Verifying Safety and Persistence Properties of Hybrid Systems Using Flowpipes and Continuous Invariants

  • Andrew Sogokon
  • Paul B. Jackson
  • Taylor T. Johnson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10227)


We propose a method for verifying persistence of nonlinear hybrid systems. Given some system and an initial set of states, the method can guarantee that system trajectories always eventually evolve into some specified target subset of the states of one of the discrete modes of the system, and always remain within this target region. The method also computes a time-bound within which the target region is always reached. The approach combines flow-pipe computation with deductive reasoning about invariants and is more general than each technique alone. We illustrate the method with a case study concerning showing that potentially destructive stick-slip oscillations of an oil-well drill eventually die away for a certain choice of drill control parameters. The case study demonstrates how just using flow-pipes or just reasoning about invariants alone can be insufficient. The case study also nicely shows the richness of systems that the method can handle: the case study features a mode with non-polynomial (nonlinear) ODEs and we manage to prove the persistence property with the aid of an automatic prover specifically designed for handling transcendental functions.


Hybrid System Safety Property Reachable State Drill String Discrete Transition 
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.



The authors wish to thank to the anonymous reviewers for their careful reading and valuable suggestions for improving this paper.


  1. 1.
  2. 2.
    Akbarpour, B., Paulson, L.C.: MetiTarski: an automatic theorem prover for real-valued special functions. J. Autom. Reason. 44(3), 175–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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). doi: 10.1007/3-540-57318-6_30 CrossRefGoogle Scholar
  4. 4.
    Berz, M., Makino, K.: Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliab. Comput. 4(4), 361–369 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blanchini, F.: Set invariance in control. Automatica 35(11), 1747–1767 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carter, R.A.: Verification of liveness properties on hybrid dynamical systems. Ph.D. thesis, University of Manchester, School of Computer Science (2013)Google Scholar
  7. 7.
    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
  8. 8.
    Clarke, E.M., Fehnker, A., Han, Z., Krogh, B.H., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. Int. J. Found. Comput. Sci. 14(4), 583–604 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975). doi: 10.1007/3-540-07407-4_17 CrossRefGoogle Scholar
  10. 10.
    Donzé, A., Maler, O.: Systematic simulation using sensitivity analysis. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 174–189. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-71493-4_16 CrossRefGoogle Scholar
  11. 11.
    Duggirala, P.S., Mitra, S.: Abstraction refinement for stability. In: Proceedings of 2011 IEEE/ACM International Conference on Cyber-Physical Systems, ICCPS, pp. 22–31, April 2011Google Scholar
  12. 12.
    Duggirala, P.S., Mitra, S.: Lyapunov abstractions for inevitability of hybrid systems. In: HSCC, pp. 115–124. ACM, New York (2012)Google Scholar
  13. 13.
    Eggers, A., Ramdani, N., Nedialkov, N.S., Fränzle, M.: Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods. Softw. Syst. Model. 14(1), 121–148 (2015)CrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Fulton, N., Mitsch, S., Quesel, J.-D., Völp, M., Platzer, A.: KeYmaera X: an axiomatic tactical theorem prover for hybrid systems. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS (LNAI), vol. 9195, pp. 527–538. Springer, Cham (2015). doi: 10.1007/978-3-319-21401-6_36 CrossRefGoogle Scholar
  16. 16.
    Ghorbal, K., Platzer, A.: Characterizing algebraic invariants by differential radical invariants. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 279–294. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54862-8_19 CrossRefGoogle Scholar
  17. 17.
    Ghorbal, K., Sogokon, A., Platzer, A.: A hierarchy of proof rules for checking differential invariance of algebraic sets. In: D’Souza, D., Lal, A., Larsen, K.G. (eds.) VMCAI 2015. LNCS, vol. 8931, pp. 431–448. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46081-8_24 Google Scholar
  18. 18.
    Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 190–203. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-70545-1_18 CrossRefGoogle Scholar
