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Prognostics and Health Management: data-driven methodsand decision under uncertainty
Integration of condition monitoring data and uncertainty reduction
William Fauriat, Post-doc fellow, Safety and Risk team
CentraleSupélec – Laboratoire de Génie Industriel
April 2019
Theoretical and applied contextData integration
Examples and discussion
Outline
1 Theoretical and applied context: PHM, CBM, Life-cycle management
2 Data integration and uncertainty reduction
3 Examples of data integration and uncertainty reduction
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 2 / 23
Theoretical and applied contextData integration
Examples and discussion
Main objectivesData integration in PHM/CBMUncertainty reduction
Applied context and main objectives
Life-cycle management /time-varying reliability
Prognostics and HealthManagement / CBM
Optimal decision underuncertainty / Sequentialdecision making
Structural reliability
pf =∫∫
R<SfR(r)fS(s)drds
Objectives
Data Integration
Data → Prediction (UQ)→ Decision
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 3 / 23
Theoretical and applied contextData integration
Examples and discussion
Main objectivesData integration in PHM/CBMUncertainty reduction
Data integration in PHM/CBM
Condition Monitoring
Data
Direct CM Indirect CM
Event / Historical data
(+prior knowledge)
Data acquisition step
Problem context/boundaries, FMEA, etc.
Health Index
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 4 / 23
Theoretical and applied contextData integration
Examples and discussion
Main objectivesData integration in PHM/CBMUncertainty reduction
Data integration in PHM/CBM
Condition Monitoring
Data
Direct CM Indirect CM
Stochasticprocessmodel
Regressionor Soft
Computing
Covariate-basedmodel
Regressionor Soft
Computing
Event / Historical data
(+prior knowledge)
Bayesian
Filtering
Physics-basedmodel
Random-valuemodel
Continuousstate
Discretestate
Discretestate
Binarystate
Data processing step
Current or futur state (or belief) prediction
Data acquisition step
(+thershold) (failed/safe)
Continuousstate
(+thershold)
Binarystate
(failed/safe)
Problem context/boundaries, FMEA, etc.
(Gamma, Brownian (ARIMA, ANN MM,...) SVM, FuzzySets,...)
(KF, PF, HMM) (PropHazardM,...) (Exp, Weibull,...)(MultVar ARIMAANN, SVM,...)
Health Index
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 4 / 23
Theoretical and applied contextData integration
Examples and discussion
Main objectivesData integration in PHM/CBMUncertainty reduction
Data integration in PHM/CBM
Condition Monitoring
Data
Direct CM Indirect CM
Stochasticprocessmodel
Regressionor Soft
Computing
Covariate-basedmodel
Regressionor Soft
Computing
Event / Historical data
(+prior knowledge)
Bayesian
Filtering
Physics-basedmodel
Random-valuemodel
Continuousstate
Discretestate
Discretestate
Binarystate
Data processing step
Current or futur state (or belief) prediction
Data acquisition step
(+thershold) (failed/safe)
Continuousstate
(+thershold)
Binarystate
(failed/safe)
Problem context/boundaries, FMEA, etc.
Systemstate
(or belief)
Cost
model
Maintenance policy optimization
Decision step
Renewal Theory SeqDec:MDP
Policy π : S->A
Actionmodel
(incl. inspection)
π� = argmina E[L(s,a)]
π� = argmaxπ E[Σkγ
kR(sk,πk)]
oror
Forward in times a L or R
(Gamma, Brownian (ARIMA, ANN MM,...) SVM, FuzzySets,...)
(KF, PF, HMM) (PropHazardM,...) (Exp, Weibull,...)(MultVar ARIMAANN, SVM,...)
