## 3 Results
\right] \tag{48}\]  
Similar to the Example 1, the FOTFs of the diagonal elements of decoupled matrix are approximated as follows:  
\[q_{11}=\frac{6.4911e^{-s}}{7.1079s^{0.8714}+1} \tag{49}\]  
\[q_{22}=\frac{-9.8693e^{-3s}}{3.3615s^{1.788}+5.2645s^{0.8462}+1} \tag{50}\]  
According to Table 1, the proposed fractional controllers are obtained for each loop:  
\[g_{c1}=0.2882\frac{1}{s^{0.1286}}\Big{(}1+\frac{1}{7.1079s^{0.8714}}\Big{)} \tag{51}\]  
\[g_{c2}=-0.1212\frac{1}{s^{0.1538}}\Big{(}1+\frac{1}{5.2645s^{0.8462}}+0.6385s^{ 0.9418}\Big{)}\frac{1}{1.1s+1} \tag{52}\]  
The performances of the proposed method are compared with those of the previous works including the simplified decoupling Smith predictor (SDSP) [15] and simplified filter Smith predictor proposed by [PERSON] et al. [22]. The sequential unit step changes in the set-points are made to the 1 st and 2 nd loop at t = 0 (min) and t = 100 (min), respectively. The closed-loop responses to the set-point changes, the control signals of loop 1 and 2 are shown in Figure 7a,b and Figure 9a,b, respectively. The performance indices are summarized in Table 3. From the figures, it is obvious that the F-SDSP and SDSP methods have almost the same performance. However, when considering Table 3, the proposed method still gives better performance indices. In comparison with the [PERSON] approach, the F-SDSP has far better responses in both set-point changes and disturbance rejection. The robustness analysis of those methods is illustrated in Figure 8. In this case, the uncertainties of 10% of gains are applied to the [PERSON] approach while the other approaches still keep the same uncertainties as the previous example, i.e. the uncertainties are decreased by 50% at high frequencies and increased by 20% in a low-frequency range. The maximum \(\mu\) values shown in Table 3 prove that the proposed controller has the smallest value that means the best robustness among those approaches. Figure 7: (**a**) Closed-loop responses to the unit step changes in the set-point of loop 1 in Example 2*. (**b**) Closed-loop responses to the unit step changes in the set-point of loop 2 in Example 2*. (* [PERSON] in all Figures 7–9 is used instead of “Simplified filter Smith predictor proposed by [PERSON] et al. [22]”). **Example 3**. : _Vinante and Luyben (VL) column_  
The transfer function matrix for a VL column introduced by [PERSON] [25] has the following form:  
\[\mathbf{G}(s)=\left[ \begin{array}{cc}\frac{-2\,2\,e^{-s}}{7s+1}&\frac{13\, \mathrm{d}^{-0.3s}}{7s+1}\\ \frac{-2\,8\,e^{-1.8s}}{9.5s+1}&\frac{4\,\mathrm{d}^{-0.35s}}{9.2s+1}\end{array} \right] \tag{53}\]  
The simplified decoupling matrix can be easily obtained by using Equation (10):  
\[\mathbf{D}(s)=\left[ \begin{array}{cc}1&0.591\\ \frac{0.651(9.2s+1)e^{-1.45s}}{9.5s+1}&1\end{array} \right] \tag{54}\]  
Similar to the previous examples, the diagonal elements \(q_{11}\) and \(q_{22}\) are approximated to the fractional forms as Equation (17) or (18). The obtained results are as the following equations:  
\[q_{11}=\frac{-1.3629e^{-s}}{6.6757s^{0.97}+1} \tag{55}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tuning Method** & **IAE** & **ITAE** & **TV** & \(\mu[\mathbf{M}]\) \\ \hline F-SDSP & 9.221 & 631.207 & 0.9587 & 0.2291 \\ SDSP & 10.963 & 667.088 & 0.7720 & 0.3091 \\ Santos & 13.294 & 881.098 & 1.2976 & 0.5750 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The performance indices for the wood and berry (WB) column. Figure 8: The SSV plots of robust stability in Example 2*. Figure 9: (**a**) The control signal in loop 1 in Example 2*. (**b**) The control signal in loop 2 in Example 2*.