Approximating the nearest stable discrete-time system

TL;DR

This paper introduces a novel characterization of stable matrices and develops an efficient projected gradient method to find the nearest stable matrix to an unstable one, aiding in stabilizing discrete-time systems.

Contribution

It provides a new matrix characterization for stability and a fast optimization algorithm to approximate the nearest stable matrix, improving stability computations.

Findings

The new characterization simplifies the stability condition.

The proposed method efficiently finds locally optimal stable matrices.

The approach outperforms existing methods in speed and accuracy.

Abstract

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 $A=S^{-1}UBS$, 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.