A novel invasive weed optimization (IWO) variant called chaotic adaptive invasive weed optimization (CAIWO) is proposed and applied for the optimization of nonuniform circular antenna arrays. A chaotic search method has been combined into the modified IWO with adaptive dispersion, where the seeds produced by a weed are dispersed in the search space with standard deviation specified by the fitness value of the weed. To evaluate the performance of CAIWO, several representative benchmark functions are minimized using various optimization algorithms. Numerical results demonstrate that the proposed approach improves the performance of the algorithm significantly, in terms of both the convergence speed and exploration ability. Moreover, the scheme of CAIWO is employed to find out an optimal set of weights and antenna element separation to obtain a radiation pattern with maximum sidelobe level (SLL) reduction with different numbers of antenna element under two cases with different purposes. The design results obtained by CAIWO have comfortably outperformed the published results obtained by other stateoftheart metaheuristics in a statistically meaningful way.
In several applications, such as mobile communication and spatial detection, techniques require antennas that have high directive radiation pattern, which cannot be achieved by a single element antenna. Antenna arrays are formed to circumvent such problems by combining many individual antenna elements in particular electrical and geometrical configurations. The primary design objective of antenna array geometry is to determine the locations of array elements that jointly produce a radiation pattern to resemble the desired pattern as nearly as possible. Poor design may result in a polluted electromagnetic environment. This will also result in wastage of power, which is a vital aspect in wireless devices that run on batteries. The classical derivativebased optimizations of designing antenna arrays are not effective as they are prone to getting local optima and strongly sensitive to initialization. Due to these inherent shortcomings of the classical technique, many modern metaheuristics approaches were tried to achieve optimized sidelobe level (SLL) and null control from the designed arrays [
Circular arrays have become popular in recent years over array over other array geometries because they have the capability to perform the scan in all directions without a considerable change in the beam pattern and provide 360° azimuth coverage [
Amongst all evolutionary algorithms (EAs) described in various articles, invasive weed optimization has emerged as one of the most powerful tools for solving the real world optimization problems [
The rest of the paper is organized an follows: Section
The original IWO algorithm was initially proposed by Mehrabian and Lucas in [
Standard deviation over the course of the run.
Then, the position of the new seed can be given as follows:
One important advantage of the IWO is that it allows all of possible candidates to participate in the reproduction process. From step (II) in Section
But there are also some shortcomings of IWO, the most prominent one is concerning the way the seeds produced by a plant are dispersed in the search spaces. As mentioned in Section
In order to overcome the drawbacks of IWO, a chaotic search and adaptive dispersion mechanism are integrated into the IWO algorithm. This variant IWO is named CAIWO.
Chaotic search methods have a greater ability to escape from the local minima. Therefore, the CAIWO algorithm has a less chance of premature convergence compared to original IWO. There are many chaotic maps such as logistic map, sinusoidal map, and tent map. We chose the sinusoidal map to improve the performance of IWO based on the discussion in the literature [
The sinusoidal map or the sinusoid sequence is defined by
The bifurcation diagram for sinusoidal map.
The adaptive dispersion mechanism is that the SD of the current generation distribute linearly among the weeds as weed with the highest fitness achieves the lowest SD and the lowest fitness achieves highest SD, which can be represented by (
The goal of the optimization algorithm is to minimize
Initially, pioneering solutions are initialized in the search space range of
Transform variable
Apply the sinusoidal sequence to transform
Transform
Evaluate each weed, sort, and rank them according to their fitness in the colony.
Sum the number of current generation and calculate the SD of each weed with respect to theirs ranking in the colony using (
Distribute the newly generated seeds using sinusoidal map in the neighborhood of the parent weed. If the chaotically distributed seed has a better fitness than the previous seed, keep the better one. Otherwise, the chaotic search is continued. The algorithm is guaranteed to converge much faster by taking advantage of the local search superiorities of chaotic search.
Rank the seeds again and exclude those with lower fitness to reach the maximum number of seeds
Continue from step 3 until maximum number of iterations is reached or a criterion is satisfied.
To verify its effectiveness, CAIWO has been applied to classical benchmark functions. All simulations are conducted in a Windows 7 Professional OS using 12core processors with Intel Xeon (
Sphere function:
Rastrigin function:
Ackley function:
Griewank function:
Test functions.
Function name  Search range  Dimension 

