六、
将等式两边分别作 \(z\) 变换得:
\[C(z) - 6z^{-1}C(z) + 8z^{-2}C(z) = z^{-2}R(z) \Rightarrow C(z) = R(z)\frac{z^{-2}}{1-6z^{-1}+8z^{-2}} = \frac{z}{(z-1)(z^2-6z+8)}\]
\[\Rightarrow c(k) = \frac{1}{3}\cdot 1^k - \frac{1}{2}\cdot 2^k + \frac{1}{6}\cdot 4^k\]
七、
(1)
\[G(s) = \frac{0.5}{s(s^2+s+1)}\]
相角范围: \(-90° \to -270°\)
\[-90° - \arctan\frac{\omega_x^2}{1-\omega_x^2} = -180° \Rightarrow \omega_x = 1 \Rightarrow \mathrm{Re} = -0.5\]
(2)
A 从 0 到无穷时, \(-\dfrac{1}{N(A)}: 0 \to -1\)
稳定自振
(3)
\[\mathrm{Re} = -0.5 = -\frac{1}{N(A)} \Rightarrow A = 6.37, \ \omega = 1\]