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GATE 2015 EE – Question 65

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 2 marks · Multiple choice

The open loop poles of a third order unity feedback system are at 0, $-1$, $-2$. Let the frequency corresponding to the point where the root locus of the system transits to unstable region be K. Now suppose we introduce a zero in the open loop transfer function at $-3$, while keeping all the earlier open loop poles intact. Which one of the following is TRUE about the point where the root locus of the modified system transits to unstable region?

  1. It corresponds to a frequency greater than K
  2. It corresponds to a frequency less than K
  3. It corresponds to a frequency K
  4. Root locus of modified system never transits to unstable region

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Correct answer: (D) Root locus of modified system never transits to unstable region

Explanation

The original system has 3 poles and no zeros, so its locus goes off along asymptotes at 60°, 180° and 300°, and it crosses into the right half plane. With a zero at $-3$ there are 3 poles and 1 zero, so only 2 branches go to infinity, along asymptotes at $\pm 90°$. The asymptote centre is $\frac{(0 - 1 - 2) - (-3)}{2} = 0$, so the two branches approach the imaginary axis without crossing it. The locus never enters the unstable region.