Residual life prediction method considering nonlinear random correlation

Abstract

The invention belongs to the technical field of residual life prediction, and discloses a residual life prediction method considering nonlinear random correlation, and the specific technical scheme is as follows: firstly, considering the influence of measurement errors and multi-index random correlation, establishing a degradation state space model based on the multi-index nonlinear random correlation, and establishing a degradation state space model based on the multi-index nonlinear random correlation; on the basis of an expectation maximization algorithm, a forward extended Kalman filtering algorithm and a reverse Rauch-Tong-Striebel smoothing algorithm for jointly estimating unknown parameters and implicit states are used for deducing and calculating residual life distribution of a system in a competitive failure mode; the correctness and effectiveness of the residual life model with multi-degradation index correlation are verified through numerical experiments, the adaptability and superiority of the proposed method compared with other prediction methods are verified by taking an XJTU-SY data set as an example, the prediction precision is high, and the practicability is good.

Description

Technical Field

[0001] The present invention belongs to the technical field of remaining useful life prediction, and particularly relates to a remaining useful life prediction method considering non - linear stochastic correlation. Background Art

[0002] Currently, how to improve the reliability and safety of intelligent high - end equipment during operation is a key issue that the intelligent manufacturing industry endeavors to solve. In practical applications, degradation is an inevitable natural phenomenon, and key components in the system need to operate safely and stably throughout their entire life cycle. However, due to the influence of aging and operating environment, their quality and performance will gradually decline. In recent years, with the development of information sensing technology, the degradation state of system components can be monitored in real time. Prognostics and Health Management (PHM) has received extensive attention currently because it can evaluate the reliability of the system under actual operating conditions. Remaining useful life (RUL) prediction, as an important part of PHM, plays an important role in ensuring the normal operation of the system and formulating reasonable maintenance decisions.

[0003] Current research related to PHM usually assumes that the health state of the system can be simply reflected by a single indicator variable. This indicator may be a physical health indicator directly representing the physical performance of the system, or a virtual health indicator indirectly reflecting the trend of system performance degradation. For degradation modeling and RUL prediction, using a single indicator usually cannot accurately describe the potential degradation mechanism of the system, resulting in inaccurate and unreliable RUL prediction results. Therefore, it is necessary to use multiple indicators to characterize the health state of the system and further accurately predict its remaining useful life. For example, when studying the crack propagation of a gear component, it is reflected by two degradation indicators: crack length and width. Or bearing degradation can be characterized by wear and crack indicators. The degradation of an aero - engine is reflected by monitoring data such as temperature, pressure, rotational speed, and flow rate sensors. Since multiple indicators characterize a complex system, considering the stochastic correlation between multiple indicators makes the RUL prediction of the system more complex. In engineering practice, the stochastic correlation between indicators is usually complex and non - linear, and it is necessary to consider a generalized non - linear model to model the stochastic correlation relationship between indicator degradations, making the constructed degradation model more generally applicable. Therefore, it is necessary to deeply study the degradation modeling and RUL prediction problems of multi - indicator systems with non - linear stochastic correlation.

[0004] In recent years, the reliability assessment and remaining useful life (RUL) prediction of systems considering multiple performance degradations have received extensive attention from scholars at home and abroad, and relevant research results have emerged in an endless stream. For the correlation modeling of multi-index systems, for example, Zavieh studied the multi-degradation modeling of friction corrosion and fatigue failure of stainless steel. Vonder Ohe et al. studied the multi-degradation composite mechanism model of friction and corrosion under the surface action of static and cyclic loads of metals. There are also a large number of domestic research results. For example, the teams led by Professor Huang Hongzhong of the University of Electronic Science and Technology, Professor Tang Jiayin of Southwest Jiaotong University, and Professor Si Shubin of Northwestern Polytechnical University have all studied the modeling of multi-variable correlated degradation processes and reliability assessment. The teams led by Professor Hu Changhua, Professor Lu Ningyun of Nanjing University of Aeronautics and Astronautics, and Professor Chen Youling of Chongqing University have successively studied the RUL prediction problem under multiple degradation variables. The research based on multiple degradation indicators mostly focuses on the reliability assessment problem of the system. Usually, it is assumed that the multiple degradation quantities are independent of each other; or have the same marginal distribution, and the correlation relationship is described by a multi-dimensional random distribution; or it is considered that the correlation structure between the indicators is consistent, and a multi-variable Copula function is used to establish a correlated degradation model, and then the system reliability or life characteristics are further calculated. However, the existing research does not explicitly reflect the correlation relationship between the indicators in the process of multi-index degradation modeling, nor does it give an accurate closed mathematical form of the correlation. Therefore, it is of great research significance to construct a reasonable non-linear correlation model to describe the degradation process of multi-index systems. Summary of the Invention

[0005] To solve the technical problems existing in the prior art, the present invention provides a remaining useful life prediction method considering non-linear stochastic correlation, which studies the non-linear stochastic correlation degradation modeling and RUL prediction problems of multi-index degradation systems. First, considering the non-linear stochastic correlation between multiple indicators, a degradation state space model of the multi-index system is established. Secondly, the extended Kalman filter algorithm and the expectation maximization algorithm are used to jointly estimate the hidden degradation state and unknown parameters. Then, under the competing failure modes, according to the definition of the first passage time, the remaining life distribution of the system is calculated, and the XJTU-SY data set is taken as an example for experimental verification and analysis.

