CN112765787B - Degradation modeling and service life prediction method considering performance index clustering in dynamic environment - Google Patents
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Abstract
The invention relates to a degradation modeling and service life prediction method considering performance index clustering in a dynamic environment, and belongs to the technical field of degradation modeling and service life prediction. Firstly, collecting test data, then establishing a performance index degradation model, estimating unknown parameters in the model, and finally predicting the service life and reliability. The method comprises the following specific steps: the method comprises the following steps: collecting test data; step two: establishing a degradation model; step three: estimating parameters, and updating the model in real time; step four: and predicting the service life and the reliability. The phenomenon that each index of a multi-performance index product shows clustering due to the influence of common potential factors is subjected to mathematical modeling, so that the degradation modeling of the product with the phenomenon is more practical, and the correlation among the indexes is better explained.
Description
Technical Field
The invention relates to a degradation modeling and service life prediction method considering performance index clustering in a dynamic environment, and belongs to the technical field of degradation modeling and service life prediction.
Background
With the development of science and technology, many products are designed to have higher reliability and longer service life, so that it is difficult to perform service life test and acquire product failure data, and thus the traditional reliability modeling and service life prediction method based on the failure data becomes difficult to adapt to practical requirements. The improvement of the sensor technology level and the rise of the fault prediction and health management technology enable the acquisition of a large amount of data related to the performance degradation of the product, and the product degradation modeling and life prediction technology based on the degradation data is also widely concerned and researched.
At present, most degradation modeling methods only aim at the condition that a product has one performance index, and actually, some products have two or more key performance indexes, and considering that correlation possibly exists among the performance indexes, the existing degradation modeling methods with multiple performance indexes mainly have the following three types: 1) describing joint distribution 2 of the degradation amount or the degradation increment of each index by using a common multidimensional distribution function), connecting edge distribution functions of the degradation amount or the degradation increment of each index by using a Copula function to obtain a joint distribution function 3), and trying to establish a model capable of directly describing the relation among each index.
Before introducing the invention, a summary is firstly made on the current research situation of multi-performance index product degradation modeling at home and abroad:
a) common multidimensional distribution function method
The method aims to use the commonly used multidimensional distribution as the joint distribution of the degradation amount or degradation increment of each performance index of the product. The multidimensional distribution used most is a multivariate normal distribution function because its joint distribution form is easily obtained, so that the reliability expression is also easily obtained. Wang et al [ Wang P, code DW. Reliability prediction based on degradation modeling for systems with multiple degradation measures [ C ]. Annual Symposium degradation and Maintainability, 2004-RAMS, Los Angeles, CA, USA,2004 ] studied the degradation modeling problem of a system composed of multiple components, described the joint distribution of the degradation amounts of each index at different time points with time-varying multivariate normal distribution, and further evaluated the Reliability. Their research results show that incorrect independence assumptions underestimate the reliability of the system when there is a correlation between multiple performance indicators. When modeling and analyzing Accelerated Degradation Test Data, Pan et al [ Pan J, Wang XY, Chen WH, Xu SW, Qian P, Liu HJ. Statistical Analysis on additive Degradation Test Data Based on Multiple Performance Parameters [ J ]. Advanced Materials Research 2012,430 and 432 ] describe the joint distribution of the Degradation amount of each Performance index of the product at different observation time points by adopting multivariate normal distribution. Jin et al [ Jin G, materials D.measurement planning for planning test design based on the bivariate Wiener process [ C ]. Quality and Reliability Engineering International 2014; 30: 1215- > 1231 ] use a binary wiener process to model the degradation of a product with two performance indicators, assuming that the independent increments of the two performance indicators obey a binary normal distribution. Pan et al [ Pan Z, Balakrishnan N.reliability modeling of degradation of products with multiple performance characteristics based on gamma processes [ J ]. Reliability Engineering & System Safety 2011; 96(8) 949-. The following limitations exist in describing the degradation process of the performance index of the product by using the multidimensional distribution: (1) each performance index is required to meet the same degradation rule, namely, the individual degradation models of each index are consistent; (2) the degradation amount or degradation increment of each index is required to follow the same distribution. In practice, however, in many cases, a plurality of performance indexes of a product do not necessarily follow the same degradation rule, so that the degradation amount or the degradation increment of the product cannot be described by the same distribution, and at this time, it is difficult to describe the degradation process of all the performance indexes by using this method, and a Copula function can be used to perform degradation modeling.
