1 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Application de la modélisation par Réseaux Bayésiens à la sûreté de fonctionnement
Philippe WEBER
Centre de Recherche en Automatique de Nancy (CRAN), UMR 7039 CNRS - Nancy Université
2, rue Jean Lamour F-54519 Vandoeuvre-lès-Nancy Cedex – FranceEmail: [[email protected]]
Présentation disponible sur HALhttp://http://hal.archiveshal.archives--ouvertes.frouvertes.fr
2 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
The objective is to define maintenance / control strategy of the Equipment
In regard toorganisational resources
financial resources and technical: diagnosis, reliability and
performances of the system
Several types of strategies are available
Decision aid methodbased on Bayesian
Networks model
Manufacturing systems
3 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
The objective is to define maintenance / control strategy of the Equipment
In regard toorganisational resources
financial resources and technical: diagnosis, reliability and
performances of the system
Several types of strategies are available
Decision aid methodbased on Bayesian
Networks model
Manufacturing systems
4 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
BayesianBayesian Networks Networks modelmodel in in ReliabilityReliability AnalysisAnalysis
DynamicDynamic BayesianBayesian Networks Networks modelmodel in in ReliabilityReliability AnalysisAnalysis aandnd DiagnosisDiagnosis
BayesianBayesian Networks in Networks in Risk analysis of socio-technical systems
Conclusion
Outline
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X1 up down0.2 0.8
X2 State 1 State 2 State 3
up 0.7 0.1 0.2down 0.1 0.6 0.3
X1
Conditional Probability TableCPT
p(X2|X1)?
Bayesian Network model of component
Discrete random variable
variable states
a priori distribution
( ) ( ) ( )( ) ( )downXpdownX1 StateXp
upXpupX1 StateXp 1 StateXp
=⋅==+
=⋅====
112
1122
The marginal probability X2 is computed as follows
( ) 22.08.01.02.002 =⋅+⋅== .7 1 StateXp
Static Bayesian Networks
X1 X2
knowledge
With the a priori knowledge
6 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
X1 up down0.2 0.8
X2 State 1 State 2 State 3
up 0.7 0.1 0.2down 0.1 0.6 0.3
X1
Conditional Probability TableCPT
p(X2|X1)?
Bayesian Network model of component
Discrete random variable
variable states
a priori distribution
The marginal probability X1 is computed with the Bayes theorem as follows
The propagation of this probability through the BN is based on inference algorithms
Static Bayesian Networks
X1 X2
knowledge
( ) ( ) ( )( )1 StateX
X1 StateXX1 StateXX
2
12121 =
==⋅====
p
uppuppupp
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Root Node
X0
p(X1 | X0)
State 1 State 2State 1 0.6 0.4State 2 0.8 0.2
p(X2 | X1)
p(X1)
up down0.1 0.9
p(X0)
p(X0)
State 1 State 2up 0.1 0.9
down 0.7 0.3
Bayesian Network model of component
X1
X2
Static Bayesian Networks
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p(X2 | X1)
p(X1)
p(X(X22=state1)=0.7*0.6+0.3*0.8==state1)=0.7*0.6+0.3*0.8=0.660.66p(X(X22=state2)=0.7*0.4+0.3*0.2==state2)=0.7*0.4+0.3*0.2=0.340.34
up down0 1
Hard evidenceHard evidence
p(X1 | X0)
p(X0)
p(X0) 11
00
11
00
11
00
Bayesian Network model of component
Root Node
X0
X1
X2
State 1 State 2up 0.1 0.9
down 0.7 0.3
State 1 State 2State 1 0.6 0.4State 2 0.8 0.2
Static Bayesian Networks
p(X(X11=state1)==state1)=0.70.7p(X(X11=state2)==state2)=0.30.3
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p(X2)
up down0 1
Hard evidenceHard evidence
p(X1 | X0)
p(X0)
p(X0) 11
00
11
00
Bayesian Network model of component
Root Node
X0
X1
X2
State 1 State 2up 0.1 0.9
down 0.7 0.3
Static Bayesian Networks
State 1 State 20 1
Hard evidenceHard evidence 11
00
p(X(X11=state1)==state1)=0.82350.8235p(X(X11=state2)==state2)=0.17650.1765
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p(X2)
up down0 1
Hard evidenceHard evidence
p(X1 | X0)
p(X0)
p(X0) 11
00
11
00
Bayesian Network model of component
Root Node
X0
X1
X2
State 1 State 2up 0.1 0.9
down 0.7 0.3
Static Bayesian Networks
State 1 State 20.2 0.8
soft evidencesoft evidence 11
00
p(X(X11=state1)==state1)=0.780.78p(X(X11=state2)==state2)=0.220.22
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Définition de la structure des TPC de FE
Fonction i affectée par la défaillance du composent
…défaillance n
…………
…défaillance 1
…Etat de fonctionnement
Mode de défaillance n de la fonction n
…Mode de défaillance 1 de la fonction i
Fonctionnement de la fonction i
Distribution des effets d’une défaillance sur plusieurs modes de défaillance ou de fonctionnement
Composant 0.6
11
00
Distribution des effets de la Distribution des effets de la défaillance 1 sur le défaillance 1 sur le
fonctionnement de la fonction ifonctionnement de la fonction ipropagation dpropagation d’’incertitudeincertitude
0.1 0.2
Apprentissage sur Historiques de GMAOHistoriques de GMAO
Bayesian Network model of component Static Bayesian Networks
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A) Architecture parallèle
( ) ( ) ( )RBi
iDF OKSpHSFpSp ===−= ∏=
3
2
