FB2-2 24th Fuzzy System Symposium (Osaka, September 3-5, 2008) On the Infimum and Supremum of T-S Inference Method Using Functional Type

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1 FB2-2 24th Fuzzy System Symposium (Osaka, September 3-5, 2008) SIRMs T-S On the Infimum and Supremum of T-S Inference Method Using Functional Type SIRMs Connected Fuzzy Inference Method 1,2 Hirosato Seki 1 Hiroaki Ishii 1 Graduate School of Information Science and Technology, Osaka University 2 Japan Society for the Promotion of Science (JSPS) Abstract: Fuzzy inference has played significant role in many applications Although T-S inference method and simplified fuzzy inference method are currently mostly used, the problem is that the number of fuzzy rules becomes very huge and so the setup and adjustment of fuzzy rules become difficult On the other hand, Yubazaki et al have proposed single input rule modules connected fuzzy inference method (SIRMs method, for short) whose final output is obtained by summarizing the product of the importance degrees and the inference results from single input fuzzy rule module Moreover, we have proposed functional type SIRMs method in which consequent part of functional type SIRMs method is generalized to a function though the consequent part of ordinary SIRMs methods is real number This paper shows the conditions for infimum and supremum of conventional fuzzy inference method using single input type fuzzy inference method, from the view point of fuzzy inference 1 - T-S [1] [2] if-then if then [3] [6] 1 if-then (SIRMs, Single Input Rule Modules) SIRMs [7] [8], [9] SIRMs SIRMs T-S SIRMs [7] [10] SIRMs T-S SIRMs 2 SIRMs SIRMs [3] [6] 1 if-then 829

2 n 1 n 1 1 SIRMs SIRMs [8], [9] Rules-1 :{x 1 = A 1 j y 1 = fj 1(x 1)} m1 Rules-i : {x i = A i j y i = fj i(x i)} mi Rules-n : {x n = A n j y n = fj n(x n)} mn (1) x i y i A i j x i f i j (x i) i = 1, 2,,n i Rules-i j =1, 2,,m i m i x 0 i Rules-i j h i j (2) y 0 i (3) y 0 i = h i j = A i j(x 0 i ) (2) m i h i jf i j(x 0 i ) m i h i j (3) x i (i =1, 2,,n)( Rules-i) w i SIRMs y 0 y 0 i y 0 = n w i yi 0 (4) SIRMs T-S [8] 1 1 T-S x 1 = A 1 j 1, = A 2 j 2 y = f(x 1, ) (i,j) = a i j + bi j x 1 + c i j (5) 1: T-S A 2 1 A 2 2 A 2 n A 1 f(x 1,) (1,1) f(x 1,) (1,2) f(x 1,) (1,n) x 1 A 1 2 f(x 1,) (2,1) f(x 1,) (2,2) f(x 1,) (2,n) A 1 m f(x1,x2) (m,1) f(x 1,) (m,2) f(x 1,) (m,n) 2: A 2 1 A 2 2 A 2 n A 1 1 y (1,1) y (1,2) y (1,n) x 1 A 1 2 y (2,1) y (2,2) y (2,n) A 1 m y (m,1) y (m,2) y (m,n) i = 1, 2,,m 1; j = 1, 2,,n 1 a i j + a i+1 j+1 = ai+1 j + a i j+1 (6) b i j = b i j+1 (7) c i j = c i+1 j (8) i =1, 2,,m; j =1, 2,,n w 1 f 1 i (x 1 )+w 2 f 2 j ( )=f(x 1, ) (i,j) (9) w 1,w 2,f 1 i (x 1),f 2 j () (1) SIRMs (1) SIRMs f i j (x i) yj i SIRMs SIRMs [7] 2 2 i =1, 2,,m 1; j =1, 2,,n 1 y (i,j) + y (i+1, j+1) = y (i, j+1) + y (i+1, j) (10) i = 1, 2,,m; j =1, 2,,n w 1 y 1 i + w 2 y 2 j = y (i,j) (11) w 1,w 2,yi 1,y2 j SIRMs 2 SIRMs min max SIRMs [7], [10] 830

