﻿ 舰载导弹测发控设备剩余寿命预测概率方法
 舰船科学技术  2020, Vol. 42 Issue (11): 161-164    DOI: 10.3404/j.issn.1672-7649.2020.11.033 PDF

Probability method of residual useful life prediction on test launch and control system based on parameter monitoring results
SU Xiao-dan
Military Representative of Navy Equipment in Beijing, Beijing 100854, China
Abstract: It presents the problem concerning probability prediction on the residual useful life of the test and launch control equipment for missile which has parameter fault. In the case of any finite number of deterministic parameters, given all deterministic parameters of the system have the same failure intensity and recovery strength, the method of prediction on the residual useful life of test and launch control equipment is derived based on parameter monitoring results, obtaining a solution to this problem. The practical implementation example of the obtained solution is given.
Key words: missile     test launch and control     system state probability     parameter fault     residual useful life     probability prediction
0 引　言

1 问题的提出与主要假设

1.1 问题的求解

 $\begin{split}{C_0}\left( {t + \Delta t} \right) =& C\left( {{A_0}} \right) + C\left( {{B_1}} \right) =\\ &\left[ {{C_0}\left( t \right) + \Delta {C_s}\Delta t} \right]\left( {1 - \lambda \Delta t} \right) + {C_1}\left( t \right)\mu \Delta t\text{。}\end{split}$ (1)

 $\begin{split}{C_0}\left( {t + \Delta t} \right) - {C_0}\left( t \right) =& - \lambda \Delta t{C_0}\left( t \right) + \mu \Delta t{C_1}\left( t \right) + \\ &\Delta {C_s}\Delta t - \lambda \Delta t\Delta {C_s}\text{。}\end{split}$ (2)

 ${\rm d}{C_0}\left( t \right)/{\rm d}t = - \lambda {C_0}\left( t \right) + \mu {C_1}\left( t \right) + \Delta {C_s}{\text{。}}$

 $\begin{split} {C_k}\left( {t + \Delta t} \right) =& C\left( {{A_k}} \right) + C\left( {{B_{k + 1}}} \right) + C\left( {{C_{k - 1}}} \right) = \\ &\left[ {{C_k}\left( t \right) + \Delta {C_s}\Delta t} \right]\left[ {1 - \left( {\lambda + \mu } \right)\Delta t} \right] +\\ &{C_{k + 1}}\left( t \right)\mu \Delta t +{C_{k - 1}}\left( t \right)\lambda \Delta t + \lambda \Delta {C_h}\Delta t \text{。}\end{split}$ (3)

 ${\rm d}{C_k}/{\rm d}t = \lambda {C_{k - 1}}\left( t \right) - \left( {\lambda + \mu } \right){C_k}\left( t \right) + \mu {C_{k + 1}}\left( t \right) + \Delta {C_s} + \lambda \Delta {C_h}{\text{。}}$

 $\begin{split} {C_m}\left( {t + \Delta t} \right) =& C\left( {{A_m}} \right) + C\left( {{C_{m - 1}}} \right) = \\ &\left[ {{C_m}\left( t \right) + \Delta {C_s}\Delta t} \right]\left[ {1 - \mu \Delta t} \right] +\\ & \left[ {{C_{m - 1}}\left( t \right) + \Delta {C_h}} \right]\lambda \Delta t\text{。} \end{split}$ (4)

 ${\rm d}{C_m}/{\rm d}t = \lambda {C_{m - 1}}\left( t \right) - \mu {C_m}\left( t \right) + \Delta {C_s} + \lambda \Delta {C_h}{\text{。}}$

 $\begin{split} &{\rm d}{C_0}\left( t \right)/{\rm d}t = - \lambda {C_0}\left( t \right) + \mu {C_1}\left( t \right) + \Delta {C_s}\text{，}\\ & {\rm d}{C_k}/{\rm d}t =\lambda {C_{k - 1}}\left( t \right) - \left( {\lambda + \mu } \right){C_k}\left( t \right) + \mu {C_{k + 1}}\left( t \right) + \Delta {C_s} + \lambda \Delta {C_h}\text{，}\\ &{\rm d}{C_m}/{\rm d}t = \lambda {C_{m - 1}}\left( t \right) - \mu {C_m}\left( t \right) + \Delta {C_s} + \lambda \Delta {C_h}\text{。}\\[-10pt] \end{split}$ (5)

 ${{A}}\left( s \right) \cdot X\left( s \right) = {B^{\left( c \right)}}\left( s \right){\text{。}}$ (6)

 ${A_{00}}\left( s \right) \!=\! - \left( {s \!+\! \mu } \right);{A_{kk}}\left( s \right) \!=\! - \left( {s \!+\! \lambda \!+\! \mu } \right),k \!=\! 1,2, \cdots ,m \!-\! 1;$
 ${A_{k,k - 1}}\left( s \right) \!=\! \lambda ;k \!=\! 1,2, \cdots ,m;{A_{k,k + 1}}\left( s \right) \!=\! \mu ,k \!=\! 1,2, \cdots ,m \!-\! 1\text{。}$

