A state-space system is observable when its internal state can be reconstructed from the measured output over a finite interval, assuming the input is known.

For

$$\dot x=Ax+Bu,$$

$$y=Cx+Du,$$

with $n$ state variables, form the observability matrix

$$\mathcal O=\begin{bmatrix}C\CA\CA^2\\vdots\CA^{n-1}\end{bmatrix}.$$

The pair $(A,C)$ is observable exactly when

$$\operatorname{rank}(\mathcal O)=n.$$

The rows describe how different state directions become visible in present and future output behavior.

If a mode is unobservable, different internal states can produce indistinguishable measurements. No estimator using those measurements alone can reconstruct that hidden component.

Observability is therefore a structural question about dynamics and sensing. It determines what information the measurements can contain before any particular observer or filter is designed.