## 5 Results and Discussion  
To improve the precision and accuracy of the pneumatic actuator's cylinder stroke while maintaining the appropriate position, a CFOPID controller was tested in SIMULINK, as illustrated in Figure 12. As a result, the suggested controller improved the transient responsiveness while minimizing the system's overall performance overshoot, rise time, and settling time. Comparisons were made between the performance of the proposed controller and the FOPID controller, which was also tested. As determined by PSO, Table 4 provides a summary of the optimal parameter values for both the CFOPID and the FOPID controllers. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Criteria** & \(\mathbf{K_{p}}\) & \(\mathbf{K_{i}}\) & \(\mathbf{\lambda}\) & \(\mathbf{K_{d}}\) & \(\mathbf{\mu}\) & \(\mathbf{K_{p}1}\) \\ \hline CFOPID & 50 & 48.6378 & 1.0141 & 50 & 0.25 & 39.8685 \\ FOPID & 50 & 50 & 0.817 & 50 & 0.25 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The optimal values of the controllers. Figure 11: Flowchart of the standard PSO algorithm. Figure 12: Simulink diagram for simulation CFOPID. Table 5 shows a summary of how well the CFOPID and FOPID controllers worked. In this research, controllers were ranked according to how well they managed the IPA positioning system. The CFOPID model that was optimized using PSO yielded the best results in terms of performance (Ts, Tr, and OS%). Figure 13 shows a comparison of the CFOPID and FOPID controllers' responses, revealing that the CFOPID controller results in a significantly lower overshot (OS%) in the system response (at 0.1587) compared to the FOPID controller (at 1.1619). Furthermore, in terms of reference signal tracking, the CFOPID beats all other controllers evaluated in this study. The performance of the suggested controller can be assessed by applying a varying setpoint to its input. A comparison of the capabilities of CFOPID controllers and FOPID controllers is presented in Figure 14. This unequivocally demonstrates that the CFOPID controller was superior to the FOPID controller in terms of performance. A significant improvement has been made to the CFOPID controller's trajectory tracking capacity, and several performance indicators have been improved as well, including the transient response of the system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Criteria** & **Tr (s)** & **Ts (s)** & **OS\%** & **ISE** \\ \hline CFOPID & 0.008 & 0.0098 & 0.1587 & 0.0001205 \\ FOPID & 0.0133 & 0.0510 & 1.1619 & 0.002699 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The optimal values of the CFOPID and FOPID controllers. Figure 13: The responses of CFOPID and FOPID controllers. Additionally, the outcomes are contrasted with those from studies that used the generalized predictive controller (GPC) presented in [36], the model predictive controller (MPC) presented in [37], and the predictive functional controller (PFC) with the novel observer method presented in [14] for the same system in order to show the superiority of the CFOPID controller. Table 6 displays the results obtained from using these controllers. Table 6 gives more evidence that the controller that is being presented is superior to those that have been proposed in earlier publications. In comparison to the methods examined in [14; 36; 37], the settling time and rise time are found to be shorter when employing the suggested CFOPID controller. Nonetheless, a slight increase in the overshoot is observed. The ability to control a ball for the IPABB system is demonstrated by the analytical findings for position tracking of a pneumatic actuator. The CFOPID controller was chosen as the inner loop control because it has a good and reliable response. The outer loop controller is a PID controller, as detailed in Section 3. The PSO algorithm can be used to tune the values of a PID controller. The gains values are: \(\mathrm{K_{p}}=10\), \(\mathrm{K_{i}}=1\), and \(\mathrm{K_{d}}=50\).