Permanent magnet synchronous motor bearing fault diagnosis method based on motor current analysis method

Abstract

A kind of permanent magnet synchronous motor bearing fault diagnosis method based on motor current analysis method, first acquisition permanent magnet synchronous motor stator U, V phase current signal, calculate W phase current;Three-phase current is obtained by space vector dimension reduction two-phase current under rectangular coordinate system, and two-phase current is carried out vector normalization processing;Based on the mechanism of action of bearing fault to current signal and the characteristics of current signal, the signal after space vector dimension reduction is carried out variation modal decomposition, the approximate entropy of modal component is calculated and the feature matrix is formed;The improved standard artificial bee colony algorithm makes the bee colony type mutually transform according to the optimal solution, and changes the bee colony initialization population generation rule, the optimal classification model is obtained by using the optimized adaptive variable type algorithm to optimize machine learning model parameter;Using optimal model, the test set in sample set is classified according to fault type, and compared with label to obtain test set accuracy rate;The present application has the advantages of high accuracy, high diagnostic efficiency and the like.

Description

Technical Field

[0001] This invention belongs to the field of bearing fault diagnosis technology, specifically relating to a method for diagnosing bearing faults in permanent magnet synchronous motors based on motor current analysis. Background Technology

[0002] In recent years, driven by the continuous development of CNC machine tools, electronic manufacturing equipment, industrial robots, medical equipment, and transportation, the servo motor market has maintained a growth trend. Permanent magnet synchronous motors, with their advantages of high power density, high efficiency, high torque-to-inertia ratio, wide-range constant power speed ratio, low vibration, and low noise, have gradually replaced DC servo motors as the main development target for contemporary high-performance servo motors. Bearing failure, as the most common type of mechanical failure in motors, has always been a major research focus for many scholars. Therefore, fast, accurate, and reliable methods for diagnosing motor bearing failures are of great significance for the service life, overall performance, and safe and stable operation of mechanical equipment.

[0003] Currently, most diagnostic methods for motor bearings rely on invasive methods to collect vibration, temperature, and sound signals (Song Xiangjin, Zhao Wenxiang. A review of rolling bearing fault diagnosis methods based on AC motor signal feature analysis [J]. Proceedings of the CSEE, 2022.). However, these signals require additional sensors installed outside the mechanical equipment for collection. Installing additional sensors not only increases diagnostic costs, but also, given that permanent magnet synchronous motors are mostly used in CNC machine tools, industrial robots, medical equipment, etc., in many cases, due to limitations such as reliability, number of mechanical equipment, sensor detection costs, and operational safety, it is impossible to install sensors outside the motor or mechanical equipment for fault signal collection. Furthermore, commonly used vibration and sound signals are easily affected by external environmental factors, resulting in excessive environmental noise mixed into the collected signals, thus affecting the signal processing process and the final fault diagnosis results (Kang Wei, Zhu Yongsheng, et al. Weak fault feature extraction of rolling bearings based on CSES and MED [J]. Vibration, Testing and Diagnosis, 2021.). Therefore, in order to more conveniently, quickly, accurately, and reliably diagnose motor bearing fault types, it is urgent to develop a non-invasive method for collecting fault signals for motor bearing fault diagnosis. Summary of the Invention

[0004] In order to overcome the shortcomings of the prior art, the present invention aims to provide a method for detecting bearing faults in permanent magnet synchronous motors based on servo driver current signals. This method is less affected by the external environment, has low detection cost, is convenient and fast, safe and reliable, and has high accuracy.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for diagnosing bearing faults in a permanent magnet synchronous motor based on motor current analysis includes the following steps:

[0007] Step 1: Simultaneously acquire the stator U and V phase current signals of the permanent magnet synchronous motor and calculate the W phase current. For the obtained U, V and W three-phase currents, use the space vector preprocessing method to perform data preprocessing operations on the three-phase currents.

[0008] Step 2: Based on the mechanism of bearing failure on current signal and the characteristics of three-phase current of permanent magnet synchronous motor, the variational mode decomposition method is used to decompose the current signal. The number of mode decompositions K is set according to the corresponding operating conditions, and the bandwidth α of the decomposed mode components is determined according to prior knowledge.

[0009] Step 3: Perform mode decomposition on the current data after space vector preprocessing in Step 1 according to the parameters determined in Step 2 to obtain sample data containing fault information with dimensions increased to a specified number.