Sphere 

30 
Rastrigin 

30 
Ackley 

30 
Griewank 

30 
The performance of CAIWO is compared with MIWO and classical IWO. The simulation results such as fitness of best run, fitness of worst run, and the standard deviation of fitness are shown in Table
Simulation results on benchmark functions.
Benchmark function  Algorithm  Fitness of best run  Fitness of worst run  Average value  Standard deviation 

Sphere function  IWO  2.6 
7.11 
3.97 
5 
MIWO  7.93 
2.44 
1.40 
2.00  
CAIWO  1.65 
1.93 
7.14 
1.62  


Rastrigin function  IWO  0.044136  0.994962  0.797145  0.154781 
MIWO  0.095132  1.241246  0.872624  0.139673  
CAIWO  1.78 
1.34 
8.12 
1.87 



Ackley function  IWO  0.083654  0.099234  0.087016  0.001684 
MIWO  0.085449  0.256521  0.854496  0.256521  
CAIWO  1.88 
9.51 
6.5 
3.12 



Griewank function  IWO  0.049237  0.150065  1.02 
1.01 
MIWO  0.024603  0.113173  8.93 
2.22  
CAIWO  1.18 
9.74 
6.57 
4.74 
Convergence characteristic of IWO, MIWO, and CAIWO over four benchmarks. (a) Sphere. (b) Rastrigin. (c) Ackley. (d) Griewank.
The
Geometry of a nonuniform circular antenna array with
The array factor in the
When the peak of the array is in
Then, the array factor can be simplifies as [
The first and most important parameter in antenna pattern synthesis is the normalized sidelobe level that is desired to be as low as possible. In this section, the CAIWO algorithm will be applied to determine the electrical and geometrical structure of circular antenna array for obtaining the radiation pattern with minimum SLL in two different cases. Three circular array antennas with
The main goal of synthesis of antenna array in case one is to generate the radiation pattern with minimum sidelobe level for a specific first null beam width (FNBW). We have incorporated the maximum sidelobe level in addition to the average side lobe in the fitness function to ensure maximum directivity of the antenna arrays.
The following objective functions represent these above requirements in a mathematical form [
To illustrate the superiority of the CAIWO algorithm, three instantiations of the circular array antenna design problem are solved by using the CAIWO algorithm with five other stateoftheart algorithms, namely MIWO, classical IWO, DE, PSO, and real coded GA. This DE variant is called DE/rand/1/bin and is the most widely used one in DE literature.
The three instantiations are arrays with 8, 10, and 12 elements. The FNBW is assumed to be a constant, corresponding to a uniform circular array with a uniform
Problem description.
Problem number  Numbers of array elements  FNBW 

1  8  70.27 
2  10  55.85 
3  12  46.26 
The control parameters for CAIWO were set through a series of parameter tuning experiments. The parameters for IWO, MIWO, PSO, DE, and GA were set following the guidelines provided in [
Figure
Design variables obtained with CAIWO algorithm.
Number of elements  FNBW 

Normalized 

8  70.27  0.5600, 1.0000  0.4423, 0.3059 
1.0000, 0.5734  0.6405, 0.7325  
0.9509, 0.6363  0.2203, 0.1938  
0.5503, 0.5000  0.3925, 0.8590  


10  55.85  0.5001, 1.0000  0.5823, 0.3462 
0.9253, 1.0000  0.41608, 0.7275  
0.5332, 0.5313  0.7032, 0.0737  
0.9433, 0.8756  0.3758, 0.3450  
0.5567, 0.5078  0.4751, 0.9033  


12  46.26  0.6448, 1.0000  0.4958, 0.4722 
1.0000, 1.0000  0.0010, 0.4733  
0.6294, 0.5000  0.6587, 0.8100  
0.5000, 0.9564  0.4700, 0.4122  
0.5532, 0.7578  0.2060, 0.4956  
1.0000, 0.5967  0.5455, 0.9976 
Design figures of merit obtained in the best (out of 50) runs the six algorithms on three design instances.
Number of elements  Algorithms  Mean SLL (dB)  Max SLL (dB)  Average null depth (dB)  Directivity (dB) 

