## 3 Results and Discussions
### Simulation Study
#### 3.1.1 First-Order Process Model  
This subsection employs a MATLAB-based simulation to analyse a first-order system using the process model given in Equation (8). The simulation provides valuable insight into dynamic behaviour by inputting relevant system parameters and utilizing MATLAB tools. The transient response parameters are obtained using MATLAB commands. The outcomes of this study contribute significantly to the field and may interest those seeking to enhance their understanding of the topic. Table 2 provides a specific time interval for set-point changes, which helps numerically evaluate the controller's performance. Figure 3 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 4, and the disturbance-rejection performance is given in Figure 5. The transfer function of the first-order system is given as follows:  
\[G(s)=\frac{1}{5s+1}. \tag{8}\]  
The comparison of the performance of PID and PI-PD controllers through first-order simulations reveals significant differences. Specifically, the PID controller exhibits an overshoot of 0.7406%, with a settling time of 98.3306 s and a rise time of 66.4899 s. In contrast, the PI-PD controller has a higher overshoot of 3.7814%, with a shorter settling time of 117.7727 s and a shorter rise time of 32.3616 s, and while PID controllers exhibit less overshoot and longer settling times, they compensate for this with relatively long rise times. On the other hand, PI-PD controllers prioritize faster rise times at the expense of higher overshoot and longer settling times. The most appropriate selection between these controllers depends on the specific application requirements. It necessitates consideration of the trade-offs between overshoot, settling time, and rise time to achieve the desired system performance. For fractional controllers, FOPID and FOPI-FOPD controllers exhibit different performance characteristics. The FOPID controller demonstrates a relatively short settling time of 86.9470 s, a fast rise time of 30.8176 s, and an impressive zero overshoot. On the other hand, the FOPI-FOPD controller also boasts zero overshoot but has a long settling time of 138.7361 s and a slow rise time of 68.7531 s. This comparison highlights the trade-offs between the two controllers. The FOPI-FOPD had a slower and steady rise time, which led to zero overshoot, which in turn produced a more stable result (non-oscillatory output). The optimal selection of these sub-regulators depends on the specific application requirements, and it is essential to consider the desired balance between overshoot, settling time, and rise time to achieve optimal system performance. In evaluating complex ordering, each variant has different performance characteristics. The COPI-PD and PI-COPD controllers have overshoot values of 3.7814% and 3.7642%, settling times of 117.7864 s and 117.6919 s, and rise times of 32.3687 s and 31.7754 s, respectively. The PID controller has less overshoot than the PI-PD controller but has a longer settling time, compromising the overshoot and response time. FOPID controllers prioritize fast response without overshoot, while FOPI-FOPD controllers emphasize stability with more extended settling and rise times. Several violations can be observed in controllers with complex orders. COPI-FOPD minimizes the overshoot, FOPI-COPD emphasizes stability without overshoot, and COPI-COPD achieves minimal overshoot. In contrast, COPI-FOPD has a significantly lower overshoot of 0.0318%, coupled with a longer settling time of 139.1953 s and a rise time of 69.9659 s. FOPI-COPD features zero overshoot, a settling time of 138.5352 s, and a rise time of 68.5506 s, emphasizing stability and a fast response. COPI-COPD achieves a minimum overshoot of 0.0014% with settling and rise times of 138.9981 s and 69.7455 s, respectively.