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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 by (5.4k points)
edited ago by

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