## 4 Illustrative Examples
### Comparison with the MOMI Tuning Method for PI Controllers  
The performance of the proposed MOMI PID\({}_{\text{b}}\) method was compared with the MOMI PI method for IPs [1] on four process models. Note that this PI controller also includes the reference weighting factor \(b\) on the proportional term. The following IPs were chosen:  
\[\begin{array}{l}G_{PI}(s)=\frac{e^{-0.5s}}{s(1+4s)}\\ G_{P2}(s)=\frac{e^{-0.5s}}{s(1+2s)^{2}}\\ G_{P3}(s)=\frac{e^{-0.5s}}{s(1+s)^{4}}\\ G_{P4}(s)=\frac{e^{-4s}}{s(1+0.5s)}\.\end{array} \tag{46}\]  
First, the parameters for the PI controller were acquired. To that end, the characteristic areas \(A_{0}-A_{2}\), presented in Table 2, were calculated. The PI controller parameters for two different values of \(b\) (\(b=0\) and \(b=0.5\)) were then calculated via the following Equations [1]:  
\[\begin{array}{l}K_{I}=0.5A_{0}\Big{(}1-b^{2}\Big{)}K_{P}{}^{2},\\ K_{P}=\frac{-A_{1}+\sqrt{A_{1}{}^{2}+\xi}}{\xi},\\ \xi=\Big{(}1-b^{2}\Big{)}\Big{(}A_{0}A_{2}-A_{1}{}^{2}\Big{)},\end{array} \tag{47}\]  
using the calculated characteristic areas from Table 4. The calculated PI controller parameters for all four IP models are presented in Table 5.  
Figure 11: Bode plots (for a different value of \(b\)) for all tested processes.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \(A_{0}\) & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) \\ \hline \(G_{P1}\) & 1 & 4.51 & 18.170 & 72.703 & 290.813 \\ \(G_{P2}\) & 1 & 4.51 & 14.170 & 38.663 & 97.973 \\ \(G_{P3}\) & 1 & 4.51 & 12.170 & 25.643 & 46.592 \\ \(G_{P4}\) & 1 & 4.51 & 10.295 & 15.895 & 18.721 \\ \hline \end{tabular}
\end{table}
Table 6: The calculated characteristic areas for the process models with the additional filter time constant \(T_{F}\) = 0.01 s.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(A_{0}\) & \(A_{1}\) & \(A_{2}\) \\ \hline \(G_{P1}\) & 1 & 4.5 & 18.125 \\ \(G_{P2}\) & 1 & 4.5 & 14.125 \\ \(G_{P3}\) & 1 & 4.5 & 12.125 \\ \(G_{P4}\) & 1 & 4.5 & 10.25 \\ \hline \end{tabular}
\end{table}
Table 4: The calculated characteristic areas for the process models.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & \(A_{0}\) & \(A_{1}\) & \(A_{2}\) \\ \hline \(G_{P1}\) & 1 & 4.5 & 18.125 \\ \(G_{P2}\) & 1 & 4.5 & 14.125 \\ \(G_{P3}\) & 1 & 4.5 & 12.125 \\ \(G_{P4}\) & 1 & 4.5 & 10.25 \\ \hline \end{tabular}
\end{table}
Table 5: The calculated PI controller parameters.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & \(A_{0}\) & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) \\ \hline \(G_{P1}\) & 1 & 4.51 & 18.170 & 72.703 & 290.813 \\ \(G_{P2}\) & 1 & 4.51 & 14.170 & 38.663 & 97.973 \\ \(G_{P3}\) & 1 & 4.51 & 12.170 & 25.643 & 46.592 \\ \(G_{P4}\) & 1 & 4.51 & 10.295 & 15.895 & 18.721 \\ \hline \end{tabular}
\end{table}
Table 7: The calculated PID\({}_{b}\) controller parameters.  
Figure 12: The PI and the \(\text{PID}_{b}\) controller closed-loop responses for the process \(G_{\text{PI}}\). Note that, for clarity, the controller output response is zoomed in for the reference step change.  
Figure 13: The PI and the \(\text{PID}_{b}\) controller closed-loop responses for the process \(G_{\text{P2}}\). Note that, for clarity, the controller output response is zoomed in for the reference step change.  
Figure 14: The PI and the PID\({}_{b}\) controller closed-loop responses for the process \(G_{P3}\). Note that, for clarity, the controller output response is zoomed in for the reference step change.  
Figure 15: The PI and the PID\({}_{b}\) controller closed-loop responses for the process \(G_{P4}\). Note that, for clarity, the controller output response is zoomed in for the reference step change.