A Dynamic Data-driven Hierarchical Bayesian Degradation Model for ...
Structural Damage Growth Prediction via Integration of Finite Element Method and Bayesian Estimation Approaches
Yuhang Liu1, Qi Shuai2, Shiyu Zhou1, Jiong Tang2
1. University of Wisconsin-Madison 2. University of Connecticut
1
Outline • Research Motivation • Literature Review • Structural Damage Growth Prediction Based on DDDAS Frame Work Model Formulation Gibbs Sampling Algorithm Model Selection
• Numerical and Experimental Studies • Conclusion & Future Work DDDAS 2016
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Research Motivation • Structural system failure identification and prediction play a critical role in ensuring safety and reliability • Most damages cannot be observed directly, rather causes the changes in structural properties • Accurately describing the growth of structural weakness leads better maintenance strategies and enhances the remaining useful life DDDAS 2016
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Literature Review • System damage diagnosis has been broadly and systematically studied over years and has made the transition from research to practice • System damage prognosis is an on-going research topic • Existing methods can be categorized into two main groups: Physicbased approach and Data-drive approach Physic-based approach Differential equations Data-drive approach Stochastic process Hidden Markov Model (HMM) DDDAS 2016
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Physic-based Approach Physic-based models mainly describe the damage evolving by differential equations • A defect propagation model by mechanistic modeling approach for RUL estimation of bearings. Li, et al. (1999) • Crack growth estimation based on Paris’ Law. Shankar, et al (2009) You and Mahadevan (2012) • Random disturbances and parameters are considered in the fatigue crack propagation model. Nicholas (2008), Krenk et al. (2008)
Inherent Drawbacks • • • •
Highly case specific Difficult to build up accurate models Uncertainties may be ignored Heavy computational load
DDDAS 2016
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Data-driven Approach Data-drive models adopt generic statistical methods to describe the damage progression • Less dependent on physical principles • Broadly apply to various systems • Relatively low computation cost Stochastic Process Model Model the sensing signal as stochastic process for degradation path prediction. Firoozeh, et al. (2010), Ye, et al. (2013) Leave the prognosis only on data level State Space Model Unobservable degradation status are defined as hidden states using HMM. Rammohan, et al. (2005), Zaidi, et al. (2011) Assumptions in HMM often violates the realistic Unreliable for long term prognosis Inherent Drawbacks: Performs poorly on explaining the change of structural properties DDDAS 2016
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Research Objective Flexibility
Engineering Insides
Computational Cost
Physic-based
Poor
Good
High
Data-driven
Good
Poor
Low
Propose an efficient dynamic data-driven method, which takes advantage of both the physical finite element model (FEM) and the data driven Bayesian framework, to tackle the structural damage growth prediction.
DDDAS 2016
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Structural Damage Growth Prediction Based on DDDAS Frame Work *
DDDAS 2016
* F. Darema, “Dynamic Data Driven Applications Systems: A New Paradigm for Application Simulations and Measurements. Computational Science.” Int’l Conf. on Computational Science (ICCS), LNCS, 3038, 662–669, 2004.
8
Hierarchical Bayesian Degradation Model-Level 1
Finite Element Physical Model (First Principle)
Natural Frequencies Mean Structure
DDDAS 2016
Experiment Measurements Natural Frequencies
𝒇𝒇𝒕𝒕𝒕𝒕 ~𝓝𝓝(𝜼𝜼𝒊𝒊 (𝜽𝜽𝒕𝒕 ), 𝝈𝝈𝟐𝟐𝒇𝒇 ), 𝒊𝒊 = 𝟏𝟏, 𝟐𝟐 … , 𝑵𝑵𝒇𝒇 , 𝒕𝒕 = 𝟏𝟏, 𝟐𝟐, . . , 𝑻𝑻 𝜼𝜼𝒊𝒊 𝜽𝜽𝒕𝒕 = 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 𝜽𝜽𝒕𝒕 , 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 = 𝒇𝒇𝟎𝟎𝟎𝟎 9
Hierarchical Bayesian Degradation Model-Level 2
Finite Element Physical Model (First Principle)
Trend of Structural Prop. 𝜽𝜽𝒕𝒕 ′𝐬𝐬 Structure
Polynomial structure of time
DDDAS 2016
Underlying trend of 𝜽𝜽𝒕𝒕 as function of time 𝒕𝒕
𝜽𝜽𝒕𝒕 ~𝓝𝓝(𝜸𝜸𝒕𝒕 , 𝝈𝝈𝟐𝟐𝜽𝜽 ) 𝜸𝜸𝒕𝒕 = 𝜷𝜷𝑻𝑻 𝒙𝒙𝒕𝒕
𝒙𝒙𝒕𝒕 = 𝟏𝟏, 𝒕𝒕, 𝒕𝒕𝟐𝟐 ⋯
𝑻𝑻
10
Hierarchical Bayesian Degradation Model-Summary Posterior Distribution ∝ Likelihood × Prior Distribution
𝒇𝒇𝒕𝒕𝒕𝒕 ~𝓝𝓝(𝜼𝜼𝒊𝒊 (𝜽𝜽𝒕𝒕 ), 𝝈𝝈𝟐𝟐𝒇𝒇 ), 𝒊𝒊 = 𝟏𝟏, 𝟐𝟐 … , 𝑵𝑵𝒇𝒇 , 𝒕𝒕 = 𝟏𝟏, 𝟐𝟐, . . , 𝑻𝑻 • Mean Structure 𝜼𝜼𝒊𝒊 𝜽𝜽𝒕𝒕 = 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 𝜽𝜽𝒕𝒕 , 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 = 𝒇𝒇𝟎𝟎𝟎𝟎 • Trend of Structural Prop. 𝜽𝜽𝒕𝒕 ~𝓝𝓝(𝜸𝜸𝒕𝒕 , 𝝈𝝈𝟐𝟐𝜽𝜽 ) • 𝜽𝜽𝒕𝒕 ′𝐬𝐬 Structure 𝜸𝜸𝒕𝒕 = 𝜷𝜷𝑻𝑻 𝒙𝒙𝒕𝒕 • Natural Frequencies
Level1
Polynomial structure of time Prior Distribution of Coefficients 𝜷𝜷 Prior Distribution of 𝝈𝝈𝟐𝟐𝒇𝒇 Prior Distribution of 𝝈𝝈𝟐𝟐𝜽𝜽
DDDAS 2016
𝑻𝑻
𝒙𝒙𝒕𝒕 = 𝟏𝟏, 𝒕𝒕, 𝒕𝒕𝟐𝟐 ⋯ 𝜷𝜷~𝓜𝓜𝓜𝓜(𝒃𝒃, 𝝈𝝈𝟐𝟐𝜷𝜷 𝑰𝑰), 𝝈𝝈𝟐𝟐𝒇𝒇 ~𝓣𝓣𝓣𝓣 𝒂𝒂𝒇𝒇 , 𝒃𝒃𝒇𝒇 𝝈𝝈𝟐𝟐𝜽𝜽 ~𝓣𝓣𝓣𝓣 𝒂𝒂𝜽𝜽 , 𝒃𝒃𝜽𝜽
Prior Dist.
