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

TL;DR

This paper characterizes the $ ext{L}_p$-gain of stable discrete-time linear switching systems using switching storage functions, introduces algorithms for minimal realization, and applies these tools to approximate $ ext{L}_2$-gain with practical examples.

Contribution

It provides a novel characterization of $ ext{L}_p$-gain via switching storage functions and algorithms for minimal realizations of systems with mode-dependent state dimensions.

Findings

Exact $ ext{L}_p$-gain characterization for stable systems

Algorithms for computing minimal rectangular realizations

Approximation methods for $ ext{L}_2$-gain and quadratic storage functions

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 $p \geq 1$, 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.