Consider the system defined by x$=$ Ax $+$ Bu y $=$ Cx$,,$ where A $=0123,$ B $=01,$ C $=(10).$ a$)(5$ marks$)$ Check the controllability and observability of the system. b$)(18$ marks$)$ Design a state feedback plus integrator of the form u $=$Kx $+$ KI Z t $0($yd$($ $)$ y$($ $))$d$,$ in order to have the closed$-$loop eigenvalues at $2+$ j$,2$ j$,10,$ and the steady state error equal to zero for a unit step input. What is the performance $($in terms of settling$-$time and overshoot$)$ that we are aiming at by imposing these eigenvalues for the closed loop system. c$)(10$ marks$)$ Design a state observer with the following performance: $*1$ seconds Settling$-$time, $*2\%$ Overshoot. d$)(20$ marks$)$ Design a discrete$-$time state feedback controller of the form uk $=$Kxk$,$ with the sampling period T $=0\mathrm{.}1$s$,$ in order to achieve the following performance for the equivalent continuoustime closed loop system: $*1$ seconds Settling$-$time, $*2\%$ Overshoot.