Application of chaotic colony predation algorithm in electromagnetics

The colony predation algorithm (CPA) is a new meta-heuristic algorithm that mimics the predation of animals to improve the radiation properties of antenna arrays. To overcome CPA’s problems of low accuracy and fast convergence, we propose an improved CPA called the chaotic colony predation algorithm (CCPA). The performance of CCPA was investigated in three parts. First, CCPA was tested with four benchmark functions. Then, CCPA was applied to equally spaced linear arrays to suppress the peak side-lobe level (PSLL) and place nulls in the desired directions. Finally, CCPA was used for a pattern synthesis of equally spaced linear arrays to reduce the PSLL under various constraints. Our results show that CCPA performed competitively with other well-known algorithms. Thus, CCPA is a promising option for solving electromagnetic optimization problems.


Introduction
The most critical function of an antenna is to transmit and receive electromagnetic waves, making an antenna the most fundamental component of modern communication systems.Antennas are applied to the communication network, which contains various wireless communication devices, including smartphones.In long-range satellite communications or radar detection, only high-gain and narrow-beamwidth antenna arrays can overcome the problem of the low gain of an individual antenna [1].The design of antenna arrays has been a challenge.Moreover, an antenna array plays a crucial role in determining the overall performance of a wireless system.Additionally, the pattern synthesis of an antenna array can significantly enhance its performance.The pattern synthesis method of antenna arrays aims to gain an appropriate current excitation and elements placement to yield the desired radiation pattern.Generally, inside the linear antenna array pattern synthesis, two different types of linear antenna arrays (LAAs) are a uniform array with equal element spacing and a nonuniform array.
In recent years, with the research on the pattern synthesis of linear antenna arrays, traditional methods such as Taylor [2] and Chebyshev [3] have been shown to be unsuitable for solving multidimensional nonlinear optimization problems.As a result, global optimization methods have been introduced to study these nonuniform LAAs.The peak side-lobe level (PSLL) obtained by these methods is lower than that obtained by traditional methods.For example, Tabu search (TS) was shown to be effective in designing LAA [4].In the context of optimizing linear arrays that are not uniformly spaced, a particle swarm optimization (PSO) algorithm was used to reduce the side-lobe level (SLL) [5][6][7].Taguchi's method (TM) was proposed to design LAA [8].Biogeography-based optimization (BBO) is another proposed method for obtaining an LAA with a low SLL and null placements [9,10].Ant colony optimization (ACO) was developed as a method to reduce SLLs in thinned array synthesis [11].To minimize the SLL of LAA, researchers employed the symbiotic organism search (SOS) [12].In addition, the differential evolution (DE) algorithm was used to reduce the side-lobe level of the planar array [13].Similarly, the artificial bee colony (ABC) algorithm reduced the level of LAA side lobes by optimizing the distance between antenna elements [14].To reduce the side-lobe level of a low-profile multisubarray antenna, researchers proposed a genetic algorithm (GA) [15].Moreover, with in-depth research on meta-heuristic algorithms, the cat swarm optimization (CSO), wind-driven optimization (WDO), grey wolf optimization (GWO), and ant lion optimization (ALO) were all applied to LAA pattern synthesis [16][17][18][19].A practical approach consisting of two steps (TSA) was proposed for the synthesis of LAA [20], and a new and enhanced version of the fruit-fly optimization (FFO) algorithm was developed for synthesizing antenna arrays [21].The flower pollination algorithm (FPA) [22], invasive weed optimization (IWO) [23], and enhanced firefly algorithm (EFA) [24] were also applied to synthesize LAA.Further, an improved genetic algorithm (IGA) was studied to synthesize linear aperiodic arrays [25].The application of global optimization algorithms has significantly impacted the development of electromagnetic technology.
In these algorithms, owing to the problem of the algorithm strategy or algorithm mechanism, the solution results and solution accuracy differ, and the number of algorithm parameters and solution time present a series of major problems that limit the current algorithm.However, an excellent algorithm can better balance the transformation of global optimization and local optimization of the global optimization algorithm, which can bring more accurate solution results and a faster solution speed.
The proposed colony predation algorithm (CPA) [29] is based on the mechanism of colony predation behavior, aiming to achieve a balancing strategy of global and local optimization.This paper also introduces a new algorithm called the chaotic colony predation algorithm (CCPA), which combines CPA with a chaotic algorithm to improve its performance.Section 2 discusses the concepts of CPA, and section 3 focuses on CCPA.Section 4 details the evaluation of the effectiveness of CCPA using four benchmark functions, and section 5 provides a performance analysis.Section 6 provides a summary of the research findings.

