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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



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





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







Data


Stochasticprocessmodel

Regressionor Soft

Computing

Covariate-basedmodel

Regressionor Soft

Computing


(+prior knowledge)

Bayesian

Filtering

Physics-basedmodel

Random-valuemodel

Continuousstate

Discretestate

Discretestate

Binarystate

Data processing step

Current or futur state (or belief) prediction


(+thershold) (failed/safe)

Continuousstate

(+thershold)

Binarystate

(failed/safe)


(Gamma, Brownian (ARIMA, ANN MM,...) SVM, FuzzySets,...)

(KF, PF, HMM) (PropHazardM,...) (Exp, Weibull,...)(MultVar ARIMAANN, SVM,...)

Health Index







Data



Regressionor Soft

Computing

Covariate-basedmodel

Regressionor Soft

Computing


(+prior knowledge)

Bayesian

Filtering

Physics-basedmodel

Random-valuemodel

Continuousstate

Discretestate

Discretestate

Binarystate


Current or futur state (or belief) prediction



Continuousstate

(+thershold)

Binarystate

(failed/safe)


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



Health Index





Uncertainty reduction��ion Monitoring

Data

Dir��

Indir��


Regressionor Soft��

ng

�� iate

-basedmodel

Regressionor Soft��

ng


(+prior knowledge)

Bayesian

Filtering

Physics-basedmodel

Random-valuemodel��

nuousstate

Discretestate

Discretestate

Binarystate


��ent or futur state (or belief) prediction



��nuous

state(+thershold)

Binarystate

(failed/safe)


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



Health Index

Valueof

Information

Priorinformation

UncertaintyReduction

Bayesia

n u

pdating




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))





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.)





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





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





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





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





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






‘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)






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




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




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.





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.





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





VoI on inspection-based CBM

To be presented in PHM Paris 2019




Conclusion




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








Theoretical issue: resolution of the (maintenance) sequential decisionproblem

Theoretical issue: formulation / computation of VoI for sequentialdecision making

Validation / error (on prior knowledge, on models)




Thank You!




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.


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