# Physics-Guided Neural Network-Based Feedforward Control for Seamless Pipe Manufacturing Process
## 2 Materials and Methods
### Mathematical Models
#### 2.1.1 Hydraulic Actuator
Parameters \(\sigma\), \(F_{c0}\), \(F_{s0}\), \(c_{s}\) from Equation (9) and parameters \(C_{i}\), \(C_{e}\) from Equations (4) and (5) are unknown and need to be determined. One method of determining these parameters is to use least squares fitting based on experimentally recorded data [26]. For this study, the system response was recorded as the piston speed for step  
\[\frac{dx}{dt}=v. \tag{2}\]0 Characteristic curves of the directional control valve. When the billet meets the piercing point, the grip of the rolls is sufficient to continue the advancement of the workpiece against the retarding effect imposed by the piercing point [1]. The smooth operation of the piercing process can be completed if the following condition is satisfied [27]:  
\[\frac{dx}{dt}=v. \tag{2}\]1  
which relates the axial components of the friction of roll (\(T\)), the positive pressure of roll (\(P\)), and the resistance of the plug nose (\(Q^{\prime}\)). The internal dynamics of the process within the mill have been extensively studied in the existing force and stress analysis literature (Refs. [1; 27; 28; 29; 30]). In this work, a simplified model to capture the dynamics outside the piercing mill is proposed. This model is primarily based on two key factors that dominate inside the mill, namely friction and material characteristics described by the stress-strain relationship, from which it is known that the resistive force of the material will initially increase in response to the pushing force until it reaches its yield stress and begins to deform. When the material starts to move, a further increase in the pushing force will cause the material to progress through the entrance of the mill, which will increase the friction force, eventually leading to a steady-state velocity corresponding to the current applied pushing force. Therefore, the following model for the resistive force as observed from the controller's perspective outside the mill is proposed:  
\[F_{ext}= \begin{cases}F_{p}\,&F_{p}<F_{yield}\\ F_{yield}+\mathcal{F}(\Delta F)\,&F_{p}\geq F_{yield}\end{cases} , \tag{11}\]  
\[\Delta F=F_{p}-F_{yield}\, \tag{12}\]  
where \(F_{p}=F_{act}-F_{int}\) is the applied pushing force, and \(F_{yield}\) is the pushing force threshold after which the material starts to deform. Function \(\mathcal{F}\) represents the resistive force delay, through which the lag between the increase in pushing force and the corresponding rise in friction (which occurs as the workpiece velocity gradually increases) is modeled. This delay will be presented with the transfer function  
\[\frac{\mathcal{F}(s)}{\Delta F(s)}=\frac{1}{\tau_{ext}s+1}, \tag{13}\]  
where time constant \(\tau_{ext}\) can be determined based on experimental data. Figure 3: Action of rotary piercing mill on the round billet. Figure 4 presents an example of pushing force setpoint tracking, where the external resistive force is modeled using the parameters \(\tau_{ext}=0.002\) s, \(F_{yield}=4000\) N.  
It can be seen that the displayed resistive force qualitatively follows the typical stress profile during the material's deformation, while the delay of the resistive force relative to the pushing force ensures the material's change in momentum as long as the pushing force continues to increase.