Flat-plate permanent magnet synchronous linear motor multi-objective optimization method, device and medium

Abstract

The invention discloses a multi-objective optimization method and equipment for a flat-plate permanent magnet synchronous linear motor and a medium, and the method comprises the steps: taking the average thrust and thrust fluctuation of the flat-plate permanent magnet synchronous linear motor as optimization objectives, analyzing design parameters influencing the flat-plate permanent magnet synchronous linear motor, and determining the value range of the flat-plate permanent magnet synchronous linear motor; designing and constructing a sample library through a Latin hypercube test; establishing an RF model based on the sample library; and the RF model is optimized based on an improved NSGA-II multi-target algorithm. The improved content comprises the steps of initializing a population by adopting logic mapping, and selecting elite individuals by combining non-dominated sorting and crowding distance calculation; two-point crossover, binary crossover simulation, random variation and adaptive variation strategies are adopted, so that the global search and local development capability of the algorithm is enhanced; in combination with a PID adjustment mechanism, an individual updating mode is adjusted according to an individual fitness value and a distance from the Pareto frontier, and the algorithm search efficiency and the global optimization capability are improved. After optimization, the motor thrust peak-to-peak value is greatly reduced, and the thrust fluctuation is remarkably improved.

Description

Technical Field

[0001] The present invention relates to the optimization of permanent magnet synchronous linear motors, and particularly to a multi-objective optimization method, device, and storage medium for flat permanent magnet synchronous linear motors. Background Art

[0002] Flat permanent magnet synchronous linear motors play a crucial role in numerous fields due to their unique advantages. In the field of high-speed precision machining, they can directly convert electrical energy into linear motion mechanical energy, avoiding energy losses and transmission errors caused by intermediate transmission links, and greatly improving machining accuracy and efficiency; in maglev train systems, they provide a powerful and precise driving force for the stable operation of trains, ensuring high-speed and smooth train travel.

[0003] However, the thrust fluctuation problem existing in flat permanent magnet synchronous linear motors severely restricts the improvement of their performance and wide application. Thrust fluctuation will cause vibration and noise during the operation of the motor, not only reducing the stability and reliability of the system, but also having a negative impact on machining accuracy. In high-precision machining scenarios, it may lead to a decline in product quality. Additionally, if the average thrust of the motor is insufficient, it cannot meet some application scenarios with high power requirements, restricting its promotion in specific fields.

[0004] Currently, traditional motor optimization methods have many limitations. Taking the finite element model and analytical method as examples, although they can analyze and optimize the motor performance to a certain extent, the calculation cost is extremely high, requiring a large amount of computing resources and time, and the accuracy is relatively low, making it difficult to accurately reflect the actual operating conditions of the motor. Therefore, a multi-objective optimization method based on surrogate models has been proposed for motor optimization. In the multi-objective optimization of motors, NSGA-II is a classic and effective algorithm that can find a set of Pareto optimal solutions. However, when dealing with complex search spaces, NSGA-II may encounter local optimal traps. In complex multi-objective optimization problems, NSGA-II may fall into local optima, thereby reducing the global exploration ability. In high-dimensional optimization problems or multi-modal problems, the convergence speed of NSGA-II may be slow, making it difficult to quickly find high-quality Pareto front solutions.

[0005] Therefore, in the process of multi-objective optimization of flat permanent magnet synchronous linear motors, that is, maximizing the average thrust and minimizing the thrust fluctuation, how to avoid multi-objective optimization from falling into local optima, improve the global search ability, and quickly find a high-quality set of motor parameter solutions is a difficult problem that those skilled in the art are committed to solving. Summary of the Invention

[0006] The embodiment of the present application provides an innovative multi-objective optimization method for a flat permanent magnet synchronous linear motor. By comprehensively applying various technical means, a complete and efficient optimization system is formed from parameter selection, sample library construction, model establishment to the implementation of the optimization algorithm, effectively improving the performance of the motor and solving the problem that traditional optimization methods are prone to fall into local optima during the multi-objective optimization process.

[0007] The embodiment of the present application provides a multi-objective optimization method for a flat permanent magnet synchronous linear motor, including the following steps:

[0008] Step S1: Taking the average thrust and thrust ripple of the flat permanent magnet synchronous linear motor as optimization objectives, analyzing the design parameters that affect the average thrust and thrust ripple, and determining the value ranges of the average thrust and thrust ripple;

[0009] Step S2: Constructing a sample library through Latin hypercube experimental design;

[0010] Step S3: Establishing a random forest surrogate model based on the sample library;

[0011] Step S4: Improving the NSGA-II algorithm, introducing a PID adjustment amount and a zero-output disturbance term to update individuals; and using the improved NSGA-II algorithm to perform multi-objective optimization on the random forest surrogate model, and obtaining the optimal design parameters of the flat permanent magnet synchronous linear motor through optimization by the improved NSGA-II algorithm.

[0012] Preferably, in the step S1, the design parameters that affect the average thrust and thrust ripple include the side tooth width, side tooth height, left end tooth width, right end tooth width, and permanent magnet width of the flat permanent magnet synchronous linear motor.

[0013] Preferably, the step S2 specifically includes: adopting Latin hypercube experimental design, selecting several sample points in the sample space, and calculating the objective function values of the average thrust and thrust ripple of each sample point through numerical simulation. The sample points and the objective function values together form a sample library.

