Method and system for predicting electromagnetic force of electromagnetic system based on Gaussian process regression hybrid model

Abstract

The invention discloses an electromagnetic system electromagnetic force prediction method and system based on a Gaussian process regression hybrid model, and the method comprises the steps: building a simplified physical model, inputting a system state into the simplified physical model, and outputting a reference prediction value; constructing a covariance kernel function, wherein the covariance kernel function is used for representing a residual error between a predicted value and a true value of the simplified physical model; establishing a Gaussian process regression model, wherein the Gaussian process regression model learns a residual error by means of a covariance kernel function; training data is utilized to train the Gaussian process regression model; combining the simplified physical model and the trained Gaussian process regression model to form a hybrid model; and inputting a new system state into the hybrid model, and adding a reference predicted value output by the simplified physical model and a residual predicted value output by the Gaussian process regression model by the hybrid model to obtain a final predicted value.

Description

A method and system for predicting electromagnetic force in electromagnetic systems based on a Gaussian process regression hybrid model. Technical Field

[0001] This invention relates to the field of electromagnetic system technology, and in particular to a method and system for predicting electromagnetic forces in electromagnetic systems based on a Gaussian process regression hybrid model. Background Technology

[0002] Existing electromagnetic force prediction methods for electromagnetic systems cannot simultaneously meet the three core requirements for real-time control of electromagnetic systems (such as magnetic suspension systems): (a) High fidelity: The electromagnetic force characteristics generated by the combined effects of complex physical phenomena such as core nonlinearity, cross-magnetization, permanent magnet rod bias effect, and closed magnetic circuit of the yoke must be accurately reproduced. The error between the model prediction and the actual physical values ​​(or the high-fidelity finite element simulation reference values) must be limited to a suitable range.

[0003] (b) High computational efficiency (real-time performance): A single prediction must be completed within milliseconds (ms) or even microseconds (µs) to meet the real-time decision-making requirements of control systems with bandwidths of up to hundreds of hertz or even higher.

[0004] (c) Physical interpretability and generalization ability: The model used by the prediction method should not be a pure black box. Its internal structure should be able to utilize known physical laws so that it can still maintain reliable prediction ability and a certain extrapolation robustness in the entire working space (including sparse data regions) even under the condition of limited training data.

[0005] Therefore, there is an urgent need for an electromagnetic force / torque prediction method that meets the requirements of high fidelity, high computational efficiency, physical interpretability, and generalization ability. Summary of the Invention

[0006] This invention provides a method and system for predicting electromagnetic forces in electromagnetic systems based on a Gaussian process regression hybrid model, in order to solve the technical problems mentioned in the background art.

[0007] To achieve the above objectives, the technical solution of this invention is implemented as follows: This invention provides a method for predicting the electromagnetic force of an electromagnetic system based on a Gaussian process regression hybrid model, comprising the following steps: S1, constructing a simplified physical model based on the first principles of electromagnetism, inputting the new system state into the simplified physical model, and outputting a baseline prediction value; the new system state includes a new pose vector and a new current vector; S2, constructing a covariance kernel function, which is used to characterize the residual between the baseline prediction value and the true value of the simplified physical model, and is designed as a combination of the current kernel function and the pose kernel function; then constructing... S3. Build a Gaussian process regression model. The Gaussian process regression model is used to learn the residual between the predicted value and the true value of the simplified physical model by means of the covariance kernel function; S4. Train the Gaussian process regression model using training data to obtain the trained Gaussian process regression model; Combine the simplified physical model and the trained Gaussian process regression model to obtain a Gaussian process regression hybrid model; S5. Input the new system state into the Gaussian process regression hybrid model, and add the baseline predicted value output by the simplified physical model to the residual predicted value output by the trained Gaussian process regression model to obtain the final predicted value.

[0008] Further, S1 specifically includes the following steps: S11, based on the fundamental laws of electromagnetic fields, construct an analytical expression for the magnetic induction intensity generated by the magnetic source at any point in space, and then superimpose the magnetic fields generated by all magnetic sources to construct the entire electromagnetic system at the field point. The total magnetic field generated at the location The analytical expression; S12, based on the total magnetic field The analytical expression was derived, and the equivalent surface current density of the permanent magnet rod in the experimental model was analyzed. S13. The surface integral of the Lorentz force gives the analytical expressions for the electromagnetic force and torque acting on the permanent magnet rod within the experimental model; S14. Based on the analytical expressions for the electromagnetic force and torque, the integrals of the analytical expressions for the electromagnetic force and torque are solved using the Gauss-Legend numerical integration method to construct a self-consistent field analytical numerical model; S15. The new system state is input into the self-consistent field analytical numerical model to obtain the baseline predicted values ​​containing the main physical effects of the system. .

