CN116522269B

CN116522269B - Fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction

Info

Publication number: CN116522269B
Application number: CN202310771620.4A
Authority: CN
Inventors: 涂建远; 章进武
Original assignee: Xiamen Juyan Electronic Technology Co ltd
Current assignee: Xiamen Juyan Electronic Technology Co ltd
Priority date: 2023-06-28
Filing date: 2023-06-28
Publication date: 2023-09-19
Anticipated expiration: 2043-06-28
Also published as: CN116522269A

Abstract

The application provides a fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction, which belongs to the traditional signal processing field and the fault diagnosis field, and is used for extracting components with frequency intervals close to each other in non-stationary fault signals, and converting the signal reconstruction into a sparse signal recovery problem based on compressed sensing by using a sparse reconstruction technology, so that accurate estimation and diagnosis of the fault signals are realized. The application comprises the following steps: 1) Constructing a sparse optimization framework of joint optimization based on a multi-frequency modulation mode mathematical model of a non-stationary signal; 2) Designing a relaxation compressed sensing based on Lp norm constraint, and restricting the smoothness of estimated parameters and ensuring the sparsity of recovered signals; 3) And sparse estimation of fault signals in a noise environment is carried out by using a regularized FOCUSS algorithm. The scheme can realize high-quality estimation of the multi-component non-stationary fault signal and ensure sparsity of an optimization algorithm solution.

Description

Fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction

Technical Field

The application belongs to the traditional signal processing field and the fault diagnosis field, and particularly relates to a fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction.

Background

In the field of fault diagnosis, fault diagnosis signals are generally generated due to abnormal situations of a system or a device. Such abnormal conditions may include sensor failure, component damage, system errors, and the like. Fault signals typically appear as abrupt or changing signals. Such variations may occur over time, in terms of amplitude, frequency, period, or other characteristics of the signal, as opposed to stationary signals. A non-stationary signal is a signal whose statistical properties change over time, and is typically composed of a plurality of non-stationary modes. Unlike stationary signals, signal characteristics cannot be uniquely determined using single-domain information in the time or frequency domain, and the increasingly stringent application requirements of non-stationary signals cannot be met. Therefore, researchers propose a time-frequency joint analysis method, which uses time-frequency distribution to describe the relationship between signal frequency and time, and more precisely describe the energy distribution, instantaneous frequency and instantaneous amplitude information of the signal.

The traditional time-frequency analysis method has Short-time Fourier transform (Short-Time Fourier transform, STFT), wavelet transform (Wavelet Transforms, WT) and S transform which are easy to realize and meet the linearity rule for the multi-component signals, so that the method is widely applied. However, limited by the Heisenberg uncertainty principle, linear time-frequency analysis methods generally cannot achieve both higher time and frequency resolution, and particularly cannot perform efficient analysis on strongly time-varying signals. Researchers have proposed various improved time-frequency representation methods for improving the resolution, for example, the synchronous compression transformation (Synchrosqueezing Transforms, SST) proposed by Daubechies combines post-processing technologies such as continuous wavelet transformation and rearrangement, and the like, so that the aggregation of time-frequency distribution, namely the concentration degree of signal energy, is greatly improved. In addition, some researchers have introduced demodulation operators and higher order approximation operators to develop a series of SST-based time-frequency analysis methods. However, SST-based methods cannot directly extract instantaneous frequencies and also face difficulties in analyzing strongly time-varying signals.

