CN112949026B - Age and state dependence considered degradation equipment residual life prediction method - Google Patents

Age and state dependence considered degradation equipment residual life prediction method Download PDF

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CN112949026B
CN112949026B CN202110068209.1A CN202110068209A CN112949026B CN 112949026 B CN112949026 B CN 112949026B CN 202110068209 A CN202110068209 A CN 202110068209A CN 112949026 B CN112949026 B CN 112949026B
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司小胜
庞哲楠
胡昌华
裴洪
李天梅
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Rocket Force University of Engineering of PLA
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Abstract

The invention discloses a method for predicting the residual life of degradation equipment considering age and state dependence. Then, the degradation state and unknown parameters are adaptively estimated and updated using a Kalman Filter (KF) and Expectation Maximization (EM) algorithm. Deducing a residual life distribution analysis form which is based on the existing observation value and can be updated in real time based on the estimated degradation state and the model parameters; the residual life prediction accuracy is effectively improved.

Description

Age and state dependence considered degradation equipment residual life prediction method
Technical Field
The invention belongs to the technical field of reliability engineering, and particularly relates to a degradation equipment residual life prediction method considering age and state dependence.
Background
As one of the key issues of Predictive and Health Management (PHM), estimating the remaining life (RUL) of a degraded device may provide a useful information reference for state-based maintenance decisions, optimal detection intervals, and spare part ordering. To obtain an accurate estimate of the RUL distribution, a suitable degradation model is needed that adequately characterizes the actual degradation process.
Nonlinearity and randomness are important factors that must be considered when estimating RUL in a random degradation model framework, and random process models are typically used to describe the randomness of degradation. At the same time, individual variability is also an important factor to consider in the estimation of RUL. Individual variability may be defined as different individuals in the same batch of equipment exhibiting different rates of degradation due to different operating environments and health conditions. Furthermore, due to the complexity of the system or the high cost of directly observing the degradation state, implicit or partially observable degradation processes are often encountered in engineering practice. The above-mentioned problems have been studied to some extent in the prior art, and it is noted that in the prior art and the invention, the degradation process is aimed at a device which depends only on age, but there is still a large number of degradation processes which are not only affected by the age of the device but also on the health state of the device itself. Taking fatigue crack propagation as an example, the degradation rate at the initial stage of the fatigue crack is low. As fatigue cracks grow, the stress strength will gradually increase, resulting in an increased degradation rate. Currently, there is no related study of a degradation apparatus remaining life prediction method that considers both age and state in an individual difference and implicit degradation state.
Disclosure of Invention
The invention aims to provide a degradation device with age and state dependence in a degradation process, and provides a degradation device residual life prediction method with age and state dependence in consideration of the individual variability of the device and the influence of an implicit degradation state.
The technical scheme adopted by the invention is as follows:
a method for predicting remaining life of a degraded device considering age and state dependence, comprising the steps of:
step 1, a degradation model is established, dependency of equipment degradation, age and state is described by using a drift function, and individuation difference and measurement error between equipment in the degradation process are considered;
step 2, on the basis of the step 1, a state space model is established, and the relation between the implicit degradation state and the observed value is described;
step 3, estimating an implicit degradation state by using a Kalman filtering algorithm, and estimating parameters in a state space model by using an expected maximization algorithm;
step 4, deducing a residual life distribution analysis form which is based on the existing observation value and can be updated in real time based on the degradation state and the model parameters estimated in the step 3;
and 5, when a new observed value is generated, updating parameters in the state space model in real time, and substituting the parameters into an expression of the residual life distribution of the equipment, so that life prediction of the degraded equipment is realized.
Preferably, in the step 1, the degradation model building process is as follows:
let X (t) denote the degradation amount of the sample at time t, then the degradation process { X (t), t.gtoreq.0 } based on the diffusion process can be expressed as:
dX(t)＝μ(X(t),t；θ)dt+σ B dB(t) (1)
where μ (X (t), t; θ) represents a nonlinear drift coefficient function while depending on age and state, μ (X, t; θ) =ax+bh (t; ζ), where θ= (a, b, ζ), and h (t) is a function with respect to time t, ζ is an unknown parameter vector in the function h (t; ζ); wherein, the parameter a is a fixed constant, and the parameter b is a random effect coefficient representing individual variability, namely b:
Figure BDA0002904895300000031
μ b is the mean value of parameter b,/->
Figure BDA0002904895300000032
Is the variance of parameter b; sigma (sigma) B Is a diffusion coefficient, θ is a parameter vector, B (t) is a standard Brownian motion, and B (t): N (0, t); without loss of generality, when t=0, X (0) =0;
the relation between the uncertain observed value and the potential degradation state at the moment t is described by { Y (t), t is more than or equal to 0}, and the expression is as follows:
Y(t)＝X(t)+ε (2)
for the random degradation process given by equation (1), in the first time sense, the lifetime T of the device can be defined as:
T＝inf{t:X(t)≥ω|X(0)＜ω} (3)
wherein εFor measuring errors, the service life T is a random variable, and omega is a failure threshold value of equipment; the probability density function of the hit T can be f T (t) represents;