[0009] Furthermore, the total magnetic field in S11 The analytical expression is: ;in, Indicates the total magnetic field. This refers to any point in the workspace of an electromagnetic system, often simply called a field point. Indicates field point The total magnetic field; i represents the current magnetic source number; This represents the Euler rotation matrix that transforms the coordinate system of the i-th magnetic source to the absolute coordinate system; This indicates the field point. Representation in the coordinate system of the i-th magnetic source; Represents the excitation current of the i-th magnetic source. The generated magnetic field; Represents the equivalent surface magnetization current of the i-th magnetic core. The generated magnetic field; Represents the excitation current of the i-th magnetic source. At the scene The generated magnetic field; Represents the equivalent surface magnetization current of the i-th magnetic core. At the scene The generated magnetic field; , The calculation formulas are as follows: ;in, The magnetic field distribution function for a unit excitation current; The magnetic field distribution function per unit magnetizing current; Number of turns per unit length; It is the length of the i-th magnetic source; and These are the inner and outer diameters of the i-th magnetic source, respectively; Represents radius And the axial coordinate is The single-turn circular current at the field point The generated magnetic field; The magnetic fields along the x, y, and z axes are: ;in, , , They represent Magnetic fields along the x, y, and z axes; The vacuum permeability; It is the first user-defined variable, and ; It is a second user-defined variable, and Modulus ; and These are the first and second kind of complete elliptic integrals, respectively; the analytical expressions for the electromagnetic force and torque acting on the permanent magnet in S12 are as follows: ;in, Representation of the electromagnetic force on a permanent magnet rod in an absolute coordinate system; This represents the electromagnetic torque acting on the permanent magnet rod in the body coordinate system. This represents the Euler rotation matrix from the body coordinate system to the absolute coordinate system; This represents the total magnetic field in the body coordinate system. This represents the Euler rotation matrix from the absolute coordinate system to the body coordinate system. The center of mass of the permanent magnet points to the surface element. Position vector; Represents the cylindrical surface of the permanent magnet rod Find the integral.

[0010] Further, S2 specifically includes the following steps: S21, the residual between the predicted value and the true value of the simplified physical model. Perform physical heuristic decomposition to reduce residuals Decomposed into a pose-dependent coefficient matrix With the current vector Constructed set of current basis functions The product of these terms yields the residual. The decomposition formula; where the true value is the numerical result of high-fidelity finite element simulation. It is a vector composed of the currents of all magnetic sources; Let be the pose vector of the permanent magnet rod within the workspace of the electromagnetic system, and In the formula These represent pitch angle and yaw angle respectively; T represents transpose; S22, based on residuals The decomposition of the residual Expanding each component yields the expanded expression of the residual components; the expanded expression of the residual components contains the coefficient functions of different current basis functions; S23, for the coefficient function of each current basis function, two core statistical assumptions based on physical intuition are introduced: Assumption 1: It is assumed that the coefficient functions of different current basis functions are statistically uncorrelated; Assumption 2: It is assumed that the coefficient functions of all current basis functions share the same normalized pose correlation structure, but each has its own independent signal variance. To characterize the difference in the contribution intensity of different physical terms to the residual; S24, based on the two core statistical assumptions based on physical intuition, the covariance kernel function of the residual is derived; the covariance kernel function is derived from the current kernel function. Pose kernel function The tensor product is formed; S25, construct a Gaussian process regression model, which is used to learn the residuals between the predicted and true values ​​of the simplified physical model by means of the covariance kernel function.

[0011] Furthermore, the residual in S21 The decomposition formula is: The expanded expression for the residual components in S22 is: ;in, Residual The kth component; Represents the coefficient matrix The OK, for A learnable vector that depends solely on the pose vector. The coefficient function of the current basis function; Represents the set of current basis functions The j-th current basis function in S23; the expression for assumption 1 in S23 is as follows: ;in, To express the covariance; Indicates for any For all cases, Cov=0 holds true; , Indicates an index; Indicates index The corresponding pose vector; the expression for assumption 2 in S23 is as follows: ;in, This represents the pose kernel function.

[0012] Furthermore, the expression for the covariance kernel function in S2 is as follows: ;in, Represents the variance of the global pose signal; Represents an arbitrary input current vector; the The calculation formula is as follows: The The calculation formula is as follows: ;in, For anisotropic Mahalanobis distance, It is a modified Bessel function of the second kind. These are hyperparameters that control the smoothness of the function; It is a gamma function.

[0013] Further, S3 specifically includes the following steps: S31, collecting training data; S32, using the quasi-Newton algorithm L-BFGS and maximizing the log marginal likelihood to solve for the optimal hyperparameter set of the Gaussian process regression model. Thus, the trained Gaussian process regression model is obtained; among which, the optimal hyperparameter set is... The expression is as follows: In the formula, This represents the logarithmic marginal likelihood function; This represents the hyperparameter vector to be optimized. The definition of is: ;in, Let be the signal variance vector of the current basis function. This represents the signal variance of the nth current basis function. Represents the set of real numbers. for The feature length scale vector of the dimensional pose space; S33. Combine the simplified physical model with the trained Gaussian process regression model to obtain the Gaussian process regression hybrid model.

