Symbolic Analysis of Hybrid Systems Involving Numerous Discrete Changes Using Loop Detection

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10107)


Hybrid systems are dynamical systems that include both continuous and discrete changes. Some hybrid systems involve a large or infinite number of discrete changes within an infinitesimal-width region of phase space. Systems with sliding mode are typical examples of such hybrid systems. It is difficult to analyze such hybrid systems through ordinary numerical simulation, since the time required for simulation increases in proportion to the number of discrete changes. In this paper, we propose a method to symbolically analyze such models involving numerous discrete changes by detecting loops and checking loop invariants of the model’s behavior. The method handles parameterized hybrid systems and checks inclusion of parameterized states focusing on the values of a switching function that dominate the dynamics of sliding mode. We implemented the main part of the method in our symbolic hybrid system simulator HyLaGI, and conducted analysis of example models.


Hybrid systems Sliding mode Loop invariants Verification Symbolic analysis 


  1. 1.
    Aljarbouh, A., Caillaud, B.: Chattering-free simulation of hybrid dynamical systems with the Functional Mock-Up Interface 2.0. In: Proceedings of 1st Japanese Modelica Conference, Linköping Electronic Conference Proceedings, vol. 124, no. 013, pp. 95–105 (2016)Google Scholar
  2. 2.
    Betsuno, K., Matsumoto, S., Wakatsuki, Y., Ueda, K.: Analysis of hybrid systems involving numerous discrete changes using loop detection. In: Proceedings of 30th Annual Conference of the Japanese Society for Artificial Intelligence, 1F3-4 (2016). (in Japanese)Google Scholar
  3. 3.
    Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and its Applications. Kluwer Academic, Boston (1988)CrossRefMATHGoogle Scholar
  4. 4.
    Henzinger, T.: The theory of hybrid automata. In: Proceedings of LICS 1996, pp. 278–292. IEEE Computer Society Press (1996)Google Scholar
  5. 5.
    Lee, E.A.: Constructive models of discrete and continuous physical phenomena. IEEE Access 2, 797–821 (2014)CrossRefGoogle Scholar
  6. 6.
    Lunze, J.: Handbook of Hybrid Systems Control: Theory, Tools, Applications. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Matsumoto, S., Kono, F., Kobayashi, T., Ueda, K.: HyLaGI: symbolic implementation of a hybrid constraint language HydLa. Electron. Notes Theoret. Comput. Sci. 317, 109–115 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mosterman, P.J., Zhao, F., Biswas, G.: Sliding mode model semantics and simulation for hybrid systems. In: Antsaklis, P., Lemmon, M., Kohn, W., Nerode, A., Sastry, S. (eds.) HS 1997. LNCS, vol. 1567, pp. 218–237. Springer, Heidelberg (1999). doi: 10.1007/3-540-49163-5_12 CrossRefGoogle Scholar
  9. 9.
    Platzer, A., Quesel, J.-D.: KeYmaera: a hybrid theorem prover for hybrid systems (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 171–178. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-71070-7_15 CrossRefGoogle Scholar
  10. 10.
    Ueda, K., Matsumoto, S., Takeguchi, A., Hosobe, H., Ishii, D.: HydLa: a high-level language for hybrid systems. In: Proceedings of 2nd Workshop on Logics for System Analysis (LfSA 2012, affiliated with CAV 2012), pp. 3–17 (2012)Google Scholar
  11. 11.
    Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
  12. 12.
    Wakatsuki, Y., Matsumoto, S., Ueda, K.: Introduction of LTL model checking to a hybrid constraint system HyLaGI. In: Proceedings of 30th Annual Conference of the Japanese Society for Artificial Intelligence, 1F3-1 (2016). (in Japanese)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Kenichi Betsuno
    • 1
  • Shota Matsumoto
    • 1
  • Kazunori Ueda
    • 1
  1. 1.Department of Computer Science and EngineeringWaseda UniversityTokyoJapan

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