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 lineardynamic 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.
Definition
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. 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.
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.
Related concepts
Observability index
The observability index of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied:, where
Unobservable subspace
The unobservable subspace of the linear system is the kernel of the linear map given bywhere 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:
Detectability
A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable. Detectability conditions are important in the context of sensor networks. Nonlinear observers 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
Consider the continuous linear time-variant system 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:
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
Example
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 .
Nonlinear systems
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. Note: Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar, Kou, Elliot and Tarn, and Singh.
Observability may also be characterized for steady state systems, 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.
