## 4 Experimental Results
The comparative results indicate that the tuning algorithm with the highest performance indices was the GWO algorithm with an ITAE cost function, achieving superior settling time and rise time metrics. In contrast, the ACO and FPA algorithms, which employed an ISE cost function, also delivered commendable results, particularly in overshoot and overall stability. Figure 4 illustrates the step response curves for each algorithm with their respective best cost functions. The graph demonstrates that the GWO algorithm provides a smoother response with minimal overshoot and rapid settling time. The ACO algorithm shows a slightly higher overshoot but maintains a quick rise time, while the FPA algorithm balances overshoot and settling time. These step response curves highlight the effectiveness of each optimization algorithm in tuning the FOPID controller, with the GWO algorithm excelling in minimizing transient response characteristics. The response curves indicate that the system exhibits characteristics of an under-damped response without multiple oscillations. Upon closer examination, it is evident that the GWO tuning results in a higher overshoot, whereas the establishment time remains relatively consistent across all algorithms. Table 13 presents the tuning results, detailing the integral (\(\lambda\)) and differential (\(\mu\)) orders along with the proportional (\(K_{p}\)), integral (\(K_{i}\)), and derivative (\(K_{d}\)) constants for each algorithm. The ACO and FPA algorithms utilized the ISE cost function, while the GWO algorithm employed the ITAE cost function. These results underscore the specific tuning parameters that yielded the optimal performance for each respective algorithm. The bar graph in Figure 5 demonstrates that the parameters of the system response using the Grey Wolf Optimizer (GWO) algorithm tend to have greater overshoot and a comparatively longer settling time than the ACO and FPA algorithms. Specifically, the GWO algorithm resulted in an overshoot of 33.2651%, a settling time of 10.7198 s, and a rise time of 1.79475 s. In contrast, the ACO algorithm achieved an overshoot of 19.3879%, a \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **Cost Function** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & \(\lambda\) & \(\mu\) \\ \hline ACO & ISE & 95 & 20 & 35 & 0.3 & 1.1 \\ GWO & ITAE & 50 & 31.8391 & 50 & 0.809 & 0.7729 \\ FPA & ISE & 73.3803 & 83.971 & 23.4439 & 2.4309 \(\times\) 10\({}^{-6}\) & 1.4011 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Results of tuned values. Figure 4: Step response curves of tuned systems with FOPID control. settling time of 20.6258 s, and a rise time of 1.79475 s. The FPA algorithm displayed similar performance to ACO, with an overshoot of 18.7534%, a settling time of 20.7844 s, and a rise time of 1.70798 s. These results indicate that while the GWO algorithm tends to produce higher overshoot, it achieves better settling times than ACO and FPA. The ACO and FPA algorithms show similar values for overshoot, rise time, and settling time, highlighting their comparable performance in tuning the system response. Figure 6 presents the values of the ITAE and ISE indices, highlighting the best results achieved by each algorithm. It is evident that the indices for the GWO algorithm are significantly higher compared to those for the ACO and FPA algorithms. Specifically, the ITAE index value for the ACO algorithm is lower than that of the FPA algorithm, indicating a more efficient performance in terms of settling time and rise time. Conversely, the ISE value for the ACO algorithm is higher, reflecting a higher overshoot as shown in Table 10, where the ACO algorithm using the ITAE cost function resulted in an overshoot of 48.5889%. In comparison, the FPA algorithm, as seen in Table 12, achieved a lower overshoot of 36.2235% with the ITAE cost function. Figure 5: Parameters of the system response with the average result of each algorithm. Figure 6: Comparison of ITAE/ISE indices of the best results by meta-heuristic algorithm.