Consider the continuous$-$time system

:

$($

x$\_($t$)=$ Ax$($t$)+$ Bu$($t$)$;

y$($t$)=$ Cx$($t$)+$ Du$($t$)$;

where

A $=$

$15$

$03$

; B $=$

$1$

$0$

; C $=[11]$; D $=0$:

a$.$ Assume that all the states are explicitly available for this sub$-$question. First check whether

the system is controllable or not. Then determine $($if possible$)$ a control law with poleplacement

techniques such that the poles of the closed$-$loop system are located at $(3$;$3).$

$($Hint: Is the system stabilizable?$)$

b$.$ Now we would like to design a state observer for the system. Is it possible to place the

observer poles to any point in complex plane? Furthermore, is the given system detectable?

Determine $($if possible$)$ a state observer, using pole$-$placement techniques such that the

observer poles are placed at $(15$;$20).($You can choose to do this step by hand or with

Matlab$).$

c$.$ Show that the system can be stabilized with the designed observer$-$controller pair. Derive

the dynamics of the extended system $($e$.$g$.$ nd the expression of

x$\_>$ e$\_>>$

$),$ where the

state estimation error is denoted with e $=$ x $^$x and $^$x denotes the state estimate vector.