Inhalt des Dokuments
Preprint 04-2017
Computing nearest stable matrix pairs
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