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为解决多变量模糊系统中的“维数灾”问题,该文提出了一般二叉树型分层模糊系统,简化了多输入复杂模糊系统的分析,并研究了它的通用逼近性。利用分片线性函数理论和中值定理,证明了该二叉树型分层模糊系统具有通用逼近性,并得到了该通用逼近性的充分条件。相对于一般模糊系统,分层模糊系统大大减少了系统的规则数。仿真结果表明:在同样的逼近精度下,二叉树型分层模糊系统的规则数比一般模糊系统减少了10倍左右,从而大大简化了系统的设计。
In order to solve the problem of “dimensionality disaster” in multivariable fuzzy systems, this paper presents a generalized binary tree hierarchical fuzzy system, which simplifies the analysis of multi-input complex fuzzy systems and studies its general approximation. By using the theory of piecewise linear function and the theorem of median value, it is proved that the binary tree hierarchical fuzzy system has general approximation and the sufficient conditions for the universal approximation are obtained. Compared with the general fuzzy system, hierarchical fuzzy system greatly reduces the number of rules of the system. Simulation results show that under the same approximation accuracy, the number of rules of binary tree hierarchical fuzzy system is reduced by about 10 times than that of general fuzzy system, which greatly simplifies the system design.