Fault Detection
and Isolation

A system that watches itself.

Every engineered machine that matters — an aircraft, a pacemaker, an electric-vehicle traction motor — does two things at once. It performs its job, and it quietly checks that it is still capable of performing its job. That second task is called Fault Detection and Isolation, or FDI.

The goal is simple to state: decide, from the signals already flowing through the machine, whether something is wrong, and if so, what. Detection is the yes-or-no; isolation is the pointing-finger. A good FDI scheme answers both in time to matter.

The trick is that faults hide. They look like disturbances. They look like noise. They look, sometimes, like perfectly normal behavior at a slightly different operating point. Everything in FDI is an attempt to build a signal — a residual — that stays small when the system is healthy and grows loud when it is not.

Model-based methods — the classical backbone.

If you have a mathematical model of the machine, you can use it as a ruler against which reality is measured. Everything in this family follows the same recipe: predict, compare, decide.

2.1Observer-based approaches

An observer is a model of the plant that runs in parallel with the real thing. It takes the same inputs u, produces an estimated output ŷ, and corrects itself using the difference between measured and predicted output. The residual is that difference.

When the model is right and nothing is broken, r stays near zero (noise aside). When something breaks — a sensor drifts, an actuator saturates, a parameter changes — r grows. The observer gain L is chosen to make the estimate converge quickly without amplifying noise too much.

Flavors worth knowing

• LuenbergerThe original deterministic observer. Gain L placed by pole assignment. Clean, cheap, assumes the model is right.
• Unknown Input Observer (UIO)Designed to make the residual blind to a known class of disturbances — measurement offsets, load torque — while still sensitive to the faults you care about. Pays in observer order.
• Sliding Mode Observer (SMO)Uses discontinuous correction to force the state error onto a surface. Robust to matched uncertainty; popular in sensorless PMSM control for back-EMF reconstruction.
• High-gain observerDrives the estimate fast by making L large. Aggressive, noise-sensitive, used where fast detection matters more than a clean residual.
• Extended / Unscented Kalman FilterThe stochastic cousins. The same idea with a noise model; the residual “innovation” is the basis for statistical tests.

2.2Parity space methods

An observer is dynamic — it has state. Parity space methods are static: they stack measurements over a short window and look for algebraic relations that must hold if everything is working.

If the plant is y = Cx + Du + noise, then for any vector v chosen so that vᵀC = 0, the quantity vᵀ(y − Du) must equal vᵀ · noise. That number is the parity residual. It cannot know what the state is; it only knows that the numbers on each side of an equation disagree.

The art is picking v so that each residual reacts to a specific fault and not others. Line up several such residuals and you get a structured residual set: a table where each column of ones and zeros is a fingerprint.

2.3Parameter estimation

Some faults do not show up as a step in the output. They show up as drift in a physical parameter — stator resistance climbing because a winding is heating, inductance falling because a magnet is demagnetizing, a capacitor’s ESR rising over years of service. Watching the output waits for the consequence. Watching the parameter catches the cause.

The standard tool is Recursive Least Squares (RLS) or the Extended Kalman Filter, run online on the system equation. The estimated parameter is tracked like a state; deviations from its nominal value are the fault indicator.

2.4Analytical redundancy — the unifying idea

Step back from the three families above. What do they share? All of them generate residuals by comparing measured signals against a prediction produced by a model. This is analytical redundancy: using the system’s own equations in place of a duplicated sensor.

Data-driven methods — pattern over physics.

Sometimes a first-principles model is out of reach. The machine is too complex, the faults too many, the parameters too uncertain. Data-driven FDI trades the model for examples of what normal looks like, and flags whatever falls outside.

3.1Multivariate statistics

Collect a matrix of historical measurements from healthy operation. Most of that variability lives in a small number of directions — the rest is noise. Principal Component Analysis (PCA) finds those directions. A new measurement is normal if it projects into roughly the same region.

Two statistics do most of the work:

• Hotelling’s T²Measures how far the projection onto the principal subspace is from the center. Detects unusual operating points within the normal directions.
• SPE / Q statisticMeasures the residual energy orthogonal to the principal subspace. Detects novel behavior the model has never seen.

3.2Signal processing

Some faults announce themselves in the frequency content of a signal. A cracked rotor bar in an induction motor imprints sidebands around the supply frequency. A bearing defect produces tone families tied to the geometry of the race. A loose mount modulates the carrier with the shaft speed.

The toolkit — FFT, wavelet transform, Hilbert-Huang, envelope analysis, cepstrum — is about getting the fault signature out of whatever it’s hiding inside.

3.3Machine learning

When the signature is too messy or too multidimensional for a spectral rule of thumb, classifiers take over. SVMs, random forests, gradient-boosted trees, and one-class methods learn a decision boundary between healthy and faulty — or, more realistically, between healthy and everything that is not healthy.

