Lithium ion battery two-stage sorting method based on entropy weight method and fuzzy C-means clustering

Abstract

The invention relates to a lithium ion battery two-stage sorting method based on an entropy weight method and fuzzy C-means clustering, and belongs to the technical field of battery sorting. The method comprises the following steps: S1, calculating SOH and RUL of the battery according to a capacity degradation curve of the lithium ion battery; s2, performing standardization processing on the SOH and the RUL of the battery, and calculating information entropy and information entropy redundancy of the SOH and the RUL through an entropy weight method; s3, index weights of SOH and RUL are calculated according to the information entropy redundancy, comprehensive evaluation values of the battery samples are further calculated, and low-performance batteries are preliminarily screened according to the comprehensive evaluation values; and S4, completing re-sorting of the remaining batteries through fuzzy C-means clustering. According to the method, the problem of poor battery performance consistency caused by direct sorting can be solved, so that the resource utilization rate of subsequent echelon utilization batteries is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of battery sorting, and relates to a two-stage lithium-ion battery sorting method based on an entropy weight method and fuzzy C-means clustering. Background Art

[0002] Lithium-ion batteries are widely used in new energy vehicles and energy storage systems due to their high energy density and strong charge retention capabilities. When lithium-ion power battery packs cannot meet the driving safety and mileage requirements of new energy vehicles, they need to be retired. In the early days of new energy vehicle operation, the number of retired batteries was very small, but due to the limitations of early battery technology and their service life, the number of retired power batteries will increase significantly in the next two years. If retired batteries are not handled in a timely manner, serious economic losses and environmental pollution will occur. Secondly, the better-performing batteries among retired batteries can still be used in small-scale energy storage and portable power sources, improving resource utilization.

[0003] Proper battery sorting can reduce the mutual impact between individual cells and prevent rapid aging caused by overcharging and discharging of certain batteries, thereby extending the service life of the entire battery pack. At the same time, for retired batteries, precise sorting technology can assess their residual value and promote the battery's cascade utilization. The treatment of retired lithium-ion batteries is divided into two categories: disassembly and recycling, and cascade utilization. Cascade utilization focuses on testing the performance of retired battery packs and reusing them in areas with lower requirements. Disassembly and recycling, on the other hand, involves the rational disassembly of batteries that do not meet the requirements for cascade utilization, extracting internal metals and other materials for recycling and reuse. Factors such as overcharging and discharging during the use of lithium-ion batteries can directly affect the battery's service life. Therefore, effective sorting based on the battery's characteristic parameters is required before cascade utilization or disassembly and recycling.

[0004] However, capacity regeneration directly affects the calculation of the SOH metric during battery sorting. Sorting batteries solely based on the SOH metric can lead to the reorganization and reuse of batteries with significantly different performance, accelerating the aging of battery packs. Furthermore, the weighting of various sorting metrics can also directly affect the sorting results. Therefore, it is necessary to design a new lithium-ion battery sorting method to address these issues. Summary of the Invention

[0005] In view of this, the purpose of the present invention is to provide a two-stage sorting method for lithium-ion batteries based on entropy weight method and fuzzy C-means clustering, which can solve the problems of interference from capacity regeneration phenomenon, randomness of sorting index weights and engineering implementation difficulties in existing battery sorting technology, effectively deal with capacity regeneration phenomenon, obtain reasonable index weight distribution, and thus improve the battery sorting effect.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] A two-stage lithium-ion battery sorting method based on entropy weight method and fuzzy C-means clustering specifically comprises the following steps:

[0008] S1: Calculate the battery's state of health (SOH) and remaining useful life (RUL) based on the capacity degradation curve of the lithium-ion battery;

[0009] S2: Normalize the battery's SOH and RUL, and calculate the information entropy and information entropy redundancy of SOH and RUL using the entropy weight method;

[0010] S3: Calculate the index weights of SOH and RUL based on the information entropy redundancy, further calculate the comprehensive evaluation value of the battery samples, and preliminarily screen low-performance batteries based on the comprehensive evaluation value;

[0011] S4: The remaining batteries are sorted again by fuzzy C-means clustering (FCM). Further, in step S1, the expression for calculating the SOH of the battery is:

[0012] SOH=C t / C0×100%

[0013] Among them, C t and C0 represent the capacity corresponding to time t and the initial rated capacity respectively.

