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Linear Systems Question (Several Parts)

  1. Nov 17, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the complete response of the system represented by




    and identify the zro-state and zero-input response components. Find the system transfer function and the system impulse response.

    2. Relevant equations

    3. The attempt at a solution

    I thought I solved for the complete response, but apparently what I got was just the zero-state response. I thought that by taking the laplace transform of the whole function, I will get the complete response. I then remembered that I need to have both zero-state and zero-input responses to get complete response, but I don't know how to find zero-input response.

    Here's what I did:







    At this point I thought this was already the complete response only in it's transformed for, so now I took the inverse laplace and got:


    which is what the answer is for the zero-state response, not complete response.
    So I tried figure out the zero-input response, but I just don't even know where to start. I looked everywhere and I can't seem to find the way to find it. Once I find the zero-input response, I will probably be able to solve the rest by myself.

    Can anyone please tell me how to find the zero-input response?
  2. jcsd
  3. Nov 18, 2007 #2
    Why can't you solve for the initial conditions analytically? Like plug in zero for t in your current solution and then solve for the initial conditions? For example:

    The first initial condition is y(0) = 1;

    [tex]y(0) = 1 -( (0) + 1 )e^{(0)} = 1 - 1[/tex]

    So add 1 to the equation to get satisfy the first condition.

    For the second one if you plug in zero into the derivative of your equation you will get -2, so add 4t to the equation so that when you take the derivative you get +4 and it won't mess with the first condition.

    This could be horribly wrong, but you can test it easily enough in matlab by taking the new function of t:

    [tex]y(t) = 2 + 4t -(t + 1)e^{-t}[/tex]

    and graphing it and comparing vs. step() of the original (zero initial condition) function.
    Last edited: Nov 19, 2007
  4. Nov 19, 2007 #3
    Thanks for the reply, I figured out the solution.
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