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2nd Order Control system PD controller

  1. Apr 27, 2016 #1
    Hi,

    This question on PD control is from a practice quiz.

    1. The problem statement, all variables and given/known data
    upload_2016-4-28_10-3-23.jpeg

    If you can't see it- the question asks to find values for Kp and Kd such that the system achieves 5% OS and has a settling time Ts of 3s.
    Cs = 3
    Cd = 2
    m = 5


    2. Relevant equations
    ω_n^2/(s^2 + 2ζw_n + ω_n^2) - 2nd order form

    Overall Transfer Function I found:
    Y/R = CG/1+CG = [ (Kp + Kds)/m ] / [ s^2 + ((Cd + Kd)/m) + ((Cs + Kp)/m)]

    Ts = 4/σ (2% of final value); where σ = ζω_n

    3. The attempt at a solution
    From the transfer function I found above i noticed that it is not exactly in the 2nd order form. However I was told that we still set

    ω_n = SQRT[ ((Cs + Kp)/m) ] {1}
    because when we add a derivative controller it only adds a zero which effects the numerator, not the poles on the denominator.

    firstly we know for 5%OS → ζ≈0.7
    from Ts = 3 = 4/ζω_n
    → ω_n = 4/3ζ = 40/21

    from eqn {1} - solving for Kp
    Kp = (ω_n^2)m - Cs = (5)(40/21)^2 - 3 = 15.14

    At this point I stopped because their answers were Kp = 17.93 and Kd = 12.12.
    Could you explain if the process i used is incorrect because I cant understand what I am doing wrong.

    Thank you
    (btw I just joined Physics Forums like an hour ago)
     

    Attached Files:

  2. jcsd
  3. Apr 29, 2016 #2

    donpacino

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    Gold Member

    There are often many ways to solve control problems. Plot your solution. Does it meet the design criteria?
     
  4. Apr 29, 2016 #3
    hi, can someone please have a look at my working to see if im on the right track, I spent many hours on it and im not even sure if this is the correct way.

    donpacino, Im a beginner in Matlab, Im not sure how a plot would verify my results.
     
  5. May 2, 2016 #4

    donpacino

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    Gold Member

    run a simulation to see if your KP and KI values meet the criteria for rise time and settling time.

    There are often multiple solutions that will work for problems like this
     
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