3 1 3 1 A i 1,A i 2,A i 3 (i =1, 2) 2 3 SIRMs Rules-1, 2 min(= ) x 1 = A 1 1 y 1 =1 15 1=1 Rules-1 = x 1 = A 1 2 y 1 =2 25 2=2 x 1 = A 1 3 y 1 =3 35 3=3 = A 2 1 y 2 =1 2 3=1 Rules-2 = = A 2 2 y 2 = =15 = A 2 3 y 2 =1 2 3=1 (12) max(= ) 3 3: A 2 1 A 2 2 A 2 3 A x 1 A A A A A 1 i i i : A i 1,A i 2 Inference result x 1 Supremum 3: Simplified fuzzy inference method Infimum SIRMs x 1 = A 1 1 y 1 =1 15 1=15 Rules-1 = x 1 = A 1 2 y 1 =2 25 2=25 x 1 = A 1 3 y 1 =3 35 3=35 = A 2 1 y 2 =1 2 3=3 Rules-2 = = A 2 2 y 2 = =35 = A 2 3 y 2 =1 2 3=3 (13) 1 3 SIRMs (12), (13) 3 w 1 = w 2 =05 [7], [10] 1 min max SIRMs SIRMs 3 Inference result SIRMs [10] 3 i =1, 2,,m; j =1, 2,,n n m y (i,j) w 1 y (i,j) + w 2 y (i,j) (14) x 1 2: min(= ) SIRMs 831

4 2 x 1 y 2 y = y (1,1) A 1 1(x 1 )A 2 1( )+ +y (m,n) A 1 m(x 1 )A 2 n( ) A 1 1 (x 1)A 2 1 ()+ + A 1 m(x 1 )A 2 n( ) (15) min SIRMs SIRMs x 1 = A 1 1 y 1 = y (1,i) = α 1 Rules-1 = x 1 = A 1 m y 1 = y (m,i) = α m = A 2 1 y 2 = y (i,1) = β 1 Rules-2 = = A 2 n y 2 = y (i,n) = β n (16) x 1 SIRMs z (3), (4), (16) z = w 1 α1a 1 1(x 1 )+ + α m A 1 m(x 1 ) A 1 1 (x 1)+ + A 1 m(x 1 ) +w 2 β1a 2 1( )+ + β n A 2 n( ) A 2 1 ()+ + A 2 n( ) (17) (19) y z 0 (16) SIRMs (16) SIRMs 2 i =1, 2,, m; j =1, 2,, n 3 y (i,j) w 1 α i + w 2 β j (20) 2 1 SIRMs SIRMs (20) (12) SIRMs = = = = = = = = =2 (21) (21) (12) SIRMs 3 [10] y z y z y (1,1) A 1 1(x 1 )A 2 1( )+ +y (m,n) A 1 m(x 1 )A 2 n( ) = A 1 1 (x 1)A 2 1 ()+ + A 1 m(x 1 )A 2 n( ) { w 1 α1a 1 1(x 1 )+ + α m A 1 m(x 1 ) A 1 1 (x 1)+ + A 1 m(x 1 ) +w 2 β1a 2 1( )+ + β n A 2 } n( ) A 2 1 ()+ + A 2 n( ) (18) (18) 4 i =1, 2,,m; j =1, 2,,n n m y (i,j) w 1 y (i,j) + w 2 y (i,j) (22) max(= ) SIRMs 4 T-S y (1,1) A 1 1(x 1 )A 2 1( )+ +y (m,n) A 1 m(x 1 )A 2 n( ) w 1 {α 1 A 1 1(x 1 )+ + α m A 1 m(x 1 )} {A 2 1( )+ + A 2 n( )} w 2 {β 1 A 2 1( )+ + β n A 2 n( )} {A 1 1(x 1 )+ + A 1 n(x 1 )} = A 1 1(x 1 )A 2 1( ){y (1,1) w 1 α 1 w 2 β 1 } +A 1 1(x 1 )A 2 2( ){y (1,2) w 1 α 1 w 2 β 2 } + +A 1 m(x 1 )A 2 n( ){y (m,n) w 1 α m w 2 β n } (19) [7] [9] T-S (6) (8) SIRMs SIRMs T-S [8] 832

