Centroid Root Locus

Question

I can't figure out how to find the root locus centroid for the poles of this simple equation in a positive feedback system.

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

I have read in many places that the centroid is found by this formula:

$$ c=\frac{sum(P)-sum(Z)}{p-z} $$

The transfer function poles are: $$ p1=-1.5+\frac{\sqrt{5}}{2} \\ p2=-1.5-\frac{\sqrt{5}}{2} $$

So the centroid should be:

$$ c=\frac{-1.5+\frac{\sqrt{5}}{2}-1.5-\frac{\sqrt{5}}{2}-0}{2-1}=-3 $$

However, in this video https://youtu.be/eyb_RzOnfGY?t=559 , and also Matlab says that the right answer for the centroid is c=-1. What I am doing wrong in the calculation? Please!

Answer 1

Wouldn't the centroid be at -3? Ill put it in MATLAB and see what I get (dont have enough rep to add comments) EDIT: This seems right to me, I noticed you only have H(s), do you know if there was any G(s)? I tried to click the link to the video and it wouldn't load, best of luck.

$$ c = \frac{-1.5 + \frac{\sqrt5}{2} -1.5 - \frac{\sqrt5}{2} - 0}{1}$$

$$ c = \frac{-3}{1} = -3$$

Answer 2

Centroid is -3. However, since the relative degree of this system is 1 you don't need to compute the centroid, for we know there is only one branch that goes to infinity, and it goes to whether $+\infty$ or $-\infty$ in the real line. What has made you confused about -1 is that this point is one of the breakpoints but not the centroid.