Switched stability of nonlinear systems via SOS-convex Lyapunov functions and semidefinite programming

Amir Ali Ahmadi, Raphaël M. Jungers

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

4 Scopus citations

Abstract

We introduce the concept of sos-convex Lyapunov functions for stability analysis of discrete time switched systems. These are polynomial Lyapunov functions that have an algebraic certificate of convexity, and can be efficiently found by semidefinite programming. We show that convex polynomial Lyapunov functions are universal (i.e., necessary and sufficient) for stability analysis of switched linear systems. On the other hand, we show via an explicit example that the minimum degree of an sos-convex Lyapunov function can be arbitrarily higher than a (non-convex) polynomial Lyapunov function. (The proof is omitted.) In the second part, we show that if the switched system is defined as the convex hull of a finite number of nonlinear functions, then existence of a non-convex common Lyapunov function is not a sufficient condition for switched stability, but existence of a convex common Lyapunov function is. This shows the usefulness of the computational machinery of sos-convex Lyapunov functions which can be applied either directly to the switched nonlinear system, or to its linearization, to provide proof of local switched stability for the nonlinear system. An example is given where no polynomial of degree less than 14 can provide an estimate to the region of attraction under arbitrary switching.

Original languageEnglish (US)
Title of host publication2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages727-732
Number of pages6
ISBN (Print)9781467357173
DOIs
StatePublished - Jan 1 2013
Externally publishedYes
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

All Science Journal Classification (ASJC) codes

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

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