CPA
CPA is a recently developed meta-heuristic algorithm that draws inspiration from nature and has been applied to solve global optimization problems.The colony predation algorithm, proposed in [29], is based on the behavior of animals that live in groups and hunt together to increase the number of prey caught and ensure their safety.This behavior often occurs in wolves, hyenas, lions, piranhas, and other animals that live in groups because the colony predation behavior of herd animals can ensure the continuation of more of their kind.
When predators find prey, they often isolate the prey first, then surround it, and finally pursue it successfully.In the search for prey and hunting, when the energy consumption is greater than the value of the prey, the predator selectively abandons the current target and chooses another, and the individual calls for support from the nearest peer.This action enables the predator to catch more food and consume less.Accordingly, the mathematical model of CPA mainly aims to imitate these behaviors.The next section of this paper describes the mathematical model of CPA from the view of the predator population and individuals.

Dispersion and encirclement behavior
When the predator population hunts its prey, it employs both dispersal and encirclement strategies on the prey, with r 1 controlling the execution of these two strategies and r 1 being a random number of [0,1].The mathematical model of these two strategies is as follows: When r 1 is greater than 0.5, the encirclement strategy is employed; otherwise, the dispersal strategy is implemented.Here, ( ) X t 1 r + represents the position of a population, X r best is the position of the prey, S is the strength of the prey, and r 2 and l are random numbers of [0,1].ub is the upper bound, and lb is the lower bound.D represents the distance from the current individual to the prey; and l tan 4 ⎛ ⎝ ⎞ ⎠ p represents the curve that encircles the hunter.The formulas for S and D are as follows: ( ) where w = 9, N denotes the count of individuals, t signifies the current count of assessments, r 3 is a random number in the range [0,1], and ( ) X t r denotes the current population of hunters.

Supporting and searching behavior
In addition to dispersing and encircling food, predator populations also require the ability to locate food sources and provide support to other populations in their predatory activities.r 4 is the determining factor for supporting the nearest individual and locating food; these behaviors can be expressed as follows: where r 5 and r 6 are random numbers of [0,1].
In addition to population-level interactions, individuals engage in communication and collaboration to enhance predation probability.This behavior can be expressed as follows: The variable r 7 is constrained within the interval [0, 1], ( ) is the individual searching for food; X 1 1 and X 2 1 denote the two nearest positions to the prey in the j-th dimension, where jä1,2,K, dim; and ( ) X t 1 j i 1 + denotes the most recently updated position of the individual.
Figure 1 shows the mechanism for updating the position of the search agent in a 2D search space while accounting for the impact of the predator leader and other predators.To illustrate the predation behavior among predators, we depict the positions of the predators and their updated positions in the 2D search space in figure 1, which details the relationship between the predator and prey and how partners support the predator by asking for help.The final predators stay in a random position within a circle defined by them.The gray circle represents the last location.

The chaotic colony predation algorithm
The proposed chaotic colony predation algorithm (CCPA) is a fusion of two improvement points based on CPA: the initialization of logistic-tent chaotic mapping [30] and the logarithmic factor in equation (10).This section details the logarithmic factor and specific expressions of logistic-tent double mapping.