1
1
F1=F1=HSHS F2=F2=HSHS
ERER SS33==HSHS
Diagramme de Fiabilité
Arbre de Défaillances
Réseau Bayésien
Modèle de fiabilité d’un système à 2 composants (les variables sont à 2 états : OK et HS)
Bayesian Network model of system Static Bayesian Networks
13 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
( ) ( ) ( )RBi
iDF OKSpOKFpSp ==== ∏=
3
2
1
F1=F1=HSHS F2=F2=HSHS
ERER SS33==HSHS
B) Architecture série
Arbre de Défaillances
Réseau Bayésien
Modèle de fiabilité d’un système à 2 composants (les variables sont à 2 états : OK et HS)
Diagramme de Fiabilité
Bayesian Network model of system Static Bayesian Networks
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Modèle de fiabilité d’un système complexe
CMP1
CMP2
CMP3
CMP4
CMP5
La représentation sous la forme d’un arbrede défaillances n’est pas adaptée car CMP1, CMP2 et CMP5 ne peuvent pas être factorisés
En RB le modèle est structuré sous la forme d’un graphe et les calculs sont réalisés par inférence
ER = CMP3.CMP4+CMP1.CMP2 +CMP1.CMP5.CMP4 +CMP2.CMP5.CMP3
Bayesian Network model of system Static Bayesian Networks
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Dans un cadre de modélisation réaliste, les composants peuvent subir plusieurs défaillances ce qui conduit à plusieurs modes de défaillance ! Le Modèle de fiabilité de ce type de système ne peut plus reposer sur un arbre de défaillances (car le modèle n’est plus booléen)
Prise en compte des dépendancesPrise en compte des dépendances
Codage directe d’équation complexe dans la TPC avec F2 multimodale
Bayesian Network model of system Static Multimodal Bayesian Networks
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Structuration du modèle sous la forme d’un graphe avec dépendance des branches et F2 multimodale
Bayesian Network model of system Static Multimodal Bayesian Networks
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DynamicDynamic BayesianBayesian Networks in Networks in SystemSystem ReliabilityReliability AnalysisAnalysis
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System reliability
)(tRS The probability that no failure occurred during the interval [0, t])(tSλ Failure rate of the system at time t
−= ∫t
SS dtttR0
)(exp)( λ
Problem statement
The probability that a failure occurred between t and t+dt is approximated by
When the system is composed with several components
Then the failure rate is defined for each component)(tnλ
dttp nn ⋅= )(λ
Markov Chain is a classic solution to model this so rt of system Reliability when failure rates are constant
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Example of application to reliability
Problem statement
1
2
3
pp1212
pp1313
operational states
2 failure states
4
pp2424
Operational states and failure states represent a system up or down
( ){ }∑
∈==
2,1)(
iitS sXptR
)( MCt
Tt X
dt
dXPI −⋅=
The reliability computation needs the solution of a differential equation system
20 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Unfortunately Markov Processare not enough sufficient to model real systems
Therefore Markov Process are extended to model more realistic problem as
Degradation results in parameters are time-variant (SMP) Phase Type Distribution
Exogenous constraint results in a conditional behaviourMarkov Switching Model
Moreover in practice the complexity of the system leads to a combinatorial explosion of states resulting in a Markov Chain with a great size
Then Dynamic Bayesian Networks (DBN) are proposed as a more synthetic model to represent these stochastic processes
Problem statement
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2 Time 2 Time slices slices A Dynamic Bayesian Network (DBN) is a BN extension including temporal dimensionality
Red arcs represent the temporal dependence between different time slices
Defining these impacts as transition-probabilities between the states of the variable X at time (k-1) and (k)
The DBN compute the behaviour of the probability distribution over the stats of the variable X
X(k) up downup 0.9 0.1
down 0 1X(k-1)
X(k-1) X(k)
inter-time slices CPT
Bayesian Network model of component Dynamic Bayesian Networks
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Then the inter-time slices CPTis a Markov Chain model
Starting from an observed situation at time k=0, the probabilitydistribution over the states at the next time is computed
using successive inferences
(the variable Xk is considered as the new observation of Xk-1 through a time feedback)
a time feedbacka time feedback
inter-time slices CPT
X(k) up downup 0.9 0.1
down 0 1X(k-1)
X(k-1) X(k)
Bayesian Network model of component Dynamic Bayesian Networks
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Then the reliability is given by simulation
Application in Reliability of component Dynamic Bayesian Network / Markov Chain modelBayesian Network / Markov Chain model
))(()( upkXPkRn ==
WEBER P., JOUFFE L. Reliability modelling with Dynamic Bayesian Networks. 5th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS'03), Washington, D.C., USA, 9-11 juin, 2003.
