## 1 Introduction  
With the rapid development of the economy and the advancement of urban industrialization, the discharge of industrial wastewater and domestic wastewater is becoming increasingly severe. The contradiction between the supply and demand of water has been severely exacerbated by the shortage of available water caused by water pollution and the increasing water demand and has become a common problem facing the world. Wastewater treatment plants (WWTPs) are industrial systems widely used in petrochemical and residential life fields. They can reduce water pollution, promote wastewater recycling, greatly reduce industrial water demand, and avoid environmental pollution. However, WWTPs are very complex time-varying dynamic systems and their internal reaction processes are influenced by the influent flow rate, pollutant load, and unknown inflow components, exhibiting characteristics such as nonlinearity, strong coupling, and strong disturbance [1]. Therefore, research on the control of wastewater treatment plants is of great significance for improving operational performance and improving effluent quality. To meet the treatment efficiency standards of WWTPs, there have been many ground-breaking research developments in recent years. Traditional control strategies such as on-off switch control, PID control, PI feedforward control, and their combination forms have been widely employed to regulate the concentrations of nitrate nitrogen and dissolved oxygen (DO) [2; 3]. However, due to the inability of the above controller parameters to adaptively change online, the control performance of the WWTP system will be affected when it is subjected to external disturbances such as influent flow. In response to this problem, a fuzzy control algorithm [4] is proposed to control the DO concentration. Compared with traditional switch control laws, the control deviation is improved by 60% and the energy consumption is reduced by more than 40%. Since then, more and more fuzzy control strategies [5; 6] have been applied to WWTPs, which have greatly improved control effectiveness and reduced energy consumption. Nevertheless, these tactics are challenging to implement in real life and mostly rely on the experience of experts. To improve the control accuracy, some model-based control algorithms have been applied to WWTPs, such as model predictive control (MPC) [7], adaptive control [8], and adaptive sliding-mode control [9]. Although it is challenging to create a precise mathematical model of a WWTP due to its strong nonlinear and non-stationary dynamic properties, model-based control strategies frequently use a linearization method near the operating point, using a reduced-order model to approximate the actual model. This will cause some dynamic characteristics of the high-order system to be neglected, leading to unmodeled dynamic problems. Therefore, it is challenging to obtain satisfactory results using model-based control systems in industrial practice. To address the above issues, neural networks have been applied to the identification and control of unknown dynamic systems in WWTPs. [PERSON] et al. [10] proposed a multivariable PI control algorithm based on a feedforward neural network model and an adaptive fuzzy neural network control algorithm [11]. [PERSON] et al. [12; 13; 14; 15] proposed nonlinear model predictive control strategies for WWTPs which have shown good performance in robustness and energy conservation. However, control strategies based on neural networks require a large amount of historical data for training and, for most WWTPs, due to limitations in measurement equipment, it is not possible to provide a large amount of data that can be used to train models. In a broad sense, a neural network is the model of a system. When the system changes, the network needs to be retrained and the determination of network nodes and hidden layers requires prior knowledge of the controlled object. Therefore, control strategies based on neural networks still cannot avoid unmodeled dynamic problems.