The logarithmic factor
In optimization problems, one balances global and local searches.In the early stage, the global search is emphasized.In a later stage, the weight of the local search is increased so that the algorithm converges quickly.This method balances the area and accuracy of the optimization process.In equation (2), the value of a increases exponentially and converges in the middle.This result is not conducive to finding the optimization value.Therefore, a is modified in equation (10) and named the logarithmic factor, and it is incorporated into CCPA: A comparison of a-values found in CPA with those found in CCPA is shown in figure 2. The value of a in CCPA decreases.Its larger value in the early stage is suitable for a global search, and its smaller value in the later stage is excellent for a local search.This method provides an outstanding dynamic balance between the two types of searches.

The logistic-tent chaotic mapping
In the field of antenna array synthesis, pseudo-random sequences are widely used.The mathematical definition of the logistic-tent chaotic algorithm depends on its initial values and parameters.It generates many random chaotic sequences, which provides a good model for finding high-quality pseudo-random sequences [30].The traditional pseudo-random sequences are replaced by logistic-tent chaotic mapping, and its mapping distribution chart is shown in figure 3. The map distribution histogram is shown in figure 4, and the formula for logistic-tent mapping is given in equation (11).
where X represents system variables, r is the control parameter, and and 4 show that the initial solution is uniform in the solution space when this mapping is used for initialization.

Test on benchmark functions
The performance of CCPA was used to test four standard benchmark functions [17,28].The optimal values obtained by CCPA and other popular algorithms in the four benchmark functions were also compared.Table 1 provides a thorough explanation of these functions, together with the 2D graphics illustrated in figure 5. Test functions F1 and F2 were characterized by unimodality, whereas the remaining test functions exhibited multimodality.
To study the whole optimization performance of CCPA in detail, we compared the optimal solutions of the three popular algorithms of GA, PSO, and GWO in the benchmark function.For a fair comparison, the population size and the maximum number of iterations were set to the same values for all four algorithms.In these tests, they were set to 100 and 1000, respectively.All four algorithms were executed 50 times independently to obtain the results.
Table 2 shows that CCPA outperformed PSO, GA, and GWO on test functions F1, F2, F3, and F4.Additionally, from a statistical view, CCPA showed excellent performance on the four tested functions, indicating that CCPA is an excellent optimization algorithm for optimizing nonlinear and multidimensional problems.

Pattern synthesis of linear antenna arrays
The 2 N isotropic array elements were placed symmetrically along the x-axis, as shown in figure 6.
The array factor (AF) of symmetric antenna arrays is given by equation (12): where k is the wave number.The parameters / k 2 , p l = I , n , n j and x n are the current excitation amplitude, phase, and position of the array element, respectively, and q is the azimuth angle.

