Scalable Stabilization of Large Scale Discrete-time Linear Systems via the 1-norm

This article reconsiders the stabilizing controller synthesis problem for discrete–time linear systems with a focus on systems of large scale. In this case, existing solutions are either not scalable, and thus, not tractable, or conservative. This motivates us to exploit finite–time control Lyapunov functions (CLFs), i.e., a relaxation of the standard CLF concept, to obtain a nonconservative and scalable synthesis method. The main idea is to employ Minkowski functions of a particular family of polytopic sets, which includes the hyper–rhombus induced by the 1–norm, as candidate finite–time CLFs. This choice results in explicit periodic vertex–interpolation control laws, which are globally stabilizing. The vertex–control laws can be computed offline using distributed optimization, in a scalable fashion, while the actual control law comes in an explicit, distributed form. Large scale illustrative examples demonstrate the effectiveness of the proposed approach.