## 4 Illustrative Examples
### MOMI PID\({}_{b}\) Controller  
In this section, the proposed method without a reference filter has been tested on some common process models. The chosen IPs were a delayed second-order process, a pure integrator plus time-delayprocess, a fourth-order process, and a non-minimum phase process. These processes are represented by the following transfer functions:  
\[ \begin{array}{l}G_{P1}(s)=\frac{e^{-0.2s}}{s(1+s)(1+0.2s)}\\ G_{P2}(s)=\frac{e^{-s}}{s}\\ G_{P3}(s)=\frac{1}{s(1+s)^{4}}\\ G_{P4}(s)=\frac{1-2s}{s(1+s)^{2}}\.\end{array}  \tag{45}\]  
First, the a-priori chosen PID controller filter time constant \(T_{F}\) = 0.01 s was added to Processes (45). Then, using Expression (8), the characteristic areas \(A_{0}\) to \(A_{4}\) were calculated. The values are presented in Table 2.  
The PID\({}_{\text{b}}\) controller parameters for different values of \(b\) (\(b\) = 0, \(b\) = 0.5, and \(b\) = 0.7) were calculated from Equations (21) and (24) using the calculated characteristic areas from Table 2. The calculated parameters for all selected IP models are presented in Table 3.  
The calculated controller parameters were applied to a PID\({}_{\text{b}}\) controller. Figures 7-10 show the closed-loop responses for the input disturbance step change (\(d\) = 1) and for the reference step change (\(r\) = 1). From the experimental results, it can be seen that the proposed tuning method provided relatively fast and stable closed-loop responses. Additionally, all responses were highly damped with only minor overshoots. The closed-loop responses are in accordance with the magnitude optimum criterion. Exceptions include the sluggish disturbance-rejection responses at \(b\) = 0.7. As expected, the best tracking performance was achieved with \(b\) = 0.7, and the best disturbance-rejection performance was achieved with \(b\) = 0 (see also the illustrative example in Section 2.3). \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(A_{0}\) & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) \\ \hline \(G_{P1}\) & 1 & 1.41 & 1.5141 & 1.5365 & 1.541 \\ \(G_{P2}\) & 1 & 1.01 & 0.5101 & 0.17177 & 0.04338 \\ \(G_{P3}\) & 1 & 4.01 & 10.0401 & 20.1004 & 35.201 \\ \(G_{P4}\) & 1 & 4.01 & 7.0401 & 10.0704 & 13.1007 \\ \hline \end{tabular}
\end{table} 
Table 2: 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 c c c c c c} \hline  & \multicolumn{3}{c}{\(b\) = 0} & \multicolumn{3}{c}{\(b\) = 0.5} & \multicolumn{3}{c}{\(b\) = 0.7} \\ \hline  & \(K_{P}\) & \(K_{I}\) & \(K_{D}\) & \(K_{P}\) & \(K_{I}\) & \(K_{D}\) & \(K_{P}\) & \(K_{I}\) & \(K_{D}\) \\ \hline \(G_{P1}\) & 2.6269 & 1.3991 & 1.4661 & 2.2778 & 1.1385 & 1.418 & 1.8731 & 0.63473 & 1.365 \\ \(G_{P2}\) & 1.128 & 0.47494 & 0.33942 & 1.0017 & 0.3258 & 0.30958 & 0.89095 & 0.18652 & 0.28414 \\ \(G_{P3}\) & 0.37914 & 0.04554 & 0.57814 & 0.3454 & 0.03499 & 0.55729 & 0.30728 & 0.02075 & 0.53411 \\ \(G_{P4}\) & 0.33133 & 0.03935 & 0.39492 & 0.2785 & 0.02479 & 0.34757 & 0.24024 & 0.01344 & 0.31634 \\ \hline \end{tabular}
\end{table} 
Table 3: The calculated PID\({}_{\text{b}}\) controller parameters. Figure 8: The process \(G_{P1}\) closed-loop responses for a different values of \(b\). Note that, for clarity, the controller output response is zoomed in for the reference step change. Figure 7: The process \(G_{P1}\) closed-loop responses for different values of \(b\). Note that, for clarity, the controller output response is zoomed in for the reference step change. The Bode plots of the closed-loop amplitude gains (between the reference and the process output) are shown in Figure 11. This figure shows that all the processes have a flat amplitude low-frequency response, which is in accordance with the MO criterion (see Figure 2). Moreover, the reference  
Figure 10: The process \(G_{\mathcal{P}4}\) closed-loop responses for a different values of \(b\). Note that, for clarity, the controller output response is zoomed in for the reference step change.