A Bayesian sparse representation method for mechanical fault redundancy based on lifting wavelet dictionary

Abstract

This invention discloses a Bayesian sparse representation method for redundant lifting wavelet dictionaries of mechanical faults, belonging to the field of mechanical fault diagnosis. By employing a lifting method, the invention designs an adaptive lifting wavelet dictionary. A multi-population search strategy is integrated into a quantum genetic optimization algorithm. A predictor is designed with the objective function of maximizing the kurtosis index of the wavelet-decomposed detail signal as the target function. Then, an updater is designed with the objective function of minimizing the reconstruction error. This effectively targets the local features of the signal and accurately extracts the signal's transient fault characteristic information. The invention is suitable for mechanical equipment fault diagnosis.

Description

Technical Field

[0001] The present invention belongs to the field of mechanical fault non-stationary transient signal analysis, and specifically relates to a mechanical fault redundant lifting wavelet dictionary Bayesian sparse representation method Background Art

[0002] Mechanical fault signals are typical non-stationary signals. These signals are characterized by a finite duration and time-varying frequency, meaning their frequency changes over time. Traditional analysis methods, based on Fourier transforms, describe the global characteristics of signals. However, these methods often ignore local information and fail to describe the signal's time-varying characteristics.

[0003] Therefore, employing effective signal processing techniques to analyze vibration signals and reveal fault characteristics is a key task in mechanical fault diagnosis. Fourier transforms, wavelet transforms, and empirical mode decomposition are all commonly used methods in fault diagnosis. Sparse representation offers significant advantages over traditional diagnostic methods. By constructing a reasonable dictionary model, this method can approximate the signal using a linear combination of a small number of dictionary atoms. Furthermore, the ability of dictionary atoms to effectively match instantaneous fault characteristics is a key factor influencing the effectiveness of sparse representation.

[0004] Currently, there are two main approaches to constructing dictionaries for sparse representation: one is to use a preset base dictionary, such as the Laplace wavelet dictionary, and the other is to use a learned dictionary, such as the k-singular value decomposition dictionary and the multi-scale learned dictionary. The redundant lifting wavelet dictionary of the present invention is a new dictionary construction method. By designing different lifting operators, the lifting method modifies the characteristics of the classical wavelet filter to obtain a biorthogonal wavelet with tight support, symmetry, and impact that best matches the transient fault characteristics. Using the flexible biorthogonal wavelet as an atom, the lifted wavelet dictionary is constructed, enhancing the matching between the dictionary and the signal characteristics and intuitively characterizing the transient fault characteristics. Summary of the Invention

[0005] The purpose of the present invention is to overcome the above-mentioned existing technical defects and provide a mechanical fault redundant lifting wavelet dictionary Bayesian sparse representation method to enhance the ability to capture the transient impact characteristics of the signal.

[0006] The technical problem of the present invention is solved as follows:

[0007] A Bayesian sparse representation method for redundant lifting wavelet dictionary of mechanical faults, comprising the following steps:

[0008] Step 1. Collect mechanical vibration signal f;

[0009] Step 2. According to the intrinsic structural characteristics of the mechanical vibration signal, the signal f is transformed into flexible biorthogonal wavelet atoms through the lifting method, and an adaptive lifting wavelet dictionary is designed;

[0010] Determine the objective function K of the lifting wavelet predictor p :

[0011]

[0012] in, and σ represent the mean and variance of the detail signal d, E{} represents the mathematical expectation; therefore, the problem is transformed into making the objective function K under the constraints p Solve the maximum optimization problem; the coefficients of the predictor satisfy the following relationship:

[0013]

[0014] Where N represents the predictor length, r is the predictor coefficient sequence number, and according to the dual lifting principle, p r =p -r+1 ;

[0015] Step 2.1. Specify the number of iterations of the Bayesian biorthogonal method to be 65, the evolutionary generation of the multi-population quantum genetic optimization algorithm to be 100, the number of populations to be 5, the number of individuals to be 800, and the binary length to be 10; the number of adaptive lifting wavelet dictionary predictors to be 6, the number of updaters to be 6, and the length of atoms in the dictionary to be 16;

