CN111046564A - Method for predicting residual life of two-stage degraded product - Google Patents
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Abstract
The invention discloses a residual life prediction method of a two-stage degradation product, belonging to the field of residual life prediction in prediction and health management, and mainly comprising the following steps: modeling a degradation process, estimating model parameters and predicting residual life; the modeling of the degradation process is to establish a two-stage degradation model by utilizing a nonlinear Wiener process; the model parameter estimation comprises: collecting historical degradation data; estimating model parameters based on a unit maximum likelihood estimation method; statistical analysis of random parameter distribution; the remaining life prediction includes: acquiring 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 service life of the two-stage degraded product, is favorable for ensuring the operational reliability of the product, reduces the maintenance cost and avoids safety accidents.
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 degraded product.
Background
The residual life prediction is the core content of the prediction and health management technology, is the effective time interval left by the product at the current time when the product loses the 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.
The prediction of remaining life has been of great interest and intensive research over the last decade. Among them, the Wiener process has been developed because it can describe non-monotonic degradation trajectory and has good mathematical characteristics. It is worth noting that in most life prediction methods based on Wiener processes, the degradation rate is generally fixed and does not change over time. However, in practical engineering, the degradation trace of many products usually has significant degradation rate variation due to the variation of external operating conditions and internal mechanisms, and exhibits 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 degraded products, the conventional life prediction method has low prediction accuracy.
In order to improve the prediction accuracy, many researchers have proposed a two-stage degradation model, but the existing method mainly focuses on whether the change points are random or whether individual differences are considered, and neglects the derivation of the analytical solution of degradation nonlinearity and lifetime of each stage. Each stage of the existing degradation model is a linear Wiener process, and as can be seen from fig. 2, in practice, the degradation process of each stage presents a nonlinear characteristic due to changes of load, internal state and external environment. Therefore, it is more reasonable to study the residual life prediction method of the two-stage degradation product based on the nonlinear Wiener process.
Disclosure of Invention
In view of the above, the present invention provides a new method for predicting remaining life of a two-stage degraded product, which makes up for the deficiencies of the prior art and can effectively predict the remaining life of the two-stage degraded product.
In order to achieve the purpose, the invention provides the following technical scheme:
a method for predicting the residual life of a two-stage degradation product comprises three contents of degradation process modeling, model parameter estimation and residual life prediction, and specifically comprises the following steps: the method comprises the following steps: modeling a degradation process, namely segmenting from a degradation variable point, and establishing a two-stage degradation model by utilizing a nonlinear Wiener process model; step two: estimating model parameters, namely estimating unknown parameters of the model by collecting historical degradation data and utilizing a two-stage unit MLE method; step three: predicting the residual life, namely deducing an approximate analytic solution of first-arrival time distribution by utilizing space-time transformation based on the two-stage degradation model obtained in the step one; step four: on the basis of the first arrival time distribution result obtained in the step three, considering the degradation rate of each stage as a random variable, and based on a total probability law, obtaining a first arrival time distribution function considering individual differences; step five: under the condition that the variable point does not appear, according to the characteristics of a Wiener process, a transition probability density function for transferring from an initial degradation state to a variable point degradation state can be deduced, and further, a state transition density function considering individual difference is obtained based on a total probability law; step six: based on the results obtained in the fourth step and the fifth step, based on the general probability law and the properties of Gaussian distribution, a life probability density function under the first arrival time concept of the two-stage degradation model can be deduced; and substituting the estimated value of the model parameter obtained in the second step into a life probability density function, thereby realizing the life prediction of the two-stage degraded product.
Further, 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 amount of degradation of the product, X (0) and X (τ) represent the state of degradation at the initial time and at the point of change, respectively,
and
is drift of each stageFunction, σ1And σ2Is the diffusion coefficient of each stage, B (t) is the standard Brownian motion;
further, in the second step, model unknown parameter estimation is performed by the proposed two-stage unit MLE method, which specifically comprises the following processes: firstly, historical degradation data of N degradation products are collected, and a degradation increment training data set delta X is establishednWherein m isnRepresenting the measured number of the nth product, Δ tn，j-1＝tn，j-tn，j-1(j＝1，..，mn) Indicating the detection interval.