[0006] To achieve the above object, the technical solution adopted by the present invention is as follows: a remaining useful life prediction method considering non-linear stochastic correlation, and the specific steps are as follows:

[0007] Step 1: Considering the influence of measurement errors and multi-index stochastic correlation, a degradation state space model based on multi-index non-linear stochastic correlation is established; assuming that the component has n degradation indicators, and each degradation indicator corresponds to its own different degradation state, a corresponding degradation model is established for the i-th degradation indicator:

[0008]

[0009] wherein, is the degradation amount of the i-th index at time t, k represents the drift coefficient of the i-th index at time t, k-1 represents the drift process of the i-th index; is the diffusion coefficient of index i, ξ k = B k - B k-1 where B k represents the Brownian motion process at time t, k represents the diffusion process of the i-th index;

[0010] Considering the correlation between various indexes, stochastic correlation modeling and analysis are carried out, and it is reflected by the drift coefficient of the Wiener process:

[0011]

[0012] wherein, a qi (i, q = 1...n) represents the correlation coefficient between index q and index i, represents the degradation state of the q-th index at time k - 1, is a non-linear parameter;

[0013] After substituting Equation (2) into Equation (1), when i = 1..n, expanding it gives:

[0014]

[0015] Let, After arranging Equation (3), we get:

[0016]

[0017] Further combining and simplifying Equation (4) gives:

[0018]

[0019] where D is the correlation coefficient matrix,

[0020] Finally, the degradation state equation is obtained:

[0021]

[0022] wherein, ξ k is the process noise of the multi-index system, that is, ξ k ~ MVN(0, Q k ), Q k ​​​It represents the process noise covariance of the multi-index system, and MVN represents the multivariate normal distribution.

[0023] After considering the measurement error, the measurement equation corresponding to the multi-index system is defined as:

[0024] Y k =Z k +V k (7)

[0025] where, Y k represents the measurement data of the multi-index system, and V k represents the measurement noise, that is, V k ~MVN(0, R k ), R k represents the variance of the measurement noise, and diag() represents the diagonal matrix, represents the measurement noise of index n.

[0026] Step 2: Apply the extended Kalman filter method to achieve online update of the state of the degraded state space model. Use the extended Kalman filter method to linearize the nonlinear function. Let Substitute the device performance degradation function f(Z k ) at time t k-1 into the Taylor expansion at the predicted value of the degradation state to obtain:

[0027]

[0028] where, f′() is the first derivative of the nonlinear function, represents the estimated value of the degradation state at time k-1, and O is the higher-order infinitesimal. Ignoring the higher-order infinitesimal term and substituting Equation (8) into Equation (6), the state equation of the degraded device after linearization is:

[0029]

[0030] where, F k-1 represents the first derivative of the nonlinear function of the degradation state , and ≈ represents approximation.

[0031] The extended Kalman filter algorithm is used for degradation state estimation, and this algorithm includes two stages: prediction and update;

[0032] The prediction stage is:

[0033]

[0034] The update stage is:

[0035]

[0036] K k = P k|k-1 [P | k-1 + R k -1 (13)

[0037] P | k = P | k-1 - K k P | k-1 (14)

[0038] where denotes the estimated value of the degradation state at time k, K k denotes the state gain at time k, P | k-1 denotes the predicted covariance matrix at time k - 1, P | k denotes the estimated covariance matrix at time k.

[0039] According to equations (10) - (14), under the condition of the observation sequence Y k at time t 1:k , the posterior estimate of the degradation state Z k is a multivariate normal distribution, that is θ k denotes the unknown parameters of the multi - index degradation model.

[0040] Step 3: Use the Expectation - Maximization - Extended Kalman Filter algorithm for simultaneous estimation of the degradation state and parameters. This filtering algorithm includes two stages: prediction and update, to achieve adaptive estimation of the parameters of the degradation state space model; for the unknown parameters of the proposed model, use the Expectation - Maximization - Extended Kalman Filter algorithm for parameter estimation. The unknown parameters are If the observed data Y 1:k of the device degradation state is known, based on equations (7) and (9), the joint log - likelihood function lnf(Z k , Y 1:k ) of the device performance degradation state Z k and the observed data Y 1:k is obtained as follows:[[]]

[0041] lnf(Z k , Y 1:k ; θ) = lnf(Z k ; θ) + lnf(Y 1:k |Z k ; θ) (15)

[0042] where f(Z k ; θ) represents the probability density function of the system degradation state, f(Y 1:k |Z k ​; θ) represents the probability density function of the measurement data; ln() represents the logarithmic function.

