Model Checking of Hybrid Systems: From Reachability Towards Stability

  • Andreas Podelski
  • Silke Wagner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3927)


We call a hybrid system stable if every trajectory inevitably ends up in a given region. Our notion of stability deviates from classical definitions in control theory. In this paper, we present a model checking algorithm for stability in the new sense. The idea of the algorithm is to reduce the stability proof for the whole system to a set of (smaller) proofs for several one-mode systems.


Model Check Hybrid System Jump Condition Strong Attractor Hybrid Automaton 
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., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid Automata. An Algorithmic Approach to the Specification and Verification of Hybrid Systems. In: Hybrid Systems: Computation and Control (1993)Google Scholar
  2. 2.
    Branicky, M.S.: Stability of hybrid systems: State of the art. In: Conference on Decision and Control (1997)Google Scholar
  3. 3.
    Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. In: Trans. on Automatic Control (1998)Google Scholar
  4. 4.
    Biere, A., Artho, C., Schuppan, V.: Liveness checking as safety checking. In: Formal Methods for Industrial Critical Systems (FMICS) (2002)Google Scholar
  5. 5.
    Bradley, A., Manna, Z., Sipma, H.B.: Linear Ranking with Reachability. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Bradley, A., Manna, Z., Sipma, H.B.: The Polyranking Principle. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, Springer, Heidelberg (2005)Google Scholar
  7. 7.
    Chutinan, A., Fehnker, A., Han, Z., Kapinski, J., Kumar, R., Krogh, B.H., Stursberg, O.: CheckMate,
  8. 8.
    Clarke, E.M., Fehnker, A., Han, Z., Krogh, B., Stursberg, O., Theobald, M.: Verification of Hybrid Systems Based on Counterexample-Guided Abstraction Refinement. In: Garavel, H., Hatcliff, J. (eds.) ETAPS 2003 and TACAS 2003. LNCS, vol. 2619, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Cook, B., Podelski, A., Rybalchenko, A.: Termination Proofs for Systems Code. In: Submitted to Conference on Programming Language Design and Implementation (PLDI) (2006)Google Scholar
  10. 10.
    Colon, M., Sankaranarayanan, S., Sipma, H.: Linear invariant generation using non-linear constraint solving. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
  12. 12.
    Frehse, G.: Phaver,
  13. 13.
    Henzinger, T.A.: The Theory of Hybrid Automata. In: Logic in Computer Science (LICS) (1996)Google Scholar
  14. 14.
    Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: Algorithmic analysis of nonlinear hybrid systems. In: Automatic Control (1998)Google Scholar
  15. 15.
    Henzinger, T., Ho, P.-H., Wong-Toi, H.: HyTech,
  16. 16.
    Holzbaur, C.: clp(Q,R),
  17. 17.
    Liberzon, D.: Switching in Systems and Control. Birkhäuser, Basel (2003)CrossRefMATHGoogle Scholar
  18. 18.
    Liberzon, D., Agrachev, A.A.: Lie-algebraic stability criteria for switched systems. In: Control and Optimization (2001)Google Scholar
  19. 19.
    Liberzon, D., Hespanha, J.P., Morse, A.S.: Stability of switched systems: a Lie-algebraic condition. In: Systems and Control Letters (1999)Google Scholar
  20. 20.
    Liberzon, D., Margaliot, M.: Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions, Systems and Control Letters (to appear)Google Scholar
  21. 21.
    Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonliear Systems. World Scientific Pub Co Inc, Singapore (1990)CrossRefMATHGoogle Scholar
  22. 22.
    Papachristodoulou, A., Prajna, S.: On the Construction of Lyapunov Functions using the Sum of Squares Decomposition. In: Conference on Decision and Control (CDC) (2002)Google Scholar
  23. 23.
    Pettersson, S.: Analysis and Design of Hybrid Systems. Ph.D. Thesis, Chalmers University of Technology, Göteborg, Sweden (1999)Google Scholar
  24. 24.
    Podelski, A., Rybalchenko, A.: A complete Method for the Synthesis of Linear Ranking Functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, Springer, Heidelberg (2004)Google Scholar
  25. 25.
    Podelski, A., Rybalchenko, A.: Transition invariants. In: Logic in Computer Science (LICS) (2004)Google Scholar
  26. 26.
    Podelski, A., Rybalchenko, A.: Transition Predicate Abstraction and Fair Termination. In: Principles of Programming Language (POPL) (2005)Google Scholar
  27. 27.
    Prajna, S., Jadbabaie, A.: Safety Verification of Hybrid Systems Using Barrier Certificates. In: Hybrid Systems: Computation and Control (2004)Google Scholar
  28. 28.
    Ramsey, F.P.: On a problem of formal logic. In: Proc. of the London Mathematical Society 30 (1930)Google Scholar
  29. 29.
    Ratschan, S., She, Z.: HSolver,
  30. 30.
    Ratschan, S., She, Z.: Safety Verification of Hybrid Systems by Constraint Propagation Based Abstraction Refinement. In: Hybrid Systems: Computation and Control (2005)Google Scholar
  31. 31.
    Rybalchenko, A.: RankFinder,
  32. 32.
    Sankaranarayanan, S., Sipma, H., Manna, Z.: Constructing Invariants for Hybrid Systems. In: Hybrid Systems: Computation and Control (2004)Google Scholar
  33. 33.
    Tiwari, A.: Termination of linear programs. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, Springer, Heidelberg (2004)CrossRefGoogle Scholar
  34. 34.
    Tiwari, A., Ruess, H., Saidi, H., Shankar, N.: Automatic Generation of Invariants. In: Margaria, T., Yi, W. (eds.) ETAPS 2001 and TACAS 2001. LNCS, vol. 2031, Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Andreas Podelski
    • 1
  • Silke Wagner
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations