GATE 2017 EC (EC2) – Question 56

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

A unity-feedback control system has open-loop transfer function $G(s)=2(s+1)/(s^3+ks^2+2s+1)$. The value of k for which the system oscillates at 2 rad/s is ________.

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Correct answer: 0.74 to 0.76

Explanation

For unity negative feedback, the characteristic equation is $1+G(s)=0$, giving $s^3+ks^2+4s+3=0$.

An oscillation at 2 rad/s requires a root at s=2j. Substitution gives $(-8j)-4k+8j+3=0$, hence $3-4k=0$.

Thus **k=0.75**. As a check, the polynomial factors as $(s+0.75)(s^2+4)$, leaving an imaginary pair at ±2j and a stable real pole.

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