Beyond Zeno: Get on with It!

  • Haiyang Zheng
  • Edward A. Lee
  • Aaron D. Ames
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3927)


In this paper we propose a technique to extend the simulation of a Zeno hybrid system beyond its Zeno time point. A Zeno hybrid system model is a hybrid system with an execution that takes an infinite number of discrete transitions during a finite time interval. We argue that the presence of Zeno behavior indicates that the hybrid system model is incomplete by considering some classical Zeno models that incompletely describe the dynamics of the system being modeled. This motivates the systematic development of a method for completing hybrid system models through the introduction of new post-Zeno states, where the completed hybrid system transitions to these post-Zeno states at the Zeno time point. In practice, simulating a Zeno hybrid system is challenging in that simulation effectively halts near the Zeno time point. Moreover, due to unavoidable numerical errors, it is not practical to exactly simulate a Zeno hybrid system. Therefore, we propose a method for constructing approximations of Zeno models by leveraging the completed hybrid system model. Using these approximation, we can simulate a Zeno hybrid system model beyond its Zeno point and reveal the complete dynamics of the system being modeled.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zhang, J., Johansson, K.H., Lygeros, J., Sastry, S.: Zeno hybrid systems. Int. J. Robust and Nonlinear Control 11(2), 435–451 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ames, A.D., Sastry, S.: Sufficient conditions for the existence of zeno behavior. In: 44th IEEE Conference on Decision and Control and European Control Conference ECC (2005)Google Scholar
  3. 3.
    Ames, A.D., Tabuada, P., Sastry, S.: On the stability of Zeno equilibria. To appear in Hybrid Systems: Computation and Control (2006)Google Scholar
  4. 4.
    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, Phoenix, AZ (1999)Google Scholar
  5. 5.
    Ames, A.D., Sastry, S.: Blowing up affine hybrid systems. In: 43rd IEEE Conference on Decision and Control (2004)Google Scholar
  6. 6.
    Mosterman, P.: An overview of hybrid simulation phenomena and their support by simulation packages. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 165–177. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Ames, A.D., Zheng, H., Gregg, R.D., Sastry, S.: Is there life after zeno? taking executions past the breaking (zeno) point. In: Sumbitted to the 2006 American Control Conference (2006)Google Scholar
  8. 8.
    van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Lecture Notes in Control and Information Sciences 251. Springer, Heidelberg (2000)MATHGoogle Scholar
  9. 9.
    Lygeros, J.: Lecture Notes on Hybrid Systems. ENSIETA 2-6/2/2004 (2004)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Burden, R.L., Faires, J.D.: Numerical analysis, 7th edn. Brroks/Cole (2001)Google Scholar
  12. 12.
    Shampine, L.F., Gladwell, I., Brankin, R.W.: Reliable solution of special event location problems for odes. ACM Trans. Math. Softw. 17(1), 11–25 (1991)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Park, T., Barton, P.I.: State event location in differential-algebraic models. ACM Transactions on Modeling and Computer Simulation (TOMACS) 6(2), 137–165 (1996)MATHCrossRefGoogle Scholar
  14. 14.
    Esposito, J.M., Kumar, V., Pappas, G.J.: Accurate event detection for simulating hybrid systems. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 204–217. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Haiyang Zheng
    • 1
  • Edward A. Lee
    • 1
  • Aaron D. Ames
    • 1
  1. 1.Center for Hybrid and Embedded Software Systems (CHESS), Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA

Personalised recommendations