Bounding the distance of a controllable system to an uncontrollable one

Abstract

Let $(A,B)$ be a pair of matrices representing a time-invariant linear system $\dot x(t)=Ax(t)+Bu(t)$ under block-similarity equivalence. In this paper we measure the distance between a controllable pair of matrices $(A,B)$ and the nearest uncontrollable one. A bound is obtained in terms of singular values of the controllability matrix $C(A,B)$ associated to the pair. This bound is not simply based on the smallest singular value of $C(A,B)$ contrary to what one may expect. Also a lower bound is obtained using geometrical techniques expressed in terms of the singular values of a matrix representing the tangent space of the orbit of the pair $(A,B)$.