[0010] Step 4: Based on the characteristic that the current signal in the time domain graph is in the form of a periodic sine wave, it is believed that the similarity of the current loop images obtained by vector synthesis of two-phase currents under the same working conditions and bearing conditions should be close. However, the current loops of different bearing conditions should be different due to the influence of fault conditions. Therefore, the approximate entropy is used as the classification feature set to reflect the difference between the probability of mutual approximation of the current loop images represented by m adjacent points in the sample sequence and the probability of mutual approximation of the current loop images represented by m+1 points.

[0011] Step 5: Improve the standard Artificial bee colony algorithm (ABC) and propose an adaptive variable type Artificial bee colony algorithm (AVTABC), so that the initial population consists entirely of scout bees, and the three types of bee colonies switch between each other according to the optimal solution.

[0012] Step 6: Using the validation set accuracy as the fitness function, the AVTABC algorithm described above is used to optimize the key parameters in the machine learning classification model based on the globally optimal fitness value, so as to obtain the model parameters with the best classification performance.

[0013] Step 7: Input the test set into the optimal classification model for testing, and calculate the accuracy of the diagnosis based on the labels of the bearing classification.

[0014] The beneficial effects of this invention are as follows:

[0015] This invention enables non-invasive signal acquisition of mechanical equipment by collecting the current signal of the motor servo driver, without the need for additional external sensors. This acquisition method is convenient and quick to operate, highly safe and reliable, has low diagnostic costs, and is less affected by external factors, thereby simplifying the subsequent fault data processing and improving diagnostic results.

[0016] This invention is based on a data preprocessing method for dimensionality reduction and normalization of current signal spatial vectors and a feature dimensionality enhancement and approximate entropy extraction method for mode decomposition. It is simple to operate, can better reflect the influence of current signals on operating conditions and fault conditions, and has good feature classification effect. It can effectively solve the problem that the influence of mechanical faults on current signals is not obvious, and further improve the fault classification accuracy.

[0017] This invention proposes an artificial bee colony algorithm (AVTABC) that can adaptively switch bee colony types based on the optimal solution. This method improves the algorithm's global optimization ability and convergence speed, and further enhances the overall diagnostic efficiency and effectiveness of the algorithm. Attached Figure Description

[0018] Figure 1 This is a flowchart of the present invention.

[0019] Figure 2 This is a trajectory diagram of the two-phase current vector synthesis after spatial vector dimensionality reduction in an embodiment of the present invention.

[0020] Figure 3 This is an iterative diagram of optimizing a Gaussian kernel SVM model using the AVTABC algorithm based on data from a 1200rpm variable load sample set, according to an embodiment of the present invention.

[0021] Figure 4 This is the confusion matrix after model classification based on 1200rpm variable load sample test set data in an embodiment of the present invention. Detailed Implementation

[0022] The present invention will now be described in further detail with reference to the accompanying drawings.

[0023] Reference Figure 1 A method for diagnosing bearing faults in permanent magnet synchronous motors based on motor current analysis includes the following steps:

[0024] Step 1: Collect the U and V phase currents from the stator current in the permanent magnet synchronous motor servo driver, and calculate the value of the W phase current according to the formula as follows:

[0025] I W =-I U -I V (1)

[0026] In the formula, I UI V I W These represent the three-phase currents U, V, and W, respectively.

[0027] For the obtained U, V, and W three-phase currents, a space vector preprocessing method is used to perform data preprocessing operations on the three-phase currents; the specific data preprocessing method is as follows:

[0028] First, the three-phase currents in the three-dimensional coordinate system are transformed into two-phase currents in the rectangular coordinate system using a space vector dimension reduction method. The transformation formula is as follows:

[0029]

[0030] In the formula I α I β These represent the two-phase currents after the dimension reduction of the three-phase current vectors U, V, and W, respectively.

