# 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  
The reduced-order model of a hydraulic actuator incorporating the proposed method for external force modeling can now be derived. This simplified model is crucial in forming the basis of the inversion-based feedforward control method. Following standard practice [24; 25], we introduce the load pressure as a measure of the actuation force,  
\[p_{L}=p_{A}-p_{B}\,, \tag{14}\]  
and the load flow (assuming \(|Q_{A}|=|Q_{B}|\)),  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]  
where \(K\) is the valve flow coefficient. The chamber pressures can be expressed through the load pressure as follows (assuming zero pressure in the tank):  
\[p_{A}=\frac{p_{S}+p_{L}}{2},\quad p_{B}=\frac{p_{S}-p_{L}}{2}. \tag{16}\]  
The load flow equation can be further simplified by linearizing it around an operating point \((\hat{u},\hat{p}_{L})\):  
\[Q_{L}=\hat{Q}_{L}+C_{q}(u-\hat{u})-C_{qp}(p_{L}-\hat{p}_{L}), \tag{17}\]  
where  
\[C_{q}=\frac{\partial Q_{L}}{\partial u}\Big{|}_{\hat{p}_{L}}=K\sqrt{\frac{1}{ 2}(p_{S}-\hat{p}_{L})}\,, \tag{18}\]  
Figure 4: Pushing force setpoint tracking in the presence of an external resistive force. \(\tau_{ext}=0.002\) s, \(F_{yield}=4000\) N.  
\[C_{qp}=\frac{\partial Q_{L}}{\partial p_{L}}\Big{|}_{\dot{a}}=-\frac{K\dot{a}}{2 \sqrt{2(p_{S}-\hat{p}_{L})}}. \tag{19}\]  
Equation (17) reduces to  
\[Q_{L}=C_{q}u-C_{qp}(p_{L}-\hat{p}_{L}), \tag{20}\]  
because \(\hat{Q}_{L}=C_{q}\dot{a}\). If, instead of the analytical flow model (15), the flow curves provided by the manufacturer are used (as shown in Figure 1), it is necessary to utilize the full form of the linearized flow Equation (17) without prior simplification (20). Since this approach is also applied in the simulation model used in this study, the full form of the flow Equation (17) will be considered in the following text, with the coefficients \(C_{q}\) and \(C_{qp}\) determined numerically using the curves from Figure 1.  
By applying these relations, the hydraulic continuity Equations (4) and (5) can be combined into a single equation related to the load pressure (neglecting leakage flows):  
\[\dot{p}_{L}=\frac{4\beta}{V_{t}}\big{(}\hat{Q}_{L}+C_{q}(u-\hat{u})-C_{qp}(p_{ L}-\hat{p}_{L})-\bar{A}v\big{)}, \tag{21}\]  
where \(V_{t}\) is the total hydraulic actuator volume, \(V_{t}=V_{A0}+V_{B0}\), and \(\bar{A}\) is the average effective piston area, \(\bar{A}=0.5(A_{A}+A_{B})\). Note that, for a symmetric actuator (double-rod cylinders), this approach is justified in calculating the actuation force as  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]0  
However, for non-symmetrical actuators, this method introduces an error due to the unequal surfaces of the piston. Despite this flaw, the advantages of this approach and its applicability to non-symmetrical actuators (which will be shown later) significantly outweigh the error. Therefore, we will use Equation (22) for the calculation of the actuation force in our reduced-order model. The equation describing the change in external force over time can be considered under the reasonable assumption that the operating point is well above the pushing force threshold \(F_{yield}\). Thus, based on Equations (11)-(13), the proposed model is reduced to  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]1  
Finally, the equation describing the motion of the piston retains its form described with Equation (1):  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]2  
Combining Equations (23) and (24), one can obtain  
\[Q_{L}=Ku\sqrt{\frac{1}{2}(p_{S}-\text{sign}(u)p_{L})}\,, \tag{15}\]3  
Now, we introduce a further simplification without loss of generality.