• • • •
Likelihood
Level2
11
Parameter Estimation- Gibbs Sampling 𝑝𝑝 𝜷𝜷 𝓕𝓕 = ⨌ 𝑝𝑝 𝚯𝚯, 𝜷𝜷, 𝜎𝜎𝑓𝑓2 , 𝜎𝜎𝜃𝜃2 , 𝑼𝑼 𝓕𝓕 𝑑𝑑𝚯𝚯𝑑𝑑𝜎𝜎𝑓𝑓2 𝑑𝑑𝜎𝜎𝜃𝜃2 𝑑𝑑𝑼𝑼
• Multiple integral hard to compute • Maximum Likelihood Estimation can be tedious Gibbs Sampling Algorithm Initialize 𝜃𝜃𝑡𝑡 0 , 𝜷𝜷
𝟎𝟎
, 𝜎𝜎𝑓𝑓2
0
For 𝒌𝒌 in iteration 𝟏𝟏: 𝑲𝑲
, 𝜎𝜎𝜃𝜃2
0
, 𝑢𝑢𝑖𝑖 𝟎𝟎
𝒌𝒌−𝟏𝟏 𝜃𝜃𝑡𝑡 𝑘𝑘 ~𝑝𝑝 𝜃𝜃𝑡𝑡 𝚯𝚯−t , 𝜷𝜷
𝜷𝜷
𝒌𝒌
𝜎𝜎𝑓𝑓2
~𝑝𝑝 𝜷𝜷 𝚯𝚯
𝑘𝑘
𝜎𝜎𝜃𝜃2
End DDDAS 2016
𝐤𝐤
𝒌𝒌−𝟏𝟏
, 𝜎𝜎𝜃𝜃2
, 𝜎𝜎𝜃𝜃2
𝑘𝑘−1
~𝑝𝑝 𝜎𝜎𝑓𝑓2 𝚯𝚯 𝐤𝐤 , 𝜎𝜎𝜃𝜃2
𝑘𝑘
~𝑝𝑝 𝜎𝜎𝜃𝜃2 𝚯𝚯
𝑢𝑢𝑖𝑖 𝒌𝒌 ~𝑝𝑝 𝑢𝑢𝑖𝑖 𝚯𝚯
𝐤𝐤
𝐤𝐤
, 𝜎𝜎𝜃𝜃2
𝑘𝑘
, 𝜎𝜎𝑓𝑓2
, 𝜷𝜷
, 𝜷𝜷 𝑘𝑘
𝑘𝑘−1
, 𝜎𝜎𝑓𝑓2
𝑘𝑘−1
, 𝜎𝜎𝑓𝑓2
𝑘𝑘−1
, 𝜎𝜎𝑓𝑓2 𝑘𝑘
𝑘𝑘−1
𝒌𝒌
𝒌𝒌
, 𝜷𝜷
, 𝑼𝑼
, 𝑼𝑼
, 𝑼𝑼 𝒌𝒌
, 𝑼𝑼
𝒌𝒌−𝟏𝟏 𝒌𝒌−𝟏𝟏
𝒌𝒌−𝟏𝟏
𝒌𝒌−𝟏𝟏
, 𝓕𝓕
, 𝓕𝓕
, 𝓕𝓕
, 𝓕𝓕
, 𝐔𝐔−𝑖𝑖𝒌𝒌−𝟏𝟏 , 𝓕𝓕 12
Illustrative Examples � Gibbs Sampling of 𝜷𝜷
Simulation Data
150 100 𝑓𝑓1 50 0 0 400 𝑓𝑓2 200 0 0
20
20
𝑓𝑓3500
0 0
20
𝜃𝜃𝑡𝑡 1.2
1
0.8 0.6
40
60
40
80
60
40
80
60
80
Time
𝛾𝛾𝑡𝑡 vs 𝛾𝛾�𝑡𝑡
2 5 1.5 2 0 1 �1 1 𝛽𝛽 0.5 -5 0 0 0.5 0.2 2 1 0.05 0.02 0.2 0.1 �2 0 𝛽𝛽 0 -0.05 -0.02 -0.5 -0.2 -0.1 -2 -1 0 -3 0.05 0.02 0.01 0.2 x 10-4 2 1 5 �3 0 𝛽𝛽 0 -0.05 -0.02 -0.01 -0.2 -2 -1 -5 0
0.6
6000
050 5 000 0.4 0.2 0.1 4 1 0.1 0.5
2000
4000
6000
2000 2000 2000
4000 4000 4000
6000 6000 6000
2000 2000 2000
4000 4000 4000
6000 6000 6000 Iteration
�2 𝜎𝜎 𝜃𝜃
0.05 0.05 0.5 0.2 0.1 2
2000
4000
6000 Iteration
000 000
1 � 𝛾𝛾𝑡𝑡 − 𝛾𝛾�𝑡𝑡 𝑇𝑇
0.5
𝑻𝑻
𝛾𝛾𝑡𝑡 = 𝜷𝜷 𝒙𝒙𝒕𝒕
𝑡𝑡
0.4 0.3 0.2
0.2
0.1 50
4000
�210 𝜎𝜎 50 10 10 𝑓𝑓20
�𝟐𝟐 Gibbs Sampling of 𝝈𝝈
Mean Error
0.4
0 0
2000
100 40 20 15 15 20
100
Time
0
20
40
60
80
Time
Bayesian Estimated function of 𝜽𝜽𝒕𝒕 True function of 𝜽𝜽𝒕𝒕
As data accumulated, the estimation of 𝜷𝜷 converges to the true value quickly. DDDAS 2016
13
Model Selection • What is the degree of polynomials ? • Model Selection Criteria: AIC, BIC, Bayesian Factor BF 𝑀𝑀𝑖𝑖 = 𝑝𝑝 𝑦𝑦 𝑀𝑀𝑖𝑖 /𝑝𝑝 𝑦𝑦 𝑀𝑀0 BIC 𝑀𝑀𝑖𝑖 = −2 log 𝑝𝑝 𝒚𝒚 𝑀𝑀𝑖𝑖 + 2𝑘𝑘 𝑀𝑀𝑖𝑖 × log 𝑛𝑛 AIC 𝑀𝑀𝑖𝑖 = −2 log 𝑝𝑝 𝒚𝒚 𝑀𝑀𝑖𝑖 + 2𝑘𝑘 𝑀𝑀𝑖𝑖
• Bayesian Factor - depends heavily on the choice of prior • BIC - advocated for descriptive purpose of the existing data • AIC - primary goal of the modelling application is predictive DDDAS 2016
14
Simulation Study
• A Fixed-Fixed uniform beam with 60 elements set up in FEM • 𝑬𝑬 = 𝟐𝟐. 𝟏𝟏 × 𝟏𝟏𝟏𝟏𝟖𝟖 𝒌𝒌𝒌𝒌𝒌𝒌, 𝑳𝑳 = 𝟐𝟐. 𝟓𝟓𝟓𝟓𝟓𝟓, 𝑰𝑰 = 𝟑𝟑. 𝟒𝟒𝟒𝟒 × 𝟏𝟏𝟏𝟏−𝟖𝟖 𝒎𝒎𝟒𝟒 , 𝑨𝑨 = 𝟔𝟔. 𝟒𝟒𝟒𝟒 × 𝟏𝟏𝟏𝟏−𝟒𝟒 𝒎𝒎𝟐𝟐 and 𝝆𝝆 = 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 𝐤𝐤𝐤𝐤 𝒔𝒔𝟐𝟐 /𝐦𝐦 • Stiffness loss happen in the 1st element • First three natural frequencies are measured • Two underlying trends are considered DDDAS 2016
15
Simulation Study (1)-Polynomial Trend 𝑓𝑓1135
1.1
𝜃𝜃
1
130
0.9
130
125 0.4 𝑓𝑓2380
0.8
0.5
0.6
0.7
0.8
0.9
1
360
0.7 0.6
340 0.4 720
𝑓𝑓3
0.5 0.4
𝑓𝑓1135
700
a)
0
20
40
𝑁𝑁𝑓𝑓 3 DDDAS 2016
60
80
𝑡𝑡
ℳ𝒩𝒩 𝒃𝒃, 𝜎𝜎𝛽𝛽2 𝑰𝑰
𝜎𝜎𝛽𝛽2
1012
𝒃𝒃
0.5
0.6
0.7
0.8
0.9
1
340 0 𝑓𝑓 720
3
700
0.5
0.6
0.7
0.8
𝒯𝒯𝒯𝒯 𝑎𝑎𝑓𝑓 , 𝑏𝑏𝑓𝑓
0
𝑎𝑎𝑓𝑓
𝑏𝑏𝑓𝑓
0
1
1
0
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
𝑡𝑡
