Soft-sensing of effluent total phosphorus using adaptive recurrent fuzzy neural network with Gustafson-Kessel clustering.

• A soft-sensing method based on GK-ARFNN is proposed. • The GK clustering algorithm is used to develop the initial fuzzy rule base. • The recursive link is introduced into the FNN to improve dynamic mapping ability. • A hierarchical adaptive second-order optimization algorithm is developed. To address the issue of soft-sensing of effluent total phosphorus in wastewater treatment processes (WWTPs), a soft-sensing system based on an adaptive recursive fuzzy neural network with Gustafson-Kessel (GK) clustering and hierarchical adaptive second-order optimization algorithm (HAS) is proposed in this paper. In GK-ARFNN, first, the GK clustering algorithm was utilized to cluster the input–output dataset. Thus, the establishment of the initial fuzzy rule base and the determination of the parameter value of the fuzzy set membership function was realized. Then, the recursive layer was added into FNN to improve the dynamic mapping ability of the system. Finally, the HAS algorithm was developed based on the improved Levenberg-Marquardt (LM) optimization algorithm, and all the free parameters of the GK-ARFNN were adjusted online using HAS to improve the generalization capability and prediction accuracy of the soft-sensing system. In addition, the convergence of the proposed GK-ARFNN algorithm was also analyzed in this paper, which can ensure the effectiveness of the solutions to modelling issues for practical industrial processes. The simulation results demonstrate that the GK-ARFNN-based soft-sensing system introduced in this paper achieved satisfactory accuracy in the prediction of effluent total phosphorus in WWTPs. The source codes of GK-ARFNN and some competitors can be downloaded from https://github.com/hyitzhb/GK-ARFNN. [ABSTRACT FROM AUTHOR]