CN111046564B - Residual life prediction method for two-stage degraded product - Google Patents
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Abstract
The invention discloses a method for predicting the residual life of a two-stage degradation product, which belongs to the field of prediction and prediction of the residual life in health management, and mainly comprises the following steps: modeling a degradation process, estimating model parameters and predicting the residual life; the degradation process modeling is to build a two-stage degradation model by utilizing a nonlinear Wiener process; the model parameter estimation includes: historical degradation data collection; model parameter estimation based on a unit maximum likelihood estimation method; statistical analysis of random parameter distribution; the remaining life prediction includes: acquiring the first arrival time distribution; deducing a state transition probability function; and predicting the residual life of the product based on the estimated parameters. The invention can effectively predict the residual life of the two-stage degraded product, is beneficial to ensuring the running reliability of the product, reducing the maintenance cost and avoiding the safety accident.
Description
Technical Field
The invention belongs to the field of prediction and health management, and relates to a method for predicting the residual life of a two-stage degradation product.
Background
The residual life prediction is core content of a prediction and health management technology, refers to an effective time interval left by the fact that a product distance at the current moment loses a specified function, is an important index for reflecting the reliability of the product, and has important significance for practically guaranteeing the operation safety, reliability and economy of the product.
Residual life prediction has been widely focused and studied in recent decades. Among them, the Wiener process has been developed for a long term because it can describe a non-monotonic degradation trajectory and has good mathematical properties. It is worth noting that in most Wiener process-based life prediction methods, the degradation rate is typically fixed and does not change over time. However, in practical engineering, there is often a significant degradation rate variation in the degradation trace of many products due to variations in external operating conditions and internal mechanisms, exhibiting two-stage degradation phenomena such as high performance capacitors, LCDs, liquid coupling devices, light emitting diodes, batteries, bearings, etc., as shown in fig. 2. For such two-stage degradation products, conventional life prediction methods have low prediction accuracy.
To improve the accuracy of predictions, many researchers have proposed a two-stage degradation model for this, but the existing methods focus mainly on whether the variability is random or whether individual variability is considered, neglecting the derivation of analytical solutions for degradation nonlinearities and lifetime for each stage. While each stage of the existing degradation model is a linear Wiener process, as can be seen from fig. 2, in practice, the degradation process of each stage exhibits a nonlinear characteristic due to changes in load, internal state, and external environment. Therefore, it is more reasonable to study the residual life prediction method of the two-stage degraded product based on the nonlinear Wiener process.
Disclosure of Invention
In view of the above, the present invention aims to provide a new method for predicting the residual life of a two-stage degradation product, which overcomes the defects of the prior art and can effectively predict the residual life of the two-stage degradation product.
In order to achieve the above purpose, the present invention provides the following technical solutions:
the method for predicting the residual life of the two-stage degraded product comprises three parts of contents of degradation process modeling, model parameter estimation and residual life prediction, and specifically comprises the following steps of: step one: modeling a degradation process, namely segmenting from a degradation transformation point, and establishing a two-stage degradation model by using a nonlinear Wiener process model; step two: estimating model parameters, namely estimating model unknown parameters by collecting historical degradation data and utilizing a two-stage unit MLE method; step three: predicting the residual life, and deriving an approximate analytic solution of the first-time distribution by utilizing space-time transformation based on the two-stage degradation model obtained in the step one; step four: taking the first arrival time distribution result obtained in the step three as a basis, taking the degradation rate of each stage as a random variable, and obtaining a first arrival time distribution function taking individual variability into consideration based on a total probability law; step five: under the condition that the variable points do not appear, according to the Wiener process characteristics, a transition probability density function for transitioning from an initial degradation state to the variable point degradation state can be deduced, and then, based on a total probability law, a state transition density function considering individual differences is obtained; step six: based on the results obtained in the step four and the step five, a life probability density function under the first arrival time concept of the two-stage degradation model can be deduced based on the total probability law and the Gaussian distribution property; substituting the estimated value of the model parameter obtained in the second step into a life probability density function, so that the life prediction of the two-stage degraded product is realized.
In the first step, segmentation is carried out from a change point, each stage carries out degradation modeling by utilizing a nonlinear Wiener process, and the constructed two-stage degradation model is as follows:
wherein: x (t) represents the performance degradation amount of the product, X (0) and X (τ) represent the degradation states at the initial time and the change point respectively,and->Is a drift function of each stage, sigma 1 Sum sigma 2 Is the diffusion coefficient of each stage, B (t) is the standard Brownian motion;
in the second step, model unknown parameter estimation is performed by the proposed two-stage unit MLE method, which specifically comprises the following steps: first, collecting historical degradation data of N degradation products, and establishing a training data set delta X of degradation increment n Wherein m is n Representing measurements of the nth productNumber, deltat n，j-1 ＝t n，j -t n，j-1 (j＝1，..，m n ) Representing the detection interval.
