(3)
\[20\lg\frac{10}{0.01} - 40\lg\frac{\omega_c}{2} = 0 \Rightarrow \omega_c\]
\[\gamma = 180° + (\omega_c) = 180° - 90° - \arctan 0.5\omega_c - \arctan 0.1\omega_c\]
七、
(1)
\[U(s)\frac{1}{s} = X_2(s)\]
\[X_2(s)\frac{1}{s} = X_1(s)\]
\[Y(s) = k_1 X_1 + k_2 X_2\]
\[\Rightarrow \dot{x}_1 = x_2, \dot{x}_2 = u \Rightarrow \dot{x} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} x + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u, \quad y = [k_1 \; k_2] x\]
(2)
\[P_c = [B \; AB] = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\]
满秩，可控
\[P_o = \begin{bmatrix} C \\ CA \end{bmatrix} = \begin{bmatrix} k_1 & k_2 \\ 0 & k_1 \end{bmatrix}\]
\[|P_o| = k_1^2 \neq 0 \Rightarrow k_1 \neq 0\]
八、
\[P_c = [B \; AB \; A^2 B] = \begin{bmatrix} 0 & 0 & 10 \\ 0 & 10 & 90 \\ 10 & 100 & 990 \end{bmatrix}\]
满秩，系统完全可控
\[P_c = [B \; AB \; A^2 B] = \begin{bmatrix} 0 & 0 & 10 \\ 0 & 10 & 90 \\ 10 & 100 & 990 \end{bmatrix}\]
设 K=[k1 k2 k2 ]
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