[0014] Preferably, the Latin hypercube experimental design realizes the uniform distribution of samples by dividing equal probability intervals in a multi-dimensional space and selecting samples in each probability interval. The specific process is as follows:

[0015] For each variable X i (i = 1, 2,..., d), defining its range as [a i , b i , where d is a positive integer;

[0016] Dividing the range [a i , b i of each variable into m samplem equal-width intervals sample where m is a positive integer, and the width Δx of each interval i is given by:

[0017]

[0018] Randomly select a point x in each interval i,j :

[0019] x i,j = a i +(j - 1)·Δx i + u i,j ·Δx i

[0020] where u i,j is a uniform random number on [0, 1], and j = 1, 2,..., m sample ;

[0021] Randomly permute the sampling points of each variable to ensure that each interval is only used once in each dimension; combine the permuted sampling points into m sample sample points to form a sampling matrix X LHS :

[0022]

[0023] where π i is a random permutation of the i-th variable.

[0024] Preferably, step S3 specifically includes:

[0025] Extract samples from the sample library to form a training subset and a test subset;

[0026] Use the training subset to train a random forest surrogate model to obtain appropriate model parameters; during training, the random forest constructs multiple decision trees and randomly samples features and data to reduce overfitting and improve the generalization ability of the model;

[0027] After training, test the model with the test subset and measure the prediction performance of the model by calculating the coefficient of determination and / or the root mean square error.

[0028] Preferably, in step S4, the NSGA-II algorithm is improved by introducing a PID adjustment amount and a zero-output perturbation term to update individuals, specifically including:

[0029] Generate an initial population uniformly distributed in the search space through a logical mapping mechanism; calculate the objective function values of individuals, and select elite individuals through non-dominated sorting and crowding distance calculation for subsequent crossover and mutation operations;

[0030] At different stages of the algorithm, different crossover and mutation strategies are adopted; in the early iteration, two-point crossover is used for crossover to enhance the global search ability, and random mutation is used for mutation to explore new regions; in the later iteration, simulated binary crossover and adaptive mutation are switched to strengthen the local development ability;

[0031] The update weight is determined by calculating the individual fitness and the distance from the Pareto front, and combined with the PID adjustment amount and the zero-output perturbation term, the individual is updated to further improve the search efficiency and global optimization ability of the algorithm.

[0032] Preferably, the specific method of selecting elite individuals by non-dominated sorting and crowding distance calculation is as follows:

[0033] Perform non-dominated sorting on the population, stratify the individuals according to their quality, calculate the crowding distance of the individuals, and preferentially select the elite individuals with large crowding distance and high non-dominated level to enter the next generation, effectively maintaining the population diversity and guiding the population to evolve towards the Pareto front.

[0034] Preferably, in step S4, the improved NSGA-II algorithm is used to perform multi-objective optimization on the random forest surrogate model, and the optimal design parameters of the flat permanent magnet synchronous linear motor are obtained by optimizing with the improved NSGA-II algorithm, specifically including:

[0035] Judge whether the model reaches the set population generation; if it reaches the set population generation, enter the next step; otherwise, continue to iterate the newly obtained population until the set population generation is reached;

[0036] Export the finally optimized Pareto front, select the finally optimized motor parameters according to the requirements, draw the motor model, and perform numerical simulation and solution on the motor.

[0037] In a second aspect, an embodiment of the present application further provides an electronic device, including a processor, a memory, and a communication bus, wherein the processor and the memory complete communication with each other through the communication bus;

[0038] The memory is used to store a computer program;

[0039] The processor is used to execute the program stored in the memory to implement the above-mentioned multi-objective optimization method for the flat permanent magnet synchronous linear motor.

[0040] In a third aspect, an embodiment of the present application further provides a computer-readable storage medium, storing a computer program, and when the computer program is executed by a processor, the above-mentioned multi-objective optimization method for the flat permanent magnet synchronous linear motor is implemented.

[0041] One or more technical solutions provided in the embodiments of the present application have at least the following technical effects or advantages:

[0042] 1. An improved NSGA-II multi-objective algorithm is proposed. The population is initialized by logical mapping, and elite individuals are selected by combining non-dominated sorting and crowding distance calculation. At different stages of the algorithm, two-point crossover, simulated binary crossover, random mutation, and adaptive mutation strategies are adopted respectively to enhance the global search and local development capabilities of the algorithm. A PID control mechanism is introduced to adjust the individual update method according to the individual fitness value and the distance from the Pareto front, improving the search efficiency of the algorithm.

[0043] 2. The improved NSGA-II algorithm is combined with the RF model. The RF model is used to establish a model with a certain amount of samples, avoiding the problem of long time-consuming finite element simulation of multiple sample points. The improved NSGA-II algorithm is fused for multi-objective optimization, quickly improving the parameter search accuracy until convergence and enhancing the optimization efficiency of the algorithm.

[0044] 4. A non-linear relationship between design variables and optimization objectives is established through a surrogate model, solving the accuracy problem caused by the sample size.

[0045] 5. Latin square experimental design is used for sampling to meet the uniformity of sample points in space, making the subsequent results stable.

[0046] 6. The RF model and the improved NSGA-II algorithm are combined for multi-objective optimization, avoiding falling into local optima during the optimization process, improving the running speed of the optimization algorithm, and enhancing the robustness of the solution set. Description of the Drawings

[0047] Figure 1 It is the flowchart of the multi-objective optimization method for the flat permanent magnet synchronous linear motor provided in Embodiment 1 of the present application;

[0048] Figure 2 It is the specific flowchart of the improved NSGA-II intelligent optimization algorithm based on the PID adjustment mechanism in Embodiment 1 of the present application;

[0049] Figure 3 It is the solution comparison diagram of the improved NSGA-II intelligent optimization algorithm based on the PID adjustment mechanism and the traditional NSGA-II on the ZDT1 test function in Embodiment 1 of the present application;

[0050] Figure 4 It is the solution comparison diagram of the improved NSGA-II intelligent optimization algorithm based on the PID adjustment mechanism and the traditional NSGA-II on the ZDT2 test function in Embodiment 1 of the present application;

[0051] Figure 5This is a comparison graph of the improved NSGA-II intelligent optimization algorithm based on the PID adjustment mechanism and the traditional NSGA-II in the ZDT3 test function in the first embodiment of this application.

