Measure of quality of finite-dimensional linear systems: A frame-theoretic view
Provides a new theoretical perspective on control system quality for control theorists, but the results are theoretical and incremental.
This paper extends the binary notion of controllability to a quantitative measure of quality for linear control systems, showing that standard quality measures (trace, minimum eigenvalue, determinant of controllability Gramian) are optimized when the system generates a tight frame. A new frame-theoretic quality measure is proposed.
A measure of quality of a control system is a quantitative extension of the classical binary notion of controllability. In this article we study the quality of linear control systems from a frame-theoretic perspective. We demonstrate that all LTI systems naturally generate a frame on their state space, and that three standard measures of quality involving the trace, minimum eigenvalue, and the determinant of the controllability Gramian achieve their optimum values when this generated frame is tight. Motivated by this, and in view of some recent developments in frame-theoretic signal processing, we propose a natural measure of quality for continuous time LTI systems based on a measure of tightness of the frame generated by it and then discuss some properties of this frame-theoretic measure of quality.