tailieunhanh - Fuzzy Systems Part 4

Tham khảo tài liệu 'fuzzy systems part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Control of T-S Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Function 51 0 C1 Ể I 01I T I S0 S1 Sr L 1 e1I 0rI. .-I. .V. 0 Ci -02 Pi0i 0 Cij 0i0j. 37 38 Here note that the relaxation variables S0 Si Nị and sij are in Rncxln and should satisfy that 0 Ai AT and 0 Bij BTj . Further with some algebraic manipulations the constraint 0 N ỡ t can be rewritten as follows r 0 N ỡ t N X i Ni NT i 1 . - . 4 Sổi2Nii 2 XffiffjNij X l -N i-1 39 i 1 i 1 j 1 j i 1 r r r where No So St N PA - So Sị N - Ai AT - Si ST Nj - Si Sj Bij B ji . Hence the condition in 35 becomes for all m e 1 r A 0 To X i hi rT i 1 i-1 X 2 Xj X j 40 i 1 i 1 j 1 j i 1 r r r where r0 4-N0 4-s0- 0 jrf A-Ni c- i s0-si Ị 4 4 - A 4 a . A 1 y s . sj 41 ịOịj Cj -Nij Cj Si Sjy Bij Eji As a result the condition in 40 boils down to 0 I 01I - 0rl jC. l Ỡ1I - 0rl T 42 where A rr0 4 i 4 c r2 21 1 1 a2 . 43 4 41 r-l Ar 52 Fuzzy Systems Proposition 1 The condition in 33 subject to C1 and C2 holds if the following conditions hold for all i j e 1 r j i 0 L 0 Ai Ai 0 Bij 0i. 44 Case 2 Continuous-time Consider the following condition with quadratic dependence 0i t and 0i t . . z. 0 L 0 t 0 t Ế Lo 2 i l lt Ýởi L LT Li i 1 i 1 i 1 r i-1 r 2 Xj X j i 1 j 1 j i 1 r i-1 z - . r 2 Xj X j i 1 j 1 j i 1 subject to for i j e 1 r j i C1 XPi 1 c2 0 p Pi C3 0 0i0j C4 p ởị vi. i 1 45 46 By the S-procedure Boyd et al 1994 and Finsler s lemma de Oliveira Skelton 2001 Fang et al 2004 the condition in Case 2 can be written as follows 0 L 0 t ở t - N ớ t ớ t 47 where 0 N ớ t ớ t is given by N 0 t 0 t C1 CT 2C2i Ai AT 2 2 C3ij Eij ST XC4i Zi ZT 48 i 1 i 1j 1 jVi i 1 in which C C2i C3iJ and C4i are from C1 C2 C3 and C4 respectively 0 C1 r IÌ T r 11 Ớ1I -I 0rI. H J S1 - Sr I 01I 0rI 49 0 C2i -6Ị2 Pịỡị 0 C3ij ỡi0j 0 C4i -ê pi Vị 9i - PịVị. 50 Here note that the multiplier variables S0 Si Ai Zi and Hjj are in Rncxn and should satisfy that 0 Ai At 0 Zi ZT and 0 Bij T . Further with some algebraic manipulations the constraint 0 N 0 t can be .