## 4 Results and Discussion  
### Polynomial Regression Analysis and Response Surface Analysis  
During the bending phase, both the polynomial regression analysis and the response surface analysis show that two specific features significantly predict changes in the condition of the hydraulic press caused by external disturbances. The prediction surface plots in Figure 8 reflect the influence of the input features \(v_{C}\) and \(F_{C}\) on the tendency and sensitivity of the hydraulic press response error \(\Delta X_{C}\) in the bending phase. Moreover, the positive gradient indicates that a higher value of features \(v_{C}\) and \(F_{C}\) is accompanied by an increase in the response error \(\Delta X_{C}\). Furthermore, the analysis reveals that the influence of feature \(v_{C}\) on the response error \(\Delta X_{C}\) demonstrates a more pronounced gradient compared to feature \(F_{C}\). In addition, the relationship between the response error \(\Delta X_{C}\) and feature \(v_{C}\) shows a linear correlation. However, the relationship between the response error \(\Delta X_{C}\) and feature \(F_{C}\) follows a third-degree polynomial pattern. Furthermore, as the load on the hydraulic press increases (simultaneously increases force \(F_{C}\)), the response becomes more sensitive, especially noticeable at higher values of \(v_{C}\) and \(F_{C}\). In such scenarios, the curve gradient is evidently steeper compared to scenarios in which the hydraulic press operates under lower load conditions. The most appropriate polynomial function for the bending phase is a third-order two-dimensional function, where the hydraulic cylinder force \(F_{C}\) as a feature has a significant dynamic influence on the hydraulic press response error \(\Delta X_{C}\), as shown in Equation (14). \[\Delta X_{n}=\beta_{0}+\beta_{1}F_{n}+\beta_{2}v_{n}+\beta_{3}F_{n}^{2}+\beta_ {4}F_{n}v_{n}+\beta_{5}F_{n}^{3}+\beta_{6}F_{n}^{2}v_{n} \tag{14}\]  
A comparative analysis is carried out to highlight differences between the results of the experimental results in cases of initial friction conditions (a) and changed friction conditions (b), where the response error of the hydraulic press ranges in between +0.75 mm and +3.49 mm. Observing the response surface plots, the feature \(F_{C}\) exhibits a shallower tendency and lower sensitivity compared to feature \(v_{C}\) with respect to the response error \(\Delta X_{C}\). Moreover, the gradient difference in comparison to cases (a) and (b) is minor. However, the gradient in case (b) is naturally higher for both features \(v_{C}\) and \(F_{C}\), as this scenario accounts for the combined influence of the forming process and changed friction conditions, in contrast to case (a), which considers the influence of the forming process. Moreover, in case (b), a greater response error \(\Delta X_{C}\) is observed due to the inclusion of the friction effects of the components. Overall, the evaluation metrics RMSE, \(R^{2}\) and SSE show valuable insights into the predictive performance of the regression models under changed conditions during the bending phase, as shown in Table 2. The coefficient of determination \(R^{2}\) shows respectable values of 0.9920 and 0.9924 for regression models built with data from experiments (a) and (b). The evaluation metric RMSE in cases (a) and (b) suggests a strong correlation between the predicted and observed values, resulting in high performance accuracy and robustness. The RMSE metric indicates a similarity with the results reflected by the \(R^{2}\) metric. Considering the evaluation metric SSE, it is noticeable that regression models (a) and (b) result in low confidence intervals. In addition, the regression models show high performance prediction ability for the bending phase.