[0043] Since ξ in the state space model k and V k both follow Gaussian distributions, there are the following relationships:

[0044]

[0045] where, | | represents the determinant of a matrix, () T represents the transpose of a matrix, represents the summation of a matrix from j = 1 to j = k, () -1 represents matrix inversion.

[0046] Substituting equations (16) and (17) into equation (15), the log-likelihood function lnf(Z k ,Y 1:k ; θ) is:

[0047]

[0048] To obtain the mathematical expectation E: By solving the expectation of the likelihood function f(Z k ,Y 1:k ; θ) with respect to the hidden state Z k , that is, E[ln f(Z k ,Y 1:k ; θ)]; Let G = E[ln f(Z k ,Y 1:k ; θ)],

[0049]

[0050] where, tr() represents the trace of a matrix,

[0051] To calculate the conditional mathematical expectation of the likelihood function, the RTS filtering algorithm is used to obtain:

[0052]

[0053] where, represents the estimated state value of the backward filtering at the (j - 1)th moment, P j-1|k represents the estimated state covariance of the backward filtering at the (j - 1)th moment, P j,j-1|k represents the estimated state cross-covariance of the backward filtering at the jth and (j - 1)th moments, F j-1|k represents the first-order derivative of the nonlinear function of the degraded state at the (j - 1)th moment.

[0054] Substituting equations (20)-(22) into equation (19) gives:

[0055]

[0056]

[0057] The maximum likelihood function is as follows:

[0058]

[0059] where arg max() represents finding the maximum value, represents the value of the unknown parameter at the (l + 1)-th iteration at time k.

[0060] Let the partial derivative of the log-likelihood function in Equation (24) be 0 for estimation, and we get:

[0061]

[0062] where, represents the partial derivative of G with respect to the degradation state for derivation.

[0063] For It is obtained by solving the non-linear equation (25). On this basis, the correlation D matrix and parameters are further obtained. By finding the maximum value of the likelihood function, the noise parameters R l+1 and Q l+1 are:

[0064]

[0065] Iteratively solve Equations (25)-(27) until θ (l+1) -θ (l) is less than the given threshold to stop the iteration, that is, the adaptive estimation of the parameters of the proposed model is realized.

[0066] Step 4: Use multiple indicators for characterization. As long as a certain indicator exceeds the corresponding failure threshold, the system fails. According to the definition of the first passage time, in the competing failure modes, at the current time t k The RULL of the system is:

[0067] L = inf{l > 0: Z 1 (k + l) ≥ w 1 , or..., or Z i (k + l) ≥ w i , or..., i = 1, 2,..., n|Z k , θ k} (28)

[0068] where inf{} represents the infimum, and k + l in Z i (k + l) represents the time tk+l , where \(i\) represents the number of indicators, and \(w\) i is the failure threshold of indicator \(i\), and \(\theta\) k is a parameter in the state space model; \(or\) represents the logical relationship of \(or\) in mathematics.

[0069] According to the definition in Equation (28), the RUL distribution function of the competing failure mode system is:

[0070]

[0071] where \(z\) i,k is the monitored value of indicator \(i\) at time \(k\), and \(\pi\) k (\(\Delta Z\) 1 , \(\Delta Z\) 2 ,... \(\Delta Z\) n |Z k , \(\theta\) k ) is the joint probability density function of the multi - indicator degradation state; \(\Delta Z\) n represents the increment of the \(n\) - indicator degradation state, represents the integral function.

[0072] To obtain the system degradation state distribution, according to the properties of the stochastic system, solve the degradation state \(Z\) of the multi - indicator at time \(t\) of the system k to get: k+l

[0073]

[0074] where \(e\) () is the exponential function, represents the summation function;

[0075] Define \(P\) k+l \(=\) cov{\(Z\) k+l , \(Z\) k+l};

[0076] Using the properties of the stochastic system state evolution equation, obtain the mean value of \(Z\) k+l and covariance \(P\) k+l :

[0077]

[0078] where \(Q\) λ+1 \(=\) cov{\(\xi\) λ+1 , \(\xi\) λ+1} is the variance of the degradation system noise, and \(e\) D(k+l-(λ+1)) is the state transition matrix;

[0079] On this basis, obtain the RUL probability density function of the system through the finite - difference formula ​

[0080]

[0081] Among them, represents the partial derivative of the remaining life distribution function.

[0082] At time t k for the observed sequence Y 1:k under the condition, using a filtering algorithm for estimation, the degradation state Z k is obtained, and the posterior estimate is a multivariate normal distribution, that is After considering the measurement error, at time t k the RUL probability density function f k (l k |θ k , Z k , Y 1:k ) is:

[0083]

[0084] Among them, h(Z k |θ k , Y 1:k ) is the probability density function of the hidden state.

