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Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion

  • Thao Dang
  • David Salinas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5643)

Abstract

This paper is concerned with the problem of computing the image of a set by a polynomial function. Such image computations constitute a crucial component in typical tools for set-based analysis of hybrid systems and embedded software with polynomial dynamics, which found applications in various engineering domains. One typical example is the computation of all states reachable from a given set in one step by a continuous dynamics described by a differential or difference equation. We propose a new algorithm for over-approximating such images based on the Bernstein expansion of polynomial functions. The images are stored using template polyhedra. Using a prototype implementation, the performance of the algorithm was demonstrated on two practical systems as well as a number of randomly generated examples.

Keywords

Hybrid System Polynomial System Multivariate Polynomial Polynomial Optimization Polynomial Optimization Problem 
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.

References

  1. 1.
    Asarin, E., Dang, T., Girard, A.: Hybridization methods for the analysis of nonlinear systems. Acta Informatica 43(7), 451–476 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boyd, S., Vandenberghe, S.: Convex optimization. Cambridge Uni. Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Clauss, F., Yu, C.I.: Application of symbolic approach to the Bernstein expansion for program analysis and optimization. Program. Comput. Softw. 30(3), 164–172 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dang, T.: Approximate reachability computation for polynomial systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 138–152. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Dang, T., Salinas, D.: Computing set images of polynomias. Technical report, VERIMAG (June 2008)Google Scholar
  6. 6.
    Fotiou, I.A., Rostalski, P., Parrilo, P.A., Morari, M.: Parametric optimization and optimal control using algebraic geometriy methods. Int. Journal of Control 79(11), 1340–1358 (2006)CrossRefzbMATHGoogle Scholar
  7. 7.
    Garloff, J.: Application of Bernstein expansion to the solution of control problems. University of Girona, pp. 421–430 (1999)Google Scholar
  8. 8.
    Garloff, J., Jansson, C., Smith, A.P.: Lower bound functions for polynomials. Journal of Computational and Applied Mathematics 157, 207–225 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garloff, J., Smith, A.P.: An improved method for the computation of affine lower bound functions for polynomials. In: Frontiers in Global Optimization. Series Nonconvex Optimization and Its Applications. Kluwer Academic Publ., Dordrecht (2004)Google Scholar
  10. 10.
    Garloff, J., Smith, A.P.: A comparison of methods for the computation of affine lower bound functions for polynomials. In: Jermann, C., Neumaier, A., Sam, D. (eds.) COCOS 2003. LNCS, vol. 3478, pp. 71–85. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Girard, A., Le Guernic, C., Maler, O.: Efficient Computation of Reachable Sets of Linear Time-Invariant Systems with Inputs. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 257–271. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    He, F., Yeung, L.F., Brown, M.: Discrete-time model representation for biochemical pathway systems. IAENG Int. Journal of Computer Science 34(1) (2007)Google Scholar
  13. 13.
    Henrion, D., Lasserre, J.B.: Gloptipoly: Global optimization over polynomials with matlab and sedumi. In: Proceedings of CDC (2002)Google Scholar
  14. 14.
    Jolliffe, I.T.: Principal Component Analysis. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  15. 15.
    Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations. Oxford Applied Mathematics and Computer Science. Oxford Uni. Press, Oxford (1987)zbMATHGoogle Scholar
  16. 16.
    Mourrain, B., Pavone, J.P.: Subdivision methods for solving polynomial equations. Technical report, INRIA Research report, 5658 (August. 2005)Google Scholar
  17. 17.
    Platzer, A., Clarke, E.M.: The Image Computation Problem in Hybrid Systems Model Checking. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 473–486. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    Sankaranarayanan, S., Sipma, H., Manna, Z.: Scalable analysis of linear systems using mathematical programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 25–41. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Sankaranarayanan, S.: Mathematical analysis of programs. Technical report, Standford, PhD thesis (2005)Google Scholar
  21. 21.
    Sankaranarayanan, S., Dang, T., Ivancic, F.: Symbolic Model Checking of Hybrid Systems using Template Polyhedra. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 188–202. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Seidel, H.-P.: Polar forms and triangular B-spline surfaces. In: Blossoming: The New Polar-Form Approach to Spline Curves and Surfaces, SIGGRAPH 1991 (1991)Google Scholar
  23. 23.
    Tchoupaeva, I.: A symbolic approach to Bernstein expansion for program analysis and optimization. In: Duesterwald, E. (ed.) CC 2004. LNCS, vol. 2985, pp. 120–133. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  24. 24.
    Tiwari, A., Khanna, G.: Nonlinear systems: Approximating reach sets. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 600–614. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Vandenberghe, S., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49–95 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Thao Dang
    • 1
  • David Salinas
    • 1
  1. 1.VerimagGièresFrance

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