The signal decomposition is an important direction of non-stationary signal analysis, and the time-frequency analysis method based on the signal decomposition does not need any prior information and can realize self-adaptive decomposition according to the signal characteristics. A representative time domain decomposition method is empirical mode decomposition (Empirical Mode Decomposition, EMD), which breaks through the limitations of the Heisenberg uncertainty principle and can obtain high resolution time-frequency distribution. In practical applications, however, EMD is sensitive to noise, prone to modal aliasing, and lacks a tight mathematical basis. In IEEE TRANSACTIONS ON SIGNAL PROCESSING (Gilles, J. Empirical Wavelet Transform [ J ]. Signal Processing, IEEE Transactions on, 2013, 61 (16): 3999-4010.), gilles proposed empirical wavelet transform (Empirical Wavelet Transform, EWT), which incorporates the adaptive decomposition concept of EMD method and the tight support framework of wavelet transform theory, provides a brand-new adaptive time-frequency analysis concept for signal processing, and overcomes the problem of modal aliasing caused by discontinuous time-frequency scale of signals. Meanwhile, the method has complete and reliable mathematical theory basis, has low calculation complexity, and can also overcome the problems of over-enveloping and under-enveloping in an EMD method. Subsequently, the variational pattern decomposition (Variational Mode Decomposition, VMD) proposed by dragomimetski and Zosso converts the pattern decomposition problem into a variational problem, enabling an adaptive frequency domain subdivision of the signal and an efficient separation of the components. However, these methods also cannot separate non-stationary components that are closely spaced or overlapping in frequency spacing. Recently, the non-chirped mode estimation (NCME) proposed by Tu et al adopts a sparse model, extracts signal components one by constraining the smoothness of estimation parameters, can directly estimate the instantaneous amplitude and instantaneous frequency of a signal, and can construct high-quality time-frequency distribution. However, just as NCME uses a greedy matching algorithm to achieve extraction one by one, it is not possible to accurately estimate signals with cross-mode or transient pulses. Chen et al propose a variational nonlinear frequency modulation mode decomposition (the Variational Nonlinear Chirp Mode Decomposition, VNCMD) based on VMD, which employs a joint optimization technique to solve the demodulation problem, enabling simultaneous extraction of signal modes. But this approach tends to make the transition to the time-varying amplitude signal smooth.

Numerous studies have shown that a sparse reconstruction framework can be used for high-precision signal estimation. Document Conference Record-Asilomar Conference on Signals, systems and Computers (Sward J, brynfsson J, jakobsson A, et al spark semi-parametric chirp estimation [ C)]i/Conference Record-Asilomar Conference on Signals, systems and computers IEEE 2015) proposes a LASSO (Least Absolute Shrinkage and Selection Operator) -based chirp estimation framework that is robust to noise. Literature spark Semi-Parametric Estimation of Harmonic Chirp Signals (Sward J, brynfsson J, J)akobsson A, et al. Sparse Semi-Parametric Estimation of Harmonic Chirp Signals[J]IEEE Transactions on Signal Processing, 2016, 64 (7): 1798-1807.) estimation of the signal can also be achieved in case the number of signal components is unknown. In addition, sparse signal reconstruction algorithms based on compressed sensing have been widely studied, which is itself a difficult NP problem, and Candides and Donoho proposeConvex compressed perceived restoration under norms is a milestone work and research into this framework has resulted in a rich tool for optimizing restoration. Therefore, most of the restoration algorithms at present are mainly based on +.>Convex under the norm optimizes compressed sensing. However, is->Norms can only be and +.>The result of the norm optimisation is equivalent and +.>The difference between the norm and the true thin fluffer is large.

Disclosure of Invention

The present application aims to solve the above-mentioned problems in the prior art, and to establish a relaxed compressed sensing framework constrained under Lp norms, in contrast to the non-convex problem under Lp norms, which can ensure sparsity and accuracy of recovered signals without such severe conditions.

