(2)
显然由图可知，当系统稳定时，其根应均在 x 轴左半边。系统的闭环特征方程为：
\[1 + G(s)H(s) = 0 \Rightarrow s^2 + 5s + K^*(s-2) = 0\]
由劳斯判据可知：
| \(s^2\) | \(1\) | \(-2K^*\) |
|---|---|---|
| \(s^1\) | \(5+K^*\) | |
| \(s^0\) | \(-2K^*(5+K^*)\) |
显然 \(-5 < K^* < 0 \Rightarrow 0 < K < 2\)
三、
\[G_0(s) = \frac{k}{s(s+1)\left(\dfrac{s}{2}+1\right)}, \quad \frac{0-20\lg k}{\lg 10 - \lg 1} = -20 \Rightarrow k = 10\]
\[\Rightarrow G_0(s) = \frac{10}{s\left(\dfrac{s}{2}+1\right)(s+1)}\]
\[G_c(s) = \frac{(5s+1)(1.25s+1)}{(50s+1)(0.125s+1)}\]
\[\Rightarrow G(s) = \frac{10(5s+1)(1.25s+1)}{s(0.5s+1)(50s+1)(0.125s+1)(s+1)}\]
由分析可知\(\omega_c\)必在1和2之间