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1 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 神戸市西区学園西町 3-1 (2012 年 3 月 30 日受理 ) C 2012 UMDS Research Association

2 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)))

3 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 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

4 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

5 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 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 ).

6 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

7 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

8 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

9 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

10 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 ) L. A. Zadeh, Fuzzy algorithms, Information and Control, 12, pp , ) E. H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEE 121, No. 12, pp , ),,, vol. 9, no. 5, pp , ), SIRMs,, vol. 10, no. 3, pp , ) 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.

11 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 33 7) IV--, pp.9 13, ) M. Mizumoto, Fuzzy Conditional lnference under Max- Composition, Information Sciences, 27(2), , ) T. Mitsuishi, K. Wasaki, J. Kawabe, N. P. Kawamoto, Y. Shidama, Fuzzy optimal control in L 2 space, In: Proc. 7th IFAC Symposium Artificial Intelligence in Real-Time Control, pp , ) T. Mitsuishi, J. Kawabe, K. Wasaki and Y. Shidama, Optimization of Fuzzy Feedback Control Determined by Product-Sum-Gravity Method, Journal of Nonlinear and Convex Analysis, Vol. 1, No. 2, pp , ) T. Mitsuishi, Y. Shidama, Optimal Control Using Functional Type SIRMs Fuzzy Reasoning Method, In: T. Honkela et al. (Eds.): ICANN 2011, Part II, LNCS 6792, pp , Springer, Heidelberg ) T. Mitsuishi, T. Terashima, T. Homma, Y. Shidama, Fuzzy Approximate Reasoning Using Single Input Rule Modules in L Space, Proc. of IEEE AFRICON, CD-ROM, ) R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York, ) F. Riesz, B. Sz.-Nagy, Functional Analysis, Dover Publications, New York, ) N. Dunford and J.T. Schwartz, Linear Operators Part I: General Theory, John Wiley & Sons, New York, ) G. Metcalfe, N. Olivetti, D. Gabbay, Proof Theory for Fuzzy Logics, Springer-Verlag, 2009.

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