Remaining Useful Life Prediction for Mixed Stochastic Deteriorating Equipment Based on Fractional Brownian Motion Process
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摘要: 在实际工程中, 设备往往是由多个不同类型元件或部件构成的集合体, 其总体性能退化程度是由内部多种随机退化过程综合影响下的结果. 不同于现有文献主要采用无记忆效应的单一线性或非线性形式随机过程模型来描述设备的真实退化, 首先建立一种基于分数布朗运动(Fractional Brownian motion, FBM)的混合随机退化模型, 用以刻画退化过程中的记忆效应与长期依赖性; 进一步, 在退化模型里同时引入双随机效应, 用以描述不同设备之间的退化差异性, 并基于弱收敛性理论推导得到首达时间(First hitting time, FHT)意义下剩余寿命(Remaining useful life, RUL)概率密度函数(Probability density function, PDF)的近似解析表达形式; 然后, 给出一种共性参数离线估计和随机参数实时更新的策略, 进而实现了剩余寿命的实时预测; 最后, 通过数值仿真例子和陀螺仪的实际退化数据, 验证了该方法的有效性和具有潜在的工程应用价值.Abstract: As a result of the interactive influence of a variety of internal random degradation processes, the overall performance of industrial equipment composed of multiple types of components usually deteriorates with its usage. Unlike most of the existing methods which describe the actual degradation of equipment via a single linear or nonlinear stochastic process model without memory effect, a new mixed stochastic degradation model based on Fractional Brownian motion (FBM) is proposed in this paper. Firstly, FBM is adopted to reflect the memory effect and long-term dependence of the degradation process. Then, double random effects are integrated into the degradation model to depict the variability between different units. Under the concept of the first hitting time (FHT), an approximate analytical expression of the probability density function (PDF) of the remaining useful life (RUL) is derived based on the weak convergence theory. Besides, a strategy of offline estimation of universal parameters and online update of random parameters is given to further realize real-time RUL prediction. Finally, numerical simulation examples and a case study of the degradation data of the gyroscope are provided to verify the effectiveness and potential engineering application value of the proposed method.
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图 2 待预测设备实时监测退化数据
Fig. 2 Real-time monitoring degradation data of the equipment for prediction
图 4 4种方法在各个时间点处RUL预测的对比
Fig. 4 Comparison of RUL prediction by four methods at each time
图 5 四种方法在第5、8、11、13个时间点RUL预测结果的对比
Fig. 5 Comparison of RUL prediction by four methods at 5, 8, 11, 13th time
图 7 陀螺仪退化模型随机参数的实时更新过程
Fig. 7 Real-time updating process of random parameters of gyroscope degradation model
图 8 在不同时间点处四种方法预测RUL的PDF对比
Fig. 8 Comparison of RUL＇s PDFs by four prediction methods at each time
表 1 四种模型参数的先验估计值
Table 1 The parameters＇ prior estimates ofthe four models
参数 本文方法 模型1 模型2 模型3 ${\mu _\lambda }$ 1.0754 2.0251 — — ${\mu _\alpha }$ 0.2547 — 3.1547 3.3643 $\sigma _\lambda ^2$ $4.56\times10^{-4}$ 0.1618 — — $\sigma _\alpha ^2$ $3.47\times10^{-4}$ — 0.0176 0.0103 $\rho$ −0.4793 — — — ${\sigma ^2}$ 0.53362 0.51362 0.58362 0.54112 $\beta$ 0.1547 — 0.6035 0.6101 $H$ 0.8472 — — 0.8130 $\ln L\left( \Theta \right)$ 57.462 46.328 48.633 51.278 ${\rm{AIC}}$ −98.924 −86.656 −89.266 −92.556 ${\rm{BIC}}$ −87.724 −82.456 −83.666 −85.556 
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表 2 陀螺仪退化模型参数的先验估计值
Table 2 A parameters＇ prior estimate of the gyroscope degradation model
参数 本文方法 模型1 模型2 模型3 ${\mu _\lambda }$ 0.0151 0.1348 — — ${\mu _\alpha }$ 0.0373 — 0.1503 0.1671 $\sigma _\lambda ^2$ $2.34\times 10^{-6}$ 0.0028 — — $\sigma _\alpha ^2$ $7.82 \times 10^{-6}$ — $5.76\times 10^{-4}$ $4.76\times 10^{-4}$ $\rho$ $-0.4503$ — — — ${\sigma ^2}$ 0.00192 0.00582 0.00422 0.00272 $\beta$ 0.2114 — 0.1342 0.1742 $H$ 0.8011 — — 0.7903 $\ln L\left( \Theta \right)$ 41.871 31.013 34.337 38.699 ${\rm{AIC}}$ −67.742 −56.026 −60.674 −67.398 ${\rm{BIC}}$ −70.942 −57.226 −62.274 −69.398
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