Completeness And Separability Of Fuzzy Valued Function Space And Approximation Capability Of Consequent Direct Link Type Hybrid Hierarchy Fuzzy Systems

When a class of fuzzy value functions constitute a metric space, the completeness and sep-arability is an important problem that must be considered to discuss the approximation of fuzzy systems.A fuzzy system is a map which is built through measuring data or digital sensor to get the input and output. which imitates the human brain of using fuzzy information to finish fuzzy inference. Although it doesn’t depend on precise mathematical model, it has the function of logical reasoning, numerical calculation, and strong approximation capability of nonlinear ability.The content of this paper consists of four primary parts:In the first part:As an introduction, we introduced the background, the current research situation of fuzzy network systems and the preliminaries.In the second part:Through introducing two induced operators, Firstly, we study the com-pleteness and separability of a class of integrable fuzzy value function space, and prove that the class of tK-integrable fuzzy value functions is a metric space. And then, the integral transforma-tion theorems and tK-integrable Borel-Cantelli Lemma are applied to study the completeness of the space, furthermore, its separability is discussed by means of the approximation of fuzzy valued simple functions and fuzzy valued Bernstein polynomials. The results show that the space of the tK-integrable fuzzy valued functions constitutes a complete separable metric space in the sense of the tK-integral norm.In the third part:To avoid the number of rules of general hybrid fuzzy system grows expo-nentially with the input variables, The methods of the consequent direct link type hierarchy and its inference rules are introduced. Through implementting hierarchy for some input variables of the generalized hybrid fuzzy system, the input and output expressions of the generalized hybrid fuzzy systems after hierarchy and the calculation formula of the total number of fuzzy rules are obtained.In the final part:using the K-integral norm (a metric) and a piecewise linear function, we can prove that the generalized hybrid fuzzy system after hierarchy has the approximation to a class of integrable functions.then, the approximation process of the consequent direct link type hierarchy generalized hybrid fuzzy systems to the integrable functions are given by an example. The results show that the hierarchical method can not only make the total number of fuzzy rules of the original system greatly reduced, but also make the system after hierarchy has still the approximation.