8  CAIWO  −13.158  −10.01  −46.88  9.67 
MIWO  −10.75  −2.17  −46.25  9.14  
IWO  −9.488  −3.61  −34.6  8.27  
DE  −10.34  −1.35  −32.95  8.87  
PSO  −10.23  −1.17  −40.6  8.78  
GA  −9.16  −1.71  −42.2  8.08  


10  CAIWO  −16.95  −11.42  −50.16  12.48 
MIWO  −13.78  −4.65  −46.35  11.75  
IWO  −11.35  −5.52  −32.8  9.93  
DE  −12.46  −4.07  −34.4  10.79  
PSO  −12.15  −4.07  −31.1  10.55  
GA  −10.65  −2.18  −30  9.40  


12  CAIWO  −18.76  −15.43  −41.015  12.85 
MIWO  −14.23  −5.16  −43.05  12.25  
IWO  −10.99  −4.34  −41.7  9.86  
DE  −11.66  −4.17  −31.4  10.39  
PSO  −11.61  −4.75  −43.55  10.37  
GA  −10.12  −3.68  −32.8  9.21 
Normalized radiation patterns for circular arrays of different number of elements obtained using six different optimization algorithms. (a) For number of elements
Convergence characteristic of six algorithms over three instances of the circular array design problem. (a) For number of elements
From Figure
From Tables
The goal of this case is to obtain the radiation pattern with minimum SLL and narrower beam width. This is done by manipulating the excitation currents and positions of elements. This case is considered similar to that reported in literatures [
In order to reduce the mutual coupling effects between elements, an additional term is added in the objective function (
In this case, the values of the element amplitudes are allowed to vary between
Along with the CAIWO results, the optimized array geometry and current excitation weights obtained using PSO [
Comparisons of results obtained by CAIWO with other algorithms.
Number of elements  Algorithms  Max SLL (dB)  HPBW (°)  ADR  Directivity (dB)  Circumferences 

8  CAIWO  −13.0225  17.19  1.4974  9.6365  9.1210 
IWO  −8.1667  14.9  4.6579  8.0011  8.3999  
BBO  −12.24  18.6  5.9595  9.43  9.0710  
SA  −12.00  26.60  3.2383  10.7201  5.8750  
PSO  −10.7996  32  2.7910  7.8269  4.4931  
GA  −9.811  32  3.9417  7.6245  4.4054  


10  CAIWO  −14.96  16.04  1.9514  10.2658  9.2333 
IWO  −10.8961  14.9  4.8786  9.5974  9.2637  
BBO  −13.95  16.60  2.6185  10.2032  9.2318  
SA  −13  18.4  3.2179  10.8722  8.0214  
PSO  −12.307  24.34  1.9755  8.6048  5.9029  
GA  −9.811  32  2.8140  9.8037  6.0886  


12  CAIWO  −16.50  13.75  2.3018  10.7494  10.5351 
IWO  −11.0042  11.46  7.6718  9.9301  12.0781  
BBO  −14.372  14.8  2.2676  10.8448  10.6453  
SA  −13.91  19.60  3.2639  10.7069  7.9523  
PSO  −13.670  21.2  2.5265  9.3667  7.1419  
GA  −11.83  20.8  4.1439  10.2209  7.7724 
Comparison of results obtained by CAIWO with other algorithms for
PSO [ 

