Fault detection and diagnosis in engineering systems basic concepts with simple examples
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Fault Detection and Diagnosis in Engineering Systems Basic concepts with simple examples. Janos Gertler George Mason University Fairfax, Virginia. Outline. What is a fault What is diagnosis Diagnostic approaches Model - free methods Principal component approach Model - based methods

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Fault Detection and Diagnosisin Engineering SystemsBasic concepts with simple examples

Janos Gertler

George Mason University

Fairfax, Virginia


  • What is a fault

  • What is diagnosis

  • Diagnostic approaches

    • Model - free methods

    • Principal component approach

    • Model - based methods

    • Systems identification

  • Application example: car engine diagnosis

What is a fault

  • Fault: malfunction of a system component

    - sensor fault- bias

    - actuator fault- parameter change

    - plant fault- leak, etc.

  • Symptom: an observable effect of a fault

  • Noise and disturbance: nuissances that may affect the symptoms

What is a fault

actuator command

actuator leak sensor faults


sensor readings

Sensor fault: reading is different from true value

Actuator fault: valve position is different from command

Plant fault: leak

What is fault diagnosis

  • Fault detection: indicating if there is a fault

  • Fault isolation: determining where the fault is

    Detection + Isolation = Diagnosis

  • Fault identification:

    • Determining the size of the fault

    • Determining the time of onset of the fault

Model-free methods

  • Fault-tree analysis

    - cause-effect trees analysed backwards

  • Spectrum analysis

    - fault-specific frequencies in sound, vibration, etc

  • Limit checking

    - checking measurements against preset limits

Limit checking


l1 l2 l3

s1 s2 s3

y1 y2 y3

y1 y2 y3

S1 faultoffnormalnormal




High/low flowoffoffoff

Limit checking

  • Easy to implement

  • Requires no design


  • To accommodate “normal” variations, must have limited fault sensitivity

  • Has limited fault specificity (symptom explosion)

Principal Component Approach

  • Modeling phase: based on normal data

    - determine the subspace where normal data exists (representation space, RepS)

    - determine the spread (variances) of data in the RepS

  • Monotoring phase: compare observations to representation space

    - if outside RepS, there are faults

    - if inside RepS but outside thresholds, abnormal operating conditions

Principal Component Approach

uflow y1 = u

y2 = u

y1 y2


Representation space


Normal spread



Principal component modeling

Centered normalised measurementsx(t) = [x1(t) … xn(t)]’

Data matrix:X = [ x(1) x(2) … x(N)]

Covariance matrix:R = XX’/N

Compute eigenvalues 1 … n and eigenvectors q1 … qn

q1 … qk , kn, belonging to nonzero 1 … k ,, span RepS

1 … k are the variances in the respective directions

Principal Components – Residual Space

Residual Space (ResS):

complement of Representation Space, spanned by the e-vectors qk+1 … qn , belonging to (near) - zero e-values

Residual = (Observation) – (Its projection on RepS)

Residuals exist in ResS

ResS provides isolation information

- directional property (fault-specific response directions)

- structural property (fault-specific Boolean structures)

Residual Space – Directional Property



y1 y2

y1 y2




Repres. Space




on u



on y1 on y2

Residual Space

Residual Space – Structural Property

u u u

r2 r3


y1 y2 y1 y2 y1 y2

r1, r2, r3 : residuals obtained by projection

uy1y2 Structure matrix

r1 0 1 1

r2 1 1 0 Fault codes

r3 1 0 1

Model-Based Methods

faults f(t)

disturbances d(t) noise n(t)

outputs y(t)

inputs u(t) parameters 

Complete model: y(t) = f[u(), f(), d(), n(), ]

Nominal model: y^(t) = f[u(), ]

Models are: static/dynamic


Obtaining Models

  • First principle models

  • Empirical models

    - “classical” systems identification

    - principal component approach

    - neuronets

Analytical Redundancy

d(t)f(t) n(t)

u(t) y(t)



e(t)RESIDUAL r(t)



MODEL y^(t)

Primary residuals: e(t) = y(t) – y^(t)

Processed residuals: r(t)

Analytical redundancy



u(t) y(t)





Residual Properties

  • Detection properties

    - sensitive to faults

    - insensitive to disturbances (disturbance decoupling)

