CN113158471B - A Remaining Lifetime Prediction Method for Degraded Equipment Considering Measurement Uncertainty - Google Patents

Abstract

The invention discloses a method for predicting the remaining life of degraded equipment considering measurement uncertainty, comprising the following steps: S1: collecting discrete measurement time points and degradation quantity sets of degraded equipment, and establishing a performance degradation model; S2: according to the performance degradation model , to estimate the parameters of the performance degradation model; S3: based on the performance degradation model and parameter estimation results, perform remaining life prediction. The present invention provides a method for predicting the remaining life of self-adaptive degraded equipment considering measurement uncertainty, aiming at the randomness of degraded equipment in the degraded process of actual equipment, fully considering the error of feature extraction in the degraded process of equipment, not only can be used for The remaining life of such equipment can be accurately predicted and analyzed, and it can also be used as an effective analysis tool in the life cycle.

Description

A Remaining Lifetime Prediction Method for Degraded Equipment Considering Measurement Uncertainty

technical field

The invention belongs to the technical field of reliability engineering, and in particular relates to a method for predicting the remaining life of degraded equipment considering measurement uncertainty.

Background technique

With the continuous development of production technology, modern industrial equipment is developing in the direction of large-scale, complex and intelligent. However, large and complex industrial equipment will be affected by various environmental factors during long-term operation, and the performance of the equipment will also change accordingly. After a period of accumulation, when a certain threshold is reached, the equipment will be damaged, manifested as a change in the output parameters of the equipment, such as deterioration of component performance, wear of mechanical components, and aging of insulating materials. Accumulated to a certain extent, the equipment will eventually fail. Once an accident occurs due to such a failure, the loss of personnel and property and even the damage to the environment are often immeasurable. If the life of equipment can be monitored and evaluated in the early stages of performance degradation to determine the best time for equipment maintenance and formulate corresponding maintenance plans, this can improve equipment reliability, reduce equipment operation risks, and reduce operating costs. Therefore, it is the basis and core content of prediction and health management (PHM) to establish the evolution model of degradation law through the measurement data of equipment degradation, and then realize the prediction of remaining useful life (RUL) of equipment. The life prediction results provide a scientific basis for maintenance decisions and spare parts replacement.

In engineering practice, it is often impractical or expensive to accurately measure the hidden degradation state of equipment. Furthermore, the measurement data related to the implied degraded state of equipment obtained through sensor condition monitoring is inevitably affected by factors such as noise, interference, and unreasonable measuring instruments. In this case, the measured data obtained are unreasonable and can only partially reflect the degradation state of the equipment. In order to describe the influence of measurement uncertainty and accurately describe the degradation of equipment, it is necessary to establish the relationship between the potential degradation state and uncertain measurement data.

In fact, based on the adaptive Wiener process framework, a degradation model considering the uncertainty measurement is established. Further, based on the logarithmic transformation and the inverse Gaussian characteristic, the analytical form of the equipment life distribution by considering the uncertain measurement adaptive Wiener process is deduced theoretically, and the analytical solution of the remaining life is deduced. Therefore, the remaining life prediction of stochastic degraded equipment considering the adaptive Wiener process of uncertain measurement is obtained, so as to improve the accuracy of remaining life prediction.

Contents of the invention

The object of the present invention is to solve the problem of remaining life prediction, and propose a method for predicting the remaining life of degraded equipment considering measurement uncertainty.

The technical solution of the present invention is: a method for predicting the remaining life of degraded equipment considering measurement uncertainty includes the following steps:

S1: Collect discrete measurement time points and degradation quantity sets of degraded equipment, and establish a performance degradation model;

S2: Estimate the parameters of the performance degradation model according to the performance degradation model;

S3: Based on the performance degradation model and parameter estimation results, perform remaining life prediction.

Further, step S1 includes the following sub-steps:

S11: Collect the discrete measurement time point 0=t ₀ <t ₁ <…<t _i of the degraded equipment and the set of degradation quantities Y _1:i ={y ₁ ,y ₂ ,…,y _i } to obtain the corresponding set of degraded states X _1:i ＝{x ₁ ,x ₂ ,…, _xi }, where y ₁ ,y ₂ ,…,y _i represent the degradation amount of degraded equipment at time t ₁ ,…,t _i , x ₁ ,x ₂ ,..., _xi represent the degraded state of the degraded equipment at time t ₁ ,...,t _i ;

S12: Establish a performance degradation equation between the degradation state set X _1:i ={x ₁ ,x ₂ ,…, _xi } and the degradation amount set Y _1:i ={y ₁ ,y ₂ ,…,y _i } y _i = _xi +ε _i , as a performance degradation model, where ε _i represents the random measurement error at time t _i .

