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Designing a PID controller - control systems engineering

  1. Oct 12, 2012 #1
    1. The problem statement, all variables and given/known data

    I am given the transfer function
    Gp (s) = 1/(s+16)
    and I have to design a controller of the form
    Gc (s) = K(1 + 1/T*s)
    (which is a PID controller)

    To meet the following requirements:

    Maximum steady state error to a steadily changing desired position = 5%

    Maximum percentage overshoot to a step change in desired position = 4.4%

    Maximum settling time allowed following a step change in desired position = 0.5sec

    2. Relevant equations

    %OS = %overshoot
    ζ = damping ratio
    ω = natural frequency

    %OS = 100e^(-pi(ζ/sqrt(1-ζ^2)))

    Settling time = 4/ζ

    3. The attempt at a solution

    First I did the calculations knowing the settling time and overshoot and found
    ζ = 0.705
    ω = 11.3464
    although I don't know how useful that working is.

    Then to make progress on the controller I modified the original equation:
    Gc (s) = K(1 + 1/T*s)
    to
    Gc (s) = K((s + Z)/s)
    where Z = 1/T

    I derived the open loop function by doing
    G(s) = Gc(s)*Gp(s)
    = (K(s + Z))/(s(s+16))

    And with the steady state error limit I tried to get some progress done on that but couldn't get very far there either.

    Ess = lim(s->0) (1/(1+G(s)))
    5% = 1/(1 + K*Z)

    Any help here would really be appreciated, I feel like I'm missing something really simple here and it's starting to get frustrating.

    Cheers.
     
  2. jcsd
  3. Oct 13, 2012 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    Your controller is a PI, not a PID.

    What is the overall closed-loop transfer function of your system, including the controller?

    Also, settling time = 4/ζω.
     
    Last edited: Oct 13, 2012
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