MTTFn
1=λ
tp n ∆≅ λ12
X(k) up down
up 1- p 12 p 12
down 0 1X(k-1)
24 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
BEN SALEM A., BOUILLAUT L., AKNIN P., WEBER P. Dynamic Bayesian Networks for classification of rail defects. IEEE 4th International Conference on Intelligent Systems Design and Applications (ISDA 2004), Budapest, Hungary, August 26-28, 2004.
Application in Reliability of component Dynamic Bayesian Network / Markov model of order nBayesian Network / Markov model of order n
))(()( upkXPkRn ==
Open problem: to learn the parameter !
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Dynamic Bayesian Network / semi Markov ProcessBayesian Network / semi Markov Process
))(()( upkXPkRn ==
System with time variant failure rate System with time variant failure rate ((weibullweibull) )
The parameter in the CPT The parameter in the CPT are indexed by timeare indexed by time
50,5.2,)(1
1 ==⋅=−
ηβη
βλ β
β
t
t
Application in Reliability of component
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Markov Switching Model Markov Switching Model
Dynamic Bayesian Network / Markov Switching ModelBayesian Network / Markov Switching Model
A process changes its behavior according to the state of A process changes its behavior according to the state of exogenous constraints representing functioning conditions, exogenous constraints representing functioning conditions, maintenance events …maintenance events …
The exogenous constraint is represented by an external variable The exogenous constraint is represented by an external variable UUkk
U(k)=U(k)=αα U(k)=U(k)=ββ
OK
f)(1 α=λ f
OK
f)(1 β=λ f
Application in Reliability of component
)( MCt
Tt X
dt
dXPI −⋅=
Analytic solution Analytic solution
DiscretDiscret simulationsimulation
WEBER P., MUNTEANU P., JOUFFE L. Dynamic Bayesian Networks modelling the dependability of systems with degradations and exogenous constraints. 11th IFAC Symposium on Information Control Problems in Manufacturing, INCOM'04. Salvador-Bahia, Brazil, April 5-7th, (2004).
27 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Dynamic Bayesian Network / Markov Switching ModelBayesian Network / Markov Switching Model
))(()( upkXPkRn ==
Application in Reliability of component
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X(k-1) X(k)
U(k-1) Y(k)
Hidden Process
Observations
Dynamic Bayesian Network / IOHMMBayesian Network / IOHMM
Application in Reliability of component
In the previous slides the stochastic processes are supposed to be completely observable. In practice this is seldom the reality because the physical degradations of a component result in a change of its state which is observed only through a variation in the component functionality.
Parameter estimation needs many data !!!Parameter estimation needs many data !!!
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Dynamic Bayesian Network Bayesian Network -- Factorized MC modelFactorized MC model
Application in Reliability of system
The reliability of component can be modelled as a DBN as presented beforeIf the components are independent the DBN allows to merge the models through a factorised form
X1(k-1) X1(k)
U(k-1)
Y3(k)
X2(k-1) X2(k)
X3(k-1) X3(k)
Y2(k)
Y1(k)
IOHMMIOHMMIOHMMIOHMM
X2(k-2)
MarkovMarkov 22
1/2Markov1/2Markov
30 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application in Reliability of system
The reliability of component can be modelled as a DBN as presented beforeIf the components are independent the DBN allows to merge the models through a factorised form
X1(k-1) X1(k)
U(k-1)
Y3(k)
X2(k-1) X2(k)
X3(k-1) X3(k)
Y2(k)
Y1(k)
IOHMMIOHMMIOHMMIOHMM
X2(k-2)
MarkovMarkov 22
1/2Markov1/2Markov
S(k)
31 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
V2
V3
V1
The method is applied to a classical example of reliability analysis.
Three valves are used to distribute or not a fluid.
Every valves have two failure modes • remains closed (RC)• remains opened (RO)
WEBER P., JOUFFE L. Reliability modelling with Dynamic Bayesian Networks. 5th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS'03), Washington, D.C., USA, 9-11 juin, 2003.
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application 1 in Reliability
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V1 Av.V2 RCV3 Av.
V1 Av.V2 Av.V3 RC
V1 RCV2 RCV3 Av.
V1 Av.V2 RCV3 RC
V1 RCV2 Av.V3 RC
s6
s2
s23
s24
s12
V1 RCV2 Av.V3 Av.
V1 Av.V2 Av.V3 Av.
V1 Av.V2 ROV3 Av.
V1 ROV2 Av.V3 Av.
V1 ROV2 ROV3 Av.
