Observability

In control theory, 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. The concept of observability was introduced by American-Hungarian engineer Rudolf E. Kalman for linear dynamic systems.

Definition

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 from the system's outputs it is possible to determine the behaviour of the entire system. If a system is not observable, this means the current values of some of its states cannot be determined through output sensors. This implies that their value is unknown to the controller (although they can be estimated through various means).

For time-invariant linear systems in the state space representation, there is a convenient test to check if a system is observable. Consider a SISO system with states (see state space for details about MIMO systems). If the row rank of the following observability matrix