(2)
由 \(-\dfrac{1}{N(X)} = G(j\omega)\)，得
\[
\begin{cases}
\mathrm{Re}(G) = \mathrm{Re}\left(\dfrac{-1}{N}\right) \\
\mathrm{Im}(G) = \mathrm{Im}\left(\dfrac{-1}{N}\right)
\end{cases}
\Rightarrow
\begin{cases}
\dfrac{\pi h}{4} = \dfrac{20}{\omega(1+0.01\omega^2)} \\
\dfrac{\pi X}{4}\sqrt{1-\left(\dfrac{h}{X}\right)^2} = \dfrac{2}{1+0.01\omega^2}
\end{cases}
\]
由①得：
\[
h = \dfrac{80}{\pi\omega(1+0.01\omega^2)}，当\omega \geq 20时，h \leq \dfrac{0.8}{\pi}0.255
\]
由①、②得：
\[
\dfrac{10\sqrt{X^2-h^2}}{\omega} = h \Rightarrow h = \sqrt{\dfrac{X^2}{\dfrac{\omega^2}{100}+1}}
\]
当 \(\omega \geq 20, X \leq 0.7\) 时，\(h \leq 0.313\)
所以h的范围是 \(0<h\leq0.255\)
五、
(1)
开环脉冲传函为：
\[
G(z) = Z\left[\dfrac{1-e^{-Ts}}{s}\cdot\dfrac{k}{2s+1}\right] = k(1-z^{-1})Z\left[\dfrac{k}{s(2s+1)}\right] = k(1-z^{-1})Z\left[\dfrac{1}{s}-\dfrac{1}{s+0.5}\right]
\]
\[
= k\dfrac{z-1}{z}\left[\dfrac{z}{z-1}-\dfrac{z}{z-0.95}\right] = k - k\dfrac{z-1}{z-0.95} = \dfrac{0.05k}{z-0.95}
\]
闭环脉冲传函为：
\[
\Phi(z) = \dfrac{G(z)}{1+G(z)} = \dfrac{0.05k}{z-0.95+0.05k}
\]
(2)
闭环特征方程为 \(D(z) = z - 0.95 + 0.05k = 0\)