Approximating the nearest stable discrete-time system

In this paper, we consider the problem of stabilizing discrete-time linear systems by computing a nearby stable matrix to an unstable one. To do so, we provide a new characterization for the set of stable matrices. We show that a matrix A is stable if and only if it can be written as A=S-1UBS, where S is positive definite, U is orthogonal, and B is a positive semidefinite contraction (that is, the singular values of B are less or equal to 1). This characterization results in an equivalent non-convex optimization problem with a feasible set on which it is easy to project. We propose a very efficient fast projected gradient method to tackle the problem in variables (S,U,B) and generate locally optimal solutions. We show the effectiveness of the proposed method compared to other approaches.