discretization of continuous systems

Consider a continuous-time system with the following state spacerepresentation

(1)

where $x(t)\in {ℝ}^n$, $u(t)\in {ℝ}^r$ and $y(t)\in {ℝ}^m$ are the statevector, input vector and output vector of the system,respectively; $A\in {ℝ}^{n\times n}$, $B\in {ℝ}^{n\times r}$,$C\in {ℝ}^{m\times n}$ and $D\in {ℝ}^{m\times r}$ are the constantreal or complex matrices.

Suppose that the sampling interval is $\tau$. By using the stepinvariance transform or the zero-order hold (ZOH), i.e.,, discretizing the systemin (1) gives a discrete-time model,

(2)

where $x(k\tau )=x(t)|{}_{t=k\tau }$,$y(k\tau )=y(t)|{}_{t=k\tau }$, and

$$G_{\tau }:={\mathrm{e}}^{A\tau },F_{\tau }:={\int }_0^{\tau }{\mathrm{e}}^{At}dtB.$$(3)