[0085] Compared with the prior art, the specific beneficial effects of the present invention are as follows: The non-linear stochastic correlation state space model proposed by the present invention can solve the actual correlation modeling problem. Secondly, the model can better describe the degradation of a multi-index system, and can intuitively consider the stochastic correlation between multiple indexes, and can solve the problem of inaccurate estimation caused by implicit correlation modeling in a multi-index system, providing a research basis for correlation modeling. BRIEF DESCRIPTION OF THE DRAWINGS

[0086] Figure 1 is the structural flowchart of the present invention.

[0087] Figure 2 is the construction diagram of the degradation paths of three degradation indexes, Figure 2 (a) is the construction diagram of the degradation path with w 1 = 0.975, Figure 2 (b) is the construction diagram of the degradation path with w 2 = 0.992, Figure 2 (c) is the construction diagram of the degradation path with w 3 = 0.987.

[0088] Figure 3 is the schematic diagram of the parameter estimation result, Figure 3 (a) is the schematic diagram of the correlation parameter, Figure 3 (b) is the schematic diagram of the noise parameter,Figure 3 (c) is a schematic diagram of non - linear parameters.

[0089] Figure 4 It is a remaining useful life prediction diagram based on three degradation models. Figure 4 (a) is the remaining useful life prediction diagram without considering the correlation of the remaining useful life. Figure 4 (b) is the remaining useful life prediction diagram considering linear correlation of the remaining useful life. Figure 4 (c) is the remaining useful life prediction diagram of the present invention considering non - linear correlation of the remaining useful life.

[0090] Figure 5 It is a comparison diagram of the means of different models and the true values.

[0091] Figure 6 It is a comparison diagram of the prediction errors of different models.

[0092] Figure 7 It is a vibration signal diagram. Figure 7 (a) is the original horizontal vibration signal diagram of bearing 3 - 4. Figure 7 (b) is the original vertical vibration signal diagram of bearing 3 - 4.

[0093] Figure 8 It is a degradation state estimation diagram of the degradation index. Figure 8 (a) is the degradation state estimation diagram of the vertical vibration index constructed for bearing 3 - 4. Figure 8 (b) is the degradation state estimation diagram of the horizontal vibration index constructed for bearing 3 - 4.

[0094] Figure 9 It is a schematic diagram of the system RUL under competing failure modes.

[0095] Figure 10 It is a comparison diagram of the means of bearing 3 - 4 and the true values based on the present method. Detailed implementation manners

[0096] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0097] As Figure 1 shown, a remaining useful life prediction method considering non - linear stochastic correlation is provided. Assume that the component has n degradation indices, and each degradation index corresponds to its own different degradation state. Then, a corresponding degradation model is established for the i - th degradation index:

[0098]

[0099] Among them, is the degradation amount of the i-th index at time t, k k-1 represents the drift coefficient of the i-th index at time t, k-1 k represents the drift process of the i-th index; is the diffusion coefficient of index i, and ξ k = B k - B k-1 where B k represents the Brownian motion process at time t, k qi represents the diffusion process of the i-th index.

[0100] For multi-degradation index modeling, considering the correlation between various indexes, the present invention conducts stochastic correlation modeling and analysis, considering that the degradation process of the index not only depends on its own degradation state but also is affected by other indexes, which is reflected by the drift coefficient of the Wiener process:

[0101]

[0102] where a qi (i, q = 1...n) represents the correlation coefficient between index q and index i, represents the degradation state of the q-th index at time k - 1, is a non-linear parameter;

[0103] After substituting Equation (2) into Equation (1), when i = 1..n, expanding gives:

[0104]

[0105] Let, After organizing Equation (3), we get:

[0106]

[0107] Further combining and simplifying Equation (4) gives:

[0108]

[0109] where D is the correlation coefficient matrix,

[0110] Finally, the degradation state equation is obtained:

[0111]

[0112] Among them, ξ k is the multi-index system process noise, that is, ξ k ~ MVN(0, Qk ),Q k represents the process noise covariance of the multi-index system, and MVN represents the multivariate normal distribution.

[0113] Due to external environmental influences such as noise, considering measurement errors, the measurement equation corresponding to the multi-index system is:

[0114] Y k =Z k +V k (7)

[0115] where Y k represents the measurement data of the multi-index system, and V k represents the measurement noise, that is, V k ~MVN(0, R k ), R k represents the variance of the measurement noise, and diag() represents the diagonal matrix, represents the measurement noise of index n.

[0116] Since traditional Kalman filtering has linear Gaussianity, it is difficult to handle non-linear degradation processes. Therefore, the present invention adopts the extended Kalman filtering method to realize the online update of the state of the implicit proportional degradation model.

