Consider the satellite antenna positioning system studied in assignment $1.$

It is desired to design a closed loop position controller for the antenna system.

The system model is given in the Simulink file 'motor$\_$antenna$\_$project.slx$\text{'}.$

$1.$ Select a Proportional controller $\backslash ($ K$($s$)=$K$\_\{$p$\}\backslash ).$

a$)$ Analyze the stability of the system as a function of the controller gain.

b$)$ Design the controller gain to have a closed loop system with an underdamped response and a maximum overshoot of $\backslash (9\mathrm{.}5\backslash \%\backslash ).$

c$)$ Simulate the response of the system to a step reference with amplitude $10$ cm $.$

d$)$ Find the corresponding settling time.

e$)$ Simulate the response of the system to a unit$-$step wind disturbance.

$2.$ Select a PI controller.

a$)$ Analyze the stability of the system as a function of the controller gains.

b$)$ Design the PI controller such that the closed loop system has an under$-$damped step response with

$-\backslash (9\mathrm{.}5\backslash \%\backslash )$ overshoot

$-\backslash (2\backslash \%\backslash )$ settling time $\backslash ($ t$\_\{$s$\}\backslash$leq $0\mathrm{.}5\backslash$mathrm$\{$sec$\}\backslash )$

c$)$ Simulate the response of the system to a step reference $($amplitude $\backslash (=10\backslash$mathrm$\{$~cm$\}\backslash ))$ and check the design specifications.

d$)$ Simulate the response of the system to a unit$-$step wind disturbance.

e$)$ Discuss the results.