Array antenna pattern optimization method and system based on multi-constraint convex optimization

Abstract

This invention provides a method and system for optimizing the radiation pattern of an array antenna based on multi-constraint convex optimization, relating to the field of antenna array optimization technology. The method includes: constructing an array topology containing multiple candidate element positions based on initial geometric parameters, and setting existence variables and initial phase design variables for each candidate element position; mapping the existence variables to a continuous impedance distribution based on a continuous interpolation function to obtain pseudo-impedance values; substituting the pseudo-impedance values ​​and initial phase design variables into the electromagnetic field integral equation of the method of moments to calculate the surface current distribution, and then calculating the radiated energy in the main lobe direction, the peak radiated energy in the side lobe region, and the current total number of elements, and determining the radiation efficiency index based on this, with the goal of minimizing its negative value; updating the design variables based on a gradient optimization algorithm; determining whether the convergence condition is met; if so, converting to a sparse element layout scheme and a phase excitation scheme to generate the optimized array antenna radiation pattern; otherwise, continuing iterative optimization.

Description

Technical Field

[0001] This invention relates to the field of antenna array optimization technology, specifically to an array antenna pattern optimization method and system based on multi-constraint convex optimization. Background Technology

[0002] Due to their high directivity and power efficiency, array antennas are widely used in radar, communications, and other fields. By controlling the excitation amplitude and phase of the array antenna, a radiation pattern of a specific shape can be formed to meet different application requirements such as main lobe pointing, side lobe suppression, and null formation. With the development of phased array technology, the scale of array antennas is increasing, leading to a sharp increase in system cost, weight, and power consumption. To reduce system complexity, sparse array technology reduces costs by reducing the number of array elements. However, sparsification usually leads to a deterioration in radiation pattern performance. How to maintain main lobe gain and effectively suppress side lobes while reducing the number of array elements has become a key problem in array antenna radiation pattern optimization. Existing sparse array synthesis methods mainly use intelligent optimization algorithms such as genetic algorithms and particle swarm optimization. However, these methods have low computational efficiency, are difficult to handle multiple constraints, and cannot guarantee obtaining the global optimal solution.

[0003] In the prior art, the method, apparatus, and medium for reconfigurable pure-phase sparse array synthesis based on Consensus-ADMM optimization, disclosed in CN113221377A, transforms the sparse array synthesis problem into a linear constraint form by introducing auxiliary variables and solves it using the consistent alternating direction multiplier method. Under the premise of satisfying the main lobe and side lobe constraints of the target beammap, it achieves pure phase control of the sparse array and can obtain a sparse array where the selected number of antennas equals the number of phase shifters. However, this method only uses minimizing the number of antennas as the objective function when constructing the sparse array synthesis problem, failing to adequately consider the geometric layout and electromagnetic performance coupling relationship of the array. Furthermore, its solution process relies on multiple iterative searches after transforming non-convex constraints into linear constraints, neglecting the mutual coupling effect between array elements, making it difficult to guarantee the coordinated optimization of radiation performance and hardware cost in practical engineering applications.

[0004] The information disclosed in the background section is only intended to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide an array antenna pattern optimization method and system based on multi-constraint convex optimization to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A method and system for optimizing the radiation pattern of an array antenna based on multi-constraint convex optimization, the specific steps of which include:

[0008] S1. Obtain the initial geometric and electrical parameters of the array antenna. Based on the initial geometric parameters, construct an initial array topology containing multiple candidate array element positions. Set existence variables and initial phase design variables for each candidate array element position and summarize them to construct initial design variables.

[0009] S2, based on a preset continuous interpolation function, the existence variable is processed to map the discrete state of whether the array element exists or not into a continuous impedance distribution, and the pseudo impedance value corresponding to the position of each candidate array element is obtained.

[0010] S3, Substitute the pseudo impedance value and initial phase design variables into the electromagnetic field integral equation constructed based on the method of moments to calculate the surface current distribution of the array antenna under the initial design variables; Calculate the radiated energy in the main lobe direction, the peak radiated energy in the side lobe region, and the current total number of array elements based on the surface current distribution;

[0011] S4. The radiation efficiency index is determined based on the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the current total number of array elements. Under the constraint conditions, the objective function is to minimize the negative value of the radiation efficiency index, and the initial design variables are updated based on the gradient optimization algorithm.

[0012] S5. Determine whether the updated design variables meet the preset convergence conditions: if so, convert them into the final sparse array element layout scheme and phase excitation scheme, and generate the optimized array antenna pattern; otherwise, use the updated design variables as the initial design variables for the next iteration and continue to optimize until the convergence conditions are met.

[0013] Furthermore, the initial geometric parameters specifically include the total number of array elements, the elliptic axis ratio of each concentric elliptical ring, the azimuth angle of the array elements on each ring, the spacing between adjacent rings, and the maximum aperture size;

[0014] The electrical parameter is specifically the operating frequency of the array antenna;

[0015] Based on the initial geometric parameters, an initial array topology containing the positions of all candidate array elements is constructed within the design area. Specifically, this involves: determining the array region boundary of the concentric elliptical ring array based on the maximum aperture size; determining the ratio of the major axis to the minor axis of each concentric elliptical ring based on the elliptical axis ratio; determining the radial spacing between each ring based on the spacing between adjacent rings; and determining the specific position coordinates of each candidate array element on each ring based on the azimuth angle of the array elements on each ring, thereby enabling the array to... In-plane generation includes The initial concentric elliptical ring array topology at the candidate array element positions;

[0016] Based on the electrical parameters, electromagnetic calculation conditions are configured for the initial array topology, specifically: the electromagnetic wave number is determined according to the operating frequency of the array antenna, and the same fixed excitation amplitude is set for the position of each candidate array element.

[0017] An existence variable is set for each candidate array element position, wherein the existence variable takes values ​​within the range of The continuous variable is used to characterize the probability of setting an array element at the candidate array element position. A value of 1 indicates that the array element will definitely be set, and a value of 0 indicates that the array element will definitely not be set. At the beginning of the optimization, the existence variable of each candidate array element position is set to 0.5, so that it evolves in the two directions of 0 or 1 in the subsequent optimization process.

[0018] An initial phase design variable is set for each candidate element position. For candidate element positions where the existence variable is 0, the initial phase design variable is set to 0; for candidate element positions where the existence variable is non-zero, the initial phase design variable is set within a preset phase range. A random value is generated internally to serve as the initial phase design variable for the candidate array element position.

[0019] Furthermore, when the existence variable is 1, the corresponding candidate element position is a metallic material state; when the existence variable is 0, the corresponding candidate element position is a non-conductive material state.