Further, step S2 includes the following sub-steps:

S21: According to the performance degradation model, establish the likelihood function l(θ|Y) of the degraded equipment for the unknown parameters θ=(k ² ,σ ² ,γ ² ,v,α), where k ² represents the diffusion of the adaptive drift term The square of the coefficient, ^σ2 represents the square of the diffusion coefficient of the adaptive Wiener, ^γ2 represents the variance of the measurement error, v represents the drift coefficient, α represents the time exponential power of the nonlinear degradation, and Y represents the monitoring data considering the measurement uncertainty;

S22: According to the likelihood function l(θ|Y) of the degraded equipment, the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α and v’s Maximum likelihood function, where Y represents the measurement data of the degraded equipment corresponding to the unknown parameter θ, k represents the diffusion coefficient of the adaptive drift term, σ represents the diffusion coefficient of the adaptive Wiener, and γ represents the measurement error;

S23: Using the multi-dimensional search method, according to the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α and the maximum likelihood function of v, k ² is sequentially obtained , σ ² , γ ² , the maximum likelihood estimates of α and v to complete the parameter estimation.

Further, in the step S21, the expression of the likelihood function l(θ|Y) of the degraded equipment is:

Among them, θ represents unknown parameters, Y represents monitoring data considering measurement uncertainty, N represents the number of degraded equipment, M represents the number of monitoring points of a degraded device, Y _i represents the monitoring data of a certain device, Σ represents Y The variance of _i , v represents the drift coefficient, S _i represents the time interval between two monitoring points;

In step S22, the expression of the maximum likelihood function of the drift coefficient v is:

In step S22, the expression of the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α is:

Among them, k represents the diffusion coefficient of the adaptive drift item, σ represents the diffusion coefficient of the adaptive Wiener, γ ² represents the variance of the measurement error, Y _i represents the monitoring data of a certain device, Σ _i represents the variance of Y _i , and Ω _i represents the variance of The variance of Y _i when considering the adaptive Wiener.

Further, step S3 includes the following sub-steps:

S31: At the discrete measurement time point 0=t ₀ <t ₁ <...<t _i , according to the performance degradation equation y _i = _xi +ε _i , the degradation state set X _1:i ={x ₁ ,x ₂ ,... , _xi } and parameter estimation results to transform the performance degradation model;

S32: Utilize the Kalman filter algorithm to estimate the degradation state according to the transformed performance degradation model, and complete the life prediction.

Further, in step S31, the calculation formula for transforming the performance degradation model is:

Among them, x _i represents the degradation state of the degraded equipment at the discrete measurement time t _i , x _i-1 represents the degradation state of the degraded device at the discrete measurement time t _i-1 , v represents the drift coefficient, ΔS _i represents the nonlinear degradation time interval, and η represents The noise term caused by the time-varying uncertainty of the degradation process, y _i represents the monitoring data when the measurement error is considered, and ε _i represents the measurement error.

Further, in step S32, the Kalman filter algorithm is used to estimate the degraded state and update it during the degraded state estimation process. The calculation formula of the degraded state estimation is:

表示通过退化量集合Y_1:i对退化状态x_i估计的期望，P_i|i表示通过退化量集合Y_1:i对退化状态x_i估计的方差，

表示上一步通过退化量集合Y_1:i对退化状态x_i估计的期望，P_i|i-1表示上一步通过退化量集合Y_1:i对退化状态x_i估计的方差，K(i)表示滤波增益，

表示在t_i-1时刻监测数据估计值的均值，v表示漂移系数，ΔS_i表示非线性退化时间间隔，y(i)表示考虑测量误差时的监测数据，γ²表示测量误差的方差，P_i-1|i-1表示在t_i-1时刻监测数据估计值的方差，π_i表示噪声项η的方差；in,

Indicates the expectation of the degraded state x _i estimated by the set of degenerate quantities Y _1:i , P _i|i represents the variance estimated by the set of degenerated quantities Y _1:i for the degraded state x _i ,

Indicates the expectation of the degradation state x _i estimated by the degradation quantity set Y ₁ :i in the previous step, P _i|i-1 represents the variance of the degradation state x _i estimated by the degradation quantity set Y _1:i in the previous step, K(i) Indicates the filter gain,

Indicates the mean value of the estimated value of the monitoring data at time t _i-1 , v indicates the drift coefficient, ΔS _i indicates the nonlinear degradation time interval, y(i) indicates the monitoring data when the measurement error is considered, γ ² indicates the variance of the measurement error, P _i-1|i-1 represents the variance of the estimated value of the monitoring data at time t _i-1 , and π _i represents the variance of the noise term η;

The calculation formula for updating is:

P _i|i = (1-K(i))P _i|i-1 .

