A note on linear transformations which leave controllable multiinput descriptor systems controllable

Abstract

Consider a generalized linear dynamical system E x ̇ = Ax + Bu , where x ∈ C n , u ∈ C m , and E, A, B are matrices of appropriate sizes with entries in C. This system, or the matrix triple ( E, A, B), is called controllable if det( αE − βA) is not a zero polynomial in α, β and ( αE − βA, B) is of full rank for all ( α, β) ∈ C (0, 0). Let f be a linear transformation on C n× n × C n× m , the linear space of all matrix pairs ( A, B). In an earlier paper, Mehrmann and Krause attempted to prove that, if f is of the form X at UXV, and rank f( αE − βA, B) = n for all ( α, β) ∈ C 2 (0, 0) and all controllable systems ( E, A, B), then U, V are nonsingular matrix with V in some lower block triangular form. In this paper, we correct an error contained in this result and discuss whether the corrected result can be generalized in such a way that no restrictions are placed on the form of f.