An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation

• A new algebraic formula for the Riemann–Liouville derivative is derived.
• The input can be unknown or known with noises.
• It can be used to estimate a derivative with an arbitrary order in discrete noisy case.
• An error bound for noisy errors is provided, which is useful for the selection of the design parameter.

This paper aims at designing a digital fractional order differentiator for a class of signals satisfying a linear differential equation to estimate fractional derivatives with an arbitrary order in noisy case, where the input can be unknown or known with noises. Firstly, an integer order differentiator for the input is constructed using a truncated Jacobi orthogonal series expansion. Then, a new algebraic formula for the Riemann–Liouville derivative is derived, which is enlightened by the algebraic parametric method. Secondly, a digital fractional order differentiator is proposed using a numerical integration method in discrete noisy case. Then, the noise error contribution is analyzed, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.