[0052] Figure 6 This is a comparison graph of the improved NSGA-II intelligent optimization algorithm based on the PID adjustment mechanism and the traditional NSGA-II in the ZDT4 test function in the first embodiment of this application.

[0053] Figure 7 This is a schematic diagram of the optimized result of the flat permanent magnet synchronous linear motor by the improved NSGA-II intelligent optimization algorithm based on the PID adjustment mechanism in the first embodiment of this application. Detailed implementation manners

[0054] In the embodiments of this application, by providing an innovative multi-objective optimization method for flat permanent magnet synchronous linear motors, a variety of technical means are comprehensively used. From parameter selection, sample library construction, model establishment to the implementation of the optimization algorithm, a complete and efficient optimization system is formed, effectively improving the performance of the motor and solving the problem that traditional optimization methods are prone to fall into local optima during the multi-objective optimization process.

[0055] The technical solutions in the embodiments of this application to solve the above technical problems are generally as follows:

[0056] In terms of optimization objectives and parameter selection, the edge tooth width (ETw), edge tooth height (ETh), left end tooth width (UT1), right end tooth width (UT2), and permanent magnet width (Wpm) are selected as the key parameters to be optimized. These five parameters have a significant impact on the performance of the flat permanent magnet synchronous linear motor. With the maximization of the average thrust and the minimization of the thrust fluctuation as the clear optimization objectives, it aims to achieve high efficiency and stability of the motor during operation.

[0057] The construction of the sample library adopts the Latin hypercube experiment design. Through this method, 250 sample points are selected in the sample space. This design method can ensure the uniform distribution of the sample points in the space, covering various possibilities in the parameter space to the greatest extent. Subsequently, numerical simulation is carried out on each sample point to calculate the corresponding objective function values, that is, the numerical values of the average thrust and the thrust fluctuation. These sample points and their objective function values together constitute the sample library, providing a rich and representative data basis for the subsequent model construction, and helping to improve the accuracy and reliability of the model.

[0058] Build a random forest surrogate model. Extract samples from the sample library and use these sample data to construct a random forest surrogate model for the objective function. The establishment process mainly includes several steps: data preparation, model training, and model evaluation. First, prepare the dataset and divide it into a training set and a test set. Next, use the training dataset to train the random forest surrogate model and select appropriate model parameters. During the training process, the random forest constructs multiple decision trees and randomly samples the features and data to reduce overfitting and improve the generalization ability of the model. After training, calculate R 2 (coefficient of determination), RMSE (root mean square error) to measure the prediction performance of the model.

[0059] The core of this application is to propose an improved NSGA-II multi-objective algorithm to optimize the random forest surrogate model. The main optimization steps include initializing the population, non-dominated sorting, dynamic crossover and mutation, and population update based on the PID adjustment mechanism. The logical mapping initialization method is adopted, and the initial population uniformly distributed in the search space is generated by means of the logical mapping formula, providing a wider search starting point for the algorithm. At the same time, calculate the individual objective function values, select elite individuals through non-dominated sorting and crowding distance calculation, and lay a good foundation for population evolution. Perform non-dominated sorting on the population, stratify the individuals according to their quality, calculate the crowding distance of the individuals, and preferentially select the elite individuals with large crowding distance and high non-dominated levels to enter the next generation, effectively maintaining the population diversity and guiding the population to evolve towards the Pareto front. At different stages of the algorithm, different crossover and mutation strategies are adopted. In the early iteration, two-point crossover is used for crossover to enhance the global search ability, and random mutation is used for mutation to explore new regions; in the later iteration, simulated binary crossover and adaptive mutation are switched to strengthen the local development ability. Determine the update weight by calculating the individual fitness and the distance from the Pareto front, and combine the PID adjustment mechanism and the zero-output perturbation term to update the individuals, further improving the search efficiency and global optimization ability of the algorithm.

[0060] Through the above optimization method, this application can achieve multi-objective optimization of the average thrust and thrust fluctuation of the flat permanent magnet synchronous linear motor, providing strong support for the efficient application of the motor in various fields. The improved NSGA-II algorithm inherits the advantages of the genetic algorithm and obtains the optimal value through the collective evolution of the population, which is conducive to obtaining the global optimal solution of the optimization and avoiding falling into the local optimum. When solving multi-objective problems, it can obtain the Pareto optimal solution set and has the advantages of fast running speed and good convergence of the solution set.

[0061] To better understand the above technical solutions, the above technical solutions will be described in detail below in combination with the accompanying drawings of the specification and specific implementation manners.

[0062] Example 1

[0063] Figure 1 This is a flowchart of the multi-objective optimization method for a flat permanent magnet synchronous linear motor provided in the first embodiment of this application. The multi-objective optimization method for the flat permanent magnet synchronous linear motor includes the following steps:

[0064] Step S1: Taking the average thrust and thrust ripple of the flat permanent magnet synchronous linear motor as optimization objectives, analyzing the design parameters that affect the average thrust and thrust ripple, and determining the value ranges of the optimization parameters.

[0065] Among them, the average thrust F avg is defined as the average value of the thrust of the motor within one electrical cycle, and the calculation formula is:

[0066]

[0067] where F(t) represents the thrust at time t, and T represents one electrical cycle.

[0068] The thrust ripple F rip is defined as the ratio of the difference between the maximum thrust and the minimum thrust to the average thrust of the motor within one electrical cycle, and the calculation formula is:

[0069]

[0070] where F max represents the maximum thrust within one electrical cycle, and F min represents the minimum thrust within one electrical cycle.

