Remaining Useful Life Prediction for Two-Phase Nonlinear Degrading Systems with Three-Source Variability
<p>The simulated degradation paths.</p> "> Figure 2
<p>The estimated changing time of 100 degradation paths.</p> "> Figure 3
<p>The online degradation path.</p> "> Figure 4
<p>The online updating process. (<bold>a</bold>) The parameter updating. (<bold>b</bold>) The underlying degradation state updating.</p> "> Figure 5
<p>The RUL prediction results. (<bold>a</bold>) PDFs of the RUL. (<bold>b</bold>) The mean RUL of the four methods.</p> "> Figure 6
<p>Performance evaluation of the RUL prediction. (<bold>a</bold>) MSE of the estimated RUL. (<bold>b</bold>) AE of the estimated RUL. (<bold>c</bold>) PDFs of the estimated RUL at time 40.</p> "> Figure 7
<p>Relative capacitance variability of the capacitors with time.</p> "> Figure 8
<p>Online parameter updating processes of Capacitor 1.</p> "> Figure 9
<p>PDFs of RUL prediction for Capacitor 1. (<bold>a</bold>) Our method. (<bold>b</bold>) Lin’s method. (<bold>c</bold>) Chai’ method. (<bold>d</bold>) Zheng’s method.</p> "> Figure 10
<p>Performance evaluation based on capacitance degradation data of Capacitor 1. (<bold>a</bold>) MSE of the predicted RUL. (<bold>b</bold>) AE of the predicted RUL. (<bold>c</bold>) RE of the predicted RUL.</p> "> Figure 11
<p>Comparison of the estimated RUL at different months. (<bold>a</bold>) PDFs of the estimated RUL at the fifth month. (<bold>b</bold>) PDFs of the estimated RUL at the sixth month. (<bold>c</bold>) PDFs of the estimated RUL at the seventh month.</p> ">
Abstract
:1. Introduction
- (1)
- A two-phase nonlinear Wiener process-based degradation model is formulated, where the drift coefficient and the measurement error of each phase are assumed to be random variables to describe the three-source variability.
- (2)
- Taking into account the nonlinearity, the uncertainty of the degradation state at the changing point as well as the three-source variability simultaneously, the approximate analytical expressions of RUL estimation are derived under the concept of the first passage time (FPT).
- (3)
- Based on the historical degradation observations of multiple units from the same batch, the offline parameter estimation is conducted by the MLE method. Subsequently, with the newly obtained degradation data of the certain operating device, the random drift coefficients and the underlying degradation states are real-time updated by combining the Bayesian rule, state-space model, and KF technique.
- (4)
- Finally, a numerical simulation and a practical case study about the degradation data of the high-voltage pulse capacitors are implemented to verify the effectiveness and applicability of the proposed approach.
2. Degradation Modeling Description
3. RUL Estimation with Three-Source Variability
3.1. RUL Estimation with Temporal Variability and Unit-to-Unit Variability
3.2. RUL Estimation Considering Three-Source Variability
4. Model Parameter and Degradation State Estimation
4.1. Offline Parameter Estimation
- if ,
4.2. Changing Point Detection
4.3. Online Implicit State Updating
5. Case Study
5.1. Numerical Example
5.2. Practical Application
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Abbreviations | |
PHM | Prognostics and health management |
RUL | Remaining useful life |
Probability density function | |
CDF | Cumulative distribution function |
CM | Condition monitoring |
KF | Kalman filtering |
EM | Expectation maximization |
ELM | Extreme learning machine |
FPT | First passage time |
MSE | Mean square error |
AE | Absolute error |
HI | health indicator |
Notation | |
Actual degradation state at time | |
Initial value | |
Time of the changing point | |
Degradation state at the changing time | |
Drift coefficient of the first phase | |
Drift coefficient of the second phase | |
Diffusion coefficients of the first phase | |
Diffusion coefficients of the second phase | |
Nonlinear function of the first phase | |
Nonlinear function of the second phase | |
Standard Brown motion | |
Measurement process | |
Measurement errors of each phase | |
Failure threshold | |
Lifetime | |
PDF of the lifetime | |
PDF of the RUL | |
CDF of the standard normal distribution | |
PDF of the standard normal distribution | |
Transition probability density function | |
Unknown model parameter vector | |
Number of the tested devices from the same batch | |
Historical data of devices from the same batch | |
Actual degradation state data of devices from the same batch | |
n-th device | |
n-th device | |
Measured increment vector | |
n-th device | |
n-th device | |
n-th device | |
Mean and standard deviation of | |
Appendix A. Proof of Theorem 1
Appendix B. Proof of Theorem 2
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Size | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1.3279 | 0.3370 | 0.0450 | 0.2963 | 1.4994 | 1.6560 | 0.2851 | 0.0972 | 0.3967 | 1.4027 | 50.5800 | 1.6302 | |
1.2559 | 0.2500 | 0.0518 | 0.2983 | 1.5002 | 1.4794 | 0.2557 | 0.0973 | 0.4000 | 1.4016 | 49.6180 | 2.1199 | |
1.2243 | 0.2311 | 0.0505 | 0.2995 | 1.5001 | 1.4865 | 0.2482 | 0.1041 | 0.4024 | 1.4017 | 49.7400 | 1.9923 | |
True value | 1.2 | 0.2 | 0.05 | 0.3 | 1.5 | 1.5 | 0.25 | 0.08 | 0.4 | 1.4 | 50 | 2 |
Metric | TMSE | MAE | CRA |
---|---|---|---|
Our method | 1.2030 × 104 | 0.8634 | 0.9653 |
Lin’s method [33] | 2.3221 × 104 | 3.7415 | 0.8795 |
Chai’s method [37] | 3.1273 × 104 | 3.9529 | 0.8976 |
Zheng’s method [25] | 6.9231 × 104 | 8.3726 | 0.8037 |
Variable | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
value | 0.0381 | 0.0087 | 0.1614 | 0.0497 | 2.0800 | 0.2232 | 0.0407 | 0.2274 | 0.0344 | 1.5217 | 6.2500 | 0.8292 |
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Cui, X.; Lu, J.; Han, Y. Remaining Useful Life Prediction for Two-Phase Nonlinear Degrading Systems with Three-Source Variability. Sensors 2024, 24, 165. https://doi.org/10.3390/s24010165
Cui X, Lu J, Han Y. Remaining Useful Life Prediction for Two-Phase Nonlinear Degrading Systems with Three-Source Variability. Sensors. 2024; 24(1):165. https://doi.org/10.3390/s24010165
Chicago/Turabian StyleCui, Xuemiao, Jiping Lu, and Yafeng Han. 2024. "Remaining Useful Life Prediction for Two-Phase Nonlinear Degrading Systems with Three-Source Variability" Sensors 24, no. 1: 165. https://doi.org/10.3390/s24010165
APA StyleCui, X., Lu, J., & Han, Y. (2024). Remaining Useful Life Prediction for Two-Phase Nonlinear Degrading Systems with Three-Source Variability. Sensors, 24(1), 165. https://doi.org/10.3390/s24010165