Consider the control surface actuator with dynamics

i $$$($t$)+$ c $$$($t$)+$ k$($t$)=$ $($t$),$ $(0)=$ $0,$$(0)=$ $0,(2)$

where i$,$ k$,$ c in R are constant parameters, $($t$)$ denotes the control input. The output y$($t$)=$

h$($t$),$ where h in R is another constant parameter.

i$)$ Find the state space representation of the dynamics where state $($t$)=[$$($t$),$$($t$)]$T$.$

ii$)$ Consider PI control L$[$$($t$)]=$ H$($s$)$L$[$V $($t$)],$ where V $($t$)$ denotes the voltage applied to

the actuator and

H$($s$)=$ kp $+$ ki

s $.(3)$

Here kp$,$ kd are known parameters. Let feedback gain K$($s$)=$ ko$.$ Compute the loop gain,

the sensitivity transfer function and the complementary sensitivity transfer function