GATE 2016 EC (EC2) – Question 55

Control Systems · Transient and Steady-State Analysis of LTI Systems · 2 marks · Numerical answer

The feedback system shown has $G(s)=1/(s^2+2s)$. Its closed-loop step response should have minimum settling time and no overshoot. The required gain k is ________.

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Correct answer: 0.95 to 1.05

Explanation

Unity negative feedback gives characteristic polynomial $s^2+2s+k$. Compare it with $s^2+2\zeta\omega_ns+\omega_n^2$: $\omega_n=\sqrt k$ and $\zeta=1/\sqrt k$.

No overshoot requires $\zeta\ge1$, hence k≤1. For 0<k<1 the slow pole is $-1+\sqrt{1-k}$, which moves left as k increases. The fastest nonoscillatory response is the critically damped boundary **k=1**.

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