CN109766518B - Uncertain accelerated degradation modeling and analyzing method considering sample individual difference - Google Patents

Uncertain accelerated degradation modeling and analyzing method considering sample individual difference Download PDF

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CN109766518B
CN109766518B CN201811524488.2A CN201811524488A CN109766518B CN 109766518 B CN109766518 B CN 109766518B CN 201811524488 A CN201811524488 A CN 201811524488A CN 109766518 B CN109766518 B CN 109766518B
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李晓阳
吴纪鹏
康锐
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Beihang University
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Abstract

The invention relates to an uncertain accelerated degradation modeling and analyzing method considering sample individual difference, which introduces an uncertain theory (uncertain theory) into the field of accelerated degradation modeling, and divides cognitive uncertainty in accelerated degradation data into two dimensions: and (3) the cognitive uncertainty of the time dimension and the cognitive uncertainty of the sample dimension, and a corresponding reliability and life evaluation function are deduced. In addition, an uncertainty statistical analysis method for the proposed model parameter estimation is presented that minimizes the sum of squares of the difference between the acquired cumulative confidence and the assumed uncertainty distribution based on the least squares principle, and that quantifies the cognitive uncertainty in the time dimension and the sample dimension using objective measures rather than subjective measures.

Description

Uncertain accelerated degradation modeling and analyzing method considering sample individual difference
Technical Field
The invention relates to the technical field of reliability analysis of accelerated performance degradation tests, in particular to an uncertain accelerated degradation modeling and analyzing method considering sample individual differences.
Background
In modern society, with the development of economy and technology, the reliability and the life span level of many products are higher and higher. For these products with high reliability and long life, it is difficult to obtain enough failure data for reliability evaluation by using the traditional reliability test based on failure data, so researchers have proposed an Accelerated Degradation Test (ADT) method. By applying stress which is severe to normal operation, the ADT can accelerate the degradation process of the product, acquire accelerated degradation data within limited test time, and perform reliability and service life evaluation under normal stress.
Generally, there are two types of factors that contribute to uncertainty in the accelerated degradation data: 1) the volatility of the time dimension, which results from the volatility of the degenerate path over time; 2) inter-individual variability of sample dimensions, which results from the intrinsic heterogeneity of different sample individuals. The currently commonly used accelerated Degradation model based on probability theory includes a Degradation path model (Josephlu C, Meeker W. Using Degradation measurements to Estimate a Time-to-failure distribution [ J ]. Technometrics,1993,35(2):161- & ltSUB & gt 174.) and a Stochastic process model (Ye Z, Xiem. Stochartic modeling and analysis of Degradation for a high-level reproducible process [ J ]. Applied Stochartic Models in uncertainty & Industry,2015,31(1):16-32.), which are often treated as random uncertainties. The stochastic process model is widely used because it can better describe the randomness of the degradation process, but some unsolved problems still exist.
For example, in the wiener process model, almost all sample trajectories are continuous but not L ipchitz, which results in a degradation increment for the degradation process that may be infinite in a finite time.
Secondly, according to the law of large numbers, the accelerated degradation model based on probability is very suitable for the situation that sufficient samples exist. However, in practical applications, most ADTs do not provide enough samples for the following reasons: in the time dimension, the number of detections of ADT data may be quite insufficient. Due to the limitation of testing means and capability, some products are difficult to collect enough ADT data in the ADT process, and only a small amount of detection data can be obtained for each test sample within a limited testing time; in the sample dimension, the number of test samples in the ADT is usually small due to the limitation of cost or test resources, which results in the insufficient information provided by the test sample dimension. Here, the insufficient number of detections in the time dimension and the insufficient number of samples in the sample dimension are collectively referred to as a small sample problem. The probability-based accelerated degradation model treats uncertainty due to volatility in the time dimension and inter-individual variability in the sample dimension as random uncertainty, however, small sample problems can lead to difficulty in providing sufficient information to learn the population, resulting in the generation of cognitive uncertainty (epigenetic uncertainties). In this case, the probabilistic accelerated degradation model based on the law of large numbers no longer applies, and the above-mentioned two-dimensional uncertainty should be regarded as a cognitive uncertainty rather than a random uncertainty.
