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 [email protected] Oct/2013

mailto:[email protected]

September 2013

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September 2013

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September 2013

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September 2013

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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

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

11

Model

Measurements

Bayesian Filter

State Estimates


September 2013

12

Model

Measurements

Bayesian Filter

State Estimates

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

• Physics-of-failure or empirical.


September 2013

13

Model

Measurements

Bayesian Filter

State Estimates

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


September 2013

14

Model

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)


kk

kkk

C

BA

xy

uxx

11

)(

),( 11

kk

kkk

g

f

xy

uxx

Linear Non-Linear

+ Noise

kkk

kkkk

C

BA

wxy

vuxx

111

),(

),,( 111

kkk

kkkk

g

f

wxy

vuxx

Discrete Time State Space Models (Refresher)


xk+1 xk xk-1

yk-1 yk yk+1

uk-2 vk-2

wk-1

... ...

uk-1 vk-1

wk

uk vk

wk+1

B

C

A

B

C

A

B

C

Interpretation as Hidden Markov Model (HMM)


),(

),,( 111

kkk

kkkk

g

f

wxy

vuxx

)|(

),|( 11

kk

kkk

p

p

xy

uxx

k

k

k

k

w

v

w

v

~

~11


Example:


kkk

kkk

wxcy

vxax

11

),0(~

),0(~

2

2

1

wk

vk

Nw

Nv

1kxa

kxc

2

v

2

w

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):


)(

)()|()|(

bp

apabpbap

daapabp

apabpbap

)()|(

)()|()|(

Posterior Prior



),|(

),|()|(),|(

1:01:1

1:01:11:0:1

kkk

kkkkkkkk

p

ppp

uyy

uyxxyuyx

)(

)()|()|(

bp

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



),ˆ|(

),ˆ|()|(),,ˆ|(

11|1

11|1

11|1

kkkk

kkkkkk

kkkkkp

ppp

uxy

uxxxyuyxx

)(

)()|()|(

bp

apabpbap ?),ˆ|( 1,1|1 kkkkkp uyxx


• Kalman Filter (KF)

kkk

kkkk

C

BA

wxy

vuxx

111

),0(~

),0(~

w

v

w

v

N

N

k

k

vxx

uxx

T

kkkkk

AAPP

BA

kkkk 1|11|

11|11|ˆˆ

1|1||

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

CK

k

kkkkkkkk

xxx

xyxx

prediction

update

1)(1|1|

wxx

TT

k CCPCPKkkkk

Wait! Some more refreshing would be good before we

continue...

Short Refresher

September 2013

25

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

26

𝐸 𝑌 = 𝐸 𝑋1) + 𝐸(𝑋2

𝑌 = 𝑋1 + 𝑋2

𝑃𝑌 = 𝑃𝑋1+ 𝑃𝑋2 ( and independent) 𝑋1 𝑋2


• Linear Gaussian Case

• Optimal Solution: Kalman Filter (KF)

kkk

kkkk

C

BA

wxy

vuxx

111

),0(~

),0(~

w

v

w

v

N

N

k

k

vxx

uxx

T

kkkkk

AAPP

BA

kkkk 1|11|

11|11|ˆˆ

1|1||

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

CK

k

kkkkkkkk

xxx

xyxx

prediction

update

1)(1|1|

wxx

TT

k CCPCPKkkkk


• Kalman Filter (KF) - prediction

vxx

uxx

T

kkkkk

AAPP

BA

kkkk 1|11|

11|11|ˆˆ

prediction


• Kalman Filter (KF) - prediction

)],,([ˆ1111| kkkkkk fE vuxx

vxx

uxx

T

kkkkk

AAPP

BA

kkkk 1|11|

11|11|ˆˆ

111 kkkk BA vuxx

prediction

Recalling the model (a priori info):

More general interpretation:


• Kalman Filter (KF) - update

1|1||

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

CK

k

kkkkkkkk

xxx

xyxxupdate

Recalling Model:



Some more notation:

1|1||

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

CK

k

kkkkkkkk

xxx

xyxxupdate

1|1|ˆˆ

kkkk Cxy

)],ˆ([ˆ1|1| kkkkkk gE wxy kkk C wxy

More generally:



1|1||

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

K

k

kkkkkkkk

xxx

yyxxupdate



Even more notation:

1|1||

)ˆ(ˆˆ1|1||

kkkkkkCPKPP

K

k

kkkkkkkk

xxx

yyxxupdate

1|ˆ~

kkkk yyy

ky~ Innovation!