[0014] Furthermore, the expression for the logarithmic marginal likelihood in S32 is as follows: ;in, Represents the residual observation vector of the training data; The covariance kernel function is calculated as follows. Covariance matrix; The variance representing additive white Gaussian noise, for 3D identity matrix.

[0015] Further, S4 specifically includes the following steps: S41, inputting the new system state into the Gaussian process regression mixture model to obtain the predicted mean and predicted variance of each force / torque component; the predicted variance of each force / torque component is used to quantify the uncertainty of the stress / torque component prediction; wherein, the expression for the predicted mean is as follows: In the formula, This represents the predicted mean of the k-th force / torque component; This indicates a new system status input. Indicates the new current vector; This represents the new pose vector; The distance between the test point and the training set samples in the training data The cross-covariance vector; the expression for the prediction variance is as follows: ;in, This represents the prediction variance of the k-th force / torque component; S42. The autocovariance scalar of the test points; S43. The residual prediction value is obtained based on the predicted mean value of each force / torque component; S444. The baseline prediction value output by the simplified physical model is added to the residual prediction value output by the trained Gaussian process regression model to obtain the final prediction value.

[0016] In another aspect, the present invention provides an electromagnetic force prediction system for an electromagnetic system, configured to execute the above-described electromagnetic force prediction method for an electromagnetic system, wherein the electromagnetic force / torque prediction system for an electromagnetic system includes: a simplified physical model for real-time calculation of a reference prediction value; a basis function calculation module for real-time calculation of the value of the current basis function; and a coefficient lookup and interpolation module, which pre-stores a pre-calculated coefficient matrix. It is used to obtain the coefficient matrix based on real-time input and through interpolation. The corresponding coefficients; the residual calculation module, used to linearly combine the value of the current basis function with the obtained coefficients to obtain the residual prediction value; the adder, used to add the benchmark prediction value and the residual prediction value to output the final prediction value.

[0017] The beneficial effects of the present invention are as follows: 1. The present invention discloses an electromagnetic force prediction method for electromagnetic systems, which has the following advantages: High fidelity: The present invention controls the error between the model prediction (i.e. the final predicted value) and the real physics (or the numerical results of high-fidelity finite element simulation) within a reasonable range.

[0018] Extremely high data and computational efficiency: Since the Gaussian process only needs to learn the smoothed residuals, rather than the entire complex function, the amount of high-fidelity training data required is significantly reduced compared to pure black-box models, lowering data acquisition costs. Furthermore, while offline training involves hyperparameter optimization, online prediction costs are extremely low.

[0019] Excellent generalization and extrapolation robustness: This invention incorporates global physical laws described by a simplified physical model. Therefore, in operating conditions not covered by training data, its predictions will not be completely out of control like those of a pure black box model. Instead, it can make reasonable inferences along physical trends, significantly improving the reliability of its application in the global workspace.

[0020] 2. The self-consistent field analytical numerical model in this invention is a physical model that considers multiple factors such as the iron core and cross-magnetization coupling. However, it does not consider the residual between the baseline predicted value and the true value. Therefore, this invention uses a Gaussian process regression model to learn the residual between the baseline predicted value and the true value of the simplified physical model. This changes the traditional approach of non-parametric Bayesian regression models learning from scratch or using a simple constant mean function, transforming the learning objective from a complex "total mapping" to a smoother "residual mapping" with smaller variance, greatly improving data efficiency and model convergence.

[0021] 3. This invention designs a physical heuristic covariance kernel function. The covariance kernel function is not a general radial basis function, but rather forces the physical decoupling of input variables. This allows the Gaussian process to learn non-parametrically while its function space is constrained to a physically intuitive subspace. Attached Figure Description

[0022] Figure 1 is a cross-sectional view of the magnetic suspension system in an embodiment of the present invention; Figure 2 is a diagram of the magnetic source (coil) array arrangement structure of the magnetic suspension system in an embodiment of the present invention; Figure 3 is a schematic diagram of the axial and vertical magnetic source coordinate system when viewed from the front in an embodiment of the present invention; Figure 4 is a schematic diagram of the axial and lateral magnetic source coordinate system when viewed from above in an embodiment of the present invention.

[0023] Explanation of the reference numerals in the attached diagram: 0, Coil 0; 1, Coil 1; 2, Coil 2; 3, Coil 3; 4, Coil 4; 5, Coil 5; 6, Coil 6; 7, Coil 7; 8, Coil 8; 9, Coil 9. Detailed Implementation

[0024] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many other different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.

