## 5 Experimental Verification
### The FDOB Experiment Based on Acceleration and Position Dual-Loop Control  
Figure 7 presents the time-domain curves of the MEMS accelerometer and the virtual accelerometer in different frequencies. Below 2 Hz, the MEMS accelerometer's signal contains significant drift and noise, while the virtual accelerometer's signal is much better with no drift and less noise, which means the modified complementary is valid. Above 2 Hz, the signals of the MEMS accelerometer and the virtual one are both good. Although they may have different amplitudes in the same frequency, it does not matter because we can identify the last transfer function with spectrum fitting. Figure 8 exhibits their open-loop bode responses from 1 Hz to 1 KHz. The blue line represents the response measured from the real platform, while the red line represents the fitting one from the math model. Obviously, the matching degree of fitting is very high. And the specific identified transfer functions are as follows. \[\widetilde{G}_{a}(s)=\frac{0.0022s^{2}}{0.007s^{2}+0.0185s+1}\cdot\frac{1}{0.00 04s+1} \tag{26}\]  
\[\widetilde{F}_{a}(s)=\frac{0.0022s^{2}}{0.007s^{2}+0.04s+1}\cdot\frac{1}{0.00 04s+1}\cdot\frac{1}{0.003s+1} \tag{27}\]  
\(\widetilde{G}_{a}(s)\) is the highly approximate model of the original acceleration model based on the MEMS accelerometer, which has a nearly 1 KHz bandwidth and is very suitable for closed-loop feedback control. \(\widetilde{F}_{a}(s)\) with no more than 100 Hz bandwidth, is the rebuilt acceleration model based on the virtual accelerometer, which is more suitable for DOB control. Figure 6: Experimental apparatus. To measure the closed-loop DS of the OCS, firstly, we should design the acceleration controller \(C_{a}\), which is usually designed as a lag controller. \[C_{a}=150\cdot\frac{0.0007s^{2}+0.0185s+1}{s(1+0.00077s)} \tag{28}\]  
After the closed-loop acceleration has been designed, we can add the FDOB. According to Equation (12), \(C_{f}\) should be designed as follows. \[C_{f-ideal}(s)=\frac{0.007s^{2}+0.04s+1}{0.0022s^{2}}\cdot(0.003s+1)\cdot(0.00 04s+1) \tag{29}\]  
Figure 8: The open-loop bode response of acceleration. (**a**) \(\widetilde{G}_{a}\) based on the MEMS accelerometer; (**b**) \(\widetilde{F}_{a}\) based on the virtual accelerometer. Figure 7: The time-domain curves of the MEMS accelerometer and the virtual accelerometer. In Equation (29), since the order of numerator is higher than the one of denominator, it cannot be accomplished in physics. The actually used controller is presented as Equation (30). \[C_{f}(s)=\frac{0.007s^{2}+0.04s+1}{0.0022s^{2}}\cdot\frac{(0.003s+1)}{(0.001s+1)} \tag{30}\]  
From Equation (11), we know the improvement of DS is brought by the numerator, so we focus on its value after compensation. \[ \begin{array}{l}1-C_{f}\widetilde{F}_{a}\\ =1-[\frac{0.007s^{2}+0.04s+1}{0.0022s^{2}}\cdot\frac{(0.003s+1)}{(0.001s+1)}] \cdot[\frac{0.0022s^{2}}{0.007s^{2}+0.04s+1}\cdot\frac{1}{0.0004s+1}\cdot\frac{ 1}{0.003s+1}]\\ =1-\frac{1}{(0.001s+1)(0.0004s+1)}=\frac{(4e-7)s^{2}+0.0014s}{(0.001s+1)(0.0004s +1)}\end{array}  \tag{31}\]  
Equation (31) is a high-pass filter as Figure 9. Compared with the traditional pure ADLC method, the low-frequency improvement is very apparent and as the frequency goes up, the promotion becomes smaller and smaller until there is no improvement, which is in line with forecasts. Figure 10 presents the measured improvement of DS brought by EDOB and EDOB to the pure ADLC method. Obviously, the DS of the pure ADLC is not satisfied. Although the EDOB method could enhance the DS in low and medium frequencies, the low-frequency improvement is insufficient because of the MEMS accelerometer's drift and much noise. Fortunately, the proposed FDOB has solved the problem. Compared with pure ADLC method, FDOB could extremely promote the low-frequency DSA and the maximum improvement could reach \(-30\) dB in 1.8 Hz.