Problem $5.$ Consider the segway from Homework $4.$

We seek to design the following controller to stabilize

the segway upright while driving with a constant

reference velocity v$\_($r$)$ :

u$\_($s$)=$K$\_($v$)($v$\_($C$)-$v$\_($r$))+$K$\_(\backslash$phi $)\backslash$phi $+$K$\_(\backslash$omega $)\backslash$omega $.$

The linearized state space model is given by the input

tilde$($u$)\_($s$),$ states tilde$($v$)\_($C$),$tilde$(\backslash$phi $),$widetilde$(\backslash$omega $),$ output tilde$(\backslash$phi $),$ and matrices:

A$=[[-0\mathrm{.}390,-2\mathrm{.}648,0\mathrm{.}076,0,1$

$1\mathrm{.}056,18\mathrm{.}438,-0\mathrm{.}206]]$

$1\mathrm{.}056$

$0,$B$=[[0\mathrm{.}402]]$

$0$

$-1\mathrm{.}088,$

C$=[[0,1,0]],$D$=0$

whereas the controller of the linearized system is:

tilde$($u$)\_($s$)=$K$\_($v$)$tilde$($v$)\_($C$)+$K$\_(\backslash$phi $)$tilde$(\backslash$phi $)+$K$\_(\backslash$omega $)$widetilde$(\backslash$omega $)=-$Kx$.$

aK$=[-$K$\_($v$)-$K$\_(\backslash$phi $)-$K$\_(\backslash$omega $)]$ such

that the poles of the closed$-$loop system are s$\_(1)=-3,$

s$\_(2)=-5,$ and s$\_(3)=-16.$

cL$=[[$L$\_($v$),$L$\_(\backslash$phi $),$L$\_(\backslash$omega $)]]^($T$)$ such that the poles

of the observer are s$\_(1)=-2,$s$\_(2)=-3,$ and s$\_(3)=-4.$

Submit your codes.