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

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.

Computing the nearest stable matrix pairs

“…However, parts of Lemma and Theorem also hold for singular Q with an extra assumption on the eigenvectors of the pencil z E −( J − R ) Q , as stated in the next result. This result is a generalization of lemma 2 in the work of Gillis et al for DH matrices with singular Q .…”

“…For this reason, we reformulate the nearest stable matrix pair problem into an equivalent optimization problem with a simpler feasible set. For this, we extend the idea of a DH matrix from the work of Gillis et al to DH pairs.…”

“…This problem is closely related to the classical state‐feedback stabilization problem in descriptor control problems with f = B u for some control function u ; see, for example, other works, where one chooses feedbacks u = K x so that the closed‐loop descriptor system as follows: is asymptotically stable. However, the perturbations are restricted only in A , whereas in derivative feedback, one also allows feedbacks E + B G ; see the works of Bunse‐Gerstner et al Instead of the feedback stabilization problem, we extend the nearest stable matrix problem of other works to matrix pencils, and we study the following problem.…”

“…This reformulation uses the fact that the set of stable matrices can be characterized as the set of dissipative Hamiltonian (DH) matrices . We recall from the work of Gillis et al that a matrix is called a DH matrix if A can be factored as A =( J − R ) Q for some such that J T =− J , R = R ≽0, that is, positive semidefinite, and Q = Q T ≻0, that is, positive definite. In this way, A is stable if and only if A is a DH matrix; see also other works …”

“…It is also clear that the set is highly nonconvex; see the work of Orbandexivry et al, and thus, it is very difficult to find a globally optimal solution of the nearest stable matrix pair problem. To address this challenge, we follow the strategy suggested by Gillis et al for matrices and reformulate the problem of computing the nearest stable matrix pair by extending the concept of DH matrices to DH matrix pairs (DH pairs) . The DH parametrization/representation of a stable LTI system can be useful in other applications.…”