Input-to-state stability of discontinuous dynamical systems with an observer-based control application

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Abstract

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems adopting Filippov's solution concept and using non-smooth ISS Lyapunov functions. The main motivation for adopting non-smooth ISS Lyapunov functions is that ``multiple Lyapunov functions'' are commonly used in the stability theory for hybrid systems. We will show that the existence of a non-smooth (but Lipschitz continuous) ISS Lyapunov function for a discontinuous system implies ISS. Next, we will prove an ISS interconnection theorem for two discontinuous dynamical systems that both admit an ISS Lyapunov function. The interconnection will be shown to be globally asymptotically stable under a small gain condition. The developed ISS theory will be applied to observer-based controller design for a class of piecewise linear systems using an observer structure proposed by the authors. The LMI-based design of the state feedback and the observer can be performed separately.
Original languageEnglish
Title of host publicationProceedings of the 10th international conference on Hybrid systems: computation and control (HSCC 2007) 3 - 5 April 2007, Pisa, Italy
EditorsA. Bemporard, A. Bicchi, G. Buttazzo
Place of PublicationBerlin
PublisherSpringer
Pages259-272
ISBN (Print)978-3-540-71492-7
DOIs
Publication statusPublished - 2007
Event10th International Conference on Hybrid Systems: Computation and Control (HSCC 2007) - Pisa, Italy
Duration: 3 Apr 20075 Apr 2007
Conference number: 10
http://cse.lab.imtlucca.it/~bemporad/hybrid/hscc07/

Publication series

NameLecture Notes in Computer Science
Volume4416
ISSN (Print)0302-9743

Conference

Conference10th International Conference on Hybrid Systems: Computation and Control (HSCC 2007)
Abbreviated titleHSCC 2007
CountryItaly
CityPisa
Period3/04/075/04/07
Internet address

Fingerprint

Dynamical systems
Lyapunov functions
Hybrid systems
State feedback
Linear systems
Controllers

Cite this

Heemels, W. P. M. H., Weiland, S., & Juloski, A. (2007). Input-to-state stability of discontinuous dynamical systems with an observer-based control application. In A. Bemporard, A. Bicchi, & G. Buttazzo (Eds.), Proceedings of the 10th international conference on Hybrid systems: computation and control (HSCC 2007) 3 - 5 April 2007, Pisa, Italy (pp. 259-272). (Lecture Notes in Computer Science; Vol. 4416). Berlin: Springer. https://doi.org/10.1007/978-3-540-71493-4_22
Heemels, W.P.M.H. ; Weiland, S. ; Juloski, A. / Input-to-state stability of discontinuous dynamical systems with an observer-based control application. Proceedings of the 10th international conference on Hybrid systems: computation and control (HSCC 2007) 3 - 5 April 2007, Pisa, Italy. editor / A. Bemporard ; A. Bicchi ; G. Buttazzo. Berlin : Springer, 2007. pp. 259-272 (Lecture Notes in Computer Science).
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Heemels, WPMH, Weiland, S & Juloski, A 2007, Input-to-state stability of discontinuous dynamical systems with an observer-based control application. in A Bemporard, A Bicchi & G Buttazzo (eds), Proceedings of the 10th international conference on Hybrid systems: computation and control (HSCC 2007) 3 - 5 April 2007, Pisa, Italy. Lecture Notes in Computer Science, vol. 4416, Springer, Berlin, pp. 259-272, 10th International Conference on Hybrid Systems: Computation and Control (HSCC 2007), Pisa, Italy, 3/04/07. https://doi.org/10.1007/978-3-540-71493-4_22

Input-to-state stability of discontinuous dynamical systems with an observer-based control application. / Heemels, W.P.M.H.; Weiland, S.; Juloski, A.

Proceedings of the 10th international conference on Hybrid systems: computation and control (HSCC 2007) 3 - 5 April 2007, Pisa, Italy. ed. / A. Bemporard; A. Bicchi; G. Buttazzo. Berlin : Springer, 2007. p. 259-272 (Lecture Notes in Computer Science; Vol. 4416).

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

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T1 - Input-to-state stability of discontinuous dynamical systems with an observer-based control application

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

AU - Weiland, S.

AU - Juloski, A.

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N2 - In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems adopting Filippov's solution concept and using non-smooth ISS Lyapunov functions. The main motivation for adopting non-smooth ISS Lyapunov functions is that ``multiple Lyapunov functions'' are commonly used in the stability theory for hybrid systems. We will show that the existence of a non-smooth (but Lipschitz continuous) ISS Lyapunov function for a discontinuous system implies ISS. Next, we will prove an ISS interconnection theorem for two discontinuous dynamical systems that both admit an ISS Lyapunov function. The interconnection will be shown to be globally asymptotically stable under a small gain condition. The developed ISS theory will be applied to observer-based controller design for a class of piecewise linear systems using an observer structure proposed by the authors. The LMI-based design of the state feedback and the observer can be performed separately.

AB - In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems adopting Filippov's solution concept and using non-smooth ISS Lyapunov functions. The main motivation for adopting non-smooth ISS Lyapunov functions is that ``multiple Lyapunov functions'' are commonly used in the stability theory for hybrid systems. We will show that the existence of a non-smooth (but Lipschitz continuous) ISS Lyapunov function for a discontinuous system implies ISS. Next, we will prove an ISS interconnection theorem for two discontinuous dynamical systems that both admit an ISS Lyapunov function. The interconnection will be shown to be globally asymptotically stable under a small gain condition. The developed ISS theory will be applied to observer-based controller design for a class of piecewise linear systems using an observer structure proposed by the authors. The LMI-based design of the state feedback and the observer can be performed separately.

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BT - Proceedings of the 10th international conference on Hybrid systems: computation and control (HSCC 2007) 3 - 5 April 2007, Pisa, Italy

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PB - Springer

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Heemels WPMH, Weiland S, Juloski A. Input-to-state stability of discontinuous dynamical systems with an observer-based control application. In Bemporard A, Bicchi A, Buttazzo G, editors, Proceedings of the 10th international conference on Hybrid systems: computation and control (HSCC 2007) 3 - 5 April 2007, Pisa, Italy. Berlin: Springer. 2007. p. 259-272. (Lecture Notes in Computer Science). https://doi.org/10.1007/978-3-540-71493-4_22