Sketching Root Locus: n=2, m=1, Angle?

In summary: Having left the startpoint (at some distance from the startpoint) the roots will no longer spread symmetrically because they then "can sense" other poles (repelling) and zeros (attracking).Recommendation: Invent some sets of poles and zeroes and "play" with them on your screen. See what the root-curves will do in different combinations. Confirm the rules in #2.
  • #1
Tekneek
70
0
The characteristic equation is 1+K(S+1)/S2 Below is the root locus diagram. I don't get why there are two branches when there is only one pole, at 0. Does it count as having two poles even if it is the same because of s^2 ? If it does then why doesn't the angle of departure make sense?

number of poles(n) = 2
number of zeros(m) = 1

angle = (2h+1)/n-m * 180 = keep getting the same angle, 180

The angle certainly does not look like 180 as it departs from its pole...
95whgh.jpg
 
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  • #2
Tekneek said:
The characteristic equation is 1+K(S+1)/S2 Below is the root locus diagram. I don't get why there are two branches when there is only one pole, at 0. Does it count as having two poles even if it is the same because of s^2 ? If it does then why doesn't the angle of departure make sense?

number of poles(n) = 2
number of zeros(m) = 1

angle = (2h+1)/n-m * 180 = keep getting the same angle, 180

The angle certainly does not look like 180 as it departs from its pole...

Yes, there are two poles, s2 = (s+0)(s+0)

The mentioned angle is the angle between the two branches (180°), not the angle between a branch and the real-axis.

I don't recognize this formula: angle = (2h+1)/n-m * 180°.

Rule 1) If the number of poles in the same point = n, then the angle between the n branches = 360°/n.

Rule 2) There will be a root-curve on the real-axis at a point, if the number of real (zeroes+poles) to the right of that point is odd.
 
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  • #3
Hesch said:
Yes, there are two poles, s2 = (s+0)(s+0)

The mentioned angle is the angle between the two branches (180°), not the angle between a branch and the real-axis.

I don't recognize this formula: angle = (2h+1)/n-m * 180°.

Rule 1) If the number of poles in the same point = n, then the angle between the n branches = 360°/n.

Rule 2) There will be a root-curve on the real-axis at a point, if the number of real (zeroes+poles) to the right of that point is odd.

What does it mean by root-curve on the real axis? Also how would I know the root locus moves in a circular pattern?
 
  • #4
Tekneek said:
What does it mean by root-curve on the real axis?

Look at your attached figure: There is a root-curve (actually 2) on the real axis (imaginary part = 0) for s < -1, because there is 1 zero and 2 poles to the right of all points on the real axis when s < -1.

Tekneek said:
Also how would I know the root locus moves in a circular pattern?

Poles will repel roots and zeros will attrack roots as the amplification in a control-loop is increased. Say you have three poles in the same point. If not the roots should leave these poles in a mutual angle of 120°, what should they do instead, and why?

Having left the startpoint (at some distance from the startpoint) the roots will no longer spread symmetrically because they then "can sense" other poles (repelling) and zeros (attracking).

Recommendation: Invent some sets of poles and zeroes and "play" with them on your screen. See what the root-curves will do in different combinations. Confirm the rules in #2.

Examples here:

https://www.google.com/search?q=roo...KcsgH064D4CQ&ved=0CAcQ_AUoAQ&biw=1366&bih=635
 
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1. What is the purpose of sketching a root locus for n=2, m=1, angle?

The root locus is a graphical representation of the poles and zeros of a system in the complex plane. Sketching the root locus allows us to analyze the stability and performance of a control system by examining how the locations of the poles and zeros change as certain parameters vary.

2. How do the angles in the root locus affect the stability of the system?

The angles in the root locus correspond to the phase of the system's transfer function. A system is considered stable if the phase remains within the range of -180 degrees to 180 degrees. If the root locus crosses the imaginary axis at a point where the phase is outside of this range, the system will become unstable.

3. What does the number of poles (n) and zeros (m) indicate in a root locus plot?

The number of poles and zeros determine the complexity of the root locus plot. For n=2 and m=1, the root locus will consist of two branches, each starting at the poles and ending at the zero. The location of these branches and their behavior will provide information about the stability and performance of the system.

4. How can the root locus be used to improve the performance of a control system?

The root locus plot allows us to see how changes in system parameters, such as the gain or the location of a zero, affect the stability and performance of the system. By manipulating these parameters, we can adjust the root locus to meet the desired performance criteria, such as faster response time or reduced overshoot.

5. Is it possible to have a stable system with n=2, m=1, angle?

Yes, it is possible to have a stable system with these parameters. The stability of the system is dependent on the location of the poles and zeros in the complex plane, not solely on the number of poles and zeros. As long as the root locus does not cross the imaginary axis at a point where the phase is outside of the stable range, the system will be stable.

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