The poles and zeros of a linear time-varying system
type: Item
instance of: scholarly article
title: The poles and zeros of a linear time-varying system
zbMATH Open document ID: 0633.93017
DOI: 10.1016/0024-3795(88)90168-1
author: E. W. Kamen
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: The author considers linear time-varying systems of the form \[ (z^ n+\sum^{n-1}_{i=0}a_ i(k)z^ i)y(k)=(\sum^{n}_{i=0}b_ i(k)z^ i)u(k) \] where \(z: f(k)\mapsto f(k+1)\) denotes the shift operator and \(a_ i(k)\), \(b_ i(k)\in {\mathbb{C}}\), resp. for the continuous case \[ (D^ n+\sum^{n-1}_{i=0}a_ i(t)D^ i)y(t)=(\sum^{n}_{i=0}b_ i(t)D^ i)u(t) \] where D: f(t)\(\mapsto \dot f(t)\) denotes the differential operator and \(a_ i(\cdot)\), \(b_ i(\cdot)\) are differentialble a suitable number of times. The polynomials above are defined over certain skew polynomial rings which are naturally induced by the shift resp. differential operator. Via a facorization of the polynomials Kamen presents a new approach to define poles and zeros of time-varying systems. He shows that the poles can be computed by solving certain nonlinear equations. The theory is applied to study (uniform) asymptotic stability.
Mathematics Subject Classification ID: 93B25
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 93D20
zbMATH DE Number: 4031495
zbMATH Keywords: time-dependent
zbMATH Keywords: continuous-time
reviewed by: Achim Ilchmann
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90168-1
OpenAlex ID: W2052577179
cites work: Differential polynomials and division algebras
cites work: Time-varying polynomial matrix systems
cites work: Algebraic Theory of Linear Time-Varying Systems
cites work: A Transfer-Function Approach to Linear Time-Varying Discrete-Time Systems
cites work: Representation and realization of operational differential equations with time-varying coefficients
description: scientific article; zbMATH DE number 4031495