360
b)
680 0.4
125 0 𝑓𝑓2380
0.9
𝜃𝜃
1
680 0
c)
𝒯𝒯𝒯𝒯 𝑎𝑎𝜃𝜃 , 𝑏𝑏𝜃𝜃 𝑎𝑎𝜃𝜃
𝑏𝑏𝜃𝜃
1
1
𝒩𝒩 𝜏𝜏𝑖𝑖 , 𝜎𝜎𝑢𝑢2 𝜏𝜏𝑖𝑖
𝜎𝜎𝑢𝑢2
0
1012
0 0
80
16
Simulation Study (1)-Polynomial Trend 𝛽𝛽1
𝛾𝛾 = 𝛽𝛽 𝑇𝑇 𝑥𝑥𝑡𝑡
𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 DDDAS 2016
𝛽𝛽2
𝛽𝛽3
𝜎𝜎𝜃𝜃2
𝜎𝜎𝑓𝑓2
17
Simulation Study (1)-Polynomial Trend
AIC Comparison for Different Models in Polynomial Trend
Degree of
𝜎𝜎𝑓𝑓2 = 0.012
Log-Likelihood
AIC
Log-Likelihood
AIC
1
-531.65
1067.30
-83.67
171.34
2
-525.42
1056.84
-78.18
162.36
3
-526.86
1061.72
-79.47
166.94
4
-527.23
1064.46
-80.05
170.10
Polynomials
DDDAS 2016
𝜎𝜎𝑓𝑓2 = 12
18
Simulation Study (1)-Polynomial Trend
Comparison of Fittings for Different Amount of Data 𝑇𝑇 = 10
𝜃𝜃𝑡𝑡1.2 1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
a) 20
40
60
80
𝑡𝑡
0 0
𝑇𝑇 = 80
𝜃𝜃𝑡𝑡1.2
1
0 0
DDDAS 2016
𝑇𝑇 = 30
𝜃𝜃𝑡𝑡1.2
b)
𝛾𝛾 𝑡𝑡 𝛾𝛾�𝑡𝑡 95% C. I 𝛾𝛾�𝑡𝑡
c)
20
40
60
80
𝑡𝑡
0 0
20
40
60
80
𝑡𝑡
19
Simulation Study (2)-Beta function Trend 𝑓𝑓1135
1.1
𝜃𝜃
1
130
0.9
130
125 0.4 𝑓𝑓2380
0.8 0.7
0.5
0.6
0.7
0.8
0.9
1
360
0.6
340 0.4 𝑓𝑓3720
0.5 0.4
𝑓𝑓1135
700
a)
0
20
40
𝑁𝑁𝑓𝑓 3 DDDAS 2016
60
80
𝑡𝑡
ℳ𝒩𝒩 𝒃𝒃, 𝜎𝜎𝛽𝛽2 𝑰𝑰
𝜎𝜎𝛽𝛽2
1012
𝒃𝒃
0.5
0.6
0.7
0.8
0.9
1
340 0 𝑓𝑓3720 700
0.5
0.6
0.7
0.8
𝒯𝒯𝒯𝒯 𝑎𝑎𝑓𝑓 , 𝑏𝑏𝑓𝑓
0
𝑎𝑎𝑓𝑓
𝑏𝑏𝑓𝑓
0
1
1
0
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
𝑡𝑡
360
b)
680 0.4
125 0 𝑓𝑓2380
0.9
1
𝜃𝜃
c)
680 0
𝒯𝒯𝒯𝒯 𝑎𝑎𝜃𝜃 , 𝑏𝑏𝜃𝜃 𝑎𝑎𝜃𝜃
𝑏𝑏𝜃𝜃
1
1
𝒩𝒩 𝜏𝜏𝑖𝑖 , 𝜎𝜎𝑢𝑢2 𝜏𝜏𝑖𝑖
𝜎𝜎𝑢𝑢2
0
1012
0 0
80
20
Simulation Study (2)-Beta function Trend 𝛽𝛽1 𝛽𝛽2
𝛾𝛾 = 𝛽𝛽 𝑇𝑇 𝑥𝑥𝑡𝑡
𝛽𝛽3 𝛽𝛽4
𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 DDDAS 2016
𝜎𝜎𝜃𝜃2
𝜎𝜎𝑓𝑓2
21
Simulation Study (2)-Beta function Trend
AIC Comparison for Different Models in Polynomial Trend
Degree of
𝜎𝜎𝑓𝑓2 = 0.012
Log-Likelihood
AIC
Log-Likelihood
AIC
1
-546.55
1097.1
-113.58
231.16
2
-538.17
1082.34
-93.46
192.92
3
-536.42
1080.84
-87.15
182.3
4
-535.91
1081.82
-86.24
182.48
Polynomials
DDDAS 2016
𝜎𝜎𝑓𝑓2 = 12
22
Simulation Study (2)-Beta function Trend
Comparison of Fittings for Different Amount of Data 𝑇𝑇 = 10
𝜃𝜃𝑡𝑡1.2
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
b)
a)
0 0
20
40
60
80
𝑡𝑡
𝑇𝑇 = 80
𝜃𝜃𝑡𝑡1.2
1
1
DDDAS 2016
𝑇𝑇 = 30
𝜃𝜃𝑡𝑡1.2
0 0
20
40
60
80
𝑡𝑡
0 0
𝛾𝛾 𝑡𝑡 𝛾𝛾�𝑡𝑡 95% C. I 𝛾𝛾�𝑡𝑡
c) 20
40
60
80
𝑡𝑡
23
Simulation Study-Fitting vs # Data
Beta function Trend
Polynomial Trend 𝑀𝑀𝐸𝐸𝑇𝑇1.4
𝑀𝑀𝐸𝐸𝑡𝑡2.5 b)
a)
1.2
2
1 1.5
0.8
0.6
0.4
1
Mean Error
Mean Error 0.5
0.2
0 10
DDDAS 2016
20
30
40
50
60
70
80
𝑇𝑇
0 10
20
30
40
50
60
70
80
𝑇𝑇
24
Experimental Study Fixed-Fixed Beam Experiment Material: Aluminum Young’s modulus: 68.9 Gpa Density: 2700 𝑘𝑘𝑘𝑘/𝑚𝑚3
Size: length 510 mm, width 19.05 mm, thickness, 4.76 mm Added mass: 2.7 g each , total 10 masses are added Added mass
DDDAS 2016
Accelerometer
25
Experimental Study – Beta-function Trend 𝐟𝐟𝟏𝟏
#mass 0 1
𝐟𝐟𝟐𝟐
92
𝐟𝐟𝟑𝟑
498
89.5
𝛉𝛉
1219
489
1200
0.46
0
87.5
481
1180
0.30
74
3
85
473
1161
0.22
75
4
83
466
1151
0.17
76
5
