Error analysis of Jacobi derivative estimators for noisy signals

Abstract

International audienceRecent algebraic parametric estimation techniques (see \cite{garnier,mfhsr}) led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal (see \cite{num0,num}). In this paper, we extend such differentiation methods by providing a larger choice of parameters in these integrals: they can be reals. For this, %as in \cite{num0,num}, the extension is done via a truncated Jacobi orthogonal series expansion. Then, the noise error contribution of these derivative estimations is investigated: after proving the existence of such integral with a stochastic process noise, their statistical properties (mean value, variance and covariance) are analyzed. In particular, the following important results are obtained: \begin{description} \item[$a)$] the bias error term, due to the truncation, can be reduced by tuning the parameters, \item[$b)$] such estimators can cope with a large class of noises for which the mean and covariance are polynomials in time (with degree smaller than the order of derivative to be estimated), \item[$c)$] the variance of the noise error is shown to be smaller in the case of negative real parameters than it was in \cite{num0,num} for integer values. \end{description} Consequently, these derivative estimations can be improved by tuning the parameters according to the here obtained knowledge of the parameters' influence on the error bounds

Synthesis on a class of algebraic differentiators and application to nonlinear observation

“…The objective is to estimate the n th order derivative of x using x ̟ . For this purpose, we apply a class of algebraic differentiators involving Jacobi polynomials, which were introduced in [7,8] using a recent algebraic parametric method (see [13] for other algebraic differentiators).…”

“…An error bound based on the integral formula given in (5) was proposed in [13] (p. 90) for this kind of noise errors. Moreover, it was shown that the Jacobi differentiator D (n) κ,µ,T,q x ̟ (t 0 − T ξ) can eliminate a (n − 1) th order structured perturbation.…”

“…[3][4][5][6][7][8][9][10][11][12][13]). The obtained so-called algebraic differentiators are divided into two classes: model-based differentiators and model-free differentiators.…”

“…For this purpose, different errors for the Jacobi differentiator have been studied in [9][10][11], respectively. In particular, the definition domain of certain design parameters has been extended to negative values which can reduce the estimation errors, especially the time-delay.…”

“…In particular, the definition domain of certain design parameters has been extended to negative values which can reduce the estimation errors, especially the time-delay. However, the proposed algorithms with negative design parameters' values were subject to numerical errors (see [10,11]). …”