V1 ROV2 Av.V3 RO
V1 Av.V2 Av.V3 RO
V1 Av.V2 ROV3 RO
V1 ROV2 ROV3 RO
s1
s3
s4
s5
s9
s26
s16
s19
s18
V1 RO.V2 RCV3 Av.
V1 Av.V2 RCV3 RO.
V1 RO.V2 Av.V3 RC
V1 Av.V2 RO.V3 RC
V1 RO.V2 RCV3 RO.
V1 RC.V2 RCV3 RO.
V1 RO.V2 RCV3 RC.
V1 RO.V2 RO.V3 RC
V1 RC.V2 ROV3 RC.
s7 s8s13s14
V1 RC.V2 ROV3 Av.
s15
V1 RCV2 ROV3 RO
s17
V1 RC.V2 AVV3 RO.
s20
s11 s10s22
s21
s25
Markov Chain
The Markov Chain is defined from 27 states
729729 parametersparameters
Application 1
75 different 75 different from 0from 0
a graphical a graphical
representation of this representation of this
model is difficultmodel is difficult
33 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
the nodes
• 6 nodes are described the 3 valves
• 3 nodes are described the system state
the CPTs
• 3 small MC
• and the logic of the failures propagation
ReliabilityReliability
58 parameters are equal to 58 parameters are equal to 1173 different from 073 different from 0143143 parametersparameters
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application 1 in Reliability
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The inter time slices CPT represents a small MC
OK
RO
RC
0.20.2
0.10.1
Equivalent toEquivalent to
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application 1 in Reliability
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ReliabilityReliability = = TrueTrue
The behavior of the probability representing• The System Remains Close (light blue) • The System Remains Open (blue)• The System Reliability (red)
The results after successive inferences
)(tRS
Application 1 in Reliability
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Process modelling approach in real application
• Methodology– Process and flow based approach
• Functional/Dysfunctional reasoning• Hierarchical structure
– Elaboration of the probabilistic network• Formalism: BN/DBN• Generic rules to transform the process model into
a probabilistic one
But the Bayesian Network (BN)needs Acyclic structure
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To Control
To transform
To measure
PLC
Sensor
Machine
Position order
input product finished product
finished product
Product report
Probabilistic network development (2)
The cycle problem
Cycle
Cycle
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Probabilistic network development (2)
The translation with specification
► No cycle
To measure
To Control
To transform
PLC
Sensor
Machine
Order(k+1)
input product (k )finished product (k+1)
finished product (k)
Product report
Flow decomposition
(k+1)
39 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
V
R1
R2
Qi Ti
Qo To
H P
H sensor
T sensor
Water heater (Physical process)
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application 2 in Reliability
WEBER P., JOUFFE L., Complex system reliability modelling with Dynamic Object Oriented Bayesian Networks (DOOBN). Reliability Engineering and SystemSafety, Volume 91, Issue 2, February 2006, Pages 149-162 (Selected Papers Presented at QUALITA 2003).
MULLER A., WEBER P., BEN SALEM A. Process model-based Dynamic Bayesian Networks for Prognostic. IEEE 4th International Conference on Intelligent Systems Design and Applications (ISDA 2004), Budapest, Hungary, August 26-28, 2004.
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0.0
0.2
0.4
0.6
0.8
1.0
0 500 1000 1500 2000
P(state1) P(state 2)
P(state 3)
P(state 4)
HEATING RESISTOR (k)
1
2 λ1
4
λ2
3 λ4 λ3
λ5
R1 R1 andand R2 maxR2 max
R1 or R2 downR1 or R2 down
Action Action to emplace to emplace
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43 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Water heater (Process model)
To regulate theinput water flow
rate
Valve
Input Water Flow
To heat water
Heatingresistors
To measure thewater temperature
Temperaturesensor
To measure thewater level
Level sensor
To distribute water
Water pipe
To control the'Water heater'
process
PLC
Water level report
Water temperature report
Water distributed& heated
To store water
Tank
Water to heatWater to heat Water to distributeWater to distribute
Position order
Water to distribute (k) Water distributed& heated (k)Heating order
E.E.
E.E. E.E.
E.E.
