CN107862134A - A kind of Wiener process reliability analysis methods for considering autocorrelation measurement error - Google Patents
A kind of Wiener process reliability analysis methods for considering autocorrelation measurement error Download PDFInfo
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Abstract
The present invention proposes a kind of Wiener process reliability analysis methods for considering autocorrelation measurement error, to solve the problems, such as that existing performance degradation analysis method has a strong impact on reliability assessment precision.Its step is：A sample input experiment is randomly extracted from a collection of product, gathers properties of product degraded data；According to the actual performance degenerative process of Wiener process description products, the Wiener process performance degradation models for considering autocorrelation measurement error are established；The unknown parameter of Performance Degradation Model is estimated using likelihood function；The reliability of product is analyzed according to the unknown parameter of estimation.The present invention is the linear Wiener processes Degradation Reliability analysis method for having AR (1) measurement error, is effectively improved Evaluation accuracy；It is stronger with general and generality, its applicability；The Reliability Function analytical expression under given degradation failure threshold value is given, the reliability assessment for later product provides foundation.
Description
Technical field
The present invention relates to the technical field of small sample fail-safe analysis, is related to a kind of reliability based on Performance Degradation Data
Analysis method, more particularly to a kind of Wiener process reliability analysis methods for considering autocorrelation measurement error.
Background technology
Become increasingly conspicuous in fields such as Aero-Space, weaponrys, the characteristics of product small sample, highly reliable, long-life, this is right
Convectional reliability analysis method based on the out-of-service time brings huge challenge.Performance degradation modeling is that solve such production
One of key technology of product fail-safe analysis, and the focus and difficult point of Reliability Engineering area research at present.
Often influenceed by a variety of enchancement factors in view of the performance degradation process of product, managed based on random process simultaneously
It is the emphasis that current scholars study by row degradation modeling is entered.In numerous theory of random processes, because Wiener processes are clear and definite
Physical significance and good mathematical property, have become it is a kind of extensively and one of most widely used degradation model in engineering.
Meanwhile in engineering in practice, can inevitably there is measurement error in the Performance Degradation Data for measuring to obtain.Therefore, having must
Establish the performance degradation analysis model for considering measurement error.
However, in the degradation models for considering measurement error all at present, assume that measurement error is mutually only at different moments
It is vertical, and obey same normal distribution.But this is not inconsistent with many actual conditions, particularly when tested between when being spaced shorter,
Autocorrelation between follow-on test measurement error often be can not ignore, and independence assumption will have a strong impact on fail-safe analysis essence
Degree.In engineering in practice, the measurement error in Performance Degradation Data generally has autocorrelation, and can not be ignored sometimes, and shows
Assume that the measurement error between the different test moment is separate in some performance degradation analysis methods, and obey same normal state
Distribution, can have a strong impact on reliability assessment precision, particularly when tested between when being spaced shorter.
The content of the invention
The technical problem of reliability assessment precision is had a strong impact on for existing performance degradation analysis method, the present invention proposes
A kind of Wiener process reliability analysis methods for considering autocorrelation measurement error, it is contemplated that the autocorrelation between measurement error,
And first-order autoregression process is introduced as measurement error item, while consider the individual difference between sample, it is effectively improved
Assess estimation；With generality, existing degradation model is considered as a kind of its special case, is commented so as to effectively improve reliability
Estimate precision.
In order to achieve the above object, the technical proposal of the invention is realized in this way：One kind considers autocorrelation measurement error
Wiener process reliability analysis methods, its step is as follows：
Step 1：M sample input experiment is randomly extracted from a collection of product, gathers properties of product degraded data；
Step 2：According to the actual performance degenerative process of Wiener process description products, establish and consider that autocorrelation measurement misses
The Wiener process performance degradation models of difference；
Step 3：With reference to the properties of product degraded data drawn in step 1, using likelihood function to being drawn in step 2
The unknown parameter of Performance Degradation Model estimated；
Step 4：The unknown parameter of the estimation drawn according to step 3 is analyzed the reliability of product.
It is described collection properties of product degraded data method be：By in m sample input experiment, divide for i-th of sample
Not in niThe individual test momentPlace carries out performance degradation measurement, and record obtains corresponding Performance Degradation DataAnd degradation valuesSo as to obtain the Performance Degradation Data y of m sample
=(y1,y2,…,ym)T；Wherein, yijIt is i-th of sample in j-th of moment tijThe test performance degradation values at place, i=1,2 ...,
M, j=1,2 ..., ni, niFor the test moment number of i-th of sample.
