Fault Detection using Parameter Estimation applied to a Winding

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Fault Detection using Parameter Estimation applied to
a Winding Machine
Philippe WEBER, Sylviane GENTIL
Laboratoire d’Automatique de Grenoble
UMR-CNRS 5528-UJF
Suivi paramètrique pour la surveillance d’installations complexesFault Detection using Parameter Estimation
applied to a Winding Machine
Aussoi 97IAR 1997
1
Outline
Objective
Identification
Choice of a model structure
and an estimation method
Consequence of a fault on parameter estimates
Actuator fault example
Diagnosis
Residual generation for detection
Residual fuzzyfication
Signature table generation for isolation
Symptom Aggregation
Isolation Function by fuzzy inference
Isolation Function by distance computation
Diagnosis task architecture
Application
Aussoi 97IAR 1997
2
Objective
Diagnosis using Parameter estimation
applied to a complex process!
8No physical parameters
8No additive signal !
8A fast and simple estimation
method
Aussoi 97IAR 1997
3
Choice of a model structure
and an estimation method
8 ARMAX MISO discrete model
e(k)
u1(k-d1)
...
ui(k-di)
...
uI(k-dI)
A(q ) ⋅ y ( k ) =
C
A
−1
B1
A
+
Bi
A
+
+
I
∑B
i =1
i
( q −1 ) ⋅ u i ( k − d i ) + C ( q −1 ) ⋅ e ( k )
y(k)
☺ transfer function parameter estimation
+
☺ low complexity
and no biased parameters
BI
A
8 RELSE with forgetting factor achieved by Orthonormal Transformation
on line variance and parameter estimation
good numerical properties
Aussoi 97IAR 1997
4
Consequence of a fault on parameter estimates
Two kinds of faults
y1=F1 u1 + F2 y2
y2=F3 u2
u1
y1
F1
process fault
variation of static or dynamic characteristics
F2
y2
u2
F3
actuator or sensor fault
a global perturbation of all parameters
and variance increase
Es1
Es2
Aussoi 97IAR 1997
5
Residual generation for detection
The reference model
[
Es l = Θ 1
Θp
...
The tracking model
]
+
-
long horizon
estimate vector with λ = 1
[
Es s = θ1
... θ p
]
short horizon
estimate vector with λ = 0.99
RESIDUALS
rj = Θ j − θ j
{ } = E {Θ } − E {θ }
E rj
j
j
0 if no fault occurred !
Aussoi 97IAR 1997
6
Actuator fault example
Two different forgetting factors
fault occurred at k=80
1.8
λ=0.98
1.7
λ=1
1.6
1.5
1.4
1.3
50
55
60
65
70
75
80
85
90
95
100
k
sampling period
Aussoi 97IAR 1997
7
Residual fuzzyfication
Probability density
No detection
σ1
σ ≤σ
2
Variance
σ
2
1
=σ
1
2
Θj
+σ
2
rj
≤ σ2
σ
2
θj
2
2
2


= σ + σ 
θj 
 Θj
2
σ2
False detection
µp
µZ
rj
The membership function of the fuzzy
sets
POSITIVE residual and ZERO residual :
1
rj
0.35σ1
2σ2
Uncertainty
2



σ
r
−
0
.
35
⋅
j
1



µ P ( r j ) = m in  1, m ax 0 ,
 2 ⋅ σ 2 2 − 0.35 ⋅ σ 1 2  



µ ( r ) = 1− µ ( r )
Z
j
P
j
Membership functions
Aussoi 97IAR 1997
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Signature table generation for isolation
The signature table D(n,h) defines the relation
between faults and short estimate vector Eshs
y1=F1 u1 + F2 u2
y2=F3 u1 + F4 u2
model m1
model m2
}
y1=F5 u1 + F6 y2
Fault on
D(n,h)
Es1s
Es2s
Es3s
u1
u2
y1
y2
Sg1
Sg2
Sg3
Sg4
1
1
1
0
1
1
0
1
1
0
1
1
model m3
isolation
Aussoi 97IAR 1997
9
Symptom Aggregation
FAULT
a global perturbation of residuals for the model mh
The symptom sm h is based on the residual
vector :
 h
h 
r
...
r
1