To deal with and deal with the problems mentioned above, the present invention introduces Uncertain theory (Uncertain theory) into the field of accelerated degradation modeling, which measures the probability of occurrence of An event by modeling Reliability (non-frequency) in the case where the sample size is small and the probability theory is no longer applicable, which is a theory different from probability theory and widely applied in many fields (L iu B. Uncertain risk Analysis and Uncertain Reliability Analysis, Ucertain RiskAnalysis and Ucertain Reliability Analysis [ J ]. Journal of Ucertain Systems,2010,4(3):163 and 170; Huang M, Ren L ee L H, model and algorithm for 4P L with Uncertain Analysis time [ J ]. Information Sciences Analysis, International transport [ J.: 2016,330J. 12J. 12 J. Information Sciences Analysis, 76J. (IEEE) and J. 1. simulation).
Disclosure of Invention
Aiming at the problem of cognitive uncertainty caused by small samples with time dimension and sample dimension existing in an accelerated degradation test, the invention provides an uncertain accelerated degradation modeling and analyzing method considering sample individual difference, which specifically comprises the following steps:
1) based on the uncertain theory, carrying out uncertain accelerated degradation modeling on the obtained data
Introducing an uncertain theory into accelerated degradation modeling, and establishing an uncertain accelerated degradation model as follows:
X(s,t)＝e(s)Λ(t)+σC(Λ(t)),e(s)～Nu(μe(s),σe(s)),Λ(t)＝tβ(1)
β>0,σ>0,μe(s)>0,σe(s)>0
where s represents the stress level, t represents time, X (s, t) represents the performance degradation process, which is related to the stress level s and time t, e(s) is the drift coefficient, obeying a mean value μe(s) standard deviation of σeUncertain normal distribution of(s), i.e. e(s) -Nu(μe(s),σe(s)),μe(s)>0,σe(s)>0；Λ(t)＝tβIs a time scale transformation function and is a monotone increasing function of t, when β is equal to 1, X (s, t) is a linear degradation process, otherwise, X (s, t) is a nonlinear degradation process, sigma is a diffusion coefficient, C (Λ (t)) obeys uncertain normal distribution (N)u) I.e., C (Λ (t)) -Nu(0,Λ(t))。
The drift coefficient e(s) in equation (1), as a degradation rate in accelerated degradation modeling, is a function related to the stress level s, i.e., a life-time stress correlation model, as shown in the following equation:
e(sl)＝exp(α0+α1sl)＝aexp(bsl),a>0,b>0 (2)
α therein0And α1Is an unknown model parameter, and α ═ exp (α)0)，b＝α1。
Since e(s) follows an uncertain normal distribution, it is assumed that in equation (2), the unknown parameter a follows a mean value μaVariance is
Figure BDA0001904058050000031
Uncertain normal distribution of (i.e. a to N)u(μa,σa),μa>0,σa>0, then the formula (2) is converted into mueAnd σeThe function for the stress level s is as follows:
ln(μe)＝ln(μa)+bsl,ln(σe)＝ln(σa)+bsl, (3)
μe>0,μa>0,σe>0,σa>0
wherein s islIs the ith normalized acceleration stress level, and for different acceleration stress types, there are different calculation formulas:
Figure BDA0001904058050000032
wherein S is0，SlAnd SHRespectively indicate normalStress level, first acceleration stress level, and highest stress level. Selecting an Arrhenius model when the acceleration stress is temperature; when the acceleration stress is electric stress, a Power lawmodel is selected.
Thus, the performance degradation process X (s, t) in equation (1) follows an uncertain normal distribution, X (s, t) -Nu(μe(s)Λ(t),(σe(s) + σ) Λ (t)), where the deterministic degradation law is characterized by the expectation of X (s, t) in equation (1), i.e., E (X (s, t)) ═ μ ″e(s) Λ (t), wherein the time-dimension variability and the detection times are insufficient and characterized by a fraction of the standard deviation of X (s, t), i.e. the standard deviation Std [ σ C (Λ (t))1 ═ σ Λ (t), of σ C (Λ (t)), and the sample-dimension inter-individual variability and the sample number are insufficient and characterized by another fraction of the standard deviation of X (s, t), i.e. the standard deviation Std [ e(s) Λ (t)1 ═ σ Λ (t), of e(s) Λ (t)e(s) Λ (t) then the uncertain normal distribution of X (s, t) is:
Figure BDA0001904058050000041
wherein phit() Representing the uncertainty distribution, X (s, t) is an uncertainty process with independent increments, whose degradation increment Δ X (s, t) follows the following uncertainty normal distribution: Δ X (s, t) -Nu(μe(s)ΔΛ(t),ΔΛ(t)(σe(s) + σ)), where Δ Λ (t) ═ t + Δ t)β-tβI.e. by
Figure BDA0001904058050000042
2) Estimating the unknown parameters of the established uncertain accelerated degradation model by adopting an uncertain statistical analysis method
And estimating the unknown parameters by adopting an uncertain statistical analysis method based on a Constant Stress Accelerated Degradation Test (CSADT) or a Step Stress Accelerated Degradation Test (SSADT).
