Improved Polynomial Fuzzy Modeling and Controller with Stability Analysis for Nonlinear Dynamical Systems

Table 3

Slack expressions defined on sub-regions and obtained polynomial membership functions for Example 2.

Sub-region	Slack expression	Membership function	Membership function	Membership function
(𝑆𝑘)	𝐿𝑆𝑘(𝐱)	𝑝1(𝑥1)	𝑝2(𝑥1)	𝑝3(𝑥1)
𝑆1: −10<𝑥1<−8	(𝑥1+10)(𝑥1+8)	0.9241−0.0351(𝑥1+9)−0.0074(𝑥1+9)^2	0.0744+0.0201(𝑥1+9)+0.0074(𝑥1+9)^2	0.0015−0.015(𝑥1+9)
𝑆2: −9<𝑥1<−7	(𝑥1+9)(𝑥1+7)	0.8808−0.0525(𝑥1+8)−0.01(𝑥1+8)^2	0.1167+0.0513(𝑥1+8)+0.01(𝑥1+8)^2	0.0025+0.0012(𝑥1+8)
𝑆3: −8<𝑥1<−6	(𝑥1+8)(𝑥1+6)	0.8176−0.0746(𝑥1+7)−0.0118(𝑥1+7)^2	0.1783+0.0726(𝑥1+7)+0.0118(𝑥1+7)^2	0.0041+0.002(𝑥1+7)
𝑆4: −7<𝑥1<−5	(𝑥1+7)(𝑥1+5)	0.7311−0.0983(𝑥1+6)−0.0114(𝑥1+6)^2	0.2622+0.095(𝑥1+6)+0.0114(𝑥1+6)^2	0.0067+0.0033(𝑥1+6)
𝑆5: −6<𝑥1<−4	(𝑥1+6)(𝑥1+4)	0.6225−0.1175(𝑥1+5)−0.0072(𝑥1+5)^2	0.3665+0.1121(𝑥1+5)+0.0059(𝑥1+5)^2	0.011+0.0054(𝑥1+5)+0.0013(𝑥1+5)^2
𝑆6: −5<𝑥1<−3	(𝑥1+5)(𝑥1+3)	−0.125(𝑥1+4)	0.982+0.1162(𝑥1+4)−0.0021(𝑥1+4)^2	0.018+0.0088(𝑥1+4)+0.0021(𝑥1+4)^2
𝑆7: −4<𝑥1<−2	(𝑥1+4)(𝑥1+2)	0.3775−0.1175(𝑥1+3)+0.0072(𝑥1+3)^2	0.5932+0.1033(𝑥1+3)−0.0105(𝑥1+3)^2	0.0293+0.0142(𝑥1+3)+0.0033(𝑥1+3)^2
𝑆8: −3<𝑥1<−1	(𝑥1+3)(𝑥1+1)	0.2689−0.0983(𝑥1+2)+0.0114(𝑥1+2)^2	0.6837+0.0757(𝑥1+1)−0.0165(𝑥1+1)^2	0.0474+0.0226(𝑥1+2)+0.0051(𝑥1+2)^2
𝑆9: −2<𝑥1<0	𝑥1+2𝑥1	0.1842−0.0746(𝑥1+1)+0.0118(𝑥1+1)^2	0.7399+0.0395(𝑥1+1)−0.0192(𝑥1+1)^2	0.0759+0.0351(𝑥1+1)+0.0074(𝑥1+1)^2
𝑆10: −1<𝑥1<1	(𝑥1+1)(𝑥1−1)	0.1192−0.0525𝑥1+0.01(𝑥1)^2	0.7616−0.02(𝑥1)^2	0.1192+0.0525𝑥1+0.01(𝑥1)^2
𝑆11: 0<𝑥1<2	𝑥1(𝑥1−2)	0.0755−0.0354(𝑥1−1)+0.0073(𝑥1−1)^2	0.7421−0.0397(𝑥1−1)−0.0188(𝑥1−1)^2	0.1824+0.0751(𝑥1−1)+0.0115(𝑥1−1)^2
𝑆12: 1<𝑥1<3	(𝑥1−1)(𝑥1−3)	0.0472−0.0224(𝑥1−2)+0.0053(𝑥1−2)^2	0.685−0.0761(𝑥1−2)−0.0169(𝑥1−2)^2	0.2678+0.0985(𝑥1−2)+0.0116(𝑥1−2)^2
𝑆13: 2<𝑥1<4	(𝑥1−2)(𝑥1−4)	0.0289+0.0145(𝑥1−3)+0.0035(𝑥1−3)^2	0.594+0.1031(𝑥1−3)−0.011(𝑥1−3)^2	0.3771−0.1176(𝑥1−3)+0.0075(𝑥1−3)^2
𝑆14: 3<𝑥1<5	(𝑥1−3)(𝑥1−5)	0.015−0.0081(𝑥1−4)+0.0023(𝑥1−4)^2	0.985−0.1209(𝑥1−4)−0.0023(𝑥1−4)^2	0.129(𝑥1−4)
𝑆15: 4<𝑥1<6	(𝑥1−4)(𝑥1−6)	0.013−0.0053(𝑥1−5)+0.0011(𝑥1−5)^2	0.1842−0.0746(𝑥1−5)+0.0118(𝑥1−5)^2	0.6221+0.1176(𝑥1−5)−0.0073(𝑥1−5)^2
𝑆16: 5<𝑥1<7	(𝑥1−5)(𝑥1−7)	0.0064−0.0031(𝑥1−6)	0.1842−0.0746(𝑥1−6)+0.0118(𝑥1−6)^2	0.7314+0.0981(𝑥1−6)−0.0117(𝑥1−6)^2
𝑆17: 6<𝑥1<8	(𝑥1−6)(𝑥1−8)	0.0044−0.003(𝑥1−7)	0.1842−0.0746(𝑥1−7)+0.0118(𝑥1−7)^2	0.8177+0.0744(𝑥1−7)−0.0116(𝑥1−7)^2
𝑆18: 7<𝑥1<9	(𝑥1−7)(𝑥1−9)	0.0022−0.0014(𝑥1−8)	0.1177−0.0509(𝑥1−8)+0.012(𝑥1−8)^2	0.8801+0.0523(𝑥1−8)−0.012(𝑥1−8)^2
𝑆19: 8<𝑥1<10	(𝑥1−8)(𝑥1−10)	0.0017+0.0011(𝑥1−9)	0.0739+0.0366(𝑥1−9)+0.0075(𝑥1−9)^2	0.9244−0.0355(𝑥1−9)−0.0075(𝑥1−9)^2