Consider the double integrator system:

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

Design an LQG controller to stabilize the system and reject constant input disturbances. Choose the control weights such that the state$-$feedback loop has sensitivity bandwidth approximately $1ra\frac{d}{s}.$ Specify the corresponding controller $C_c(s)$ and demonstrate the LTR procedure following the steps of Problem $2.$