A method for online prediction of equipment failures in small sample sizes

Abstract

This invention discloses a method for online prediction of equipment faults with a small sample size, including the following prediction steps: S1, Data Processing: By optimizing the threshold function in the WTD denoising algorithm and introducing BIC to evaluate the impact of the number of decomposition layers on the complexity of WTD, an improved WTD algorithm is proposed for online filtering of noise in fault signals. This invention proposes an online prediction model for equipment faults with a small sample size, and verifies the effectiveness and reliability of the model using rolling bearing life cycle vibration data. BIC can accurately find the optimal number of decomposition layers in the WTD algorithm, providing a basis for improving the parameter settings of the WTD model. The improved WTD algorithm has excellent denoising effect, ensuring the reliability of fault data. The improved MEST algorithm and dual CSFI algorithm can effectively convert fault signals into fault degree indicators, providing high-quality data support for subsequent fault prediction.

Description

Technical Field

[0001] This invention relates to the field of equipment failure prediction technology, specifically to an online failure prediction method for small-sample equipment. Background Technology

[0002] Fault prediction technology is based on fault mechanism analysis, assumes fault occurrence patterns, utilizes historical degradation data, deeply mines potential fault information, and constructs physical or data-driven prediction models to predict the degree of fault. It is the foundation for equipment health status assessment, remaining service life analysis, and condition-based maintenance. Equipment degradation exhibits significant differences, making it difficult to extract features from early-stage faults and difficult to reflect the current equipment health status from later-stage faults. Furthermore, the amount of fault data in recent times is insufficient to meet the requirements for fitting accuracy, increasing the difficulty of fault prediction. Small sample size, fast convergence, and high-precision online fault prediction technology has become a research hotspot in the field of fault prediction.

[0003] However, current fault prediction technologies have poor adaptability and are ineffective in predicting faults in equipment with high fault complexity, small sample data volume, and strong prediction timeliness. Summary of the Invention

[0004] This invention provides a method for online prediction of equipment faults with a small sample size, which can effectively solve the problems mentioned in the background art regarding the poor adaptability of current fault prediction technologies and their poor performance in predicting faults of equipment with high fault complexity, small sample data, and strong prediction timeliness.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a method for online prediction of equipment faults in a small sample size, comprising the following prediction steps:

[0006] S1. Data Processing: By optimizing the threshold function in the WTD noise reduction algorithm and introducing BIC to evaluate the impact of the number of decomposition layers on the complexity of WTD, an improved WTD algorithm is proposed for online noise filtering in fault signals.

[0007] S2. Fault severity identification: A nonparametric model of the system or equipment is constructed through MEST. An estimated vector is obtained by optimal reconstruction estimation of the observation vector and the historical memory matrix. The difference between the estimated vector and the observation vector is used to reflect the fault severity, and CSFI is introduced for smoothing.

[0008] S3. Online Fault Prediction: By using the non-statistical analysis method of TSFM, random data fluctuations are eliminated, and gradient descent is introduced to update the smoothing factor online. An adaptive sliding time window is introduced to dynamically extract time series data, thereby improving the fitting ability of TSFM.

[0009] S4. Experimental Analysis: Verify the effectiveness and feasibility of the equipment failure prediction model under small sample conditions;

[0010] S5. Data Processing and Analysis: The complexity of the improved WTD algorithm with different decomposition levels is evaluated using a preset BIC.

[0011] S6. Fault severity identification and analysis: The wavelet decomposition coefficients obtained by the improved WTD are used as the observation variables of the improved MEST. The sampling frequency, health status and degradation status are set to identify and analyze the fault severity. The data is processed by dual CSFI to eliminate the "sharp points" in the curve where the derivative does not exist, and the smoothed fault severity points are obtained.

[0012] S7. Online Fault Test Analysis: Preset adaptive sliding time window, adaptive smoothing factor, learning factor, maximum number of training iterations, and minimum allowable error. Input the smoothed bearing fault severity value into the online fault prediction model to obtain the trend of adaptive smoothing factor change, adaptive sliding time window length change, and prediction error change.

[0013] S8. Summary of Prediction Results: Summarize the effectiveness of the online prediction model in fault prediction.

[0014] 2. The online fault prediction method for small-sample equipment according to claim 1, characterized in that, in step S1, an improved WTD algorithm is used to filter out noise in the fault signal online, the principle of which is as follows:

[0015]

[0016] In the formula, λ is the threshold;

[0017] ω j,k These are the wavelet coefficients of the fault signal;

[0018] To estimate wavelet coefficients;

[0019] j is the decomposition scale, and 1≤j≤J, where J is the maximum scale;

[0020] sgn() is a symbolic function;

[0021] This threshold function is continuous in the wavelet domain, when ω j,k →λ - hour, When ω j,k →λ + hour,

[0022] The choice of threshold λ should satisfy:

[0023]

[0024] In the formula, N represents the signal length;

[0025] σ j The standard deviation of the j-th layer of Gaussian white noise is expressed as:

[0026]

[0027] In the formula, Cd j,k This represents the high-frequency component of the wavelet decomposition at the j-th level.

[0028] p is the number of wavelet coefficients at this scale;

[0029] WTD believes the fault signal exists in the low-frequency range. j,k In the high-frequency range, noise exists in Cd. j,k middle;

[0030] Since the amplitude of the noise follows a Gaussian distribution, the high-frequency component Cd is decomposed to the maximum number of layers. j,k To evaluate the model's complexity using data, the BIC (Browser Inference Function) is introduced to assess the model's complexity:

[0031] BIC=qln(N)-2ln(L) (4)

[0032] In the formula, q is the number of model parameters;

[0033] N is the number of samples;

[0034] L is the maximum likelihood function that follows a Gaussian distribution, i.e.:

[0035]

[0036] According to the above technical solution, in step S2, the estimated vector and the observation vector are specifically calculated as follows:

[0037] Suppose that at a certain time t, n interrelated variables are observed in the equipment, and let them be denoted as observed variables X. t ,Right now

[0038] X t =[x t,1 ,x t,2 …x t,n ] T (6)

[0039] In the formula, x t,n Let be the observed value of the state variable at time t;

[0040] Construct a historical memory matrix D with m historical moments and n associated state variables, i.e.

