基于调控参数建模的混合模糊系统的逼近分析

Approximation Analysis of the Hybrid Fuzzy System Based on the Modeling of a Control Parameter

模糊系统是通过规则来描述人类语言信息,它是处理不确定信息的一种重要工具.常见的模糊系统为Mamdani模糊系统和Takagi-Sugeno(T-S)模糊系统.本文首先通过引入调控参数统一两类系统并建立广义的混合模糊系统.其次,利用多元函数带皮亚诺余项的泰勒展式证明混合模糊系统在无穷范数下对连续可微函数具有逼近性,进而借助拉格朗日型余项的Hessian矩阵证得该系统具有二阶逼近精度.最后,通过选取样本点分析说明基于调控参数建模的混合模糊系统的逼近效果优于单独的Mamdani或T-S模糊系统.

A fuzzy system describes human language information by fuzzy rules,it is an important tool to deal with the uncertain information. Mamdani fuzzy system and Takagi-Sugeno （T-S） fuzzy system are two kinds of common systems. Firstly, the two types of systems are unified by introducing a control parameter so as to establish a generalized hybrid fuzzy system in this paper. Secondly, the hybrid fuzzy system can approximate to a continuous differentiable function with respect to infinite norm through Taylor formula with Peano remainder term of multivariate function, and certify that this system also has the second order approximation accuracy by means of Hessian matrix in Lagrange remainder term. Finally, the results show that the approximation effect of the hybrid fuzzy system based on the modeling of control parameters is better than the separate Mamdani or T-S fuzzy system by choosing sample points analysis.