# Approximating the nearest stable discrete-time system

arXiv: https://arxiv.org/abs/1802.08033 · HTML: https://mlsanity.com/paper/1802.08033/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2018.

## Built on by (most cited first, 6 of 6)

- [Stabilizing reinforcement learning control: A modular framework for optimizing over all stable behavior](https://mlsanity.com/paper/2310.14098/) (2023)
- [Stability Preserving Data-driven Models With Latent Dynamics](https://mlsanity.com/paper/2204.11744/) (2022)
- [Inference of Continuous Linear Systems from Data with Guaranteed Stability](https://mlsanity.com/paper/2301.10060/) (2023)
- [Physical Backdoor Trigger Activation of Autonomous Vehicle using Reachability Analysis](https://mlsanity.com/paper/2303.13992/) (2023)
- [Optimal Centered Active Excitation in Linear System Identification](https://mlsanity.com/paper/2604.05518/) (2026)
- [A Novel Schur-Decomposition-Based Weight Projection Method for Stable State-Space Neural-Network Architectures](https://mlsanity.com/paper/2605.14489/) (2026)

## Abstract (arXiv)

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.