xlijRepresents the j-th performance detection value, t, of the i-th sample at the l-th acceleration stress levellijIs the corresponding detection time, l is 1,2, …, k, i is 1,2, …, nl,j＝1,2,…,ml. Wherein k represents the number of acceleration stress levels; n islSample size at the first acceleration stress level, mlRepresenting the number of performance tests for the ith sample at the ith acceleration stress level, let Δ Λ be a misaligned time interval, i.e.
Figure BDA0001904058050000043
And Δ xlijIs the corresponding performance degradation increment, i.e. Δ xlij＝xli(j+1)-xlij. In the uncertain accelerated degradation model composed of equations (1) and (3), the unknown parameter vector is 9 ═ μa,σa,b,σ,β),μa>0,σa>0,b>0,σ>0,β>0。
3) Product reliability and life assessment under normal operating stress
a) First-through time distribution acquisition
Life t of product performance degradation processωDefined as the Time (FHT) at which its degradation process First crosses a given failure threshold omega, i.e. the First Time of failure
tω＝inf{tω≥0|X(s,t)＝ω} (7)
For incremental uncertainty procedure, the uncertainty distribution of its FHT is as follows:
Figure BDA0001904058050000051
where γ (-) represents the uncertain distribution of FHT and z represents time.
b) Reliability and life assessment
Obtaining the distribution of the corresponding service life of the proposed model according to the extreme value theorem in the uncertain theory:
Figure BDA0001904058050000052
it is believed that the reliability distribution is:
Figure BDA0001904058050000053
wherein R isB(t) is the confidence reliability under the uncertainty measure.
At the same time, an uncertain distribution of its confident reliable lifetime (B L (α)) was also obtained:
Figure BDA0001904058050000054
further, the uncertain statistical analysis method based on the Constant Stress Accelerated Degradation Test (CSADT) in the step 2) is as follows:
a) build confidence
As can be seen from equation (6), the j-th degradation incremental variable, Δ x, under the l-th acceleration stresslj＝(Δxli1,…,Δxlij,…),i＝1,2,…,nlIs an uncertain variable following an uncertain normal distribution, where Δ xlijIs Δ xljObserved values at the ith sample. Calculating Deltax from the obtained accelerated degradation dataljAll observed values of (a) xlij。
From the perspective of risk analysis, it is reasonable to assume that the larger the degradation increment is, the larger the corresponding cumulative confidence is, and therefore, an approximate median rank/average rank formula commonly used in statistics is adopted to obtain all observed values Δ x of the kth degradation increment variable under the ith stresslijDegree of confidence αlij：
Figure BDA0001904058050000061
Wherein i represents an ith non-repeating observation of a jth degradation increment variable at the ith acceleration stress; n is a radical ofljRepresenting the number of unequal observations, with an upper limit of nl(ii) a A and B take different values, so that different approximate median rank/average rank formulas can be obtained, as follows:
F1:αlij＝i/(Nlj+1),i＝1,...,Nlj(13)
F2:αlij＝(i-0.5)/Nlj,i＝1,...,Nlj(14)
F3:αlij＝(i-0.3)/(Nlj+0.4),i＝1,...,Nlj(15)
F4:αlij＝(i-0.5)/(Nlj+0.25),i＝1,...,Nlj(16)
for all observed values of the jth degradation incremental variable under the ith acceleration stress, if two or more observed values have the same size, the corresponding credibility is also the same; in a Constant Stress Accelerated Degradation Test (CSADT), the number of samples may be different at each stress level.
b) Estimating unknown parameters
Obtaining the reliability of each performance degradation increment observed value based on the formulas (13) to (16), performing parameter estimation by adopting a least square principle, and minimizing the square sum of the difference between the assumed distribution and the obtained reliability, namely
Figure BDA0001904058050000062
Wherein Q ispRepresenting the corresponding objective function, theta, after obtaining confidence levels using equations (13) - (16)pRepresenting the corresponding unknown parameter vector αlijRepresenting the confidence level corresponding to the performance degradation increment.
c) Optimizing parameter evaluation results
According to formula (17), QpThe smaller the corresponding obtained parameter estimation result, the more appropriate it is if all Q's are calculatedpThe following formula is satisfied, i.e. the corresponding parameter estimation result is used as the final parameter estimation result:
θfinal＝{θ|Qfinal＝min{Qp}},p＝1,2,3,4 (18)
therein, 9finalRepresenting the final parameter estimation result, QfinalRepresenting the minimum objective function Q.