Further, in step S32, the specific method for predicting the remaining life is: taking the time when the degradation state estimation result reaches the predetermined failure threshold for the first time as the starting time of the remaining life of the degraded equipment.

The beneficial effects of the present invention are: the present invention provides a method for predicting the remaining life of self-adaptive degraded equipment considering measurement uncertainty, and fully considers the feature extraction of equipment during the degraded process in view of the randomness of degraded equipment in the actual degraded process It can not only accurately predict and analyze the remaining life of such equipment, but also can be used as an effective analysis tool in the life cycle to provide a strong theoretical basis for maintenance management decisions such as equipment spare parts ordering, so as to achieve efficient and reasonable Equipment management, avoiding waste, so this method has good engineering application value.

Description of drawings

Fig. 1 is the flowchart of remaining life prediction method;

Figure 2 is a diagram of the capacity degradation process of four groups of lithium batteries;

Figure 3 is the fitting effect diagram of lithium battery degradation under the two methods;

Figure 4 is a probability density distribution diagram of the remaining life of the lithium battery under the two methods;

Fig. 5 is the absolute error diagram of remaining life prediction of two methods;

Fig. 6 is the relative error graph of remaining life prediction of two methods;

Figure 7 is a graph of the mean square error of remaining life prediction for the two methods.

Detailed ways

Embodiments of the present invention will be further described below in conjunction with the accompanying drawings.

Before describing the specific embodiments of the present invention, in order to make the scheme of the present invention more clear and complete, at first the abbreviations and key term definitions that appear in the present invention are explained:

Wiener process: Adaptive Wiener process.

As shown in Figure 1, the present invention provides a method for predicting the remaining life of degraded equipment considering measurement uncertainty, comprising the following steps:

S1: Collect discrete measurement time points and degradation quantity sets of degraded equipment, and establish a performance degradation model;

S2: Estimate the parameters of the performance degradation model according to the performance degradation model;

S3: Based on the performance degradation model and parameter estimation results, perform remaining life prediction.

In the embodiment of the present invention, step S1 includes the following sub-steps:

S11: Collect the discrete measurement time point 0=t ₀ <t ₁ <…<t _i of the degraded equipment and the set of degradation quantities Y _1:i ={y ₁ ,y ₂ ,…,y _i } to obtain the corresponding set of degraded states X _1:i ＝{x ₁ ,x ₂ ,…, _xi }, where y ₁ ,y ₂ ,…,y _i represent the degradation amount of degraded equipment at time t ₁ ,…,t _i , x ₁ ,x ₂ ,..., _xi represent the degraded state of the degraded equipment at time t ₁ ,...,t _i ;

S12: Establish a performance degradation equation between the degradation state set X _1:i ={x ₁ ,x ₂ ,…, _xi } and the degradation amount set Y _1:i ={y ₁ ,y ₂ ,…,y _i } y _i = _xi +ε _i , as a performance degradation model, where ε _i represents the random measurement error at time t _i .

In the embodiment of the present invention, step S2 includes the following sub-steps:

S21: According to the performance degradation model, establish the likelihood function l(θ|Y) of the degraded equipment for the unknown parameters θ=(k ² ,σ ² ,γ ² ,v,α), where k ² represents the diffusion of the adaptive drift term The square of the coefficient, ^σ2 represents the square of the diffusion coefficient of the adaptive Wiener, ^γ2 represents the variance of the measurement error, v represents the drift coefficient, α represents the time exponential power of the nonlinear degradation, and Y represents the monitoring data considering the measurement uncertainty;

S22: According to the likelihood function l(θ|Y) of the degraded equipment, the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α and v’s Maximum likelihood function, where Y represents the measurement data of the degraded equipment corresponding to the unknown parameter θ, k represents the diffusion coefficient of the adaptive drift term, σ represents the diffusion coefficient of the adaptive Wiener, and γ represents the measurement error;

S23: Using the multi-dimensional search method, according to the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α and the maximum likelihood function of v, k ² is sequentially obtained , σ ² , γ ² , the maximum likelihood estimates of α and v to complete the parameter estimation.

In the embodiment of the present invention, in the step S21, the expression of the likelihood function l(θ|Y) of the degraded equipment is:

Among them, θ represents unknown parameters, Y represents monitoring data considering measurement uncertainty, N represents the number of degraded equipment, M represents the number of monitoring points of a degraded device, Y _i represents the monitoring data of a certain device, Σ represents Y The variance of _i , v represents the drift coefficient, S _i represents the time interval between two monitoring points;

In step S22, the expression of the maximum likelihood function of the drift coefficient v is:

In step S22, the expression of the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α is:

Among them, k represents the diffusion coefficient of the adaptive drift item, σ represents the diffusion coefficient of the adaptive Wiener, γ ² represents the variance of the measurement error, Y _i represents the monitoring data of a certain device, Σ _i represents the variance of Y _i , and Ω _i represents the variance of The variance of Y _i when considering the adaptive Wiener.

