Design Method of Sparse Array Antenna Based on Improved Pelican Optimization Algorithm

Abstract

The present invention discloses a design method for sparse array antennas based on an improved pelican optimization algorithm, including: using the improved POA to obtain the optimal solution of the array antenna pattern synthesis design model under the design objective. By leveraging the advantages of the triangular walk strategy in balancing global and local searches, after the stage of moving towards the prey (exploration stage) and the stage of spreading wings on the water (exploitation stage) of the POA, the triangular walk strategy is added, so that after the exploitation stage, the triangular walk strategy is used to recreate an exploitation stage, which can effectively solve the imbalance problem between exploration and exploitation, improve the local search accuracy of the algorithm, jump out of the local optimum, and at the same time accelerate the convergence speed to find the best solution. The design method of the present invention can effectively suppress the sidelobe level of the sparse antenna array without broadening the main lobe width, and form deep nulls in the specified position direction to achieve the required array antenna pattern.

Description

Technical Field

[0001] The present invention belongs to the technical field of mobile communication antenna design, and particularly relates to a design method of a sparse array antenna based on an improved pelican optimization algorithm. Background Art

[0002] With the development of technology, the requirements for antenna performance are increasing day by day. In a traditional uniform linear array, the element spacing is fixed. Although this can achieve functions such as directional signal transmission and reception to a certain extent, there are also limitations. For example, the sidelobe level of a uniform linear array is relatively high, which will lead to an increase in interference. When receiving a signal, in addition to the useful signal in the main lobe direction, the sidelobes will also receive interference signals from other directions; when transmitting a signal, the sidelobes will also cause energy waste and may generate unnecessary interference to other directions. Moreover, in many application scenarios, there are certain limitations on the cost and complexity of the device. If a traditional array wants to meet performance requirements such as high resolution, it often requires a large number of elements, which increases the cost, volume, and weight of the device, etc.

[0003] By non-uniformly distributing the positions of the elements, a sparse linear array can effectively reduce the sidelobe level. Under the same antenna aperture requirements, a sparse linear array can reduce the number of elements, which is very beneficial for reducing costs, reducing the volume and complexity of the device. At the same time, it can optimize the antenna pattern without increasing the number of elements. In fields such as radar detection and signal coverage in communication, it can make the signal better focus on the target direction, reduce interference from other directions, and thus meet the development needs of modern high-precision, high-cost-performance, and miniaturized devices.

[0004] The Pelican Optimization Algorithm (POA) is a new type of swarm intelligence optimization algorithm. The inspiration for the pelican optimization algorithm is mainly based on a series of natural behaviors of pelicans in nature such as living and hunting. Pelicans are large, long-billed, gregarious birds with a particularly large throat, similar to a bag, and usually feed on fish. They often cooperate with each other to hunt. Once they identify the prey, they will fly down from a high altitude, rush towards the prey, and then spread their wings on the water surface, forcing the fish to swim towards the shallow water area, which helps them capture the prey. The mathematical model describing the pelican optimization algorithm has three stages: initialization, the stage of moving towards the prey, and the stage of spreading wings on the water. In the stage of moving towards the prey, the pelican will first identify and lock the prey, and then drive the prey towards the shallow water area. In the stage of spreading wings on the water, the pelican will rush towards the prey, then push the prey up to the water surface, and finally store the prey in the bag in its throat, and then continue to catch the next prey.

[0005] Some studies have shown that compared with the Particle Swarm Optimization (PSO), Ant Lion Optimization Algorithm (ALO), and Sparrow Search Algorithm (SSA), POA has higher precision in the optimization process and can effectively converge to the optimal solution of the problem. At the same time, POA also has strong robustness and adaptability, and it can solve optimization problems in various fields, such as heterogeneous wireless sensor networks, microgrid energy management, and automatic facial emotion recognition, etc.

