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Concept# Observability

Summary

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 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 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.
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 and only if the column rank of the observability matrix, defined as
is equal to , then the system is observable. The rationale for this test is that if columns 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 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.

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