A method for modeling degradation process based on wiener principle and gaussian process regression

Abstract

The application relates to the technical field of degradation reliability modeling, and provides a degradation process modeling method based on the Wiener principle and Gaussian process regression, which comprises the following steps: according to a workpiece vibration signal measured at actual equal-interval sampling time points, obtaining a data set through feature extraction and dimension reduction, describing the data set by using a Wiener process considering random effects of a drift parameter, obtaining an analytical form of a Wiener kernel, introducing Gaussian process regression to obtain a Gaussian regression model, performing parameter estimation on the Gaussian process regression model by using maximum likelihood estimation based on a training set, and predicting a test set by using the Gaussian regression model to obtain the effect of reliability evaluation and life prediction of equipment. The application realizes the combination of a data-driven method and a physical mechanism, the data information is more thoroughly mined, and the prediction accuracy of the Wiener kernel in the Gaussian process regression is obviously improved.

Description

Technical Field

[0001] This invention relates to the field of degradation reliability modeling technology, and in particular to a degradation process modeling method based on Wiener's principle and Gaussian process regression. Background Technology

[0002] Traditional reliability assessment methods are based on the two-state assumption that products can only exist in two states: normal and faulty. However, for modern high-reliability, long-life products, obtaining failure data becomes increasingly difficult due to constraints in testing time and cost. Therefore, reliability modeling methods based on performance degradation have emerged. These methods aim to fully explore the statistical information in reliability testing, establish corresponding physical or mathematical models to evolve the performance degradation patterns, and then conduct reliability assessments and lifetime predictions. Degradation modeling is a crucial foundation for performance degradation reliability assessments and remaining lifetime predictions. In recent years, stochastic process-based and data-driven modeling methods have gained significant popularity among scholars both domestically and internationally.

[0003] However, existing data-driven methods such as deep learning, support vector machines, and Gaussian process regression are suitable for large datasets, but their accuracy is low when the sample size is small. On the other hand, methods based on stochastic processes are essentially Markov processes, which only consider the correlation between adjacent data points and have a low degree of data mining. Summary of the Invention

[0004] In view of this, the present invention provides a degradation process modeling method based on Wiener's principle and Gaussian process regression to solve the problems of existing technologies that are only applicable to big data and have low data correlation, as well as the problem that high accuracy can still be obtained when the sample size is small.

[0005] This invention provides a degradation process modeling method based on Wiener's principle and Gaussian process regression, comprising:

[0006] S1 processes the workpiece vibration signals obtained from the actual equally spaced sampling times to obtain the training set G and the test set G. * Training set G = {x, y}, test set G * ={x * ,y * Let x be the actual set of equally spaced sampling training time points, x = {x1, x2, ..., x}. M Let y be the set of actual training set degradation values ​​corresponding to the actual equally spaced sampling times, y = {y1, y2, ..., y} M}, where M is the total number of samples in the training set;

[0007] S2 establishes a Wiener degradation model based on Wiener's principle that takes into account the random effects of drift parameters;

[0008] S3 introduces the Wiener degeneracy model into the Gaussian process to obtain the Gaussian process model;

[0009] S4 uses the negative log-likelihood function to obtain the parameter estimates of the Gaussian process model based on the training set, and uses the training set and the Gaussian process model to perform posterior distribution prediction to obtain the Gaussian regression model; the prediction results of the Gaussian regression model are verified using the test set.

[0010] Optionally, a judgment method can be used to process the vibration signal of the workpiece to obtain a training set and a test set.

[0011] Optionally, the Wiener degradation model is obtained based on Wiener's principle, which takes into account the random effects of the drift parameters:

[0012] y'(n)=x0+θω(n)+σB(n)+ε; (1)

[0013] Where y'(n) is the degradation measurement at time n, x0 is the initial degradation value, θ is the drift parameter, ω(n) is the drift function, σ is the divergence coefficient, B(n) is the standard Brownian motion, and ε is white noise, which follows a normal distribution and is denoted as:

[0014] ε~N(0,λ 2 (2)

[0015] Where λ is the standard deviation;

[0016] Introduce a random effect into the drift parameter θ, such that the drift parameter θ follows a normal distribution, denoted as:

[0017]

[0018] Where, μ θ Let θ be the mean. Let θ be the variance;

[0019] Brownian motion σB(n) represents the change in vibration signal caused by changes in working conditions encountered by the workpiece during operation, and its distribution law is denoted as:

[0020] σB(n)~N(0,σ 2 n). (4)

[0021] Optionally, the drift function includes a linear function, an exponential function, or a power function.

