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GATE 2025 EE – Question 23

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 1 mark · Multiple choice

The Nyquist plot of a strictly stable $G(s)$ having the numerator polynomial as $(s-3)$ encircles the critical point $-1$ once in the anti-clockwise direction. Which one of the following statements on the closed-loop system shown in figure, is correct?

Diagram for GATE 2025 EE question 23
  1. The system stability cannot be ascertained.
  2. The system is marginally stable.
  3. The system is stable.
  4. The system is unstable.

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Show answer and explanation

Correct answer: (D) The system is unstable.

Explanation

$G(s)$ is strictly stable, so it has no right-half-plane poles ($P=0$). A stable open-loop system gives a stable closed loop only if the Nyquist plot does not encircle $-1$ at all. Any encirclement of $-1$ (here one) means the closed-loop system has right-half-plane poles, so it is unstable. (The RHP zero at $s=3$ only makes the plot non-minimum-phase and does not change this conclusion.)