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OBSERVABILITY

- a system is observable iff the system state x(t) can be found by observing the input u and output y over a period of time from x(t) to x(t+h).

- If an input is to be observable it must be detectable in the output. For example consider the following state equations.

• This often happens when a system has elements that are decoupled, or when a pole and zero cancel each other.

• Observability can be verified formally for an LTI system with the following relationship.

• Another theorem for testing observability is given below. If any of the states satisfies the equation it is unobservable.

• Yet another test for observability is,

• If a system in unobservable, it is possible to make it observable by changing the model.

• A pole-zero cancellation is often the cause of the loss of observability.

• If all unstable modes are observable, the system is detectable.