流通科学大学論集 - 経済 情報 政策編 - 第 21 巻第 1 号,23-33(2012) SIRMs SIRMs Fuzzy fuzzyapproximate approximatereasoning reasoningusing using Lukasiewicz Łukasiewicz logical Logical operations Operations Takashi Mitsuishi * SIRMs SIRMs SIRMs SIRMs I. Zadeh 1;2) Mamdani 3) Mamdani -- 7) t- t- 16) t-t- 8) single input rule modules SIRMs IF-THEN 4;5) SIRMs T-S IF-THEN SIRMs 6) * 流通科学大学商学部 651-2188 神戸市西区学園西町 3-1 (2012 年 3 月 30 日受理 ) C 2012 UMDS Research Association
24 三石貴志 SIRMs t- t- 9;10;11;12) IF-THEN IF-THEN SIRMs II. SIRMs 1. R n n. ẋ(t) =f(x(t),u(t)). (1) f : R n R R n f M f > 0 (v 1,v 2 ) R n R f(v 1,v 2 ) M f (v 1 + v 2 + 1) x(t) u(t) u(t) =ρ(x(t)) r x 0 B r = {x R n ; x r} T 10) 1 ρ : R n R, x 0 B r. ẋ(t) =f(x(t),ρ(x(t)))
ウカシェヴィッツの多値論理演算を用いた SIRMs ファジィ推論法 25 x(0) = x 0 [0,T] x(t, x 0,ρ) (t, x 0 ) [0,T] B r x(t, x 0,ρ). r 2 > 0 { } Φ= ρ : R n R;, sup ρ(u) r 2 u R n, ab. a t [0,T],x 0 B r ρ Φ x(t, x 0,ρ) r 1. r 1 = e Mf T r +(e Mf T 1)(r 2 + 1). (2) b ρ 1,ρ 2 Φ. t [0,T], x 0 B r x(t, x 0,ρ 1 ) x(t, x 0,ρ 2 ) elf (1+Lρ 1 )t 1 1+L ρ1 sup u [ r 1,r 1] n ρ 1 (u) ρ 2 (u), (3) L f, L ρ1 f, ρ 1 2. 1 u(t) = ρ(x(t)) t [0,T] x = (x 1,x 2,,x n )=(x 1 (t),x 2 (t),,x n (t)) = x(t) ρ IF-THEN SIRMs M F ρ M F ρ F T J = w(x(t, ζ, ρ F ),ρ F (x(t, ζ, ρ F )))dtdζ (4) B r 0 w : R n R R J ρ F B r T J M 4 M F 1 M {F k } k N MF k F M(k ) sup ρ F k(x) ρ F (x) 0(k ) (5) x [ r 1,r 1] n T F M J = w(x(t,ζ,ρ F ),ρ F (x(t,ζ,ρ F )))dtdζ B r 0 1. M
26 三石貴志 J M (t, ζ) [0,T] B r 5 1b 56 1a lim x(t, ζ, ρ F k) x(t, ζ, ρ F) =0 (6) k lim ρ F k(x(t, ζ, ρ F k)) = ρ F(x(t,ζ,ρ F )) (7) k w : R n R R 67 14;15) J M 3. (Single Input Rule Modules) 1 u(t) =ρ(x(t)) SIRMs SIRM-1 : {R 1 j : if x 1 = A 1 j then y = C 1 j } m1 SIRM-i : {R i j : if x i = A i j then y = C i j} mi (8) SIRM-n : {R n j : if x n = A n j then y = C n j } mn n x 1,x 2,...x n m i (i =1, 2,...,n) y x SIRMs A i j (x i) Cj i(y) (i =1, 2,...,n; j =1, 2,...,m i) SIRM-i j A i j Ci j x i y A i = (A i 1,A i 2,...,A i m i ), C i =(C i 1,C i 2,...,C i m i ) (i =1, 2,...,n), A = (A 1, A 2,...,A n ), C =(C 1, C 2,...,C n ). A i (i =1, 2,...,m i ) SIRM-i A C (A, C) SIRMs8
ウカシェヴィッツの多値論理演算を用いた SIRMs ファジィ推論法 27 4. SIRMs t [0,T] x = (x 1,x 2,...,x n ) R n (A, C)SIRMs8 ρ SIRMs 4) 1: SIRM-i j Rj i α i j(x i,y)=a i j(x i ) C i j(y) (j =1, 2...,m i ; i =1, 2,...,n). A i j(x i ) C i j(y) =(A i j(x i )+C i j(y) 1) 0, C i j (y) (1 Ai j (x i)) 0 0 2: 1 SIRM-i Rj i m i β i (x i,y)= αj(x i i,y) j =1, 2 (i =1, 2,...,n). α i 1(x i,y) α i 2(x i,y)=(α i 1(x i,y)+α i 2(x i,y)) 1, 3: SIRM-i yβi (x i,y)dy γ i (x i )= βi (x i,y)dy. SIRMs x 1,x 2,...,x n ( SIRM-i) d i (i =1, 2,,n) SIRM-i d i (i =1, 2,,n) 1 A C d =(d 1,d 2,...,d n ) 4: d ρ ACd (x) = d i γ i (x i ).
