Consider a state-variable model of a system

$\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -\alpha & – 2 \beta \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ \alpha \end{bmatrix} r$

$y=\begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$
where $y$ is the output, and $r$ is the input. The damping ratio $xi$ and the undamped natural frequency $\omega _n$ (rad/sec) of the system are given by

1. $\xi = \frac{\beta}{\sqrt{\alpha}}$; $\omega_n = \sqrt{\alpha}$
2. $\xi = \sqrt{\alpha}$; $\omega_n = \frac{\beta}{\sqrt{\alpha}}$
3. $\xi = \frac{\sqrt{\alpha}}{\beta}$; $\omega_n = \sqrt{\beta}$
4. $\xi = \sqrt{\beta}$; $\omega_n = \sqrt{\alpha}$
in new
edited