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Probably Approximate Safety Verification of Hybrid Dynamical Systems

  • Bai XueEmail author
  • Martin Fränzle
  • Hengjun Zhao
  • Naijun Zhan
  • Arvind Easwaran
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11852)

Abstract

In this paper we present a method based on linear programming that facilitates reliable safety verification of hybrid dynamical systems subject to perturbation inputs over the infinite time horizon. The verification algorithm applies the probably approximately correct (PAC) learning framework and consequently can be regarded as statistically formal verification in the sense that it provides formal safety guarantees expressed using error probabilities and confidences. The safety of hybrid systems in this framework is verified via the computation of so-called PAC barrier certificates, which can be computed by solving a linear programming problem. Based on scenario approaches, the linear program is constructed by a family of independent and identically distributed state samples. In this way we can conduct verification of hybrid dynamical systems that existing methods are not capable of dealing with. Some preliminary experiments demonstrate the performance of our approach.

Keywords

Hybrid systems Probably approximately safe Linear program 

Notes

Acknowledgements

Bai Xue was funded by CAS Pioneer Hundred Talents Program under grant No. Y8YC235015, NSFC under grant No. 61872341 and 61836005. Martin Fränzle was funded by Deutsche Forschungsgemeinschaft through grant FR 2715/4. Hengjun Zhao was funded by NSFC under grant No. 61702425. Naijun Zhan was funded by NSFC under grant No. 61625206 and 61732001. Arvind Easwaran was supported by the Energy Research Institute (ERI@N), NTU, Singapore.

