On the Real Stability Radius of Sparse Systems

TL;DR

This paper investigates the robustness of sparse linear time-invariant systems by formulating the stability radius as an optimization problem, deriving optimality conditions, and developing algorithms to analyze stability under structured perturbations.

Contribution

It introduces a novel framework for computing the real stability radius of sparse systems, linking optimal perturbations to eigenvectors and providing a gradient-based solution method.

Findings

Characterized optimal perturbations via eigenvector relations.

Developed a convergence-guaranteed gradient/Newton algorithm.

Provided structural insights into sparse network stability.

Abstract

In this paper, we study robust stability of sparse LTI systems using the stability radius (SR) as a robustness measure. We consider real perturbations with an arbitrary and pre-specified sparsity pattern of the system matrix and measure their size using the Frobenius norm. We formulate the SR problem as an equality-constrained minimization problem. Using the Lagrangian method for optimization, we characterize the optimality conditions of the SR problem, thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system. Further, we use the Sylvester equation based parametrization to develop a penalty based gradient/Newton descent algorithm which converges to the local minima of the optimization problem. Finally, we illustrate how our framework provides structural insights into the robust stability of sparse networks.