# Physics-Guided Neural Network-Based Feedforward Control for Seamless Pipe Manufacturing Process
## 3 Results
### Final Experiments  
#### 3.2.1 Constant Setpoint Tracking with Benchmark Neural Network  
The first experiment was chosen to verify the applicability of the benchmark neural network in real time and its integration with the existing controller structure on the PLC. In this experiment, a comparison was performed with a force PI controller, which is the standard type of controller for this application in industrial settings. It is important to note that, when using the benchmark neural network to compute the feedforward part of the control signal, the feedback component remains present in the form of a PI controller to ensure overall robustness in real industrial applications. 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 constant setpoint for the pushing force (1000 N). * The second cylinder operates in velocity mode, tracking step changes in speed. Due to the limited stroke of the cylinder, and to cover a wide range of speeds during a single stroke, the durations at specific speeds are kept short, resulting in a speed profile, as shown in Figure 19, that resembles a linear change in velocity. Although the system is designed to operate at approximately constant speeds and forces, this experiment was selected with the aim to test the response of the cylinder, maintaining a constant force, to disturbances in the form of speed variations generated by the second cylinder. The speed range covered was from 0 to 20 mm/s, and, to gain insights into the advantages of using a neural network as a feedforward, the same desired speed profile was requested from the second cylinder during both tests, with and without the feedforward. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Model** & **Physical Model (ID)** & **Standalone NN** & **PGNN** \\ \hline
**MSE (\%)** & \(2.49\cdot 10^{-2}\) & \(2.42\cdot 10^{-2}\) & \(2.33\cdot 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: MSE in % for different models in Figure 18.  
Figure 18: Comparison of different models: the actual system output (\(u_{\text{Actual}}\)) is compared with the identified physical model (\(u_{\text{Phys-ID}}\)), standalone neural network (\(u_{\text{NN}}\)), and the total PGNN output (\(u_{\text{PGNN}}\)). The speeds achieved during the tests are shown in Figure 19, where it can be observed that the speed profile was approximately the same during both experiments. Figure 20 presents a comparative analysis of the results for the maintenance of a constant force setpoint in the presence of velocity disturbances, both with and without the feedforward. In the first case, a standard PI controller was used, while, in the second case, the benchmark neural network acted as the feedforward, determining the dominant part of the control signal based on the desired force value and the current speed. The fluctuations around the desired force value, as a result of the continuously increasing piston speed, are clearly visible in Figure 20. However, the improved tracking of the desired value can be observed when the neural network-based feedforward is used. This improvement can also be quantified by the integral of the squared error (ISE), which, in the mentioned cases, amounts to  
\[\left(\text{ISE}\right)_{\text{PI}}=150.44\,\ \ \ \left(\text{ISE}\right)_{ \text{Benchmark\, NN}}=134.75\]  
Figure 19: Piston speeds during constant setpoint tracking, with different controller structures: PI controller; benchmark neural network feedforward. Figure 20: Comparison of constant force setpoint tracking with varying piston speeds, with different controller structures: PI controller; benchmark neural network feedforward. Figure 21 shows the calculated control signal for the experiment with the benchmark neural network-based feedforward.