[0020] A tangential interpolation function is used to construct a continuous mapping relationship between the pseudo impedance values ​​at each candidate array element position and the existence variables. The expression of the tangential interpolation function is as follows:

[0021]

[0022]

[0023] In the formula, Indicates the first The pseudo impedance value at each candidate array element position, This is the index of the candidate array element position; This is the equivalent impedance value of the metallic material. This represents the equivalent impedance value of a non-conductive material. The preset penalty factor, ; For the penalty function; For the first The existence variable of the positions of candidate array elements;

[0024] The range of values ​​is determined by the tangential interpolation function. The existence of a variable is continuously mapped to a range of values ​​within The pseudo impedance values ​​are obtained, thus the pseudo impedance values ​​corresponding to the positions of each candidate array element are obtained, so that when the existence variable is 0, it corresponds to the high impedance state of non-conductive materials, when the existence variable is 1, it corresponds to the zero impedance state of metallic materials, and when the existence variable is an intermediate value, it corresponds to the intermediate impedance state between the two.

[0025] Furthermore, the surface current distribution of the array antenna under the initial design variables is calculated based on the following specific logic:

[0026] An electromagnetic field solution model for the array antenna is established based on the electric field integral equation. The Green's function is set according to the electromagnetic wave number. The pseudo impedance values ​​of each candidate array element position are substituted into the impedance boundary conditions to form a global impedance matrix containing the pseudo impedance matrix. At the same time, an excitation voltage vector is constructed based on the initial phase design variables of each candidate array element position and the fixed excitation amplitude.

[0027] The electric field integral equation is discretized into a system of linear equations using the Galerkin method:

[0028]

[0029] In the formula, This represents the mutual impedance matrix associated with the array geometry; It is a diagonal impedance matrix composed of the pseudo-impedance values ​​of each candidate array element position; Let be the surface current distribution vector to be solved; The excitation voltage vector;

[0030] By solving the system of linear equations, the calculation results The surface current distribution at each candidate element location of the array antenna, as an initial design variable.

[0031] Furthermore, based on the surface current distribution obtained from the solution, the radiated electric field of the array antenna at each observation point in the far field region is calculated using the equivalent dipole model, thereby determining the radiated energy at each observation point;

[0032] The radiation energy at the pointing angle of the main lobe is determined as the radiation energy in the direction of the main lobe.

[0033] use The norm approximation method processes the radiative energy in the sidelobe region to obtain the peak radiative energy in the sidelobe region, where... In norm The larger the value, the closer the calculation result is to the true maximum value of the sidelobe region;

[0034] The existence variables of each candidate array element position are statistically analyzed. Candidate array element positions with existence variables greater than a preset threshold are determined as valid array elements. The total number of all valid array elements is counted as the current total number of array elements.

[0035] Furthermore, the formula for calculating the radiation effectiveness index is as follows:

[0036]

[0037] In the formula, Radiation efficiency index; Energy radiated in the direction of the main lobe. This represents the peak radiation energy in the sidelobe region. This represents the current total number of array elements. This is a preset penalty factor for the number of array elements, used to balance the weight between radiation performance and the number of array elements.

[0038] Furthermore, the negative value of the radiation effectiveness index is used as the objective function, and the objective function is a convex function;

[0039] Set constraints, specifically including: the radiated energy in the main lobe direction is not lower than a preset threshold. ,Right now The current total number of array elements does not exceed the preset maximum value. ,Right now The constraints constitute a convex set.

[0040] Under the condition that the constraints are met, the objective function is minimized as the optimization objective. The gradient optimization algorithm is used to solve the convex optimization problem model. By iteratively updating the existence variables and phase design variables, the objective function is gradually reduced until convergence, and the updated existence variables and phase design variables are obtained.

[0041] Furthermore, the specific execution process of S5 is as follows:

[0042] Obtain the existence variables and phase design variables after each iteration update, calculate the change in design variables between two adjacent iterations, and denote the maximum change in the existence variables as... The maximum change in the phase design variable is denoted as The formula it is based on is as follows:

[0043]

[0044]

[0045] In the formula, The maximum change of an existing variable. The maximum change of the phase design variable; The index is the number of iterations. ; Indicates the first After the first iteration update Existence variables of candidate array element positions Indicates the first After the first iteration update Phase design variables for the positions of candidate array elements; The index of the candidate array element position. , This represents the total number of candidate array element positions.

[0046] Preset existence variable convergence threshold Convergence threshold for phase design variables ,when and If the current iteration satisfies the convergence condition, it is determined that the convergence condition is not met; otherwise, it is determined that the convergence condition is not met.

[0047] When the convergence condition is met, the existence variables of the current iteration are converted into the final sparse layout scheme of the array elements. Specifically, for each candidate array element position, if... If the candidate element position is reserved, then the element is determined to be a reserved element. If the candidate element position is determined, the element will be removed, and the phase design variable for the current iteration will be... As the final phase excitation value for each retained array element, a phase excitation scheme is generated. Based on the sparse array element layout scheme and the phase excitation scheme, an optimized array antenna pattern is generated.

[0048] When the convergence condition is not met, the updated existence variable will be... and phase design variables The optimization continues until the convergence condition is met, and the initial values ​​are used as the starting points for the next iteration.

[0049] The present invention also provides an array antenna pattern optimization system based on multi-constraint convex optimization, for performing the above-described array antenna pattern optimization method based on multi-constraint convex optimization, including:

[0050] The initial parameter configuration module is used to obtain the initial geometric and electrical parameters of the array antenna, construct an initial array topology containing multiple candidate array element positions based on the initial geometric parameters, set existence variables and initial phase design variables for each candidate array element position and summarize them to construct the initial design variables;

[0051] The impedance mapping modeling module is used to process the existence variables based on a preset continuous interpolation function, mapping the discrete state of the existence or non-existence of array elements to a continuous impedance distribution, and obtaining the pseudo impedance value corresponding to the position of each candidate array element.

[0052] The current distribution solution module is used to substitute the pseudo impedance value and the initial phase design variables into the electromagnetic field integral equation constructed based on the method of moments to calculate the surface current distribution of the array antenna under the initial design variables; and calculates the radiated energy in the main lobe direction, the peak radiated energy in the side lobe region, and the current total number of array elements based on the surface current distribution.

[0053] The gradient optimization solution module is used to determine the radiation efficiency index based on the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the current total number of array elements. Under the constraint conditions, the objective function is to minimize the negative value of the radiation efficiency index, and the initial design variables are updated based on the gradient optimization algorithm.

[0054] The iterative convergence control module is used to determine whether the updated design variables meet the preset convergence conditions. If so, it converts the final array element sparse layout scheme and phase excitation scheme to generate the optimized array antenna pattern. Otherwise, the updated design variables are used as the initial design variables for the next iteration and optimization continues until the convergence conditions are met.

