1. Introduction
In recent years, the spaceborne array antenna has attracted increasingly extensive attention due to its beamforming, beam steering and high-gain capabilities. For array antenna pattern synthesis, the main concern is to search for the appropriate excitation amplitude and phase to obtain the desired radiation pattern. Various approaches, for example, traditional mathematical techniques and optimization algorithms, have been put forward to the array antenna pattern synthesis problem [
1,
2,
3,
4]. Normally, Taylor weighting method and Chebyshev weighting method are commonly used to achieve sidelobe suppression. Nonetheless, the amplitude excitation range satisfying Taylor or Chebyshev distribution is too large to be controlled, leading to the difficulty in its application in practical engineering, whereas the global optimization algorithm can effectively limit the range of amplitude variation. Owing to the limitation of such a device design, in Wilkinson circuit, the signal combination efficiency will be affected when there is a large signal power difference. This study controls the maximum variation range of amplitude to 20 dB in this project. Undoubtedly, if the amplitude change is too small, the sidelobe suppression will be very poor. In practice, this paper has achieved the attenuation control by using attenuators. Low sidelobe level is one of the key technical indicators in the antenna pattern, which can effectively improve the communication quality by facilitating the signal-to-noise ratio, reducing the influence of the clutter signal outside the main beam and enhancing the anti-interference ability of the entire system [
5,
6,
7,
8].
Genetic algorithm (GA) and particle swarm optimization (PSO) are the first two popular and classical optimization algorithms, which have been successfully applied to antenna synthesis for their high efficiency and simplicity [
9,
10,
11,
12]. However, these two algorithms have the disadvantage of premature convergence in solving multi-parameter optimization problems. Therefore, an increasing number of improved algorithms based on classical GA and PSO have been employed to the antenna synthesis problem [
13,
14], such as differential evolution algorithm (DE) [
15], moth flame optimization algorithm (MFO) [
16], fruit-fly optimization algorithm (FOA) [
17], invasive weed optimization algorithm (IWO) [
18], grey wolf optimization algorithm (GWO) [
19], mayfly optimization algorithm (MOA) [
20], compressed sensing (CS) [
21], biogeography-based optimization (BBO) [
22], firefly algorithm (FA) [
23], ant colony optimization (ACO) [
24], and so on. Despite the fact that these aforementioned algorithms can achieve favorable results in array antenna synthesis, the increase in parameters will lead to slower calculation speed and lower algorithm efficiency. Furthermore, these solutions easily fall into a local optimal solution when the number of phased array elements increases. Therefore, further improvement is needed.
The whale optimization algorithm (WOA) [
25] is type of simple but effective swarm optimization algorithm. The research concept originates from the hunting behavior of humpback whales, which was proposed by Australian researchers Seyedali Mirjalili and Andrew Lewis in 2016. WOA has advantages of simple structure and fewer parameters, so it has a faster solving speed and higher precision in the process of solving multivariate functions. In 2018, Zhang applied WOA to the synthesis of wide-sided linear aperiodic arrays with minimum sidelobe level under uniform excitation [
26]. It can be clearly seen from these publications that WOA significantly superior to several standard algorithms, both in terms of convergence speed and convergence precision. Therefore, this study investigates the future WOA-based enhancements.
To further improve the global search ability and solution accuracy of the WOA, an improved whale optimization algorithm (IWOA) has been proposed in this paper and and applied to the synthesis of low-sidelobe planar array antennas. An inertia weight varying with iteration times is added to change the speed of whale position update, and the convergence speed in the later stage can be improved by adjusting the weight dynamically. In addition, in order to enable whales to develop more diverse search paths update their positions, the idea of variable spiral search is introduced, which can adjust the spiral shape of whales dynamically during search, increase the ability of whales to explore unknown areas, and then improve the global search ability of the algorithm. Through several typical test functions, compared with other algorithms, the superiority of IWOA algorithm in convergence speed and solution accuracy is verified.
Finally, the proposed IWOA algorithm is applied to the pattern synthesis for the space-borne planar phased array antenna. The model is established under the constraints of the practical engineering and the high-gain and low-sidelobe requirements are achieved through the amplitude excitation of each array element calculated by this algorithm. The experimental and simulation results prove the superiority and effectiveness of the proposed IWOA algorithm.
The portions of this paper are structured as follows.