b) Copula function method
The Copula function can connect marginal distributions of degradation amounts or degradation increments to obtain a joint distribution function, and it has no limitation on the marginal distribution of a single variable. At present, the Copula function is widely applied to the financial field and the biological statistics field, and based on the characteristic that the Copula function is strong to process correlation among random variables, the Copula function is introduced into the application degradation modeling field in recent years to better solve the multi-index joint modeling problem with complex correlation.
Sari et al [ Sari JK, Newby MJ, Brombacher AC, Tang LC. Bivariate constant stress degradation model LED lighting system Reliability evaluation with two-stage modeling [ C ] Quality and Reliability Engineering International 2009; 25(8) 1067-1084,2009 ] the binary degradation modeling problem was processed using a generalized linear regression model and a binary Copula function. Pan et al [ Pan Z, Balakrishnan N, Sun Q, Zhou J. binary degradation analysis of products based on Wiener processes and copulas [ J ]. Journal of Statistical Computation and Simulation 2013; 83(7) 1316-; 85(2) 405-; 65(2) 624-. Li and Xue [ Li X, Xue P. multivariable storage degradation modeling based on unpopular function [ J ]. Advances in Mechanical Engineering 2014 ] assumes that the degradation process of each index is a wiener process with time scale change, and a multidimensional Frank Copula function is adopted to construct the joint distribution of n individual performance index degradation quantities. Xu et al [ Xu D, Xing M, Wei Q, Qin Y, Xu J, Chen Y, Kang R.failure after viewer modification and reliability evaluation of product based on video-copula and accessed degradation data [ J ]. Mechanical Systems and Signal Processing 2018; the degradation modeling and service life prediction of the intelligent electric energy meter are researched, the basic error of the intelligent electric energy meter is determined by four related performance indexes, the degradation process of each performance index is described by Brownian motion with drift, and the edge reliability of each index is connected into the overall reliability by using a vine Copula method. Jiang et al [ Jiang C, Zhang W, Han X, et al. A Vine-Copula-Based Reliability Analysis Method for Structures with multimedia correction [ J ]. Journal of Mechanical Design 2015; 137(6) 386-405, a structural reliability analysis method is provided, a vine Copula function is introduced to solve the reliability evaluation problem of a multivariable system with correlation, the probability density function of multidimensional variables is connected to obtain the joint distribution probability density, and the joint density integration is carried out to obtain the structural reliability of the system.
Although the Copula function can handle complex correlation more flexibly, there are many disadvantages: (1) how to select a suitable Copula function is still a problem; (2) most Copula functions are complex in form and difficult to operate practically.
(3) Other methods
In addition to focusing on describing the correlation of the degradation process of each index and connecting their marginal distributions to obtain a joint distribution, there have been some studies attempting to build a model that can directly describe the relationship between performance indexes. For example, Xu et al [ Xu a, Shen L, Wang B, Tang y.on modifying bivariate wiener degradation process.ieee transformations on Reliability 2018; 67(3) 897-906 ] a binary degradation model is established based on the wiener process, in which the drift terms of the two indexes share a random variable, thereby reflecting the dependency relationship of the two indexes. Yousefi et al [ Yousefi N, Coit DW, Song s.reliability analysis of systems conditioning clusters of dependent adaptation components. reliability Engineering and System Safety 2020 ] states that in a complex System, when components are present in a shared environment, random factors such as temperature, wind speed, polluted environment, etc. affect the degraded paths of all components simultaneously, and thus they exhibit a "clustering" behavior and are therefore dependent on each other. This model, which takes into account the "clustering" effect, can provide richer information for maintenance strategies.