In the embodiment of the present invention, step S3 includes the following sub-steps:

S31: At the discrete measurement time point 0=t ₀ <t ₁ <...<t _i , according to the performance degradation equation y _i = _xi +ε _i , the degradation state set X _1:i ={x ₁ ,x ₂ ,... , _xi } and parameter estimation results to transform the performance degradation model;

S32: Utilize the Kalman filter algorithm to estimate the degradation state according to the transformed performance degradation model, and complete the life prediction.

In the embodiment of the present invention, in step S31, the calculation formula for transforming the performance degradation model is:

Among them, x _i represents the degradation state of the degraded equipment at the discrete measurement time t _i , x _i-1 represents the degradation state of the degraded device at the discrete measurement time t _i-1 , v represents the drift coefficient, ΔS _i represents the nonlinear degradation time interval, and η represents The noise term caused by the time-varying uncertainty of the degradation process, y _i represents the monitoring data when the measurement error is considered, and ε _i represents the measurement error.

In the embodiment of the present invention, in step S32, the Kalman filter algorithm is used to estimate the degraded state and update it during the degraded state estimation process. The calculation formula of the degraded state estimation is:

K(i)＝P _i|i-1 (P _i|i-1 +γ ² ) ^-1

P _i|i-1 ＝P _i-1|i-1 +π _i

表示通过退化量集合Y_1:i对退化状态x_i估计的期望，P_i|i表示通过退化量集合Y_1:i对退化状态x_i估计的方差，

表示上一步通过退化量集合Y_1:i对退化状态x_i估计的期望，P_i|i-1表示上一步通过退化量集合Y_1:i对退化状态x_i估计的方差，K(i)表示滤波增益，

表示在t_i-1时刻监测数据估计值的均值，v表示漂移系数，ΔS_i表示非线性退化时间间隔，y(i)表示考虑测量误差时的监测数据，γ²表示测量误差的方差，P_i-1|i-1表示在t_i-1时刻监测数据估计值的方差，π_i表示噪声项η的方差；in,

Indicates the expectation of the degraded state x _i estimated by the set of degenerate quantities Y _1:i , P _i|i represents the variance estimated by the set of degenerated quantities Y _1:i for the degraded state x _i ,

Indicates the expectation of the degradation state x _i estimated by the degradation quantity set Y ₁ :i in the previous step, P _i|i-1 represents the variance of the degradation state x _i estimated by the degradation quantity set Y _1:i in the previous step, K(i) Indicates the filter gain,

Indicates the mean value of the estimated value of the monitoring data at time t _i-1 , v indicates the drift coefficient, ΔS _i indicates the nonlinear degradation time interval, y(i) indicates the monitoring data when the measurement error is considered, γ ² indicates the variance of the measurement error, P _i-1|i-1 represents the variance of the estimated value of the monitoring data at time t _i-1 , and π _i represents the variance of the noise term η;

The calculation formula for updating is:

P _i|i = (1-K(i))P _i|i-1 .

In the embodiment of the present invention, in step S32, the specific method for predicting the remaining life is: taking the time when the degradation state estimation result reaches the predetermined failure threshold for the first time as the starting time of the remaining life of the degraded equipment.

Detailed ways

Next, the method of the present invention is verified on the lithium battery capacity degradation data with measurement uncertainty. This set of data sets was obtained through charge and discharge experiments at room temperature, and recorded changes in battery state information (including capacity) with charge and discharge cycles. Due to complex aging mechanisms, the capacity of lithium batteries decreases with charge and discharge cycles. Figure 2 shows the degradation of the four sets of batteries #5, #6, #7 and #18 of NASA. It can be seen from the figure that the capacity of each battery has a downward trend over time. Due to the influence of measurement uncertainty, the value of the monitored degradation data Y _1:i may exceed the failure threshold before the actual failure time.

A. Establish a performance degradation model that can describe the adaptive Wiener process of the equipment considering the measurement uncertainty

In engineering practice, it is often impractical or expensive to accurately measure the hidden degradation state of equipment. Furthermore, the measurement data related to the implied degraded state of equipment obtained through sensor condition monitoring is inevitably affected by factors such as noise, interference, and unreasonable measuring instruments. In this case, the measured data obtained are unreasonable and can only partially reflect the degradation state of the equipment. To describe the effect of measurement uncertainty, the relationship between potential degradation states and uncertain measurement data can be considered.

