An example of Zero robustness in Piecewise Affine Systems

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

Abstract

An example is presented in order to illustrate that nominally exponentially stable discrete-time PieceWise Affine (PWA) systems can have zero robustness [1] to arbitrarily small additive disturbances. It is shown that this is mainly due to the absence of a continuous Lyapunov function. The importance of this issue cannot be overstated since nominally stabilizing controllers are affected by perturbations when applied in practice. Moreover, many of the methods for synthesizing stabilizing state-feedback controllers for discrete-time PWA systems employ discontinuous (piecewise quadratic) Lyapunov functions, for example see [2, 3]. The discrete-time input-to-state stability framework [4] is used in order to develop an a posteriori robustness test, which boils down to solving a finite number of linear programming problems. The test can be used to check whether a specific nominally stable PWA system, possibly with a discontinuous Lyapunov function, has some (inherent) robustness to additive disturbances or not.
Original languageEnglish
Title of host publicationProceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands
Place of PublicationHeeze, Netherlands
Pages43-43
Publication statusPublished - 2006
Event25th Benelux Meeting on Systems and Control, March 13-15, 2006, Heeze, The Netherlands - Heeze, Netherlands
Duration: 13 Mar 200615 Mar 2006

Conference

Conference25th Benelux Meeting on Systems and Control, March 13-15, 2006, Heeze, The Netherlands
CountryNetherlands
CityHeeze
Period13/03/0615/03/06

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Lyapunov functions
Controllers
State feedback
Linear programming

Cite this

Lazar, M., & Heemels, W. P. M. H. (2006). An example of Zero robustness in Piecewise Affine Systems. In Proceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands (pp. 43-43). Heeze, Netherlands.
Lazar, M. ; Heemels, W.P.M.H. / An example of Zero robustness in Piecewise Affine Systems. Proceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands. Heeze, Netherlands, 2006. pp. 43-43
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title = "An example of Zero robustness in Piecewise Affine Systems",
abstract = "An example is presented in order to illustrate that nominally exponentially stable discrete-time PieceWise Affine (PWA) systems can have zero robustness [1] to arbitrarily small additive disturbances. It is shown that this is mainly due to the absence of a continuous Lyapunov function. The importance of this issue cannot be overstated since nominally stabilizing controllers are affected by perturbations when applied in practice. Moreover, many of the methods for synthesizing stabilizing state-feedback controllers for discrete-time PWA systems employ discontinuous (piecewise quadratic) Lyapunov functions, for example see [2, 3]. The discrete-time input-to-state stability framework [4] is used in order to develop an a posteriori robustness test, which boils down to solving a finite number of linear programming problems. The test can be used to check whether a specific nominally stable PWA system, possibly with a discontinuous Lyapunov function, has some (inherent) robustness to additive disturbances or not.",
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Lazar, M & Heemels, WPMH 2006, An example of Zero robustness in Piecewise Affine Systems. in Proceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands. Heeze, Netherlands, pp. 43-43, 25th Benelux Meeting on Systems and Control, March 13-15, 2006, Heeze, The Netherlands, Heeze, Netherlands, 13/03/06.

An example of Zero robustness in Piecewise Affine Systems. / Lazar, M.; Heemels, W.P.M.H.

Proceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands. Heeze, Netherlands, 2006. p. 43-43.

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

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Lazar M, Heemels WPMH. An example of Zero robustness in Piecewise Affine Systems. In Proceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands. Heeze, Netherlands. 2006. p. 43-43