Parameters for Root Locus Method using Lead Compensator

[hurliehoo]

Hi everyone, I would please like some suggestions regarding where my error is, designing a lead compensator with the root locus method for a unity feedback control system where G = [12.5] / [2.5 1 2500]. I want to set the overshoot zeta = 0.5 for an overshoot of 16%, and settling time Ts = 0.06s.

Using these parameters I get a desired pole at -66.665 + j115.467.

I think I understand the application of the root-locus method, ie summing the angles of the poles and zeros with the new pole etc, and I've tried it with a couple of different values for compensator poles and zeros using these parameters (eg a = 66.665, b = 290.65). I always get the correct Ts, but an overshoot of 46% instead of 16%.

 

[swraman]

how are you factoring in the overshoot into your calculation for the new poles?

 

[hurliehoo]

how are you factoring in the overshoot into your calculation for the new poles?

Hi, thanks for the reply.

Basically I'm using this formula to find roots at

and Ts = 4 / ( zeta * w(0) ) = 0.06

using w(0) = 133.33 because zeta = 0.5 for a 16% overshoot due to

Then I calculate the angle to the desired pole at -66.665 + j115.467 from the first singularity at -0.2+j31.362 (original system) = -128.41

From the second singularity -0.2-j31.362 = -114.32

This gives a total of -242.73. It needs to adjust by +62.73 to get an odd multiple of 180. I then add for example a zero at -66.665 on the real axis, to add 90.

-242.73 + 90 = -152.73 so a pole of angle 27.27 will produce an odd multiple of 180. Putting a pole at -290.65 on the real axis does this.

... At least I think it should do ... the system doesn't give an overshoot anywhere near 16% though, and I can't understand why.