In the first stage of parameter estimation, the degradation model parameters are estimated by using the degradation data of each test sample
According to the nature of the standard Brownian motion, Δ XnFollowing a normal distribution, the log-likelihood function for the parameter Θ is expressed as:
wherein: m isaAnd mbRepresenting the number of measurements in the first and second phases, satisfying ma+mb＝mn，
Correlating α to log likelihood function1，n，α2，n(n＝1，...，N)，
And
so that it is equal to 0, the corresponding parameter estimate is obtained:
wherein, firstly, the 'FMINSEARCH' in MATLAB is utilized to search the parameter β1And β2The estimated value is obtained by substituting the formula. Then, the maximum log-likelihood function is obtained, and the value of the variable point can be obtained.
In a second stage of parameter estimation, using each sample estimated in the first stage
And
and
and carrying out statistical analysis to obtain the hyper-parameters corresponding to the random parameter distribution.
Further, in the third step, based on the space-time transformation, referring to the lifetime derivation process of the nonlinear Wiener process, the first-arrival time distribution function is expressed as:
wherein: f. ofT(t | τ) is the lifetime probability density function at time t, xτIs the amount of degradation at the point of change;
further, in step four, based on the result of step three, considering individual differences, the degradation rate of each stage is regarded as a random variable, i.e. set
And
according to the law of general probability, the first-arrival time distribution function considering individual variability is expressed as:
(1)0＜t≤τ
(2)t＞τ
further, in the fifth step, 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 has not reached the failure threshold, i.e., the lifetime is greater than the transition point time. Therefore, the state transition probability entering the second stage is defined as:
mτ(xτ)＝Pr{X(τ)＝xτ|X(0)＝x0，T＞τ}Pr{T＞τ}
wherein: m isτ(xτ) Representing slave state x0Transition to State xτThe transition probability density of (2).
According to Wiener process characteristics and the law of total probability, the transition probability density function considering 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 first arrival time concept can be further expressed as:
if t is greater than 0 and less than or equal to tau, fTThe expression of (t | τ) is the same as the result of step four, if t > τ, based on the nature of gaussian distribution, the integral in the above equation can be solved, and further the lifetime probability density function can be obtained as follows:
wherein: a ═ A1-A2，B＝B1-B2
Wherein: phi (-) and phi (-) represent the cumulative distribution function and probability density function of a standard normal distribution, respectively,
and substituting the parameters estimated in the second step into the life probability density function obtained in the sixth step to realize the residual life prediction of the two-stage product.
The core idea and principle of the invention are as follows:
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 the difference of individuals and the uncertainty of a variable point degradation state, and deduces an approximate analytical solution of the residual life under the first arrival time concept.
The invention has the beneficial effects that:
1) compared with the existing two-stage degradation model, the two-stage degradation model is constructed by adopting a nonlinear Wiener process, 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 independently and effectively estimate the parameters of each sample and is not limited by random parameter distribution; 3) through space-time transformation, an approximate analytic solution of first-arrival time distribution of the two-stage degradation model is obtained, and 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 purpose, technical scheme and beneficial effect of the invention more clear, the invention provides the following drawings for explanation:
FIG. 1 is a flow chart of the residual life prediction for a two-stage degraded product of the present invention;
FIG. 2 is a schematic diagram of the degradation curve of the two-stage degraded product of the present invention;
FIG. 3 is a graph of simulated degradation trajectories;
FIG. 4 is a schematic illustration of remaining life prediction;
FIG. 5 is a schematic diagram of a predicted expected and actual remaining life comparison.
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 illustrative of the invention only and are not limiting upon the scope of the invention.
The implementation process of this embodiment specifically includes: the model is established in the degradation process, the model parameter estimation and the residual life prediction are carried out, a numerical simulation is used for explanation based on an MATLAB tool, 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, and as shown in the figure, the method specifically includes the following steps:
the method comprises the following steps: and modeling a degradation process. Fig. 2 is a schematic diagram of the degradation of a two-stage degradation product, and it can be seen from the diagram that the degradation process of the product shows a two-stage degradation phenomenon. Thus, segmentation is performed at the degradation transition point, utilizing two non-linear Wi with different degradation rateseneAnd (3) carrying out degradation modeling on the r process, wherein the constructed two-stage degradation model is as follows:
wherein: x (t) represents the amount of degradation of the product, X (0) and X (τ) represent the state of degradation at the initial time and at the point of change, respectively,
and
is a drift function of each stage, σ1And σ2Is the diffusion coefficient of each stage, and B (t) is the standard Brownian motion. To express the difference between individuals, let
And
but these values are fixed values for each sample.
