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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.
A special case of this result appeared first in 1963 in a paper by Elmer G. Gilbert, [1] and was later expanded to the current PBH test with contributions by Vasile M. Popov in 1966, [3] [4] Vitold Belevitch in 1968, [5] and Malo Hautus in 1969, [5] who emphasized its applicability in proving results for linear time-invariant systems.
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.
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 ...
While controllability and observability improvements for internal circuit elements definitely are important for test, they are not the only type of DFT. Other guidelines, for example, deal with the electromechanical characteristics of the interface between the product under test and the test equipment.
In control theory, we may need to find out whether or not a system such as ˙ = + () = + is controllable, where , , and are, respectively, , , and matrices for a system with inputs, state variables and outputs.
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).
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