On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

• Autores: Nicolas Gillis, Punit Sharma
• Localización: Automatica: A journal of IFAC the International Federation of Automatic Control, ISSN 0005-1098, Vol. 85, 2017, págs. 113-121
• Idioma: inglés
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• Resumen

  • 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.