## 2 EDOB and FDOB
### The FDOB Control  
In OCS, the CCD's low-frequency signal is very valuable. On the contrary, although the accelerometer has a bad low-frequency signal, it has a high bandwidth (around 1000 Hz) nearly with no delay. It is reasonable to combine the CCD with the MEMS accelerometer to get a fusion acceleration. The basic principle of the proposed complementary filter is shown in Figure 2.  
From Figure 2, we get  
\[\begin{array}{lll}S_{out}&=&\left[\frac{C(s)}{1+C(s)}H_{low-bandwidth}+ \frac{1}{1+C(s)}H_{high-bandwidth}\right]\cdot S_{in}\\ &=&\left[B_{closed}\cdot H_{low-bandwidth}+B_{restrain}\cdot H_{high-bandwidth}\right]\cdot S_{in}\end{array} \tag{5}\]  
The closed-loop transfer function \(B_{closed}\) is a low-pass filter which is to extract the low-bandwidth sensor's signal, while the restrain transfer function \(B_{restrain}\) is a high-pass one to filter the high-bandwidth sensor's signal. If \(H_{low-bandwidth}=H_{high-bandwidth}\), the fusion output is equal to the real motion status. However, even if \(H_{low-bandwidth}\
eq H_{high-bandwidth}\), it does not matter because we can also calculate the real model with spectrum fitting. This is equivalent to rebuilding the platform's model and producing a virtual sensor.  
Suppose the platform's new acceleration model is \(F_{a}(s)\) under the complementary filter method, \(F_{a}(s)/G_{a}(s)\) represents the transfer characteristic of the virtual accelerometer. Depending on \(F_{a}(s)\), we build a new DOB structure called FDOB, which is shown in Figure 3.  
The closed-loop acceleration is given as follows  
\[a=uG_{a}+\theta_{d}S^{2} \tag{6}\]  
\[u=(a_{ref}-a)C_{a}-(a\frac{F_{a}}{G_{a}}-u\widetilde{F}_{a})C_{f} \tag{7}\]  
After substitution,  
\[a-\theta_{d}S^{2}=(a_{ref}-a)C_{a}G_{a}-[aF_{a}-(a-\theta_{d}S^{2})\widetilde {F}_{a}]C_{f} \tag{8}\]  
Figure 2: The basic principle of the complementary filter. \(H_{high-bandwidth}\) and \(H_{low-bandwidth}\) are respectively the transfer characteristics of the high-bandwidth and low-bandwidth sensors. \(C(s)\) is a designed open-loop filter. \(S_{in}\) and \(S_{out}\) are respectively the real motion status and the fusion output.  
So,  
\[a=\frac{C_{a}G_{a}}{1+C_{a}G_{a}+C_{f}(F_{a}-\widetilde{F}_{a})}a_{ref}+\frac{(1- C_{f}\widetilde{F}_{a})s^{2}}{1+C_{a}G_{a}+C_{f}(F_{a}-\widetilde{F}_{a})}\theta_{d} \tag{9}\]  
Similarly, we set \(a_{ref}=0\), the acceleration loop DS of the FDOB method is as follows.  
\[E_{FDOB}=\frac{a}{\theta_{d}s^{2}}=\frac{1-C_{f}\widetilde{F}_{a}}{1+C_{a}G_{a }+C_{f}(F_{a}-\widetilde{F}_{a})} \tag{10}\]  
Compared with Equation (4), the compensation object has changed to be \(\widetilde{F}_{a}(s)\). Different from directly using the accelerometer's data, the fusion acceleration is better with no drift and less noise in low frequency, which means we can observe a more accurate outer disturbance and the low-frequency DS would be much more improved than by the EDOB method.  
Unfortunately, the FDOB as a feedforward way would not solve all the problems. Due to the existing of residual noise, the effect of compensation would be poor and even terrible in lower frequency, which is faced by all the feedforward methods. Considering that a feedback loop could get a converged result and naturally suppress noise's influence, we attempt to build a virtual velocity loop between the acceleration loop and the position loop.