Control Systems HW: Adjust K for A=2, B=1, C=3, a=3, b=1

[Hessami]

Homework Statement

I have a transfer function in the laplace domain, and I have been asked to:
"Develop Relationships in terms of the constant parameters and adjust K such that the constants are: A = 2, B = 1, C = 3, a = 3, b = 1"

After finding this relationship, I am asked to convert it back to time domain, find damped natural frequency, and adjust K for critical Damping.

I drew up this graph in MSpaint, sorry about the terrible quality, but it should hopefully be enough to get the general drift. As far as i can tell it's a graph of the response (laplacian?) against the undamped natural frequency multiplied by time, and it looks like a decaying cosine function. (at least the original does!).

Homework Equations

Transfer function (Laplace Domain) = K/[(s+a)(s+b-A.B)+K.C.B]

The Attempt at a Solution

I derived the Transfer Equation from a block diagram that I was given, and I am confident that I got that part correct.

I'm not sure what this question means by "develop relationships in terms of the constant parameters and adjust K such that the constants are... etc" , because as far as I can tell, the constants are not actually related to each other, and the Value of K has no bearing on what the other constants will be, right?

Following this, the rest should be fairly straightforward:
1) Convert back to time domain using the Laplace transform table,

2) Find my damped natural frequency from measuring periods on the graph.

3)Arranging the equation into

​

(D^2)/(omega^2) + D(2.E/omega) + 1

​

where D = differential operator, omega = undamped natural freq, E = damping ratio.
Substituting natural frequency found in 3, and then making E = 1 for critical damping, to find the adjusted K value.TL;DR -
What is being asked in the phrase "Develop Relationships in terms of the constant parameters and adjust K such that the constants are: A = 2, B = 1, C = 3, a = 3, b = 1"?

Should I substitute the constants into the equation, and find the equations for s?

Please help me! I've been stumped on this all week and can only feel myself getting stupider!

 

[CEL]

How did you plot this response without knowing the value of K?
For critical damping, the denominator must have two equal real roots. Since all other values are known, you can calculate the value of K such that the denominator has the form $$(s+s_1)^2$$.
For other values of K, you must have other constraints.

 

[Hessami]

Thanks for your post CEL :)

Ah, sorry, I wasn't very clear.

The graph that I drew here was given to me as part of the question, but I didn't have access to a scanner when i made that post, so I copied it out by drawing it in paint.

My question was mostly based around the "develop relationships" part. I don't quite understand what kind of relationship they are asking for?

Also, at this point in the question, the damping ratio is not necessarily critical.

Am I right in stating that the denominator of the transfer function represents the characteristic equation of the output?

What does solving for the roots of this equation actually achieve? I'm planning to read up on the Kreyszig book to try and get a better hold on all of this from a more mathematical perspective. Any other suggestions as to how I should go about understanding the concepts?

 

[CEL]

Thanks for your post CEL :)

Ah, sorry, I wasn't very clear.

The graph that I drew here was given to me as part of the question, but I didn't have access to a scanner when i made that post, so I copied it out by drawing it in paint.

My question was mostly based around the "develop relationships" part. I don't quite understand what kind of relationship they are asking for?

Also, at this point in the question, the damping ratio is not necessarily critical.

Am I right in stating that the denominator of the transfer function represents the characteristic equation of the output?

What does solving for the roots of this equation actually achieve? I'm planning to read up on the Kreyszig book to try and get a better hold on all of this from a more mathematical perspective. Any other suggestions as to how I should go about understanding the concepts?

Your wave is of the form $$A+Be^{\sigma t}cos(\omega_d t+\phi)$$
Measuring the time interval between two peaks of your graphic you have the period of your damped natural frequency $$\omega_d$$.
A is the value when time goes to infinity.
The envelope of your wave is $$A+Be^{\sigma t}$$. Measuring the value at two peaks and replacing the value of the time, you can determine $$\sigma$$.
So you have the roots of your characteristic polynomial and can evaluate K.