A Hierarchical Approach for the Synthesis of Stabilizing Controllers for Hybrid Systems

  • Janusz Malinowski
  • Peter Niebert
  • Pierre-Alain Reynier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6996)


We consider a discretization based approach to controller synthesis of hybrid systems that allows to handle non-linear dynamics. In such an approach, states are grouped together in a finite index partition at the price of a non-deterministic over approximation of the transition relation. The main contribution of this work is a technique to reduce the state explosion generated by the discretization: exploiting structural properties of ODE systems, we propose a hierarchical approach to the synthesis problem by solving it first for sub problems and using the results for state space reduction in the full problem. A secondary contribution concerns combined safety and liveness control objectives that approximate stabilization.


Hybrid System Control Objective Inverted Pendulum Hierarchical Approach Atomic Proposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur, R., Dang, T., Ivancic, F.: Counterexample-guided predicate abstraction of hybrid systems. Theor. Comput. Sci. 354(2), 250–271 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alur, R., Henzinger, T.A., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. Proceedings of the IEEE (88), 971–984 (2000)Google Scholar
  3. 3.
    Asarin, E., Dang, T., Girard, A.: Hybridization methods for the analysis of nonlinear systems. Acta Inf. 43(7), 451–476 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brandfield, J., Stirling, C.: Modal mu-calculi. In: The Handbook of Modal Logic, pp. 721–756. Elsevier, Amsterdam (2006)Google Scholar
  5. 5.
    Cassez, F., David, A., Fleury, E., Larsen, K., Lime, D.: Efficient on-the-fly algorithms for the analysis of timed games. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 66–80. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Clarke, E.M., Fehnker, A., Han, Z., Krogh, B.H., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. Int. J. Found. Comput. Sci. 14(4), 583–604 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cleaveland, R., Steffen, B.: A linear-time model-checking algorithm for the alternation-free modal mu-calculus. In: FMSD, pp. 48–58. Springer, Heidelberg (1993)Google Scholar
  8. 8.
    Donzé, A., Maler, O.: Systematic simulation using sensitivity analysis. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 174–189. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research. LNCS, vol. 2500. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  10. 10.
    Henzinger, T.A., Horowitz, B., Majumdar, R., Wong-Toi, H.: Beyond hytech: Hybrid systems analysis using interval numerical methods. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 130–144. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Henzinger, T.A., Jhala, R., Majumdar, R.: Counterexample-guided control. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 886–902. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Liu, X., Smolka, S.A.: Simple linear-time algorithms for minimal fixed points. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 53–66. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Moor, T., Davoren, J.M., Raisch, J.: Learning by doing systematic abstraction refinement for hybrid control synthesis. In: IEE Proc. Control Theory & Applications, Special issue on hybrid systems, vol. 153 (2006)Google Scholar
  14. 14.
    Perk, S., Moor, T., Schmidt, K.: Controller synthesis for an i/o-based hierarchical system architecture. In: International Workshop on Discrete Event Systems (WODES), pp. 474–479. IEEE, Los Alamitos (2008)Google Scholar
  15. 15.
    Tiwari, A., Khanna, G.: Series of abstractions for hybrid automata. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 465–478. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Tomlin, C., Lygeros, J., Sastry, S.: Computing controllers for nonlinear hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 238–255. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Janusz Malinowski
    • 1
  • Peter Niebert
    • 1
  • Pierre-Alain Reynier
    • 1
  1. 1.LIF, Université de Provence & CNRS, UMR 6166France

Personalised recommendations