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Finding transfer function gain without root locus

Finding transfer function gain without root locus

Finding transfer function gain without root locus

(OP)
Hi,

Having a bit of trouble with this, assuming a proportional controller is used, I need to determine the gain to achieve a damping ratio of 0.5, for the following transfer function: G(s) = -4(s+0.4) / s^2+1.6s+14.

I need to solve this without using root locus. The furthest I've been able to go is determining the zeros and poles of the transfer function as -0.4, and -1.6+/-3.66i respectively. Some additional data is given: "Data that may be required: (s-2.38)(s+4.14-2.60i)(s+4.14-2.60i)=s^3+5.9s^2+4.24s-56.82" I think I recognise the left hand side of that equation coming from the characteristic equation of a state space model but I'm really not sure how I could use this.

Can anyone lend a hand? Thanks.

RE: Finding transfer function gain without root locus

(OP)
IRstuff: good thing that this is a posting regarding my personal interests then and nothing to do with my studies :)

RE: Finding transfer function gain without root locus

I am assuming you provided the open loop transfer function.
If the closed loop transfer function had a critically damped characteristic equation then it would be (s^2+2ζωs+(ω^2+k))=(s^2+2λs+λ^2)
but since in this example you want the desired characteristic equation to be
(s^2+2ζωs+(ω^2+k))=(s^2+λs+λ^2)
Now there are two equations and two unknowns, k an λ.
2ζω=λ=1.6
ω^2+k=λ^2
solve for k an λ
What bothers me is that I calculate a negative k.
Is this a realistic example?
If ω=sqrt(14) then
ζ=.214
Adding gain will only make the closed loop damping factor smaller.

Peter Nachtwey
Delta Computer Systems
http://www.deltamotion.com

RE: Finding transfer function gain without root locus

Who uses the root locus? It really is a waste of time to learn because....
Most control systems use more than the proportional gain.
The instructors always assume the open loop transfer function is known.
The results, the calculated gain doesn't rarely results in a usable response.
In 30+ years of controls I have never used the root locus.
The instructors are just wasting the students time and money.

The only place where I could see the root locus being used is for a mechanical governor or a hydraulic valve that uses a spring for control.
However, I doubt it because I have NEVER seen a mechanical system with a transfer function attached.



Peter Nachtwey
Delta Computer Systems
http://www.deltamotion.com
http://forum.deltamotion.com/

RE: Finding transfer function gain without root locus

Root locus methods are not used in industrial automation as most controls are based on reduced order and linearized system descriptions.

For example, distillation columns, stirred reactors, pH control with mixing/holding tanks, aircraft and ship board steering, satellite controls especially for multiple objects, paper machines... all are all high order processes. The various studies that requiring identification of the dominant interactions are done 5-10 years prior to the design phase.

Are root locus methods, Routh-Hurwitz commonly used in the design phase. The short answer is no. In part the control systems commonly dealt with are reduced to 2nd order systems intentionally. You don't need much analysis for that and the control schemes are easily developed.

It is a useful tool, but only when you need it. Were an engineer present a root locus plot to a project manager...it would be a good time to start looking for new employment...



RE: Finding transfer function gain without root locus

Quote:


In part the control systems commonly dealt with are reduced to 2nd order systems intentionally.
Yes, this would make a good topic. With modern tools one can easily solve higher order systems.

Quote:


You don't need much analysis for that and the control schemes are easily developed.
But the tools exist now. However there are the practical aspects. Just because you can in a Mathematica work sheet doesn't mean you should or it is the best way.
This point is why the topic would make a good thread.

Quote:


It is a useful tool, but only when you need it. Were an engineer present a root locus plot to a project manager...it would be a good time to start looking for new employment.
And the young engineer would be thinking why did they waste my time and money learning root-locus.

Peter Nachtwey
Delta Computer Systems
http://www.deltamotion.com
http://forum.deltamotion.com/

RE: Finding transfer function gain without root locus

Quote:


Here is a higher order servo system, with model, being tested,
I read it all. That isn't too bad for a college student although I didn't agree with somethings and the techniques are not the ones I would use.
At least he didn't waste time with root locus but anybody that mentions Z-N for motion control hasn't really tried it. It doesn't work.
2.1.5 about pole placement being inadequate is wrong.
What was good is his table of frequencies, amplitudes and speeds. So many of my customers screw that up because they haven't done the math ( physics ).

Peter Nachtwey
Delta Computer Systems
http://www.deltamotion.com
http://forum.deltamotion.com/

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