## 4 Results and Discussion
### Expert System Adaptive Control Evaluation
The validation of the PR-MPC mechanism integrated into the expert system has also been conducted using new scenarios that were not included in the learning phase of the regression models, as shown in Figure 12. Here, we evaluate the regression model's efficiency in predicting the response error \(\Delta X_{C}\) and the expert system's efficiency in adapting the control signal in real time. Hydraulic press cycles 3 and 4 showcase changed hydraulic cylinder velocities in phases and introduce new forming process intensities. Hydraulic press cycle 3 showcases lower hydraulic cylinder velocity during fast forward movement (\(v_{C}=20\) mm/s) in comparison to hydraulic press cycle 4, where \(v_{C}=50\) mm/s. Consequently, in hydraulic press cycle 4, the response error \(\Delta X_{C}\) of the hydraulic press is higher compared to hydraulic press cycle 3. Furthermore, the PR-MPC mechanism integrated into the expert system demonstrates comparable results in minimizing the response error \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Phases**} & \multicolumn{5}{c}{**Absolute Error of the Residuals [mm]**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** \\ \hline Cycle 1 (initial response) & 0.993 & 2.038 & 3.142 & 1.082 & 3.081 \\ Cycle 1 (improved response) & 0.016 & 0.192 & 0.231 & 0.055 & 0.074 \\ Cycle 2 (initial response) & 1.215 & 3.089 & 2.431 & 1.099 & 1.756 \\ Cycle 2 (improved response) & 0.101 & 0.083 & 0.221 & 0.073 & 0.097 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Absolute error of response error \(\Delta X_{C}\) in validation scenarios. for existing scenarios in the observation area and enhances the hydraulic press's response to the reference. In the bending phase, hydraulic press cycle 3 (\(v_{\text{C}}\) = 10 mm/s) showcases a lower response error in comparison to hydraulic press cycle 4 (\(v_{\text{C}}\) = 20 mm/s) with the initial control system. However, the PR-MPC mechanism integrated into the expert system demonstrates the hydraulic press's ability to adapt to the effects of sheet metal bending, which caused higher response errors \(\Delta X_{\text{C}}\). Despite this, the measured displacement of the hydraulic cylinder with the active improved control system closely matches the reference, indicating effective adaptation. In the leveling phase, the hydraulic press's response error \(\Delta X_{\text{C}}\) in hydraulic press cycle 3 and 4 shows similar results; moreover, the PR-MPC mechanism integrated into the expert system improves and decreases the response error \(\Delta X_{\text{C}}\) to the reference displacement values in both cases. Observing the fast backward movement of the hydraulic cylinder, the hydraulic press error \(\Delta X_{\text{C}}\) in hydraulic press cycle 3 shows higher values compared to hydraulic press cycle 4. The improved control system effectively adapts the hydraulic press's behavior, reducing the response error and successfully bringing it closer to the reference value. Table 6 shows the absolute response error \(\Delta X_{\text{C}}\) in new validation scenarios across phases, with comparisons between the initial and improved responses for both hydraulic press cycles 3 and 4. The values reflect a significant reduction in the hydraulic press response error \(\Delta X_{\text{C}}\) after the PR-MPC mechanism's integration into the control system. In the (1) holding phase, the response error \(\Delta X_{\text{C}}\) for hydraulic press cycle 3 is 0.861 mm in the initial control system. The response error \(\Delta X_{\text{C}}\) decreases to 0.069 mm after the control