Computing nearest stable matrix pairs

Abstract

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair $(E,A)$, minimize the Frobenius norm of $(\hat{I}{”}_E,\hat{I}{”}_A)$ such that $(E+\hat{I}{”}_E,A+\hat{I}{”}_A)$ is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair $(E,A)$ is DH if $A=(J-R)Q$ with skew-symmetric $J$, positive semidefinite $R$, and an invertible $Q$ such that $Q^{T}E$ is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.

19 pages, 4 figures