Disclosure of Invention
(1) The purpose of the invention is as follows:
most of the existing multi-performance index degradation modeling researches are fuzzy in source explanation of correlation among indexes, and a multi-performance index degradation model is inevitably deviated from a real situation only from the data driving angle, so that a service life prediction result is inaccurate. For some multi-performance index products, the correlation between the performance indexes of the products is that the degradation rates of the products have correlation, and the correlation between the degradation rates is caused by the influence of common potential factors (the common potential factors can be the same component or the same product intrinsic parameters) and the influence of a common dynamic environment. If these common potential factors are called "degradation classes", then the performance indicators of the products are affected by them and have a phenomenon of higher or lower baseline degradation rate, which is called "clustering" phenomenon. Aiming at the products with the phenomena, the invention provides a multi-performance index product degradation modeling and service life prediction method considering index clustering, which is more suitable for the real situation.
(2) Technical scheme
The invention discloses a degradation modeling and life prediction method considering performance index clustering in a dynamic environment, and the overall technical scheme is shown in figure 1. Firstly, collecting test data, then establishing a performance index degradation model, estimating unknown parameters in the model, and finally predicting the service life and the reliability, wherein the specific steps are as follows:
the method comprises the following steps: collecting test data
For a product with a plurality of performance indexes, product performance degradation data and corresponding environmental profile data are collected through tests or engineering practice. The method comprises the following steps: 1) presetting a sampling interval time delta t according to an actual situation, wherein the delta t is as small as possible; 2) determining the number of products participating in the test, the number of performance indexes of each product and the sensitive environmental stress of the product; 3) In order to obtain balanced test data, the degradation amount of each performance index of each product is measured and recorded every delta t, and meanwhile, the sensitive environmental stress value of the product is collected and recorded by using a sensor.
Step two: building a degradation model
Assuming that a certain product shares d performance indexes, under a dynamic environment, the degradation amount of the product at the time t can be expressed as the sum of an initial degradation amount, a degradation rate cumulative effect term and Brownian motion:
wherein X (s) (0) The product is the product with the s (s ═ 1,2, …, d) performance index degraded at the initial time.
B (t) is the standard Brownian motion.
σ (s) Inconsistency and instability in the product degradation process are described for the diffusion parameters of the s-th individual performance index of the product. Generally do not change with time and changing conditions, so the diffusion parameter is generally a constant, σ (s) B(t)～N(0,[σ (s) ] 2 t)。
r (s) And (t) is a degradation rate function of the s-th performance index of the product, which is the product of the baseline degradation rate of the s-th performance index and the environmental stress function, and the specific mathematical expression is shown as the formula (2). Integral of
Is a degradation rate integration term, where u is an integration variable, and the integration interval is [0, t]。
(2) Where z (t) represents the environmental stress level at time t, which is a scalar if the degradation of the product is affected by only one environmental stress, and a multi-dimensional vector otherwise.
Is the action form of the environmental stress on the s-th performance index, and can be specifically given according to the type of the environmental stress. For example, in the case of temperature stress,
may be given by the arrhenius model; for electrical stress, it can be given by an inverse power-rate model. r is 0 (s) The baseline degradation rate, which represents the product's performance index, is a linear combination of the baseline degradation rates of all "degradation classes":
θ q means a baseline degradation rate of the q (q ═ 1,2, …, w) th "degradation class", w is the total number of "degradation classes", and w and d satisfy w ≦ d 1 ,θ 2 ,…,θ w ) T ～N(μ θ , Σ θ ) In which μ θ ,Σ θ Mean vector and covariance matrix of theta, sigma, respectively θ The baseline degradation rates for different "degradation classes" are considered to be independent of each other. Alpha is alpha q (s) Is a weight parameter, α q (s) θ q Indicating the magnitude of the contribution of the qth "degradation class" to the baseline degradation rate of the s-th performance indicator.