[0055] Compared with the prior art, the beneficial effects of the present invention are:

[0056] 1. This invention simultaneously assigns existence variables and phase design variables to each candidate array element position, and maps the discrete array element existence state to a continuous impedance distribution based on the tangential interpolation function, thereby achieving coupled optimization of array element sparsity and phase excitation. Compared with the existing technology that only optimizes a single dimension with the minimum number of antennas as the objective function, this invention can simultaneously balance the three dimensions of main lobe gain, side lobe suppression and array element number during the optimization process, overcoming the shortcomings of traditional methods that are difficult to balance radiation performance and hardware cost.

[0057] 2. This invention substitutes the parameterized impedance model and phase design variables into the electromagnetic field integral equation constructed based on the method of moments. By solving the linear equation system, the surface current distribution is accurately calculated. The method of moments can strictly consider the electromagnetic coupling between array elements, avoiding the calculation error caused by the traditional sparse array synthesis method that ignores mutual coupling by using the pattern product theorem. This makes the optimization results closer to the real radiation performance in actual engineering applications.

[0058] 3. This invention constructs a radiation performance index based on the radiated energy in the main lobe direction, the peak radiated energy in the side lobe region, and the current total number of array elements. This index integrates the gain performance, interference suppression capability, and lightweighting level of the array antenna into a single quantitative evaluation indicator. By using a preset array element number penalty factor to balance radiation performance and hardware cost, this index provides a scientific evaluation basis for array optimization under multiple constraints.

[0059] 4. This invention uses the negative value of the radiation efficiency index as the objective function, and constructs a convex optimization problem model by combining the main lobe radiation energy threshold constraint, the upper limit constraint of the total number of array elements, and the range constraint of the design variable values. Convex optimization has the characteristic that the local optimal solution is the global optimal solution. Compared with heuristic algorithms such as genetic algorithms and particle swarm algorithms, this invention can guarantee the acquisition of the global optimal solution, with higher computational efficiency and more stable convergence. Attached Figure Description

[0060] Figure 1 This is a schematic diagram of the overall method flow of the present invention;

[0061] Figure 2 Dual Y-axis images of radiant energy in the main lobe direction, peak radiant energy in the side lobe region, and radiative efficiency index.

[0062] Figure 3 A 3D scatter plot of the radiation energy in the main lobe direction, the current total number of array elements, and the radiation efficiency index;

[0063] Figure 4 This is a schematic diagram of the overall system modules of the present invention. Detailed Implementation

[0064] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to specific embodiments.

[0065] It should be noted that, unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0066] Example:

[0067] Please see Figures 1-3 The present invention provides a technical solution:

[0068] The array antenna pattern optimization method based on multi-constraint convex optimization includes the following steps:

[0069] S1. Obtain the initial geometric and electrical parameters of the array antenna. Based on the initial geometric parameters, construct an initial array topology containing multiple candidate array element positions. Set existence variables and initial phase design variables for each candidate array element position and summarize them to construct initial design variables.

[0070] In this embodiment, the initial geometric parameters specifically include the total number of array elements, the elliptic axis ratio of each concentric elliptical ring, the azimuth angle of the array elements on each ring, the spacing between adjacent rings, and the maximum aperture size.

[0071] The electrical parameter is specifically the operating frequency of the array antenna;

[0072] Based on the initial geometric parameters, an initial array topology containing the positions of all candidate array elements is constructed within the design area. Specifically, this involves: determining the array region boundary of the concentric elliptical ring array based on the maximum aperture size; determining the ratio of the major axis to the minor axis of each concentric elliptical ring based on the elliptical axis ratio; determining the radial spacing between each ring based on the spacing between adjacent rings; and determining the specific position coordinates of each candidate array element on each ring based on the azimuth angle of the array elements on each ring, thereby enabling the array to... In-plane generation includes The initial array topology is a concentric elliptical ring structure with candidate array element positions; this configuration can adapt to the array design requirements of various geometric shapes such as circles, ellipses and even rectangles, and has good versatility.

[0073] Based on the electrical parameters, electromagnetic calculation conditions are configured for the initial array topology. Specifically, the electromagnetic wave number is determined according to the operating frequency of the array antenna. This wave number will be used for the calculation of the Green's function in the subsequent method of moments solution. The same fixed excitation amplitude is set for the positions of each candidate array element to simplify the optimization complexity and ensure the core feature of pure phase control.

[0074] An existence variable is set for each candidate array element position, wherein the existence variable takes values ​​within the range of The continuous variable is used to characterize the probability of setting up an array element at the candidate array element position. A value of 1 indicates that the array element will definitely be set up, and a value of 0 indicates that the array element will definitely not be set up. At the beginning of the optimization, the existence variable of each candidate array element position is set to 0.5, so that it is in an intermediate state and can evolve in the two directions of 0 or 1 in the subsequent optimization process. This continuous relaxation process is the key to achieving gradient optimization solution, because discrete 0-1 variables cannot be directly differentiated, while continuous variables can be iteratively updated through gradient information.

[0075] An initial phase design variable is set for each candidate element position. For candidate element positions where the existence variable is 0, the initial phase design variable is set to 0 to avoid invalid phase calculations. For candidate element positions where the existence variable is non-zero, the initial phase design variable is set within a preset phase range. A random value is generated within the array to serve as the initial phase design variable for the candidate array element position. Random initialization helps to break the symmetry, avoid the optimization getting stuck in local optima, and increase the algorithm's exploration capability.

[0076] Step S1, through systematic parameter configuration and initialization design, establishes the basic framework for the optimization problem: It accurately generates the position coordinates of candidate array elements based on the initial geometric parameters of the concentric elliptical ring array, enabling the topology to flexibly adapt to various geometric shapes such as circles, ellipses, and rectangles, demonstrating good universality; it determines the electromagnetic wave number based on the operating frequency and sets the same fixed excitation amplitude for all candidate array elements, simplifying the optimization complexity while ensuring the core characteristics of pure phase control; and it sets the existence variable as... By initializing the continuous variables in the interval to 0.5, the discrete 0-1 sparse matrix problem is cleverly relaxed into a differentiable continuous optimization problem, laying the foundation for the application of gradient optimization algorithms. At the same time, the phase design variables are randomly initialized for candidate matrix elements with non-zero existence variables, which effectively breaks the symmetry in the early stage of optimization, enhances the global exploration capability of the algorithm, avoids the local optimum trap, and thus ensures the stability, convergence and solution quality of the subsequent iteration process at the starting point of optimization.

[0077] S2, based on a preset continuous interpolation function, the existence variable is processed to map the discrete state of whether the array element exists or not into a continuous impedance distribution, and the pseudo impedance value corresponding to the position of each candidate array element is obtained.

