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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 diagnosis in engineering systems basic concepts with simple examples

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
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 fault1
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
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
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
Limit checking


l1 l2 l3

s1 s2 s3

y1 y2 y3

y1 y2 y3

S1 fault off normal normal

Leak3 normal normal off

Leak2 normal off off

Leak1 off off off

High/low flow off off off

Limit checking1
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
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 approach1
Principal Component Approach

uflow y1 = u

y2 = u

y1 y2


Representation space


Normal spread



Principal component modeling
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
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
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
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
Obtaining Models

  • First principle models

  • Empirical models

    - “classical” systems identification

    - principal component approach

    - neuronets

Analytical redundancy
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 redundancy1
Analytical redundancy



u(t) y(t)





Residual properties
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 properties1
Residual Properties

  • Isolation properties

    - selectively sensitive to faults

     structured residuals perfect

     directional residuals decoupling

     “optimal” residuals

Residual generation
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
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 methods1
Linear Residual Generation Methods

Under identical conditions

(same plant, same response specification)

the various methods lead to

identical residual generators

Dynamic consistency relations
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 relations1
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
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
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
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

Gm fleet experiment
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
Fault sensitivities – GM fleet experiment

Critical fault sizes for detection and diagnosis

(fleet experiment)

Thr Iac Egr Map Maf

detection 2% 10% 12% 5% 2%

diagnosis 6% 20% 17% 7% 8%