# An Analytical Design of Simplified Decoupling Smith Predictors for Multivariable Processes
## 3 Results  
In this section, three examples are considered to demonstrate the performances of the proposed method in comparison with those of other well-known methods. **Example 1**. : _Wood and Berry (WB) distillation column._  
Figure 6: DSP structure with multiplicative output uncertainty. Figure 7: \(\mathbf{M}-\mathbf{\Delta}\) structure for robustness analysis. The WB column is one of the most representative two-input two-output (TITO) process models widely used for evaluating the performance of the multi-loop controllers. The original matrix transfer function of WB can be found in [30] and expressed as Equation (54):  
\[\mathbf{G}(s)=\left[ \begin{array}{cc}\frac{12.8e^{-s}}{16.7s+1}&\frac{-18.9e^ {-3s}}{21s+1}\\ \frac{6.6e^{-7s}}{10.9s+1}&\frac{-19.4e^{-3s}}{14.4s+1}\end{array} \right]. \tag{54}\]  
The matrix elements of the simplified decoupler for a \(2\times 2\) process can be calculated by the equations in Table 1:  
\[d_{21}=-\frac{g_{21}}{g_{22}}=-\frac{6.6e^{-7s}}{10.9s+1}\bigg{(}-\frac{14.4s+ 1}{19.4e^{-3s}}\bigg{)}\Rightarrow d_{21}=-\frac{0.34(14.4s+1)e^{-4s}}{10.9s+1} \tag{55}\]  
\[d_{12}=-\frac{g_{12}}{g_{11}}=\bigg{(}\frac{18.9e^{-3s}}{21s+1}\bigg{)}\bigg{(} \frac{16.7s+1}{12.8e^{-s}}\bigg{)}\Rightarrow d_{12}=-\frac{1.477(16.7s+1)e^ {-2s}}{21s+1}. \tag{56}\]  
Therefore, the simplified decoupling matrix can be easily obtained:  
\[\mathbf{D}(s)=\left[ \begin{array}{cc}1&\frac{1.477(16.7s+1)e^{-2s}}{21s+1} \\ \frac{0.34(14.4s+1)e^{-4s}}{10.9s+1}&1\end{array} \right]. \tag{57}\]  
For the purpose of controller design, the diagonal elements of the decoupled matrix must be calculated by the equations in Table 2 for a \(2\times 2\) case:  
\[q_{11}=g_{11}-\frac{g_{12}g_{21}}{g_{22}}=\frac{12.8e^{-s}}{16.7s+1}-\bigg{(} \frac{18.9e^{-3s}}{21s+1}\bigg{)}\bigg{(}\frac{6.6e^{-7s}}{10.9s+1}\bigg{)} \bigg{(}\frac{14.4s+1}{19.4e^{-3s}}\bigg{)} \tag{58}\]  
\[q_{22}=g_{22}-\frac{g_{12}g_{21}}{g_{11}}=-\frac{19.4e^{-3s}}{14.4s+1}+\bigg{(} \frac{18.9e^{-3s}}{21s+1}\bigg{)}\bigg{(}\frac{6.6e^{-7s}}{10.9s+1}\bigg{)} \bigg{(}\frac{16.7s+1}{12.8e^{-s}}\bigg{)}. \tag{59}\]  
Since it is impossible to design an analytical PID controller based on these equations, the modified PSO algorithm presented in Section 2 was adopted in order to reduce \(q_{11},\ q_{22}\) in Equations (58) and (59) to the standard form of Equation (11). The approximation procedure of \(q_{11}\) is as follows:  
* Plotting the step response of \(q_{11}\) (Figure 8) and recording its output for training. * Based on the step response, the constraints of the tuned parameters in Equation (11) are predetermined, including \(K_{\min},\ K_{\max},\ \mathrm{and}\ \tau_{\max}\). * Running the proposed PSO algorithm according to Figure 4 to find the best approximation. It is similar for the approximation of \(q_{22}\). The obtained results are the following equations:  
\[q_{11}=\frac{6.4444(9.8668s+1)e^{-s}}{(17.5491s+1)(3.2015s+1)} \tag{60}\]  
\[q_{22}=\frac{-9.7497(8.7203s+1)e^{-3s}}{(14.0079s+1)(3.1501s+1)}. \tag{61}\]  
Figures 8 and 9 also illustrate the comparison of the responses of approximation models of the proposed method with the CM method of [PERSON] and [PERSON] [5] in time and frequency domains. It can be seen that the step responses of the solution transfer functions using PSO were better approximated; that is, they were as close as possible to the step responses of the original models. Further, the fitness function, Equation (14), guaranteed the minimum error of these two outputs. Actually, the purpose of using the PSO algorithm was to propose a novel and simple approach to seeking approximation  
Figure 11: The control signal in loop 1 of the WB column (**a**). The control signal in loop 2 of the WB column (**b**). Figure 10: Closed-loop responses to the unit step changes in the set-point of loop 1 (**a**). Closed-loop responses to the unit step changes in the set-point of loop 2 (**b**).