A Characterization of the Distance Between Controllable and Uncontrollable LTI Systems

The controllability distance for a linear time-invariant (LTI) system is defined as the norm of the smallest perturbation rendering the system uncontrollable. This is a widely used concept in control theory and provides a measure of the robustness of a system. Previous investigations have shown that the controllability distance can be characterized by a optimization problem involving singular values of extended matrices. This characterization has been established for general first-order systems and a certain class of higher-order systems. In this paper, we develop an analogous characterization of the controllability distance for a more general family of LTI systems, where controllability is formulated in a behavioral framework.