# LQR calculation of damping/ natrual frequency?

Yes.. this is a homework-- I think I've figured out part of the problem but can't find any info on how to do the second. I would appreciate a hint or
clue..
The system:
A = [0 1; 1 0] B = [0; 1]
Q = [q 0; 0 q] R = [r 0; 0 r]
I have to solve the problem symbollically.
I have determined the P matrix, and Kc as a function of (r,q) r,q undetermined constants
I have
Kc = [ g g]
where g = -1+ sqrt[1-(q/r)]
and P = [w f; 0 f]
where:
w = r(1+sqrt(1-q/r))-q f = r(1+sqrt(1-q/r))
How can I find omega and zi for the closed loop system?
I do not know for certain if my expressions for P matrix and Kc are correct or not.
Perhaps a better question is can I feed matlab symbolic expressions into it's LQR function to check my work? I tried using symbolic math to compute all of this in matlab, but had some troubles since I'm not that familiar with symbolic package in matlab.
Now part of the problem is to compute damping factor and natural freq. The prof didn't give us ANY notes or info on how to compute these values for an LQR design.
Thanks,
Bo
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I would convert this to a transfer function in the s domain.

I don't know if this is even relevant

I don't think this is either.

The controller can be expressed by g+g*s in the s domain.
I would then make the closed loop transfer function
T(s)=Gc(s)*Gp(s)/(1+Gc(s)*Gp(s)) and simplify so the characteristic equation is in the denominator.
Then the coefficients for s should determin zeta and omega.

One can subtitute this in for g after most of the work is done.

What is this for?

Isn't Kc = [ Kp Kd ]? It had better be. In your case it looks like Kp and Kd are both equal to g which really doesn't make a lot of sense.

This is easy enough to do by hand.

No problem after you simplify T(s)
The

If finding the natural frequency and damping factor of the CLTF then who cares how the gains were calculated. That is a different problem. I think that LQR stuff was added to confuse the issue. What you really need ist he open loop transfer function and the gains.
If I understand this correctly it is a 5 minute problem.
Peter Nachtwey
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wrote:

Peter,
Thanks for the reply and sorry in taking so long to get back to you... I eventually figured out what to do and what the prof was wanting. Seemingly, your approach is 'too realistic' for my prof. ie. all he wants is derivations/proofs, not practicality :)
Thanks,
Bo

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I am a little short of derivations and proofs. I have to make things really work. So what did your professor think is the right way? Was I right about the LQR stuff confusing the issue?
Peter Nachtwey
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No. It was definitely part of the problem/solution. Setting my LQR derived d.e. = 2nd order 'std' d.e. and then solving for wn and damping factor was the correct method.... I ended up with a B and was happy about it--even though it wrecked my 4.0. Reflected more on the prof than on me I felt...
Bo