# Structured real radius of controllability for higher order LTI systems

## TL;DR

This paper introduces a method to compute the radius of controllability for higher order LTI systems by relating it to the low rank approximation of a Toeplitz structured matrix, enhancing robustness analysis.

## Contribution

It establishes a novel link between controllability radius and structured low rank approximation, providing an effective computational approach for higher order systems.

## Key findings

- Method effectively computes controllability radius.
- Approach outperforms benchmark numerical methods.
- Numerical examples validate the approach.

## Abstract

In this paper, we consider the problem of computing the nearest uncontrollable (C-uncontrollable) system to a given higher order system. The distance to the nearest uncontrollable system, also termed as the radius of controllability, is a good measure of gauging the numerical robustness of the given system with respect to controllability. Here, we invoke the equivalence of C-controllability of a higher order system with full rank property of a certain Toeplitz structured matrix. This enables us to pose the problem of computing the radius of controllability as equivalent to the problem of computing the nearest structured low rank approximation of this Toeplitz structured matrix. Through several numerical examples and comparison with the benchmark numerical problem, we illustrate that our approach works well.