0.7765  0.3928  0.6069  0.8446  1  0.7015  0.9321  0.3583 

0.359  0.5756  0.2494  0.7638  0.6025  0.8311  0.7809  0.3308  


GA [ 

0.3289  0.2537  0.7849  1  0.9171  0.5183  0.6176  0.4612 

0.1739  0.3144  0.662  0.7425  0.6297  0.8929  0.4633  0.5267  


SA [ 

0.3047  0.484  0.7751  0.9867  0.3371  0.4422  0.4067  0.6807 

0.9997  0.7743  0.9042  0.5652  0.8056  0.7818  0.5848  0.4594  


BBO [ 

1  0.6736  0.1678  1  0.9088  0.6553  0.7571  1 

0.6341  1  1.8892  0.8456  0.5693  1.1639  1.3329  1.6367  


IWO 

0.6188  0.3016  0.3394  0.337  0.2093  0.9749  0.3675  0.7029 

1.6076  0.3132  1.1011  1.243  0.2924  1.9749  0.7915  1.0762  


CAIWO 

0.9406  0.6999  0.6677  0.9998  0.9294  0.6882  0.8101  0.9710 

0.5747  1.0263  1.7891  0.9507  0.5899  1.1968  1.3859  1.6077 
Comparison of results obtained by CAIWO with other algorithms for
PSO [ 

1  0.7529  0.7519  1  0.5062  1  0.7501  0.7524  1  0.5067 

0.317  0.9654  0.3859  0.9654  0.3185  0.3164  0.9657  0.3862  0.965  0.3174  


GA [ 

0.9545  0.4283  0.3392  0.9074  0.8086  0.4533  0.5634  0.6015  0.7045  0.5948 

0.3641  0.4512  0.275  1.6373  0.6902  0.9415  0.4657  0.2898  0.6456  0.3282  


SA [ 

0.692  0.5679  0.5937  0.6703  0.9693  0.6014  0.3575  0.302  0.5908  0.9718 

0.6221  0.988  0.7777  0.9934  0.6217  0.9514  0.7626  0.598  0.7655  0.941  


BBO [ 

1  1  1  0.3819  0.897  1  0.7679  0.8899  0.7246  1 

0.5301  1.0603  1.3264  1  0.4307  0.4408  1.5276  1.3255  1  0.5904  


IWO 

0.2851  0.5493  0.7386  0.1722  0.8401  0.2874  0.3323  0.7330  0.4616  0.4811 

0.0100  0.5291  0.9957  1.2975  1.7320  0.5507  0.4642  0.7686  1.2974  1.6184  


CAIWO 

0.9941  0.9842  0.9518  0.5123  0.9233  0.9858  0.6593  0.9196  0.7336  0.9997 

0.5910  1.0471  1.2862  1.0304  0.4570  0.4422  1.4190  1.3552  1.0090  0.5962 
Comparison of results obtained by CAIWO with other algorithms for
PSO [ 

0.9554  0.6441  0.7109  0.7769  1  1  0.3958  0.7162  0.6746  0.7695  0.9398  0.6145 

0.2569  0.8509  0.6607  0.7057  0.854  0.3734  0.1609  0.8321  0.6464  0.7079  0.833  0.26  


GA [ 

0.2064  0.5416  0.2246  0.6486  0.7212  0.7913  0.5277  0.3495  0.5125  0.4475  0.5233  0.8553 

0.4936  0.4184  1.4474  0.7577  0.4204  0.5784  0.452  0.8872  0.7514  0.4202  0.4223  0.7234  


SA [ 

0.6231  0.399  0.3418  0.6054  0.9444  0.738  0.6741  0.3001  0.4311  0.5435  0.4195  0.9795 

0.8315  0.791  0.6699  0.8087  0.7347  0.5331  0.4777  0.896  0.4874  0.8657  0.3461  0.5105  


BBO [ 

1  0.6501  0.6224  0.502  0.554  1  0.6683  0.7234  0.441  0.5123  0.4793  1 

0.6704  1  1.3046  0.8081  1  0.431  0.6183  1.1574  1.3465  0.6551  1  0.6539  


IWO 

0.8558  0.9912  0.4590  0.3337  0.7796  0.5693  0.1545  0.7573  0.3417  0.1292  0.7849  0.3379 

0.5683  1.6654  0.2651  1.4680  1.1137  0.5021  0.3406  0.8223  0.7608  0.1292  1.6655  1.7793  


CAIWO 

0.8996  0.7367  0.5923  0.4529  0.5664  0.9989  0.6111  0.7397  0.4340  0.5125  0.5364  0.9023 