82
463
1145
0.14
77
6
80
457
1134
0.12
78
7
79
455
1123
0.11
79
8
76
448
1117
0.10
80
𝑓𝑓2490 𝑅𝑅 2 = 0.93
𝑓𝑓31200 𝑅𝑅 2 = 0.94
485
88
1190
480
86
1180
475 84
1170
470
82
465
80
460
1160 1150 1140
455
78 76 0.1
1
2
𝑓𝑓190 𝑅𝑅 2 = 0.88
DDDAS 2016
𝐭𝐭
0.2
0.3
𝑢𝑢𝑖𝑖 = 76.06 𝑣𝑣𝑖𝑖 = 33.12 0.4
𝜃𝜃
450 0.1
0.2
0.3
𝑢𝑢𝑖𝑖 = 445.3 𝑣𝑣𝑖𝑖 = 105.3 0.4
𝜃𝜃
1130 1120 0.1
0.2
0.3
𝑢𝑢𝑖𝑖 = 1107 𝑣𝑣𝑖𝑖 = 219.5 0.4
𝜃𝜃
26
Experimental Study – Beta-function Trend 𝛽𝛽1
0.4
𝛽𝛽2
0.3 0.2 0.1
𝛽𝛽3
0 -0.1 0 𝑢𝑢1
𝑢𝑢2 𝑢𝑢3 DDDAS 2016
20
40
60
80
𝜎𝜎𝜃𝜃2
𝜎𝜎𝑓𝑓2
27
Conclusion & Future Work Conclusion • Proposed a dynamic data-driven hierarchical Bayesian degradation model for weakness growth estimation. • Effectively recover the true polynomial trend and approximate the beta-function trend accurately. • Validate through simulation and experimental studies
Future Work • Adaptive sensor tuning for data collection • Change point detection for online signals • Decision making on system management
DDDAS 2016
28
ACKNOWLEDGEMENT The financial support of this work is provided by Air Force Office of Scientific Research under the program Dynamic Data Driven Application Systems (DDDAS)
DDDAS 2016
29
Yuhang Liu1, Qi Shuai2, Shiyu Zhou1, Jiong Tang2
1. University of Wisconsin-Madison 2. University of Connecticut
1
Outline • Research Motivation • Literature Review • Structural Damage Growth Prediction Based on DDDAS Frame Work Model Formulation Gibbs Sampling Algorithm Model Selection
• Numerical and Experimental Studies • Conclusion & Future Work DDDAS 2016
2
Research Motivation • Structural system failure identification and prediction play a critical role in ensuring safety and reliability • Most damages cannot be observed directly, rather causes the changes in structural properties • Accurately describing the growth of structural weakness leads better maintenance strategies and enhances the remaining useful life DDDAS 2016
3
Literature Review • System damage diagnosis has been broadly and systematically studied over years and has made the transition from research to practice • System damage prognosis is an on-going research topic • Existing methods can be categorized into two main groups: Physicbased approach and Data-drive approach Physic-based approach Differential equations Data-drive approach Stochastic process Hidden Markov Model (HMM) DDDAS 2016
4
Physic-based Approach Physic-based models mainly describe the damage evolving by differential equations • A defect propagation model by mechanistic modeling approach for RUL estimation of bearings. Li, et al. (1999) • Crack growth estimation based on Paris’ Law. Shankar, et al (2009) You and Mahadevan (2012) • Random disturbances and parameters are considered in the fatigue crack propagation model. Nicholas (2008), Krenk et al. (2008)
Inherent Drawbacks • • • •
