Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems

Amir Ali Ahmadi, Pablo A. Parrilo

Research output: Chapter in Book/Report/Conference proceedingConference contribution

76 Scopus citations

Abstract

We relax the monotonicity requirement of Lyapunov's theorem to enlarge the class of functions that can provide certificates of stability. To this end, we propose two new sufficient conditions for global asymptotic stability that allow the Lyapunov functions to increase locally, but guarantee an average decrease every few steps. Our first condition is non-convex, but allows an intuitive interpretation. The second condition, which includes the first one as a special case, is convex and can be cast as a semidefinite program. We show that when non-monotonic Lyapunov functions exist, one can construct a more complicated function that decreases monotonically. We demonstrate the strength of our methodology over standard Lyapunov theory through examples from three different classes of dynamical systems. First, we consider polynomial dynamics where we utilize techniques from sum-of-squares programming. Second, analysis of piecewise affine systems is performed. Here, connections to the method of piecewise quadratic Lyapunov functions are made. Finally, we examine systems with arbitrary switching between a finite set of matrices. It will be shown that tighter bounds on the joint spectral radius can be obtained using our technique.

Original languageEnglish (US)
Title of host publicationProceedings of the 47th IEEE Conference on Decision and Control, CDC 2008
Pages614-621
Number of pages8
DOIs
StatePublished - Dec 1 2008
Externally publishedYes
Event47th IEEE Conference on Decision and Control, CDC 2008 - Cancun, Mexico
Duration: Dec 9 2008Dec 11 2008

Publication series

NameProceedings of the IEEE Conference on Decision and Control
ISSN (Print)0191-2216

Other

Other47th IEEE Conference on Decision and Control, CDC 2008
CountryMexico
CityCancun
Period12/9/0812/11/08

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

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    Ahmadi, A. A., & Parrilo, P. A. (2008). Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems. In Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008 (pp. 614-621). [4739402] (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2008.4739402