故 \(e_{ss} = 0\)

3.

信号流图

回路：\(L_1=-G_1G_2\), \(L_2=-G_2\), \(\Delta = 1-L_1-L_2\)

前向通路：\(P_1=G_1G_2\), \(\Delta_1=1\), \(P_2=1\), \(\Delta_2=1\)

故 \(\dfrac{C(s)}{R(s)} = \dfrac{P_1+P_2}{\Delta} = \dfrac{1+G_1G_2}{1+G_2+G_2G_1}\)

4.

四、

(1)

设系统的开环传递函数 \(G(s) = \dfrac{K}{s(s+1)(s+2)}\)

开环极点：\(P_1=0, P_2=-1, P_3=-2\)

实轴上的根轨迹：\([-1,0], (-\infty,-2]\)

分离点：\(\dfrac{dG(s)}{ds}=0 \Rightarrow 3s^2+6s+2=0 \Rightarrow s=-0.42\)

渐近线：\(\sigma_a = \dfrac{-1+(-2)}{3-0} = -1\)，\(\varphi_a = \dfrac{(2k+1)\pi}{3-0}\) (\(k=0,1,2\))

与虚轴的交点：闭环特征方程 \(D(s)=s^3+3s^2+2s+K\) 令 \(s=j\omega\) 代入