System equivalence for AR-systems over rings -- with an application to delay-differential systems
type: Item
instance of: scholarly article
title: System equivalence for AR-systems over rings -- with an application to delay-differential systems
zbMATH Open document ID: 0951.93015
DOI: 10.1007/PL00009851
author: L. C. G. J. M. Habets
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 2000-12-19T00:00:00Z
review text: Auto-regressive (AR) systems with defining equations described by matrices over an arbitrary integral domain \(\mathcal{R}\) are studied in this paper. The results form a generalization of AR-systems over \({\mathbb{R}}[s]\) or \({\mathbb{R}}[s, s^{-1}]\) in a way reminiscent of the extension of the theory of state space systems over fields to the ring case. In the behavioral approach to dynamical systems, a system is described as a triple \((T,W,\mathcal{B})\), where \(T\) is the time-axis, \(W\) is the space in which the signals take their values and \(\mathcal{B}\) (the behavior) is a subspace of the signal space \(W^{T}\). The behavior is then considered as the set of all time-trajectories which satisfy the governing laws of the system. The set-up in this paper is as follows: Let \( \mathcal{R}\) be an integral domain and let \( \mathcal{M}\) be a module over \( \mathcal{R}\), consisting of all (one-variable) trajectories under consideration. Each ring element is associated with an operator acting on the elements of the module \( \mathcal{M}\). This set-up differs from the behavioral approach in that a time-axis \(T\) and a space \(W\) are not explicitly described. Instead, \( {\mathcal{M}}^{q}\) is taken as the signal space, allowing endowment with a richer structure. The goal of the paper is to characterize all matrices over \( \mathcal{R}\) describing the same behavior for a given module \( \mathcal{M}\) over \( \mathcal{R}\). This problem is tackled by defining a ring extension \( \mathcal{R}_{M}\) of \( \mathcal{R}\) which explicitly depends on \( \mathcal{M}\). It turns out that \( \mathcal{M}\) is also a module over \( \mathcal{R_M}\) and that a system \( (P, {\mathcal{M}}^q)\) over \( \mathcal{R}\), where \(P \in {\mathcal{R}}^{p \times q}\), can also be considered as a system over \( \mathcal{R_M}.\) This allows greater freedom of manipulation, provided \( \mathcal{R}\) is indeed properly contained in \( \mathcal{R_M}\). A sufficient condition for equal…
Mathematics Subject Classification ID: 93B25
Mathematics Subject Classification ID: 34K35
zbMATH DE Number: 1348035
zbMATH Keywords: autoregressive systems (AR-systems)
zbMATH Keywords: systems over rings
zbMATH Keywords: delay-differential systems
zbMATH Keywords: behavioral approach
zbMATH Keywords: system equivalence
reviewed by: Frieda Theron
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1348035