The loop transfer function of a closed-loop control system is given as :

$G(s)H(s)=\frac{k(s+1)}{s(s+2)(s+3)}$

The centroid of the asymptotes in root locus is :

Explanation:

To determine the centroid of the asymptotes in the root locus of the given loop transfer function, we need to first understand the parameters involved. The given loop transfer function of the closed-loop control system is:

G(s) H(s) = k(s + 1) / [s(s + 2)(s + 3)]

The root locus method involves plotting the roots of the characteristic equation of the closed-loop system as the gain "k" varies from 0 to ∞. The centroid of the asymptotes is a point on the real axis where the asymptotes of the root locus intersect the real axis. This point is important as it provides insights into the system's stability and behavior.

The formula for finding the centroid (σ) of the asymptotes is given by:

σ = (sum of poles - sum of zeros) / (number of poles - number of zeros)

From the given transfer function:

Poles: s = 0, s = -2, s = -3

Zeros: s = -1

Number of poles (P) = 3

Number of zeros (Z) = 1

Sum of poles = 0 + (-2) + (-3) = -5

Sum of zeros = -1

Substituting these values into the formula for the centroid:

σ = (-5 - (-1)) / (3 - 1)

σ = (-5 + 1) / 2

σ = -4 / 2

σ = -2

Therefore, the centroid of the asymptotes in the root locus is at (-2, 0).

Correct Option Analysis:

The correct option is:

Option 3: (-2, 0)

This option correctly identifies the centroid of the asymptotes for the given loop transfer function