Input-to-state stability and interconnections of discontinuous dynamical systems

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Abstract

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov's solution concept, that is appropriate for `open' systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise 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 piecewise 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
Pages (from-to)3079-3086
Number of pages8
JournalAutomatica
Volume44
Issue number12
DOIs
Publication statusPublished - 2008

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

Cite this

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title = "Input-to-state stability and interconnections of discontinuous dynamical systems",
abstract = "In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov's solution concept, that is appropriate for `open' systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise 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 piecewise 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.",
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Input-to-state stability and interconnections of discontinuous dynamical systems. / Heemels, W.P.M.H.; Weiland, S.

In: Automatica, Vol. 44, No. 12, 2008, p. 3079-3086.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Heemels, W.P.M.H.

AU - Weiland, S.

PY - 2008

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