Scenario $4$: The input, mass B is replaced by a motor which can provide a force $\backslash (($F$)\backslash )$ to lift up mass A $(1$ kg$\backslash ()\backslash )$ along the plane by distance $\backslash ((\backslash$mathrm$\{$L$\}=3\backslash$mathrm$\{$~m$\})\backslash )$

$($a$)$ Set up the new system equation

$($b$)$ Given that a sensor is used to measure the position of the mass $($y$),$ a proportional $($K$)$ controller is used to control the force input $($based on the position error$)$ such as the position of the mass reaches the desired position $\backslash (\backslash$left$($y$\_\{$d$\}=3\backslash$mathrm$\{$~m$\}\backslash$right$)\backslash ).$ Simulate the position of the mass over time by using different K values $\backslash ((\backslash$mathrm$\{$K$\}=0\mathrm{.}3,0\mathrm{.}5\backslash ),$ and $0\mathrm{.}8).$ Explain whether or not the mass can reach the desired position at steady state

$($c$)$ Modify the system $($e$.$g$.$ adding a damper to the system, change the controller, etc$)$ to enhance the performance of the system