24.2: Observability

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It turns out it is more natural to think in terms of "unobservability" as reflected in the following definition.

Definition 24.1

A state \(q\) of a finite dimensional dynamic system is said to be unobservable over \([0, T)\) if, with \(x(0) = q\) and for every \(u(t)\) over \([0, T)\), we get the same \(y(t)\) as we would with \(x(0) = 0, i.e. an unobservable initial condition cannot be distinguished from the zero initial condition. The dynamic system is called unobservable if it has an unobservable state, and otherwise it is called observable.

The initial state \(x(0)\) can be uniquely determined from input/output measurements iff the system is observable (prove this). This can be taken as an alternate definition of observability.