I read this and this Wikipedia pages, but both of them are explaining continuous-time systems. My question is about discrete-time case.
For example, given the state-space equations of the second order, single input, single output discrete-time system:
$$ \begin{array}{lccrccr}
\mathbf{x}[n+1] &=& \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} & \mathbf{x}[n]
&+& \begin{bmatrix} b_{1} \\ b_{2} \end{bmatrix} & u[n]
\\
y[n] &=& \begin{bmatrix} c_{1} & c_{2} \end{bmatrix} & \mathbf{x}[n]
&+& d & u[n]
\end{array} $$
How do I find the transfer function (i.e.; $H(z) = \dfrac{Y(z)}{U(z)}$) of this system?
Best Answer
For a MIMO system given in the state-space form $$x[n+1]= A x[n] + B u[n]$$ $$ y[n] = C x[n] + D u [n] $$
The transfer matrix is given by $H(z) = C (zI-A)^{-1} B + D$, and $Y(z) = H(z) U(z)$.
See the derivation here.
Note for the SISO case, it degenerates the the same thing as a scalar system.