Wikipedia / Observability
Wikipedia / Observability
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. 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.
Definition
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). 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 systemss in the state space representation, there are convenient tests to check whether a system is observable. Consider a SISO state variables (see state space for details about MIMO systems) given by
Observability matrix
If and only if the column rank of the observability matrix, defined as
. Observability is a sufficient and necessary condition for the design of continuous-time state observers.
Related concepts
Observability index
The observability index of a linear time-invariant discrete system is the smallest natural number, where
Unobservable subspace
The unobservable subspace of the linear system is the kernel of the linear map given by can also be written as
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.
Functional observability
Functional observability is a property that extends the classical notion of observability for cases in which full-state observability is not possible or required (due to lack of measurement signals and sensor placement). Rather than requiring full-state reconstruction, functional observability establishes the condition under which a linear functional can still be estimated using solely information from the output signals., a system is functionally observable if and only if
Functional observability is an important concept because it determines the sufficient and necessary condition under which a functional observer (also known as a Darouach observer . Under certain conditions, functional observability and output controllability are mathematical duals) are equivalent under a system transformation.
Linear time-varying systems
Consider the continuous linear time-variant system
to within an additive constant vector which lies in the null space defined by
is the state-transition matrix.
is nonsingular.
defined as above has the following properties:
is symmetric
satisfies the linear matrix differential equation
satisfies the equation
Observability matrix generalization
is nonsingular.
and a positive integer k such that
is defined recursively as
Example
and matrices .
Nonlinear systems
are to be smooth vector fields.
to be the space containing all repeated Lie derivatives, where
Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar, Kou, Elliot and Tarn, and Singh.
There also exist an observability criteria for nonlinear time-varying systems.
Static systems and general topological spaces
. Just as observability criteria are used to predict the behavior of Kalman filters 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.