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

Consider a discrete-time linear time-invariant descriptor system Ex(k+1)=Ax(k) for k ∈ 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 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.