A convolution algebra of delay-differential operators and a related problem of finite spectrum assignability
type: Item
instance of: scholarly article
title: A convolution algebra of delay-differential operators and a related problem of finite spectrum assignability
zbMATH Open document ID: 0954.93007
DOI: 10.1007/PL00009859
author: Heide Gluesing-Luerssen
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 2000-04-09T00:00:00Z
review text: A well-known algebraic model for time-delay systems with commensurate delays is used. Namely, each polynomial \[ p=\sum^L_{j=0} \sum^N_{i=0}p_{ij}s^iz^j \in\mathbb{R}[s,z] \] of two variables with real coefficients gives rise to the delay-differential operator \[ \left(p\left ({d\over dt}, \sigma \right) w\right)(t)= \sum^L_{j=0} \sum^N_{i=0} p_{ij}w^{(i)} (t-j), \] which is considered as an operator on the space of real valued infinitely differentiable functions defined on \(\mathbb{R}\). Within this model, the ring \[ {\mathcal H}= \left\{{p \over r}:p,r\in \mathbb{R}[s,z],\;r\neq 0,\text{ Ker} r\left({d\over dt}, \sigma\right)\subseteq\text{Ker} p\left({d\over dt},\sigma \right) \right \} \] and its various subrings (in which no polynomials in \(z^{-1}\) are allowed) are studied. It was proved in a previous paper by the author that \({\mathcal H}\) is a Bézout domain. In the reviewed paper, it is shown that the ring \({\mathcal H}\) and its subrings can be interpreted as algebras of distributions. In the last section, the problem of coefficient assignability via dynamic state feedback of systems of the form \(\dot x=A(\sigma)x+ B(\sigma) u\) is discussed. Here \(A(\sigma)\) and \(B(\sigma)\) are point delay matrices of sizes \(n\times n\) and \(n\times m\), respectively. It is proved that the system is coefficient assignable (in a weak sense) if and only if it is spectrally controllable, i.e., \[ \text{rank} \bigl[sI-A(e^{-s}),\;B(e^{-s})\bigr]= n \] for all complex numbers \(s\).
reviewed by: Leiba Rodman
Mathematics Subject Classification ID: 93B25
Mathematics Subject Classification ID: 93B55
Mathematics Subject Classification ID: 93C23
Mathematics Subject Classification ID: 93C80
zbMATH DE Number: 1428465
zbMATH Keywords: spectrum assignability
zbMATH Keywords: algebraic model
zbMATH Keywords: time-delay systems
zbMATH Keywords: commensurate delays
zbMATH Keywords: delay-differential operator
zbMATH Keywords: algebras of distributions
zbMATH Keywords: dynamic state feedback
MaRDI profile type: Publication
full work available at URL: pl00009859
OpenAlex ID: W1986239379
description: scientific article; zbMATH DE number 1428465