[0025] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0026] This invention uses a magnetic suspension system as an example to predict the electromagnetic force / torque of the magnetic suspension system. The magnetic source is a coil. First, the structure of the magnetic suspension system is shown in Figure 1. The system contains a coil array with coils in various orientations, including eight radial coils and two axial coils. The eight radial coils consist of four vertical coils and four lateral coils. The arrangement of the coils in the array is shown in Figure 2. Coil 0 and coil 9 are axial coils (hollow core coils); coils 1 to 8 are radial coils (iron core coils). Based on their main functions, coils 1, 3, 5, and 7 provide vertical force and are called vertical coils; coils 2, 4, 6, and 8 provide lateral force and are called lateral coils. The absolute coordinate system of the magnetic suspension system... Defined as: origin Located at the center of symmetry of the magnetic suspension system; The axis is along the horizontal airflow direction; The axis is vertically upward; The axis lies in the horizontal plane, and... The axes are perpendicular, and the three axes follow the right-hand rule.

[0027] The experimental model is located at the center of the magnetic suspension system. The rocket-like structure in the middle of Figures 3 and 4 is shown in the model, and a permanent magnet is embedded within it. The model's body coordinate system... Defined as: origin The center of mass of the experimental model coincides with the center of mass of the embedded permanent magnet rod (usually designed to be consistent with the center of mass of the experimental model). The shaft is set along the axis of the experimental model and points towards the tail. The axis is perpendicular to the plane of symmetry of the experimental model and points to the right side of the fuselage; The axis lies in the plane of symmetry, and... The axis is vertical and points upward.

[0028] Coordinate system of each coil in the magnetic levitation system Defined as: the origin of each coordinate system is defined at the geometric center of the corresponding coil; The shaft is set along the axial direction of the corresponding coil, where coil 0 (0) and coil 9 (9) are... The axis points to the right, with coil 1 and coil 8 pointing to the right. The shaft is vertically upward; coil 1 and coil 8. axial direction and absolute coordinate system The axes remain consistent, with coil 0 (0) and coil 9 (9) aligned. The axis points from the front to the back; The direction of the axis is determined by the right-hand rule.

[0029] This application provides a method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model, comprising the following steps: S1, constructing a simplified physical model based on the first principles of electromagnetism, inputting a new system state into the simplified physical model, and outputting a baseline prediction value; the new system state includes a new pose vector and a new current vector; S2, constructing a covariance kernel function, which is used to characterize the residual between the baseline prediction value and the true value of the simplified physical model, and is designed as a combination of a current kernel function and a pose kernel function; then constructing a Gaussian process regression model, which is used to learn the residual between the baseline prediction value and the true value of the simplified physical model with the help of the covariance kernel function; S3, training the hyperparameters of the Gaussian process regression model using training data to obtain a trained Gaussian process regression model; combining the simplified physical model and the trained Gaussian process regression model to obtain a Gaussian process regression hybrid model; S4, inputting the new system state into the Gaussian process regression hybrid model, adding the baseline prediction value output by the simplified physical model to the residual prediction value output by the trained Gaussian process regression model to obtain the final prediction value.

[0030] The present invention has the following advantages: High fidelity: The present invention controls the error between the model prediction (i.e. the final predicted value) and the real physics (or the numerical result of high-fidelity finite element simulation) within a reasonable range.

[0031] Extremely high data and computational efficiency: Since the Gaussian process only needs to learn the smoothed residuals, rather than the entire complex function, the amount of high-fidelity training data required is significantly reduced compared to pure black-box models, lowering data acquisition costs. Furthermore, while offline training involves hyperparameter optimization, online prediction costs are extremely low.

[0032] Excellent generalization and extrapolation robustness: This invention incorporates global physical laws described by a simplified physical model. Therefore, in operating conditions not covered by training data, its predictions will not be completely out of control like those of a pure black box model. Instead, it can make reasonable inferences along physical trends, significantly improving the reliability of its application in the global workspace.

[0033] In some embodiments, S1 specifically includes the following steps: S11. Based on the fundamental law of electromagnetic fields, namely the Biot-Savart law, construct an analytical expression for the magnetic induction intensity generated by a single coil in the coil array at any point in space. Then, superimpose the magnetic fields generated by all the coils in the coil array to construct the magnetic field intensity of the entire electromagnetic system array at the field point. The total magnetic field generated at the location The analytical expression; where, field point S12, based on the total magnetic field, represents the field points of the coil array in the absolute coordinate system. The analytical expression was derived, and the equivalent surface current density of the permanent magnet rod in the experimental model was analyzed. S13. The surface integral of the Lorentz force gives the analytical expressions for the electromagnetic force and torque acting on the permanent magnet rod within the experimental model; S14. Based on the analytical expressions for the electromagnetic force and torque, the integrals of the analytical expressions for the electromagnetic force and torque are solved using the Gauss-Legend numerical integration method to construct a self-consistent field analytical numerical model; S15. The new system state is input into the self-consistent field analytical numerical model to obtain the baseline predicted values ​​containing the main physical effects of the system. .