Step three: estimating parameters and updating the model in real time
Assuming that N product individuals, the individual performance index of each product individual and the degradation signal observed value under m times of observation are collected in the first step as samples, and converting the original samples into degradation increment samples in a differential mode:
the serial numbers k, s and i respectively represent the kth product, the s individual performance index and the ith observation; t is t i Indicating the time corresponding to the ith observation,
the s individual performance index of the k product at the i observation time t i The observed value of the degraded signal of (a) is,
and the difference between the degradation amount of the s-th performance index of the kth product at the i +1 th observation time and the degradation amount of the s-th performance index of the kth product at the i-th observation time is represented.
The base line degradation rate of the degradation curve of the s-th performance index of the kth product
Considering an intermediate variable, since the wiener process has independent increments, there are:
wherein z (t) i ) Indicates the environmental stress of the product at the ith observation time t i The following values.
Then the s-th performance index's degradation increment sample set x (s) ：
The corresponding likelihood function is:
by maximizing the likelihood function, the likelihood function can be obtained
Is estimated as
When s is 1,2, …, d, the estimated values of the d groups are obtained. Writing the obtained estimated values of the d groups of baseline degradation rates into a matrix form:
and as the input of factor analysis, the factor analysis result can be obtained:
parameters in equation (8)
Can be derived from a factorial analysis method, wherein gamma (s) Mean value of baseline degradation rates, θ ', representing the s-th performance indicator' q (q-1, 2, …, w) is an independent identically distributed random variable with mean 0 and variance 1, and coefficients
Is represented by theta' q The degree of influence on the s-th performance index. This result is different from the result of the formula (3), but for r 0 (s) Has no influence on the reduction of (2).
Step four: and predicting the service life and the reliability.
After parameter estimation, the reliability of the product can be predicted by combining a performance degradation threshold and a future environmental profile. Definition of T (s) First crossing its failure threshold D for the amount of degradation of the s-th performance indicator (s) Time (first wearing time):
T (s) ＝inf{t＞0,X (s) (t)≥D (s) } (9)
the reliability of the s-th performance index at the time t (t is greater than or equal to 0) can be represented as the probability that the maximum value of the degradation amount of the degradation curve in the [0, t ] interval is smaller than the failure threshold value:
wherein v is [0, t]At any time in the interval, u represents the integrand
Of an integration interval of [0, v ]]。
Meanwhile, the problem can be converted into the standard wiener process which passes through the curve boundary g (s) (v) The probability of (a) of (b) being,
R (s) (t)＝P{B(v)＜g (s) (v),0≤v≤t} (11)
wherein, the curve boundary g (s) (v) As a result of the transformation of equation (10), for,
finally, according to Daniels [ H.E.Daniels.applying the first crossing-time diversity for a curved boundary, Bernoulli 2(2) (1996), 133-]The boundary tangent method mentioned above can obtain a predetermined θ ' (θ ' ═ θ ' 1 ,θ′ 2 ,…, θ′ w ) First time of wear T under the condition (s) Of (a) a probability density function f (s) (t|θ′)，
At the same time, the reliability expression can be obtained,
in the above formula, u is a function f (s) (t | theta') over an integration interval of [0, t]。
Given θ', the reliability functions of the indexes are independent of each other, so that the reliability of the product can be obtained according to a total probability formula:
wherein the probability density function p (θ ') of θ' can be expressed as
And finally, drawing a curve according to the reliability model, and predicting the service life of the product.
(3) Advantages of the invention
The phenomenon that each index of a multi-performance index product shows clustering due to the influence of common potential factors is subjected to mathematical modeling, so that the degradation modeling of the product with the phenomenon is more practical, and the relevance among the indexes is better explained.
