# On a frame theoretic measure of quality of LTI systems

## TL;DR

This paper introduces a novel measure of control system quality based on frame theory, showing that optimal controllability corresponds to tight frames in the controllability matrix.

## Contribution

It links classical controllability measures to frame theory, proposing a new quality measure for LTI systems rooted in tight frame properties.

## Key findings

- Optimal control measures occur when the controllability matrix columns form a tight frame.
- The proposed measure aligns with classical measures like trace, eigenvalues, and determinant.
- Provides a new perspective connecting control theory and frame theory.

## Abstract

It is of practical significance to define the notion of a measure of quality of a control system, i.e., a quantitative extension of the classical notion of controllability. 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 from a tight frame. Motivated by this, and in view of some recent developments in frame theoretic signal processing, we provide a measure of quality for LTI systems based on a measure of tightness of the columns of the reachability matrix .