校正后系统的相角裕度为
\[
\gamma' = 180° + \angle G_c(\omega_c')G_0(\omega_c')
\]
\[
= \left(90° + \arctan\frac{\omega_c'}{0.2} + \arctan\frac{\omega_c'}{20} - \arctan\frac{\omega_c'}{0.02} - \arctan\frac{\omega_c'}{6.32}\right.
\]
\[
\left. - \arctan\frac{\omega_c'}{\omega_{c4}} - \arctan\frac{\omega_c'}{50}\right)\Bigg|_{\omega_c'=2}
\]
令 \(\gamma' = \gamma = 58.3°\)，解得
\[
\omega_{c4} = 9.07\ \text{rad/s}
\]
滞后超前校正网络的开环传递函数为
\[
G_c(s) = \frac{10\left(\dfrac{1}{0.2}s+1\right)\left(\dfrac{1}{6.32}s+1\right)}{\left(\dfrac{1}{0.02}s+1\right)\left(\dfrac{1}{9.07}s+1\right)}
\]
校正后系统的开环传递函数为
\[
G(s) = G_0(s)G_c(s) = \frac{20\left(\dfrac{1}{0.2}s+1\right)\left(\dfrac{1}{20}s+1\right)}{s\left(\dfrac{1}{0.02}s+1\right)\left(\dfrac{1}{6.32}s+1\right)\left(\dfrac{1}{9.07}s+1\right)\left(\dfrac{1}{50}s+1\right)}
\]
其开环对数幅频渐近特性曲线如图 6-47 所示。
MATLAB 验证：待校正系统的开环 Bode 图如图 6-48 所示，单位阶跃响应如图 6-50 所示，测得
\[
\omega_c = 1.85\ \text{rad/s}, \quad \gamma = 60.6°, \quad e_{ss}(\infty) = 0.5
\]
\[
\sigma\% = 7\%, \quad t_p = 1.29\ \text{s}, \quad t_s = 2.05\ \text{s}\ (\Delta = 2\%)
\]
已校正系统的开环 Bode 图如图 6-49 所示，单位阶跃响应如图 6-51 所示，测得
\[
\omega_c' = 1.89\ \text{rad/s}, \quad \gamma' = 59.4°, \quad e_{ss}(\infty) = 0.05
\]
\[
\sigma\% = 12\%, \quad t_p = 1.42\ \text{s}, \quad t_s = 7.04\ \text{s}\ (\Delta = 2\%)
\]
图 6-48 待校正系统的开环 Bode 图(MATLAB)
图 6-49 已校正系统的开环 Bode 图(MATLAB)
MATLAB 文本：exe616.m
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