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

François Boulier (1) , Anja Korporal (2) , François Lemaire (3) , Wilfrid Perruquetti (4, 2) , Adrien Poteaux (1) , Rosane Ushirobira (2)

1 CALFOR - Calcul Formel
2 NON-A - Non-Asymptotic estimation for online systems
3 CEPN - Centre d'Economie de l'Université Paris Nord
4 Centrale Lille

This paper provides a contribution to the parameter esti-mation 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 sen-sitive. 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 re-cent 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. In Engineering, a wide variety of information is not directly obtained through measurement. Various parameters or internal variables are unknown or not mea-sured. In addition, sensor signals are very often distorted and tainted by mea-surement noises. To simulate, control or supervise such processes, and to extract information conveyed by the signals, a system has to be identified and parame-ters and variables must be estimated. Most of traditional estimation methods are related to asymptotic statistics. However, there exist some difficulties that have been long known as inherent to these existing methods. Among them, two im-portant limitations can be pointed out: these methods apply essentially to linear systems and they are noise sensitive due to the use of numerical derivation. The parameter estimation problem has been tackled by many different approaches in control theory. Algebraic techniques to this end were notably introduced in the works by M. Fliess et al. [8, 15, 7, 9, 6] and inspired for instance, algebraic methods for the parameter estimation of a multi-sinusoidal waveform signal from noisy data [22].

Domaines: Calcul formel [cs.SC]; Automatique / Robotique

François Boulier, Anja Korporal, François Lemaire, Wilfrid Perruquetti, Adrien Poteaux, et al.. An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data. Computer Algebra in Scientific Computing, Sep 2014, Warsaw, Poland. pp.28 - 43. ⟨hal-01114187⟩