to utilize the state monitoring information, it is assumed that the device is at discrete time point 0=t 0 ＜t 1 ＜t 2 ＜L＜t k Monitoring to t k By the time a set of degradation measurements is recorded as Y 1:k ＝{y 1 ,y 2 ,L,y k}, wherein yk ＝Y(t k ) Representing t k The observed value at the moment, the real degradation state corresponding to the observed value is expressed as X 1:k ＝{x 1 ,x 2 ,L,x k}, wherein xk ＝X(t k ) The method comprises the steps of carrying out a first treatment on the surface of the Defining t according to the concept of the time of arrival k Time of day, remaining life L k Is that
L k ＝inf{l k >0:X(l k +t k )≥ω|x k ＜ω} (4)
The probability density function is expressed as
Figure BDA0002904895300000033
Preferably, in the step 2, the state space model building process is as follows:
the residual life estimation takes into account the influence of individual variability and measurement errors during degradation, and therefore only the observed value Y 1:k Available, actual degradation state x k Cannot be directly utilized;
random effect parameter presence update procedure b k ＝b k-1 +α, wherein α:
Figure BDA0002904895300000034
b 0 ＝b:/>
Figure BDA0002904895300000035
the initial distribution is obtained by using the observation value to obtain posterior distribution of the parameter b; at the same time, the state equation and the measurement equation are converted into discrete time equations for state estimation when new observation data is available, and then the state equationThe equation (1) and the observation equation (2) may be at discrete points in time t k The transition at k=1, 2, l is a state space model as follows:
Figure BDA0002904895300000041
wherein ,
Figure BDA0002904895300000042
v k ＝σ B [B(t k )-B(t k-1 )]，ε k for epsilon at t k A specific value of time; { v k } k≥1 and {εk } k≥1 Is an independent co-distributed noise sequence, and v k :/>
Figure BDA0002904895300000043
ε k :/>
Figure BDA0002904895300000044
Let z k ＝x k -ax k Then x k ＝z k /(1-a)＝βz k The method comprises the steps of carrying out a first treatment on the surface of the Thus, a new potential degradation state z k And a random parameter b k Can be regarded as a new implicit state, requiring dependency on the observed value Y 1:k Estimating; based on the state space model (5), a new implicit state can be estimated by using Kalman filtering, and the state space model (5) can be rewritten as:
Figure BDA0002904895300000045
wherein ,sk ∈R 2×1 ，η k ∈R 2×1 ，A k ∈R 2×2 ，C∈R 1×2 ，η k :N(0,Q k ) In particular, to
Figure BDA0002904895300000046
Preferably, in step 3, implicit degradation state estimation:
based on a state space model (6), implicit states are estimated using a Kalman filtering algorithm, first s is defined k Based on t k The conditional expectation and variance of the time-of-day available observations are as follows:
Figure BDA0002904895300000047
Figure BDA0002904895300000048
wherein ,
Figure BDA0002904895300000049
Figure BDA0002904895300000051
accordingly, the expected sum variance obtained by one-step prediction is respectively
Figure BDA0002904895300000052
Figure BDA0002904895300000053
According to the definition above, the Kalman filtering algorithm may be summarized as follows:
(1) State estimation
Figure BDA0002904895300000054
Figure BDA0002904895300000055
Figure BDA0002904895300000056
Figure BDA0002904895300000057
(2) Variance update
P k|k ＝P k|k-1 -K(k)Cr′ k|k-1 P k|k-1
Wherein, the initial state of the Kalman filtering is set as follows:
Figure BDA0002904895300000058
from formula (6), s k The conditional probability density function of (2) is a bivariate gaussian distribution, i.e. s k :
Figure BDA0002904895300000059
Then, the potential degradation state z k And random effect parameter b k The posterior distribution of (a) is the posterior distribution of (b) with time t k Relatedly, based on the property of the double-variable Gaussian distribution, it is possible to obtain
Figure BDA00029048953000000510
Figure BDA00029048953000000511
Figure BDA00029048953000000512
wherein ,
Figure BDA0002904895300000061
Figure BDA0002904895300000062
wherein ,
Figure BDA0002904895300000063
preferably, in step 3, adaptive parameter estimation:
representing unknown parameters in the state space model (6) as vectors
Figure BDA0002904895300000064
By iterative calculation and maximization of conditional expectation of log likelihood function, expectation maximization algorithm can generate parameter estimation sequence converging to maximum likelihood estimation of parameter, let ∈ ->
Figure BDA0002904895300000065
Estimated value representing unknown parameters in jth iteration of the expectation maximization algorithm, +.>
Figure BDA0002904895300000066
Representing a conditional expectation operator;
the expectation maximization algorithm comprises the steps of:
(1) Calculate the j+1st iteration value
Figure BDA0002904895300000067
wherein ,l(s1:k ,Y 1:k |Θ)＝lnp(s 1:k ,Y 1:k |Θ)；
(2) Will be
Figure BDA0002904895300000068
Maximizing to obtain a new estimated value of theta
Figure BDA0002904895300000069
To be used for
Figure BDA00029048953000000610
Repeating the above two steps until meeting the convergence condition;
for the state space model (4), t is cut off k Time of day, implicit degradation state sequence s 1:k And observing sequence Y 1:k The joint log likelihood function of (2) can be expressed as:
Figure BDA00029048953000000611
next, the conditional expectation of the log-likelihood function (14) is calculated, and it is possible to obtain:
Figure BDA0002904895300000071
to calculate
Figure BDA0002904895300000072
The conditional expectation of each term in equation (17) needs to be deduced; first, the definition is as follows:
Figure BDA0002904895300000073
Figure BDA0002904895300000074
Figure RE-GDA0003020202560000075
the method can be obtained through complex algebraic operation:
Figure BDA0002904895300000076
wherein ,
Figure BDA0002904895300000077
Figure BDA0002904895300000078
Figure BDA0002904895300000079
obviously, for the purpose of calculation
Figure BDA00029048953000000710
Need to get +.>
Figure BDA00029048953000000711
P i|k ，P i-1|k and Mi|k The method comprises the steps of carrying out a first treatment on the surface of the These conditions are expected to be calculated by a kalman filter algorithm and an RTS smoothing algorithm; the specific steps of the RTS smoothing algorithm are as follows: />
(1) Estimating the hidden state according to the Kalman filtering algorithm to obtain
Figure BDA00029048953000000712
and Pk|k ；
(2) Backward smoothing iterations, for i=k, k-1, l,1, backward recursion
Figure BDA0002904895300000081
Figure BDA0002904895300000082
Figure BDA0002904895300000083
(3) Initialization of
M k|k ＝[I-K(k)C]A k-1 P k-1|k-1 (20)
(4) Updating covariance matrix
