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GATE 2024 IN – Question 45

Control Systems · Time response and steady-state error · 2 marks · Multiple choice

A system has $G(s)=\omega_n^2/(s^2+2\xi\omega_n s+\omega_n^2)$. Choose the ranges of $\xi$ and $\omega_n$ (rad/s) such that its poles lie in the shaded region of the s-plane shown.

Diagram for GATE 2024 IN question 45
  1. $\xi\ge1/2,\ \omega_n\ge2$
  2. $\xi\ge1/4,\ \omega_n\ge2$
  3. $\xi\ge1/2,\ \omega_n\ge\sqrt3$
  4. $\xi\ge1/4,\ \omega_n\ge\sqrt3$

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Correct answer: (A) $\xi\ge1/2,\ \omega_n\ge2$

Explanation

For complex second-order poles, distance from the origin is $\omega_n$, and damping ratio is the cosine of the angle from the negative real axis. The boundary at 120° from the positive real axis gives $\xi=\cos60^\circ=1/2$. The excluded circle has radius 2.

Thus the intended bounds are $\xi\ge1/2,\omega_n\ge2$, **A**.