ph 

aggregation
AND
T-norm
OR
T-conorm
min
Arithmetic
mean
The symptom is defined by two
complementary fuzzy sets :
Gp Globally perturbed
Np Not perturbed
max
The membership functions for each model are computed by an aggregation
1
µ G p ( sm h ) = h
p
ph
∑µ
j =1
P
( r jh )
Aggregated symptom vector
[
Sp = µ G p ( s m 1 )... µ G p ( s m H )
and
µ (s ) = 1 − µ (s )
Np
mh
Gp
mh
]
Aussoi 97IAR 1997
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Isolation function by fuzzy inference (in single fault case)
Sp
µFn(Sp) = 0
match ?
no fault Fn
0 < µFn(Sp) < 1 fault Fn suspicion
Sgn
µFn(Sp) = 1
D(n,h)
Es1s
Es2s
Es3s
Sg1
Sg2
Sg3
Sg4
1
1
1
0
1
1
0
1
1
0
1
1
fault Fn
The rule for Sg2
Np
Gp
IF
sm1 is Gp
and sm2 is Gp
and sm3 is Np
THEN the fault is F2
and is a T-norm computed by a min operator, the rule becomes :
µF2(Sp) = min {µGp(sm1) , µGp(sm2) , µNp(sm3)}
Aussoi 97IAR 1997
11
Isolation function by distance computation
The Isolation Function F(Sp,Sgn) represents a degree of likeness
between Sp and the fault signature Sgn
F(Sp, Sgn) = 0
m2
no fault n
0 < F(Sp , Sgn) < 1 fault n suspicion degree
Sg1(1,1)
1
F(Sp , Sgn) = 1
fault n
D1
General formula by distance computation
Sp(0.7,0.2)
F ( Sp, Sgn ) = 1 −
D2
1
ω
H
( ∑ | µ G p ( s mh ) − D ( n , h )| )
q
h =1
Sg2(1,0)
1 1 
D (n, h) = 

1 0 
1
m1
Hamming distance q=1 ω=space dimension
1
F ( Sp , Sg n ) = 1 −
H
H
∑ |µ
h =1
Gp
( s mh ) − D ( n , h )|
Aussoi 97IAR 1997
12
1
q
Isolation function by distance computation
No fault
[
Sp = 0
0
]
1
F ( Sp , Sg 2 ) = 1 − [ 0 − 1 + 0 − 0 ] = 0.5
2
eliminate insensitive dimensions
1
F ( Sp , Sg n ) = 1 −
Wn
1
1 1 
D (n, h) = 

