$.($a$)$ Convert $($manually$)$ the following digital transfer function into its state space representation:

$0\mathrm{.}82+1$

$()=21\mathrm{.}2+0\mathrm{.}4$

Provide the detail derivations and the values of state space parameter matrices.

$($b$)$ Use MATLAB to find the state space model and compare it with your manual achievement.