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Thread: determining the unit ramp response of a transfer function

  1. #1
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    determining the unit ramp response of a transfer function

    Hello so I have a question about determining some values analytically for the response of a transfer function or a lag of the form: 1/(ts+1) where t is just a variable. I know what the response will look like. I know its initial value will be 0 but I am having a problem figuring out what its final value will be and also where the pole will be in terms of the variable. I know for instance that I can take the limit of that same transfer function at inf and 0 to find initial and final values for its step response. Also I know that t will be roughly at the location of the pole of the transfer function. Can I use the same principles with a ramp response?
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  2. #2
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    Re: determining the unit ramp response of a transfer function

    if I understand you have have a system transfer function

    $H(s) = \dfrac{\alpha}{s+\alpha},~\alpha = \dfrac 1 u$

    (I changed $t$ to $u$ to avoid confusion with the time variable $t$ )

    you input a unit ramp and wish to determine the output in the time domain.

    Is this correct?

    $X(t) = t,~0 \leq t$

    $X(s) = \dfrac {1}{s^2}$

    $Y(s) = \dfrac{\alpha}{s^2(s+\alpha)}= \dfrac{1}{s^2}-\dfrac{1}{\alpha s}+\dfrac{1}{\alpha (\alpha +s)}$

    by inspection we can see the output is a unit ramp minus a unit step scaled by $\dfrac{1}{\alpha}$ plus a damped exponential scaled by $\dfrac{1}{\alpha}$

    $y(t) = t + \dfrac{1}{\alpha}\left(e^{-\alpha t}-1\right)$

    It should be easy enough to plug some values of $t$ in

    Have I misunderstood the question?
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  3. #3
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    Re: determining the unit ramp response of a transfer function

    No you haven't misunderstood. Thanks for making the substitution that helped. Thanks for the help! Assuming that 1/s^2 is the Laplace of the ramp function?
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  4. #4
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    Re: determining the unit ramp response of a transfer function

    Quote Originally Posted by vysero View Post
    Assuming that 1/s^2 is the Laplace of the ramp function?
    yes
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