An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems

Abstract

In this paper, we further develop the study of rings of integro-differential-delay operators considered as noncommutative polynomial algebras satisfying standard calculus identities. Within the algebraic analysis approach, we show that transformations and reductions of linear differential time-delay systems can be interpreted as homomorphisms and isomorphisms of finitely presented left modules over an algebra of integro-differential-delay operators. In particular, we show how Fiagbedzi-Pearson's transformation can be found again and generalized. This transformation maps the solutions of a first-order differential linear system with state and input delays to the solutions of a purely state-space linear system. Fiagbedzi-Pearson's transformation reduces to the well-known Artstein's reduction when the system has no state delay and yields an isomorphism of the solution spaces.