Large-Scale and Global Maximization of the Distance to Instability

TL;DR

This paper develops algorithms for maximizing the distance to instability of matrices, improving robustness analysis for dynamical systems, with methods applicable to both small and large matrices.

Contribution

It introduces a globally convergent algorithm for small matrices and a subspace framework with superlinear convergence for large matrices.

Findings

Global convergence proven for small matrices

Subspace framework reduces computational complexity

Superlinear convergence rate achieved

Abstract

The larger the distance to instability from a matrix is, the more robustly stable the associated autonomous dynamical system is in the presence of uncertainties and typically the less severe transient behavior its solution exhibits. Motivated by these issues, we consider the maximization of the distance to instability of a matrix dependent on several parameters, a nonconvex optimization problem that is likely to be nonsmooth. In the first part we propose a globally convergent algorithm when the matrix is of small size and depends on a few parameters. In the second part we deal with the problems involving large matrices. We tailor a subspace framework that reduces the size of the matrix drastically. The strength of the tailored subspace framework is proven with a global convergence result as the subspaces grow and a superlinear rate-of-convergence result with respect to the subspace dimension.