Health Index
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 4 / 23
Theoretical and applied contextData integration
Examples and discussion
Main objectivesData integration in PHM/CBMUncertainty reduction
Uncertainty reduction������ion Monitoring
Data
Dir��� �
Indir��� �
Stochasticprocessmodel
Regressionor Soft������
ng
�� ��iate
-basedmodel
Regressionor Soft������
ng
Event / Historical data
(+prior knowledge)
Bayesian
Filtering
Physics-basedmodel
Random-valuemodel�����
nuousstate
Discretestate
Discretestate
Binarystate
Data processing step
����ent or futur state (or belief) prediction
Data acquisition step
(+thershold) (failed/safe)
�����nuous
state(+thershold)
Binarystate
(failed/safe)
Problem context/boundaries, FMEA, etc.
Systemstate
(or belief)
����model
Maintenance policy optimization
Decision step
Renewal Theory SeqDec:MDP
Policy π : S->A
Actionmodel
(incl. inspection)
π* = argmina E[L(s,a)]
π* = argmaxπ E[ΣkγkR(sk,πk)]
oror
Forward in times a L or R
(Gamma, Brownian (ARIMA, ANN MM,...) SVM, FuzzySets,...)
(KF, PF, HMM) (PropHazardM,...) (Exp, Weibull,...)(MultVar ARIMAANN, SVM,...)
Health Index
Valueof
Information
Priorinformation
UncertaintyReduction
Bayesia
n u
pdating
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 5 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Bayesian updating
Use data to update prior knowledge
Updating / Learning rule
p(h|D) =p(D|h)p(h)
∑
hp(D|h)p(h) (1)
p(D|h) is the ‘observation model’ which connects observations (here D)to the value/state h for which we seek the distribution (prediction)
h can be a probability (e.g. population data) to be updated usingpseudo-random failure data (and e.g. binomial likelihood)
h can be a state (X(t)) to be updated using a measurement modelY (t) = X(t) + ε or p(Y (t)|X(t))
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 6 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Data Types
Direct CM (signal, individual component): X(t)
Indirect CM (signal, individual component): Y(t)
Health Index: X(t) = Φ(Y(t))
Event data (population): sample of times to failure {Ti}Historical data: Run to failure paths [X(t), t ∈ [0, T ]]
Covariate data Z(t) (speed, temperature, load, etc.)
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 7 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Prediction tools: direct CM data
Prediction: p(X(t + h)|Θ, X(τ), τ ∈ [0, t])
Stochastic process model: {X(t), t ∈ [0, τ ]} : continuous (Brownian,Gamma, etc.) + threshold xth or discrete state (Markov Chain)Using statistical inference
Regression or soft computing: X(t + k∆t) = Φ(Xt−1, Xt−2, ..., Xt−p)Time-series (ARIMA), Linear regression, Neural Networks, SVM, Fuzzymodels, etc.Using learning/optimization algorithms and methods
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 8 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Prediction tools: indirect CM data
Estimation: p(X(t)|Θ, Y(τ), Z(τ), τ ∈ [0, t])
Prediction: p(X(t + h)|Θ, Y(τ), Z(τ), τ ∈ [0, t])
Regression or soft computing X(t + k∆t) = Φ(Yt−1, Yt−2, ..., Yt−p, Z)Using learning/optimization algorithms and methods
Physics-based modeling X = Φ(Y, Z) and study of p(X = 0) orp(X < xth)
Bayesian filtering (Kalman filter, Particle Filter) (continuous) or HMMmodel (discrete): p(Xt+1|Xt) and p(Xt |Yt)Using statistical inference or physics-based modeling
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 9 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Prediction tools: population/distribution/R.V.