[0006] In the field of mobile communication antenna design, when using evolutionary algorithms such as the Runner-root Algorithm (RRA) and the Multi-Verse Optimization Algorithm (MVO) to solve the antenna pattern synthesis problem, if deep nulls are placed in the target direction, the sidelobe level will increase correspondingly or the main lobe width will broaden, and it cannot meet the ideal requirements for both the sidelobe level and the deep nulls. Moreover, they lack exploration in the middle and late stages of optimization and are prone to falling into local optima, resulting in an increase in the sidelobe level or the deep null level of the target pattern. However, POA has an optimization process of identifying and locking prey, so in the early exploration, it can obtain a more accurate position, and in the middle and late stages, it conducts local neighborhood search to find a better position in a more precise manner. Therefore, it can greatly improve the global and local search capabilities and is a better algorithm for solving the antenna pattern synthesis problem.

[0007] However, similar to other intelligent swarm optimization algorithms, POA also has problems such as premature convergence, imbalance between exploration and exploitation, and lack of population diversity. Summary of the Invention

[0008] Aiming at the defects of the existing technology, the purpose of the present invention is to propose a method for obtaining the optimal solution of the array antenna pattern synthesis problem through an improved pelican optimization algorithm and thus obtaining the best design scheme. This method adds a triangular walking strategy to POA, improving the algorithm's convergence precision and convergence speed. By optimizing the positions of the array antenna elements, on the premise that the main lobe width does not broaden, a sidelobe level lower than the index requirement and deep nulls in a specific direction are formed to optimize and solve the complex array antenna pattern synthesis problem.

[0009] The technical solution of the present invention is as follows:

[0010] A method for designing a sparse array antenna based on an improved pelican optimization algorithm, which includes:

[0011] Set up a sparse array antenna pattern synthesis design model containing design objectives;

[0012] Obtain the optimal solution of the design model under the design objective through the improved POA;

[0013] Among them, the design objective is: by optimizing the element spacing of the array antenna, while ensuring that the main lobe width of the antenna array remains unchanged, form a sidelobe level lower than the expected value in its sidelobe region, and form deep nulls at specified positions;

[0014] The improved POA contains the following two stages of moving towards the prey and spreading wings on the water, as well as the pelican's position update model after adding the triangular wandering strategy:

[0015]

[0016]

[0017] Among them, respectively represent the new states of the pelican in the stage of moving towards the prey, the stage of spreading wings on the water, and the triangular wandering stage. X i represents the new position of the pelican at the t-th iteration of each stage. rand in all the above formulas represents a random number from 0 to 1, and all F i represents the objective function value.

[0018] In formula (1), x i,j represents the state of the i-th pelican in the j-dimensional space, p j represents the position of the prey in the j-dimensional space, and I is a random number from 1 to 2.

[0019] In formula (3), R is a constant equal to 0.2, t represents the current iteration number, and T represents the maximum iteration number; the coefficient represents the radius of the neighborhood of the group members, and local search is performed near each member to converge to a better solution. This coefficient is effective for the exploitation ability of POA and can be closer to the optimal global solution. In the initial iteration, the value of this coefficient is larger, so a larger area around each member is considered. As the number of algorithm iterations increases, decreases, resulting in a smaller neighborhood radius for each member. This enables us to scan the area around each member in the population with smaller and more precise step sizes, so that POA can converge to a solution closer to the global optimum according to the usage concept.

[0020] In formulas (5) and (6), C represents the centroid, X z and X k represent the values generated around X i , W represents the number of walking steps, and R mRepresents a random number from 1 to m, where m represents the number of problem variables.

[0021] According to some specific embodiments of the present invention, the design model is set as follows:

[0022] The model is an unequally spaced linear array composed of 2N ideal point sources, and the excitation current amplitudes in the linear array are centrosymmetric and the excitation phases are all zero. The expression of the array antenna pattern is as follows:

[0023]

[0024] Among them, F(θ) represents the expression of the array antenna pattern, k = 2π / λ represents the wave number, d is the spacing between array elements, λ is the free space wavelength, and θ is the angle between the ray direction and the array axis.

[0025] According to some specific embodiments of the present invention, in the improved POA, the fitness of each individual is obtained through the following fitness function value:

[0026]

[0027] Among them, fit is the fitness function value, PSL is the Peak Sidelobe Level (abbreviation: PSL), TPSL is the Target Peak Sidelobe Level (abbreviation: TPSL), NSLL is the Null Sidelobe Level (abbreviation: NSLL), α and β are weight coefficients, and α + β = 1, F(θ i ) represents the array antenna pattern distribution function, i represents the current i-th deep null point, and N 0 is the number of deep null points.

