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Reachability Analysis of Nonlinear Systems Using Conservative Approximation

  • Eugene Asarin
  • Thao Dang
  • Antoine Girard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2623)

Abstract

In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C 2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.

Keywords

Nonlinear System Hybrid System Autonomous System Piecewise Linear Interpolation Error 
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.

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References

  1. 1.
    R. Alur, C. Courcoubetis, N. Halbwachs, T. A. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis and S. Yovine. The Algorithmic Analysis of Hybrid Systems, Theoretical Computer Science 138, 3–34, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    H. Anai and V. Weispfenning. Reach Set Computations Using Real Quantifier Elimination, Hybrid Systems: Computation and Control, in M. D. Di Benedetto and A. Sangiovanni-Vincentelli (Eds), 63–75 LNCS 2034, Springer-Verlag, 2001.CrossRefGoogle Scholar
  3. 3.
    E. Asarin, T. Dang and A. Girard. Reachability Analysis of Nonlinear Systems using Conservative Approximations, Technical Report IMAG Oct 2002, Grenoble http://www-verimag.imag.fr/~tdang/piecewise.ps.gz.
  4. 4.
    E. Asarin and T. Dang and O. Maler. d/dt: A tool for Verification of Hybrid Systems, Computer Aided Verification, Springer-Verlag, LNCS, 2002.Google Scholar
  5. 5.
    M. D. Di Benedetto and A. Sangiovanni-Vincentelli. Hybrid Systems: Computation and Control, LNCS 2034, Springer-Verlag, 2001.CrossRefGoogle Scholar
  6. 6.
    O. Botchkarev and S. Tripakis. Verification of Hybrid Systems with Linear Differential Inclusions Using Ellipsoidal Approximations, Hybrid Systems: Computation and Control, in B. Krogh and N. Lynch (Eds), 73–88 LNCS 1790, Springer-Verlag, 2000.CrossRefGoogle Scholar
  7. 7.
    A. Chutinan and B. H. Krogh. Verification of Polyhedral Invariant Hybrid Automata Using Polygonal Flow Pipe Approximations, Hybrid Systems: Computation and Control, in F. Vaandrager and J. van Schuppen (Eds), 76–90 LNCS 1569, Springer-Verlag, 1999.CrossRefGoogle Scholar
  8. 8.
    T. Dang and O. Maler. Reachability Analysis via Face Lifting, Hybrid Systems: Computation and Control, in T. A. Henzinger and S. Sastry (Eds), 96–109 LNCS 1386 Springer-Verlag, 1998.Google Scholar
  9. 9.
    J. Della Dora, A. Maignan, M. Mirica-Ruse, and S. Yovine. Hybrid Computation, Proc. of ISSAC’01, 2001.Google Scholar
  10. 10.
    J. Dieudonné. Calcul Infinitésimal, Collection Méthodes, Hermann Paris, 1980.Google Scholar
  11. 11.
    A. Girard. Approximate Solutions of ODEs Using Piecewise Linear Vector Fields, Proc. CASC’02, 2002.Google Scholar
  12. 12.
    A. Girard. Detection of Event Occurence in Piecewise Linear Hybrid Systems, Proc. RASC’02, December 2002, Nottingham, UK.Google Scholar
  13. 13.
    M. R. Greenstreet and I. Mitchell. Reachability Analysis Using Polygonal Projections, Hybrid Systems: Computation and Control, in F. Vaandrager and J. van Schuppen (Eds), 76–90 LNCS 1569 Springer-Verlag, 1999.CrossRefGoogle Scholar
  14. 14.
    M. Greenstreet and C. Tomlin. Hybrid Systems: Computation and Control, LNCS, Springer-Verlag, 2002.zbMATHGoogle Scholar
  15. 15.
    L. C. G. J. M. Habets and J. H. van Schuppen. Control of Piecewise-Linear Hybrid Systems on Simplices and Rectangles, Hybrid Systems: Control and Computation, in M. D. Di Benedetto and A. Sangiovanni-Vincentelli (Eds), 261–273 LNCS 2034, Springer-Verlag, 2001.CrossRefGoogle Scholar
  16. 16.
    T. A. Henzinger, P.-H. Ho and H. Wong-Toi. HyTech: A Model Checker for Hybrid Systems, Software Tools for Technology Transfer 1, 110–122, 1997.zbMATHCrossRefGoogle Scholar
  17. 17.
    T. A. Henzinger, P.-H. Ho, and H. Wong-Toi. Analysis of Nonlinear Hybrid Systems, IEEE Transactions on Automatic Control 43, 540–554, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    J. Hubbard and B. West. Differential Equations: A Dynamical Systems Approach, Higher-Dimensional Systems, Texts in Applied Mathematics, 18, Springer Verlag, 1995.Google Scholar
  19. 19.
    G. Lafferriere, G. Pappas, and S. Yovine. Reachability computation for linear systems, Proc. of the 14th IFAC World Congress, 7–12 E, 1999.Google Scholar
  20. 20.
    , K. Larsen, P. Pettersson, and W. Yi. Uppaal in a nutshell, Software Tools for Technology Transfert 1, 1997.Google Scholar
  21. 21.
    T. H. Marshall. Volume formulae for regular hyperbolic cubes, Conform. Geom. Dyn., 25–28, 1998.Google Scholar
  22. 22.
    I. Mitchell and C. Tomlin. Level Set Method for Computation in Hybrid Systems, Hybrid Systems: Computation and Control, in B. Krogh and N. Lynch, 311–323 LNCS 1790, Springer-Verlag, 2000.CrossRefGoogle Scholar
  23. 23.
    P. Saint-Pierre. Approximation of Viability Kernels and Capture Basin for Hybrid Systems, Proc. of European Control Conference ECC’01, 2776–2783, 2001.Google Scholar
  24. 24.
    O. Stursberg, S. Kowalewski and S. Engell. On the generation of Timed Approximations for continuous systems, Mathematical and Computer Modelling of Dynamical Systems 6–1, 51-70, 2000.Google Scholar
  25. 25.
    A. Puri and P. Varaiya. Verification of Hybrid Systems using Abstraction, Hybrid Systems II, in P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry (Eds), LNCS 999, Springer-Verlag, 1995.Google Scholar
  26. 26.
    S. Yovine. Kronos: A Verification Tool for Real-time Systems, Software Tools for Technology Transfer 1, 123–133, 1997.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Eugene Asarin
    • 1
  • Thao Dang
    • 1
  • Antoine Girard
    • 2
  1. 1.VERIMAGGièresFrance
  2. 2.LMC-IMAGGrenobleFrance

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