[0031] For the two-phase currents mentioned above, their vector magnitudes are calculated using the following formula:

[0032]

[0033] Based on the obtained current vector magnitude, the two-phase signals are vector normalized using the following formula:

[0034]

[0035] Step 2: Based on the mechanism of bearing faults affecting current signals and the characteristics of three-phase current in a permanent magnet synchronous motor, specifically that for a three-phase balanced motor system, even harmonics are eliminated in the current signal after Fourier transform, and the current signal mainly represents the fundamental frequency and its odd harmonic modulation components; according to the phase modulation principle of bearing faults on motor phase currents, it can be known that the fault information of the stator current is distributed on both sides of the current fundamental frequency and its odd harmonic modulation components. Therefore, mode decomposition is adopted to enhance the feature dimension. The components after mode decomposition represent the current fundamental frequency and its odd harmonics, respectively, as described below:

[0036] Step 2.1: Based on the characteristics of the current signal as a superposition of the fundamental frequency and odd harmonic components, and the Shannon sampling theorem which states that in order to recover the signal without distortion, the sampling frequency should be greater than or equal to twice the highest frequency in the required signal spectrum, the number of mode decompositions K under different operating conditions can be calculated according to the following formula, combining the characteristics of the current signal and the Shannon sampling theorem.

[0037]

[0038] In the formula f e The fundamental frequency of the current; F s K is the sampling frequency; K is the number of decompositions;

[0039] In the above formula, the fundamental frequency of the current is f e The calculation formula is as follows:

[0040]

[0041] In the formula, n is the motor speed; p is the number of pole pairs of the rotating magnetic field of the motor;

[0042] Step 2.2: According to prior knowledge, the penalty factor α in mode decomposition determines the bandwidth of the IMF component. The common value range is 1000 to 3000. Here, we choose 2500.

[0043] Step 3: Based on the K value determined in Step 2, decompose the original signal into K modal components. Each modal component contains a center frequency. These K center frequencies, from low to high, represent the fundamental frequency of the current and the modulation components of its odd harmonics. At this point, each modal component contains information about the bearing's current state. The modal decomposition process is as follows:

[0044] Step 3.1: Variational Mode Decomposition. First, the intrinsic mode components are defined as an amplitude-modulated and frequency-modulated signal u. k (t), whose expression is as follows:

[0045]

[0046] In the formula A k (t) represents the instantaneous amplitude; t represents phase; t represents time.

[0047]

[0048] In the formula ω k (t) is u k The instantaneous frequency of (t);

[0049] Step 3.2: For each u k Performing a Hilbert transform on (t) yields its analytic signal as shown below:

[0050]

[0051] In the formula, δ(t) represents the Dirac distribution, and j is the imaginary unit;

[0052] Step 3.3: Add an exponential term to the signal obtained in Step 3.2 to adjust the center frequency band of each estimated modal component spectrum to the corresponding baseband:

[0053]

[0054] In the formula, The estimated center frequency for each analytical signal;

[0055] Step 3.4: Calculate the squared gradient norm 2 of the demodulated signal obtained in Step 3.3, and obtain the variational constraint model:

[0056]

[0057] In the formula, K is the number of IMF components; To find the partial derivative with respect to t, {u k}={u1,....u k} represents the final components obtained from the decomposition; {ω k}={ω1,....,ω k} represents the set of actual center frequencies of each IMF component; x(t) represents the original signal;

[0058] Step 3.5: By introducing a quadratic penalty factor α and a Lagrange multiplier λ(t), the above variational constrained model can be converted into an unconstrained model:

[0059]

[0060] The update formula for λ(t) is as follows:

[0061]

[0062] In the formula, τ represents the noise tolerance of the signal; n represents the number of iterations;

[0063] Step 3.6: By continuously updating using the alternating direction multiplier method, the optimal solution for each component is finally obtained; for each optimal solution obtained and the given precision ε, it is determined whether the stopping condition is met. If the stopping condition is met, the update ends; otherwise, the loop continues, and K components are output; the stopping condition expression is as follows:

[0064]

[0065] Step 4: Based on the characteristic that current signals exhibit a periodic sine wave form, meaning that the current loop images obtained after vector uniqueness processing for two-phase currents under the same operating conditions and bearing states should have similarity, but different bearing states are affected by fault conditions, and their current loops should have certain differences. Therefore, approximate entropy is used as the classification feature set to reflect the difference between the probability of mutual approximation of the time-domain images represented by m adjacent points of the sample sequence and the probability of mutual approximation of the time-domain images represented by m+1 points. The approximate entropy is calculated for the K IMF mode components after decomposition of each sample, forming a feature vector matrix. The specific implementation method is as follows:

[0066] Step 4.1: Assume that each sample is an N-dimensional time series x(1), x(2), ..., x(N) composed of continuous points. Define the relevant parameters m and r of the algorithm, where m is an integer representing the embedding dimension and r is a real number representing the similarity tolerance coefficient.