  19. 19.
    Henzinger, T.A.: The Theory of Hybrid Automata, pp. 278–292. IEEE Computer Society Press, Washington, DC (1996)Google Scholar
  20. 20.
    Immler, F.: Verified reachability analysis of continuous systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 37–51. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46681-0_3 Google Scholar
  21. 21.
    Kong, S., Gao, S., Chen, W., Clarke, E.: dReach: \(\delta \)-reachability analysis for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 200–205. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46681-0_15 Google Scholar
  22. 22.
    Koymans, R.: Specifying real-time properties with metric temporal logic. Real-Time Syst. 2(4), 255–299 (1990)CrossRefGoogle Scholar
  23. 23.
    Lin, Y., Stadtherr, M.A.: Validated solutions of initial value problems for parametric ODEs. Appl. Numer. Math. 57(10), 1145–1162 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, J., Lv, J., Quan, Z., Zhan, N., Zhao, H., Zhou, C., Zou, L.: A calculus for hybrid CSP. In: Ueda, K. (ed.) APLAS 2010. LNCS, vol. 6461, pp. 1–15. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-17164-2_1 CrossRefGoogle Scholar
  25. 25.
    Liu, J., Zhan, N., Zhao, H.: Computing semi-algebraic invariants for polynomial dynamical systems. In: EMSOFT, pp. 97–106. ACM (2011)Google Scholar
  26. 26.
    Lygeros, J., Johansson, K.H., Simić, S.N., Zhang, J., Sastry, S.S.: Dynamical properties of hybrid automata. IEEE Trans. Autom. Control 48(1), 2–17 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Maidens, J.N., Arcak, M.: Reachability analysis of nonlinear systems using matrix measures. IEEE Trans. Autom. Control 60(1), 265–270 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Maidens, J.N., Arcak, M.: Trajectory-based reachability analysis of switched nonlinear systems using matrix measures. In: CDC, pp. 6358–6364, December 2014Google Scholar
  29. 29.
    Makino, K., Berz, M.: Cosy infinity version 9. Nucl. Instrum. Methods Phys. Res., Sect. A 558(1), 346–350 (2006)CrossRefGoogle Scholar
  30. 30.
    Matringe, N., Moura, A.V., Rebiha, R.: Generating invariants for non-linear hybrid systems by linear algebraic methods. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 373–389. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15769-1_23 CrossRefGoogle Scholar
  31. 31.
    Mitrohin, C., Podelski, A.: Composing stability proofs for hybrid systems. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 286–300. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-24310-3_20 CrossRefGoogle Scholar
  32. 32.
    Möhlmann, E., Hagemann, W., Theel, O.: Hybrid tools for hybrid systems – proving stability and safety at once. In: Sankaranarayanan, S., Vicario, E. (eds.) FORMATS 2015. LNCS, vol. 9268, pp. 222–239. Springer, Cham (2015). doi: 10.1007/978-3-319-22975-1_15 CrossRefGoogle Scholar
  33. 33.
    Möhlmann, E., Theel, O.: Stabhyli: a tool for automatic stability verification of non-linear hybrid systems. In: HSCC, pp. 107–112. ACM (2013)Google Scholar
  34. 34.
    Navarro-López, E.M., Carter, R.: Hybrid automata: an insight into the discrete abstraction of discontinuous systems. Int. J. Syst. Sci. 42(11), 1883–1898 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Navarro-López, E.M., Carter, R.: Deadness and how to disprove liveness in hybrid dynamical systems. Theor. Comput. Sci. 642(C), 1–23 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Navarro-López, E.M., Suárez, R.: Practical approach to modelling and controlling stick-slip oscillations in oilwell drillstrings. In: Proceedings of the 2004 IEEE International Conference on Control Applications, vol. 2, pp. 1454–1460. IEEE (2004)Google Scholar
  37. 37.
    Nedialkov, N.S.: Interval tools for ODEs and DAEs. In: SCAN (2006)Google Scholar
  38. 38.
    Neher, M., Jackson, K.R., Nedialkov, N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45(1), 236–262 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Nishida, T., Mizutani, K., Kubota, A., Doshita, S.: Automated phase portrait analysis by integrating qualitative and quantitative analysis. In: Proceedings of the 9th National Conference on Artificial Intelligence, pp. 811–816 (1991)Google Scholar
  40. 40.
    Paulson, L.C.: MetiTarski: past and future. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 1–10. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-32347-8_1 CrossRefGoogle Scholar
  41. 41.
