Optimal fuzzy interpolation system with representation of error estimation (Q2704871)

The authors consider the following single-input and single-output fuzzy system defined in a closed ball of an inner product space with a reproducing kernel: NEWLINE\[NEWLINEy: U'\to V';\quad U',V'\subseteq S_{rk},\tag{1}NEWLINE\]NEWLINE where the inner product space \(S_{rk}\) has a reproducing kernel \(R(x,y)\) satisfying the condition NEWLINE\[NEWLINEu(x)= (u(y), R(x,y))\qquad\forall u\in S_{rk}.NEWLINE\]NEWLINE For any function \(u(x)\in U:= \{u|u(x)\leq\rho, \rho> 0\}\), taking the well-known expression \(e_U(x)\) as the best approximation of the fuzzy system (1) to the function \(u(x)\) and using a Zadeh type calculating method for compound propositions, an optimal fuzzy interpolation system with an error estimation formula is established.NEWLINENEWLINENEWLINETo convey the effectiveness of the results obtained, two illustrating examples are indicated.