Equally spaced linear array
To see how CCPA would perform in real-world problems, we applied it to classical LAA antenna array design problems.The model was an equally spaced LAA composed of 2 N isotropic elements.In the subsequent two cases, CCPA was used to optimize linear antenna arrays with equally spaced and excitation amplitudes.In LAAs  Table 1.Benchmark function and their properties.x fx 0, 0, ...,0 , x fx 0, 0, ...,0 , = =  The fitness function is given by equation (14): where ui q and li q are the regions of SLL that need to be optimized, i q D = q− .li q k q is the angle of the null, and AF (θ) is the array factor given by equation (13).
In the first example, a 20-element LAA was optimized using CCPA to suppress the PSLL and place nulls in specific directions.Its purpose is to suppress the maximum SLL in the region [ ]   0 , 76 q = and [ ] 104 , 180 , q =   along with nulls at 76 q =  and 104 .q =  For CCPA, the population size was 20, and the number of iterations for each run was 500.Since the array is symmetric, only 10 elements had to be optimized, and the search region was [0,1].The optimized excitation amplitudes are given in table 3. A comparison of the obtained SLLs for antennas optimized using different methods is shown in table 4. The optimized radiation pattern obtained via CCPA is illustrated in figure 7.For comparison purposes, radiation patterns generated by the backtracking search optimization algorithm (BSA) could be plotted [20], and FPA [18] and CPA were also plotted.Figure 8 shows the 3D radiation before and after CCPA optimization.
As shown in table 4, CCPA obtained better results than FPA, BSA, and CPA.The value of the maximum SLL obtained by CCPA was −31.57dB, which is 2.35 dB, 0.26 dB, and 1.61 dB lower than that of the BSA, FPA, and CPA arrays, respectively.The null level obtained by CCPA was much lower than that obtained by BSA, FPA, and CPA.Finally, the null level in the direction of 104°was −143.3 dB, which is 9.1 dB, 22.4 dB, and 17.3 dB lower than that of BSA, FPA, and CPA, respectively.
In the second example, a 20-element LAA was subjected to CCPA optimization for maximum SLL reduction and null placement.The optimization aimed to suppress the maximum SLL in the region where [ ] 0 , 82 q =   and [ ] 98 , 180 .q =   The double nulls were placed in the direction of 64°and 76°.The population was set to 30, and the maximum number of iterations was set to 1,000.Other parameters for optimization were the same as those in the previous example.
Table 5 provides the normalized excitation amplitudes acquired using CCPA.In table 6, the outcomes of the CCPA and other well-known algorithms are contrasted.The array patterns of BBO [22], GWO [27], CPA, and CCPA-optimized [29] arrays with null at 64°and 76°directions are shown in figure 9.Moreover, figure 10 shows the 3D radiation before and after CCPA optimization.Table 6 shows that CCPA obtained the best results.CCPA enabled the placement of the nulls in the desired directions as deep as −110.74dB.A comparative analysis was performed on the peak SLL, and the null depths obtained by CCPA, BBO [9], GWO [26], and CPA are shown in table 6.The peak SLL obtained using CCPA was −30.18 dB, which is 3.47 dB lower than that obtained using the BBO [9] array.The peak is 1.74 dB and 1.09 dB lower than that of the GWO [26] array and CPA array, respectively.

Unequally spaced linear array
In recent years, there have been many studies on unequally spaced linear antenna arrays [14].Compared with equidistant antenna arrays, nonuniform arrays can reduce the cost and complexity of the feed network.Compared with equally spaced linear antenna arrays, an unequally spaced array optimizes the distance between adjacent elements, and excitations are constant (i.e., I 1, n = 0 n j = ).Thus, equation ( 12) is modified to equation (15).Figure 11 illustrates a generally linear array consisting of (2 N+1) elements that are positioned along the x-axis.
If the antenna array element is (2N + 1), the array factor is modified by equation ( 16): A general linear antenna array with (2 N+1) elements placed on the x-axis is depicted in figure 13.
To avoid mutual coupling and grating lobe, we must ensure that the distance between elements is not too long or too short; therefore, antenna placement optimization must satisfy equation (17): where x i and x j represent the positions of any two different elements of the antenna.The fitness function for designing an antenna array is also the same function described in the literature, which was found by comprehensive learning particle swarm optimization (CLPSO) [28].It is as follows: is the decision vector that describes the coordinates of the array elements, S denotes the area that excludes the main beam, BW c is the calculated FNBW, BW d is the desired FNBW, I is a penalty factor (106), and t is the beamwidth tolerance.
In the first case, the number of array elements in this unequally spaced array was 10.The objective of the linear array synthesis was to suppress the level of the PSLL.The desired first zero beamwidth (FNBW) was set to 23°, the array was uniformly excited and arranged with half-wavelength spacing, and the beamwidth tolerance was set at ±5% [28].The population size and the maximum number of iterations were set to 40 and 1000, respectively.The array factor equation is given by equation (15).The position of elements obtained using CCPA optimization is shown in table 7. Table 8 presents a comparison of SLL obtained using the CCPA and other competing algorithms.Figure 12 illustrates the radiation patterns of the four arrays obtained using various algorithms.Figure 13 shows the 3D radiation before and after CCPA optimization.
Table 8 summarizes the optimized layouts of the arrays corresponding to the results presented.The results show that when we used CCPA, a 10-element array achieved a PSLL of −19.10 dB, which is approximately  0.03 dB lower than that obtained using GWO [26] and CLPSO [28] and about 0.05 dB and 0.1 dB lower than that of WDO [17] and CPA, respectively.In the second case, the design objective was to suppress the PSLL for symmetrical arrays of 17 elements.The minimum element spacing was set to 0.5 .
l The population size and the number of iterations were fixed to compare with other algorithms.Their values were 100 and 1000, respectively.The same problem was handled in [26].Table 9 presents the optimized element positions obtained using CCPA, while table 10 provides a comparison of the PSLLs obtained using CCPA and other widely used algorithms.The patterns obtained with the GWO [26], TSA [20], and IGA [25] are plotted in figure 14. Figure 15 shows the 3D radiation before and after CCPA optimization.It is clear from table 10 that CCPA achieved a peak SLL of −20.448 dB, which is 0.126 dB and 0.105 dB lower compared to IGA [25] and CPA, respectively, and 0.142 dB and 0.153 dB higher than those of TSA [20] and GWO [26], respectively.