Drawings
FIG. 1 shows a flow chart of the method of the present invention.
Fig. 2 is an explanatory diagram showing the relationship between key components and performance indexes in the simulation object DT830 according to the present invention.
FIG. 3(a) is a graph illustrating a temperature stress profile of a simulation case according to the present invention.
FIG. 3(b) is a diagram illustrating a relative humidity profile of a simulation case according to the present invention.
FIG. 3(c) is a schematic diagram showing a salt spray concentration profile of a simulation example of the present invention.
FIG. 4 is a graph showing the degradation signal of a sample in the case of simulation according to the present invention.
FIG. 5 is a graph showing the reliability of product life prediction obtained by the present invention and a K-M curve.
Detailed Description
The feasibility of the proposed model is verified by simulation tests. The simulated object is a DT830 type digital multimeter which comprises five measurement gears: direct Current Voltage (DCV), Direct Current (DCA), Direct Current Resistance (DCR), Alternating Current Voltage (ACV), Alternating Current (ACA), every measurement gear all has a plurality of different range. When the measurement error of any measurement gear of the multi-purpose meter exceeds a certain range, the multi-purpose meter is considered to be invalid, and therefore the measurement error under the highest measurement range of the five gears is used as five performance indexes of the multi-purpose meter for simulation. The five performance indexes are actually influenced by the degradation of two groups of key common components in the multimeter, so that the five performance indexes have degradation tendency, namely 1) voltage division/shunt resistance; 2) integrated chip ICL7160 and LCD. The voltage dividing/dividing resistors can be subdivided into two groups, wherein one group controls a direct current voltage level (DCV), an alternating current voltage level (ACV) and a direct current resistor level (DCR), and the other group controls a direct current level (DCA) and an alternating current level (ACA). The integrated chip and the LCD are used for reading, digital-to-analog conversion and display of electric signals, and therefore all five performance indexes are influenced. The relationship between these two key components and each performance index is shown in fig. 2.
In the context of the foregoing multi-table, it can be considered that five performance indicators of the multi-table are affected by three "degradation classes" together to have correlation, and the baseline degradation rates of the three "degradation classes" are set as follows: theta.theta. 1 ～N(1.56,0.24),θ 2 ～N(1.87, 0.38),θ 3 N (1.49,0.3), and the baseline degradation rates for the five performance indicators can be expressed as:
wherein the respective weight coefficient settings are shown in table 1.
TABLE 1 weight coefficient settings
Assuming that the multimeter is in service in a marine environment, the environmental stresses of sensitivity include: temperature (denoted as T) and relative humidity (denoted as S) 1 ) And salt spray concentration (denoted as S) 2 ) The environmental stress profile is shown in fig. 3(a), 3(b) and 3 (c).
Action form of environmental stress on index baseline degradation rate
Can be expressed as:
wherein the parameter E a (s) Represents activation energy, k B Representing boltzmann's constant. E a (s) /k B 、C 1 (s) 、C 2 (s) Are model parameters, set as shown in table 2 below.
TABLE 2 environmental Effect related parameter settings
Finally, the wiener process diffusion coefficient and the failure threshold of each performance index are set, as shown in table 3
TABLE 3 wiener Process diffusion coefficients and failure threshold settings for each index
Based on the above parameter settings and stress profile settings, assuming that 50D 830 samples were subjected to a degradation test for a period of 720 days, the amount of degradation for each of their performance indicators was measured once a day. And generating simulation data. FIG. 4 is a simulated degradation curve for one of the samples.
The application steps and methods of the present invention will be described in detail below:
the method comprises the following steps: collecting test data
Test data were collected by simulation testing.
Step two: building a degradation model
And fitting the product performance degradation process by adopting a degradation model considering performance index clustering in a dynamic environment.