[0078] In this embodiment, when the existence variable is 1, the corresponding candidate array element position is in a metallic material state, and the equivalent impedance value is... When the existence variable takes the value of 0, the corresponding candidate array element position is a non-conductive material state, and the equivalent impedance value is... ;

[0079] A tangential interpolation function is used to construct a continuous mapping relationship between the pseudo impedance values ​​at each candidate array element position and the existence variables. The expression of the tangential interpolation function is as follows:

[0080]

[0081]

[0082] In the formula, Indicates the first The pseudo impedance value at each candidate array element position, This is the index of the candidate array element position; This is the equivalent impedance value of the metallic material. This represents the equivalent impedance value of a non-conductive material. The preset penalty factor, ; For the penalty function; For the first The existence variable of the positions of candidate array elements;

[0083] For this formula, the dependent variable Used to characterize the The pseudo impedance value at each candidate array element position physically represents the degree to which that position impedes the propagation of electromagnetic waves. The larger the value, the higher the equivalent impedance at that location, and the closer it is to the insulating state of a non-conductive material, making it difficult for electromagnetic waves to pass through. This corresponds to an existence variable. The value approaches 0, meaning that no array element is set at that position; The smaller the value (approaching 0), the lower the equivalent impedance at that location, and the closer it is to the ideal conductor state of a metallic material, allowing electromagnetic waves to pass through smoothly. This corresponds to the existence variable. Approaching 1, that is, the array element is set at this position; the intermediate impedance value corresponds to the existence variable being in a transitional state between 0 and 1, and the design variable is prompted to converge towards both ends through a penalty mechanism during the optimization process;

[0084] Penalty function By adjusting the existence variable The weights in the tangential interpolation function indirectly affect the pseudo impedance value. ,when hour, , making The term with the highest weight maps the pseudo-impedance value to the high impedance state of a non-conductive material; when hour, , making The term is completely suppressed, and the pseudo-impedance value is mapped to the zero-impedance state of the metallic material. The penalty factor Control the rate at which the intermediate value converges towards both ends. The larger the value, the stronger the penalty for the intermediate value. During the optimization process, the existence variable tends to quickly polarize to 0 or 1. This is in line with the discrete decision-making requirements of array elements in engineering practice, which requires either to retain or remove them, and avoids the design ambiguity caused by a large number of intermediate impedance states.

[0085] This expression uses a tangential interpolation function to construct a continuous mapping, so that the continuous change of the existence variable from 0 to 1 can smoothly correspond to the continuous transition of the pseudo impedance value from high impedance to zero impedance, ensuring the differentiability required for gradient optimization. In addition, the introduction of the penalty function causes the intermediate value to be penalized during the optimization process, which encourages the design variable to converge to 0 or 1. Thus, after the optimization converges, a clear binary array element layout scheme can be obtained. This design not only meets the requirement of gradient optimization algorithm for continuous variables, but also ensures the physical realizability of the final result.

[0086] S3, Substitute the pseudo impedance value and initial phase design variables into the electromagnetic field integral equation constructed based on the method of moments to calculate the surface current distribution of the array antenna under the initial design variables; Calculate the radiated energy in the main lobe direction, the peak radiated energy in the side lobe region, and the current total number of array elements based on the surface current distribution;

[0087] In this embodiment, the specific logic for calculating the surface current distribution of the array antenna under the initial design variables is as follows:

[0088] An electromagnetic field solution model for the array antenna is established based on the electric field integral equation, according to the electromagnetic wave number. Define the Green function ,in, This represents the position vector of the field point, i.e., the spatial coordinates of the observation point. This represents the position vector of the source point, i.e., the spatial coordinates of the current source. The imaginary unit is used to describe the propagation characteristics of the electromagnetic field generated by a point source in free space. The pseudo impedance values ​​of each candidate array element position are substituted into the impedance boundary conditions to form a global impedance matrix containing the pseudo impedance matrix. At the same time, the excitation voltage vector is constructed according to the initial phase design variables of each candidate array element position and the fixed excitation amplitude.

[0089] The electric field integral equation is discretized into a system of linear equations using the Galerkin method:

[0090]

[0091] In the formula, This represents the mutual impedance matrix associated with the array geometry; It is a diagonal impedance matrix composed of the pseudo-impedance values ​​of each candidate array element position; Let be the surface current distribution vector to be solved; The excitation voltage vector;

[0092] By solving the system of linear equations, the calculation results As an initial design variable, the surface current distribution at each candidate element position of the array antenna is considered. This current distribution takes into account the material properties of each candidate element position and the mutual coupling effect between elements. The material properties are reflected by pseudo impedance values, and the mutual coupling effect is reflected by mutual impedance matrix.

[0093] This step establishes an electromagnetic field solution model for the array antenna based on the electric field integral equation. First, the free-space Green's function is set according to the electromagnetic wave number determined in S2. Then, the pseudo-impedance values ​​obtained from step 2 for each candidate array element position are substituted into the impedance boundary conditions, thereby incorporating the material properties of each candidate array element position into the electromagnetic field solution model in the form of impedance. When the existence variable approaches 1, the pseudo-impedance value approaches 0, corresponding to the zero impedance state of metallic materials, where current can pass freely. When the existence variable approaches 0, the pseudo-impedance value approaches infinity, corresponding to the high impedance state of non-conductive materials, where current is blocked. Based on this, the Galerkin method is used to discretize the electric field integral equation, forming a model containing a global mutual impedance matrix and pseudo-impedance. The system of linear equations for the impedance matrix is ​​used, where the mutual impedance matrix is ​​determined by the array geometry and accurately reflects the electromagnetic coupling between the array elements. The pseudo-impedance matrix is ​​a diagonal matrix with the pseudo-impedance values ​​of each candidate array element position as its diagonal elements, used to characterize the constraint of material properties on the current distribution at each position. Simultaneously, an excitation voltage vector is constructed based on the initial phase design variables and fixed excitation amplitude at each candidate array element position. The excitation voltage amplitude at each candidate array element position is equal to the fixed excitation amplitude, and the phase is equal to the corresponding initial phase design variable. For candidate array element positions where the existence variable is 0, the phase design variable is 0, and the corresponding position of the excitation voltage vector is also 0. The surface current distribution vector is obtained by solving this system of linear equations.