In the first stage of parameter estimation, degradation model parameters are estimated by using degradation data of each test sampleΔX according to the characteristics of standard Brownian motion n Obeying the normal distribution, the log-likelihood function for the parameter Θ is expressed as:
wherein: m is m a And m b Representing the number of measurements of the first stage and the second stage, satisfying m a +m b ＝m n ，
Regarding a to log likelihood function 1，n ，α 2，n (n＝1，...，N)，And->And (3) making the bias equal to 0, and obtaining corresponding parameter estimated values:
wherein: the "FMINSEARCH" in MATLAB is used to search the parameter beta 1 And beta 2 Substituting the above formula to obtain the estimated value. And then solving the maximum log likelihood function to obtain the value of the variable point.
In the second phase of parameter estimation, each sample estimated by the first phase is usedAnd->And->And carrying out statistical analysis to obtain the super-parameters corresponding to the random parameter distribution.
Further, in step three, based on the space-time transformation, referring to the lifetime derivation process of the nonlinear Wiener process, the first time distribution function is expressed as:
wherein: f (f) T (t|τ) is a life probability density function at time t, x τ Is the amount of degradation at the change point;
further, in the fourth step, based on the result of the third step, the degradation rate of each stage is regarded as a random variable, i.e., setAnd->According to the law of total probability, the first-arrival time distribution function taking into account individual variability is expressed as:
(1)0＜t≤τ
(2)t＞τ
further, in the fifth step, if the transformation point does not appear, it is difficult to accurately acquire the degradation state at the transformation point. The occurrence of the second stage degradation process means that the first stage degradation does not reach the failure threshold, i.e. the lifetime is greater than the transition point time. The probability of a state transition into the second phase is defined as:
m τ (xτ)＝Pr{X(τ)＝x τ |X(0)＝x 0 ，T＞τ}Pr{T＞τ}
wherein: m is m τ (x τ ) Representing slave state x 0 Transition to state x τ Is used for the transition probability density of (1).
According to Wiener process characteristics and the law of total probability, a transition probability density function taking into account the influence of random effects can be expressed as:
further, in step six, based on the results of step four and step five, the lifetime probability density function under the concept of the time of arrival can be further expressed as:
if t is more than 0 and less than or equal to tau，f T The expression of (t|τ) is the same as the result of step four, and if t > τ, based on the gaussian distribution property, the integral in the above expression can be solved, and the life probability density function can be obtained as follows:
wherein: a=a 1 -A 2 ，B＝B 1 -B 2
Wherein: phi (·) and phi (·) represent the cumulative distribution function and probability density function of the standard normal distribution respectively,
substituting the parameters estimated in the second step into the life probability density function obtained in the sixth step, so that the residual life prediction of the two-stage product can be realized.
The core ideas and principles of the invention are:
the invention provides a two-stage degradation model based on a nonlinear Wiener process, which carries out model unknown parameter estimation by a two-stage unit MLE method, considers individual variability and uncertainty of a variable point degradation state, and deduces an approximate analysis solution of the residual life under the concept of first arrival time.
The beneficial effects of the invention are as follows:
1) Compared with the existing two-stage degradation model, the method adopts the nonlinear Wiener process to construct the two-stage degradation model, so that the two-stage degradation process is more reasonably described; 2) The model is subjected to parameter estimation by a two-stage unit MLE method, and the method can be used for estimating the parameters of each sample independently and effectively without being limited by random parameter distribution; 3) The approximate analytic solution of the first time distribution of the two-stage degradation model is obtained through space-time transformation, so that the service life of the two-stage degradation model can be effectively represented; 4) The uncertainty of the degradation state at the variable point and the difference between individuals are comprehensively considered, and the accuracy of life prediction is improved.
Drawings
In order to make the purposes, technical schemes and beneficial effects of the invention clearer, the invention provides the following drawings for description:
FIG. 1 is a flow chart of the residual life prediction of a two-stage degraded product of the present invention;
FIG. 2 is a schematic diagram of a degradation curve of a two-stage degradation product of the present invention;
FIG. 3 is a simulated degradation trace plot;
FIG. 4 is a schematic diagram of residual life prediction;
FIG. 5 is a schematic diagram showing the comparison of predicted and actual remaining life.
Detailed Description
Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be understood that the preferred embodiments are presented by way of illustration only and not by way of limitation.