1) Stochastic degradation process modeling considering measurement uncertainty

The degradation model based on the adaptive Wiener process is a typical nonlinear stochastic model describing the random degradation of equipment, and this type of model has been widely used in the degradation modeling of mechanical wear and corrosion. Generally, the degradation model {X(t),t≥0} based on the Wiener process can be described as:

where v(t) is the time-varying drift rate following the Wiener process, v ₀ >0 is the initial drift rate, k is the diffusion coefficient for adaptive drift, and W(t) is the standard Brownian independent of B(t) sports. S(t; α) is a function with a parameter α that increases monotonically with time. In this paper, aiming at the situation where measurement uncertainty exists in the degradation process, based on Equation (45) and the monitoring data during the degradation process, a degradation model between the potential degradation state and the uncertain measurement data is established. Specifically, it can be expressed as:

Y(t)=X(t)+ε, (2)

Wherein, ε is a random measurement error, and it is assumed that at any time t, ε is an independent and identically distributed Gaussian distribution, and ε～N(0,γ ² ). It is further assumed that ε and B(t) are independent of each other. The above assumptions are widely used in the field of degradation modeling and life estimation.

2) Definition of remaining life

To enable remaining lifetime estimation, the lifetime of a device is defined by the notion of time-to-first-arrival of stochastic degradation processes. In other words, once the stochastic degradation process reaches a predetermined failure threshold for the first time, the device is considered invalid and requires maintenance before it can be used again. Based on the concept of time-to-first-arrival, the lifetime of a device can be defined as:

T=inf{t:X(t)≥w|X(0)<w}, (3)

where w is a preset fault threshold, usually determined by some industry standards, such as amplitude and gyro drift.

Based on this, under the above framework, the main goal of the present invention is to realize the prediction of the remaining life of service equipment based on real-time monitoring degradation data, and to update the remaining life distribution after acquiring new degradation monitoring data. Assuming that the discrete monitoring time point for obtaining degradation measurement data is 0=t ₀ <t ₁ <...<t _i , let y _i =Y(t _i ) represent the degradation amount at time t _i . Therefore, the set of all degradation measurement data up to time t _i can be expressed as Y _1:i ={y ₁ ,y ₂ ,…,y _i }, and the set of corresponding degradation states is X _1:i ={x ₁ , x ₂ ,..., _xi }, where x _i =X(t _i ). From formula (2), the measurement equation at time t _i can be further described as y _i = _xi +ε _i , where ε _i is the realization of independent and identical distribution of ε.

L _i =inf{l _i >0:X(l _i +t _i )≥w}, (4)

和

The probability density function and cumulative distribution function corresponding to the remaining life L _i are respectively

and

B. Parameter estimation

In order to realize the estimation of the unknown parameters of the degradation model θ=(k ² ,σ ² ,γ ² ,v,α), the method of maximum likelihood estimation is adopted, assuming that there are N measuring devices, and the measurement time of the i-th device is t ₁ ,t ₂ ,...t _M , and the corresponding measurement data is {Y _i (t _j )=y _i,j ,i=1,2,...N,j=1,2,...M}. From formula (2), it can be seen that the measurement of the i-th device at time t _j can be expressed as

Wherein, ε _i,j is the measurement error, and ε _i,j ~N(0,γ ² ).

For convenience, take the measurement data of the i-th device as an example to study the likelihood function corresponding to the measurement data. Let t=(t ₁ ,t ₂ ,…,t _M )’, y _i =y _i,j ,j=1,2,…,M, according to the independence assumption and the independent increment property of the standard Brownian motion, we know that y _i is a multivariate normal distribution with mean and variance characteristics as follows:

Wherein D and Q are all a matrix of M * M, and I _M is a M ^th order identity matrix, specifically as follows:

Therefore, for the measurement data of the i-th device, we have

y _i ～N(u,k ² D+σ ² Q+γ ² I _M ), (8)

The likelihood function of all measurement data Y corresponding to θ is

Further, the first-order partial derivative with respect to v is obtained from formula (9), we have

The profile likelihood function of k ² , σ ² , γ ² and α with respect to the maximum likelihood estimate of v is

Based on this, the maximum likelihood estimates of k ² , σ ² , γ ² and α can be obtained by maximizing the profile function (11) by using the multidimensional search method. Then, by substituting the maximum likelihood estimates of k ² , σ ² , γ ² and α back into formula (10), the maximum likelihood estimates of v can be obtained, as shown in Table 1.

