Subtleties in robust stability of discrete-time piecewise affine systems

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Abstract

In this paper we consider (inherent) robustness ofdiscrete-time piecewise affine (PWA) systems. We demonstrate, via examples, that globally exponentially stable discrete-time PWA systems may have no robustness. More precisely, we show that the exponential stability property cannot prevent thatarbitrarily small additive disturbances keep the state trajectory far from the origin. Mathematically speaking, this means that the system is not input-to-state stable with respect to arbitrarily small disturbances. The non-robustness property is related to the absence of a continuous Lyapunov function. Theseresults issue a warning regarding existing stability analysis and synthesis methods for PWA systems that rely on discontinuous Lyapunov functions, as no robustness might be present. In many cases though, the search for Lyapunov functions in discrete-time PWA systems employs discontinuous Lyapunov functions (e.g. piecewise quadratic ones). To establish robustness(or non-robustness) of nominally stable PWA systems in these cases, when a continuous Lyapunov function is not known, robustness tests based on discontinuous Lyapunov functions are needed. Such tests are proposed in this article.
Original languageEnglish
Title of host publicationProceedings of the 2007 American Control Conference (ACC 2007) 9-13 July 2007, New York, New York, USA
Place of PublicationPiscataway, New Jersey, USA
PublisherInstitute of Electrical and Electronics Engineers
Pages3464-3469
ISBN (Print)1-4244-0988-8
DOIs
Publication statusPublished - 2007
Event2007 American Control Conference (ACC 2007), July 11-13, 2007, New York, NY, USA - Marriott Marquis Hotel at Times Square, New York, NY, United States
Duration: 11 Jul 200713 Jul 2007
http://acc2007.a2c2.org/

Conference

Conference2007 American Control Conference (ACC 2007), July 11-13, 2007, New York, NY, USA
Abbreviated titleACC 2007
CountryUnited States
CityNew York, NY
Period11/07/0713/07/07
Internet address

Fingerprint

Lyapunov functions
Asymptotic stability
Robust stability
Trajectories

Cite this

Lazar, M., Heemels, W. P. M. H., & Teel, A. R. (2007). Subtleties in robust stability of discrete-time piecewise affine systems. In Proceedings of the 2007 American Control Conference (ACC 2007) 9-13 July 2007, New York, New York, USA (pp. 3464-3469). Piscataway, New Jersey, USA: Institute of Electrical and Electronics Engineers. https://doi.org/10.1109/ACC.2007.4282881
Lazar, M. ; Heemels, W.P.M.H. ; Teel, A.R. / Subtleties in robust stability of discrete-time piecewise affine systems. Proceedings of the 2007 American Control Conference (ACC 2007) 9-13 July 2007, New York, New York, USA. Piscataway, New Jersey, USA : Institute of Electrical and Electronics Engineers, 2007. pp. 3464-3469
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Lazar, M, Heemels, WPMH & Teel, AR 2007, Subtleties in robust stability of discrete-time piecewise affine systems. in Proceedings of the 2007 American Control Conference (ACC 2007) 9-13 July 2007, New York, New York, USA. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, USA, pp. 3464-3469, 2007 American Control Conference (ACC 2007), July 11-13, 2007, New York, NY, USA, New York, NY, United States, 11/07/07. https://doi.org/10.1109/ACC.2007.4282881

Subtleties in robust stability of discrete-time piecewise affine systems. / Lazar, M.; Heemels, W.P.M.H.; Teel, A.R.

Proceedings of the 2007 American Control Conference (ACC 2007) 9-13 July 2007, New York, New York, USA. Piscataway, New Jersey, USA : Institute of Electrical and Electronics Engineers, 2007. p. 3464-3469.

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

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AB - In this paper we consider (inherent) robustness ofdiscrete-time piecewise affine (PWA) systems. We demonstrate, via examples, that globally exponentially stable discrete-time PWA systems may have no robustness. More precisely, we show that the exponential stability property cannot prevent thatarbitrarily small additive disturbances keep the state trajectory far from the origin. Mathematically speaking, this means that the system is not input-to-state stable with respect to arbitrarily small disturbances. The non-robustness property is related to the absence of a continuous Lyapunov function. Theseresults issue a warning regarding existing stability analysis and synthesis methods for PWA systems that rely on discontinuous Lyapunov functions, as no robustness might be present. In many cases though, the search for Lyapunov functions in discrete-time PWA systems employs discontinuous Lyapunov functions (e.g. piecewise quadratic ones). To establish robustness(or non-robustness) of nominally stable PWA systems in these cases, when a continuous Lyapunov function is not known, robustness tests based on discontinuous Lyapunov functions are needed. Such tests are proposed in this article.

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Lazar M, Heemels WPMH, Teel AR. Subtleties in robust stability of discrete-time piecewise affine systems. In Proceedings of the 2007 American Control Conference (ACC 2007) 9-13 July 2007, New York, New York, USA. Piscataway, New Jersey, USA: Institute of Electrical and Electronics Engineers. 2007. p. 3464-3469 https://doi.org/10.1109/ACC.2007.4282881