The application provides a fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction, which comprises sparse optimization of non-stationary signals, a joint optimization mode and relaxed compressed sensing based on Lp norm constraint; specific:

the sparse optimization of the non-stationary signals is realized, the signal estimation is converted into the parameter estimation problem by utilizing a multi-mode nonlinear frequency modulation model of the signals, the smoothness of constraint estimation parameters of a sparse optimization framework is constructed, and the instantaneous amplitude and the instantaneous frequency of the signals can be directly estimated by adjusting the sparsity of the weight parameter balance estimation parameters and the fitting degree of measured values;

the joint optimization mode is to identify all components of the signal in the fitting item of the optimization model at the same time, and ensure the accuracy of parameter estimation by limiting the residual noise energy;

the relaxed compressed sensing based on the Lp norm constraint controls the smoothness of the instantaneous amplitude of the estimated signal by minimizing the Lp norm of the second derivative of the estimated parameter, flexibly controls the sparsity by adjusting the parameter p epsilon (0, 1), has better robustness to outliers and noise and better retains the structural characteristics of the signal while maintaining the sparsity.

Further, the sparse optimization of the nonstationary signal adopts a joint optimization mode to directly estimate all components of the signal and construct high-quality time-frequency distribution, and then a proper parameter p is selected to control the sparsity and the accuracy of the reconstructed signal through an Lp norm constraint relaxation compressed sensing optimization algorithm.

Further, the fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction comprises the following steps:

(1) Constructing a sparse reconstruction framework of joint optimization based on a multi-frequency modulation mode mathematical model of a non-stationary signal;

(2) Designing a relaxation compressed sensing based on Lp norms, and guaranteeing sparse flexibility of a reconstructed signal;

(3) And sparse estimation of fault signals in a noise environment is carried out by using a regularized FOCUSS algorithm.

Further, in step (1), the non-stationary signal is first represented as a mathematical model of the multi-frequency modulation mode, and the frequency modulation signal estimation is converted into a parameter estimation problem. Then, the sparsity of the second derivative of the estimation parameter is restrained to enhance the smoothness of the instantaneous amplitude, and a joint optimization technology is adopted to construct a sparse reconstruction frame, so that the instantaneous amplitude and the instantaneous frequency of the signal can be directly estimated, the signal components can be simultaneously extracted, and the high-quality time-frequency distribution can be constructed;

further, in the step (2), the sparse optimization model in the step (1) is designed into a relaxed compressed sensing sparse reconstruction framework based on Lp norm constraint, so that the sparsity can be flexibly controlled by adjusting the parameter p, the robustness to outliers and noise is better, and the structural characteristics of signals are better reserved while the sparsity is maintained.

Further, in the step (3), when the optimization model in the step (2) is solved by adopting the FOCUSS algorithm, a weighting function is constructed by using the result obtained by the last iteration estimation, so that the energy of the new estimation result obtained by the current iteration is more concentrated. However, since pseudo-inverse operation tends to be ill-conditioned in the presence of noise, not only noise cannot be suppressed but also noise can be severely amplified, resulting in failure to obtain an effective estimate. The influence of noise is reduced by introducing regularization term, so that the reconstruction system is more robust in a noise environment.

Compared with the prior art, the application has the following beneficial effects:

1. the method can directly estimate the instantaneous amplitude and the instantaneous frequency of the signal, simultaneously extract all components of the signal, and can construct high-quality time-frequency distribution.

2. And designing the relaxed compressed sensing based on Lp norm constraint, and constraining the smoothness of estimation parameters, so that the accuracy of estimation is improved while the sparsity of a recovered signal is ensured.

3. And sparse estimation of fault signals in a noise environment is performed by using a regularized FOCUSS algorithm, so that the robustness of a reconstruction system is effectively improved.

Drawings

FIG. 1 is a flow chart of the algorithm of the present application.

Description of the embodiments

In order to make the technical problems, technical schemes and beneficial effects to be solved more clear, the application is further described in detail below with reference to the accompanying drawings and embodiments.

Referring to fig. 1, the embodiment of the application provides a fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction, which comprises the following steps:

step (1): and constructing a sparse reconstruction framework of joint optimization based on a multi-frequency modulation mode mathematical model of the non-stationary signal.

