Mathematical Methods in Robust Control of Discrete-Time Linear Stochastic Systems (eBook)

Preface 5
Contents 7
1 Elements of probability theory 11
1.1 Probability spaces 11
1.2 Random variables 13
1.2.1 Definitions and basic results 13
1.2.2 Integrable random variables. Expectation 13
1.2.3 Independent random variables 15
1.3 Conditional expectation 15
1.4 Markov chains 17
1.4.1 Stochastic matrices 17
1.4.2 Markov chains 17
1.5 Some remarkable sequences of random variables 20
1.6 Discrete-time controlled stochastic linear systems 22
1.7 The outline of the book 27
1.8 Notes and references 29
2 Discrete-time linear equations defined by positive operators 30
2.1 Some preliminaries 30
2.1.1 Convex cones 30
2.1.2 Minkovski seminorms and Minkovski norms 32
2.2 Discrete-time equations defined by positive linear operators on ordered Hilbert spaces 36
2.2.1 Positive linear operators on ordered Hilbert spaces 36
2.2.2 Discrete-time affine equations 40
2.3 Exponential stability 42
2.4 Some robustness results 51
2.5 Lyapunov-type operators 54
2.5.1 Sequences of Lyapunov-type operators 54
2.5.2 Exponential stability 57
2.5.3 Several special cases 64
2.5.4 A class of generalized Lyapunov-type operators 66
2.6 Notes and references 67
3 Mean square exponential stability 68
3.1 Some representation theorems 69
3.2 Mean square exponential stability. The general case 77
3.3 Lyapunov-type criteria 85
3.4 The case of homogeneous Markov chain 86
3.5 Some special cases 88
3.5.1 The periodic case 88
3.5.2 The time-invariant case 93
3.5.3 Another particular case 95
3.6 The case of the systems with coefficients depending upon .t and .t-1 98
3.7 Discrete-time affine systems 104
3.8 Notes and references 110
4 Structural properties of linear stochastic systems 111
4.1 Stochastic stabilizability and stochastic detectability 111
4.1.1 Definitions and criteria for stochastic stabilizability and stochastic detectability 111
4.1.2 A stability criterion 115
4.2 Stochastic observability 119
4.3 Some illustrative examples 129
4.4 A generalization of the concept of uniform observability 131
4.5 The case of the systems with coefficients depending upon .t, .t-1 134
4.6 A generalization of the concept of stabilizability 136
4.7 Notes and references 137
5 Discrete-time Riccati equations of stochastic control 138
5.1 An overview on discrete-time Riccati-type equations of stochastic control 138
5.2 A class of discrete-time backward nonlinear equations 142
5.2.1 Several notations 142
5.2.2 A class of discrete-time generalized Riccati equations 143
5.3 A comparison theorem and several consequences 147
5.4 The maximal solution 148
5.5 The stabilizing solution 155
5.6 The Minimal Solution 161
5.7 An iterative procedure to compute the maximal solution and the stabilizing solution of DTSGRE 165
5.8 Discrete-time Riccati equations of stochastic control 173
5.8.1 The maximal solution and the stabilizing solution of DTSRE-C 173
5.8.2 The case of DTSRE-C with definite sign of weighting matrices 177
5.8.3 The case of the systems with coefficients depending upon .t and .t-1 180
5.9 Discrete-time Riccati filtering equations 184
5.10 A numerical example 188
5.11 Notes and references 190
6 Linear quadratic optimization problems 191
6.1 Some preliminaries 191
6.1.1 A brief discussion on the linear quadratic optimization problems 191
6.1.2 A usual class of stochastic processes 193
6.1.3 Several auxiliary results 193
6.2 The problem of the linear quadratic regulator 199
6.3 The linear quadratic optimization problem 202
6.4 The linear quadratic problem. The affine case 210
6.4.1 The problem setting 211
6.4.2 Solution of the problem OP 1 212
6.4.3 On the global bounded solution of (6.11) 214
6.4.4 The solution of the problem OP 2 217
6.5 Tracking problems 222
6.6 Notes and references 227
7 Discrete-time stochastic H2 optimal control 228
7.1 H2 norms of discrete-time linear stochastic systems 229
7.1.1 Model setting 229
7.1.2 H2-type norms 230
7.1.3 Systems with coefficients depending upon .t and .t-1 231
7.2 The computation of H2-type norms 231
7.2.1 The computations of the norm G 2 and the norm ˜ G˜ 2 232
7.2.2 The computation of the norm |||G|||2 241
7.2.3 The computation of the H2 norms for the system of type (7.1) 246
7.3 Some robustness issues 250
7.4 H2 optimal controllers. The case with full access to measurements 252
7.4.1 H2 optimization 252
7.4.2 The case of systems with coefficients depending upon .t and .t-1 258
7.5 The H2 optimal control. The case with partial access to measurements 260
7.5.1 Problem formulation 260
7.5.2 Some preliminaries 262
7.5.3 The solution of the H2 optimization problems 265
7.6 H2 suboptimal controllers in a state estimator form 270
7.7 An H2 filtering problem 279
7.8 A case study 287
7.9 Notes and references 288
8 Robust stability and robust stabilization of discrete-time linear stochastic systems 291
8.1 A brief motivation 291
8.2 Input–output operators 293
8.3 Stochastic version of bounded real lemma 302
8.3.1 Stochastic bounded real lemma. The finite horizon time case 303
8.3.2 The bounded real lemma. The infinite time horizon case 306
8.3.3 An H8-type filtering problem 316
8.4 Robust stability. An estimate of the stability radius 322
8.4.1 The small gain theorems 322
8.4.2 An estimate of the stability radius 328
8.5 The disturbance attenuation problem 331
8.5.1 The problem formulation 331
8.5.2 The solution of the disturbance attenuation problem. The case of full state measurements 333
8.5.3 Solution of a robust stabilization problem 339
8.6 Notes and references 340
Bibliography 341
Abbreviations 347
Index 348