Extremal storage functions and minimal realizations of discrete-time linear switching systems

Abstract

We study the ${\mathcal{L}}_p$ induced gain of discrete-time linear switching systems with graph-constrained switching sequences. We first prove that, for stable systems in a minimal realization, for every $p\ge 1$, the ${\mathcal{L}}_p$-gain is exactly characterized through switching storage functions. These functions are shown to be the $p$th power of a norm. In order to consider general systems, we provide an algorithm for computing minimal realizations. These realizations are \emph{rectangular systems}, with a state dimension that varies according to the mode of the system. We apply our tools to the study on the of ${\mathcal{L}}_2$-gain. We provide algorithms for its approximation, and provide a converse result for the existence of quadratic switching storage functions. We finally illustrate the results with a physically motivated example.

To appear in proceedings of the CDC 2016 - proofs added here