[0117] When applying the extended Kalman filtering method, it is necessary to linearize the non-linear function first. First, let Based on the basic properties of the power series, the device performance degradation function f(Z k ) at time t k-1 is Taylor-expanded at the predicted value of the degradation state , and we can get:

[0118]

[0119] where f′() is the first derivative of the non-linear function, represents the estimated value of the degradation state at time k-1, and O is a higher-order infinitesimal. Ignoring the higher-order infinitesimal terms and substituting Equation (8) into Equation (6), the state equation of the degraded device after linearization is:

[0120]

[0121] where F k-1 represents the first derivative of the non-linear function of the degradation state , and ≈ represents approximation.

[0122] Next, the extended Kalman filtering is used for degradation state estimation. This algorithm includes two stages: prediction and update.

[0123] 1). Prediction:

[0124]

[0125] 2). Update:

[0126]

[0127] K k =P | k-1 [P | k-1 +R k -1 (13)

[0128] P k|k =P k|k-1 -K k P k|k-1 (14)

[0129] Among them, represents the estimated value of the degradation state at time k, K k represents the state gain at time k, P k|k-1 represents the covariance matrix predicted at time k - 1, P k|k represents the covariance matrix estimated at time k.

[0130] According to equations (10)-(14), under the condition of the observation sequence Y k at time t 1:k , the posterior estimate of the degradation state Z k is a multivariate normal distribution, that is θ k represents the unknown parameters of the multi-index degradation model.

[0131] For the unknown parameters of the proposed model, the present invention adopts the expectation maximization - extended Kalman filter algorithm for parameter estimation. The unknown parameters of the present invention are If the observed data Y 1:k of the device degradation state is known, based on equations (7) and (9), the joint log-likelihood function of the device performance degradation state Z k and the observed data Y 1:k can be obtained:

[0132] ln f(Z k ,Y 1:k ; θ) = ln f(Z k ; θ) + ln f(Y 1:k |Z k ; θ) (15)

[0133] Among them, f(Z k ; θ) represents the probability density function of the system degradation state, f(Y 1:k |Z​k ; θ) represents the probability density function of the measurement data; ln() represents the logarithmic function.

[0134] Since ξ in the state space model k and V k both follow Gaussian distributions, the following relationships hold:

[0135]

[0136]

[0137] where, | | represents the determinant of a matrix, () T represents the transpose of a matrix, represents the summation of matrices from j = 1 to j = k, () -1 represents matrix inversion.

[0138] Substituting equations (16) and (17) into equation (15), the log-likelihood function can be obtained as:

[0139]

[0140] 1), Calculate the mathematical expectation:

[0141] By solving the expectation of the likelihood function f(Z k , Y 1:k ; θ) with respect to the hidden state Z k , we can get E[ln f(Z k , Y 1:k ; θ)]. Let G = E[ln f(Z k , Y 1:k ; θ)],

[0142]

[0143] where, tr() represents the trace of a matrix.

[0144] Since there are many hidden variables in equation (19) and it cannot be directly maximized, in order to calculate the conditional mathematical expectation of the likelihood function, the RTS filtering algorithm can be used to obtain:

[0145]

[0146] where, represents the estimated state value of the backward filtering at time j - 1, P j-1|k represents the estimated state covariance of the backward filtering at time j - 1, P j,j-1|k represents the estimated cross-covariance of the states of the backward filtering at times j and j - 1, F j-1|k represents the degenerate state at time j - 1 The first derivative of the non-linear function.

[0147] Substituting Eqs. (20)-(22) into Eq. (19) gives:

[0148]

[0149] 2) Maximize the likelihood function:

[0150]

[0151] where argmax() represents finding the maximum value, represents the value of the unknown parameter at the (l + 1)-th iteration at time k.

[0152] After estimating by setting the partial differential of the log-likelihood function to 0, we get:

[0153]

[0154] where, represents the partial derivative of G with respect to the degradation state Take the partial derivative.

[0155] For It is obtained by solving the non-linear equation (25). On this basis, the correlation D matrix and parameters are further obtained. By finding the maximum value of the likelihood function, the noise parameters R l+1 and Q l+1 are:

[0156]

[0157] Iteratively solve Eqs. (25)-(27) until θ (l+1) -θ (l) is less than the given threshold to stop the iteration, that is, the adaptive estimation of the parameters of the proposed model is realized.

[0158] For a multi-index system, multiple indicators are used for characterization. Usually, the system failure mode is competing failure. The competing failure mode is that as long as a certain indicator exceeds the corresponding failure threshold among multiple degradation indicators, the system fails. Specifically, according to the definition of the first passage time, in the competing failure mode, at the current time t k The RULL of the system is:

[0159] L = inf{l > 0: Z 1 (k + l) ≥ w 1 , or..., or Z i (k + l) ≥ w i , or..., i = 1, 2,..., n|Z k , θ k}(28)

[0160] Among them, inf{} represents the infimum, and Z i (k + l) where k + l represents time t k+l , i represents the number of indicators, w i is the failure threshold of indicator i, and θ k is a parameter in the state - space model; or represents the logical relationship of "or" in mathematics.