[0071] The design parameters that affect the average thrust and thrust ripple include but are not limited to the edge tooth width (ETw), edge tooth height (ETh), left end tooth width (UT1), right end tooth width (UT2), and permanent magnet width (Wpm) of the flat permanent magnet synchronous linear motor.

[0072] Step S2: Construct a sample library through Latin hypercube experimental design.

[0073] Sampling in the sample space through the Latin hypercube of experimental design, and using numerical simulation to calculate the objective function values of the average thrust and thrust ripple of each sample point. Save the sample points and their corresponding objective function values in the sample point database to jointly form a sample library.

[0074] A good experimental design should satisfy both spatial uniformity and projection uniformity while meeting the requirements. The purpose is to predict the response of the entire design area as accurately as possible with less sample point information. The Latin hypercube experimental design realizes the uniform distribution of samples by dividing equal probability intervals in the multi-dimensional space and selecting samples within each interval.

[0075] For each variable X i(i = 1, 2, ..., d), define its range as [a i , b i . Among them, d represents the dimension of the design variable, and a i and b i respectively represent the lower bound and the upper bound of the i-th variable.

[0076] Divide the range [a i , b i of each variable into m sample equal-width intervals. m sample represents the number of sample points (which is also the number of divided intervals). The width Δx i of each interval is:

[0077]

[0078] Randomly select a point x i,j in each interval:

[0079] x i,j = a i + (j - 1)·Δx i + u i,j ·Δx i

[0080] Among them, u i,j is a uniform random number on [0, 1], and j = 1, 2, ..., m sample . x i,j represents the sampling point of the i-th variable in the j-th interval.

[0081] Randomly permute the sampling points of each variable to ensure that each interval is only used once in each dimension. Combine the permuted sampling points into m sample sample points to form a sampling matrix X LHS :

[0082]

[0083] Among them, π i is the random permutation of the i-th variable.

[0084] Latin Hypercube Sampling (LHS) ensures uniform coverage in each dimension through interval division and random permutation, thereby improving the sampling efficiency.

[0085] Step S3: Establish a Random Forest (RF) surrogate model based on the sample library.

[0086] First, data sampling is carried out. Samples are extracted from the sample library and divided into a training subset and a test subset. Specifically, from the original dataset in the sample library, the random forest generates multiple training subsets and test subsets through sampling with replacement. The number of samples in each training subset and test subset is equal to the size N of the original dataset sample .

[0087] The training subset is used to train the random forest surrogate model, and appropriate model parameters are selected. During the training process, the random forest constructs multiple decision trees and randomly samples features and data to reduce overfitting and improve the generalization ability of the model

[0088] Each decision tree is constructed based on a randomly sampled training subset. When constructing each decision tree, for each splitting node, from all M feat features, m feat features (m feat < M feat , m feat , M feat are all positive integers) are randomly selected, and the feature that gives the best splitting effect is chosen from them. The growth of the decision tree is usually not restricted by pruning until all nodes are pure or the minimum number of samples is reached

[0089] After training is completed, the model is tested through the test subset, and R 2 (coefficient of determination), RMSE (root mean square error) are calculated to measure the prediction performance of the model

[0090] The expression of R 2 (coefficient of determination) is as follows

[0091]

[0092] where n data is the number of data points, y i is the true model response value, is the predicted value of the approximation model, is the average value of the true response values, calculated as When the value of R 2 is closer to 1, it indicates that the global accuracy of the model is higher

[0093] Root Mean Square Error (RMSE), which is an index used to evaluate the prediction error of the model. It represents the average difference between the predicted value and the actual value. The calculation formula is as follows

[0094]

[0095] where n data represents the number of data points; y iRepresents the actual value; Represents the predicted value; Represents the square of the prediction error for each data point.

[0096] For each data point, calculate the predicted value The difference from the actual value (y i ). Square the difference of each prediction error to avoid positive and negative errors canceling each other out. Sum all the squared error values and then divide by the number of data points n data , to obtain the mean squared error. Take the square root of the mean squared error to get the RMSE.

[0097] The random forest regression tree forms the predicted value of the random forest surrogate model by taking the average of the decision trees h(X in , θ), where X in represents the input features and θ represents the parameters of the decision tree (including split features, split thresholds, etc.). The generalization error of the random forest model is defined as the gap between the predicted values and the true values for all possible data points. The average prediction generalization error of the model can be expressed as

[0098]

[0099] where, represents taking the expectation with respect to the parameters θ of the decision tree; represents taking the expectation with respect to the distribution of the sample points (X in , Y), and Y represents the target variable.

[0100] The random forest consists of n tree independent decision trees, n tree is a positive integer, and its model predicted value is the average of all decision tree predicted values:

[0101]

[0102] The generalization error PE * (tree) is described by the following formula:

[0103]

[0104] For θ, we can obtain Then the generalization error of the random forest can be expressed as:

[0105] PE * (forest) ≤ ρ tree PE * (tree)

[0106] where, ρ treeRepresents the average correlation between decision trees; PE * (tree) is the average generalization error of a single decision tree.

[0107] Step S4: Improve the NSGA-II algorithm by introducing the PID adjustment amount and the zero-output perturbation term to update the individuals, so as to further improve the search efficiency and global optimization ability of the algorithm; and use the improved NSGA-II algorithm to perform multi-objective optimization on the RF model, and obtain the optimal design parameters of the flat permanent magnet synchronous linear motor through the optimization of the improved NSGA-II algorithm. Figure 2 Is the specific flowchart of the improved NSGA-II intelligent optimization algorithm based on the PID adjustment mechanism.