[0028] According to some specific embodiments of the present invention, the design method includes:

[0029] S1: Based on the described design model, perform POA initialization on the solution space of its design goal according to the following model;

[0030] x i,j = l j + rand·(u j - l j ), i = 1, 2,..., N, j = 1, 2,..., m (10)

[0031] Each pelican individual is randomly generated between the upper and lower bounds of the given problem. Among them, x i,jis the value of the j-th variable specified by the i-th candidate solution, N is the number of population members, m is the number of problem variables, rand is a random number in the interval [0, 1], and u j and l j represent the upper and lower position boundaries of the pelican, respectively.

[0032] S2: After initialization, in the current iteration, calculate the fitness function value of each pelican in the population, and select the pelican individual with the smallest fitness function value in the population as the current best individual X best , which is considered the most suitable hunting position for the pelican, i.e., the optimal solution. Its fitness function value is f best ; At the same time, set the fitness function value of the prey to prepare for the next hunting.

[0033] f best = min(f(X i )) (11)

[0034]

[0035] S3: Enter the stage of moving towards the prey. By comparing the fitness function values of the pelican and the prey, use formula (1) to update the new state of the pelican, and use formula (2) to update the position of the pelican;

[0036] S4: Enter the stage of spreading wings on the water. Based on the prey position information obtained in the stage of moving towards the prey, the pelican will fly to the water surface, flap its wings, and drive the prey into the shallow water area. At this time, use formula (3) to update the new state of the pelican in the stage of spreading wings on the water, and then compare it with the objective function value, and use formula (4) to determine whether to update the new position of the pelican.

[0037] S5: Enter the stage of triangular wandering. Based on the last updated position of the pelican in the stage of spreading wings on the water, update the position of the pelican in the stage of triangular wandering through the pelican position update model described by formulas (5) and (6);

[0038] S6: After the iterative update of the above three stages is completed, calculate the fitness function values of all pelican individuals in the population, then sort them from small to large according to the fitness function values, select the pelican with the smallest fitness function value, update the best individual of the pelican, and determine the position of the next pelican;

[0039] S7: After each iterative update, judge whether the algorithm reaches the maximum number of iterations. If so, the algorithm ends and outputs the optimal solution X best . Otherwise, after increasing the number of iterations, return to S2 and continue the iteration.

[0040] According to some specific embodiments of the present invention, the individual fitness is obtained through the following calculation model:

[0041]

[0042] Among them, fit is the fitness function value, PSL is the Peak Sidelobe Level (abbreviated as PSL), TPSL is the Target Peak Sidelobe Level (abbreviated as TPSL), NSLL is the Null Sidelobe Level (abbreviated as NSLL), α and β are weight coefficients, and satisfy α + β = 1, F(θ i ) represents the array antenna pattern distribution function, i represents the current i-th deep null, N 0 is the number of deep nulls.

[0043] According to some specific embodiments of the present invention, the selection method of the weight coefficients α and β is as follows: According to the degree of closeness of the pattern in the design area to the design target, the weight coefficient is increased for the area where the search is slow. Through experiments, the values of α and β are obtained.

[0044] The present invention has the following beneficial effects:

[0045] In the design method of the present invention, an improved POA is used for the synthesis design of the array antenna pattern. By adding a triangular wandering strategy, the relative position relationship between multiple pelican individuals is effectively utilized to optimize the positions of the pelican individuals, enhance the fine search ability of the POA in the local search area, help prevent the algorithm from falling into local optima, facilitate the algorithm to quickly converge to the global optimal solution, and at the same time enhance the local exploration ability of the algorithm in the middle and late stages, so as to obtain better target sidelobe levels and deep null levels.

[0046] The present invention adopts the method of adding a triangular wandering strategy to the POA. In the original POA, in the entire exploration and exploitation process, there are only two stages: moving towards the prey and spreading wings on the water. The exploitation stage is less, which may lead to an imbalance between exploration and exploitation and cause the algorithm to fail to find a better solution. By adding triangular wandering after the two update stages of the algorithm, the balance relationship between algorithm exploration and exploitation can be effectively balanced, enabling the algorithm to jump out of the local optimum and possibly find a better solution.

