Transfer Function of a State Space Model

The transfer function can be defined as the $z$ transform of the impulse response:

$$H(z) \isdef \sum_{n=0}^{\infty} h(n) z^{-n} = D + \sum_{n=1}^{\in... ...z^{-n} = D + z^{-1}C \left[\sum_{n=0}^{\infty} \left(z^{-1}A\right)^n\right] B$$

Using the closed-form sum of a matrix geometric series,^E.4we obtain

$$\fbox{$\displaystyle H(z) = D + C \left(zI - A\right)^{-1}B$} \protect$$ (E.5)

Note that if there are $p$ inputs and $q$ outputs, $H(z)$ is a $p\times q$ transfer-function matrix (or ``matrix transfer function'').