The foundation considers that the method for the Wiener process performance degradation models of autocorrelation measurement error is：
It is described consider autocorrelation measurement error Wiener process performance degradation models be：
Wherein, X (t) is that product is described in t Wiener process forms in the product of initial amount of degradation X (0)=0
Actual performance degenerative process, Y (t) represent the performance degradation amount that product obtains in the measurement of t；β is coefficient of deviation, andN () is normal distribution, μbAnd σbRespectively coefficient of deviation β average and standard deviation；Λ=Λ (t, θ) is
On time t continuous strictly monotone increasing function, θ is the unknown parameter in function Λ=Λ (t, θ)；σ ＞ 0 are diffusion coefficient；B
(Λ) is extensive calibrations Wiener processes, makes i-th of sample in j-th of moment tijThe test value Λ at placeij=Λ (tij, θ), i=
1,2 ..., m, j=1,2 ..., ni；
For the testing timeB(Λi1) and performance degradation increment Between independently of each other, and B (Λi1)~N (0, Λi1), B (Λij)-B
(Λi(j-1))~N (0, Λij-Λi(j-1))；
Measurement error ε (t) meets first order autoregressive model
Measurement error ε (t) average and covariance has following property：
Wherein,For auto-correlation coefficient, andRandom errorσeFor random error eiStandard deviation；
γ0For autoregressive process second order away from, and
Stochastic variable coefficient of deviation β, extensive calibrations Wiener process B () and measurement error ε (t) are separate.
Use the method analyzed based on degraded data determine the detailed process of function Λ form for：For m sample
Performance Degradation Data yij=y (tij)=Y (tij), yijFor moment tijThe test performance degradation values at place, tijExist for i-th of sample
J-th of moment；The mean degradation path data of sample is obtained first Then, using phase
The function answered, such as linear function, exponential function or exponential function, mean degradation path data is fitted respectively；Finally adopt
Λ expression formula is used as by the use of the functional form being fitted.
The method that the unknown parameter to Performance Degradation Model is estimated is：
Step 1：Define the unknown parameter battle array of Performance Degradation ModelI-th of sample difference
In niPerformance degradation measurement y is obtained at the individual test momentiObey niTie up normal distribution：yi~MN (μbΛi,Σi), wherein, i-th
The test moment battle array of sampleThe covariance matrix of i-th of sample performance degraded dataAndWith
Step 2：To the parameter Reparameterization in unknown parameter battle array Θ, orderWithThen have
Step 3：Combination product Performance Degradation Data, the log-likelihood function for obtaining sample are：
In formula,
Step 4：The log-likelihood function of sample is sought respectively on parameter μbWithSingle order local derviation, and make its be equal to 0, connection
It is vertical to obtain parameter μbWithMaximum-likelihood estimation：
By parameter μbWithMaximum-likelihood estimation bring the log-likelihood function of sample into, obtain on unknown parameterEdge log-likelihood function：
Step 5：Edge log-likelihood function is maximized by multidimensional search algorithm, obtains unknown parameter battle array's
Maximum-likelihood estimationThen parameter μ is carried it intobWithThe formula of Maximum-likelihood estimation can obtain parameter
Maximum-likelihood estimationWithMeanwhile parameterAnd σ2Maximum-likelihood estimation beWithFinally, may be used
Obtain parameterMaximum-likelihood estimation be
The method that the reliability of the product is analyzed is：
Step 1. reliablity estimation：
If the failure threshold of product is Df, then it is defined as according to hit time concept, life of product T：
T=inf { t:X(t)≥Df| X (0) ＜ Df,
Then life of product T distribution function is
In formula, Φ () is Standard Normal Distribution.
Reliability Function of the product in preset time t be：
Step 2. mean time to failure, MTTF is estimated：
Product mean time to failure, MTTF is：
Beneficial effects of the present invention：
1. suffer from the influence of measurement error for actual performance degradation testing, and in follow-on test measurement error from
The situation that correlation be can not ignore sometimes, the present invention propose a kind of linear Wiener processes with AR (1) measurement error and moved back
Change analysis method for reliability, be effectively improved Evaluation accuracy.
2. there is the present invention general and generality, existing Wiener processes degradation model to be considered as the one of the invention
Kind special case, so that the applicability of the present invention is stronger.
3. The present invention gives the Reliability Function analytical expression under given degradation failure threshold value, be later product can
Assessed by property and provide foundation.