[0108] In the improved NSGA-II algorithm, the generation of the initial population adopts the logical mapping method to ensure that the population is evenly distributed in the search space. Specifically, during initialization, individuals with a random uniform distribution are generated through the logical mapping mechanism. The distribution of individuals in the objective function space can effectively improve the comprehensiveness of the search and avoid blindness within the search space.

[0109] The Logistic Map is a chaotic map often used to generate pseudo-random number sequences. The chaotic nature of the Logistic Map can provide a uniform and dispersed population distribution to a certain extent, which is beneficial for the generation of the initial population in evolutionary algorithms.

[0110] The formula of the Logistic Map is as follows:

[0111] Z n+1 = μZ n ·(1 - Z n )

[0112] Where Z n is the value of the nth iteration, and the initial value Z0 is a randomly selected number, usually in the interval (0, 1). μ is the control parameter of the Logistic Map, and usually μ = 4 is selected to ensure that the map exhibits chaotic behavior. Z n+1 is the value of the next iteration.

[0113] Through this mapping, we can generate a chaotic sequence Z, and then use this sequence to initialize the individuals in the population. The specific process:

[0114] 1) Set parameters: Determine the initial value Z0 (randomly selected) and the control parameter μ of the Logistic Map.

[0115] 2) Generate a chaotic sequence: Generate a chaotic sequence of length n pop using the Logistic Map formula where n pop is the population size.

[0116] Z n+1 = μZ n ·(1 - Z n )

[0117] 3) Mapping to the search space: Map the chaotic sequence Z n to the search space of the problem.

[0118] 4) Construct the initial population: Repeat the above steps to generate N pop individuals to form the initial population N pop is a positive integer.

[0119] 5) Evaluation of the initial population: Calculate the fitness value of each individual and prepare to enter the next non - dominated sorting and crowding distance calculation.

[0120] The population of each generation is processed by the non - dominated sorting method. First, sort the individuals in the population according to the objective function values and select the non - dominated solution set. Second, calculate the crowding distance of each individual to measure the diversity of the population. Finally, select a certain number of elite individuals for subsequent crossover and mutation operations.

[0121] In calculating the non - dominated rank, for each individual p in the population, calculate its dominance situation relative to other individuals. If individual p is not dominated by other individuals, it belongs to the first non - dominated rank (i.e., Front 1). Continue to judge the non - dominated ranks of other individuals in the same way until a non - dominated rank is assigned to each individual. If individual p is not inferior to q in all objectives and is superior to q in at least one objective, then p dominates q, satisfying (such that) p < q. Calculation of the non - dominated rank:

[0122]

[0123] where, F i represents the i - th non - dominated front, P represents the current population, and ∈ is the attenuation coefficient.

[0124] Within the same non - dominated rank, the crowding distance is used to measure the density of each individual in the objective space. Individuals with larger crowding distances are preferentially retained to ensure the diversity of the population. For an individual i belonging to the same non - dominated front, its crowding distance on objective m is expressed as The formula for calculating the crowding distance is as follows:

[0125]

[0126] where, f m (i + 1) and f m(i - 1) are the objective values of the adjacent individuals before and after individual i on the m-th objective, and are the maximum and minimum values on this objective. For boundary individuals (i.e., individuals with the maximum or minimum value on a certain objective), their crowding distance is set to infinity (∞) to ensure that boundary solutions are retained.

[0127] The total crowding distance d i is:

[0128]

[0129] where M obj is the number of objective functions.

[0130] According to the non - dominated sorting result, individuals with a lower non - dominated rank are selected first until the size of the next - generation population is satisfied. If the number of individuals in a certain non - dominated rank is too large and exceeds the remaining population positions, the individuals in this rank are sorted according to the crowding distance, and individuals with a larger crowding distance are selected.

[0131] The selected elite individuals will be used for crossover and mutation operations to generate the next - generation population. Through non - dominated sorting and crowding distance calculation, the diversity of the population and the coverage of the objective space are ensured.

[0132] According to the progress of the current iteration, a dynamic operator selection mechanism is adopted. In the early iterations of the algorithm, a two - point crossover operation with strong global search ability is used to ensure that the population explores the entire search space. In the later iterations, it is switched to the SBX simulated binary crossover operation with strong local exploitation ability to improve the accuracy of the solution set. In addition, under the adjustment of the PID controller, the mutation amplitude and direction of the mutation operation will be dynamically adjusted to improve the adaptability of the algorithm and avoid local optima.

[0133] The crossover operation is one of the core operations in the evolutionary algorithm. It generates new offspring individuals by exchanging the gene information of parent individuals, thereby exploring new solution spaces. The dynamic crossover operation in the improved algorithm is divided into two stages.

[0134] Global search stage (early iterations), a two - point crossover with strong global search ability is adopted. This method can explore a large range within the search space, thereby increasing the diversity of the population and avoiding getting trapped in local optima early. In the two - point crossover operation process, two crossover points (positions) c1 and c2 are randomly selected. The gene segments between these two positions are exchanged. Two new offspring individuals are generated.

[0135] In the local development stage (late iteration), simulated binary crossover (SBX) with strong local development ability is adopted. SBX can perform fine local search within the solution space to help find better solutions. In the SBX crossover operation process, for two parent individuals x1 and x2, two offspring individuals y1 and y2 are generated according to the formula. The parameter β SBX controls the degree of the crossover operation and is dynamically adjusted according to the current iteration number.

[0136] The formula of SBX:

[0137] y p,i = 0.5 × [(1 + β SBX )x 1,i + (1 - β SBX )x 2,i

[0138] y q,i = 0.5 × [(1 - β SBX )x 1,i + (1 + β SBX )x 2,i

[0139] Among them, the calculation of β SBX is as follows:

[0140]

[0141] In the formula, μ c is the control parameter, and u is a uniform random number on [0, 1].

