For the system

x$=$

$21$

$11$

x $+$

$1$

$2$

u

y $=$

$11$

x

$($a$)$ design a full$-$dimensional state estimator with eigenvalues at $\{2$j$2\}.$ Provide the state

equation for the estimator. $($Note: the state equation for the estimator should have both

u and y as inputs, and it should contain the estimated state vector $$x$.).$

$($b$)$ design a REDUCED$-$dimensional $($i$.$e$.,$ reduced$-$order$)$ state estimator with an eigenvalue

at $3.$ Let z be the state variable for the reduced$-$order estimator. Provide the z$$ state

equation that contains y and u as inputs, and provide the output equation that gives the

full$-$state estimate $$x $($and is written in terms of the y and z variables$).$