CN107145645B - Method for predicting residual life of non-stationary degradation process with uncertain impact - Google Patents

Method for predicting residual life of non-stationary degradation process with uncertain impact Download PDF

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CN107145645B
CN107145645B CN201710257820.2A CN201710257820A CN107145645B CN 107145645 B CN107145645 B CN 107145645B CN 201710257820 A CN201710257820 A CN 201710257820A CN 107145645 B CN107145645 B CN 107145645B
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state
degradation
impact
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CN107145645A (en
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徐正国
王豆
柯晓杰
陈积明
秦刚华
谢尉扬
胡伯勇
张震伟
孙优贤
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Zhejiang University ZJU
Zhejiang Energy Group Research Institute Co Ltd
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Zhejiang Energy Group Research Institute Co Ltd
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Abstract

The invention provides a method for predicting the residual life of a non-stationary degradation process with uncertain impact, and belongs to the technical field of reliability engineering. The present invention combines the impact effect with a continuous degradation process to more reasonably characterize the actual degradation process. In the state estimation stage, the invention provides a new online state estimation algorithm to identify the real system state and provide necessary support for the subsequent residual life prediction. In the parameter estimation phase, the invention adopts an expectation-maximization algorithm to obtain unknown parameters in the model. Aiming at the residual life prediction, the invention considers the uncertainty of state estimation and the influence of impact effect at the same time, and gives a probability density function and an accumulated probability density function of residual life prediction distribution by an analytical expression. The model provided by the invention is more in line with the actual degradation condition, can obtain a more accurate online residual life prediction result, and has great significance for engineering fault prediction and health management.

Description

Method for predicting residual life of non-stationary degradation process with uncertain impact
Technical Field
The invention belongs to the technical field of reliability engineering, and relates to a method for predicting the residual life of a non-stable degradation process with uncertain impact.
Background
For most operating industrial equipment, equipment degradation and failure phenomena are unavoidable. The online running equipment is monitored in real time and the health state of the equipment is evaluated in time, so that the residual service life of the equipment is predicted, the operation and maintenance cost can be obviously reduced, and the running safety of the equipment is ensured. At present, a prediction method based on physical modeling is difficult to adapt to the complex situation of the existing industrial equipment. The data-driven modeling method based on the random process well takes the advantages of the sensing technology and is more and more popular among researchers. A large number of different predictive models are proposed and show good predictive results for different operating degradation characteristics. However, most of the existing work results focus on the situation of predicting the steady degradation of the object, and the problem of the occurrence of the impact phenomenon in the operation process is rarely considered. In fact, due to the internal and external influences of the operation of the plant (such as part defects, loads, temperatures, etc.), the phenomenon of shock may occur at any time during the whole operation of the plant. Such shock phenomena cause irreversible damage to the health of the equipment, accelerate the degradation process and, in the worst case, cause the equipment to be shut down. If such phenomena are not taken into account in the prediction model, the obtained prediction results are optimistic, and the results are obviously unacceptable for key systems. Although there has been some research work on impact and degradation processes, there has been no good solution in the field of remaining life prediction.
Disclosure of Invention
Aiming at the technical situation in the prior art, the invention aims to solve the problem that the equipment has an impact phenomenon in the operation process, and a model which can reasonably describe the characteristics of a degradation process is constructed according to the obtained performance degradation data to realize the online high-accuracy prediction of the residual life of the equipment.
The concept of the present invention will now be explained as follows:
the invention provides a degradation-impact system model based on a wiener process, which combines an impact effect with a continuous degradation process, thereby more reasonably depicting an actual degradation process. In the state estimation stage, the invention provides a new online state estimation algorithm to identify the real system state and provide necessary support for the subsequent residual life prediction. In the parameter estimation phase, the invention adopts an expectation-maximization algorithm to obtain unknown parameters in the model. Aiming at the residual life prediction, the invention considers the uncertainty of state estimation and the influence of impact effect at the same time, and gives a probability density function and an accumulated probability density function of residual life prediction distribution by an analytical expression. The model provided by the invention is more suitable for the actual degradation condition, and can obtain a more accurate online residual life prediction result.