[0022] Optionally, when the drift function is an exponential function, the Wiener exponential function degradation model is:

[0023] y'(n)=x0+θq n +σB(n)+ε; (5)

[0024] Where q is the parameter of the exponential function.

[0025] Optionally, based on the Wiener exponential function degradation model, the column vector y' = [y1, y2, ..., y] of the degradation test quantities is obtained. M ] T The column vector y' of the Wiener exponential function degenerates test quantity follows a multivariate normal distribution, denoted as:

[0026]

[0027] Ω=σ 2 Q+λ 2 I M (7)

[0028] Q = [min{t i ,t j}] 1≤i,j≤M (8)

[0029]

[0030] Among them, I M is an M-order identity matrix; t is a column vector of the exponential mapping of the training set time. t' is the transpose of t. Σ is the Wiener covariance matrix; t i Let t be the i-th element of column vector t. j Let be the j-th element of column vector t, where 1 ≤ i ≤ M and 1 ≤ j ≤ M.

[0031] Optionally, a Gaussian process model can be obtained based on the training set:

[0032] Y(n)=f(n)+ε; (10)

[0033] Where Y is the Gaussian degradation test quantity; f(n) is the Gaussian degradation quantity at time n following a Gaussian process, expressed as:

[0034] f(n)~GP(m(n),k f (n,n')); (11)

[0035] Where m(n) represents the mean function of the Gaussian process; k f (n,n') represents the covariance function of the Gaussian process, where n and n' represent time points, respectively; they are expressed as:

[0036] m(n) = E[f(n)]; (12)

[0037] k f (n,n')=E[(f(n)-m(n))(f(n')-m(n'))]; (13)

[0038] Based on the Gaussian process model, the degradation test quantity Y of the Gaussian process is obtained, and Y follows a multivariate normal distribution:

[0039] Y~MVN(m(t),K f (t,t)+λ 2 I M (14)

[0040] Where m(t) is the mean vector of Y, K f (t,t) is the covariance matrix of the Gaussian process;

[0041] Substituting formula (6) into formula (14), we obtain the mean function:

[0042] m(t) = x0 + μ θ t; (15)

[0043] Substituting formula (9) into formula (14), we obtain the Wiener kernel:

[0044]

[0045] Where, k ij The covariance matrix K f Elements in (t,t); m = M;

[0046] The parameters of the Gaussian process model are obtained based on m(t) and the Wiener kernel:

[0047] Θ={x0,μ θ ,σ θ ,σ,λ,q}; (17)

[0048] According to formula (14), the negative log-likelihood function NLML is obtained:

[0049]

[0050] Based on the training set, the parameter values ​​of the Gaussian process model are obtained by minimizing the negative log-likelihood function:

[0051]

[0052] Optionally, the posterior distribution can be predicted using the training set and the Gaussian process model to obtain the joint prior distribution of the training set and the Gaussian regression degradation test data:

[0053]

[0054] Among them, t * y is a column vector representing the exponential mapping of the test set time; * The column vector is the test set degradation amount corresponding to the equally spaced sampling time.

[0055] The posterior distribution of the predicted degradation values ​​is obtained as follows:

[0056] P(y * |t * ,t,y)=N(y * |m * ,Σ * ); (twenty one)

[0057] Obtain the Gaussian regression model:

[0058] m * =m(t) * )+K f (t,t * )[K f (t,t)+λ 2 I M ] -1 (ym(t)); (22)

[0059] Σ * =K f (t * ,t * )-K f (t,t * ) T [K f (t,t)+λ 2 I M ] -1 K f (t,t * ); (twenty three)

[0060] Where, m * Σ is the mean vector of the degraded predicted values; * This is the covariance matrix of the degraded predicted values.

[0061] The advantages of this invention compared to the prior art are:

[0062] 1. This invention combines the Wiener process covariance matrix and the Gaussian process regression algorithm to achieve a combination of data-driven methods and physical mechanisms;

[0063] 2. In the Wiener process that considers the random effects of drift parameters, this invention, based on existing stochastic process methods, takes into account global correlation, resulting in a more thorough mining of data information and improved accuracy of prediction results;

[0064] 3. The solution of the present invention, as demonstrated by experiments, shows that the Wiener kernel significantly improves the prediction accuracy of Gaussian process regression in the modeling and prediction of mechanical product degradation. Attached Figure Description

[0065] To more clearly illustrate the technical solutions in this invention, the accompanying drawings used in the description of the embodiments or prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0066] Figure 1 This is a flowchart of the degradation process modeling method based on Wiener's principle and Gaussian process regression provided by the present invention;

[0067] Figure 2 This is a schematic diagram illustrating the prediction effect of the Wiener kernel provided by the present invention;

[0068] Figure 3 This is a schematic diagram illustrating the prediction effect of the classical radial basis kernel in Gaussian process regression provided by the present invention.

