unobservability

Unobservability refers to the property of a system or model whereby certain variables or aspects cannot be determined from observed data alone. In control theory and systems theory, unobservability means that, given the inputs and measured outputs over time, some state variables cannot be inferred with certainty. When a system is described by x' = Ax + Bu, y = Cx + Du, it is observable if the only state x that yields zero outputs for all time is x = 0. Equivalently, the observability matrix [C; CA; CA^2; ...; CA^{n-1}] has full rank equal to the state dimension. If not, the dimension of the unobservable subspace is positive, and those state components cannot be reconstructed from measurements, regardless of the input.

In practice, unobservability limits state estimation and controller design. The Kalman decomposition separates the state into observable and unobservable parts, and efficient estimators or controllers are designed for the observable subspace. If the unobservable part influences future outputs only through input channels, increasing sensing or excitation may reduce practical uncertainty, but true unobservability cannot be removed by sensors alone.

Outside control theory, unobservability arises in physics as quantities or regions that cannot be directly measured, such as inside opaque media or behind horizons, and in statistics as latent or hidden variables that are not identified by the data. Distinguishing observable from unobservable components is central to model building and to interpreting what inferences are valid.