A Characterization of Lyapunov Inequalities for Stability of Switched Systems

Jungers, Raphaël;Ahmadi, Amir Ali;Parrilo, Pablo A.;Roozbehani, Mardavij
(2017) IEEE Transactions on Automatic Control — Vol. 62, p. 3062-3067 (2017)

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Authors
  • Jungers, RaphaëlUCLouvain
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  • Ahmadi, Amir AliPrinceton University, Princeton, NJ, USA
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  • Parrilo, Pablo A.Institute of Technology, Cambridge, MA, USA
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  • Roozbehani, MardavijInstitute of Technology, Cambridge, MA, USA
    Author
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 15 years. We prove in this note that a family of language-theoretic 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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Jungers, R., Ahmadi, A. A., Parrilo, P. A., & Roozbehani, M. (2017). A Characterization of Lyapunov Inequalities for Stability of Switched Systems. IEEE Transactions on Automatic Control, 62, 3062-3067. https://doi.org/10.1109/tac.2017.2671345 (Original work published 2017)