Computing nearest stable matrix pairs
For control theory and dynamical systems, this provides a practical algorithm to stabilize matrix pairs, addressing a known bottleneck in computing nearest stable pairs.
The paper introduces a reformulation of the nearest stable matrix pair problem using dissipative Hamiltonian matrix pairs, enabling a fast gradient method to compute a nearby stable approximation. The method efficiently finds stable pairs with minimal Frobenius norm perturbation.
In this paper, we study the nearest stable matrix pair problem: given a square matrix pair $(E,A)$, minimize the Frobenius norm of $(Δ_E,Δ_A)$ such that $(E+Δ_E,A+Δ_A)$ 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 $Q^TE$ 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.