[0041]

[0042] The m observation vectors X in the historical memory matrix D obsLinear weighting yields the estimated vector X. est ,Right now

[0043] X est =DW=w1X1+w2X2…w m X m (8)

[0044] In the formula, W = [w1, w2…w m ] T Let X be an m-dimensional weight vector, representing the input observation vector X. obs The similarity to the historical memory matrix D, i.e.

[0045]

[0046] In the formula, This is a non-linear operator used to replace the product operation in ordinary matrices;

[0047] D T With X obs The Mahalanobis distance (MD) between them is used as a non-linear operator in MEST, i.e.

[0048]

[0049] In the formula, ∑ -1 It is the inverse matrix of the covariance matrix of a multidimensional random variable;

[0050] The more similar two state matrices are, the smaller their MD is;

[0051] The greater the difference between the two state matrices, the greater the result of their nonlinear operation.

[0052] Substituting equation (9) into equation (8), we obtain the final expression for the MEST model estimation vector:

[0053]

[0054] By comparing the observation vector X obs With the estimated vector X est The difference between them yields the residual value ε, which reflects the degree of equipment failure, i.e.:

[0055] ε=X est -X obs (12)

[0056] By comparing the application ranges of various fault indicators, the root mean square (RMS) is selected to reflect the degree of fault.

[0057] By finding n dimensions X est With X obsThe root mean square value (RMSV) of the residual ε yields the indicator DR, which reflects the degree of equipment failure.

[0058]

[0059] The equipment failure severity index DR obtained using MEST is composed of multiple discrete points. The curve composed of DR contains multiple "cusps" where the derivative does not exist. CSFI is introduced for smoothing.

[0060] According to the above technical solution, in S3, the TSFM expression is:

[0061]

[0062] In the formula, DR t This is the original sequence data;

[0063] Let be the adaptive smoothing factor for the i-th training iteration in the (t+T)-th prediction;

[0064] This is a first-order smoothing value;

[0065] It is a quadratic smoothed value;

[0066] If T represents the prediction time. Let represent the predicted value of the i-th training iteration at time t+T. Then the prediction formula is:

[0067]

[0068] in:

[0069]

[0070] By comparing the applicability of various gradient descent algorithms, stochastic gradient descent (SGD) is selected to adaptively update the smoothing factor. Then the i-th training adaptive smoothing factor of the (t+T)-th prediction. The expression is:

[0071]

[0072] In the formula, For the (t+T)th prediction, the (i-1)th training adaptive smoothing factor is used.

[0073] β is the learning factor;

[0074] This is the i-th training prediction value for the (t+T)-th prediction.

[0075] DR t+T This is the (t+T)th actual value;

[0076] Let be the partial derivative of the training loss function for the (t+T)th prediction with respect to α:

[0077]

[0078] In the formula, The length of the i-th training time series data predicted in the (t+T)-th time is the adaptive sliding time window.

[0079] Draw a diagram illustrating the sliding time window, where Data_mode_i represents the training data for the i-th training iteration, and Data_test_i represents the test data for the i-th training iteration.

[0080]

[0081] In the formula, μ is the adjustment factor;

[0082] When time series data cannot fully reflect fault information, i.e., the partial derivative of the loss function... If it is always in the same direction, then It should continue to grow or decrease. It can increase by a factor of 1+μ or decrease by a factor of 1-μ.

[0083] When time series data can partially reflect fault information, i.e., the partial derivative of the loss function If they are not all in the same direction, then Length depends on direction, When it is the right time, Length increase, When it is negative, The length has been shortened;

[0084] The variance value VARV is used to represent the final prediction error.

[0085]

[0086] In the formula, len(T) is the number of fault severity index data in the (t+T)th prediction.

[0087] According to the above technical solution, the online fault prediction process in step S3 is as follows:

[0088] Step 1: For the i-th training iteration of the (t+T)-th prediction, the prediction model first uses the preset sliding time window from the (i-1)-th iteration. Divide the data into training data (Data_mode_i) and test data (Data_test_i) for the i-th training iteration, and then divide the data from the (i-1)-th training iteration... and Substitute into equation (15) to predict the i-th training iteration. value;

[0089] Step 2: Substitute into equation (18) to calculate the loss function for the i-th training iteration. If the maximum number of iterations (maxtrans) is satisfied or less than the minimum error (minerror), then training stops. If the above conditions are not met, then the adaptive smoothing factor is updated according to equations (17) and (19). and adaptive sliding time window Repeat Step 1 to begin the (t+T)th prediction and the (i+1)th training iteration.

[0090] Step 3: Obtain the results from Step 2 Substitute into equation (15) to obtain the final result. The value is calculated, and the prediction error e is calculated according to equation (20).

[0091] According to the above technical solution, in step S4, bearing performance degradation data is used as the verification object. The sampling frequency is set to 25.6 kHz / min, the radial force is 12 kN, the rotation speed is 2100 rpm, and the operation time is 157.44 s. The vibration signal in the horizontal direction is selected to reflect the degree of failure of the tested bearing.

[0092] Based on the above technical solution, the WTD effect of different number of subdivisions in S5 is compared as follows:

[0093] The BIC value is minimized when the wavelet decomposition level j = 7.

[0094] When the number of wavelet decomposition layers j < 7, the effect of improving WTD noise reduction is enhanced as j increases.

[0095] When the wavelet decomposition level j > 7, the effect of improving WTD noise reduction is not significantly enhanced as j increases.

[0096] When j=7, the improved WTD algorithm can ensure that the noise reduction effect meets the requirements and effectively prevent the problem of excessive model complexity caused by excessive accuracy.

[0097] The model complexity is evaluated based on the fact that the noise amplitude follows a Gaussian distribution. Therefore, the high-frequency part of the maximum decomposition layer, Cd, is used. j,k Output and plot the amplitude distribution of Gaussian white noise;

[0098] When j < 7, improve the high-frequency part of the maximum decomposition layer Cd in WTD. j,k If the noise follows a Gaussian distribution, then improving the threshold in WTD does not destroy the distribution characteristics of Gaussian white noise, meaning the wavelet decomposition is insufficient.

