control//fault diagnosis//fault isolation//fault signature matrix

A fault signature matrix is a table with one row per fault and one column per residual, holding a 1 where that residual reacts to that fault, and it is the working tool of fault isolation: when an alarm fires, the pattern of residuals that moved is matched against the rows to name the suspect. It is read like a detective reads alibis.

A fault signature matrix is a table with one row per fault and one column per residual, holding a 1 where that residual reacts to that fault, and it is the working tool of fault isolation: when an alarm fires, the pattern of residuals that moved is matched against the rows to name the suspect. It is read like a detective reads alibis.

Take a centrifugal pump driven by a motor, with four measurements: shaft speed ω\omegaω, flow qqq, pressure rise Δp\Delta pΔp and motor current III. Three residuals come from three models: r1r_1r1​ compares the measured pressure with the pump curve evaluated at ω\omegaω and qqq; r2r_2r2​ compares the current with the motor model at ω\omegaω and qqq; r3r_3r3​ compares the flow with the piping curve evaluated at Δp\Delta pΔp (centrifugal pump). In this simplified model the matrix is

Fault r1r_1r1​ r2r_2r2​ r3r_3r3​

F1, pressure sensor biased 1 0 1

F2, flow sensor biased 1 1 1

F3, impeller worn 1 1 0

F4, pipe partly blocked 0 0 1

F5, motor electrical fault 0 1 0

If r1r_1r1​ and r3r_3r3​ complain and r2r_2r2​ does not, the suspect is the pressure sensor, the one measurement both use and the motor model does not. A blockage moves the operating point along the pump curve, so pump and motor still agree with their models and only the piping relation protests. Every row has a 1, so every fault is detectable, and all five rows differ, so three residuals separate five faults.

The matrix says what can be told apart before a single fault happens.

Identical rows are faults the instrumentation cannot distinguish, and the remedy is a new sensor or relation that adds a column where they differ; isolation demands residuals with different signatures, and the matrix is where that demand is checked.

The table assumes one fault at a time. F1 and F5 together give (1, 1, 1), the row of F2; a double fault can impersonate a single one, and only more columns or a reading of each residual's magnitude and sign resolve it.

A 1 means can react, which is weaker than will cross the threshold: a small fault seen by a residual with low sensitivity leaves a 0 in practice, so real patterns are matched to the nearest row, ideally weighted by how likely each fault is. On site the suspect is then confirmed by the cheapest independent check: a calibrated test gauge on the same tapping settles a pressure transmitter in minutes.

Building the matrix from a system's equations, choosing which relations become residuals, is the subject of structural analysis; for a handful of components it is done by hand, as above.