Problem$-1(25$ pts$)$: Consider the mass$-$damper system shown in Figure $1$:

Where, $\backslash ($ m$,$ b$,$ F $\backslash ),$ and $\backslash ($ x $\backslash )$ are the mass of the system, damping coefficient of the system, force applied to the system, and position of the system, respectively.

a$)(5$ pts$)$ Obtain the transfer function from $\backslash ($ F$(\backslash$mathrm$\{$~s$\})\backslash )$ to $\backslash ($ X$(\backslash$mathrm$\{$~s$\})\backslash )$ for the system.

b$)(10$ pts $)$ For the system parameters shown below;

$\backslash [$

$\backslash$begin$\{$array$\}\{$c$\}$

m$=1\backslash$mathrm$\{$~kg$\}\backslash \backslash$

b$=5\backslash$mathrm$\{$Ns$\}/\backslash$mathrm$\{$m$\}$

$\backslash$end$\{$array$\}$

$\backslash ]$

Design a feedback control system as shown in Figure $2.$

Here, $\backslash ($ C$($s$)\backslash )$ is the controller to be designed, and $\backslash ($ G$($s$)\backslash )$ is the plant transfer function you obtained in part a$.$ The closed$-$loop system must produce the unit$-$step response shown in Figure $3.$

c$)(5$ pts$)$ Obtain the steady$-$state error for the closed$-$loon designed in put $\backslash ($ b $\backslash )$ for unit step input and unit ramp input.

d$)(5$ pts $)$ Find the system type for the closed$-$loop system