## 4 Examples and Comparisons
### Comparisons with Other Methods
#### 3.3.3 Case 3  
The third comparison was performed on second-order process with time-delay  
\[G_{P3}(s)=\frac{e^{-s}}{s(1+s)^{2}}. \tag{38}\]  
On the basis of the given process model, areas were determined from Equation (9):  
\[A_{0}=1,\;A_{1}=3,\;A_{2}=5.5. \tag{39}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Ms14** & **Ms20** & **TA** & ** ALI** \\ \hline \(K_{P}\) & 0.332 & 0.577 & 0.766 & 0.48 \\ \(K_{I}\) & 0.021 & 0.129 & 0.187 & 0.092 \\ \(b\) & 0.6 & 0.39 & 0.32 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 7: PI controller parameters for Case 2.  
Figure 14: Closed-loop responses of G\({}_{\text{P2}}\) process. PI controllers were tuned by MOMI, Ms14, Ms20, TA, and ALI methods.  
Then, the controller was designed according to Equations (12) and (14):  
\[K_{P}=0.181,\ K_{I}=0.012,\ b=0.5. \tag{40}\]  
Once more, the proposed MOMI method was compared with some of the presented tuning methods. The calculated PI controller parameters for the presented methods are given in Table 8.  
Figure 15 compares closed-loop responses for tracking and for disturbance-rejection between mentioned tuning rules applied to 2-DOF PI controllers.  
The fastest 5% settling time for reference tracking was obtained by the MOMI method. The ITSE criterion for reference tracking (see Table 4) gave the best value for the TA method, but the MOMI method remained close. Again, the TA method had some undershoot. Similar results were obtained with the ITSE criterion for disturbance-rejection (Table 5).  
**Case 4**  
The last comparison was performed on high-order process  
\[G_{\text{P4}}(s)=\frac{1}{s(1+0.25s)}^{8}. \tag{41}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Ms14** & **Ms20** & **TA** \\ \hline \(K_{P}\) & 0.095 & 0.154 & 0.291 \\ \(K_{I}\) & 0.0026 & 0.0104 & 0.023 \\ \(b\) & 0.6 & 0.48 & 0.32 \\ \hline \hline \end{tabular}
\end{table}
Table 8: PI controller parameters for Case 3.  
Figure 15: Closed-loop responses of G\({}_{\text{P3}}\) process. PI controllers were tuned by MOMI, Ms14, Ms20, and TA methods.  
On the basis of the given process model, areas were determined from Equation (9):  
\[A_{0}=1,\ A_{1}=2,\ A_{2}=2.25. \tag{42}\]  
Then, the controller was designed according to Equations (12) and (14):  
\[K_{P}=0.27,\ K_{I}=0.028,\ b=0.5. \tag{43}\]  
Once more, the proposed MOMI method was compared with some of the presented tuning methods. The calculated PI controller parameters for the presented methods are given in Table 9.  
Figure 16 compares closed-loop responses for tracking and disturbance-rejection between the mentioned tuning rules applied to 2-DOF PI controllers.  
The fastest 5% settling time for reference change was obtained by the MOMI method. Similarly, the MOMI method gave the lowest ITSE values for reference change and disturbance-rejection.  
Experiment results showed that the proposed MOMI method gave very good closed-loop responses for a broad range of process models when compared to some other methods. The disturbance-rejection and tracking responses were relatively fast and stable, with relatively small overshoots. As mentioned before, the obtained tracking or disturbance-rejection performance can be additionally improved by appropriately modifying controller parameter \(b\).  
Figure 16: Closed-loop responses of G\({}_{\text{P4}}\) process. PI controllers were tuned by MOMI, Ms14, and Ms20 methods.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Ms14** & **Ms20** \\ \hline \(K_{P}\) & 0.136 & 0.215 \\ \(K_{I}\) & 0.005 & 0.021 \\ \(b\) & 0.66 & 0.45 \\ \hline \hline \end{tabular}
\end{table}
Table 9: PI controller parameters for Case 4.