## 3 Modified Disturbance Observer  
The reason for the occurrence of integral saturation is that there is too much drift that cannot be effectively rejected by the DOB when the MEMS accelerometer detects movements at low frequency. The key to avoiding integral saturation is that more drift of the accelerometer needs to be rejected. Therefore, we propose a creative MDOB structure as shown in Figure 4, where we change the node of feedforward by moving the output node of DOB controller \(C_{f}\) from the outer loop of the nominal plant to its inner loop. Moreover, in order to adapt to this change, we then propose a corresponding new simple but effective way to design the MDOB controller, which enormously simplifies the design process of the MDOB controller and effectively improves both drift rejection and disturbance suppression performance. Figure 3: A structure of disturbance observer in acceleration feedback control (AFC). The mathematical analyses in Figure 4 are as follows:  
\[a=\theta_{d}s^{2}+uG_{a} \tag{3}\]  
\[u=u_{c}-(a+a_{drift}-u_{c}\widetilde{\mathcal{C}}_{a})C_{f} \tag{4}\]  
\[u_{c}=(a_{ref}-a-a_{drift})C_{a} \tag{5}\]  
After substitution, it is easy to obtain the following transfer function of the AFC with the MDOB:  
\[a=\frac{C_{a}G_{a}(1+\widetilde{G}_{a}C_{f})}{1+G_{a}C_{f}+C_{a}G_{a}(1+ \widetilde{G}_{a}C_{f})}a_{ref}+\frac{1}{1+G_{a}C_{f}+C_{a}G_{a}(1+\widetilde{G }_{a}C_{f})}s^{2}\theta_{d}-\frac{G_{a}C_{f}+C_{a}G_{a}\widetilde{G}_{a}C_{f}}{ 1+G_{a}C_{f}+C_{a}G_{a}(1+\widetilde{G}_{a}C_{f})}a_{drift} \tag{6}\]  
Due to \(G_{a}\approx\widetilde{G}_{a}\), Equation (6) can also be simplified as Equation (7). As shown in Equation (7), the disturbance transfer function and drift transfer function can be described as Equations (8) and (9), respectively. Compared with the acceleration feedback closed loop without the DOB or MDOB shown in Equation (1), \(T_{drift}\) is totally changed by the recommended MDOB. This means that the drift rejection ability can be greatly improved by designing the MDOB controller. Beyond this, the tracking performance is not influenced by introducing the MDOB into the AFC, as shown in Equations (1) and (7). \[a=\frac{C_{a}G_{a}}{1+C_{a}G_{a}}a_{ref}+\frac{1}{(1+C_{a}G_{a})(1+\widetilde{ G}_{a}C_{f})}s^{2}\theta_{d}-\frac{\widetilde{G}_{a}C_{f}}{1+\widetilde{G}_{a}C_{f}} a_{drift} \tag{7}\]  
\[T_{dis}(s)=\frac{1}{(1+C_{a}G_{a})(1+\widetilde{G}_{a}C_{f})}s^{2}\theta_{d} \tag{8}\]  
\[T_{drift}(s)=\frac{\widetilde{G}_{a}C_{f}}{1+\widetilde{G}_{a}C_{f}}a_{drift} \tag{9}\]  
Focusing on the changes in the disturbance transfer function and drift transfer function with the MDOB, we should design a MDOB controller that is beneficial for both drift rejection and disturbance suppression. A new type of MDOB controller \(C_{f}\), which is a causal system and might be a very suitable one, is depicted in Equation (10). It is clear that there is only one parameter \(K\) that needs to be determined, meaning the design process will be enormously simplified. After the design, the overall drift rejection ability with the MDOB is shown in Equation (11). It is an inertial element and its inertia time constant is \(\frac{1}{K}\). The disturbance transfer function with the MDOB controller is shown in Equation (12), and the disturbance suppression ability at low frequency can be enhanced from constant. \[C_{f}=\frac{K}{s}\widetilde{G}_{a}^{-1} \tag{10}\]  
Figure 4: Control structure of modified disturbance observer (MDOB) in AFC. \[a=\theta_{d}s^{2}+uG_{a} \tag{3}\]0  
\[a=\theta_{d}s^{2}+uG_{a} \tag{3}\]1  
where \(\widetilde{G}_{a}{}^{-1}\) denotes the inverse of nominal plant \(\widetilde{G}_{a}\). In general, the bandwidth of the AFC loop is designed as up to 200 Hz, hence the cut-off frequency of its drift rejection is also about 200 Hz. According to the comparison of Equations (1) and (2), the drift rejection ability with the DOB is even worse when the DOB is introduced. In other words, the cut-off frequency of the drift rejection with the DOB is larger than 200 Hz.