Discrete-Time Markov Jump Linear Systems

Contents......Page 8
1.1 Introduction......Page 12
1.2 Some Examples......Page 15
1.3 Problems Considered in this Book......Page 19
1.4 Some Motivating Remarks......Page 22
1.5 A Few Words On Our Approach......Page 23
1.6 Historical Remarks......Page 24
2.1 Some Basics......Page 26
2.2 Auxiliary Results......Page 29
2.3 Probabilistic Space......Page 31
2.4.1 Stability and the Lyapunov Equation......Page 32
2.4.2 Controllability and Observability......Page 34
2.4.3 The Algebraic Riccati Equation and the Linear-Quadratic Regulator......Page 37
2.5 Linear Matrix Inequalities......Page 38
3.1 Outline of the Chapter......Page 40
3.2 Main Operators......Page 41
3.3.1 Main Result......Page 47
3.3.2 Examples......Page 48
3.3.3 Proof of Theorem 3.9......Page 52
3.3.4 Easy to Check Conditions for Mean Square Stability......Page 56
3.4.1 Main Results......Page 59
3.4.2 Wide Sense Stationary Input Sequence......Page 60
3.4.3 The l[sub(2)]-disturbance Case......Page 66
3.5.1 Definitions and Tests......Page 68
3.5.2 Stabilizability with Markov Parameter Partially Known......Page 70
3.6.1 Main Results......Page 74
3.6.2 An Application of Almost Sure Convergence Results......Page 77
3.7 Historical Remarks......Page 80
4.1 Outline of the Chapter......Page 82
4.2.1 Problem Statement......Page 83
4.2.2 The Optimal Control Law......Page 85
4.3.1 Definition of the Problems......Page 89
4.3.2 The Markov Jump Linear Quadratic Regulator Problem......Page 91
4.3.3 The Long Run Average Cost......Page 92
4.4.1 Preliminaries and the H[sub(2)]-norm......Page 93
4.4.2 The H[sub(2)]-norm and the Grammians......Page 94
4.4.4 Connection Between the CARE and the H[sub(2)]-control Problem......Page 97
4.5.1 Preliminaries......Page 101
4.5.2 Auxiliary Result......Page 102
4.5.3 The Optimal Control Law......Page 105
4.5.4 An Application to a Failure Prone Manufacturing System......Page 107
4.6 Historical Remarks......Page 110
5.1 Outline of the Chapter......Page 112
5.2 Finite Horizon Filtering with θ(k) Known......Page 113
5.3 Infinite Horizon Filtering with θ(k) Known......Page 120
5.4.1 Preliminaries......Page 124
5.4.2 Optimal Linear Filter......Page 125
5.4.3 Stationary Linear Filter......Page 128
5.5.2 Problem Formulation......Page 130
5.5.3 LMI Formulation of the Filtering Problem......Page 135
5.5.4 Robust Filter......Page 138
5.6 Historical Remarks......Page 139
6.1 Outline of the Chapter......Page 141
6.2.1 Preliminaries......Page 142
6.2.2 A Separation Principle......Page 143
6.3.1 Preliminaries......Page 146
6.3.2 Definition of the H[sub(2)]-control Problem......Page 147
6.3.3 A Separation Principle for the H[sub(2)]-control of MJLS......Page 149
6.4 Historical Remarks......Page 151
7.1 Outline of the Chapter......Page 153
7.2.1 The General Problem......Page 154
7.2.2 H[sub(∞)] Main Result......Page 155
7.3.1 Sufficient Condition......Page 158
7.3.2 Necessary Condition......Page 161
7.4 Recursive Algorithm for the H[sub(∞)]-control CARE......Page 172
7.5 Historical Remarks......Page 176
8.1.1 Optimal Control for a Solar Thermal Receiver......Page 177
8.1.2 Optimal Policy for the National Income with a Multiplier–Accelerator Model......Page 179
8.1.3 Adding Noise to the Solar Thermal Receiver problem......Page 181
8.2 Robust Control via LMI Approximations......Page 183
8.2.1 Robust H[sub(2)]-control......Page 184
8.2.2 Robust Mixed H[sub(2)]/H[sub(∞)]-control......Page 192
8.2.3 Robust H[sub(∞)]-control......Page 197
8.3.1 Algorithm......Page 198
8.3.2 H[sub(∞)]-control for the UarmII Manipulator......Page 199
8.4 Examples of Linear Filtering with θ(k) Unknown......Page 207
8.4.1 Stationary LMMSE Filter......Page 208
8.4.2 Robust LMMSE Filter......Page 209
8.5 Historical Remarks......Page 211
A.1 Duality Between the Control and Filtering CARE......Page 212
A.2 Maximal Solution for the CARE......Page 217
A.3.1 Connection Between Maximal and Stabilizing Solutions......Page 225
A.3.2 Conditions for the Existence of a Stabilizing Solution......Page 226
A.4 Asymptotic Convergence......Page 235
B.1.1 Proof of Theorem 5.9 and Lemma 5.11......Page 238
B.1.2 Stationary Filter......Page 241
B.2 Robust Filter......Page 245
C. Auxiliary Results for the H[sub(2)]-control Problem......Page 257
References......Page 265
Notation and Conventions......Page 278
F......Page 284
L......Page 285
Q......Page 286
U......Page 287