# An Analytical Design of Simplified Decoupling Smith Predictors for Multivariable Processes
## 3 Results
The robustness of the controller was evaluated by inserting the perturbation uncertainties of \(\pm 10\%\) and \(\pm 20\%\) in the process gain, time constant, and time delay into the actual process, simultaneously, whereas the controller settings were those provided for the nominal process. The simulation results of the model mismatch for various tuning methods of two cases are given in Tables 5 and 6, respectively. In the first case (\(\pm 10\%\)), the responses of all tuning methods were still stable, but the proposed method had the best performance in general. In the latter (\(\pm 20\%\)), however, the responses of [PERSON]'s and [PERSON]'s methods became unstable, while [PERSON] et al. 's method and the DSP structure still gave acceptable results. However, Table 6 shows that it is obvious that the proposed method had the better performance. **Example 2**. : _Vinante and Luyben (VL) column._  
The transfer function matrix for a VL column introduced by [PERSON] [32] has the following form:  
\[\mathbf{G}(s)=\left[ \begin{array}{cc}\frac{-2.2e^{-s}}{7s+1}&\frac{1.3e^{-0.3 s}}{7s+1}\\ \frac{-2.8e^{-1.8s}}{9.5s+1}&\frac{4.3e^{-0.35s}}{9.2s+1}\end{array} \right]. \tag{62}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**WB (+20\%)**} & \multicolumn{2}{c}{**WB (-20\%)**} \\ \cline{2-4}  & **IAE** & **TV** & **IAE** & **TV** \\ \hline Proposed & 14.5501 & 1.1279 & 11.45 & 0.6961 \\ [PERSON] & 15.6505 & 1.2815 & \(2.8933\times 10^{9}\) & \(2.2552\times 10^{8}\) \\ [PERSON] et al. & 18.3324 & 0.9344 & \(1.7633\times 10^{10}\) & 367.7749 \\ [PERSON] et al. & 12.7908 & 1.440 & 15.4013 & 1.1324 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Robustness analysis for the WB column under \(\pm 20\%\) parametric uncertainty in all parameters. Figure 12: The SSV plots for robust stability in the WB column. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**WB (+10\%)**} & \multicolumn{2}{c}{**WB (-10\%)**} \\ \cline{2-5}  & **IAE** & **TV** & **IAE** & **TV** \\ \hline Proposed & 12.5540 & 0.9323 & 10.8035 & 0.7429 \\ [PERSON] & 13.6015 & 1.0957 & 13.5207 & 0.8999 \\ [PERSON] et al. & 16.1987 & 0.8659 & 14.7554 & 0.4872 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Robustness analysis for the WB column under \(\pm 10\%\) parametric uncertainty in all parameters. This plant was chosen to evaluate the proposed method because of the special case in \(d_{12}\). According to Table 1, \(d_{12}\) is a non-causal element (\(d_{12}=0.591e^{0.75}\)), which means it cannot satisfy the realizability problem. In this case, the steady state gain, \(K=0.591\), of this element can be used instead. The simplified decoupling matrix can be easily obtained by using Table 1 (\(2\times 2\) process):  
\[\mathbf{D}(s)=\left  
In this simulation study, the performance of proposed method was compared with the simplified decoupling method ([PERSON] and [PERSON] [5]) as well as the centralized inverted decoupling method ([PERSON] et al. [8]). The implementing diagram for VL column is shown in Figure 10, Appendix A. The resulting multi-loop PID controllers by the proposed and other methods were tabulated and are shown in Table 7. For the sequential unit step changes in the set-points, Figure 14(a),b compare the closed-loop time responses afforded by the proposed and other methods. The proposed controller showed superior responses with faster settling times and no overshoots compared with the other methods. The sequential unit step changes considered as disturbances were also investigated for the first and second loops. The values of the performance indices (Table 7) confirmed the superior performance of the proposed controller for the load changes as well as the set-point changes. The control signals of the two loops are presented in the Figure 15(a),b.