Fundamental Theory of Control of Systems Involving a Kronecker Product of Matrices

Abstract

This paper presents a criterion for the equivalence relation of an n-th order nonhomogeneous differential equation to its companion vector equation, and then presents a necessary and sufficient condition for the equivalence of the general first-order non-homogeneous equation to the scalar differential equation. Next a Kronecker product first-order system is formulated by embedding two companion vector equations of different dimensions, and the relationship between the scalar differential equations of different orders to their companion Kronecker product systems is obtained. A necessary and sufficient condition for their linear equivalence is presented. Finally a set of sufficient conditions is given for the controllability of the Kronecker product first-order system in terms of the fundamental matrix solutions. Several interesting examples will be presented that highlight the necessity of these investigations.