Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals.

Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs. (This definition is slanted towards the state space representation.) Less formally, this means that you can watch your sensors and figure out what's going on inside the system, eventually, although eventually can be a very long time.

For linear systems in the state space representation, there is a convenient test to check if a system is observable. Using the standard notation, if the rank of the observability matrix

is equal to p, then the system is observable.