## 6 Conclusions  
1. Tension modeling was conducted based on [PERSON]'s law and rolling theories. The model reflects tension control properties such as variable rigidity, tension coupling, and positional disturbances. This demonstrated that high-precision tension control is very difficult and the existing hydraulic control methods are not suitable.
2. In consideration of the mill properties, the piston sides of the left and right tension cylinders are connected together, and the back pressure is controlled by a constant back pressure control device. In this way, the relief flows are greatly reduced, which is beneficial for reducing fluctuations in the back pressure. Furthermore, because of the existence of back pressure, the use of proportional pressure relief valves enables the avoidance of dead and nonlinear zones while working, ultimately resulting in the achievement of high-precision tension control.
3. The feedforward control for servovalves can ensure the effective tracking of rolling speed regardless of forward and backward slip. The surplus flow is absorbed by the  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Dimensions/mm** & **Pass** & **Left Tension/kN** & **Right Tension/kN** & **Rolling Force** & **Rolling Speed** & **Exit Thickness** & **Reduction Rate** \\  & & & & **/kN** & **/m**s\({}^{-1}\) & **/mm** & **/\%** \\ \hline \multirow{4}{*}{\(3.6\times 200\times 650\)} & 1 & 48 & 45 & 1460 & 0.15 & 2.768 & 23.11 \\  & 2 & 42 & 44 & 1623 & 0.15 & 2.067 & 25.33 \\ \cline{1-1}  & 3 & 40 & 38 & 1537 & 0.15 & 1.556 & 24.72 \\ \cline{1-1}  & 4 & 33 & 36 & 1590 & 0.20 & 1.197 & 23.06 \\ \cline{1-1}  & 5 & 30 & 28 & 1256 & 0.4 & 1.017 & 15.05 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cold-rolling schedule for high-strength steel.  
Figure 18: The actual tension control effect of the tension cylinders. (**a**) The control effect of left tension; (**b**) the control effect of right tension.  
corresponding proportional pressure valve in the closed-loop tension control. This method is simple, reliable, and has strong robustness.
4. Based on a CRPM with 50 kN hydraulic tension, cold-rolling experiments were carried out on a typical metallic material. The dynamic tension control accuracy was controlled within \(\pm\)0.2 kN, when the maximum rolling speed reached 0.4 m/s and the maximum single pass reduction reached 37%. Compared with the existing techniques, the proposed method has obvious advantages.
5. The hydraulic platform and the control method developed in this paper can serve as a helpful reference for the designs of hydraulic servo systems such as electro-hydraulic load simulators with positional disturbances.
6. The prerequisite for the effective use of the method proposed in this article is that the set rolling speed is known. Although the exit and entry speeds of the specimen are directly affected by forward and backward slip, the two factors are bounded. Therefore, the redundant flow generated by the feedforward of the servovalves is also bounded and can be adequately absorbed by the proportional relief valve, and an ideal tension control effect is achieved. For more complex working conditions, a strategy combining the optimization of hydraulic schematic design, the proposal of excellent control strategies and advanced algorithms, and improving the performance of the system components may be effective in further significantly suppressing the disturbances from extraneous forces.  
Conceptualization, G.W. and J.D.; methodology, J.S.; software, G.W. and Y.G.; validation, G.W. and J.S.; formal analysis, J.D.; investigation, G.W.; writing--original draft preparation, G.W.; project administration, G.W.; funding acquisition, J.S. All authors have read and agreed to the published version of the manuscript.  
Project (U21A20117, 52074085) supported by the National Natural Science Foundation of China.  
Not applicable.  
The data presented in this study are available in article.  
The authors declare no conflict of interest.