Step (1-1) non-stationary signal mathematical model:

a non-stationary signal typically contains a plurality of modalities, the mathematical model of which can be expressed as:

(1)

in the formula (1), the components are as follows,indicates the number of frequency modulation modes contained in the non-stationary signal, < >>Is->The instantaneous amplitude of the individual modes, and />Respectively +.>Instantaneous frequency and initial phase of each mode. Considering the unavoidable errors in estimating the unknown parameters, the instantaneous frequency can be rewritten as the sum of the estimated value and the error, i.e. < ->. In this embodiment we assume the instantaneous amplitude +.>And instantaneous frequency->Are not negative.

In connection with the triangular formula, formula (1) can also be expressed as:

(2)

wherein ,

(3)

(4)

instantaneous amplitudeAnd instantaneous frequency increment->Can be worry about> and />And (3) obtaining:

(5)

(6)

thus, the frequency modulation signal estimation can be converted into and />Parameter estimation of (a)Problems.

Step (1-2) constructing a sparse reconstruction framework of joint optimization:

based on nonlinear frequency modulation mode estimation, a joint optimization method is used for identifying signal modes simultaneously, and the following sparse optimization model is constructed:

(7)

wherein the first term ensures accuracy of the parameter estimation by limiting residual noise energy and the second term uses the second derivativeNorms to enhance the smoothness of the instantaneous amplitude, penalty coefficients +.>For adjusting the relative weight between the cost function and the desired sparsity. Equation (7) can be represented in a simplified form as a matrix:

(8)

wherein ,

is->Block diagonal matrix of (2), wherein->Is a second order differential matrix:

it can be seen that the minimization problem of the above formula can be seen as a generalized LASO problem, which can be converted to a general LASO problem for further solution. Due toIs a matrix of line full rank, first find an AND/OR>Orthogonal matrix->Constructing a full-rank square matrix ++>We can then transform the variable to +.>The optimization problem can be equivalently:

(9)

order theThen->So the above formula is about->Solution of (2)Is given by the following linear regression:

(10)

substituting the above formula into formula (10) to obtain the estimationIs described in the general LASSO equation:

(11)

wherein ,，/>，/>is a unitary matrix, and->. Solution to the above equation we will further discuss in the following section, finally we can get the solution to equation (9) by inverse transformation: />。

The signal estimation is completed in an iterative modeThen, using the instantaneous frequency information contained in (3) and (4), updating the instantaneous frequency increment in combination with the arctangent demodulation technique:

(12)

in general, the instantaneous frequency is a smooth function, and in order to avoid numerical errors caused by discrete time sampling, the instantaneous frequency increment can be modified by a low-pass filter:

(13)

wherein ,,/>filter parameters->Is a second order differential matrix. Thus, the instantaneous frequency can be updated as:

(14)

then use the estimated instantaneous frequencyTo update dictionary matrix->. The algorithm will iteratively update in this way until the stopping criterion is reached.

Step (2): relaxed compressed sensing based on Lp norm constraints:

the compressed sensing theory can sample signals at a rate lower than the nyquist sampling rate by using a specific measurement matrix, meanwhile, the compression of the signals is completed, then, the high-precision original signal reconstruction is realized based on the measured value of the signals, and the reconstruction of sparse signals can also be directly realized. Algorithm for signal reconstruction by minimizing vector according to sparse characteristics of signalThe norm, i.e., the number of non-zero terms in the vector, solves the minimization model:

(15)

whileThe problem of norm minimization is a typical NP-hard problem, and current algorithms cannot guarantee convergence to global optimum, are not easy to implement, and are relatively unstable. Donoho and Chen et al demonstrate +.A condition where the transformation matrix is uncorrelated with the observation matrix>Norms and +.>The solution of the norms is equivalent, so that the problem of difficult optimization of NP can be translated into +.>The norm convex optimization problem, or called the Base Pursuit (BP) problem:

(16)

can be conveniently converted into a linear programming problem to solve,convex optimization of norms the compressed sensing framework becomes the main method of sparse reconstruction, and for noisy situations, the problem is generalized to the problem of base-track denoising (Basis Pursuit Denoise, BPDN):

