## 3 Results and Discussions
### Experimental Study
#### 3.2.2 Pressure Process Model
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Controllers** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & \(\lambda\) & \(\mu\) & \(\beta\) & \(\gamma\) & **OS (\%)** & **Settling Time (\(t_{s}\), Seconds)** & **Rise Time (\(t_{r}\), Seconds)** \\ \hline PID & 0.5 & 0.5 & \(-\)0.01 & - & - & - & - & 6.5841 & 163.0095 & 18.5776 \\ PI-PD & 0.5 & 0.5 & \(-\)0.01 & - & - & - & 0.3905 & 175.9587 & 38.7020 \\ FOPID & 0.5 & 0.5 & \(-\)0.01 & 0.98 & 0.02 & - & - & 4.0610 & 162.1264 & 25.0510 \\ FOPI-FOPD & 0.5 & 0.5 & \(-\)0.01 & 0.98 & 0.02 & - & - & 0.2565 & 178.9595 & 40.4344 \\ COPI-PD & 0.5 & 0.5 & \(-\)0.01 & 0.98 & - & 0.01 & - & 0.2876 & 176.0390 & 38.7466 \\ PI-COPD & 0.5 & 0.5 & \(-\)0.01 & - & 0.02 & - & 0.01 & 7.5495 \(\times\) 10\({}^{-12}\) & 177.5542 & 83.0930 \\ COPI-FOPD & 0.5 & 0.5 & \(-\)0.01 & 0.98 & 0.02 & 0.01 & - & 0 & 180.4267 & 86.5732 \\ FOPI-COPD & 0.5 & 0.5 & \(-\)0.01 & 0.98 & 0.02 & - & 0.01 & 0 & 180.2877 & 84.9259 \\ COPI-COPD & 0.5 & 0.5 & \(-\)0.01 & 0.98 & 0.02 & 0.01 & 0.01 & 2.1894 \(\times\) 10\({}^{-11}\) & 180.4000 & 86.4863 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Controller parameters and transient response for real-time pressure process plant. In evaluating complex-order controllers, each variant has demonstrated unique performance characteristics. COPI-PD has exhibited a moderate overshoot of 0.2876%, with a settling time of 176.0390 s and a rise time of 38.7466 s. PI-COPD has shown a nearly imperceptible overshoot of 7.5495 \(\times\) 10\({}^{-12}\)%, achieving a settling time of 177.5542 s and a rise time of 83.0930 s. COPI-FOPD has excelled in overshoot elimination with a value of zero percent, coupled with a settling time of 180.4267 s and a rise time of 86.5732 s. FOPI-COPD has emerged as a standout performer, exhibiting zero percent overshoot, a settling time of 180.2877 s, and a rise time of 84.9259 s, making it the preferred choice. Although COPI-COPD has also achieved a negligible overshoot of 2.1894 \(\times\) 10\({}^{-11}\)% with a settling time of 180.4 s and a rise time of 86.4863 s, the comprehensive assessment positions FOPI-COPD as the superior controller, showcasing optimal performance across the overshoot, settling time, and rise time among the complex-order controllers evaluated. After a comprehensive examination, a clear trend emerges, indicating that the proposed complex sequential controllers, specifically the FOPI-COPD, consistently surpass their competitors and demonstrate the pinnacle of overall performance. This recurring advantage is not a singular event but a recurring theme throughout various set-point tracking scenarios. The outstanding performance of complex sequencers, particularly the FOPI-COPD, provides substantial evidence of their adeptness in controlling changes in diverse set points. This empirical validation positions these controllers as the foremost choice for applications prioritizing accuracy, adaptability, and superior overall performance. By  
Figure 16: Pressure process model’s tracking response with compared controllers. Figure 15: Pressure process model’s step response with compared controllers. nature, complex sequential controllers, especially the FOPI-COPD, have proven to be robust tools in set-point tracking, improving control dynamics and ensuring higher performance based on dynamic operating requirements. As per the analysis conducted on the array of control structures, the FOPI-COPD controller consistently emerges as the best performing controller, showcasing superior overall performance compared to its counterparts. This recurring trend underscores the robust interference suppression capabilities inherent in the FOPI-COPD controller. The empirical evidence from this analysis positions the FOPI-COPD controller as the optimal choice for applications where minimizing interference effects is a crucial performance criterion.