[0016] Step 2.2. Take the objective function K p As the fitness function, the multi-population quantum genetic optimization algorithm is integrated into it to solve the predictor P = [p -N / 2+1 ,…,p1,…,p N / 2 ];

[0017] Step 2.3. Generate the approximation signal of the next layer through the updater and determine the reconstruction error J of the updater metric U for:

[0018]

[0019] Where U represents the updater, and Reconstructed signals when detail signal d=0 are Even sequence samples and odd sequence samples, s (0) Indicates that the even sequence samples are not updated, d (0) Indicates that the odd sequence samples are not updated; when the detail signal d=0, and It can be expressed as follows:

[0020]

[0021]

[0022] Where “*” represents the convolution operation, P is the predictor, and s is the unupdated signal;

[0023] Let λ be the Lagrangian operator, the lifting and dual lifting principles constructed by lifting wavelet, the objective function J U Convert to J u (u,λ);

[0024]

[0025] in, represents the updater step size;

[0026] To make J u (u,λ) is the smallest, and u is calculated for each of them. j The partial derivative with respect to λ is:

[0027]

[0028]

[0029] Solve the above equation to obtain the coefficients According to the symmetry of the updater coefficients, the updater operator of the adaptive feature is finally obtained

[0030] Step 2.4. Use the adaptive lifting wavelet to construct atoms that match the transient fault feature structure and form an adaptive lifting wavelet dictionary; that is:

[0031]

[0032] Among them, {s p1 ,s p2 ,...,s pN The elements in} are atoms in the adaptive lifting wavelet dictionary;

[0033] Step 3. Specify the maximum number of decompositions n of the vibration signal f using the Bayesian biorthogonal method max , the number of decompositions of the current Bayesian biorthogonal method is n = 0, and the support set S is established (0) And set the initial state to empty The initial residual signal is the mechanical vibration signal, namely R 0 f = f, execute the orthogonal matching pursuit algorithm to select the initial atomic component j from the adaptive lifting wavelet dictionary D; obtain the threshold Th through the Bayesian method j, and filter the elements in component j; any element greater than the threshold Th j All elements less than the threshold Th are retained j The elements are discarded and the filtered components are added to the support set S (t) =S (t-1) ∪j (t) , where j (t) is the support set after Bayesian screening;

[0034] Step 4. Complete the double orthogonality through the Gram-Schmidt orthogonalization process, that is, the selected optimal atom is always orthogonal to the residue; at the same time, it is also orthogonal to the atoms that do not belong to the selected support set; update the sparse representation coefficients based on the obtained double orthogonal atoms; determine whether the Bayesian double orthogonal algorithm meets the iteration termination condition n = n max ; If yes, go to step 5, otherwise, update the residual signal Let n=n+1 and go to step 3;

[0035] Step 5. Sum the projections of the residual signals generated by the Bayesian orthogonal matching pursuit decomposition on the best matching atom. The projection sum signal is the final reconstructed signal f 重构 ;

[0036] Step 6: Reconstruct the signal f 重构 Perform envelope spectrum analysis to identify mechanical failures.

[0037] The advantages and benefits of this invention are as follows: Fixed-base dictionaries require the manual specification of numerous parameters for base atoms, resulting in a poor match between the atoms and the characteristic structures of real faults. To achieve optimal matching between atoms and fault features, a lifting method is employed, incorporating a multi-population quantum genetic algorithm to construct a lifting wavelet dictionary, enhancing the dictionary's compatibility with signal features. This method can effectively extract faults even when the fault signal is submerged in ambient noise. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 A flow chart of the method of the present invention;

[0039] Figure 2 Time domain waveform of vibration signal;

[0040] Figure 3 Schematic diagram of decomposition and reconstruction of adaptive lifting wavelet;

[0041] Figure 4 Waveform diagram of the simulation signal;

[0042] Figure 5 Adaptive lifting of wavelet scaling functions and wavelet functions;

[0043] Figure 6Non-adaptive lifting wavelet scaling function and wavelet function;

[0044] Figure 7 Reconstruct the signal time domain waveform and envelope spectrum. DETAILED DESCRIPTION

[0045] The invention is further described below with reference to the accompanying drawings and embodiments.