Brief description of the drawings
In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing
There is the required accompanying drawing used in technology description to be briefly described, it should be apparent that, drawings in the following description are only this
Some embodiments of invention, for those of ordinary skill in the art, on the premise of not paying creative work, can be with
Other accompanying drawings are obtained according to these accompanying drawings.
Fig. 1 is the flow chart of the present invention.
Fig. 2 is the curve map of the Performance Degradation Data of laser in instantiation.
Fig. 3 is the curve map of the average behavior degraded data of laser in instantiation.
Fig. 4 is the Reliability Function figure of laser in instantiation.
Embodiment
Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete
Site preparation describes, it is clear that described embodiment is only part of the embodiment of the present invention, rather than whole embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained every other under the premise of creative work is not paid
Embodiment, belong to the scope of protection of the invention.
As shown in figure 1, a kind of Wiener process reliability analysis methods for considering autocorrelation measurement error, its step is such as
Under：
Step 1：M sample input experiment is randomly extracted from a collection of product, gathers properties of product degraded data.
Assuming that m sample input experiment is randomly extracted from a collection of product, for i-th of sample respectively in niIndividual test
MomentPlace carries out performance degradation measurement, and record obtains corresponding Performance Degradation Data
And degradation valuesSo as to obtain the Performance Degradation Data y=(y of m sample1,y2,…,ym)T.Its
In, yijIt is i-th of sample in j-th of moment tijThe test performance degradation values at place, i=1,2 ..., m, j=1,2 ..., ni, ni
For the test moment number of i-th of sample.
Step 2：Establish the Wiener process performance degradation models for considering autocorrelation measurement error.
1. the determination of actual performance degenerative process.
Make X (t) represent actual performance amount of degradation of the product in t, due to product performance degradation process by internal and
The influence of outside a variety of environmental factors, can be regarded as a random process.Here using a kind of Wiener commonly used in the world
The actual performance degenerative process of product is described in process form, specific as follows
X (t)=X (0)+β Λ+σ B (Λ)
In formula, X (0) is initial amount of degradation, it is assumed that X (0)=0；β is coefficient of deviation, in order to describe the otherness between product,
Assuming that β is normal random variable, i.e.,N () is normal distribution, μbAnd σbRespectively coefficient of deviation β average and mark
It is accurate poor；Λ=Λ (t, θ) is the continuous strictly monotone increasing function on time t, and θ is the unknown ginseng in function Λ=Λ (t, θ)
Number；σ ＞ 0 are diffusion coefficient；B (Λ) is extensive calibrations Wiener processes, makes i-th of sample in j-th of moment tijThe test at place
Value Λij=Λ (ti,jθ), i=1,2 ..., m, j=1,2 ..., ni.From definition, for the testing timeB(Λi1) and performance degradation increment
Between independently of each other, wherein,
B(Λi1)~N (0, Λi1)
B(Λij)-B(Λi(j-1))~N (0, Λij-Λi(j-1))。
2. consider measurement error item ε (t).
In engineering in practice, because the influence of the factors such as measuring instrument, artificial, noise, measurement error can not be ignored.And pass
System performance degradation analysis method assumes that measurement error ε (t) is independent identically distributed normal distribution.However, the hypothesis have ignored
Autocorrelation between measurement error, it will sometimes have a strong impact on the precision of Reliability Assessment.Therefore, in order to consider to survey
Measure the autocorrelation between error, it is assumed that measurement error ε (t) meets first order autoregressive model AR (1), i.e.,
And measurement error ε (t) average and covariance have following property：
In above formula,For auto-correlation coefficient, andRandom errorσeFor random error eiStandard
Difference；γ0For autoregressive process second order away from, and
3. establish the Wiener process performance degradation models for considering autocorrelation measurement error.
On the basis analyzed more than, the Wiener process performance degradation models for considering autocorrelation measurement error are：
Wherein, Y (t) represents the performance degradation amount that product obtains in the measurement of t；Stochastic variable β, B () and ε (t)
Independently of each other；Coefficient of deviation β mean μ in modelbAnd standard deviation sigmab, diffusion coefficient σ, θ, auto-correlation coefficientAnd random error
eiStandard deviation sigmaeIt is unknown parameter.
4. function Λ form determines.
At present, function Λ form determines mainly to analyze two methods by Analysis of Failure Mechanism and based on degraded data,
Because the Analysis of Failure Mechanism difficulty of product is larger.Therefore, it is more using the method based on degraded data analysis, specific mistake in engineering
Cheng Wei：For the Performance Degradation Data of m sample, y is madeij=y (tij)=Y (tij), it is assumed that each sample is when identical is tested
Measured at quarter, be such as unsatisfactory for condition, then can carry out interpolation processing.The mean degradation path data of sample is obtained firstThen, using corresponding function, such as linear function, exponential function, exponential function,
Mean degradation path data is fitted respectively；Finally using expression formula of the best functional form of fitting as Λ.
