## 5 Simulation Experiment
### Loading Force Tracking
#### 5.2.3 Comparative Results of Different Outer Loop Controllers  
There are two purposes for the design of the DOB inner loop compensator. One is to suppress the disturbance including the extraneous force and friction. The other is to compensate the model mismatch between the nominal model and the controlled plant, and to make the controlled plant behave like the nominal model in a certain frequency range. Through the above analysis, the DOB compensated plant can be treated as \(P_{\text{n}}(s)\) with perturbation but no disturbance. Therefore, we design an outer loop PID controller to compare with the performance of the designed H\(\infty\) controller. The parameters of the PID controller for this experiment are proportional gain 2.48, integral gain 55.78 and differential gain 0. This PID controller and the designed H\(\infty\) controller have almost the same rise time of the step response of the nominal model \(P_{\text{n}}(s)\). Then, we measure the output of the command loading force on the controlled plant modeled by AMESim and \(P_{\text{n}}(s)\) separately, compare the output of the same controller under a different model/plant, and compare the output of the different controllers under the same model. The command loading forces are \(1+4\sin(i\times 2\pi t)\) (kN), where \(i=5,10,20,30,40,50,60,70,80,90,100\). In the AMESim model, the perturbation parameters take their nominal value and the motion disturbance always exerts on it. The maximum magnitudes of the output results are listed in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Outer loop controller** & \multicolumn{3}{c}{**PID Controller**} & \multicolumn{3}{c}{**H\(\infty\) Controller**} \\ \hline
**Frequency** & \begin{tabular}{c} **Output of** \\ \(P_{\text{n}}(s)\) **(N)** \\ \end{tabular} & \begin{tabular}{c} **Output of** \\ **AMESim (N)** \\ \end{tabular} & \begin{tabular}{c} **Relative** \\ **Error (N)** \\ \end{tabular} & \begin{tabular}{c} **Output of** \\ \(P_{\text{n}}(s)\) **(N)** \\ \end{tabular} & \
\begin{tabular}{c} **Output of** \\ **AMESim (N)** \\ \end{tabular} &
\begin{tabular}{c} **Relative** \\ **Error (N)** \\ \end{tabular} \\ \hline
5 Hz & 4984 & 4992 & 8 & 4999 & 5005 & 6 \\
10 Hz & 4980 & 5004 & 24 & 4998 & 5010 & 12 \\
20 Hz & 4983 & 5005 & 22 & 4996 & 5013 & 17 \\
30 Hz & 4994 & 5043 & 49 & 4991 & 5037 & 46 \\
40 Hz & 5010 & 5100 & 90 & 4984 & 5075 & 91 \\
50 Hz & 5031 & 5171 & 140 & 4975 & 5100 & 125 \\
60 Hz & 5058 & 5255 & 197 & 4964 & 5130 & 166 \\
70 Hz & 5082 & 5352 & 270 & 4951 & 5137 & 186 \\
80 Hz & 5120 & 5450 & 330 & 4936 & 5120 & 184 \\
90 Hz & 5157 & 5535 & 378 & 4919 & 5090 & 171 \\
100 Hz & 5237 & 5600 & 363 & 4900 & 5050 & 150 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative results of different outer loop controllers. Figure 22: The loading performance of small command loading force, (**a**) command of \(200+100\)sin\(10\pi t\) (N); (**b**) command of \(500+500\)sin\(10\pi t\) (N). As seen from the data in Table 3, the performance of the PID controller for the controlled plant is good below 50 Hz. The outputs of \(P_{\rm n}(s)\) and the controlled plant modeled by AMESim are close, and they are very similar with the results of the H\(\alpha\) controller. That is because the model mismatch and the disturbance below 50 Hz have been compensated by the DOB compensator, and this result is consistent with our design of the \(Q\)-filter. The output error between \(P_{\rm n}(s)\) and the controlled plant with PID increases with a frequency above 50 Hz, and the magnitude of the output has a trend of divergence. For the H\(\alpha\) controller, the output error between \(P_{\rm n}(s)\) and the controlled plant increases a little but starts to decrease about 70 Hz; meanwhile, the magnitude of the output has a trend of convergence. That means the designed H\(\alpha\) controller indeed guarantees system robustness in high-frequency regions while ensuring the tracking performance.