Design a state feedback controller law u $=-$kx to yield a $20\mathrm{.}8\%$ overshoot and settling time

of $4$ seconds for the system represented by the following transfer function:

$()=(+4)$

$(+1)(+2)(+5)$

The state space representation is given by$,$

$=[$

$510$

$021$

$001$

$]+[$

$0$

$0$

$1$

$]()$

$=[110]$

a$.$ Check the controllability

b$.$ Determine the transformation matrix $=(,)(,)1$

c$.$ Determine the controller values for the original system

d$.$ Write the state space form provided an input $$ to the system:

$=()+,=$

e$.$ Use MATLAB to find the closed$-$loop transfer function. Have you met your design

requirements? Why or why not?