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arXiv · 1609.05779

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

Matthew Philippe·Ray Essick·Geir Dullerud·Raphaël Jungers
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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.

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Matthew Philippe, Ray Essick, Geir Dullerud, Raphaël Jungers. 2016-09-19. Extremal storage functions and minimal realizations of discrete-time linear switching systems. https://arxiv.org/abs/1609.05779

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