Figure BDA0002904895300000084
In order to reduce the complexity of the parameter estimation algorithm,
Figure BDA0002904895300000085
can be divided into three parts, the first part only contains the initial value of the Kalman filtering algorithm +.>
Figure BDA0002904895300000086
Can be expressed as:
Figure BDA0002904895300000087
respectively regarding the formula (22) as s 0 and P0|0 The partial derivative is calculated and is set to 0, so that the estimated value of the j+1st iteration can be obtained:
Figure BDA0002904895300000088
the second part contains the parameter vector in the state equation
Figure BDA0002904895300000089
Can be expressed as:
Figure BDA00029048953000000810
unknown parameters
Figure BDA00029048953000000811
The estimation can be performed by a section likelihood function method, and in order to facilitate the solution of the partial derivative, the formula (24) is rewritten as:
Figure BDA0002904895300000091
/>
assuming that ζ is fixed, the formula (25) is respectively related to
Figure BDA0002904895300000092
and />
Figure BDA0002904895300000093
Taking the partial derivative and letting it be 0, one can get:
Figure BDA0002904895300000094
Figure BDA0002904895300000095
substituting equations (26) and (27) into (25) to obtain a section likelihood function about xi, and obtaining an estimated value of j+1th iteration of xi by using a search algorithm;
the third part contains the parameter vector Θ in the observation equation 3 ＝{β,σ ε -can be expressed as:
Figure BDA0002904895300000096
unknown parameter theta 3 ＝{β,σ ε Estimation method and apparatus
Figure BDA0002904895300000097
Similarly, assuming β is fixed, formula (28) is given for +.>
Figure BDA0002904895300000098
Taking the partial derivative and letting it be 0, one can get:
Figure BDA0002904895300000099
substituting formula (29) into formula (28) to maximize the cross-sectional likelihood function with respect to beta, thereby obtaining an estimated value
Figure BDA00029048953000000910
Since β=1/(1-a), it is possible to obtain +.>
Figure BDA00029048953000000911
Thus, maximize Q
Figure BDA00029048953000000912
By maximizing +.>
Figure BDA00029048953000000913
and />
Figure BDA00029048953000000914
Obtained.
Preferably, in step 4, for the degradation model (1) equation, while taking into account the personalized differences between the devices and the measurement errors, t is defined according to the remaining lifetime definition (4) equation k Probability density function of time remaining life
Figure BDA0002904895300000101
Can be expressed as:
Figure BDA0002904895300000102
wherein ,
Figure BDA0002904895300000103
the invention has the beneficial effects that: the invention utilizes a nonlinear diffusion process to describe a degradation process of degradation equipment with age and state dependent simultaneously, and provides a degradation modeling and residual life prediction method which simultaneously considers individual variability and implicit degradation state of the equipment. First, by constructing a state space model, the relationship between the implicit degradation state and the observed value is described. Then, the degradation state and unknown parameters are adaptively estimated and updated using a Kalman Filter (KF) and Expectation Maximization (EM) algorithm. Based on the estimated degradation state and model parameters, a residual life distribution analysis form which is based on the existing observation value and can be updated in real time is deduced; the prediction accuracy of the residual life is effectively improved.
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In order to more clearly illustrate the embodiments of the invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, it being obvious that the drawings in the following description are only some embodiments of the invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a graph of gyroscope degradation under step acceleration stress;
FIG. 2 is a graph of probability density of remaining life of a # 2 bearing;
FIG. 3 is a graph of probability density of remaining life of a 3# bearing;
fig. 4 is a 4# bearing remaining life probability density function.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments of the present invention.
Thus, the following detailed description of the embodiments of the invention, as presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The invention specifically provides a degradation equipment residual life prediction method considering age and state dependence, which comprises the following steps:
step 1, a degradation model is established, dependency of equipment degradation, age and state is described by using a drift function, and individuation difference and measurement error between equipment in the degradation process are considered;
the degradation model building process is as follows:
let X (t) denote the performance degradation of the sample at time t, then the degradation model { X (t), t.gtoreq.0 } based on the diffusion process can be expressed as:
dX(t)＝μ(X(t),t；θ)dt+σ B dB(t) (1)
where μ (X (t), t; θ) represents a nonlinear drift coefficient function while depending on age and state, μ (X, t; θ) =ax+bh (t; ζ), where θ= (a, b, ζ), and h (t) is a function with respect to time t, ζ is an unknown parameter vector in the function h (t; ζ); wherein, the parameter a is a fixed constant, and the parameter b is a random effect coefficient representing individual variability, namely b:
Figure BDA0002904895300000121
μ b is the mean value of parameter b,/->
Figure BDA0002904895300000122
Is the variance of parameter b; sigma (sigma) B Is a diffusion coefficient, θ is a parameter vector, B (t) is a standard Brownian motion, and B (t): N (0, t); without loss of generality, when t=0, X (0) =0;
in addition, perfect measurement of the potential degradation state is often difficult to achieve, and measurement errors caused by external interference, noise, non-ideal instruments and other factors are inevitably generated in the measurement process. In this case, the observation only partially reflects the potential degradation state. To characterize the effect of measurement uncertainty, the relationship between the uncertainty observations and the potential degradation state at time t is described by { Y (t), t.gtoreq.0 } as follows:
Y(t)＝X(t)+ε (2)
for the random degradation process given by equation (1), in the first time sense, the lifetime T of the device can be defined as:
T＝inf{t:X(t)≥ω|X(0)＜ω} (3)
where ε is the measurement error, life T is a random variable, ω is the failure threshold of the device, and is typically determined by industry standards and expert experience. Probability Density Function (PDF) of lifetime T is available f T (t) represents.
