# Physics-Guided Neural Network-Based Feedforward Control for Seamless Pipe Manufacturing Process
## 2 Materials and Methods
### Mathematical Models
#### 2.1.3 Linearized Reduced-Order Actuator Model with External Force
Since, during force control operation, the cylinder piston moves continuously in one direction (\(v>0\)), a friction force model that does not consider the static component can be adopted, bypassing the issue with the sign\((\cdot)\) function:  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]4Combining Equations (21), (23), and (25), and the fact that \(F_{p}=p_{L}\bar{A}-F_{int}\), the following system of equations is obtained:  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]5  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]6  
For simpler notation, the previous equations can be written as follows:  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]7  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]8  
where the introduced coefficients have the following values:  
\[ \begin{split} C_{1}&=\frac{4\beta}{V_{t}}C_{qp}\,\ \ \ C_{2}=\frac{4\beta}{V_{t}}C_{q}\,\ \ \ C_{3}=\frac{4\beta}{V_{t}}\bar{A}\frac{\tau_{ext}}{m}\,\\ C_{4}&=\frac{1}{\tau_{ext}}+\frac{\sigma}{m}\,\ \ \ C_{5}= \frac{\bar{A}}{\tau_{ext}}\,\ \ \ C_{6}=\frac{F_{c0}}{\tau_{ext}},\\ C_{7}&=\frac{4\beta}{V_{t}}(\hat{Q}_{L}-C_{q}\hat {u}+C_{qp}\hat{p}_{L}).\end{split}  \tag{31}\]  
From Equation (29), \(F_{ext}\) can be expressed and differentiated to obtain  
\[\hat{F}_{ext}=\frac{1}{C_{3}}(C_{2}\hat{u}-\hat{p}_{L}-C_{1}\hat{p}_{L}), \tag{32}\]  
after which it can be inserted into Equation (30) to obtain  
\[\frac{1}{C_{3}}(C_{2}\hat{u}-\hat{p}_{L}-C_{1}\hat{p}_{L})+\frac{C_{4}}{C_{3} }(C_{2}u+C_{7}-\hat{p}_{L}-C_{1}p_{L})=C_{5}p_{L}-C_{6}. \tag{33}\]  
The notation can be further simplified by introducing coefficients  
\[ \begin{split} A_{1}&=C_{1}+C_{4}\,\ \ \ A_{0}=C_{5}C_{3}+C_{4}C_{1},\\ B_{1}&=C_{2}\,\ \ \ B_{0}=C_{4}C_{2}\,\ \ \ C=C_{6}C_{3}+C_{4}C_{7}.\end{split}  \tag{34}\]  
in order to obtain the final equation that relates the valve opening and the load pressure for the linearized reduced-order actuator model with the external force:  
\[\hat{p}_{L}+A_{1}\hat{p}_{L}+A_{0}p_{L}=B_{1}\hat{u}+B_{0}u+C. \tag{35}\]  
Figure 5 presents a comparison of the system's response to a step valve opening obtained using a fully nonlinear simulation (mathematical model described in Section 2.1.1) and the reduced-order linearized model (35). For the operating point, the nominal values were determined and are shown in Table 3.  
Figure 5 shows that using Equation (35) results in a slightly larger pushing force compared to the fully nonlinear model. This error is due to the linearization of the flow around the nominal point and the use of the reduced-order model. It should be noted that the accuracy of the model (35) improves as the surface areas on both sides of the cylinder piston become more similar. In reference [25], excellent agreement between the full and reduced-order hydraulic actuator models was achieved, but with a surface area ratio of  
0.94. The cylinder considered in this study has a surface area ratio of 0.51, which is at the other extreme, making perfect agreement between the full and reduced models impossible. From Figure 5, it can be seen that, for the nominal valve opening, the force calculation error is 33%, while, for smaller valve openings, it is 45%, and, for larger openings, it drops to 20%. This error arises from the reduced-order model's inherent simplifications, particularly in the approximation of the load pressure using Equation (14). If this model were solely used for model-based control system synthesis--such as model predictive control (MPC) deployed on real hardware--these deviations would lead to degraded performance due to the missing dynamics and inaccuracies typical of reduced-order models. However, in the context of this study, the model is not intended to serve as the sole basis for control but rather as a component of a physics-guided neural network (PGNN)-based feedforward controller.