According to the invention concept, the invention provides an online prediction method for the residual life of electromechanical equipment under the condition of two-stage degradation, which comprises the following steps:
step 1: establishing a performance degradation model for describing a non-stationary degradation process with impact, wherein the performance degradation model is a degradation-impact system model based on a wiener process;
step 2: estimating the system state by adopting an online state estimation algorithm comprising two stages;
and step 3: estimating model parameters of the degradation-impact system on line by adopting an expectation maximization algorithm;
and 4, step 4: and after the system state estimation and the parameter estimation are finished, obtaining a distribution expression of the residual service life of the equipment by using the updated system state, the estimated parameters and the measured values, and predicting the residual service life of the equipment.
Based on the above scheme, the following implementation manner can be specifically adopted for each step:
the step 1 is as follows:
step 1.1: the degradation-impact model is expressed as follows:
Figure BDA0001273889770000021
yk＝xk+νk
where k denotes the kth State monitoring Point, tkIndicates the time, x, corresponding to the kth state monitoring pointkRepresenting the level of degradation, i.e. the state of degradation of the system, eta represents the drift of the degradation of the system, tauk＝tk-tk-1Denotes the sampling interval, and σ denotes the diffusion coefficient of the wiener Process, BETA (τ)k)～N(0,τk) Representing standard brownian motion, N (μ, Σ) represents a normal distribution with a mean μ and a variance Σ; y iskRepresents tkThe state of the moment is monitored and measured, and the noise v is measuredkSatisfy vkN (0, R); in the model, S represents the damage to the system caused by the impact process and is a constant; assuming that the arrival of the impact is a Poisson process, the arrival rate is λ, C (t)k) Represents until tkThe total number of impacts occurring at a time, i.e. for all deltat, the probability of the total number of impacts occurring within the deltat time being n
Figure BDA0001273889770000031
Step 1.2: when the equipment is in the running state, the monitoring system continuously acquires the state monitoring data of the equipment and records the state monitoring data as y1:k＝{y1,y2,...,yk}。
The step 2 is as follows:
step 2.1: the online state estimation algorithm is divided into two stages of state prediction and state updating; in the state prediction phase, at the kth state monitoring point, based on the previous time tk-1The state estimation result and the degradation-impact model of (1) to obtain tkThe state prediction equation at the moment:
Figure BDA0001273889770000032
Pk|k-1＝Pk-1|k-1+Qk
wherein the content of the first and second substances,
Figure BDA0001273889770000033
Figure BDA0001273889770000034
as an estimate of the drift of the degradation,
Figure BDA0001273889770000035
representing an estimated value of damage to the system caused by the impact process; in the state prediction phase, it is noted that it is necessary to determine whether a shock has occurred in the last sampling period, and thus determine the state transition equation of the system.
Step 2.2: in the state updating stage, after a new state monitoring value is obtained, the hidden state of the system is updated based on Kalman filtering by using the information:
Figure BDA0001273889770000036
Kk＝Pk|k-1(Pk|k-1+Rk)-1
Figure BDA0001273889770000037
Pk|k＝Pk|k-1-KkPk|k-1
in the formula: kkTo Kalman gain, Pk|k-1Is an error covariance matrix, RkTo measure noise;
according to the recursive characteristics of the equations in the above two steps, the state of the system is estimated in real time.