Step three: parameter estimation
Historical data for the first 360 time points (360 historical data for 50 products) were used for parameter estimation. Based on the parameter estimation method provided by the invention, the parameter to be estimated { E ] in the model a (s) /k B ,C 1 (s) ,C 2 (s) ,σ (s) Where s is 1,2,3,4,5 and an intermediate variable r k,0 (s) The estimated values of s-1, 2,3,4,5, k-1, 2, …,50, where the former estimated results are shown in table 4,
TABLE 4 estimation results of the parameters to be estimated
After factor analysis, the baseline degradation rate of each index can be expressed as:
wherein theta' 1 ,θ′ 2 ,θ′ 3 The mean value is 0, and the covariance is the ternary normal distribution of the three-dimensional unit matrix.
Step four: reliability prediction and verification
The unknown parameters and the threshold value { D (s) And s is substituted into the probability density function formula (13) to further calculate the product reliability according to the reliability models (14) and (15). And comparing with a Kaplan-Meier (K-M) reliability prediction method based on failure time, verifying the prediction precision, wherein the failure data is shown in a table 5:
table 5 time to failure data
As shown in fig. 5, the black curve for reliability prediction based on the degradation model provided by the present invention is highly consistent with the red circled polygonal line for reliability prediction based on the K-M method, and the root mean square error thereof is 0.0217, and the feasibility of the method provided by the present invention can be verified from the result.
Claims (3)
1. A degradation modeling and life prediction method considering performance index clustering in a dynamic environment is characterized by comprising the following specific steps:
the method comprises the following steps: collecting test data
For a product with a plurality of performance indexes, product performance degradation data and corresponding environmental profile data are collected through tests or engineering practice;
step two: establishing a degradation model
And (3) setting a common d individual performance indexes of a certain product, wherein the degradation quantity of the product at the time t is expressed as the sum of the initial degradation quantity, a degradation rate cumulative effect term and Brownian motion under a dynamic environment:
wherein X (s) (0) The degradation amount of the performance index of the product at the initial moment is the s (s is 1,2, …, d);
b (t) is standard Brownian motion;
σ (s) inconsistency and instability in the product degradation process are described for the diffusion parameters of the s-th individual performance index of the product; generally do not change with time and changing conditions, so the diffusion parameter is a constant, σ (s) B(t)～N(0,[σ (s) ] 2 t)；
r (s) (t) is a degradation rate function of the s-th performance index of the product, which is the product of the baseline degradation rate of the s-th performance index and an environmental stress function, and the specific mathematical expression is shown as the formula (2); integral of
Is a degradation rate integration term, where u is an integration variable, and the integration interval is [0, t]；
Where z (t) represents the environmental stress level at time t, which is a scalar if the degradation of the product is affected by only one environmental stress, and whether it is a scalar or notThen, it is a multi-dimensional vector;
the action form of the environmental stress on the s-th individual performance index is specifically given according to the type of the environmental stress; r is 0 (s) The baseline degradation rate, which represents the product's performance index, is a linear combination of the baseline degradation rates of all "degradation classes":
θ q a baseline degradation rate representing the q (q-1, 2, …, w) th "degradation class", w being the total number of "degradation classes", and w and d satisfying w ≦ d 1 ,θ 2 ,…,θ w ) T ～N(μ θ ,Σ θ ) In which μ θ ,Σ θ Mean vector and covariance matrix of theta, sigma, respectively θ Considered as a diagonal matrix, i.e. the baseline degradation rates of different "degradation classes" are considered to be independent of each other; alpha (alpha) ("alpha") q (s) Is a weight parameter, α q (s) θ q Representing the contribution of the qth 'degradation class' to the baseline degradation rate of the s-th performance indicator;
step three: estimating parameters and updating the model in real time
In the first step, N product individuals, individual performance indexes of each product individual d and a degradation signal observation value under m times of observation are collected as samples, and the original samples are converted into degradation incremental samples in a differential mode:
the serial numbers k, s and i respectively represent the kth product, the s-th individual performance index and the ith observation; t is t i Indicating the time corresponding to the ith observation,
the s individual performance index of the k product at the i observation time t i The observed value of the degraded signal of (a) is,
the difference between the degradation amount of the s individual performance index of the kth product at the i +1 th observation time and the degradation amount of the s individual performance index of the kth product at the i th observation time is represented;
the base line degradation rate of the degradation curve of the s-th performance index of the kth product
Considering an intermediate variable, since the wiener process has independent increments, there are:
wherein z (t) i ) Indicating the environmental stress of the product at the ith observation time t i Taking the value of;
then the s-th performance index's degradation increment sample set x (s) ：
The corresponding likelihood function is:
maximizing the likelihood function to obtain
Is estimated as
Taking s as 1,2, …, d respectively to obtain estimated values of d groups; writing the obtained estimated values of the d groups of baseline degradation rates into a matrix form:
and as the input of factor analysis, obtaining the factor analysis result:
parameters in equation (8)
Derived from a factorial analysis method in which gamma is (s) Mean value of baseline degradation rates, θ ', representing the s-th performance indicator' q (q-1, 2, …, w) is an independent identically distributed random variable with mean 0 and variance 1, and coefficients
Is represented by theta' q The degree of influence on the s-th performance index; although the result is different from the result of the formula (3), the result is different from the result of the formula (3) in the case of r 0 (s) The reduction of (a) has no effect;
step four: predicting the service life and reliability;
predicting the reliability of the product by combining a performance degradation threshold and a future environment profile; definition of T (s) For the first time that the degradation amount of the s-th individual performance index passes through the failure threshold value D (s) Time of (2):
T (s) ＝inf{t＞0,X (s) (t)≥D (s) } (9)
the reliability of the s-th individual performance index at the time t (t is more than or equal to 0) is represented as the probability that the maximum value of the degradation amount of the degradation curve in the [0, t ] interval is less than the failure threshold value:
wherein v is [0, t]At any time in the interval, u represents the integrand
Of integration interval of [0, v ]]；
Simultaneous conversion to the standard wiener process across the curve boundary g (s) (v) The probability of (a) of (b) being,
R (s) (t)＝P{B(v)＜g (s) (v),0≤v≤t} (11)
wherein the curve boundary g (s) (v) As a result of the modification of equation (10), for,
finally, according to Daniels [ H.E.Daniels.applying the first crossing-time diversity for a curved boundary, Bernoulli 2(2) (1996), 133-]The boundary tangent method mentioned above yields a predetermined θ ' (θ ' ═ θ ' 1 ,θ′ 2 ,…,θ′ w ) First time of wearing T under the condition (s) Probability density function f (s) (t|θ′)，
And at the same time, obtaining a reliability expression,
in the above formula, u is a function f (s) (t | theta') over an integration interval of [0, t]；
Given θ', the reliability functions of the indexes are independent of each other, so that the reliability of the product is obtained according to a total probability formula:
wherein the probability density function p (theta ') of theta' is expressed as
And finally, drawing a curve according to the reliability model, and predicting the service life of the product.
2. The degradation modeling and life prediction method considering performance index clustering in a dynamic environment as claimed in claim 1, wherein: the specific method in the step 1 is as follows: 1) presetting a sampling interval time delta t according to an actual situation; 2) determining the number of products participating in the test, the number of performance indexes of each product and the sensitive environmental stress of the product; 3) in order to obtain balanced test data, the degradation amount of each performance index of each product is measured and recorded every delta t, and meanwhile, the sensitive environmental stress value of the product is collected and recorded by using a sensor.
3. The degradation modeling and life prediction method considering performance index clustering in a dynamic environment as claimed in claim 1, wherein: in the formula (2), with respect to the temperature stress,
given by the arrhenius model; for the electrical stress, it is given by the inverse power-rate model.
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