[0034] In some embodiments, the total magnetic field in S11 The analytical expression is: (1) Among them, Indicates the total magnetic field. This refers to any point in the workspace of an electromagnetic system, often simply called a field point. Indicates field point The total magnetic field; i represents the current coil number; This represents the Euler rotation matrix that transforms the coordinate system of the i-th coil to the absolute coordinate system; This indicates the field point. The representation of the i-th coil in the coordinate system, and This represents the mapping from the absolute coordinate system to the coordinate system of the i-th coil; Represents the excitation current of the i-th coil. At the scene The generated magnetic field; Represents the equivalent surface magnetization current of the i-th magnetic core. At the scene The magnitude of the generated magnetic field is determined by the magnetization of the iron core. Decide; Represents the excitation current of the i-th coil. Opposing point The generated magnetic field; Represents the equivalent surface magnetization current of the i-th magnetic core. Opposing point The generated magnetic field; the summation of the second term in formula (1) takes into account the magnetization effect of the iron core of coil 1 to coil 8; , The calculation formulas are as follows: (2) Among them, The magnetic field distribution function for a unit excitation current; The magnetic field distribution function per unit magnetizing current; Number of turns per unit length; It is the length of the i-th coil; and These are the inner and outer diameters of the i-th coil, respectively. Represents radius And the axial coordinate is The single-turn circular current at the field point The generated magnetic field; The magnetic fields along the x, y, and z axes are: (3) Among them, , , They represent Magnetic fields along the x, y, and z axes; The vacuum permeability; It is the first user-defined variable, and ; It is a second user-defined variable, and Modulus ; and These are the first and second kind of complete elliptic integrals, respectively. For a coil array containing multiple iron cores, the magnetization state of each core depends not only on the current in its own winding but also on the magnetic field generated by the magnetization of other cores, i.e., the cross-magnetization effect. A self-consistent field analytical numerical model uses a self-consistent field iterative algorithm to handle this coupling problem. It is assumed that the iron cores are made of a nonlinear soft magnetic material with a permeability of... The magnetization is uniform and axial, and passes through the demagnetization factor. Considering the demagnetizing effect, the magnetization of the iron core is... Internal magnetic field strength With external incentive field The relationship between them is: (4) Given the first magnetization in step iteration The applied field can be calculated using formula (2). Then, the updated magnetization is obtained by solving formula (4). The iteration continues until the magnetization of all iron cores converges to a stable value.

[0035] The analytical expressions for the electromagnetic force and torque acting on the permanent magnet in S12 are as follows: (5) Among them, Representation of the electromagnetic force on a permanent magnet rod in an absolute coordinate system; This represents the electromagnetic torque acting on the permanent magnet rod in the body coordinate system. This represents the Euler rotation matrix from the body coordinate system to the absolute coordinate system; This represents the overall magnetic field in the body coordinate system. This represents the Euler rotation matrix from the absolute coordinate system to the body coordinate system. The center of mass of the permanent magnet points to the surface element. Position vector; Represents the cylindrical surface of the permanent magnet rod Find the integral.

[0036] In some embodiments, S2 specifically includes the following steps: S21, using physical heuristics to analyze the residuals between the predicted values ​​and the true values ​​of the simplified physical model. Perform physical heuristic decomposition to reduce residuals Decomposed into a pose-dependent coefficient matrix With the current vector Constructed set of current basis functions The product of Represent the set of real numbers and obtain the residuals. The decomposition formula; where the true value is the numerical result of high-fidelity finite element simulation. Let be the vector consisting of the currents in all the coils. This refers to the current in coil 9 of coil number 9; Let be the pose vector of the permanent magnet rod within the workspace of the electromagnetic system, and In the formula These represent pitch angle and yaw angle respectively; T represents transpose; S22, based on residuals The decomposition of the residual Expanding each component yields the expanded expression of the residual components; the expanded expression of the residual components contains the coefficient functions of different current basis functions; S23, for the coefficient function of each current basis function, two core statistical assumptions based on physical intuition are introduced: Assumption 1: Statistical independence of the coefficient functions of the current basis functions, that is, assuming that the coefficient functions of different current basis functions are statistically uncorrelated; Assumption 2: Heterogeneity of signal variance, that is, assuming that the coefficient functions of all current basis functions share the same normalized pose correlation structure, but each has its own independent signal variance. To characterize the difference in the contribution intensity of different physical terms to the residuals; S24, based on the two core statistical assumptions based on physical intuition introduced, the covariance kernel function of the residuals is derived, denoted as... The covariance kernel function is derived from the current kernel function. Pose kernel function The tensor product is formed; S25, construct a Gaussian process regression model, which is used to learn the residuals between the predicted and true values ​​of the simplified physical model by means of the covariance kernel function.

