Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time...
Transcript of Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time...
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Tutorial – PHM Conference 2013
Discrete Time Bayesian Estimation for Failure Prognosis
Bruno P. Leão [email protected] Oct/2013
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September 2013
2
By Brian & Jaclyn Drum (Flickr) [CC-BY-2.0], via Wikimedia Commons
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September 2013
3
By Kicior99 (Own work) [CC-BY-SA-3.0-2.5-2.0-1.0], via Wikimedia Commons
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September 2013
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By USAF [Public domain], via Wikimedia Commons
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September 2013
5
[Public domain], via Wikimedia Commons
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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.
![Page 7: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/7.jpg)
Discrete Time Bayesian Estimation
Popular in PHM Community
Different types of filters:
• Kalman Filter (and variations for non-linear problems)
• Particle Filter
• Other
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Failure Prognosis
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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)
![Page 10: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/10.jpg)
Bayesian Estimation for Prognosis
Discrete Time
Bayesian Filter Monte Carlo or
Alternative
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Bayesian Filtering for Prognosis
September 2013
11
Model
Measurements
Bayesian Filter
State Estimates
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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.
![Page 13: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/13.jpg)
Bayesian Filtering for Prognosis
September 2013
13
Model
Measurements
Bayesian Filter
State Estimates
• Sensor measurements that provide info on degradation state and/or evolution
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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
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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
![Page 16: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/16.jpg)
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
![Page 17: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/17.jpg)
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
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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
![Page 19: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/19.jpg)
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
![Page 20: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/20.jpg)
What is the problem we want to solve?
Discrete Time Bayesian Estimation
?),|( 1:0:1 kkkp uyx
?),ˆ|( 1,1|1 kkkkkp uyxx
iteratively
![Page 21: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/21.jpg)
How can we solve it?
Using Bayes rule (refresher):
Discrete Time Bayesian Estimation
)(
)()|()|(
bp
apabpbap
daapabp
apabpbap
)()|(
)()|()|(
Posterior Prior
![Page 22: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/22.jpg)
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
![Page 23: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/23.jpg)
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
![Page 24: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/24.jpg)
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...
![Page 25: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/25.jpg)
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
![Page 26: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/26.jpg)
Short Refresher
September 2013
26
𝐸 𝑌 = 𝐸 𝑋1) + 𝐸(𝑋2
𝑌 = 𝑋1 + 𝑋2
𝑃𝑌 = 𝑃𝑋1+ 𝑃𝑋2 ( and independent) 𝑋1 𝑋2
![Page 27: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/27.jpg)
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
![Page 28: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/28.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - prediction
vxx
uxx
T
kkkkk
AAPP
BA
kkkk 1|11|
11|11|ˆˆ
prediction
![Page 29: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/29.jpg)
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:
![Page 30: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/30.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
)ˆ(ˆˆ1|1||
kkkkkkCPKPP
CK
k
kkkkkkkk
xxx
xyxxupdate
![Page 31: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/31.jpg)
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:
![Page 32: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/32.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
)ˆ(ˆˆ1|1||
kkkkkkCPKPP
K
k
kkkkkkkk
xxx
yyxxupdate
![Page 33: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/33.jpg)
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!
![Page 34: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/34.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
1|1||
~ˆˆ1||
kkkkkkCPKPP
K
k
kkkkkk
xxx
yxxupdate
![Page 35: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/35.jpg)
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
![Page 36: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/36.jpg)
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
![Page 37: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/37.jpg)
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
![Page 38: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/38.jpg)
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
![Page 39: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/39.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF) - update
T
kk
kkkkkk
KPKPP
K
kkkkk yxx
yxx
1||
~ˆˆ1||
update
![Page 40: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/40.jpg)
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
![Page 41: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/41.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
,~ μ
b
aN ?)|( bap
![Page 42: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/42.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
,~ μ
b
aN
b
a
μ
μμ
bba
aba
![Page 43: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/43.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
b
a
μ
μμ
bba
aba
bbababa μbμμ 1
|
babababa 1
|
Schur Complement
![Page 44: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/44.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
)~|(),,ˆ|( 11|1 kkkkkkk pp yxuyxx
ky~ky
ky1|1
ˆ kkx
1ku
![Page 45: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/45.jpg)
Discrete Time Bayesian Estimation
• Kalman Filter (KF)
– Schur Complement
k
k
y
x
b
a~
?)~|( kkp yx
![Page 46: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/46.jpg)
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|
![Page 47: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/47.jpg)
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|
![Page 48: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/48.jpg)
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
![Page 49: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/49.jpg)
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
![Page 50: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/50.jpg)
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
![Page 51: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/51.jpg)
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
![Page 52: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/52.jpg)
• Unscented Transform (UT)
Discrete Time Bayesian Estimation
Sigma-Points },{ )(i
i w
X
![Page 53: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/53.jpg)
• Unscented Transform (UT)
nonlinear transformation
X T
Discrete Time Bayesian Estimation
![Page 54: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/54.jpg)
N
i
i
iwT1
)(ˆ
N
i
T
ii
i
TT TTwP1
)( )ˆ)(ˆ(ˆ
T
• Unscented Transform (UT)
Discrete Time Bayesian Estimation
![Page 55: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/55.jpg)
Discrete Time Bayesian Estimation
• Unscented KF (UKF)
1|1
1|1ˆ
kkP
kk
x
xSP
selection
Initial SP set
fk(.) gk(.)
![Page 56: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/56.jpg)
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
![Page 57: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/57.jpg)
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
![Page 58: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/58.jpg)
Discrete Time Bayesian Estimation
• Unscented KF (UKF)
1|1
1|1ˆ
kkP
kk
x
xSP
selection
Initial SP set
fk(.) gk(.)
kkP yx
![Page 59: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/59.jpg)
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
![Page 60: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/60.jpg)
Discrete Time Bayesian Estimation
• Sigma-Point Kalman Filter (SPKF)
– Unscented KF (UKF)
– Central Difference KF (CDKF)
– Cubature KF (CKF)
– ...
![Page 61: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/61.jpg)
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
![Page 62: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/62.jpg)
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)(
![Page 63: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/63.jpg)
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
![Page 64: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/64.jpg)
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)(
![Page 65: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/65.jpg)
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
![Page 66: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/66.jpg)
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)
![Page 67: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/67.jpg)
Bayesian Estimation for Prognosis
Discrete Time
Bayesian Filter Monte Carlo or
Alternative
![Page 68: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/68.jpg)
68
Sample Application
![Page 69: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/69.jpg)
69
Sample Application
![Page 70: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/70.jpg)
70
Sample Application
![Page 71: Discrete Time Bayesian Estimation for Failure …...Tutorial – PHM Conference 2013 Discrete Time Bayesian Estimation for Failure Prognosis Bruno P. Leão leao@ge.com Oct/2013 September](https://reader030.fdocuments.in/reader030/viewer/2022040203/5e9101a9d4392b4c1d2a9356/html5/thumbnails/71.jpg)
• 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
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