# Control Systems - Root Locus

## Homework Statement

Given the transfer function:
$$R(s)=\frac{K}{s(s+2)(s^2+4s+5)}$$

Find the following (needed to sketch the root locus:
a) Number of branches
b) Symmetry
c) Starting and ending points
d) behavior at infinity (asymptotes)
e) Real axis breakaway and break-in points.
f) calculation of jw-axis crossings

## The Attempt at a Solution

a)number of branches
The number of branches of the root locus equals the number of closed loop poles.
-poles cause the denominator to go to 0.
Therefore there is a pole at 0, -2,, and by quadratic eqns -2 (plus minus) j.
so, if im not mistaken there are 4 poles at 0,-2,-2+j and -2-j. This means that there will be four branches????
Yet, when I plot the root locus in matlab I seem to have three branches????? b)symmetry
My text book just says "the root locus is symmetrical about the real axis". This is true from the image but how can I show this mathematically?

c)Starting and ending points
again my textbook is vague/ maybe i just dont understand it very well. "The root locus begins at the finite and infinite poles of G(s)H(s) and ends at the infinite zeroes of G(s)H(s)"
To me this means that the branches begin at the poles from part a), which is seen in the picture - but there are no zeroes so how can there be an end point?

d) Behavior at infinity
??????????Help

e)Real axis breakaway and break-in points
breaks away at a point where the gain (K) is max
Breaks in where the gain is min.

I dont understand this as teh gain should very between 0 and infinity????

f) jw axis crossings
I'm thinking that a routh table may be necessary here, thoughts???

Thanks, I am fairly stuck here - so any advice will be helpful.