Consider the single input single output linear system

x$\_=$ Ax $+$ bu

y $=$ cx

where A $=$

$$

$00$

$01$

$$

; b $=$

$$

$1$

$1$

$$

; c $=($c$1$; c$2)$

$($a$)$ Show that the system is controllable.

$($b$)$ Write down an algebraic expression for an input u$($t$),0$ t $1$ such that x$(1)=$

$$

$1$

$2$

$$

when

x$(0)=$

$$

$0$

$0$

$$

$.$

You may write M for the controllability Gramian

R $1$

$0$ eA$(1)$BBT eAT $(1)$d$$

$($c$)$ Find a nonzero row vector c $=($c$1$; c$2)$ such that the system is not observable.

Assuming this particular vector c that you found, write down an initial state x$0$ which cannot be distinguished

from zero by processing the output of the system.