Phase-Only Pattern Synthesis for Spaceborne Array Antenna Based on Improved Mayfly Optimization Algorithm

Abstract

A new optimization algorithm—Improved Mayfly Optimization Algorithm (IMOA)—is proposed in this paper to fulfill the low sidelobe level (SLL) design requirements of the spaceborne array antenna. MOA is a new heuristic algorithm inspired by the flying behavior and mating process of mayflies. It has a unique speed updating system with great convergence, strong stability, fast solution speed, and high precision. Based on the MOA, IMOA not only introduces the adaptive inertial weight factor to enhance the search ability, but also uses the Levy flight strategy and the golden sine operator to improve the disadvantage of easily falling into the local optimal solution. Firstly, according to the antenna pattern requirements of high gain and low sidelobe, an optimization problem model is carried out. Then, the IMOA is applied to solve the problem by only controlling the phase under a given secondary amplitude distribution. Simulation results show that IMOA has great advantages in the maximum sidelobe level (MSLL) suppression and convergence speed. Finally, the EM simulations are conducted on the 528-element planar array antenna. The maximum sidelobe level suppression performances in the test are very consistent with the theoretical simulation, which verifies the feasibility and effectiveness of the proposed IMOA.

1. Introduction

The spaceborne antenna array has attracted increased attention in recent years due to its excellent beamforming, beam steering, and high gain characteristics. The fundamental concern in the synthesis of array antenna patterns is to find an adequate excitation amplitude and phase to create the required radiation pattern. To handle the complex antenna pattern synthesis problem, many approaches, including traditional mathematical methodologies and optimization algorithms, have been developed [1,2,3,4]. One of the most important technical indications in antenna design synthesis is the sidelobe level. An antenna pattern with a low sidelobe level can significantly increase communication quality by boosting the signal-to-noise ratio, decreasing the influence of the clutter signal outside the main beam, and improving the overall system’s anti-interference performance [5,6,7,8]. Controlling the excitation amplitude of an antenna element in a large-scale array antenna requires a corresponding feed network, which is hard and expensive. However, phase weighting simply requires the phase shifter, which is simple and inexpensive. As a result, using phase-only weighting for array antenna pattern synthesis is frequently requested to simplify the complexity of the array feeding network and minimize manufacturing costs [9,10].

Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) are the first two popular and classical optimization algorithms, which have been successfully applied to antenna pattern synthesis due to their high efficiency and simplicity [11,12,13,14]. However, these two algorithms have the disadvantage of premature convergence when solving multi-parameter optimization problems. Therefore, more and more improved algorithms based on the classic GA and PSO have been proposed [15,16], such as Differential Evolution Algorithm (DE) [17], Moth Flame Optimization Algorithm (MFO) [18], Fruit-Fly Optimization Algorithm (FOA) [19], Invasive Weed Optimization Algorithm (IWO) [20], Grey Wolf Optimization Algorithm (GWO) [21], Mayfly Algorithm (MA) [22], Compressed Sensing (CS) [23], Biogeography-Based Optimization (BBO) [24], Firefly Algorithm (FA) [25], Ant Colony Optimization (ACO) [26], and so on. Although above algorithms can achieve good results in array antenna synthesis, further improvement is still needed. When the number of antenna elements increases, too many parameters will lead to slow calculation speed and low efficiency of the algorithm, and their solutions will easily fall into the local optimal solution.

Mayfly Optimization Algorithm (MOA) is a new intelligent optimization algorithm proposed by Zervoudakis and Tsafarakis in 2020 [27], which was inspired by the flying behavior and mating process of mayflies, including the mayfly crossing, mutation, group gathering, wedding dance, and random walking operations. Due to its unique speed updating system, MOA has the advantages of strong stability, fast solution speed, and high precision.

