# Approximating the Real Structured Stability Radius with Frobenius Norm Bounded Perturbations

## TL;DR

This paper introduces a novel, efficient algorithm to approximate the real stability radius of large-scale linear systems under Frobenius norm-bounded perturbations, providing practical upper bounds with scalability advantages.

## Contribution

It is the first algorithm addressing the Frobenius-norm stability radius problem, leveraging eigenvalue computations for large sparse systems, and extending existing pseudospectral approximation methods.

## Key findings

- Effective in practice despite only providing upper bounds
- Maintains efficiency for large sparse systems
- First algorithm for Frobenius-norm stability radius approximation

## Abstract

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the $H_\infty$ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost mainly consists of computing the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.