Approximating the nearest stable discrete-time system
For control engineers and researchers, this work offers a computationally efficient approach to stabilize discrete-time linear systems, though it is incremental as it builds on existing optimization techniques.
The paper provides a new characterization for stable discrete-time matrices and proposes a fast projected gradient method to find the nearest stable matrix to an unstable one, achieving locally optimal solutions more efficiently than existing methods.
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.