control//state estimation//unmeasured state estimation

An estimator can recover a quantity that no sensor measures, as long as a model ties it to quantities that are measured. A car tracked by GPS gets positions only, and its Kalman filter still ends up with a good velocity; a drone's filter learns the slow drift of its gyroscope from the GPS and the magnetometer (sensor bias). In industry the same idea goes by the name of soft sensor or virtual sensor: a quantity that is expensive, slow or impossible to measure on line, read instead from the ones that are cheap.

An estimator can recover a quantity that no sensor measures, as long as a model ties it to quantities that are measured. A car tracked by GPS gets positions only, and its Kalman filter still ends up with a good velocity; a drone's filter learns the slow drift of its gyroscope from the GPS and the magnetometer (sensor bias). In industry the same idea goes by the name of soft sensor or virtual sensor: a quantity that is expensive, slow or impossible to measure on line, read instead from the ones that are cheap.

The intuition is a pattern in the errors. The filter predicts 100 m and the GPS says 103; it predicts 110 and the GPS says 114; it predicts 120 and the GPS says 125. The position error keeps growing by about a metre a step, and the only thing in the model that explains a growing position error is a velocity that is too low. The measurement of one quantity corrects the other through the model that links them, pk+1=pk+vk Δtp_{k+1}=p_k+v_k,\Delta tpk+1​=pk​+vk​Δt.

What makes it work is the correlation the model creates. Each prediction ties the errors of the coupled quantities together (if the velocity is higher than believed, the car is also further ahead than believed), and the filter keeps that tie in the off-diagonal terms of its covariance. A reading of position then moves the velocity by the share that correlation dictates; the worked numbers and the picture are in covariance propagation.

It works only for what the sensors can see, directly or through the model. A quantity whose changes leave no trace in any measurement, however long one waits, cannot be recovered: its uncertainty never shrinks. That is the condition called observability (controllability and observability).

The estimate is only as good as the link. A soft sensor inherits every error of the model that ties the hidden quantity to the measured ones, so it is trusted inside the conditions where that model was checked and re-validated against a real measurement now and then.