[0099] When j≥7, improve the high-frequency part of the maximum decomposition layer Cd of WTD. j,k Since the noise does not follow a Gaussian distribution, improving the threshold in WTD disrupts the distribution characteristics of Gaussian white noise, meaning the wavelet decomposition is sufficient. This demonstrates the feasibility of setting BIC to evaluate the number of decomposition levels and the high-frequency component Cd of the preset maximum wavelet decomposition level. j,k To assess the scientific validity of the data;

[0100] The wavelet decomposition level j=7 was set to improve the noise reduction processing of bearing performance degradation data by WTD.

[0101] According to the above technical solution, in step S6, the wavelet decomposition coefficients obtained by the improved WTD are used as the observation variables X of the improved MEST. t That is, when j = 7, X t =[Ca t,7 Cd t,7 Cd t,6 Cd t,5 Cd t,4 Cd t,3 Cd t,2 Cd t,1 The sampling frequency is set to fs = 25600Hz, or 1 second. The first 2 seconds of bearing performance degradation data are set to a healthy state, i.e., historical time m = 2, and the historical memory matrix is ​​D = [X1, X2]. Other times are set to a performance degradation state every 2 seconds, i.e., the observation vector X obs By incorporating bearing performance degradation data, a total of 77 failure severity DRs were evaluated.

[0102] Meanwhile, in order to eliminate multiple "sharp points" in the DR curve where the derivatives are missing, dual CSFI processing was used to process the data, with the number of interpolations being 10 times the original data volume, resulting in a total of 7700 smoothed fault degree points.

[0103] According to the above technical solution, in step S7, the initial value of the adaptive sliding time window length is 100, the initial value of the adaptive smoothing factor α is 0.05, the learning factor β is 0.5, the maximum number of training iterations maxtrans is 1000, and the minimum allowable error minerror is 10. -8 The smoothed DR values ​​of 7700 bearing fault severity were input into the online fault prediction model, and the trends of the adaptive smoothing factor α, the adaptive sliding time window length, and the prediction error e were compared.

[0104] Based on the above technical solution, the prediction results in step S8 are summarized as follows:

[0105] BIC can accurately determine the optimal number of decomposition layers for the WTD algorithm;

[0106] The improved WTD algorithm delivers excellent noise reduction performance;

[0107] MD's improved MEST algorithm and dual CSFI algorithm can effectively convert fault signals into fault severity indicators.

[0108] The proposed online prediction model can achieve online prediction of equipment failures.

[0109] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0110] By proposing a small-sample online fault prediction model for equipment, and verifying its effectiveness and reliability using rolling bearing lifecycle vibration data, the BIC model accurately identifies the optimal decomposition level of the WTD algorithm, providing a basis for improving the parameter settings of the WTD model. The improved WTD algorithm exhibits excellent noise reduction performance, ensuring the reliability of fault data. The improved MEST algorithm and dual CSFI algorithm by MD effectively transform fault signals into fault severity indicators, providing high-quality data support for subsequent fault prediction. The proposed online fault prediction model, which features adaptive fault data length extraction via a sliding time window and adaptive parameter updates, can continuously mine potential fault severity information in the data, enabling online prediction of equipment faults and meeting the requirements for fault prediction of equipment with high fault complexity, small sample data volume, and strong prediction timeliness. Attached Figure Description

[0111] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings:

[0112] Figure 1 This is a flowchart illustrating the steps of the online fault prediction model for equipment of the present invention;

[0113] Figure 2 This is the WTD schematic diagram of the present invention;

[0114] Figure 3 This is a schematic diagram of the sliding time window of the present invention;

[0115] Figure 4 This is a flowchart of the online fault prediction process of this invention;

[0116] Figure 5 This is a structural diagram of the bearing fatigue testing rig of the present invention;

[0117] Figure 6 This is a diagram of the bearing amplitude signal in the experiment of this invention;

[0118] Figure 7 This is a graph showing the variation of BIC values ​​in this invention;

[0119] Figure 8These are WTD effect diagrams for different decomposition layers of the present invention;

[0120] Figure 9 This is the Gaussian white noise amplitude distribution diagram of the present invention;

[0121] Figure 10 These are comparison charts of the various WTD effects of this invention;

[0122] Figure 11 This is a trend chart of the DR value variation over the entire bearing life of this invention;

[0123] Figure 12 This is a trend chart of the adaptive α value variation of the present invention;

[0124] Figure 13 This is a trend chart of the adaptive sliding time window length variation of the present invention;

[0125] Figure 14 This is a trend chart of the prediction error of this invention;

[0126] Figure 15 This is a diagram showing the predicted fitting effect of the present invention;

[0127] Figure 16 This is a flowchart illustrating the fault testing steps of the present invention. Detailed Implementation

[0128] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0129] Example: Figure 1 and 16 As shown, the present invention provides a technical solution, a method for online prediction of equipment faults in a small sample, comprising the following prediction steps:

[0130] S1. Data Processing:

[0131] Commonly used noise reduction algorithms include Empirical Mode Decomposition (EMD) and WTD.

[0132] The basic principle of EMD is to decompose the fault signal into intrinsic mode functions (IMFs) of various orders and then extract the features of the fault signal from them. However, the EMD algorithm has problems such as mode mixing and end effect in the context of a large amount of noise.

[0133] WTD mainly includes two types of wavelet decomposition: hard thresholding and soft thresholding. Although WTD has problems such as the wavelet decomposition level relying on experience, the discontinuity of the hard threshold function which can easily cause oscillations after denoising, and the deviation of wavelet coefficients in soft thresholding which can easily lead to large errors in the reconstructed signal, it is widely used in online fault prediction due to its low computational cost and high denoising efficiency. For example, by improving the wavelet threshold function, noise in the signal can be effectively filtered out, and the signal-to-noise ratio of the output signal can be improved. However, it still has problems such as the need to improve the denoising effect and how to determine the wavelet decomposition level.