In 2021, Owoola applied the MOA to pattern synthesis of uniform and sparse linear antenna array to reduce side lobes [28]. An Improved Mayfly Optimization Algorithm (IMOA) is suggested in this study and used for the antenna pattern synthesis of low sidelobe planar arrays to increase the global search ability and solution accuracy of the MOA. The IMOA is offered as an alternative to the regular MOA. On the one hand, an adaptive inertial weight factor is used to improve the algorithm’s search ability; on the other hand, the Levy flight strategy and the golden sine factor are introduced to address the shortcoming of being easily trapped in the local optimal solution, which increases population diversity and speeds up convergence. This work compares the IWOA to various algorithms and uses numerous common test functions to demonstrate its advantages in terms of convergence speed and solution correctness. Finally, the suggested IMOA is applied to an antenna synthesis model developed within the restrictions of the practical engineering project in order to meet the design objectives of high gain and low sidelobe. With a particular secondary amplitude distribution, IMOA estimates the phase excitation of each antenna element. By comparing the results of experiments and simulations, the superiority and effectiveness of the proposed IMOA in the synthesis of space-borne planar array antennas are demonstrated.

The sections of this paper are organized as follows: Section 2 introduces the mayfly optimization algorithm. Section 3 explains the improved mayfly optimization algorithm (IMOA) in detail in three aspects. Section 4 analyzes the performance comparison between IMOA and other algorithms. Section 5 gives the results of the related experiments. Section 6 concludes this paper overall.

2. Mayfly Optimization Algorithm

The Mayfly algorithm, a novel form of intelligent optimization algorithm, offers significant optimization ability and research value. It is inspired by mayfly mating activity. The optimal male and female mayfly individuals are mated to produce the optimal progeny during mating behavior. Similarly, suboptimal people are paired to produce suboptimal offspring, and so on. This process follows the rule of survival of the fittest, eventually phasing out those with inferior fitness.

Suppose the positions of the male and female mayflies in the d-dimensional space are $x=(x_1,x_2,x_3,\cdots ,x_d)$; then, the fitness function value $f\left(x\right)$ can be calculated by the position information. Assuming that the speed of the mayfly individual in the dimensional space is $x=(v_1,v_2,v_3,\cdots ,v_d)$, the flight direction of each mayfly is a dynamic interaction between individual and social flight experience, and both male and female individuals have the best position $pbest$.

2.1. Movement of Male Mayfly

Male mayflies tend to congregate in groups, and each male mayfly’s location is altered based on its own and adjacent experience. Let $x_i^t$ be the current position of the mayfly i at the tth iteration, and $v_i^t$ be the speed of the mayfly i at the tth iteration; the expression for updating the position is:

$$x_i^{t+1}=x_i^t+v_i^{t+1}$$

(1)

Considering the mayfly’s constant movement and the dancing performance at a distance above the water, its speed is updated as:

$$v_{ij}^{t+1}=v_{ij}^t+a_1{e}^{-\beta r_p^2}(pbes{t}_{ij}-x_{ij}^t)+a_2{e}^{-\beta r_g^2}(gbes{t}_j-x_{ij}^t)$$

(2)

where $v_{ij}^t$ is the speed of the mayfly i in the tth iteration in the dimension j; $x_{ij}^t$ is the position of the mayfly i in the tth iteration of the dimension j, $a_1$ and $a_2$ are the attraction coefficients of the mayflies’ swimming behavior; $pbest$ is the best position in history, $gbest$ is the global optimal position; $\beta$ is the visibility coefficient, which is used to control the visible range of the mayfly, $r_p$ represents the distance between the current position and $pbest$, and $r_g$ is the distance between the current position and $gbest$. The distance calculation formula is:

$$∥x_i-X_i∥=\sqrt{{\sum }_{j=1}^n(x_{ij}-X_{ij}{}^2}$$

(3)

The best mayflies have to continue to perform their distinctive up-and-down dance to obtain the optimal position; therefore, the best mayflies must change speed constantly. Its speed is updated as:

$$v_{ij}^{t+1}=v_{ij}^t+d·r$$

(4)

where d represents the dance coefficient, and $r\in [-1,1]$ is a random coefficient.

