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GATE 2020 IN – Question 24

Control Systems · Time response and steady-state error: Transient response, steady state error, time-delay systems · 1 mark · Multiple choice

A closed-loop system has $C(s)/R(s)=1/(s+1)$. For input $r(t)=\sin t$, its steady-state output is

  1. 1
  2. $\cos t/\sqrt2$
  3. $\sin(t+\pi/4)/\sqrt2$
  4. $\sin(t-\pi/4)/\sqrt2$

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Correct answer: (D) $\sin(t-\pi/4)/\sqrt2$

Explanation

Evaluate the frequency response at ω = 1 rad/s: $H(j)=1/(1+j)$, of magnitude $1/\sqrt2$ and phase −π/4. A stable LTI system responds to a sustained sinusoid at the same frequency, scaled and phase shifted by H. Thus $c_{ss}(t)=\sin(t-\pi/4)/\sqrt2$, D.

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