A method for solving the inverse mapping of the Kriging surrogate model of motor performance based on coordinate descent method

Abstract

The present invention discloses a method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method. First, different motor size parameter combinations are sampled, and the samples are used to construct a Kriging proxy model with multiple motor size parameters as input variables and a single motor performance as the output response. #imgabs0# Then, given the motor performance y c , #imgabs1# is used as the minimization objective function, and the range of motor size parameters in the samples from the Kriging proxy model training source is used as the feasible domain to construct the optimization model. Finally, the objective function is repeatedly minimized along the coordinate axis direction corresponding to the single component of the input variable based on the coordinate descent method, so that the objective function finally converges to 0, and the given motor performance y is obtained. c Corresponding motor size parameter combination #imgabs2#The method proposed in this invention can solve the inverse mapping of the Kriging proxy model of motor performance, and has good practicality and economy.

Description

Technical Field

[0001] The invention relates to a method for solving inverse mapping of a Kriging proxy model of motor performance based on a coordinate descent method, and belongs to the technical field of motor body simulation design. Background Art

[0002] Applications such as motor sizing optimization and sensitivity analysis often require traversing hundreds or even thousands of simulation models to calculate results. However, each simulation calculation is extremely expensive. Proxy models solve this problem by fitting the mapping relationship between parameter combinations and simulation results, saving simulation computing resources that consume a significant amount of time. However, most proxy models only implement unidirectional mapping. For example, a proxy model that fits the mapping relationship between multiple motor sizing parameters and a single motor performance can only use sizing parameter combinations as input variables and motor performance as output responses. It cannot output a qualified motor sizing parameter combination given the motor performance.

[0003] The Kriging interpolation method, systematically proposed by Georges Matheron in "Principles of Geostatistics," can be used as a regression model. It fits the mapping relationship between input variables and output responses using known sampled data, while also providing the predicted value and variance of the predicted value at any point. This method offers high approximation accuracy and generates a more accurate spatial prediction model. However, when fitting the mapping relationship between input variables and output responses, the Kriging model can only predict the output response from a given input variable; it cannot reversely predict the value of the input variable from a given output response.

[0004] In "Optimization Theory and Algorithms", Chen Baolin demonstrated the external penalty function method of converting the equality condition into the square term of the function and adding it to the objective function. This method can also be used to solve problems where the function expression is too complex to find its inverse mapping, and to convert the inverse mapping problem into an optimization model for solution. At the same time, the steepest descent method based on the Armjio criterion mentioned in the book was first proposed by Cauchy. It iterates along the negative gradient direction and has a faster convergence speed when solving unconstrained convex optimization problems. However, when the inverse mapping problem of the Kriging model is converted into an optimization problem, the model is only accurate in the sampling area, and the converted optimization problem needs to be solved in the feasible domain with the sampling area as the feasible domain.

[0005] The projected gradient method, proposed by Rosen in "The Gradient Projection Method for Nonlinear Programming," is a generalization of the steepest descent method for constrained optimization problems. It projects the gradient descent direction onto the feasible region, ensuring that the optimization variables obtained during each iteration remain within the feasible region. This allows for rapid solutions to constrained nonconvex optimization problems. However, when applied to optimization problems transformed from the inverse mapping problem of the Kriging model, the excessive number of optimization variables and the complexity of the model ultimately lead to the optimization model becoming trapped in a local optimum. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method, which can provide a motor simulation model size parameter combination corresponding to the output given performance when the motor performance value is given.

[0007] The present invention adopts the following technical solutions to solve the above technical problems:

[0008] Determine multiple motor size parameters and calculate m different motor size parameter combinations s through finite element simulation analysis method i Corresponding motor performance y i (i=1,2,...,m) as the sample data set (S,Y); then the motor size parameters are used as the multidimensional input variables X of the proxy model, and the motor performance is used as the output response y of the proxy model. The Kriging proxy model is trained by the sample data set (S,Y) Used to fit the mapping relationship between motor parameter combination and motor performance;

[0009] For a given motor performance constant y c ,by As the objective function, the variation range of motor size in the sample data set (S, Y) is taken as the feasible domain of the multidimensional variable X, and a minimization optimization model is constructed.

