# An Approach for Setting Parameters for Two-Degree-of-Freedom PID Controllers
## 3 Illustrative Examples
### Comparison with Panagopoulos' Method
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{4}{c}{**DDE\(-\)GFM**} & \multicolumn{4}{c}{**Panagopoulos’ Method**} \\ \cline{2-13}  & \(\mathbf{h_{0}}\) & \(\mathbf{h_{1}}\) & \(\mathbf{m}\) & \(\mathbf{k_{p}}\) & \(\mathbf{k_{i}}\) & \(\mathbf{k_{d}}\) & \(\mathbf{b}\) & \(\mathbf{k}\) & \(T_{i}\) & \(T_{d}\) & \(\mathbf{b_{1}}\) & \(T_{sp}\) \\ \hline \(G_{1}\) & 0.0625 & 0.5 & 0.35 & 0.6999 & 0.0863 & 1.4594 & 0.69 & 0.68 & 4.5 & 2.27 & 0 & 0.08 \\ \(G_{2}\) & 0.25 & 1 & 0.5 & 0.5495 & 0.1365 & 0.5608 & 0.5458 & 0.555 & 3.21 & 1.74 & 0 & 1.61 \\ \(G_{3}\) & 16 & 8 & 0.35 & 44.767 & 88.768 & 5.7393 & 44.384 & 43.133 & 0.189 & 0.13 & 0 & 0.81 \\ \(G_{4}\) & 0.49 & 1.4 & 0.35 & 2.1591 & 0.7389 & 1.6449 & 2.1112 & 2.27 & 1.91 & 0.98 & 0 & 0.53 \\ \(G_{5}\) & 0.25 & 1 & 0.3 & 1.6934 & 0.4118 & 1.8324 & 1.647 & 1.47 & 2.33 & 1.25 & 0 & 0.72 \\ \(G_{6}\) & 0.16 & 0.8 & 0.3 & 1.3189 & 0.2605 & 1.7107 & 1.3024 & 1.15 & 2.74 & 1.49 & 0 & 0.97 \\ \(G_{7}\) & 0.2025 & 0.9 & 0.4 & 0.9721 & 0.2149 & 1.1373 & 0.9549 & 0.982 & 3.14 & 1.73 & 0 & 1.24 \\ \(G_{8}\) & 0.64 & 1.6 & 0.85 & 0.5757 & 0.2276 & 0.3722 & 0.5691 & 0.542 & 2.07 & 0.79 & 0 & 1.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The related parameters of the DDE\(-\)GFM and Panagopoulos’ method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Process** & **Work Point in \((q,p)\) Plane** & **Control Signal of DDE–GFM and Panagopoulos’ Method** & **Time Response of DDE–GFM and Panagopoulos’ Method** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The simulation results covering the position of the work point in the \((q,p)\) plane and the control signal and time response comparing the DDE–GFM and Panagopoulos’ method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Process** & **Work Point in \((q,p)\) Plane** & **Control Signal of DDE–GTM and** & **Time Response of DDE–GTM and** \\  & **Panagopoulos’ Method** & **Panagopoulos’ Method** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Process** & **Work Point in \((q,p)\) Plane** & **Control Signal of DDE–GFM and** & **Time Response of DDE–GFM and** \\  & **Panagopoulos’ Method** & **Panagopoulos’ Method** \\ \hline \hline \end{tabular}
\end{table} 
Table S: _Cont._Further details of the simulation results can be seen in Table 6, which includes the settling time, the overshooting, the integral absolute error (IAE), and the maximum sensitivity \(M_{s}\) of the two methods. It can be seen that the settling time is shortened by 15-60%. Except for \(G_{3}\) for which the overshooting is zero, the overshooting of the other processes are all reduced by more than 70%. Furthermore, the overshooting of \(G_{5}\), \(G_{6}\), and \(G_{8}\) are even directly reduced to zero. The IAE index of the two methods for each process are approximately equal except for that of \(G_{3}\) for which the IAE index is approximately zero. In addition, the maximum sensitivity \(M_{s}\) of two methods for all the processes are equal to 2.