The continuous LTI system is

$\dot{x}=Ax+Bu$

The corresponding discrete system is

$x_k=Fx_{k-1}+Gu_{k}$

What are F and G? Usually we have two methods: use zero-order hold method to discretize it; or use approximation to a rigorous discrete system. Today I will show they are the same to the first order !!! So in the future, we are confident to directly use zero-order holding method to discretize a system as long as you want the first order accuracy!!!

1) Zero-order holding method:

$x_k=x_{k-1}+(Ax_{k-1}+Bu_k)\Delta t$

where $\Delta t=t_{k}-t_{k-1}$. Then

$x_k=(I+\Delta t A)x_{k-1}+\Delta t Bu_k$

So

>>>$F=I+\Delta t A$

>>>$G=\Delta t B$

2) Rigorous discrete system

From

$x(t)=e^{A(t-t_0)}x_0+\int_{t_0}^te^{A(t-\tau)} Bu(\tau) \mbox{d}\tau$

it is easy to see

>>>$F=e^{A\Delta t}=I+\Delta t A + \frac{1}{2}A^2\Delta t^2\cdots$

>>>$G=\left(\int_0^{\Delta t}e^{A\alpha}d\alpha\right)B=\left(\Delta t+\frac{1}{2}A\Delta t^2+\cdots\right)B$

Conclusion: Obviously, method 1 and 2 are equivalent to the first order!!!