(17)

wherein ,is the noise level of the data. Or +.>LASSO problem under norm constraint:

(18)

wherein ,is a regular term coefficient. />Convex optimization under norms compressed perceptual restoration frameworks have received extensive attention in many fields and a series of algorithms have been proposed. However, is->Norms and +.>The condition of the norm optimizing solution equivalence is not easy to judge, and the condition is only in strict condition +.>Norms and +.>The optimized result of the norm has equivalence, and the general actual signal is difficult to meet. In general, a->The difference between the norm and the true sparse solution is too large, and the sparse position cannot be distinguished, so that although the reconstructed signal approaches the original signal on the Euclidean distance as a whole, the phenomenon of position confusion exists, and unexpected artificial effects are easy to occur. One natural improvement is the relaxed compressed perceptual framework under Lp norm, which the optimization objective (11) can be rewritten as:

(19)

wherein In contrast, the non-convex problem at Lp norms does not require such harsh conditions to ensure sparsity of the recovered signal.

Step (3): regularized FOCUSS algorithm:

the FOCUSS algorithm is proposed to overcome the defects of relatively scattered energy and lower resolution inherent to the pseudo-inverse method of the minimum two norms. The core idea is to construct a weighting function by using the result obtained by the last iteration estimation, so that the energy of the new estimation result obtained by the current iteration is more concentrated. The algorithm comprises two steps: first pass through minimumThe norm obtains a sparse signal low resolution solution, and then the sparse solution is obtained through affine scale transformation. The basic FOCUSS algorithm is based on weighted least-squares solution, and its cost function is:

(20)

in the formula ,，/>is a diagonal matrix. First, initializing a full 1 array with the same dimension as the array receiving matrixReconstructing an original signal through the following iterative solving process:

1)；

2)；

3)。

when no noise exists, the basic FOCUSS algorithm can obtain a robust estimated value, however, because pseudo-inverse operation is often pathological under the condition of noise, noise cannot be suppressed, noise can be amplified seriously, and the method cannot be obtainedIs an efficient estimate of (a). D.rao et al combine the function algorithm with Lp-norm based sparsity metrics, suggesting that regularized function algorithm is used for sparse signal reconstruction in noisy environments. By introducing regularization term->Noise effects can be reduced, making the system more robust, where the cost function is defined as:

(21)

in the formula ,is a punishment parameter; />Is Lp norm>. The iterative process of the regularized FOCUSS algorithm is as follows:

1)；

2)；

3)；

4)。

the application can realize sparse compressed sensing estimation by utilizing the regularized FOCUSS algorithm, can effectively inhibit noise and improve estimation performance.

In this embodiment, the reconstruction algorithm needs to initialize parameters, where the parameters to be initialized include: signal signalTermination parameter->Penalty coefficient->Number of modes of signal->Maximum number of iterations->。

The present application is not limited to the preferred embodiments described above, but various equivalent modifications and substitutions can be made by those skilled in the art without departing from the spirit of the present application, and these modifications are included in the scope of the present application as defined in the appended claims.

Claims

1. The fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction is characterized by comprising a non-stationary signal sparse reconstruction model, a joint optimization mode and a relaxation compressed sensing based on Lp norm constraint, wherein:

the sparse reconstruction model of the nonstationary signal is characterized in that a multi-frequency modulation mode mathematical model of the signal is utilized to convert signal estimation into parameter estimation, smoothness of constraint estimation parameters of a sparse optimization framework is constructed, and instantaneous amplitude and instantaneous frequency of the signal can be directly estimated by adjusting sparsity of weight parameter balance estimation parameters and fitting degree of measured values;

the joint optimization mode is that all components of a signal are identified in a fitting item of a constructed sparse optimization framework at the same time, and the accuracy of parameter estimation is ensured by limiting residual noise energy;

the relaxed compressed sensing based on Lp norm constraint means that the smoothness of the instantaneous amplitude of an estimated signal is controlled by minimizing the Lp norm of the second derivative of the estimated parameter, the sparsity of the estimated parameter is flexibly controlled by adjusting the parameter p epsilon (0, 1), and the relaxed compressed sensing has better robustness for outliers and noise and better maintains the structural characteristics of the signal while maintaining the sparsity;

the fault diagnosis method comprises the following steps:

