## 3 The Fusion Virtual Velocity Loop  
After a virtual velocity loop is added, the ADLC system turns to be a triple-loop control(TLC) structure which is exhibited in Figure 4. The fusion method in Reference [14] is based on time domain, which would bring the CCD's time delay to the velocity no matter in low frequency or high frequency and caused a big drop of DS in relatively high frequency. The proposed complementary filter method is based on frequency domain. Since we combine the low-frequency signal of the CCD which is less affected by the delay with the high-frequency signal of the MEMS accelerometer, we minimize the impact of delay on the system.  
The whole DS without and with a virtual velocity loop is respectively as follows.  
\[\begin{array}{ll}E_{ADLC-FDOB}&=\frac{\theta}{\theta_{d}}=\frac{1-C_{f} \widetilde{F}_{a}}{1+C_{a}G_{a}+C_{f}C_{a}G_{a}\frac{1}{s^{2}}+(F_{a}- \widetilde{F}_{a})C_{f}}~{}\approx\frac{1-C_{f}\widetilde{F}_{a}}{1+C_{a}G_{a }+C_{p}C_{a}G_{a}\frac{1}{s^{2}}}\\ &=\frac{1}{1+C_{a}G_{a}}\cdot\frac{1-C_{f}\widetilde{F}_{a}}{1+C_{p}\frac{G_{a }G_{a}}{1+C_{q}G_{a}}\frac{1}{s^{2}}}\approx\frac{1}{1+C_{a}G_{a}}\cdot\frac{ 1-C_{f}\widetilde{F}_{a}}{1+C_{p}\frac{1}{s^{2}}}\end{array} \tag{11}\]  
Figure 3: The FDOB method.  
Similarly,  
\[\begin{array}{ll}E_{TLC-FDOB}&=\frac{\theta}{\theta_{d}}=\frac{1-C_{p}\bar{F}_{ a}}{1+C_{q}G_{a}+C_{p}C_{q}G_{a}\frac{1}{2}+C_{p}C_{q}C_{q}G_{a}\frac{1}{2}+C_{f}(F_{a}- \bar{F}_{a})}\\ &\approx\frac{1-C_{p}\bar{F}_{a}}{1+C_{q}G_{a}+C_{p}C_{q}G_{a}\frac{1}{2}+C_{p }C_{q}C_{q}G_{a}\frac{1}{2}}\\ &=\frac{1}{1+C_{q}G_{a}}\cdot\frac{1}{1+\frac{1}{1+\frac{C_{q}G_{a}}{1+C_{q}G_{ a}}}C_{o}}\cdot\frac{1-C_{f}\bar{F}_{a}}{1+\frac{1}{1+\frac{C_{q}G_{a}}{1+C_{q}G_{a}}C_{o}}C_{p}} \cdot\frac{1-C_{f}\bar{F}_{a}}{1+\frac{1}{1+\frac{C_{q}G_{a}}{1+C_{q}G_{a}}C_{ o}}C_{p}}\\ &\approx\frac{1}{1+C_{q}G_{a}}\cdot\frac{1}{1+C_{p}\frac{1}{3}}\cdot\frac{1-C _{f}\bar{F}_{a}}{1+C_{p}\frac{1}{3}}\end{array} \tag{12}\]  
Generally, we choose to design the open-loop transfer function to be an integral element in order to get an adequate phase and amplitude margin. So if the controllers can be designed ideally, obviously, \(|1/(1+C_{p}\cdot(1/s))|<1\). And \(|E_{TLC-FDOB}|\) would be smaller than \(|E_{ADLC-FDOB}|\), which means the whole DS would be enhanced in all frequency domain.  
As we know, the system's robustness is a key indicator of the control performance. And it refers to the system's sensitivity to parameter changes. Since the ADLC and TLC-FDOB both have a same position loop, we just need to examine the velocity sensitivity transfer function.  
The sensitivity functions of ADLC and TLC-FDOB are as follows.  
\[S_{ADLC}=\frac{\left(\frac{C_{q}(G_{a}+\Delta G_{a})}{1+C_{q}(G_{a}+\Delta G_ {a})}\cdot\frac{1}{s}-\frac{C_{q}G_{a}}{1+C_{q}G_{a}}\cdot\frac{1}{s}\right)/ \left(\frac{C_{q}G_{a}}{1+C_{q}G_{a}}\cdot\frac{1}{s}\right)}{\Delta G_{a  
Apparently, according to the above analysis, \(|S_{TLC-FDOB}|<|S_{ADLC}|\), which signifies that the FDOB and additional virtual velocity loop could also improve the robustness of the traditional ADLC structure. If the parameters of the controlled object change a little because of external interference, the system's stability of the TLC-FDOB would suffer less than that of the ADLC.