Continuity of dynamical systems: A system theoretic approach
type: Item
instance of: scholarly article
title: Continuity of dynamical systems: A system theoretic approach
zbMATH Open document ID: 0657.93011
DOI: 10.1007/BF02551406
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 1988-01-01T00:00:00Z
review text: The paper studies the convergence of time-invariant linear discrete-time systems from a ``behavioural'' point of view [see \textit{J. C. Willems}, Automatica 22, 675-694 (1986; Zbl 0628.62088); and ibid. 23, 87-115 (1987; Zbl 0628.62089)]. The systems are described by autoregressive linear equations and are identified with the solution space \(B\subset W^{{\mathbb{Z}}}\) of these equations, where \(W={\mathbb{R}}^ q\) is the corresponding signal alphabet. A family of systems \(\Sigma_{\epsilon}\), \(\epsilon >0\) is said to converge to a system \(\Sigma_ 0\) if the associated closed, linear shift-invariant subspaces \(B_{\epsilon}\) of \(W^{{\mathbb{Z}}}\) (provided with the pointwise topology) converge to the subspace \(B_ 0\). Under the assumption that the system lags are uniformly bounded it is shown that \(\Sigma_{\epsilon}\to \Sigma_ 0\) as \(\epsilon\) \(\to 0\) if and only if the system \(\Sigma_{\epsilon}\) can be described by autoregressive equations whose coefficients converge to the coefficients of a suitable system equation for \(\Sigma_ 0\).
Mathematics Subject Classification ID: 93B25
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93B27
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 93C55
zbMATH DE Number: 4074888
zbMATH Keywords: convergence
zbMATH Keywords: time-invariant linear discrete-time systems
zbMATH Keywords: autoregressive linear equations
zbMATH Keywords: time-invariant
author: Johannes W. Nieuwenhuis
author: Jan C. Willems
reviewed by: D. Hinrichsen
MaRDI profile type: Publication
cites work: Convergence of Closed Subsets in a Topological Space
cites work: Q3802944
cites work: From time series to linear system. I: Finite dimensional linear time invariant systems
cites work: Q4204049
full work available at URL: https://doi.org/10.1007/bf02551406
OpenAlex ID: W2083412220
description: scientific article; zbMATH DE number 4074888