2.2. Movement of Female Mayfly

Female mayflies and male mayflies differ in that male mayflies congregate, whereas female mayflies do not. Female mayflies, on the other hand, will fly to mate with male mayflies. Assuming that $y_i^t$ is the position of the female mayfly i in the tth iteration, its position update is expressed as:

$$y_i^{t+1}=y_i^t+v_i^{t+1}$$

(5)

The speed of the female mayfly is updated as follows:

$$v_{ij}^{t+1}=\{\begin{array}{c}{v}_{ij}^t+a_2{e}^{-\beta r_m^2(x_{ij}^t-y_{ij}^t)}if f\left(y_i\right)>f\left(x_i\right)\hspace{1em}\mathrm{(6a)}\hfill \\ v_{ij}^t+fl·r if f\left(y_i\right)\le f\left(x_i\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{(6b)}\hfill \end{array}$$

where $v_{ij}^t$ represents the speed of the mayfly, $y_{ij}^t$ represents the position of the mayfly, $r_m$ represents the distance between the male and the female mayfly, and $fl$ is the random walk coefficient, which only works when the female mayfly is under no attack.

2.3. Mayfly Mating

Mating between male and female individuals is a trait of all living things, including mayflies. The mating procedure is as follows: two parents are chosen at random from male and female populations, and the method of selecting male and female samples is the same as that of a male enticing a female.

The optimal male mayfly individual is paired with the optimal female mayfly individual in this method, and the suboptimal male mayfly individual is mated with the subpar female mayfly individual. Following mating, the ideal and suboptimal offspring are as follows:

$$offspring1=L·male+(1-L)·female$$

(6)

$$offspring2=L·female+(1-L)·male$$

(7)

where $L\in [-1,1]$.

3. Improved Mayfly Optimization Algorithm

The standard MOA algorithm has better convergence speed and solution accuracy than other swarm intelligence optimization algorithms, but the overall convergence effect and search accuracy can be further improved. Therefore, this paper proposes the IMOA to optimize and improve the strategy from the following three aspects.

3.1. Adaptive Weight

The concept of inertia weight initially appeared in the particle swarm method, which indicates that, in the iteration of particle swarm optimization, the change of particle coordinates is proportional to inertia weight. In this paper, a nonlinear inertial weighting factor is inspired by this. When the inertia weight is high, the algorithm has a strong search ability and will explore a larger region; when the inertia weight is low, the algorithm has a strong late search ability and will search only around the ideal solution. The formula for adaptive inertia weight is as follows:

$$w=sin\left(\frac{\pi ·t}{2·i{t}_{max}}\right)+1$$

(8)

where t is the current iteration number, and $i{t}_{max}$ is the maximum iteration number. The position update of the male mayfly in MOA after introducing the inertial weight factor is:

$$x_i^{t+1}=w{x}_i^t+v_i^{t+1}$$

(9)

The position update expression of the female mayfly is:

$$y_i^{t+1}=w{y}_i^t+v_i^{t+1}$$

(10)

After integrating adaptive weights, the position update operation will dynamically alter the weights based on the number of iterations, considerably improving the algorithm’s overall search ability.

3.2. Levy Flight Strategy

The Levy distribution is a non-Gaussian random process proposed by French mathematician Levy [29]. Levy flight is a random walking pattern that follows the Levy distribution. When walking in a multidimensional space, Levy flight has isotropic random directions. The mayfly’s position and speed during flight are likewise fully random, and the mayfly’s flight step length when foraging and attracting the opposite sex roughly obeys the Levy distribution, so it can be compared to the Levy flight model.

