Control Theory (EE): How to set up a transmittance given wn only?

[CDTOE]

Homework Statement

How do we set up a second-order plant transmittance with the only information available are:

One pole is at a position where the undamped natural frequency (ω_n = 0 rad/sec), and the other pole is at a position where ω_n = 2 rad/sec?

The question asks to build that transmittance, draw pole-zero plot, find the transfer function and break it down into natural and forced components, which are required to be plotted, given a step input. I only had trouble setting up the plant transmittance which is, unfortunately, the base of the whole solution.

Homework Equations

None.

The Attempt at a Solution

To set up the plant transmittance, I couldn't quite understand how I should do it. A second-order system will have a characteristic polynomial of the form (S²+2ω_nζs+ω_n²), where ζ is the damping ratio. Given this polynomial, I couldn't see how we can have two different ω_n values. If one of the poles would have ω_n, then the system should be underamped or undamped depending on the value of ζ. ω_n = 0 would mean that the said pole is on the real axis given that there is no an oscillatory part.

Given all of that, I decided that I set up a transmittance based on mild guessing, so it was:

G_P(s) = $\frac{1}{s(s+2)}$

A pole at the origin and another at -2. I couldn't do anything other than doing something that at least shows that I can draw a pole-zero plot and do all what's asked.

 

[milesyoung]

Are you sure you've given the problem statement here exactly like it was worded to you? Often people tend to post their own interpretation of the problem, which is what gives them difficulty in solving it in the first place.

For one thing, I have to admit, I've never heard of 'plant transmittance' as something which isn't synonymous with 'plant transfer function'. Your problem statement seems to indicate that determining it is something separate from determining the plant transfer function.

Isolines of natural frequency are circular arcs in the LHP with radius equal to the natural frequency, and since complex poles only come in conjugate pairs for a LTI system with real coefficients, your poles must be purely real. They must also then be located at s = 0 and s = -2, so I can't really see anything wrong with your plant transfer function. The wording of your assignment, as given, just generally seems a bit off, but that's just my opinion.

 

[CDTOE]

I don't remember how the problem was worded exactly, but I'm sure that the only information that were given to set up the plant (transfer function) was that there's a pole with ω_n=0 rad/sec and another pole with ω_n= 2 rad/sec. As for the system transfer function, it wasn't mentioned whether the system is open or closed loop, so I believe that it would be the same as the plant transfer function given that there would be no feedback (open loop system).

As for pole positions, yes, there shouldn't be a pair of complex conjugate poles given that there's a pole with ω_n=0. Given all of this, I believe, like you said, that we should be having an overdamped system with two poles one at the origin and another one at -2.

How about that?

 

[milesyoung]

So since you now have the plant transfer function, all is well?

With regards to the bit about having two different values of undamped natural frequency - for real poles you can just interpret the undamped natural frequency as a way to locate them along the real line, which is what you've done.

 

[CDTOE]

Yes, everything is good now. Thanks for help.