\(z\)-transform and Volterra-operator based approaches to controllability and observability analysis for discrete linear repetitive processes
type: Item
instance of: scholarly article
title: \(z\)-transform and Volterra-operator based approaches to controllability and observability analysis for discrete linear repetitive processes
zbMATH Open document ID: 1035.93039
DOI: 10.1023/A:1023586803743
author: Eric Rogers
author: David H. Owens
author: Michael Dymkov
author: Ivan Vasilevich Gaĭshun
author: Krzysztof Gałkowski
published in: Multidimensional Systems and Signal Processing
publication date: 2003-08-06T00:00:00Z
review text: Discrete linear repetitive processes are considered, having the state space representation \[ \begin{aligned} x(k+1,t+1) &= Ax(k+1,t)+Bu(k+1,t)+B_{0}y(k,t),\\ y(k+1,t) &= Cx(k+1,t)+Du(k+1,t)+D_{0}y(k,t), \end{aligned} \] \(k\in\mathbb{Z}^{+}\), \(t\in[0,\alpha]\subset\mathbb{Z}^{+}\); \(x(k,t)\in\mathbb{R}^{n}\) is the state vector, \(y(k,t)\in\mathbb{R}^{m}\) is the pass profile vector and \(u(k,t)\in\mathbb{R}^{l}\) is the control input vector. The initial (or boundary) conditions have the form \[ x(k,0)=\gamma(k),\;k\geq1;\quad y(0,t)=d_{t},\;0\leq t\leq\alpha \] where \(\gamma(k)\) and \(d_{t}\) are known constant \(n\)-vectors. The repetitive processes are connected with 2D linear systems, but the key difference is that in their case the duration of information propagation in one of the two directions, namely the along-the-pass direction, is finite (i.e. \(t\in[0,\alpha]\)). This difference imposes the necessity to develop a distinct system theory for repetitive processes. The aim of this paper is to introduce and to study different concepts of controllability and observability for discrete linear repetitive processes. A repetitive process is said to be pass controllable if there exists a control input sequence which drives the process to produce any pre-specified pass profile \(y_{k_{f}}=[y_{k_{f}}(0), y_{k_{f}}(1),\dots,y_{k_{f}}(\alpha)]\) on some pass index \(k_{f}\), for any initial conditions. If the pre-specified pass profile occurs on a pre-specified pass index \(k_{f}\) the process is said to be \textit{pass profile controllable}; this property has been studied in a previous paper written by three of the authors. A repetitive process is said to be stongly pass controllable if \(\exists\) a pass index \(k_{f}\) and a moment \(t_{f}\in[0,\alpha]\) such that for any bounded sequence of initial conditions there exists a control input sequence which drives the process to produce any pre-specified bounded state sequences \(x(k,t_{f}),\;k\geq k_{f}+1\) and pa…
reviewed by: Valeriu Prepeliţă
Mathematics Subject Classification ID: 93C35
Mathematics Subject Classification ID: 93B17
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B07
zbMATH DE Number: 1960150
zbMATH Keywords: repetitive process
zbMATH Keywords: controllability
zbMATH Keywords: observability
zbMATH Keywords: z-transform
zbMATH Keywords: Volterra operator
zbMATH Keywords: discrete processes
zbMATH Keywords: rank tests
zbMATH Keywords: strong pass controllability
MaRDI profile type: Publication
full work available at URL: a:1023586803743
OpenAlex ID: W1605282905
description: scientific article; zbMATH DE number 1960150