The step 3 is as follows:
step 3.1: let the unknown parameter set be θ ═ η, σ, μ0,∑0S }, where μ0，∑0Mean and variance, respectively, representing the initial system state; obtaining the distribution of the measurement noise R through the prior knowledge of the used sensor or the offline analysis of historical operating data;
step 3.2: at the kth state monitoring point, the u-th iteration on the unknown parameter set is based on the result of the u-1 th iteration and the estimation result of the current hidden state, and the process is as follows:
Figure BDA0001273889770000041
wherein
Figure BDA0001273889770000042
For the estimated value of the unknown parameter set in the u-th iteration at the kth state monitoring point,
Figure BDA0001273889770000043
is composed of
Figure BDA0001273889770000044
The conditions of (a) are expected to be,
Figure BDA0001273889770000045
is composed of
Figure BDA0001273889770000046
An equivalent expression of (a);
Figure BDA0001273889770000047
to include all state estimation results
Figure BDA0001273889770000048
And all condition monitoring measurements y0:kA joint log-likelihood function of (a); based on Bayes' theorem and Markov characteristics, obtaining
Figure BDA0001273889770000049
The mathematical analysis of (2) expresses:
Figure BDA00012738897700000410
from said degradation-impact model, a conditional distribution, i.e. [ y ], is derived for the system variablesi|xi,θ]～N(xi,R)，[xi|xi-1,θ]～N(xi-1+ητi,σ2τi)，x0|θ～N(μ0,∑0)，
Figure BDA00012738897700000411
Figure BDA00012738897700000412
In the above formula, xiRepresents the system state of a normal continuous degradation process without shock, and
Figure BDA00012738897700000413
then this represents the sampling period
Figure BDA00012738897700000414
To
Figure BDA00012738897700000415
Systems in which shock phenomena occurA system state; thus, the set of state monitoring points where the impact occurs is denoted as M ═ M1,m2,...mm](ii) a Then pair
Figure BDA00012738897700000416
In the expression of (1), the state value x at the time 0 to k0:kCalculating an expectation; then
Figure BDA00012738897700000417
Is shown as
Figure BDA0001273889770000051
Figure BDA0001273889770000052
Step 3.3: using Kalman smoothing algorithms to process the desired solving problem, i.e. computing
Figure BDA0001273889770000053
And is marked as
Figure BDA0001273889770000054
The variance of (A) is denoted as Pi|kCovariance of the smoothed State Pi,i-1|kAs follows:
Figure BDA0001273889770000055
Figure BDA0001273889770000056
Pi,i-1|k＝Gi-1Pi|i+GiGi-1(Pi+1,i|k-Pi|i)
Figure BDA0001273889770000057
when i ═ k, there are
Figure BDA0001273889770000058
Based on the smoothing algorithm, the condition expectation formula is rewritten into
Figure BDA0001273889770000059
Figure BDA0001273889770000061
Figure BDA0001273889770000062
Step 3.4: after obtaining the system state smoothing value, the expectation of the condition is maximized, and the unknown parameter set θ is obtained as { η, σ, μ ═ η, σ, and0,∑0and S, analyzing a solution after u iterations:
Figure BDA0001273889770000063
Figure BDA0001273889770000064
Figure BDA0001273889770000065
Figure BDA0001273889770000066
Figure BDA0001273889770000067
the step 4 is as follows:
step 4.1: according to the degradation-impact model, adding impact process satisfying Poisson process with arrival rate of lambdaWriting the predicted degradation track x (t) into a system degradation process
Figure BDA0001273889770000068
After obtaining the system degradation and impact law, it is used to predict the remaining life, which is defined by the first arrival time (FTP), i.e. at the time tkThe random variable of remaining life is defined as RUL ═ inf { l: x (t)k+l)≥ω|x(tk) Where l is the implementation of a random variable for remaining life, ω is a predefined threshold;
step 4.2: for the degradation-impact model, the measurement data y are monitored on the basis of all states1:kConsidering the arrival rate lambda of the impact process, the failure threshold is omega, tkProbability density function f (l | y) of remaining life distribution at time1:kλ) and the cumulative probability density function F (l | y)1:kλ), is expressed as:
Figure BDA0001273889770000071
Figure BDA0001273889770000072
in the formula: phi [ ] is the cumulative distribution function of the standard normal distribution;
thus, analytical expressions of a probability density function and an accumulated probability density function of the residual life of the online prediction equipment are obtained.
The method for predicting the residual life of the unstable degradation process with the uncertain impact can be used for processing the unstable degradation process with the impact, and has good expandability on different occasions. Such as when the impact effect is 0, to a generally smooth degradation process. By constructing a degradation-impact model capable of reasonably describing the characteristics of the degradation process, a more accurate prediction effect can be obtained. The method provides powerful data support for subsequent equipment health management, is particularly valuable for high-reliability equipment maintenance management, and has wide prospects in the aspect of practical engineering application.
