## 5 Simulation Results and Discussion
### Case 2 (3 \(\times\) 3) Shell Heavy Oil Fractionator  
According to [PERSON] et al. [54] and [PERSON] and [PERSON] [55], the Shell heavy oil fractionator (SHOF) is a multivariable process involving loop interactions and temporal delays. This kind of distillation column separates the crude oil from the mixture by making use of the variations in boiling points. The MIMO transfer function of the process is  
\[M(s)=\begin{bmatrix}\frac{4.05^{-27s}}{50s+1}&\frac{17.75^{-28s}}{60s+1}&\frac{ 5.88e^{-27s}}{50s+1}\\ \frac{5.39e^{-38s}}{50s+1}&\frac{5.72e^{-28s}}{60s+1}&\frac{5.90e^{-27s}}{40s+1 }\\ \frac{4.38e^{-20s}}{33s+1}&\frac{4.42e^{-22s}}{44s+1}&\frac{7.20}{19s+1}\\ \end{bmatrix} \tag{47}\]  
Decouplers can be created using Equation (3) as  
\[D(s)=\begin{bmatrix}1&\frac{(-88.5s+1.77)e^{-s}}{243s+4.05}&-1.45\\ \frac{(-323.4s+5.39)e^{-4s}}{286s+5.72}&1&\frac{(-345s+6.90)e^{-s}}{215.6s+5.39 }\\ \frac{(-83.2s+4.38)e^{-20s}}{237.6s+7.2}&\frac{(-83.98e+4.42)e^{-22s}}{316.8s+7.2}&1\\ \end{bmatrix} \tag{48}\]  
Figure 10: Singular values of the sensitivity function: Equation (45).  
Figure 9: Singular values of the sensitivity function: Equation (44).  
The FOPDT models are generated from Equations (9) through (11) and are presented as such in [29].  
\[g_{11}^{*}(s)=\frac{-1.19e^{-16.75s}}{1.95s+1} \tag{49}\]  
\[g_{22}^{*}(s)=\frac{-0.875e^{-11.57s}}{1.86s+1} \tag{50}\]  
\[g_{33}^{*}(s)=\frac{-4.8e^{-0.007s}}{25.26s+1} \tag{51}\]  
Moreover, Equation (36) defines the optimization problem from which the decentralized FOPI controller is derived.  
\[C(s)=\begin{bmatrix}10.5+\frac{1.8}{s}&0&0\\ 0&13.8+\frac{3.9}{s}&0\\ 0&0&9.1+\frac{4.2}{s}\end{bmatrix} \tag{52}\]  
As seen from Figures 11-13, the primary goal is to control the flow rate on top-draw, side-draw, and bottom reflux duty, respectively, in order to keep the top-end-point composition (y1), side-end-point composition (y2), and bottom reflux temperature (y3) at acceptable values. The proposed controller is compared with a conventional PID controller. According to the proposed coast function, less overshoot, rise time, and settling time are required to meet the design criteria. In addition, input and output disturbances are used to validate the disturbance rejection. A step signal in the form of a disturbance is injected into the process input (at 100 s) and output (at 140 s). Figures 11-13 show the reference tracking and their corresponding controller outputs. The design specifications are met. The optimal controller variables are presented in Table 3. Additionally, the comparisons of the performance indexes are presented in the same table. The proposed MFPOA-based FOPID controller records the minimum values of \(M_{P}\), \(T_{r}\), and \(T_{s}\). Therefore, it can be concluded that the proposed MFPOA-based FOPID controller can decrease error and meet the design goals.  
Figure 11: Response of Equation (49).  
Figure 12: Response of Equation (50).  
system's stability regions as given by Equations (39) and (40). The stable region is the area that the curve encloses. The figures suggest that the suggested MFPOA-based FOPID controller has more stable zones. Additionally, the peak value is below 1.5, meaning it is desirable.  
Figure 16: Singular values of the sensitivity function: Equation (51).  
Figure 14: Singular values of the sensitivity function: Equation (49).  
Figure 15: Singular values of the sensitivity function: Equation (50).