Center-of-gravity fuzzy systems based on normal fuzzy implications

In this paper, the construction and approximation problem of a single input and single output (SISO) fuzzy system with normal implication and center-of-gravity defuzzifier is discussed. First, the method of a non-singleton fuzzifier for the input variable and the concept of an adaptive universe for the output fuzzy set are proposed. Then, by using this method and this concept of an adaptive universe, SISO fuzzy systems based on center-of-gravity defuzzifier and normal implications such as the Kleene–Dienes implication or the Lukasiewicz implication are constructed. The constructed fuzzy systems have the general form S ¯ ( x ) = A i ∗ ( x ) f ( x i ) + A i + 1 ∗ ( x ) f ( x i + 1 ) , with A i ∗ ( x ) + A i + 1 ∗ ( x ) = 1 , and, furthermore, they are universal approximators. The sufficient conditions for the proposed fuzzy systems to be universal approximators are also obtained. To illustrate the universal property, an example is also given.