[0067] Step 4.2: Reconstruct the m-dimensional vectors X(1), X(2), ..., X(N-m+1), where X(i) = [x(i), x(i+1), ..., x(i+m-1)];

[0068] Step 4.3: Define the distance d[x(i),x(j)] between x(i) and the other vectors x(j) as the maximum difference between their corresponding elements, i.e.:

[0069]

[0070] Step 4.4: When 1≤i≤N-m+1, for each value of i, count the number of distances d less than r according to the given similarity tolerance coefficient r, and calculate the ratio of this number to the total number of distances Nm, denoted as [missing information]. Right now:

[0071]

[0072] Step 4.5: [Regarding...] Find the logarithm, then find the average, denoted as:

[0073]

[0074] In the formula Φ m (r) is the approximate entropy value with embedding dimension m obtained;

[0075] Step 4.6: Solve for the approximate entropy of this time series:

[0076] ApEn = Φ m (r)-Φ m+1 (r) (18)

[0077] Step 5: Improve the standard Artificial bee colony algorithm (ABC) and propose an adaptive variable type Artificial bee colony algorithm (AVTABC). Set the initial population to consist entirely of scout bees, and the three types of bee colonies can be converted to each other according to the optimal solution.

[0078] Step 6: Using the training set validation accuracy as the fitness function, optimize the key parameters in the machine learning classification model using the improved AVTABC algorithm to obtain the model parameters that achieve the best classification performance, as follows:

[0079] Step 6.1: Initialize the population, with a total population of s and a maximum number of hired bees of m;

[0080] Step 6.2: Set the initial s bees to be scout bees. Select several better solutions from the optimal solutions found by these scout bees, convert these scout bees into mercenary bees, and convert the other bees into follower bees.

[0081] Step 6.3: Calculate the training set validation accuracy at each optimal solution and obtain the model parameters corresponding to the highest accuracy.

[0082] Step 6.4: After each iteration, repeat the above steps (Steps 6.1-6.3) until the maximum number of iterations is reached, and output the optimal model parameters;

[0083] Step 7: Based on the optimized model parameters from Step 6, set up the model, input the test set into the model, compare the output results with the true labels, calculate the test set accuracy, and verify the feasibility of the diagnostic method.

[0084] The present invention will be further described in detail below with reference to the embodiments.

[0085] In this embodiment, to verify the feasibility of using current signals to diagnose motor bearing fault types, an Huichuan 750W permanent magnet synchronous motor was used for experimental verification. The motor has 5 pole pairs, and the motor rotation frequency is 5 times the current fundamental frequency, which can be obtained from equation (6). There are 15 experimental conditions, namely 5 speed conditions and 3 load conditions, and 5 sets of data are collected for each condition. The data acquisition system is used to collect motor current data under 15 conditions, namely normal, outer ring fault, inner ring fault and rolling element fault. The sampling frequency is set to 6250Hz, the length of each sample data is 7200, the number of samples under each condition is 60, and the ratio of training set to test set is 3:1.

[0086] For the three-phase currents U, V, and W in the decomposed sample, a space vector dimensionality reduction method is used to transform them into two-phase currents in a rectangular coordinate system. After obtaining the vector magnitudes of the two-phase current signals, vector normalization is performed on the two-phase currents. At this point, the two-phase signals can effectively reflect the fault characteristics, and the current loop diagram can be obtained by plotting them. Figure 2 The diagram shows the current loop diagrams for four bearing states under no-load and 1200rpm conditions in this embodiment.

[0087] Considering that the maximum speed under experimental conditions is 2500 rpm, the minimum value of the number of decompositions K can be obtained from equation (5) as 8. Therefore, we set α = 2500 and K = 8 in the variational mode decomposition parameters.

[0088] A sample set was created using variable load data under operating conditions of 1200 rpm as an example. The specific details are shown in Table 1.

[0089] Table 1. Sample Set Description

[0090]

[0091] For each sample in the sample set, perform variational mode decomposition according to step 3. After each sample is decomposed, eight 7200*1 IMF components can be obtained. For all IMF components of all the above samples, calculate the approximate entropy according to step 4. Finally, a 720*8 dimensional feature matrix can be obtained. After randomly shuffling the order, 75% is taken as the training set and 25% is taken as the test set.