Table 1

C. Remaining life prediction considering the influence of measurement uncertainty

First, only the potential degenerate process {X(t),t≥0} is considered. In order to incorporate the current degradation state and the subsequent update mechanism of remaining life prediction, assuming that the system is still operating normally at time t _i , the potential degradation state is X(t _i )= _xi ( _xi <w). Therefore, for l≥t _i , given x _i , according to the Markov property of the Wiener process, the trajectory of the degradation process from time t _i changes with time is

In this case, if l is the first arrival time of the stochastic process {X(l),l≥t _i }, then according to the formula (4) of the remaining life, lt _i corresponds to the remaining life of the equipment at time t _i . Therefore, the transformation t=lt _i is adopted for formula (12), where t>0, then the degeneration process {X(t),t≥0} becomes:

首次穿过阈值w_i＝w-x_i的时间，其中

且

也就是说，在t_i时刻，Therefore, the remaining lifetime at time t _i is equal to the random process

The time when the threshold w _i =wx _i is crossed for the first time, where

and

That is to say, at time t _i ,

噪声部分能近似为一个标准Brownian运动B₀(ψ(t))，具体如下in

The noise part can be approximated as a standard Brownian motion B ₀ (ψ(t)), as follows

就需要首先得到

而为了推出

就需要证明随机过程仍然是一个标准Brownian运动过程，这一结论由以下引理保证。in order to derive

need to first get

And in order to launch

It is necessary to prove that the random process is still a standard Brownian motion process, and this conclusion is guaranteed by the following lemma.

Lemma 1: Given t _i , the random process {D(t),t≥0} (where D(t)=B(t+t _i )-B(t _i ) for any t≥0) is still A standard Brownian motion process, where {B(t),t≥0} is the standard Brownian motion.

仍为Wiener过程。因此，在给定当前退化状态x_i(x_i≤w)的前提下，条件剩余寿命分布

可以通过以下结论得到。According to the relevant theory of stochastic process, the time when the Wiener process reaches a certain fixed threshold for the first time obeys the inverse Gaussian distribution. From the previous derivation, it can be proved that

Still a Wiener process. Therefore, given the current degradation state x _i ( _xi ≤ w), the conditional remaining life distribution

It can be obtained through the following conclusions.

Theorem 1: For the remaining life defined by the degradation models (11) and (14), given the degradation state x _i ( _xi ≤ w) at the current time t _i , the conditional remaining life distribution at time t _i can be concluded as follows:

而且预测的剩余寿命仅依赖当前的退化状态x_i。为了实现在不确定测量状态下剩余寿命预测，需要先基于Y_1:i估计设备的退化状态x_i及其分布，以刻画测量不确定性对状态估计的影响。The above conclusions only consider the potential degradation process

Furthermore, the predicted remaining lifetime depends only on the current degradation state x _i . In order to realize the remaining life prediction under the uncertain measurement state, it is necessary to estimate the degradation state _xi and its distribution of the equipment based on Y _1:i first, so as to describe the influence of measurement uncertainty on state estimation.

In order to estimate the degradation state of the equipment, the degradation state equation and the measurement equation are transformed into discrete time equations at the monitoring moment. Then the transformed degradation model can be obtained at discrete time points t _i , i=1, 2,...:

其方差

ε_i是ε在t_i时刻的实现，因而进一步有η～N(0,π_i)和ε_i～N(0,γ²)。in

its variance

ε _i is the realization of ε at time t _i , so there are further η～N(0,π _i ) and ε _i ～N(0,γ ² ).

和p_i|i＝var(x_i|Y_1:i)分别为通过测量Y_1:i对退化状态x_i估计的期望和方差。此外，定义

和p_i|i-1＝var(x_i|Y_1:i-1)分别为一步预测的期望和方差。因此，在t_i时刻，基于Kalman滤波的潜在退化状态估计和更新过程如下：According to the established model (17), the estimation of potential degraded state can be realized by using Kalman filtering technique. First, define

and p _i|i = var(x _i |Y _1:i ) are the expectation and variance of the estimated degradation state x _i by measuring Y _1: i, respectively. Additionally, define

and p _i|i-1 = var( _xi |Y _1:i-1 ) are the expectation and variance of one-step prediction respectively. Therefore, at time t _i , the estimation and update process of potential degraded state based on Kalman filter is as follows:

State estimation:

Variance update:

P _i|i = (1-K(i))P _i|i-1 , (19)

在这种情况下，由于测量不确定性，对退化状态的估计也具有不确定性，由其后验分布x_i|v,Y_1:i描述。为了在剩余寿命预测中融入以上退化估计的不确定性，因而有Applying the above Kalman filtering algorithm, the posterior estimation of the implicit degraded state x _i based on Y _1:i obeys the Gaussian distribution, and can be expressed as

In this case, due to the measurement uncertainty, the estimate of the degraded state also has uncertainty, described by its posterior distribution x _i |v,Y _1:i . In order to incorporate the uncertainty of the above degradation estimates in the remaining life prediction, there is

Further, the following conclusions are given to calculate the integral in equation (20).