5 SIRMs T-S SIRMs T-S 5, T-S min SIRMs ɛ i,ζ j T-S SIRMs i =1, 2,, m; j =1, 2,, n a i j + b i jx 1 + c i j w 1 ɛ i + w 2 ζ j (23) SIRMs T-S 6 1 T-S max SIRMs η i,θ j T-S SIRMs i =1, 2,, m; j =1, 2,, n a i j + b i jx 1 + c i j w 1 η i + w 2 θ j (24) SIRMs T-S 3 T-S min max min max T-S SIRMs min max T-S SIRMs x 1 = A 1 1 y 1 = f(x 1 ) (1,i) = ɛ 1 Rules-1 = x 1 = A 1 m y 1 = f(x 1 ) (m,i) = ɛ m = A 2 1 y 2 = f( ) (i,1) = ζ 1 Rules-2 = = A 2 n y 2 = f( ) (i,n) = ζ n (25) x 1 = A 1 1 y 1 = Rules-1 = x 1 = A 1 m y 1 = n f(x 1 ) (1,i) = η 1 n f(x 1 ) (m,i) = η m m = A 2 1 y 2 = f( ) (i,1) = θ 1 Rules-2 = m = A 2 n y 2 = f( ) (i,n) = θ n (26) 3, 4 SIRMs T-S T-S min max SIRMs T-S 5 SIRMs T-S [1] T Takagi and M Sugeno: Fuzzy identification of systems and its applications to modeling and control, IEEE Trans Syst Man Cybern, VolSMC-15, No1, pp , 1985 [2], :,, Vol2, No3, pp , 1990 [3] N Yubazaki, J Yi, M Otani, and K Hirota: SIRMs dynamically connected fuzzy inference model and its applications, Proc IFSA 97, Vol3, pp , Prague, Czech, 1997 [4] J Yi, N Yubazaki, and K Hirota: Upswing and stabilization control of inverted pendulum and cart system by the SIRMs dynamically connected fuzzy inference model, Proc 1999 IEEE Int Conf Fuzzy Syst, Vol1, pp , Seoul, Korea, 1999 [5] J Yi, N Yubazaki, and K Hirota: A proposal of 833

6 SIRMs dynamically connected fuzzy inference model for plural input fuzzy control, Fuzzy Sets Syst, Vol125, No1, pp79 92, 2002 [6] J Yi, N Yubazaki, and K Hirota: A new fuzzy controller for stabilization of parallel-type double inverted pendulum system, Fuzzy Sets Syst, Vol126, No1, pp , 2002 [7] H Seki, H Ishii, and M Mizumoto: On the property of single input rule modules connected type fuzzy reasoning method, Proc 2007 IEEE Int Conf Fuzzy Syst, pp , London, UK, 2007 [8] H Seki, H Ishii, and M Mizumoto: On the generalization of single input rule modules connected type fuzzy reasoning method, IEEE Trans Fuzzy Syst, 2007 (to appear) [9] H Seki and H Ishii: On the monotonicity of functional type SIRMs connected fuzzy reasoning method and T-S reasoning method, Proc 2008 IEEE World Congress on Computational Intelligence, pp58 63, Hong Kong, 2008 [10] H Seki and H Ishii: On the infimum and supremum of simplified fuzzy inference method, Proc 2008 IEEE Conference on Soft Computing in Industrial Applications, pp , Hokkaido, Japan, TEL: FAX: hseki@istosaka-uacjp 834