北方工业大学 2012 年自动控制原理真题答案
一、
(1)
\[G(s) = \frac{k}{s(0.05s+1)}, \Phi(s) = \frac{20k}{s^2+20s+20k}\]
\[\xi = 0.5 \Rightarrow 2\xi\omega_n = 20, \omega_n^2 = 20k \Rightarrow k = 20\]
(2)
\[t_s = \frac{3.5}{\xi\omega_n} = 0.35s\]
故 60 秒后实际心速是 1 次每秒
\[h_{max} = 1 + \sigma\% = 1 + e^{-\frac{\pi\xi}{\sqrt{1-\xi^2}}}\]
二、
求出每一方框的传递函数
\[\dot{e}(t) = kr(t) - c(t)\]
\[T_2\dot{x}_2(t) + x_2(t) = \tau\dot{e}(t) + e(t)\]
\[T_1x_1(t) + x_1(t) = k_2n(t) + k_3x_2(t)\]
\[\dot{c}(t) = n(t) + x_1(t)\]
\[\Rightarrow E(s) = \frac{1}{s}[kR(s) - C(s)]\]
\[X_2(s) = \frac{\tau s + 1}{T_2 s + 1}E(s)\]
\[X_1(s) = \frac{k_2}{T_1 s + 1}N(s) + \frac{k_3}{T_1 s + 1}X_2(s)\]
\[C(s) = \frac{N(s)}{s} + \frac{X_1(s)}{s}\]
如图所示: