# Problem inverting Z transform for DT LTI system

1. Nov 12, 2009

### goc9000

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 $$\geq$$ 0 and 0 otherwise.

NOTE: The system is causal.

2. Relevant equations

I have calculated the z-domain response to be:

$$y(z) = \frac{z^2-3z+3}{(z-1)(z-3)}$$

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:

$$y(t) = -\frac{1}{2} + \frac{1}{2}3^t$$

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:

$$y(z) = 1 + \frac{z}{(z-1)(z-3)}$$

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

$$y(t) = \delta(t) -\frac{1}{2} + \frac{1}{2}3^t$$

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? :)