[0094] The beneficial effects of this step are as follows: First, the electromagnetic field solution model based on the electric field integral equation and Green's function can accurately describe the propagation law of electromagnetic waves in free space, laying a rigorous physical foundation for the calculation of surface current distribution. Second, by substituting pseudo-impedance values ​​into the impedance boundary conditions, the discrete decision of array element sparsity is cleverly integrated into the continuous electromagnetic field solution process, enabling the optimization algorithm to iteratively update existence variables through gradient information. Third, the introduction of the mutual impedance matrix realizes the rigorous consideration of the mutual coupling effect between array elements, avoiding the calculation error caused by the traditional sparse array synthesis method ignoring mutual coupling by using the pattern product theorem. Finally, the linear equation system formed by discretization using the Galerkin method has good numerical stability, and the surface current distribution obtained by the solution comprehensively considers the material properties of each candidate array element position and the electromagnetic coupling effect between array elements, providing an accurate data foundation for the subsequent accurate calculation of the radiation energy in the main lobe direction and the peak radiation energy in the side lobe region.

[0095] Based on the surface current distribution obtained from the solution, the radiated electric field of the array antenna at each observation point in the far field region is calculated by using the equivalent dipole model, and then the radiated energy at each observation point is determined.

[0096] The radiation energy at the pointing angle of the main lobe is determined as the radiation energy in the direction of the main lobe.

[0097] use The norm approximation method processes the radiative energy in the sidelobe region to obtain the peak radiative energy in the sidelobe region, where... In norm The larger the value, the closer the calculation result is to the true maximum value of the sidelobe region;

[0098] The existence variables of each candidate array element position are statistically analyzed. Candidate array element positions with existence variables greater than a preset threshold are determined as valid array elements. The total number of all valid array elements is counted as the current total number of array elements.

[0099] Based on the obtained surface current distribution, each array element is equivalent to a dipole radiation source using an equivalent dipole model. The radiated electric field generated by each element in the far-field region is calculated according to the current distribution, and the total radiated electric field at each observation point is obtained through vector superposition, thereby determining the radiated energy at each observation point. Based on this, the radiated energy at the preset main lobe pointing angle is determined as the radiated energy in the main lobe direction, used to measure the gain performance of the array antenna in the target direction. For the sidelobe region, a... The norm approximation method processes the radiant energy at all sidelobe observation points by selecting a larger norm. The value makes the calculation result approach the maximum radiation energy of the sidelobe region, thereby obtaining the peak radiation energy of the sidelobe region, which is used to measure the anti-interference capability of the array antenna; at the same time, the existence variables of each candidate array element position are statistically analyzed, and the candidate array element positions with existence variables greater than the preset threshold are determined as valid array elements. The total number of all valid array elements is counted as the current total number of array elements, which is used to evaluate the hardware cost and complexity of the array.

[0100] Its beneficial effects are as follows: First, the radiated electric field calculation method based on the equivalent dipole model can accurately reflect the phase superposition relationship between each array element, providing a theoretical basis for the accurate calculation of radiated energy; second, the comparison between the radiated energy in the main lobe direction and the peak radiated energy in the side lobe region directly reflects the beam directivity and sidelobe suppression capability of the array antenna; third, The norm approximation method avoids the error caused by improper selection of discrete sampling points when directly calculating the sidelobe maximum value. This is achieved by adjusting... The value can achieve a balance between computational accuracy and computational efficiency; finally, the effective array element statistical method based on threshold determination can transform continuous existence variables into discrete array element layout schemes, which not only ensures the differentiability of the optimization process, but also ensures the physical realizability of the final result, providing accurate basic data for the subsequent calculation of radiation efficiency index.

[0101] S4. The radiation efficiency index is determined based on the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the current total number of array elements. Under the constraint conditions, the objective function is to minimize the negative value of the radiation efficiency index, and the initial design variables are updated based on the gradient optimization algorithm.

[0102] In this embodiment, the formula for calculating the radiation effectiveness index is as follows:

[0103]

[0104] In the formula, Radiation efficiency index; Energy radiated in the direction of the main lobe. This represents the peak radiation energy in the sidelobe region. This represents the current total number of array elements. A preset element quantity penalty factor is used to balance the weight between radiation performance and element quantity in this embodiment. ,Will Setting the value to 0.5 will reduce the penalty for the total number of array elements. Providing a moderate penalty when the number of array elements increases can effectively suppress the cost increase caused by excessive growth in the number of array elements, while avoiding suppressing the optimization space for improving radiation performance by moderately increasing the number of array elements due to excessive penalty, thus achieving a balanced trade-off between radiation performance and hardware cost.

[0105] For this formula, the dependent variable The comprehensive radiation performance of an array antenna is a comprehensive index that integrates main lobe gain performance, side lobe suppression capability, and hardware cost (i.e., number of array elements) into a single metric for quantitative evaluation. A higher value indicates that the array antenna can achieve higher main lobe radiation energy per unit sidelobe radiation energy and per unit number of array elements. This means higher main lobe gain and lower sidelobe interference are achieved with fewer array elements, resulting in better overall performance; conversely, a lower value indicates lower performance. The smaller the value, the worse the overall performance of the array antenna, which may result in insufficient main lobe gain, excessive side lobes, or excessive array elements leading to cost waste.

[0106] When it increases, The gain increases accordingly because the main lobe gain is the main performance indicator of an array antenna. Higher main lobe energy means stronger directivity and a greater effective range, which meets the beam focusing requirements of engineering applications. When it increases, The sidelobes decrease accordingly because excessively high sidelobes can introduce interference signals, reducing the anti-interference capability and signal-to-noise ratio of the array antenna. In engineering, it is desirable to have the sidelobes as low as possible. When it increases, This decreases accordingly because the number of array elements directly determines the array's hardware cost, power consumption, and system complexity. Therefore, the number of array elements should be minimized while still meeting performance requirements. Thus, the formula... By penalizing redundant array elements, the optimization process tends to achieve better radiation performance with fewer array elements.

[0107] This formula directly reflects the trade-off between main lobe gain and side lobe suppression in the form of a ratio, which aligns with the core objective of array antenna pattern optimization. Secondly, it introduces a penalty factor for the number of array elements. right By applying an exponential penalty and incorporating hardware costs into the performance evaluation system, when a moderate increase in the number of array elements can significantly improve the main lobe gain or reduce the sidelobes... It can still maintain a high level, but when the number of array elements increases excessively while the performance improvement is limited, The amplification effect of the term will suppress The value guides the optimization algorithm to automatically find the best balance between radiation performance and hardware cost.