[0010] For the minimization optimization model constructed above, given a random multi-dimensional initial optimization variable X 0 , randomly select the multidimensional variable X 0 A component of , the value of the component that minimizes the objective function is obtained by the one-dimensional projected gradient method, and then the initial proxy model input variable X is updated 0 The value of this component in the new stage agent model input variable X is obtained 1 , and the process of updating the components of the proxy model input variable X is repeated until the objective function of the optimization model converges to the global minimum value 0, and the motor performance constant y is obtained. c Corresponding motor size parameter combinations as input variables of the proxy model

[0011] As a preferred solution of the present invention, the sample data set (S, Y) required for the Kriging proxy model training is obtained by performing finite element model simulation analysis and calculation on the motor through Latin hypercube sampling. Any sample s in S i =(s i1 ,s i2 ,...,s in ) is an n-dimensional vector, corresponding to n motor size parameters.

[0012] As a preferred solution of the present invention, the Kriging proxy model needs to perform maximum value normalization preprocessing on the sample data set during training.

[0013] As a preferred solution of the present invention, the basis functions of the Kriging proxy model include but are not limited to constant basis functions, linear basis functions, and quadratic polynomial basis functions.

[0014] As a preferred solution of the present invention, the Kriging proxy model inverse mapping does not require the original mapping to be a bijection in the traditional sense, but is a generalized inverse mapping that does not require a one-to-one correspondence.

[0015] As a preferred solution of the present invention, the hyperparameter tuning process of the Kriging proxy model is completed by a genetic algorithm.

[0016] As a preferred solution of the present invention, the objective function of the minimization optimization model is the square of the difference between the given performance value and the Kriging proxy model, that is, feasible region It is determined by the range of sample values ​​in the sample data set S of the constructed Kriging surrogate model, that is, the j-th component X of the variable X j The feasible range is [X j min ,X j max ],in The final optimized model is as follows:

[0017]

[0018] As a preferred solution of the present invention, the motor performance includes but is not limited to motor torque, loss, etc.

[0019] As a preferred solution of the present invention, the component updating process of the proxy model input variable X is specifically performed as follows:

[0020] For a randomly selected sequence number j, the jth component of the new stage input variable X' is solved by the one-dimensional projected gradient method As a new component, the other components remain unchanged compared with the original input variable X, that is, where X=(X1,X2,...,X n ), X'=(X1',X2',...,X n ').

[0021] As a preferred solution of the present invention, the step length search process of a single component in the one-dimensional projected gradient method follows the Armjio criterion.

[0022] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0023] 1. The present invention samples different motor parameter combinations, trains the motor performance Kriging proxy model through the samples, and uses the function Indicates that X is the input variable representing the motor parameter combination, and the function returns the predicted value of the motor performance by the Kriging model. Then, given the required performance y c Under the premise of The objective function is the optimization model with the sample area collected for Kriging surrogate model training as the feasible domain. When the target value of the optimization model is 0, the optimal solution is the motor parameter combination corresponding to the given required performance. This method is very simple and easy to understand. It can effectively obtain the inverse mapping result while ensuring the accuracy of the Kriging surrogate model in the sample area.

[0024] 2. The present invention solves the optimization problem after the inverse mapping problem of the Kriging model is transformed based on the coordinate descent method, that is, in each stage, a component of the optimization variable is randomly selected, and the solution is obtained by the one-dimensional projection gradient method along the coordinate axis direction corresponding to the component, until the final optimization target value converges to 0. The optimal solution finally obtained is the value obtained by the inverse mapping of the Kriging model, and when the value of the initial optimization variable is changed, the value of the optimal solution finally obtained will also change. Then, multiple groups of motor size parameter combinations corresponding to given performance values ​​can be provided by changing the initial optimization variable value multiple times. The optimization model after the inverse mapping problem of the Kriging model solved based on the coordinate descent method can effectively obtain the optimal solution that makes the objective function converge to 0. The coordinate descent method converts the optimization model solving process of high-dimensional optimization variables into an optimization model of multiple one-dimensional optimization variables, simplifies the solution process, and has strong flexibility. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1It is a flow chart of a method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method of the present invention;

[0026] Figure 2 This is a structural diagram of a δ-type permanent magnet flat wire motor with a stator and 48 slots and a rotor and 8 poles according to an embodiment;

[0027] Figure 3 It is the distribution of the number of iterations of 100 repeated experiments to solve the surrogate model torque inverse mapping optimization model;

[0028] Figure 4 This is the distribution of the number of iterations of 100 repeated experiments to solve the proxy model loss inverse mapping optimization model. DETAILED DESCRIPTION

[0029] The embodiments of the present invention are described in detail below, and examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be interpreted as limiting the present invention.

