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arXiv · 1608.08311

A Characterization of Lyapunov Inequalities for Stability of Switched Systems

Raphael M. Jungers·Amirali Ahmadi·Pablo Parrilo·Mardavij Roozbehani
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Abstract

We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past fifteen years. We prove in this note that a family of languagetheoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.

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Raphael M. Jungers, Amirali Ahmadi, Pablo Parrilo, Mardavij Roozbehani. 2016-08-30. A Characterization of Lyapunov Inequalities for Stability of Switched Systems. https://doi.org/10.1109/tac.2017.2671345

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