State Variable Analysis

$\dot{X}=AX+BU$

$Y=CX+DU$

$sX(s)=AX(s)+BU(s)$

$\Rightarrow (sI-A)X(s)=BU(s)$

$\Rightarrow X(s)=(sI-A)^{-1}BU(s)$

$Y(s)=CX(s)+DU(s)$

$\Rightarrow Y(s)=C(sI-A)^{-1}BU(s)+DU(s)$

$\Rightarrow Y(s)=[C(sI-A)^{-1}B+D]U(s)$

$\Rightarrow \frac{Y(s)}{U(s)}=C(sI-A)^{-1}B+D$

$\frac{Y(s)}{U(s)}=C(sI-A)^{-1}B$