[0161] According to the definition in Equation (28), the RUL distribution function of the competing failure - mode system is:

[0162]

[0163] Among them, z i,k is the monitored value of indicator i at time k, and π k (ΔZ 1 , ΔZ 2 ,... ΔZ n |Z k , θ k ) is the joint probability density function of the multi - indicator degradation state; ΔZ n represents the increment of the n - indicator degradation state, represents the integral function.

[0164] In order to obtain the system degradation - state distribution, according to the properties of the stochastic system, solve the degradation state Z k of the multi - indicators at time t k+l to get:

[0165]

[0166] Among them, e () is the exponential function, represents the summation function;

[0167] Define P k+l = cov{Z k+l , Z k+l};

[0168] Using the properties of the stochastic - system state - evolution equation, obtain the mean value of Z k+l and the covariance P : k+l :

[0169]

[0170] Among them, Q λ+1 = cov{ξ λ+1 , ξ λ+1} is the variance of the degradation - system noise, and e D(k+l-(λ+1)) is the state - transition matrix;

[0171] Based on this, the probability density function of the system's RUL is obtained through the finite difference formula

[0172]

[0173] wherein, represents the partial derivative of the remaining useful life distribution function.

[0174] At time t k under the condition of the observation sequence Y 1:k , the degradation state Z is estimated by using the filtering algorithm, and the posterior estimate is a multivariate normal distribution, that is k After considering the measurement error, the probability density function f of the system's RUL at time t k k (l k |θ k ,Z k ,Y 1:k ) is as follows: 1:k ) is:

[0175]

[0176] wherein, h(Z k |θ k ,Y 1:k ) is the probability density function of the hidden state.

[0177] To prove the correctness and effectiveness of the proposed RUL prediction method based on multi-index non-linear stochastic correlation of the present invention, numerical experiments and the XJTU-SY dataset are used for verification.

[0178] Numerical experiment: According to the multi-index non-linear correlation model proposed in equations (6) and (7), simulated degradation data is generated to construct three degradation indices. The parameters of the proposed model are set as a 11 =1, a 22 =0.990, a 33 =0.998 a 12 =0.011,, a 13 =0.004 a 21 =0.002, a 22 =0.007 a 31 =0.020,

[0179] Figure 2 The degradation paths of the three indices are given. This group contains 515 sampling points, and the fault thresholds corresponding to the three indices are w 1 =0.975, w 2= 0.992, w 3 = 0.987. The extended Kalman filter is used for filtering, and the degraded states after the estimation of the three indicators are as Figure 2 shown. The comparison between the results of the parameter estimation based on the expectation maximization and the true parameter values is as Figure 3 shown.

[0180] From Figure 2 it can be seen that as the monitoring time increases, when the extended Kalman filter algorithm is used for state estimation, the degraded states estimated by the three indicators can well track the true trajectory. From Figure 3 it can be seen that the parameter estimation results in the proposed model all converge to the vicinity of the true values. To prove the correctness of the remaining useful life prediction proposed in the present invention, numerical experiments are used for verification under the competing failure modes. Taking the 400th to 515th data as an example, the system RUL prediction results (M2) are shown in Figure 4 , and are compared with those without considering the correlation and the linear correlation . The remaining useful life prediction results without considering the correlation and the remaining useful life prediction results considering the linear correlation are as Figure 4 shown.

[0181] From Figure 4 it can be seen that as the monitoring time increases, the probability density functions of the remaining useful life of different models become sharper and sharper, meaning that the variance gradually becomes smaller. The comparison between the remaining useful life means of different models and the true values is as shown in 5.

[0182] From Figure 5 it can be seen that as the monitoring time goes by, the remaining useful life mean of the model proposed in the present invention is closer to the true value than other models. The comparison of the mean-square errors (MSE) of the remaining useful life of different prediction methods is as Figure 6 shown.

[0183] From Figure 6 it can be seen that as the monitoring time continues to increase, the MSE of the M2 model proposed in the present invention is smaller than that of M0 and M1, and the prediction results are more accurate. Therefore, considering the non-linear correlation of the multi-index degraded states can improve the prediction performance of the system.

[0184] To further verify the applicability of the method proposed in the present invention, the XJTU-SY bearing dataset is used for verification. In this experiment, bearings 3-4 are randomly selected for verification and analysis. The original horizontal and vertical vibration signals of bearings 3-4 are as Figure 7 shown. Next, the root mean square eigenvalue is used to construct the vertical vibration and horizontal vibration indicators for bearings 3-4, as shown in Figure 8 . The correlation between the two indicators is analyzed, and then the two indicators are used for degradation modeling and remaining useful life prediction.

