Rule number and approximation of the hybrid fuzzy system based on binary tree hierarchy

摘要

To effectively avoid internal rule explosion of a fuzzy system or computer memory overflow caused by increased input variables, a hybrid fuzzy system is established by unifying the Takagi–Sugeno and the Mamdani fuzzy systems based on a binary tree hierarchical method. This method can greatly reduce the total number of rules within the system. Firstly, a calculation formula of the total number of rules for the hybrid fuzzy system is given, by comparing with other layered systems, the total number of rules based on the binary tree hierarchy has the largest decline. Secondly, a new K-integral norm is redefined by introducing a K-quasi-subtraction operator. Using the piecewise linear function the approximation capability of the hybrid fuzzy system after hierarchy to a kind of integrable functions is studied. Finally, the binary tree hierarchical structure expressions of the hybrid fuzzy system are given through two simulation examples.