# Physics-Guided Neural Network-Based Feedforward Control for Seamless Pipe Manufacturing Process
## 2 Materials and Methods
### Mathematical Models
#### 2.1.1 Hydraulic Actuator  
In the following text, the mass of the moving parts is denoted with \(m\) (the mass of the piston and the accompanying parts moved by the piston), the piston's velocity with \(v\), and the piston's position (measured from the midpoint of the actuator's stroke) with \(x\). The equations describing the movement of the piston are as follows:  
\[m\frac{dv}{dt}=F_{act}-F_{int}-F_{ext}\, \tag{1}\]  
\[\frac{dx}{dt}=v. \tag{2}\]  
The actuation force \(F_{act}\) is obtained on the basis of the pressure difference in the cylinder chambers:  
\[F_{act}=p_{A}A_{A}-p_{B}A_{B}, \tag{3}\]  
where \(A_{A}\) and \(A_{B}\) are the piston-side area and rod-side area, respectively. The equations that describe the pressure change in the cylinder chambers are  
\[\frac{V_{A}dp_{A}}{dt}+A_{A}v=Q_{A}-C_{i}(p_{A}-p_{B}), \tag{4}\]  
\[\frac{V_{B}dp_{B}}{dt}-A_{B}v=-Q_{B}+C_{i}(p_{A}-p_{B})-C_{e}p_{B}, \tag{5}\]  
where \(\beta\) is the bulk modulus of the hydraulic oil. The volumes of the chambers depend on the position of the piston:  
\[V_{A}=V_{A0}+A_{A}x, \tag{6}\]  
\[V_{B}=V_{B0}-A_{B}x. \tag{7}\]  
The coefficients \(C_{i}\) and \(C_{e}\) account for fluid leaks between the chambers and the external environment. Many hydraulic actuators are designed with high-quality seals, which reduce the external leakage to a level that does not significantly affect the performance, making it negligible unless the system operates under conditions where seal degradation is a concern. Internal leakage primarily accounts for the pressure drop due to minor fluid penetration between the actuator chambers, influenced by factors such as cylinder wear--related to the hydraulic cylinder's life cycle--temperature-dependent expansion, and the fluid viscosity. To obtain a reduced-order model that is suitable for real-time integration with data-driven methods due to its simplicity, external leakage is neglected in this study as this simplifies the model without compromising the accuracy. Internal leakage will be considered in the identification process of the fully nonlinear actuator model but is also neglected in the reduced-order modeling since its contribution to the actuator behavior is minor in well-maintained systems with intact seals. This approach is standard in many hydraulic actuator modeling methods (e.g., [24; 25]), particularly in linearized models used for control system synthesis. The flows \(Q_{A}\) and \(Q_{B}\) are determined based on the opening of the directional valve (\(u\)) and the pressure difference at the inlet and outlet of the valve (\(\Delta p\)):  
\[Q=Q(u,\Delta p). \tag{8}\]  
This relationship is described by the manufacturer's specifications and is illustrated in Figure 1.  
The primary task of the controller is to determine the valve opening \(u\) in order to ensure a constant material pressing force \(F_{act}\). The term \(F_{int}\) in Equation (1) models the internal resistances to the movement of the piston that occur within the cylinder itself. In general, this term represents a function of the piston's velocity and position and may depend on how the cylinder is oriented (e.g., if the cylinder is positioned vertically, this term also includes the gravitational force acting on the piston). The equation modeling the internal resistances to the piston's movement takes the form of the Stribeck friction curve [24]:  
\[F_{int}=F_{int0}+\sigma v+\text{sgn}(v)\left[F_{c0}+F_{s0}\text{exp}\left(- \frac{|v|}{c_{s}}\right)\right], \tag{9}\]  
where \(\sigma\) is the parameter for viscous friction, \(F_{c0}\) is the parameter for Coulomb friction, and \(F_{s0}\) and \(c_{s}\) are the parameters for static friction. The first component in Equation (9) models a constant load, which is independent of the direction of the piston's velocity. This resistance component does not exist, as this work considers horizontally positioned cylinders.