[0014] The expression for calculating the RUL of a battery is:

[0015] RUL=Cycle t -Cycle EOL

[0016] Among them, Cycle t and Cycle EOL They represent the number of cycles at time t and the number of cycles at the end of life respectively.

[0017] Furthermore, in step S2, the SOH and RUL of the battery are standardized, and the expressions are:

[0018]

[0019] Among them, w ij and y ij Represent the original and standardized index data of the jth battery sample, min(.) and max(.) are the minimum and maximum function operations, respectively. j Indicates the battery's SOH or RUL, both are positive indicators.

[0020] The information entropy and information entropy redundancy of SOH and RUL are calculated by entropy weight method, including: calculating y ij In the jth indicator (A j ) in the proportion (p ij ), the calculation formula is:

[0021]

[0022] Among them, S represents the total amount of information, when y ij When all are 0, then define p ij =1 / S;

[0023] The information entropy of the jth indicator (e j ) is calculated as:

[0024] The information entropy redundancy of the jth indicator (d j ) is calculated as: j =1-e j .

[0025] Furthermore, in step S3, the comprehensive evaluation value of the battery sample is calculated, and the expression is:

[0026]

[0027] Among them, Score i represents the comprehensive evaluation value of the i-th battery, ω j Indicates the indicator weight of battery SOH or RUL, d j It represents the information entropy redundancy of battery SOH or RUL, where R is the number of indicators.

[0028] Low-performance batteries are preliminarily screened based on the comprehensive evaluation value. Specifically, adaptive screening is performed using a percentage threshold, that is, batteries with the lowest percentage threshold comprehensive evaluation value are eliminated.

[0029] Furthermore, in step S4, the remaining batteries are sorted again by fuzzy C-means clustering, which specifically includes: constructing the objective function (F m (U,V)):

[0030]

[0031] 0<u ab <1,1≤a≤C,1≤b≤N

[0032] Among them, m is the fuzzy weighted index, U is the membership matrix, V is the cluster center matrix, N and C are the number of samples and the number of categories respectively, u ab Indicates the membership degree of the b-th battery sample to the a-th category, st is the constraint condition; (Dab ) 2 Represents the distance from the battery sample to the cluster center, and its expression is as follows:

[0033]

[0034] Among them, x b and v a denote the center of the b-th battery sample and the a-th category respectively, M is a symmetric positive definite matrix, (·) T Represents the transpose operation of the matrix;

[0035] Initialize the membership matrix, set the number of iterations l = 1, update the cluster center, first, find F m v a The partial derivative of and set it equal to 0:

[0036]

[0037] Then, D ab Substituting the expression into the above formula, we can further obtain:

[0038]

[0039] Since M is a symmetric positive definite matrix, further simplification yields the cluster center of the lth iteration as follows:

[0040]

[0041] Update the membership matrix, combine the st condition, and transform F m Convert to Lagrange multiplier method:

[0042]

[0043] Among them, L λ is the objective function that transforms the original constrained optimization problem into an unconstrained optimization problem, λ b is the Lagrange multiplier, corresponding to the constraint condition of the bth sample The multiplier of

[0044] Convert the above formula to u ab Find the partial derivative and set it equal to 0 to solve u ab The expression is:

[0045]

[0046] Combined with the st constraint, we can further obtain λ b The expression is:

[0047]

[0048] λ b Substitute the expression into u ab In the expression of , the membership degree of the lth iteration is:

[0049]

[0050] Set the iteration termination condition and make classification decisions; the termination condition expression is as follows:

[0051]

[0052] Among them, L max is the maximum number of iterations, ε u It is the membership convergence threshold, which is generally preset to a very small positive number.

