New Improved Fractional Order Differentiator Models Based on Optimized Digital Differentiators

Table 3

Mathematical models of half differentiators based on 3rd order optimized operators.

Mathematical models of half differentiators based on 3rd order optimized operators
Algorithm	Transfer function in pole-zero form
PSO	𝐺PSO_3rd_3()(𝑧)=1.0747𝑧−0.7422)(𝑧+0.3892)(𝑧+0.2492(𝑧−0.1853)(𝑧+0.2354)(𝑧+0.433)
	𝐺PSO_3rd_6()(𝑧)=1.0748𝑧−0.9033)(𝑧−0.2948)(𝑧+0.2512)(𝑧+0.2397)(𝑧+0.383)(𝑧+0.4194(𝑧−0.6398)(𝑧+0.4678)(𝑧+0.3939)(𝑧+0.2476)(𝑧+0.223)(𝑧−0.008066)
	𝐺PSO_3rd_9()(𝑧)=1.0748𝑧−0.9502)(𝑧−0.5926)(𝑧+0.4445)(𝑧+0.3965)(𝑧+0.3815)(𝑧+0.2517)(𝑧+0.2468)(𝑧+0.2316)(𝑧−0.1144(𝑧−0.8073)(𝑧−0.3441)(𝑧+0.3865)(𝑧+0.4144)(𝑧+0.483)(𝑧+0.25)(𝑧+0.2412)(𝑧+0.2159)(𝑧+0.04502)
GA	𝐺GA_3rd_3()(𝑧)=1.0393𝑧+0.259)(𝑧−0.2665)(𝑧−0.4826(𝑧+0.4564)𝑧^2−0.2763𝑧+0.2576
	𝐺GA_3rd_6(𝑧(𝑧)=1.0393𝑧−0.824)(𝑧+0.4138)(𝑧+0.1912)(𝑧−0.05418)^2−0.3657𝑧+0.244)(𝑧+0.5368)(𝑧+0.2948)𝑧^2−0.6608𝑧+0.1659𝑧^2−0.1389𝑧+0.27)
	𝐺GA_3rd_9()𝑧(𝑧)=1.0393𝑧−0.8991)(𝑧+0.4866)(𝑧+0.3107)(𝑧+0.163)(𝑧−0.008699^2−0.6199𝑧+0.1803𝑧^2−0.22𝑧+0.2639(𝑧−0.6505)(𝑧+0.5643)(𝑧+0.3957)(𝑧+0.2343)(𝑧−0.1531)𝑧^2−0.4068𝑧+0.2363𝑧^2−0.1005𝑧+0.2715
PSO-GA hybrid optimization	𝐺HYBR_3rd_3()(𝑧)=1.0761𝑧−0.7193)(𝑧+0.4597)(𝑧+0.3395(𝑧−0.1263)(𝑧+0.2642)(𝑧+0.5308)
	𝐺HYBR_3rd_6()(𝑧)=1.0762𝑧−0.8834)(𝑧−0.2504)(𝑧+0.2871)(𝑧+0.3543)(𝑧+0.4436)(𝑧+0.5127(𝑧−0.6142)(𝑧+0.5698)(𝑧+0.4702)(𝑧+0.3298)𝑧^2+0.2994𝑧+0.02545
	𝐺HYBR_3rd_9()(𝑧)=1.0762𝑧−0.9309)(𝑧−0.5656)(𝑧+0.5446)(𝑧+0.4752)(𝑧+0.4387)(𝑧+0.3588)(𝑧+0.3251)(𝑧+0.2413)(𝑧−0.0382(𝑧−0.7858)(𝑧−0.3041)(𝑧+0.2951)(𝑧+0.3455)(𝑧+0.4532)(𝑧+0.5055)(𝑧+0.5848)𝑧^2+0.3459𝑧+0.04169