# An Analytical Design of Simplified Decoupling Smith Predictors for Multivariable Processes
## 3 Results
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
\begin{tabular}{c} **Tuning** \\ **Method** \\ \end{tabular} & **Loops** & \(\mathbf{K_{ci}}\) & \(\mathbf{\tau_{Hi}}\) & \(\mathbf{\tau_{Di}}\) & \(\mathbf{\tau_{Fi}}\) & \(\mathbf{\lambda_{i}}\) & \(\mathbf{\mu[\mathrm{M}]}\) & **IAE** & **TV** \\ \hline \multirow{2}{*}{Proposed} & 1 & \(-3.3143\) & 6.3129 & 0 & 0 & 1.5 & \multirow{2}{*}{0.3046} & \multirow{2}{*}{3.4382} & \multirow{2}{*}{8.7549} \\  & 2 & 2.1543 & 8.5344 & 0 & 0 & 1.5 & & \\ \hline \multirow{2}{*}{[PERSON] et al.} & 1 & \(-1.666\) & 7.0 & \(-\) & \(-\) & \(-\) & \(-\) & 0.4107 & 4.5255 & 11.295 \\  & 2 & 1.067 & 9.1983 & \(-\) & \(-\) & \(-\) & \(-\) & & \\ \hline \multirow{2}{*}{[PERSON] and [PERSON] (2013)} & 1 & \(-2.885\) & 6.638 & \(-\) & \(-\) & 0.7 & \multirow{2}{*}{0.3504} & \multirow{2}{*}{3.7690} & \multirow{2}{*}{11.347} \\  & 2 & 2.144 & 8.223 & \(-\) & \(-\) & 0.4 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Controller parameters and resulting performance indices for the VL column. Figure 14: Step and Bode responses of original \(q_{22}\) and its approximations (VL column). Figure 16: _Cont._  
Figure 15: Closed-loop responses to the unit step changes in the set-point of loop 1 (VL column) (a). Closed-loop responses to the unit step changes in the set-point of loop 2 (VL column) (b). For the robustness study, the SSV analysis is illustrated in Figure 17 and the perturbation uncertainties of \(\pm 20\%\) were inserted in all three parameters (time constants, gain, and time delay) of the nominal model, simultaneously. As shown in Table 8, the controller settings of the proposed method provided superior robust performance compared with the others. **Example 3. **_Ogunn  
Figure 16: The control signal in loop 1 of the VL column (**a**). The control signal in loop 2 of the VL column (**b**). Figure 17: The SSV plots for robust stability in the VL column. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**VL (+20\%)**} & \multicolumn{2}{c}{**VL (-20\%)**} \\ \cline{2-5}  & **IAE** & **TV** & **IAE** & **TV** \\ \hline Proposed & 3.8897 & 10.8287 & 4.2337 & 10.6356 \\ [PERSON] et al. & 7.9795 & 12.0647 & 5.8493 & 10.0969 \\ [PERSON] and [PERSON] (2013) & 5.1319 & 15.8333 & 4.6460 & 11.6270 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Robustness analysis for the VL column under \(\pm 20\%\) parametric uncertainty in all parameters. A well-known multiproduct distillation column for the separation of a binary ethanol-water mixture studied by many researchers was considered [5; 33]. The open-loop transfer function matrix is given by Equation (66):  
\[\mathbf{G}(s)=\left[\begin{array}{cc} \begin{array}{cc}0.66e^{-2.6s}&-0.61e^ {-3.5s}\\ \hline 6.7s+1&8.64s+1\\ 1.11e^{-6.5s}&-2.36e^{-3s}\\ \hline 3.25s+1&5s+1\\ -34.68e^{-9.2s}&46.2e^{-9.4s}\\ \hline 8.15s+1&10.9s+1\end{array} & \begin{array}{cc}\begin{array}{cc}-0.0049e^{-s} \\ \hline 9.06s+1\\ -0.01e^{-1.2s}\\ \hline 7.09s+1\end{array}\\ \hline 8.07(11.61s+1)e^{-s}\\ \hline 3.89s+1)(18.8s+1)\end{array} \right]. \tag{66}\]  
Using Table 1 for the \(3\times 3\) case, it can be seen that the equations for calculations of \(d_{ij}\) were quite complex and that it was not as easy to directly derive the decoupler matrix as the two previous examples. In this case, the proposed PSO algorithm was also used as described step by step in Example 1 to obtain each element of the decoupler matrix. The final obtained result was as follows:  
\[\mathbf{D}(s)=\left[ \begin{array}{cc}1&0.7549\frac{(0.8337s+1)}{0.533s+1}e^ {-0.9s}&0.0062e^{-0.5233s}\\ 0.3905e^{-2.71s}&-0.0014\\ 19.8247\frac{(13.8874s+1)}{8.7956s+1}e^{-8.2s}&-23.1741\frac{(2.0113s+1)}{5.5 621s+1}e^{-8.4s}&1\end{array} \right].