## 4 Parameters Design and Performance Analysis  
### The Design of \(Q_{1}\) and the Performance Improvement with EOB  
Firstly, we should determine the form of the original position controller \(C_{p}\) in Figure 2. According to the previous analysis, the inner loop as the controlled object was close to 1. It is easy to verify whether \(C_{p}=k=\frac{\pi}{4\tau}\) can stabilize the platform with PM more than 45\({}^{\circ}\) and GM more than 6 dB. The crossover frequency and gain frequency of the outer loop can be defined as \(\omega_{c}\) and \(\omega_{g}\). We can easily get \(\omega_{c}=\frac{\pi}{4\tau}\) and \(\omega_{g}=\frac{\pi}{2\tau}\). To investigate the stability of the system, the open-loop transfer function was presented as follows. \[G_{open}=C_{p}\frac{1}{s}e^{-\tau s} \tag{13}\]  
After adding the EOB, the open-loop transfer can be changed to Equation (14). \[ \begin{array}{ll}G^{\prime}_{open}&=C^{\prime}_{p}\frac{1}{s}e^{-\tau s}\\ &=\frac{sQ_{1}C_{p}^{-1}+1}{1-Q_{1}e^{-\tau s}}\cdot(C_{p}\frac{1}{s}e^{-\tau s} )\end{array}  \tag{14}\]  
In Equation (14), the low-pass filter \(Q_{1}\) may be set as a simple first-order form as follows. \[Q_{1}=\frac{1}{1+Ts} \tag{15}\]  
To make the equivalent controller \(C^{\prime}_{p}\) have no effect on the stability margin, two restrictions should be met: (1) \(\arg[L(j\omega_{c})]\geq 0\); (2) \(-20\log\left[L(j\omega_{g})\right]\geq 0\), where \(L(s)\) should be set as follows. \[L(s)=\frac{sQ_{1}C_{p}^{-1}+1}{1-Q_{1}e^{-as}}=\frac{ks+(Ts+1)}{(Ts+1)-e^{-as}} \tag{16}\]  
According to Appendix A, we know that the phase and gain loss of \(L(s)\) is inevitable. In order to meet the stability condition of \(PM>45^{\circ}\) and \(GM>6\)dB, we should reduce the controller gain appropriately. Although it would cause the close-loop bandwidth to drop slightly, it is acceptable because the low-frequency ability of suppressing errors is more important than the bandwidth. With the EOB structure, the previous controller \(C_{p}\) is substituted by \(0.8C_{p}\), accompanied with \(T=6\alpha\) and \(\alpha=\tau\). Comparing Equations (6) and (7) with Equations (1) and (2), it can be found that the promotion of the closed-loop performance depends on the amplitude of \(1-Q_{1}e^{-as}\). \[\left|1-Q_{1}e^{-j\tau\omega}\right|^{2}=1+\frac{1+2T\omega\sin\left(\tau \omega\right)-2\cos\left(\tau\omega\right)}{1+\left(T\omega\right)^{2}} \tag{17}\]  
Obviously, if \(\omega\to 0\), then \(\left|1-Q_{1}e^{-j\tau\omega}\right|^{2}\to 0\), which means that the improvement is huge at the very low frequency. Since \(1+2T\omega\sin\left(\tau\omega\right)\) always increases and \(2\cos\left(\tau\omega\right)\) decreases when \(\tau\omega\) changes from 0 to 0.5\(\pi\), it is evident that \(\omega_{o}\in\left(0,\ \omega_{g}\right)\) exists, which makes \(1+2T\omega_{o}\sin\left(\tau\omega_{o}\right)=2\cos\left(\tau\omega_{o}\right)\). When \(\omega<\omega_{o}\), \(\left|1-Q_{1}e^{-j\tau\omega}\right|^{2}<1\) the system performance would be enhanced. The following discussion is on the amplification of \(\left|1-Q_{1}e^{-j\tau\omega}\right|^{2}\) for errors. Equation (17) can be rewritten as follows. \[\left|1-Q_{1}e^{-j\tau\omega}\right|^{2}=1+\frac{1+2T\omega\sin\left(\tau \omega\right)-2\cos\left(\tau\omega\right)}{1+\left(T\omega\right)^{2}}\leq 1+ \frac{1+2T\omega}{1+\left(T\omega\right)^{2}}\textless 3 \tag{18}\]  
It means that the amplification of the errors at the medium frequency is limited. Moreover, actually, \(\left|1-Q_{1}e^{-j\tau\omega}\right|^{2}\) cannot reach the boundary value in Equation (18) and approaches 0 when \(\omega\) is big enough. In this paper, the CCD's delay is 0.02 s (two frames 100 Hz sampling rate). Substitute all the parameters to \(1-Q_{1}e^{-\alpha s}\) and the Simulation is shown in Figure 7. The performance improvement was big at the low frequency and even reached \(-20\) dB at 0.1 Hz. The peak in the middle frequency from 2 Hz to 20 Hz was consistent with the previous analysis, but it was very small and not at the low frequency for tracking.