Drawings
FIG. 1 State estimation and measurements of a degradation-impact System model
FIG. 2 comparison of remaining Life prediction results for degradation-impact System model and Si model with truth values
FIG. 3 residual life prediction mean square error of degradation-impact system model and Si model
Detailed Description
The present invention will now be further described with reference to the accompanying drawings, and some of the principles have been described in detail above, and will not be described again here. The following example illustrates the specific operation steps and verifies the performance of the proposed model using a real case based on milling machine data.
The milling machine data records the operational degradation process of cutting metal material with the milling cutter. The case contains 16 different operating conditions, and the variables include cutting depth, material properties, feed amount and the like. In each case, a number of cutting passes are included and the termination of the device reaching a failure threshold is made, using as an example a case in which the signal of its acoustic sensor is used as an indicator of degradation. In order to better verify the performance of the model, another effective adaptive model (denoted as Si model) is used as a comparison. In order to quantify the residual life prediction results under different prediction models, Mean Square Error (MSE) is introduced as a measurement index, and Total Mean Square Error (TMSE) is calculated to compare the performance difference of the MSE and the TMSE.
Step 1: establishing a performance degradation model describing a non-stationary degradation process with impact
Step 1.1: the degradation-impact model is expressed as follows:
Figure BDA0001273889770000081
yk＝xk+νk
where k denotes the kth State monitoring Point, tkIndicates the time, x, corresponding to the kth state monitoring pointkRepresents the level of degradation, i.e. the state of degradation of the system,eta represents the degradation drift of the system, tauk＝tk-tk-1Denotes the sampling interval, and σ denotes the diffusion coefficient of the wiener Process, BETA (τ)k)～N(0,τk) Representing standard brownian motion, N (μ, ∑) represents a normal distribution with mean μ and variance Σ. y iskRepresents tkThe state of the moment is monitored and measured, and the noise v is measuredkSatisfy vkN (0, R). In the present model, S represents the damage to the system caused by the impact process and is a constant. Assuming that the arrival of the impact is a Poisson process, the arrival rate is λ, C (t)k) Represents until tkThe total number of impacts occurring at a time, i.e., for all deltat,
Figure BDA0001273889770000082
in the present model, η, σ, S are unknown and need to be estimated.
Step 1.2: when the equipment is in the running state, the monitoring system continuously acquires state monitoring data about the equipment, namely a signal of the sound wave sensor in the example, which can be recorded as y1:k＝{y1,y2,...,yk}。
Step 2: estimating system state using a new online state estimation algorithm comprising two stages
Step 2.1: the new state estimation algorithm is divided into two stages of one-step state prediction and state updating. In the state prediction phase, at the kth state monitoring point, based on the previous time tk-1The following prediction equation can be obtained from the state estimation result and the degradation-impact model.
Figure BDA0001273889770000091
Pk|k-1＝Pk-1|k-1+Qk
Wherein the content of the first and second substances,
Figure BDA0001273889770000092
during the state prediction phase, it is noted that it is necessary to determine whether or not the last sampling period has been reachedAnd (4) generating impact, and determining a state transition equation of the system according to the impact.
Step 2.2: in the state updating stage, after a new state monitoring value is obtained, more information is obtained and can be used for updating the hidden state of the system.
Figure BDA0001273889770000093
Kk＝Pk|k-1(Pk|k-1+Rk)-1
Figure BDA0001273889770000094
Pk|k＝Pk|k-1-KkPk|k-1
According to the recursive nature of the equations in the above two steps, the state of the system can be estimated in real time.
And step 3: on-line estimation of model parameters
Step 3.1: the unknown parameter set is denoted as θ ═ η, σ, μ0,∑0S }, where μ0，∑0Respectively representing the mean and variance of the initial system state. The distribution of the measurement noise R is assumed to be known and can be obtained by a priori knowledge of the technical parameters of the sensors used or by off-line analysis of historical operating data.
Step 3.2: at the kth state monitoring point, the u-th iteration on the unknown parameter set is based on the u-1 th iteration result and the estimation result of the current hidden state. This process can be expressed as follows.
Figure BDA0001273889770000095
Wherein
Figure BDA0001273889770000096
To include all state estimation results
Figure BDA0001273889770000097
And all condition monitoring measurements y0:kThe joint log-likelihood function of (a). Based on Bayes' theorem and Markov characteristics, the method can obtain
Figure BDA00012738897700001015
The mathematical expression of (a) is as follows.