Step 3：With reference to the properties of product degraded data drawn in step 1, using likelihood function to being drawn in step 2
The unknown parameter of Performance Degradation Model estimated.
1. define relevant parameter and data matrix.OrderFor unknown parameter battle array in model, i-th
Sample is respectively in niPerformance degradation measurement y is obtained at the individual test momentiObey niTie up normal distribution：yi~MN (μbΛi,Σi), its
InThe covariance matrix of i-th of sample performance degraded dataAndWith
2. Reparameterization.
In order to simplify estimation procedure, Reparameterization, orderWith
Then have
3. define likelihood function.
Combination product Performance Degradation Data, the log-likelihood function for obtaining sample are
In formula,
4. determine marginal likelihood function.
The log-likelihood function of sample is sought respectively on parameter μbWithSingle order local derviation, and make its be equal to 0, simultaneous can obtain
Parameter μbWithMaximum-likelihood estimation：
By parameter μbWithMaximum-likelihood estimation bring the log-likelihood function of sample into, obtain on unknown parameterEdge log-likelihood function be：
5. unknown parameter is estimated.
Edge log-likelihood function is maximized by multidimensional search algorithm, location parameter can be obtainedIt is very big
Possibility predicationThen parameter μ is carried it intobWithThe formula of Maximum-likelihood estimation can to obtain parameter very big
Possibility predicationWithMeanwhile parameterAnd σ2Maximum-likelihood estimation beWithFinally, can obtain
ParameterMaximum-likelihood estimation be
Step 4：The unknown parameter of the estimation drawn according to step 3 is analyzed the reliability of product.
1. reliablity estimation.
If the failure threshold of product is Df, then it is defined as according to hit time concept, life of product T：
T=inf { t:X(t)≥Df| X (0) ＜ Df,
Then life of product T distribution function is
Φ () is Standard Normal Distribution in formula.
Reliability Function of the product in Given Life t be：
2. mean time to failure, MTTF is estimated.Product mean time to failure, MTTF is
Experimental verification is carried out to the present invention below by certain laser performance degradation experiment data：
Laser has the characteristics of high reliability long life, and to carry out fail-safe analysis to it, Fig. 2 gives 10 certain laser
The Performance Degradation Data of device operating current increased percentage at 80 DEG C of temperature, test interval are 250 hours, and experiment is extremely
Terminate within 4000 hours, failure threshold DfFor 30%.
Determine that method can obtain the mean degradation path data of laser, such as Fig. 3 by the form of function Λ in step 2
It is shown.Apparent from Fig. 3, the performance degradation path of laser has obvious linear feature, therefore function Λ shape
Formula is linear, i.e. Λ=Λ (t, θ)=t, wherein, θ=1.Therefore, the Wiener Process Characters for considering autocorrelation measurement error are established
Can degradation model.
Show that unknown parameter estimated result is as shown in table 1 in model by the method for the step 3 in Fig. 1.
The unknown parameter estimated result of model in the instantiation of table 1
Finally, the reliability of product can be calculated by Reliability Function, and then obtains reliability curves, as a result such as Fig. 4 institutes
Show.So as to carry out fail-safe analysis to product, such as the reliability of life of product t=10000 hours is：
Product mean time to failure, MTTF is：
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention, all essences in the present invention
God any modification, equivalent substitution and improvements made etc., should be included in the scope of the protection with principle.
Claims (6)
1. a kind of Wiener process reliability analysis methods for considering autocorrelation measurement error, it is characterised in that its step is as follows：
Step 1：M sample input experiment is randomly extracted from a collection of product, gathers properties of product degraded data；
Step 2：According to the actual performance degenerative process of Wiener process description products, establish and consider autocorrelation measurement error
Wiener process performance degradation models；
Step 3：With reference to the properties of product degraded data drawn in step 1, using likelihood function to the property that is drawn in step 2
The unknown parameter of energy degradation model is estimated；
Step 4：The unknown parameter of the estimation drawn according to step 3 is analyzed the reliability of product.
