## 4 Experimental Results  
The following figures depict the responses of flow and temperature after receiving the unit step function input in Simulink using Z-N PID (Figures 16 and 17), PSO PID and SSO PID (Figures 18 and 19), PSO fuzzy PID and SSO fuzzy PID (Figures 20 and 21), and SSO fuzzy PI (Figures 22 and 23). The results are compared in Table 6.  
Figure 19: Temperature versus time plot for PID tuned with PSO and SSO.  
Figure 18: Flow versus time plot for PID tuned with PSO and SSO.  
Figure 21: Temperature versus time plot for PID tuned with SSO and SSO fuzzy.  
Figure 20: Flow versus time plot for PID tuned with SSO and SSO fuzzy.  
Figure 23: Temperature versus time plot for PI tuned with SSO and SSO fuzzy.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & ***ITAE (200 s)** \\ \hline Z-N PID (flow controller) & 2.0743 & 0.1200 & 0.0300 & 1148 \\ Z-N PID (temp controller) & 0.2158 & 0.1200 & 0.0300 & 150 \\ PSO PID (flow controller) & 0.8536 & 0.3810 & 0.0620 & 95.79 \\ PSO PID (temp controller) & 0.2410 & 0.4168 & 0.0063 & 15.21 \\ SSO PID (flow controller) & 0.9857 & 0.8732 & 0.3767 & 19.42 \\ SSO PID (temp controller) & 0.0419 & 0.7976 & 0.1098 & 3.468 \\ PSO fuzzy PID (flow controller) & 0.8536 & 0.3810 & 0.0620 & 50.09 \\ PSO fuzzy PID (temp controller) & 0.2410 & 0.4168 & 0.0063 & 10.63 \\ SSO fuzzy PID (flow controller) & 0.9857 & 0.8732 & 0.3767 & 15.79 \\ SSO fuzzy PID (temp controller) & 0.0419 & 0.7976 & 0.1098 & 4.23 \\ SSO fuzzy PI (flow controller) & 0.9857 & 0.8732 & – & 14.68 \\ SSO fuzzy PI (temp controller) & 0.0419 & 0.7976 & – & 3.316 \\ \hline \hline \end{tabular}
\end{table}
Table 6: \(K_{p},K_{i},K_{d}\), and ITAE of different tuning methods. *ITAE: Integral of absolute error multiplied by time.  
Figure 22: Flow versus time plot for PI tuned with SSO and SSO fuzzy.  
From the above-mentioned figures and table, we can clearly realize the advantages and disadvantages of the controllers designed by various methods. The performances ordered from high to low correspond to SSO fuzzy PI, SSO fuzzy PID, SSO PID, PSO fuzzy PID, PSO PID, and Z-N PID. From Figures 16 and 17, it can be seen that the response time for Z-N PID to converge to 1 is the slowest, nearly 200 s, and thus, it is not suitable for practical applications where the response time is emphasized. The experimental results of PSO PID are not always excellent in the simulation process and are easily limited to the local optimum. Moreover, its run time is longer than that of SSO, but obviously, the time required to converge to 1 has dropped to approximately 50 s, as shown in Figures 18-21.  
SSO PID performs not only with faster run time, but also with a higher probability of obtaining an excellent solution. Its settling time is shorter than that of PSO, as shown in Figures 18 and 19. By combining the fuzzy theory with PSO and SSO, it was found that the experimental results are both more optimized than the original ones as can be seen from Figures 20-23 and the values of the integral of absolute error multiplied by time (ITAE) in Table 6. PSO fuzzy PID, SSO fuzzy PID, and SSO fuzzy PI not only inherit the advantages of PSO PID and SSO PID, but also change dynamically in real time. The PID gain parameters are adjusted by the proposed strategies, and hence, the settling time is shortened successfully. Moreover, SSO fuzzy PI performs better than SSO fuzzy PID in run time and ITAE value, as shown in Figures 22 and 23 and Table 6. Therefore, SSO fuzzy PI stands out among all strategies.