$3$ Controllable Canonical Form

Consider a linear time$-$invariant plant with state$-$space realization:

$$

$$

$$

$$x$1$

$$x$2$

$$x$3$

y

$$

$$

$=$

$$

$$

$$

$0100$

$0010$

$2341$

$1010$

$$

$$

$$

$$

$$

$$

x$1$

x$2$

x$3$

u

$$

$$

$$

$($a$)$ Find the transfer function H$($s$)$ associated with this plant.

$($b$)$ Suppose we use a proportional feedback controller

u $=$ k$($y $+$ r$)$

where r is an external reference signal. Find the possible values of the controller gain k that

will stabilize the plant.