Consider the double integrator system:

$x^{}=Ax+Bu,y=Cx+Du$;$A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=0.$

a$.$ Design an observer$-$based controller to stabilize the system and specify the corresponding controller $C_c(s).$ Use $(-1,-1)$ and $(-10,-10)$ as closed loop poles for the state feedback and observer poles that are $1x$ and $10$x the state feedback poles. Describe the properties of the closed$-$loop input disturbance sensitivity $(I+C_c(s)P(s){)}^{-1}$ 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 $(I+C_c(s)P(s){)}^{-1}$ and compare it with the sensitivity of the statefeedback solution of Problem $1.$