Further, the uncertain statistical analysis method based on the Step Stress Accelerated Degradation Test (SSADT) in the step 2) is as follows:
in the Step Stress Accelerated Degradation Test (SSADT), the number of samples was the same at each stress levelI.e., n 1-n 2- … … -n, all test specimens were started from the lowest acceleration stress and gradually increased to the highest acceleration stress, i.e., S1→S2→…→Sk。
The process of performance degradation in SSADT is represented as follows:
Figure BDA0001904058050000071
wherein [ tl-1,tl) Represents the time zone in which the test specimen was subjected to the test under the first accelerating stress.
a) Sorting SSADT data into groups by stress level
b) Converting SSADT data into CSADT data
According to equation (6), for the ith sample at the ith acceleration stress level,
Figure BDA0001904058050000078
Figure BDA0001904058050000077
obey mean value of
Figure BDA0001904058050000072
Standard deviation of
Figure BDA0001904058050000073
Is not determined to be normal, i.e.
Figure BDA0001904058050000074
Wherein
Figure BDA0001904058050000075
μelAnd σelThe degradation rate e at the first acceleration stress levellMean and standard deviation of. At this time, x'lijIs CSADT data, and
Figure BDA0001904058050000076
is the corresponding transition time.
c) Parameter estimation
After SSADT data is converted into CSADT data, an uncertain statistical analysis method which is the same as a Constant Stress Accelerated Degradation Test (CSADT) is adopted, and a parameter estimation result of the established model under the SSADT can be obtained.
Compared with the prior art, the invention has the following beneficial effects:
1. according to the source of cognitive uncertainty, the invention divides the cognitive uncertainty in the accelerated degradation data into two dimensions: cognitive uncertainty in the time dimension and cognitive uncertainty in the sample dimension.
2. Based on an uncertain theory, the invention constructs a new uncertain accelerated degradation model and deduces a corresponding reliability and service life evaluation function. In the proposed model of uncertain accelerated degradation, the cognitive uncertainties in both the time dimension and the sample dimension are clearly quantified by different variables.
3. The invention also provides an uncertain statistical analysis method for the proposed model parameter estimation, which is based on the least square principle, minimizes the sum of squares of the difference between the obtained accumulated credibility and the assumed uncertain distribution, and adopts objective measurement rather than subjective measurement to quantify the cognitive uncertainty of the time dimension and the sample dimension.
Drawings
FIG. 1 is a flow chart of the method for modeling and evaluating uncertain accelerated degradation established by the present invention.
FIG. 2 is a flow chart of an uncertain statistical analysis method under a constant stress accelerated degradation test established by the invention.
Fig. 3 is a graph showing stress relaxation constant stress accelerated degradation test data of a certain connector in example 1 of the present invention.
Fig. 4 is a graph showing the deterministic degradation law research-degradation rate trend with stress level change in the case of the stress relaxation constant stress accelerated degradation test of a certain connector in example 1 of the present invention.
Fig. 5 is a graph of uncertainty research-actual data VS simulation data in the case of the stress relaxation constant stress accelerated degradation test of a certain connector in example 1 of the present invention.
Fig. 6 is a graph of the trend of the deterministic degradation law research-time VS stress level VS stress relaxation change in the stress relaxation constant stress accelerated degradation test case of a certain connector in the embodiment 1 of the present invention.
Fig. 7 is a graph showing the reliability of assurance against normal operating stress and the reliability life of assurance against stress relaxation constant stress accelerated degradation test case of a certain connector according to example 1 of the present invention.
FIG. 8 is a graph of light intensity step stress accelerated degradation test data for a certain L ED in example 2 of the present invention.
Fig. 9 is a graph of deterministic degradation law study-degradation rate versus stress level trend in the light intensity step-by-step stress accelerated degradation test case of a certain L ED in example 2 of the present invention.
FIG. 10 is a graph of uncertainty research-actual data VS simulation data in the light intensity step stress accelerated degradation test case of a certain L ED in example 2 of the present invention.
Fig. 11 is a graph of the trend of the deterministic degradation law study-time VS stress level VS stress relaxation change in the light intensity step stress accelerated degradation test case of a certain L ED in example 2 of the present invention.