Lemma 2: Given the current degenerate state x _i and all measurements Y _1:i have

Based on Theorem 1 and Lemma 2, using the total probability formula, the integral problem involved in the calculation formula (20) can be analyzed, and the following remaining life prediction results under uncertain measurement can be obtained.

Theorem 2: For the remaining life defined in the degradation models (11) and (14), given all uncertain measurements Y _1:i up to the current time t _i , the following conclusions hold about the remaining life prediction at time t _i :

in,

According to the estimated parameter values, the remaining life distribution of the equipment can be obtained by the adaptive Wiener process considering the uncertain measurement.

In order to verify the effectiveness of the proposed method in predicting the results of remaining life, the adaptive Wiener process method without considering the measurement uncertainty is defined as method 1, and at the same time, the adaptive Wiener process method considering the measurement uncertainty proposed in this paper is defined as It is method 2, and it is beneficial to predict, verify and compare the monitoring data of the No. 5 lithium battery.

(1) Comparison of remaining life prediction results of adaptive Wiener process considering measurement uncertainty

The degradation prediction fitting effect and remaining life prediction results obtained by the two methods based on the test data are shown in Fig. 3 and Fig. 4 . From Figure 3, it can be seen that the fitting effect of method 2 is better than that of method 1. Figure 4 shows that the probability density function of remaining life predicted by method 2 can well cover the true value of remaining life, and the predicted mean value is close to the actual remaining life at each monitoring point, and the results are significantly better than those predicted by method 1.

(2) Relative error comparison of the two methods

In order to verify that the method in this paper can improve the accuracy of RUL prediction, the absolute error, relative error and mean square error corresponding to each monitoring point of RUL predicted by the two models are given. It can be seen from Figure 5, Figure 6 and Figure 7 that the absolute error, mean square error and relative error of method 2 are relatively smaller than method 1. The mean squared error of Approach 1 will experience some large fluctuations due to fluctuations in the degraded data. In contrast, Method 2 proposed in this paper takes into account the impact of measurement uncertainty of degraded data, effectively avoids large fluctuations in the mean square error of RUL prediction, and shows better stability.

The working principle and process of the present invention are as follows: the present invention belongs to the technical field of reliability engineering, and relates to a method for predicting the remaining life of self-adaptive degraded equipment considering measurement uncertainty. The method comprises the following steps: establishing a performance degradation model capable of describing the adaptive Wiener process of the equipment considering the measurement uncertainty; estimating model parameters and remaining life distribution parameters; predicting the random degradation of the equipment considering the adaptive Wiener process of the measurement uncertainty remaining life.

The beneficial effects of the present invention are: the present invention provides a method for predicting the remaining life of adaptive degraded equipment considering measurement uncertainty, and fully considers the feature extraction of equipment in the degraded process in view of the randomness of degradation in the actual equipment degraded process It can not only accurately predict and analyze the remaining life of such equipment, but also can be used as an effective analysis tool in the life cycle to provide a strong theoretical basis for maintenance management decisions such as equipment spare parts ordering, so as to achieve efficient and reasonable Equipment management, avoiding waste, so this method has good engineering application value.

Those skilled in the art will appreciate that the embodiments described here are to help readers understand the principles of the present invention, and it should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical revelations disclosed in the present invention without departing from the essence of the present invention, and these modifications and combinations are still within the protection scope of the present invention.

Claims (6)

1. A method for predicting the remaining life of degraded equipment considering measurement uncertainty, characterized in that it comprises the following steps: S1: Collect discrete measurement time points and degradation quantity sets of degraded equipment, and establish a performance degradation model; S2: Estimate the parameters of the performance degradation model according to the performance degradation model; S3: Based on the performance degradation model and parameter estimation results, perform remaining life prediction; The step S2 includes the following sub-steps: S21: According to the performance degradation model, establish the likelihood function l(θ|Y) of the degraded equipment for the unknown parameters θ=(k ² ,σ ² ,γ ² ,v,α), where k ² represents the diffusion of the adaptive drift term The square of the coefficient, ^σ2 represents the square of the diffusion coefficient of the adaptive Wiener, ^γ2 represents the variance of the measurement error, v represents the drift coefficient, α represents the time exponential power of the nonlinear degradation, and Y represents the monitoring data considering the measurement uncertainty; S22: According to the likelihood function l(θ|Y) of the degraded equipment, the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α and v’s Maximum likelihood function, where Y represents the measurement data of the degraded equipment corresponding to the unknown parameter θ, k represents the diffusion coefficient of the adaptive drift term, σ represents the diffusion coefficient of the adaptive Wiener, and γ represents the measurement error; S23: Using the multi-dimensional search method, according to the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α and the maximum likelihood function of v, k ² is sequentially obtained , σ ² , γ ² , the maximum likelihood estimates of α and v, complete the parameter estimation; In the step S21, the expression of the likelihood function l(θ|Y) of the degraded equipment is:

Among them, θ represents unknown parameters, Y represents monitoring data considering measurement uncertainty, N represents the number of degraded equipment, M represents the number of monitoring points of a degraded device, Y _i represents the monitoring data of a certain device, Σ represents Y The variance of _i , v represents the drift coefficient, S _i represents the time interval between two monitoring points; In the step S22, the expression of the maximum likelihood function of the drift coefficient v is:

In the step S22, the expression of the profile likelihood function l(k,σ,γ,α|Y,v) of k ² , σ ² , γ ² and α is:

Among them, k represents the diffusion coefficient of the adaptive drift item, σ represents the diffusion coefficient of the adaptive Wiener, γ ² represents the variance of the measurement error, Y _i represents the monitoring data of a certain device, Σ _i represents the variance of Y _i , and Ω _i represents the variance of The variance of Y _i when considering the adaptive Wiener. 2. The method for predicting the remaining life of degraded equipment considering measurement uncertainty according to claim 1, wherein said step S1 comprises the following sub-steps: S11: Collect the discrete measurement time point 0=t ₀ <t ₁ <…<t _i of the degraded equipment and the set of degradation quantities Y _1:i = {y ₁ ,y ₂ ,…,y _i } to obtain the corresponding set of degraded states X _1:i ＝{x ₁ ,x ₂ ,…, _xi }, where y ₁ ,y ₂ ,…,y _i represent the degradation amount of degraded equipment at time t ₁ ,…,t _i , x ₁ ,x ₂ ,..., _xi represent the degraded state of the degraded equipment at time t ₁ ,...,t _i ; S12: Establish the performance degradation equation between the degradation state set X _1:i ={x ₁ ,x ₂ ,…, _xi } and the degradation amount set Y _1:i ={y ₁ ,y ₂ ,…,y _i } y _i = _xi +ε _i , as a performance degradation model, where ε _i represents the random measurement error at time t _i . 3. The method for predicting the remaining life of degraded equipment considering measurement uncertainty according to claim 1, wherein said step S3 comprises the following sub-steps: S31: Within the discrete measurement time point 0=t ₀ <t ₁ <...<t _i , according to the performance degradation equation y _i = _xi +ε _i , the degradation state set X _1:i ={x ₁ ,x ₂ ,... , _xi } and parameter estimation results to transform the performance degradation model; S32: Utilize the Kalman filter algorithm to estimate the degradation state according to the transformed performance degradation model, and complete the life prediction. 4. The method for predicting the remaining life of degraded equipment considering measurement uncertainty according to claim 3, characterized in that, in the step S31, the calculation formula for transforming the performance degradation model is:

Among them, x _i represents the degradation state of the degraded equipment at the discrete measurement time t _i , x _i-1 represents the degradation state of the degraded device at the discrete measurement time t _i-1 , v represents the drift coefficient, ΔS _i represents the nonlinear degradation time interval, and η represents The noise term caused by the time-varying uncertainty of the degradation process, y _i represents the monitoring data when the measurement error is considered, and ε _i represents the measurement error. 5. The method for predicting the remaining life of degraded equipment considering measurement uncertainty according to claim 3, characterized in that, in the step S32, the Kalman filter algorithm is used to estimate the degraded state, and during the degraded state estimation process Update, the calculation formula for degraded state estimation is:

K(i)＝P _i|i-1 (P _i|i-1 +γ ² ) ^-1 P _i|i-1 ＝P _i-1|i-1 +π _i in,

Indicates the expectation of the degraded state x _i estimated by the set of degenerate quantities Y _1:i , P _i|i represents the variance estimated by the set of degenerated quantities Y _1:i for the degraded state x _i ,

Indicates the expectation of the degradation state x _i estimated by the degradation quantity set Y ₁ :i in the previous step, P _i|i-1 represents the variance estimated by the degradation quantity set Y _1:i on the degradation state x _i in the previous step, K(i) Indicates the filter gain,

Indicates the mean value of the estimated value of the monitoring data at time t _i-1 , v indicates the drift coefficient, ΔS _i indicates the nonlinear degradation time interval, y(i) indicates the monitoring data when the measurement error is considered, γ ² indicates the variance of the measurement error, P _i-1|i-1 represents the variance of the estimated value of the monitoring data at time t _i-1 , and π _i represents the variance of the noise term η; The calculation formula for updating is: P _i|i = (1-K(i))P _i|i-1 . 6. The method for predicting the remaining life of degraded equipment considering measurement uncertainty according to claim 3, characterized in that, in the step S32, the specific method for predicting the remaining life is: the first time the degraded state estimation result reaches The time of the predetermined failure threshold is used as the starting time of the remaining life of the degraded equipment.

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