Then, the lifetime under the first arrival time concept can 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 typically a constant.
Step two: and estimating model parameters. A degradation trace for a number of sample samples was first generated using MATLAB, as shown in fig. 3. Constructing a degradation incremental training set for each group of degradation data
In the first stage of parameter estimation, the degradation model parameters are estimated by using the degradation data of each test sample
According to the nature of the standard Brownian motion, Δ XnAnd obtaining a log-likelihood function of the parameter theta according to normal distribution:
wherein: m isaAnd mbRepresenting the number of measurements in the first and second phases, satisfying ma+mb＝mn，
Correlating α to log likelihood function1，n，α2，n(n＝1，...，N)，
And
so that it is equal to 0, the corresponding parameter estimate is obtained:
wherein, firstly, the 'FMINSEARCH' in MATLAB is utilized to search the parameter β1And β2The estimated value is obtained by substituting the formula. Then, the maximum log-likelihood function is obtained, and the value of the variable point can be obtained.
This results in each sample transition point. In a second stage of parameter estimation, using each sample estimated in the first stage
And
and
statistical analysis was performed, and assuming that the change point obeyed a normal distribution, the mean parameter of the normal distribution obeyed the change point was 49.93, and the variance was 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 the space-time transformation, the following can be obtained:
wherein:
similarly, it can be obtained that the first-arrival time distribution function of the two-stage degradation model is expressed as:
wherein: f. ofT(t | τ) is the lifetime probability density function at time t, xτIs the amount of degradation at the point of change.
Step four: considering individual differences, the degradation rate of each stage is regarded as a random variable, i.e. set
And
according to the following theorem:
based on the law of total probability, the first-arrival time distribution function considering individual differences 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 has not reached the failure threshold, i.e., the lifetime is greater than the transition point time. Therefore, the state transition probability entering the second stage is defined as:
mτ(xτ)＝Pr{X(τ)＝xτ|X(0)＝x0，T＞τ}Pr{T＞τ}
wherein: m isτ(xτ) Representing slave state x0Transition to State xτThe transition probability density of (2).
According to Wiener process characteristics and the law of total probability, the transition probability density function considering the influence of random effects can be expressed as:
step six: based on the results of step four and step five, the lifetime probability density function under the first arrival time concept can be further expressed as:
if t is greater than 0 and less than or equal to tau, fTThe expression of (t | τ) is the same as the result of step four, if t > τ, based on the nature of gaussian distribution, the integral in the above equation can be solved, and further the lifetime probability density function can be obtained as follows:
wherein: a ═ A1-A2，B＝B1-B2
Wherein: phi (-) and phi (-) represent the cumulative distribution function and probability density function of a 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 remaining life probability density function of the two-stage product, as shown in fig. 4, and fig. 5 is a comparison of the predicted expectation of the remaining life obtained by the method of the present invention with the actual remaining life, and results of the single-stage nonlinear degradation model and the two-stage linear degradation model. As can be seen from the life prediction result, the method can effectively predict the residual life of the two-stage degraded product.
Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.
Claims (7)
1. The method for predicting the residual life of the two-stage degraded product is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: modeling a degradation process, namely segmenting from a degradation variable point, and establishing a two-stage degradation model by utilizing a nonlinear Wiener process model;
step two: estimating model parameters, namely estimating unknown parameters of the model by collecting historical degradation data and utilizing a two-stage unit MLE method;
step three: predicting the residual life, namely deducing an approximate analytic solution of first-arrival time distribution by utilizing space-time transformation based on the two-stage degradation model obtained in the step one;
step four: on the basis of the first arrival time distribution result obtained in the step three, considering the degradation rate of each stage as a random variable, and obtaining a first arrival time distribution function considering the individual difference on the basis of a total probability law;
step five: under the condition that the change point does not appear, deducing a transition probability density function for transferring from the initial degradation state to the change point degradation state according to the characteristics of a Wiener process, and further obtaining a state transition density function considering individual difference based on a total probability law;
step six: deducing a life probability density function under the first arrival time concept of the two-stage degradation model based on the total probability law and the properties of Gaussian distribution based on the results obtained in the fourth step and the fifth step; and substituting the estimated value of the model parameter obtained in the second step into a life probability density function, thereby realizing the life prediction of the two-stage degraded product.
