Advertisement

Non-Standard Zeno-Free Simulation Semantics for Hybrid Dynamical Systems

  • Ayman AljarbouhEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11847)

Abstract

Geometric-Zeno behavior is one of the most challenging problems in the analysis and simulation of hybrid systems. Geometric-Zeno solutions involve an accumulation of an infinite number of discrete events occurring in a finite amount of time. In models that exhibit geometric-Zeno, discrete events occur at an increasingly smaller distance in time, converging to a limit point according to a geometric series. In practice, simulating models of hybrid systems exhibiting geometric-Zeno is highly challenging, in that the simulation either halts or produces faulty results, as the time interval lengths are decreasing to arbitrary small positive numbers. Although many simulation tools for hybrid systems have been developed in the past years, none of them have a Zeno-free semantic model. All of them produce faulty results when simulating geometric-Zeno models. In this paper, we propose a non-standard Zeno-free mathematical formalism for detecting and eliminating geometric-Zeno during simulation. We derive sufficient conditions for the existence of geometric-Zeno behavior based on the existence of a non-standard contraction map in a complete metric space. We also provide methods for carrying solutions from pre-Zeno to post-Zeno. We illustrate the concepts with examples throughout the paper.

Keywords

Hybrid systems Modeling and simulation Zeno behavior Simulation tools Non-standard analysis Model verification and completeness 

Notes

Acknowledgements

This work is supported by the European project ITEA3 MODRIO under contract No 6892, and the Grant ARED of Brittany Regional Council.