In the process of finding the best answer, the MOA method is prone to falling into the local optimal value. The Levy flight strategy is implemented into the male mayfly’s speed update operation to make the solution hunting process more active and varied. The male mayfly’s revised speed update formula is as follows:

$$v_{ij}^{t+1}=v_{ij}^t+a_1{e}^{-\beta r_p^2}(Levy\oplus pbes{t}_{ij}-x_{ij}^t)+a_2{e}^{-\beta r_g^2}(gbes{t}_j-x_{ij}^t)$$

(11)

where $Levy$ represents the Levy flight factor. The expression of the Levy distribution is:

$$Levy(s,\lambda )\approx s^{-\lambda }$$

(12)

where s is a random step size; $\lambda$ is a random number of $(1,3]$.

3.3. Golden Sine Factor

The Golden Sine Algorithm is introduced in 2017 as a new meta-heuristic algorithm that exploits the link between the sine function and the unit circle in mathematics for computational iterative optimization [30]. After introducing the golden section number, the solution space can be reduced during the location updating process, and then the area that may generate the ideal solution will be sought, which can considerably enhance search efficiency and achieve a better balance between search and development.

The core of the Golden Sine Algorithm is its position update formula. First, s individuals are randomly generated. The position of the ith ($i=1,2,3,\cdots ,s$) individual in the d dimension solution space in the tth iteration is $X_i^t=(X_{i1},X_{i2},\cdots ,X_{id})$, and $P_i^t=(P_{i1},P_{i2},\cdots ,P_{id})$ represents the optimal position of the tth generation individual. Then, each individual is updated using the following formula:

$$X_i^{t+1}=X_i^t|sin\left(r_1\right)|+r_{2}sin\left(r_1\right)|c_1·P_i^t-c_2·X_i^t|$$

(13)

$$c_1=a·\tau +b·(1-\tau )$$

(14)

$$c_2=a·(1-\tau )+b·\tau$$

(15)

$$\tau =\frac{\sqrt{5}-1}{2}$$

(16)

where $r_1$ and $r_2$ are random numbers, $r_1\in [0,2\pi ]$, $r_2\in [0,\pi ]$, $c_1$ and $c_2$ are coefficients calculated through the golden ratio, which can drive the search agent closer to the target value, and a and b are the initial values of the golden section. The golden sine is divided into standard sine intervals, where the period of the sine function is $2\pi$. In order to enable the population to traverse the searched space of each dimension in the entire period, here we take $a=\pi$, $b=-\pi$.

MOA incorporates the golden sine factor to boost search speed. The golden sine section coefficient is employed to generate offspring in the mating and reproduction of mayflies, and the better formula for offspring generation can be stated as:

$$offspring1=x_i^1|sin\left(r_1\right)|+r_{2}sin\left(r_1\right)|c_1·pbes{t}_x-c_2·x_i^t|$$

(17)

$$offspring2=y_i^1|sin\left(r_1\right)|+r_{2}sin\left(r_1\right)|c_1·pbes{t}_y-c_2·x_i^t|$$

(18)

where $pbes{t}_x$ and $pbes{t}_y$ represent the optimal position under the current number of iterations.

3.4. The Flow of IMOA

The algorithm flow chart of the IMOA is shown in Figure 1.

The pseudo code of IMOA is described in Algorithm  1.

Algorithm 1:improved mayfly optimization algorithm

1: Parameter initialization: Population number $N1,N2,Nc$, Iterations $t,Dim$;   2: Randomly generate initial position X;   3: $X^{*}$= the best search agent;   4: $t=1$;   5: whilet < Maximum number of iterations do   6:     Calculate the fitness of each search agent;   7:     Define the weights w(t) and the golden sine operator;   8:     $t=1$;   9:     for i to $N1$ do  10:         Update $r_g$,$r_p$ by Equation (3)  11:         if Fitness(male)>Fitness(gbest) then  12:            Update the speed of the male mayfly by Equation (12);  13:             else if Fitness(male) ≤ Fitness(gbest) then;  14: Update the speed of the male mayfly by Equation (4);  15:         end if  16:         Update the position by Equation (10);  17:         Update the fitness;  18:     end for  19:     for i to $N2$ do  20:         Update $r_m$ Equation (3)  21:         if Fitness(male)>Fitness(female) then  22:            Update the speed of the female mayfly by Equation (6a)  23:            else if Fitness(male) ≤ Fitness(female) then  24:            Update the speed of the female mayfly by Equation (6b);  25:         end if  26:         Update the position by Equation (11);  27:         Update the fitness;  28:     end for  29:     Ranking the fitness of males and females;  30:     $k=1$;  31:     for k to $Nc/2$ do  32:         Update the position of the offspring by Equations (18) and (19);  33:     end for  34:     Update the position of the global best;  35:     Calculate the fitness of each search agent;  36:     $t=t+1$;  37: end while  38: $Return{X}^{\ast }$;