The implementation process of the embodiment specifically includes: the degradation process modeling, model parameter estimation and residual life prediction are performed by utilizing numerical simulation based on MATLAB tools, and the effect of the invention is shown by combining with the attached drawings. FIG. 1 is a flow chart of life prediction of a two-stage degraded product, as shown, the method specifically includes the steps of:
step one: and modeling a degradation process. Fig. 2 is a schematic diagram of the degradation of a two-stage degradation product, from which it can be seen that the degradation process of the product exhibits a two-stage degradation phenomenon. Thus, segmentation at the degradation transition point, utilizing two nonlinear Wi with different degradation rates e n e And (3) carrying out degradation modeling in the r process, wherein the constructed two-stage degradation model is as follows:
wherein: x (t) represents the performance degradation amount of the product, X (0) and X (τ) represent the degradation states at the initial time and the change point respectively,and->Is a drift function of each stage, sigma 1 Sum sigma 2 Is the diffusion coefficient of each stage, and B (t) is the standard brownian motion. In order to express the differences between individuals, let +.>And->But for each sample these values are fixed values.
The lifetime under the concept of the time of arrival can then be expressed as:
T＝inf{t：X(t)≥w|X(0)≤w}
wherein: w is a failure threshold, the value of which is determined by relevant industry standards or expert experience, and is usually a constant.
Step two: and (5) estimating model parameters. A degradation trajectory of a plurality of sampled samples is first generated using MATLAB, as shown in fig. 3. Constructing a degradation incremental training set for each set of degradation dataIn the first stage of parameter estimation, degradation model parameters are estimated by using degradation data of each test sampleΔX according to the characteristics of standard Brownian motion n Obeying normal distribution, and obtaining a log-likelihood function of the parameter Θ:
wherein: m is m a And m b Representing the number of measurements of the first stage and the second stage, satisfying m a +m b ＝m n ，
Regarding a to log likelihood function 1，n ，α 2，n (n＝1，...，N)，And->And (3) making the bias equal to 0, and obtaining a corresponding parameter estimated value:
wherein: the "FMINSEARCH" in MATLAB is used to search the parameter beta 1 And beta 2 Substituting the above formula to obtain the estimated value. And then solving the maximum log likelihood function to obtain the value of the variable point.
Thus, each sample variation point can be obtained. In the second phase of parameter estimation, each sample estimated by the first phase is usedAnd->And->Statistical analysis is performed, and assuming that the variation point obeys the normal distribution, the mean parameter of the normal distribution obeys the variation point is 49.93, and the variance is 0.98.
Step three: and predicting the residual life. Referring to the lifetime derivation process of the nonlinear Wiener process, if the degradation process is:
based on space-time transformation, it is possible to:
wherein:
similarly, the first-time distribution function of the two-stage degradation model is expressed as:
wherein: f (f) T (t|τ) is a life probability density function at time t, x τ Is the amount of degradation at the change point.
Step four: considering individual variability, the degradation rate of each stage is regarded as a random variable, i.e. setAnd->According to the following theorem:
based on the law of total probability, the first-arrival time distribution function taking into account individual variability is expressed as:
(1)0＜t≤τ
(2)t＞τ
step five: if the change point does not appear, it is difficult to accurately acquire the degradation state at the change point. The occurrence of the second stage degradation process means that the first stage degradation does not reach the failure threshold, i.e. the lifetime is greater than the transition point time. The probability of a state transition into the second phase is defined as:
m τ (x τ )＝Pr{X(τ)＝x τ |X(0)＝x 0 ，T＞τ}Pr{T＞τ}
wherein: m is m τ (x τ ) Representing slave state x 0 Transition to state x τ Is used for the transition probability density of (1).
According to Wiener process characteristics and the law of total probability, a transition probability density function taking into account the influence of random effects can be expressed as:
step six: based on the results of step four and step five, the life probability density function under the concept of the first time can be further expressed as:
if t is more than 0 and less than or equal to tau, f T The expression of (t|τ) is the same as the result of step four, and if t > τ, based on the gaussian distribution property, the integral in the above expression can be solved, and the life probability density function can be obtained as follows:
wherein: a=a 1 -A 2 ，B＝B 1 -B 2
Wherein: phi (·) and phi (·) represent the cumulative distribution function and probability density function of the standard normal distribution respectively,
substituting the parameters estimated in the second step into the life probability density function obtained in the sixth step to obtain a residual life probability density function of the two-stage product, wherein the residual life probability density function is shown in fig. 4, and fig. 5 is a comparison between the residual life prediction expectation obtained by the method of the invention and the results of the real residual life, the single-stage nonlinear degradation model and the two-stage linear degradation model. From the result of life prediction, the method can effectively predict the residual life of the two-stage degradation product.
Finally, it is noted that the above embodiments are only for illustrating the technical solution of the present invention and not for limiting the same, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications and equivalents may be made thereto without departing from the spirit and scope of the present invention, which is intended to be covered by the claims of the present invention.