[0108] Table 1: Statistical Table of Radiation Efficiency Index

[0109]

[0110] Combination Figure 2 , Figure 3 It can be seen that the radiation efficiency index is significantly positively correlated with the radiation energy in the main lobe direction. Figure 2 When the radiant energy in the main lobe direction increases, the radiative efficacy index also shows an upward trend. However, changes in the peak radiant energy in the side lobe regions are negatively correlated with the radiative efficacy index; a decrease in the peak radiant energy in the side lobe regions will drive an increase in the radiative efficacy index. These coupled changes form a clear inverse and positive correlation in the dual Y-axis images. Figure 3 In the 3D scatter distribution, the radiation efficiency index shows a spatial upward trend with the increase of radiation energy in the main lobe direction, while the influence of the current total number of array elements on the radiation efficiency index is affected by the array element number penalty factor. Regulation, in Under the current settings, when the total number of array elements is moderately increased within the preset maximum value, the radiation efficiency index can still maintain a high level. However, if the total number of array elements increases excessively, it will... The amplification effect of the term leads to a decrease in the radiation efficiency index, and the scatter points in space exhibit a characteristic of shifting towards the low exponent region as the total number of array elements exceeds the threshold; overall, the numerical variation pattern in Table 1 is consistent with... Figure 2 , Figure 3 The visualization features are highly consistent, verifying that the radiation efficiency index formula can effectively quantify the coupling relationship between main lobe gain, side lobe suppression and the number of array elements. It also reflects that the convex optimization model of this scheme can achieve a dynamic balance among the three under the constraints of the main lobe radiation energy threshold and the upper limit of the total number of array elements, allowing the radiation efficiency index to converge to the optimal range. This provides an intuitive visualization reference and quantitative basis for parameter adjustment of array antenna pattern optimization.

[0111] The negative value of the radiation effectiveness index is used as the objective function, and the objective function is a convex function;

[0112] Set constraints, specifically including: the radiated energy in the main lobe direction is not lower than a preset threshold. ,Right now The current total number of array elements does not exceed the preset maximum value. ,Right now The constraints constitute a convex set.

[0113] Under the condition that the constraints are met, the objective function is minimized as the optimization objective. The gradient optimization algorithm is used to solve the convex optimization problem model. By iteratively updating the existence variables and phase design variables, the objective function is gradually reduced until convergence, and the updated existence variables and phase design variables are obtained.

[0114] This step first sets two core constraints based on engineering application requirements: the radiated energy in the main lobe direction is not lower than a preset threshold to ensure that the optimization result meets basic directional requirements and avoids insufficient radar detection range or communication coverage due to excessively low main lobe gain; the total number of array elements does not exceed a preset maximum value to ensure that the optimized array hardware cost, power consumption, and system complexity are controlled within an acceptable range; among these, the preset threshold... The method for determining the minimum main lobe gain is as follows: based on the link budget requirements of the radar or communication system, and combined with parameters such as operating frequency, transmit power, and receive sensitivity, the minimum main lobe gain required is calculated backwards, and then converted into the corresponding main lobe directional radiated energy threshold; a preset maximum value is then used. The determination method is as follows: it is determined comprehensively based on factors such as system hardware cost budget, phase shifter quantity limit, and structural size constraints, and is usually equal to the number of available RF channels or phase shifters; on this basis, the negative value of the radiation efficiency index is used as the objective function, which together with the above constraints constitutes a convex optimization problem model. Since the objective function is a convex function and the constraints form a convex set, this optimization problem has the characteristic that the local optimum is the global optimum; when solving this convex optimization problem using the gradient optimization algorithm, the existence variables and phase design variables are iteratively updated to gradually reduce the objective function until convergence, and finally the updated existence variables and phase design variables are obtained.

[0115] This step incorporates practical constraints from engineering applications directly into the optimization model by setting a main lobe radiation energy threshold and an upper limit on the total number of array elements, ensuring the optimization results are engineering-feasible. Secondly, the objective function and constraints are jointly constructed into a convex optimization problem model, leveraging the inherent characteristics of convex optimization to guarantee a globally optimal solution, avoiding the pitfalls of heuristic algorithms such as genetic algorithms and particle swarm optimization that easily get trapped in local optima. Thirdly, gradient-based optimization methods can efficiently iterate using the gradient information of the objective function and constraints, exhibiting higher computational efficiency compared to heuristic algorithms, and are particularly suitable for array antenna optimization problems with a large number of candidate array elements. Finally, the solution process for convex optimization problems has good numerical stability and convergence guarantees, enabling the acquisition of the optimal solution satisfying the constraints within a relatively small number of iterations.

[0116] S5. Determine whether the updated design variables meet the preset convergence conditions: if so, convert to the final sparse array element layout scheme and phase excitation scheme, and generate the optimized array antenna pattern; otherwise, use the updated design variables as the initial design variables for the next iteration and continue to optimize until the convergence conditions are met.

[0117] In this embodiment, the specific execution process of S5 is as follows:

[0118] Obtain the existence variables and phase design variables after each iteration update, calculate the change in design variables between two adjacent iterations, and denote the maximum change in the existence variables as... The maximum change in the phase design variable is denoted as The formula it is based on is as follows:

[0119]

[0120]

[0121] In the formula, The maximum change of an existing variable. The maximum change of the phase design variable; The index is the number of iterations. ; Indicates the first After the first iteration update Existence variables of candidate array element positions Indicates the first After the first iteration update Phase design variables for the positions of candidate array elements; The index of the candidate array element position. , This represents the total number of candidate array element positions.

[0122] Preset existence variable convergence threshold Convergence threshold for phase design variables ,when and If the current iteration satisfies the convergence condition, it is determined that the convergence condition is not met; otherwise, it is determined that the convergence condition is not met.

[0123] This formula calculates the changes in design variables for the positions of all candidate array elements between two consecutive iterations and takes the maximum value as the convergence criterion. The existence variables and phase design variables are updated after each iteration, and the maximum change in the existence variables is obtained by comparing the absolute value of the difference between the current iteration and the previous iteration. and the maximum change of phase design variables Preset convergence threshold and ,when and When the convergence threshold is reached, it indicates that the changes in the design variables for all candidate array elements have fallen below the preset threshold, meaning the optimization process has stabilized and the convergence condition is met; otherwise, iterative updates continue. This convergence judgment strategy effectively avoids insufficient optimization caused by premature termination of iterations, while also preventing the waste of computational resources caused by prolonged iterations. Its beneficial effects are: using the maximum change as the convergence criterion comprehensively reflects the convergence status of all design variables in the entire array, avoiding misjudgments caused by focusing only on individual variable changes; by setting convergence thresholds for existence variables and phase design variables separately, the convergence accuracy of the two different types of variables can be independently controlled, adapting to the different convergence characteristics of sparse layouts and phase optimizations; this judgment method is computationally simple, easy to implement, and has good numerical stability, enabling timely termination of iterations while ensuring solution accuracy, achieving a balance between optimization efficiency and solution accuracy.

