# Efficient Design of Discrete Fractional-Order Differentiators Using Nelder–Mead Simplex Algorithm

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## Abstract

Applications of fractional-order operators are growing rapidly in various branches of science and engineering as fractional-order calculus realistically represents the complex real-world phenomena in contrast to the integer-order calculus. This paper presents a novel method to design fractional-order differentiator (FOD) operators through optimization using Nelder–Mead simplex algorithm (NMSA). For direct discretization, Al-Alaoui operator has been used. The numerator and the denominator terms of the resulting transfer function are further expanded using binomial expansion to a required order. The coefficients of *z*-terms in the binomial expansions are used as the starting solutions for the NMSA, and optimization is performed for a minimum magnitude root-mean-square error between the ideal and the proposed operator magnitude responses. To demonstrate the performance of the proposed technique, six simulation examples for fractional orders of half, one-third, and one-fourth, each approximated to third and fifth orders, have been presented. Significantly improved magnitude responses have been obtained as compared to the published literature, thereby making the proposed method a promising candidate for the design of discrete FOD operators.

## Keywords

Fractional calculus Fractional-order differentiator Al-Alaoui operator Nelder–Mead simplex algorithm## References

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