Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time...

Tutorial – PHM Conference 2013

Discrete Time Bayesian Estimation for Failure Prognosis

Bruno P. Leão leao@ge.com Oct/2013

September 2013

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September 2013

By Kicior99 (Own work) [CC-BY-SA-3.0-2.5-2.0-1.0], via Wikimedia Commons

September 2013

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September 2013

[Public domain], via Wikimedia Commons

GE Global Research

Lead Scientist Brazil Technology Center

• Control and Automation Engineering degree from Federal University of Minas Gerais, UFMG (2004)

• M.Eng. degree in Aeronautical and Mechanical

Engineering from Aeronautics Institute of

Technology, ITA (2006)

• D.Sc. degree in Electronics Engineering and

Computer Science from ITA (2011)

• Since 2012 he is part of GE GRC Brazil – Smart

Systems CoE

• 8 years experience in the aeronautical industry

(Embraer) as Research Leader, Researcher and

Systems Engineer

• A number of PHM related papers and US Patent

Bruno P. Leão, D.Sc.

Discrete Time Bayesian Estimation

Popular in PHM Community

Different types of filters:

• Kalman Filter (and variations for non-linear problems)

• Particle Filter

• Other

Failure Prognosis

Failure Prognosis task in 2 steps:

• Estimate parameters related to degradation state and its trend from measured data

• Extrapolate degradation using estimated parameters to yield RUL

Θ ~ p(Θ|Ψ)

RUL ~ p(RUL|Θ)

Bayesian Estimation for Prognosis

Discrete Time

Bayesian Filter Monte Carlo or

Alternative

Bayesian Filtering for Prognosis

September 2013

Measurements

Bayesian Filter

State Estimates

September 2013

Measurements

Bayesian Filter

State Estimates

• A priori info. Used to get a priori state estimates.

• Physics-of-failure or empirical.

September 2013

Measurements

Bayesian Filter

State Estimates

• Sensor measurements that provide info on degradation state and/or evolution

September 2013

Measurements

Bayesian Filter

State Estimates

• A posteriori estimates • State vector will usually

contain degradation and trend parameters.

• Fixed parameters (e.g. for degradation trend model) may be estimated

Discrete Time State Space Models (Refresher)

),( 11

Linear Non-Linear

+ Noise

),,( 111

Discrete Time State Space Models (Refresher)

xk+1 xk xk-1

yk-1 yk yk+1

uk-2 vk-2

... ...

uk-1 vk-1

Interpretation as Hidden Markov Model (HMM)

),,( 111

),|( 11

xk+1 xk xk-1

yk-1 yk yk+1

... ...

uk-1 uk

),|( 11 kkkp uxx

)|( 11 kkp xy )|( kkp xy )|( 11 kkp xy

),|( 1 kkkp uxx

Example:

What is the problem we want to solve?

?),|( 1:0:1 kkkp uyx

?),ˆ|( 1,1|1 kkkkkp uyxx

iteratively

How can we solve it?

Using Bayes rule (refresher):

)()|()|(

apabpbap

daapabp

apabpbap

)()|()|(

Posterior Prior

),|()|(),|(

1:01:1

1:01:11:0:1

kkkkkkkk

uyxxyuyx

)()|()|(

apabpbap ?),|( 1:0:1 kkkp uyx

),ˆ|(

),ˆ|()|(),,ˆ|(

kkkkkk

kkkkkp

uxxxyuyxx

)()|()|(

apabpbap ?),ˆ|( 1,1|1 kkkkkp uyxx

• Kalman Filter (KF)

kkkk 1|11|

11|11|ˆˆ

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

kkkkkkkk

prediction

update

1)(1|1|

k CCPCPKkkkk

Wait! Some more refreshing would be good before we

continue...

Short Refresher

September 2013

𝐸 𝑌 = 𝐴𝐸 𝑋

𝑃𝑋 = 𝐸 𝑋 − 𝐸 𝑋 𝑋 − 𝐸 𝑋 𝑇

𝑃𝑌 = 𝐸 𝐴𝑋 − 𝐴𝐸 𝑋 𝐴𝑋 − 𝐴𝐸 𝑋 𝑇

𝑃𝑌 = 𝐴 𝐸 𝑋 − 𝐸 𝑋 𝑋 − 𝐸 𝑋 𝑇 𝐴𝑇

𝑃𝑌 = 𝐴 𝑃𝑋𝐴𝑇

𝑌 = 𝐴𝑋

Short Refresher

September 2013

𝐸 𝑌 = 𝐸 𝑋1) + 𝐸(𝑋2

𝑌 = 𝑋1 + 𝑋2

𝑃𝑌 = 𝑃𝑋1+ 𝑃𝑋2 ( and independent) 𝑋1 𝑋2

• Linear Gaussian Case

• Optimal Solution: Kalman Filter (KF)

kkkk 1|11|

11|11|ˆˆ

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

kkkkkkkk

prediction

update

1)(1|1|

k CCPCPKkkkk

• Kalman Filter (KF) - prediction

kkkk 1|11|

11|11|ˆˆ

prediction

• Kalman Filter (KF) - prediction

)],,([ˆ1111| kkkkkk fE vuxx

kkkk 1|11|

11|11|ˆˆ

111 kkkk BA vuxx

prediction

Recalling the model (a priori info):

More general interpretation:

• Kalman Filter (KF) - update

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

kkkkkkkk

xyxxupdate

Recalling Model:

Some more notation:

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

kkkkkkkk

xyxxupdate

1|1|ˆˆ

kkkk Cxy

)],ˆ([ˆ1|1| kkkkkk gE wxy kkk C wxy

More generally:

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

kkkkkkkk

yyxxupdate

Even more notation:

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

kkkkkkkk

yyxxupdate

kkkk yyy

ky~ Innovation!

