Problem $1.(20$ points$)$

$($a$)$ Construct a Simulink model of the helicopter control system shown in the figure below.

Choose some reasonable parameters and plot the actual angular velocity as a function of time, assuming that

the desired angular velocity is zero, $(t)=0,$ and that the top$-$rotor torque is non$-$zero, $T_t(t)=bu(t).$ Give

your plot for three different values of $K$ and discuss how the behavior varies with $K.$

$($b$)$ Modify the model of part $($a$)$ to replace the Controller $($the simple scale$-$by$-$K actor$)$ with the alternative

controller shown in the figure below.

This alternative controller is called a proportional$-$integrator $($PI$)$ controller. It has two parameter $K_1$ and $K_2.$

Experiment with the values of these parameters, give some plots of the behavior with the same inputs as in

part $($a$),$ and discuss the behavior of this controller in contrast to the one of part $($a$).$

The question is from course Embedded and Cyber Physical System.