[0053] The beneficial effects of the present invention are as follows: the present invention constructs an efficient and reliable two-stage sorting model for lithium-ion batteries. First, the two sorting indicators of health state (SOH) and remaining useful life (RUL) are calculated based on the battery capacity degradation curve; secondly, the information entropy and information entropy redundancy of SOH and RUL are calculated by the entropy weight method, and the weights of the two sorting indicators are further obtained; then, the comprehensive evaluation value of the battery is calculated, and low-performance batteries are preliminarily screened; finally, fuzzy C-means clustering (FCM) is used to complete the re-sorting of the remaining batteries, which can effectively deal with the capacity regeneration phenomenon and obtain a reasonable indicator weight distribution, thereby improving the battery sorting effect (the sorting effect of the proposed model is evaluated by calculating the standard deviation (SD) and coefficient of variation (CV) of battery groups of the same category). The sorting model constructed by the present invention can solve the problem of poor battery performance consistency caused by direct sorting, thereby improving the resource utilization rate of subsequent cascade utilization of batteries. Through experimental comparative analysis, it can be seen that the fuzzy C-means clustering adopted by the present invention has better sorting effect than the common hard clustering method, and improves the consistency of battery SOH and RUL.

[0054] Other advantages, objects, and features of the present invention will be described in part in the following description and, in part, will be apparent to those skilled in the art upon examination of the following description or may be learned from practice of the present invention. The objects and other advantages of the present invention may be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below with reference to the accompanying drawings, in which:

[0056] Figure 1 This is a flow chart of the two-stage lithium-ion battery sorting method based on the entropy weight method and fuzzy C-means clustering of the present invention;

[0057] Figure 2 The results of the preliminary screening of NASA batteries and Oxford batteries using the entropy weight method;

[0058] Figure 3 Final sorting results for NASA batteries;

[0059] Figure 4 This is the final sorting result of Oxford battery;

[0060] Figure 5 The standard deviation and coefficient of variation of different categories of batteries after sorting for NASA batteries;

[0061] Figure 6 The standard deviation and coefficient of variation of different categories of batteries after Oxford battery sorting. DETAILED DESCRIPTION

[0062] The following describes the embodiments of the present invention by means of specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic illustrations of the basic concept of the present invention, and the following embodiments and features in the embodiments can be combined with each other without conflict.

[0063] See also Figures 1 to 6 , an embodiment of the present invention provides a two-stage sorting method for lithium-ion batteries based on entropy weight method and fuzzy C-means clustering. The simulation software used in this embodiment is MATLAB2023b, and the battery test data of the National Aeronautics and Space Administration (NASA) and Oxford University (Oxford) are used. The retired single cells are simulated by the data of different charge and discharge cycles of the battery, and the battery sorting research is further carried out. The capacity of the selected battery is 70% to 80% of the rated capacity. NASA batteries use constant current and constant voltage charging and constant current discharging, and Oxford batteries use constant current charging and discharging. The flow chart of the model proposed in this embodiment is as follows Figure 1 As shown, the method specifically includes the following steps:

[0064] Step S1: Calculate the battery's state of health (SOH) and remaining useful life (RUL) based on the lithium-ion battery's capacity degradation curve. The battery's SOH is usually defined as the ratio of the current capacity to the rated capacity, and the RUL is usually defined as the difference in the number of cycles from the current moment to the end of life (EOL):

[0065] SOH=C t / C0×100%

[0066] RUL=Cycle t -Cycle EOL

[0067] Among them, C t and C0 represent the capacity corresponding to time t and the initial rated capacity respectively, Cycle t and Cycle EOL They represent the number of cycles at time t and the number of cycles at the end of life respectively.

[0068] Step S2: Standardize the original SOH and RUL of the battery, and calculate the information entropy and information entropy redundancy of SOH and RUL by entropy weight method, which specifically includes the following steps:

[0069] S21: Standardize the original SOH and RUL of the battery:

[0070]

[0071] Among them, w ij and y ij Represents the original and standardized index data of the i-th sample, min(.) and max(.) are the minimum and maximum function operations, respectively. j Indicates the battery's SOH and RUL, both are positive indicators.

[0072] S22: Calculate y ij In the jth indicator (A j ) in the proportion (p ij ):

[0073]

[0074] Among them, S represents the total amount of information, when y ij When all are 0, then define p ij =1 / S.

[0075] S23: For the jth indicator, its information entropy (e j ) is calculated as follows:

[0076]

[0077] S24: Information entropy redundancy (d j ) reflects the discrete degree of the indicator, and its expression is as follows:

[0078] d j =1-e j

[0079] When d j The larger , the greater the difference of the j-th indicator, thus providing more information.

