## 3 Designing a New Hydraulic Platform and Control Method
### Control Strategy for the Left and Right Tension Control Devices
#### 3.2.1 The Feedforward Control Method for Servovalves  
First, the speed and direction of rolling can be determined according to the rolling schedule. The next step is to calculate the feedforward signals for servovalves. The actual flow depends on the displacement of the spool and the valve pressure drop. For different values of valve pressure drop, the flow may be calculated by the square root function for a sharp-edged orifice [18]:  
\[Q=Q_{\rm N}\frac{x_{\rm v}}{x_{\rm max}}\sqrt{\frac{\Delta P}{\Delta P_{\rm N}}} \tag{16}\]  
where \(\Delta P\) is the actual valve pressure drop; \(x_{\rm v}\) is the displacement of the spool; \(x_{\rm max}\) is the maximum displacement of the spool; \(\Delta P_{\rm N}\) is the rated valve pressure drop; \(Q_{\rm N}\) is the rated flow. With regard to the entry side servovalve, \(\Delta P=P_{1}\), where \(P_{1}\) is the actual pressure of valve port A. With regard to the exit side servovalve, \(\Delta P=P_{\rm s}-P_{1}\), where \(P_{\rm s}\) is the working supply pressure of the valve. Note that \(\Delta P\) in Equation (16) is the total pressure drop including P\(\rightarrow\)A and B\(\rightarrow\)T (or P\(\rightarrow\)B and A\(\rightarrow\)T), 50% each. In the newly designed system,  
Figure 5: The tension control strategy diagram. the servovalve is used as a two-edged valve, so the valve pressure drop is simply P\(\rightarrow\)A or B\(\rightarrow\)T. Therefore, the relation between valve flow and the displacement of the spool is  
\[Q=Q_{\text{N}}\frac{x_{\text{v}}}{x_{\text{max}}}\sqrt{\frac{\Delta P}{0.5 \Delta P_{\text{N}}}} \tag{17}\]  
In addition, the displacement of the spool is proportional to the electric command signal. We obtain  
\[\frac{S_{\text{v}}}{S_{\text{vmax}}}=\frac{Q}{Q_{\text{N}}}\sqrt{\frac{0.5 \Delta P_{\text{N}}}{\Delta P}} \tag{18}\]  
where \(S_{\text{v}}\) is the electric command signal; \(S_{\text{vmax}}\) is the maximum electric command signal. In Equation (18), the parameters such as \(\Delta P_{\text{N}}\), \(Q_{\text{N}}\), and \(S_{\text{vmax}}\) can be easily determined from the valve manual. The calculation of \(\Delta P\) requires \(P_{\text{s}}\) and \(P_{1}\). \(P_{\text{s}}\) and \(P_{1}\) can be obtained from pressure sensors. Therefore, \(S_{\text{v}}\) can be calculated when we obtain \(Q\), which is related to rolling speed, forward slip, and backward slip. The forward slip value in the cold-rolling process is usually less than 5%, so the electric command signal for the exit side servovalve is set as  
\[S_{\text{vh}}=S_{\text{vmax}}\frac{1.05 vA}{Q_{\text{N}}}\sqrt{\frac{0.5\Delta P _{\text{N}}}{\Delta P}} \tag{19}\]  
where \(A\) is the ring area of the tension cylinder. In this way, the valve flow certainly exceeds the required input flow. The excess flow is absorbed by the exit side proportional pressure relief valve, and the relief flow satisfies  
\[Q_{\text{ph}}<S_{\text{vmax}}\frac{0.05 vA}{Q_{\text{N}}}\sqrt{\frac{0.5\Delta P _{\text{N}}}{\Delta P}} \tag{20}\]  
In this way, the adjusted relief flow range is much smaller than that in [7] under the same working conditions. The backward slip value in the cold-rolling process is usually less than 45%, so the electric command signal for the entry side servovalve is set as  
\[S_{\text{vH}}=S_{\text{vmax}}\frac{0.55 vA}{Q_{\text{N}}}\sqrt{\frac{0.5\Delta P _{\text{N}}}{\Delta P}} \tag{21}\]  
In this way, the valve flow is certainly less than the actual output flow from the ring side of the entry side tension cylinder. The excess flow is absorbed by the corresponding proportional pressure relief valve to keep the tension stable, and the relief flow satisfies  
\[Q_{\text{pH}}<S_{\text{vmax}}\frac{0.45 vA}{Q_{\text{N}}}\sqrt{\frac{0.5\Delta P _{\text{N}}}{\Delta P}} \tag{22}\]  
In this way, the adjusted relief flow range is much smaller than that in [7] under the same working conditions. All in all, the adjusted flow ranges of the proportional pressure relief valves are markedly reduced by the feedforward flow effect of the servovalves.