The main object of the present invention is to estimate and update the RUL distribution of a single device in operation based on real-time observations of the degradation process. To utilize the state monitoring information, it is assumed that the device is at discrete time point 0=t 0 ＜t 1 ＜t 2 ＜L＜t k Monitoring to t k By the time a set of degradation measurements is recorded as Y 1:k ＝{y 1 ,y 2 ,L,y k}, wherein yk ＝Y(t k ) Representing t k Observations of time of day. The true degradation state corresponding to this is denoted as X 1:k ＝{x 1 ,x 2 ,L,x k}, wherein xk ＝X(t k ). Defining t according to the concept of the time of arrival k Time of day, remaining life L k Is that
L k ＝inf{l k >0:X(l k +t k )≥ω|x k ＜ω} (4)
Its PDF is expressed as
Figure BDA0002904895300000131
Step 2, on the basis of the step 1, a state space model is established, and the relation between the implicit degradation state and the observed value is described;
for a practical system, our goal is to at a specific point in time t k Estimating remaining lifetime (RUL), which is the kth monitored data from the start time, and the degradation state is x k ＝X(t k ). To obtain the monitoring t of the system k The remaining lifetime (RUL) at the moment is introduced as followsAnd (5) managing.
Theory 1 for the degradation process described by equation (1), if t k Time degradation observation x k ＝X(t k ) What t k The Probability Density Function (PDF) of the time remaining lifetime (RUL) estimate is equivalent to a random process { U (l) k ),l k Pass through constant threshold ω k ＝ω-x k Wherein U (l) k )＝X(l k +t k )-x k And U (0) =0. Random procedure { U (l) k ),l k 0 may be expressed as
dU(l k )＝μ(X(l k +t k ),l k +t k ；θ)dl k +σ B dB(l k )
Given random effect parameters b and t k Implicit degradation observations of time x k Residual life of
Figure BDA0002904895300000132
The PDF of (C) can be approximated as +.>
Figure RE-GDA0003020202560000133
wherein ,
Figure RE-GDA0003020202560000134
represented as
Figure RE-GDA0003020202560000135
Figure BDA0002904895300000136
For simplicity, let eta (u, l k ；θ)＝μ(X(l k +t k ),l k +t k ；θ)＝μ(U(l k )+x k ,l k +t k The method comprises the steps of carrying out a first treatment on the surface of the θ). Then
Figure BDA0002904895300000137
and />
Figure BDA0002904895300000138
Can be expressed as
Figure BDA0002904895300000141
In the above derivation, it is assumed that no random effect exists, and an implicit degradation state can be directly observed. However, there are individual differences in the different devices and it is often not possible to measure the potential degradation state perfectly in practical engineering.
For the effect of individual variability and measurement errors in the degradation process in the estimation of the remaining lifetime (RUL), in this case only the observed value Y 1:k Available, actual degradation state x k Cannot be directly utilized. The main purpose of the invention is based on the observed value Y 1:k Deriving Probability Density Function (PDF) for remaining life (RUL)
Figure BDA0002904895300000142
And the remaining life (RUL) profile may be updated based on the new state monitoring data.
Random effect parameter b presence update procedure b k ＝b k-1 +α, wherein α:
Figure BDA0002904895300000143
b 0 ＝b:/>
Figure BDA0002904895300000144
the initial distribution is used to obtain a posterior distribution of the parameter b. Meanwhile, in order to recognize an implicit degradation state, it is necessary to convert a state equation and a measurement equation into a discrete time equation in order to perform state estimation when new observation data is available. Then, the state equation (1) and the observation equation (2) may be at discrete time points t k ，k＝1,2,The transition at L is to a state space model as follows:
Figure BDA0002904895300000145
wherein ,
Figure BDA0002904895300000146
v k ＝σ B [B(t k )-B(t k-1 )]，ε k for epsilon at t k A specific value of time; { v k } k≥1 and {εk } k≥1 Is an independent co-distributed noise sequence, and v k :/>
Figure BDA0002904895300000147
ε k :/>
Figure BDA0002904895300000148
Let z k ＝x k -ax k Then x k ＝z k /(1-a)＝βz k The method comprises the steps of carrying out a first treatment on the surface of the Thus, a new potential degradation state z k And a random parameter b k Can be regarded as a new implicit state, requiring dependency on the observed value Y 1:k Estimating; let z k ＝x k -ax k Then x k ＝z k /(1-a)＝βz k The method comprises the steps of carrying out a first treatment on the surface of the Thus, a new potential degradation state z k And random parameter b k Can be regarded as a new implicit state, requiring dependency on the observed value Y 1:k Estimating; based on the state space model (5), a new implicit state can be estimated by using Kalman filtering, and the state space model (5) can be rewritten as:
Figure BDA0002904895300000151
wherein ,sk ∈R 2×1 ，η k ∈R 2×1 ，A k ∈R 2×2 ，C∈R 1×2 ，η k :N(0,Q k ) In particular, to
Figure BDA0002904895300000152
Step 3, estimating an implicit degradation state by using a Kalman filtering algorithm, and estimating parameters in a state space model by using an expected maximization algorithm;
implicit degradation state estimation:
based on a state space model (6), implicit states are estimated using a Kalman filtering algorithm, first s is defined k Based on t k The conditional expectation and variance of the time-of-day available observations are as follows:
Figure BDA0002904895300000153
Figure BDA0002904895300000154
wherein ,
Figure BDA0002904895300000155
Figure BDA0002904895300000156
accordingly, the expected sum variance obtained by one-step prediction is respectively
Figure BDA0002904895300000157
Figure BDA0002904895300000158
According to the definition above, the Kalman filtering algorithm may be summarized as follows:
(1) State estimation
Figure BDA0002904895300000161
Figure BDA0002904895300000162
