1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

First Order Control System

  1. Aug 18, 2012 #1
    Hi all, got a Control question here, and I'm struggling with what I assume is a simple algebraic step. Thanks in advance!

    1. The problem statement, all variables and given/known data
    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.
    2. Relevant equations

    3. 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
     

    Attached Files:

  2. jcsd
  3. Aug 18, 2012 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    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! :-)
     
  4. Aug 18, 2012 #3
    Yes mate, thats 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 :)
     
  5. Aug 18, 2012 #4

    rude man

    User Avatar
    Homework Helper
    Gold Member

    Righto! Let's take the general expression a/(bs+c+1). Now, divide numerator and denominator by c+1. What do you get?
     
  6. Aug 20, 2012 #5
    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!
     
  7. Aug 20, 2012 #6

    rude man

    User Avatar
    Homework Helper
    Gold Member

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

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

    ?
     
  8. Aug 21, 2012 #7
    Yeah, I can't do it. Have been trying for a bit now, wasting far too much time on something this simple!
     
  9. Aug 21, 2012 #8
    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
     
  10. Aug 21, 2012 #9

    rude man

    User Avatar
    Homework Helper
    Gold Member

    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook