A Characterization of Lyapunov Inequalities for Stability of Switched Systems

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 ha...

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Bibliographic Details
Published inIEEE transactions on automatic control Vol. 62; no. 6; pp. 3062 - 3067
Main Authors Jungers, Raphael M., Ahmadi, Amir Ali, Parrilo, Pablo A., Roozbehani, Mardavij
Format Journal Article
LanguageEnglish
Published New York IEEE 01.06.2017
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects
Asymptotic stability
Discrete time systems
Encapsulation
Inequalities
Linear matrix inequalities
Lyapunov methods
set theory
stability
Stability criteria
Switched systems
Switches
switching systems (control)
Theorems
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Summary: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.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2017.2671345