Implementing PI Controller using Forward Difference Approximation

$\displaystyle u(t)$	$\displaystyle =K \left\{e(t)+\frac{1}{\tau_i}\int_0^t e(t)dt\right\}$	(5.29)
On taking the Laplace transform, we obtain
$\displaystyle u(t)$	$\displaystyle =K\left\{1+\frac 1{\tau_i s}\right\}e(t)$	(5.30)
By mapping controller given in equation 5.30 to the discrete time domain using forward difference formula, we get
$\displaystyle u(n)$	$\displaystyle =K\left\{1+\frac{T_s}{\tau_i}\frac{1}{z-1}\right\}e(n)$	(5.31)
On cross multiplying, we get
$\displaystyle (z-1)u(n)$	$\displaystyle =K\left\{(z-1)+\frac{T_s}{\tau_i}\right\}e(n)$	(5.32)
We divide by and then by using shifting theorem, we get
$\displaystyle u(n)-u(n-1)$	$\displaystyle =K\left\{e(n)-e(n-1)+\frac{T_s}{\tau_i}e(n-1)\right\}$	(5.33)
The PI controller is usually written as
$\displaystyle u(n)$	$\displaystyle =u(n-1)+s_0 e(n)+s_1e(n-1)$	(5.34)
where
$\displaystyle s_0$	$\displaystyle = K$	(5.35)
$\displaystyle s_1$	$\displaystyle =K\left(-1+\frac{T_s}{\tau_i}\right)$	(5.36)