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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.
Controllability and observability are dual aspects of the same problem. Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.
As in the finite-dimensional case, observability is the dual notion of controllability. In the infinite-dimensional case there are several different notions of observability which in the finite-dimensional case coincide. The three most important ones are: Exact observability (also known as continuous observability), Approximate observability,
Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal.
The observability and controllability of a system are mathematical duals (i.e., as controllability provides that an input is available that brings any initial state to any desired final state, observability provides that knowing an output trajectory provides enough information to predict the initial state of the system).
The term is borrowed from control theory, where the "observability" of a system measures how well its state can be determined from its outputs. Similarly, software observability measures how well a system's state can be understood from the obtained telemetry (metrics, logs, traces, profiling). The definition of observability varies by vendor:
Camouflage observability could also be a useful concept, including defining metrics for observability and then advertising based on those metrics. (Although like the software case, this focuses on some numerical measure of observability rather a hard binary choice of observable vs. unobservable).
That is called the discrete Observability Gramian. We can easily see the correspondence between discrete time and the continuous time case, that is, if we can check that W d c {\displaystyle {\boldsymbol {W}}_{dc}} is positive definite, and all eigenvalues of A {\displaystyle {\boldsymbol {A}}} have magnitude less than 1 {\displaystyle 1} , the ...
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