 $\sum\limits_{k = 1}^{m + 1} {{A_k}{X_k} = {B^{\left( c \right)}}}{\text{。}}$ (7)

 $\begin{split}&{{ A}_1} = \left[ {\begin{array}{*{20}{c}} { - \left( {s + \lambda } \right)} \\ \lambda \\ {{O_{m - 1}}} \end{array}} \right]\text{，}{{ A}_k} = \left[ {\begin{array}{*{20}{c}} {{O_{k - 2}}} \\ \mu \\ { - \left( {s + \lambda + \mu } \right)} \\ \lambda \\ {{O_{m - k}}} \end{array}} \right]\text{，}\\ &{{ A}_2} = \left[ {\begin{array}{*{20}{c}} \mu \\ { - \left( {s + \lambda + \mu } \right)} \\ \lambda \\ {{O_{m - 2}}} \end{array}} \right]\text{，}{{ A}_{m + 1}} = \left[ {\begin{array}{*{20}{c}} {{O_{m - 1}}} \\ \mu \\ { - \left( {s + \lambda } \right)} \end{array}} \right]\text{。}\end{split}$

 $X_k^ * = C_{k - 1}^ * = {M_k}{A_k}\prod\limits_{j = 1\atop j \ne k}^{m + 1} {{R_j}} {B^{\left( c \right)}}{\text{，}}$ (8)

 $\bar C\left( t \right) = \sum\limits_{k = 1}^m {{C_k}\left( t \right){P_k}\left( t \right)}{\text{。}}$ (9)

 ${P_k}\left( t \right) = \sum\limits_{i = 1}^n {\frac{{{F_k}\left( {{s_i}} \right)}}{{{{G'}_k}\left( {{s_i}} \right)}}\exp \left( {{s_i}t} \right)} \text{。}$ (10)

1.2 算例

 ${A_1}{X_1} + {A_2}{X_2} = {B^{\left( c \right)}}{\text{。}}$

 $\begin{split}&C_0^ * \left( s \right) = \frac{{\left( {s + 2\mu } \right)\Delta {C_s} + \lambda \mu {C_h}}}{{{s^2}\left( {s + \lambda + \mu } \right)}}\text{，}\\ &C_1^ * \left( s \right) = \frac{{\left( {s + 2\lambda } \right)\Delta {C_s} + \lambda \left( {s + \lambda } \right){C_h}}}{{{s^2}\left( {s + \lambda + \mu } \right)}}\text{。}\end{split}$

 $\begin{split} \bar C\left( t \right) =& \frac{{2\Delta {C_s} + \lambda {C_h}}}{{{{\left( {\lambda + \mu } \right)}^2}}}\left[ {{\lambda ^2} + {\mu ^2} - \lambda \left( {\lambda - \mu } \right){e^{ - \left( {\lambda + \mu } \right)t}}} \right]t - \\ & \frac{{\left[ {\left( {\lambda - \mu } \right)\Delta {C_s} - \lambda \mu {C_h}} \right]}}{{{{\left( {\lambda + \mu } \right)}^3}}}\left[ {\lambda - \mu - 2\lambda \left( {\lambda - \mu } \right){e^{ - \left( {\lambda + \mu } \right)t}}} \right]\times\\ &\left( {1 - {e^{ - \left( {\lambda + \mu } \right)t}}} \right){\text{。}}\\[-10pt] \end{split}$ (11)

2 结　语

 [1] 胡昌华, 马清亮, 郑建飞编著. 导弹测试与发射控制技术[M]. 北京: 国防工业出版社, 2015年9月. [2] 司小胜, 胡昌华. 数据驱动的设备剩余寿命预测理论及应用[M]. 北京: 国防工业出版社, 2016. [3] 胡昌华, 樊红东, 王兆强. 设备剩余寿命预测与最优维修决策[M]. 北京: 国防工业出版社 2019 年1月. [4] 蔡忠义 陈云翔. 考虑测量误差和随机效应的设备剩余寿命预测[J]. 系统工程与电子技术, 2019, 41(7): 1410-1416. [5] 司小胜, 胡昌华, 张琪, 等. 不确定退化测量数据下的设备剩余寿命估计[J]. 电子学报, 2015, 43(1): 30-35. DOI:10.3969/j.issn.0372-2112.2015.01.006 [6] 郑建飞, 胡昌华, 司小胜, 等. 考虑不确定测量和个体差异的非线性随机退化系统剩余寿命估计[J]. 自动化学报, 2017, 43(2): 259-270. [7] 李建民. 基于灰色理论的舰船装备剩余寿命预测模型[J]. 舰船电子工程, 2009, 29(3): 99-102. DOI:10.3969/j.issn.1627-9730.2009.03.029 [8] В.В. Пицык Вероятностное прогнозирование остаточного ресурса измерительной техники по результатам параметрического контроля. Измерительная техника. 2016No3, 12−15 Probability method of residual useful life prediction on test and launch control equipment based on parameter monitoring results [J]. Measuring technique, 2016, (3): 12−15.