0
$\begingroup$

I am looking at the following system: $x(n) = A x(n-1) + b$ where x and b are vectors and A is a matrix. How can I derive the stability and causality conditions for such a system using Z transform?

If I apply Z transform to each element of x, I have: $X(z) = A z^{-1} X(z) + b$ Therefore, $(I - Az^{-1})X(z) = b$. Then, $X(z) = (I - Az^{-1})^{-1} b$.

Is this correct? What are the conditions for stability and causality?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Is $b$ time-varying, or did you perhaps mean $x_n = A x_{n-1} + b u_n$, where $u_n$ is the system input? $\endgroup$
    – TimWescott
    Sep 30, 2020 at 17:29
  • $\begingroup$ This looks so very much like the canonical state-space representation (aside from $b$ not being a constant multiplier to an input $u_n$) that I'm wondering -- is it homework? $\endgroup$
    – TimWescott
    Sep 30, 2020 at 17:30

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.