On interconnections of discontinuous dynamical systems : an input-to-state stability approach

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Abstract

Abstract—In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting non-smooth ISS Lyapunov functions. The main motivation for investigating non-smooth ISS Lyapunov functions is the success of "multiple Lyapunov functions" in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of DDS. It is proven that the existence of a non-smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a non-smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities
Original languageEnglish
Title of host publicationProceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana
Place of PublicationPiscataway
PublisherInstitute of Electrical and Electronics Engineers
Pages109-114
ISBN (Print)978-1-4244-1498-7
DOIs
Publication statusPublished - 2007

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Lyapunov functions
Dynamical systems
Continuous time systems
Open systems
Linear matrix inequalities
Hybrid systems
Linear systems
Large scale systems

Cite this

Heemels, W. P. M. H., & Weiland, S. (2007). On interconnections of discontinuous dynamical systems : an input-to-state stability approach. In Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana (pp. 109-114). Piscataway: Institute of Electrical and Electronics Engineers. https://doi.org/10.1109/CDC.2007.4434742
Heemels, W.P.M.H. ; Weiland, S. / On interconnections of discontinuous dynamical systems : an input-to-state stability approach. Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana. Piscataway : Institute of Electrical and Electronics Engineers, 2007. pp. 109-114
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Heemels, WPMH & Weiland, S 2007, On interconnections of discontinuous dynamical systems : an input-to-state stability approach. in Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana. Institute of Electrical and Electronics Engineers, Piscataway, pp. 109-114. https://doi.org/10.1109/CDC.2007.4434742

On interconnections of discontinuous dynamical systems : an input-to-state stability approach. / Heemels, W.P.M.H.; Weiland, S.

Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana. Piscataway : Institute of Electrical and Electronics Engineers, 2007. p. 109-114.

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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Heemels WPMH, Weiland S. On interconnections of discontinuous dynamical systems : an input-to-state stability approach. In Proceedings of the 46th Conference on Decision and Control (CDC 2007) 12-14 December 2007, New Orleans, Louisiana. Piscataway: Institute of Electrical and Electronics Engineers. 2007. p. 109-114 https://doi.org/10.1109/CDC.2007.4434742