Theorem

A linear state observer exists for the system $(A,B,C,D)$ if and only if $(C,A)$ is Detectable. Furthermore, this observer is given by $$\begin{align*} \dot{w}(t)&=(A+LC)w(t)+\begin{bmatrix}-L & B+LD\end{bmatrix}\begin{bmatrix}y(t) \\ u(t)\end{bmatrix}\\ \hat{x}(t)&=Iw(t)+\begin{bmatrix}0 & 0\end{bmatrix}\begin{bmatrix}y(t) \\ u(t)\end{bmatrix} \end{align*}$$or more simply $$\dot{\hat{x}}(t)=A\hat{x}(t)+Bu(t)+L(C\hat{x}(t)+Du(t)-y(t))$$where $L$ is chosen s.t. $A+LC$ is Hurwitz.