Consider the following system,which is given in diagonal form:

$$x $=$

$$

$$

$100$

$020$

$003$

$$

$$ x $+$

$$

$$

$1$

$1$

$0$

$$

$$ u$.(3)$

y $==[101]$ x $(4)$

$($a$)(5$ points$)$ Show that the system is neither controllable nor observable by finding the uncontrollable and unobserv$-$

able states.

$($b$)(5$ points$)$ Determine the transfer function of the system and the order of a minimal state$-$space realization. How

can this be related to the controllable and observable states of the system?

$1$