复模糊值函数的积分及其性质

Differential Coefficient and Properties of Complex Fuzzy-valued Function

复模糊值函数理论在模糊控制中是广泛存在的,讨论复模糊值函数积分的性质有重要的理论和实际意义。本文首先介绍了模糊数的概念、运算规则及复模糊值函数的表达式f=(x)=((x),(x)),在新的序关系的意义下给出复模糊值函数f=(x)=(1(x),2(x))Riemann积分的定义。在此基础上给出了复模糊值函数的r-截集的概念,利用r-截集把复模糊值函数转化为区间值函数,用扩张原理给出了复模糊值函数积分表达式,并讨论了复模糊值函数积分的性质,得出了复模糊值函数积分具有区间可加性、不等式性、对实系数和复系数具有线性性质等结论。

The value function of complex fuzzy is widespread in fuzzy control; discussions of complex fuzzy value function integral properties have important theoretical and practical significance. In this paper, firstly, the concept and the operation rules of fuzzynumber and the expression of complex fuzzy value function f〖DD（-＊3〗~^（x）=（f^1（x）,f^2（x）） are introduced, and the Riemann integraldefinition of the complex fuzzy value function f~^（x）=（f^1（x）,~2（x））is given under the new order relation of meaning. Basedon the definition of r-cut set is given, the value function of complex fuzzy is turned to interval valued function using r-cut set, then complex fuzzy value function integral expression is given with extension principle. In addition, the properties of integral fuzzy-valued function are discussed, and interval additivity, inequality sex and linearity on the real coefficient and complex coefficient of the com- plex fuzzy value function integral are obtained.