Figure BDA0002904895300000163
Figure BDA0002904895300000164
/>
(2) Variance update
P k|k ＝P k|k-1 -K(k)Cr′ k|k-1 P k|k-1
Wherein, the initial state of the Kalman filtering is set as follows:
Figure BDA0002904895300000165
from formula (6), s k The conditional probability density function of (2) is a bivariate gaussian distribution, i.e. s k :
Figure BDA0002904895300000166
Then, the potential degradation state z k And random effect parameter b k The posterior distribution of (a) is the posterior distribution of (b) with time t k Relatedly, based on the property of the double-variable Gaussian distribution, it is possible to obtain
Figure BDA0002904895300000167
Figure BDA0002904895300000168
Figure BDA0002904895300000169
wherein ,
Figure BDA00029048953000001610
Figure BDA00029048953000001611
wherein ,
Figure BDA00029048953000001612
and (3) adaptive parameter estimation:
for convenience, the unknown parameters in the model (6) are represented as vectors
Figure BDA00029048953000001613
Estimating Θ using the Expectation Maximization (EM) algorithm provides a basic framework for solving the Maximum Likelihood Estimation (MLE) problem of the existence of implicit states.
By iterative calculation and maximization of conditional expectation of log likelihood function, expectation maximization algorithm can generate parameter estimation sequence converging to maximum likelihood estimation of parameter, let
Figure BDA0002904895300000171
Estimated value representing unknown parameters in the jth iteration of the expectation maximization algorithm, ++>
Figure BDA0002904895300000172
Representing a conditional expectation operator.
In summary, the EM algorithm includes the following two steps:
(1) E, step E: calculate the j+1st iteration value
Figure BDA0002904895300000173
wherein ,l(s1:k ,Y 1:k |Θ)＝ln p(s 1:k ,Y 1:k |Θ)；
(2) M steps: will be
Figure BDA0002904895300000174
Maximizing to obtain a new estimated value of theta
Figure BDA0002904895300000175
To be used for
Figure BDA0002904895300000176
The above two steps are iterated repeatedly until the convergence condition is satisfied. />
For the state space model (6), t is cut off k Time of day, implicit degradation state sequence s 1:k And observing sequence Y 1:k The joint log likelihood function of (2) can be expressed as:
Figure BDA0002904895300000177
next, the conditional expectation of the log-likelihood function (16) is calculated, and it is possible to obtain:
Figure BDA0002904895300000181
to calculate
Figure BDA0002904895300000182
The conditional expectation of each term in equation (17) needs to be deduced; first, the definition is as follows:
Figure BDA0002904895300000183
Figure BDA0002904895300000184
Figure RE-GDA0003020202560000184
the method can be obtained through complex algebraic operation:
Figure BDA0002904895300000186
wherein ,
Figure BDA0002904895300000187
Figure BDA0002904895300000188
Figure BDA0002904895300000189
obviously, for the purpose of calculation
Figure BDA00029048953000001810
Need to get +.>
Figure BDA00029048953000001811
P i|k ，P i-1|k and Mi|k The method comprises the steps of carrying out a first treatment on the surface of the These conditions are expected to be calculated by a kalman filter algorithm and an RTS smoothing algorithm; the specific steps of the RTS smoothing algorithm are as follows:
(1) Estimating the hidden state according to the Kalman filtering algorithm to obtain
Figure BDA00029048953000001812
and Pk|k ；
(2) Backward smoothing iterations, for i=k, k-1, l,1, backward recursion
Figure BDA0002904895300000191
Figure BDA0002904895300000192
Figure BDA0002904895300000193
(3) Initialization of
M k|k ＝[I-K(k)C]A k-1 P k-1|k-1 (20)
(4) Updating covariance matrix
Figure BDA0002904895300000194
In order to reduce the complexity of the parameter estimation algorithm,
Figure BDA0002904895300000195
can be divided into three parts, the first part only contains the initial value of the Kalman filtering algorithm +.>
Figure BDA0002904895300000196
Can be expressed as:
Figure BDA0002904895300000197
respectively regarding the formula (22) as s 0 and P0|0 The partial derivative is calculated and is set to 0, so that the estimated value of the j+1st iteration can be obtained:
Figure BDA0002904895300000198
the second part contains the parameter vector in the state equation
Figure BDA0002904895300000199
Can be expressed as:
Figure BDA00029048953000001910
unknown parameters
Figure BDA00029048953000001911
The estimation can be performed by a section likelihood function method, and in order to facilitate the solution of the partial derivative, the formula (24) is rewritten as: />
Figure BDA0002904895300000201
Assuming that ζ is fixed, the formula (25) is respectively related to
Figure BDA0002904895300000202
and />
Figure BDA0002904895300000203
Taking the partial derivative and letting it be 0, one can get:
Figure BDA0002904895300000204
Figure BDA0002904895300000205
substituting equations (26) and (27) into (25) to obtain a section likelihood function about xi, and obtaining an estimated value of j+1th iteration of xi by using a search algorithm;
the third part contains the parameter vector Θ in the observation equation 3 ＝{β,σ ε -can be expressed as:
Figure BDA0002904895300000206
unknown parameter theta 3 ＝{β,σ ε Estimation method and apparatus
Figure BDA0002904895300000207
Similarly, assuming β is fixed, formula (28) is given for +.>
Figure BDA0002904895300000208
Taking the partial derivative and letting it be 0, one can get:
Figure BDA0002904895300000209
substituting formula (29) into formula (28) to maximize the cross-sectional likelihood function with respect to beta, thereby obtaining an estimated value
Figure BDA00029048953000002010
Since β=1/(1-a), it is possible to obtain +.>
Figure BDA00029048953000002011
Thus, maximize
Figure BDA00029048953000002012
By maximizing +.>
Figure BDA00029048953000002013
and />
Figure BDA00029048953000002014
Obtained.