[0069] Figure 4 This is a schematic diagram illustrating the prediction effect of the classical Marton kernel in Gaussian process regression provided by the present invention. Detailed Implementation

[0070] In the following description, specific details such as particular system architectures and techniques are set forth for illustrative purposes and not for limitation, in order to provide a thorough understanding of the embodiments of the invention. However, those skilled in the art will understand that the invention can be implemented in other embodiments without these specific details. In other instances, detailed descriptions of well-known systems, apparatuses, circuits, and methods are omitted so as not to obscure the description of the invention with unnecessary detail.

[0071] The following will describe in detail, with reference to the accompanying drawings, a degradation process modeling method based on Wiener's principle and Gaussian process regression according to the present invention. The specific method includes:

[0072] S1 processes the bearing vibration signals obtained from the actual equally spaced sampling times to obtain the training set G and the test set G. * Training set G = {x, y}, test set G * ={x * ,y * Let x be the actual set of equally spaced sampling training time points, x = {x1, x2, ..., x}. M Let y be the set of actual training set degradation values ​​corresponding to the actual equally spaced sampling times, y = {y1, y2, ..., y}. M}, where M is the total number of samples in the training set;

[0073] S2 establishes a Wiener degradation model based on Wiener's principle that takes into account the random effects of drift parameters;

[0074] S3 introduces the Wiener degeneracy model into the Gaussian process to obtain the Gaussian process model;

[0075] S4 uses the negative log-likelihood function to obtain the parameter estimates of the Gaussian process model based on the training set, and uses the training set and the Gaussian process model to perform posterior distribution prediction to obtain the Gaussian regression model; the prediction results of the Gaussian regression model are verified using the test set.

[0076] Optionally, a judgment method can be used to process the vibration signal of the bearing to obtain a training set and a test set.

[0077] Optionally, the Wiener degradation model is obtained based on Wiener's principle, which takes into account the random effects of the drift parameters:

[0078] y'(n)=x0+θω(n)+σB(n)+ε; (1)

[0079] Where y'(n) is the degradation measurement at time n, x0 is the initial degradation value, θ is the drift parameter, ω(n) is the drift function, σ is the divergence coefficient, B(n) is the standard Brownian motion, and ε is white noise, which follows a normal distribution and is denoted as:

[0080] ε~N(0,λ 2 (2)

[0081] Where λ is the standard deviation;

[0082] Introduce a random effect into the drift parameter θ, such that the drift parameter θ follows a normal distribution, denoted as:

[0083]

[0084] Where, μ θ Let θ be the mean. Let θ be the variance;

[0085] The change in vibration signal caused by the change in operating conditions encountered by the bearing during operation is represented by Brownian motion σB(n), and its distribution law is denoted as:

[0086] σB(n)~N(0,σ 2 n). (4)

[0087] Optionally, the drift function includes a linear function, an exponential function, or a power function.

[0088] Optionally, when the drift function is an exponential function, the Wiener exponential function degradation model is:

[0089] y'(n)=x0+θq n +σB(n)+ε; (5)

[0090] Where q is the parameter of the exponential function.

[0091] Optionally, based on the Wiener exponential function degradation model, the column vector y' = [y1, y2, ..., y] of the degradation test quantities is obtained. M ] T The column vector y' of the Wiener exponential function degenerates test quantity follows a multivariate normal distribution, denoted as:

[0092]

[0093] Ω=σ 2 Q+λ 2 I M (7)

[0094] Q = [min{t i ,t j}] 1≤i,j≤M (8)

[0095]

[0096] Among them, I M is an M-order identity matrix; t is a column vector of the exponential mapping of the training set time. t' is the transpose of t. Σ is the Wiener covariance matrix; t i Let t be the i-th element of column vector t. j Let be the j-th element of column vector t, where 1 ≤ i ≤ M, 1 ≤ j ≤ M, and i ≠ j.