Claims (4)
1. The method for predicting the residual life of the two-stage degradation product is characterized by comprising the following steps of: the method comprises the following steps:
step one: modeling a degradation process, namely segmenting from a degradation transformation point, and establishing a two-stage degradation model by using a nonlinear Wiener process model;
in the first step, segmentation is carried out from a change point, each stage carries out degradation modeling by utilizing a nonlinear Wiener process, and the constructed two-stage degradation model is as follows:
wherein: x (t) represents the performance degradation amount of the product, X (0) and X (τ) represent the degradation states at the initial time and the change point respectively,and->Is a drift function of each stage, sigma 1 Sum sigma 2 Is the diffusion coefficient of each stage,b (t) is standard Brownian motion;
step two: estimating model parameters, namely estimating model unknown parameters by collecting historical degradation data and utilizing a two-stage unit MLE method;
in the second step, model unknown parameter estimation is carried out by the proposed two-stage unit MLE method, and the method specifically comprises the following steps:
first, collecting historical degradation data of N degradation products, and establishing a training data set delta X of degradation increment n Wherein m is n Represents the number of measurements, Δt, of the nth product n,j-1 ＝t n,j -t n,j-1 (j＝1,…,m n ) Representing a detection interval;
in the first stage of parameter estimation, degradation model parameters are estimated by using degradation data of each test sample
ΔX according to the characteristics of standard Brownian motion n Obeying the normal distribution, the log-likelihood function for the parameter Θ is expressed as:
wherein: m is m a And m b Representing the number of measurements of the first stage and the second stage, satisfying m a +m b ＝m n ；
Regarding a to log likelihood function 1n ,α 2n (n＝1,...,N)，And->And (3) making the bias equal to 0, and obtaining corresponding parameter estimated values:
wherein: the "FMINSEARCH" in MATLAB is used to search the parameter beta 1 And beta 2 Substituting the estimated value into the above formula to obtain an estimated value; obtaining the maximum log likelihood function to obtain the value of the variable point;
in the second phase of parameter estimation, each of the first phase estimates is utilizedOf individual samplesAnd->And->Carrying out statistical analysis to obtain super parameters corresponding to the random parameter distribution;
step three: predicting the residual life, and deriving an approximate analytic solution of the first-time distribution by utilizing space-time transformation based on the two-stage degradation model obtained in the step one;
step four: taking the first arrival time distribution result obtained in the step three as a basis, taking the degradation rate of each stage as a random variable, and obtaining a first arrival time distribution function taking individual variability into consideration based on a total probability law;
in the fourth step, based on the result of the third step, considering individual variability, the degradation rate of each stage is regarded as a random variable, namelyAnd->According to the law of total probability, the first-arrival time distribution function taking into account individual variability is expressed as:
when 0<t is less than or equal to τ:
when t > τ:
step five: under the condition that the variable points do not appear, deducing a transition probability density function for transitioning from an initial degradation state to a variable point degradation state according to the Wiener process characteristics, and further obtaining a state transition density function considering individual differences based on a total probability law;
step six: based on the results obtained in the step four and the step five, deducing a life probability density function under the concept of the first arrival time of the two-stage degradation model based on the total probability law and the Gaussian distribution property; substituting the estimated value of the model parameter obtained in the second step into a life probability density function, so that the life prediction of the two-stage degraded product is realized.
2. The method for predicting remaining life of a two-stage degradation product according to claim 1, wherein: in the third step, based on space-time transformation, referring to a life derivation process of a nonlinear Wiener process, the first time distribution function is expressed as:
wherein: f (f) T (t|τ) is a life probability density function at time t, x τ Is the amount of degradation at the change point.
3. The method for predicting remaining life of a two-stage degradation product according to claim 2, wherein: in the fifth step, the state transition probability of entering the second stage is defined as follows:
m τ (x τ )＝Pr{X(τ)＝x τ |X(0)＝x 0 ,T>τ}Pr{T>τ}
wherein: m is m τ (x τ ) Representing slave state x 0 Transition to state x τ Is a transition probability density of (2);
according to Wiener process characteristics and the law of total probability, a transition probability density function taking into account the influence of random effects can be expressed as:
4. a method for predicting the remaining life of a two-stage degradation product according to claim 3, wherein: in step six, based on the results of step four and step five, the lifetime probability density function under the concept of the first time can be further expressed as:
if 0 is<t≤τ，f T The expression of (t|τ) is the same as the result of step four, if t>Based on the property of Gaussian distribution, the integral in the above equation can be solved, and then the life probability density function can be obtained as follows:
wherein: a=a 1 -A 2 ，B＝B 1 -B 2
Wherein phi (·) and phi (·) represent the cumulative distribution function and the probability density function of the standard normal distribution respectively,
substituting the parameters estimated in the second step into the life probability density function obtained in the sixth step, so that the residual life prediction of the two-stage product can be realized.
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