Exact discretization

Discretization of a continuous system. Assumption: $u[k]=u(t),kT\le t\le (k+1)T$

Proof at page=126

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We want to discretize a system on state space form

Link to original

To this

$$\begin{array}{rl}x[k+1]& =A_{d}x[k]+B_{d}u[k]\\ y[k]& =C_{d}x[k]+D_{d}u[k]\end{array}$$

Where

$$A_d=e^{AT},B_d=\left({\int }_0^T{e}^{A\tau }d\tau \right)B,C_d=C,D_d=D$$

Where T is the sample time or step size, and $T>0$. The approximation gets better as $T\to 0$.