2. the Wiener process reliability analysis methods according to claim 1 for considering autocorrelation measurement error, its feature
It is, the method that properties of product degraded data is gathered in the step 1 is：By in m sample input experiment, tried for i-th
Sample is respectively in niThe individual test momentPlace carries out performance degradation measurement, and record obtains corresponding performance degradation number
According toAnd degradation valuesSo as to obtain the performance degradation number of m sample
According to y=(y1,y2,…,ym)T；Wherein, yijIt is i-th of sample in j-th of moment tijThe test performance degradation values at place, i=1,
2 ..., m, j=1,2 ..., ni, niFor the test moment number of i-th of sample.
3. the Wiener process reliability analysis methods according to claim 1 for considering autocorrelation measurement error, its feature
It is, the method that the Wiener process performance degradation models for considering autocorrelation measurement error are established in the step 2 is：
It is described consider autocorrelation measurement error Wiener process performance degradation models be：
Wherein, X (t) is that the true of product is described in t Wiener process forms in the product of initial amount of degradation X (0)=0
Performance degradation process, Y (t) represent the performance degradation amount that product obtains in the measurement of t；β is coefficient of deviation, andN () is normal distribution, μbAnd σbRespectively coefficient of deviation β average and standard deviation；Λ=Λ (t, θ)
For the continuous strictly monotone increasing function on time t, θ is the unknown parameter in function Λ=Λ (t, θ)；σ ＞ 0 are diffusion system
Number；B (Λ) is extensive calibrations Wiener processes, makes i-th of sample in j-th of moment tijThe test value Λ at placeij=Λ (tij,
θ), i=1,2 ..., m, j=1,2 ..., ni；
For the testing timeB(Λi1) and performance degradation increment Between independently of each other, and B (Λi1)~N (0, Λi1), B (Λij)-
B(Λi(j-1))~N (0, Λij-Λi(j-1))；
Measurement error ε (t) meets first order autoregressive model AR (1)：
Measurement error ε (t) average and covariance has following property：
Wherein,For auto-correlation coefficient, andRandom errorσeFor random error eiStandard deviation；γ0For
The second order of autoregressive process away from, and
Stochastic variable coefficient of deviation β, extensive calibrations Wiener process B () and measurement error ε (t) are separate.
4. the Wiener process reliability analysis methods according to claim 3 for considering autocorrelation measurement error, its feature
Be, use the method analyzed based on degraded data determine the detailed process of function Λ form for：For the performance of m sample
Degraded data yij=y (tij)=Y (tij), yijFor moment tijThe test performance degradation values at place, tijIt is i-th of sample at j-th
Moment；The mean degradation path data of sample is obtained firstThen, using corresponding
Function, such as linear function, exponential function or exponential function, mean degradation path data is fitted respectively；Finally using plan
Expression formula of the functional form got togather as Λ.
5. the Wiener process reliability analysis methods according to claim 1 for considering autocorrelation measurement error, its feature
It is, the method that the unknown parameter to Performance Degradation Model is estimated is：
Step 1：Define the unknown parameter battle array of Performance Degradation ModelI-th of sample is respectively in niIt is individual
Test at the moment and obtain performance degradation measurement yiObey niTie up normal distribution：yi~MN (μbΛi,Σi), wherein, i-th sample
Test moment battle arrayThe covariance matrix of i-th of sample performance degraded dataAndWith
Step 2：To the parameter Reparameterization in unknown parameter battle array Θ, orderWithThen have
Step 3：Combination product Performance Degradation Data, the log-likelihood function for obtaining sample are：
In formula,
Step 4：The log-likelihood function of sample is sought respectively on parameter μbWithSingle order local derviation, and make its be equal to 0, simultaneous can
Obtain parameter μbWithMaximum-likelihood estimation：
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6. the Wiener process reliability analysis methods according to claim 1 for considering autocorrelation measurement error, its feature
It is, the method that the reliability of the product is analyzed is：
Step 1. reliablity estimation：
If the failure threshold of product is Df, then it is defined as according to hit time concept, life of product T：
T=inf { t:X(t)≥Df| X (0) ＜ Df,
Then life of product T distribution function is
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In formula, Φ () is Standard Normal Distribution.
Reliability Function of the product in preset time t be：
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</mtd>
</mtr>
</mtable>
</mfenced>
Step 2. mean time to failure, MTTF is estimated：
Product mean time to failure, MTTF is：
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| CN113378309A (en) * | 2021-01-28 | 2021-09-10 | 河南科技大学 | Rolling bearing health state online monitoring and residual life prediction method |
| CN113033002A (en) * | 2021-03-26 | 2021-06-25 | 中国人民解放军火箭军工程大学 | Unbiased estimation method for model parameters of complex wiener degradation process |
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