Comparison of the methods for discrete approximation of the fractional-order operator

TL;DR

This paper compares various discretization methods for fractional-order differentiators and systems, highlighting the effectiveness of the continued fraction expansion over the Muir expansion through mathematical analysis and simulation results.

Contribution

It introduces and evaluates alternative discretization techniques for fractional-order operators, specifically comparing CFE and Muir expansion methods.

Findings

CFE outperforms PSE in accuracy and effectiveness

Muir expansion is nearly unusable for practical purposes

CFE has restrictions on time step selection

Abstract

In this paper we will present some alternative types of discretization methods (discrete approximation) for the fractional-order (FO) differentiator and their application to the FO dynamical system described by the FO differential equation (FDE). With analytical solution and numerical solution by power series expansion (PSE) method are compared two effective methods - the Muir expansion of the Tustin operator and continued fraction expansion method (CFE) with the Tustin operator and the Al-Alaoui operator. Except detailed mathematical description presented are also simulation results. From the Bode plots of the FO differentiator and FDE and from the solution in the time domain we can see, that the CFE is a more effective method according to the PSE method, but there are some restrictions for the choice of the time step. The Muir expansion is almost unusable.