[0030] In the process of motor parameter design, simulation analysis is expensive. The proxy model establishes a one-way mapping relationship between motor size parameters and motor performance, saving most of the time spent on simulation analysis. At the same time, motor parameter design often requires determining the size parameters of the motor based on the actual application performance requirements of the motor. Under the premise of using the proxy model to reduce the cost of simulation analysis, the inverse mapping between the motor performance and the motor size parameters of the proxy model is obtained. The technical solution adopted by the present invention is to establish a Kriging proxy model in advance with motor size parameters as input variables and motor performance as output response, and then convert the inverse mapping problem into the solution of the optimization model, and simplify the solution process based on the coordinate descent method, so as to finally obtain the motor parameter combination corresponding to the given motor performance.

[0031] Taking a δ-type permanent magnet flat wire motor with 48 stator slots and 8 rotor poles as an example, the implementation process of the present invention is described in detail. Figure 2 This is the structural diagram of the motor.

[0032] The overall implementation process of the present invention is as follows Figure 1 As shown in the figure, first, multiple motor size parameters are determined as input variables to build the Kriging proxy model, and the motor size parameters and performance sample data sets are collected through experimental design to establish the Kriging proxy model. Used to predict motor performance; then for a given motor performance constant y c , the squared difference between the Kriging surrogate model prediction function and the given value As the objective function, the variation range of motor dimensions in the sample data set is taken as the feasible domain of the multidimensional variable X, and a minimization optimization model is constructed. Finally, the optimization model is solved based on the coordinate descent method. At each stage, a component of the optimization variable is randomly selected and solved along the coordinate axis corresponding to the component using the one-dimensional projected gradient method until the final optimization target value converges to 0.

[0033] In the Kriging model inverse mapping solution strategy proposed in the invention, first, the motor parameter input variables and the motor performance output response of the constructed Kriging proxy model are determined. In this embodiment, the 11 selected motor parameter input variables and initial parameter values ​​are shown in Table 1. The motor performance as the output response is the torque mean during the stable period of the motor operating condition or the mean total motor loss. The total loss includes the motor core loss, eddy current loss and copper loss.

[0034] Table 1 Main dimensions and parameters of the δ-type permanent magnet flat wire motor with 48 stator slots and 8-pole rotor

[0035] Serial number Input variable description Variable designation Initial value 1 Stator inner diameter mm 127.6 2 Stator slot width mm 4 3 Stator slot height mm 14 4 slot opening mm 1.5 5 Tooth depth mm 1 6 <![CDATA[V-shaped magnet L1 angle]]> ° 53 7 <![CDATA[Thickness of L1 permanent magnet]]> mm 3.5 8 <![CDATA[Width of L1 permanent magnet]]> mm 14.75 9 <![CDATA[Width of the magnetic bridge of the L1 permanent magnet]]> mm 3 10 <![CDATA[Thickness of L2 permanent magnet]]> mm 3 11 <![CDATA[Width of L2 permanent magnet]]> mm 6

[0036] By using Latin hypercube sampling, changing the value of the input variable, and performing finite element simulation analysis, the initial data set (S, Y) of the motor performance is obtained. Any sample s in S i =(s i1 ,s i2 ,...,s in ) is an n-dimensional vector (n is 11), corresponding to the 11 motor size parameters in the table. In this embodiment, a total of m groups of data are collected (m is 100).