Step 4, deducing a residual life distribution analysis form which is based on the existing observation value and can be updated in real time based on the degradation state and the model parameters estimated in the step 3
To calculate t k Time PDF
Figure BDA0002904895300000211
The full probability formula can be utilized as follows: />
Figure BDA0002904895300000212
wherein ,p(sk |Y 1:k ) Is s k |Y 1:k Is the same as the condition expectation of p (z k |b k ,Y 1:k) and p(bk |Y 1:k ) Z respectively k |b k ,Y 1:k and bk |Y 1:k Is desirable for the conditions of (2).
To derive the distribution of remaining life, the following quotients are introduced.
The quotients 2 there is Z: N (μ, σ) 2 ) And ω, β, L, M εi, Q εi + Can obtain
Figure RE-GDA0003020202560000211
From this, the Probability Density Function (PDF) of the remaining life in the time-of-arrival sense can be derived using a full probability formula, the specific remaining life expression being as follows:
for the degradation process (1), the individual variability and measurement errors between devices are taken into account; according to the definition of remaining life (RUL), L k ＝inf{l k >0:X(l k +t k )≥ω|x k < ω }; then t k Probability Density Function (PDF) of time remaining life (RUL)
Figure BDA0002904895300000214
Can be expressed as:
Figure BDA0002904895300000215
wherein ,
Figure BDA0002904895300000221
Figure BDA0002904895300000222
Figure BDA0002904895300000223
Figure BDA0002904895300000224
Figure BDA0002904895300000225
and (3) proving: PDF with residual life can be obtained according to the quotation 1
Figure BDA0002904895300000226
Can be approximated as
Figure RE-GDA0003020202560000215
Will x k ＝βz k Substituted into the above formula to obtain
Figure BDA0002904895300000228
And simplifying the method, and the result is as follows: />
Figure BDA0002904895300000229
wherein ,
Figure BDA00029048953000002210
Figure BDA00029048953000002211
Figure BDA00029048953000002212
from formula (12), z k |b k ,Y 1:k :
Figure BDA00029048953000002213
Based on the sum of the full probability formulas, using the lemma 2, we can get:
Figure BDA00029048953000002214
wherein ,
Figure BDA00029048953000002215
about b k Is a function of (2).
Further, from the formula (12) and the formula (13), b is known k |Y 1:k :
Figure BDA0002904895300000231
Based on the sum of the full probability formulas, using the lemma 2, we can get:
Figure BDA0002904895300000232
wherein ,
Figure BDA0002904895300000233
Figure BDA0002904895300000234
/>
Figure BDA0002904895300000235
Figure BDA0002904895300000236
the syndrome is known.
Example application analysis
Taking degradation data of a certain type of bearing as an example, the effectiveness of the method provided by the invention is verified. The test data for the four samples are shown in figure 1.
The last 25 observations of the 2 nd-4 th bearing are used as a test data set, and the rest of degradation data are used as training data sets. According to the parameter estimation method proposed in step 3, an estimated value of an unknown parameter in the model can be obtained, as shown in table 1.
TABLE 1 degradation model parameter estimation
Figure BDA0002904895300000241
To illustrate the effectiveness of this method in residual life (RUL) prediction, samples may be subjected to residual life (RUL) prediction based on initial estimates of model parameters in table 1. In the RUL prediction process, once new degradation observations are available, the model parameters are updated, resulting in a more accurate residual life (RUL) distribution. In fig. 2-4, a Probability Density Function (PDF) of the remaining life (RUL), a predicted average remaining life (RUL), and an actual remaining life (RUL) at that time are shown for comparison.
For three bearings, the predicted residual life of the method is very close to the actual residual life, and the probability density function of the predicted residual life becomes higher and sharper along with the updating of model parameters, which shows that the uncertainty of the predicted result is gradually reduced.
The effectiveness and the superiority of the equipment degradation modeling and residual life prediction method considering age and state dependence are further verified.
The foregoing is merely illustrative of the present invention and not restrictive, and other modifications and equivalents thereof may occur to those skilled in the art without departing from the spirit and scope of the present invention.