[0097] Optionally, a Gaussian process model can be obtained based on the training set:

[0098] Y(n)=f(n)+ε; (10)

[0099] Where Y is the Gaussian degradation test quantity; f(n) is the Gaussian degradation quantity at time n following a Gaussian process, expressed as:

[0100] f(n)~GP(m(n),k f (n,n')); (11)

[0101] Where m(n) represents the mean function of the Gaussian process; k f (n,n') represents the covariance function of the Gaussian process, where n and n' represent time n and n', respectively; and are expressed as:

[0102] m(n) = E[f(n)]; (12)

[0103] k f (n,n')=E[(f(n)-m(n))(f(n')-m(n'))]; (13)

[0104] Based on the Gaussian process model, the degradation test quantity Y of the Gaussian process is obtained, and Y follows a multivariate normal distribution:

[0105] Y~MVN(m(t),K f (t,t)+λ 2 I M (14)

[0106] Where m(t) is the mean vector of Y, K f (t,t) is the covariance matrix of the Gaussian process;

[0107] Substituting formula (6) into formula (14), we obtain the mean function:

[0108] m(t) = x0 + μ θ t; (15)

[0109] Substituting formula (9) into formula (14), we obtain the Wiener kernel:

[0110]

[0111] Where, k ij The covariance matrix K f Elements in (t,t); m = M.

[0112] The parameters of the Gaussian process model are obtained based on m(t) and the Wiener kernel:

[0113] Θ={x0,μ θ ,σ θ ,σ,λ,q}; (17)

[0114] According to formula (14), the negative log-likelihood function NLML is obtained:

[0115]

[0116] Based on the training set, the parameter values ​​of the Gaussian process model are obtained by minimizing the negative log-likelihood function:

[0117]

[0118] Optionally, the posterior distribution can be predicted using the training set and the Gaussian process model to obtain the joint prior distribution of the training set and the Gaussian regression degradation test data:

[0119]

[0120] Among them, t * y is a column vector representing the exponential mapping of the test set time; * The column vector is the test set degradation amount corresponding to the equally spaced sampling time.

[0121] The posterior distribution of the predicted degradation values ​​is obtained as follows:

[0122] P(y * |t * ,t,y)=N(y * |m * ,Σ * ); (twenty one)

[0123] Obtain the Gaussian regression model:

[0124] m * =m(t) * )+K f (t,t * )[K f (t,t)+λ 2 I M ] -1 (ym(t)); (22)

[0125] Σ * =K f (t * ,t * )-K f (t,t * ) T [K f (t,t)+λ 2 I M ] -1 K f (t,t * ); (twenty three)

[0126] Where, m * Σ is the mean vector of the degraded predicted values; * This is the covariance matrix of the degraded predicted values.

[0127] To facilitate understanding of the present invention, the method of the present invention is illustrated in detail below with examples. However, the present invention can also be applied to other embodiments different from this one. Therefore, the scope of protection of the present invention is not limited to the following examples.

[0128] Rolling bearings are high-reliability mechanical components, and their degradation experimental data is enormous. First, the vibration signals of the rolling bearings are feature-extracted and fused to obtain easily processed data signals. Furthermore, since the failure phase of rolling bearings is much shorter than the normal operating phase, the data signals need to be processed in stages, eliminating the data signals from the first stage to improve the training accuracy of the model. To date, there has been much research on the two-stage degradation process in degradation reliability studies, with commonly used stage judgment rules including the "3σ method" and the "SIC detection method." After eliminating the data signals from the first stage, the data is divided into training and test sets in a 2 / 3 and 1 / 3 ratio, respectively.

[0129] Based on the training and testing sets, the parameter values ​​of the Gaussian process model for rolling bearings were obtained using the modeling method of this invention, as shown in Table 1:

[0130] Table 1. Parameter values ​​of the Gaussian process model for rolling bearings

[0131]

[0132] The Gaussian regression model obtained using the method of this invention is used for prediction. Figure 2 This is a schematic diagram illustrating the prediction effect of the Wiener kernel provided by the present invention; Figure 3 This is a schematic diagram illustrating the prediction effect of the classical radial basis kernel in Gaussian process regression provided by the present invention. Figure 4 This is a schematic diagram illustrating the prediction performance of the classical Marton kernel in Gaussian process regression provided by this invention. (See attached diagram.) Figure 2-4 As can be seen, the method in this embodiment achieves time-varying prediction of the degradation process of rolling bearings, and the prediction accuracy is improved compared with the traditional Gaussian process regression method. Furthermore, it incorporates the physical mechanism contained in the Wiener process, giving the model parameters a real physical meaning, which can also play a certain guiding role in improving the reliability of mechanical parts. Moreover, by combining the prediction results of this model with some knowledge of life prediction and product reliability, the remaining life and time-varying reliability of mechanical parts can be predicted in real time to solve more practical engineering problems.

