## 3 Results  
In this section, three examples of the well-known TITO processes are considered to demonstrate the performances of the proposed method in comparison with those of other existing methods. Figure 3: \(\mathbf{M}\)-\(\Delta\) structure with multiplicative output uncertainties. **Example 1**. : _Heavy oil fractionator_  
It is a \(2\times 2\) process given in [20]. The open-loop transfer function matrix is as follow (43) where time constants and delays are expressed in minutes. \[\mathbf{G}(s)=\left[ \begin{array}{cc}\frac{4.05e^{-27s}}{27s+1}&\frac{1.77e^{ -28s}}{60s+1}\\ \frac{5.39e^{-18}}{50s+1}&\frac{5.27e^{-24s}}{60s+1}\end{array} \right] \tag{41}\]  
The decoupler matrix is easily obtained using the Equation (10) and the result is as follow:  
\[\mathbf{D}(s)=\left[ \begin{array}{cc}1&-\frac{0.437(27s+1)}{60s+1}e^{-s}\\ \frac{0.9423(60s+1)}{(50s+1)}e^{-4s}&1\end{array} \right] \tag{42}\]  
Then, using (11) and (12) to calculate the diagonal elements of the decoupled matrix. However, in this case, the approximation technique using the PSO algorithm [15] is addressed to derive the fractional order transfer functions. The results are obtained by the following equations:  
\[q_{11}=\frac{2.3979e^{-27s}}{15.1333s^{1.1334}+6.9815s+1} \tag{43}\]  
\[q_{22}=\frac{3.3877e^{-14s}}{45.6092s^{0.9967}+1} \tag{44}\]  
Using the above fractional transfer functions and the proposed controller for each case, the fractional controllers are derived as follows:  
\[g_{c1}=0.0622\Big{(}1+\frac{1}{4.166s}+5.7077s^{0.1334}\Big{)}\frac{1}{7s+1} \tag{45}\]  
\[g_{c2}=0.6732\frac{1}{s^{0.0033}}\Big{(}1+\frac{1}{45.6092s^{0.9967}}\Big{)} \tag{46}\]  
In this example, the proposed method is compared with those of inverted decoupling internal model control with filter (IDIMC-F) and centralized inverted decoupling control (ID-K) which are proposed by [PERSON] [20]. The closed-loop responses for the sequential step changes in the set-points of loops 1 and 2 are shown in Figure 4a,b. The performance indices of the three methods are summarized in Table 2.  
Figure 4: (**a**) Closed-loop responses to the unit step changes in the set-point of loop 1 in Example 1. (**b**) Closed-loop responses to the unit step changes in the set-point of loop 2 in Example 1. From Figure 4a,b, it can be seen that the proposed controller has familiar response compared to the IDIMC-F method and far better than the ID-K one in loop 1, however, in loop 2 the proposed method has faster rising time and shorter settling time in comparison with the others. Disturbance rejection performance by considering the mutual effects of sequential changes on loops 1 and 2 is much improved in the proposed method. As a result, the performance indices including TV, IAE, and ITAE of the proposed one are superior to the others. Figure 5a,b illustrate the manipulated variables of both loops with respect to time and they indicate that the proposed method has smoother signals. The robust stability of the system is investigated using Equation (42) with a multiplicative output uncertainty, where \(\mathbf{W}_{o}(s)  
The well-known WB column is used for evaluating the performances of the proposed method. The transfer function matrix of WB can be found in [24] and expressed as Equation (49):  
\[\mathbf{G}(s)=\left[ \begin{array}{cc}\frac{12.8e^{-s}}{16.7s+1}&\frac{-18.9e^{ -3s}}{21s+1}\\ \frac{6.66e^{-s}}{10.9s+1}&\frac{-19.4e^{-3s}}{14.4e+1}\end{array} \right] \tag{47}\]  
The matrix of the simplifier decoupler for this process can be obtained by Equation (10):  
\[\mathbf{D}(s)=\left[ \begin{array}{cc}1&\frac{1.477(16.7s+1)e^{-2s}}{21s+1}\\ \frac{0.34(14.4s+1)e^{-4s}}{10.9s+1}&1\end{array}