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Discretize using a forward-Euler scheme

  1. Jan 15, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider the differential equation
    \begin{equation}
    y'''-y''=u
    \end{equation}
    Discretize (1) using a forward-Euler scheme with sampling period
    \begin{equation}
    \Delta=1
    \end{equation}
    and find the transfer function between u(k) and y(k)
    2. Relevant equations
    The Euler method is
    $$
    y_{n+1}=y_n+hf(x_n,y_n)
    $$

    3. The attempt at a solution
    Laplace transform of (1) yields
    $$
    s^3Y(s)-s^2Y(s)=U(s)
    $$
    From my teacher I know that
    $$
    s=\frac{z-1}{\Delta}
    $$
    Using this formula on the Laplace transform of (1) yields
    $$
    \bigg(\frac{z-1}{\Delta}\bigg)^3y_{k}-\bigg(\frac{z-1}{\Delta}\bigg)^2{y_k}=u_k
    $$
    Substituting (2) in this equation yields
    $$
    (z-1)^3y_k-(z-1)^2y_k=u_k
    $$
    $$
    y_{k+3}-y_{k+2}=u_k
    $$
    Now I want to find the transfer function between u(k) and y(k) but I don't see and y(k).
    Can somebody please help me? I have my exam tomorrow!
     
    Last edited: Jan 15, 2016
  2. jcsd
  3. Jan 15, 2016 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I can't follow the step
    to $$
    y_{k+3}-y_{k+2}=u_k $$Could you explain why this doesn't work out to e.g. $${y_k \over u_k}\ = \ {1\over (z−1)^3 − (z−1)^2 } {\rm\quad ?} $$
     
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