# Physics-Guided Neural Network-Based Feedforward Control for Seamless Pipe Manufacturing Process
## 3 Results
### Final Experiments
#### 3.2.2 Varying Setpoint Tracking with Different Feedforward Structures  
The second experiment was chosen with the aim of comparatively demonstrating the application of different feedforward structures in real time under conditions that correspond to operational use. The test setup was as follows.  
* Both grippers (Figure 3) are engaged, creating a rigid mechanical connection between the two pistons.
* The first cylinder operates in force mode, following a ramp setpoint for the pushing force, with a randomized profile within the operational force range.
* The second cylinder operates in velocity mode, maintaining a constant speed of approximately 40 mm/s, which corresponds to the operating conditions of the real system. Three control system structures were tested: 1. A PI controller without feedforward; 2. A PI controller with neural network-based feedforward that has access to both speed and desired force information (benchmark neural network feedforward); 3. A PI controller with a physics-guided neural controller with neural network-based feedforward that has access only to the desired force (PGNN feedforward).  
Figure 22 presents a comparative analysis of the force ramp tracking using the three controllers mentioned. It is evident that the use of the PI controller alone is insufficient in tracking the time-varying reference signal.  
Figure 21: Valve opening during constant force setpoint tracking with varying piston speeds for the benchmark neural network-based feedforward controller.  
As seen in the figure, both feedforwards result in the good tracking of the desired force value, except in the section where the reference value begins to decrease. This indicates that the PI controller's gains are not optimally tuned in the presence of the feedforward, with their high values leading to overshoot.  
Nevertheless, the use of both feedforwards resulted in good setpoint tracking, with the following integral of the squared error values for the part of the experiment shown in Figure 22:  
\[(\mathrm{ISE})_{\mathrm{PI}}=20.3985\,\quad(\mathrm{ISE})_{\mathrm{Benchmark\,NN}} =3.9848\,\quad(\mathrm{ISE})_{\mathrm{FGNN}}=3.6953\]  
It is important to note how the PGNN-based feedforward compares to the benchmark network, which is significantly more training-demanding.  
The final output of the system is also influenced by the quality of the second cylinder's tracking of the desired velocity. Given the mechanical connection between the cylinders, there is a mutual influence between them, even though they operate in different modes (force and velocity). The achieved velocity profile during the observed part of the experiment is shown in Figure 23.  
It is observed that the quality of the tracking of the desired velocity is somewhat lower; however, it should be noted that velocity regulation was not the focus of this work. The task of the second cylinder was to provide an approximate expected velocity during the production process, without strict requirements regarding oscillations and steady-state errors. Furthermore, as previously mentioned, from the perspective of the force control system of the first cylinder, the variable motion represents a disturbance that must be addressed by maintaining the desired force. Therefore, the variable motion of the entire system during the experiment supports the examination of the robustness of the force controller.  
Figure 22: Comparison of varying force setpoint tracking performance with different controller structures: PI controller; benchmark neural network feedforward; physics-guided neural network (PGNN) feedforward.