Optimization of Integer Order Integrators for Deriving Improved Models of Their Fractional Counterparts

Table 5

Mathematical models of proposed half integrators based on 2nd and 3rd order PSO optimized DIs.

Mathematical models of half integrators based on 2nd and 3rd order PSO optimized DIs
Name of PSO optimized DIs	Transfer functions of FOIs derived by indirect discretization of CFE
𝐻PSO_2nd_opt(𝑧)	𝐻2nd-fo-2()(𝑧)=0.9302𝑧−0.1825)(𝑧+0.5869(𝑧−0.7422)(𝑧+0.5604)
	𝐻2nd-fo-4()(𝑧)=0.93011𝑧−0.003372)(𝑧+0.5633)(𝑧+0.6071)(𝑧−0.6396(𝑧−0.9034)(𝑧−0.293)(𝑧+0.5788)(𝑧+0.5566)
	𝐻2nd-fo-6()()(𝑧)=0.93011𝑧−0.8073)(𝑧−0.3427)(𝑧+0.616𝑧+0.5758)(𝑧+0.5587)(𝑧+0.04899()()𝑧+0.5936)(𝑧+0.5649)(𝑧+0.5556𝑧−0.5924)(𝑧−0.9502)(𝑧−0.1107
𝐻PSO_3rd_opt(𝑧)	𝐻3rd-fo-3()(𝑧)=0.93046𝑧−0.1852)(𝑧+0.2354)(𝑧+0.433(𝑧−0.742)(𝑧+0.3892)(𝑧+0.2492)
	𝐻3rd-fo-6()()(𝑧)=0.93038𝑧−0.6395)(𝑧+0.4678)(𝑧+0.3939𝑧+0.2476)(𝑧+0.223)(𝑧−0.008033()()𝑧−0.9031)(𝑧−0.2946)(𝑧+0.2512𝑧+0.2397)(𝑧+0.383)(𝑧+0.4194