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Preprint 04-2017

Computing nearest stable matrix pairs

Source file is available as :   Portable Document Format (PDF)

Author(s) : Nicolas Gillis , Volker Mehrmann , Punit Sharma

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 04-2017

MSC 2000

93D09 Robust stability
65F15 Eigenvalues, eigenvectors

Abstract :
In this paper, we study the nearest stable matrix pair problem: given a square matrix pair $(E,A)$, minimize the Frobenius norm of $(\Delta_E,\Delta_A)$ such that $(E+\Delta_E,A+\Delta_A)$ is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set byintroducing 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.

Keywords : dissipative Hamiltonian system, distance to stability, convex optimization

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