    Platzer, A.: Differential dynamic logic for hybrid systems. J. Autom. Reason. 41(2), 143–189 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Platzer, A., Clarke, E.M.: Computing differential invariants of hybrid systems as fixedpoints. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 176–189. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-70545-1_17 CrossRefGoogle Scholar
  44. 44.
    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
  45. 45.
    Podelski, A., Wagner, S.: Model checking of hybrid systems: from reachability towards stability. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 507–521. Springer, Heidelberg (2006). doi: 10.1007/11730637_38 CrossRefGoogle Scholar
  46. 46.
    Podelski, A., Wagner, S.: Region stability proofs for hybrid systems. In: Raskin, J.-F., Thiagarajan, P.S. (eds.) FORMATS 2007. LNCS, vol. 4763, pp. 320–335. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-75454-1_23 CrossRefGoogle Scholar
  47. 47.
    Podelski, A., Wagner, S.: A sound and complete proof rule for region stability of hybrid systems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 750–753. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-71493-4_76 CrossRefGoogle Scholar
  48. 48.
    Prabhakar, P., Garcia Soto, M.: Abstraction based model-checking of stability of hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 280–295. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_20 CrossRefGoogle Scholar
  49. 49.
    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). doi: 10.1007/978-3-540-24743-2_32 CrossRefGoogle Scholar
  50. 50.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Richardson, D.: Some undecidable problems involving elementary functions of a real variable. J. Symb. Logic 33(4), 514–520 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Sankaranarayanan, S.: Automatic invariant generation for hybrid systems using ideal fixed points. In: HSCC, pp. 221–230 (2010)Google Scholar
  53. 53.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. FMSD 32(1), 25–55 (2008)zbMATHGoogle Scholar
  54. 54.
    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
  55. 55.
    Sogokon, A., Jackson, P.B.: Direct formal verification of liveness properties in continuous and hybrid dynamical systems. In: Bjørner, N., de Boer, F. (eds.) FM 2015. LNCS, vol. 9109, pp. 514–531. Springer, Cham (2015). doi: 10.1007/978-3-319-19249-9_32 CrossRefGoogle Scholar
  56. 56.
    Sogokon, A., Jackson, P.B., Johnson, T.T.: Verifying safety and persistence properties of hybrid systems using flowpipes and continuous invariants. Technical report, Vanderbilt University (2017)Google Scholar
  57. 57.
    Strzeboński, A.W.: Cylindrical decomposition for systems transcendental in the first variable. J. Symb. Comput. 46(11), 1284–1290 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Taly, A., Tiwari, A.: Deductive verification of continuous dynamical systems. In: Kannan, R., Kumar, K.N. (eds.) FSTTCS. LIPIcs, vol. 4, pp. 383–394. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Wadern (2009)Google Scholar
  59. 59.
    Tiwari, A.: Generating box invariants. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 658–661. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-78929-1_58 CrossRefGoogle Scholar
  60. 60.
    Wang, S., Zhan, N., Zou, L.: An improved HHL prover: an interactive theorem prover for hybrid systems. In: Butler, M., Conchon, S., Zaïdi, F. (eds.) ICFEM 2015. LNCS, vol. 9407, pp. 382–399. Springer, Cham (2015). doi: 10.1007/978-3-319-25423-4_25 CrossRefGoogle Scholar
  61. 61.
    Xue, B., Easwaran, A., Cho, N.J., Fränzle, M.: Reach-avoid verification for nonlinear systems based on boundary analysis. IEEE Trans. Autom. Control (2016)Google Scholar
  62. 62.
    Zhao, H., Yang, M., Zhan, N., Gu, B., Zou, L., Chen, Y.: Formal verification of a descent guidance control program of a lunar lander. In: Jones, C., Pihlajasaari, P., Sun, J. (eds.) FM 2014. LNCS, vol. 8442, pp. 733–748. Springer, Cham (2014). doi: 10.1007/978-3-319-06410-9_49 CrossRefGoogle Scholar
  63. 63.
    Zhao, H., Zhan, N., Kapur, D.: Synthesizing switching controllers for hybrid systems by generating invariants. In: Liu, Z., Woodcock, J., Zhu, H. (eds.) Theories of Programming and Formal Methods. LNCS, vol. 8051, pp. 354–373. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39698-4_22 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute for Software Integrated SystemsVanderbilt UniversityNashvilleUSA
  2. 2.Laboratory for Foundations of Computer ScienceUniversity of EdinburghEdinburghScotland, UK

Personalised recommendations