3.5. Algorithm Complexity Analysis

Assuming that the computation complexity of IMOA is $T\left(n\right)$, the dimension of the search space is denoted as D, the number of male mayflies is $N_1$, the number of female mayflies is $N_2$, the number of offspring is $N_3$, and the maximum number of iterations is denoted as $T_{max}$. The complexity of the initialization process is $O\left(1\right)$, and the time complexity of the standard MOA is $T\left(n\right)=O(1+T_{max}(N_{1}D+N_{2}D+N_{3}D))$.

IMOA adds adaptive inertia weights, Levy flight strategies, and golden sine factors, so the complexity will increase correspondingly. The complexity of adaptive inertia weight is $O\left(1\right)$, the complexity of Levy flight is $O\left(N_1\right)$, and the complexity of golden sine factor is $O\left(1\right)$. Since the golden sine factor is nested in the mating loop, the time complexity added by IMOA is $O\left(T_{max}(1+N_1+N_3)\right)$, the complexity after filtering low-order terms is $T\left(n\right)=O\left(T_{max}(N_1(D+1)+N_{2}D+N_3(D+1))\right)$. Overall, IMOA has a slight increase in complexity, but not much.

4. Performance Analysis

Because numerous factors must be optimized, reliability analysis is required prior to array antenna pattern synthesis. According to Wolpert’s “no free lunch” (NFL) theorem [31], there is no algorithm in the world that can solve all issues in all sectors. As a result, we use four typical test functions to validate the IMOA’s efficiency. These four functions are well-known and have served as reference functions for optimization algorithms such as GA and PSO.

Table 1. Test functions and specific information.

Function Name	Expression	Search Space	Dim	Fmin
Sphere	$F_1={\sum }_{i=1}^d{x}_i^2$	[−10,10]	30	0
Rosenbrock	$F_2={\sum }_{i=1}^{d-1}[100{(x_{i+1}-x_i^2)}^2+(1-x_i){}^2]$	[−30,30]	30	0
Quartic	$F_3={\sum }_{i=1}^{d}i{x}_i^4+random[0,1)$	[−1.28,1.28]	30	0+random noise
Ackley	$F_4=20+e-20exp[-0.2\sqrt{\frac{1}{d}{\sum }_{i=1}^d{x}_i^2}]-exp\left[\frac{1}{d}{\sum }_{i=1}^{d}cos(2\pi x_i\right)]$	[−32,32]	30	0

displays the benchmark functions.

Using the above four standard test functions, the GA, PSO, DE, IWO, WOA, standard MOA, and IMOA are simulated and compared in 30 dimensions. The computed $f\left(x\right)$ is defined as the solution’s fitness value. In the aforementioned algorithm, the population size is set to 40, and the maximum number of iterations is 500.

Table 2. Parameter setups of different algorithms.

Algorithm	Values of the Parameters
GA [13]	$P_c=0.8,P_m=0.08$
PSO [32]	$C1=1.5,C2=2.0$
DE [15]	$F=0.5,CR=0.1$
IWO [33]	${\sigma }_{i}nitial=0.05,{\sigma }_{f}inal=0.01$
MOA [20]	$male=20,female=20$, $a_1=1,a_2=1.5,\beta =2,d=5$
WOA [26]	$b=1,r=[0,1],l=[-1,1],p=[0,1]$
IWOA	$male=20,female=20$, $a_1=1,a_2=1.5,\beta =2,d=5,a=\pi ,b=-\pi$

displays the parameters of various algorithms. Furthermore, to eliminate random bias, tests are independently repeated 30 times.