[0037] This invention designs a physically heuristic covariance kernel function. Instead of a general radial basis function, the covariance kernel function forces a physical decoupling of the input variables: it directly encodes the structural prior of the residual function into the mathematical form of the covariance matrix. This allows the Gaussian process to learn non-parametrically while its function space is constrained to a physically intuitive subspace.

[0038] In some embodiments, the residual in S21 The decomposition formula is: (6) The expansion expression of the residual components in S22 is: (7) Among them, Residual The kth component; Represents the coefficient matrix The OK, for A learnable vector that depends solely on the pose vector. The coefficient function of the current basis function; Represents the set of current basis functions The j-th current basis function in S23; the expression for assumption 1 in S23 is as follows: (8) Among them, To express the covariance; Indicates for any For all cases, Cov=0 holds true; , Indicates an index; Indicates index The corresponding pose vector; the expression for assumption 2 in S23 is as follows: (9) Among them, This represents the pose kernel function. This assumption allows different current basis functions to have different contribution strengths, which is more consistent with the physical reality of magnetic suspension systems.

[0039] In some embodiments, the expression for the covariance kernel function in S2 is as follows: (10) Among them, Represents the variance of the global pose signal; Represents an arbitrary input current vector; the The calculation formula is as follows: (11) The above The calculation formula is as follows: (12) Among them, For anisotropic Mahalanobis distance, It is a modified Bessel function of the second kind. These are hyperparameters that control the smoothness of the function; It is a gamma function.

[0040] In some embodiments, S3 specifically includes the following steps: S31, collecting training data; S32, solving for the optimal hyperparameter set of the Gaussian process regression model using the quasi-Newton algorithm L-BFGS and maximizing the logarithmic marginal likelihood. Thus, the trained Gaussian process regression model is obtained; among which, the optimal hyperparameter set is... The expression is as follows: In equation (13), Represents the logarithmic marginal likelihood function This represents the hyperparameter vector to be optimized. The definition of is: (14) Among them, Let be the signal variance vector of the current basis function. Let represent the signal variance of the current basis function of the nth coil in the coil array. Represents the set of real numbers. for The feature length scale vector of the dimensional pose space; S33. Combine the simplified physical model with the trained Gaussian process regression model to obtain the Gaussian process regression hybrid model.

[0041] In some embodiments, the expression for the logarithmic marginal likelihood in S32 is as follows: (15) Among them, Represents the residual observation vector of the training data; The covariance kernel function is calculated as follows. Covariance matrix; The variance representing additive white Gaussian noise, for The formula embodies the inherent bias-variance tradeoff mechanism of Bayesian nonparametric models: the first term (data fitting term) penalizes the deviation between the model's predictions and the observed data, ensuring accuracy; the second term (complexity penalty term) increases with the determinant of the covariance matrix, penalizing model complexity, thus automatically reflecting Occam's razor principle to prevent overfitting.

[0042] The residual observation vector of the training data It follows a zero-mean multivariate Gaussian distribution, expressed as follows: (16) Among them, The sign representing the distribution, i.e., the residual observation vector. The system follows a zero-mean multivariate Gaussian distribution. In some embodiments, S4 specifically includes the following steps: S41, inputting the new system state into the Gaussian process regression mixture model to obtain the predicted mean and predicted variance of each force / torque component; the predicted variance of each force / torque component is used to quantify the uncertainty of the stress / torque component prediction; wherein, the expression for the predicted mean is as follows: In equation (17), This represents the predicted mean of the k-th force / torque component; This indicates a new system status input. Indicates the new current vector; This represents the new pose vector; The distance between the test point and the training set samples in the training data The cross-covariance vector; the expression for the prediction variance is as follows: (18) Among them, This represents the prediction variance of the k-th force / torque component; is the autocovariance scalar of the test points.

[0043] S42. Calculate the residual prediction value based on the predicted mean of each force / torque component; S43. Add the baseline prediction value output by the simplified physical model to the residual prediction value output by the trained Gaussian process regression model to obtain the final prediction value.

[0044] The prediction uncertainty (i.e., prediction variance) output by this invention is no longer a black-box global scalar. It can be traced back and decomposed into different physical contributions (e.g., uncertainties corresponding to different current cross-coupling terms). This provides interpretable uncertainty information that can be used to design risk-aware or adaptively robust control laws.

[0045] A second aspect of the present invention also provides an electromagnetic force prediction system for an electromagnetic system, configured to execute the above-described electromagnetic force prediction method for an electromagnetic system, wherein the electromagnetic force / torque prediction system for an electromagnetic system includes: a simplified physical model for real-time calculation of a reference prediction value; a basis function calculation module for real-time calculation of the value of the current basis function; and a coefficient lookup and interpolation module, which pre-stores a pre-calculated coefficient matrix. It is used to obtain the coefficient matrix based on real-time input and through interpolation. The corresponding coefficients; the residual calculation module, used to linearly combine the value of the current basis function with the obtained coefficients to obtain the residual prediction value; the adder, used to add the benchmark prediction value and the residual prediction value to output the final prediction value.

