On the Dimensions of Controllability Subspaces: A Characterization via Polynomial Matrices and Kronecker Invariants

Abstract

The controllability subspaces of a pair $(A,B)$, instrumental in the formulation of the geometric theory of decoupling, are shown to have a natural analog in terms of the kernel of the singular pencil of matrices $(\lambda I - A; - B)$. In addition the pencil of matrices leads directly to the multivariable canonical form of Brunovsky. The possible dimensions of controllability subspaces are shown to be completely determined by a set of invariants of the pencil of matrices. The minimum dimension of controllability subspaces which contain arbitrary subspaces of the image of B is ascertained, and a construction for such subspaces is given.