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?

  • 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
    Commented 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
    Commented Sep 30, 2020 at 17:30


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