[0133] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be included within the protection scope of the present invention.

Claims

1. A degradation process modeling method based on Wiener's principle and Gaussian process regression, characterized in that, include: S1 processes the workpiece vibration signals obtained from actual, equally spaced sampling times to obtain a training set. G and test set G * ; training set G ={ x , y }, test set G * ={ x * , y * }, x This is the actual set of equally spaced sampling training time points. x= , y This represents the set of actual training set degradation values ​​corresponding to the actual equally spaced sampling times. y ={ y 1, y 2,…, y M }, M It is the total number of samples in the training set; S2 establishes a Wiener degradation model based on Wiener's principle that takes into account the random effects of drift parameters; S3 introduces the Wiener degeneracy model into the Gaussian process to obtain the Gaussian process model; S4 uses the negative log-likelihood function to obtain the parameter estimates of the Gaussian process model based on the training set, and uses the training set and the Gaussian process model to perform posterior distribution prediction to obtain the Gaussian regression model. The expression for Wiener's degradation model is: y' ( n ) = x 0+ ω ( n )+ σB ( n )+ ε (1) in, y' ( n )yes n Degradation measurement at time, x 0 is the initial value for degradation. For drift parameters, ω ( n ) is the drift function. σ It is the divergence coefficient. B ( n () is standard Brownian motion; ε It is white noise. ε It follows a normal distribution, denoted as: (2) in, Standard deviation; Give drift parameters Introducing random effects makes the drift parameter It follows a normal distribution, denoted as: (3) in, for The mean, for The variance; Using Brownian motion σB ( n This represents the change in vibration signal caused by changes in working conditions encountered by the workpiece during operation, and its distribution pattern is denoted as: (4) When the drift function is an exponential function, the Wiener exponential function degradation model is as follows: (5) in, q These are the parameters of the exponential function; Based on the Wiener exponential function degradation model, the column vector of degradation test quantities is obtained. y' =[ y 1, y 2,…, y M ] T It follows a multivariate normal distribution, denoted as: y' ~MVN( x 0+ ) (6) (7) (8) （9） in, yes M An identity matrix of order 1; t For the column vector of the exponential mapping of the training set time, t = [ ] T ; for t transpose, =[ ]; This is the Wiener covariance matrix; column vector t The i One element, column vector t The j elements, 1≤ i ≤ M ,1≤ j ≤ M ; Gaussian process model obtained from training set: Y ( )= f ( )+ ε （10） in, Y It is a Gaussian degradation test quantity; f ( It follows a Gaussian process. n The Gaussian degradation at time t is expressed as: （11） in, The mean function representing a Gaussian process; The covariance function representing a Gaussian process. ; respectively represented as: （12） ；（13） Based on the Gaussian process model, the degradation test quantities of the Gaussian process are obtained. Y , Y Follows a multivariate normal distribution: ；（14） in, m ( t )for Y The mean vector, The covariance matrix of the Gaussian process; Substituting formula (6) into formula (14), we obtain the mean function: （15） Substituting formula (9) into formula (14), we obtain the Wiener kernel: （16） in, Covariance matrix Elements in; m = M ; based on m ( t The parameters of the Gaussian process model are obtained using the Wiener kernel. （17） According to formula (14), the negative log-likelihood function is obtained. NLML : （18） Based on the training set, the parameter values ​​of the Gaussian process model are obtained by minimizing the negative log-likelihood function: （19）。 2. The degradation process modeling method according to claim 1, characterized in that, The vibration signal of the bearing is processed using a judgment method to obtain the training set and the test set.

3. The degradation process modeling method according to claim 2, characterized in that, By using the training set and the Gaussian process model to predict the posterior distribution, the joint prior distribution of the training set and the Gaussian regression degradation test data is obtained: ； (20) in, A column vector representing the exponential mapping of the test set time; The column vector is the test set degradation amount corresponding to the equally spaced sampling time. The posterior distribution of the predicted degradation values ​​is obtained as follows: ；(21) Obtain the Gaussian regression model: ；(22) ；(23) in, This is the mean vector of the predicted degradation values; Let be the covariance matrix of the degraded predicted values.

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