Table 3. Results of algorithms on basic benchmark functions.

Function ID	Statistics	GA	PSO	DE	IWO	MOA	WOA	IWOA
F1	Best	5.9459 × ${10}^{-02}$	9.0916$\times {10}^{-03}$	1.5094 × ${10}^{-06}$	2.5677 × ${10}^{-05}$	8.091 × ${10}^{-13}$	3.2727 × ${10}^{-94}$	0.0000 × ${10}^{+00}$
	Average	1.7310 × ${10}^{-01}$	1.6365 × ${10}^{-02}$	2.9594 × ${10}^{-06}$	3.4267 × ${10}^{-05}$	1.674 × ${10}^{-10}$	1.2064 × ${10}^{-81}$	0.0000 × ${10}^{+00}$
	Std.	2.5419 × ${10}^{-01}$	5.2516 × ${10}^{-02}$	1.1558 × ${10}^{-06}$	4.7039 × ${10}^{-06}$	3.7614 × ${10}^{-10}$	5.2016 × ${10}^{-81}$	0.0000 × ${10}^{+00}$
F2	Best	2.0635 × ${10}^{+02}$	2.6832 × ${10}^{+01}$	2.4624 × ${10}^{+01}$	2.5779 × ${10}^{+01}$	3.1467 × ${10}^{-01}$	2.6852 × ${10}^{+01}$	3.2867 × ${10}^{-03}$
	Average	4.009 × ${10}^{+02}$	6.3317 × ${10}^{+01}$	5.2010 × ${10}^{+01}$	2.6677 × ${10}^{+01}$	2.4533 × ${10}^{+01}$	2.7647 × ${10}^{+01}$	1.2652 × ${10}^{-02}$
	Std.	1.5590 × ${10}^{+02}$	3.0563 × ${10}^{+01}$	3.6656 × ${10}^{+01}$	0.6031 × ${10}^{+00}$	2.9960 × ${10}^{+01}$	0.4685 × ${10}^{+00}$	2.9483 × ${10}^{-03}$
F3	Best	2.4553 × ${10}^{-01}$	5.5778 × ${10}^{-01}$	6.8942 × ${10}^{-02}$	5.7910 × ${10}^{-03}$	7.9499 × ${10}^{-02}$	3.8572 × ${10}^{-03}$	4.5898 × ${10}^{-04}$
	Average	5.0547 × ${10}^{-01}$	1.6257 × ${10}^{+00}$	1.1957 × ${10}^{-01}$	2.6554 × ${10}^{-02}$	2.0486 × ${10}^{-02}$	3.0396 × ${10}^{-02}$	5.2412 × ${10}^{-03}$
	Std.	1.2738 × ${10}^{-01}$	3.8376 × ${10}^{-01}$	3.0263 × ${10}^{-02}$	8.8917 × ${10}^{-02}$	7.7150 × ${10}^{-03}$	3.9468 × ${10}^{-03}$	8.3869 × ${10}^{-04}$
F4	Best	3.2840 × ${10}^{-01}$	1.9277 × ${10}^{+00}$	2.9891 × ${10}^{-03}$	3.9968 × ${10}^{-14}$	1.2840 × ${10}^{-01}$	7.6815 × ${10}^{-16}$	8.8818 × ${10}^{-16}$
	Average	7.3548 × ${10}^{-01}$	2.7495 × ${10}^{+00}$	4.7469 × ${10}^{-03}$	5.7495 × ${10}^{-14}$	5.2548 × ${10}^{-01}$	4.7962 × ${10}^{-15}$	8.8818 × ${10}^{-16}$
	Std.	2.1209 × ${10}^{-01}$	5.7580 × ${10}^{-01}$	1.0955 × ${10}^{-03}$	8.5984 × ${10}^{-15}$	3.1209 × ${10}^{-01}$	2.158 × ${10}^{-15}$	0.0000 × ${10}^{+00}$

