Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs.
In control theory, the observability and controllability of a linear system are mathematical duals.
The concept of observability was introduced by the Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems.
[1][2] A dynamical system designed to estimate the state of a system from measurements of the outputs is called a state observer for that system, such as Kalman filters.
Consider a physical system modeled in state-space representation.
A system is said to be observable if, for every possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors).
On the other hand, if the system is not observable, there are state trajectories that are not distinguishable by only measuring the outputs.
For time-invariant linear systems in the state space representation, there are convenient tests to check whether a system is observable.
state variables (see state space for details about MIMO systems) given by If and only if the column rank of the observability matrix, defined as is equal to
of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied:
is the set of continuous functions from
can also be written as [3] Since the system is observable if and only if
The following properties for the unobservable subspace are valid:[3] A slightly weaker notion than observability is detectability.
A system is detectable if all the unobservable states are stable.
[4] Detectability conditions are important in the context of sensor networks.
[5][6] Consider the continuous linear time-variant system Suppose that the matrices
to within an additive constant vector which lies in the null space of
In fact, it is not possible to distinguish the initial state for
defined as above has the following properties: The system is observable in
are analytic, then the system is observable in the interval [
Consider a system varying analytically in
, and since this matrix has rank = 3, the system is observable on every nontrivial interval of
Define the observation space
to be the space containing all repeated Lie derivatives, then the system is observable in
, where Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,[10] Kou, Elliot and Tarn,[11] and Singh.
[12] There also exist an observability criteria for nonlinear time-varying systems.
[13] Observability may also be characterized for steady state systems (systems typically defined in terms of algebraic equations and inequalities), or more generally, for sets in
[14][15] Just as observability criteria are used to predict the behavior of Kalman filters or other observers in the dynamic system case, observability criteria for sets in
are used to predict the behavior of data reconciliation and other static estimators.
In the nonlinear case, observability can be characterized for individual variables, and also for local estimator behavior rather than just global behavior.