Control System Problem

  • Thread starter lazyaditya
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  • #1
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Homework Statement


Since this is a third order system and there will be a zero in its transfer function i am confused that how the natural frequency and K will be linked ? Please do help i really get confused in these type of problems.


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1 st image in the attachments is the attempt, and second image is the question.

Homework Statement





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The Attempt at a Solution

 

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  • #2
rude man
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Assume Gc = (s+a)/(s+b) as you have done.

Then the open-loop gain is GGc = (k/s2)[(s+a)/(s+b)] = n/d.

Expand n + d into s3 + cs2 + ds + e

and equate coefficients of like powers of s with

(s2 + 2ζωns + ωn2)(s + αωn)
which you also have to expand as above.

By equating coefficients of like powers of s you get 3 equations with 3 unknowns (a, b and α).
Solve for a and b.

(k was not given numerically so you can assume any value which will agree with one of the four answer choices).
 
  • #3
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Why do you have used (s + αωn) i mean how do you have "αωn" as a root of the characteristic equation ?
 
  • #4
rude man
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Why do you have used (s + αωn) i mean how do you have "αωn" as a root of the characteristic equation ?
Good question.

α is a new, dimensionless variable for a 3rd-order system. If you compare the inverse-Laplace transform for the 3rd-order chas. equation with the same xfr function for the 2nd order system, i.e. without the extra (s + αωn) term, you would get a similar time response to a delta function input except for a modified coefficient in front of the sine term, plus a second, non-sinusoidal, term. The second term decays as exp(-αωnt) whereas the sinusoidal part decays as exp(-ζωnt), same as for the 2nd-order system. The argument of the sine is the same for both 2nd and 3rd order systems
= (ωn√(1 - ζ2)t + ψ).

And the phase angle ψ(3rd order) = ψ(2nd order) - a term including α.

So the bottom-line answer is that the two systems behave somewhat similarly if for the 3rd order system you retain the 2nd order expression multiplied by (s + αωn).

The complete time response expression is a mess to write out & I'm not going to do it here, with or without the extra (s + αωn) in the chas. equation. I suggest you get hold of a very extensive Laplace transform table which includes the time responses to both cases.
 
  • #5
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I am still not very much clear about the part the root of 3 rd order system including natural frequency.
 

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  • #6
rude man
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I am still not very much clear about the part the root of 3 rd order system including natural frequency.
Alpha is not arbitrarily chosen. You get 3 equations to solve for a, b and alpha.

Can you get hold of a really good Laplace transform table?

BTW your photos are 90 degrees twisted and very hard to read.
 
  • #7
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sorry for the photos.
 
  • #8
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I got 3 equations but what value of "k" should i take ?
 
  • #9
rude man
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I got 3 equations but what value of "k" should i take ?
Like I said, a constant that will make one of your answer choices correct. I think it was dumb of them not to give you a value for k. It was obviously an incompletely stated problem.

I haven't done the work and don't want to, so here you're a bit on your own.

EDIT: wait, did you solve for a(k) and b(k), and if so, what did you get?
 
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  • #11
rude man
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OK, so then a = 5 - 125/k = (5k - 125)/k and
b(k) = k/5

So if you start with b(k), choice (a) would seem closest: b(k) = 9.9, k = 49.5, then a(k) ~ 2.5.

As I said, I think it was dumb of them not to have given you k numerically since it certainly determines a and b.
 
  • #12
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Thanks a lot and sorry for such a late reply, since i wasn't able to check my thread for a long time.But how did you come to conclusion of "k" being 49.5 ?
 
  • #13
rude man
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b(k) = k/5 so
k = 5b(k)
But b = 9.9
Therefore k = 5*9.9 = 49.5
 
  • #14
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can i ask another question over here only regarding electromagnetic fields or should i post another thread ?
 
  • #15
rude man
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can i ask another question over here only regarding electromagnetic fields or should i post another thread ?
I would start a new thread since control systems and e-m are very different disciplines.
 
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