## 5 Simulation Experiment and Result Analysis  
### Transfer Function of Feedforward Controller  
According to the dynamic and static characteristics analysis and modeling of each channel of the heat exchanger in Section 2, the transfer function of the controlled object can be obtained as follows: \(G_{PC}(s)=\frac{1.9}{105.7s^{2}+25.7s+1}e^{-5.14s}\); The transfer function of the main disturbance, namely the hot fluid flow to the outlet temperature channel, is: \(G_{PD1}(s)=\frac{-1.35}{105.7s^{2}+25.7s+1}e^{-5.14s}\); The transfer function of the temperature channel at the inlet of the hot fluid is: \(G_{PD2}(s)=\frac{0.464}{37.66s+1}\); The transfer function of the temperature channel at the refrigerant inlet is: \(G_{PD3}(s)=\frac{0.536}{105.7s^{2}+25.7s+1}e^{-5.14s}\). In this paper, the transfer function of the secondary loop flow regulating valve is: \(G_{\rm{P2}}(s)=\frac{1}{s+1}\), PID controller parameter: \(k_{p}=0.3\), \(k_{i}=0.00001\), \(k_{d}=0.1\), so: \(G_{\rm{G2}}(s)=\left(0.3+\frac{0.00001}{s}+0.1s\right)\).  
Then, according to Equation (58), it can be obtained:  
\[G_{\rm{P2}}^{\prime}(s)=\frac{G_{\rm{G2}}(s)G_{\rm{P2}}(s)}{1+G_{\rm{G2}}(s)G _{\rm{P2}}(s)}=\frac{0.1s^{2}+0.3s+0.00001}{1.1s^{2}+1.3s+0.00001} \tag{61}\]  
The transfer function of the hot fluid flow to feedforward controller 1 of the outlet temperature channel is:  
\[G_{ff1}(s)=-\frac{G_{PD1}(s)}{G_{\rm{P2}}^{\prime}(s)G_{\rm{PC}}(s)}=\frac{0.7 81s^{2}+0.923s+0.0000071}{0.1s^{2}+0.3s+0.00001} \tag{62}\]  
Since the constant term in the numerator and denominator is too small, it is omitted here; the above formula can then be simplified as follows:  
\[G_{ff1}(s)=\frac{0.781s+0.923}{0.1s+0.3} \tag{63}\]  
Similarly, the transfer function of feedforward controller 2 from the inlet temperature of the hot fluid to the outlet temperature channel is:  
\[G_{ff2}(s)=-\frac{G_{PD2}(s)}{G_{\rm{P2}}^{\prime}(s)G_{\rm{PC}}(s)}\approx-0. 244\frac{165.68s^{2}+34.51s+1.3}{3.766s^{2}+11.398s+0.3} \tag{64}\]  
The transfer function of feedforward controller 3 from the refrigerant inlet temperature to the outlet temperature channel is:  
\[G_{ff3}(s)=-\frac{G_{PD3}(s)}{G_{\rm{P2}}^{\prime}(s)G_{\rm{PC}}(s)}=-\frac{0. 3102s+0.3666}{0.1s+0.3} \tag{65}\]  
According to the structural analysis of the feedforward control system, we can know that the control law of the feedforward controller is determined by the disturbance channel and the control channel in general. The heat exchanger and this kind of object contain aperiodic properties and over-damping properties, so they can be used to express the first- or second-order capacity lag, which according to the characteristics of the control system, can be connected with a pure lag to approximate. At this point:  
\[G_{PD}(s)=\frac{K_{2}}{T_{2}s+1}e^{-\tau_{2}s} \tag{66}\]  
\[G_{PC}(s)=\frac{K_{1}}{T_{1}s+1}e^{-\tau_{1}s} \tag{67}\]  
Then:  
\[G_{ff}(s)=-\frac{G_{PD}(s)}{G_{PC}(s)}=-K_{f}\frac{T_{1}s+1}{T_{2}s+1}e^{-\tau _{f}s} \tag{68}\]where, \(K_{f}=\frac{K_{2}}{K_{1}}\), is the static feedforward amplification coefficient; \(\tau_{f}=\tau_{2}-\tau_{1}\).  
When there is little difference between \(\tau_{1}\) and \(\tau_{2}\), Equation (68) can be simplified as follows:  
\[G_{ff}(s)=-K_{f}\frac{T_{1}s+1}{T_{2}s+1} \tag{69}\]  
At this point, according to Equation (69), we can simplify the transfer function of feedforward controller 1 in this paper, namely Formula (70), into:  
\[G_{ff1}(s)=\frac{0.781s+0.923}{0.1s+0.3}=-(-3.07)\frac{0.847s+1}{0.33s+1} \tag{70}\]  
where: \(K_{f}=-3.07\), \(T_{1}=0.847\), \(T_{2}=0.33\).  
Similarly, the transfer function of feedforward controller 3 is simplified as follows:  
\[G_{ff3}(s)=-\frac{0.3102s+0.3666}{0.1s+0.3}=-1.22\frac{0.847s+1}{0.33s+1} \tag{71}\]