[0092] The validation method employed was five-fold cross-validation. The classification model used was a support vector machine (SVM) model with a Gaussian kernel. The validation set accuracy was used as the fitness function. The AVTABC algorithm was employed to optimize the penalty coefficient C and the width g of the Gaussian kernel in the SVM model. After obtaining the optimal parameters, the model parameters were modified, and the optimal model was derived. Figure 3 The diagram shows the iterative process of optimizing SVM parameters using the AVTABC algorithm in this embodiment.

[0093] The fault types were classified into five variable load sample sets under five speed conditions and three variable speed sample sets under three load conditions using the method described above. The results are shown in Table 2. Figure 4 The image shows the confusion matrix after classifying the test set using the optimized model in this embodiment.

[0094] Table 2. Accuracy of the 5-speed variable load sample set

[0095]

[0096] This invention proposes a method for bearing fault diagnosis based on the current signal of a permanent magnet synchronous motor driver. First, the three-phase motor signal undergoes spatial vector dimensionality reduction and vector magnitude normalization preprocessing. Then, variational mode decomposition is used to enhance the feature dimension, and approximate entropy values ​​are calculated as classification features. This method is convenient and fast, and can effectively reflect the fault characteristics in the current signal. Simultaneously, an improved adaptive variable-type artificial bee colony algorithm is used to optimize key parameters in the machine learning classification model. This optimization is highly efficient, has strong global search capabilities, is less prone to getting trapped in local optima, and can improve the model's classification performance to a certain extent. When diagnosing samples under varying operating conditions, this method exhibits high accuracy, good stability, and minimal susceptibility to external environmental influences. It also effectively addresses practical engineering challenges such as the difficulty of installing external sensors, high diagnostic costs, and the impact of the diagnostic process on equipment safety.

Claims

1. A method for bearing fault diagnosis of permanent magnet synchronous motor based on motor current analysis method, characterized in that, The method comprises the following steps: Step 1: simultaneously collecting stator U-phase and V-phase current signals of the permanent magnet synchronous motor, and calculating W-phase current, and performing data preprocessing on the obtained U-phase, V-phase and W-phase currents by using a space vector preprocessing method; Step 2: according to the action mechanism of bearing faults on current signals and the characteristics of three-phase currents of the permanent magnet synchronous motor, the current signals are decomposed by using a variational mode decomposition method, the number K of mode decompositions is set according to corresponding working conditions, and the bandwidth a of the mode components after decomposition is determined according to prior knowledge; Step 3: the current data after the space vector preprocessing in step 1 are mode-decomposed according to the parameters determined in step 2, and sample data containing fault information are obtained by increasing the dimension to a specified number; Step 4: based on the characteristics that the current signals in the time domain graph are in the form of a periodic sinusoidal wave, it is considered that the similarity of current loop images obtained by vector synthesis of two-phase currents of the same bearing state under the same working condition should be close, but the current loops of different bearing states are affected by faults and should be different, therefore, approximate entropy is used as a classification feature set to reflect the difference between the mutual approximation probability of the current loop images represented by m adjacent points and the mutual approximation probability of the current loop images represented by m+1 points; Step 5: an adaptive variable type artificial bee colony algorithm AVTABC is proposed by improving the standard artificial bee colony algorithm, so that all the scout bees are generated in the initial population, and the three types of bee colonies are switched according to the optimal solution; Step 6: the accuracy rate of the verification set is used as the fitness function, the key parameters in the machine learning classification model are optimized based on the global optimal fitness value by using the AVTABC algorithm, so as to obtain the model parameters with the optimal classification effect; the specific process is as follows: Step 6.1: initialize the population, the total number is s, and the upper limit of the employed bees is m; Step 6.2: set the initial s bees as all scout bees, select a number of better solutions from the optimal solutions searched by the scout bees, convert the part of the scout bees into employed bees, and convert the other bees into follower bees; Step 6.3: calculate the training set verification accuracy rate at each optimal solution, and obtain the model parameters corresponding to the highest accuracy rate; Step 6.4: after each iteration is completed, repeat steps 6.1-6.3 until the maximum number of iterations is reached, and output the optimal model parameters; Step 7: input the test set into the optimal classification model for testing, and calculate the accuracy rate of diagnosis according to the labels of the bearing classification.

