where

$\backslash [$

P$($s$)=\backslash$frac$\{2\}\{$s$\}$

$\backslash ]$

and the system has the following characteristics:

$($C$1)$ The input disturbance $\backslash ($ d$\_\{$i$\}($t$)\backslash )$ has significant energy in the frequency band $\backslash ([0,5]\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash ).$

$($C$2)$ The measurement noise $\backslash ($ n$($t$)\backslash )$ has significant energy in the frequency band $[15,500]$ rad$/$s$.$

$($C$3)$ The reference signal $\backslash ($ r$($t$)\backslash )$ has significant energy in the frequency band $[0,20]$ rad$/$s$.$

You are required to synthesize proper controller transfer functions $\backslash ($ F$($s$)\backslash )$ and $\backslash ($ C$($s$)\backslash )$ to meet the following specific design goals while taking into account the above system characteristics:

$($P$1)$ Zero steady$-$state errors for ramp type output disturbances $\backslash ($ d$\_\{$o$\}($t$)\backslash )$

$($P$2)$ The response of the closed$-$loop system for step reference input $\backslash ($ r$($t$)\backslash )$ has no oscillations.

$($P$3)$ The closed$-$loop system should follow the reference signal well in the frequency band specified in $($C$3).$

To solve this problem, you may want to follow the following procedure:

a$)$ Determine the correct structure of feedback controller $\backslash ($ C$($s$)\backslash )$ that is needed to meet the steady$-$state performance requirement P$1.$

b$)$ Determine the suitable desired pole locations of the closed$-$loop system that take into account the system characteristics $($C$1)-($C$3).$ To receive full credit, you need to justify your answer as well.

c$)$ Determine the parameters of the feedback controller $\backslash ($ C$($s$)\backslash )$ to place the closed$-$loop poles at the desired locations.

d$)$ Determine a suitable filter transfer function $\backslash ($ F$($s$)\backslash )$ so that $($P$2)$ and $($P$3)$ are satisfied.