﻿ 基于退化曲线相似性的剩余使用寿命估计方法
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 应用科技  2018, Vol. 45 Issue (5): 82-87, 90  DOI: 10.11991/yykj.201709007 0

### 引用本文

LI Jinsong, ZHANG Huijuan, YANG Zhong, et al. Remaining useful life estimation based on similarity of degenerate curves[J]. Applied Science and Technology, 2018, 45(5), 82-87, 90. DOI: 10.11991/yykj.201709007.

### 文章历史

1. 南京航空航天大学 自动化学院，江苏 南京 211106;
2. 航空机电综合航空科技重点实验室 电子工程部，江苏 南京 211106

Remaining useful life estimation based on similarity of degenerate curves
LI Jinsong1, ZHANG Huijuan2, YANG Zhong1, LI Xiaoming2, ZHANG Huibin1
1. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China;
2. Electronic Engineering Department, Aviation Key Laboratory of Science and Technology on Aero Electromechanical System Integration, Nanjing 211106, China
Abstract: In order to estimate the remaining useful life (RUL) of key equipment in the area of prognostic and health management (PHM), an RUL estimation method was presented based on the similarity match of degenerate curves. In the offline training phase, the linear regression method was used to fuse multi-dimensional sensor data into one-dimensional health index (HI), and then fit the HI degradation curve. And further, extract a large number of fixed-length HI curves by stepwise method, then reduce the number of curves by K-Means++ clustering algorithm. These curves were related with the degenerate mode of the equipment, implying the RUL information. In the online estimation phase, the curve was compared with the test unit, and the matching curves were used to estimate the RUL of test unit. The validity of the method was verified by testing the failure data set of aviation turbofan engine.
Keywords: prognostic and health management    remaining useful life    health indicator    degenerate curves    linear regression    polynomial fitting    K-Means++ cluster    Euclidean distance

1 基于退化曲线的RUL估计概述

2 曲线提取

2.1 HI轨迹转换

 ${\delta _{{\rm{HI}}}}({t}) = (l - t)/{R_{{\rm{th}}}},1 \leqslant t \leqslant l$ (1)

N个训练单元所有时间点传感器数据和实际健康度进行线性回归处理[2]，计算回归参数。各时间点健康度可由式（1）估计得出：

 ${\hat \delta _{{\rm{HI}}}} = \alpha + {{\beta}} {{{x}}^{\rm{T}}} = \alpha + \sum {_1^m} {\beta _i}{x_i}$ (2)

2.2 HI子轨迹提取

 ${ {\mathcal {{T}}^i}}= {[{{t}}_1^i,{t}_2^i, \cdots ,{t}_{{l_i} - {l_s} + {\rm{1}}}^i]^{\rm{T}}} = \left[ {\begin{array}{*{20}{c}} {t_1^i}& \ldots &{t_{{l_s}}^i} \\ \vdots & \vdots & \vdots \\ {t_{{l_i} - {l_s} + 1}^i}& \cdots &{t_{{l_i}}^i} \end{array}} \right]$

K-means聚类是一种基于划分的聚类方法，其目的是将n个点划分到K个聚类，使得每个点都属于离它最近的均值点对应的聚类，算法过程如下：

1）从数据集中随机选择K个点作为初始聚类中心；

2）计算剩余数据到各聚类中心之间的距离，并将其分配给最近的类；

3）计算上一步得到聚类中每一类的图心，作为新的聚类中心；

4）重复步骤2）~3）直到新的聚类中心等于原聚类中心。

1）从数据集中随机选择1个点作为聚类中心；

2）计算剩余数据到已有均值点之间的最小距离 $D(x)$

3）增选剩余数据中 $D(x)$ 最大的点为聚类中心；

4）重复步骤2）~3）直到选择出共K个聚类中心。

K-Means++算法提高了聚类效果的稳定性，本文使用此算法将HI子轨迹聚为 $p$ 类，选择每一类的中心点作为参考曲线 ${{S}}$ 的HI轨迹参数 ${{s}}$

