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International Workshop on Hybrid Systems: Computation and Control

HSCC 2006: Hybrid Systems: Computation and Control pp 318–332Cite as

Approximate Abstraction of Stochastic Hybrid Automata

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3927))

Abstract

This paper discusses a notion of approximate abstraction for linear stochastic hybrid automata (LSHA). The idea is based on the construction of the so called stochastic bisimulation function. Such function can be used to quantify the distance between a system and its approximate abstraction. The work in this paper generalizes our earlier work for jump linear stochastic systems (JLSS). In this paper we demonstrate that linear stochastic hybrid automata can be cast as a modified JLSS and modify the procedure for constructing the stochastic bisimulation function accordingly. The construction of quadratic stochastic bisimulation functions is essentially a linear matrix inequality problem. In this paper, we also discuss possible extensions of the framework to handle nonlinear dynamics and variable rate Poisson processes. As an example, we apply the framework to a chain-like stochastic hybrid automaton.

This research is supported by the National Science Foundation Presidential Early CAREER (PECASE) Grant 0132716.

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References

  1. Pola, G., Bujorianu, M., Lygeros, J., Benedetto, M.D.: Stochastic hybrid models: an overview. In: Proc. IFAC Conf. Analysis and Design of Hybrid Systems, St. Malo, IFAC (2003)

    Google Scholar 

  2. Hespanha, J.P.: Stochastic hybrid systems: applications to communication networks. In: [21], pp. 387–401

    Google Scholar 

  3. Hu, J., Wu, W.C., Sastry, S.: Modeling subtilin production in bacillus subtilis using stochastic hybrid systems. In: [21], pp. 417–431

    Google Scholar 

  4. Glover, W., Lygeros, J.: A stochastic hybrid model for air traffic control simulation. In: [21], pp. 372–386

    Google Scholar 

  5. Hu, J., Lygeros, J., Sastry, S.: Towards a theory of stochastic hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 160–173. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  6. Oksendal, B.: Stochastic differential equations: an introduction with applications. Springer, Berlin (2000)

    MATH  Google Scholar 

  7. Ghosh, M.K., Arapostathis, A., Marcus, S.: Ergodic control of switching diffusions. SIAM Journal on Control and Optimization 35(6), 1952–1988 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davis, M.H.A.: Markov models and optimization. Chapman and Hall, London (1993)

    Book  MATH  Google Scholar 

  9. Bujorianu, M.L., Lygeros, J., Bujorianu, M.C.: Bisimulation for general stochastic hybrid systems. In: [22], pp. 198–214

    Google Scholar 

  10. Strubbe, S., van der Schaft, A.J.: Bisimulation for communicating piecewise deterministic Markov processes. 623–639

    Google Scholar 

  11. Ying, M., Wirsing, M.: Approximate bisimilarity. In: Rus, T. (ed.) AMAST 2000. LNCS, vol. 1816, pp. 309–322. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  12. Girard, A., Pappas, G.J.: Approximate bisimulation for constrained linear systems. to appear in the Proceedings of the IEEE Conf. Decision and Control (2005)

    Google Scholar 

  13. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. Theoretical Computer Science 318(3), 323–354 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. van Breugel, F., Worrell, J.: An algorithm for quantitative verification of probabilistic transition systems. In: Proc. of CONCUR, Aalborg, pp. 336–350. Springer, Heidelberg (2001)

    Google Scholar 

  15. Julius, A.A., Girard, A., Pappas, G.J.: Approximate bisimulation for a class of stochastic hybrid systems. submitted to the American Control Conference 2006 (2005)

    Google Scholar 

  16. Prajna, S., Jadbabaie, A., Pappas, G.J.: Stochastic safety verification using barrier certificates. In: Proc. 43rd IEEE Conference on Decision and Control, Bahamas, IEEE Computer Society Press, Los Alamitos (2004)

    Google Scholar 

  17. Prajna, S., Papachristodoulou, A., Seiler, P., Parillo, P.A.: SOSTOOLS and its control application. In: Positive polynomials in control, Springer, Heidelberg (2005)

    Google Scholar 

  18. Neogi, N.A.: Dynamic partitioning of large discrete event biological systems for hybrid simulation and analysis. 463 – 476

    Google Scholar 

  19. Hespanha, J.P.: Polynomial stochastic hybrid systems. 322–338

    Google Scholar 

  20. Löfberg, J.: http://control.ee.ethz.ch/~joloef/yalmip.php

  21. Alur, R., Pappas, G.J. (eds.): HSCC 2004. LNCS, vol. 2993. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  22. Morari, M., Thiele, L. (eds.): HSCC 2005. LNCS, vol. 3414. Springer, Heidelberg (2005)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. Electrical and Systems Engineering, University of Pennsylvania, 200 South 33rd Street, Philadelphia, PA, 19104, USA

    A. Agung Julius

Authors
  1. A. Agung Julius
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Editor information

Editors and Affiliations

  1. Center for Control, Dynamical Systems and Computation, University of California, 93106, Santa Barbara, CA, USA

    João P. Hespanha

  2. SRI International, 333 Ravenswood Ave, Menlo Park, CA, USA

    Ashish Tiwari

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Julius, A.A. (2006). Approximate Abstraction of Stochastic Hybrid Automata. In: Hespanha, J.P., Tiwari, A. (eds) Hybrid Systems: Computation and Control. HSCC 2006. Lecture Notes in Computer Science, vol 3927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11730637_25

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  • DOI: https://doi.org/10.1007/11730637_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-33170-4

  • Online ISBN: 978-3-540-33171-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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