Solving Control System Problems: Natural Frequency and K Linkage Explained

[lazyaditya]

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.

Homework Equations

The Attempt at a Solution

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

 

[rude man]

Assume G_c = (s+a)/(s+b) as you have done.

Then the open-loop gain is GG_c = (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 + 2ζω_ns + ω_n²)(s + αω_n)
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).

 

[lazyaditya]

Why do you have used (s + αωn) i mean how do you have "αωn" as a root of the characteristic equation ?

 

[rude man]

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 + αω_n) 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(-αω_nt) whereas the sinusoidal part decays as exp(-ζω_nt), same as for the 2nd-order system. The argument of the sine is the same for both 2nd and 3rd order systems
= (ω_n√(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 + αω_n).

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 + αω_n) in the chas. equation. I suggest you get hold of a very extensive Laplace transform table which includes the time responses to both cases.

 

[lazyaditya]

I am still not very much clear about the part the root of 3 rd order system including natural frequency.

 

[rude man]

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.

 

[lazyaditya]

sorry for the photos.

 

[lazyaditya]

I got 3 equations but what value of "k" should i take ?

 

[rude man]

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?

 

[lazyaditya]

I have posted the Image.

 

[rude man]

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.

 

[lazyaditya]

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 ?

 

[rude man]

b(k) = k/5 so
k = 5b(k)
But b = 9.9
Therefore k = 5*9.9 = 49.5

 

[lazyaditya]

can i ask another question over here only regarding electromagnetic fields or should i post another thread ?

 

[rude man]

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.