Highly case specific Difficult to build up accurate models Uncertainties may be ignored Heavy computational load
DDDAS 2016
5
Data-driven Approach Data-drive models adopt generic statistical methods to describe the damage progression • Less dependent on physical principles • Broadly apply to various systems • Relatively low computation cost Stochastic Process Model Model the sensing signal as stochastic process for degradation path prediction. Firoozeh, et al. (2010), Ye, et al. (2013) Leave the prognosis only on data level State Space Model Unobservable degradation status are defined as hidden states using HMM. Rammohan, et al. (2005), Zaidi, et al. (2011) Assumptions in HMM often violates the realistic Unreliable for long term prognosis Inherent Drawbacks: Performs poorly on explaining the change of structural properties DDDAS 2016
6
Research Objective Flexibility
Engineering Insides
Computational Cost
Physic-based
Poor
Good
High
Data-driven
Good
Poor
Low
Propose an efficient dynamic data-driven method, which takes advantage of both the physical finite element model (FEM) and the data driven Bayesian framework, to tackle the structural damage growth prediction.
DDDAS 2016
7
Structural Damage Growth Prediction Based on DDDAS Frame Work *
DDDAS 2016
* F. Darema, “Dynamic Data Driven Applications Systems: A New Paradigm for Application Simulations and Measurements. Computational Science.” Int’l Conf. on Computational Science (ICCS), LNCS, 3038, 662–669, 2004.
8
Hierarchical Bayesian Degradation Model-Level 1
Finite Element Physical Model (First Principle)
Natural Frequencies Mean Structure
DDDAS 2016
Experiment Measurements Natural Frequencies
𝒇𝒇𝒕𝒕𝒕𝒕 ~𝓝𝓝(𝜼𝜼𝒊𝒊 (𝜽𝜽𝒕𝒕 ), 𝝈𝝈𝟐𝟐𝒇𝒇 ), 𝒊𝒊 = 𝟏𝟏, 𝟐𝟐 … , 𝑵𝑵𝒇𝒇 , 𝒕𝒕 = 𝟏𝟏, 𝟐𝟐, . . , 𝑻𝑻 𝜼𝜼𝒊𝒊 𝜽𝜽𝒕𝒕 = 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 𝜽𝜽𝒕𝒕 , 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 = 𝒇𝒇𝟎𝟎𝟎𝟎 9
Hierarchical Bayesian Degradation Model-Level 2
Finite Element Physical Model (First Principle)
Trend of Structural Prop. 𝜽𝜽𝒕𝒕 ′𝐬𝐬 Structure
Polynomial structure of time
DDDAS 2016
Underlying trend of 𝜽𝜽𝒕𝒕 as function of time 𝒕𝒕
𝜽𝜽𝒕𝒕 ~𝓝𝓝(𝜸𝜸𝒕𝒕 , 𝝈𝝈𝟐𝟐𝜽𝜽 ) 𝜸𝜸𝒕𝒕 = 𝜷𝜷𝑻𝑻 𝒙𝒙𝒕𝒕
𝒙𝒙𝒕𝒕 = 𝟏𝟏, 𝒕𝒕, 𝒕𝒕𝟐𝟐 ⋯
𝑻𝑻
10
Hierarchical Bayesian Degradation Model-Summary Posterior Distribution ∝ Likelihood × Prior Distribution
𝒇𝒇𝒕𝒕𝒕𝒕 ~𝓝𝓝(𝜼𝜼𝒊𝒊 (𝜽𝜽𝒕𝒕 ), 𝝈𝝈𝟐𝟐𝒇𝒇 ), 𝒊𝒊 = 𝟏𝟏, 𝟐𝟐 … , 𝑵𝑵𝒇𝒇 , 𝒕𝒕 = 𝟏𝟏, 𝟐𝟐, . . , 𝑻𝑻 • Mean Structure 𝜼𝜼𝒊𝒊 𝜽𝜽𝒕𝒕 = 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 𝜽𝜽𝒕𝒕 , 𝒖𝒖𝒊𝒊 + 𝒗𝒗𝒊𝒊 = 𝒇𝒇𝟎𝟎𝟎𝟎 • Trend of Structural Prop. 𝜽𝜽𝒕𝒕 ~𝓝𝓝(𝜸𝜸𝒕𝒕 , 𝝈𝝈𝟐𝟐𝜽𝜽 ) • 𝜽𝜽𝒕𝒕 ′𝐬𝐬 Structure 𝜸𝜸𝒕𝒕 = 𝜷𝜷𝑻𝑻 𝒙𝒙𝒕𝒕 • Natural Frequencies
Level1
Polynomial structure of time Prior Distribution of Coefficients 𝜷𝜷 Prior Distribution of 𝝈𝝈𝟐𝟐𝒇𝒇 Prior Distribution of 𝝈𝝈𝟐𝟐𝜽𝜽
DDDAS 2016
𝑻𝑻
𝒙𝒙𝒕𝒕 = 𝟏𝟏, 𝒕𝒕, 𝒕𝒕𝟐𝟐 ⋯ 𝜷𝜷~𝓜𝓜𝓜𝓜(𝒃𝒃, 𝝈𝝈𝟐𝟐𝜷𝜷 𝑰𝑰), 𝝈𝝈𝟐𝟐𝒇𝒇 ~𝓣𝓣𝓣𝓣 𝒂𝒂𝒇𝒇 , 𝒃𝒃𝒇𝒇 𝝈𝝈𝟐𝟐𝜽𝜽 ~𝓣𝓣𝓣𝓣 𝒂𝒂𝜽𝜽 , 𝒃𝒃𝜽𝜽
Prior Dist.