displays the numerical statistical findings of 30 independent experiments calculated using various techniques. The three statistical indicators are ideal value, average value, and variance, which are used to assess the algorithm’s optimization accuracy, average accuracy, and robustness, in that order. The best outcomes are indicated in bold for each function. The three IMOA indicators listed above clearly outperform the other algorithms in solving the four test functions, as seen in Table 3.

Correspondingly, it can be seen from the convergence curves of different algorithms shown in Figure 2a–d that the proposed IMOA has faster convergence speed and better solution results in solving the above four test functions. However, the performance of the IMOA in the sidelobe suppression problem for antenna synthesis still needs to be further evaluated.

5. Application of IMOA in the Array Antenna Pattern Synthesis

5.1. Signal Model of the Planar Phased Array Antenna

Consider the planar phased array composed of N elements shown in Figure 3, its far-field radiation pattern is:

$$E(\theta ,\varphi )=\sum _{n=1}^{N}g(\theta ,\varphi )w_n^{*}{e}^{jk(x_{n}sin\theta cos\varphi +y_{n}sin\theta sin\varphi )}$$

(19)

$$=g(\theta ,\varphi )\sum _{n=1}^N{\left|w\right|}_n{e}^{jk(x_{n}sin\theta cos\varphi +y_{n}sin\theta sin\varphi )+j{\phi }_n}$$

(20)

where $g(\theta ,\varphi )$ is the element pattern function, $k=2\pi /\lambda$, $\lambda$ is the wavelength, $(x_n,y_n)$ is the coordinate position of the nth array element, and $\left|w_n\right|·e^{j{\phi }_n}$ is the amplitude excitation factor and phase excitation factor of the corresponding array element.

This paper applies the IMOA to actual engineering projects. In large-scale spaceborne array antennas, the method to control the excitation amplitude of the array antenna units by designing the corresponding feed network is complicated, difficult, and expensive. However, phase weighting generally only needs to be controlled by a phase shifter, which is simple and convenient, without additional cost. However, it is difficult to achieve a particularly low sidelobe only through phase modulation. In order to achieve the low sidelobe level, the secondary amplitude distribution is first obtained by binarizing the Taylor distribution; then, the desired sidelobe suppression is achieved only by optimizing the phase of the excitation. The fitness function can be expressed as:

$$F_{MSLL}={\alpha }_1\left|G_0\right.-G\left.{}_{des}\right|+{\alpha }_2\left|MSL{L}_{max}\right.-MSLL\left.{}_{des}\right|$$

(21)

where $G_0$ is the calculated gain of the mainlobe pointing angle, $G_{des}$ is the designed gain of the mainlobe pointing angle, $MSL{L}_{max}$ is the calculated maximum sidelobe level among the entire pattern range, $MSL{L}_{des}$ is the designed maximum sidelobe level, ${\alpha }_1$ and ${\alpha }_2$ are two weight coefficients, and ${\alpha }_1+{\alpha }_2=1$ in the design.

5.2. Results Analysis

In order to endure high radiation power from the transmitter [34,35], the Tx antenna employs a type of the crossed dipole which is made of copper. Every two opposite arms form a linearly-polarized dipole. Circularly-polarized radiation is realized through adjusting the length of arms of the two dipoles. The antenna is fed by a balun through the open slit which is 1/4 wavelength. The Simulation Gain is 7.1 dB. The design diagram and radiation pattern of the antenna element are shown as Figure 4 and Figure 5

In the 528-element planar array antenna model, the secondary amplitude distribution is first obtained by binarizing the Taylor distribution. The secondary amplitude distribution ratio is 1:0.7, which can be seen in Figure 6. The frequency is set to 18.75 GHz, and 528 elements with different unit patterns are imported through the simulation software HFSS. Then, GA, PSO, DE, IWO, WOA, and standard MOA and IMOA are carried out to optimize the phase excitation to achieve sidelobe suppression. Figure 7 is the planar array antenna. The parameter design of each algorithm is shown in Table 2. Each algorithm is iterated 2000 times and repeated 20 times independently to ensure the reliability of the experiment. It can be seen from Figure 8 that the IMOA is significantly better than other algorithms in terms of convergence speed, and the final solution is also better than other algorithms.

