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Conic Abstractions for Hybrid Systems

  • Sergiy Bogomolov
  • Mirco GiacobbeEmail author
  • Thomas A. Henzinger
  • Hui KongEmail author
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10419)

Abstract

Despite researchers’ efforts in the last couple of decades, reachability analysis is still a challenging problem even for linear hybrid systems. Among the existing approaches, the most practical ones are mainly based on bounded-time reachable set over-approximations. For the purpose of unbounded-time analysis, one important strategy is to abstract the original system and find an invariant for the abstraction. In this paper, we propose an approach to constructing a new kind of abstraction called conic abstraction for affine hybrid systems, and to computing reachable sets based on this abstraction. The essential feature of a conic abstraction is that it partitions the state space of a system into a set of convex polyhedral cones which is derived from a uniform conic partition of the derivative space. Such a set of polyhedral cones is able to cut all trajectories of the system into almost straight segments so that every segment of a reach pipe in a polyhedral cone tends to be straight as well, and hence can be over-approximated tightly by polyhedra using similar techniques as HyTech or PHAVer. In particular, for diagonalizable affine systems, our approach can guarantee to find an invariant for unbounded reachable sets, which is beyond the capability of bounded-time reachability analysis tools. We implemented the approach in a tool and experiments on benchmarks show that our approach is more powerful than SpaceEx and PHAVer in dealing with diagonalizable systems.

Keywords

Affine system Hybrid system Reachability analysis Conic abstraction Discrete abstraction 
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Notes

Acknowledgments

This work was partly supported by the Austrian Science Fund (FWF) under grants S11402-N23 (RiSE/SHiNE) and Z211-N23 (Wittgenstein Award) and by the ARC project DP140104219 (Robust AI Planning for Hybrid Systems).

