SCIENCE CHINA Information Sciences, Volume 64 , Issue 1 : 112205(2021) https://doi.org/10.1007/s11432-019-2724-5

## Online remaining-useful-life estimation with a Bayesian-updated expectation-conditional-maximization algorithm and a modified Bayesian-model-averaging method

• AcceptedNov 1, 2019
• PublishedDec 18, 2020
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### Abstract

Online remaining-useful-life (RUL) estimation is an effective method with respect to ensuring the safety of complex-huge systems. Generally, current methods assume a specific degradation model when degradation values are observed in the initial degradation phase. However, this assumption may not always be robust enough owing to the often-ambiguous inherent incipient-degradation characteristic. Therefore, besides model-parameter uncertainty, the uncertainty of the degradation model is worth examining in online RUL estimations. In this paper, a Bayesian-updated expectation-conditional-maximization (ECM) algorithm is adopted to address the uncertainty of prior parameters, and a modified Bayesian-model-averaging method is used to deal with the uncertainty of the degradation model. Then, simulation studies are conducted to analyze the effectiveness of the proposed fusion algorithm. Results suggest that the Bayesian-updated ECM algorithm and modified Bayesian-model-averaging method effectively address the associated uncertainties of model parameters and the degradation model itself. Finally, we apply the proposed fusion algorithm to predict the RUL of a gyroscope.

### Acknowledgment

This work was supported by National Key RD Program of China (Grant No. 2018YFB1306100) and National Natural Science Foundation of China (Grant Nos. 61922089, 61773386, 61833016, 61903376, 61673311).

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• Figure 1

The algorithm of RUL calculation for model ${M_c}$.

• Figure 2

(Color online) Results of Bayesian-updated ECM algorithm for model ${M_x}$. (a) Simulated trajectory with IID errors; (b) predicted RUL of model ${M_x}$; (c) estimated ${\mu~&apos;_{x0}}$ of model ${M_x}$; (d) estimated ${\mu~&apos;_{x1}}$ of model ${M_x}$.

• Figure 3

(Color online) Results of Bayesian-updated ECM algorithm for model ${M_y}$. (a) Simulated trajectory with Brownian errors; (b) predicted RUL of model ${M_y}$; (c) estimated ${\mu~&apos;_{y0}}$ of model ${M_y}$; (d) estimated ${\mu~&apos;_{y1}}$ of model ${M_y}$.

• Figure 4

(Color online) Effectiveness of modified Bayesian-model-averaging method for trajectory with IID errors. protectłinebreak (a) Modified Bayesian model averaging for trajectory with IID errors; (b) original Bayesian model averaging for trajectory with IID errors; (c) the obtained model probability of the modified Bayesian model averaging for model ${M_x}$; (d) the obtained model probability of the original Bayesian model averaging for model ${M_x}$.

• Figure 5

(Color online) Effectiveness of modified Bayesian model averaging for trajectory with Brownian errors. protectłinebreak (a) Modified Bayesian model averaging for trajectory with Brownian errors; (b) original Bayesian model averaging for trajectory Brownian errors; (c) obtained model probability of modified Bayesian model averaging for model ${M_y}$; (d) obtained model probability of original Bayesian model averaging for model ${M_y}$.

• Figure 6

Illustration of a deformed motor bearing.

• Figure 7

(Color online) Effectiveness of the proposed algorithm. (a) The observed drift data of the gyroscope; (b) estimated RUL of different prior parameters; (c) estimated RUL of different prior probabilities; (d) estimated RUL of our method.

• Table 1

Table 1Distributions of the selected simulation parameters

 Simulation parameter Simulation distribution ${\theta'_x}$ $N(0.02,1~\times~{10^{~-~6}})$ Exponential degradation with IID errors ${\beta'_x}$ $N(0.01,1~\times~{10^{~-~6}})$ $\sigma_x^2$ 0.1 ${\theta'_y}$ $N(0.02,1~\times~{10^{~-~6}})$ Exponential degradation with Brownian errors ${\beta'_y}$ $N(0.01,1~\times~{10^{~-~6}})$ $\sigma_y^2$ 0.0004
• Table 2

Table 2Prior-distribution assumption for Bayesian-updated ECM algorithm

 Model Parameter Bayesian method [28] Our method Distribution 1 Distribution 2 Distribution 3 ${\theta~'_x}$ $N(0.04,1~\times~{10^{~-~6}})$ $N(0.04,1~\times~{10^{~-~6}})$ $N(0.1,1~\times~{10^{~-~6}})$ $N(0.2,1~\times~{10^{~-~6}})$ ${M_x}$ ${\beta~'_x}$ $N(0.02,1~\times~{10^{~-~6}})$ $N(0.02,1~\times~{10^{~-~6}})$ $N(0.05,1~\times~{10^{~-~6}})$ $N(0.1,1~\times~{10^{~-~6}})$ $\sigma~_x^2$ 0.01 0.01 0.01 0.01 ${\theta~'_y}$ $N(0.04,1~\times~{10^{~-~6}})$ $N(0.04,1~\times~{10^{~-~6}})$ $N(0.1,1~\times~{10^{~-~6}})$ $N(0.2,1~\times~{10^{~-~6}})$ ${M_y}$ ${\beta~'_y}$ $N(0.02,1~\times~{10^{~-~6}})$ $N(0.02,1~\times~{10^{~-~6}})$ $N(0.05,1~\times~{10^{~-~6}})$ $N(0.1,1~\times~{10^{~-~6}})$ $\sigma~_y^2$ 0.004 0.004 0.004 0.004
• Table 3

Table 3Prior-distribution assumption for observed drift data

 Model中国 Parameter Distribution 1 Distribution 2 Distribution 3 ${\theta~'_x}$ $N(0.0025,1~\times~{10^{~-~6}})$ $N(0.005,1~\times~{10^{~-~6}})$ $N(0.0075,1~\times~{10^{~-~6}})$ ${M_x}$ ${\beta~'_x}$ $N(0.001,1~\times~{10^{~-~6}})$ $N(0.002,1~\times~{10^{~-~6}})$ $N(0.003,1~\times~{10^{~-~6}})$ $\sigma~_x^2$ 0.01 0.01 0.01 ${\theta~'_y}$ $N(0.0025,1~\times~{10^{~-~6}})$ $N(0.005,1~\times~{10^{~-~6}})$ $N(0.0075,1~\times~{10^{~-~6}})$ ${M_y}$ ${\beta~'_y}$ $N(0.001,1~\times~{10^{~-~6}})$ $N(0.002,1~\times~{10^{~-~6}})$ $N(0.003,1~\times~{10^{~-~6}})$ $\sigma~_y^2$ 0.00004 0.00004 0.00004

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