1 0 
F(Sp,Sg2)
∑ {| µ
h =1
[
Sp = 0
Sg1
Sp
H
[
Sp = 0.7
Sg2
1
0
Gp
( smh ) − D ( n , h ) |⋅ D ( n , h ) }
]
F ( Sp , Sg 1 ) = 0
F ( Sp , Sg 2 ) = 0
0 .2
F ( Sp , Sg 1 ) = 0.45
F ( Sp , Sg 2 ) = 0.7
]
Useful too for multiple fault analysis
Aussoi 97IAR 1997
13
Diagnosis task architecture
u
Process fault
Actuator fault
Sensor fault
y
Process
Two horizon
Es lh
Es c
h
Residual
generation
-
r jh
Knowledge
Aggregation
∑
∑
Membership
functions
Signature
table Sg n
Fuzzification
Symptom vector Sp
Isolation
F(Sp,Sg n )
Sensor or actuator
fault indicators
Aussoi 97IAR 1997
14
Application
After off line parameter estimation:
A1(q-1).T1(k)=B12(q-1). Ω2(k-d12)+B1u1(q-1).u1(k-d1u1)
A2(q-1).Ω2(k)= B2u2(q-1).u2(k-d2u2)
A3(q-1).T3(k)=B32(q-1). Ω2(k-d32)+B3u3(q-1).u3(k-d3u3)
Ω1
Ω3
Ω2
T1
Redundant transfer function generation:
A11(q-1).T1(k)=B11u2(q-1).u2(k-d11u2)+B11u1(q-1).u1(k-d1u1)
A33(q-1).T3(k)=B33u2(q-1).u2(k-d33u2)+B33u3(q-1).u3(k-d3u3)
T3
M1
M2
u1
u2
u3
Bias on
D
Es1
Es2
Es3
Es 4
Es5
T1
SgT1
1
0
0
1
0
Ω2
SgΩ2
1
1
1
0
0
T3
SgT3
0
0
1
0
1
u1
Sgu1
1
0
0
1
0
u2
Sgu2
0
1
0
1
1
u3
Sgu3
0
0
1
0
1
Aussoi 97IAR 1997
M3
15
Aussoi 97IAR 1997
Fault suspicion degree
using fuzzy rules
1
0.5
0
0
1
µ(FT1)
100
200
300
400
500
0.5
0
0
1
µ(FΩ2)
100
200
300
400
500
0.5
0
0
1
µ(FT3)
100
200
300
400
500
0.5
0
0
1
µ(Fu1)
100
200
300
400
500
0.5
0
0
1
µ(Fu2)
100
200
300
400
500
0.5
16
0
0
µ(Fu3)
100
200
300
400
500
Aussoi 97IAR 1997
Fault isolation function evolution
1
0.5
0
0
1
F(Sp,SgT1)
50
100
150
200
250
300
350
400
450
500
0.5
0
0
1
F(Sp ,SgΩ2)
50
100
150
200
250
300
350
400
450
500
0.5
0
0
1
F(Sp , SgT3)
50
100
150
200
250
300
350
400
450
500
0.5
0
0
1
F(Sp , Sgu1)
50
100
150
200
250
300
350
400
450
500
0.5
0
0
1
F(Sp , Sgu2)
50
100
150
200
250
300
350
400
450
500
0.5
17
0
0
F(Sp , Sgu3)
50
100
150
200
250
300
350
400
450
500
Conclusion
Fault detection and isolation of sensor or actuator FAULTS
• transfer function redundancy
better isolation performance
• fuzzy residual evaluation
taking uncertainty into
account
• symptom aggregation
allows comparison with
the fault signature table
• the result with a classical fuzzy inference system in a single fault case
method validation
• the result with distance computation for classification
possibility of multiple faults isolation
Aussoi 97IAR 1997
18
Future work
Parameter estimation method with adaptive forgetting factor
using information like
condition number,
prediction error,
to solve the non-persistent excitation problem
Find an optimal signature table
Closed loop application
Aussoi 97IAR 1997
19
PRBS
Step input
5
Step input +fault
5
5
x 10
10
9
9
9
8
8
8
7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
0
50
100
150
200
50
100
150
200
0
0
0.5
0.5
0.5
0.45
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.3
0.25
0.25
0.25
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
0
50
100
150
200
0
0
50
100
150
200
0
0
0.01
0.009
0.009
0.009
0.008
0.008
0.008
0.007
0.007
0.007
0.006
0.006
0.006
0.005
0.005
0.005
0.004
0.004
0.004
0.003
0.003
0.003
0.002
0.002
0.002
0.001
0.001
0.001
0
0
50
100
150
200
0
0
50
100
150
200
0
0
2
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
-1
0
50
100
150
200
-1
0
50
100
150
200
-1
0
200
50
100
150
200
50
100
150
200
50
100
150
200
Parameters
2
150
Prediction error
0.01
0.01
100
Variance
0.4
0.35
50
conditioning number
Aussoi 97IAR 1997
x 10
10
x 10
10
On line conditioning number evolution
20

Fault Detection using Parameter Estimation applied to a Winding

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