References

  1. 1.
    Henzinger, T.: The theory of hybrid automata. In: Proceedings of IEEE Symposium on Logic in Computer Science, pp. 278–292 (1996)Google Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T., Ho, P., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(1), 3–34 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_18 CrossRefGoogle Scholar
  4. 4.
    Dang, T., Maler, O.: Reachability analysis via face lifting. In: Henzinger, T.A., Sastry, S. (eds.) HSCC 1998. LNCS, vol. 1386, pp. 96–109. Springer, Heidelberg (1998). doi: 10.1007/3-540-64358-3_34 CrossRefGoogle Scholar
  5. 5.
    Kloetzer, M., Belta, C.: Reachability analysis of multi-affine systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 348–362. Springer, Heidelberg (2006). doi: 10.1007/11730637_27 CrossRefGoogle Scholar
  6. 6.
    Prabhakar, P., Viswanathan, M.: A dynamic algorithm for approximate flow computations. In: HSCC, pp. 133–142 (2011)Google Scholar
  7. 7.
    Lal, R., Prabhakar, P.: Bounded error flowpipe computation of parameterized linear systems. In: 2015 International Conference on Embedded Software (EMSOFT 2015), Amsterdam, Netherlands, 4–9 October 2015, pp. 237–246 (2015)Google Scholar
  8. 8.
    Kong, H., Bogomolov, S., Schilling, C., Jiang, Y., Henzinger, T.A.: Safety verification of nonlinear hybrid systems based on invariant clusters. In: HSCC, ser. (HSCC 2017), pp. 163–172. ACM, New York (2017)Google Scholar
  9. 9.
    Chutinan, A., Krogh, B.H.: Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In: Vaandrager, F.W., Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 76–90. Springer, Heidelberg (1999). doi: 10.1007/3-540-48983-5_10 CrossRefGoogle Scholar
  10. 10.
    Asarin, E., Bournez, O., Dang, T., Maler, O.: Approximate reachability analysis of piecewise-linear dynamical systems. In: Lynch, N., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 20–31. Springer, Heidelberg (2000). doi: 10.1007/3-540-46430-1_6 CrossRefGoogle Scholar
  11. 11.
    Kurzhanski, A., Varaiya, P.: Ellipsoidal techniques for reachability analysis: internal approximation. Syst. Contr. Lett. 41(3), 201–211 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Botchkarev, O., Tripakis, S.: Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations. In: Lynch, N., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 73–88. Springer, Heidelberg (2000). doi: 10.1007/3-540-46430-1_10 CrossRefGoogle Scholar
  13. 13.
    Stursberg, O., Krogh, B.H.: Efficient representation and computation of reachable sets for hybrid systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 482–497. Springer, Heidelberg (2003). doi: 10.1007/3-540-36580-X_35 CrossRefGoogle Scholar
  14. 14.
    Girard, A.: Reachability of uncertain linear systems using zonotopes. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 291–305. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-31954-2_19 CrossRefGoogle Scholar
  15. 15.
    Girard, A., Guernic, C., Maler, O.: Efficient computation of reachable sets of linear time-invariant systems with inputs. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 257–271. Springer, Heidelberg (2006). doi: 10.1007/11730637_21 CrossRefGoogle Scholar
  16. 16.
    Guernic, C., Girard, A.: Reachability analysis of hybrid systems using support functions. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 540–554. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02658-4_40 CrossRefGoogle Scholar
  17. 17.
    Jiang, Y., Song, H., Wang, R., Gu, M., Sun, J., Sha, L.: Data-centered runtime verification of wireless medical cyber-physical system. IEEE Trans. Ind. Inform. PP(99), 1 (2016)Google Scholar
  18. 18.
    Jiang, Y., Zhang, H., Li, Z., Deng, Y., Song, X., Gu, M., Sun, J.: Design and optimization of multiclocked embedded systems using formal techniques. IEEE Trans. Ind. Electron. 62(2), 1270–1278 (2015)CrossRefGoogle Scholar
  19. 19.
    Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: HyTech: a model checker for hybrid systems. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 460–463. Springer, Heidelberg (1997). doi: 10.1007/3-540-63166-6_48 CrossRefGoogle Scholar
  20. 20.
    Frehse, G.: Phaver: algorithmic verification of hybrid systems past hytech. Int. J. Softw. Tools Technol. Transfer 10(3), 263–279 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Batt, G., Belta, C., Weiss, R.: Temporal logic analysis of gene networks under parameter uncertainty. Trans. Autom. Contr. 53(Special Issue), 215–229 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Alur, R., Dang, T., Ivančić, F.: Progress on reachability analysis of hybrid systems using predicate abstraction. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 4–19. Springer, Heidelberg (2003). doi: 10.1007/3-540-36580-X_4 CrossRefGoogle Scholar
  23. 23.
    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). doi: 10.1007/3-540-45873-5_36 CrossRefGoogle Scholar
  24. 24.
    Tiwari, A.: Abstractions for hybrid systems. Formal Methods Syst. Des. 32(1), 57–83 (2008)CrossRefzbMATHGoogle Scholar
  25. 25.
    Roohi, N., Prabhakar, P., Viswanathan, M.: Hybridization based CEGAR for hybrid automata with affine dynamics. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 752–769. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49674-9_48 CrossRefGoogle Scholar
  26. 26.
    Sogokon, A., Ghorbal, K., Jackson, P.B., Platzer, A.: A method for invariant generation for polynomial continuous systems. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 268–288. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49122-5_13 CrossRefGoogle Scholar
  27. 27.
    Asarin, E., Dang, T., Girard, A.: Hybridization methods for the analysis of nonlinear systems. Acta Informatica 43(7), 451–476 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Henzinger, T., Wong-Toi, H.: Linear phase-portrait approximations for nonlinear hybrid systems. Hybrid Syst. III, 377–388 (1996)Google Scholar
  29. 29.
    Frehse, G., Guernic, C., Donzé, A., Cotton, S., Ray, R., Lebeltel, O., Ripado, R., Girard, A., Dang, T., Maler, O.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22110-1_30 CrossRefGoogle Scholar
  30. 30.
    Frehse, G.: PHAVer: algorithmic verification of hybrid systems past HyTech. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 258–273. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-31954-2_17 CrossRefGoogle Scholar
  31. 31.
    Doyen, L., Henzinger, T.A., Raskin, J.-F.: Automatic rectangular refinement of affine hybrid systems. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 144–161. Springer, Heidelberg (2005). doi: 10.1007/11603009_13 CrossRefGoogle Scholar
  32. 32.
    Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic press, Amsterdam (2012)zbMATHGoogle Scholar
  33. 33.
    Kong, H., Bartocci, E., Bogomolov, S., Grosu, R., Henzinger, T.A., Jiang, Y., Schilling, C.: Discrete abstraction of multiaffine systems. In: Cinquemani, E., Donzé, A. (eds.) HSB 2016. LNCS, vol. 9957, pp. 128–144. Springer, Cham (2016). doi: 10.1007/978-3-319-47151-8_9 Google Scholar
  34. 34.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Scalable analysis of linear systems using mathematical programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 25–41. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-30579-8_2 CrossRefGoogle Scholar
  35. 35.
    GLPK (GNU linear programming kit). www.gnu.org/software/glpk
  36. 36.
    Frehse, G., Kateja, R., Le Guernic, C.: Flowpipe approximation and clustering in space-time. In: Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control, pp. 203–212. ACM (2013)Google Scholar
  37. 37.
    Fehnker, A., Ivančić, F.: Benchmarks for hybrid systems verification. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 326–341. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24743-2_22 CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Australian National UniversityCanberraAustralia
  2. 2.IST AustriaKlosterneuburgAustria

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