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

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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 of the present work is the development of an explicit and effective numerical algorithm for computing the required transformation matrix and its inverse

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