control//controllability and observability//observability

Observability is the property of a system with given sensors that says whether the measurements, combined with the model over a stretch of time, contain the information needed to reconstruct the whole state, and it is the question to ask while choosing sensors, before any filter is written. A drone with only an accelerometer and a gyroscope estimates its tilt well, because gravity tells it where down is, but it cannot know its heading: gravity points the same way whether it faces north or east. A magnetometer makes heading observable, and so do a GPS and some horizontal acceleration, which leave the heading's trace in how the measured velocity changes. Absolute position is never observable from an inertial unit alone (dead reckoning). (The observability of software operations, logs and traces, borrowed the word; here it is a property of a model and its sensors.)

Observability is the property of a system with given sensors that says whether the measurements, combined with the model over a stretch of time, contain the information needed to reconstruct the whole state, and it is the question to ask while choosing sensors, before any filter is written. A drone with only an accelerometer and a gyroscope estimates its tilt well, because gravity tells it where down is, but it cannot know its heading: gravity points the same way whether it faces north or east. A magnetometer makes heading observable, and so do a GPS and some horizontal acceleration, which leave the heading's trace in how the measured velocity changes. Absolute position is never observable from an inertial unit alone (dead reckoning). (The observability of software operations, logs and traces, borrowed the word; here it is a property of a model and its sensors.)

Three words get mixed up. Measurable means a sensor gives the quantity directly. Observable means it can be rebuilt from measurements and model over time: a velocity from successive positions, a gyroscope's bias from its disagreement with the accelerometer, the wind on a hovering drone from the tilt it holds against it (virtual sensor). Identifiable is the same question asked of the model's parameters, and it needs the system to have moved enough while it was watched (persistent excitation). The test is the rank of the observability matrix; a direction of the state it misses, an unobservable direction, produces exactly the same measurements as zero, and it drifts freely while the filter reports nothing.

Observability comes before the algorithm.

What leaves no trace in the measurements is recovered by no algorithm, however expensive: no Kalman filter estimates it, no network learns it, and a thousand flight hours without a magnetometer still contain no heading. When the answer is no, the next step is a sensor or an experiment, never a better filter.

Observable does not mean well estimated. The rank is yes or no, and quality is a degree: velocity taken as the difference of two positions from a sensor with 10 cm of noise at 100 Hz carries about 14 m/s of noise. The degree of observability is the smallest singular value of the observability matrix (SVD); near zero, the worst direction is barely seen and its noise is hugely amplified.

It can depend on what the system does. In hover, a horizontal accelerometer bias and a small tilt error give the same reading, and only a manoeuvre separates them. For a nonlinear model the rank test is applied to the Jacobians along a trajectory (linearization), so observability belongs to the flight as much as to the sensors.

Detectability is the weaker condition a filter actually needs: unobservable parts that die out by themselves can stay unseen without the estimate diverging.

The same question returns under other names. Two faults with the same signature cannot be told apart (fault isolation); visual-inertial odometry has four unobservable directions, absolute position and the rotation about gravity (visual-inertial odometry). Decision theory uses the word more coarsely, instant by instant: a problem is fully observable when the sensors give the whole state at every moment, as in chess, and partially observable otherwise (POMDP).

Its mirror is controllability; the version that asks only for a few target variables is functional observability.