[0134] Therefore, by optimizing the threshold function in WTD and introducing BIC to evaluate the impact of the number of decomposition layers on the complexity of WTD, an improved WTD algorithm is proposed for online noise filtering in fault signals. The principle is as follows:

[0135]

[0136] In the formula, λ is the threshold;

[0137] ω j,k These are the wavelet coefficients of the fault signal;

[0138] To estimate wavelet coefficients;

[0139] j is the decomposition scale, and 1≤j≤J, where J is the maximum scale;

[0140] sgn() is a symbolic function;

[0141] This threshold function is continuous in the wavelet domain, when ω j,k →λ - hour, When ω j,k →λ + hour,

[0142] The choice of threshold λ should satisfy:

[0143]

[0144] In the formula, N represents the signal length;

[0145] σ j The standard deviation of the j-th layer of Gaussian white noise is expressed as:

[0146]

[0147] In the formula, Cd j,k This represents the high-frequency component of the wavelet decomposition at the j-th level.

[0148] p is the number of wavelet coefficients at this scale;

[0149] like Figure 2 As shown, WTD believes the fault signal exists in the low-frequency portion Ad. j,k In the high-frequency range, noise exists in Cd. j,k middle;

[0150] Since the amplitude of the noise follows a Gaussian distribution, the high-frequency component Cd is decomposed to the maximum number of layers. j,k To evaluate the model's complexity, BIC (Block Identity) is introduced to assess the model's complexity, ensuring that the model's complexity is effectively reduced while maintaining denoising accuracy.

[0151] BIC=qln(N)-2ln(L) (4)

[0152] In the formula, q is the number of model parameters;

[0153] N is the number of samples;

[0154] L is the maximum likelihood function that follows a Gaussian distribution, i.e.

[0155]

[0156] S2, Fault severity identification:

[0157] The core idea of ​​MEST is to construct a nonparametric model of a system or equipment. By optimally reconstructing the observation vector and the historical memory matrix, the estimated vector is obtained. The difference between the estimated vector and the observation vector is used to reflect the degree of failure of the system or equipment. Compared with neural networks, it has the advantages of low computational cost, clear physical meaning of the model, and simple model structure.

[0158] Suppose that at a certain time t, n interrelated variables are observed in the equipment, and let them be denoted as observed variables X. t ,Right now

[0159] X t =[x t,1 ,x t,2 …x t,n ] T (6)

[0160] In the formula, x t,n Let be the observed value of the state variable at time t;

[0161] Construct a historical memory matrix D with m historical moments and n associated state variables, i.e.

[0162]

[0163] The m observation vectors X in the historical memory matrix D obs Linear weighting yields the estimated vector X. est ,Right now

[0164] X est =DW=w1X1+w2X2…w m X m (8)

[0165] In the formula, W = [w1, w2…w m ] T Let m be an m-dimensional weight vector, representing the input observation vector.

[0166] X obs The similarity to the historical memory matrix D, i.e.

[0167]

[0168] In the formula, These are non-linear operators used to replace product operations in ordinary matrices, avoiding... The resulting irreversible phenomenon expands the scope of application of formula (9);

[0169] To improve the multidimensional data processing capability of the MEST algorithm, D T With X obs The Mahalanobis distance (MD) between them is used as a non-linear operator in MEST, i.e.

[0170]

[0171] In the formula, ∑ -1 Let be the inverse of the covariance matrix of a multidimensional random variable. Intuitively, it can be seen that the more similar the two state matrices are, the smaller their MD is.

[0172] The greater the difference between the two state matrices, the greater the result of their nonlinear operation.

[0173] Substituting equation (9) into equation (8), we can obtain the optimal MEST model estimation vector.

[0174] The final expression is

[0175]

[0176] By comparing the observation vector X obs With the estimated vector X est The difference between them can be used to intuitively obtain the residual value ε, which reflects the degree of equipment failure.

[0177] ε=X est -X obs (12)

[0178] By comparing the application ranges of various fault indicators, the root mean square (RMS) is selected to reflect the degree of fault, as shown in Table 1:

[0179] Table 1 Error Index Table

[0180]

[0181] Wherein, VARV is the variance value;

[0182] RMSV stands for root mean square.

[0183] SF is the shape factor;

[0184] MF is the marginal factor;

[0185] E represents energy;

[0186] SE is the Shannon entropy;

[0187] RE stands for the entropy of benevolence and righteousness.

[0188] TE represents the entropy of Chalis;

[0189] By finding n dimensions X est With X obs The RMSV of the residual ε yields the DR index, which reflects the degree of equipment failure.

[0190]

[0191] Since the equipment failure severity index DR obtained by MEST is composed of multiple discrete points, the curve composed of DR inevitably contains multiple "cusps" where the derivatives do not exist, which greatly increases the difficulty of failure prediction. CSFI can directly use the smoothness of the connection points of each node and the connection conditions to construct a relational expression, determine the undetermined coefficients, construct an interpolation polynomial, and complete the operation of smoothing the curve with "cusps". It has the advantages of simple construction, small amount of calculation and high reliability. Therefore, CSFI is introduced for smoothing.

[0192] S3. Online Fault Prediction:

[0193] TSFM is a non-statistical analysis method that can eliminate random data fluctuations, increase the importance of recent data in prediction, and conform to the objective laws of data in fault prediction. Furthermore, due to its small data requirements, simple calculation process, and convenient model construction, it is particularly suitable for online fault prediction in the short or medium to long term. However, its lack of ability to identify data inflection points and the overly subjective selection of smoothing factors severely limit its fault prediction effectiveness. Therefore, gradient descent is introduced to update the smoothing factor online, and an adaptive sliding time window is introduced to dynamically extract time series data to improve the fitting ability of TSFM. The TSFM expression is:

[0194]

[0195] In the formula, DRt This is the original sequence data;

[0196] Let be the adaptive smoothing factor for the i-th training iteration in the (t+T)-th prediction;

[0197] This is a first-order smoothing value;

[0198] It is a quadratic smoothed value;

[0199] If T represents the prediction time. Let represent the predicted value of the i-th training iteration at time t+T. Then the prediction formula is:

[0200]

[0201] in:

[0202]

[0203] By comparing the applicability of various gradient descent algorithms, stochastic gradient descent (SGD) is selected to adaptively update the smoothing factor. As shown in Table 2:

[0204] Table 2 Comparison of Gradient Descent Algorithms

[0205]

[0206] Among them, BGD stands for Batch Gradient Descent (BGD), and Momentum-SGD stands for Momentum Stochastic Gradient Descent (SGDM).