Claims (4)

1. A method for predicting remaining life of a degraded device in consideration of age and state dependence, comprising the steps of:
step 1, a degradation model is established, dependency of equipment degradation, age and state is described by using a drift function, and individuation difference and measurement error between equipment in the degradation process are considered;
step 2, on the basis of the step 1, a state space model is established, and the relation between the implicit degradation state and the observed value is described;
step 3, estimating an implicit degradation state by using a Kalman filtering algorithm, and estimating parameters in a state space model by using an expected maximization algorithm;
step 4, deducing a residual life distribution analysis form which is based on the existing observation value and can be updated in real time based on the degradation state and the model parameters estimated in the step 3;
step 5, when a new observation value is generated, updating parameters in the state space model in real time, and substituting the parameters into an expression of the residual life distribution of the equipment so as to realize life prediction of the degraded equipment;
in the step 1, the degradation model building process is as follows:
let X (t) denote the degradation amount of the sample at time t, then the degradation process { X (t), t.gtoreq.0 } based on the diffusion process can be expressed as:
dX(t)＝μ(X(t),t；θ)dt+σ B dB(t) (1)
where μ (X (t), t; θ) represents a nonlinear drift coefficient function while depending on age and state, μ (X, t; θ) =ax+bh (t; ζ), where θ= (a, b, ζ), and h (t) is a function with respect to time t, ζ is an unknown parameter vector in the function h (t; ζ); wherein, the parameter a is a fixed constant, and the parameter b is a random effect coefficient representing individual variability, namely b:
Figure FDA0004186237640000011
μ b is a ginsengMean value of number b>
Figure FDA0004186237640000012
Is the variance of parameter b; sigma (sigma) B For diffusion coefficients, θ is a parameter vector, B (t) is a standard Brownian motion, and B (t): n (0, t); without loss of generality, when t=0, X (0) =0;
the relation between the uncertain observed value and the potential degradation state at the moment t is described by { Y (t), t is more than or equal to 0}, and the expression is as follows:
Y(t)＝X(t)+ε (2)
for the random degradation process given by equation (1), in the first time sense, the lifetime T of the device can be defined as:
T＝inf{t:X(t)≥ω|X(0)＜ω} (3)
wherein epsilon is a measurement error, the service life T is a random variable, and omega is a failure threshold of the equipment; the probability density function of the hit T can be f T (t) represents;
to utilize the state monitoring information, it is assumed that the device is at discrete time point 0=t 0 ＜t 1 ＜t 2 ＜…＜t k Monitoring to t k By the time a set of degradation measurements is recorded as Y 1:k ＝{y 1 ,y 2 ,…,y k}, wherein yk ＝Y(t k ) Representing t k The observed value of the moment, the real degradation state corresponding to the observed value is expressed as X 1:k ＝{x 1 ,x 2 ,…,x k}, wherein xk ＝X(t k ) The method comprises the steps of carrying out a first treatment on the surface of the Defining t according to the concept of the time of arrival k Time of day, remaining life L k Is that
L k ＝inf{l k >0:X(l k +t k )≥ω|x k ＜ω} (4)
The probability density function is expressed as
Figure FDA0004186237640000021
In the step 2, the state space model building process is as follows:
residual life estimationThe influence of individual variability and measurement errors during degradation is taken into account, so that only the observed value Y 1:k Available, actual degradation state x k Cannot be directly utilized;
random effect parameter presence update procedure b k ＝b k-1+α, wherein ,
Figure FDA0004186237640000022
the initial distribution is obtained by using the observation value to obtain posterior distribution of the parameter b; at the same time, the state equation and the measurement equation are converted into discrete-time equations for state estimation when new observation data is available, and then the state equation (1) and the observation equation (2) can be obtained at discrete time points t k K=1, 2, …), where the transition is to a state space model, as follows:
Figure FDA0004186237640000023
wherein ,
Figure FDA0004186237640000024
v k ＝σ B [B(t k )-B(t k-1 )]，ε k for epsilon at t k A specific value of time; { v k } k≥1 and {εk } k≥1 Is an independent and equidistributed noise sequence, and +.>
Figure FDA0004186237640000025
Let z k ＝x k -ax k Then x k ＝z k /(1-a)＝βz k The method comprises the steps of carrying out a first treatment on the surface of the Thus, a new potential degradation state z k And random parameter b k Can be regarded as a new implicit state, requiring dependency on the observed value Y 1:k Estimating; based on the state space model (5), a new implicit state can be estimated by using Kalman filtering, and the state space model (5) can be rewritten as:
Figure FDA0004186237640000031
wherein ,sk ∈R 2×1 ，η k ∈R 2×1 ，A k ∈R 2×2 ，C∈R 1×2 ，η k :N(0,Q k ) In particular, to
Figure FDA0004186237640000032
2. A method for predicting remaining life of a degraded device taking into account age and state dependence as claimed in claim 1, wherein in step 3, the degraded state estimation is implicit:
based on a state space model (6), implicit states are estimated using a Kalman filtering algorithm, first s is defined k Based on t k The conditional expectation and variance of the time-of-day available observations are as follows:
Figure FDA0004186237640000033
Figure FDA0004186237640000034
wherein ,
Figure FDA0004186237640000035
Figure FDA0004186237640000036
accordingly, the expected sum variance obtained by one-step prediction is respectively
Figure FDA0004186237640000037
/>
Figure FDA0004186237640000038
According to the definition above, the Kalman filtering algorithm may be summarized as follows:
(1) State estimation
Figure FDA0004186237640000041
Figure FDA0004186237640000042
Figure FDA0004186237640000043
Figure FDA0004186237640000044
(2) Variance update
P kk ＝P kk-1 -K(k)Cr k ′ k-1 P kk-1
Wherein, the initial state of the Kalman filtering is set as follows:
