## 3 Results and Discussions
### Experimental Study
#### 3.2.1 Flow Process Model
the comparison between FOPID and FOPI-FOPD shows that FOPI-FOPD is better than FOPID as it has zero overshoot. However, it has a lower settling time of 239.3677 s and a rise time value of 79.5915 s, which is smaller for FOPID. FOPI-FOPD shows excellent interference rejection performance with no overshoot. The analysis conducted on complex sequential controllers such as COPI-PD, PI-COPD, COPI-FOPD, FOPI-COPD, and COPI-COPD revealed that FOPI-COPD showed superior performance compared to the others. FOPI-COPD achieved a settling time of 241.8741 s with a rise time of only 81.9517 s and no overshoot, indicating high accuracy and stability. These results set FOPI-COPD apart from another similar controller that exhibits non-zero overshoot values, emphasizing its exceptional performance in achieving precise and stable control. These performance comparisons highlight the importance of considering multiple performance parameters while assessing controllers and suggest that traditional controllers, such as PID, may suffer from overshoot. In contrast, fractional controllers like FOPI-FOPD and FOPI-COPD offer improved overshoot minimization, settling time, and rise time. These findings suggest that FOPI-COPD is a compelling option for applications where precise control is paramount. A salient observation concentrates from this comprehensive study, with the proposed complex-order controller consistently emerging as the epitome of effectiveness, outperforming its competitors in terms of overall performance. This consistent superiority is a recurring theme throughout the configuration value tracking scenario, with the nuanced adaptability showcased by the complex-order controller underscoring its prow  
Figure 12: Flow process model’s step response with compared controllers. Figure 13: Flow process model’s tracking response with compared controllers. the optimal choice for applications where precise control and adaptability to varied configurations are pivotal performance benchmarks. The complex-order controller establishes itself as a stalwart in configuration value tracking, promising enhanced control dynamics and an elevated performance standard in the face of evolving operational requirements. In the context of disturbance rejection, a real-world analysis is presented where a deliberate disturbance value of 6 is introduced into the controller structure after 150 s. This mirrors the dynamic nature of external disturbances in a process plant. Figure 14 illustrates the intricacies of this scenario, providing a clear visual representation of the proficiency of each controller structure in mitigating the effects of disturbances. These controllers demonstrate their adeptness through dynamic adjustments to the output, aligning it with the set point, as shown in the step response diagram captured in Figure 12. Notably, the proposed complex-order controller consistently outperforms competitors in interference suppression, showcasing superior overall performance. All of the above consistently better performances over other controllers underscore the robustness of the proposed complex-order controllers in dealing with disturbances, positioning them as the optimal choice for applications where disturbance rejection is a critical performance criterion. The empirical evidence from this evaluation highlights the potential of complex-order controllers in providing resilient and effective solutions for managing external disturbances, further solidifying their candidacy for applications where precision and adaptability to disturbances are paramount considerations.