Sketching Root Locus of System: K Varies 0 to ∞

[weavertri]

Hello guys, I need help to sketch the root locus of the system when K is varying from 0 to infinity. (K>=0)
The system:
http://www.freeimagehosting.net/newuploads/anspe.jpg

I'm stuck. I really don't know how proceed.
Thanks in advance.

 

[TimeToShine]

Plot the poles and zeroes (There are no zeroes here). Use the root locus drawing rules.

 

[weavertri]

yeah, that's what I'm doing... the problem is: K is not a constant. to use the root locus drawing rules the characteristic equation of the closed loop system must be written as 1 + GH = 0. All that I got in this case is
(s+2)(s+5)(s+K) + 10K = 0.

 

[weavertri]

I guess I finally did it.
my characteristic equation is (s+2)(s+5)(s+K) + 10K = 0.
we expand to s³ + 7s² + 10s + Ks² 7Ks + 20K = 0
the C.E. must be written as 1 + K*F(s) = 0, then

s³ + 7s² + 10s + K(s² + 7s + 20) = 0

1 + K(s² + 7s + 20)/( s³ + 7s² + 10s) = 0



this is equal to

1 + K(s + 3.5 - 2.8i)(s + 3.5 + 2.8i)/(s(s+2)(s+5)) = 0


now we can sketch the root locus where the zeros are (z1= 3.5 + 2.8i and z2 = 3.5 - 2.8i) and the poles p1=0, p2=-2 and p3=-5.

:)

 

[rezeew]

I understand your struggle with sketching the root locus of a system when K is varying from 0 to infinity. The root locus is a plot of the locations of the closed-loop poles as a function of the gain parameter K. It is a useful tool for analyzing the stability and performance of a system.

To sketch the root locus, we need to follow these steps:

1. Identify the open-loop transfer function of the system. From the given information, the open-loop transfer function can be represented as G(s) = K/(s+2)(s+3).

2. Determine the poles and zeros of the open-loop transfer function. In this case, we have two poles at s = -2 and s = -3 and no zeros.

3. Plot the poles and zeros on the s-plane. This will give us the starting point for our root locus plot.

4. Determine the asymptotes of the root locus. As K approaches infinity, the root locus will approach the asymptotes. The number of asymptotes is equal to the number of poles minus the number of zeros, which in this case is 2-0=2. The asymptotes can be calculated using the formula: θ = (2k+1)*π/n, where θ is the angle of the asymptote, k is the index of the asymptote (ranging from 0 to n-1), and n is the number of asymptotes.

5. Plot the asymptotes on the s-plane. These will be straight lines passing through the centroid of the poles and zeros.

6. Determine the breakaway and break-in points. These are the points where the root locus branches off or converges to the poles. In this case, there are no breakaway or break-in points since the poles are real and distinct.

7. Plot the root locus branches. These branches represent the movement of the closed-loop poles as K varies from 0 to infinity. The branches start from the poles and move towards the asymptotes.

8. Determine the intersection points of the root locus with the imaginary axis. These points correspond to the values of K where the closed-loop poles are on the imaginary axis. This will give us the stability boundaries of the system.

9. Finally, sketch the complete root locus by connecting all the branches and intersection points. The root locus will have a shape similar to a spiral or a curve.

I hope this helps you in sketching the root locus