New Improved Fractional Order Differentiator Models Based on Optimized Digital Differentiators

Table 2

Mathematical models of half differentiators based on 2nd order optimized operators.

Mathematical models of half differentiators based on 2nd order optimized operator
Algorithm	Transfer functions in pole-zero form
PSO	𝐺PSO_2nd_2()(𝑧)=1.0751𝑧−0.7416)(𝑧+0.5604(𝑧+0.5869)(𝑧−0.1823)
	𝐺PSO_2nd_4()(𝑧)=1.0752𝑧−0.9026)(𝑧−0.2927)(𝑧+0.5788)(𝑧+0.5566(𝑧−0.003298)(𝑧+0.5633)(𝑧+0.6072)(𝑧−0.6391)
	𝐺PSO_2nd_6()(𝑧)=1.0752𝑧−0.9494)(𝑧−0.5919)(𝑧+0.5936)(𝑧+0.5649)(𝑧+0.5556)(𝑧−0.1105(𝑧−0.8067)(𝑧−0.3424)(𝑧+0.616)(𝑧+0.5758)(𝑧+0.5587)(𝑧+0.04903)
GA	𝐺GA_2nd_2()(𝑧)=1.1017𝑧−0.4323)(𝑧+0.8014(𝑧+0.7299)(𝑧−0.06811)
	𝐺GA_2nd_4(𝑧)=1.1019(𝑧−0.5249)(𝑧+0.7494)(𝑧+0.8142)(𝑧−0.1481)(𝑧−0.3713)(𝑧+0.6849)(𝑧+0.7921)(𝑧+0.07968))
	𝐺GA_2nd_6()(𝑧)=1.1019𝑧−0.01236)(𝑧−0.3426)(𝑧−0.5511)(𝑧+0.7145)(𝑧+0.7874)(𝑧+0.8176(𝑧+0.8069)(𝑧+0.757)(𝑧+0.666)(𝑧−0.4701)(𝑧−0.1822)(𝑧+0.1305)
PSO-GA hybrid; optimization	𝐺HYBR_2nd_2()(𝑧)=1.1039𝑧−0.5974)(𝑧+0.7144(𝑧+0.7261)(𝑧−0.08744)
	𝐺HYBR_2nd_4()(𝑧)=1.1042𝑧+0.7226)(𝑧+0.7126)(𝑧−0.7375)(𝑧−0.1918(𝑧−0.5068)(𝑧+0.7157)(𝑧+0.7347)(𝑧+0.08783)
	𝐺HYBR_2nd_6()(𝑧)=1.1042𝑧−0.4648)(𝑧−0.7778)(𝑧+0.729)(𝑧+0.7164)(𝑧+0.7121)(𝑧−0.01829(𝑧−0.6543)(𝑧−0.238)(𝑧+0.7136)(𝑧+0.7213)(𝑧+0.7384)(𝑧+0.1412)