## 3 Modified Disturbance Observer
By contrast, the cut-off frequency of the drift rejection with the MDOB is much smaller for constant \(K\), which means that it can bring much better drift rejection ability as shown in \[a=\theta_{d}s^{2}+uG_{a} \tag{3}\]2  
Under the MFC structure as shown in Figure 2, the disturbance suppression ability of the position loop is depicted as Equation (14), and the disturbance suppression ability of the position loop with the MDOB is shown in Equation (15). Through the comparison of Equations (14) and (15), the disturbance suppression ability of the position loop is changed by introducing the MDOB to the MFC, and the improvement can be represented as \(\left|1/(1+\widetilde{G}_{a}C_{f})\right|<1\). \[a=\theta_{d}s^{2}+uG_{a} \tag{3}\]3  
\[a=\theta_{d}s^{2}+uG_{a} \tag{3}\]4  
\[ \begin{array}{ll}T_{dis\_MFC+MDOB}&=\frac{1}{[1+C_{f}G_{a}+C_{c}G_{a}(1+ \widetilde{G}_{a}C_{f})]+\frac{1}{s}C_{v}C_{a}G_{a}(1+\widetilde{G}_{a}C_{f})+ \frac{1}{s^{2}}C_{v}C_{v}C_{a}G_{a}(1+\widetilde{G}_{a}C_{f})}\theta_{d}\\ &=\frac{1}{(1+C_{a}G_{a})(1+\widetilde{G}_{a}C_{f})+\frac{1}{s}C_{c}G_{a}(1+ \widetilde{G}_{a}C_{f})+\frac{1}{s^{2}}C_{p}C_{v}C_{a}G_{a}(1+\widetilde{G}_{ a}C_{f})}\theta_{d}\\ &=\frac{1}{1+C_{a}G_{a}+\frac{1}{s}C_{v}C_{a}G_{a}+\frac{1}{s^{2}}C_{p}C_{v}C_ {a}G_{a}}\cdot\frac{1}{(1+\widetilde{G}_{a}C_{f})}\theta_{d}\end{array}  \tag{15}\]  
According to the previous analyses, we can obtain the enhanced part of the disturbance suppression ability of the MFC with a MDOB controller, which is equal to \(\left|1/(1+\widetilde{G}_{a}C_{f})\right|\), as shown in Figure 6. In fact, the drift rejection performance determines whether the enhanced part of the disturbance suppression ability can be achieved. Except for the multiple integration elements to offset the multiple differential elements in \(\widetilde{G}_{a}\), the MDOB controller even includes a further  
Figure 5: The comparison of overall drift rejection. integration element. If there is too much drift left when the MDOB is utilized, it will be extremely easy for the multiple integration elements of the MDOB controller to lead to integral saturation. Fortunately, benefiting from the improvement in drift rejection ability with the MDOB, much more drift produced by the insensitivity at low frequency of the MEMS accelerometer is rejected. As a result, the possibility of the occurrence of integral saturation becomes much lower, and this means that we can enhance the disturbance suppression ability at low frequency, which was somewhat difficult to achieve before. In addition, it is obvious that if constant \(K\) increases, the disturbance suppression ability will be better but the drift rejection ability will become worse according to Equations (11) and (12). Therefore, there is a trade-off between disturbance suppression and drift rejection which depends on the specific characters of different types of MEMS accelerometers. The robustness of a system refers to the sensitivity of the system to parameter changes, which is also an important factor for evaluating the control performance. Moreover, the sensitivity functions of the AFC and the AFC with the MDOB are as follows:  
\[S_{AFC}=\frac{\left(\frac{C_{a}(G_{a}+\Delta G_{a})}{1+C_{a}(G_{a}+\Delta G_{ a})}.\frac{1}{s}-\frac{C_{a}G_{a}}{1+C_{a}G_{a}}.\frac{1}{s}\right)/\left(\frac{C_{a}G_{a}}{1+ C_{a}G_{a}}.\frac{1}{s}\right)}{\Delta G_{a}/G_{a}}=\frac{1}{1+C_{a}(G_{a}+ \Delta G_{a})}\approx\frac{1}{1+C_{a}G_{a}} \tag{16}\]  
\[ \begin{array}{ll}S_{AFC+MDOB}&=\frac{(D^{\prime}-D)/D}{\Delta G_{a}/G_{a}}\\ &=\frac{1}{1+(G_{a}+\Delta G_{a})C_{f}+C_{a}(G_{a}+\Delta G_{a})(1+\widetilde {G}_{a}C_{f})}\\ &\approx\frac{1}{(1+C_{a}G_{a})(1+\widetilde{G}_{a}C_{f})},\end{array}