A note on approximating the nearest stable discrete-time descriptor system with fixed rank
It provides the first method for stabilizing discrete-time descriptor systems with fixed rank, which is a niche problem in control theory.
This paper addresses the problem of stabilizing discrete-time descriptor systems by finding a nearby stable system with fixed rank, reformulating the nonconvex problem into an optimization with a simple feasible set and using block coordinate descent. Numerical examples demonstrate the algorithm's effectiveness.
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.