[0046] This example constructs an atom dictionary based on the characteristics of mechanical vibration signals using a redundancy lifting method. A multi-population quantum genetic optimization algorithm is incorporated into the solution of the lifting wavelet predictor and updater to obtain information that accurately represents the transient fault characteristics of the signal. Using the obtained atoms, the fault signal is reconstructed using a Bayesian biorthogonal method. Finally, time-frequency analysis of the reconstructed signal verifies the effectiveness and robustness of this method for mechanical fault diagnosis in the presence of noise and interference.

[0047] The present invention integrates the multi-population quantum genetic optimization algorithm into the determination process of the lifting wavelet predictor, with the objective function K P As the fitness function, a predictor that adaptively matches the signal features is constructed. By defining the measure of the updater reconstruction error J U This generates an approximation signal for the next layer, enabling it to more accurately represent features of the original signal beyond the detail signal. By combining the selection of an overcomplete atom dictionary in sparse representation, the present invention constructs an adaptive lifting wavelet dictionary. A Bayesian biorthogonal method is used to obtain the atoms that best match the residual signal in the adaptive lifting wavelet dictionary.

[0048] Based on the above ideas, this embodiment provides a Bayesian sparse representation fault diagnosis method using a redundant lifting wavelet dictionary structure. This method diagnoses bearings in mechanical equipment. The experimental machine specifications are shown in Table 1. The machine speed during the experiment was 1436 rpm, the data sampling frequency was 12800 Hz, and the data length was 4096 points. Table 1 lists the bearing parameters.

[0049] Table 1 Bearing specifications of experimental machinery and equipment

[0050]

[0051] The schematic flow chart of the method described in this embodiment is as follows Figure 1 As shown, the following steps are included:

[0052] Step 1. Collect the vibration signal f through the acceleration sensor, and its time domain waveform is as follows: Figure 2 As shown. Figure 2 It can be seen that due to the presence of noise and interference components, it is difficult to observe obvious impact signals in the time domain waveform of the vibration signal, making it difficult to accurately identify mechanical faults.

[0053] Step 2. According to the characteristics of the mechanical vibration signal, the lifting method is first used to change the characteristics of the classical wavelet filter by designing different lifting operators to obtain the symmetric biorthogonal wavelet that best matches the transient fault characteristics. The objective function K of the lifting wavelet predictor is P Defined as:

[0054]

[0055] in and σ represent the mean and variance of the detail signal d, and E{} represents the mathematical expectation. Therefore, the problem is transformed into making the objective function K under the constraints p Solve the maximum optimization problem. The coefficients of the predictor satisfy the following relationship:

[0056]

[0057] Where N represents the length of the predictor and r is the number of the predictor coefficient. And according to the dual lifting principle, p r =p -r+1 .

[0058] Step 2.1. This invention combines the strong versatility and high efficiency of the quantum genetic optimization algorithm, incorporates the multi-population concept into it, and proposes a multi-population quantum genetic optimization algorithm to improve global search capabilities and promote frequent information exchange. The number of iterations of the Bayesian biorthogonal model is specified to be 65, the evolutionary generation of the multi-population quantum genetic optimization algorithm is specified to be 100, the number of populations is specified to be 5, the number of individuals is specified to be 800, and the binary length is specified to be 10. The number of adaptive lifting wavelet dictionary predictors is specified to be 6, the number of updaters is specified to be 6, and the length of atoms in the dictionary is specified to be 16. The objective function K is used as the p As the fitness function, the multi-population quantum genetic optimization algorithm is integrated into it to solve the predictor P = [p -N / 2+1 ,…,p1,…,p N / 2 ];

[0059] Step 2.2. The updater generates the approximation signal of the next layer. In order to make the approximation signal more accurately represent the features of the original signal except the detail signal, the reconstruction error J of the updater is defined as U for:

[0060]

[0061] in and Reconstructed signals when detail signal d=0 are Even sequence samples and odd sequence samples of s (0) Indicates that the even sequence samples are not updated, d (0)Indicates that the odd sequence samples are not updated; when the detail signal d=0. and It can be expressed as follows:

[0062]

[0063]

[0064] In the above formula, “*” represents the convolution operation, and P is the predictor;