2. The method of predicting remaining life of a two-stage degraded product according to claim 1, characterized in that: 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 amount of degradation of the product, X (0) and X (τ) represent the state of degradation at the initial time and at the point of change, respectively,
and
is a drift function of each stage, σ1And σ2Is the diffusion coefficient of each stage, and B (t) is the standard Brownian motion.
3. The method of predicting remaining life of a two-stage degraded product according to claim 1, characterized in that: 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 processes:
firstly, historical degradation data of N degradation products are collected, and a degradation increment training data set delta X is establishednWherein m isnRepresenting the measured number of the nth product, Δ tn，j-1＝tn，j-tn，j-1(j＝1，...，mn) Indicating a detection interval;
in the first stage of parameter estimation, the degradation model parameters are estimated by using the degradation data of each test sample
According to the nature of the standard Brownian motion, Δ XnFollowing a normal distribution, the log-likelihood function for the parameter Θ is expressed as:
wherein: m isaAnd mbRepresenting the number of measurements in the first and second phases, satisfying ma+mb＝mn，
Correlating α to log likelihood function1，n，α2，n(n＝1，...，N)，
And
so that it is equal to 0, the corresponding parameter estimate is obtained:
wherein, firstly, the 'FMINSEARCH' in MATLAB is utilized to search the parameter β1And β2Substituting the formula into the formula to obtain an estimated value; then, the maximum log-likelihood function is solved, and the value of the variable point can be obtained;
in a second stage of parameter estimation, using each sample estimated in the first stage
And
and
and carrying out statistical analysis to obtain the hyper-parameters corresponding to the random parameter distribution.
4. The method of predicting remaining life of a two-stage degraded product according to claim 1, characterized in that: in the third step, based on space-time transformation, referring to the lifetime derivation process of the nonlinear Wiener process, the first-arrival time distribution function is expressed as:
wherein: f. ofT(t | τ) is the lifetime probability density function at time t, xτIs the amount of degradation at the point of change.
5. The method of predicting remaining life of a two-stage degraded product according to claim 1, characterized in that: in step four, based on the result of step three, the individual differences are considered, and the degradation rate of each stage is taken as a random variable, i.e. the degradation rate is set
And
according to the law of general probability, the first-arrival time distribution function considering individual variability is expressed as:
when t is more than 0 and less than or equal to tau:
when t > τ:
6. the method of predicting remaining life of a two-stage degraded product according to claim 1, characterized in that: in step five, the state transition probability entering the second stage is defined as:
mτ(xτ)＝Pr{X(τ)＝xτ|X(0)＝x0，T＞τ}Pr{T＞τ}
wherein: m isτ(xτ) Representing slave state x0Transition to State xτA transition probability density of (a);
according to Wiener process characteristics and the law of total probability, the transition probability density function considering the influence of random effects can be expressed as:
7. the method of predicting remaining life of a two-stage degraded product according to claim 1, characterized in that: in step six, based on the results of step four and step five, the lifetime probability density function under the first arrival time concept can be further expressed as:
if t is greater than 0 and less than or equal to tau, fTThe expression of (t | τ) is the same as the result of step four, if t > τ, based on the nature of gaussian distribution, the integral in the above equation can be solved, and further the lifetime probability density function can be obtained as follows:
wherein: a ═ A1-A2，B＝B1-B2
Wherein phi (-) and phi (-) respectively represent a cumulative distribution function and a probability density function of a standard normal distribution,
and substituting the parameters estimated in the second step into the life probability density function obtained in the sixth step to realize the residual life prediction of the two-stage product.
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