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 ${\mu~'_{x0}}$ of model ${M_x}$; (d) estimated ${\mu~'_{x1}}$ 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 ${\mu~'_{y0}}$ of model ${M_y}$; (d) estimated ${\mu~'_{y1}}$ 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.
| 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 |
| 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 | |
| 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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