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{lunate} Fn, u ε{lunate} Fm, yε{lunate} Fp, 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. © 1992.