[0037] The Kriging proxy model is trained using the motor performance dataset, and the dataset is preprocessed with maximum value normalization during the training process. The 11 motor size parameters in the table above are used as the multidimensional input variables X of the proxy model, and the motor performance is used as the output response y of the Kriging proxy model. The Kriging proxy model is obtained by training the sample dataset (S, Y) It is used to fit the mapping relationship between multiple motor parameters and motor performance and predict motor performance. The Kriging proxy model prediction expression can be given by the following formula:

[0038]

[0039] Among them, the input variable X is an n-dimensional vector, f(X) is the basis function of the regression model,

[0040] f(X)=[f1(X),f2(X),…,f p (X)] T (2)

[0041] In this example, when f(X) is a constant basis function, that is, p=1, f1(X)=1, r(X) is the correlation vector between X and S,

[0042]

[0043] It is defined as follows, where θ is a hyperparameter and the value of θ is determined by the genetic algorithm so that the correlation coefficient between the predicted value of the surrogate model and the actual value is maximized.

[0044]

[0045] β * and γ * is the weight vector of the corresponding function,

[0046] β * =(F T R -1 F) -1 F T R -1 Y (5)

[0047] γ * =R -1 (Y-Fβ * ) (6)

[0048] Where F and R are

[0049]

[0050] For a given motor performance constant y c In this example, the given value of torque is 120Nm and the given value of total loss is 2500W. As the objective function, the feasible region It is determined by the range of sample values ​​in the sample data set S of the constructed Kriging surrogate model, that is, the j-th component X of the variable X j The feasible range is [X j min ,X j max ],in The optimization model transformed from the inverse mapping problem of the Kriging surrogate model is as follows:

[0051]

[0052] Given a random multidimensional initial surrogate model input variable Randomly select the multidimensional variable X 0A component of , the value of the component that minimizes the objective function is obtained by the one-dimensional projected gradient method, and then the initial proxy model input variable X is updated 0 The value of this component in the new stage agent model input variable X 1 , and the process of updating the components of the proxy model input variable X is repeated until the objective function of the optimization model converges to the global minimum value 0 or the total number of iterations in all stages is greater than or equal to the maximum number of iterations k max , and get the motor performance constant y c Corresponding motor size parameter combinations as input variables of the proxy model The objective function converges to 0 in the termination condition and the algorithm stops when the objective function is less than ε (ε is set to 1e-5 in this example).

[0053] When determining the component value that minimizes the objective function, assume that the sequence number j is randomly selected and the j-th component of the input variable X' in the new stage is solved by the one-dimensional projected gradient method. As a new component, the other components remain unchanged compared with the original input variable X, that is, where X=(X1,X2,...,X n ), X'=(X1',X2',...,X n ').

[0054] Solving the optimization model using the coordinate descent method was compared with solving the multi-variable optimization model directly using the projected gradient method. The experiment was repeated 100 times with varying initial optimization variables. Tables 2 and 3 show the results obtained by directly solving the optimization model for the torque and loss inverse mapping using the gradient projection method, respectively. In this case, the gradient projection method updates all components of X simultaneously in each iteration. It can be seen that directly using the gradient projection method, which updates all components of X in each iteration, has a high probability of causing the objective function to not converge to zero. Across the 100 repeated experiments, the direct solution resulted in convergence of the objective function to zero in only 50% to 60% of the cases.

[0055] Table 2 Direct solution results of the surrogate model torque inverse mapping

[0056] Iteration interval Frequency Average time (seconds) k≤10 21 1.17 10<k≤20 25 2.40 k>20 10 3.90 No convergence 44 3.10 total 100 259.92

[0057] Table 3 Direct solution results of proxy model loss inverse mapping

[0058] Iteration interval Frequency Average time (seconds) k≤10 13 2.40 10<k≤20 19 4.62 k>20 22 9.45 No convergence 46 5.72 total 100 589.81

[0059] When solving the optimization model after the inverse mapping problem of motor performance is transformed based on the coordinate descent method, only the value of one component of X is changed when updating the optimization variable at each stage, which simplifies the solution process and can effectively make the optimization model objective function converge to 0. Similar to directly solving the inverse mapping problem, the initial optimization variable is also changed, and the experiment is repeated 100 times with the number of iterations. The results are shown in Tables 4 and 5. In this case, k refers to the sum of the number of iterations of the one-dimensional projected gradient method in all stages to solve the optimization model. Compared with the direct use of the projected gradient method, Figure 3 and Figure 4 The following are the distributions of the number of iterations for 100 repeated experiments, respectively, for solving the inverse mapping of torque and loss motor performance. Solving the optimization model using the coordinate descent method updates only one X component at each stage, rather than updating all components simultaneously. This approach not only shortens the total time required to solve the optimization model but also significantly increases the percentage of repeated experiments where the optimization model objective function converges to zero. The percentage of optimization model objective functions that fail to converge to zero for the inverse mapping of torque and loss remains below 5%.

