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GATE 2022 EC – Question 22

Control Systems · Bode and Root-Locus Plots · 1 mark · Multiple choice

The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure. Note that $K$ is varied from 0 to $\infty$.

Select the transfer function $G(s)$ that results in the root-locus plot of the closed-loop system as shown in the figure.

Diagram for GATE 2022 EC question 22
  1. $G(s)=\dfrac{1}{(s+1)^5}$
  2. $G(s)=\dfrac{1}{s^5+1}$
  3. $G(s)=\dfrac{s-1}{(s+1)^6}$
  4. $G(s)=\dfrac{s+1}{s^6+1}$

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Show answer and explanation

Correct answer: (A) $G(s)=\dfrac{1}{(s+1)^5}$

Explanation

The asymptotes are at $36^\circ,108^\circ,180^\circ,252^\circ,324^\circ$, which is 5 asymptotes, so $n-m=5$. They meet at the centroid $\sigma=-1$. For $G(s)=\frac{1}{(s+1)^5}$, $n-m=5$ with all poles at $-1$, so the centroid is $-1$. Option B has centroid $0$, and C and D have $n-m=5$ but the wrong centroid or unstable structure.