## 4 Simulation Analysis
### Simulation
#### 4.2.3 Control Effect with the Proposed Method  
Using the proposed method, proportional relief valves with a nominal size of 6 mm and rated flow of 40 L/min are selected. Assuming that the forward slip \(S_{\text{h}}=0.01\) and the backward slip \(S_{\text{H}}=0.1\), the tension control results from using the proposed control strategy are shown in Figure 14, where \(Q_{\text{sh}}\) is the flow through the exit servovalve; \(Q_{\text{ch}}\) is the flow into the ring side of the exit cylinder; \(Q_{\text{ph}}\) is the flow through the exit proportional relief  
Figure 12: The tension control effect with method 2 under accurate prediction of \(S_{\text{h}}\) and \(S_{\text{H}}\). (**a**) The control effect of forward tension; (**b**) the control effect of backward tension.  
Figure 13: The tension control effect with method 2 under inaccurate prediction of \(S_{\text{h}}\) and \(S_{\text{H}}\). (**a**) The control effect of forward tension; (**b**) the control effect of backward tension.  
value; \(Q_{\rm SH}\) is the flow through the entry servovalve; \(Q_{\rm CH}\) is the flow out of the ring side of the entry cylinder; \(Q_{\rm pH}\) is the flow through the entry proportional relief valve. From the working principle of the hydraulic system, \(Q_{\rm ph}=Q_{\rm sh}-Q_{\rm ch}\), \(Q_{\rm pH}=Q_{\rm CH}-Q_{\rm SH}\). The largest forward tension error is about \(-0.25\) kN, and the largest backward tension error is about \(-0.27\) kN.  
Assuming that forward slip \(S_{\rm h}=0.04\) and backward slip \(S_{\rm H}=0.4\), the tension control results obtained using the proposed method are shown in Figure 15. Even though the values for forward and backward slip have changed greatly, the tension control effect maintains high accuracy. Comparing Figure 14c with Figure 15c, with the increase in the forward slip, \(Q_{\rm ch}\) increases and \(Q_{\rm sh}\) remains unchanged, resulting in \(Q_{\rm ph}\) decreasing. Comparing Figure 14d with Figure 15d, it can be seen that \(Q_{\rm CH}\) decreases with the increase in backward slip and \(Q_{\rm SH}\) remains unchanged, resulting in \(Q_{\rm pH}\) decreasing. Therefore, with a wide range of forward and backward slip, the flow through the two proportional relief valves also changes but is always within the adjustable flow range of the proportional relief valves, and the valves with a small nominal size have a high response speed. Therefore, accuracy in tension control is guaranteed.  
Figure 14: The simulation results for tension control with the proposed method when \(S_{\rm h}=0.01\) and \(S_{\rm H}=0.1\). (**a**) The control effect of forward tension; (**b**) the control effect of backward tension; (**c**) the flow curves on the exit side; (**d**) the flow curves on the entry side.  
Compared with method 1, in the proposed method, the proportional relief valves require a much smaller flow adjustment range and a smaller nominal size, which allows them to have a faster response speed. In addition, the tension control is not affected by the fluid inertia effect present in method 1, so there is better tension control when using the proposed method.  
Compared with method 2, the proposed method uses proportional pressure valves for tension control and is not dependent on the accurate prediction of forward and backward slip. Even when the forward and backward slip vary over a wide range, good tension control accuracy is maintained, and the method exhibits stronger robustness.