FIG. 12 is a graph of confidence reliability versus confidence lifetime for a certain L ED light intensity step stress accelerated degradation test case under normal operating stress in example 2 of the present invention.
Detailed Description
The stress relaxation constant stress accelerated degradation test data of a certain electric connector and the light intensity stepping stress accelerated degradation test data of a certain L ED are taken as actual cases to develop the work of determinacy degradation rule, uncertainty quantification and reliability and service life evaluation, and the implementation technical scheme provided above is explained and explained.
Example 1
Stress relaxation constant stress accelerated degradation test (SR CSADT Case) for certain electrical connector
The data of the SR CSADT Case are shown in FIG. 3, and the basic information of the test is shown in Table 1 below.
TABLE 1 SR CSADT Case basic information
Figure BDA0001904058050000091
As can be seen from table 1 and fig. 3, in this case, the number of detections and the number of samples under each acceleration stress are 12, 11, 11 and 6, 6, 6, respectively, and there is a cognitive uncertainty caused by the problem of small samples in both the time dimension and the sample dimension, so that the proposed method is suitable for modeling and evaluation.
The method comprises the following specific steps:
1. based on the uncertain theory, carrying out uncertain accelerated degradation modeling on the obtained data
As can be seen from the analysis in the previous paragraph, the CSADT data obtained by the case is suitable for carrying out deterministic degradation rules and non-deterministic modeling by the non-deterministic accelerated degradation model composed of the formulas (1) and (3). Since the acceleration stress is a temperature stress, an Arrenhius model is selected as a formula for characterizing the relationship between the performance degradation process and the acceleration stress in the stress-life correlation model, wherein the temperature is expressed in kelvin (K).
2. Estimating the unknown parameters of the established uncertain accelerated degradation model by adopting an uncertain statistical analysis method
This case is CSADT, so the estimation of the model parameters can be done directly according to fig. 2.
a) Build confidence
First, all the performance degradation increments are calculated, and the j degradation increment variable, deltax, under the l acceleration stress is calculatedlj＝(Δxli1,…,Δxlij,…),i＝1,2,…,nlSort from small to large, then according to F1–F4(equations (13) - (16)) obtain the reliability thereof. For example, when l is 1 and j is 1, NljIf 6, use F1–F4The obtained credibility is respectively:
TABLE 2 Performance degradation increment correspondence confidence acquisition
Figure BDA0001904058050000101
b) Estimation of unknown parameters
According to the above confidence level and equation (17), the parameter estimation result is obtained as follows:
TABLE 3 SR CSADT Case parameter estimation results
Figure BDA0001904058050000102
c) Parameter estimation result optimization
As can be seen from Table 3, using F1Obtaining the parameter estimation corresponding to the reliability is that the obtained objective function Q is minimum, namely Qfinal＝min{Ql}＝Q1And therefore its corresponding parameter estimation result as the final parameter estimation result, i.e. thetafinal＝θ1。
3. Deterministic degradation rule and uncertainty analysis
And analyzing the deterministic degradation rule of the actual case data and the uncertainty contained in the actual case data based on the proposed uncertain accelerated degradation model and the obtained parameter estimation result.
For deterministic degradation laws, μ in equation (1) can be employede(s) Λ (t) and equation (3), the results are shown in fig. 4 and 5, and for the uncertainty study, 500 simulations were performed at each acceleration stress using the proposed uncertain accelerated degradation model and the obtained parameter estimation results, and these simulation data were put in the same graph as the actual data shown in fig. 3, as shown in fig. 6, it can be seen from the results of fig. 4-6 that the proposed uncertain accelerated degradation model is suitable for the case of the deterministic degradation law and uncertainty study in a small sample.
4. Product reliability and life assessment under normal operating stress
Based on the obtained parameter evaluation results and equations (10) and (11), the product reliability and lifetime evaluation of the case under normal working stress is obtained, and the results are shown in fig. 7.
Example 2
Light intensity step stress accelerated degradation test for a certain L ED (L ED SSADT Case)
The L ED SSADT Case data is shown in FIG. 8, which is the basic information for the assay, as shown in Table 4 below.
Table 4L ED SSADT Case basic information
Figure BDA0001904058050000111
As can be seen from table 4 and fig. 8, in this case, the number of detections and the number of samples under each acceleration stress are 7, 4, 2, and 5, respectively, and there is a cognitive uncertainty caused by the problem of small samples in both the time dimension and the sample dimension, and thus the proposed method is suitable for modeling and evaluation.