~ˆˆ1||

kkkkkkCPKPP

kkkkkk

yxxupdate

~ˆˆ1||

kkkkkkCPKPP

kkkkkk

yxxupdate

Now let’s look into the Kalman gain:

1)(1|1|

k CCPCPKkkkk

kkP yx

~ˆˆ1||

kkkkkkCPKPP

kkkkkk

yxxupdate

We are almost there... Let’s just manipulate the covariance equation a bit:

1|1|1|1|

kkkkkkkk

CPCCPCPCPK TT

k xwxxx

kkP yx

~ˆˆ1||

kkkkkkCPKPP

kkkkkk

yxxupdate

k kkkkkkkPPPCPK yxyyxx

kkPP yy

~ˆˆ1||

kkkkkkCPKPP

kkkkkk

yxxupdate

k kkkkkkkkkPPPPPCPK yxyyyyxx

kkkkkk

kkkkk yxx

~ˆˆ1||

update

)],ˆ([ˆ

)],,([ˆ

kkkkkk

– Schur Complement

aN ?)|( bap

bbababa μbμμ 1

babababa 1

Schur Complement

)~|(),,ˆ|( 11|1 kkkkkkk pp yxuyxx

ˆ kkx

?)~|( kkp yx

,~~ μ

ˆ1|kkx

yxx 1|

kkk kkkkkPP yxμ yyxyx

kkkkkkkkkPPPPP xyyyxxyx

1~| 1|

Schur Complement

ˆ1|kkx

yxx 1|

kkk kkkkkPP yxμ yyxyx

kkkkkkkkkPPPPP xyyyxxyx

1~| 1|

Schur Complement

ˆ1|kkx

yxx 1|

• Extended KF (EKF)

11|11|

),(),(

),ˆ(ˆ

))ˆ((ˆˆ

kkkkkkkk

11|11|

),(),(

),ˆ(ˆ

))ˆ((ˆˆ

kkkkkkkk

11|11|

),(),(

),ˆ(ˆ

))ˆ((ˆˆ

kkkkkkkk

• Unscented Transform (UT)

Sigma-Points },{ )(i

nonlinear transformation

TT TTwP1

)( )ˆ)(ˆ(ˆ

• Unscented KF (UKF)

selection

Initial SP set

fk(.) gk(.)

selection

Initial SP set

fk(.) gk(.)

selection

Initial SP set

fk(.) gk(.)

y 1|ˆ

selection

Initial SP set

fk(.) gk(.)

kkP yx

)],ˆ([ˆ

)],,([ˆ

kkkkkk

• Sigma-Point Kalman Filter (SPKF)

– Unscented KF (UKF)

– Central Difference KF (CDKF)

– Cubature KF (CKF)

– ...

Kalman Filter Particle Filter

Propagation of mean and covariance

Propagation of complete distribution

Approximation by two

first statistical moments Point-mass

approximation

“Gaussian” assumption No Gaussian assumption

Analytical, Linearization, Unscented Transform

Monte Carlo

Lower computational cost

Higher computational cost

• Particle Filters (PF)

– Sampling Importance Resampling (SIR)

(e.g. ) )ˆ|()ˆ|( )(

k fXq xxx

)ˆ|(~ˆ )(

k Xq xx

Define w(m) so that

(e.g. )

kkk wp1

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

k p yxx

(e.g. ) )ˆ|()ˆ|( )(

k fXq xxx

)ˆ|(~ˆ )(

k Xq xx

Define w(m) so that

(e.g. )

kkk wp1

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

k p yxx

• i.e.: update the state vector for each particle based on previous values and model

• analogous to prediction in KF

(e.g. ) )ˆ|()ˆ|( )(

k fXq xxx

)ˆ|(~ˆ )(

k Xq xx

Define w(m) so that

(e.g. )

kkk wp1

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

k p yxx

(e.g. ) )ˆ|()ˆ|( )(

k fXq xxx

)ˆ|(~ˆ )(

k Xq xx

Define w(m) so that

(e.g. )

kkk wp1

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

k p yxx

• i.e.: update the weight for each particle based on the likelihood of the current measurement

• analogous to update in KF

(e.g. ) )ˆ|()ˆ|( )(

k fXq xxx

)ˆ|(~ˆ )(

k Xq xx

Define w(m) so that

(e.g. )

kkk wp1

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

k p yxx

Bayesian Estimation for Prognosis

Discrete Time

Bayesian Filter Monte Carlo or

Alternative

Sample Application

• References

– ANDERSON, B. D. O.; MOORE, J. B. Optimal Filtering. Mineola: Dover Publications, 2005.

– CHEN, Z. Bayesian filtering: from Kalman filters to particle filters, and beyond. Hamilton: McMaster University, 2003.

– JULIER, S. J.; UHLMANN, J. K. Unscented filtering and nonlinear estimation. IEEE Review, v.92, n.3, p.401-422, Mar. 2004.

– DOUCET, A.; JOHANSEN, A. M. A tutorial on particle filtering and smoothing: fifteen years later, In: CRISAN, D.; ROZOVSKY, B. (eds.) Handbook of nonlinear filtering, Cambridge: Cambridge University Press, 2009.

– ORCHARD, M. E. A particle filtering-based framework for on-line fault diagnosis and failure prognosis. 2007. 138f. Thesis (Doctor of Philosophy in Electrical and Computer Engineering) – Georgia Tech, Atlanta.

– AN, D.; CHOI, J.-H.; KIM, N.H. A Tutorial for Model-based Prognostics Algorithms based on Matlab Code. Proceedings of the International Conference of the PHM Society, 2012.

Thank you.

Bruno P. Leão leao@ge.com

Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time...

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