control//fault diagnosis//fault isolation

Fault isolation is the step of fault diagnosis that says which fault occurred, by designing several residuals that each react to a different subset of faults and reading which of them have moved; it turns *something is wrong* into *replace the pressure transmitter*, which is what the technician called to the plant at three in the morning can act on. Detecting is comparatively easy. Isolating is where the design effort goes.

Fault isolation is the step of fault diagnosis that says which fault occurred, by designing several residuals that each react to a different subset of faults and reading which of them have moved; it turns something is wrong into replace the pressure transmitter, which is what the technician called to the plant at three in the morning can act on. Detecting is comparatively easy. Isolating is where the design effort goes.

The usual device is a set of structured residuals. Each residual is built from a relation that uses some sensors and models and leaves others out, so a fault in what it leaves out cannot move it. A pump with four measurements (speed, flow, pressure rise, motor current) yields one residual from the pump curve, one from the motor model, one from the piping curve; a biased pressure sensor upsets the two relations that use pressure and leaves the third alone. The pattern of reactions across residuals is recorded in the fault signature matrix. A fault is detectable if at least one residual reacts to it and two faults are isolable if their patterns differ.

Two faults with the same signature cannot be told apart with those sensors.

This is an observability limit of the instrumentation, the fault version of observability: no cleverer algorithm separates them, and the fix is another sensor or another independent relation. Ask whether the faults that matter are isolable before choosing any method.

Reality is noisier than the binary pattern. A residual with low sensitivity to a small fault may not cross its threshold, and the observed pattern then matches no row; the practical answer is the closest row, or each candidate weighted by its prior probability, which is Bayes' rule applied to suspects.

Multiple faults deceive. Two faults acting together can produce the pattern of a third, a combination no single-fault table anticipates; more residuals help, and so does reading magnitude and direction as well as yes or no. In a hexacopter the direction of the torque error points at the motor that lost thrust, which a yes or no per residual could never say.

Data-driven monitoring isolates more weakly: after a Q statistic alarm, a contribution plot ranks the variables that add most to it, which says where to look rather than what broke.

Isolation earns its cost when the response depends on the fault (which spare to order, which sensor to ignore, which degraded mode to enter); if every fault leads to the same shutdown, detection is enough.