[0047] The present invention applies an improved pelican optimization algorithm to the synthesis of the array antenna pattern. By optimizing the element spacing of the array antenna, under the premise that the main lobe width is not broadened, the sidelobe level is effectively suppressed, forming a sidelobe level lower than the index requirement and deep nulls in specific directions.

[0048] The present invention first applies the Pelican Optimization Algorithm to the design of array antenna pattern synthesis problems, and in view of the problems of slow convergence speed and easy entrapment in local optima when the Pelican Optimization Algorithm is used to solve array antenna pattern synthesis problems, an improved Pelican Optimization Algorithm is proposed. Based on the imbalance problem between the exploration and exploitation stages in the Pelican Optimization Algorithm, a triangular random walk strategy is added to the exploitation stage of the Pelican Optimization Algorithm to balance the relationship between the two, jump out of the local optimum, and find the best solution.

[0049] The improved Pelican Optimization Algorithm used in the present invention has strong generality and portability, can be applied to optimization problems in related fields, and can also be combined with other algorithms. It not only provides new ideas and methods for array antenna pattern synthesis problems, but also effectively expands the application depth and breadth of the Pelican Optimization Algorithm. Description of the Drawings

[0050] Figure 1 The pattern obtained by the multi-version optimization algorithm in Example 1.

[0051] Figure 2 The pattern obtained by the original Pelican Optimization Algorithm in Example 1.

[0052] Figure 3 The convergence curve graph obtained by the original Pelican Optimization Algorithm in Example 1.

[0053] Figure 4 The pattern obtained by the improved Pelican Optimization Algorithm in Example 1.

[0054] Figure 5 The convergence curve graph obtained by the improved Pelican Optimization Algorithm in Example 1.

[0055] Figure 6 The pattern obtained by the Roulette Root Algorithm in Example 2.

[0056] Figure 7 The pattern obtained by the original Pelican Optimization Algorithm in Example 2.

[0057] Figure 8 The convergence curve graph obtained by the original Pelican Optimization Algorithm in Example 2.

[0058] Figure 9 The pattern obtained by the improved Pelican Optimization Algorithm in Example 2.

[0059] Figure 10 The convergence curve graph obtained by the improved Pelican Optimization Algorithm in Example 2. Detailed Implementation Manner

[0060] The present invention is described in detail below in conjunction with the embodiments and drawings, but it should be understood that the embodiments and drawings are only used to exemplify the present invention and do not constitute any limitation on the protection scope of the present invention. All reasonable changes and combinations within the scope of the inventive concept of the present invention fall within the protection scope of the present invention.

[0061] According to the technical method of the present invention, some specific implementation methods of the sparse array antenna design method based on the improved Pelican optimization algorithm include the following steps:

[0062] S0 sets the comprehensive design model of the array antenna pattern. The model is an unequally spaced linear array composed of 2N ideal point sources, and the excitation current amplitude in the linear array is centrally symmetric and the excitation phase is zero. The pattern expression is shown in equation (14). The design goal is to optimize the element spacing of the array antenna, form a sidelobe level lower than the expected value in the sidelobe area while ensuring that the main lobe width of the array remains unchanged, and form a deep zero point at the specified position:

[0063]

[0064] Wherein, F(θ) represents the expression of the array antenna pattern, k=2πfλ represents the wave number, d is the spacing between array elements, λ is the free space wavelength, and θ is the angle between the ray direction and the array axis.

[0065] The above design model can fully reduce optimization variables, lower calculation costs, and obtain excellent design solutions.

[0066] S1: Based on the design model of S0, POA is initialized for the solution space of its design target.

[0067] According to some specific embodiments of the present invention, the initialization includes: setting the initial evolutionary generation t=1, setting the population size N, the maximum number of iterations of the algorithm T, the dimension m of the problem, and the feasible solution space of the problem, and randomly generating a set of initial solutions:

[0068]

[0069] Where N is the population size, X is the population matrix of pelicans, and X i It is the i-th pelican.