1|1||

~ˆˆ1||

kkkkkkCPKPP

K

k

kkkkkk

xxx

yxxupdate



1|1||

~ˆˆ1||

kkkkkkCPKPP

K

k

kkkkkk

xxx

yxxupdate

Now let’s look into the Kalman gain:

1)(1|1|

wxx

TT

k CCPCPKkkkk

kkP yx

1

kPy



1|1||

~ˆˆ1||

kkkkkkCPKPP

K

k

kkkkkk

xxx

yxxupdate

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

1|1|1|1|

1)(

kkkkkkkk

CPCCPCPCPK TT

k xwxxx

kkP yx

1

kPy

T

kkP yx



1|1||

~ˆˆ1||

kkkkkkCPKPP

K

k

kkkkkk

xxx

yxxupdate


T

k kkkkkkkPPPCPK yxyyxx

1

||1|

1

kkPP yy



1|1||

~ˆˆ1||

kkkkkkCPKPP

K

k

kkkkkk

xxx

yxxupdate


T

k kkkkkkkkkPPPPPCPK yxyyyyxx

11

||1|

kK T

kK



T

kk

kkkkkk

KPKPP

K

kkkkk yxx

yxx

1||

~ˆˆ1||

update



1

1||

1|1|

1111|

1||

~ˆˆ

)],ˆ([ˆ

)],,([ˆ

kkk

kkkkk

PPK

KPKPP

K

gE

fE

k

T

kk

kkkkkk

kkkkkk

kkkkkk

yyx

yxx

yxx

wxy

vuxx



– Schur Complement

,~ μ

b

aN ?)|( bap




,~ μ

b

aN

b

a

μ

μμ

bba

aba




b

a

μ

μμ

bba

aba

bbababa μbμμ 1

|

babababa 1

|

Schur Complement




)~|(),,ˆ|( 11|1 kkkkkkk pp yxuyxx

ky~ky

ky1|1

ˆ kkx

1ku




k

k

y

x

b

a~

?)~|( kkp yx




,~~ μ

y

xN

k

k

0

ˆ1|kkx

μ

kkk

kkkk

PP

PP

yxy

yxx 1|




kkk kkkkkPP yxμ yyxyx

~ˆ 1

1|~|

kkkkkkkkkPPPPP xyyyxxyx

1~| 1|

Schur Complement

0

ˆ1|kkx

μ

kkk

kkkk

PP

PP

yxy

yxx 1|

kK


• Extended KF (EKF)

v

xx

x

xx

xx

ux

x

ux

uxx

T

k

kk

k

kk

kkkkk

kkk

kk

kkk

kk

fP

fP

f

1|11

1|1

1|11

1|

ˆ1

11

ˆ1

11

11|11|

),(),(

),ˆ(ˆ

1|

1|

1||

ˆ

1|1||

)(

))ˆ((ˆˆ

kk

kkk

kkkkP

gKPP

gK

k

kk

kkkkkkkk

x

xx

xxx

x

xyxx

• Unscented Transform (UT)


Sigma-Points },{ )(i

i w

X


nonlinear transformation

X T


N

i

i

iwT1

)(ˆ

N

i

T

ii

i

TT TTwP1

)( )ˆ)(ˆ(ˆ

T




• Unscented KF (UKF)

1|1

1|1ˆ

kkP

kk

x

xSP

selection

Initial SP set

fk(.) gk(.)



1|1

1|1ˆ

kkP

kk

x

xSP

selection

Initial SP set

fk(.) gk(.)

1|

1|ˆ

kk

P

kk

x

x



1|1

1|1ˆ

kk

P

kk

x

xSP

selection

Initial SP set

fk(.) gk(.)

kP

kk

y

y 1|ˆ



1|1

1|1ˆ

kkP

kk

x

xSP

selection

Initial SP set

fk(.) gk(.)

kkP yx



1

1||

1|1|

1111|

1||

~ˆˆ

)],ˆ([ˆ

)],,([ˆ

kkk

kkkkk

PPK

KPKPP

K

gE

fE

k

T

kk

kkkkkk

kkkkkk

kkkkkk

yyx

yxx

yxx

wxy

vuxx


• 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)

Draw

(e.g. ) )ˆ|()ˆ|( )(

1

)(

1

m

kk

m

k fXq xxx

)ˆ|(~ˆ )(

1

)( m

k

m

k Xq xx

Define w(m) so that

(e.g. )

N

m

m

k

m

kkk wp1

)()(

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

1

)( m

kk

m

k

m

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

kk

m

k p yxx

Nw m

k 1)(




Draw

(e.g. ) )ˆ|()ˆ|( )(

1

)(

1

m

kk

m

k fXq xxx

)ˆ|(~ˆ )(

1

)( m

k

m

k Xq xx

Define w(m) so that

(e.g. )

N

m

m

k

m

kkk wp1

)()(

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

1

)( m

kk

m

k

m

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

kk

m

k p yxx

Nw m

k 1)(

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

• analogous to prediction in KF




Draw

(e.g. ) )ˆ|()ˆ|( )(

1

)(

1

m

kk

m

k fXq xxx

)ˆ|(~ˆ )(

1

)( m

k

m

k Xq xx

Define w(m) so that

(e.g. )

N

m

m

k

m

kkk wp1

)()(

:1 )ˆ()|(ˆ xxyx

)ˆ|( )()(

1

)( m

kk

m

k

m

k gww xy

Resample:

Reset weights:

)|(ˆ~ˆ )(

kk

m

k p yxx

Nw m

k 1)(

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

• analogous to update in KF

Bayesian Estimation for Prognosis

Discrete Time

Bayesian Filter Monte Carlo or

Alternative

68

Sample Application

69

Sample Application

70

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 [email protected]

mailto:[email protected]

discrete time bayesian estimation for failure …...tutorial – phm conference 2013 discrete time...

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