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Calculating the transfer function from Laplace Transforms

  1. Sep 15, 2012 #1
    1. The problem statement, all variables and given/known data
    A Control system for a spacecraft platform is governed by the following equations:

    d²p/dt² + 2 dp/dt + 4p = θ
    v1 = r - p
    dθ/dt = .6*v2
    v2 = 7*v1

    All the variavles v1, v2, r, p, θ are functions in the time domain

    Determine the system transfer function P(s)/R(s)

    2. Relevant equations

    d²x/dt² -> Laplace -> s² * X(s)
    dx/dt -> Laplace -> s * X(s)
    x(t) -> Laplace -> X(s)

    3. The attempt at a solution

    So first I took the Laplace of all the given equations:

    d²p/dt² + 2 dp/dt + 4p = θ -> s²*P(s) +2s*P(s) + 4*P(s) = θ(s) )
    dθ/dt = .6*v2 -> θ(s) = .6*V2(s)

    v1 = r - p -> V1(s) = R(s) - P(s)
    v2 = 7*v1 -> V2(s) = 7*V1(s)

    I now have four equations I can substitute:
    1. s²*P(s) +2s*P(s) + 4*P(s) = θ(s)
    2. θ(s) = .6*V2(s)
    3. V1(s) = R(s) - P(s)
    4. V2(s) = 7*V1(s)

    I substitute Eq 3 into Eq 4 and I obtain V2(s) = 7*R(s) - 7*P(s)
    then I substitute my new V2(s) equation into Eq 2, which gives me

    θ(s) = 4.2*R(s) - 4.2*P(s)

    Which I then substitute into Eq 1.

    I get a Transfer Function

    P(s)/R(s) = 4.2/(s³ + 2s² + 8.2s)

    I think I did all my calculations right, but the solution in the book is this:
    ItMBm.jpg

    Can somebody tell me what I did wrong?
     
    Last edited: Sep 15, 2012
  2. jcsd
  3. Sep 15, 2012 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    You'll want to double check this part :wink:
     
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