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Static Analysis for State-Space Reduction of Polygonal Hybrid Systems

  • Gordon Pace
  • Gerardo Schneider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4202)

Abstract

Polygonal hybrid systems (SPDI) are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions. The reachability problem as well as the computation of certain objects of the phase portrait, namely the viability, controllability and invariance kernels, for such systems is decidable. In this paper we show how to compute another object of an SPDI phase portrait, namely semi-separatrix curves and show how the phase portrait can be used for reducing the state-space for optimizing the reachability analysis.

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References

  1. [AD94]
    Alur, R., Dill, D.L.: A theory of timed automata. Theoretical Computer Science 126, 183–235 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. [ALQ+01a]
    Aubin, J.-P., Lygeros, J., Quincampoix, M., Sastry, S., Seube, N.: Towards a viability theory for hybrid systems. In: European Control Conference (2001)Google Scholar
  3. [ALQ+01b]
    Aubin, J.-P., Lygeros, J., Quincampoix, M., Sastry, S., Seube, N.: Viability and invariance kernels of impulse differential inclusions. In: Conference on Decision and Control, vol. 40. IEEE, Los Alamitos (2001)Google Scholar
  4. [APSY02]
    Asarin, E., Pace, G., Schneider, G., Yovine, S.: SPeeDI: a verification tool for polygonal hybrid systems. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, p. 354. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. [ASY01]
    Asarin, E., Schneider, G., Yovine, S.: On the decidability of reachability for planar differential inclusions. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, p. 89. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. [ASY02]
    Asarin, E., Schneider, G., Yovine, S.: Towards computing phase portraits of polygonal differential inclusions. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, p. 49. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. [Aub01]
    Aubin, J.-P.: The substratum of impulse and hybrid control systems. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 105–118. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. [DV95]
    Deshpande, A., Varaiya, P.: Viable control of hybrid systems. In: Antsaklis, P.J., Kohn, W., Nerode, A., Sastry, S.S. (eds.) HS 1994. LNCS, vol. 999, pp. 128–147. Springer, Heidelberg (1995)Google Scholar
  9. [FvDFH96]
    Foley, J.D., van Dam, A., Feiner, S.K., Hughes, J.F.: Computer graphics (2nd ed. in C): principles and practice. Addison-Wesley Longman Publishing Co., Inc., Boston (1996)Google Scholar
  10. [HKPV95]
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? In: STOC 1995, pp. 373–382. ACM Press, New York (1995)CrossRefGoogle Scholar
  11. [LPY01]
    Lafferriere, G., Pappas, G., Yovine, S.: Symbolic reachability computation of families of linear vector fields. Journal of Symbolic Computation 32(3), 231–253 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. [MP93]
    Maler, O., Pnueli, A.: Reachability analysis of planar multi-linear systems. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 194–209. Springer, Heidelberg (1993)Google Scholar
  13. [MS00]
    Matveev, A., Savkin, A.: Qualitative theory of hybrid dynamical systems. Birkhäuser, Boston (2000)MATHGoogle Scholar
  14. [PS]
    Pace, G., Schneider, G.: SPeeDI + , http://www.cs.um.edu.mt/speedi
  15. [PS06]
    Pace, G., Schneider, G.: Static analysis of SPDIs for state-space reduction. Technical Report 336, Department of Informatics, University of Oslo (April 2006)Google Scholar
  16. [Sch02]
    Schneider, G.: Algorithmic Analysis of Polygonal Hybrid Systems. PhD thesis, VERIMAG – UJF, Grenoble, France (July 2002)Google Scholar
  17. [Sch04]
    Schneider, G.: Computing invariance kernels of polygonal hybrid systems. Nordic Journal of Computing 11(2), 194–210 (2004)MATHMathSciNetGoogle Scholar
  18. [SJSL00]
    Simić, S., Johansson, K., Sastry, S., Lygeros, J.: Towards a geometric theory of hybrid systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, p. 421. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. [SP02]
    Saint-Pierre, P.: Hybrid kernels and capture basins for impulse constrained systems. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, p. 378. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gordon Pace
    • 1
  • Gerardo Schneider
    • 2
  1. 1.Dept. of Computer Science and AIUniversity of MaltaMsidaMalta
  2. 2.Dept. of InformaticsUniversity of OsloOsloNorway

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