The method comprises the following specific steps:
1. determining theory, and performing uncertain accelerated degradation modeling on obtained data
As can be seen from the analysis in the previous paragraph, the SSADT data obtained in this embodiment is suitable for the deterministic degradation law and the non-deterministic modeling of the non-deterministic accelerated degradation model composed of equations (1) and (3). Since the acceleration stress is a temperature stress, an Arrenhius model is selected as a formula for characterizing the relationship between the performance degradation process and the acceleration stress in the stress-life correlation model, wherein the temperature is expressed in kelvin (K).
2. Estimating the unknown parameters of the established uncertain accelerated degradation model by adopting an uncertain statistical analysis method
This case is under SSADT, so the SSADT data is first converted to CSADT data, and then the model parameters are estimated according to fig. 2.
a) Converting SSADT data into CSADT data
According to
Figure BDA0001904058050000124
And
Figure BDA0001904058050000121
the SSADT data will be converted to CSADT data.
b) Build confidence
Calculating all performance degradation increments, and calculating the j degradation increment variable, delta x under the l acceleration stresslj＝(Δxli1,…,Δxlij,…),i＝1,2,…,nlSort from small to large, then according to F1–F4(equations (11) - (16)) obtain the reliability thereof. For example, when l is 1 and j is 1, NljIf 3, use F1–F4The obtained credibility is respectively:
TABLE 5 Performance degradation increment correspondence confidence acquisition
Figure BDA0001904058050000122
c) Estimation of unknown parameters
According to the above confidence level and equation (17), the parameter estimation result is obtained as follows:
table 6L ED SSADT Case parameter estimation results
Figure BDA0001904058050000123
d) Parameter estimation result optimization
As can be seen from Table 6, using F1The acquisition confidence is that the corresponding parameter estimate is that the resulting objective function Q is minimal, i.e. Qfinal＝min{Ql}＝Q1And therefore its corresponding parameter estimation result as the final parameter estimation result, i.e. thetafinal＝θ1。
3. Deterministic degradation rule and uncertainty analysis
And analyzing the deterministic degradation rule of the actual case data and the uncertainty contained in the actual case data based on the proposed uncertain accelerated degradation model and the obtained parameter estimation result.
For deterministic degradation laws, μ in equation (1) can be employede(s) Λ (t) and equation (3), the results are shown in FIGS. 9 and 10, and for uncertainty studies, the proposed uncertainty accelerated degradation model and the obtained parameter scores were usedThe results were estimated to be 500 simulations at each acceleration stress, and these simulation data were put in the same graph as the actual data shown in fig. 8, see fig. 11. As can be seen from the results of fig. 9-11, the proposed uncertain accelerated degradation model is suitable for the deterministic degradation law and uncertainty study for this case in small samples.
4. Product reliability and life assessment under normal operating stress
Based on the obtained parameter evaluation results and equations (10) and (11), the product reliability and lifetime evaluation of the case under normal working stress is obtained, and the result is shown in fig. 12.
Although exemplary embodiments of the present invention have been described for illustrative purposes, those skilled in the art will appreciate that various modifications, additions, substitutions and the like can be made in form and detail without departing from the scope and spirit of the invention as disclosed in the accompanying claims, all of which are intended to fall within the scope of the claims, and that various steps in the various sections and methods of the claimed product can be combined together in any combination. Therefore, the description of the embodiments disclosed in the present invention is not intended to limit the scope of the present invention, but to describe the present invention. Accordingly, the scope of the present invention is not limited by the above embodiments, but is defined by the claims or their equivalents.