E.E. = Electrical Energy
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application 2 in Reliability
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Water heater (Probabilistic model - DBN)
1
2 λ1
4
λ2
3 λ4 λ3
λ5
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application 2 in Reliability
45 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
To provide Warm Water
HD1 Order T=50°C
AD2 water input pressure and Ti
AD4 WATER HEATER PROCESS
RHD water output temperature T and flow rate Qo
AD1 Electric Power
AD3 system parameters temperature T and level H
To transform Pressure
to Qi
HD Order V
AD2 water input pressure and Ti
AD4-1 VALVE V
RHD Qi
AD1 Electric Power
AD Ti To transform
H to Qo
AD4-4 WATER PIPE
RHD water output flow rate Qo
HD1 Order T=50°C
To transform Qi to H
Ti to T
AD Qi
AD4-3 TANK, HEATING RESISTOR
RHD water output temperature T
AD1 Electric Power
To control V and P
AD4-2 COMPUTER, SENSORS
HD Order P
AD3 system parameters temperature T and level H
AD1 Electric Power
AD water level H
RHD V
RHD P
A1
A2
A3
A4
RHD water level H
Water heater (SADT model and OODBN model)
Dynamic Bayesian Network Dynamic Bayesian Network -- Factorized MC modelFactorized MC model
Application 2 in Reliability
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AD Ti To heat water
from Ti to T
AD Qi
AD4-3 HEATING RESISTOR
RHD water temperature T
AD1 Electric Power
HD Order P
A3
AD4-3 TANK
To stock water
Qi to H
RHD water level H
A31
A32
AD water level H
to heat water
to stock water
AD Ti
AD Qi RHD level H
RHD temperature T
AD Electric Power HD Order P
EF to stock water AD Qi
TANK
RHD level H
EF to heat water AD Ti
HEATING RESISTOR
AD Electric Power
HD Order P
AD water level H
RHD temperature T
Application 2 in Reliability
47 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
EF to heat water AD Ti
HEATING RESISTOR
AD Electric Power
HD Order P
AD water level H
RHD temperature T
EF to stock water AD Qi
TANK
RHD level H
RHD water level H
0
0.10.20.30.40.50.60.70.80.9
1
0 400 800 1200 1600 2000
P(correct)P(incorrect)
EF to heatwater(k)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 500 1000 1500 2000
P(state1)P(state2)
P(state3)P(state4)
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RHD water output temperature T (k)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 500 1000 1500 2000
P(correct) P(incorrect)
RHD water output flow rate Qo (k)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0 400 800 1200 1600 2000
P(correct) P(incorrect)
To transform Pressure
to Qi
HD Order V
AD2 water input pressure and Ti
AD4-1 VALVE V
RHD Qi
AD1 Electric Power
AD Ti To transform
H to Qo
AD4-4 WATER PIPE
RHD water output flow rate Qo
HD1 Order T=50°C
To transform Qi to H
Ti to T
AD Qi
AD4-3 TANK, HEATING RESISTOR
RHD water output temperature T
AD1 Electric Power
To control V and P
AD4-2 COMPUTER, SENSORS
HD Order P
AD3 system parameters temperature T and level H
AD1 Electric Power
AD water level H
RHD V
RHD P
A1
A2
A3
A4
RHD water level H
Application 2
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R(k)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000 1200 1400 1600 1800 2000
To provide Warm Water
HD1 Order T=50°C
AD2 water input pressure and Ti
AD4 WATER HEATER PROCESS
RHD water output temperature T and flow rate Qo
AD1 Electric Power
AD3 system parameters temperature T and level H
Application 2 in Reliability
50 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
DynamicDynamic BayesianBayesian Networks in Networks in DiagnosisDiagnosis & & ReliabilityReliability AnalysisAnalysis
51 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Dynamic Bayesian Networks in System Diagnosis
The diagnosis is composed of three stages:
• Classically, decision making is realized by an elementary logic
Nevertheless, in this case, when multiple faults, false alarms and missing detections occur, the faults can not be isolated
• In the spirit of (Isermann, 1994), fault isolation performance can increase through the integration of other knowledge in the diagnosis
residuals generation residuals
evaluation decision making
uj Fn
rjmi
52 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Problem statementProblem statement
Increasing effectiveness of modelIncreasing effectiveness of model --based fault diagnosisbased fault diagnosis
Computed by means of stochastic process model, reliability analyComputed by means of stochastic process model, reliability analysis sis define define the the a prioria priori behavior of the probabilities distribution behavior of the probabilities distribution over the over the functioning and malfunctioning and mal--functioning states of the systemfunctioning states of the system
with the integration of reliability analysiswith the integration of reliability analysis
In fault diagnosis the decision is then based on the fusion of information coming from residuals evaluation and an a prioria prioribehaviorbehavior computed by a computed by a probabilistic model of reliability
The probabilistic model of reliability must take into account must take into account the observations on the system, the observations on the system, this is new in reliability analysisthis is new in reliability analysis!?!?
Bayesian Networks (BN) are investigated to compute the decision => BN are able to model dynamic and probabilistic problems
53 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
FDI decision making
a) b) D(n,j) F1 Fn FN D(n,j) F1 Fn FN
u1 1 0 0 u1 0 1 1
uj 0 1 0 uj 1 0 1
uJ 0 0 1 uJ 1 1 0
u1
uj
uJ
1F
nF
NF
u1
uj
uJ
1F
nF
NF
The fault is the cause of the residual deviationThe fault is the cause of the residual deviation
The BN Structure is defined directly by the incidence matrix DD
A fault is modelled as a random variable Fndefined over defined over two states
{not Occurred, Occurred}
A symptom is represented also as ujdefined over the states
{not detected, detected}
WEBER P., THEILLIOL D., AUBRUN C., EVSUKOFF A.G., Increasing effectiveness of model-based fault diagnosis: A Dynamic Bayesian Network design for decision making. 6th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Beijing, P.R. China (30/08/2006), pp. 109-114.
