The GATE Grind

GATE 2020 EC – Question 50

Control Systems · Bode and Root-Locus Plots · 2 marks · Multiple choice

The characteristic equation of a system is

$$s^3+3s^2+(K+2)s+3K=0.$$

In the root locus plot for the given system, as $K$ varies from 0 to $\infty$, the break-away or break-in point(s) lie within

  1. $(-1,0)$.
  2. $(-2,-1)$.
  3. $(-3,-2)$.
  4. $(-\infty,-3)$.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $(-1,0)$.

Explanation

Rewrite as $1+\frac{K(s+3)}{s(s+1)(s+2)}=0$. The break points satisfy $\frac{dK}{ds}=0$ with $K=-\frac{s(s+1)(s+2)}{s+3}$, which gives $s^3+6s^2+9s+3=0$ with roots near $-0.47$, $-1.5$ and $-3.7$. The real-axis locus exists only on $[-1,0]$ and $[-3,-2]$, so only $-0.47$ is a valid break point, and it lies in $(-1,0)$.