Control System Problem

Quiz 5: Root Locus

Name:_____________________________

Here is a proportional control system.

The root locus for this system is shown below.

Determine the root locus gain for the maximum damping ratio for the pair of complex poles.

First, we have to determine where the pole is that has the maximum damping ratio. Draw a line tangent to the root locus as shown below. The point at the smallest angle off the horizontal is shown with a black dot, and the tangent line is in orange. The tangent of the smallest angle (largest damping ratio)is:

Smallest angle = tan^-1(4.5/10) = 24.2^o

Largest damping ratio = cos(24.2^o) = 0.91

To determine the gain, get the pole distances and zero distances and use the relationship:

Root locus gain = Product of pole distances/Product of zero distances.

Note, there are four open loop poles and one open loop zero. If you do the math, you should get something like: (Very rough approximations for the lengths, starting with the distance from the pole at -8)

Approximate RL Gain = 6x3x2x1/1.5 = 24

The actual root locus gain is 24. You should get something in that vicinity. The actual line you need to use are shown below in purple above.

Determine the maximum damping ratio.

We did that in the first part and found that the maximum damping ratio was 0.91.

Determine if there is any real pole slower than the complex poles for the gain you found.

Here is the root locus with all of the poles for a root locus gain of 24 marked with a black dot. Note that there is a pole near s = -9, which is much further into the LHP. That's a pole with a much faster decay than the complex poles with a real part at s = -2.1. The pole at -9 has a time constant of .11 seconds which the pole at -2.1 has a time constant for the decay of 1/2.1 or a little less than .5 seconds. (That's the time constant of decay of the envelope of the oscillations.)

However, there is another pole right at 2 = -2 and that's just a little bit slower decay than the complex pole envelope decay time constant. You might have had trouble getting that exactly, but you shold have realized they're about the same.

Will this system ever become unstable? If not, explain why. If it will, explain why. Use any appropriate root locus rules to support your explanation.

Well, there are four poles and just one zero. That means there are three poles going to infinity and two of them go at plus or minus sixty degrees. That's going to take them into the RHP for some sufficiently large gain value, so you're going to be able to make the system unstable. Don't be mislead into thinking that the asymptotes are at plus and minus ninety degrees. Tain't so. You should be able to figure out that it's going to be unstable even though the plot doesn't seem to go far enough (actually to a high enough gain). There's no doubt about it, and the conclusion is inescapable.