• • • •
Likelihood
Level2
11
Parameter Estimation- Gibbs Sampling 𝑝𝑝 𝜷𝜷 𝓕𝓕 = ⨌ 𝑝𝑝 𝚯𝚯, 𝜷𝜷, 𝜎𝜎𝑓𝑓2 , 𝜎𝜎𝜃𝜃2 , 𝑼𝑼 𝓕𝓕 𝑑𝑑𝚯𝚯𝑑𝑑𝜎𝜎𝑓𝑓2 𝑑𝑑𝜎𝜎𝜃𝜃2 𝑑𝑑𝑼𝑼
• Multiple integral hard to compute • Maximum Likelihood Estimation can be tedious Gibbs Sampling Algorithm Initialize 𝜃𝜃𝑡𝑡 0 , 𝜷𝜷
𝟎𝟎
, 𝜎𝜎𝑓𝑓2
0
For 𝒌𝒌 in iteration 𝟏𝟏: 𝑲𝑲
, 𝜎𝜎𝜃𝜃2
0
, 𝑢𝑢𝑖𝑖 𝟎𝟎
𝒌𝒌−𝟏𝟏 𝜃𝜃𝑡𝑡 𝑘𝑘 ~𝑝𝑝 𝜃𝜃𝑡𝑡 𝚯𝚯−t , 𝜷𝜷
𝜷𝜷
𝒌𝒌
𝜎𝜎𝑓𝑓2
~𝑝𝑝 𝜷𝜷 𝚯𝚯
𝑘𝑘
𝜎𝜎𝜃𝜃2
End DDDAS 2016
𝐤𝐤
𝒌𝒌−𝟏𝟏
, 𝜎𝜎𝜃𝜃2
, 𝜎𝜎𝜃𝜃2
𝑘𝑘−1
~𝑝𝑝 𝜎𝜎𝑓𝑓2 𝚯𝚯 𝐤𝐤 , 𝜎𝜎𝜃𝜃2
𝑘𝑘
~𝑝𝑝 𝜎𝜎𝜃𝜃2 𝚯𝚯
𝑢𝑢𝑖𝑖 𝒌𝒌 ~𝑝𝑝 𝑢𝑢𝑖𝑖 𝚯𝚯
𝐤𝐤
𝐤𝐤
, 𝜎𝜎𝜃𝜃2
𝑘𝑘
, 𝜎𝜎𝑓𝑓2
, 𝜷𝜷
, 𝜷𝜷 𝑘𝑘
𝑘𝑘−1
, 𝜎𝜎𝑓𝑓2
𝑘𝑘−1
, 𝜎𝜎𝑓𝑓2
𝑘𝑘−1
, 𝜎𝜎𝑓𝑓2 𝑘𝑘
𝑘𝑘−1
𝒌𝒌
𝒌𝒌
, 𝜷𝜷
, 𝑼𝑼
, 𝑼𝑼
, 𝑼𝑼 𝒌𝒌
, 𝑼𝑼
𝒌𝒌−𝟏𝟏 𝒌𝒌−𝟏𝟏
𝒌𝒌−𝟏𝟏
𝒌𝒌−𝟏𝟏
, 𝓕𝓕
, 𝓕𝓕
, 𝓕𝓕
, 𝓕𝓕
, 𝐔𝐔−𝑖𝑖𝒌𝒌−𝟏𝟏 , 𝓕𝓕 12
Illustrative Examples � Gibbs Sampling of 𝜷𝜷
Simulation Data
150 100 𝑓𝑓1 50 0 0 400 𝑓𝑓2 200 0 0
20
20
𝑓𝑓3500
0 0
20
𝜃𝜃𝑡𝑡 1.2
1
0.8 0.6
40
60
40
80
60
40
80
60
80
Time
𝛾𝛾𝑡𝑡 vs 𝛾𝛾�𝑡𝑡
2 5 1.5 2 0 1 �1 1 𝛽𝛽 0.5 -5 0 0 0.5 0.2 2 1 0.05 0.02 0.2 0.1 �2 0 𝛽𝛽 0 -0.05 -0.02 -0.5 -0.2 -0.1 -2 -1 0 -3 0.05 0.02 0.01 0.2 x 10-4 2 1 5 �3 0 𝛽𝛽 0 -0.05 -0.02 -0.01 -0.2 -2 -1 -5 0
0.6
6000
050 5 000 0.4 0.2 0.1 4 1 0.1 0.5
2000
4000
6000
2000 2000 2000
4000 4000 4000
6000 6000 6000
2000 2000 2000
4000 4000 4000
6000 6000 6000 Iteration
�2 𝜎𝜎 𝜃𝜃
0.05 0.05 0.5 0.2 0.1 2
2000
4000
6000 Iteration
000 000
1 � 𝛾𝛾𝑡𝑡 − 𝛾𝛾�𝑡𝑡 𝑇𝑇
0.5
𝑻𝑻
𝛾𝛾𝑡𝑡 = 𝜷𝜷 𝒙𝒙𝒕𝒕
𝑡𝑡
0.4 0.3 0.2
0.2
0.1 50
4000
�210 𝜎𝜎 50 10 10 𝑓𝑓20
�𝟐𝟐 Gibbs Sampling of 𝝈𝝈
Mean Error
0.4
0 0
2000
100 40 20 15 15 20
100
Time
0
20
40
60
80
Time
Bayesian Estimated function of 𝜽𝜽𝒕𝒕 True function of 𝜽𝜽𝒕𝒕
As data accumulated, the estimation of 𝜷𝜷 converges to the true value quickly. DDDAS 2016
13
Model Selection • What is the degree of polynomials ? • Model Selection Criteria: AIC, BIC, Bayesian Factor BF 𝑀𝑀𝑖𝑖 = 𝑝𝑝 𝑦𝑦 𝑀𝑀𝑖𝑖 /𝑝𝑝 𝑦𝑦 𝑀𝑀0 BIC 𝑀𝑀𝑖𝑖 = −2 log 𝑝𝑝 𝒚𝒚 𝑀𝑀𝑖𝑖 + 2𝑘𝑘 𝑀𝑀𝑖𝑖 × log 𝑛𝑛 AIC 𝑀𝑀𝑖𝑖 = −2 log 𝑝𝑝 𝒚𝒚 𝑀𝑀𝑖𝑖 + 2𝑘𝑘 𝑀𝑀𝑖𝑖
• Bayesian Factor - depends heavily on the choice of prior • BIC - advocated for descriptive purpose of the existing data • AIC - primary goal of the modelling application is predictive DDDAS 2016
14
Simulation Study
• A Fixed-Fixed uniform beam with 60 elements set up in FEM • 𝑬𝑬 = 𝟐𝟐. 𝟏𝟏 × 𝟏𝟏𝟏𝟏𝟖𝟖 𝒌𝒌𝒌𝒌𝒌𝒌, 𝑳𝑳 = 𝟐𝟐. 𝟓𝟓𝟓𝟓𝟓𝟓, 𝑰𝑰 = 𝟑𝟑. 𝟒𝟒𝟒𝟒 × 𝟏𝟏𝟏𝟏−𝟖𝟖 𝒎𝒎𝟒𝟒 , 𝑨𝑨 = 𝟔𝟔. 𝟒𝟒𝟒𝟒 × 𝟏𝟏𝟏𝟏−𝟒𝟒 𝒎𝒎𝟐𝟐 and 𝝆𝝆 = 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 𝐤𝐤𝐤𝐤 𝒔𝒔𝟐𝟐 /𝐦𝐦 • Stiffness loss happen in the 1st element • First three natural frequencies are measured • Two underlying trends are considered DDDAS 2016
