Costa / Fragoso / Marques Discrete-Time Markov Jump Linear Systems

1. Auflage 2006
ISBN: 978-1-84628-082-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 286 Seiten

Reihe: Probability and Its Applications

This will be the most up-to-date book in the area (the closest competition was published in 1990) This book takes a new slant and is in discrete rather than continuous time

Autoren/Hrsg.

Costa, O. L. V.

Fragoso, M. D.

Marques, R. P.

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;1 Markov Jump Linear Systems;12
3.1;1.1 Introduction;12
3.2;1.2 Some Examples;15
3.3;1.3 Problems Considered in this Book;19
3.4;1.4 Some Motivating Remarks;22
3.5;1.5 A Few Words On Our Approach;23
3.6;1.6 Historical Remarks;24
4;2 Background Material;26
4.1;2.1 Some Basics;26
4.2;2.2 Auxiliary Results;29
4.3;2.3 Probabilistic Space;31
4.4;2.4 Linear System Theory;32
4.5;2.5 Linear Matrix Inequalities;38
5;3 On Stability;40
5.1;3.1 Outline of the Chapter;40
5.2;3.2 Main Operators;41
5.3;3.3 MSS: The Homogeneous Case;47
5.4;3.4 MSS: The Non-homogeneous Case;59
5.5;3.5 Mean Square Stabilizability and Detectability;68
5.6;3.6 Stability With Probability One;74
5.7;3.7 Historical Remarks;80
6;4 Optimal Control;82
6.1;4.1 Outline of the Chapter;82
6.2;4.2 The Finite Horizon Quadratic Optimal Control Problem;83
6.3;4.3 In.nite Horizon Quadratic Optimal Control Problems;89
6.4;4.4 The H2-control Problem;93
6.5;4.5 Quadratic Control with Stochastic 2-input;101
6.6;4.6 Historical Remarks;110
7;5 Filtering;112
7.1;5.1 Outline of the Chapter;112
7.2;5.2 Finite Horizon Filtering with .(k) Known;113
7.3;5.3 Infinite Horizon Filtering with (k) Known;120
7.4;5.4 Optimal Linear Filter with .(k) Unknown;124
7.5;5.5 Robust Linear Filter with .(k) Unknown;130
7.6;5.6 Historical Remarks;139
8;6 Quadratic Optimal Control with Partial Information;142
8.1;6.1 Outline of the Chapter;142
8.2;6.2 Finite Horizon Case;143
8.3;6.3 Infinite Horizon Case;147
8.4;6.4 Historical Remarks;152
9;7 H-Control;154
9.1;7.1 Outline of the Chapter;154
9.2;7.2 The MJLS H-like Control Problem;155
9.3;7.3 Proof of Theorem 7.3;159
9.4;7.4 Recursive Algorithm for the H-control CARE;173
9.5;7.5 Historical Remarks;177
10;8 Design Techniques and Examples;178
10.1;8.1 Some Applications;178
10.2;8.2 Robust Control via LMI Approximations;184
10.3;8.3 Achieving Optimal H-control;199
10.4;8.4 Examples of Linear Filtering with .(k) Unknown;208
10.5;8.5 Historical Remarks;212
11;A Coupled Algebraic Riccati Equations;214
11.1;A.1 Duality Between the Control and Filtering CARE;214
11.2;A.2 Maximal Solution for the CARE;219
11.3;A.3 Stabilizing Solution for the CARE;227
11.4;A.4 Asymptotic Convergence;237
12;B Auxiliary Results for the Linear Filtering Problem with .(k) Unknown;240
12.1;B.1 Optimal Linear Filter;240
12.2;B.2 Robust Filter;247
13;C Auxiliary Results for the H2-control Problem;260
14;References;268
15;Notation and Conventions;282
16;Index;288

Leseproben

1 Markov Jump Linear Systems (p.1)

One of the main issues in control systems is their capability of maintaining an acceptable behavior and meeting some performance requirements even in the presence of abrupt changes in the system dynamics. These changes can be due, for instance, to abrupt environmental disturbances, component failures or repairs, changes in subsystems interconnections, abrupt changes in the operation point for a non-linear plant, etc.

Examples of these situations can be found, for instance, in economic systems, aircraft control systems, control of solar thermal central receivers, robotic manipulator systems, large flexible structures for space stations, etc. In some cases these systems can be modeled by a set of discrete-time linear systems with modal transition given by a Markov chain.

This family is known in the specialized literature as Markov jump linear systems (from now on MJLS), and will be the main topic of the present book. In this first chapter, prior to giving a rigorous mathematical treatment and present specific definitions, we will, in a rather rough and nontechnical way, state and motivate this class of dynamical systems.

1.1 Introduction
Most control systems are based on a mathematical model of the process to be controlled. This model should be able to describe with relative accuracy the process behavior, in order that a controller whose design is based on the information provided by it performs accordingly when implemented in the real process.

As pointed out by M. Kac in [148], "Models are, for the most part, caricatures of reality, but if they are good, then, like good caricatures, they portray, though perhaps in a distorted manner, some of the features of the real world."

This translates, in part, the fact that to have more representative models for real systems, we have to characterize adequately the uncertainties. Many processes may be well described, for example, by time-invariant linear models, but there are also a large number of them that are subject to uncertain changes in their dynamics, and demand a more complex approach.

If this change is an abrupt one, having only a small influence in the system behavior, classical sensitivity analysis may provide an adequate assessment of the effects.

On the other hand, when the variations caused by the changes significantly alter the dynamic behavior of the system, a stochastic model that gives a quantitative indication of the relative likelihood of various possible scenarios would be preferable.

Over the last decades, several different classes of models that take into account possible different scenarios have been proposed and studied, with more or less success. To illustrate this situation, consider a dynamical system that is, in a certain moment, well described by a model G1.

Suppose that this system is subject to abrupt changes that cause it to be described, after a certain amount of time, by a di.erent model, say G2. More generally we can imagine that the system is subject to a series of possible qualitative changes that make it switch, over time, among a countable set of models, for example, {G1, G2, . . . , GN}.