[0185] First, the degradation state and parameter estimation are carried out using the measurement data of two indicators. The actual degradation state results are as Figure 8 shown. According to the proposed models (6) and (7), the extended Kalman filter - expectation maximization algorithm is used for parameter estimation, and the results are shown in Table 1. According to the structural characteristics of the bearing system, in the competing failure mode, as long as one of the two indicators exceeds the threshold, the system fails. Taking the monitoring time from 1400 to 1515 as an example, the corresponding remaining useful life prediction results are as Figure 9 shown.

[0186] Table 1 Parameter Estimation Results of XJTU - SY

[0187]

[0188] As can be seen from Figure 9 - 10 , in the competing failure mode, the shape of the RUL distribution of the bearing system becomes sharper as the monitoring time increases, and the true value of the system gradually approaches the mean value. To further prove the applicability of the proposed prediction model, the method of the present invention is compared with other prediction methods. Here, the root mean square error (RMSE) and mean absolute error (MAE) error metrics are used to evaluate the effectiveness of the prediction methods, and the comparison error results are shown in Table 2.

[0189] Table 2 Comparison of Different Prediction Methods

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[0191] As can be seen from Table 2, compared with other prediction methods, the RMSE and MAE prediction errors of the model proposed in the present invention are the smallest. The method based on deep learning requires a considerable amount of time and data volume to train the RUL prediction model, and the neural network model is essentially a black box and cannot intuitively reflect the correlation between indicators. Since the method proposed in the present invention does not require pre - training of the model and can explicitly reflect the non - linear stochastic correlation between multiple indicators, it provides a new way for correlation modeling.

[0192] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the scope of the present invention.

Claims

1. A method for predicting remaining life considering nonlinear random correlation, characterized in that: The specific steps are as follows: Step 1: Considering the influence of measurement error and multi-index random correlation, a degradation state space model based on multi-index nonlinear random correlation is established; Step 2: Use the extended Kalman filter method to achieve online update of the state of the degraded state space model, including two stages: prediction and update; Step 3: Use the expectation maximization-extended Kalman filter algorithm to estimate the degradation state and parameters simultaneously, and realize the adaptive estimation of the parameters of the degradation state space model; Step 4: Use multiple indicators to characterize. As long as one indicator exceeds the corresponding failure threshold, the system is judged to have failed. According to the definition of the first arrival time, the remaining service life of the system at the current moment is determined under the competitive failure mode.

2. The method for predicting remaining life considering nonlinear random correlation according to claim 1, characterized in that: In step 1, assuming that the component has n degradation indicators, each degradation indicator corresponds to a different degradation state, then the corresponding degradation model is established for the i-th degradation indicator: in, t k The degradation amount of the i-th indicator at time, Indicates t k-1 The drift coefficient of the i-th indicator at the moment, Represents the drift process of the i-th indicator; is the diffusion coefficient of index i, ξ k =B k -B k-1 , B k Indicates t k The Brownian motion process at each moment, represents the diffusion process of the i-th indicator; Considering the correlation between various indicators, random correlation modeling and analysis are performed, which is reflected by the drift coefficient of the Wiener process: Among them, a qi (i,q=1...n) represents the correlation coefficient between index q and index i, represents the degradation state of the qth index at time k-1, is a nonlinear parameter; Substituting equation (2) into equation (1), when i = 1..n, we can get: make, Arranging formula (3) yields: Further merging and simplifying equation (4) yields: Among them, D is the correlation coefficient matrix, that is, Finally, the degenerate state equation is obtained: in, ξ k is the process noise of the multi-index system, that is, k ~MVN(0,Q k ), Q k represents the process noise covariance of the multi-index system, MVN represents the multivariate normal distribution; After considering the measurement error, the measurement equation corresponding to the multi-index system is defined as: Y k =Z k +V k (7) Among them, Y k Represents the measurement data of the multi-index system, V k represents the measurement noise, i.e., V k ~MVN(0,R k ), R k represents the variance of the measurement noise, diag() represents a diagonal matrix, represents the measurement noise of index n.

3. The method for predicting remaining life considering nonlinear random correlation according to claim 2, characterized in that: In step 2, the extended Kalman filter method is used to linearize the nonlinear function. t k-1 The equipment performance degradation function f(Z k-1 ) in the degraded state predicted value Taylor expansion is performed at: Where f′() is the first-order derivative of the nonlinear function, represents the estimated value of the degraded state at time k-1, O is a high-order infinitesimal, and the high-order infinitesimal terms are ignored. Substituting equation (8) into equation (6), the state equation of the degraded device after linearization is: in, F k-1 Indicates a degraded state The first derivative of the nonlinear function of , ≈ indicates approximation; The extended Kalman filter algorithm is used for degradation state estimation, which includes two stages: prediction and update; The prediction phase is: The update phases are: K k =P k|k-1 [P k|k-1 +R k ] -1 (13) P k|k =P k|k-1 -K k P k|k-1 (14) in, represents the estimated value of the degradation state at time k, K k represents the state gain at time k, P kk-1 represents the covariance matrix predicted at time k-1, P kk represents the covariance matrix estimated at time k; According to equations (10)-(14), at t k Time observation sequence Y 1:k Under the condition, the degraded state Z k The posterior estimate is a multivariate normal distribution, that is, θ k represents the unknown parameters of the multi-index degradation model.