15
Simulation Study (1)-Polynomial Trend 𝑓𝑓1135
1.1
𝜃𝜃
1
130
0.9
130
125 0.4 𝑓𝑓2380
0.8
0.5
0.6
0.7
0.8
0.9
1
360
0.7 0.6
340 0.4 720
𝑓𝑓3
0.5 0.4
𝑓𝑓1135
700
a)
0
20
40
𝑁𝑁𝑓𝑓 3 DDDAS 2016
60
80
𝑡𝑡
ℳ𝒩𝒩 𝒃𝒃, 𝜎𝜎𝛽𝛽2 𝑰𝑰
𝜎𝜎𝛽𝛽2
1012
𝒃𝒃
0.5
0.6
0.7
0.8
0.9
1
340 0 𝑓𝑓 720
3
700
0.5
0.6
0.7
0.8
𝒯𝒯𝒯𝒯 𝑎𝑎𝑓𝑓 , 𝑏𝑏𝑓𝑓
0
𝑎𝑎𝑓𝑓
𝑏𝑏𝑓𝑓
0
1
1
0
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
𝑡𝑡
360
b)
680 0.4
125 0 𝑓𝑓2380
0.9
𝜃𝜃
1
680 0
c)
𝒯𝒯𝒯𝒯 𝑎𝑎𝜃𝜃 , 𝑏𝑏𝜃𝜃 𝑎𝑎𝜃𝜃
𝑏𝑏𝜃𝜃
1
1
𝒩𝒩 𝜏𝜏𝑖𝑖 , 𝜎𝜎𝑢𝑢2 𝜏𝜏𝑖𝑖
𝜎𝜎𝑢𝑢2
0
1012
0 0
80
16
Simulation Study (1)-Polynomial Trend 𝛽𝛽1
𝛾𝛾 = 𝛽𝛽 𝑇𝑇 𝑥𝑥𝑡𝑡
𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 DDDAS 2016
𝛽𝛽2
𝛽𝛽3
𝜎𝜎𝜃𝜃2
𝜎𝜎𝑓𝑓2
17
Simulation Study (1)-Polynomial Trend
AIC Comparison for Different Models in Polynomial Trend
Degree of
𝜎𝜎𝑓𝑓2 = 0.012
Log-Likelihood
AIC
Log-Likelihood
AIC
1
-531.65
1067.30
-83.67
171.34
2
-525.42
1056.84
-78.18
162.36
3
-526.86
1061.72
-79.47
166.94
4
-527.23
1064.46
-80.05
170.10
Polynomials
DDDAS 2016
𝜎𝜎𝑓𝑓2 = 12
18
Simulation Study (1)-Polynomial Trend
Comparison of Fittings for Different Amount of Data 𝑇𝑇 = 10
𝜃𝜃𝑡𝑡1.2 1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
a) 20
40
60
80
𝑡𝑡
0 0
𝑇𝑇 = 80
𝜃𝜃𝑡𝑡1.2
1
0 0
DDDAS 2016
𝑇𝑇 = 30
𝜃𝜃𝑡𝑡1.2
b)
𝛾𝛾 𝑡𝑡 𝛾𝛾�𝑡𝑡 95% C. I 𝛾𝛾�𝑡𝑡
c)
20
40
60
80
𝑡𝑡
0 0
20
40
60
80
𝑡𝑡
19
Simulation Study (2)-Beta function Trend 𝑓𝑓1135
1.1
𝜃𝜃
1
130
0.9
130
125 0.4 𝑓𝑓2380
0.8 0.7
0.5
0.6
0.7
0.8
0.9
1
360
0.6
340 0.4 𝑓𝑓3720
0.5 0.4
𝑓𝑓1135
700
a)
0
20
40
𝑁𝑁𝑓𝑓 3 DDDAS 2016
60
80
𝑡𝑡
ℳ𝒩𝒩 𝒃𝒃, 𝜎𝜎𝛽𝛽2 𝑰𝑰
𝜎𝜎𝛽𝛽2
1012
𝒃𝒃
0.5
0.6
0.7
0.8
0.9
1
340 0 𝑓𝑓3720 700
0.5
0.6
0.7
0.8
𝒯𝒯𝒯𝒯 𝑎𝑎𝑓𝑓 , 𝑏𝑏𝑓𝑓
0
𝑎𝑎𝑓𝑓
𝑏𝑏𝑓𝑓
0
1
1
0
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
𝑡𝑡
360
b)
680 0.4
125 0 𝑓𝑓2380
0.9
1
𝜃𝜃
c)
680 0
𝒯𝒯𝒯𝒯 𝑎𝑎𝜃𝜃 , 𝑏𝑏𝜃𝜃 𝑎𝑎𝜃𝜃
𝑏𝑏𝜃𝜃
1
1
𝒩𝒩 𝜏𝜏𝑖𝑖 , 𝜎𝜎𝑢𝑢2 𝜏𝜏𝑖𝑖
𝜎𝜎𝑢𝑢2
0
1012
0 0
80
20
Simulation Study (2)-Beta function Trend 𝛽𝛽1 𝛽𝛽2
𝛾𝛾 = 𝛽𝛽 𝑇𝑇 𝑥𝑥𝑡𝑡
𝛽𝛽3 𝛽𝛽4
𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 DDDAS 2016
𝜎𝜎𝜃𝜃2
𝜎𝜎𝑓𝑓2
21
Simulation Study (2)-Beta function Trend
AIC Comparison for Different Models in Polynomial Trend
Degree of
𝜎𝜎𝑓𝑓2 = 0.012
Log-Likelihood
AIC
Log-Likelihood
AIC
1
-546.55
1097.1
