Consider the LTIC system $$\begin{align*} \dot{x}(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t) \end{align*}$$ We know only the input and the output (i.e. $u,y$) and we want to approximate the state asymptotically in time. So, we want a $\hat{x}$ s.t. $$\lim_{ t \to \infty } \lVert x(t)-\hat{x}(t) \rVert =0$$The function $\hat{x}$ is an “estimator” or “observer” which itself is an LTIC system. So we wish to find an LTIC system $$\begin{align*} \dot{w}(t)&=Mw(t)+\begin{bmatrix}N & P \end{bmatrix}\begin{bmatrix}y(t) \\ u(t)\end{bmatrix}\\ \hat{x}(t)&=Qw(t)+\begin{bmatrix}R & S\end{bmatrix}\begin{bmatrix}y(t) \\ u(t)\end{bmatrix} \end{align*}$$s.t. $\lim_{ t \to \infty }\lVert x(t)-\hat{x}(t) \rVert=0,\forall x(0),w(0),u(t)$.