Design of control system by root-locus method 1

[fahimperacha]

I am trying to achieve a specific performance criterion of a LTI higher order system approximated to second order system by root locus method. The criteria is 2% Percentage overshoot and minimum settling time. The process transfer function is

(s+2)/s^2(s+9)

and the controller is a PID controller whose transfer function is

kd(s+1-j1)(s+1+j1)/s.

The question is to find the value of gain kd at which that above performance criteria is met.

I tried to find for a more simpler system whose root locus was just a straight line cutting the real axis at a certain point on the left side of s-plane and i was successful, because we can find the angle of line representing the required performance criteria by cos^-1(damping coefficient) and the area under the intersection of that line and the root-locus, can give the value of Kd.

But, if the root locus is more curvy, we cannot determine the exact value of Kd manually by drawing roughly on a piece of paper without matlab?

Please reply with a possible solution of this issue.

 

[Skogsgurra]

Is this a real controller? Your coefficients are way off anything you are likely to see IRL. Tell us what your small and your large time constants are. Sometimes, you can simply ignore the D part.

Gunnar Englund

http://www.gke.org

--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

 

[electricpete]

I’m pretty sure it is a textbook problem, not a real-world tuning problem (how many times do you know the process transfer function... and process and feedback transfer function have such simple numbers for poles and zero’s).

But you didn’t ask for answer, just for general input, so I don’t see a problem.

The open loop transfer function is
K*H(s)*G(s) = kd(s+1-j1)(s+1+j1)/s * (s+2)/s^2(s+9)

For a given s in the complex plane, the angle of the open loop transfer function is sum of angles (s-sp) minus sum of angles (s-sz) where sp and sz are zeroes.

The general approach to analyse pole zero diagram for stability without benefit of computer plotting is to look for places where the angle of the open loop transfer function is -180 +/- n*2*pi and magnitude = 1.

FWIW, my opinion is same as yours... the open loop transfer function looks too complicated to do much without help of computer. If you have matlab, use rlocusplot


=====================================
(2B)+(2B)' ?

 

[xnuke]

Given the system and controller above, the specified performance criteria of 2% overshoot maximum can't be met no matter what the value of kd. (I had a much longer explanation of why, but since this totally seems like a school problem as the others have stated, I won't post it.)

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.