Figure BDA0001273889770000101
According to the regression-impact model proposed by the invention, some condition distributions on system variables, namely y, can be obtainedi|xi,θ～N(xi,R)，xi|xi-1,θ～N(xi-1+ητi,σ2τi)，x0|θ～N(μ0,∑0)，
Figure BDA0001273889770000102
In the above formula, xiRepresents the system state of a normal continuous degradation process without shock, and
Figure BDA0001273889770000103
then this represents the sampling period
Figure BDA0001273889770000104
To
Figure BDA0001273889770000105
The system state of the shock phenomenon occurs therebetween. Thus, the set of state monitoring points where the impact occurs is denoted as M ═ M1,m2,...mm]. Then, to
Figure BDA0001273889770000106
X in the expression0:kTo find the expectation, then
Figure BDA0001273889770000107
Can be expressed as
Figure BDA0001273889770000108
Figure BDA0001273889770000109
Step 3.3: using Kalman smoothing algorithms to process the desired solving problem, i.e. computing
Figure BDA00012738897700001010
(is described as
Figure BDA00012738897700001011
)，
Figure BDA00012738897700001012
Variance of (note as P)i|k) Covariance of the smoothed State Pi,i-1|kAs follows.
Figure BDA00012738897700001013
Figure BDA00012738897700001014
Pi,i-1|k＝Gi-1Pi|i+GiGi-1(Pi+1,i|k-Pi|i)
Figure BDA0001273889770000111
In particular, when i ═ k, there are
Figure BDA0001273889770000112
Based on the smoothing algorithm, the condition expectation formula can be rewritten into
Figure BDA0001273889770000113
Figure BDA0001273889770000114
Figure BDA0001273889770000115
Step 3.4: after obtaining the system state smoothing value, the expectation of the condition is maximized, and then the unknown parameter set θ ═ η, σ, μ can be obtained0,∑0And S } is solved.
Figure BDA0001273889770000116
Figure BDA0001273889770000117
Figure BDA0001273889770000118
Figure BDA0001273889770000119
Figure BDA0001273889770000121
According to the step 2 and the step 3, for each state monitoring point, system hidden state estimation and model unknown parameter set identification are carried out, and the result is shown in fig. 1, and meanwhile, an actual state monitoring measurement sequence is also shown.
And 4, step 4: obtaining distribution expression of residual life of equipment
Step 4.1: according to the degradation-impact model provided by the invention, the impact process satisfying the Poisson process with the arrival rate of lambda is addedIn the system degradation process, the predicted degradation track can be written as
Figure BDA0001273889770000122
After the system degradation and impact rule is obtained, the residual life can be predicted. The remaining life of the system is defined by the time of first arrival (FTP), i.e. at time tkThe random variable of remaining life is defined as RUL ═ inf { l: x (t)k+l)≥ω|x(tk) Where l is the implementation of a random variable for remaining life, and ω is a predefined threshold. In this example, the failure threshold is set to 0.42.
Step 4.2: for the degradation-impact model proposed by the invention, the measurement data y are monitored on the basis of all states1:kConsidering the arrival rate lambda of the impact process, the failure threshold is omega, tkThe probability density function and cumulative probability density function of the remaining life distribution at the time may be expressed as:
Figure BDA0001273889770000123
Figure BDA0001273889770000124
the following table shows the total mean square error calculation of the remaining life prediction.
TABLE 1 Total mean square error of milling machine data residual life prediction results
Model (model) Degradation-impact system model Si model
TMSE(×1010) 2.8127 4.7242
Fig. 1 shows the state estimation results and the measured values of the degradation-impact system model. Fig. 2 shows probability density curves of the distribution of the predicted remaining life for 9200, 9400, 9600, 9800 and 10000 state monitoring points of the degradation-impact system model and the Si model, and a comparison with the true value. From the figure, the prediction results of the two models are relatively small due to the larger uncertainty of the actual case, but the prediction curve of the degradation-impact system model provided by the invention can be better close to the real residual life compared with the Si model. The remaining life prediction mean square error and the total mean square error calculation results of the degradation-impact system model and the Si model are respectively given in FIG. 3 and Table 1. Obviously, the total predicted mean square error of the degradation-impact system model is smaller than that of the Si model, which well proves the superiority of the model provided by the invention on the prediction performance for the degradation condition. Finally, it should be noted that in the Si model, the degradation drift of the system is extended to a random variable rather than a constant in the model proposed by the present invention, which means that there is a great prediction flexibility in many scenarios. However, for non-stationary degradation processes with impacts, the degradation-impact system model can obtain better residual life prediction results than the Si model. This clearly illustrates well that it is reasonable and necessary to use the degradation-impact model proposed by the present invention to deal with such non-stationary degradation processes.