[0070] S2: After initialization, in the current iteration, the fitness function value of each pelican in the population is calculated, and the pelican individual with the smallest fitness function value in the population is selected as the current best individual X through the following model: best , and it is considered that this is the most suitable position for the pelican to hunt, that is, the optimal solution. Its fitness function value is f best ; At the same time, the fitness function value of the prey is set to prepare for the next hunting.

[0071] f best = min(f(X i )) (16)

[0072]

[0073] S3: Further, during the iterative calculation process, calculate the fitness function values of all individuals. The fitness function value of each individual can be calculated by Equation (16);

[0074]

[0075] where fit is the fitness function value, PSL is the Peak Sidelobe Level (abbreviated as PSL), TPSL is the Target Peak Sidelobe Level (abbreviated as TPSL), NSLL is the Null Sidelobe Level (abbreviated as NSLL), α and β are weight coefficients, and satisfy α + β = 1, F(θ i ) represents the array antenna pattern distribution function, i represents the current i-th deep null, and N 0 is the number of deep nulls. The selection method of the weight coefficients is as follows: According to the degree of the pattern in each design area approaching the design target, the weight coefficient can be appropriately increased in the area with slow evolution. After repeated experiments, the values of α and β are determined.

[0076] S4: Enter the stage of moving towards the prey. By comparing the fitness function values of the pelican and the prey, use Equation (1) to update the new state of the pelican, and at the same time use Equation (2) to update the position of the pelican;

[0077] S5: Enter the stage of spreading wings on the water. Based on the prey position information obtained in the stage of moving towards the prey, the pelican will fly to the water surface, flap its wings, and drive the prey into the shallow water area. At this time, use Equation (3) to update the new state of the pelican in the stage of spreading wings on the water, and then compare it with the objective function value, and use Equation (4) to determine whether to update the new position of the pelican.

[0078] S6: Enter the stage of triangular wandering. Based on the last updated position of the pelican in the stage of spreading wings on the water, update the position of the pelican in the stage of triangular wandering through the pelican position update model described by Equations (5) and (6);

[0079] S7: After the iterative update of the above three stages is completed, calculate the fitness function values of all pelican individuals in the population, then sort them from small to large according to the fitness function values, select the pelican with the smallest fitness function value, update the best individual of the pelican, and determine the position of the next pelican;

[0080] S8: After each iterative update, determine whether the algorithm has reached the maximum number of iterations. If so, end the algorithm and output the optimal solution X best . Otherwise, after increasing the number of iterations, return to S2 and continue the iteration.

[0081] Embodiment 1

[0082] The present invention will be described below in conjunction with simulation experiments.

[0083] 1. Experimental content:

[0084] Perform pattern synthesis design on a sparse linear array with current amplitude centrosymmetry and zero phase described by Equation (8), and use experimental examples to verify the effect of the algorithm proposed by the present invention for the pattern synthesis problem of sparse array antennas.

[0085] Experiment 1: 2N = 20, excitation current I n = 1, and compare it with the 20-element array in the paper "Linear Antenna Array Pattern Synthesis Using Multi-Verse Optimization Algorithm" published by Raghuvanshi, Anoop. Among them, the main lobe width 2θ 0 = 16°, the sidelobe regions are θ = [0°, 82°] and θ = [98°, 180°], and the deep nulls are placed at the positions where θ is equal to 64°, 76°, 104°, and 116°. The population size is 30 for both, and the number of iterations is also 1000 for both.

[0086] 2. Experimental results:

[0087] Figure 1 is the array pattern optimized by the multi-version optimization algorithm, where the highest level in the sidelobe region is -22.06 dB. At the positions where θ is equal to 64° and 76°, the deep null values are -70.00 dB and -78.11 dB respectively, and the main lobe width is 18.2°. Figure 2 and Figure 3 is the pattern of the sparse array antenna optimized by the original pelican algorithm and the convergence curve of the optimization result. Among them, the highest level in the sidelobe region is -23.35 dB. At the positions where θ is equal to 64° and 76°, the deep null values are -86.33 dB and -82.62 dB respectively, and the main lobe width is 18.2°. Figure 4 and Figure 5It is the radiation pattern of a sparse array antenna optimized by the improved pelican algorithm and the convergence curve of the optimization result. The highest level in the sidelobe region is -23.54 dB. At the positions where θ equals 64° and 76°, the deep null values are -88.85 dB and -83.77 dB respectively, and the main lobe width is 17.6°. It can be seen from this that the peak sidelobe level of the improved pelican optimization algorithm is 0.19 dB lower than that of the original pelican optimization algorithm and 1.48 dB lower than that of the multi-version optimization algorithm. Moreover, at specific deep null positions, the deep null values are lower and the main lobe width is narrower. This verifies the effectiveness of the algorithm proposed in the present invention.

