How Do You Determine the Transfer Function of a First Order Control System?

AI Thread Summary
The discussion centers on determining the transfer function of a first-order control system regulating water level in a tank. The governing equation is established, incorporating flow rates and the tank's cross-sectional area. The user struggles to express the transfer function in the standard form H(s)/Hi(s) = u/(1 + Ts) to identify the steady-state gain and time constant. Guidance is provided on simplifying the expression by dividing the numerator and denominator appropriately. The conversation emphasizes the importance of algebraic manipulation in deriving the desired transfer function format.
pobatso
Messages
16
Reaction score
0
Hi all, got a Control question here, and I'm struggling with what I assume is a simple algebraic step. Thanks in advance!

Homework Statement


A closed loop control system governs the level of water in a tank (H(s)) to meet a target height (Hi(S)). The flow of water into the tank is controlled by a transducer that feeds the current level of the tank into a differencing junction that works out the error (H(s)-Hi(s)). The flow rate of water pumped in is proportional to this error, with gain K.

The flow out of the tank is also constrained by a linearized flow restrictor, with flow out equal to the height/constant (Qd=H(s)/R).

There is also an additional flow into the tank from a separate pipe, with flow rate Qd.

The question is to find the transfer function, time constant and steady state gain. I've attached a diagram.

Homework Equations



The Attempt at a Solution



So far I've gotten as far as the governing equation:
Qi + Qd - Qo = A.dH(t)/Dt
Laplace: Qi(s) + Qd(s) - Qo(s) = A.s.H(s)

Where Qi = Flow in
Qd = Additional disturbance flow
Qo = Flow out
A = XSection area of tank

Using the information about the individual components this goes to:

K.Hi(s) - K.H(s) + Qd(s) - H(s)/R = A.s.H(s)

The correct way to describe transfer function (Checked with answer booklet):

H(s)=(R.K.Hi + R.Qd(s)) / (R.k +R.A.s + 1)

But I can't get the hang of expressing it in a way that would allow me to get the specific time function, ie I can't arrange it into form H(s)/Hi(s)=u/(1+Ts) where u is the SS Gain and T is the time constant.

Any help with this step would be fab!

Regards
 

Attachments

  • Capture.PNG
    Capture.PNG
    6.8 KB · Views: 569
Physics news on Phys.org
You have the right equation, so why can't you form H/Hi?

Is you problem the fact that you have a transform of the form a/(bs + c + 1)? Surely you know how to change that to the form d/(es+1)? High school algebra! :-)
 
Yes mate, that's the one. I know that I'm being a bit of a moron with this one, but its just left me. Watching youtube vids as well to help bring it back, but I think just having one example from someone else with one of these that is related to this topic area as well would be a great help, even if it is stuff I once covered in what feels like a very long time ago :)
 
pobatso said:
Yes mate, that's the one. I know that I'm being a bit of a moron with this one, but its just left me. Watching youtube vids as well to help bring it back, but I think just having one example from someone else with one of these that is related to this topic area as well would be a great help, even if it is stuff I once covered in what feels like a very long time ago :)

Righto! Let's take the general expression a/(bs+c+1). Now, divide numerator and denominator by c+1. What do you get?
 
Think that's it! Know its pretty stupid, but I can't get any sense out of the division. Not a good sign to get to 2nd year Uni without learning algebraic division :/ Thanks again!
 
pobatso said:
Think that's it! Know its pretty stupid, but I can't get any sense out of the division. Not a good sign to get to 2nd year Uni without learning algebraic division :/ Thanks again!

How about a/(bs + c+1) = d/(es+1) where

d = a/(c+1)
e = b/(c+1)

?
 
Yeah, I can't do it. Have been trying for a bit now, wasting far too much time on something this simple!
 
Have gotten something like:

(R^2(K^2.Hi + K.Qd(s)) - R(K.Hi + Qd(s))) / (R^2(A.K.s + K^2) - R.A.s -1
 
pobatso said:
Using the information about the individual components this goes to:

K.Hi(s) - K.H(s) + Qd(s) - H(s)/R = A.s.H(s)

Move all the H(s) terms to the left-hand side of this equation (which you yourself correctly derived). Then form H(s)/Hi(s). You must realize that there are really two transfer functions: H(s)/Hi(s) and H(s)/Qd(s). The problem ask you for the former only, so disregard the latter.

Then reduce the denominator of H(s)/Hi(s) to es+1 as I've shown you.

Don't give up!
 
Back
Top