Conclusions
In this study, a novel swarm intelligence-based optimization algorithm named CCPA was designed.Four benchmark functions were used to test the effectiveness of CCPA, and the results were compared with CPA and GWO.Several electromagnetic (EM) optimization problems were tested, including equally spaced linear antenna array and unequally spaced linear antenna array synthesis.Despite a slight widening of the main lobe in the second example, the CPA and CCPA consistently outperformed other comparison algorithms in equally spaced linear antenna arrays.The performance of CPA in unequally spaced linear antenna arrays was worse than that of other related algorithms.Nevertheless, the performance of the chaotic CPA, namely CCPA, was on part with that of popular algorithms, even surpassing their performance.The results indicate that the performance of CCPA is better than CPA in antenna array synthesis and that CCPA has the potential for future extensions in optimizing the excitation amplitude of the feed source in multi-beam arrays.This extension presented here is a   promising approach for optimizing time-modulated multi-beam arrays, broadening the application scope of CCPA in various electromagnetic fields.

where r 4
represents a random number within the range of [−2, 2], P r nearest denotes the position of the nearest predator within the support group, X r rand is a newly generated population position formed randomly, and D 1 denotes the displacement distance of a random population.The formula for X r rand and D 1 are as follows:

Figure 2 .
Figure 2. Comparison of a-values in CPA and CCPA.

6Figure 5 .
Figure 5. Illustration of the 2D versions of the test functions.

Figure 7 .
Figure 7. Radiation pattern of 20-element LAA optimized using CCPA compared with other algorithms.

Figure 8 .
Figure 8. 3D radiation patterns of 20-element LAA with null at 104°before and after CCPA optimization: (a) before and (b) after.

Figure 9 .
Figure 9. Radiation pattern of CCPA-optimized 20-element LAA with null at 64°and 76°compared to other optimization methods.

Figure 10 .
Figure 10.3D radiation patterns of 20-element LAA with null at 64°and 76°before and after CCPA optimization: (a) before and (b) after.

Figure 13 .
Figure 13.3D radiation patterns of 10-element LAA before and after CCPA optimization: (a) before and (b) after.

Figure 15 .
Figure 15.3D radiation patterns of 17-element LAA before and after CCPA optimization: (a) before and (b) after.

Table 2 .
Results comparison with 100 independent runs.

Table 3 .
Optimum amplitude excitations obtained using FPA for 20-element LAA with null at 104°.

Table 4 .
Comparison of results of different algorithms for 20-element LAA with null at 104°.

Table 6 .
Comparison of results of different algorithms for 20-element LAA with null at 64°and 76°.

Table 7 .
The positions of the 10-element liner array obtained using CCPA optimization.

Table 8 .
The PSLLs of the 10-element unequally spaced linear array optimized by three different optimization techniques.

Table 9 .
The positions of the 17-element array obtained using CCPA optimization.

Table 10 .
The PSLLs of the 17-element unequally spaced linear array optimized by five different optimization techniques.