## 1 Introduction  
At present, one of the main purposes of designing control strategies in the field of industrial processes is to improve energy efficiency. More than 80% of the world's energy use today involves heat transfer processes [1]. The heat exchanger is a common piece of heat transfer equipment in the chemical production process. It is also called a heat exchanger, which uses the heat exchange between cold and hot fluids to ensure that the temperature of the liquid after heating meets the needs of the process so that production can be carried out smoothly [2]. Choosing a reasonable control scheme can not only control the outlet temperature of the heat exchanger but also improve heat transfer efficiency and provide economic benefits [3; 4; 5; 6; 7]. The performance of the control system has a great influence on the economic efficiency of the process, and the control performance can usually be improved by applying advanced control algorithms [8; 9; 10]. The most commonly used control strategy in the field of industrial process control is proportional-integral-derivative (PID) control. Since 1936, experts and scholars have proposed many PID tuning methods [11]. But how to determine the appropriate PID parameters is still a question worthy of discussion [12]. In industrial control, the appropriate PID adjustment method is generally selected according to the model type, such as the first-order plus dead time (FOPDT) model [13; 14], integrator plus dead time (IPDT) model [15; 16], etc. In industrial production, the PID control scheme is generally used to control the system. However, for the process dynamic model with a time-varying and large lag, the traditional PID control can't eliminate the obvious overshoot in the control, has a long adjustment time, and can't solve the problem of time-varying control model parameters. Since 1970, model predictive control (MPC) has been proposed and developed, and it has a good control effect on complex optimization problems [17]. However, when dealing with nonlinear, uncertain, and time-varying control systems, MPC control alone is still relatively laborious. Furthermore, MPC is not as easy to implement as PID control. Therefore, it is very necessary to find a way to effectively combine MPC and PID control. [PERSON] et al. combined neural network predictive control (NNPC) and fuzzy control in the outlet temperature control of heat exchangers. The simulation results demonstrate the effectiveness and superiority of the complex control structure, which combines NNPC and an auxiliary fuzzy controller [18]. [PERSON] et al. applied balancing-based adaptive control (BBAC) to a simple heat distribution system and showed that the method ensured better anti-jamming capability compared with the traditional PI controller [19]. [PERSON] et al. proposed a predictive control strategy based on a simplified model-based heat exchange network approach [20]. [PERSON] et al. proposed an advanced technique based on artificial neural network model predictive control (NNMPC) and compared it with two PIDs and a linear MPC controller in simulation [21]. [PERSON] et al. took the time-varying parameters of the heat exchanger as parameter uncertainty and adopted robust MPC to deal with the process with uncertainty [22]. In Table 1, the main findings of the studies are provided. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**References** & **Type of Heat** & **Control Scheme** & **Result** \\ \hline \multirow{6}{*}{[PERSON] et al. [18]} & \multirow{6}{*}{Tubular} & The NN predictive control & The advantage of this approach is that it is not  
The predictive functional control (PFC) algorithm, as the third generation of the MPC control algorithm, differs from the traditional MPC algorithm in that it is a structured control input [23]. It introduces the concept of basis function, and its output response can be obtained through offline calculation in advance, thereby reducing the workload of the computer that needs to be solved online and improving work efficiency.