Problem $6$

Consider the feedback system as shown in Figure $2$:

a$.$ Determine conditions on $K_P,K_D,$ and $$ so that the closed$-$loop system is asymptotically stable.

b$.$ Specialize the conditions from point $($a$)$ to the following two separate cases:

i$.=0$

ii$.K_P=K_D$

c$.$ Let $d(t)=0$ and $r(t)=u(t).$ For $0,$ do there exist values of $K_P,K_D$ such that $y()=r()=1?$

d$.$ Let $r(t)=0,d(t)=sin(t),$ and $0,$ namely, let the user's command be zero but let the disturbance be a persistent sinusoid$.$ If $K_P,K_D$ are selected to satisfy the stabilizing conditions from point $($a$),$ what is $y()?$