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Universal approximation theorem for uninorm-based fuzzy systems modeling. (English) Zbl 1040.93043
The fundamental result of this study concerns the universal approximation property of fuzzy systems endowed with arbitrary uninorms. Uninorms are two-argument functions $$U:[0,1]^2\to [0,1]$$ satisfying the following properties: (a) identity $$U(g,x)=x,(b)$$ commutativity $$U(x,y)=U(y,x)$$, (c) associativity $$U(x,U (y,z))= U(U(x, y),z)$$, (d) monotonicity $$U(x,y)\leq U(x',y')$$ for all $$x\leq x'$$ and $$y\leq y'$$. For $$g=0$$ the uninorm becomes a certain s-norm while for $$g=1$$ the definition coincides with the one for any t-norm. Furthermore the authors introduce a relevancy transformation $$h:[0,1]^2\to[0,1]$$ such that (a) $$h(1,x)=x$$; $$h(0,x)= g$$; (b) if $$y<y'$$ then $$h(x,y)\leq h(x,y')$$, and (c) if $$x<x'$$ then $$h(x,y)\leq h(x',y)$$ for $$y\leq g$$ and $$h(x,y)\geq h(x',y)$$ for $$y\geq g$$. The property of universal approximation is derived for the Mamdani type of model with the fuzzy relation of the fuzzy system being computed in the following form $R(x,y)=U \biggl( h\bigl(A_1(x), B_1(y) \bigr), \dots,h \bigl(A_N(x), B_N(y)\bigr)\biggr).$ Here the model is based on a collection of $$N$$-rules of the form “if $$A_i$$ then $$B_i$$” involving fuzzy sets $$A_i$$ and $$B_i$$ defined in the input and output space, respectively.

##### MSC:
 93C42 Fuzzy control/observation systems 93A30 Mathematical modelling of systems (MSC2010)
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