[0124] When the convergence condition is met, the existence variables of the current iteration are converted into the final sparse layout scheme of the array elements. Specifically, for each candidate array element position, if... If the candidate element position is reserved, then the element is determined to be a reserved element. If the candidate element position is determined, the element will be removed, and the phase design variable for the current iteration will be... As the final phase excitation value for each retained array element, a phase excitation scheme is generated. Based on the sparse array element layout scheme and the phase excitation scheme, an optimized array antenna pattern is generated.

[0125] When the convergence condition is not met, the updated existence variable will be... and phase design variables The optimization continues until the convergence condition is met, and the initial values ​​are used as the starting points for the next iteration.

[0126] Step S5 calculates the maximum change in the existence variables and phase design variables of all candidate array element positions between two adjacent iterations and compares it with a preset convergence threshold, thus achieving precise termination control of the optimization process. Using the maximum change as the convergence criterion comprehensively reflects the convergence state of all design variables in the entire array, avoiding misjudgments that might occur if only individual variable changes are considered or average values ​​are used, ensuring that the design variables for all candidate array element positions are stable. By setting independent convergence thresholds for existence variables and phase design variables respectively, it can flexibly adapt to the different convergence characteristics of sparse layout and phase optimization variables, achieving fine-tuning of convergence accuracy. The threshold-based conversion mechanism transforms continuous design variables into discrete array element sparse layout schemes and phase excitation schemes, ensuring both the differentiability of the optimization process and the physical realizability of the final result. This convergence judgment strategy can terminate iterations in a timely manner while ensuring solution accuracy, avoiding waste of computational resources due to prolonged iterations and preventing insufficient optimization due to premature termination, achieving a balance between optimization efficiency and solution accuracy, and providing a reliable convergence control method for obtaining array antenna pattern optimization results that meet engineering requirements.

[0127] Please see Figure 4 An array antenna pattern optimization system based on multi-constraint convex optimization includes:

[0128] The initial parameter configuration module is used to obtain the initial geometric and electrical parameters of the array antenna, construct an initial array topology containing multiple candidate array element positions based on the initial geometric parameters, set existence variables and initial phase design variables for each candidate array element position and summarize them to construct the initial design variables;

[0129] The impedance mapping modeling module is used to process the existence variables based on a preset continuous interpolation function, mapping the discrete state of the existence or non-existence of array elements to a continuous impedance distribution, and obtaining the pseudo impedance value corresponding to the position of each candidate array element.

[0130] The current distribution solution module is used to substitute the pseudo impedance value and the initial phase design variables into the electromagnetic field integral equation constructed based on the method of moments to calculate the surface current distribution of the array antenna under the initial design variables; and calculates the radiated energy in the main lobe direction, the peak radiated energy in the side lobe region, and the current total number of array elements based on the surface current distribution.

[0131] The gradient optimization solution module is used to determine the radiation efficiency index based on the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the current total number of array elements. Under the constraint conditions, the objective function is to minimize the negative value of the radiation efficiency index, and the initial design variables are updated based on the gradient optimization algorithm.

[0132] The iterative convergence control module is used to determine whether the updated design variables meet the preset convergence conditions. If so, it converts the final array element sparse layout scheme and phase excitation scheme to generate the optimized array antenna pattern. Otherwise, the updated design variables are used as the initial design variables for the next iteration and optimization continues until the convergence conditions are met.

[0133] The above formulas are all dimensionless calculations. The formulas are derived from software simulations based on a large amount of collected data to obtain the most recent real-world results. The preset parameters in the formulas are set by those skilled in the art according to the actual situation.

[0134] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented in software, the above embodiments can be implemented, in whole or in part, as a computer program product. Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented by electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution.

[0135] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment, depending on actual needs.

[0136] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A method for array antenna pattern optimization based on multi-constraint convex optimization, characterized in that, include: S1. Obtain the initial geometric and electrical parameters of the array antenna. Based on the initial geometric parameters, construct an initial array topology containing multiple candidate array element positions. Set existence variables and initial phase design variables for each candidate array element position and summarize them to construct initial design variables. S2, based on a preset continuous interpolation function, the existence variable is processed to map the discrete state of whether the array element exists or not into a continuous impedance distribution, and the pseudo impedance value corresponding to the position of each candidate array element is obtained. S3. Substitute the pseudo impedance value and the initial phase design variables into the electromagnetic field integral equation constructed based on the method of moments to calculate the surface current distribution of the array antenna under the initial design variables. Calculate the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the total number of current array elements based on the surface current distribution. S4. The radiation efficiency index is determined based on the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the current total number of array elements. Under the constraint conditions, the objective function is to minimize the negative value of the radiation efficiency index, and the initial design variables are updated based on the gradient optimization algorithm. S5, determine whether the updated design variables meet the preset convergence conditions: if so, convert to the final array element sparse layout scheme and phase excitation scheme, and generate the optimized array antenna pattern. Otherwise, the updated design variables will be used as the initial design variables for the next iteration and optimization will continue until the convergence condition is met.

2. The method of claim 1, wherein the method is based on multi-constraint convex optimization. The initial geometric parameters specifically include the total number of array elements, the elliptic axis ratio of each concentric elliptical ring, the azimuth angle of the array elements on each ring, the spacing between adjacent rings, and the maximum aperture size. The electrical parameter is specifically the operating frequency of the array antenna; Based on the initial geometric parameters, an initial array topology containing the positions of all candidate array elements is constructed within the design area. Specifically, this involves: determining the array region boundary of the concentric elliptical ring array based on the maximum aperture size; determining the ratio of the major axis to the minor axis of each concentric elliptical ring based on the elliptical axis ratio; determining the radial spacing between each ring based on the spacing between adjacent rings; and determining the specific position coordinates of each candidate array element on each ring based on the azimuth angle of the array elements on each ring, thereby enabling the array to... In-plane generation includes The initial concentric elliptical ring array topology at the candidate array element positions; Based on the electrical parameters, electromagnetic calculation conditions are configured for the initial array topology, specifically: the electromagnetic wave number is determined according to the operating frequency of the array antenna, and the same fixed excitation amplitude is set for the position of each candidate array element. An existence variable is set for each candidate array element position, wherein the existence variable takes values ​​within the range of The continuous variable is used to characterize the probability of setting an array element at the candidate array element position. A value of 1 indicates that the array element will definitely be set, and a value of 0 indicates that the array element will definitely not be set. At the beginning of the optimization, the existence variable of each candidate array element position is set to 0.5, so that it evolves in the two directions of 0 or 1 in the subsequent optimization process. An initial phase design variable is set for each candidate element position. For candidate element positions where the existence variable is 0, the initial phase design variable is set to 0; for candidate element positions where the existence variable is non-zero, the initial phase design variable is set within a preset phase range. A random value is generated internally to serve as the initial phase design variable for the candidate array element position.