Claims (1)

1. A method for predicting the residual life of a non-stationary degradation process with uncertain impact is characterized by comprising the following steps:
step 1: establishing a performance degradation model for describing a non-stationary degradation process with impact, wherein the performance degradation model is a degradation-impact system model based on a wiener process;
step 2: estimating the system state by adopting an online state estimation algorithm comprising two stages;
and step 3: estimating model parameters of the degradation-impact system on line by adopting an expectation maximization algorithm;
and 4, step 4: after the system state estimation and the parameter estimation are finished, a distribution expression of the residual service life of the equipment is obtained by using the updated system state, the estimated parameters and the measured values, and the residual service life of the equipment is predicted;
the step 1 is as follows:
step 1.1: the degradation-impact model is expressed as follows:
Figure FDA0002679355350000011
yk＝xk+νk
where k denotes the kth State monitoring Point, tkIndicates the time, x, corresponding to the kth state monitoring pointkRepresenting the level of degradation, i.e. the state of degradation of the system, eta represents the drift of the degradation of the system, tauk＝tk-tk-1Denotes the sampling interval, σ denotes the diffusion coefficient of the wiener process, B (τ)k)～N(0,τk) Representing standard brownian motion, N (μ, Σ) represents a normal distribution with a mean μ and a variance Σ; y iskRepresents tkThe state of the moment is monitored and measured, and the noise v is measuredkSatisfy vkN (0, R); in the model, S represents the damage to the system caused by the impact process and is a constant; assuming that the arrival of the impact is a Poisson process, the arrival rate is λ, C (t)k) Represents until tkThe total number of impacts occurring at a time, i.e. for all deltat, the probability of the total number of impacts occurring within the deltat time being n
Figure FDA0002679355350000012
Step 1.2: when the equipment is in the running state, the monitoring system continuously acquires the state monitoring data of the equipment and records the state monitoring data as y1:k＝{y1,y2,...,yk}；
The step 2 is as follows:
step 2.1: the online state estimation algorithm is divided into two stages of state prediction and state updating; in the state prediction phase, at the kth state monitoring point, based on the previous time tk-1The state estimation result and the degradation-impact model of (1) to obtain tkThe state prediction equation at the moment:
Figure FDA0002679355350000021
Pk|k-1＝Pk-1|k-1+Qk
wherein the content of the first and second substances,
Figure FDA0002679355350000022
Figure FDA0002679355350000023
as an estimate of the drift of the degradation,
Figure FDA0002679355350000024
representing an estimated value of damage to the system caused by the impact process;
Figure FDA0002679355350000025
and Pk|k-1Respectively representing the mean and variance of the prediction states;
step 2.2: in the state updating stage, after a new state monitoring value is obtained, the hidden state of the system is updated based on Kalman filtering by using the information:
Figure FDA0002679355350000026
Kk＝Pk|k-1(Pk|k-1+Rk)-1
Figure FDA0002679355350000027
Pk|k＝Pk|k-1-KkPk|k-1
in the formula: kkFor Kalman gain, RkMeasuring the variance of the noise for the kth state monitoring point;
estimating the state of the system in real time according to the recursive characteristics of the equations in the two steps;
the step 3 is as follows:
step 3.1: let the unknown parameter set be θ ═ η, σ, μ0,∑0S }, where μ0，∑0Mean and variance, respectively, representing the initial system state; obtaining the distribution of variance R of measurement noise through the prior knowledge of the used sensor or the offline analysis of historical operating data;
step 3.2: at the kth state monitoring point, the u-th iteration on the unknown parameter set is based on the result of the u-1 th iteration and the estimation result of the current hidden state, and the process is as follows:
Figure FDA0002679355350000028
wherein
Figure FDA0002679355350000029
For the estimated value of the unknown parameter set in the u-th iteration at the kth state monitoring point,
Figure FDA0002679355350000031
is composed of
Figure FDA0002679355350000032
The conditions of (a) are expected to be,
Figure FDA0002679355350000033
is composed of
Figure FDA0002679355350000034
An equivalent expression of (a);
Figure FDA0002679355350000035
to include all state estimation results
Figure FDA0002679355350000036
And all condition monitoring measurements y0:kA joint log-likelihood function of (a); based on Bayes' theorem and Markov characteristics, obtaining
Figure FDA0002679355350000037
The mathematical analysis of (2) expresses:
Figure FDA0002679355350000038
from said degradation-impact model, a conditional distribution, i.e. [ y ], is derived for the system variablesi|xi,θ]～N(xi,R)，[xi|xi-1,θ]～N(xi-1+ητi,σ2τi)，x0|θ～N(μ0,∑0)，
Figure FDA0002679355350000039
Figure FDA00026793553500000310
In the above formula, xiRepresents the system state of a normal continuous degradation process without shock, and
Figure FDA00026793553500000311
then this represents the sampling period
Figure FDA00026793553500000312
To
Figure FDA00026793553500000313
The shock phenomenon appears in the middleThe system state of the image; thus, the set of state monitoring points where the impact occurs is denoted as M ═ M1,m2,...mm](ii) a Then pair
Figure FDA00026793553500000314
In the expression of (1), the state value x at the time 0 to k0:kCalculating an expectation; then
Figure FDA00026793553500000315
Is shown as
Figure FDA00026793553500000316
Step 3.3: using Kalman smoothing algorithms to process the desired solving problem, i.e. computing
Figure FDA00026793553500000317
And is marked as
Figure FDA00026793553500000318
Figure FDA00026793553500000319
The variance of (A) is denoted as Pi|kCovariance of the smoothed State Pi,i-1|kAs follows:
Figure FDA00026793553500000320
Figure FDA0002679355350000041
Pi,i-1|k＝Gi-1Pi|i+GiGi-1(Pi+1,i|k-Pi|i)
Figure FDA0002679355350000042
when i ═ k, there are
Figure FDA0002679355350000043
Based on the smoothing algorithm, the condition expectation formula is rewritten into
Figure FDA0002679355350000044
Figure FDA0002679355350000045
Figure FDA0002679355350000046
Step 3.4: after obtaining the system state smoothing value, the expectation of the condition is maximized, and the unknown parameter set θ is obtained as { η, σ, μ ═ η, σ, and0,∑0and S, analyzing a solution after u iterations:
Figure FDA0002679355350000047
Figure FDA0002679355350000048
Figure FDA0002679355350000049
Figure FDA0002679355350000051
Figure FDA0002679355350000052
the step 4 is as follows:
step 4.1: according to the degradation-impact model, an impact process meeting a Poisson process with an arrival rate of lambda is added into a system degradation process, and a predicted degradation track x (t) is written into
Figure FDA0002679355350000053
After obtaining the system degradation and impact law, the system degradation and impact law is used for predicting the remaining life, and the remaining life of the system is defined by the first arrival time FTP, namely at the moment tkThe random variable of remaining life is defined as RUL ═ inf { l: x (t)k+l)≥ω|x(tk) Where l is the implementation of a random variable for remaining life, ω is a predefined threshold;
step 4.2: for the degradation-impact model, the measurement data y are monitored on the basis of all states1:kConsidering the arrival rate lambda of the impact process, the failure threshold is omega, tkProbability density function f (l | y) of remaining life distribution at time1:kλ) and the cumulative probability density function F (l | y)1:kλ), is expressed as:
Figure FDA0002679355350000054
Figure FDA0002679355350000061
in the formula: phi [ ] is the cumulative distribution function of the standard normal distribution;
thus, analytical expressions of a probability density function and an accumulated probability density function of the residual life of the online prediction equipment are obtained.
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