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Problem inverting Z transform for DT LTI system

  1. Nov 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Hello all,

    As part of a Control Theory assignment I have to compute the step response for the discrete-time LTI system given by the following state-space matrices:

    A = \[ \left( \begin{array}{ccc}
    0 & 1 \\
    0 & 3 \end{array} \right)\]

    B = \[ \left( \begin{array}{ccc}
    0 \\
    1 \end{array} \right)\]

    C = \[ \left( \begin{array}{ccc}
    1 & 0 \end{array} \right)\]

    X_0 = \[ \left( \begin{array}{ccc}
    1 \\
    1 \end{array} \right)\]

    The input is u(t) = 1(t), i.e. 1 for t [tex]\geq[/tex] 0 and 0 otherwise.

    NOTE: The system is causal.

    2. Relevant equations

    I have calculated the z-domain response to be:

    [tex]y(z) = \frac{z^2-3z+3}{(z-1)(z-3)}[/tex]

    3. The attempt at a solution

    To get the time-domain response I am supposed to compute the inverse Z transform via the residue theorem, and here I run into a little problem: if I run y(z) through the algorithm as it is, I get the time-domain response:

    [tex]y(t) = -\frac{1}{2} + \frac{1}{2}3^t[/tex]

    which is indeed the correct system response except for the point t=0, where y(t)=0 but the correct value should be 1 (this can be seen by multiplying X0 with C - a MATLAB simulation also agrees).

    What's funnier is that if I just rearrange y(z) a bit:

    [tex]y(z) = 1 + \frac{z}{(z-1)(z-3)}[/tex]

    and compute the inverse Z transforms for the two terms separately, I get:

    [tex]y(t) = \delta(t) -\frac{1}{2} + \frac{1}{2}3^t[/tex]

    which is the "real" correct response.

    I'm obviously doing something wrong since the result of a calculation really should not depend on the form of the terms... Can anyone help? :)
  2. jcsd
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