[0080] Step S3: Calculate the index weights of SOH and RUL based on the information redundancy, further calculate the comprehensive evaluation value of the battery sample, and preliminarily screen low-performance batteries based on the comprehensive evaluation value, specifically including the following steps:

[0081] S31: Calculate the index weights of SOH and RUL by information entropy redundancy (ω j ):

[0082]

[0083] Among them, d j It represents the information entropy redundancy of battery SOH or RUL, where R is the number of indicators.

[0084] S32: Multiply the SOH and RUL by the corresponding indicator weights, and then sum them up to obtain the comprehensive evaluation value (Score) of the battery:

[0085]

[0086] Among them, Score i represents the comprehensive evaluation value of the i-th battery.

[0087] S33: Low-performance batteries are preliminarily screened out based on the comprehensive evaluation values. In order to adapt to the characteristics and distribution of different battery data sets, a percentage threshold is used for adaptive screening, that is, batteries with the lowest percentage threshold comprehensive evaluation values ​​are eliminated.

[0088] Figure 2 The results of the preliminary sorting of NASA batteries and Oxford batteries using the entropy weight method are shown. Figure 2 The SOH of the initially screened NASA batteries was less than 72%, and their RULs were less than 10 cycles. The SOH of the initially screened Oxford batteries was less than 73.5%, and their RULs did not exceed 8 cycles. These results demonstrate that the entropy weight method can effectively screen out low-performance batteries, thereby improving the reliability of subsequent battery sorting.

[0089] Step S4: The remaining batteries are sorted again by fuzzy C-means clustering (FCM), and the sorting effect of the proposed model is evaluated by calculating the standard deviation (SD) and coefficient of variation (CV), and comparing the silhouette coefficient (SC) and CH index (Calinski-Harabaz Index, CHI) of K-means clustering (K-means) and hierarchical clustering (HC). The specific steps include:

[0090] S41: Construct the objective function of fuzzy C-means clustering for residual battery sorting:

[0091]

[0092] 0<u ab <1,1≤a≤C,1≤b≤N

[0093] Among them, m is the fuzzy weighted index, U is the membership matrix, V is the cluster center matrix, N and C are the number of samples and the number of categories respectively, u ab Indicates the membership degree of the b-th battery sample to the a-th category, st is the constraint condition, (D ab ) 2 Represents the distance from the battery sample to the cluster center, and its expression is as follows:

[0094]

[0095] Among them, x b and v a denote the center of the b-th battery sample and the a-th category respectively, M is a symmetric positive definite matrix, (·) T Represents the transpose operation of a matrix.

[0096] S42: Initialize the membership matrix, set the number of iterations l = 1, update the cluster center, first, calculate F m v a The partial derivative of and set it equal to 0:

[0097]

[0098] Then, D ab Substituting the expression into the above formula, we can further obtain:

[0099]

[0100] Since M is a symmetric positive definite matrix, further simplification yields the cluster center of the lth iteration as follows:

[0101]

[0102] S43: Update the membership matrix, combine the st condition, and transform F m Convert to Lagrange multiplier method:

[0103]

[0104] Convert the above formula to u ab Find the partial derivative and set it equal to 0 to solve u ab The expression is:

[0105]

[0106] Combined with the st constraint, we can further obtain λ b The expression is:

[0107]

[0108] λ b Substitute the expression into u ab In the expression of , the membership degree of the lth iteration is:

[0109]

[0110] S44: Set the iteration termination condition and make classification decision. ab Tolerance (ε u >0) and the maximum number of iterations L max When any of the above conditions holds, the fuzzy C-means clustering stops iterating, otherwise l=l+1 and returns to S42. The termination condition expression is as follows:

[0111]

[0112] Figure 3 and Figure 4 These are the results of re-sorting NASA batteries and Oxford batteries by fuzzy C-means clustering. Figure 3It can be seen that the data points of the 4th category NASA battery are relatively dense, indicating that its SOH and RUL are more consistent. The number of batteries in categories 1, 2, and 3 is large, and there is a small amount of battery overlap between different categories, while the distance between category 3 and category 4 is large, reflecting the obvious battery performance grading. Oxford batteries use an interval statistical counting method, that is, 100 charge and discharge cycles of the battery are counted as 1 cycle. Therefore, the SOH and RUL of Oxford batteries are both obvious linear decreasing processes. Compared with NASA batteries, the classification interval of Oxford batteries is more obvious.

