## 2 Experimental and Methodology
### Working Area of Hydraulic Press and Design of Experiment  
The phenomenon of energy conversion from hydraulic energy to the output energy produced by a hydraulic cylinder shows that the working range of a hydraulic press depends on two crucial parameters: hydraulic cylinder velocity \(v_{C}\) and hydraulic cylinder force \(F_{C}\), as shown in Figure 4. The upper and lower limits of the input parameters are determined based on the working limits of the hydraulic press during the sheet metal bending cycle, i.e., the bending, leveling, and moving phases. Preliminary research results have shown that a force of \(F_{C}=14.4\) kN is required on the piston rod of the hydraulic cylinder to successfully perform the bending of the workpiece with a thickness of \(s_{0}=5\) mm, which corresponds to 88% of the hydraulic press's nominal capacity. The upper and lower limits of the input parameters \(F_{C}=[F_{C,L},F_{C,L}]\) and \(v_{C}=[v_{C,L},v_{C,L}]\) determine the working area of the hydraulic prism \(\mathrm{WR}=\{(v_{C,L}\,,F_{C,L}),\,(v_{C2}\,,F_{C2}),\,\ldots,\,(v_{Cn}\,,F_{C,L})\}\). During the bending phase, the overall energy of the hydraulic cylinder is mainly affected by the intensity of the hydraulic cylinder force \(F_{C}\), which is generated to overcome the specimen deformation. However, both the kinetic and potential energy of a hydraulic cylinder are required to deform a specimen into a desired shape. In addition, in the movement phase, the hydraulic energy is converted to perform rapid forward and backward movements of the hydraulic cylinder, mainly requiring high kinetic energy. In the leveling phase, the hydraulic cylinder is semi-static, so mainly potential energy is required to overcome the resistance of the specimen, while the kinetic energy used is minimal. The subset of observation and validation points represents the experimentally measured hydraulic cylinder velocity \(v_{C}\) and the hydraulic cylinder force \(F_{C}\). The diagram of the observational and validation area illustrates various scenarios used for learning and validation purposes in regression models to determine their prediction accuracy, as shown in Figure 5. These scenarios are described as observational and validation points, representing different conditions of the hydraulic press, exposed to varying hydraulic cylinder force \(F_{C}\), hydraulic cylinder velocity \(v_{C}\), and frictional conditions. A subset of observation points is selected for the regression modeling, since it is hardly possible to anticipate the entire working area of the hydraulic press by setting the observation area \(\mathrm{OA}=\{(v_{C1}\,,F_{C1}),\,(v_{C2}\,,F_{C2}),\ldots,\,(v_{Cn}\,,F_{Cn})\}\), where \(n\ \in\ \mathbb{R}\). Here, each point represents a combination of parameters measured in the time domain, such as the displacement of hydraulic cylinder \(X_{C}\), the pressure in the chambers of hydraulic cylinders \(p_{A}\) and \(p_{B}\), the force of hydraulic cylinder \(F_{C}\), and the position of hydraulic valve spool \(X_{V}\). Next, the engineering knowledge is utilized to develop an expert system with a PR-MPC mechanism and embedded intelligence to detect changing conditions and readjust the control signal for better performance of the hydraulic press, which is further explained in Section 3. Confirmation of the effectiveness of the expert system is carried out in the validation phase as some of the observation points and additional validation points are selected to compare the response of the hydraulic press with and without the adaptive control module. The effectiveness of this method is measured as the response error \(\Delta X_{C}\), determined by the difference between the measured and referenced value of the hydraulic cylinder movement, as indicated in Equation (1). The determined validation area is \(\text{VA}=\{(v_{\text{C1}}\,,F_{\text{C1}}),(v_{\text{C2}}\,,F_{\text{C2}}),\, \ldots,(v_{\text{Cm}}\,,F_{\text{Cm}})\}\), where \(m\,\in\,R\). Figure 4: Hydraulic press working area.