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Control Engineering

  1. Apr 21, 2009 #1
    1. The problem statement, all variables and given/known data
    If the steady-state error of the proportional-plus-derivative-plus acceleration system is to be less than 10% determine a suitable value for Kp.

    2. Relevant equations
    The input of the system is: v
    The output of the system is: y

    The transfer function of the system including the controller is:
    (23(Kp+4s+s^2)) / (s^3+19s^2+48s+96+23Kp)

    3. The attempt at a solution
    With steady state: s = 0
    Therefore: y steady state = v x (23Kp / (96 + 23Kp))
    The error of the system is: v-y
    Therefore: error = v - (23Kp / (96+23Kp))v
    error = 96v / (96 + 23Kp)

    How can I correctly substitute the error of 10% now in this equation.
    I think (v-y)/y = 0.1 [error]

    But this will leave 3 variables in the equation...

    Thanks in advance!
     
  2. jcsd
  3. Apr 23, 2009 #2
    Hi,

    The error is a ratio. So when you get to

    error = ( 96 / ( 96 + 23 Kp ) ) v,

    The error in percentage is:

    %error = ( 96 / ( 96 + 23 Kp ) ) < 10%

    There you can then solve for the minimum Kp.
     
  4. Apr 26, 2009 #3
    Cheers.

    So that means that:
    error = v - y
    error [%] = ((v-y)/v)*100%

    So:

    error = ( 96 / ( 96 + 23 Kp ) ) v
    error [%] = ( 96 / ( 96 + 23 Kp ) ) * 100%

    ( 96 / ( 96 + 23 Kp ) ) * 100% < 10%
    ( 96 / ( 96 + 23 Kp ) ) < 0.1

    Then I get: Kp > 37.565

    Is that correct? Thanks in advance!
     
    Last edited: Apr 26, 2009
  5. Apr 26, 2009 #4
    Hi, you must have done something wrong in the algebra, because Kp should be positive.

    ( 96 / ( 96 + 23 Kp ) ) < 0.1

    Conceptually, when the denominator is 960, you will have exactly 0.1 error.
    As Kp increases, the error decreases further. When you solve it using algebra
    you can verify that Kp to give you a denominator that is greater than 960.
     
  6. Apr 27, 2009 #5
    I see, I just edited my reply before your reply.

    Thanks a lot!
     
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