[0113] S45: Calculate the standard deviation (SD) and coefficient of variation (CV) of SOH and RUL of batteries of the same category to analyze the battery sorting effect of the proposed model:

[0114]

[0115] Where Y represents the battery's SOH or RUL, q and k are the index and total number of similar batteries, respectively. CV reflects the degree of dispersion of the indicator relative to the mean (AV). CV < 0.1 is generally considered to indicate good data consistency.

[0116] The standard deviation and coefficient of variation of various battery groups after NASA battery sorting are as follows Figure 5 and as shown in Table 1.

[0117] Table 1 Performance difference analysis of NASA battery sorting

[0118]

[0119] As shown in Table 1, in NASA batteries, SD SOH The maximum value of CV appears in the fourth type of battery, indicating that the SOH distribution of the fourth type of battery is more dispersed. SOH The maximum values ​​of all appear in the third type of battery, reflecting that its SOH consistency is the worst among the five types of batteries, while the CV of the fifth type of battery is SOH Since the data range of RUL is much larger than that of SOH, the value of RUL evaluation index will also be larger than the corresponding value of SOH. RUL The maximum value appears in the first category battery, indicating that its RUL inconsistency is more obvious, while the coefficient of variation of the RUL of the fifth category battery is the lowest. At the same time, the coefficient of variation of the SOH of the fifth category battery is also the lowest, further reflecting that the SOH and RUL of the high-performance battery pack (fifth category) selected by the entropy weight method and fuzzy C-means clustering hybrid model are both highly consistent.

[0120] The standard deviation and coefficient of variation of various battery groups after Oxford battery sorting are as follows Figure 6 and as shown in Table 2.

[0121] Table 2 Performance difference analysis of Oxford battery sorting

[0122]

[0123] As shown in Table 2, among Oxford batteries, the SD of various types of batteries SOH and CV SOH The SOH values ​​for Category 1 batteries are relatively close, while those for Category 5 batteries are relatively dispersed. The coefficient of variation for the RUL of Category 5 batteries is the smallest, at 0.0639, indicating the best RUL consistency for Category 5 batteries.

[0124] S46: Comparative analysis of the sorting effects of the proposed model, hierarchical clustering (HC), and K-means clustering (K-means) on different battery data sets. Evaluations were performed using intra-class aggregation and inter-class distance, and the corresponding silhouette coefficient (SC) and CH index (CHI) were calculated:

[0125]

[0126] Where n is the total number of batteries of all categories. q and d q They represent the average distance between battery q and other batteries in the same cluster (cohesion) and the average distance between all batteries in the nearest neighbor cluster (separation). TR(·) represents the trace of the matrix, B k and W k They represent the inter-cluster scatter matrix and the intra-cluster scatter matrix respectively. b , c b and c are the number of samples in the bth cluster, the center of the bth cluster, and the center of all battery data, respectively. A larger CHI indicates a smaller covariance of data within a category and a clearer boundary between categories. A closer SC is to 1, the better the clustering effect.

[0127] The CH index and silhouette coefficient of different clustering algorithms after battery sorting are shown in Table 3.

[0128] Table 3 CHI and SC values ​​of different clustering methods As can be seen from Table 3, FCM has the highest silhouette coefficient and CHI for NASA batteries and Oxford batteries, with silhouette coefficients of 0.707 and 0.686, and CHIs of 595.296 and 273.146, respectively. This further shows that after sorting by FCM, the same group of batteries has a smaller cohesive distance, and the separation distances of different groups of batteries are larger. Secondly, in NASA batteries, the evaluation indicators of FCM are significantly improved. Compared with hierarchical clustering, the CHI of FCM is improved by 15.75%, and the silhouette coefficient of FCM is improved by 10.47%. The above results show that in the sorting applications of different batteries, the fuzzy C-means clustering adopted by the present invention has better sorting effect than the common hard clustering method, and improves the consistency of battery SOH and RUL.

[0129] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solutions, which should all be included in the scope of the claims of the present invention.

