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GATE 2019 EE – Question 41

Control Systems · State space model, Solution of state equations of LTI systems · 2 marks · Multiple choice

Consider a state-variable model of a system

$$\begin{bmatrix}\dot x_1\\\dot x_2\end{bmatrix}=\begin{bmatrix}0&1\\-\alpha&-2\beta\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}+\begin{bmatrix}0\\\alpha\end{bmatrix}r,\qquad y=[1\ \ 0]\begin{bmatrix}x_1\\x_2\end{bmatrix}$$

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. $\xi=\dfrac\beta{\sqrt\alpha};\ \omega_n=\sqrt\alpha$
  2. $\xi=\sqrt\alpha;\ \omega_n=\dfrac\beta{\sqrt\alpha}$
  3. $\xi=\dfrac{\sqrt\alpha}\beta;\ \omega_n=\sqrt\beta$
  4. $\xi=\sqrt\beta;\ \omega_n=\sqrt\alpha$

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Correct answer: (A) $\xi=\dfrac\beta{\sqrt\alpha};\ \omega_n=\sqrt\alpha$

Explanation

The characteristic equation is $\det(sI-A)=s^2+2\beta s+\alpha$. Comparing with $s^2+2\xi\omega_ns+\omega_n^2$ gives $\omega_n=\sqrt\alpha$ and $\xi=\frac{\beta}{\sqrt\alpha}$.