Linear operators preserving certain equivalence relations originating in system theory

Abstract

Let F be C or R . A finite-dimensional linear time-invariant system is described in state-space form by [xdot] = Ax + Bu, y = Cx + Du, and is identified with the matrix 4-tuple ( A, B, C, D), where x ϵ F n, u ϵ F m, yϵ F p, and A, B, C, D, are matrices of appropriate sizes and with entries in F . For fixed n, m, p, let M be the linear space of all systems ( A, B, C, D). Equivalence relations ∼ can be defined on M based on the possibility of changes of basis inthe state space, the input space, or the output space, and the possibility of state feedback and/or output feedback. We characterize those nonsingular linear operators φ on M that satisfy φ( X) ∼ φ( Y) whenever X ∼ Y.