2. The method of claim 1, wherein, The formula for calculating the W-phase current in step 1 is as follows: (1) wherein , , respectively represent the U, V, W three-phase currents; The data preprocessing method used for the obtained U-phase, V-phase and W-phase currents is as follows: Firstly, the three-phase currents in the three-dimensional coordinate system are converted into two-phase currents in the rectangular coordinate system by using the space vector dimension reduction method, and the conversion formula is as follows: (2) In the formula , respectively represent two-phase currents after dimension reduction of U, V, W three-phase current vectors. For the above two-phase currents, the vector module length is calculated, and the calculation formula is as follows: (3) For the obtained current vector module length, the two-phase signals are subjected to vector normalization processing, and the calculation formula is as follows: (4)。 3. The method of claim 1, wherein, The specific process of step 2 is as follows: Step 2.1: according to the Shannon sampling theorem, the sampling frequency should be greater than or equal to 2 times the highest frequency in the required signal spectrum, therefore, the number K of mode decompositions under different working conditions is calculated according to the following formula: (5) where f e is the fundamental frequency of the current; F s is the sampling frequency; K is the number of modal decompositions; In the above formula, the current fundamental frequency f e The calculation formula is as follows: (6) Where n is the motor speed; p is the pole pair number of the motor rotating magnetic field; Step 2.2: The penalty factor alpha in modal decomposition determines the bandwidth of the IMF component according to prior knowledge, and the value range is 1000~3000.

4. The method of claim 1, wherein, Step 3: The modal decomposition process is as follows: Step 3.1: The variational modal decomposition first defines the intrinsic modal component as an amplitude-frequency modulation signal, and the expression is as follows: (7) wherein is the instantaneous amplitude; is the phase; t is time; (8) wherein is instantaneous frequency; Step 3.2: For each The Hilbert transform is performed to obtain its analytic signal as follows: (9) wherein is a Dirac distribution, j is the imaginary unit; Step 3.3: Add an exponential term to the signal obtained in step 3.2 to adjust the center frequency band of the estimated modal component spectrum to the corresponding baseband: (10) In the formula, is the estimated center frequency for each resolved signal; Step 3.4: Calculate the gradient two-norm square of the demodulation signal obtained in step 3.3, and obtain the variational constraint model: (11) Where K is the number of modal decomposition; {u k}={u 1，…. u k} as the final component; as the actual center frequency of each IMF component; x(t) as the original signal; Step 3.5: Introducing a quadratic penalty factor a and a Lagrange multiplier The above variational constrained model can be converted to an unconstrained model: (12) In the formula The update formula for is as follows: (13) In the formula is the noise tolerance of the signal; n is the number of iterations; Step 3.6: The optimal solution of each component is finally obtained by constantly updating using the alternating direction multiplier method; for the optimal solution of each component obtained and the given precision , it is judged whether the stop condition is met, and if the stop condition is met, the updating is ended, otherwise the loop is continued, and the K components obtained are output; the stop condition expression is as follows: (14)。 5. The method of claim 1, wherein, Step 4: The specific process is as follows: Step 4.1: Assume that each sample is an N-dimensional time series composed of continuous points x(1), x(2), …, x(N), and define the related parameters m and r of the algorithm, where m is an integer, representing the embedding dimension, and r is a real number, representing the coefficient of the similarity tolerance; Step 4.2: Reconstruct the m-dimensional vector X(1), X(2), …, X(N-m+1), where X(1)=[ x(1), x(1+1), …, x(1+m-1)], X(2)=[ x(2), x(2+1), …, x(2+m-1)], … and so on; Step 4.3: Define the distance d[x(i), x(j)] between x(i) and the rest of the vector x(j) as the maximum difference between the corresponding elements, that is: (15) Step 4.4: When For each value of i, the number of distances d that are less than the given similarity tolerance factor r is counted, and the ratio to the total number of distances N-m is computed, denoted as That is: (16) Step 4.5: [Regarding...] Find the logarithm, then find the average, denoted as: (17) In the formula That is, the approximate entropy value of the embedded dimension m is obtained. Step 4.6: Solve the approximate entropy of the time series: (18)。

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