Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão [email protected] Oct/2013
Apr 04, 2020
Tutorial – PHM Conference 2013
Discrete Time Bayesian Estimation for Failure Prognosis
Bruno P. Leão [email protected] Oct/2013
September 2013
2
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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
Bayesian Filtering for Prognosis
September 2013
12
Model
Measurements
Bayesian Filter
State Estimates
• A priori info. Used to get a priori state estimates.
• Physics-of-failure or empirical.
Bayesian Filtering for Prognosis
September 2013
13
Model
Measurements
Bayesian Filter
State Estimates
• Sensor measurements that provide info on degradation state and/or evolution
Bayesian Filtering for Prognosis
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)
Discrete Time Bayesian Estimation
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)
Discrete Time Bayesian Estimation
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)
Discrete Time Bayesian Estimation
),(
),,( 111
kkk
kkkk
g
f
wxy
vuxx
)|(
),|( 11
kk
kkk
p
p
xy
uxx
k
k
k
k
w
v
w
v
~
~11
Interpretation as Hidden Markov Model (HMM)
Discrete Time Bayesian Estimation
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
Interpretation as Hidden Markov Model (HMM)
Example:
Discrete Time Bayesian Estimation
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?
Discrete Time Bayesian Estimation
?),|( 1:0:1 kkkp uyx
?),ˆ|( 1,1|1 kkkkkp uyxx
iteratively
How can we solve it?
Using Bayes rule (refresher):
Discrete Time Bayesian Estimation
)(
)()|()|(
bp
apabpbap
daapabp
apabpbap
)()|(
)()|()|(
Posterior Prior
How can we solve it?
Discrete Time Bayesian Estimation
),|(
),|()|(),|(
1:01:1
1:01:11:0:1
kkk
kkkkkkkk
p
ppp
uyy
uyxxyuyx
)(
)()|()|(
bp
apabpbap ?),|( 1:0:1 kkkp uyx
How can we solve it?
Discrete Time Bayesian Estimation
),ˆ|(
),ˆ|()|(),,ˆ|(
11|1
11|1
11|1
kkkk
kkkkkk
kkkkkp
ppp
uxy
uxxxyuyxx
)(
)()|()|(
bp
apabpbap ?),ˆ|( 1,1|1 kkkkkp uyxx
Discrete Time Bayesian Estimation
• 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
Discrete Time Bayesian Estimation
• 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
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - prediction
vxx
uxx
T
kkkkk
AAPP
BA
kkkk 1|11|
11|11|ˆˆ
prediction
Discrete Time Bayesian Estimation
• 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:
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
)ˆ(ˆˆ1|1||
kkkkkkCPKPP
CK
k
kkkkkkkk
xxx
xyxxupdate
Recalling Model:
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
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:
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
)ˆ(ˆˆ1|1||
kkkkkkCPKPP
K
k
kkkkkkkk
xxx
yyxxupdate
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
Even more notation:
1|1||
)ˆ(ˆˆ1|1||
kkkkkkCPKPP
K
k
kkkkkkkk
xxx
yyxxupdate
1|ˆ~
kkkk yyy
ky~ Innovation!
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
~ˆˆ1||
kkkkkkCPKPP
K
k
kkkkkk
xxx
yxxupdate
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
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
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
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
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
~ˆˆ1||
kkkkkkCPKPP
K
k
kkkkkk
xxx
yxxupdate
We are almost there... Let’s just manipulate the covariance equation a bit:
T
k kkkkkkkPPPCPK yxyyxx
1
||1|
1
kkPP yy
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
~ˆˆ1||
kkkkkkCPKPP
K
k
kkkkkk
xxx
yxxupdate
We are almost there... Let’s just manipulate the covariance equation a bit:
T
k kkkkkkkkkPPPPPCPK yxyyyyxx
11
||1|
kK T
kK
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
T
kk
kkkkkk
KPKPP
K
kkkkk yxx
yxx
1||
~ˆˆ1||
update
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
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
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
,~ μ
b
aN ?)|( bap
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
,~ μ
b
aN
b
a
μ
μμ
bba
aba
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
b
a
μ
μμ
bba
aba
bbababa μbμμ 1
|
babababa 1
|
Schur Complement
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
)~|(),,ˆ|( 11|1 kkkkkkk pp yxuyxx
ky~ky
ky1|1
ˆ kkx
1ku
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
k
k
y
x
b
a~
?)~|( kkp yx
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
,~~ μ
y
xN
k
k
0
ˆ1|kkx
μ
kkk
kkkk
PP
PP
yxy
yxx 1|
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
kkk kkkkkPP yxμ yyxyx
~ˆ 1
1|~|
kkkkkkkkkPPPPP xyyyxxyx
1~| 1|
Schur Complement
0
ˆ1|kkx
μ
kkk
kkkk
PP
PP
yxy
yxx 1|
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
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
Discrete Time Bayesian Estimation
• 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
Discrete Time Bayesian Estimation
• 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
Discrete Time Bayesian Estimation
• 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)
Discrete Time Bayesian Estimation
Sigma-Points },{ )(i
i w
X
• Unscented Transform (UT)
nonlinear transformation
X T
Discrete Time Bayesian Estimation
N
i
i
iwT1
)(ˆ
N
i
T
ii
i
TT TTwP1
)( )ˆ)(ˆ(ˆ
T
• Unscented Transform (UT)
Discrete Time Bayesian Estimation
Discrete Time Bayesian Estimation
• Unscented KF (UKF)
1|1
1|1ˆ
kkP
kk
x
xSP
selection
Initial SP set
fk(.) gk(.)
Discrete Time Bayesian Estimation
• Unscented KF (UKF)
1|1
1|1ˆ
kkP
kk
x
xSP
selection
Initial SP set
fk(.) gk(.)
1|
1|ˆ
kk
P
kk
x
x
Discrete Time Bayesian Estimation
• Unscented KF (UKF)
1|1
1|1ˆ
kk
P
kk
x
xSP
selection
Initial SP set
fk(.) gk(.)
kP
kk
y
y 1|ˆ
Discrete Time Bayesian Estimation
• Unscented KF (UKF)
1|1
1|1ˆ
kkP
kk
x
xSP
selection
Initial SP set
fk(.) gk(.)
kkP yx
Discrete Time Bayesian Estimation
• Unscented KF (UKF)
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
Discrete Time Bayesian Estimation
• Sigma-Point Kalman Filter (SPKF)
– Unscented KF (UKF)
– Central Difference KF (CDKF)
– Cubature KF (CKF)
– ...
Discrete Time Bayesian Estimation
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
Discrete Time Bayesian Estimation
• 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)(
Discrete Time Bayesian Estimation
• 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)(
• i.e.: update the state vector for each particle based on previous values and model
• analogous to prediction in KF
Discrete Time Bayesian Estimation
• 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)(
Discrete Time Bayesian Estimation
• 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)(
• i.e.: update the weight for each particle based on the likelihood of the current measurement
• analogous to update in KF
Discrete Time Bayesian Estimation
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)(
• Particle Filters (PF)
– Sampling Importance Resampling (SIR)
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.
Discrete Time Bayesian Estimation