A. The continuous—time case 275
10.1 Problem formulation 275
10.2 Basic assumptions 278
10.3 The solution 279
10.3.1 The solution under normalizing conditions 281
10.3.2 The plan of the proof 283
10.4 Redheffer s theorem 284
10.5 Necessity of (Cl) and (C2) 287
10.6 An auxiliary necessary condition 289
10.7 Necessity of (C3) 293
10.8 Proof of sufficiency 296
10.8.1 The disturbance estimation (DE) case 297
10.8.2 The disturbance feedforward (DF) case 300
xii CONTENTS
10.8.3 The output estimation (OE) case 301
10.8.4 The general case 302
10.9 State feedback solution 302
B. The discrete—time case 305
10.10 Problem formulation 305
10.11 Basic assumptions 306
10.12 The solution 307
10.13 Redheffer s theorem 309
10.14 Necessity of (CD1) and (CD2) 313
10.15 An auxiliary necessary condition 313
10.16 Necessity of (CD3) 315
10.17 Proof of sufficiency 318
10.17.1 The disturbance estimation (DE) case 318
10.17.2 The disturbance feedforward (DF) case 319
10.17.3 The output estimation (OE) case 320
10.17.4 The general case 320
10.18 State feedback solution 326
Notes and References 328
11 Robust Stabilization 329
A. The continuous—time case 329
11.1 Problem formulation and some prerequisites 329
11.2 Optimal solution to the DF problem 332
11.2.1 An evaluation of 7m;n 333
11.2.2 An optimal solution 336
11.3 Robust stabilization for normalized coprime factors uncertainties . . . 343
11.3.1 An evaluation of the maximum stability margin 344
11.3.2 An optimal robustly stabilizing controller 346
11.4 Robust stabilization for multiplicative uncertainties 349
B. The discrete—time case 352
11.5 Problem formulation and some prerequisites 352
11.6 Optimal solution to the DF problem 352
11.6.1 An evaluation of 7m;n 353
11.6.2 An optimal solution 353
11.7 Robust stabilization for normalized coprime factors uncertainties . . . 359
11.7.1 An evaluation of the maximum stability margin 360
11.7.2 An optimal robustly stabilizing controller 363
11.8 Robust stabilization for multiplicative uncertainties 366
Notes and references 368
References 369
Index 375
|
| any_adam_object | 1 |
| author | Ionescu, Vlad Oară, Cristian Weiss, Martin 1928-2020 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.64 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV012522561 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:29:02Z |
| institution | BVB |
| isbn | 0471971472 |
| language | English |
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| spelling | Ionescu, Vlad Verfasser aut Generalized Riccati theory and robust control a Popov function approach Vlad Ionescu ; Cristian Oară ; Martin Weiss Chichester [u.a.] Wiley 1999 XXII, 380 S. txt rdacontent n rdamedia nc rdacarrier Commande, Théorie de la ram Linear control systems Riccati, Equation de ram Systèmes linéaires ram Vergelijking van Riccati gtt Control theory Linear systems Riccati equation Algebraische Riccati-Gleichung (DE-588)4434741-8 gnd rswk-swf Robuste Kontrolle (DE-588)4232797-0 gnd rswk-swf Algebraische Riccati-Gleichung (DE-588)4434741-8 s Robuste Kontrolle (DE-588)4232797-0 s DE-604 Oară, Cristian Verfasser aut Weiss, Martin 1928-2020 Verfasser (DE-588)119228645 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008500985&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ionescu, Vlad Oară, Cristian Weiss, Martin 1928-2020 Generalized Riccati theory and robust control a Popov function approach Commande, Théorie de la ram Linear control systems Riccati, Equation de ram Systèmes linéaires ram Vergelijking van Riccati gtt Control theory Linear systems Riccati equation Algebraische Riccati-Gleichung (DE-588)4434741-8 gnd Robuste Kontrolle (DE-588)4232797-0 gnd |
| subject_GND | (DE-588)4434741-8 (DE-588)4232797-0 |
| title | Generalized Riccati theory and robust control a Popov function approach |
| title_auth | Generalized Riccati theory and robust control a Popov function approach |
| title_exact_search | Generalized Riccati theory and robust control a Popov function approach |
| title_full | Generalized Riccati theory and robust control a Popov function approach Vlad Ionescu ; Cristian Oară ; Martin Weiss |
| title_fullStr | Generalized Riccati theory and robust control a Popov function approach Vlad Ionescu ; Cristian Oară ; Martin Weiss |
| title_full_unstemmed | Generalized Riccati theory and robust control a Popov function approach Vlad Ionescu ; Cristian Oară ; Martin Weiss |
| title_short | Generalized Riccati theory and robust control |
| title_sort | generalized riccati theory and robust control a popov function approach |
| title_sub | a Popov function approach |
| topic | Commande, Théorie de la ram Linear control systems Riccati, Equation de ram Systèmes linéaires ram Vergelijking van Riccati gtt Control theory Linear systems Riccati equation Algebraische Riccati-Gleichung (DE-588)4434741-8 gnd Robuste Kontrolle (DE-588)4232797-0 gnd |
| topic_facet | Commande, Théorie de la Linear control systems Riccati, Equation de Systèmes linéaires Vergelijking van Riccati Control theory Linear systems Riccati equation Algebraische Riccati-Gleichung Robuste Kontrolle |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008500985&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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