(1) Constructing a sparse optimization framework of joint optimization based on a multi-frequency modulation mode mathematical model of a non-stationary signal;

step (1-1) non-stationary signal mathematical model:

the mathematical model of the non-stationary signal is expressed as:

wherein ,indicates the number of frequency modulation modes contained in the non-stationary signal, < >>Is->Instantaneous amplitude of individual modes-> and />Respectively +.>Instantaneous frequency and initial phase of each modality;

in view of the unavoidable errors in estimating the unknown parameters, the instantaneous frequency can be rewritten as the sum of the estimated value and the error, that is,；

assuming instantaneous amplitudeAnd instantaneous frequency->All non-negative, combined with the triangular equation, the mathematical model of the non-stationary signal can also be expressed as:

wherein ,

thus, the frequency modulation signal estimation can be converted into and />A parameter estimation problem of (2);

step (1-2) constructing a sparse optimization framework of joint optimization:

based on nonlinear frequency modulation mode estimation, a joint optimization method is used for identifying signal modes simultaneously, and a sparse optimization model is constructed:

wherein the first term ensures accuracy of the parameter estimation by limiting residual noise energy and the second term uses the second derivativeNorms to enhance the smoothness of the instantaneous amplitude, penalty coefficients +.>For adjusting the relative weight between the cost function and the desired sparsity;

designing a relaxed compressed perceptual framework based on Lp norms:

wherein p is E (0, 1);

(3) Sparse estimation of fault signals in a noise environment is carried out by using a regularized FOCUSS algorithm;

the FOCUSS algorithm includes two steps: firstly, obtaining a sparse signal low-resolution solution through a minimum norm, and then obtaining the sparse solution through affine scale transformation.

2. The fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction of claim 1, wherein the sparse optimization of the non-stationary signal directly estimates all components of the signal by adopting a joint optimization mode and constructs high-quality time-frequency distribution, and then selects a proper parameter p through an Lp norm constraint relaxation compressed sensing optimization algorithm to control the sparsity and accuracy of the reconstructed signal.

3. The fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction of claim 1, wherein in step (1), firstly, the non-stationary signal is represented as a multi-frequency modulation mode mathematical model, and the frequency modulation signal estimation is converted into a parameter estimation problem; then, the sparsity of the second derivative of the estimation parameter is restrained to enhance the smoothness of the instantaneous amplitude, and a joint optimization technology is adopted to construct a sparse optimization framework, so that the instantaneous amplitude and the instantaneous frequency of the signal can be directly estimated, the signal components can be simultaneously extracted, and the high-quality time-frequency distribution can be constructed.

4. The fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction according to claim 1, wherein in step (2), the sparse optimization framework of step (1) is designed as a relaxed compressed sensing sparse optimization framework based on Lp norm constraint, the sparsity can be flexibly controlled by adjusting the parameter p, and the fault diagnosis method has better robustness to outliers and noise and better retains structural characteristics of signals while maintaining the sparsity.

5. The fault diagnosis method based on Lp norm non-stationary signal sparse reconstruction according to claim 1, wherein in step (3), when the reconstruction signal in step (2) is solved by adopting a facility algorithm, a weighting function is constructed by using the result obtained by the last iteration estimation, so that the new estimation result obtained by the current iteration is more concentrated in energy; the influence of noise is reduced by introducing regularization term, so that the reconstruction system is more robust in a noise environment, and the problem that effective estimation cannot be obtained because the pseudo-inverse operation is often in a pathological state under the noisy condition and noise cannot be suppressed and the noise is seriously amplified is effectively avoided.