# Measure of quality of finite-dimensional linear systems: A frame-theoretic view

## TL;DR

This paper introduces a frame-theoretic approach to quantify the quality of finite-dimensional linear control systems, showing that optimal control properties align with the tightness of the generated frame.

## Contribution

It establishes a novel connection between control system quality measures and frame theory, proposing a new measure based on frame tightness for continuous-time LTI systems.

## Key findings

- Standard quality measures are optimized when the generated frame is tight.
- The proposed measure aligns with classical measures like trace, eigenvalues, and determinant.
- Frame-theoretic properties provide insights into control system performance.

## Abstract

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.