On computing the distance to stability for matrices using linear dissipative Hamiltonian systems. (November 2017)

• Record Type: Journal Article
• Authors: Gillis, Nicolas; Sharma, Punit
• Is Part Of: Automatica. Volume 85(2017)
• Page Start: 113
• Page End: 121
• Publication Date: 2017-11
• Subjects: Dissipative Hamiltonian systems -- Distance to stability -- Convex optimization; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2017.07.047
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix A is stable if and only if it can be written as A = ( J − R ) Q, where J = − J T, R ⪰ 0 and Q ≻ 0 (that is, R is positive semidefinite and Q is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables ( J, R, Q ) : (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require O ( n 3 ) operations per iteration, where n is the size of A . We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms.