Model-free fractional order differentiator based on fractional order Jacobi orthonormal functions

Xiao-Lin Li , Yi-Ming Chen (1, 2) , Da-Yan Liu (3, 1) , Yan-Qiao Wei (3, 1) , Driss Boutat (3, 1)

1 PRISME - Laboratoire pluridisciplinaire de recherche en ingénierie des systèmes, mécanique et énergétique
2 LE STUDIUM - LE STUDIUM
3 INSA CVL - Institut National des Sciences Appliquées - Centre Val de Loire

The aim of this paper is to design an algebraic and robust fractional order differentiator to estimate both the Riemann-Liouville and the Caputo fractional derivatives with an arbitrary order of an unknown signal in noisy environment, without knowing the model defining the signal. For this purpose, a new class of fractional order Jacobi orthonormal functions is firstly introduced. Secondly, the truncated fractional order Jacobi orthonormal series expansion is applied to filter the noisy signal, whose fractional derivative is used to estimate the desired one. Thus, the obtained differentiator is exactly given by an integral formula which depends on a set of design parameters. Thirdly, by applying the generalized Taylor's formula, some error analysis is provided. In particular, error bounds are given, which permit to study the design parameters' influence. Fourthly, a digital fractional order differentiator is deduced in discrete noisy case. Finally, by comparing with two existing fractional order differentiators, numerical results are given to illustrate the accuracy and the robustness of the proposed fractional order differentiator.

Mots clés: Fractional order differentiator; Fractional order Jacobi orthonormal functions; Error analysis

Domaines: Analyse numérique [math.NA]

Xiao-Lin Li, Yi-Ming Chen, Da-Yan Liu, Yan-Qiao Wei, Driss Boutat. Model-free fractional order differentiator based on fractional order Jacobi orthonormal functions. Digital Signal Processing, 2017, 71, pp.69-82. ⟨10.1016/j.dsp.2017.09.001⟩. ⟨hal-02884925⟩