[0142] The introduction of the PID controller enables the individual positions in each generation to be dynamically adjusted according to the feedback of the historical optimal solution set. The controller calculates the mutation direction and amplitude based on the position difference between the individual and the historical optimal individual. The formula is as follows:

[0143] Δu(t) = K p · [e k (t) - e k-1 (t)] + K i · e k (t) + K d · [e k (t) - 2e k-1 (t) + e k-2 (t)]

[0144] Among them, Δu(t) is the control adjustment amount, and K p , K i , K d are the proportional, integral, and differential coefficients respectively. The usual value range is K p ∈[0.1, 1.0], K i ​​∈[0.01,0.1], K d ∈[0.5,2.0]. k (t) represents the error of the kth individual at iteration time t, which is defined as:

[0145] e k (t) = x target,k -x k (t)

[0146] where x target,k is the target position (the Pareto optimal solution of the current individual), x k (t) is the position of individual k at time t.

[0147] On the basis of PID control, the Levy flight mechanism is combined to further adjust the individual position to ensure population diversity and enhance the global search ability of the algorithm. The update formula is:

[0148] x k (t+1)=x k (t)+η·Δu(t)+(1-η)·L

[0149] Among them, L is the Levy flight factor, η is the weight factor that balances the PID control amount and the Levy flight influence, η∈[0,1]. When η is close to 1, it is more inclined to PID control, and when η is close to 0, it is more inclined to Levy flight random exploration.

[0150] The distribution of the step length L of Levy flight follows a stable distribution, as shown below:

[0151]

[0152] Where u and v are random variables that follow a normal distribution. σ is the scaling factor, defined as:

[0153]

[0154] Where Γ represents the gamma function, yve β l is the exponential parameter of the Levy flight, usually in the range of β levy ∈(0,2], to ensure sufficient jumpiness.

[0155] The introduction of the Levy flight mechanism makes it possible for individuals to jump to farther areas of the search space, thereby preventing the population from falling into local optimal solutions and enhancing the global search capability in the early stages.

[0156] The zero output factor is a parameter used to control population update. By exploring a larger range in the early stage, the zero output factor helps the population focus on more refined local exploitation in the later stage. The application of the zero output factor can be reflected by the following adjustment formula:

[0157] x k (t + 1) = x k (t + 1)+∈(t)·L

[0158] where ∈(t iter ) is a decay coefficient that gradually decreases as the number of iterations increases. The calculation formula is:

[0159]

[0160] where ∈0 is the initial value, usually set as ∈0∈[0.5, 1.0], λ is the decay rate, usually set as λ∈[0.01, 0.1], and t iter is the current number of iterations.

[0161] During the early global search, at the initial stage of the algorithm, ∈(t iter ) is larger, and the influence of Levy flight is also larger, making it more likely for individuals to jump to new search areas and enhancing global exploration. During the later local exploitation, as the number of iterations increases, ∈(t iter ) gradually decreases, causing the population to mainly focus on local exploitation in the later stage, refining the search, and improving the quality of the solution.

[0162] After the population of each generation undergoes crossover, mutation, and PID adjustment, it competes with the elite individuals of the previous generation, and the elite retention strategy of NSGA-II is used to select the next generation of the population. This process continues to iterate until the preset maximum number of iterations is reached or the convergence condition is satisfied. The finally output solution set is the Pareto optimal solution set, which serves as the solution to the multi-objective optimization problem.

[0163] To verify the superiority of the improved NSGA-II algorithm, the improved NSGA-II algorithm (lmproved NSGA-II) and the traditional NSGA-II algorithm (Traditional NSGA-II) are compared and tested on the ZDT series of test functions, where True PF represents the true Pareto front. Among them, Figure 3 shows the comparison between the improved NSGA-II algorithm and the traditional NSGA-II algorithm on the ZDT1 function (Comparison results on ZDT1). Objective function - f1 is the objective function 1 in ZDT1, and Objective function - f2 is the objective function 2 in ZDT1. Figure 4Shows the comparison between the improved NSGA-II algorithm and the traditional NSGA-II algorithm on the ZDT2 function (Comparison results on ZDT2). Objective function - f1 is the first objective function in ZDT2, and Objective function - f2 is the second objective function in ZDT2. Figure 5 Shows the comparison between the improved NSGA-II algorithm and the traditional NSGA-II algorithm on the ZDT3 function (Comparison results on ZDT3). Objective function - f1 is the first objective function in ZDT3, and Objective function - f2 is the second objective function in ZDT3. Figure 6 Shows the comparison between the improved NSGA-II algorithm and the traditional NSGA-II algorithm on the ZDT4 function (Comparison results on ZDT4). Objective function - f1 is the first objective function in ZDT4, and Objective function - f2 is the second objective function in ZDT4.

[0164] The following takes the multi-objective optimization of a specific flat permanent magnet synchronous linear motor as an example for illustration.

[0165] Aiming at optimizing the average thrust and thrust ripple of the flat permanent magnet synchronous linear motor, the edge tooth width (ETw), edge tooth height (ETh), left end tooth width (UT1), right end tooth width (UT2), and permanent magnet width (Wpm) of the flat permanent magnet synchronous linear motor are selected as design variables.

[0166] The design variables of the flat permanent magnet synchronous linear motor are marked as Figure 3 shown. The value ranges of the design variables of the flat permanent magnet synchronous linear motor are shown in Table 1.

[0167] Table 1 Value ranges of design variables

[0168]

[0169] According to the number of variables, m sample is 250 sample points are taken through Latin hypercube sampling, and 250 experimental points are generated using the LHS method. LHS divides the design space into 250 equal-probability intervals and randomly selects a sample point in each interval to ensure that the values of each variable are evenly distributed throughout the design space. In this way, the 250 generated sample points have high distribution uniformity and can better describe the variation law of the target variables in the design space.