[0207] Then the i-th training adaptive smoothing factor of the (t+T)-th prediction. The expression is:

[0208]

[0209] In the formula, For the (t+T)th prediction, the (i-1)th training adaptive smoothing factor is used.

[0210] β is the learning factor;

[0211] This is the i-th training prediction value for the (t+T)-th prediction.

[0212] DR t+T This is the (t+T)th actual value;

[0213] Let be the partial derivative of the training loss function for the (t+T)th prediction with respect to α:

[0214]

[0215] like Figure 3 As shown in the formula, The length of the i-th training time series data predicted in the (t+T)-th time is the adaptive sliding time window.

[0216] Figure 6 In this context, Data_mode_i refers to the training data for the i-th training iteration, and Data_test_i refers to the test data for the i-th training iteration.

[0217]

[0218] In the formula, μ is the adjustment factor;

[0219] Depend on Figure 6 From equations (18) and (19), it is easy to see that when time series data cannot fully reflect fault information, i.e., the partial derivative of the loss function... If it is always in the same direction, then It should continuously increase or decrease to meet the needs of extracting fault information. It can increase by a factor of 1+μ or decrease by a factor of 1-μ.

[0220] When time series data can partially reflect fault information, i.e., the partial derivative of the loss function If they are not all in the same direction, then Length depends on direction, When it is the right time, Length increase, When it is negative, The length has been shortened.

[0221] VARV in Table 1 is used to represent the final prediction error.

[0222]

[0223] In the formula, len(T) is the number of fault severity index data in the (t+T)th prediction.

[0224] like Figure 4 As shown, Step 1: For the i-th training of the t+T-th prediction, the prediction model first uses the preset sliding time window from the (i-1)-th prediction. Divide the data into training data (Data_mode_i) and test data (Data_test_i) for the i-th training iteration, and then divide the data from the (i-1)-th training iteration... and Substitute into equation (15) to predict the i-th training iteration. value;

[0225] Step 2: Substitute into equation (18) to calculate the loss function for the i-th training iteration. If the maximum number of iterations (maxtrans) is satisfied or less than the minimum error (minerror), then training stops. If the above conditions are not met, then the adaptive smoothing factor is updated according to equations (17) and (19). and adaptive sliding time window Repeat Step 1 to begin the (t+T)th prediction and the (i+1)th training iteration.

[0226] Step 3: Obtain the results from Step 2 Substitute into equation (15) to obtain the final result. The value is calculated, and the prediction error e is calculated according to equation (20).

[0227] S4. Experimental Analysis:

[0228] To further verify the effectiveness and feasibility of the equipment failure prediction model under small sample conditions, rolling bearing performance degradation data were used for analysis and verification.

[0229] like Figure 5 As shown, in order to obtain the fault data of the bearing model LDK UER204, two PCB 352C33 accelerometers were placed on the housing of the bearing under test, with an included angle of 90° between them, that is, one was placed on the horizontal axis and the other on the vertical axis. The sampling frequency was set to 25.6kHz / min, the radial force was 12kN, the rotation speed was 2100rpm, and the operation time was 157.44s. Since the load was applied in the horizontal direction, the accelerometer in this direction can more accurately reflect the degradation information of the bearing under test. Therefore, the vibration signal in the horizontal direction was selected to reflect the degree of failure of the bearing under test.

[0230] like Figure 6 As shown, in the early stage of the bearing performance degradation experiment, the amplitude change is not significant, and the bearing is in the normal operating stage, i.e. before point a, but at this time it contains a large amount of noise signal;

[0231] In the later stage of the performance degradation experiment, the bearing amplitude increases with the increase of working time. The bearing is in the performance degradation stage, that is, between point a and point c. The detected signal contains a large number of bearing performance degradation signals and noise signals.

[0232] At the end of the performance degradation test, the bearing amplitude suddenly increased, and the bearing was in the failure stage, that is, after point c.

[0233] Therefore, the key to validating the proposed model is whether it can eliminate the influence of noise signals, accurately determine the bearing performance degradation time 'a' and the failure time 'c', truly reflect the trend of bearing performance degradation, and ensure high prediction accuracy and short prediction time.

[0234] S5. Data Processing and Analysis:

[0235] like Figure 7 and 8 As shown, the complexity of the improved WTD algorithm with different decomposition levels is evaluated using a preset BIC.

[0236] The BIC value is minimized when the wavelet decomposition level j = 7.

[0237] When the number of wavelet decomposition layers j < 7, the effect of improving WTD noise reduction is enhanced as j increases.

[0238] When the wavelet decomposition level j > 7, the effect of improving WTD noise reduction is not significantly enhanced as j increases.

[0239] Therefore, when j=7, the improved WTD algorithm can not only ensure that the noise reduction effect meets the requirements, but also effectively prevent the problem of excessive model complexity caused by excessive accuracy.

[0240] like Figure 9 As shown, the Gaussian distribution of noise amplitude is used as the criterion for evaluating model complexity. Therefore, the high-frequency part of the maximum decomposition layer Cd is used. j,k Output;

[0241] When j < 7, improve the high-frequency part of the maximum decomposition layer Cd in WTD. j,k If the noise follows a Gaussian distribution, then improving the threshold in WTD does not destroy the distribution characteristics of Gaussian white noise, meaning the wavelet decomposition is insufficient.

[0242] When j≥7, improve the high-frequency part of the maximum decomposition layer Cd of WTD. j,k Since it does not follow a Gaussian distribution, improving the threshold in WTD destroys the distribution characteristics of Gaussian white noise, i.e., wavelet decomposition is sufficient.

