- 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.