Topics in Multidimensional Linear Systems Theory

By: Eva Zerz

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Paperback | 1 September 2000

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184 Pages

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The past few decades have witnessed an increasing interest in the field of multidimensional systems theory. This is concerned with systems whose trajectories depend not on one single variable (usually interpreted as time or frequency), but on several independent variables, such as the coordinates of an image.
The behavioural approach introduced by J. C. Willems provides a particularly suitable framework for developing a linear systems theory in several variables. The book deals with the classical concepts of autonomy, controllability, observability, and stabilizability. All the tests and criteria given are constructive in the sense that algorithmic versions may be implemented in modern computer algebra systems, using Gr¶bner basis techniques.
There is a close connection between multidimensional systems theory and robust control of one-dimensional systems with several uncertain parameters. The central link consists in the basic tool of linear fractional transformations. The book concludes with examples from the theory of electrical networks.

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Controllability of Multidimensional Systems	p. 1
Preliminaries and Notation	p. 2
Primeness of Multivariate Polynomial Matrices	p. 4
Controllability, Observability, and Autonomy	p. 15
Controllability as Concatenability of Trajectories	p. 22
Controllability as Parameterizability	p. 33
Co-prime Factorizations of Multivariate Rational Matrices	p. 41
Notions	p. 42
Characterizations of Primeness	p. 44
Constructing Co-prime Factorizations	p. 45
Properties of Co-prime Factorizations	p. 47
Factorization Ideals	p. 49
The Bivariate Case	p. 52
Stabilizability	p. 57
Co-primeness over Rings of Fractions	p. 57
Feedback Stabilization	p. 64
Strict System Equivalence	p. 69
System Equivalence	p. 71
Strict System Equivalence	p. 73
System Isomorphisms	p. 77
First Order Representations of Multidimensional Systems	p. 85
Preliminaries and Notation	p. 86
Output-nulling Representations	p. 89
Driving-variable Representations	p. 101
Linear Fractional Transformations	p. 105
LFT Construction	p. 108
Reduction to Trim Form	p. 114
Balancing	p. 121
LFTs and Robust Stability	p. 127
Electrical Networks	p. 141
Graph Theory	p. 141
Weight Functions	p. 142
Kirchhoff's Laws	p. 143
Linear Networks	p. 145
LFT Representations	p. 150
Bibliography	p. 157
Index	p. 163
Table of Contents provided by Blackwell. All Rights Reserved.

Topics in Multidimensional Linear Systems Theory

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