A Characterization of Lyapunov Inequalities for Stability of Switched Systems

作者：Jungers Raphael M^*; Ahmadi Amir Ali; Parrilo Pablo A; Roozbehani Mardavij

来源：IEEE Transactions on Automatic Control, 2017, 62(6): 3062-3067.

DOI：10.1109/TAC.2017.2671345

• Linear matrix inequalities
• lyapunov methods
• set theory
• stability
• switching systems (control)

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.

• 出版日期2017-6