28 三石貴志 ρ x A, C d III. SIRMs (A, C) 1. r>0 r 2 > 0 T 1 2 r 1 C[ r 1,r 1 ] C[ r 2,r 2 ] [ r 1,r 1 ][ r 2,r 2 ] ij > 0(i =1, 2,...,m; j =1, 2,...,n) 2 F ij = {µ C[ r 1,r 1 ]; 0 µ(x) 1 for x [ r 1,r 1 ], µ(x) µ(x ) ij x x for x, x [ r 1,r 1 ]} G = {µ C[ r 2,r 2 ]; 0 µ(y) 1 for y [ r 2,r 2 ]}. A i j F ij Cj i G F ij G ij F ij G C[ r 1,r 1 ] C[ r 2,r 2 ] i =1, 2,...,n j =1, 2,...,m i F ij C[ r 1,r 1 ] G 9) n m i L ( = F ij G ). L (A, C) SIRMs8 15) 2 L d i 1 4) 1 d i (i =1, 2,,n) D { } D = d =(d 1,d 2,...,d n ) R n ; i =1, 2,...,n, d i (0, 1), d i 1
ウカシェヴィッツの多値論理演算を用いた SIRMs ファジィ推論法 29 L = L D. d L (A, C,d) 3 0 δ>0 { L δ = (A, C,d) L; i =1, 2,...,n, x [ r 1,r 1 ] n, r2 r 2 β i (x i,y)dy δ }. (9) δ L 0 x 0 L δ (A, C,d) L δ 3 L δ {(A k, C k,d k )} L (A, C,d) L i =1, 2,...,n; j =1, 2,...,m i A j k i A j i = C j k i C j i = sup x i [ r 1,r 1] sup C j i y [ r 2,r 2] A j i k (xi ) A j i (x i) 0 k (y) C j i (y) 0 x [ r 1,r 1 ] n {(A k, C k,d k )} L δ (A, C,d) L i =1, 2,...,n r2 β i (x i,y)dy = lim r k 2 r2 m i r 2 r2 m i A i j(x i ) Cj(y)dy i = lim k r 2 A i j k (xi ) C i j k (y)dy δ (A, C,d) L δ. L δ L IV. 1 u(x) =ρ ACd (x) t [0,T] x =(x 1,x 2,,x n )= (x 1 (t),x 2 (t),,x n (t)) = x(t) SIRMs84. SIRMs SIRMs 9 L δ (A, C,d) ρ ACd (x) = d i r2 r 2 yβ i (x i,y)dy r2 r 2 β i (x i,y)dy
30 三石貴志 m i β i (x i,y)= A i j(x i ) Cj(y) i (i =1, 2,...,n) ρ ACd 4 (A, C,d) L δ ab. a ρ ACd [ r 1,r 1 ] n. b x [ r 1,r 1 ] n ρ ACd (x) r 2. a ρ ACd 4. α i j, β i, γ i i =1, 2,...,n; j =1, 2,...,m i x =(x i ) n,x =(x i ) n [ r 1,r 1 ] n αj(x i i,y) αj(x i i,y) = A i j(x i ) Cj(y) i A i j(x i ) Cj(y) i 1 { A i 2 j(x i ) A i j(x i ) + A i j(x i )+Cj(y) i 1 A i j(x i )+Cj(y) i 1 } A i j(x i ) A i j(x i ) ij x i x i 1 α i j ij β i i =1, 2,...,nm i 1 m i 1 m i 1 αj(x i i,y) αj(x i i,y) (m i 1) x i x i. (mi 1) m i 1 m i 1 αj(x i i,y) αm i i (x i,y) αj(x i i,y) αm i i (x i,y) = 1 m i 1 m i 1 2 αj(x i i,y)+αm i i (x i,y)+1 αj(x i