Observability
A system is observable if its full internal state can be reconstructed from the outputs over a finite time interval.
Can the Output Reveal Every State
Observability is the dual question to controllability. It asks whether, by watching the outputs over some time window, we can deduce the complete internal state. If some component of the state produces no distinguishable effect on the outputs, that component is unobservable and cannot be estimated.
The observability matrix
For a system with state matrix A and output matrix C of order n, stack C, C*A, C*A^2, up to C*A^(n-1) vertically to form the observability matrix. The system is observable if and only if this matrix has full rank n. Like the controllability test, it is a finite rank check.
Why it matters
- State-feedback control assumes the state is known, but usually only outputs are measured.
- An observer or Kalman filter reconstructs the state from outputs, and this is possible only if the system is observable.
- An unobservable unstable mode can grow without ever appearing in the measurements: dangerous and undetectable.
- Observability guides sensor placement, telling the designer which measurements are needed to see the whole state.
Detectability
The weaker counterpart to observability is detectability: every unobservable mode is stable. A detectable system may not let you reconstruct all states, but any state you cannot see decays on its own, so a stable observer can still be built. Detectability is the practical minimum for estimator design.
The estimation link
Observability guarantees that an observer's error dynamics can be given any desired poles, so the estimate converges as fast as wanted. Combined with controllability and the separation principle, this lets state estimation and state feedback be designed independently, then joined into a working output-feedback controller.
In monitoring a complex plant such as a plasma, where many internal quantities cannot be measured directly, observability analysis determines whether the chosen diagnostics suffice to infer the hidden state, or whether additional sensors are required.