## 2 Tracking Control System and Disturbance Observer  
A typical tracking control system is shown in Figure 1, which can be used to track the movement of stars and communicate between the earth and satellites. The CCD is utilized to detect the errors between the target position and pointing position. However, the disturbance under the pedestal affects the tracking accuracy. Moreover, the frame rate of the CCD is limited. In order to improve the disturbance suppression ability and tracking accuracy, a MFC and a disturbance observer are recommended in the TCS.  
The structure of the multi-loop feedback control system is shown in Figure 2[13; 17]. It is composed of a position feedback control (PFC) loop, a velocity feedback control (VFC) loop and an acceleration feedback control (AFC) loop. There are three kinds of sensors--the position sensor, velocity sensor and acceleration sensor--which are used to acquire the position information, angular velocity and angular acceleration, respectively.  
In this figure, \(G_{a}(s)\) denotes the acceleration controlled plant, \(C_{a}(s)\) denotes the acceleration controller, \(C_{p}(s)\) denotes the velocity controller, \(C_{p}(s)\) denotes the position controller, \(\theta_{d}\) denotes the disturbance angle, and \(\theta_{ref}\) denotes the given target position.  
In addition to the CCD, the MEMS accelerometer has also been widely used in target tracking systems in recent years due to its excellent performance. Due to its high bandwidth and specific dynamic characters, the transfer function of the MEMS accelerometer equals constant 1. As a result, the acceleration transfer function of AFC is depicted in Equation (1).  
\[a=\frac{C_{a}G_{a}}{1+C_{a}G_{a}}a_{ref}+\frac{1}{1+C_{a}G_{a}}s^{2}\theta_{d} -\frac{C_{a}G_{a}}{1+C_{a}G_{a}}a_{drift} \tag{1}\]  
The MEMS accelerometer can also be utilized to measure the frequency characteristic up to 1 kHz, which means high-precision nominal plant \(\widetilde{G}_{a}\approx G_{a}\) can be acquired through system identification [11; 12; 17]. However, there is massive drift caused by the insensitivity at low frequency of the MEMS accelerometer when angular acceleration is detected. The drift makes it difficult to improve the disturbance suppression ability at low frequency with the feedforward method. The harm from the drift is especially obvious when the DOB method is introduced into the TCS with the MEMS accelerometer as shown in Figure 3.  
Figure 1: A typical tracking control system.  
Figure 2: The multi-loop feedback control system.  
The acceleration transfer function in Figure 3 is depicted in Equation (2). It is clear that the drift rejection ability when the DOB is recommended is even worse than that of the AFC according to the comparison of Equations (1) and (2). When a DOB controller is designed, there are multiple integration elements which are used to offset the multiple differential elements in \(G_{a}\). As a result, considerable drift and the multiple integration of the DOB controller finally lead to integral saturation when the MEMS accelerometer works at low frequency [30].  
\[a=\frac{C_{a}G_{a}}{1+C_{a}G_{a}}a_{ref}+\frac{1-\widetilde{G}_{a}C_{f}}{1+C_{ a}G_{a}}s^{2}\theta_{d}-\frac{G_{a}C_{f}+C_{a}G_{a}}{1+C_{a}G_{a}}a_{drift}, \tag{2}\]  
where \(C_{f}\) denotes the DOB controller.  
Through Equation (2), it is clear that the ideal DOB controller is \(C_{f\_ideal}=\widetilde{G_{a}}^{-1}\). As the nominal plant \(\widetilde{G}_{a}\) of the controlled object is a strictly proper transfer function, the ideal DOB controller is a non-causal system, which can be handled with a low-pass filter [29; 30].