Control Systems - Root Locus

[Fisher92]

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

Homework Equations

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 I am 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 textbook 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 don't 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 don't 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.

 

[KataKoniK]

Thank you for your question. I would like to offer some guidance and clarification on the concepts and equations involved in finding the root locus for the given transfer function.

a) Number of branches:
Your understanding is correct. The number of branches of the root locus is equal to the number of closed loop poles. In this case, there are four poles at 0, -2, -2+j, and -2-j, so there will be four branches.

b) Symmetry:
The root locus is symmetrical about the real axis because the poles and zeros of the transfer function are symmetrical about the real axis. This can be shown mathematically by considering the conjugate pairs of complex poles and zeros. In this case, the poles at -2+j and -2-j are conjugate pairs, as well as the zeros at -2+j and -2-j. This symmetry is reflected in the root locus plot.

c) Starting and ending points:
The starting points of the root locus branches are the poles of the open-loop transfer function, while the ending points are the zeros of the open-loop transfer function. In this case, the open-loop transfer function is R(s)=K, which has no zeros. Therefore, the root locus branches will not have an ending point.

d) Behavior at infinity:
As the gain K approaches infinity, the root locus branches approach the poles at -2 and -2+j/-2-j. This can be seen in the plot as the branches get closer to these poles as K increases.

e) Real axis breakaway and break-in points:
The real axis breakaway point is where the root locus branches depart from the real axis, and the real axis break-in point is where the branches return to the real axis. In this case, the breakaway point occurs when K is equal to the maximum value that allows the closed-loop poles to remain stable, which can be found by using the Routh-Hurwitz stability criterion. The break-in point occurs when K is equal to the minimum value that allows the closed-loop poles to remain stable.

f) jw axis crossings:
To find the jw axis crossings, you can use the argument principle, which states that the number of poles and zeros of a transfer function enclosed by a contour is equal to the number of times the contour intersects the jw axis. Therefore, to find the jw axis crossings, you can plot the contour of the