One-class vs. multi-class

In practice you often have abundant healthy data and very few labeled faults. One-class SVM and Isolation Forest learn only what normal looks like and call everything else anomalous. This matches reality better than training on rare, possibly unrepresentative fault examples.

3.4Deep learning

Modern diagnostics reach for deep networks when the input is high-dimensional and the signature is not something you want to handcraft. Three patterns dominate:

• CNNs on vibrationTreat a raw or time-frequency image of vibration as an image classification problem. Learned filters discover bearing-defect patterns that would take a PhD in tribology to specify by hand.
• LSTM / Transformer on time seriesCapture long-range temporal structure — the way a drift builds up over hours of operation. Useful for wear-out and slowly-developing electrical faults.
• Autoencoders for unsupervised anomaly detectionTrain a reconstruction network on healthy data only. Whatever the network reconstructs poorly is flagged. This is the neural-network face of PCA’s SPE statistic.

Hybrid methods — the current frontier.

Pure model-based schemes are brittle in the face of unmodeled dynamics. Pure data-driven schemes are brittle outside their training distribution. The answer, everywhere in industry right now, is to combine them.

Three dominant patterns

• Residual-learningRun a physics model; train a network only on its residual. The network absorbs unmodeled dynamics without destroying the physical core.
• Physics-informed neural networks (PINNs)The loss function includes a penalty for violating known physical laws. The network cannot wander into physically nonsense regions.
• Digital twinsA calibrated high-fidelity simulator of the specific machine, updated online from fleet data. Residuals against the twin catch faults that fleet-average models would miss.

The design problems that decide everything.

Picking a residual generator is the glamorous part. Making it work in the field is about seven quieter problems.

5.1Residual generation vs. residual evaluation

Every FDI scheme splits into two independent questions. Generation: what signal do I build? Evaluation: what decision logic do I apply to it?

You can bolt a CUSUM detector onto a Luenberger observer or a PCA score. You can pair the same observer with a fixed threshold, a GLR test, or a Bayesian decision rule. Keeping the halves separate makes the design problem tractable and the options clear.

5.2Robustness vs. sensitivity — the fundamental trade

A residual should be loud when there is a fault and silent when there is not. But disturbances, noise, and model mismatch also excite it. You cannot perfectly separate the two; you can only trade them off.

The formal statement of this trade lives in the H_∞ / H₋ framework. You shape a transfer function from fault to residual (keep this large in a band where faults live) and from disturbance to residual (keep this small). The trick — and the continuing research — is that these bands often overlap.

5.3Threshold design

Once the residual exists, someone has to set a number above which an alarm is raised. The options differ in how they model the world.

• Fixed thresholdSimplest. Tune once, live with it. Breaks whenever operating conditions drift.
• Adaptive thresholdMoves with operating point — speed, load, temperature. Standard in automotive.
• CUSUMAccumulates small deviations. Catches slow drifts a fixed threshold would miss.
• GLR (Generalized Likelihood Ratio)The statistically optimal test when the fault signature is known. Outputs a time of change and magnitude estimate along with the alarm.
• BayesianPosterior probability of fault, updated every sample. Gives a principled way to trade off false alarms against missed detections given their cost.

5.4Fault isolation

Detection says something is wrong. Isolation says what. The three workhorses:

• Structured residualsThe parity-space fingerprint idea of §2.2 — each fault produces a unique binary signature across a bank of residuals.
• Directional residualsEach fault produces a residual vector pointing in a known direction in signal space. The angle of the observed residual picks out the fault.
• Dedicated observer scheme (DOS)One observer per fault, each engineered to be sensitive to exactly one fault and blind to the others. Diagnostic by construction; expensive by count.

5.5Fault identification and estimation

Beyond “which”, you often want to know how much and when it started. Identification recovers the fault signal — magnitude, time profile, severity. Tools include augmented-state observers that treat the fault as an unknown input to estimate, adaptive observers that parameterize the fault and adapt, and GLR tests that directly output the estimated fault onset and amplitude.

In automotive diagnostics, identification feeds directly into derating: a small fault triggers torque limiting; a large one triggers shutdown. The quantitative estimate determines the response.

5.6Detectability and isolability

Before you design anything, there is a prior question: are the faults even distinguishable from the outputs you have? This is a structural property of the system, not a matter of algorithmic cleverness.

A fault is detectable if it leaves a trace in the measured output that no disturbance and no healthy operating-point change can mimic. It is isolable from a second fault if the two leave different traces. Both properties have algebraic characterizations — rank conditions on the system’s transfer matrix extended with the fault and disturbance directions.

If the check fails, no observer, no classifier, and no network can fix it. The only remedy is an additional sensor.

5.7Incipient vs. abrupt faults

Not all faults behave the same way. Abrupt faults hit in a single sample — an IGBT opens, a wire breaks, a sensor saturates. Incipient faults develop slowly — a bearing wears, insulation degrades, a magnet loses flux. The signals tell different stories and call for different detectors.