 | Dragan V. Mathematical methods in robust control of discrete-time linear stochastic systems / V.Dragan, T.Morozan, A.-M.Stoica. - New York: Springer, 2010. - x, 346 p.: ill. - Bibliogr.: p.337-342. - Ind.: p.345-346. - ISBN 978-1-4419-0629-8
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Preface ......................................................... v
1 Elements of probability theory ............................... 1
1.1 Probability spaces ...................................... 1
1.2 Random variables ........................................ 3
1.2.1 Definitions and basic results .................... 3
1.2.2 Integrable random variables. Expectation ......... 3
1.2.3 Independent random variables ..................... 5
1.3 Conditional expectation ................................. 5
1.4 Markov chains ........................................... 7
1.4.1 Stochastic matrices .............................. 7
1.4.2 Markov chains .................................... 7
1.5 Some remarkable sequences of random variables .......... 10
1.6 Discrete-time controlled stochastic linear systems ..... 12
1.7 The outline of the book ................................ 17
1.8 Notes and references ................................... 19
2 Discrete-time linear equations .............................. 21
2.1 Some preliminaries ..................................... 21
2.1.1 Convex cones .................................... 21
2.1.2 Minkovski seminorms and Minkovski norms ......... 23
2.2 Discrete-time equations defined by positive
operators .............................................. 27
2.2.1 Positive linear operators on ordered Hilbert
spaces .......................................... 27
2.2.2 Discrete-time affine equations .................. 31
2.3 Exponential stability .................................. 33
2.4 Some robustness results ................................ 42
2.5 Lyapunov-type operators ................................ 45
2.5.1 Sequences of Lyapunov-type operators ............ 45
2.5.2 Exponential stability ........................... 48
2.5.3 Several special cases ........................... 55
2.5.4 A class of generalized Lyapunov-type
operators ....................................... 57
2.6 Notes and references .................................. 58
3 Mean square exponential stability ........................... 59
3.1 Some representation theorems ........................... 60
3.2 Mean square exponential stability. The general case .... 68
3.3 Lyapunov-type criteria ................................. 76
3.4 The case of homogeneous Markov chain ................... 77
3.5 Some special cases ..................................... 79
3.5.1 The periodic case ............................... 79
3.5.2 The time-invariant case ......................... 84
3.5.3 Another particular case ......................... 86
3.6 The case of the systems with coefficients depending
upon ηt and ηt-1 ....................................... 89
3.7 Discrete-time affine systems ........................... 95
3.8 Notes and references .................................. 101
4 Structural properties of linear stochastic systems ......... 103
4.1 Stochastic stabilizability and stochastic
detectability ......................................... 103
4.1.1 Definitions and criteria for stochastic
stabilizability and stochastic detectability ... 103
4.1.2 A stability criterion .......................... 107
4.2 Stochastic observability .............................. 1ll
4.3 Some illustrative examples ............................ 121
4.4 A generalization of the concept of uniform
observability ......................................... 123
4.5 The case of the systems with coefficients depending
upon ηt, ηt-1 ......................................... 126
4.6 A generalization of the concept of stabilizability .... 128
4.7 Notes and references .................................. 129
5 Discrete-time Riccati equations ............................ 131
5.1 An overview on discrete-time Riccati-type equations ... 131
5.2 A class of discrete-time backward nonlinear
equations ............................................. 135
5.2.1 Several notations .............................. 135
5.2.2 A class of discrete-time generalized Riccati
equations ...................................... 136
5.3 A comparison theorem and several consequences ......... 140
5.4 The maximal solution .................................. 141
5.5 The stabilizing solution .............................. 148
5.6 The Minimal Solution .................................. 154
5.7 An iterative procedure ................................ 158
5.8 Discrete-time Riccati equations of stochastic
control ............................................... 166
5.8.1 The maximal solution and the stabilizing
solution of DTSRE-C ............................ 166
5.8.2 The case of DTSRE-C with definite sign of
weighting matrices ............................. 170
5.8.3 The case of the systems with coefficients
depending upon ηt and ηt-1 ..................... 173
5.9 Discrete-time Riccati filtering equations ............. 177
5.10 A numerical example ................................... 181
5.11 Notes and references .................................. 183
6 Linear quadratic optimization problems ..................... 185
6.1 Some preliminaries .................................... 185
6.1.1 A brief discussion on the linear quadratic
optimization problems .......................... 185
6.1.2 A usual class of stochastic processes .......... 187
6.1.3 Several auxiliary results ...................... 187
6.2 The problem of the linear quadratic regulator ......... 193
6.3 The linear quadratic optimization problem ............. 196
6.4 The linear quadratic problem. The affine case ......... 204
6.4.1 The problem setting ............................ 205
6.4.2 Solution of the problem OP 1 ................... 206
6.4.3 On the global bounded solution of (6.11) ....... 208
6.4.4 The solution of the problem OP 2 ............... 211
6.5 Tracking problems ..................................... 216
6.6 Notes and references .................................. 221
7 Discrete-time stochastic H2 optimal control ................ 223
7.1 H2 norms of discrete-time linear stochastic systems ... 224
7.1.1 Model setting .................................. 224
7.1.2 H2-type norms .................................. 225
7.1.3 Systems with coefficients depending upon ηt
and ηt-1 ....................................... 226
7.2 The computation of H2-type norms ...................... 226
7.2.1 The computations of the norm  Gt2 and
the norm  G2 ................................... 227
7.2.2 The computation of the norm  G2 ............... 236
7.2.3 The computation of the H2 norms for the
system of type (7.1) ........................... 241
7.3 Some robustness issues ................................ 245
7.4 The case with full access to measurements ............. 247
7.4.1 H2 optimization ................................ 247
7.4.2 The case of systems with coefficients
depending upon ηt and ηt-1 ..................... 253
7.5 The case with partial access to measurements .......... 255
7.5.1 Problem formulation ............................ 255
7.5.2 Some preliminaries ............................. 257
7.5.3 The solution of the H2 optimization problems ... 260
7.6 H2 suboptimal controllers in a state estimator form ... 265
7.7 An H2 filtering problem ............................... 274
7.8 A case study .......................................... 282
7.9 Notes and references .................................. 283
8 Robust stability and robust stabilization .................. 287
8.1 A brief motivation .................................... 287
8.2 Input-output operators ................................ 289
8.3 Stochastic version of bounded real lemma .............. 298
8.3.1 Stochastic bounded real lemma. The finite
horizon time case .............................. 299
8.3.2 The bounded real lemma. The infinite time
horizon case ................................... 302
8.3.3 An H∞-type filtering problem ................... 312
8.4 Robust stability. An estimate of the stability
radius ................................................ 318
8.4.1 The small gain theorems ........................ 318
8.4.2 An estimate of the stability radius ............ 324
8.5 The disturbance attenuation problem ................... 327
8.5.1 The problem formulation ........................ 327
8.5.2 The solution of the disturbance attenuation
problem. The case of full state measurements ... 329
8.5.3 Solution of a robust stabilization problem ..... 335
8.6 Notes and references .................................. 336
Bibliography .................................................. 337
Abbreviations ................................................. 343
Index ......................................................... 345
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