Problem $4.$ Consider the state model for a DC motor, x$=01000$ Kt Jm $0$ Ke La $$ Ra La $$ x $+001$ La $$ u where x $=$ m$,$m$,$ ia T $,$ m is the shaft angle position, $$m is the shaft angle speed, and ia is the armature current. The parameters for the motor are: Ra $=0\mathrm{.}5,$ La $=0\mathrm{.}015$ H$,$ Jm $=0\mathrm{.}2$ kg $$m$2,$ and Ke $=$ Kt $=0\mathrm{.}4.$ There are three sensors available for possible outputs: an encoder to measure m y$1=100$ x $+0$u$,$ a tachometer to measure the speed $$m$.$ y$2=010$ x $+0$u and a hall effect sensor to measure the armature current ia$,$ y$3=001$ x $+0$u $($a$.)$ Design a state feedback controller such that the dominant poles correspond to a settling time of $1$ sec and percent overshoot of $10\%.$ The other two pole may be placed as real valued pole with factor of at least $4$ times larger than the real part of the dominant poles. $($b$.)$ Check the observability for all three possible sensor locations. Are any of the options not feasible for observer design? $($c$.)$ Design an observer for y$1$ such that the observer has closed loop poles of $-30,-35,$ a