On a frame theoretic measure of quality of LTI systems

A measure of quality of a control system, i.e., a quantitative extension of the classical notion of controllability is a practically relevant idea. In this article we demonstrate that the three standard measures of quality involving the trace, minimum eigenvalue, and the determinant of the controllability grammian achieve their optimum values when the columns of the controllability matrix form a tight frame. Motivated by this, and in view of some recent developments in frame theoretic signal processing, we propose a natural measure of quality for LTI systems based on a measure of tightness of the columns of the reachability matrix. A sufficient condition in terms of the proposed measure of quality for the classical boolean notion of controllability has been provided.