[0046] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Furthermore, the technical solutions of the various embodiments of the present invention can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model, characterized in that, The process includes the following steps: S1. Construct a simplified physical model based on the first principles of electromagnetism, input the new system state into the simplified physical model, and output the baseline prediction value; the new system state includes the new pose vector and the new current vector of the system. S2. Construct a covariance kernel function, which is used to characterize the residual between the baseline prediction and the true value of the simplified physical model. It is designed as a combination of the current kernel function and the pose kernel function. Then, construct a Gaussian process regression model. The Gaussian process regression model learns the residual between the baseline prediction and the true value of the simplified physical model with the help of the covariance kernel function. S3. Train the hyperparameters of the Gaussian process regression model using training data to obtain the trained Gaussian process regression model. The simplified physical model is combined with the trained Gaussian process regression model to obtain a Gaussian process regression hybrid model; S4, the new system state is input into the Gaussian process regression hybrid model, and the baseline prediction value output by the simplified physical model is added to the residual prediction value output by the trained Gaussian process regression model to obtain the final prediction value.

2. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 1, characterized in that, S1 specifically includes the following steps: S11. Based on the fundamental laws of electromagnetic fields, construct an analytical expression for the magnetic induction intensity generated by a magnetic source at any point in space. Then, superimpose the magnetic fields generated by all magnetic sources to construct the magnetic field strength of the entire electromagnetic system at the field point. The total magnetic field generated at the location The analytical expression; S12, based on the total magnetic field The analytical expression was derived, and the equivalent surface current density of the permanent magnet rod in the experimental model was analyzed. S13. The surface integral of the Lorentz force gives the analytical expressions for the electromagnetic force and torque acting on the permanent magnet rod within the experimental model; S14. Based on the analytical expressions for the electromagnetic force and torque, the integrals of the analytical expressions for the electromagnetic force and torque are solved using the Gauss-Legend numerical integration method to construct a self-consistent field analytical numerical model; S15. The new system state is input into the self-consistent field analytical numerical model to obtain the baseline predicted values ​​containing the main physical effects of the system. 。 3. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 2, characterized in that, The total magnetic field in S11 The analytical expression is: ;in, Indicates the total magnetic field. This refers to any point in the workspace of an electromagnetic system, often simply called a field point. Indicates field point The total magnetic field; i represents the current magnetic source number; This represents the Euler rotation matrix that transforms the coordinate system of the i-th magnetic source to the absolute coordinate system; This indicates the field point. Representation in the coordinate system of the i-th magnetic source; Represents the excitation current of the i-th magnetic source. The generated magnetic field; Represents the equivalent surface magnetization current of the i-th magnetic core. The generated magnetic field; Represents the excitation current of the i-th magnetic source. At the scene The generated magnetic field; Represents the equivalent surface magnetization current of the i-th magnetic core. At the scene The generated magnetic field; 、 The calculation formulas are as follows: ;in, The magnetic field distribution function for a unit excitation current; The magnetic field distribution function per unit magnetizing current; Number of turns per unit length; It is the length of the i-th magnetic source; and These are the inner and outer diameters of the i-th magnetic source, respectively; Represents radius And the axial coordinate is The single-turn circular current at the field point The generated magnetic field; The magnetic fields along the x, y, and z axes are: ;in, 、 、 They represent Magnetic fields along the x, y, and z axes; The vacuum permeability; It is the first user-defined variable, and ； It is a second user-defined variable, and Modulus ； and These are the first and second kind of complete elliptic integrals, respectively; the analytical expressions for the electromagnetic force and torque acting on the permanent magnet in S12 are as follows: ;in, Representation of the electromagnetic force on a permanent magnet rod in an absolute coordinate system; This represents the electromagnetic torque acting on the permanent magnet rod in the body coordinate system. This represents the Euler rotation matrix from the body coordinate system to the absolute coordinate system; This represents the total magnetic field in the body coordinate system. This represents the Euler rotation matrix from the absolute coordinate system to the body coordinate system. The center of mass of the permanent magnet points to the surface element. Position vector; Represents the cylindrical surface of the permanent magnet rod Find the integral.

4. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 3, characterized in that, S2 specifically includes the following steps: S21, the residual between the predicted value and the true value of the simplified physical model. Perform physical heuristic decomposition to extract residuals Decomposed into a pose-dependent coefficient matrix With the current vector Constructed set of current basis functions The product of these terms yields the residual. The decomposition formula; where the true value is the numerical result of high-fidelity finite element simulation. It is a vector composed of the currents of all magnetic sources; Let be the pose vector of the permanent magnet rod within the workspace of the electromagnetic system, and In the formula These represent pitch angle and yaw angle respectively; T represents transpose; S22, based on residuals The decomposition of the residual The components are expanded to obtain the expanded expressions of the residual components; the expanded expressions of the residual components contain the coefficient functions of different current basis functions; S23, for the coefficient functions of each current basis function, two core statistical assumptions based on physical intuition are introduced: Assumption 1: It is assumed that the coefficient functions of different current basis functions are statistically uncorrelated; Assumption 2: It is assumed that the coefficient functions of all current basis functions share the same normalized pose correlation structure, but each has its own independent signal variance. To characterize the difference in the contribution intensity of different physical terms to the residual; S24, based on the two core statistical assumptions based on physical intuition, the covariance kernel function of the residual is derived; the covariance kernel function is derived from the current kernel function. Pose kernel function The tensor product is formed; S25, construct a Gaussian process regression model, which is used to learn the residuals between the predicted and true values ​​of the simplified physical model by means of the covariance kernel function.

5. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 4, characterized in that, The residual in S21 The decomposition formula is: The expanded expression for the residual components in S22 is: ;in, Residual The kth component; Represents the coefficient matrix The OK, for A learnable vector that depends solely on the pose vector The coefficient function of the current basis function; Represents the set of current basis functions The j-th current basis function in S23; the expression for assumption 1 in S23 is as follows: ;in, To express the covariance; Indicates for any For all cases, Cov=0 holds true; 、 Indicates an index; Indicates index The corresponding pose vector; the expression for assumption 2 in S23 is as follows: ;in, This represents the pose kernel function.

6. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 5, characterized in that, The expression for the covariance kernel function in S2 is as follows: ;in, Represents the variance of the global pose signal; Represents an arbitrary input current vector; the The calculation formula is as follows: The The calculation formula is as follows: ;in, For anisotropic Mahalanobis distance, It is a modified Bessel function of the second kind. These are hyperparameters that control the smoothness of the function; It is a gamma function.

7. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 6, characterized in that, S3 specifically includes the following steps: S31, collecting training data; S32, solving for the optimal hyperparameter set of the Gaussian process regression model using the quasi-Newton algorithm L-BFGS and the method of maximizing the log marginal likelihood. Thus, the trained Gaussian process regression model is obtained; among which, the optimal hyperparameter set is... The expression is as follows: In the formula, This represents the logarithmic marginal likelihood function; This represents the hyperparameter vector to be optimized. The definition of is: ;in, Let be the signal variance vector of the current basis function. This represents the signal variance of the nth current basis function. Represents the set of real numbers. for The feature length scale vector of the dimensional pose space; S33. Combine the simplified physical model with the trained Gaussian process regression model to obtain the Gaussian process regression hybrid model.

8. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 7, characterized in that, The expression for the logarithmic marginal likelihood in S32 is as follows: ;in, Represents the residual observation vector of the training data; The covariance kernel function is calculated as follows. Covariance matrix; The variance representing additive white Gaussian noise, for 3D identity matrix.

9. The method for predicting electromagnetic force in an electromagnetic system based on a Gaussian process regression hybrid model according to claim 8, characterized in that, S4 specifically includes the following steps: S41, inputting the new system state into the Gaussian process regression mixture model to obtain the predicted mean and predicted variance of each force / moment component; the predicted variance of each force / moment component is used to quantify the uncertainty of the stress / moment component prediction; wherein, the expression for the predicted mean is as follows: In the formula, This represents the predicted mean of the k-th force / torque component; This indicates a new system status input. Indicates the new current vector; This represents the new pose vector; The distance between the test point and the training set samples in the training data The cross-covariance vector; the expression for the prediction variance is as follows: ;in, This represents the prediction variance of the k-th force / torque component; S42. The autocovariance scalar of the test points; S43. The residual prediction value is obtained based on the predicted mean value of each force / torque component; S444. The baseline prediction value output by the simplified physical model is added to the residual prediction value output by the trained Gaussian process regression model to obtain the final prediction value.

10. An electromagnetic force prediction system for an electromagnetic system, characterized in that, The electromagnetic system electromagnetic force prediction method according to any one of claims 1 to 9 is configured to be executed, wherein the electromagnetic system electromagnetic force / torque prediction system includes: a simplified physical model for real-time calculation of a reference prediction value; a basis function calculation module for real-time calculation of the value of the current basis function; and a coefficient lookup and interpolation module, which pre-stores a pre-calculated coefficient matrix. It is used to obtain the coefficient matrix based on real-time input and through interpolation. The corresponding coefficients; the residual calculation module, used to linearly combine the value of the current basis function with the obtained coefficients to obtain the residual prediction value; the adder, used to add the benchmark prediction value and the residual prediction value to output the final prediction value.

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