[0170] The generated sample point matrix form is as follows:

[0171]

[0172] Among them, x i = [x i1 , x i2 ,..., x id T represents the d design parameters of the i-th sample point, where d is the dimension of the design variable.

[0173] Import the 250 generated experimental points into the finite element analysis (FEA) model, and perform motor performance simulations for each experimental point respectively. Obtain the corresponding optimization objective values f1(x) (average thrust) and f2(x) (thrust fluctuation) through finite element calculations. The calculation results correspond one-to-one with the experimental points, forming a complete data set.

[0174] Utilize the experimental points generated by Latin Hypercube Sampling (LHS) and their corresponding finite element simulation objective values to construct a training data set (X train , Y train ). Among them, the input feature matrix represents the d key parameters in the experimental design, and the objective value matrix Y train contains two objective function values: average thrust F avg and thrust fluctuation F rip .

[0175] The training process of the random forest model includes three steps. The first step is to draw subsamples from the training data set with replacement to construct multiple decision trees, and each tree only uses part of the samples and features. The second step is to split the nodes of each tree through random feature selection to ensure the diversity of the model. The third step is to integrate the regression results of all decision trees and take the average as the final prediction value of the model.

[0176] After training is completed, input the test set data into the random forest regression model to obtain the prediction values of the model. The test data comes from independent sample points separated from the training set and is used to evaluate the generalization ability of the model. The model performance is quantitatively evaluated through the following two metrics:

[0177] Coefficient of determination R 2 :

[0178]

[0179] Among them, y i is the true value, is the predicted value, is the mean of the true values, m test is the number of test samples. R 2 ​It reflects the interpretability of the model for the target variable, with a value range of [0, 1]. The closer the value is to 1, the better the model fitting effect.

[0180] Root Mean Square Error (RMSE):

[0181]

[0182] RMSE measures the deviation between the predicted value and the true value. The smaller the value, the higher the prediction accuracy of the model.

[0183] Table 2 Model Performance

[0184]

[0185] The improved NSGA-II algorithm is selected to conduct multi-objective optimization design on the model. During the optimization process, the results of each iteration need to undergo crossover and mutation operations. The solutions with better performance are retained by the elite strategy and evaluated together with more new solutions in the next iteration. The optimal result shows that when the number of iterations approaches 200 times, the improved NSGA-II has good convergence. This verifies the rationality of the optimal method proposed in this embodiment. Table 3 shows the comparison between the optimized motor and the motor with the original structure after 200 iterations of the two objectives. The motor thrust before and after optimization is as Figure 7 shown.

[0186] Table 3 Comparison Before and After Optimization

[0187]

[0188]

[0189] Compared with the original structure, the average thrust of the motor with optimal parameters has decreased by 3.91%. As the motor operates, the thrust ripple of the optimal motor gradually becomes stable, the peak-to-peak thrust has decreased by 89.89%, the thrust fluctuation has dropped to 10.52% of the original motor, and the final thrust fluctuation rate is 2.43%.

[0190] Embodiment 2

[0191] Based on the same concept, this embodiment also provides an electronic device, which includes a processor, a memory, and a communication bus. Among them, the processor and the memory complete communication with each other through the communication bus.

[0192] Among them, the memory stores a computer program executable by the processor, and the processor is used to execute the program stored in the memory to implement the multi-objective optimization method of the flat permanent magnet synchronous linear motor as described in Embodiment 1.

[0193] The communication bus mentioned in the above electronic device may be a Peripheral Component Interconnect (PCI) bus, an Extended Industry Standard Architecture (EISA) bus, or the like. This communication bus can be divided into an address bus, a data bus, a control bus, and the like.

[0194] The memory may include a Random Access Memory (RAM), and may also include a non-volatile memory, such as at least one disk memory. Optionally, the memory may also be at least one storage device located far from the aforementioned processor.

[0195] The aforementioned processor may be a general-purpose processor, including a Central Processing Unit (CPU), a Network Processor (NP), etc., and may also be a Digital Signal Processor (DSP), an Application Specific Integrated Circuit (ASIC), a Field Programmable Gate Array (FPGA for short), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components.

[0196] Embodiment III

[0197] Based on the same concept, this embodiment also provides a computer-readable storage medium storing a computer program, and when the computer program is executed by a processor, it implements the multi-objective optimization method of the flat permanent magnet synchronous linear motor as described in Embodiment I.

[0198] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk memories, CD-ROMs, optical memories, etc.) containing computer-usable program codes.

[0199] The present application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, as well as the combination of flows and / or blocks in the flowcharts and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing devices generate a device for implementing the functions specified in Figure 1 one process or multiple processes and / or blocks Figure 1 one block or multiple blocks.

[0200] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to work in a particular manner, such that the instructions stored in the computer-readable memory produce a manufacture including an instruction device that implements the functions specified in one or more processes and / or blocks Figure 1 in one or more processes and / or blocks Figure 1 specified in the function.

[0201] These computer program instructions can also be loaded onto a computer or other programmable data processing apparatus, such that a series of operational steps are performed on the computer or other programmable apparatus to produce a computer-implemented process, whereby the instructions executed on the computer or other programmable apparatus provide steps for implementing the functions specified in one or more processes and / or blocks Figure 1 in one or more processes and / or blocks Figure 1 specified in the function.

[0202] Although the preferred embodiments of the present application have been described, those skilled in the art can make additional changes and modifications once they learn of the basic creative concepts. Therefore, the appended claims are intended to be construed to include the preferred embodiments as well as all changes and modifications falling within the scope of the present application.

