A unified canonical form for controllable and uncontrollable linear dynamical systems†

Abstract

A linear transformation is proposed which will transform an arbitrary (constant) linear dynamical system p into a certain standard (canonical) form. This particular canonical form coincides with the well-known phase-variable canonical form [l]-[5] for the case of completely controllable, scalar input, linear dynamical systems and coincides with a canonical form recently proposed by Luenberger [6] and Wonham [7] in the case of completely controllable, vector input, linear dynamical systems. For linear dynamical systems which are not completely controllable, the canonical form proposed herein displays explicitly: (i) the sub-system of p which is completely controllable and (ii) the sub-system of p which is completely uncontrollable. The explicit identification of these two sub-systems permits us to effectively implement the important fundamental stabilization theorem for constant linear dynamical systems and also a useful theorem on spectrum controllability for linear dynamical systems. An important feature...