3. The method of claim 2, wherein: When the existence variable is 1, the corresponding candidate element position is a metallic material state; when the existence variable is 0, the corresponding candidate element position is a non-conductive material state. A tangential interpolation function is used to construct a continuous mapping relationship between the pseudo impedance values ​​at each candidate array element position and the existence variables. The expression of the tangential interpolation function is as follows: In the formula, Indicates the first The pseudo impedance value at each candidate array element position, This is the index of the candidate array element position; This is the equivalent impedance value of the metallic material. This represents the equivalent impedance value of a non-conductive material. The preset penalty factor, ; For the penalty function; For the first The existence variable of the positions of candidate array elements; The range of values ​​is determined by the tangential interpolation function. The existence of a variable is continuously mapped to a range of values ​​within The pseudo impedance values ​​are obtained, thus the pseudo impedance values ​​corresponding to the positions of each candidate array element are obtained, so that when the existence variable is 0, it corresponds to the high impedance state of non-conductive materials, when the existence variable is 1, it corresponds to the zero impedance state of metallic materials, and when the existence variable is an intermediate value, it corresponds to the intermediate impedance state between the two.

4. The method of claim 3, wherein: The specific logic underlying the calculation of the surface current distribution of the array antenna under the initial design variables is as follows: An electromagnetic field solution model for the array antenna is established based on the electric field integral equation. The Green's function is set according to the electromagnetic wave number. The pseudo impedance values ​​of each candidate array element position are substituted into the impedance boundary conditions to form a global impedance matrix containing the pseudo impedance matrix. At the same time, an excitation voltage vector is constructed based on the initial phase design variables of each candidate array element position and the fixed excitation amplitude. The electric field integral equation is discretized into a system of linear equations using the Galerkin method: In the formula, This represents the mutual impedance matrix associated with the array geometry; It is a diagonal impedance matrix composed of the pseudo-impedance values ​​of each candidate array element position; Let be the surface current distribution vector to be solved; The excitation voltage vector; By solving the linear equations, the calculated results The surface current distribution of each candidate array element position of the array antenna under the initial design variable.

5. The method of claim 4, wherein: Based on the surface current distribution obtained from the solution, the radiated electric field of the array antenna at each observation point in the far field region is calculated by using the equivalent dipole model, and then the radiated energy at each observation point is determined. The radiation energy at the pointing angle of the main lobe is determined as the radiation energy in the direction of the main lobe. Adopt The norm approximation method is used to process the radiation energy in the sidelobe region to obtain the peak radiation energy in the sidelobe region, wherein The value in the norm The greater the value is, the closer the calculation result is to the real maximum value in the sidelobe region. The existence variables of each candidate array element position are statistically analyzed. Candidate array element positions with existence variables greater than a preset threshold are determined as valid array elements. The total number of all valid array elements is counted as the current total number of array elements.

6. The array antenna pattern optimization method based on multi-constraint convex optimization according to claim 1, characterized in that: The formula for calculating the radiation effectiveness index is as follows: In the formula, Radiation efficiency index; Energy radiated in the direction of the main lobe. This represents the peak radiation energy in the sidelobe region. This represents the current total number of array elements. This is a preset penalty factor for the number of array elements, used to balance the weight between radiation performance and the number of array elements.

7. The method of claim 6, wherein: The negative value of the radiation effectiveness index is used as the objective function, and the objective function is a convex function; The constraint condition comprises: the main lobe direction radiation energy is not lower than a preset threshold , i.e. ; the total number of current array elements does not exceed a preset maximum value , i.e. ; The constraints constitute a convex set; Under the condition that the constraints are met, the objective function is minimized as the optimization objective. The gradient optimization algorithm is used to solve the convex optimization problem model. By iteratively updating the existence variables and phase design variables, the objective function is gradually reduced until convergence, and the updated existence variables and phase design variables are obtained.

8. The method of claim 1, wherein: The specific execution process of S5 is as follows: Obtain the existence variables and phase design variables after each iteration update, calculate the change in design variables between two adjacent iterations, and denote the maximum change in the existence variables as... The maximum change in the phase design variable is denoted as The formula it is based on is as follows: In the formula, The maximum change of an existing variable. The maximum change of the phase design variable; The index is the number of iterations. ; Indicates the first After the first iteration update Existence variables of candidate array element positions Indicates the first After the first iteration update Phase design variables for the positions of candidate array elements; The index of the candidate array element position. , This represents the total number of candidate array element positions. a preset existence variable convergence threshold and a phase design variable convergence threshold when and the current iteration satisfies the convergence condition, otherwise, it is determined that the convergence condition is not satisfied; When the convergence condition is met, the existence variables of the current iteration are converted into the final sparse layout scheme of the array elements. Specifically, for each candidate array element position, if... If the candidate element position is reserved, then the element is determined to be a reserved element. If the candidate element position is determined, the element will be removed, and the phase design variable for the current iteration will be... As the final phase excitation value for each retained array element, a phase excitation scheme is generated. Based on the sparse array element layout scheme and the phase excitation scheme, an optimized array antenna pattern is generated. When the convergence condition is not met, the updated existence variable will be... and phase design variables The optimization continues until the convergence condition is met, and the initial values ​​are used as the starting points for the next iteration.

9. A multi-constraint convex optimization-based array antenna pattern optimization system, used to execute the multi-constraint convex optimization-based array antenna pattern optimization method according to any one of claims 1-8, characterized in that, include: The initial parameter configuration module is used to obtain the initial geometric and electrical parameters of the array antenna, construct an initial array topology containing multiple candidate array element positions based on the initial geometric parameters, set existence variables and initial phase design variables for each candidate array element position and summarize them to construct the initial design variables; The impedance mapping modeling module is used to process the existence variables based on a preset continuous interpolation function, mapping the discrete state of the existence or non-existence of array elements to a continuous impedance distribution, and obtaining the pseudo impedance value corresponding to the position of each candidate array element. The current distribution solution module is used to substitute the pseudo impedance value and the initial phase design variables into the electromagnetic field integral equation constructed based on the method of moments to calculate the surface current distribution of the array antenna under the initial design variables. Calculate the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the total number of current array elements based on the surface current distribution. The gradient optimization solution module is used to determine the radiation efficiency index based on the radiation energy in the main lobe direction, the peak radiation energy in the side lobe region, and the current total number of array elements. Under the constraint conditions, the objective function is to minimize the negative value of the radiation efficiency index, and the initial design variables are updated based on the gradient optimization algorithm. The iterative convergence control module is used to determine whether the updated design variables meet the preset convergence conditions. If so, it is converted into the final array element sparse layout scheme and phase excitation scheme to generate the optimized array antenna pattern. Otherwise, the updated design variables will be used as the initial design variables for the next iteration and optimization will continue until the convergence condition is met.

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