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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.
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.
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).
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.
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,
with observability matrix. Here it is important to note, that the observability matrix and the system matrix are transposed: and A T. Ackermann's formula can also be applied on continuous-time observed systems.
The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros.
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.