4. The method for predicting remaining life considering nonlinear random correlation according to claim 3, characterized in that: In step 3, the expectation maximization-extended Kalman filter algorithm is used to estimate the unknown parameters of the proposed model. The unknown parameters are If the observed data Y of the equipment degradation state is known 1:k , based on equations (7) and (9), we can get the equipment performance degradation state Z k With the observed data Y 1:k The joint log-likelihood function ln f(Z k ,Y 1:k ;θ): ln f(Z k ,Y 1:k θ)zln f(Z k Nθ)+ln f(Y 1:k |Z k Nθ) (15) Among them, f(Z k ; θ) represents the probability density function of the system degradation state, f(Y 1:k |Z k ; θ) represents the probability density function of the measured data; ln() represents the logarithmic function; Since ξ k and V k All obey Gaussian distribution, then there is the following relationship: Among them, || represents the determinant of the matrix, () T represents the transpose of a matrix, represents the sum of the matrices from j = 1 to j = k, () -1 represents matrix inversion; Substituting equations (16) and (17) into equation (15), we get the log-likelihood function ln f(Z k ,Y 1:k ; θ) is: To obtain the mathematical period E: By using the likelihood function f(Z k ,Y 1:k ;θ) Solve for the hidden state Z k The expectation of E[ln f(Z k ,Y 1:k ; θ)]; let G = E[ln f(Z k ,Y 1:k ;θ)], Among them, tr() means finding the trace of the matrix; In order to calculate the conditional mathematical expectation of the likelihood function, the RTS filtering algorithm is used: in, represents the estimated state value of the backward filtering at time j-1, P j-1|k represents the estimated state covariance of the backward filter at time j-1, P j,j-1|k represents the estimated state cross-covariance of the backward filtering at time j and j-1, F j-1|k Indicates the degenerate state at time j-1 The first derivative of the nonlinear function of ; Substituting equations (20)-(22) into equation (19), we obtain: The function that maximizes the likelihood is: Among them, arg max() means to find the maximum value. represents the unknown parameter value of the l+1th step of iteration at time k; Let the partial differential of the log-likelihood function of formula (24) be 0 and estimate it: in, Indicates the degenerate state of G Find partial derivatives; for By solving the nonlinear equation (25), we can further obtain the correlation matrix D and Parameters, by finding the maximum value of the likelihood function, noise parameter R l+1 and Q l+1 for: Iterate and solve equations (25)-(27) until θ (l+1) -θ (l) When the value is less than a given threshold, the iteration is stopped, and the adaptive estimation of the parameters of the proposed model is achieved.

5. The method for predicting remaining life considering nonlinear random correlation according to claim 4, characterized in that: In step 4, in the competitive failure mode, the current time t k The RULL of the system is: L=inf{l>0:Z1(k+l)≥w1,or...,orZ i (k+l)≥w i ,or...,i=1,2,...,n|Z k ,θ k } (28) Among them, inf{} represents the infimum, Z i (k+l) where k+l represents time t k+l , i represents the number of indicators, w i is the fault threshold of index i, θ k is a parameter in the state space model; or represents the mathematical logical relationship of or; According to the definition of formula (28), the RUL distribution function of the competitive failure mode system is: Among them, z i,k is the monitoring quantity of index i at time k, π k (ΔZ1, ΔZ2, ... ΔZ n |Z k ,θ k ) is the joint probability density function of multiple index degradation states; ΔZ n represents the increment of the degradation state of the n index, represents the integral function; In order to obtain the system degradation state distribution, according to the properties of the random system, solve the system t k The degradation state Z of multiple indicators at the moment k+l have to: Among them, e () represents the exponential function, represents the sum function; definition P k+l =cov{Z k+l ,Z k+l }; Using the properties of the state evolution equation of the random system, we can obtain Z k+l Mean and covariance P k+l : Among them, Q λ+1 =cov{ξ λ+1 ,ξ λ+1 } is the variance of the degradation system noise, e D(k+l-(λ+1)) is the state transfer matrix; On this basis, the RUL probability density function of the system is obtained through the finite difference formula in, Remaining life distribution Find partial derivatives of functions; In t k Time observation sequence Y 1:k Under the condition, the filtering algorithm is used to estimate the degradation state Z k The posterior estimate is a multivariate normal distribution, that is, After considering the measurement error, at t k The RUL probability density function f of the time system k (l k |θ k ,Z k ,Y 1:k )for: Among them, h(Z k |θ k ,Y 1:k ) is the probability density function of the hidden state.

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