三、
(1)
\[G(s) = \dfrac{4}{s(s+1)^2}\]
相角范围:\(-90°\) \(\to\) \(-270°\)
令 \(s=j\omega\) 代入得
\[G(j\omega) = -4\dfrac{2\omega^2+j\omega(1-\omega^2)}{4\omega^4+(\omega-\omega^3)^2}\]
过第三象限和第二象限
与虚轴交点:\(G(j1)=-2\)
(2)
\(N = N_+ - N_- = 0-1=-1\), \(Z = P-2N = 2\),系统不稳定
(3)
\[N(A) = \dfrac{4}{A\pi}\sqrt{1-\left(\dfrac{1}{A}\right)^2},\ A\ge1\]
当 \(A=1\) 时, \(-\dfrac{1}{N(A)}\to-\infty\),\(A\to+\infty\),\(-\dfrac{1}{N(A)}\to-\infty\)
对\(-\dfrac{1}{N(A)}\)求导，得 \(A = 1.28\),\(-\dfrac{1}{N(A)}=-1.61\) 两曲线有交点从不稳定区到稳定区，系统有自振
令两曲线实部和虚部相等，可得：\(A = 2.29\),\(f=\dfrac{\omega}{2\pi}=0.16HZ\)
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