References

  1. 1.
  2. 2.
  3. 3.
  4. 4.
  5. 5.
  6. 6.
  7. 7.
    DOMNA: a lite matlab simulator for zeno-free simulation of hybrid dynamical systems, with zeno detection and avoidance in run-time. https://bil.inria.fr/fr/software/view/2691/tab
  8. 8.
    Sevama: A simulink toolbox and simulator for zeno-free simulation of hybrid dynamical systems, with zeno detection and avoidance in run-time. https://bil.inria.fr/fr/software/view/2679/tab
  9. 9.
    Aljarbouh, A., Caillaudr, B.: Simulation for hybrid systems: chattering path avoidance. In: Proceedings of the 56th Conference on Simulation and Modelling (SIMS 56), Linkoping Electronic Conference Proceedings, vol. 119, pp. 175–185 (2015).  https://doi.org/10.3384/ecp15119175
  10. 10.
    Aljarbouh, A., Caillaudr, B.: Chattering-free simulation of hybrid dynamical systems with the functional mock-up interface 2.0. In: Proceedings of the First Japanese Modelica Conferences, Linkoping Electronic Conference Proceedings, vol. 124, pp. 95–105 (2016).  https://doi.org/10.3384/ecp1612495
  11. 11.
    Aljarbouh, A., Zeng, Y., Duracz, A., Caillaud, B., Taha, W.: Chattering-free simulation for hybrid dynamical systems semantics and prototype implementation. In: 2016 IEEE International Conference on Computational Science and Engineering (CSE) and IEEE International Conference on Embedded and Ubiquitous Computing (EUC) and 15th International Symposium on Distributed Computing and Applications for Business Engineering (DCABES), pp. 412–422, August 2016.  https://doi.org/10.1109/CSE-EUC-DCABES.2016.217
  12. 12.
    Aljarbouh, A., Caillaud, B.: On the regularization of chattering executions in real time simulation of hybrid systems. In: Cap, C. (ed.) Baltic Young Scientists Conference, p. 49. The 11th Baltic Young Scientists Conference, Universität Rostock, Tallinn, Estonia, July 2015. https://hal.archives-ouvertes.fr/hal-01246853
  13. 13.
    Alur, R., et al.: Hybrid systems the algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(1), 3–34 (1995).  https://doi.org/10.1016/0304-3975(94)00202-T. http://www.sciencedirect.com/science/article/pii/030439759400202TMathSciNetCrossRefGoogle Scholar
  14. 14.
    Alur, R., Henzinger, T.A., Ho, P.H.: Automatic symbolic verification of embedded systems. IEEE Trans. Softw. Eng. 22(3), 181–201 (1996).  https://doi.org/10.1109/32.489079CrossRefGoogle Scholar
  15. 15.
    Alur, R., Henzinger, T.A.: Modularity for timed and hybrid systems. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 74–88. Springer, Heidelberg (1997).  https://doi.org/10.1007/3-540-63141-0_6CrossRefGoogle Scholar
  16. 16.
    Ames, A.D., Abate, A., Sastry, S.: Sufficient conditions for the existence of zeno behavior. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 696–701, December 2005.  https://doi.org/10.1109/CDC.2005.1582237
  17. 17.
    Ames, A.D., Sastry, S.: Blowing up affine hybrid systems. In: 43rd IEEE Conference on Decision and Control, CDC, vol. 1, pp. 473–478, December 2004.  https://doi.org/10.1109/CDC.2004.1428675
  18. 18.
    Ames, A.D., Zheng, H., Gregg, R.D., Sastry, S.: Is there life after zeno? Taking executions past the breaking (zeno) point. In: 2006 American Control Conference, 6 pp., June 2006.  https://doi.org/10.1109/ACC.2006.1656623
  19. 19.
    Antsaklis, P.J.: Special issue on hybrid systems: theory and applications a brief introduction to the theory and applications of hybrid systems. Proc. IEEE 88(7), 879–887 (2000).  https://doi.org/10.1109/JPROC.2000.871299CrossRefGoogle Scholar
  20. 20.
    Benveniste, A., Bourke, T., Caillaud, B., Pouzet, M.: Non-standard semantics of hybrid systems modelers. J. Comput. Syst. Sci. 78(3), 877–910 (2012). http://www.sciencedirect.com/science/article/pii/S0022000011001061. In Commemoration of Amir PnueliMathSciNetCrossRefGoogle Scholar
  21. 21.
    Bourke, T., Pouzet, M.: Zélus: a synchronous language with ODEs. In: Belta, C., Ivančić, F. (eds.) HSCC - Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control, pp. 113–118, Calin Belta and Franjo Ivančić. ACM, Philadelphia, April 2013.  https://doi.org/10.1145/2461328.2461348. https://hal.inria.fr/hal-00909029
  22. 22.
    Cai, C., Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid systems: limit sets and zero dynamics with a view toward output regulation. In: Astolfi, A., Marconi, L. (eds.) Analysis and Design of Nonlinear Control Systems, pp. 241–261. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-74358-3_15CrossRefzbMATHGoogle Scholar
  23. 23.
    Camlibel, M.K., Schumacher, J.M.: On the zeno behavior of linear complementarity systems. In: Proceedings of the 40th IEEE Conference on Decision and Control, vol. 124, pp. 346–351 (2001).  https://doi.org/10.1109/.2001.980124
  24. 24.
    Egerstedt, M., Johansson, K.H., Sastry, S., Lygeros, J.: On the regularization of Zeno hybrid automata. Syst. Control. Lett. 38, 141–150 (1999). https://control.ee.ethz.ch/index.cgi?page=publications;action=details;id=2985
  25. 25.
    Eker, J., Janneck, J.W., Lee, E.A., Ludvig, J., Neuendorffer, S., Sachs, S.: Taming heterogeneity - the ptolemy approach. Proc. IEEE 91(1), 127–144 (2003).  https://doi.org/10.1109/JPROC.2002.805829CrossRefGoogle Scholar
  26. 26.
    Frehse, G., et al.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22110-1_30CrossRefGoogle Scholar
  27. 27.
    Fritzson, P.: Introduction to Modeling and Simulation of Technical and Physical Systems with Modelica. Wiley-IEEE Press, Hoboken (2011)CrossRefGoogle Scholar
  28. 28.
    Goebel, R., Teel, A.R.: Lyapunov characterization of zeno behavior in hybrid systems. In: 47th IEEE Conference on Decision and Control, CDC 2008, pp. 2752–2757, December 2008.  https://doi.org/10.1109/CDC.2008.4738864
  29. 29.
    Goldblatt, R.: Lecture on the Hyperreals: An Introduction to Nonstandard Analysis. Springer, Heidelberg (1988)zbMATHGoogle Scholar
  30. 30.
    Johansson, K.H., Lygeros, J., Sastry, S., Egerstedt, M.: Simulation of zeno hybrid automata. In: Proceedings of the 38th IEEE Conference on Decision and Control, vol. 4, pp. 3538–3543 (1999).  https://doi.org/10.1109/CDC.1999.827900
  31. 31.
    Lamperski, A., Ames, A.D.: Lyapunov-like conditions for the existence of zeno behavior in hybrid and Lagrangian hybrid systems. In: 2007 46th IEEE Conference on Decision and Control, pp. 115–120, December 2007.  https://doi.org/10.1109/CDC.2007.4435003
  32. 32.
    Lamperski, A., Ames, A.D.: On the existence of zeno behavior in hybrid systems with non-isolated zeno equilibria. In: 47th IEEE Conference on Decision and Control, CDC 2008, pp. 2776–2781, December 2008.  https://doi.org/10.1109/CDC.2008.4739100
  33. 33.
    Lamperski, A., Ames, A.D.: Lyapunov theory for zeno stability. IEEE Trans. Autom. Control 58(1), 100–112 (2013).  https://doi.org/10.1109/TAC.2012.2208292MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lee, E.A., Zheng, H.: Operational semantics of hybrid systems. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 25–53. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-31954-2_2CrossRefGoogle Scholar
  35. 35.
    Lygeros, J., Tomlin, C., Sastry, S.: Hybrid systems: modeling, analysis and control. Lecture Notes on Hybrid Systems (2008). http://www-inst.cs.berkeley.edu/~ee291e/sp09/handouts/book.pdf
  36. 36.
    Shen, J., Pang, J.S.: Linear complementarity systems: zeno states. SIAM J. Control Optim. 44(3), 1040–1066 (2005).  https://doi.org/10.1137/040612270, http://dx.doi.org/10.1137/040612270MathSciNetCrossRefGoogle Scholar
  37. 37.
    Thuan, L.Q.: Non-zenoness of piecewise affine dynamical systems and affine complementarity systems with inputs. Control Theory Technol. 12(1), 35–47 (2014).  https://doi.org/10.1007/s11768-014-0074-5. http://dx.doi.org/10.1007/s11768-014-0074-5MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Gipsa-lab, Grenoble INP, University of Grenoble AlpesGrenobleFrance

Personalised recommendations