An algebraic approach to canonical forms and invariants for linear time-invariant differential and difference systems

Abstract

This paper deals with canonical forms and invariants for linear time-invariant differential and difference systems. The signal spaces of differential systems are regarded as modules over the ring C[p] of polynomials in the differentiation operator p with complex coefficients. Systems are considered as sets of input-output pairs (u,y;) satisfying modulo equations of the form A(p)y= B(p)u, where A{p) and B(p) are matrices with entries in C[p]. It is shown that under certain conditions all S3Tstem matrices [A(p) — B(p)] associated with a given system are equivalent in the sense that they can be obtained from each other by elementary row operations. The main result is that every system matrix can be brought to a unique canonical upper triangular (CUT) form. Difference systems are considered similarly by replacing p by a shift operator. For these systems there exists a CUT-form which is strongly dependent on causality.