Observability and State Estimation

Whether the twin can infer the internal state from external diagnostics is a precise property - observability - that governs where estimates are trustworthy.

Can the state be seen?

Before trusting any estimator, one must ask whether the measurements even determine the state. Observability is that property: a system is observable if its full internal state can be reconstructed from the output history. For the breeder, magnetics alone leave the internal current profile weakly observable - the formal reason kinetic diagnostics are needed.

Linear observability (discrete):

  Observability matrix
    O = [ H ; H F ; H F^2 ; ... ; H F^{n-1} ]

  System observable  <=>  rank(O) = n (full)

Observability Gramian (informativeness):
  W_o = sum_{k=0}^{inf} (F')^k H' H F^k
  eigenvalues of W_o -> how strongly each mode is observed

Degrees of observability

Observability is not just yes/no. The observability Gramian's eigenvalues quantify how strongly each state direction is seen; small eigenvalues mark near-unobservable modes where estimates rest on the model, not data. The stack maps these weak directions so the twin knows which parts of its state are diagnostic-backed and which are model-inferred - and widens uncertainty accordingly.

# observability check + weak-direction map
O = observability_matrix(F, H)
assert matrix_rank(O) == n            # full observability?
Wo = observability_gramian(F, H)
w, V = eigh(Wo)
weak = V[:, w < w.max()*ratio]        # poorly-observed directions
# widen twin uncertainty along 'weak'; flag to MPC

Sensor placement and honesty

Observability drives diagnostic design: sensors are placed to make the safety-critical states - shape descriptors, stability margins, plug potential - strongly observable. Where coverage is inherently thin, the stack does not pretend otherwise; it reports a degraded-observability condition rather than a confident but unsupported estimate. On the burner, whole regions are only model-observable given the absence of a comparable device, which is stated plainly.

• Observability matrix rank: can the state be reconstructed at all.
• Gramian eigenvalues: how strongly each mode is measured.
• Weak directions rely on the model; uncertainty widened there.
• Drives sensor placement and honest degraded-observability flags.

Observability analysis is the bridge from raw diagnostics to trustworthy state: it tells the estimator, and the operators, exactly how much of the machine they can actually see.