Question

If the open loop transfer function of a system is $G(s)H(s) = \frac{K(s+1)}{s(s+3)(s+4)}$ then its centroid is:

• A. $(-3, 0)$
• B. $(-3.5, 0)$
• C. $(-2, 0)$
• D. $(-4, 0)$

Hint:

The centroid of root locus asymptotes is always a real number, meaning its imaginary coordinate is always $0$.
So the centroid coordinate is always in the form of $(\sigma, 0)$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem asks for the centroid of the asymptotes of the root locus for a given open-loop transfer function.

Step 2: Key Formula or Approach:
The centroid ($\sigma$) is the intersection point of the asymptotes of the root locus on the real axis, given by:
\[ \sigma = \frac{\sum \text{Real parts of Poles} - \sum \text{Real parts of Zeros}}{P - Z} \]
where $P$ is the number of open-loop poles and $Z$ is the number of open-loop zeros.

Step 3: Detailed Explanation:

• Determine the poles from the denominator of $G(s)H(s)$:
\[ s(s+3)(s+4) = 0 \implies \text{Poles at } s = 0, s = -3, s = -4 \]
The number of poles is $P = 3$.
The sum of the poles is:
\[ \sum \text{Poles} = 0 + (-3) + (-4) = -7 \]

• Determine the zeros from the numerator of $G(s)H(s)$:
\[ s+1 = 0 \implies \text{Zero at } s = -1 \]
The number of zeros is $Z = 1$.
The sum of the zeros is:
\[ \sum \text{Zeros} = -1 \]

• Substitute these values into the centroid formula:
\[ \sigma = \frac{(-7) - (-1)}{3 - 1} = \frac{-7 + 1}{2} = \frac{-6}{2} = -3 \]

• The coordinates on the s-plane are $(-3, 0)$.

Step 4: Final Answer:
The centroid is $(-3, 0)$, which corresponds to Option (A).