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
Boulier, F.: http://www.lifl.fr/~boulier/BLAD (2004)
Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Computing representations for radicals of finitely generated differential ideals. AAECC 20(1), 73–121 (2009)
Boulier, F., Lemaire, F., Regensburger, G., Rosenkranz, M.: On the Integration of Differential Fractions. In: ISSAC 2013, pp. 101–108. ACM, New York (2013)
Bronstein, M.: Symbolic Integration I. Springer (1997)
Denis-Vidal, L., Joly-Blanchard, G., Noiret, C.: System identifiability (symbolic computation) and parameter estimation (numerical computation). Numerical Algorithms 34, 282–292 (2003)
Fliess, M., Join, C., Sira-Ramírez, H.: Non-linear estimation is easy. Int. J. Modelling Identification and Control 4(1), 12–27 (2008)
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)
Fliess, M., Sira-Ramírez, H.: An algebraic framework for linear identification. ESAIM Control Optim. Calc. Variat. 9, 151–168 (2003)
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)
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)
Godfrey, K.R.: The identifiability of parameters of models used in biomedicine. Mathematical Modelling 7(9-12), 1195–1214 (1986)
Guo, L., Regensburger, G., Rosenkranz, M.: On integro-differential algebras. JPAA 218(3), 456–473 (2014)
Hairer, E.: Homepage, http://www.unige.ch/~hairer (2000)
Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)
Mboup, M.: Parameter estimation for signals described by differential equations. Applicable Analysis 88, 29–52 (2009)
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)
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)
Ritt, J.F.: Differential Algebra. American Mathematical Society Colloquium Publications, vol. 33. AMS, New York (1950)
Rosenkranz, M., Regensburger, G.: Integro-differential polynomials and operators. In: ISSAC 2008, pp. 261–268. ACM, New York (2008)
Shinbrot, M.: On the analysis of linear and nonlinear dynamical systems from transient-response data. NACA, Washington, D.C (1954)
The Ametista Group (2013), http://www.lifl.fr/Ametista
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)
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Boulier, F., Korporal, A., Lemaire, F., Perruquetti, W., Poteaux, A., Ushirobira, R. (2014). An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_3
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DOI: https://doi.org/10.1007/978-3-319-10515-4_3
Publisher Name: Springer, Cham
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