# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 4 Experimental Results
### Synchronization Control Experiment Tests  
The system frequency response performances before using the synchronization control are tested under random excitation with a peak-peak value of \(\pm\)1 mm and a frequency range of 0.1-100 Hz. The system frequency response characteristics of the dual-shaking table system are shown in Figure 17. It can be seen that there is large difference between the two shaking tables both in magnitude and phase frequency characteristics, which can cause the large synchronization error when the shaking tables move. Note that the experimental results demonstrate that the control effect of the proposed control scheme are similar in all DOFs, so this part and the next part will list only the ZDOF experimental results for simplicity.  
The step signal or random signal is given to tune the parameters of the proposed ASMC and traditional SMC, through watching the synchronization error of the two shaking tables. First, set all these parameters to 0 and then increase the parameters' values to reduce the synchronization error until the error stops reducing. The final tuned parameters are shown in Table 4 corresponding to the following experimental results.  
For the purpose of comparison, the synchronization control experiments are performed under sine signal and random signal excitations using the proposed ASMC controller and traditional SMC controller, respectively. The 2 mm 12 Hz sine signal experiment results are shown in Figure 18, from which it can be seen that the maximum synchronization error is 1.301 mm without synchronization control and the error is reduced to 0.866 mm and 0.3 mm using the traditional SMC and proposed ASMC synchronization controller, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Controller** & \(\mathbf{k_{s}}\) & \(\mathbf{\mathrm{f_{s}}}\) & \(\mathbf{c}\) & **ff** \\ \hline Traditional SMC & 6.4 & 0.95 & 0 & 0 \\ ASMC & 8.6 & 1.2 & 4 & 0.6 \\ \hline \end{tabular}
\end{table}
Table 4: The parameters of the sliding mode control based on adaptive reaching law (ASMC) and traditional sliding mode control (SMC).  
Figure 18: Sine signal synchronization control test results. (**a**) Without synchronization control; (**b**) synchronization control with standard sliding mode control (SMC); (**c**) Proposed sliding mode control based on adaptive reaching law (ASMC) synchronization control.  
The random signal excitations with amplitude of \(\pm 1\) mm and frequency range of 0.1 Hz-30 Hz are used to test different controllers. The reference signal in the time domain and frequency domain are shown in Figure 19. Also, the experimental results of the synchronization errors are shown in Figure 20. The detailed data comparisons of errors are tabulated in Table 5, which counts the maximum errors and root mean square (RMS) errors.  
From Figure 20 and Table 5, the maximum and RMS of synchronization errors are from 0.2888 mm to 0.0306 mm and from 0.0854 mm to 0.0089 mm, which is almost reduces to 10% and demonstrates that the proposed synchronization controller is feasible and efficient. Figure 21 shows the system frequency response characteristics with the proposed ASMC synchronization controller, which demonstrates that the two shaking tables with the proposed synchronization controller have similar frequency characteristics.  
Figure 19: The reference signal. (**a**) Time domain; (**b**) Frequency domain.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controller** & **Without** & **Traditional** & **Proposed** \\ \hline Maximum errors & 0.2888 mm & 0.1994 mm & 0.0306 mm \\ Root mean square (RMS) errors & 0.0854 mm & 0.056 mm & 0.0089 mm \\ \hline \hline \end{tabular}
\end{table}
Table 5: Synchronization errors under random signal.  
Figure 20: Synchronization errors under random signal. (**a**) Time domain; (**b**) Frequency domain.