54 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
not occured 1occured 0
F
u not detected detectednot occured
occuredF
Conditional Probability Table CPT
ujFn
P(uj|Fn)
Bayesian Network ParametersBayesian Network Parameters
Discrete random variable
variable states
a prioridistribution P(Fn)
FDI decision making
jc-1 jc
jb jb-1
The Bayes theorem is applied in the BN inference to compute the probability that a fault occurred according to the states of the symptoms uj
)(
)()()(
j
njn
jn up
FupFpuFp =
a priori distribution on Fault
Conditional Probability Table parameters
Online residual evaluation
uj
Fn
Fn
( )( )occurredFdnotdetecteupb
occurred notFdetectedupc
njj
njj
===
===
missing detection
false alarms
55 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
the inter-time slices CPT are equivalent to Markov Chain model of each component
Starting from an observed situation at time k=0, the probability distribution over the states is computed (simulation) using successive inferences
a time feedbacka time feedback
inter-time slices CPT
a priori Reliability ModelDynamic Bayesian Network ParametersDynamic Bayesian Network Parameters
0
0,2
0,4
0,6
0,8
1
0 50 100 150 200 250 300
)(tRn
a priori Reliability of the component n
)1( −kni
)(kni
CPT )(kni
)1( −kni up down
up 1-p12 p12
down 0 1
56 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Fusion
The a priori Reliability of the component n is used to initialise the is used to initialise the a priori distribution distribution on the fault Fn states
Hypothesis to simplify the model in this first work:
•• Only one component contribute to the Only one component contribute to the a priori distribution distribution on a fault
• A component reliability is independent from the others components states
u not occured occuredup 1 0
down 0 1F
Fn
)(kni
1u
Ju
nF
)(kni
)1( −kni
WEBER P., THEILLIOL D., AUBRUN C., EVSUKOFF A.G., Increasing effectiveness of model-based fault diagnosis: A Dynamic Bayesian Network design for decision making. 6th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Beijing, P.R. China (30/08/2006), pp. 109-114.
57 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Application in Diagnosis
Water heater (Physical process)Water heater (Physical process)
V
R1
R2
Qi Ti
Qo To
H P
H sensor
T sensor
Q sensor
The goal of the process is to assure a constant water flow rate Qowith a given controlled temperature To.
58 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
The decision DBN modelThe decision DBN model
011u3
001u2
100u1
TQHSensor faults
Incidence matrix
Application in Diagnosis
59 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
The decision DBN modelThe decision DBN model
982occured
595not occured
detectednot detectedFault T
u1
For all faults of the system, it is assumed that the probability of miss detection is fixed to 0.02 and the probability of false alarms is fixed to 0.05
Application in Diagnosis
60 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
The decision DBN modelThe decision DBN model
982occured
595not occured
detectednot detectedFault T
u1
982occured
595not occured
detectednot detectedFault Q
u2
For all faults of the system, it is assumed that the probability of miss detection is fixed to 0.02 and the probability of false alarms is fixed to 0.05
Application in Diagnosis
61 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
The decision DBN modelThe decision DBN model
982occured
595not occured
detectednot detectedFault T
u1
99.960.04occured
98.11.9not occuredOccurred
98.11.9occured
9.7590.25not occurednot occurred
detectednot detectedFault QFault H
u3
982occured
595not occured
detectednot detectedFault Q
u2
For all faults of the system, it is assumed that the probability of miss detection is fixed to 0.02 and the probability of false alarms is fixed to 0.05
Application in Diagnosis
62 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
The decision DBN modelThe decision DBN model
up dgd
λ1 down
λ2
λ3
Markov Chains
up down
λ1
up down
λ1
sensor Q
sensor T
sensor H
λ1=0.22 10-4
λ1=2 10-4
λ1=1.25 10-4
λ2=3.3 10-4
λ3=0.22 10-4
Application in Diagnosis
63 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Test scenarioTest scenario
false
alar
m o
n u1
failu
re o
n th
e se
nsor
Q
fault
-free
case
T an
d H s
enso
rs fa
ults
0 5 10 15 20 25 30 35 400
0.5
1
0 5 10 15 20 25 30 35 400
0.5
1
0 5 10 15 20 25 30 35 400
0.5
1
Samples
A B C D
0 5 10 15 20 25 30 35 400
0.5
1
u1
0 5 10 15 20 25 30 35 400
0.5
1
u2
0 5 10 15 20 25 30 35 400
0.5
1
u3
Samples
A B C D
011u3
001u2
100u1
TQHSensor faults
Incidence matrix
Application in Diagnosis
64 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Test scenarioTest scenario
0 5 10 15 20 25 30 35 400
0.5
1
0 5 10 15 20 25 30 35 400
0.5
1
0 5 10 15 20 25 30 35 400
0.5
1
Samples
A B C D
0 5 10 15 20 25 30 35 400
0.5
1
u1
0 5 10 15 20 25 30 35 400
0.5
1
u2
0 5 10 15 20 25 30 35 400
0.5
1
u3
Samples
A B C D
A B C DA B C D
0 5 10 15 20 25 30 35 400
0.5
1
Fau
lt H
0 5 10 15 20 25 30 35 400
0.5
1
Fau
ltQ
0 5 10 15 20 25 30 35 400
0.5
1
Fau
ltT
Samples
close to 0
close to 0
close to 0
Reset
false
alar
m o
n u1
failu
re o
n th
e se
nsor
Q
fault
-free
case
T an
d H s
enso
rs fa
ults
)(
)()()(
j
njn
jn uP
FuPFPuFP =
Application in Diagnosis
65 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
A safety barriers-based approach for the risk analysis of socio-technical systems
LEGER A., DUVAL C., WEBER P., LEVRAT E., FARRET R., Bayesian Network Modelling the risk analysis of complex socio technical systems. Workshop on Advanced Control and Diagnosis, ACD'2006, Nancy, France (16/11/2006).