0.5700  1.0866  1.2745  0.7590  1.0124  0.4897  0.5611  1.1737  1.3395  0.6553  1.0571  0.5562 
Normalized radiation patterns for circular arrays of different number of elements obtained using PSO GA SA BBO IWO and CAIWO. (a) For number of elements
In the 8 elements antenna array, the maximum SLL achieved by CAIWO is −13.0225 dB, the HPBW is 17.19°, and the ADR is 1.4974. Evidently, CAIWO provides better SLL, HPBW, and ADR than other techniques. The SLL obtained by CAIWO is lower by 0.7825, 1.0225, 3.2229, 3.2115 and 4.8558 dB than by the PSO, the GA, the SA, the BBO, and the IWO optimized arrays, respectively. The obtained HPBW is also narrower by 1.41, 9.42, 14.81, and 14.81° than the PSO, GA, SA, and BBO algorithms, respectively, except for the IWO algorithms. The ADR achieved by CAIWO is lower by 3.1605, 4.4621, 1.7409, 1.2936, and 2.4443 than by the IWO, BBO, SA, PSO, and GA algorithms. The best radiation pattern of the 8 elements array generated by CAIWO is plotted in Figure
The maximum SLL of 10 elements array achieved by CAIWO is −14.96 dB, the HPBW is 17.19°, and the ADR is 1.9514. Evidently, CAIWO outperforms the other techniques. The SLL obtained by CAIWO is lower by 1.01 1.96, 2.653, 5.149, and 4.0639 dB than by the PSO, the GA, the SA, the BBO, and the IWO optimized arrays, respectively. Moreover, the HPBW obtained by CAIWO is also narrower by 0.56, 2.36, 8.3, and 15.96° than the PSO, GA, SA, and BBO algorithms, respectively, except for the IWO algorithms. The ADR achieved by CAIWO is lower by 2.9272, 0.6671, 1.2665, 0.0241, and 0.8626 than by the IWO, BBO, SA, PSO, and GA algorithms. The best radiation pattern of the 10 elements array generated by CAIWO is plotted in Figure
In the 12 elements antenna array, the maximum SLL achieved by CAIWO is −16.5 dB, the HPBW is 13.75°, and the ADR is 2.3018. Once again, the CAIWO algorithm yields results that are superior to the other algorithm. The maximum SLL is better than that achieved by the other algorithms. The reduction in SLL is significant and it is lower by 2.128, 2.59, 2.83, 4.67, and 5.4958 dB than by the PSO, the GA, the SA, the BBO, and the IWO optimized arrays, respectively. The obtained HPBW is also better than that attained by the other algorithms. It is narrower by 1.05, 5.85, 7.45, and 7.05° than by the PSO, GA, SA, and BBO algorithms, respectively, except for the IWO algorithms. The ADR achieved by CAIWO is lower by 5.3700, 0.9621, 0.2247, and 1.8421 than by the IWO, SA, PSO, and GA algorithms, except for BBO algorithm. The best radiation pattern of the 12 elements array generated by CAIWO PSO, GA, SA, BBO, and IWO algorithms is plotted in Figure
This paper proposed a variant invasive weed optimization called CAIWO algorithm. The proposed algorithm takes advantages of chaotic search and modified IWO with adaptive dispersion, where the seeds produced by a weed are dispersed in the search space with standard deviation specified by the fitness value of the weed. The statistical results obtained from the four benchmark functions demonstrate the superiority of the proposed CAIWO algorithm to MIWO and classical IWO. In addition, numerical examples of circular antenna array synthesis problems have been presented. We formulated two design problems with different purposes. The first case is an optimization task on the basis of a cost fitness that takes care of the average sidelobe levels and the null control. The second case takes care of the maximum SLL reduction with the constraint on the beam width. The fitness functions of both cases are minimized under a constraint in order to avoid the mutual coupling between the array elements. The simulation results over different element number show that the CAIWO algorithm could comfortably outperform the above mentioned algorithms in circular antenna array synthesis.
The authors declare that there is no conflict of interests regarding the publication of this paper.