\[
G(s) = \frac{k_3(\tau s + 1)}{s^2(T_1 s + 1)(T_2 s + 1)}
\]
\[
\Rightarrow \frac{C(s)}{R(s)} = \frac{Kk_3(\tau s + 1)}{s^2(T_1 s + 1)(T_2 s + 1) + k_3(\tau s + 1)}
\]
三、
(1)
\(K_d = 0\) 系统为 1 型系统
\[
G(s) = \frac{k}{s(Ts+1)}
\]
\(K_p = \lim_{s \to 0} G(s) = \infty\), \(e_{ss1} = 0\)
\(K_v = \lim_{s \to 0} sG(s) = k\), \(e_{ss2} = \dfrac{1}{k}\)
\(K_a = \lim_{s \to 0} s^2 G(s) = 0\), \(e_{ss3} = \infty\)
(2)
\[
\frac{C(s)}{R(s)} = \frac{k(k_d s + 1)}{k + s(Ts+1)}, \quad E(s) = R(s) - C(s)
\]
\[
\Rightarrow \frac{E(s)}{R(s)} = \frac{Ts^2 + (1-kk_d)s}{k + s(Ts+1)}
\]
\[
e_{ss} = \lim_{s \to 0} sR(s)\frac{E(s)}{R(s)} = 0 \Rightarrow k_d = \frac{1}{k}
\]
59