## 3 Results and Discussions
### Simulation Study
#### 3.1.2 Second-Order Process Model  
In this section, a simulation based on MATLAB analyses a second-order system. The process model specified in Equation (9) enables systematically exploring the system's behaviour and dynamics. Table 3 provides the numerically evaluating performance of various controllers at a specific time interval for including set-point changes. Figure 6 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 7, and the disturbance rejection performance is given in Figure 8. The transfer function of the second-order system is given as follows. \[G(s)=\frac{2s+1}{s^{2}+3s+1}. \tag{9}\]  
In the comparison between PID controllers and PI-PD controllers, PID controllers are known for having a very low overshoot value of 3.663 \(\times\) 10\({}^{-7}\)%, which indicates precise and stable control. These controllers have a settling time of 116.8035 s and a rise time of 14.1159 s. In contrast, the overshoot value of PI-PD controllers can be as high as 0.9225%, indicating a trade-off between the overshoot and response time. The choice of which controller to use depends on the application's specific requirements. PID controllers prioritize minimal overshoot and a fast response, while PI-PD controllers offer a faster settling time at the expense of higher overshoot and longer rise time. Fractional-order controllers, particularly FOPID, feature zero overshoot and have a settling time of 113.8499 s and a rise time of 30.5718 s, highlighting their precise and stable control. It is essential to consider the unique properties of each variant when evaluating complex-order controllers. When it comes to FOPI-FOPD, it has a minimum overshoot value of 0.0547%, a slightly longer settling time of 146.5290 s, and a slightly longer rise time of 93.6863 s, which indicates a trade-off between stability and response time with the proposed COPI-COPD. The choice of these controls is determined by the balance of overshoot, settling time, and rise time required to achieve optimal system performance. Meanwhile, COPI-PD and PI-COPD controllers have similar overshoot values, settling times, and rise times. COPI-FOPD has a low overshoot value of 0.1751%, a long settling time of 147.0024 s, and a long rise time of 94.8372 s. FOPI-COPD shows stability without overshoot, with settling and rise times of 148.1435 s and 95.0625 s. The minimum overshoot value of COPI-COPD is 0.0076%, and the settling time and rise time are 148.6575 s and 96.3397 s, respectively. A comprehensive comparison highlights the multiple trade-offs between these complex-order controllers and emphasizes the need to balance specific application requirements for optimal system performance. \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 (\(t_{s}\), Seconds)** & **Rise Time (\(t_{r}\), Seconds)** \\ \hline PID & 17.5 & 4.12 & 9.46 & - & - & - & - & 3.663 \(\times\) 10\({}^{-7}\) & 116.8035 & 14.1159 \\ PI-PD & 17.5 & 4.12 & 9.46 & - & - & - & - & 0.9225 & 105.4154 & 34.6860 \\ FOPID & 17.5 & 4.12 & 9.46 & 0.98 & 0.02 & - & - & 0 & 113.8499 & 30.5718 \\ FOPI-FOPD & 17.5 & 4.12 & 9.46 & 0.98 & 0.02 & - & - & 0.0547 & 146.5290 & 93.6863 \\ COPI-PD & 17.5 & 4.12 & 9.46 & 0.98 & - & 0.01 & - & 0.9242 & 105.4284 & 34.6943 \\ PI-COPD & 17.5 & 4.12 & 9.46 & - & 0.02 & - & 0.01 & 0.8785 & 105.2664 & 33.0961 \\ COPI-FOPD & 17.5 & 4.12 & 9.46 & 0.98 & 0.02 & 0.01 & - & 0.1751 & 147.0024 & 94.8372 \\ FOPI-COPD & 17.5 & 4.12 & 9.46 & 0.98 & 0.02 & - & 0.01 & 0 & 148.1435 & 95.0625 \\ COPI-COPD & 17.5 & 4.12 & 9.46 & 0.98 & 0.02 & 0.01 & 0.01 & 0.0076 & 148.6575 & 96.3397 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Controller parameters and transient response for second-order system. PID controllers are generally more effective in reducing overshoot and providing fast, stable responses.