Task $1$: An open loop transfer function $G(s)$ that represent the dynamics of direct current $($DC$)$ electrical motor is under study. The transfer function from the armature voltage $u_a($input$)$ to the angular velocity of the motor shaft $($output$)$ is as follows:

$G(s)=\frac{(s)}{u_a(s)}=\frac{k_T}{s^{2}J{L}_a+s(b{L}_a+J{R}_a)+b{R}_a+k_b{k}_T}$

The variables, their meaning and values are provided in Table $1.$ Moreover, the armature resistance $R_a=2\mathrm{.}6.$

Table $1.$ The electrical machine parameters used in the project work.

$\backslash$table$[[$Rated speed, $n_n,3000$ rpm$],[$Armature inductance, $L_a,3$ mH$],[$Rotor Inertia, $J,80*{10}^{-6}kg{m}^2$