## 2 Problem Solve
### Robustness of Equivalent Composite Control  
From Equation (12), it is apparent that equivalent composite control modifies the closed-loop transfer function of the velocity loop, thereby affecting the controlled object of the position loop. Further analysis is needed to understand the influence of the compensator \(K_{A}(s)\) on \(G_{P}(s)\). For ease of formula derivation and to better observe the impact of the time coefficient on changes in the system's poles and zeros, we assume \(a_{1}=0\) and \(k_{v}\geq\frac{4}{b_{2}^{2}}\). As a result, the controlled object of the position loop \(\tilde{G}_{P}(s)\) can be expressed as shown in Equation (17):  
\[ \begin{split}&\tilde{G}_{P}(s)=\frac{1}{a_{2}s^{2}}\frac{(s+z_{1})(s+ z_{2})}{(s+p_{1})(s+p_{2})}\\ & p_{1}=\frac{2k_{v}}{\left(k_{v}b_{2}+\frac{1}{a_{2}}\right)+ \sqrt{\left(k_{v}b_{2}+\frac{1}{a_{2}}\right)^{2}-4k_{v}}}\\ & p_{2}=\frac{\left(k_{v}b_{2}+\frac{1}{a_{2}}\right)+\sqrt{ \left(k_{v}b_{2}+\frac{1}{a_{2}}\right)^{2}-4k_{v}}}{2}\end{split}  \tag{17}\]  
where \(z_{1}=1/b_{2}\), \(z_{2}=1/a_{2}\). As \(a_{2}\) varies, one zero and two poles will change accordingly. With the decrease of \(a_{2}\), the zero \(Z_{2}\) gradually moves away from the imaginary axis, pole \(P_{1}\) approaches the imaginary axis, and pole \(P_{2}\) moves away from the imaginary axis. The movement of zero \(Z_{2}\) and pole \(P_{1}\) will affect the stability of the system. The trajectory is illustrated in Figure 7. In the figure, \(Z_{3}\) represents the zero of the position loop controller \(K_{P}(s)\). Even if the parameters of the velocity loop do not meet \(k_{v}\geq\frac{4}{b_{2}^{2}}\), the movement trend of the zero and pole remains unchanged. This is not further elaborated here. The results are depicted in Figure 8.  
According to Equation (11), with the adoption of velocity lag compensation, the open-loop transfer function \(W_{Po}(s)\) of the position loop becomes a third-order system. It exhibits a decay rate of \(-60\) dB per decade in the low-frequency range. The stability of the system is maintained due to the presence of three dominant zeros, \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\), which introduce lag, causing \(W_{Po}(s)\) to eventually cross the real axis at a rate lower than \(-40\) dB per decade. However, as the time coefficient \(a_{2}\) decreases, although it enhances system performance, if \(a_{2}\) becomes sufficiently small, pole \(P_{1}\) may become smaller than \(Z_{1}\), \(Z_{2}\), or even \(Z_{3}\). In such cases, when \(W_{Po1}\) crosses the real axis at a rate exceeding \(-40\) dB per decade, the system becomes unstable. Figure 7: The locus of main zero and pole when \(k_{v}\geq 4/b_{2}^{2}\). From Equation (17), it is evident that \(G_{P}(s)\) has no non-negative poles. According to the Nyquist stability criterion, a system is stable as long as its Nyquist plot does not encircle the point \((-1,j0)\). Therefore, the distance \(\rho_{\text{min}}\) from the Nyquist plot to the point \((-1,j0)\) represents the true stability margin of the electro-optical equipment servo system. This value happens to be equal to the reciprocal of the maximum sensitivity of the position loop, defined as shown in Equation (18):  
\[S=\frac{d\ln W_{Pc}}{d\ln G_{\text{P1}}}=\frac{G_{\text{P1}}}{W_{Pc}}\frac{dW_{ Pc}}{dG_{\text{P1}}}=\frac{1}{1+K_{p}G_{\text{P1}}} \tag{18}\]  
where \(W_{Pc}\) represents the closed-loop transfer function of the tracking loop. The relationship between \(\rho_{\text{min}}\) and \(S(j\omega)\) is described by  
\[\rho_{\text{min}}=\left|\text{min}(1+K_{p}G_{\text{P1}})\right|=\text{max} \left|\frac{1}{1+K_{p}G_{\text{P1}}}\right|=\text{max}\left|\frac{1}{S(j\omega )}\right| \tag{19}\]  
The impact of velocity lag compensation on sensitivity is illustrated in Table 2, where \(\omega_{0}\) denotes the point of \(|S(j\omega_{0})|=1\). From Table 2, it is evident that as \(a_{2}\) decreases, the maximum sensitivity also gradually increases. According to engineering experience, sensitivity should be controlled within the range of 1.2 to 2.