An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data

  • François Boulier
  • Anja Korporal
  • François Lemaire
  • Wilfrid Perruquetti
  • Adrien Poteaux
  • Rosane Ushirobira
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8660)

Abstract

This paper provides a contribution to the parameter estimation methods for nonlinear dynamical systems. In such problems, a major issue is the presence of noise in measurements. In particular, most methods based on numerical estimates of derivations are very noise sensitive. An improvement consists in using integral equations, acting as noise filtering, rather than differential equations. Our contribution is a pair of algorithms for converting fractions of differential polynomials to integral equations. These algorithms rely on an improved version of a recent differential algebra algorithm. Their usefulness is illustrated by an application to the problem of estimating the parameters of a nonlinear dynamical system, from noisy data.

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References

  1. 1.
  2. 2.
    Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Computing representations for radicals of finitely generated differential ideals. AAECC 20(1), 73–121 (2009)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Boulier, F., Lemaire, F., Regensburger, G., Rosenkranz, M.: On the Integration of Differential Fractions. In: ISSAC 2013, pp. 101–108. ACM, New York (2013)Google Scholar
  4. 4.
    Bronstein, M.: Symbolic Integration I. Springer (1997)Google Scholar
  5. 5.
    Denis-Vidal, L., Joly-Blanchard, G., Noiret, C.: System identifiability (symbolic computation) and parameter estimation (numerical computation). Numerical Algorithms 34, 282–292 (2003)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Fliess, M., Join, C., Sira-Ramírez, H.: Non-linear estimation is easy. Int. J. Modelling Identification and Control 4(1), 12–27 (2008)CrossRefGoogle Scholar
  7. 7.
    Fliess, M., Mboup, M., Mounier, H., Sira-Ramírez, H.: Questioning some paradigms of signal processing via concrete examples. In: Silva-Navarro, G., Sira-Ramírez, H. (eds.) Algebraic Methods in Flatness, Signal Processing and State Estimation, pp. 1–21. Editiorial Lagares (2003)Google Scholar
  8. 8.
    Fliess, M., Sira-Ramírez, H.: An algebraic framework for linear identification. ESAIM Control Optim. Calc. Variat. 9, 151–168 (2003)CrossRefMATHGoogle Scholar
  9. 9.
    Fliess, M., Sira-Ramírez, H.: Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques. In: Identification of Continuous-Time Models from Sampled Data. Advances in Industrial Control, pp. 362–391 (2008)Google Scholar
  10. 10.
    Gao, X., Guo, L.: Constructions of Free Commutative Integro-Differential Algebras. In: Barkatou, M., Cluzeau, T., Regensburger, G., Rosenkranz, M. (eds.) AADIOS 2012. LNCS, vol. 8372, pp. 1–22. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  11. 11.
    Godfrey, K.R.: The identifiability of parameters of models used in biomedicine. Mathematical Modelling 7(9-12), 1195–1214 (1986)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Guo, L., Regensburger, G., Rosenkranz, M.: On integro-differential algebras. JPAA 218(3), 456–473 (2014)MATHMathSciNetGoogle Scholar
  13. 13.
    Hairer, E.: Homepage, http://www.unige.ch/~hairer (2000)
  14. 14.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)MATHGoogle Scholar
  15. 15.
    Mboup, M.: Parameter estimation for signals described by differential equations. Applicable Analysis 88, 29–52 (2009)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Noiret, C.: Utilisation du calcul formel pour l’identifiabilité de modèles paramétriques et nouveaux algorithmes en estimation de paramètres. PhD thesis, Université de Technologie de Compiègne (2000)Google Scholar
  17. 17.
    Pearson, A.E.: Explicit parameter identification for a class of nonlinear input/output differential operator models. In: Proceedings of the 31st IEEE Conference on Decision and Control, vol. 4, pp. 3656–3660 (1992)Google Scholar
  18. 18.
    Ritt, J.F.: Differential Algebra. American Mathematical Society Colloquium Publications, vol. 33. AMS, New York (1950)MATHGoogle Scholar
  19. 19.
    Rosenkranz, M., Regensburger, G.: Integro-differential polynomials and operators. In: ISSAC 2008, pp. 261–268. ACM, New York (2008)Google Scholar
  20. 20.
    Shinbrot, M.: On the analysis of linear and nonlinear dynamical systems from transient-response data. NACA, Washington, D.C (1954)Google Scholar
  21. 21.
    The Ametista Group (2013), http://www.lifl.fr/Ametista
  22. 22.
    Ushirobira, R., Perruquetti, W., Mboup, M., Fliess, M.: Algebraic parameter estimation of a multi-sinusoidal waveform signal from noisy data. In: European Control Conference, Zurich (April 2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • François Boulier
    • 1
  • Anja Korporal
    • 2
  • François Lemaire
    • 1
  • Wilfrid Perruquetti
    • 2
    • 3
  • Adrien Poteaux
    • 1
  • Rosane Ushirobira
    • 2
  1. 1.Computer Algebra GroupUniversité Lille 1, LIFL, UMR CNRS 8022France
  2. 2.Inria, Non-A teamFrance
  3. 3.École Centrale de Lille, LAGIS, UMR CNRS 8219France

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