Problem:

Given an antenna positioning system as shown below:

The schematics of the system can be seen in the following figure: The values of the parameters in the schematics can be seen below for $3$ configurations:

Schematic Parameters

Tasks:

$1.$ Derive the mathematical models and block diagrams for configuration $2$ and $3$

$2.$ Simplify the block diagrams that you get in task $1$ until they become unity feedback systems

$3.$ Design $($with reasonings$)$ PD controls on $2$ and $3$ with the following design requirements:

a$.$ Maximum overshoot $\backslash (25\backslash \%\backslash )$ b$.$ Settling time $(2\backslash \%)$ is $2$ seconds

c$.\backslash (\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$v$\}\}\backslash )($velocity constant for steady state error with ramp input$)$ is $20$

$4.$ Show the simulation results of the PD$-$controlled systems $($results of task $3$ in closed$-$loop systems$)$ for configuration $2$ and $3,$ and highlight the achievements $($compare with the simulation results of uncontrolled$/$original systems$)$