## 2 Problem Solve  
### Tracking Issues in Electro-Optical Equipment  
The closed-loop structure of the position loop in electro-optical equipment is depicted in Figure 1. In this structure, \(K_{P}(s)\) represents the controller for the position loop, and \(W_{Vc}(s)\) denotes the closed-loop transfer function of the speed loop. The open-loop transfer function of the position loop can be expressed by the following equation:  
Figure 1: Position loop of electro-optical equipment. \[W_{P_{0}}(s)=\frac{1}{s}K_{P}(s)W_{V_{C}}(s)=\frac{k_{p}}{s^{\tau}}\frac{1+d_{1}s+d _{2}s^{2}+\cdots+d_{m}s^{m}}{1+c_{1}s+c_{2}s^{2}+\cdots+c_{n}s^{n}} \tag{1}\]  
where \(\tau=1,2,3\), correspond to Type I, Type II, and Type III systems, respectively. The error transfer function of the system is given by  
\[\frac{E(s)}{\theta(s)}=\frac{1}{1+W_{P_{0}}(s)} \tag{2}\]  
According to the dynamic error coefficient method [22], the tracking error satisfies the following:  
\[E(s)=\frac{\theta(s)}{1+W_{P_{0}}(s)}=C_{0}\theta(s)+C_{1}s\theta(s)+\frac{C_ {2}}{2}s^{2}\theta(s)+\cdots \tag{3}\]  
where \(C_{i}=\left[\frac{d^{i}}{ds^{i}}\frac{E(s)}{R(s)}\right]_{s=0}\) represent the quality factors of various orders for the electro-optical equipment. Equation (1) through Equation (3) indicate that, given the open-loop transfer function, the system's tracking error can be roughly estimated using the velocity quality factor \(C_{0}\), acceleration quality factor \(C_{1}\), and jerk quality factor \(C_{2}\). The dynamic error coefficients for Type I to Type III systems are shown in Table 1.  
The frequency of the input signal \(\theta(t)\) for electro-optical equipment is typically low, generally not exceeding 1.57 rad/s [22]. It can be assumed that the input signal is entirely within the low-frequency range, i.e., lower than the first crossover frequency of the position loop. In this case, the tracking error is approximately equal to  
\[e(t)=\frac{1}{k_{p}}\bigg{(}\frac{dr(t)}{dt}\bigg{)}^{v} \tag{4}\]  
In the early designs of gyroscopes, due to limitations in the frame rate of infrared cameras and the computational capabilities of image processing chips, there was a significant lag in the deviation output. As a result, controllers could not tolerate the \(-90^{\circ}\) phase loss introduced by the integral loop, leading to the adoption of lag-lead control strategies [11]. However, with the rapid advancement in infrared imaging technology and chip-based graphics processing capabilities in recent years, the lag time of the deviation output has been greatly reduced. This reduction has made it possible to implement controllers with integral loops. \[K_{P}(s)=\frac{k_{p}\big{(}1+d_{1}{}^{\prime}s+d_{2}{}^{\prime}s^{2}+\cdots d _{m}{}^{\prime}s^{m}\big{)}}{s(1+c_{1}{}^{\prime}s+c_{2}{}^{\prime}s^{2}+ \cdots c_{n}s^{n})} \tag{5}\]  
In fact, the three main parameters that primarily affect the tracking performance of electro-optical equipment are the integral term \(\frac{1}{s}\), the open-loop gain \(k_{p}\), and the parameter \(d_{1}{}^{\prime}\) affecting the dominant zero. Therefore, the controller's form can be abbreviated to  
\[K_{P}(s)=\frac{k_{p}(1+b_{1}s)}{s} \tag{6}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \(C_{i}\) & **Type I** & **Type II** & **Type III** \\ \hline \(C_{0}\) & 0 & 0 & 0 \\ \(C_{1}\) & \(\frac{1}{k_{p}}\) & 0 & 0 \\ \(\frac{C_{2}}{2}\) & \(\frac{c_{1}-d_{1}}{k_{p}}-\frac{1}{k_{p}^{2}}\) & \(\frac{1}{k_{p}}\) & 0 \\ \(\frac{C_{3}}{3! }\) & \(\frac{1}{k_{p}^{2}}+\frac{2(c_{1}-d_{1 At this point, \(W_{P_{0}}(s)\) represents a Type II system. \(W_{P_{0}}(s)\) is capable of providing up to \(k_{p}=\omega_{P}^{2}\) system gain (with \(\omega_{P}\) as the position loop cutoff frequency), if the quality factor still cannot meet the tracking accuracy requirements, it becomes necessary to introduce coaxial tracking technology.