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.