Z transform of a state space model in discrete time

Consider a LTI state space model of the form

\[ \begin{align*} x^{n+1} &= A x^n + B u^n, \\ y^{n+1} &= C x^n + D u^n. \end{align*} \]

Applying the Z-transform yields

\[ \begin{align*} z X(z) -z x_0&= A X(z) + B U(Z), \\ Y(z) &= C X(z) + D U(z), \end{align*} \]

or equivalently

\[ \begin{align*} (z -A) X(z) &= z x_0 + B U(Z), \\ Y(z) &= C (z -A)^{-1} z x_0 + \left(C (z -A)^{-1}B + D\right) U(z). \end{align*} \]

In order to compute a transfer function, consider \(x_0 = 0\). We then get

\[ \begin{align*} Y(z) = \left(C (z -A)^{-1}B + D\right) U(z). \end{align*} \]

If \(y\) is taken to be a power-balanced output to \(u\) , the transfer function \(H(z) = y/u\) should correspond to an impedance-like (or admittance) quantity. For a power-preserving discretization, this quantity is expected to be passive (such that the poles of \(\left(C (z -A)^{-1}B + D\right)\) must have positive real part).