[0243] Therefore, this demonstrates the feasibility of setting the number of BIC evaluation decomposition levels and the high-frequency component Cd of the preset maximum wavelet decomposition level. j,k To assess the scientific validity of the data;

[0244] like Figure 10 As shown, the wavelet decomposition layer number j=7 is set, and the proposed improved WTD and improved WTD (hereinafter referred to as the referenced improved WTD) are used to perform noise reduction on the bearing performance degradation data;

[0245] Figure 10 (1), (4), (2), (5), (3), and (6) are the original time-domain diagram, the original spectrum diagram, the time-domain diagram after using the improved WTD noise reduction, the spectrum diagram after using the improved WTD noise reduction, the time-domain diagram after improving WTD noise reduction, and the frequency-domain diagram after improving WTD noise reduction, respectively.

[0246] The proposed improved WTD algorithm has a better noise reduction effect than the referenced improved WTD algorithm. It can significantly suppress the influence of noise during normal operation, while retaining the bearing failure degree information during the performance degradation and failure stages, which is helpful for accurately determining the start time of the performance degradation and failure stages.

[0247] S6. Fault severity identification and analysis:

[0248] like Figure 11 As shown, the wavelet decomposition coefficients obtained from the improved WTD are used as the observed variables X of the improved MEST. t That is, when j = 7, X t =[Ca t,7 Cd t,7 Cd t,6 Cd t,5 Cd t,4 Cd t,3 Cd t,2 Cd t,1 The sampling frequency is set to fs = 25600Hz, or 1 second. The first 2 seconds of bearing performance degradation data are set to a healthy state, i.e., historical time m = 2, and the historical memory matrix is ​​D = [X1, X2]. Other times are set to a performance degradation state every 2 seconds, i.e., the observation vector X obs By incorporating bearing performance degradation data, a total of 77 failure severity DRs were evaluated.

[0249] Meanwhile, in order to eliminate multiple "sharp points" in the DR curve where the derivatives are missing, dual CSFI processing was used to process the data, with the number of interpolations being 10 times that of the original data volume, resulting in a total of 7700 smoothed fault degree points;

[0250] Figure 11 In the middle (1), (2), and (3), respectively, there are bearing performance degradation curves, failure degree DR curves, and smoothed failure degree DR curves.

[0251] Depend on Figure 11 As can be seen from (1) and (2), when the bearing performance degrades at point a, the failure degree DR increases simultaneously. The improved MEST model can accurately determine the start time of the performance degradation stage and can increase the failure degree DR value before the failure point c, when the failure point b is about to occur, and give a warning signal.

[0252] Depend on Figure 11 As can be seen from (2) and (3), the dual CSFI algorithm can maintain the trend of the original fault degree DR curve, eliminate the cusp of the curve where the derivative does not exist, and enhance the predictability of the data.

[0253] S7. Online Fault Prediction Analysis:

[0254] The experimental hardware consisted of an i7-10875H CPU, 32GB of RAM, and an RTX 2060 graphics card. The initial values ​​for the adaptive sliding window length were 100, the adaptive smoothing factor α was 0.05, the learning factor β was 0.5, the maximum number of training iterations (maxtrans) was 1000, and the minimum allowable error (minerror) was 10. -8 The smoothed DR values ​​of 7700 bearing fault severity were input into the online fault prediction model. The trends of the adaptive smoothing factor α, the adaptive sliding time window length, and the prediction error e are shown in the figure.

[0255] like Figure 12 As shown, during the normal operation phase, i.e. before point a, the bearing failure degree DR value changes little. During this phase, the adaptive smoothing factor α, except for a brief adjustment in the early stage, quickly stabilizes at around 0.5.

[0256] During the performance degradation stage, i.e. after point a, the adaptive smoothing factor α changes significantly as the fault severity DR increases. At point b, when the bearing is about to fail, the adaptive smoothing factor α adjusts more drastically. This indicates that the proposed adaptive smoothing factor α can change in real time with the fault severity DR, accurately determine the performance degradation point a and the point b when the bearing is about to fail, and has strong optimization capabilities, thus meeting the requirements for online fault prediction.

[0257] like Figure 13 As shown, during the normal operation phase, i.e. before point a, the bearing failure degree DR value changes little, and the adaptive sliding time window length stabilizes at the initial value of about 100 during this phase.

[0258] In the performance degradation stage, i.e. after point a, as the fault degree DR increases, the performance degradation data from the normal operation stage can no longer characterize the fault degree of the bearing at this time, and may even produce negative effects. Therefore, the length of the adaptive sliding time window is rapidly shortened, and small fluctuations begin to occur at point b, the moment when the bearing is about to fail. This indicates that the proposed adaptive sliding time window length can change in real time with the fault degree DR. In the stable stage, the sliding time window length is kept stable, and in the stage of drastic change, the sliding time window length is adjusted rapidly. It can accurately determine the performance degradation moment point a and the moment when the failure will occur point b, thus meeting the requirements of online fault prediction.

[0259] like Figure 14 and 15As shown, the trend of the prediction error e is similar to the trend of the bearing failure degree DR. That is, the error gradually increases with the complexity of the failure degree DR trend. The error increases significantly when the failure is about to occur at point b. This is consistent with the trend of the adaptive smoothing factor α and the adaptive sliding time window length. The predicted value and the actual value almost completely coincide. The final prediction error e = 0.068%, which has high fitting accuracy and meets the requirements of online fault prediction.

[0260] Under these experimental conditions, the average prediction time per test is approximately 0.0277s, which is about 1.385% of the time interval between two failure severity predictions (DRs). This means that the model can provide prediction results of equipment failure severity in a short time, and the prediction time is short, which meets the requirements for failure prediction.

[0261] S8. Summary of Prediction Results:

[0262] To address the challenges of equipment fault prediction characterized by high complexity, limited sample data, and strong timeliness, a small-sample online equipment fault prediction model is proposed. The effectiveness and reliability of this model are verified using rolling bearing lifecycle vibration data. The prediction results are summarized below:

[0263] (1) BIC can accurately find the optimal number of decomposition layers for the WTD algorithm, providing a basis for improving the parameter settings of the WTD model;

[0264] (2) The improved WTD algorithm has excellent noise reduction effect, ensuring the reliability of fault data;

[0265] (3) The improved MEST algorithm and dual CSFI algorithm of MD can effectively transform fault signals into fault degree indicators, providing high-quality data support for subsequent fault prediction.