Claims

1. A two-stage lithium-ion battery sorting method based on entropy weight method and fuzzy C-means clustering, characterized in that: The method specifically comprises the following steps: S1: Calculate the battery's SOH and RUL based on the capacity degradation curve of the lithium-ion battery, where SOH represents the state of health and RUL represents the remaining service life; S2: Normalize the battery's SOH and RUL, and calculate the information entropy and information entropy redundancy of SOH and RUL using the entropy weight method; S3: Calculate the index weights of SOH and RUL based on the information entropy redundancy, further calculate the comprehensive evaluation value of the battery samples, and preliminarily screen low-performance batteries based on the comprehensive evaluation value; S4: The remaining batteries are sorted again by fuzzy C-means clustering.

2. The two-stage separation method for lithium-ion batteries according to claim 1, characterized in that: In step S1, the expression for calculating the battery's SOH is: SOH=C t / C0×100% Among them, C t and C0 represent the capacity corresponding to time t and the initial rated capacity respectively.

3. The two-stage separation method for lithium-ion batteries according to claim 1, characterized in that: In step S1, the expression for calculating the RUL of the battery is: RUL=Cycle t -Cycle EOL Among them, Cycle t and Cycle EOL They represent the number of cycles at time t and the number of cycles at the end of life respectively.

4. The two-stage separation method for lithium-ion batteries according to claim 1, characterized in that: In step S2, the battery's SOH and RUL are standardized, and the expressions are: Among them, w ij and y ij Represent the original and standardized index data of the jth battery sample, min(.) and max(.) are the minimum and maximum function operations, respectively. j Indicates the battery's SOH or RUL, both are positive indicators.

5. The two-stage separation method for lithium-ion batteries according to claim 4, characterized in that: In step S2, the information entropy and information entropy redundancy of SOH and RUL are calculated by the entropy weight method, specifically including: calculating y ij At the jth indicator A j The proportion p ij , the calculation formula is: Among them, S represents the total amount of information, when y ij When all are 0, then define p ij =1 / S; The information entropy e of the jth indicator j The calculation formula is: The information entropy redundancy d of the jth indicator j The calculation formula is: j =1-e j .

6. The two-stage separation method for lithium-ion batteries according to claim 5, characterized in that: In step S3, the comprehensive evaluation value of the battery sample is calculated, and the expression is: Among them, Score i represents the comprehensive evaluation value of the i-th battery, ω j Indicates the indicator weight of battery SOH or RUL, d j It represents the information entropy redundancy of battery SOH or RUL, where R is the number of indicators.

7. The two-stage separation method for lithium-ion batteries according to claim 1, characterized in that: In step S3, low-performance batteries are preliminarily screened based on the comprehensive evaluation values. Specifically, adaptive screening is performed using a percentage threshold, that is, batteries with the lowest percentage threshold comprehensive evaluation values ​​are eliminated.

8. The two-stage separation method for lithium-ion batteries according to claim 1, characterized in that: In step S4, the remaining batteries are sorted again by fuzzy C-means clustering, which specifically includes: constructing the objective function (F m (U,V)): Among them, m is the fuzzy weighted index, U is the membership matrix, V is the cluster center matrix, N and C are the number of samples and the number of categories respectively, u ab Indicates the membership degree of the b-th battery sample to the a-th category, st is the constraint condition; (D ab ) 2 Represents the distance from the battery sample to the cluster center, and its expression is as follows: Among them, x b and v a denote the center of the b-th battery sample and the a-th category respectively, M is a symmetric positive definite matrix, (·) T Represents the transpose operation of the matrix; Initialize the membership matrix, set the number of iterations l = 1, update the cluster center, first, find F m v a The partial derivative of and set it equal to 0: Then, D ab Substituting the expression into the above formula, we can further obtain: Since M is a symmetric positive definite matrix, further simplification yields the cluster center of the lth iteration as follows: Update the membership matrix, combine the st condition, and transform F m Convert to Lagrange multiplier method: Among them, L λ is the objective function that transforms the original constrained optimization problem into an unconstrained optimization problem, λ b is the Lagrange multiplier, corresponding to the constraint condition of the bth sample The multiplier of Convert the above formula to u ab Find the partial derivative and set it equal to 0, solve for u ab The expression is: Combined with the st constraint, we can further obtain λ b The expression is: λ b Substitute the expression into u ab In the expression of , the membership degree of the lth iteration is: Set the iteration termination condition and make classification decisions; the termination condition expression is as follows: Among them, L max is the maximum number of iterations, ε u is the membership convergence threshold.

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