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Homework Help: Laplace and pole zero diagram.

  1. Jan 7, 2013 #1
    The following diff. equation describes the functionality of a system with respect to time. However, it is not known how the system will behave when stimulated. Apply a forward Laplace transform to determine damping ratio and pole zeros. Plot a pole zero diagram and comment on stability.

    d2y/dt2 + 6dy/dt + 10y = 63dy/dt + 63x

    Putting like terms on either side of the equation:

    d2y/dt2 + 6dy/dt - 63dy/dt + 10y = 63x

    = d2y/dt2 - 57dy/dt + 10y = 63x

    Laplace transform:

    s2Y(s) - 57sY(s) + 10Y(s) = 63 .... (63 because unit impulse used to stimulate?)

    Simplify for Y(s):

    Y(s).(s2 - 57s + 10) = 63

    Solve for Y(s):

    Y(s) = 63/(s2 - 57s + 10)

    Using quadratic formula to find poles:

    s= (57 +/- √572 - 4 x 1 x 10)/(2 x 1)

    = (57 +/- 56.648)/2 = 28.5 +/- 56.648 (I was expecting a complex number!)

    I think that I am nearly there, but I suspect that I have gone wrong.
    Last edited: Jan 7, 2013
  2. jcsd
  3. Jan 7, 2013 #2

    rude man

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    Gold Member

    Factor out Y(s)F(s) = X(s) from the above transformed equation with X(s) = 1 (unit impulse input).

    Then Y(s)/X(s) = 1/F(s) and let H(s) = 1/F(s) so that now Y(s) = X(s)H(s). H(s) is now the system transfer function. Given any X(s) you can now compute Y(s) and consequently y(t).

    Note that the magnitude of the input impulse has nothing to do with pole/zero location, nor damping ratio. You only use it if you want to compute Y(s) and y(t).

    Note also that with an impulse input, all the poles & zeros of the output Y(s) are due to the system H(s) only.
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