Consider the system

x$($k $+1)=$ Ax$($k$)+$Bu$($k$),$ y$($k$)=$ Cx$($k$)$

where

A $=[0110],$ B $=[01],$ C $=[10]$

$($a$)$ Use state feedback to assign the eigenvalues of A$+$BF at $0\mathrm{.}5$j$0\mathrm{.}5.$ Plot x$($k$)$ for x$(0)=0\mathrm{.}60\mathrm{.}4$ T$.$

$($b$)$ Design an observer with eigenvalues at $(1$ j$),$ where $0.$ What is the acceptable range of $?$

$($c$)$ Use the state estimate $$ x from $($b$)$ in the linear feedback control u $=$ F$$ x$,$ where F was found in $($a$).$ For x$(0)=0\mathrm{.}60\mathrm{.}4$ T and $$ x$(0)=00$ T$,$ plot x$2($k$),$ e$2($k$)=$ x$2($k$)$ x$2($k$),$ y$($k$),$ and u$($k$)$ of the closed$-$loop system for $=0\mathrm{.}5,0\mathrm{.}2,$ and $0.$ Comment on the results.

$($d$)$ With u $=$ F$$ x $+$Nr$,$ derive a state model of the closed$-$loop system with input r and output y$.$ How would you choose N so that the system will have zero steady$-$state error to step inputs.