Consistent approximation of fractional order operators

TL;DR

This paper introduces a piecewise approximation method for fractional order operators that maintains key algebraic identities, improving accuracy and consistency in numerical implementations of fractional controllers.

Contribution

It proposes a novel piecewise model with specific parameter design procedures to ensure algebraic consistency and high approximation accuracy for fractional differintegrators.

Findings

The method preserves fundamental fractional operator equations.

It achieves higher approximation accuracy compared to existing methods.

The approach facilitates more reliable simulation and implementation of fractional controllers.

Abstract

Fractional order controllers become increasingly popular due to their versatility and superiority in various performance. However, the bottleneck in deploying these tools in practice is related to their analog or numerical implementation. Numerical approximations are usually employed in which the approximation of fractional differintegrator is the foundation. Generally, the following three identical equations always hold, i.e., $\frac{1}{s^\alpha}\frac{1}{s^{1-\alpha}} = \frac{1}{s}$, $s^\alpha \frac{1}{s^\alpha} = 1$ and $s^\alpha s^{1-\alpha} = s$. However, for the approximate models of fractional differintegrator $s^\alpha$, $\alpha\in(-1,0)\cup(0,1)$, there usually exist some conflicts on the mentioned equations, which might enlarge the approximation error or even cause fallacies in multiple orders occasion. To overcome the conflicts, this brief develops a piecewise approximate model and provides two procedures for designing the model parameters. The comparison with several existing methods shows that the proposed methods do not only satisfy the equalities but also achieve high approximation accuracy. From this, it is believed that this work can serve for simulation and realization of fractional order controllers more friendly.