i,y)+αm i i (x i,y) 1 m i 1 m i 1 α i j(x i,y)+αm i i (x i,y)+1 αj(x i i,y)+αm i i (x i,y) 1 m i 1 m i 1 αj(x i i,y) αj(x i i,y) + αi m i (x i,y) αm i i (x i,y) (mi 1) x i x i + imi x i x i = ( (mi 1) + imi ) x i x i m i β i (x i,y)= αj(x i i,y) γ i (x i ) γ i (x i ) 4r 2 3 m i β i (x i,y) β i (x i,y) δ 2
ウカシェヴィッツの多値論理演算を用いた SIRMs ファジィ推論法 31 12) d i < 1 ρ ACd (x) ρ ACd (x ) 4r 2 3 δ 2 d i γ i (x i ) γ i (x i ) m i β i (x i,y) β i (x i,y) 4r 2 3 δ 2 { mi ( (mi 1) + imi ) } x x ρ ACd [ r 1,r 1 ] n b 4 1 ρ ACd Φ [ r 1,r 1 ] n R n [ r 1,r 1 ] n R n 13) ρ ACd 1 1 x(t, x 0, ρ ACd ) 1b3 ρ ACd ρ ACd R n ρ ACd ρ ACd L δ 5 4 T J = w(x(t,ζ,ρ ACd ),ρ ACd (x(t,ζ,ρ ACd )))dtdζ B r 0 9 L δ 1 L δ 3 ρ ACd L δ i =1, 2,...,n; j =1, 2,...,m i A i k j (xi ) Cj i k (y) A i j (x i ) Cj(y) i A i k j (xi ) A i j(x i ) + Cj i k (y) C i j (y), m i m i αj i k (xi,y) αj(x i i,y) = m i m i αj i k (xi,y) 1 αj(x i i,y) 1 ρ Ak C k d k(x) ρ ACd(x) 1 n m i 3 2r δ 2 2 A i k j (xi ) A i 2 j(x i ) + r 2 r2 m i Cj i k (y) C i j (y) dy r 2 r2 m i +2r 2 y Cj i k (y) C i j (y) dy 2 + r 2 d k i d i r 2
32 三石貴志 L δ (A k, C k,d k ) (A, C,d)(k ) lim sup ρ Ak C k d k(x) ρ ACd(x) =0 x [ r 1,r 1] n k ρ ACd (A, C,d) SIRMs V. SIRMs SIRMs SIRMs 1 SIRMs 1) L. A. Zadeh, Fuzzy Sets, Information and Control, vol. 8, pp.338 353. 1965. 2) L. A. Zadeh, Fuzzy algorithms, Information and Control, 12, pp.94 102, 1968. 3) E. H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEE 121, No. 12, pp.1585 1588, 1974. 4),,, vol. 9, no. 5, pp.699 709, 1997. 5), SIRMs,, vol. 10, no. 3, pp.522 531, 1998. 6) H. Seki, H. Ishii and M. Mizumoto, On the generalization of single input rule modules connected type fuzzy reasoning method, Proc. of Joint 3rd International Conference on Soft Computing and Intelligent Systems and 7th International Symposium on advanced Intelligent Systems (SCIS&ISIS 2006), pp.30 34, 2006.
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