| Affiliation: | 1.Laboratoire de Paul Painlevé,Université de Lille 1,Villeneuve d’Ascq,France;2.Laboratory of Applied Mathematics and Metrology (L2MA),Arts et Metiers ParisTech,Lille Cedex,France;3.Laboratoire de LAGIS,école Centrale de Lille,Villeneuve d’Ascq,France;4.équipe Projet Non-A,INRIA Lille-Nord Europe Parc Scientifique de la Haute Borne 40,Villeneuve d’Ascq,France |
| Abstract: | Recent algebraic parametric estimation techniques (see Fliess and Sira-Ramírez, ESAIM Control Optim Calc Variat 9:151–168, 2003, 2008) led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal (see Mboup et al. 2007, Numer Algorithms 50(4):439–467, 2009). In this paper, we extend such differentiation methods by providing a larger choice of parameters in these integrals: they can be reals. For this, 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: | (a) | the bias error term, due to the truncation, can be reduced by tuning the parameters, |
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