[0060] Table 4. Solution results of the inverse mapping of the surrogate model torque based on the coordinate descent method

[0061] Iteration interval Frequency Average time (seconds) k≤10 58 0.88 10<k≤20 23 1.43 k>20 19 2.89 No convergence 0 - total 100 138.97

[0062] Table 5. Results of proxy model loss inverse mapping based on coordinate descent method

[0063] Iteration interval Frequency Average time (seconds) k≤10 48 2.56 10<k≤20 50 5.04 k>20 1 8.92 No convergence 1 22.01 total 100 405.89

[0064] The above embodiments are merely exemplary descriptions of the present invention and do not limit its scope of protection. Those skilled in the art may also make partial changes thereto. For example, the motor type and motor performance may be selected based on actual conditions, and the type of proxy model used to fit the mapping relationship between the motor size parameter combination and the motor performance may be selected based on actual conditions. Any form of equivalent replacement that complies with the spirit of the invention falls within the scope of protection of the present invention.

Claims

1. A method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method, characterized in that: The method is specifically as follows: Calculate m different motor size parameter combinations s i Corresponding motor performance y i , i=1,2,...,m, as the sample data set (S,Y); then the motor size parameters are used as the multidimensional input variables X of the proxy model, and the motor performance is used as the output response y of the proxy model. The Kriging proxy model is trained by the sample data set (S,Y) Used to fit the mapping relationship between motor parameter combination and motor performance; For a given motor performance constant y c ,by As the objective function, the range of motor size in the sample data set (S, Y) is taken as the feasible domain of the multidimensional variable X, and a minimization optimization model is constructed; For the minimization optimization model constructed above, given a random multi-dimensional initial optimization variable X 0 , randomly select X 0 A component of , find the value of the component that minimizes the objective function, and then update X with this value 0 , get the new stage Kriging proxy model input variable X 1 , and repeatedly update the components of the Kriging proxy model input variable X until the objective function of the minimization optimization model converges to the global minimum value 0, and the motor performance constant y is obtained. c Corresponding motor size parameter combinations as input variables of the Kriging proxy model 2. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The sample data set (S, Y) required for the Kriging proxy model training is obtained by performing finite element model simulation analysis and calculation on the motor through Latin hypercube sampling. Any sample s in S i =(s i1 ,s i2 ,...,s in ) is an n-dimensional vector, corresponding to n motor size parameters.

3. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The Kriging proxy model needs to perform maximum value normalization preprocessing on the sample data set during training.

4. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The basis functions of the Kriging proxy model include but are not limited to constant basis functions, linear basis functions, and quadratic polynomial basis functions.

5. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The inverse mapping of the Kriging proxy model does not require the original mapping to be a bijection in the traditional sense, but is a generalized inverse mapping that does not require a one-to-one correspondence.

6. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The hyperparameter tuning process of the Kriging surrogate model is completed by genetic algorithm.

7. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The objective function of the minimization optimization model is the square of the difference between the given performance value and the Kriging proxy model, that is, The feasible region is determined by the range of sample values ​​in the sample data set S of the Kriging surrogate model, that is, the j-th component X of the variable X j The feasible range is [X j min ,X j max ],in 8. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The motor performance includes but is not limited to motor torque and loss.

9. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1 is characterized in that: The component update process of the Kriging surrogate model input variable X is as follows: For a randomly selected sequence number j, the jth component of the new stage input variable X' is solved by the one-dimensional projected gradient method As a new component, the other components remain unchanged compared with the original input variable X, that is, where X=(X1,X2,...,X n ), X'=(X1',X2',...,X n ').

10. The method for solving the inverse mapping of the Kriging proxy model of motor performance based on the coordinate descent method according to claim 1, characterized in that: The value of the component that minimizes the objective function is obtained by a one-dimensional projected gradient method, wherein the search process of a single component step length in the one-dimensional projected gradient method follows the Armjio criterion.

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