Approximating the nearest stable discrete-time system

Nicolas Gillis; Michael Karow; Punit Sharma

doi:10.1016/j.laa.2019.03.014

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arXiv · 1802.08033

Approximating the nearest stable discrete-time system

Nicolas Gillis·Michael Karow·Punit Sharma

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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.

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BibTeX RIS

Nicolas Gillis, Michael Karow, Punit Sharma. 2018-02-22. Approximating the nearest stable discrete-time system. https://doi.org/10.1016/j.laa.2019.03.014

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