control//fault diagnosis//residual

2022-09-06

A residual is the difference between what a sensor measures and what a trusted source says it should measure, and it is the signal on which all of fault diagnosis is built: healthy, it is noise; faulty, it moves. In symbols,

rk=yk−y^k,r_k = y_k - \hat y_k ,rk=yk−y^k,

with yky_kyk the reading and y^k\hat y_ky^k the expectation. On a healthy system rkr_krk should be zero-mean noise of known spread with no structure; a fault shifts it, grows it or gives it a shape. Detecting a fault is deciding whether the residual is still noise, which is simple to say and the whole difficulty of the subject.

The trusted source is some form of redundancy: a second identical sensor, or a model acting as one (analytical redundancy). An observer-based residual uses an observer's predicted output as y^k\hat y_ky^k; a parameter-estimation residual tracks a physical parameter online (a pump's efficiency, a motor's resistance) and compares it with its nominal value. A running Kalman filter already produces one of the best: its innovation is white with a known covariance when the model, QQQ and RRR are right, so the NIS tests it against a chi-square and a persistent bias in it means a broken sensor or a changed system. PX4's estimator does exactly this with GPS, barometer and magnetometer.

A residual is what you see minus what you expected; detecting is deciding when it stops being noise.

A system that already runs a Kalman filter has half a diagnostic system for free, because its innovation is that residual, already scaled by its expected size.

A real residual holds more than the fault: model error, unmodelled disturbances (load, temperature, wind), noise, and synchronization error. Comparing a reading with a prediction 20 ms out of step turns every transient into a residual that means nothing (time synchronization).

The model error sets the threshold. The worse the model, the wider the band of healthy residuals, the larger the smallest fault that can be seen; the error does not vanish, it moves into lost sensitivity (error relocation).

Operating-condition normalization removes what the operating point explains. A motor temperature residual that rises every summer is July, and a good residual is computed against the expected value for this load, speed and ambient, or divided by what scales it.

A filter that adapts too quickly absorbs the fault: its estimate follows the faulty reading and the residual returns to zero. Diagnosis needs the expectation to stay loyal to the healthy model.

The word has other senses. A residual connection in a neural network is a skip path, and a regression residual is a fitting error; the common thread, a surprise that carries information, is the pattern in residual as surprise.