DUVAL C., LEGER A., WEBER P., LEVRAT E., IUNG B., FARRET R., Choice of a risk analysis method for complex socio-technical systems. European Safety and Reliability Conference, ESREL 2007, Stavanger, Norvège (25/06/2007).
66 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
A global risk analysis model
Ein 1 Ein 2 Ein 3 Ein 4 Ein 5 Ec 6 Ein 7 Ec 8
EI EI EI EI
EI
EI
ERC
ERS
ERS
PhD PhD PhD PhD
EM EM
EM EM EM EM
&
Or
&
Or
Or
Or
Or
Decision a Decision b
Organi s ational factor a
Organis ati onal factor b
Internal organis ational layer
Decisions and actions layer
Technical layer
External organi s ational layer
Natural environment layer
Organis ational factor c
Organis ational factor d
Environmental factor a
Environmental factor b
Transactional exchange Vertical exchange Horizontal e xchange
Caption Risk reduction barrier
Paté-Cornell M.E.-Murphy D.M.,‘Human and management factors in probabilistic risk analysis: the SAM approach and observations from recent applications’, Reliability Engineering and System Safety, n°53, pp. 115-126, 1996.
Necessity to establish relations between different kinds of layers in the model of the system: the technical layer (closed system) and the human/organisational layer (open system)
67 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
A global risk analysis modelThe generic global Bayesian network model structureThe generic global Bayesian network model structure
InternalInternal organisationalorganisational layer layer
68 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
A global risk analysis model
DecisionsDecisions andand actions layer actions layer
InternalInternal organisationalorganisational layer layer
DecisionsDecisions andand actions layer actions layer
69 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Application Pentane storageTransitional storage tank (product: liquid pentane)
Extremely flammable in air → storage operation made in presence of gaseous nitrogen (to prevent any reaction with air)
Safety components : actuator 1 (A1), vent hole, safety valve, pressure sensor, retention pool
Necessity of a regular and specific control for the actuator 1 and the safety valve :
- Actuator 1 has to remain open (to insure an optimal pentane output) → padlocking and regular follow-up (actions schedule) ,
- Safety valve has to be operational in case of important pressure rises → regular control of its good operating .
Nitrogen (gas)
A1 A2 A3
A4
Pentane (liquid)
Safety valve Vent hole
Pentane
Nitrogen
Pressure sensor
Retention pool
Pentane (liquid)
A1: manual actuator (opened)
A2, A3, A4: automatic actuator
70 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Studied scenario
Temperature > 30°C
Absence or lowoutput of Nitrogen
Pressure rise in the
tank
Tank explosion
Pentane puddle in the workshop
(Liquid)
Pentane cloudin the workshop
(Gas)
Fire due to the pentane
puddle
Fire due to the pentane
cloud
Safety impact due to thepentane puddle
Availability impact due to the pentane puddle
Durability impact due to the pentane puddle
Safety impact due to thepentane cloud
Availability impact due to the pentane cloud
Durability impact due to the pentane cloud
Pressure sensor
Actuator 1 padlocked(Opened)
Safetyvalve
Vent hole Retention pool
Ein/Ec
EI ERC
ERS PhD
Local EM
Liquid pentane in the tank
Global safetyimpact
Global availabilityimpact
Global durabilityimpact
Global EM
Temperature > 30°C
Absence or lowoutput of Nitrogen
Pressure rise in the
tank
Tank explosion
Pentane puddle in the workshop
(Liquid)
Pentane cloudin the workshop
(Gas)
Fire due to the pentane
puddle
Fire due to the pentane
cloud
Safety impact due to thepentane puddle
Availability impact due to the pentane puddle
Durability impact due to the pentane puddle
Safety impact due to thepentane cloud
Availability impact due to the pentane cloud
Durability impact due to the pentane cloud
Pressure sensor
Actuator 1 padlocked(Opened)
Safetyvalve
Vent hole Retention pool
Ein/Ec
EI ERC
ERS PhD
Local EM
Liquid pentane in the tank
Global safetyimpact
Global availabilityimpact
Global durabilityimpact
Global EM
Context: - a temperature upper than 30°C (in summer),- liquid pentane in the tank ,- insufficient quantity of nitrogen in the tank .
