# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 4 Experimental Results
### Tracking Control Experiment Tests  
The random signal excitations with an amplitude of \(\pm 1\) mm and frequency range of 0.1 Hz-100 Hz are performed to identify the system transfer function according to the reference signal and response signal using the RLS algorithm. The system model parameters are shown in Figure 22 and the identified discrete transfer functions are expressed as  
\[ \begin{array}{l}\hat{G}_{d}(z)=\frac{1.3052\times 10^{-3}(z^{2}-0.8966z+0.3641)(z^ {2}-1.126z+1.366)(z^{2}-3.972z+7.417)}{(z+0.7426)(z-0.7161)(z^{2}-1.856z+0.8684)(z ^{2}-1.076z+0.6268)}\\ \hat{G}_{d2}(z)=\frac{1.8497\times 10^{-4}(z-10.9)(z+0.5198)(z^{2}-1.265z+0.5542)(z ^{2}+1.681z+1.98)}{(z+0.7768)(z-0.7411)(z^{2}-1.859z+0.8713)(z^{2}-1.079z+0.725) }\end{array}  \tag{43}\]  
The feed-forward inverse transfer function according to Equation (31) can be obtained as  
\[ \begin{array}{lcl}\hat{G}_{d}^{-1}(z)&=\frac{(z+0.7426)(z-0.7161)(z^{2}-1.85 6z+0.8684)(z^{2}-1.076z+0.6268)}{1.3052\times 10^{-3}(z^{2}-0.8966z+0.3641)}\\ &\times\frac{(z^{-2}-1.126z^{-1}+1.366)(z^{2}-3.972z^{-1}+7.417)}{(1-1.126+1.3 66)^{2}(1-3.972+7.417)^{2}z^{4}}\\ \hat{G}_{d2}^{-1}(z)&=\frac{(z+0.7768)(z-0.7411)(z^{2}-1.859z+0.8713)(z^{2}-1.0 79z+0.725)}{1.8497\times 10^{-4}(z^{-0.5198})(z^{2}-1.265z+0.5542)}\\ &\times\frac{(z^{-1}-10.9)(z^{-2}+1.681z^{-1}+1.98)}{(1-10.9)^{2}(1+1.681z+1.98 )}z^{3}\end{array}  \tag{44}\]  
From Equations (43) and (44), the frequency characteristics of the identified system models and the designed inverse models are shown in Figure 23.  
Figure 21: Frequency characteristics with proposed synchronization controller. (**a**) Magnitude frequency characteristics; (**b**) Phase frequency characteristics. Figure 22: The system model parameters using recursive least squares (RLS) algorithm. (**a**) 1#_shaking table; (**b**) 2#_shaking table. Note that the \(G_{d}\) and \(G_{d2}\) in Equation (43) seem to be noticeably different even though the tables are supposed to be synchronized. The noticeable difference in Equation (43) is mainly due to the fact that the order of the identified model is relatively high and there is some difference when the frequency is relatively high, as shown in Figure 23. Especially, the difference of the phase characteristic of the two tables is noticeable. However, the \(G_{d}\) and \(G_{d2}\) have the similar characteristics when the frequency is not too high both in magnitude and phase characteristics, as shown in Figure 23 and we are concerned within the characteristics within the medium and low frequency range. So, the obtained results for \(G_{d}\) and \(G_{d2}\) in Equation (43) are correct and reasonable. To verify the proposed tracking controller shown in Figure 12, the 1.5 mm 20 Hz sine signals are given and the test results are shown in Figure 24. Note that the tracking control tests below are both with the proposed ASMC synchronization controller. Also, the frequency parameter\(f_{q}\) in Equation (37) is 48 Hz during the experimental tests. It can be seen in Figure 24a that the maximum tracking errors are 2.328 mm/2.18 mm using the traditional controller, which is even larger than the maximum of the given signal due to the large phase lag of the system. The errors reduce to 0.7435 mm/0.5042 mm with the feed-forward inverse model controller and reduce to 0.3169 mm/0.2109 mm with the proposed tracking controller, which is almost 13%/9% of that of the traditional controller. The random signal tests are performed to further verify the proposed tracking controller. The random signal used in this section is the same as in Section 4.3, shown in Figure 19. The tracking errors both in the time domain and frequency domain of the two shaking tables using different controllers are shown in Figure 25. The detailed data comparisons of tracking errors are tabulated in Table 6, which counts the maximum errors and RMS errors. From Figure 25 and Table 6, it can be obtained that the maximum tracking errors using the traditional controller are 0.735 mm/0.801 mm, which is close to the maximum reference signal.