## 3 Results and Discussions
### Simulation Study
#### 3.1.3 Third-Order Process Model  
In analysing third-order systems, transfer function models are crucial for simulating and comparing controllers, and their transfer function model is given in Equation (10). Table 4 provides the numerically evaluated performance of various controllers at a specific time interval, including set-point changes and disturbance rejection analysis. Figure 9 shows the step response of all the controllers followed by each controller structure successfully controlling its output to match the variable set-point changes in Figure 10, and the disturbance rejection performance is given in Figure 11. By focusing on the controller's transient response characteristics, transfer function models can provide insight into the controller's performance under varying conditions. This approach allows for a nuanced study of the controller's behaviour and provides a systematic means of recognizing the controller's effectiveness in tuning the system response. Therefore, transfer function models represent a methodologically rigorous path that allows for a comprehensive investigation and evaluation of controller performance within the dynamic landscape of third-order systems. The transfer function of the third-order system is given as follows. \[G(s)=\frac{400}{s^{3}+30\text{s}^{2}+200\text{s}}. \tag{10}\]  
PID controllers are known for their fast response time but have a significant overshoot of 22.1842%, allowing large deviations from the set point. The PI-PD controller, on the other hand, achieves slightly longer settling and rise times but reaches a lower overshoot value of 6.7384%, resulting in improved control accuracy. This trade-off between faster response and higher overshoot for PID controllers versus precise control of PI-PD controllers with slightly \begin{table}
\begin{tabular}{c 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 (\(t_{s}\),** & **Rise Time (\(t_{r}\),** & **Seconds)** & **Rise Time (\(t_{r}\),** & **Seconds)** \\ \hline PID & 2.4532 & 4.8789 & 0.3314 & - & - & - & - & 22.1842 & 97.8430 & 17.0763 \\ PI-PD & 2.4532 & 4.8789 & 0.3314 & - & - & - & - & 6.7384 & 88.5320 & 15.3694 \\ FOPID & 2.4532 & 4.8789 & 0.3314 & 0.98 & 0.02 & - & - & 54.8152 & 93.8416 & 11.3073 \\ FOPI-FOPD & 2.4532 & 4.8789 & 0.3314 & 0.98 & 0.02 & - & - & 15.8640 & 94.9228 & 12.0943 \\ COPI-PD & 2.4532 & 4.8789 & 0.3314 & 0.98 & - & 0.01 & - & 6.6980 & 88.5421 & 15.3753 \\ PI-COPD & 2.4532 & 4.8789 & 0.3314 & - & 0.02 & - & 0.01 & 7.3362 & 88.0580 & 15.3340 \\ COPI-FOPD & 2.4532 & 4.8789 & 0.3314 & 0.98 & 0.02 & 0.01 & - & 16.5639 & 94.7933 & 12.0374 \\ FOPI-COPD & 2.4532 & 4.8789 & 0.3314 & 0.98 & 0.02 & - & 0.01 & 15.8146 & 94.8964 & 12.0969 \\ COPI-COPD & 2.4532 & 4.8789 & 0.3314 & 0.98 & 0.02 & 0.01 & 0.01 & 16.5141 & 94.7665 & 12.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Controller parameters and transient response for third-order system. Figure 8: Second-order process model’s disturbance response with compared controllers. longer settling and rise times is noteworthy. Regarding fractional-order regulators, FOPID controllers can provide potentially faster response times but have a high overshoot value of 54.8152%. The FOPI-FOPD controller, with an overshoot value of 15.8640% and a slight compromise in rise time, is a better option for those seeking the balance between overshoot and the response time. COPI-PD and PI-COPD controllers have similar overshoot values but different settling and rise times for complex-order controllers. COPI-FOPD increases settling time to minimize the overshoot to 16.5639%, while FOPI-COPD emphasizes stability with a slightly reduced overshoot value of 15.8146%. COPI-COPD, on the other hand, minimizes the overshoot to 16.5639%. Set-point tracking involves adjusting input values at various intervals to evaluate the controller's robustness under different conditions.