Seven algorithms are selected to solve the problem of sidelobe suppression for phased array antennas, and the corresponding optimal solution of each algorithm is shown in Figure 9. As shown in Figure 9, the maximum sidelobe obtained by IMOA optimization is −25.73 dB, and the maximum sidelobe in the conventional pattern without adjusting the phase is −14.98 dB. The obvious optimization proves the superiority of the IMOA, which can also be verified by the comparison of 3D antenna pattern in Figure 10.

The optimal value, average value, and variance of 20 independent experiments are shown in

Table 4. Results of algorithms on basic benchmark functions.

Algorithms	Best MSLL (dB)	Worst MSLL (dB)	Average MSLL (dB)	Standard Deviation (dB)
Uniform Array	−14.98	−14.98	−14.98	0
GA	−18.87	−16.95	−17.68	0.77
PSO	−20.13	−17.86	−18.89	1.32
DE	−20.37	−18.16	−19.02	0.98
IWO	−21.64	−20.28	−20.96	0.56
MOA	−23.52	−21.19	−22.21	1.13
WOA	−23.08	−21.56	−22.13	0.63
IMOA	−25.73	−24.55	−25.26	0.42

. After IMOA optimization, the best MSLL reaches −25.73 dB, the worst MSLL is −24.55 dB, and the standard deviation of 20 experiments is 0.42. It can be seen that the IMOA performs better on optimization results and robustness through the comparison with other algorithms in Table 4.

In the simulation model mentioned above, the mutual coupling among the array elements are not taken into account. Therefore, it is necessary to conduct EM simulations to determine whether the above algorithm can effectively optimize the maximum sidelobe of the pattern in the presence of mutual coupling.

The EM simulations are conducted based on ANSYS Electromagnetics (HFSS). The phase excitation obtained through algorithm optimization in MATLAB is input into the 528-element planar array antenna model designed in HFSS software to verify the feasibility of the optimization algorithm in actual engineering. Figure 11 shows the comparison of the 2D beam optimization performance of each algorithm obtained through EM simulation. It can be seen that the MSLL optimized by the IMOA is −23.51 dB. Obviously, the MSLL suppression performance of the IMOA is the best compared with other algorithms. The sidelobe suppression performance of the HFSS simulation results deteriorate to a certain extent after adding coupling, but both simulations still demonstrate the effectiveness of the IMOA.

Furthermore, the 3D waveform diagrams before and after optimization of 528 elements are shown in Figure 12a,b respectively. The color of the largest sidelobe becomes lighter, indicating that the value is significantly smaller, which proves that the maximum SLLs are significantly optimized.

6. Conclusions

In this paper, an Improved Mayfly Optimization Algorithm (IMOA) is proposed for the low sidelobe design requirements of the spaceborne array antennas. On the basis of the standard MOA, the IMOA has three improvement strategies: adaptive weight, the Levy flight strategy, and the golden sine operator. The superiority of IMOA is proved by comparing with other algorithms through the classic test function simulations, and the IMOA is successfully applied to the phase-only pattern synthesis of a 528-element planar array antenna. On the maximum sidelobe level suppressing problem, IMOA has better performance than GA, PSO, DE, IWO, WOA, and the standard MOA in terms of convergence speed and the final optimization results. Finally, the practicability and effectiveness of the IMOA are verified through the electromagnetic field simulation with coupling added.