[0203] Obviously, those skilled in the art can make various changes and modifications to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application fall within the scope of the claims of the present application and their equivalent technologies, the present application is also intended to include these modifications and variations.

Claims

1. A multi-objective optimization method for a flat permanent magnet linear synchronous motor, characterized in that: The steps include: Step S1: taking the average thrust and thrust fluctuation of the flat permanent magnet synchronous linear motor as optimization targets, analyzing the design parameters that affect the average thrust and thrust fluctuation, and determining the value ranges of the average thrust and thrust fluctuation; Step S2: construct a sample library through Latin hypercube experimental design; Step S3: establishing a random forest proxy model based on the sample library; Step S4: Improve the NSGA-Ⅱ algorithm, introduce PID adjustment variables and zero output disturbance terms, and update the individuals; and use the improved NSGA-Ⅱ algorithm to perform multi-objective optimization on the random forest agent model, and obtain the optimal design parameters of the flat permanent magnet synchronous linear motor through the improved NSGA-Ⅱ algorithm.

2. The multi-objective optimization method for a planar permanent magnet linear synchronous motor according to claim 1, characterized in that: In the step S1, the design parameters affecting the average thrust and thrust fluctuation include the side tooth width, side tooth height, left end tooth width, right end tooth width, and permanent magnet width of the flat permanent magnet linear synchronous motor.

3. The multi-objective optimization method for a planar permanent magnet linear synchronous motor according to claim 1, characterized in that: The step S2 specifically includes: adopting Latin hypercube test design, selecting a number of sample points in the sample space, and calculating the objective function value of the average thrust and thrust fluctuation of each sample point through numerical simulation, and the sample points and the objective function value together constitute a sample library.

4. The multi-objective optimization method for a planar permanent magnet linear synchronous motor according to claim 3, characterized in that: The Latin hypercube test design achieves uniform distribution of samples by dividing equal probability intervals in multidimensional space and selecting samples in each probability interval. The specific process is as follows: For each variable X i (i=1,2,...,d), and define its range as [a i ,b i ], d is a positive integer; Set the range of each variable [a i ,b i ] is divided into m sample equal width intervals, m sample is a positive integer, the width of each interval is Δx i for: Randomly select a point x in each interval i,j : x i,j =a i +(j-1)·Δx i +u i,j ·Δx i Among them, u i,j is a uniform random number on [0,1], j = 1, 2, ..., m sample ; The sampling points of each variable are randomly arranged to ensure that each interval is used only once in each dimension; the arranged sampling points are combined into m sample sample points, forming a sampling matrix X LHS : Among them, π i is a random permutation of the ith variable.

5. The multi-objective optimization method for a planar permanent magnet linear synchronous motor according to claim 1, characterized in that: The step S3 specifically includes: Extract samples from the sample library to form a training subset and a test subset; Use the training subset to train the random forest proxy model to obtain appropriate model parameters; during the training process, the random forest constructs multiple decision trees and randomly samples features and data to reduce overfitting and improve the generalization ability of the model; After training, the model is tested on the test subset and the prediction performance of the model is measured by calculating the coefficient of determination and / or root mean square error.

6. The multi-objective optimization method for a planar permanent magnet linear synchronous motor according to claim 5, characterized in that: In step S4, the NSGA-II algorithm is improved, PID adjustment amount and zero output disturbance term are introduced, and individuals are updated, specifically including: Generate an initial population uniformly distributed in the search space through a logical mapping mechanism; calculate the individual objective function value, and select elite individuals through non-dominated sorting and crowding distance calculation for subsequent crossover and mutation operations; Different crossover and mutation strategies are used at different stages of the algorithm. In the early iterations, two-point crossover is used for crossover to enhance global search capabilities, and random mutation is used for mutation to explore new areas. In the later iterations, simulated binary crossover and adaptive mutation are used to enhance local development capabilities. The update weight is determined by calculating the individual fitness and the distance from the Pareto frontier. The individuals are updated in combination with the PID adjustment amount and the zero output disturbance term to further improve the search efficiency and global optimization ability of the algorithm.

7. The multi-objective optimization method for a planar permanent magnet linear synchronous motor according to claim 6, characterized in that: The above method of selecting elite individuals through non-dominated sorting and crowding distance calculation is as follows: The population is non-dominated and sorted, individuals are stratified according to their merits and demerits, and the crowding distance of individuals is calculated. Elite individuals with large crowding distance and high non-dominated level are preferentially selected to enter the next generation, effectively maintaining population diversity and guiding the population to evolve towards the Pareto frontier.

8. The multi-objective optimization method for a planar permanent magnet linear synchronous motor according to claim 7, characterized in that: In step S4, the random forest proxy model is optimized by multi-objective optimization using the improved NSGA-Ⅱ algorithm, and the optimal design parameters of the flat permanent magnet synchronous linear motor are obtained by optimizing the improved NSGA-Ⅱ algorithm, which specifically includes: Determine whether the model has reached the set population generation number; if it has reached the set population generation number, proceed to the next step; otherwise, continue to iterate and optimize the new population until the set population generation number is reached; The Pareto front obtained by the final optimization is exported, the final optimized motor parameters are selected according to the requirements, the motor model is drawn, and the motor is numerically simulated and solved.

9. An electronic device, characterized in that: It includes a processor, a memory and a communication bus, wherein the processor and the memory communicate with each other via the communication bus; The memory is used to store computer programs; The processor is used to execute the program stored in the memory to implement the multi-objective optimization method of the flat-type permanent magnet synchronous linear motor according to any one of claims 1 to 8.

10. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the multi-objective optimization method for a flat-type permanent magnet linear synchronous motor as claimed in any one of claims 1 to 8 is implemented.

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