Random-value / Distribution models:Exponential T ∼ EXP(λ), Weibull T ∼ WBL(λ, α), LognormalT ∼ LN(µ, σ)Using statistical inference methods
Covariate-based model: e.g. proportional hazard model:h(t, x) = λ(t)c(x) = λ(t)c(z(t)) withp(T > t|Z(t)) = exp(−
∫
h(u, z(u))du)Using statistical inference methods or physics-based modeling
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 10 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Resolution of the maintenance problem
Renewal theory
C∞ = limt→∞
C(t)
t=
E [C(S)]
E(S)(2)
where C(S) is the cost of the renewal cycle and S is its length
Sequential decision problem
V ∗ = maxπ
E
[
∞∑
k=0
γkR(sk , πk)
]
(3)
where R(s, a) is the reward function for taking action a when in state s and γ
is the discount factor
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 11 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Uncertainty reduction and Value of Information
From works on optimal decision making (in the 60’s)
Value of a piece of information is described by the expected gain in utilitydue to its collection
Rational decision making : “indentify[ing] a course of action (which mayor may not include experimentation) that is logically consistent with thedecision maker’s own preferences for consequences, as expressed bynumerical utilities, and with the weights he attaches to the possible statesof the world, as expressed by numerical probabilities”
Combination of Bayesian perspective and Decision theory
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 12 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Uncertainty reduction and Value of Information
‘One-shot’ decision
VoI = mina
E [L(s, a)]−Eo[mina
Es|o[L(s, a)]] (4)
0 2 4 6 8 10State value
0
0.2
0.4
0.6
0.8
1
1.2
PD
F
Prior PDFPosterior obs1Posterior obs2Change in optimal action
VoI = mina
∫
L(s, a)p(s)ds −∫
(
mina
∫
L(s, a)p(s|o)ds
)
p(o)do (5)
Sequential decision making
V ∗ = maxπ
E
[
∞∑
k=0
γkR(sk , πk)
]
(6)
VoI = V ∗(without add info) − Eadd info[V∗(with add info)] (7)
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 13 / 23
Theoretical and applied contextData integration
Examples and discussion
Bayesian updatingData integration / CBMUncertainty reduction / VoI
Uncertainty reduction and Value of Information
VoI = mina
∫
L(s, a)p(s)ds −∫
(
mina
∫
L(s, a)p(s|o)ds
)
p(o)do (8)
Requirements
Specify the connection between the state and the observation p(s|o)(observation model/uncertainty reduction): data processing / how toreduce uncertainty?
Carry out the resolution (optimization) of the decision problem
(conditional or not) mina E [L(s, a)]
Specify the action and cost models L(s, a)
Specify the prior degradation model / state distribution p(s)(unconditional / pre-posterior analysis framework)
VoI as a metric for resource prioritization: (policy improvement, system level
scheduling, sensor placement, model development)
Parameter uncertainty reduction (‘sensor placement in the wide sense’), Value of Prognostics
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 14 / 23
Theoretical and applied contextData integration
Examples and discussion
Dynamic reliability assessmentCrack propagationVoI on inspection-based CBM
Dynamic risk assessment
Prior failure probability is identified from population data
Prior is taken: πprior ∼ Beta(α, β)
Likelihood is taken as binomial with probability π and k success and n − k
failures:
(
nk
)
πk(1 − π)n−k
Posterior is: πpost ∼ Beta(α + k, β + n − k)
Pseudo random values for successes are obtained with PF on batterydegradation data
Zeng, Z., & Zio, E. (2018). Dynamic risk assessment based on statistical failure data andW. Fauriat, LGI, CentraleSupélec Data-driven PHM 15 / 23
Theoretical and applied contextData integration
Examples and discussion
Dynamic reliability assessmentCrack propagationVoI on inspection-based CBM
Dynamic risk assessment
Zeng, Z., & Zio, E. (2018). Dynamic risk assessment based on statistical failure data andcondition-monitoring degradation data. IEEE Transactions on Reliability, 67(2), 609-622.
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 16 / 23
Theoretical and applied contextData integration
Examples and discussion
Dynamic reliability assessmentCrack propagationVoI on inspection-based CBM
Crack propagation
Dynamic model is taken from literature: xt+1 = xt + eωi C(β√
xt)n∆t
Observation model converts ultrasound measurement data into anestimate of crack’s length: ln zt
d−zt= β0 + β1ln xt
d−xt+ vk
PF is used to integrate measurement data and get posterior on state
Posterior is crack’s length given measurement data and dynamic model
Myötyri, E., Pulkkinen, U., & Simola, K. (2006). Application of stochastic filtering forlifetime prediction. Reliability Engineering & System Safety, 91(2), 200-208.