Figure FDA0004186237640000045
from formula (6), s k The conditional probability density function of (2) is a bivariate gaussian distribution, i.e
Figure FDA0004186237640000046
Then, the potential degradation state z k And random effect parameter b k The posterior distribution of (a) is the posterior distribution of (b) with time t k Relatedly, based on the nature of the bivariate Gaussian distribution, one can obtain
Figure FDA0004186237640000047
wherein ,
Figure FDA0004186237640000048
Figure FDA0004186237640000049
wherein ,
Figure FDA00041862376400000410
3. a method for predicting remaining life of a degraded equipment taking into account age and state dependency as claimed in claim 2, wherein in step 3, the adaptive parameter estimates:
representing unknown parameters in the state space model (6) as vectors
Figure FDA0004186237640000051
By iterative calculation and maximization of conditional expectation of log likelihood function, expectation maximization algorithm can generate parameter estimation sequence converging to maximum likelihood estimation of parameter, let ∈ ->
Figure FDA0004186237640000052
Representing an estimate of the unknown parameter in the jth iteration of the expectation maximization algorithm,
Figure FDA0004186237640000053
representing a conditional expectation operator; />
The expectation maximization algorithm comprises the steps of:
(1) Calculate the j+1st iteration value
Figure FDA0004186237640000054
wherein ,l(s1:k ,Y 1:k |Θ)＝lnp(s 1:k ,Y 1:k |Θ)；
(2) Will be
Figure FDA0004186237640000055
Maximizing to obtain a new estimated value of theta
Figure FDA0004186237640000056
To be used for
Figure FDA0004186237640000057
Repeating the above two steps until meeting the convergence condition;
for the state space model (4), t is cut off k Time of day, implicit degradation state sequence s 1:k And observing sequence Y 1:k The joint log likelihood function of (2) can be expressed as:
Figure FDA0004186237640000058
next, the conditional expectation of the log-likelihood function (14) is calculated, and it is possible to obtain:
Figure FDA0004186237640000059
to calculate
Figure FDA0004186237640000061
The conditional expectation of each term in equation (17) needs to be deduced; head partFirst, the definition is as follows:
Figure FDA0004186237640000062
Figure FDA0004186237640000063
Figure FDA0004186237640000064
the method can be obtained through complex algebraic operation:
Figure FDA0004186237640000065
wherein ,
Figure FDA0004186237640000066
Figure FDA0004186237640000067
Figure FDA0004186237640000068
obviously, for the purpose of calculation
Figure FDA0004186237640000069
Need to get +.>
Figure FDA00041862376400000610
P i|k ，P i-1|k and Mi|k The method comprises the steps of carrying out a first treatment on the surface of the These conditions are expected to be calculated by Kalman filteringCalculating by a method and an RTS smoothing algorithm; the specific steps of the RTS smoothing algorithm are as follows:
(1) Estimating the hidden state according to the Kalman filtering algorithm to obtain
Figure FDA00041862376400000611
and Pk|k ；
(2) Backward smoothing iterations, for i=k, k-1, …,1, backward recursion
Figure FDA00041862376400000612
Figure FDA00041862376400000613
Figure FDA00041862376400000614
(3) Initialization of
M k|k ＝[I-K(k)C]A k-1 P k-1|k-1 (20)
(4) Updating covariance matrix
Figure FDA0004186237640000071
In order to reduce the complexity of the parameter estimation algorithm,
Figure FDA0004186237640000072
can be divided into three parts, the first part only contains the initial value of the Kalman filtering algorithm +.>
Figure FDA0004186237640000073
Can be expressed as:
Figure FDA0004186237640000074
respectively regarding the formula (22) as s 0 and P0|0 The partial derivative is calculated and is set to 0, so that the estimated value of the j+1st iteration can be obtained:
Figure FDA0004186237640000075
the second part contains the parameter vector in the state equation
Figure FDA0004186237640000076
Can be expressed as: />
Figure FDA0004186237640000077
Unknown parameters
Figure FDA0004186237640000078
The estimation can be performed by a section likelihood function method, and in order to facilitate the solution of the partial derivative, the formula (24) is rewritten as:
Figure FDA0004186237640000079
assuming that ζ is fixed, the formula (25) is respectively related to
Figure FDA00041862376400000710
and />
Figure FDA00041862376400000711
Taking the partial derivative and letting it be 0, one can get:
Figure FDA00041862376400000712
Figure FDA00041862376400000713
substituting equations (26) and (27) into (25) to obtain a section likelihood function about ζ, and obtaining an estimated value of j+1st iteration of ζ by using a search algorithm;
the third part contains the parameter vector Θ in the observation equation 3 ＝{β,σ ε -can be expressed as:
Figure FDA0004186237640000081
unknown parameter theta 3 ＝{β,σ ε Estimation method and apparatus
Figure FDA0004186237640000082
Similarly, assuming β is fixed, formula (28) is given for +.>
Figure FDA0004186237640000083
Taking the partial derivative and letting it be 0, one can get:
Figure FDA0004186237640000084
substituting formula (29) into formula (28) to maximize the cross-sectional likelihood function with respect to beta, thereby obtaining an estimated value
Figure FDA0004186237640000085
Since β=1/(1-a), it is possible to obtain +.>
Figure FDA0004186237640000086
Thus, maximize
Figure FDA0004186237640000087
By maximizing +.>
Figure FDA0004186237640000088
and />
Figure FDA0004186237640000089
Obtained.
4. A method for predicting the remaining life of a degradation device in consideration of age and state dependency as claimed in claim 3, wherein in step 4, for the degradation model (1) equation, while taking into consideration the personalized difference and measurement error between devices, t is defined according to the remaining life definition (4) equation k Probability density function of time remaining life
Figure FDA00041862376400000810
Can be expressed as:
Figure FDA00041862376400000811
wherein ,
Figure FDA0004186237640000091
Figure FDA0004186237640000092
Figure FDA0004186237640000093
Figure FDA0004186237640000094
Figure FDA0004186237640000095
/>
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