Computing the nearest stable matrix pairs. Issue 5 (1st February 2018)

• Record Type: Journal Article
• Authors: Gillis, Nicolas; Mehrmann, Volker; Sharma, Punit
• Is Part Of: Numerical linear algebra with applications. Volume 25:Issue 5(2018)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2018-02-01
• Subjects: convex optimization -- dissipative Hamiltonian system -- distance to stability; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2153
• Languages: English
• ISSNs: 1070-5325

Abstract

Summary: 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 matrix pairs: A matrix pair ( E, A ) is dissipative Hamiltonian 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.