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, eBook

Reihe: Probability and Its Applications

Safety critical and high-integrity systems, such as industrial plants and economic systems can be subject to abrupt changes - for instance due to component or interconnection failure, and sudden environment changes etc.

Combining probability and operator theory, Discrete-Time Markov Jump Linear Systems provides a unified and rigorous treatment of recent results for the control theory of discrete jump linear systems, which are used in these areas of application.

The book is designed for experts in linear systems with Markov jump parameters, but is also of interest for specialists in stochastic control since it presents stochastic control problems for which an explicit solution is possible - making the book suitable for course use.

From the reviews:

"This text is very well written...it may prove valuable to those who work in the area, are at home with its mathematics, and are interested in stability of linear systems, optimal control, and filtering."

Zielgruppe

Research

Autoren/Hrsg.

Costa, O.L.V.

Fragoso, M.D.

Marques, R.P.

Inhaltsverzeichnis

Preface.- Markovian Jump Linear Systems.- Background Material.- On Stability.- Optimal Control.- Linear Filtering.- Quadratic Optimal Control with Partial Information.- H2- Control.- Design Techniques and Examples.- Appendix A: Coupled Algebraic Riccati Equations.- Appendix B: Auxiliary Results for the Linear Filtering Problem with (k) Unknown.- Appendix C: Auxiliary Results for the H2 Control Problem.- Notation and Corrections.- References.

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}.