Cone Fuzzy System And Its Application In Nonlinear System Modeling

Fuzzy systems can effectively model complex nonlinear systems with uncertainties,fuzzy system can make full use of the experience knowledge of domain experts and make it easy to understand.Theoretically,the modeling precision of fuzzy model is related to the number of fuzzy rules.The more fuzzy rules,the higher the precision of the fuzzy model.However,in practice,when modeling fuzzy models for some nonlinear systems,the improvement of fuzzy model precision is limited by simply increasing the number of fuzzy rules,which,on the contrary,greatly increases the calculation amount of the entire modeling process,leading to redundancy of the fuzzy rules and overfitting of the models,so the balance and compromise between complexity and precision has become a hot issue in the research of fuzzy system identification.Therefore,how to design a simple and effective fuzzy system,improve its approximation performance and reduce its computational complexity become the main starting points of this paper.The main work of this paper includes the following aspects:(1)The fuzzy set of one-dimensional space is extended to two-dimensional space.Cone fuzzy set is proposed and introduced into the construction of fuzzy system,the Mamdani and T-S type of cone fuzzy systems based on fuzzy grid partition are obtained.The derivation process and the general design steps of cone fuzzy system are discussed in detail.The calculation of the fuzzy rule antecedents of the traditional fuzzy system needs to be realized by t-norm.The calculation of t-norm is not needed in the derivation of cone fuzzy system,and the structure and operation of the fuzzy system are simplified.The universal approximation property and approximation precision of cone fuzzy system are discussed in detail.It is proved that RPFS and TPFS has first and second order approximation precision,respectively.Simulation results show that cone fuzzy system can achieve better precision with fewer rules.(2)In order to solve the problem that the controlled plant is difficult to model in fuzzy control,the modeling methods based on Mamdani and T-S type RPFS and TPFS are proposed.The HX equation obtained by the traditional fuzzy inference modeling method is a piecewise constant coefficient nonlinear model,whereas,the models obtained by the modeling methods based on Mamdani and T-S type TPFS are piecewise constant linear model.By using the modeling method based on Mamdani and T-S type RPFS and TPFS,the second-order time-invariant free motion system is modeled,and the input-output and state-space models of the system are obtained.The simulation comparison results show that the system models based on cone fuzzy system have higher precision.(3)Aiming at the problem that it is difficult to determine the number of fuzzy rules and the structure of the fuzzy model in fuzzy system,an improved initial method of K-means algorithm which can automatically obtain the initial clustering centers and reasonably determine the clustering number(rule number)is proposed.The improved K-means algorithm is used to divide the input space in the identification process of cone fuzzy system,and the clustering center is selected as the peak point of the cone membership function.It overcomes the defect that the number of clustering in the traditional clustering algorithm must be determined in advance when used for fuzzy system modeling.Mackey-Glass chaotic time series,Box-Jenkins gas furnace data and automobile MPG data which are three benchmark problems of the system identification are simulated.The simulation results show that the cone fuzzy system identification method based on the improved K-means algorithm is simple and fast,and the approximation accuracy is high.(4)Aiming at the problem that the number of rules of fuzzy system increases exponentially with the increase of dimensionality,a hierarchical cone fuzzy system with linear increase of the number of rules with the increase of dimensionality is proposed.Two basic types of hierarchical cone fuzzy system are derived and their identification methods are given:the system structure is determined through improved K-means clustering,and then the parameters of the antecedent and consequence are optimized by genetic algorithm.In the identification and simulation of Mackey-Glass chaotic time series,Box-Jenkins gas furnace data and a nonlinear system,the advantages of hierarchical cone fuzzy system in accuracy,computational efficiency and robustness to noise are verified.