[0266] (4) The proposed online fault prediction model has the ability to continuously mine the potential fault degree information in the data and realize online prediction of equipment faults.

[0267] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for online prediction of equipment faults in a small sample size, characterized in that: The prediction steps include the following: S1. Data Processing: By optimizing the threshold function in the WTD noise reduction algorithm and introducing BIC to evaluate the impact of the number of decomposition layers on the complexity of WTD, an improved WTD algorithm is proposed for online noise filtering in fault signals. S2. Fault severity identification: A nonparametric model of the system or equipment is constructed through MEST. An estimated vector is obtained by optimal reconstruction estimation of the observation vector and the historical memory matrix. The difference between the estimated vector and the observation vector is used to reflect the fault severity, and CSFI is introduced for smoothing. S3. Online Fault Prediction: By using the non-statistical analysis method of TSFM, random data fluctuations are eliminated, and gradient descent is introduced to update the smoothing factor online. An adaptive sliding time window is introduced to dynamically extract time series data, thereby improving the fitting ability of TSFM. S4. Experimental Analysis: Verify the effectiveness and feasibility of the equipment failure prediction model under small sample conditions; S5. Data Processing and Analysis: The complexity of the improved WTD algorithm with different decomposition levels is evaluated using a preset BIC. S6. Fault severity identification and analysis: The wavelet decomposition coefficients obtained by the improved WTD are used as the observation variables of the improved MEST. The sampling frequency, health status and degradation status are set to identify and analyze the fault severity. The data is processed by dual CSFI to eliminate the "sharp points" in the curve where the derivative does not exist, and the smoothed fault severity points are obtained. S7. Online Fault Test Analysis: Preset adaptive sliding time window, adaptive smoothing factor, learning factor, maximum number of training iterations, and minimum allowable error. Input the smoothed bearing fault severity value into the online fault prediction model to obtain the trend of adaptive smoothing factor change, the trend of adaptive sliding time window length change, and the trend of prediction error change. S8. Summary of Prediction Results: Summarize the effectiveness of the online prediction model in fault prediction; In S3, the TSFM expression is: In the formula, DR t This is the original sequence data; Let be the adaptive smoothing factor for the i-th training iteration in the (t+T)-th prediction; This is a first-order smoothing value; It is a quadratic smoothed value; If T represents the prediction time. Let represent the predicted value of the i-th training iteration at time t+T. Then the prediction formula is: in: By comparing the applicability of various gradient descent algorithms, stochastic gradient descent (SGD) is selected to adaptively update the smoothing factor. Then the i-th training adaptive smoothing factor of the (t+T)-th prediction. The expression is: In the formula, For the (t+T)th prediction, the (i-1)th training adaptive smoothing factor is used. β is the learning factor; This is the i-th training prediction value for the (t+T)-th prediction. DR t+T This is the (t+T)th actual value; Let be the partial derivative of the training loss function for the (t+T)th prediction with respect to α: In the formula, The length of the i-th training time series data predicted in the (t+T)-th time is the adaptive sliding time window. Draw a diagram illustrating the sliding time window, where Data_mode_i represents the training data for the i-th training iteration, and Data_test_i represents the test data for the i-th training iteration. In the formula, μ is the adjustment factor; When time series data cannot fully reflect fault information, i.e., the partial derivative of the loss function... If it is always in the same direction, then It should continue to grow or decrease. It can increase by a factor of 1+μ or decrease by a factor of 1-μ. When time series data can partially reflect fault information, i.e., the partial derivative of the loss function If they are not all in the same direction, then Length depends on direction, When it is the right time, Length increase, When it is negative, The length has been shortened; The variance value VARV is used to represent the final prediction error. In the formula, len(T) is the number of fault severity index data in the (t+T)th prediction.

2. The method for online prediction of equipment faults with a small sample size according to claim 1, characterized in that, In step S1, the improved WTD algorithm is used to filter out noise in fault signals online. The principle is as follows: In the formula, λ is the threshold; ω j,k These are the wavelet coefficients of the fault signal; To estimate wavelet coefficients; j is the decomposition scale, and 1≤j≤J, where J is the maximum scale; sgn() is a symbolic function; This threshold function is continuous in the wavelet domain, when ω j,k →λ - hour, When ω j,k →λ + hour, The choice of threshold λ should satisfy: In the formula, N represents the signal length; σ j The standard deviation of the j-th layer of Gaussian white noise is expressed as: In the formula, Cd j,k This represents the high-frequency component of the wavelet decomposition at the j-th level. p is the number of wavelet coefficients at this scale; WTD believes the fault signal exists in the low-frequency range. j,k In the high-frequency range, noise exists in Cd. j,k middle; Since the amplitude of the noise follows a Gaussian distribution, the high-frequency component Cd is decomposed to the maximum number of layers. j,k To evaluate the model's complexity using data, the BIC (Browser Inference Function) is introduced to assess the model's complexity: BIC=qln(N)-2ln(L) (4) In the formula, q is the number of model parameters; N is the number of samples; L is the maximum likelihood function that follows a Gaussian distribution, i.e.:

