# Physics-Guided Neural Network-Based Feedforward Control for Seamless Pipe Manufacturing Process
## 2 Materials and Methods
### Experimental Setup  
The piercing process is inherently complex from the control system synthesis perspective, as it involves the coordination of multiple controllers with different functions. For instance, during a single work cycle of the two cylinders, the following sequence takes place. Initially, one cylinder operates in force mode, maintaining a constant pressing force on the material. * As the first cylinder approaches the end of its stroke, the second cylinder begins to move, matching the speed of the first to ensure a smooth transition of force. * During the overlap, when both grippers are active, both cylinders operate in force mode, providing a constant resultant force, as illustrated in Figure 7.
* Once the force has fully transitioned to the second cylinder, the first cylinder's gripper is deactivated, and the first cylinder returns to its starting position. This cycle then repeats with the roles of the two cylinders reversed. Consequently, for the successful execution of each sequence, and thus the entire continuous process, it is critical to coordinate three control subsystems: force, speed, and position control. However, since the objective of this paper is to investigate the application of different feedforward strategies based on neural networks within the force control subsystem, the other two control subsystems--position and velocity--will not be addressed here. The schematic structure of the force control system for one cylinder is presented in Figure 8.  
Figure 8: Force controller structure with neural network-based feedforward component. Figure 7: Force exchange between two cylinders during material pressing into the mill. The control system is configured as a PI regulator with feedforward, where the feed-forward component, in general, determines the dominant part of the control signal based on the desired force and the current piston speed. The role of the PI controller is to correct minor deviations from the force setpoint, generating the remaining portion of the control signal. The feedback signal for the force controller, as well as the measured piston velocity for the feedforward, are obtained from pressure sensors in the cylinder chambers and the position sensor of the piston, respectively. The control signal is provided as the desired opening of the proportional 3/4 directional control valve (DCV). In Figure 8, the feedforward is represented by a neural network symbol with two inputs: the desired force and the actual velocity. This feedforward structure corresponds to the benchmark model described in Section 3.1.1. When using the PGNN, the structure of the feedforward neural network differs, as its inputs depend not only on the current desired force but also on its previous values and the previous outputs of the network, as shown in Figure 6. Since this recurrent connection is internal to the PGNN, it is not explicitly shown in Figure 8 for simplicity. However, compared to the benchmark model, the key difference is that the PGNN does not require the current velocity value to generate its portion of the control signal. To implement the PGNN, the desired pushing force, \(F_{p,\text{DES}}\), is divided by the average piston surface area, \(\bar{A}\), to compute the desired load pressure, \(p_{L}\), which serves as an input to the PGNN. This load pressure, \(p_{L}\), is used to form the input regressor vector, allowing the PGNN to capture the system's dynamics. Furthermore, since the reference force is known in advance, the PGNN can utilize information about previous inputs, previous outputs, and future values of \(p_{L}\) to generate precise predictions. By managing the recurrent connection internally, as depicted in Figure 6, the PGNN effectively models dependencies on previous states and outputs, enabling it to predict the required valve opening with high accuracy.