## 3 Design of IPFC Controller  
### Predictive Functional Control (PFC) Algorithm  
The predictive functional control algorithm belongs to the third generation of predictive control, which has three basic characteristics of general predictive control: predictive model, rolling optimization, and feedback correction [27]. The biggest feature of PFC is the structured control input; the control input at each moment is represented by the linear combination of the pre-selected basis function family, and then the linear weighting coefficient is calculated through online optimization, and then the current control input is calculated [28]. The structure diagram of the predictive functional controller is shown in Figure 2.  
In Figure 2, \(G\) is the controlled object, \(G_{m}\) is the prediction model, \(c\) is the set output value, \(u(k)\) is the amount of control, \(d(k)\) is the interference, \(y(k+1)\) is the output of the actual system, \(y_{m}(k+1)\) is the output of the prediction model, and \(e(k+1)\) is the prediction error. In predictive functional control, the control input is a linear combination of a set of basis functions \(f_{j}(j=1,2,\cdots,N)\) related to processing characteristics and tracking setpoints, namely:  
\[u(k+i)=\sum_{j=1}^{N}\mu_{j}f_{j}(i)\quad i=0,1,\cdots,H-1 \tag{18}\]  
where, \(N\) is the number of basis functions; \(\mu_{j}\) is the linear combination coefficient; \(H\) is the prediction optimization time-domain length. Generally, the choice of basis function mainly depends on the set value and the nature of the controlled object, such as the step function, slope function, or parabola function. If the set value is constant or has a small change rate between controlled regions, select a basis function, that is, the step function. When the change rate of the set value is greater than a certain value in the controlled area, two basic functions, namely, the step function and the slope function, are selected. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **50\%Ethylene Glycol (CH\({}_{2}\)OH)\({}_{2}\)** & **Toluene (C\({}_{7}\)H\({}_{8}\))** \\ \hline Input temperature & \(-15\,^{\circ}\)C & 60 \(\,^{\circ}\)C \\ Output temperature & \(-10\,^{\circ}\)C & 20 \(\,^{\circ}\)C \\  & 1085.61 kg/m\({}^{3}\)\(\,(-15\,^{\circ}\)C) & 829.1482 kg/m\({}^{3}\)(60 \(\,^{\circ}\)C) \\ Density & 1084.62 kg/m\({}^{3}\)\((-10\,^{\circ}\)C) & 866.8158 kg/m\({}^{3}\)(20 \(\,^{\circ}\)C) \\ Specific heat capacity & 3145 J /(kg\(\,{}^{\circ}\)C)\((-15\,^{\circ}\)C) & 1818.4 J /(kg\(\,{}^{\circ}\)C)(60 \(\,^{\circ}\)C) \\  & 3165 J /(kg\(\,{}^{\circ}\)C)\((-10\,^{\circ}\)C) & 1685.5 J /(kg\(\,{}^{\circ}\)C)(20 \(\,^{\circ}\)C) \\ Working pressure & 0.3 MPa & 0.09 MPa \\ Volume flow & \(3.61\times 10^{-3}\) m\({}^{3}\)/s & \(7.5\times 10^{-4}\) m\({}^{3}\)/s \\ Heat transfer area & 20 m\({}^{2}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Operating conditions of the heat exchanger. Figure 2: PFC structure diagram. The output \(y_{m}(k)\) of the prediction model is composed of the model-free response \(y_{l}(k)\) and the forced response \(y_{f}(k)\), then the model forced response at the time \((k+i)\) is:  
\[y_{f}(k+i)=\sum_{j=1}^{N}\mu_{j}g_{j}(i) \tag{19}\]  
where, \(g_{j}(i)\) is the model output under the action of the basis function \(f_{j}(i)\). The model output at the time \((k+i)\) is:  
\[y_{m}(k+i)=y_{l}(k+i)+y_{f}(k+i) \tag{20}\]  
The model of the prediction functional takes the state space equation, and its discrete state space expression is:  
\[\left\{ \begin{array}{l}X_{m}(k)=A_{m}X_{m}(k)+B_{m}u(k)\\ y_{m}(k)=C_{m}X_{m}(k)\end{array} \right. \tag{21}\]  
where, \(X_{m}\in R^{n\times 1}\) is the state vector of the prediction model, \(y_{m}\in R^{1\times 1}\) is the prediction output of the prediction model, \(u\in R^{1\times 1}\) is the control input of the prediction model, \(A_{m}\in R^{n\times n}\), \(B_{m}\in R^{n\times 1}\), \(C_{m}\in R^{1\times n}\) are the system matrices of the prediction model.