# An Approach for Setting Parameters for Two-Degree-of-Freedom PID Controllers
## 3 Illustrative Examples
### PID Controller Parameters Analysis with Simulation  
In this subsection, the influence of the controller parameters \((m,k,h_{0},h_{1})\) will be analyzed through simulation.  
Consider a process given by \(G=\frac{1}{(s+1)^{3}}\).  
If the settling time is less than 10 s, the desired dynamic coefficient \((h_{0},h_{1})\) can be calculated from Equation (16). \(h_{1}=1.5\) and \(h_{0}=0.5625\) are chosen. The \(m\) contours are shown in Figure 4 where \(m\) is equal to 0.5, 0.6, 0.7, and 0.8 respectively.  
As shown in Figure 4, the greater the \(m\), the closer to origin the contour. Parameter \(\omega\) increases in the counter-clockwise direction. In order to analyze the influence of parameter \(m\) and parameter \(k\) on the behavior of the closed-loop time response and the frequency response, seven points A, B, C, D, E, F, and G are chosen. Among them, points C, D, E, and F are characterized by the same stability margin \(m=0.6\), and points A, B, C and G are characterized by roughly the same disturbance observer coefficient \(k\), which is big enough. The coordinates of the seven points in the \((q,p)\) plane are listed in Table 1.  
Figure 4: The \(m\) contours.  
To demonstrate the influence of \(h_{0}\) and \(h_{1}\), the same \(m\) and \(=\) different \((h_{0},h_{1})\) are chosen, and then, the \(h\) contours are depicted in Figure 11 where \(m=0.6\) and \((h_{0},h_{1})\) are respectively equal to \((0.36,1.2)\), \((0.5625,1.5)\), and \((0.81,1.8)\).  
As shown in Figure 11, the \(h\) contour is closer to the origin with the increasing of both \(h_{0}\) and \(h_{1}\). To analyze the influence of both \(h_{0}\) and \(h_{1}\), three points H, C, and I were chosen. The coordinates of these points in the \((q,p)\) plane are listed in Table 2.  
The closed-loop time response and the closed-loop frequency response of process \(G(s)\) for points H, C, and I are shown in Figures 12 and 13, which show the influence of the parameters \(h_{0}\) and \(h_{1}\). In the time response case, the load disturbance is introduced at \(t=25\) s.  
On the one hand, Figure 12 shows that the setpoint time response speeds up by increasing both \(h_{0}\) and \(h_{1}\). From Figure 13, it can be observed that the magnitude of the low-frequency range is closer to \(0\) dB. The above phenomena result from the fact that the desired dynamic is faster with greater \(h_{0}\) and  
\begin{table}
\begin{tabular}{c c c c} \hline  & **H** & **C** & **I** \\ \hline \(q\) & 1.546 & 2.128 & 2.712 \\ \(p\) & 0.0927 & 0.1106 & 0.1251 \\ \hline \end{tabular}
\end{table}
Table 2: The coordinate of points H, C, and I in the \((q,p)\) plane.  
Figure 11: The \(h\) contours.  
Figure 10: The closed-loop load disturbance frequency response of the process \(G(s)\) for points A, B, C, and G.  
\(h_{1}\). On the other hand, Figure 12 also shows that the load disturbance response returns to the target faster, and the peak is higher with greater \(h_{0}\) and \(h_{1}\).  
The simulation results of the nine points A-I have not only shown the influence of the parameters \((m,k,h_{0},h_{1})\), but also prove that the specified stability margin can be met as follows. All the closed-loop poles of points from A to I are shown in Table 3, and it can be seen that all the closed-loop poles lie in the specified sector region. From Table 3, it can be also observed that among all the \(m_{n}\) corresponding to the poles, the smallest is equal to the previous specified \(m\).