Liquid pentane → gaseous pentane → pressure rise in the tank → evacuated by the vent hole and the safety valve (in a good operating) …
Risk of pressure rise in the tank → tank explosion → fire in the workshop
71 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Bayesian network model
Organisational layer
Decisions and actions layer
Technical layer
Natural environment
Organisational layer
Decisions and actions layer
Technical layer
Natural environment
Output indicators
72 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Bayesian network model
73 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Conclusion
Bayesian Networks
– Equivalence between the Bayesian Networks and fault tree…
– Multimodal model
– Acyclic Graph constraint only
Dynamic Bayesian Networks
– Equivalence between the Dynamic Bayesian Networks and MC, ½ MC, MSM, IOHMM
–– Thanks to the factorization,Thanks to the factorization, DBN leads to a synthetic representation of complex systems
Future works Future works
– MDP application in Maintenance
– Dynamic Evidential Networks in reliability
74 / 73 S3 31-01 2008, CRAN UMR 7039 CNRS, Nancy Univer sité
Some ref.
WEBER P., JOUFFE L., Complex system reliability modelling with Dynamic Object Oriented Bayesian Networks (DOOBN). Reliability Engineering andSystem Safety, Volume 91, Issue 2, pages 149-162 (Selected Papers Presented at QUALITA 2003) February (2006).
WEBER P., SUHNER M.C., IUNG B. System approach-based Bayesian Network to aid maintenance of manufacturing process. 6th IFAC Symposium on Cost Oriented Automation, Low Cost Automation, Berlin, 8-9 octobre, (2001).
WEBER P., SUHNER M.C. An application of Bayesian Networks to the Performance Analysis of a Process. European Conference on System Dependability and Safety (ESRA 2002/lambda-Mu13), Lyon (France), 19-21 mars (2002).
WEBER P., JOUFFE L. Reliability modelling with Dynamic Bayesian Networks. 5th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS'03), Washington, D.C., USA, 9-11 juin, (2003).
WEBER P., MUNTEANU P., JOUFFE L. Dynamic Bayesian Networks modelling the dependability of systems with degradations and exogenous constraints. 11th IFAC Symposium on Information Control Problems in Manufacturing, INCOM'04. Salvador-Bahia, Brazil, April 5-7th, (2004).
MULLER A., WEBER P., BEN SALEM A. Process model-based Dynamic Bayesian Networks for Prognostic. Fourth International Conference on Intelligent Systems Design and Applications (ISDA 2004), Budapest, Hungary, August 26-28, (2004).
BOUILLAUT L., WEBER P., BEN SALEM A., AKNIN P. Use of Causal Probabilistic Networks for the improvement of the Maintenance of Railway Infrastructure. IEEE International Conference on Systems, Man and Cybernetics, Hague, Netherlands, October 10-13, (2004).
BEN SALEM A., BOUILLAUT L., AKNIN P., WEBER P. Dynamic Bayesian Networks for classification of rail defects. IEEE 4th International Conference on Intelligent Systems Design and Applications, Budapest, Hungary, August 26-28, (2004).
BEN SALEM A., MULLER A., WEBER P., Dynamic Bayesian Networks in system reliability analysis. 6th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Beijing, P.R. China (30/08/2006), pp. 481-486, (2006).
WEBER P., THEILLIOL D., AUBRUN C., EVSUKOFF A.G., Increasing effectiveness of model-based fault diagnosis: A Dynamic Bayesian Network design for decision making. 6th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Beijing, P.R. China (30/08/2006), pp. 109-114, (2006).
SIMON C., WEBER P., Bayesian Networks Implementation of the Dempster Shafer Theory to Model Reliability Uncertainty. Workshop on Bayesian Networks in Dependability (BND2006) in the First International Conference on Availiability, Reliability and Security, ARES 2006, Vienna, April 20-22, Autriche (2006), pp. 788-793, (2006).
LEGER A., DUVAL C., WEBER P., LEVRAT E., FARRET R., Bayesian Network Modelling the risk analysis of complex socio technical systems. Workshop on Advanced Control and Diagnosis, ACD'2006, Nancy, France (16/11/2006).
DUVAL C., LEGER A., WEBER P., LEVRAT E., IUNG B., FARRET R., Choice of a risk analysis method for complex socio-technical systems. European Safety and Reliability Conference, ESREL 2007, Stavanger, Norvège (25/06/2007).
References of CRAN are available with HAL
http://http://hal.archiveshal.archives--ouvertes.frouvertes.fr