2.3 曲线参数 $\tau ,\mu$ 确定

 $\left\{ \begin{gathered} d = \mathop {\min }\limits_{1 \leqslant i \leqslant l - {l_s} + 1} {\left\| {{{s}} - {{{t}}_i}} \right\|_2} \hfill \\ \rho = l - \arg \mathop {\min }\limits_{1 \leqslant i \leqslant l - {l_s} + 1} {\left\| {{{s}} - {{{t}}_i}} \right\|_2} \hfill \\ \end{gathered} \right.$ (3)

 ${{B}} = {[{{b}}_1 ,{{b}}_2 ,...,{{b}}_N ]^{\rm T}} = \left[ {\begin{array}{*{20}{c}} {d_1 } \\ \vdots \\ {d_N } \end{array}\begin{array}{*{20}{c}} {\rho _1 } \\ \vdots \\ {\rho _N } \end{array}} \right]$ (4)

 ${{B}} = \left[ {\begin{array}{*{20}{c}} {d_{1'} } \\ \vdots \\ {d_{N'} } \end{array}\begin{array}{*{20}{c}} {\rho _{1'} } \\ \vdots \\ {\rho _{N'} } \end{array}} \right]$

 $k = \arg \mathop {\min }\limits_{2 \leqslant j \leqslant N} \operatorname{var} [\rho _{1'} ,...,\rho _{j'} ]$

3 剩余寿命估计与预测性能评价

3.1 剩余寿命估计

 ${{E}} = \left[ {\begin{array}{*{20}{c}} {{e_{1,1}}}& \ldots &{{e_{1,{l_t} - {l_s} + 1}}} \\ \vdots & \vdots & \vdots \\ {{e_{q,1}}}& \cdots &{{e_{q,{l_t} - {l_s} + 1}}} \end{array}} \right]$

 ${{E'}} = \left[ {\begin{array}{*{20}{c}} {{e_1}} \\ \vdots \\ {{e_q}} \end{array}\begin{array}{*{20}{c}} {{c_1}} \\ \vdots \\ {{c_q}} \end{array}} \right]$

${e_i} < {\tau _i}$ ，说明参考曲线 ${{{S}}_i}$ 可用于估计测试单元RUL，存储 $[{c_i},{\mu _i}]$ 。假设最终存储了k类曲线的数据，以距离阈值为自变量为对参考曲线 ${{{S}}_i}$ 的预测结果进行加权处理，则测试单元预测失效时间 ${\hat t_e}$ 及预测剩余寿命 $\hat r$ 可以表示为

 $\left\{ \begin{gathered} {{\hat t}_e} = \sum\limits_{i = 1}^k {\frac{{{\tau _i}}}{{\sum\limits_{j = 1}^k {{\tau _j}} }}({c_i} + } {\mu _i}) \hfill \\ \hat r = {t_e} - {l_t} \hfill \\ \end{gathered} \right.$
3.2 预测性能评价

 $\begin{gathered} {\varepsilon _i} = {{\hat r}_i} - {r_i} \hfill \\ {\eta _i} = \left\{ \begin{gathered} \exp [({r_i} - {{\hat r}_i})/13] - 1,\;{\rm{ }}{{\hat r}_i} - {r_i} \leqslant 0 \hfill \\ \exp [({{\hat r}_i} - {r_i})/10] - 1,\;{\rm{ }}{{\hat r}_i} - {r_i} > 0 \hfill \\ \end{gathered} \right. \hfill \\ \bar \eta = \sum\limits_{i = 1}^n {{\eta _i}} /n \hfill \\ \end{gathered}$

4 案例分析

4.1 传感器数据选择

4.2 模型训练

5 结论

1）使用线性回归和多项式拟合方法将多维传感器参数融合为一维HI曲线，保留传感器退化信息的同时，简化了数据结构；

2）通过判别曲线与测试单元HI轨迹相关性，完成设备剩余寿命估计；

3）将K-means++聚类应用到上述方法中，极大提高了剩余寿命预测的稳定性。

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