Approximations of higher-order fractional differentiators and integrators using indirect discretization

Authors: RICHA YADAV, MANEESHA GUPTA

Abstract: This paper describes new approximations of fractional order integrators (FOIs) and fractional order differentiators (FODs) by using a continued fraction expansion-based indirect discretization scheme. Different tenth-order fractional blocks have been derived by applying three different s-to-z transforms described earlier by Al-Alaoui, namely new two-segment, four-segment, and new optimized four-segment operators. A new addition has been done in the new optimized four-segment operator by modifying it by the zero reflection method. All proposed half (s$^{\pm 1/2})$ and one-fourth (s$^{\pm 1/4})$ differentiator and integrator models fulfill the stability criterion. The tenth-order fractional differ-integrators (s$^{\pm \alpha })$ based on the modified new optimized four-segment rule show tremendously improved results with relative magnitude errors (dB) of $\le $ -15 dB for $\alpha $ = 1/2 and $\le $ -20 dB for $\alpha $ = 1/4 in the full range of Nyquist frequency so these have been further analyzed. The main contribution of this paper lies in the reduction of these tenth-order blocks into four new fifth-order blocks of half and one-fourth order models of FODs and FOIs. The analyses of magnitude and phase responses show that the proposed new fifth-order half and one-fourth differ-integrators closely approximate their ideal counterparts and outperform the existing ones.

Keywords: Fractional order differentiators, fractional order integrators, half and one-fourth order differentiators and integrators, continued fraction expansion

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