In control theory, observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. The observability and controllability of a linear system are mathematical duals. The concept of observability was introduced by Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems. A dynamical system designed to estimate the state of a system from measurements of the outputs is called a state observer or simply an observer for that system.
Consider a physical system modeled in state-space representation. A system is said to be observable if, for any 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). In other words, one can determine the behavior of the entire system from the system's outputs. On the other hand, if the system is not observable, there are state trajectories that are not distinguishable by only measuring the outputs.
Linear time-invariant systems
For time-invariant linear systems in the state space representation, there are convenient tests to check whether a system is observable. Consider a SISO system with state variables (see state space for details about MIMO systems) given by
If the row rank of the observability matrix, defined as
is equal to , then the system is observable. The rationale for this test is that if rows are linearly independent, then each of the state variables is viewable through linear combinations of the output variables .
The observability index of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied: , where
The unobservable subspace of the linear system is the kernel of the linear map given by
where is the set of continuous functions from to . can also be written as 
Since the system is observable if and only if , the system is observable if and only if is the zero subspace.
The following properties for the unobservable subspace are valid:
A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.
sliding mode and cubic observers can be applied for state estimation of time invariant linear systems, if the system is observable and fulfills some additional conditions.
Linear time-varying systems
Suppose that the matrices , and are given as well as inputs and outputs and for all then it is possible to determine to within an additive constant vector which lies in the null space of defined by
where is the state-transition matrix.
It is possible to determine a unique if is nonsingular. In fact, it is not possible to distinguish the initial state for from that of if is in the null space of .
Note that the matrix defined as above has the following properties:
- satisfies the equation
Observability matrix generalization
The system is observable in [,] if and only if there exists an interval [,] in such that the matrix is nonsingular.
If are analytic, then the system is observable in the interval [,] if there exists and a positive integer k such that
where and is defined recursively as
Consider a system varying analytically in and matrices
Then , and since this matrix has rank = 3, the system is observable on every nontrivial interval of .
Given the system , . Where the state vector, the input vector and the output vector. are to be smooth vector fields.
Define the observation space to be the space containing all repeated Lie derivatives, then the system is observable in if and only if .
Static systems and general topological spaces
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 . 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.
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