P-Controller Design for Steady State Error

[Spimon]

Homework Statement

So this is a bit of a two-part question and I'm unsure which part I'm not doing right (or both!).
i) Find the closed loop transfer function of the system shown
ii) Design a proportional controller for the system to give a 10% steady state error

Any help, hints, suggestions would be greatly appreciated.
Thanks!

Homework Equations

Go = 4.0
α = 0.168
β = 0.0047
Z1 = 8.9
G(sen2) = 5.33*10^-5
The reference level is 40mm (it's a twin water tank system)

The Attempt at a Solution

Firstly, the CLTF

And the P-controller

Thanks for any help :D

 

[rude man]

1. In your CLTF panel. what's with the last two equations? You didn't try to isolate the pole locations... not that there was any reason to ... but your 1st equation is correct. I'm too lazy to work thru the rest of that panel.

2. As for the k_p computation: Assuming your final T(s) is correct, just take
lim s → 0 of T(s). Using your derivation of T(s) I make that to be
T(0) = kpG₀G_sens/(β + G_sensCG₀z₁)
which you set to 0.1 & solve for k_p.

 

[Spimon]

Oh, ok. Thanks guys. Maybe it is correct, or at least fundamentally on track - I'm not so worried about the actual numbers as the method.

Rude Boy, the final 2 lines of the CLTF are the characteristic equation - just the denominator of the transfer function.
To isolate the pole locations, do I make set the numerator to equal zero and solve for S?
Similarly, to isolate zero locations I set the denominator to zero and solve for S?

Thanks! :)

 

[rude man]

Oh, ok. Thanks guys. Maybe it is correct, or at least fundamentally on track - I'm not so worried about the actual numbers as the method.

Rude Boy, the final 2 lines of the CLTF are the characteristic equation - just the denominator of the transfer function.
To isolate the pole locations, do I make set the numerator to equal zero and solve for S?
Similarly, to isolate zero locations I set the denominator to zero and solve for S?

Thanks! :)

Well, there was really no need to find the poles (yes, they are found by setting the denominator to zero) since you weren't asked to find the time response, just the steady-state error. And that, for a unity step input, is just T(0).
(Reason: step input transform is 1/s but the final-value theorem says lim s→ 0 of sT(s) so the s's cancel).

If you had been asked to find the time response to a step input you would, after finding the poles, have to do a fractional expansion of the entire T(s), includng the numerator, and then done the inverse transform on each of the terms of that expansion.

Or, if you got lucky, you might have found the entire T(s) in a table of transforms, then there would have been no need to find the poles.