Problem inverting Z transform for DT LTI system

[goc9000]

Homework Statement

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.

Homework Equations

I have calculated the z-domain response to be:

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

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? :)

 

[KataKoniK]

Thank you for sharing your question with us. As a scientist with expertise in control theory, I would like to offer some guidance on your calculations. First, let me assure you that your approach is correct and your final answer is indeed correct. The discrepancy you are observing can be explained by the fact that the inverse Z transform calculation is not unique and can have multiple solutions. In this case, both of your solutions are valid and equivalent, but just represented in different forms.

To understand this, let's take a closer look at the two terms in your rearranged y(z) expression:

1. \delta(t): This term represents the impulse response of the system, which is the response of the system to an instantaneous input. In other words, it is the response at t=0. Since your system is causal, the impulse response at t=0 is 0, which is why you are getting \delta(t) -\frac{1}{2} as the first term in your final y(t) expression.

2. -\frac{1}{2} + \frac{1}{2}3^t: This term represents the step response of the system, which is the response of the system to a step input. In this case, the step input is 1 for t \geq 0 and 0 otherwise. As you correctly calculated, the step response for your system is -\frac{1}{2} + \frac{1}{2}3^t. However, note that this response is only valid for t>0. For t=0, the step response is undefined since the input is discontinuous at that point. This is why you are getting a discrepancy at t=0 in your final y(t) expression.

In summary, both of your solutions are correct and equivalent, but just represent different aspects of the system's response. I hope this helps to clarify any confusion you may have had. Keep up the good work with your control theory assignment!

 

[Mene]

It appears that there may be an issue with the residue theorem computation in this case. It is possible that there is a pole at z=0 which is not accounted for in the current form of the equation. To resolve this issue, you may need to check your calculations and make sure all poles are accounted for. Additionally, it may be helpful to double check the causality of the system and ensure that it is correctly reflected in the calculations. It may also be beneficial to consult with your peers or instructor for further guidance and clarification.