# Discrete-time system from continuous

$G(s) = \dfrac{K}{Ts+1}$

$t = k\Delta T$

$G(s) = \dfrac{Y(s)}{U(s)} = \dfrac{K}{Ts+1} \ \Rightarrow \ Y(s)(Ts+1) = KU(s)$

$Ty'(t) + y(t) = K u(t)$

$T\dfrac{y_{k+1}-y_k}{\Delta T} + y_k = K u_k$

$T\dfrac{y_{k+1}}{\Delta T} + (1-\frac{T}{\Delta T})y_k = K u_k\ \Rightarrow \ y_{k+1} = -(1-\frac{T}{\Delta T})\frac{\Delta T}{T}y_k + K\frac{\Delta T}{T} u_k$

$y_{k+1} = -(\frac{\Delta T}{T}-1)y_k + K\frac{\Delta T}{T} u_k$