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GATE 2024 EE – Question 21

Control Systems · Transient and Steady-state analysis of linear time invariant systems · 1 mark · Multiple choice

Consider the standard second-order system of the form $\dfrac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}$ with the poles $p$ and $p^*$ having negative real parts. The pole locations are also shown in the figure. Now consider two such second-order systems as defined below:

System 1: $\omega_n=3$ rad/sec and $\theta=60^\circ$

System 2: $\omega_n=1$ rad/sec and $\theta=70^\circ$

Which one of the following statements is correct?

Diagram for GATE 2024 EE question 21
  1. Settling time of System 1 is more than that of System 2.
  2. Settling time of System 2 is more than that of System 1.
  3. Settling times of both the systems are the same.
  4. Settling time cannot be computed from the given information.

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Show answer and explanation

Correct answer: (B) Settling time of System 2 is more than that of System 1.

Explanation

The settling time is $T_s\approx\dfrac4{\zeta\omega_n}=\dfrac4{|\sigma|}$, and the real part of the poles is $\sigma=\omega_n\cos\theta$ (with $\theta$ measured from the negative real axis). System 1: $3\cos60^\circ=1.5$. System 2: $1\cdot\cos70^\circ=0.342$. A smaller $|\sigma|$ means a longer settling time, so System 2 settles more slowly.