Claims (3)

1. An uncertain accelerated degradation modeling and analyzing method considering sample individual differences specifically comprises the following steps:
1) based on the uncertain theory, carrying out uncertain accelerated degradation modeling on the obtained data
Introducing an uncertain theory into accelerated degradation modeling, and establishing an uncertain accelerated degradation model as follows:
Figure FDA0002524262980000011
wherein s represents the stress level and t representsTABLE time, X (s, t) stands for the process of performance degradation, which is related to stress level s and time t, e(s) is the drift coefficient, obeying a mean value of μe(s) standard deviation of σeUncertain normal distribution of(s), i.e. e(s) -Nu(μe(s),σe(s)),μe(s)>0,σe(s)>0；Λ(t)＝tβIs a time scale transformation function and is a monotone increasing function of t, when β is equal to 1, X (s, t) is a linear degradation process, otherwise, X (s, t) is a nonlinear degradation process, sigma is a diffusion coefficient, C (Λ (t)) obeys uncertain normal distribution (N)u) I.e., C (Λ (t)) -Nu(0,Λ(t))；
The drift coefficient e(s) in equation (1), as a degradation rate in accelerated degradation modeling, is a function related to the stress level s, i.e., a life-time stress correlation model, as shown in the following equation:
e(sl)＝exp(α0+α1sl)＝aexp(bsl),a>0,b>0 (2)
α therein0And α1Is an unknown model parameter, and a ═ exp (α)0)，b＝α1；
Since e(s) follows an uncertain normal distribution, it is assumed that in equation (2), the unknown parameter a follows a mean value μaVariance is
Figure FDA0002524262980000012
Uncertain normal distribution of (i.e. a to N)u(μa,σa),μa>0,σa>0, then the formula (2) is converted into mueAnd σeThe function for the stress level s is as follows:
Figure FDA0002524262980000013
wherein s islIs the ith normalized acceleration stress level, and for different acceleration stress types, there are different calculation formulas:
Figure FDA0002524262980000014
wherein S is0，SlAnd SHRespectively representing a normal stress level, an ith acceleration stress level, and a highest stress level; selecting an Arrhenius model when the acceleration stress is temperature; selecting a Powerlaw model when the acceleration stress is the electric stress;
thus, the performance degradation process X (s, t) in equation (1) follows an uncertain normal distribution, X (s, t) -Nu(μe(s)Λ(t),(σe(s) + σ) Λ (t)), where the deterministic degradation law is characterized by the expectation of X (s, t) in equation (1), i.e., E (X (s, t)) ═ μ ″e(s) Λ (t), and the time dimension fluctuation and the detection times are insufficient, which are determined by the standard deviation Std [ sigma C (Λ (t)) of a part of the standard deviation of X (s, t), namely sigma C (Λ (t))]σ Λ (t), and the inter-individual variability of sample dimensions and the lack of sample numbers are characterized by the other part of the standard deviation of X (s, t), namely the standard deviation Std [ e(s) Λ (t) of e(s) Λ (t)]＝σe(s) Λ (t), the uncertain normal distribution of X (s, t) is:
Figure FDA0002524262980000021
wherein phit() Representing the uncertainty distribution, X (s, t) is an uncertainty process with independent increments, whose degradation increment Δ X (s, t) follows the following uncertainty normal distribution: Δ X (s, t) -Nu(μe(s)ΔΛ(t),ΔΛ(t)(σe(s) + σ)), where Δ Λ (t) ═ t + Δ t)β-tβI.e. by
Figure FDA0002524262980000022
2) Estimating the unknown parameters of the established uncertain accelerated degradation model by adopting an uncertain statistical analysis method
Estimating unknown parameters by adopting an uncertain statistical analysis method based on a Constant Stress Accelerated Degradation Test (CSADT) or a Stepping Stress Accelerated Degradation Test (SSADT);
xlijrepresents the j-th performance detection value, t, of the i-th sample at the l-th acceleration stress levellijIs the corresponding detection time, l is 1,2, …, k, i is 1,2, …, nl,j＝1,2,…,mlWhere k represents the number of acceleration stress levels, nlSample size at the first acceleration stress level, mlRepresenting the performance test times of the ith sample at the ith acceleration stress level, and setting delta Λ as a misaligned time interval, i.e.
Figure FDA0002524262980000023
And Δ xlijIs the corresponding performance degradation increment, i.e. Δ xlij＝xli(j+1)-xlij(ii) a In the uncertain accelerated degradation model composed of the equations (1) and (3), the unknown parameter vector is θ ═ μa,σa,b,σ,β),μa>0,σa>0,b>0,σ>0,β>0；
3) Product reliability and life assessment under normal operating stress
a) First-through time distribution acquisition
Life t of product performance degradation processωDefined as the time FHT at which its degradation process first crosses a given failure threshold omega, i.e.
tω＝inf{tω≥0|X(s,t)＝ω} (7)
For incremental uncertainty procedure, the uncertainty distribution of its FHT is as follows:
Figure FDA0002524262980000031
wherein γ (·) represents the uncertain distribution of FHT, z represents time;
b) reliability and life assessment
Obtaining the distribution of the corresponding service life of the proposed model according to the extreme value theorem in the uncertain theory:
Figure FDA0002524262980000032
it is believed that the reliability distribution is:
Figure FDA0002524262980000033
wherein R isB(t) is the confidence reliability under the uncertainty measure,
at the same time, an uncertain distribution of its confidence reliability lifetime B L (α) was also obtained as follows:
Figure FDA0002524262980000034
where α represents a confidence level.