[0065] Let λ be the Lagrangian operator, the lifting and dual lifting principles constructed by lifting wavelet, the objective function J U Can be converted to J u (u,λ):

[0066]

[0067] in, represents the updater step size;

[0068] To make J u (u,λ) is the smallest, and u is calculated for each of them. j The partial derivative of λ is

[0069]

[0070]

[0071] Solving the above equations can obtain the coefficients From the symmetry of the updater coefficients, we can obtain the updater operator for adaptive signal features The decomposition and reconstruction diagram of adaptive lifting wavelet is as follows Figure 3 As shown;

[0072] In order to verify the advantages of adaptive lifting wavelet, a mechanical model fault impulse signal is constructed, where the impulse component expression is as follows:

[0073]

[0074] Where y0 and ζ are the scale factor and damping coefficient respectively, ω n =2πf n , f n is the normal operating frequency. In this study, y0, ζ, f n The impulse signal waveform is as follows: Figure 4 As shown in (a), the waveform after adding Gaussian white noise (noise intensity d = 0.1) to the impulse signal is as follows Figure 4As shown in (b), it can be found that the periodic shock that characterizes the fault is submerged by the noise in the vibration signal, and the characteristic information of the shock component cannot be found from the synthetic signal.

[0075] Adopting the adaptive lifting wavelet construction method, the predictor and updater matching the impact part characteristics of the simulation signal are designed. The number of predictor coefficients is 6, and the number of updater coefficients is 6. Using the predictor and updater that adaptively match the signal characteristics, the lifting wavelet scaling function and wavelet function that match the impact part characteristics of the simulation signal are constructed as follows: Figure 5 As shown in the figure, it can be seen that the scaling function and the wavelet function have stronger signal representation capabilities in the adaptive method. In order to verify the effectiveness of the present invention, the lifting wavelet scaling function and the wavelet function that match the impact part characteristics of the simulation signal constructed by the non-adaptive method are shown in the figure. Figure 6 shown.

[0076] Step 2.3. Use the adaptive lifting wavelet to construct atoms that match the transient fault feature structure and form an adaptive lifting wavelet dictionary. That is:

[0077]

[0078] where s p Atoms in the adaptive lifting wavelet dictionary.

[0079] Step 4. Specify the maximum number of decompositions n for the Bayesian biorthogonal method max , the number of decompositions of the current Bayesian biorthogonal method is n = 0, and the support set S is established (0) And set the initial state to empty The initial residual signal is the mechanical vibration signal, namely R 0 f = f, execute the orthogonal matching pursuit algorithm to select the initial atomic component j from the adaptive lifting wavelet dictionary D. The threshold Th is obtained by the Bayesian method j , and filter the elements in component j, all elements greater than the threshold Th j All elements less than the threshold Th are retained j The elements of are discarded. Add the filtered components to the support set S (t) =S (t-1) ∪j (t) . where j (t) is the support set after Bayesian screening. (t) is the support set after Bayesian screening.

[0080] Step 5. Ensure doubly orthogonalization through the Gram-Schmidt orthogonalization process. That is, the selected atoms are always orthogonal to the residuals, and at the same time, they are also orthogonal to atoms that are not part of the selected support set. Update the sparse representation coefficients based on the obtained biorthogonal atoms. Determine whether the Bayesian biorthogonal model satisfies the iteration termination condition n = nmax If yes, go to step 6, otherwise let n=n+1 to update the residual signal Go to step 4;

[0081] Step 6. Sum the projections of the residual signals from each decomposition on the best matching atom. The summed projection signal is the final reconstructed signal f 重构 . Perform envelope spectrum analysis on the reconstructed signal to identify mechanical faults.

[0082] The time domain waveform and envelope spectrum of the reconstructed signal are as follows Figure 7 As shown by Figure 7 (a) shows that the periodic impact of the mechanical fault has been accurately extracted, and the time interval of the periodic impact is the inverse of the characteristic frequency of the mechanical outer ring fault. 重构 Perform envelope analysis and get Figure 7 (b) shows the envelope spectrum of the reconstructed signal. The characteristic frequency f0 of the mechanical outer race fault and its multiples (2f0, 3f0) are clearly visible in the envelope spectrum. Therefore, it can be determined that the test machine has an outer race fault. The diagnostic results are consistent with the experimental plan, proving the effectiveness of the embodiment. Therefore, it can be determined that the test machine has a fault.