Section 2 introduces the WOA;
Section 3 elaborates the IWOA details from three aspects;
Section 4 analyzes the performance comparison between IWOA and other algorithms;
Section 5 gives the results of related experiments; and
Section 6 concludes this paper overall.
3. Improved Whale Optimization Algorithm
Based on the standard WOA, the proposed IWOA improves the strategy optimization in three aspects: adaptive weight, spiral coefficient optimization, and optimal neighborhood perturbation.
3.1. Adaptive Weight
The concept of inertia weight first appeared in the particle swarm algorithm, which means the change of particle coordinates is related to inertia weight in the iteration of particle swarm optimization [
27].
This inspires the addition of an inertia weight
w that modifies the position update of IWOA according to the quantity of iterations. The influence of the optimal whale position on the present individual position adjustment needs to be reduced in the early stages of the algorithm search in order to increase global search capability. The influence of the ideal whale position will steadily improve as the number of iterations increases, encouraging other whales to fast converge to the ideal whale position, allowing for an acceleration of convergence speed. The adaptive inertia weight made up of iteration number
t is chosen as follows depending on the change in the whale optimization algorithm’s update count:
The position update formula of IWOA is:
After the adaptive weight is implemented, the position update will dynamically alter the weight based on the rise in the number of iterations, allowing the best whale position to steer different whales at different times. The whale group will concentrate on the direction of the optimal position with the increase in the number of iterations, and the larger weight will make the whale position move faster, thereby speeding up the algorithm convergence speed.
3.2. Helix Factor Optimization
When the whale’s spiral position is updated, the coefficient b controls the radian of the spiral update. The coefficient b is a fixed constant in the standard whale algorithm, which makes the whale move according to a fixed spiral radian when searching for prey in a spiral. A single method of movement makes it simple to converge on a local optimal solution, hence diminishing the algorithm’s capacity for global search.
As a result, parameter
b is also designed as a variable that changes with the number of iterations, allowing the spiral curve of the whale to be dynamically adjusted during the spiral search, increasing the whales’ ability to explore unknown areas and thus improving the algorithm’s global search ability. The new adaptive weight spiral position update formula is as follows:
The helix mathematical model is used to design the parameter b. The shape of the helix is dynamically modified based on the initial helix model by introducing the number of iterations. During iterations, the spiral’s shape changes from large to small when the specified parameter b is increased.
In the early stages of the algorithm, the whales hunt for targets in a greater spiral form to explore the global ideal solution as much as possible in order to improve the algorithm’s global optimal search ability. The whales hunt for targets in a narrow spiral shape later in the algorithm to improve the precision of the optimization method.
3.3. Optimal Neighborhood Perturbation
The whales will take the optimal position of the current position as the iteration target each time when updating its position. In each iteration process, the whale’s position will only be updated when a better solution appears, which will result in higher probability of not updating position in the later iterations of evolution, and then the search efficiency of the entire algorithm will reduce.
As a result, IWOA employs the optimal neighborhood perturbation technique and searches again near the optimal position to obtain a better global value, which not only improves the algorithm’s convergence speed but also prevents the algorithm from prematurely aging. A random disturbance is generated in the best possible location to boost its search for nearby space. The neighborhood disturbance formula is as follows:
where
limits the range of Gaussian random numbers between [
, 1];
is a uniformly generated random number between [0, 1],
is the generated new location.
A greedy strategy is adopted to determine whether to retain the generated neighborhood positions, and the corresponding formula is as follows:
where
is the adaptive value of the x position. The original position will be replaced if the generated position is better, otherwise, the optimal position remains the global optimum.
3.4. IOWA Algorithm Flow
The algorithm flow chart of the IOWA algorithm is shown in
Figure 3:
The pseudo code of IWOA is shown in Algorithm 2.
Algorithm 2 IWOA |
- 1:
Parameter initialization: Population number N, Iterations T, ; - 2:
Randomly generate initial position X; - 3:
= the best position of the whale; - 4:
- 5:
whilet < Maximum number of iterations do - 6:
Calculate the fitness of each search agent; - 7:
Update the whale’s location using Equations (14) and (15); - 8:
; - 9:
for i to N do - 10:
Update and p; - 11:
; - 12:
for j to do - 13:
if then - 14:
if then - 15:
Update the whale’s location using Equation ( 11); - 16:
else - 17:
Update the whale’s location using Equation ( 10); - 18:
end if - 19:
else if then - 20:
Update the whale’s location using Equation ( 12); - 21:
end if - 22:
end for - 23:
end for - 24:
; - 25:
end while - 26:
;
|
3.5. Algorithm Complexity Analysis
The time complexity of an algorithm is a function that qualitatively reflects the method’s running time. The number and structure of the arithmetic unit influence the time complexity of an optimization algorithm.