2. The method for modeling and analyzing uncertain accelerated degradation considering sample individual differences according to claim 1, wherein the uncertain statistical analysis method based on Constant Stress Accelerated Degradation Test (CSADT) in step 2) is as follows:
a) build confidence
As can be seen from equation (6), the j-th degradation incremental variable, Δ x, under the l-th acceleration stresslj＝(Δxli1,…,Δxlij,…),i＝1,2,…,nlIs an uncertain variable following an uncertain normal distribution, where Δ xlijIs Δ xljCalculating delta x according to the obtained accelerated degradation data based on the observed value at the ith sampleljAll observed values of (a) xlij；
From the perspective of risk analysis, it is reasonable to assume that the larger the degradation increment is, the larger the corresponding cumulative confidence is, and therefore, an approximate median rank/average rank formula commonly used in statistics is adopted to obtain all observed values Δ x of the kth degradation increment variable under the ith stresslijDegree of confidence αlij：
Figure FDA0002524262980000041
Wherein i represents the j-th degradation incremental variable under the l-th acceleration stressIth non-repeating observation, NljRepresenting the number of unequal observations, with an upper limit of nlThe values of A and B are different, so that different approximate median rank/average rank formulas can be obtained, as follows:
F1:αlij＝i/(Nlj+1),i＝1,...,Nlj(13)
F2:αlij＝(i-0.5)/Nlj,i＝1,...,Nlj(14)
F3:αlij＝(i-0.3)/(Nlj+0.4),i＝1,...,Nlj(15)
F4:αlij＝(i-0.5)/(Nlj+0.25),i＝1,...,Nlj(16)
for all observed values of the jth degradation incremental variable under the ith accelerated stress, if two or more observed values have the same size, the corresponding reliability is also the same, and in the constant stress accelerated degradation test CSADT, the number of samples may be different at each stress level;
b) estimating unknown parameters
Obtaining the reliability of each performance degradation increment observed value based on the formulas (13) to (16), performing parameter estimation by adopting a least square principle, and minimizing the square sum of the difference between the assumed distribution and the obtained reliability, namely
Figure FDA0002524262980000051
Wherein Q ispRepresenting the corresponding objective function, theta, after obtaining confidence levels using equations (13) - (16)pRepresenting the corresponding unknown parameter vector, αlijRepresenting the corresponding reliability of the performance degradation increment;
c) optimizing parameter evaluation results
According to formula (17), QpThe smaller the corresponding obtained parameter estimation result, the more appropriate it is if all Q's are calculatedpThe following formula is satisfied, i.e. the corresponding parameter estimation result is used as the final parameter estimation result:
θfinal＝{θ|Qfinal＝min{Qp}},p＝1,2,3,4 (18)
wherein, thetafinalRepresenting the final parameter estimation result, QfinalRepresenting the minimum objective function Q.
3. The modeling and analyzing method for uncertain accelerated degradation considering sample individual differences according to claim 1, wherein the uncertain statistical analysis method based on Step Stress Accelerated Degradation Test (SSADT) in step 2) is as follows:
in the step stress accelerated degradation test SSADT, the number of samples is the same at each stress level, i.e., n 1-n 2- … … -n, and all test samples are gradually increased from the lowest accelerated stress to the highest accelerated stress, i.e., S1→S2→…→Sk；
The process of performance degradation in SSADT is represented as follows:
Figure FDA0002524262980000052
wherein [ tl-1,tl) Represents the time zone of the test sample under the first accelerated stress;
a) SSADT data was divided into several groups by stress level;
b) converting the SSADT data into CSADT data;
according to equation (6), for the ith sample at the ith acceleration stress level,
Figure FDA0002524262980000053
Figure FDA0002524262980000054
obey mean value of
Figure FDA0002524262980000055
Standard deviation of
Figure FDA0002524262980000056
Is not determined to be normal, i.e.
Figure FDA0002524262980000057
Wherein
Figure FDA0002524262980000058
μelAnd σelThe degradation rate e at the first acceleration stress levellMean and standard deviation of (c), at this time, x'lijIs CSADT data, and
Figure FDA0002524262980000061
is the corresponding transition time;
c) parameter estimation
After SSADT data are converted into CSADT data, an uncertain statistical analysis method which is the same as that of a constant stress accelerated degradation test CSADT is adopted, and a parameter estimation result of the established model under the SSADT can be obtained.
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