# An Approach for Setting Parameters for Two-Degree-of-Freedom PID Controllers
## 3 Illustrative Examples
### Comparison with Panagopoulos' Method  
This section will present the results compared with [PERSON] method [15] to show the superiority of the proposed tuning method. Figure 12: The closed-loop time response of process \(G(s)\) for points H, C, and I. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**The Closed-Loop Poles**} & \multicolumn{2}{c}{**The Smallest \(m_{n}\)**} \\ \hline A & \(-0.868+1.085\) & \(-0.868-0.085\)i & \(-0.632+0.089\)i & \(-0.632-0.089\)i & 0.8 \\ B & \(-0.8456+1.2079\)i & \(-0.8456-1.2079\)i & \(-0.6544+0.0939\)i & \(-0.6544-0.0939\)i & 0.7 \\ C & \(-0.8316+1.389\)i & \(-0.8316-1.389\)i & \(-0.6684+0.1071\)i & \(-0.6684-0.1071\)i & 0.6 \\ D & \(-0.9996+1.666\)i & \(-0.9996-1.1666\)i & \(-0.5004+0.2067\)i & \(-0.5004-0.2067\)i & 0.6 \\ E & \(-1.1377+1.896\)i & \(-1.1377-1.896\)i & \(-0.3623+0.152\)i & \(-0.3623-0.152\)i & 0.6 \\ F & \(-1.2462+2.0772\)i & \(-1.2462-2.0772\)i & \(-0.3947\)i & \(-0.1129\)i & 0.6 \\ G & \(-0.8145+1.6289\)i & \(-0.8145-1.6289\)i & \(-0.6855+0.1083\)i & \(-0.6855-0.1083\)i & 0.5 \\ H & \(-0.6852+1.142\)i & \(-0.6852-1.142\)i & \(-0.8148+0.2052\)i & \(-0.8148-0.2052\)i & 0.6 \\ I & \(-0.9654+1.6088\)i & \(-0.9654-1.6088\)i & \(-0.6266\) & \(-0.4426\) & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The closed-loop poles and the corresponding smallest m of all points A- I.  
Figure 13: The closed-loop setpoint frequency response of the process \(G(s)\) for points H, C, and I. The control law of [PERSON] method is as follows:  
\[\left\{ \begin{array}{l}u(t)=k(b_{1}y^{\prime}_{r}(t)-y(t))+\frac{k}{T_{i}}\int \limits_{0}^{t}(y^{\prime}_{r}(\tau)-y(\tau))d\tau\\ +kT_{d}\Big{(}-\frac{dy(t)}{dt}\Big{)}\\ y^{\prime}_{r}(s)=\frac{1}{\left(1+sT_{sp}\right)}y_{r}(s)\end{array} \right.\]  
Under the condition that \(M_{s}=2\), eight different processes [15] serve as examples. The eight processes cover high-order process, non-minimum phase process, time-delay process, and integrating process, which are listed as follows:  
\[G_{1}=\frac{1}{s(s+1)^{3}}\]  
\[G_{2}=\frac{e^{-5s}}{\left(s+1\right)^{3}}\]  
\[G_{3}=\frac{1}{(s+1)(0.2s+1)(0.04s+1)(0.008s+1)}\]  
\[G_{4}=\frac{1}{\left(s+1\right)^{4}}\]  
\[G_{5}=\frac{1}{\left(s+1\right)^{5}}\]  
\[G_{6}=\frac{1}{\left(s+1\right)^{6}}\]  
\[G_{7}=\frac{1}{\left(s+1\right)^{7}}\]  
\[G_{8}=\frac{-2s+1}{\left(s+1\right)^{3}}\]  
Table 4 collects the related parameters of the DDE\(-\)GFM, including \(h_{0}\), \(h_{1}\), \(m\), \(k_{p}\), \(k_{i}\), \(k_{d}\), and \(b\), and the related parameters of [PERSON] method, which covers \(k\), \(T_{i}\), \(T_{d}\), \(b_{1}\) and \(T_{sp}\). Table 5 presents the position of the work point in the \((q,p)\) plane of the DDE\(-\)GFM and the control signal and time response of the two methods. On the one hand, it can be seen that the work points of all the processes are positive, as well as the nearby \(q-\)axis, which can make parameter \(k\) large enough. Consequently, the estimation of the total disturbance is more precise. On the other hand, the settling time of the DDE\(-\)GFM for \(G_{3}\) and \(G_{8}\) is obviously shorter than that of [PERSON] method. Moreover, compared with [PERSON] method, the overshooting of the DDE\(-\)GFM is substantially smaller, except for \(G_{3}\) for which the overshooting is zero.