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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Janos Gertler
George Mason University
Fairfax, Virginia
 sensor fault  bias
 actuator fault  parameter change
 plant fault  leak, etc.
actuator command
actuator leak sensor faults
fault
sensor readings
Sensor fault: reading is different from true value
Actuator fault: valve position is different from command
Plant fault: leak
Detection + Isolation = Diagnosis
 causeeffect trees analysed backwards
 faultspecific frequencies in sound, vibration, etc
 checking measurements against preset limits
flow
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
BUT
 determine the subspace where normal data exists (representation space, RepS)
 determine the spread (variances) of data in the RepS
 if outside RepS, there are faults
 if inside RepS but outside thresholds, abnormal operating conditions
uflow y1 = u
y2 = u
y1 y2
y2
Representation space
Fault
Normal spread
y1
u
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 , kn, belonging to nonzero 1 … k ,, span RepS
1 … k are the variances in the respective directions
Residual Space (ResS):
complement of Representation Space, spanned by the evectors qk+1 … qn , belonging to (near)  zero evalues
Residual = (Observation) – (Its projection on RepS)
Residuals exist in ResS
ResS provides isolation information
 directional property (faultspecific response directions)
 structural property (faultspecific Boolean structures)
Residual Space – Directional Property
uflow
u
y1 y2
y1 y2
y2
residual
observation
Repres. Space
q1
y1
u
on u
q3
q2
on y1 on y2
Residual Space
u u u
r2 r3
r1
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
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
linear/nonlinear
 “classical” systems identification
 principal component approach
 neuronets
d(t)f(t) n(t)
u(t) y(t)
PLANT
+
e(t)RESIDUAL r(t)
PROCESSING

MODEL y^(t)
Primary residuals: e(t) = y(t) – y^(t)
Processed residuals: r(t)
 sensitive to faults
 insensitive to disturbances (disturbance decoupling)
 insensitive to model errors (modelerror robustness)
perfect decoupling under limited circumstances
“optimal” decoupling
 insensitive to noise
noise filtering
statistical testing
 selectively sensitive to faults
structured residuals perfect
directional residuals decoupling
“optimal” residuals
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
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
r3
on y1
r2
on u
r1
on y2
Directional residuals
 direct consistency relations
 parity relations from statespace model
 Luenberger observer
 unknown input observer
 the above with singular value decomposition
 constrained leastsquares
 Hinfinity optimization
Under identical conditions
(same plant, same response specification)
the various methods lead to
identical residual generators
y(t) = M(q)u(t) + Sf(q)f(t) + Sd(q)d(t)
q : shift operator
e(t) = y(t) – M(q)u(t) = Sf(q)f(t) + Sd(q)d(t)
r(t) = W(q)e(t) = W(q)[Sf(q)f(t) + Sd(q)d(t)]
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)]
W(q) = [f(q)d(q)][Sf(q) Sd(q)] 1
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
inverse is computed via the fault system matrix
 create reference model by identification
 reidentify system online
discrepancy indicates parametric fault
for small faults, faulteffect linearization
continuoustime model identification (noise sensitive or requires initialization)
 Principal Components are widely used in chemical plants
 reliable numerical package is available
onboard carengine diagnosis
cartocar variation (model variation robustness)
 GM: parity relations
 Ford: neuronets
 Daimler: parity relations + identification
are just simulation studies
Fleet of “identical” vehicles (Chevy Blazer) available at GM
Residual means and variances vary
increase thresholds (sacrifice sensitivity)
Only a 50% increase is necessary
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%