Systematic Simulation Using Sensitivity Analysis

  • Alexandre Donzé
  • Oded Maler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4416)


In this paper we propose a new technique for verification by simulation of continuous and hybrid dynamical systems with uncertain initial conditions. We provide an algorithmic methodology that can, in most cases, verify that the system avoids a set of bad states by conducting a finite number of simulation runs starting from a finite subset of the set of possible initial conditions. The novelty of our approach consists in the use of sensitivity analysis, developed and implemented in the context of numerical integration, to efficiently characterize the coverage of sampling trajectories.


Hybrid System Sensitivity Function Voltage Control Oscillator Expansion Function Oscillator Circuit 
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  1. [ACH+95]
    Alur, R., et al.: The algorithmic analysis of hybrid systems. Theoretical Computer Science 138, 3–34 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [ADG03]
    Asarin, E., Dang, T., Girard, A.: Reachability analysis of nonlinear systems using conservative approximation. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 20–35. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. [BCLM05]
    Branicky, M.S., et al.: Sampling-based reachability algorithms for control and verification of complex systems. In: Proc. Thirteenth Yale Workshop on Adaptive and Learning Systems (June 2005)Google Scholar
  4. [BCLM06]
    Branicky, M.S., et al.: Sampling-based planning, control and verification of hybrid systems. IEE Proceedings Control Theory and Applications 153, 575–590 (2006)CrossRefGoogle Scholar
  5. [BF04]
    Bhatia, A., Frazzoli, E.: Incremental search methods for reachability analysis of continuous and hybrid systems. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 142–156. Springer, Heidelberg (2004)Google Scholar
  6. [BL02]
    Barton, P.I., Kun Lee, C.: Modeling, simulation, sensitivity analysis, and optimization of hybrid systems. ACM Trans. Model. Comput. Simul. 12(4), 256–289 (2002)CrossRefGoogle Scholar
  7. [CK98]
    Chutinan, A., Krogh, B.H.: Computing polyhedral approximations to dynamic flow pipes. In: Proc. of the 37th Annual International Conference on Decision and Control, CDC’98, IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  8. [DM98]
    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)Google Scholar
  9. [FKR06]
    Frehse, G., Krogh, B.H., Rutenbar, R.A.: Verifying analog oscillator circuits using forward/backward abstraction refinement. In: DATE 2006: Design, Automation and Test in Europe (March 2006)Google Scholar
  10. [Gir05]
    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)Google Scholar
  11. [GP06]
    Girard, A., Pappas, G.J.: Verification using simulation. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 272–286. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. [HP00]
    Hiskens, I.A., Pai, M.A.: Trajectory sensitivity analysis of hybrid systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47(2), 204–220 (2000)CrossRefGoogle Scholar
  13. [HS74]
    Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, London (1974)zbMATHGoogle Scholar
  14. [KKMS03]
    Kapinski, J., et al.: On systematic simulation of open continuous systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 283–297. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. [LaV06]
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006), Available at zbMATHGoogle Scholar
  16. [LYL04]
    Lindemann, S.R., Yershova, A., LaValle, S.M.: Incremental grid sampling strategies in robotics. In: Proceedings Workshop on Algorithmic Foundations of Robotics, pp. 297–312 (2004)Google Scholar
  17. [SB06]
    Singer, A.B., Barton, P.I.: Bounding the solutions of parameter dependent nonlinear ordinary differential equations. SIAM Journal on Scientific Computing 27(6), 2167–2182 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [SH05]
    Serban, R., Hindmarsh, A.C.: Cvodes: the sensitivity-enabled ode solver in sundials. In: Proceedings of IDETC/CIE 2005, Long Beach, CA (Sep. 2005)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Alexandre Donzé
    • 1
  • Oded Maler
    • 1
  1. 1.VERIMAG, 2, Avenue de Vignate, 38610 GièresFrance

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