北方工业大学 2006 年自动控制原理真题答案
一、
- B 2. C 3. A 4. C 5. A（状态空间模型不唯一）
二、
(1)
\(D(s) = Ts^2 + s + k \Rightarrow \omega_n = \dfrac{1}{T}, \omega_n^2 = \dfrac{k}{T} \Rightarrow kT = 1\)
\(u = s+1 \Rightarrow s = u-1 \Rightarrow D(u) = Tu^2 + (1-2T)u + T + k - 1 \leq 0\)
根据劳斯判据
| \(u^2\) | \(T\) | \(T+k-1\) |
| \(u^2\) | \(1-2T\) | |
| \(u^0\) | \(T+k-1\) |
\(\Rightarrow 0<T<0.5, k>1\)
(2)
\(t_s = \dfrac{3.5}{\xi \omega_n} = 7T \Rightarrow 0 < t_s < 3.5\)
(3)
\(\dfrac{C(s)}{R(s)} = \dfrac{G(s)(k_0 s + 1)}{1+G(s)} = \dfrac{k(k_0 s+1)}{Ts^2+s+k}\)
\(E(s) = R(s) - C(s) = R(s)\left[1 - \dfrac{C(s)}{R(s)}\right] = \dfrac{Ts^2 + (1-kk_0)s}{Ts^2+s+k} R(s)\)
\(e_{ss} = \lim_{s \to 0} sE(s) = 0 \Rightarrow 1 - kk_0 = 0 \Rightarrow k_0 = \dfrac{1}{k}\)
三、
(1)
\(G(s) = \dfrac{k(s+3)}{s(s+2)}\)
开环极点：\(0, -2\)，开环零点：\(-3\)
实轴上的根轨迹：\((-\infty, -3], [-2, 0]\)