Table of Contents
Preface ix
List of Tables xiii
List of Figures xv
Symbols and Acronyms xvii
1 Mathematical Preliminaries 1
1.1 Stochastic Differential Equations 1
1.1.1 Existence and uniqueness of solutions 1
1.1.2 Itô's formula 6
1.1.3 Various definitions of stability 10
1.2 Generalized Lyapunov Operators 12
1.3 Basic Concepts of Stochastic Systems 17
1.3.1 Exact observability 17
1.3.2 Exact delegability 23
1.3.3 Mean square stabilization 26
1.4 Notes and References 28
2 Linear Continuous-Time Stochastic H2/H∞ Control 29
2.1 Introduction 29
2.2 Finite Horizon H2/H∞ Control 30
2.2.1 Definitions and lemmas 31
2.2.2 Finite horizon stochastic bounded real lemma (SBRL) 34
2.2.3 Finite horizon stochastic LQ control 39
2.2.4 Conditions for the existence of Nash equilibrium strategies 41
2.2.5 Main results 44
2.2.6 Unified treatment of H2, H∞ and mixed H2/H∞ control problems 52
2.3 Infinite Horizon H2/H∞ Control 55
2.3.1 Two Lyapunov-type theorems 57
2.3.2 Infinite horizon stochastic LQ control 61
2.3.3 Infinite horizon SBRL 70
2.3.4 Stochastic H2/H∞ control 76
2.4 Relationship between Stochastic H2/H∞ and Nash Game 84
2.5 Algorithm for Solving Coupled GAREs 87
2.6 Notes and References 88
3 Linear Discrete-Time Stochastic H2/H∞ Control 91
3.1 Finite Horizon H2/H∞ Control 91
3.1.1 Definitions 92
3.1.2 Two identities 94
3.1.3 Finite horizon SBRL 96
3.1.4 Discrete-time stochastic LQ control 99
3.1.5 Finite horizon H2/H∞ with (x, v)-dependent noise 100
3.1.6 Unified treatment of H2, H∞ and H2/H∞ control 103
3.1.7 A numerical example 106
3.1.8 H2/H∞ control of systems with (x, u)- and (x, u, v)-dependent noise 107
3.2 Two-Person Non-Zero Sum Nash Game 109
3.3 Infinite Horizon H2/H∞ Control 111
3.3.1 Preliminaries 112
3.3.2 Standard LQ control result 117
3.3.3 An SBRL 120
3.3.4 H2/H∞ control with (x, v)-dependent noise 125
3.3.5 Numerical algorithms 133
3.3.6 H2/H∞ control with (x, u)- and (x, u, v)-dependent noise 139
3.4 Infinite Horizon Indefinite LQ Control 141
3.5 Comments on Stochastic H2/H∞ and Nash Game 147
3.6 Notes and References 147
4 H2/H∞ Control for Linear Discrete Time-Varying Stochastic Systems 149
4.1 Stability and Uniform Detectability 149
4.2 Lyapunov-Type Theorem under Uniform Detectability 156
4.3 Exact Detectability 161
4.4 Lyapunov-Type Theorems for Periodic Systems under Exact Detectability 166
4.5 Further Remarks on LDTV Systems 168
4.6 Infinite Horizon Time-Varying H2/H∞ Control 169
4.7 Notes and References 172
5 Linear Markovian Jump Systems with Multiplicative Noise 173
5.1 Introduction 173
5.2 Finite Horizon H2/H∞ Control of Discrete-Time Markov Jump Systems 174
5.2.1 An SBRL 175
5.2.2 Results on the H2/H∞ control 179
5.2.3 Algorithm and numerical example 181
5.2.4 Unified treatment of H2, H∞ and H2/H∞ control based on Nash game 182
5.3 Infinite Horizon Discrete Time-Varying Control 187
5.3.1 Definitions and preliminaries 188
5.3.2 An SBRL 189
5.3.3 Main result 193
5.3.4 An economic example 196
5.4 Infinite Horizon Discrete Time-Invariant H2/H∞ Control 197
5.4.1 Stability, stabilization, and SBRL 199
5.4.2 Exact detectability and extended Lyapunov theorem 201
5.4.3 Main result and numerical algorithm 203
5.5 Finite Horizon H2/H∞ Control of Continuous-Time Systems 207
5.5.1 Definitions and lemmas 208
5.5.2 Nash equilibrium strategy and H2/H∞ control 211
5.6 Infinite Horizon Continuous-Time H2/H∞ Control 215
5.6.1 A moment equation 215
5.6.2 Exact observability and detectability 217
5.6.3 Comments on the H2/H∞ control 220
5.7 Notes and References 221
6 Nonlinear Continuous-Time Stochastic H∞ and H2/H∞ Controls 223
6.1 Dissipative Stochastic Systems 223
6.2 Observability and Detectability 227
6.3 Infinite Horizon H∞ Control 229
6.4 Finite Horizon Nonlinear H∞ Control 236
6.5 H∞ Control of More General Stochastic Nonlinear Systems 241
6.6 Finite Horizon H2/H∞ Control 250
6.7 Notes and References 257
7 Nonlinear Stochastic H∞ and H2/H∞ Filtering 259
7.1 Nonlinear H∞ Filtering: Delay-Free Case 259
7.1.1 Lemmas and definitions 260
7.1.2 Main results 261
7.2 Suboptimal Mixed H2/H∞ Filtering 266
7.3 LMI-Based Approach for Quasi-Linear H∞ Filter Design 268
7.4 Suboptimal Mixed H2/H∞ Filtering of Quasi-Linear Systems 274
7.5 Numerical Example 277
7.6 Nonlinear HOC Filtering: Time-Delay Case 278
7.6.1 Definitions and lemmas 278
7.6.2 Main results 285
7.7 Luenberger-Type Linear Time-Delay H2/H∞ Filtering 288
7.8 Notes and References 291
8 Some Further Research Topics in Stochastic H2/H∞ Control 293
8.1 Stochastic H2/H∞ Control with Random Coefficients 293
8.1.1 SBRL and stochastic LQ lemma 294
8.1.2 Mixed H2/H∞ control 296
8.1.3 H∞ control 298
8.1.4 Some unsolved problems 301
8.2 Nonlinear Discrete-Time Stochastic H2/H∞ Control 302
8.2.1 Dissipation, l2-gain and SBRL 303
8.2.2 Observability and detectability 306
8.2.3 Review of martingale theory 307
8.2.4 LaSalle-type theorems 308
8.2.5 Difficulties in affine nonlinear discrete H2/H∞ control 316
8.3 Singular Stochastic H2/H∞ Control 316
8.3.1 Lemma and definition 317
8.3.2 Asymptotical mean square admissibility 318
8.3.3 An illustrative example 324
8.3.4 Problems in H2/H∞ control 325
8.4 Mean-Field Stochastic H2/H∞ Control 326
8.4.1 Definition for control 326
8.4.2 Finite horizon SBRL 327
8.4.3 Mean-field stochastic LQ control 336
8.4.4 H2/H∞ control with (x, u)-dependent noise 337
8.4.5 Further research problems 341
8.5 Notes and References 341
References 343
Index 361
