## 3 The EOB and DOB-Based Indirect Feedforward Control  
### The CCD-Based EOB  
The proposed way of combining EOB and DOB could be regarded as an indirect feedforward control of model reference. The EOB structure is shown in Figure 4. The inner velocity loop has changed the velocity transfer function and it could be treated as 1 in low frequency. The given velocity \(v_{ref}\) after passing a delay element is actually an output of the inner closed-loop model. Through combining the differential of the boresight error and the model output, a composite velocity is produced, which simultaneously contains the information of the target and disturbances. In order to better understand the essence of the EOB, its equivalent structure, as shown in Figure 5, should be referred to.  
Figure 4: The EOB structure based on VDPC. \(e^{-\alpha s}\) refers to an artificially added delay and \(\alpha=\tau\). \(Q_{1}\) refers to the EOB controller.  
Figure 5: The equivalent structure of EOB. \(B_{v}=C_{v}G_{v}/(1+C_{v}G_{v})\) refers to the closed-loop transfer function. \(B_{v}\approx 1\) at a low frequency.  
In Figure 5, the EOB is split into a tracking feedforward structure and a closed-loop DOB. In the tracking feedforward, the input of \(Q_{1}\) refers to the target's velocity with the delay of CCD. Similarly, the input of \(Q_{1}\) in DOB refers to an external disturbance with a same delay. These delayed signals are valuable, which can be fed forward to the system at a low frequency, because the phase lag caused by delay would be little at the very low frequency. However, at the high frequency, the phase lag would be great, which could lead to instability of the system. Hence, to reduce the negative influence of the delayed input, the low-pass filter \(Q_{1}\) should be adopted, which would only allow the passing of low-frequency signals. Of course, because of the existing of the delay, the effect of EOB is slightly worse than the direct feedforward way based on sensors fusion, but this still can be accepted. Therefore, EOB is an incomplete feedforward way working at the low frequency.  
To get the error transfer functions easier, the EOB in Figure 4 can also be regarded as an equivalent position controller as follows.  
\[{C^{\prime}}_{p}=\frac{sQ_{1}+C_{p}}{1-Q_{1}e^{-\alpha s}} \tag{5}\]  
With \({C^{\prime}}_{p}\), the error transfer functions in Figure 4 can be derived below.  
\[\begin{array}{rl}\widehat{S}_{R}=\frac{E}{R}&=\frac{1-Q_{1}e^{-\alpha s}}{1+Q _{1}(\frac{C_{p}}{1+C_{p}}\frac{1}{C_{p}}e^{-\tau s}-e^{-\alpha s})+C_{p}\frac {C_{p}\tau Q_{1}}{1+C_{p}\tau Q_{1}}\frac{1}{e^{-\tau s}}}\\ &\approx\frac{1-Q_{1}e^{-\alpha s}}{1+C_{p}\frac{1}{\varepsilon}e^{-\tau s}} \end{array} \tag{6}\]  
\[\begin{array}{rl}\widehat{S}_{D}=\frac{E}{D}&=\frac{1}{1+C_{p}C_{p}}\cdot \frac{1}{1+\frac{sQ_{1}+C_{p}}{1-Q_{1}e^{-\tau s}}\frac{1}{1+C_{p}C_{p}}\frac{ 1}{s}e^{-\tau s}}\\ &=\frac{1}{1+C_{p}C_{p}}\cdot\frac{1-Q_{1}e^{-\alpha s}}{1+Q_{1}(\frac{C_{p}C_ {p}C_{p}}{1+C_{p}C_{p}}e^{-\tau s}-e^{-\alpha s})+C_{p}\frac{C_{p}C_{p}}{1+C_{ p}C_{p}}\frac{1}{s}e^{-\tau s}}\\ &\approx\frac{1}{1+C_{p}C_{p}}\cdot\frac{1-Q_{1}e^{-\alpha s}}{1+C_{p}\frac{1} {s}e^{-\tau s}}\end{array} \tag{7}\]  
Compared to Equations (3) and (4), the values of Equations (6) and (7) could not reach 0 due to the existence of delay. However, it could be close to 0 at the very low frequency, because \(e^{-\alpha s}\approx 1\) under this condition. Theoretically, the lower bandwidth of \(Q_{1}\) could benefit the stability of the system, but the very low frequency would reduce the effect of feedforward. In order to maximize the benefits of the feedforward while ensuring the stability, the matter of how to choose \(Q_{1}\) with guaranteed gain margin (GM) and phase margin (PM) will be discussed in Section 4.