References

  1. 1.
    Alur, R.: Formal verification of hybrid systems. In: EMSOFT 2011, pp. 273–278. IEEE (2011)Google Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) HS 1991-1992. LNCS, vol. 736, pp. 209–229. Springer, Heidelberg (1993).  https://doi.org/10.1007/3-540-57318-6_30CrossRefGoogle Scholar
  3. 3.
    Asarin, E., Maler, O.: Achilles and the tortoise climbing up the arithmetical hierarchy. J. Comput. Syst. Sci. 57(3), 389–398 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Calafiore, G.C.: Random convex programs. SIAM J. Optim. 20(6), 3427–3464 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Campi, M.C., Garatti, S.: A sampling-and-discarding approach to chance-constrained optimization: feasibility and optimality. J. Optim. Theory Appl. 148(2), 257–280 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Campi, M.C., Garatti, S., Prandini, M.: The scenario approach for systems and control design. Annu. Rev. Control 33(2), 149–157 (2009)CrossRefGoogle Scholar
  7. 7.
    Dai, L., Gan, T., Xia, B., Zhan, N.: Barrier certificates revisited. J. Symb. Comput. 80, 62–86 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Djaballah, A.: Computation of barrier certificates for dynamical hybrids systems using interval analysis. Université Paris-Saclay (2017)Google Scholar
  9. 9.
    Egyed, A.: Invited talk: a roadmap for engineering safe and secure cyber-physical systems. In: MEDI 2018, pp. 113–114 (2018)Google Scholar
  10. 10.
    Fränzle, M.: Analysis of hybrid systems: an ounce of realism can save an infinity of states. In: Flum, J., Rodriguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 126–139. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48168-0_10CrossRefGoogle Scholar
  11. 11.
    Fränzle, M., Gerwinn, S., Kröger, P., Abate, A., Katoen, J.-P.: Multi-objective parameter synthesis in probabilistic hybrid systems. In: Sankaranarayanan, S., Vicario, E. (eds.) FORMATS 2015. LNCS, vol. 9268, pp. 93–107. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-22975-1_7CrossRefzbMATHGoogle Scholar
  12. 12.
    Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. J. Satisf. Boolean Model. Comput. 1, 209–236 (2007)zbMATHGoogle Scholar
  13. 13.
    Fränzle, M., Shirmohammadi, M., Swaminathan, M., Worrell, J.: Costs and rewards in priced timed automata. In: ICALP 2018, pp. 125:1–125:14 (2018)Google Scholar
  14. 14.
    Gao, S., Avigad, J., Clarke, E.M.: Delta-decidability over the reals. In: LICS 2012, pp. 305–314 (2012)Google Scholar
  15. 15.
    Haussler, D.: Probably approximately correct learning. Computer Research Laboratory, University of California, Santa Cruz (1990)Google Scholar
  16. 16.
    Henrion, D., Lasserre, J.B., Savorgnan, C.: Approximate volume and integration for basic semialgebraic sets. SIAM Rev. 51(4), 722–743 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? J. Comput. Syst. Sci. 57(1), 94–124 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13–30 (1963)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Huang, C., Chen, X., Lin, W., Yang, Z., Li, X.: Probabilistic safety verification of stochastic hybrid systems using barrier certificates. ACM Trans. Embed. Comput. Syst. 16(5), 186:1–186:19 (2017)Google Scholar
  20. 20.
    Kong, H., Bogomolov, S., Schilling, C., Jiang, Y., Henzinger, T.A.: Safety verification of nonlinear hybrid systems based on invariant clusters. In: HSCC 2017, pp. 163–172. ACM (2017)Google Scholar
  21. 21.
    Kong, H., He, F., Song, X., Hung, W.N.N., Gu, M.: Exponential-condition-based barrier certificate generation for safety verification of hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 242–257. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39799-8_17CrossRefGoogle Scholar
  22. 22.
    Lin, W., Wu, M., Yang, Z., Zeng, Z.: Exact safety verification of hybrid systems using sums-of-squares representation. Sci. China Inf. Sci. 57(5), 1–13 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nahhal, T., Dang, T.: Test coverage for continuous and hybrid systems. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 449–462. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-73368-3_47CrossRefGoogle Scholar
  24. 24.
    Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 477–492. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24743-2_32CrossRefzbMATHGoogle Scholar
  25. 25.
    Ratschan, S.: Simulation based computation of certificates for safety of dynamical systems. In: Abate, A., Geeraerts, G. (eds.) FORMATS 2017. LNCS, vol. 10419, pp. 303–317. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-65765-3_17CrossRefzbMATHGoogle Scholar
  26. 26.
    Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation-based abstraction refinement. ACM Trans. Embed. Comput. S. 6(1), 8 (2007)CrossRefGoogle Scholar
  27. 27.
    Ratschan, S., She, Z.: Providing a basin of attraction to a target region of polynomial systems by computation of lyapunov-like functions. SIAM J. Control Optim. 48(7), 4377–4394 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rohn, J.I., Kreslova, J.: Linear interval inequalities. Linear Multilinear Algebra 38, 79–82 (1994)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sankaranarayanan, S., Chen, X., Ábrahám, E.: Lyapunov function synthesis using Handelman representations. In: NOLCOS 2013, pp. 576–581 (2013)Google Scholar
  30. 30.
    Schupp, S., et al.: Current challenges in the verification of hybrid systems. In: Berger, C., Mousavi, M.R. (eds.) CyPhy 2015. LNCS, vol. 9361, pp. 8–24. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-25141-7_2CrossRefGoogle Scholar
  31. 31.
    Shmarov, F., Zuliani, P.: ProbReach: verified probabilistic delta-reachability for stochastic hybrid systems. In: HSCC 2015, pp. 134–139 (2015)Google Scholar
  32. 32.
    Sogokon, A., Ghorbal, K., Tan, Y.K., Platzer, A.: Vector barrier certificates and comparison systems. In: Havelund, K., Peleska, J., Roscoe, B., de Vink, E. (eds.) FM 2018. LNCS, vol. 10951, pp. 418–437. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-95582-7_25CrossRefGoogle Scholar
  33. 33.
    Xue, B. Fränzle, M., Zhan, N.: Inner-approximating reachable sets for polynomial systems with time-varying uncertainties. IEEE Trans. Autom. Control (2019) Google Scholar
  34. 34.
    Xue, B., She, Z., Easwaran, A.: Under-approximating backward reachable sets by polytopes. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9779, pp. 457–476. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-41528-4_25CrossRefGoogle Scholar
  35. 35.
    Xue, B., She, Z., Easwaran, A.: Underapproximating backward reachable sets by semialgebraic sets. IEEE Trans. Autom. Control 62(10), 5185–5197 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Xue, B., Wang, Q., Zhan, N., Fränzle, M.: Robust invariant sets generation for state-constrained perturbed polynomial systems. In: HSCC 2019, pp. 128–137 (2019)Google Scholar
  37. 37.
    Zhang, Y., Yang, Z., Lin, W., Zhu, H., Chen, X., Li, X.: Safety verification of nonlinear hybrid systems based on bilinear programming. IEEE Trans. CAD Integr. Circuits Syst. 37(11), 2768–2778 (2018)CrossRefGoogle Scholar
  38. 38.
    Zuliani, P., Platzer, A., Clarke, E.M.: Bayesian statistical model checking with application to simulink/stateflow verification. In: HSCC 2010, pp. 243–252 (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Bai Xue
    • 1
    • 2
    • 3
    Email author
  • Martin Fränzle
    • 4
  • Hengjun Zhao
    • 5
  • Naijun Zhan
    • 1
    • 2
  • Arvind Easwaran
    • 6
  1. 1.State Key Laboratory of Computer ScienceInstitute of Software, CASBeijingChina
  2. 2.University of Chinese Academy of SciencesBeijingChina
  3. 3.Beijing Institute of Control EngineeringBeijingChina
  4. 4.Carl von Ossietzky Universität OldenburgOldenburgGermany
  5. 5.Southwest UniversityChongqingChina
  6. 6.Nanyang Technological UniversitySingaporeSingapore

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