[0088] The comparison among the three is shown in Table 1 below (retaining two decimal places after the decimal point):

[0089] Table 1 Comparison of simulation results of different algorithms

[0090]

[0091] Example 2

[0092] The present invention will be further described below in combination with simulation experiments.

[0093] 1. Experimental content:

[0094] For the pattern synthesis design of a sparse linear array with centrosymmetric current amplitude and zero phase described by Equation (8), experimental examples are used to verify the effect of the algorithm proposed in the present invention for the pattern synthesis problem of sparse array antennas.

[0095] Experiment 1: 2N = 32, element spacing I n = 1, and it is compared with the 32-element array in the paper "Runner-root algorithm to control sidelobe levels and null depths in linear antenna arrays" published by Subhashini, K.R. Among them, the main lobe width 2θ 0 = 6°, the sidelobe regions are θ = [0°, 87°] and θ = [93°, 180°], and the deep nulls are placed at the positions where θ equals 81° and 99°. The population size is 30 for both, and the number of iterations is also 1000 for both.

[0096] 2. Experimental results:

[0097] Figure 6 It is the array radiation pattern optimized by the runner-root algorithm, where the highest level in the sidelobe region is -18.60 dB, and the deep nulls corresponding to θ at 81° and 99° are -85.39 dB. Figure 7 and Figure 8It is the pattern and the convergence curve of the optimization result of the sparse array antenna optimized by the original pelican optimization algorithm in Embodiment 2, where the highest level in the sidelobe region is -19.74 dB, and the deep nulls corresponding to θ at 81° and 99° are -106.92 dB. Figure 9 and Figure 10 It is the pattern and the convergence curve of the optimization result of the sparse array antenna optimized by the improved pelican optimization algorithm in Embodiment 2. The highest level in its sidelobe region is -19.94 dB, and the deep nulls corresponding to θ at 81° and 99° are -117.00 dB. It can be seen from this that the peak sidelobe level of the improved pelican optimization algorithm is 0.2 dB lower than that of the original pelican optimization algorithm and 1.34 dB lower than that of the wheel root algorithm. In the comparison of the deep nulls at the positions of θ at 81° and 99°, the value obtained by the improved pelican optimization algorithm is 10.08 dB lower than that obtained by the original pelican optimization algorithm and 31.61 dB lower than that obtained by the wheel root algorithm. This further verifies the effectiveness of the algorithm proposed by the present invention.

[0098] The comparison among the three is shown in Table 2 below (retaining two decimal places after the decimal point):

[0099] Table 2 Comparison of the simulation results of different algorithms

[0100]

[0101] The above embodiments are only the preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions falling within the idea of the present invention belong to the protection scope of the present invention. It should be pointed out that for those of ordinary skill in the art, without departing from the principle of the present invention, the improvements and refinements should also be regarded as within the protection scope of the present invention.