    - insensitive to model errors (model-error robustness)

     perfect decoupling under limited circumstances

     “optimal” decoupling

    - insensitive to noise

     noise filtering

     statistical testing

Residual Properties

  • Isolation properties

    - selectively sensitive to faults

     structured residuals perfect

     directional residuals decoupling

     “optimal” residuals

Residual Generation

uflow Model:

u y1 = u + u + y1

y1 y2

y1 y2 y2 = u + u + y2

Primary residuals:

e1 = y1– u = u + y1 u y1 y2

e2 = y2 – u = u + y2 r1 1 1 0

Processed residuals: r2 1 0 1

r1 = e1 = u + y1 r3 0 1 1

r2 = e2 = u + y2

r3 = e2 – e1 = y2– y1 Structured residuals

Residual Generation

uflow Model:

u y1 = u + u + y1

y1 y2

y1 y2 y2 = u + u + y2

Primary residuals:

e1 = y1– u = u + y1

e2 = y2 – u = u + y2

Processed residuals:

r1 = e1 = u + y1

r2 = e2 = u + y2

r3 = e1 – e2 = y1– y2


on y1


on u


on y2

Directional residuals

Linear Residual Generation Methods

  • Perfect decoupling

    - direct consistency relations

    - parity relations from state-space model

    - Luenberger observer

    - unknown input observer

  • Approximate decoupling

    - the above with singular value decomposition

    - constrained least-squares

    - H-infinity optimization

Linear Residual Generation Methods

Under identical conditions

(same plant, same response specification)

the various methods lead to

identical residual generators

Dynamic Consistency Relations

  • System description:

    y(t) = M(q)u(t) + Sf(q)f(t) + Sd(q)d(t)

    q : shift operator

  • Primary residuals:

    e(t) = y(t) – M(q)u(t) = Sf(q)f(t) + Sd(q)d(t)

  • Residual transformation:

    r(t) = W(q)e(t) = W(q)[Sf(q)f(t) + Sd(q)d(t)]

Dynamic Consistency Relations

  • Response specification:

    r(t) = f(q)f(t) + d(q)d(t)

    f(q): specified fault response(structured or directional)

    d(q) : specified disturbance response (decoupling)

     W(q)[Sf(q) Sd(q)] = [f(q)d(q)]

  • Solution for square system:

    W(q) = [f(q)d(q)][Sf(q) Sd(q)] -1

Dynamic Consistency Realtions

  • Realization:

    The residual generator W(q) must be causal and stable;

    [Sf(q) Sd(q)] -1 is usually not so

    Modified specification:

    W(q) = [f(q)d(q)] (q)[Sf(q) Sd(q)] -1

    (q) : response modifier, to provide causality and stability without interfering with specification

  • Implementation:

    inverse is computed via the fault system matrix

Diagnosis via Systems Identification

  • Approach:

    - create reference model by identification

    - re-identify system on-line

     discrepancy indicates parametric fault

  • Difficulty: discrete-time model parameters are nonlinear functions of plant parameters

     for small faults, fault-effect linearization

     continuous-time model identification (noise sensitive or requires initialization)


  • Very large systems

    - Principal Components are widely used in chemical plants

    - reliable numerical package is available

  • An intermediate-size system: rain-gauge network in Barcelona, Spain (structured parity relations)

  • Aerospace: traditionally Kalman filtering


  • Mass-produced small systems:

    on-board car-engine diagnosis

    car-to-car variation (model variation robustness)

    - GM: parity relations

    - Ford: neuronets

    - Daimler: parity relations + identification

  • Many published papers “with application to”

    are just simulation studies

GM – GMU On-Board Diagnosis Project

  • OBD-II: any component fault causing emissions (CH, CO, NOX) go 50% over limit must be detected on-line

  • Pilot project: intake manifold subsystem (THR, MAP, MAF, EGR)

  • Structured parity relations based on direct identification

  • After more in-house development, this is being gradually introduced on GM cars

Filtered and integrated residual with fault

On-board report – MAP fault

GM fleet experiment

Fleet of “identical” vehicles (Chevy Blazer) available at GM

  • Collect data from 25 vehicles

  • Identify models from combined data from 5 vehicles

  • Test on data from 25 vehicles

    Residual means and variances vary

     increase thresholds (sacrifice sensitivity)

    Only a 50% increase is necessary

Fault sensitivities – GM fleet experiment

Critical fault sizes for detection and diagnosis

(fleet experiment)


detection 2%10%12% 5% 2%

diagnosis6%20%17% 7% 8%

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