-113.58
231.16
2
-538.17
1082.34
-93.46
192.92
3
-536.42
1080.84
-87.15
182.3
4
-535.91
1081.82
-86.24
182.48
Polynomials
DDDAS 2016
𝜎𝜎𝑓𝑓2 = 12
22
Simulation Study (2)-Beta function Trend
Comparison of Fittings for Different Amount of Data 𝑇𝑇 = 10
𝜃𝜃𝑡𝑡1.2
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
b)
a)
0 0
20
40
60
80
𝑡𝑡
𝑇𝑇 = 80
𝜃𝜃𝑡𝑡1.2
1
1
DDDAS 2016
𝑇𝑇 = 30
𝜃𝜃𝑡𝑡1.2
0 0
20
40
60
80
𝑡𝑡
0 0
𝛾𝛾 𝑡𝑡 𝛾𝛾�𝑡𝑡 95% C. I 𝛾𝛾�𝑡𝑡
c) 20
40
60
80
𝑡𝑡
23
Simulation Study-Fitting vs # Data
Beta function Trend
Polynomial Trend 𝑀𝑀𝐸𝐸𝑇𝑇1.4
𝑀𝑀𝐸𝐸𝑡𝑡2.5 b)
a)
1.2
2
1 1.5
0.8
0.6
0.4
1
Mean Error
Mean Error 0.5
0.2
0 10
DDDAS 2016
20
30
40
50
60
70
80
𝑇𝑇
0 10
20
30
40
50
60
70
80
𝑇𝑇
24
Experimental Study Fixed-Fixed Beam Experiment Material: Aluminum Young’s modulus: 68.9 Gpa Density: 2700 𝑘𝑘𝑘𝑘/𝑚𝑚3
Size: length 510 mm, width 19.05 mm, thickness, 4.76 mm Added mass: 2.7 g each , total 10 masses are added Added mass
DDDAS 2016
Accelerometer
25
Experimental Study – Beta-function Trend 𝐟𝐟𝟏𝟏
#mass 0 1
𝐟𝐟𝟐𝟐
92
𝐟𝐟𝟑𝟑
498
89.5
𝛉𝛉
1219
489
1200
0.46
0
87.5
481
1180
0.30
74
3
85
473
1161
0.22
75
4
83
466
1151
0.17
76
5
82
463
1145
0.14
77
6
80
457
1134
0.12
78
7
79
455
1123
0.11
79
8
76
448
1117
0.10
80
𝑓𝑓2490 𝑅𝑅 2 = 0.93
𝑓𝑓31200 𝑅𝑅 2 = 0.94
485
88
1190
480
86
1180
475 84
1170
470
82
465
80
460
1160 1150 1140
455
78 76 0.1
1
2
𝑓𝑓190 𝑅𝑅 2 = 0.88
DDDAS 2016
𝐭𝐭
0.2
0.3
𝑢𝑢𝑖𝑖 = 76.06 𝑣𝑣𝑖𝑖 = 33.12 0.4
𝜃𝜃
450 0.1
0.2
0.3
𝑢𝑢𝑖𝑖 = 445.3 𝑣𝑣𝑖𝑖 = 105.3 0.4
𝜃𝜃
1130 1120 0.1
0.2
0.3
𝑢𝑢𝑖𝑖 = 1107 𝑣𝑣𝑖𝑖 = 219.5 0.4
𝜃𝜃
26
Experimental Study – Beta-function Trend 𝛽𝛽1
0.4
𝛽𝛽2
0.3 0.2 0.1
𝛽𝛽3
0 -0.1 0 𝑢𝑢1
𝑢𝑢2 𝑢𝑢3 DDDAS 2016
20
40
60
80
𝜎𝜎𝜃𝜃2
𝜎𝜎𝑓𝑓2
27
Conclusion & Future Work Conclusion • Proposed a dynamic data-driven hierarchical Bayesian degradation model for weakness growth estimation. • Effectively recover the true polynomial trend and approximate the beta-function trend accurately. • Validate through simulation and experimental studies
Future Work • Adaptive sensor tuning for data collection • Change point detection for online signals • Decision making on system management
DDDAS 2016
28
ACKNOWLEDGEMENT The financial support of this work is provided by Air Force Office of Scientific Research under the program Dynamic Data Driven Application Systems (DDDAS)
DDDAS 2016
29
