Bounded Verification with On-the-Fly Discrepancy Computation

  • Chuchu FanEmail author
  • Sayan Mitra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9364)


Simulation-based verification algorithms can provide formal safety guarantees for nonlinear and hybrid systems. The previous algorithms rely on user provided model annotations called discrepancy function, which are crucial for computing reachtubes from simulations. In this paper, we eliminate this requirement by presenting an algorithm for computing piecewise exponential discrepancy functions. The algorithm relies on computing local convergence or divergence rates of trajectories along a simulation using a coarse over-approximation of the reach set and bounding the maximal eigenvalue of the Jacobian over this over-approximation. The resulting discrepancy function preserves the soundness and the relative completeness of the verification algorithm. We also provide a coordinate transformation method to improve the local estimates for the convergence or divergence rates in practical examples. We extend the method to get the input-to-state discrepancy of nonlinear dynamical systems which can be used for compositional analysis. Our experiments show that the approach is effective in terms of running time for several benchmark problems, scales reasonably to larger dimensional systems, and compares favorably with respect to available tools for nonlinear models.


Coordinate Transformation Discrepancy Function Lipschitz Constant Symmetric Part Reachable State 
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.


  1. 1.
    Althoff, M., Stursberg, O., Buss, M.: Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: CDC, pp. 4042–4048 (2008)Google Scholar
  2. 2.
    Angeli, D.: A lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47(3), 410–421 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Angeli, D., Sontag, E.D., Wang, Y.: A characterization of integral input-to-state stability. IEEE Trans. Autom. Control 45(6), 1082–1097 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Annpureddy, Y., Liu, C., Fainekos, G., Sankaranarayanan, S.: S-TaLiRo: a tool for temporal logic falsification for hybrid systems. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 254–257. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  5. 5.
    Bak, S., Caccamo, M.: Computing reachability for nonlinear systems with hycreate. In: Demo and Poster Session, HSCC (2013)Google Scholar
  6. 6.
    CAPD. Computer assisted proofs in dynamics (2002).
  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) CrossRefGoogle Scholar
  8. 8.
    Dang, T., Donzé, A., Maler, O., Shalev, N.: Sensitive state-space exploration. In: CDC, pp. 4049–4054 (2008)Google Scholar
  9. 9.
    Dang, T., Maler, O.: Reachability analysis via face lifting. In: Henzinger, T.A., Sastry, S.S. (eds.) HSCC 1998. LNCS, vol. 1386, pp. 96–109. Springer, Heidelberg (1998) 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) CrossRefGoogle Scholar
  11. 11.
    Duggirala, P.S., Fan, C., Mitra, S., Viswanathan, M.: Meeting a powertrain verification challenge. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 536–543. Springer, Heidelberg (2015) CrossRefGoogle Scholar
  12. 12.
    Duggirala, P.S., Mitra, S., Viswanathan, M.: Verification of annotated models from executions. In: EMSOFT, pp. 26:1–26:10 (2013)Google Scholar
  13. 13.
    Duggirala, P.S., Wang, L., Mitra, S., Viswanathan, M., Muñoz, C.: Temporal precedence checking for switched models and its application to a parallel landing protocol. In: Jones, C., Pihlajasaari, P., Sun, J. (eds.) FM 2014. LNCS, vol. 8442, pp. 215–229. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  14. 14.
    Fan, C., Duggirala, P.S., Mitra, S., Viswanathan, M.: Progress on powertrain verification challenge with C2E2. ARCH (2015)Google Scholar
  15. 15.
    Fan, C., Mitra, S.: Bounded verification with on-the-fly discrepancy computation (full version).
  16. 16.
    Frehse, G., Le 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) CrossRefGoogle Scholar
  17. 17.
    Girard, A., Pola, G., Tabuada, P.: Approximately bisimilar symbolic models for incrementally stable switched systems. IEEE Trans. Autom. Control 55(1), 116–126 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Han, Z., Mosterman, P.J.: Towards sensitivity analysis of hybrid systems using simulink. In: HSCC, pp. 95–100 (2013)Google Scholar
  19. 19.
    Huang, Z., Fan, C., Mereacre, A., Mitra, S., Kwiatkowska, M.: Invariant verification of nonlinear hybrid automata networks of cardiac cells. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 373–390. Springer, Heidelberg (2014) Google Scholar
  20. 20.
    Huang, Z., Mitra, S.: Proofs from simulations and modular annotations. In: HSCC, pp. 183–192 (2014)Google Scholar
  21. 21.
    Islam, M., DeFrancisco, R., Fan, C., Grosu, R., Mitra, S., Smolka, S.A., et al.: Model checking tap withdrawal in c. elegans (2015). arXiv preprint arXiv:1503.06480
  22. 22.
    Julius, A.A., Pappas, G.J.: Trajectory based verification using local finite-time invariance. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 223–236. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  23. 23.
    Lohmiller, W., Slotine, J.-J.E.: On contraction analysis for non-linear systems. Automatica 34(6), 683–696 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nedialkov, N.: VNODE-LP: Validated solutions for initial value problem for ODEs. Technical report, McMaster University (2006)Google Scholar
  25. 25.
    Sharma, B.B., Kar, I.N.: Design of asymptotically convergent frequency estimator using contraction theory. IEEE Trans. Autom. Control 53(8), 1932–1937 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zamani, M., Pola, G., Mazo, M., Tabuada, P.: Symbolic models for nonlinear control systems without stability assumptions. IEEE Trans. Autom. Control 57(7), 1804–1809 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-ChampaignChampaignUSA

Personalised recommendations