Fractional order differentiation by integration with Jacobi polynomials

TL;DR

This paper extends the differentiation by integration method using Jacobi polynomials to fractional orders, providing a simple, robust, and accurate differentiator for noisy signals applicable in both continuous and discrete models.

Contribution

It introduces a novel fractional order differentiator based on Jacobi polynomials and Riemann-Liouville derivatives, generalizing previous integer order methods.

Findings

The proposed differentiator accurately estimates fractional derivatives of noisy signals.

It demonstrates robustness against noise in numerical simulations.

The method is applicable to both continuous and discrete-time models.

Abstract

The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises.