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

[pobatso]

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

 

[rude man]

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! :-)

 

[pobatso]

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 :)

 

[rude man]

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?

 

[pobatso]

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!

 

[rude man]

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)

?

 

[pobatso]

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

 

[pobatso]

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

 

[rude man]

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!