## 2 Problem Solve
### Robustness of Equivalent Composite Control
Therefore, choosing a time constant such that \(a_{2}/b_{2}\ \in[2,4]\) represents a good compromise between performance and robustness. When \(a_{2}=b_{2}\), the sensitivity of the controlled object reaches 4.41, which is difficult to achieve in practical engineering [23]. Since the crossover frequency of the velocity loop is typically chosen between 40 rad/s and 60 rad/s, following the design approach in this paper, \(b_{2}\in[0.067,0.125]\) is a suitable range. In this case, \(\frac{1}{a_{2}}\in[4,7.46]\). If the cutoff frequency can be increased further, \(a_{2}\) can be set to a smaller value, leading to better tracking performance of the system. The sensitivity function profoundly influences the design of electro-optical equipment servo systems. From Equation (18), it is also evident that the sensitivity function \(|S(j\omega)|\) can represent the tracking performance of the electro-optical equipment towards the target and its suppression of disturbances. A lower sensitivity implies better tracking performance for signals at that frequency and stronger suppression of disturbances. However, sensitivity is \begin{table}
\begin{tabular}{l l l l l l} \hline \(a_{2}\) & **0.3537** & **0.2830** & **0.2122** & **0.1415** & **0.0707** \\ \hline \(a_{2}/b_{2}\) & 5 & 4 & 3 & 2 & 1 \\ \(\omega_{0}\) & 10.5 & 10.1 & 9.8 & 9.87 & 11.3 \\ \(\text{max}|S(j\omega)|\) & 1.36 & 1.39 & 1.48 & 1.84 & 4.41 \\ \hline \end{tabular}
\end{table} 
Table 2: The relationship between maximum sensitivity and \(a_{2}\). Figure 8: The locus of main zero and pole when \(k_{0}\)\(<4/b_{2}^{2}\). constrained by the Bode integral theorem and cannot be arbitrarily specified. With velocity lag compensation control, the sensitivity function satisfies the following equation:  
\[\int_{0}^{\infty}\ln\lvert S(j\omega)\rvert d\omega=-\frac{\pi}{2}\lim_{s\to \infty}sK_{P}(s)G_{P}(s)=0 \tag{20}\]  
Combining Equation (20) and Table 1, it is evident that equivalent composite control does not significantly enhance \(\omega_{0}\). The sensitivity suppressed in the low-frequency range will appear in the high-frequency range. When sensitivity exceeds 1, the system will amplify disturbances. Even if a lead-lag controller is selected, sensitivity still exhibits peaks exceeding 1. Comparing the step responses of velocity lag compensation and composite control, the simulation results are depicted in Figure 9. When \(a_{2}=0.0707\), oscillations in the system increase noticeably. However, when \(a_{2}\) decreases to 0.0354, the control system becomes unstable. The impact of \(a_{2}\) on the sensitivity function curve \(\lvert S(j\omega)\rvert\) is shown in Figure 10. While a lower \(a_{2}\) improves tracking and disturbance rejection in the low-frequency range, the system will amplify disturbances more significantly in the 10 rad/s to 100 rad/s frequency range. This phenomenon is known as the waterbed effect. In the design of electro-optical equipment servo systems, special attention must be given to the region where \(\lvert S(j\omega)\rvert>1\). This is to avoid having significant energy useful signals in this frequency band, which could otherwise lead to instability and performance issues. Figure 10: Sensitivity function changes with \(a_{2}\). Figure 9: Step response of systems corresponding to different velocity lags.