Consider the double integrator system:

$=+,=+$; $=[01$

$00],=[0$

$1],=[10],=0.$

a$.$ Design an observer$-$based controller to stabilize the system and specify the corresponding controller

$().$ Use $(-1,-1)$ and $(-10,-10)$ as closed loop poles for the state feedback and observer poles that are

$1$x and $10$x the state feedback poles. Describe the properties of the closed$-$loop input disturbance

sensitivity $(+()())1$ and compare it with the sensitivity of the state$-$feedback solution of

Problem $1.$

b$.$ Repeat $($a$)$ for the system with the RHP zero C $=[1,-0\mathrm{.}1].$ Describe the properties of the closed$-$loop

input disturbance sensitivity $(+()())1$ and compare it with the sensitivity of the state$-$

feedback solution of Problem $1$