Claims

1. A sparse array antenna design method based on an improved Pelican optimization algorithm, characterized by: include: Setting up a comprehensive design model of sparse array antenna patterns with design objectives; The design model is set up as follows: The model is an unequally spaced linear array composed of 2N ideal point sources, and the excitation current amplitude in the linear array is centrally symmetric and the excitation phase is zero. The array antenna pattern expression is: Wherein, F(θ) represents the array antenna pattern expression, k=2π / λ represents the wave number, d is the array element spacing, λ is the free space wavelength, and θ is the angle between the ray direction and the array axis; Obtaining the optimal solution of the design model under the design goal through the improved POA; In the improved POA, the fitness of each individual is obtained by the following fitness function value expression: Where fit is the fitness function value, PSL is the peak sidelobe level, TPSL is the target peak sidelobe level, NSLL is the deep zero level, α and β are weight coefficients, and α+β=1, F(θ i ) represents the array antenna pattern distribution function, i represents the current i-th deep zero point, and N0 is the number of deep zero points; The design goal is to optimize the array element spacing of the sparse array antenna, so as to form a side lobe level lower than the expected value in the side lobe area while ensuring that the main lobe width of the array antenna remains unchanged, and to form a deep null point in the specified position direction; The improved POA includes the following two stages of moving towards prey and flapping wings on water, as well as the position update model of the pelican after adding the triangular swimming strategy: in, They represent the new states of the pelican in the phase of moving towards prey, the phase of spreading wings on water, and the phase of triangular swimming; X i represents the new position of the pelican at the tth iteration of each stage; rand in all the above formulas represents a random number between 0 and 1, and all F i represents the objective function value; In formula (1), x i,j represents the state of the i-th pelican in the j-dimensional space, p j represents the position of the prey in the j-dimensional space, and I is a random number from 1 to 2; In formula (3), R is a constant equal to 0.2, t represents the current number of iterations, and T represents the maximum number of iterations; In formulas (5) and (6), C represents the center of mass, X z and X k Indicates that in X i The value generated around, W represents the number of walks, R m Represents a random number from 1 to m, where m represents the number of problem variables.

2. The design method of sparse array antenna based on improved Pelican optimization algorithm according to claim 1 is characterized in that: The following steps are involved: S1: Based on the design model, the solution space of the design target is initialized by POA according to the following model; x i,j =l j +rand·(u j -l j ),i=1,2,…,N,j=1,2,…,m(10) Each pelican is randomly generated between the upper and lower bounds of the given problem; where x i,j is the value of the jth variable specified by the ith candidate solution, N is the number of population members, m is the number of problem variables, rand is a random number in the interval [0,1], and u j and l j They represent the upper and lower position boundaries of the pelican respectively; S2: After initialization, in the current iteration, the fitness function value of each pelican in the population is calculated, and the pelican individual with the smallest fitness function value in the population is selected as the current best individual X through the following model: best , which is considered to be the most suitable hunting position for pelicans, i.e. the optimal solution; its fitness function value is f best ; At the same time, the fitness function value of the prey is set to prepare for the next hunt; f best =min(f(X i ))(11) S3: Enter the stage of moving towards the prey. By comparing the fitness function values ​​of the pelican and the prey, the new state of the pelican is updated using formula (1), and the position of the pelican is updated using formula (2); S4: Entering the stage of spreading wings on the water, based on the prey position information obtained in the stage of moving towards the prey, the pelican will fly to the water surface, flap its wings, and force the prey into the shallow water area. At this time, the new state of the pelican in the stage of spreading wings on the water is updated using formula (3), and then compared with the objective function value, and formula (4) is used to determine whether the new position of the pelican needs to be updated; S5: Entering the triangular walking stage, based on the last updated position of the pelican in the water wing-spreading stage, the pelican position update model of the triangular walking stage described in formulas (5) and (6) is used to update the pelican position in the triangular walking stage; S6: After the above three stages of iterative updates are completed, the fitness function values ​​of all pelicans in the population are calculated, and then they are sorted from small to large according to the fitness function values, and the pelican with the smallest fitness function value is selected, the best individual pelican is updated, and the position of the next pelican is determined; S7: After each iteration, determine whether the algorithm has reached the maximum number of iterations. If so, the algorithm ends and outputs the optimal solution X best ; Otherwise, after increasing the number of iterations, return to S2 and continue iterating.

3. The design method of sparse array antenna based on improved Pelican optimization algorithm according to claim 2 is characterized in that: The initialization includes: setting the initial evolutionary generation t=1, setting the population size N, the maximum number of iterations of the algorithm T, the dimension m of the problem, and the feasible solution space of the problem, and randomly generating a set of initial solutions: Where N is the population size, X is the population matrix of pelicans, and X i It is the i-th pelican.

4. The design method of sparse array antenna based on improved Pelican optimization algorithm according to claim 1, characterized in that: The weight coefficients α and β are selected in the following manner: according to the degree to which the directional diagram of the design area is close to the design target, the weight coefficient of the area with slow evolution is increased, and the values ​​of α and β are obtained through experiments.

Images