Nonlinear systems modelling based on self-organizing fuzzy neural network with hierarchical pruning scheme

Abstract In this paper, a self-organizing fuzzy neural network with hierarchical pruning scheme (SOFNN-HPS) is proposed for nonlinear systems modelling in industrial processes. In SOFNN-HPS, to strike the optimal balance between system accuracy and network complexity, an online self-organizing scheme for identifying the structure and parameters of the network simultaneously is developed. First, to enhance the characterization ability of the fuzzy rules for nonlinear systems, the asymmetric Gaussian functions that can partition the input space more flexibly are introduced as membership functions. Second, a hierarchical pruning scheme, which is designed by rule density and rule significance, is used to delete the redundant fuzzy rules while using the geometric growing criteria to generate fuzzy rules automatically, which can avoid the requirement of pre-setting the pruning threshold and prevent the mistaken deletion of significant rules. Third, an adaptive allocation strategy is adopted to set the antecedent parameters of the fuzzy rules in the learning process, which can not only adjust the region of generalized ellipsoidal basis functions for better local approximation, but also balance the accuracy of the system and the interpretability of the rule base obtained. Finally, to speed up the convergence of the estimation error, a modified recursive least square algorithm is used to update the consequent parameters of the resulting fuzzy rules online. In addition, the convergence proofs of the estimation error and the network linear parameters of SOFNN-HPS are given, and they are helpful in successfully applying the SOFNN-HPS in practical engineering. To verify the effectiveness of SOFNN-HPS, two benchmark test problems and a key water quality parameter prediction experiment in the wastewater treatment process are examined. The simulation results demonstrate that the proposed SOFNN-HPS algorithm can obtain a self-organizing fuzzy neural network with compact structure and powerful generalization performance. The source codes of SOFNN-HPS and other competitors can be downloaded from https://github.com/hyitzhb/SOFNN-HPS .