Problem $1$: PID controller and stead$-$state error

Consider a general feedback control system shown in Figure $1.$ Let $G_P(s)=\frac{1}{s(s+1)(s+2)},H(s)=1.$

Please design a P controller with gain $K_P,$ and write down the closed$-$loop transfer function.

Please design a PI controller with gains $K_P,$ and $K_{\text{I}}$ and write down the closed$-$loop transfer function.

With the designed $P$ controller, please check whether the steady$-$state error is $0$ when the input is a unit step signal, a unit ramp signal, and a unit parabolic signal, respectively. Please show your reasoning.

With the designed PI controller, please check whether the steady$-$state error is $0$ when the input is a unit step signal, a unit ramp signal, and a unit parabolic signal, respectively. Please show your reasoning.

Please design a PID controller with gains $K_p,K_{\text{I}},$ and $K_D,$ and write down the closed$-$loop transfer function.