3. The method for online prediction of equipment faults with a small sample size according to claim 1, characterized in that, In S2, the estimated vector and the observation vector are calculated as follows: Suppose that at a certain time t, n interrelated variables are observed in the equipment, and let them be denoted as observed variables X. t ,Right now X t =[x t,1 ,x t,2 …x t,n ] T (6) In the formula, x t,n Let be the observed value of the state variable at time t; Construct a historical memory matrix D with m historical moments and n associated state variables, i.e. The m observation vectors X in the historical memory matrix D obs Linear weighting yields the estimated vector X. est ,Right now X est =DW=w1X1+w2X2…w m X m (8) In the formula, W = [w1, w2…w m ] T Let X be an m-dimensional weight vector, representing the input observation vector X. obs The similarity to the historical memory matrix D, i.e. In the formula, This is a non-linear operator used to replace the product operation in ordinary matrices; D T With X obs The Mahalanobis distance (MD) between them is used as a non-linear operator in MEST, i.e. In the formula, ∑ -1 It is the inverse matrix of the covariance matrix of a multidimensional random variable; The more similar two state matrices are, the smaller their MD is; The greater the difference between the two state matrices, the greater the result of their nonlinear operation. Substituting equation (9) into equation (8), we obtain the final expression for the MEST model estimation vector: By comparing the observation vector X obs With the estimated vector X est The difference between them yields the residual value ε, which reflects the degree of equipment failure, i.e.: e=X est -X obs (12) By comparing the application ranges of various fault indicators, the root mean square (RMS) is selected to reflect the degree of fault. By finding n dimensions X est With X obs The root mean square value (RMSV) of the residual ε yields the indicator DR, which reflects the degree of equipment failure. The equipment failure severity index DR obtained using MEST is composed of multiple discrete points. The curve composed of DR contains multiple "cusps" where the derivative does not exist. CSFI is introduced for smoothing.

4. The method for online prediction of equipment faults with a small sample size according to claim 1, characterized in that, In S3, the online fault prediction process is as follows: Step 1: For the i-th training iteration of the (t+T)-th prediction, the prediction model first uses the preset sliding time window from the (i-1)-th iteration. Divide the data into training data (Data_mode_i) and test data (Data_test_i) for the i-th training iteration, and then divide the data from the (i-1)-th training iteration... and Substitute into equation (15) to predict the i-th training iteration. value; Step 2: Substitute into equation (18) to calculate the loss function for the i-th training iteration. If the maximum number of iterations (maxtrans) is satisfied or less than the minimum error (minerror), then training stops. If the above conditions are not met, then the adaptive smoothing factor is updated according to equations (17) and (19). and adaptive sliding time window Repeat Step 1 to begin the (t+T)th prediction and the (i+1)th training iteration. Step 3: Obtain the results from Step 2 Substitute into equation (15) to obtain the final result. The value is calculated, and the prediction error e is calculated according to equation (20).

5. The method for online prediction of equipment faults with a small sample size according to claim 1, characterized in that, In step S4, bearing performance degradation data is used as the verification object. The sampling frequency is set to 25.6 kHz / min, the radial force is 12 kN, the rotation speed is 2100 rpm, and the operation time is 157.44 s. The vibration signal in the horizontal direction is selected to reflect the degree of failure of the tested bearing.

6. The method for online prediction of equipment faults with a small sample size according to claim 1, characterized in that, The WTD effects of different sub-levels in S5 are compared as follows: The BIC value is minimized when the wavelet decomposition level j = 7. When the number of wavelet decomposition levels j < 7, the effect of improving WTD noise reduction is enhanced as j increases. When the wavelet decomposition level j > 7, the effect of improving WTD noise reduction is not significantly enhanced as j increases. When j=7, the improved WTD algorithm can ensure that the noise reduction effect meets the requirements and effectively prevent the problem of excessive model complexity caused by excessive accuracy. The model complexity is evaluated based on the fact that the noise amplitude follows a Gaussian distribution. Therefore, the high-frequency part of the maximum decomposition layer, Cd, is used. j,k Output and plot the amplitude distribution of Gaussian white noise; When j < 7, improve the high-frequency part of the maximum decomposition layer Cd in WTD. j,k If the noise follows a Gaussian distribution, then improving the threshold in WTD does not destroy the distribution characteristics of Gaussian white noise, meaning the wavelet decomposition is insufficient. When j≥7, improve the high-frequency part of the maximum decomposition layer Cd of WTD. j,k Since the noise does not follow a Gaussian distribution, improving the threshold in WTD disrupts the distribution characteristics of Gaussian white noise, meaning the wavelet decomposition is sufficient. This demonstrates the feasibility of setting BIC to evaluate the number of decomposition levels and the high-frequency component Cd of the preset maximum wavelet decomposition level. j,k To assess the scientific validity of the data; The wavelet decomposition level j=7 was set to improve the noise reduction processing of bearing performance degradation data by WTD.

7. The method for online prediction of equipment faults with a small sample size according to claim 1, characterized in that, In S6, the wavelet decomposition coefficients obtained from the improved WTD are used as the observation variables X of the improved MEST. t That is, when j = 7, X t =[Ca t,7 Cd t,7 Cd t,6 Cd t,5 Cd t,4 Cd t,3 Cd t,2 Cd t,1 The sampling frequency is set to fs = 25600Hz, or 1 second. The first 2 seconds of bearing performance degradation data are set to a healthy state, i.e., historical time m = 2, and the historical memory matrix is ​​D = [X1, X2]. Other times are set to a performance degradation state every 2 seconds, i.e., the observation vector X obs By incorporating bearing performance degradation data, a total of 77 failure severity DRs were evaluated. Meanwhile, in order to eliminate multiple "sharp points" in the DR curve where the derivatives are missing, dual CSFI processing was used to process the data, with the number of interpolations being 10 times the original data volume, resulting in a total of 7700 smoothed fault degree points.

8. The method for online prediction of equipment faults in a small sample size according to claim 1, characterized in that, In step S7, the initial value of the adaptive sliding time window length is 100, the initial value of the adaptive smoothing factor α is 0.05, the learning factor β is 0.5, the maximum number of training iterations maxtrans is 1000, and the minimum allowable error minerror is 10. -8 The smoothed DR values ​​of 7700 bearing fault severity were input into the online fault prediction model, and the trends of the adaptive smoothing factor α, the adaptive sliding time window length, and the prediction error e were compared.

9. The method for online prediction of equipment faults with a small sample size according to claim 1, characterized in that, In S8, the prediction results are summarized as follows: BIC can accurately determine the optimal number of decomposition layers for the WTD algorithm; The improved WTD algorithm delivers excellent noise reduction performance; MD's improved MEST algorithm and dual CSFI algorithm can effectively convert fault signals into fault severity indicators. The proposed online prediction model can achieve online prediction of equipment failures.

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