Problem $3.$ The block diagram of a control system is given in the figure:

a$)$ Draw the block diagram in Laplace domain.

b$)$ Calculate the transfer function from $\backslash (\backslash$beta $\backslash )$ to $\backslash (\backslash$omega $\backslash )$ and replace the inner loop with a single block.

c$)$ Calculate the transfer function from $\backslash ($ u $\backslash )$ to $\backslash (\backslash$varphi $\backslash )$ and replace the whole diagram with a single block.

d$)$ Based on the time$-$domain block diagram, formulate the ordinary differential equation model and write the system in state space form with states $\backslash (\backslash$varphi $\backslash )$ and $\backslash (\backslash$omega $\backslash ).$ Calculate the corresponding transfer function.

e$)$ Draw the block diagram in Laplace domain of a control loop with the system above and a PI controller. Calculate the closed$-$loop transfer function.