Claims

1. A Bayesian sparse representation method for redundant lifting wavelet dictionary of mechanical faults, comprising the following steps: Step 1. Collect mechanical vibration signal f; Step 2. According to the intrinsic structural characteristics of the mechanical vibration signal, the signal f is transformed into flexible biorthogonal wavelet atoms through the lifting method, and an adaptive lifting wavelet dictionary is designed; Determine the objective function K of the lifting wavelet predictor p : in, and σ represent the mean and variance of the detail signal d, and E{·} represents the mathematical expectation; therefore, the problem is transformed into making the objective function K p Solve the maximum optimization problem; the coefficients of the predictor satisfy the following relationship: Where N represents the predictor length, r is the predictor coefficient sequence number, and according to the dual lifting principle, p r =p -r+1 ; Step 2.

1. Specify the number of iterations of the Bayesian biorthogonal method to be 65, the evolutionary generation of the multi-population quantum genetic optimization algorithm to be 100, the number of populations to be 5, the number of individuals to be 800, and the binary length to be 10; the number of adaptive lifting wavelet dictionary predictors to be 6, the number of updaters to be 6, and the length of atoms in the dictionary to be 16; Step 2.

2. Take the objective function K p As the fitness function, the multi-population quantum genetic optimization algorithm is integrated into it to solve the predictor P = [p -N / 2+1 ,…,p1,…,p N / 2 ]; Step 2.

3. Generate the approximation signal of the next layer through the updater and determine the reconstruction error J of the updater metric U for: Where U represents the updater, and Reconstructed signals when detail signal d=0 are Even sequence samples and odd sequence samples, s (0) Indicates that the even sequence samples are not updated, d (0) Indicates that the odd sequence samples are not updated; when the detail signal d=0, and It can be expressed as follows: Where "*" represents the convolution operation, P is the predictor, and s is the unupdated signal; Let λ be the Lagrangian operator, the lifting and dual lifting principles constructed by lifting wavelet, the objective function J U Convert to J u (u,λ); in, represents the updater step size; In order to minimize Ju(u,λ), we need to calculate u j The partial derivative with respect to λ is: Solve the above equation to obtain the coefficients According to the symmetry of the updater coefficients, the updater operator of the adaptive feature is finally obtained Step 2.

4. Use the adaptive lifting wavelet to construct atoms that match the transient fault feature structure and form an adaptive lifting wavelet dictionary; that is: Among them, {s p1 ,s p2 ,...,s pN The elements in} are atoms in the adaptive lifting wavelet dictionary; Step 3. Specify the maximum number of decompositions n of the vibration signal f using the Bayesian biorthogonal method max , the number of decompositions of the current Bayesian biorthogonal method is n = 0, and the support set S is established (0) And set the initial state to empty The initial residual signal is the mechanical vibration signal, namely R 0 f = f, execute the orthogonal matching pursuit algorithm to select the initial atomic component j from the adaptive lifting wavelet dictionary D; obtain the threshold Th through the Bayesian method j , and filter the elements in component j; any element greater than the threshold Th j All elements less than the threshold Th are retained j The elements are discarded and the filtered components are added to the support set S (t) =S (t-1) ∪j (t) , where j (t) is the support set after Bayesian screening; Step 4. Complete the double orthogonality through the Gram-Schmidt orthogonalization process, that is, the selected optimal atom is always orthogonal to the residue; at the same time, it is also orthogonal to the atoms that do not belong to the selected support set; update the sparse representation coefficients based on the obtained double orthogonal atoms; determine whether the Bayesian double orthogonal algorithm meets the iteration termination condition n = n max ; If yes, go to step 5, otherwise, update the residual signal Let n=n+1 and go to step 3; Step 5. Sum the projections of the residual signals generated by the Bayesian orthogonal matching pursuit decomposition on the best matching atom. The projection sum signal is the final reconstructed signal f 重构 ; Step 6: Reconstruct the signal f 重构 Perform envelope spectrum analysis to identify mechanical failures.

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