A note on approximating the nearest stable discrete-time descriptor system with fixed rank

TL;DR

This paper introduces a novel method for approximating and stabilizing discrete-time descriptor systems with fixed rank by formulating a nonconvex optimization problem and applying a block coordinate descent algorithm.

Contribution

It is the first to address stabilizing descriptor systems with fixed rank by reformulating the problem into a tractable optimization framework.

Findings

Effective algorithm for stabilizing descriptor systems demonstrated

Successful application on multiple example systems

Provides a practical approach for fixed-rank system approximation

Abstract

Consider a discrete-time linear time-invariant descriptor system $Ex(k+1)=Ax(k)$ for $k \in \mathbb Z_{+}$. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system $\hat E x(k+1)= \hat A x(k)$ with $\text{rank}(\hat E)=r$. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.