Robust stability conditions for switched linear systems: Commutator bounds and the ojasiewicz inequality

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

Abstract

This paper discusses conditions for stability of switched linear systems under arbitrary switching, formulated in terms of smallness of appropriate commutators of the matrices generating the switched system. Such conditions provide robust variants of well-known stability conditions requiring these commutators to vanish and leading to the existence of a common quadratic Lyapunov function. The main contribution of the paper is to apply the ojasiewicz inequality to characterize the persistence of a common quadratic Lyapunov function as the matrices are perturbed so that their commutators no longer vanish but instead are sufficiently small. It is shown how known constructions of common quadratic Lyapunov functions for commuting matrices and for matrices generating nilpotent or solvable Lie algebras can be used, in conjunction with the ojasiewicz inequality, to estimate allowable deviations of the commutators from zero.

Original languageEnglish (US)
Title of host publication2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages722-726
Number of pages5
ISBN (Print)9781467357173
DOIs
StatePublished - Jan 1 2013
Event52nd IEEE Conference on Decision and Control, CDC 2013 - Florence, Italy
Duration: Dec 10 2013Dec 13 2013

Publication series

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

Other

Other52nd IEEE Conference on Decision and Control, CDC 2013
CountryItaly
CityFlorence
Period12/10/1312/13/13

ASJC Scopus subject areas

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

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  • Cite this

    Baryshnikov, Y., & Liberzon, D. M. (2013). Robust stability conditions for switched linear systems: Commutator bounds and the ojasiewicz inequality. In 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013 (pp. 722-726). [6759967] (Proceedings of the IEEE Conference on Decision and Control). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2013.6759967