Modulating functions-based fractional order differentiator for fractional order linear systems with a biased output

Abstract

This paper aims at designing a non-asymptotic fractional order differentiator for a class of fractional order linear systems with zero initial conditions, where the fractional orders can be commensurate or non-commensurate, and the output is corrupted by a non zero-mean noise. Firstly, a set of fractional differential equations are constructed, based on which the modulating functions method is applied, such that the fractional derivative with an arbitrary order of the output can be exactly given by an algebraic formula using a recursive way. In particular, this arbitrary order can be different to the fractional orders defining the considered system. Unlike the improper integral in the definition of the fractional derivatives, the obtained formula can be given by proper integrals by choosing appropriate modulating functions. Moreover, by taking an additional condition on the modulating functions, the term biasing the output can be eliminated in the obtained formula. After constructing the needed modulating functions, a digital fractional order differentiator is proposed in discrete noisy case with some error analysis. Finally, the efficiency and the robustness of the proposed fractional order differentiator is shown in numerical results.