The number of search agents, iterations, and the location update method largely affect the temporal complexity of WOA. The following is the analysis and comparison of the time complexity of WOA and IWOA:
In the standard whale optimization algorithm, the time complexity of the WOA can be obtained as by setting the size of whales in the algorithm as N, the problem dimension as D, and the maximum number of iterations as T.
The IWOA introduces three improved strategies on the basis of the original algorithm, but the number of cycles of the algorithm does not increase after adding the adaptive weight and variable spiral update strategy, which can be known from the above algorithm modification process. Thus, the time complexity of the IWOA can be obtained as . Adding the optimal neighborhood perturbation adds a cycle of traversing the population to the periphery of the algorithm, thus, the amount of computation increases by .
The increased operation amount will not cause too much computational burden to the time complexity of the entire algorithm, therefore the overall time complexity of the IWOA is the same as that of the standard WOA.
Space complexity is used to judge the memory space requirement of an algorithm. The space complexity here is mainly determined by the size of the whale and the dimension of the problem to be solved. Therefore, the IWOA does not increase the space complexity.
4. Performance Analysis
Since numerous factors are required to be optimized, the reliability analysis ought to be performed prior to array antenna pattern synthesis. According to Wolpert’s “No Free Lunch” (NFL) theorem [
28], there is no algorithm that can solve all issues of all sectors worldwide. To confirm the efficacy of the IWOA method, four well-known test functions are selected, which are also utilized as benchmark functions for GA and PSO optimization techniques. The benchmark functions are listed in
Table 1. To validate the IWOA algorithm’s efficiency, four well-known test functions are chosen, which are also used as benchmark functions for optimization strategies such as GA and PSO.
Table 1 displays the benchmark functions.
The GA, PSO, DE, IWO, MOA, standard WOA algorithms, and IWOA algorithms were simulated and compared in 30 dimensions using the aforementioned four standard test functions. The computed
can be defined as the fitness value of the solution. In the procedure described, the population size is set to 40 and the maximum number of iterations is 500.
Table 2 illustrates the parameters of several algorithms. Additionally, tests were performed 30 times separately to prevent random deviation.The population size in the above algorithm is set to the same 40, and the maximum number of iterations is 500. The parameters of different algorithms are shown in
Table 2. Furthermore, tests are independently repeated 30 times to avoid random deviation.
Table 3 presents the results of thirty different tests computed by various methods. Ideal value, average value, and variance are the three statistical indicators, which are used to evaluate the optimization accuracy, average accuracy, and robustness of the algorithm in turn. The best results are indicated in bold for each function. The aforementioned three indicators of IWOA algorithm are obviously superior to other algorithms. Therefore, compared with other algorithms, IWOA has the optimal overall performance in solving the solution problem of the above four test functions, as shown in
Table 3. As a result, compared with other algorithms, IWOA has the most satisfying overall performance in solving these four test functions listed above.
Correspondingly,
Figure 4a–d show the convergence curves of different algorithms in the process of solving the above test functions. It can be seen that the IWOA has a faster convergence and better solution result.
However, the performance of the IWOA algorithm in sidelobe suppression optimization problem needs further evaluation.
6. Conclusions
Aiming at the design requirements of low sidelobe level of spaceborne phased array antenna, an improved IWOA algorithm based on a standard WOA algorithm is proposed. It has three improvement strategies, which are adaptive weight, spiral coefficient optimization, and optimal neighborhood perturbation. The superiority of IWOA is proved by comparing with other algorithms in the simulation experiment with classical test functions as the validation criteria. In addition, the IWOA is effectively used for antenna synthesis of a 64-element planar array antenna. IWOA outperforms GA, PSO, DE, IWO, MOA, and regular WOA in terms of convergence speed and optimization outcomes when applied to the sidelobe level suppression issue. Ultimately, the practicability and effectiveness of IWOA are verified through the electromagnetic field simulation with coupling added.