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 17 / 23
Theoretical and applied contextData integration
Examples and discussion
Dynamic reliability assessmentCrack propagationVoI on inspection-based CBM
VoI on inspection-based CBM
Use of gamma model
Simple decision context
L(s, a) a = 0 a = 1
s = 0 cF cR
s = 1 0 cR
Inspection gives perfect knowledge of current condition at given time
VoI is calculated at any time by comparing prior outcome and conditionalposterior outcome
To be presented in PHM Paris 2019
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 18 / 23
Theoretical and applied contextData integration
Examples and discussion
Dynamic reliability assessmentCrack propagationVoI on inspection-based CBM
VoI on inspection-based CBM
To be presented in PHM Paris 2019
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 19 / 23
Theoretical and applied contextData integration
Examples and discussion
Conclusion
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 20 / 23
Theoretical and applied contextData integration
Examples and discussion
Conclusion / Discussion
Objective: Integration of data / uncertainty reduction in adecision-making context
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 21 / 23
Theoretical and applied contextData integration
Examples and discussion
Conclusion / Discussion
Objective: Integration of data / uncertainty reduction in adecision-making context
Application: VoI as a resource prioritization metric
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 21 / 23
Theoretical and applied contextData integration
Examples and discussion
Conclusion / Discussion
Objective: Integration of data / uncertainty reduction in adecision-making context
Application: VoI as a resource prioritization metric
Application-dependent issue: data-processing and integration: applicationof VoI framework on practical examples
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 21 / 23
Theoretical and applied contextData integration
Examples and discussion
Conclusion / Discussion
Objective: Integration of data / uncertainty reduction in adecision-making context
Application: VoI as a resource prioritization metric
Application-dependent issue: data-processing and integration: applicationof VoI framework on practical examples
Theoretical issue: resolution of the (maintenance) sequential decisionproblem
Theoretical issue: formulation / computation of VoI for sequentialdecision making
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 21 / 23
Theoretical and applied contextData integration
Examples and discussion
Conclusion / Discussion
Objective: Integration of data / uncertainty reduction in adecision-making context
Application: VoI as a resource prioritization metric
Application-dependent issue: data-processing and integration: applicationof VoI framework on practical examples
Theoretical issue: resolution of the (maintenance) sequential decisionproblem
Theoretical issue: formulation / computation of VoI for sequentialdecision making
Validation / error (on prior knowledge, on models)
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 21 / 23
Theoretical and applied contextData integration
Examples and discussion
Thank You!
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 22 / 23
Theoretical and applied contextData integration
Examples and discussion
References
Si, X. S., Wang, W., Hu, C. H., & Zhou, D. H. (2011). Remaining useful lifeestimation - a review on the statistical data driven approaches. European journalof operational research, 213(1), 1-14.
Jardine, A. K., Lin, D., & Banjevic, D. (2006). A review on machinerydiagnostics and prognostics implementing condition-based maintenance.Mechanical systems and signal processing, 20(7), 1483-1510.
Frangopol, D. M., Kallen, M. J., & Noortwijk, J. M. V. (2004). Probabilisticmodels for life-cycle performance of deteriorating structures: review and futuredirections. Progress in structural engineering and Materials, 6(4), 197-212.
Memarzadeh, M., & Pozzi, M. (2016). Value of information in sequentialdecision-making: Component inspection, permanent monitoring and system-levelscheduling. Reliability Engineering & System Safety, 154, 137-151.
Straub, D. (2014). Value of information analysis with structural reliabilitymethods. Structural Safety, 49, 75-85.
Zonta, D., Glisic, B., & Adriaenssens, S. (2014). Value of information: impact ofmonitoring on decision-making. Structural Control and Health Monitoring,21(7), 1043-1056.
Huynh, K. T., Barros, A., & Bérenguer, C. (2012). Maintenancedecision-making for systems operating under indirect condition monitoring: valueof online information and impact of measurement uncertainty. IEEETransactions on Reliability, 61(2), 410-425.
W. Fauriat, LGI, CentraleSupélec Data-driven PHM 23 / 23
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