Continuous-time Markov jump linear systems

✍ Scribed by Oswaldo Luiz do Valle Costa; Marcelo D Fragoso; Marcos G Todorov

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 295
• Series: Probability and its applications
• Category: Library

✦ Synopsis

1.Introduction.- 2.A Few Tools and Notations.- 3.Mean Square Stability.- 4.Quadratic Optimal Control with Complete Observations.- 5.H2 Optimal Control With Complete Observations.- 6.Quadratic and H2 Optimal Control with Partial Observations.- 7.Best Linear Filter with Unknown (x(t), theta(t)).- 8.H_$infty$ Control.- 9.Design Techniques.- 10.Some Numerical Examples.- A.Coupled Differential and Algebraic Riccati Equations.- B.The Adjoint Operator and Some Auxiliary Results.- References.- Notation and Conventions.- Index

✦ Table of Contents

Cover......Page 1
Continuous-Time Markov Jump Linear Systems......Page 4
Preface......Page 6
Contents......Page 9
1.1 Markov Jump Linear Systems......Page 13
1.2 Some Applications of MJLS......Page 19
1.4 Overview of the Chapters......Page 23
1.5 Historical Remarks......Page 25
2.2 Some Basic Notation and Definitions......Page 27
2.3 Semigroup Operators and Infinitesimal Generator......Page 28
2.4 The Fundamental Theorem for Differential Equations......Page 29
2.5 Continuous-Time Markov Chains......Page 31
2.6 The Space of Sequences of N Matrices......Page 35
2.7 Auxiliary Results......Page 38
2.8 Linear Matrix Inequalities......Page 41
3.2 The Models and Problem Statement......Page 44
3.3 Main Operators and Auxiliary Results......Page 47
3.4.1 MSS, StS, and the Spectrum of an Augmented Matrix......Page 55
3.4.2 Coupled Lyapunov Equations......Page 58
3.5 The L2r(Omega,F, P) and Jump Diffusion Cases......Page 63
3.5.1 The L2r(Omega,F, P) Disturbance Case......Page 64
3.5.2 The Jump Diffusion Case......Page 66
3.5.3 Summary......Page 69
3.6.1 Definitions and LMIs Conditions......Page 70
3.6.2 Mean-Square Stabilizability with theta(t) Partially Known......Page 72
3.6.3 Dynamic Output Mean-Square Stabilizability......Page 75
3.7 Historical Remarks......Page 79
4.2 Notation and Problem Formulation......Page 81
4.3 Dynkin's Formula......Page 84
4.4 The Finite-Horizon Optimal Control Problem......Page 86
4.5 The Infinite-Horizon Optimal Control Problem......Page 89
4.6 Historical Remarks......Page 91
5.2 Robust and Quadratic Mean-Square Stabilizability......Page 93
5.3 Controllability, Observability Gramians, and the H2-Norm......Page 95
5.4.1 Preliminaries......Page 98
5.4.2 Pi Exactly Known......Page 99
5.4.3 Pi Not Exactly Known......Page 101
5.5 The Convex Approach and the CARE......Page 102
5.6 Historical Remarks......Page 106
6.1 Outline of the Chapter......Page 107
6.2.1 Problem Statement......Page 108
6.2.2 Filtering Problem......Page 109
6.2.3 A Separation Principle for MJLS......Page 115
6.3.1 Problem Statement......Page 122
6.3.2 Filtering H2 Problem......Page 125
6.3.3 The Separation Principle......Page 129
6.3.4 An LMIs Approach for the H2 Control Problem......Page 133
6.4 Historical Remarks......Page 136
7.2 Preliminaries......Page 137
7.3 Problem Formulation for the Finite-Horizon Case......Page 139
7.4 Main Result for the Finite-Horizon Case......Page 141
7.5 Stationary Solution for the Algebraic Riccati Equation......Page 146
7.6.1 Auxiliary Results and Problem Formulation......Page 149
7.6.2 Solution for the Stationary Filtering Problem via the ARE......Page 157
7.7 Historical Remarks......Page 159
8.2 Description of the Problem......Page 161
8.3.1 Problem Formulation and Main Result......Page 163
8.3.2 Proof of Proposition 8.3 and Lemma 8.4......Page 166
8.4 The Hinfty Control Problem......Page 175
8.5 Static State Feedback......Page 176
8.6 Dynamic Output Feedback......Page 177
8.6.1 Main Results......Page 178
8.6.2 Analysis of Dynamic Controllers......Page 180
8.6.3 Synthesis of Dynamic Controllers......Page 186
8.6.4 Hinfty Analysis and Synthesis Algorithms......Page 189
8.7 Historical Remarks......Page 191
9.2 Stability Radius......Page 192
9.3 A Robustness Margin for Pi......Page 198
9.4 Robust Control......Page 202
9.4.1 Preliminary Results......Page 203
9.4.2 Robust H2 Control......Page 205
9.4.3 The Equalized Case......Page 208
9.4.4 Robust Mixed H2 / Hinfty Control......Page 209
9.5 Robust Linear Filtering Problem via an LMIs Formulation......Page 210
9.5.1 The LMIs Formulation......Page 211
9.5.2 Robust Filter......Page 215
9.5.3 ARE Approximations for the LMIs Problem......Page 217
9.6 Historical Remarks......Page 219
10.2 An Example on Economics......Page 222
10.3 Coupled Electrical Machines......Page 226
10.3.1 Problem Statement......Page 227
10.3.3 H2 Control......Page 229
10.3.4 Stability Radius Analysis......Page 230
10.4 Robust Control of an Underactuated Robotic Arm......Page 232
10.5 An Example of a Stationary Filter......Page 235
10.6 Historical Remarks......Page 239
A.2 Coupled Differential Riccati Equations......Page 240
A.3 Maximal Solution......Page 246
A.4 Stabilizing Solution......Page 253
A.5 Filtering Coupled Algebraic Riccati Equations......Page 255
A.6 Asymptotic Convergence......Page 257
A.7 Filtering Differential and Algebraic Riccati Equation for Unknown theta(t)......Page 261
B.2 Preliminaries......Page 266
B.3 Main Results......Page 267
Notation and Conventions......Page 272
References......Page 280
Index......Page 294