Question

The centroid of the following system is \[ G(s)H(s) = \frac{K(s+10)(s+20)}{s^3(s+100)(s+800)} \]

• A. \(290\)
• B. \(-290\)
• C. \(310\)
• D. \(-310\)

Hint:

The centroid does not need to be a point on the root locus itself; it is simply a geometric anchor point for the linear high-gain asymptotes.

Answer 1

The Correct Option is B

Concept: In the Root Locus technique, as the open-loop gain $K$ varies from 0 to infinity, the branches of the root locus tend toward infinity along straight-line paths called asymptotes. The point where these asymptotes intersect on the real axis of the s-plane is called the centroid ($\sigma$). The formula to compute the centroid is: \[ \sigma = \frac{\sum (\text{Real parts of open-loop poles}) - \sum (\text{Real parts of open-loop zeros})}{P - Z} \] where $P$ is the total number of open-loop poles and $Z$ is the total number of open-loop zeros.

Step 1: Identify the positions and values of the open-loop poles.
The poles are found by setting the denominator of $G(s)H(s)$ to zero: \[ s^3(s+100)(s+800) = 0 \] This gives the following poles: * $s = 0$ (with a multiplicity of 3) * $s = -100$ * $s = -800$ The total number of poles is $P = 5$. The sum of these poles is: \[ \sum \text{Poles} = 0 + 0 + 0 + (-100) + (-800) = -900 \]

Step 2: Identify the positions and values of the open-loop zeros.
The zeros are found by setting the numerator of $G(s)H(s)$ to zero: \[ (s+10)(s+20) = 0 \] This gives the following zeros: * $s = -10$ * $s = -20$ The total number of zeros is $Z = 2$. The sum of these zeros is: \[ \sum \text{Zeros} = (-10) + (-20) = -30 \]

Step 3: Substitute the sums into the centroid formula.
\[ \sigma = \frac{(-900) - (-30)}{5 - 2} \] \[ \sigma = \frac{-900 + 30}{3} = \frac{-870}{3} = -290 \] The asymptotes intersect on the real axis at exactly $-290$, which matches Option (B).