Rigorous Particle Swarm Optimization Algorithm ...

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INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY, VOL.12, NO.6, NOVEMBER 2017

Rigorous Particle Swarm Optimization Algorithm Using Fuzzy Logic for Patterns Antennas Arrays Synthesis Brahimi Mohamed*, Kadri Boufeldja Smart Grids & Renewable Energies Laboratory University Tahri Mohamed of Bechar, 08000, Algeria. E-mail: [email protected]

Abstract- This paper proposes the implementation of a rigorous optimization algorithm, called fuzzy particle swarm optimization (FPSO), obtained by the hybridization between particle swarm algorithm and a fuzzy controller for the synthesis of linear and planar antenna arrays. This algorithm uses the fuzzy controller to adjust its parameters such as, the inertia weight and learning factors during the evolution process. The proposed algorithm has been applied to sixteen-elements antenna array and compared to the standard particle swarm optimization. Simulation results show that the optimal pattern of the antenna array is able to approach the desired pattern. A comparative study has been performed with published results found in the literature which illustrates the effectiveness of the proposed method for both linear and planar arrays. Index Terms- Fuzzy controller, linear array, particle swarm optimization, planar array, synthesis.

I. INTRODUCTION Antennas arrays find various applications in communications systems [1-2]. They are capable of increasing the antenna gain and radiation capability, suppressing the side lobe and forming nulls towards a specific direction [3]. Because of the complexity in synthesis problem, analytical methods are not generally used for designing a thinned array. Therefore, global optimization tools such as genetic algorithms (GA), particle swarm optimization (PSO) are used to solve these problems [3-4]. Particles swarm optimization (PSO) is an evolutionary algorithm based on the

swarm intelligence [5-6], which can be used to solve complex global optimization problems. Currently, the algorithm and its variations are applied to many practical problems [7-8]. It has many outstanding advantages, like fast convergence, simple computation and easy implementation. However, similar to other evolution algorithms, the basic version of PSO algorithm suffers from problems such as prematurity, limited searching scope and trend to converge to local extremes. To overcome these problems, many researchers have employed methods to adapt PSO parameters. For this latter, a fuzzy controller was designed to adjust the parameters of the PSO algorithm. We present in this paper the synthesis of the complex radiation pattern of a linear and planar antenna array with probe feed by only optimizing the amplitude excitation. The desired radiation pattern is specified by a narrow beam pattern with a beam width of 8 degrees and maximum side lobe level of (-20 dB). II. THE PARTICLE SWARM OPTIMIZATION ALGORITHM Particle swarm optimization is similar in some ways to genetic algorithms (GA) and other evolutionary algorithms, but requires less computational bookkeeping and generally fewer lines of code, including the fact that the basic algorithm is very easy to understand and implement [8-11]. It was originally developed by a social psychologist (James Kennedy) and an electrical engineer (Russell Eberhart) in 1995 [10].

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In PSO, the particles are “flown” through the problem space by following the current optimum particles. Each particle keeps track of its coordinates in the associated problem space with the best solution (fitness) that it has achieved so far. This implies that each particle has a memory, which allows it to remember the best position on the feasible search space that it has ever visited. This value is commonly called previous best (pbest). Another best value and tracked by the particle swarm optimizer is the best value obtained so far by any particle in the neighborhood of the particle. This location is commonly called global best (g-best). The basic concept behind the PSO technique consists of changing the velocity (or accelerating) of each particle toward its p-best and the g-best positions at each time step. This means that each particle tries to modify its current position and velocity according to the distance between its current position and p-best, and the distance between its current position and g-best. In its canonical form, particle swarm optimization is modeled as follows: Vi t 1  .Vi t  C1.R1.(Pbesti  X it )  C2 .R2 .(Gbest  X it )

(1)

X it 1  X it  V i t 1

(2)

Where ω is the inertia weight, C1 and C2 are the learning factors and R1, R2 are random numbers in the range [0,1]. The first part of (1) representing the previous velocity, which provides the necessary momentum for particles to roam across the search space. The second part, known as the “cognitive” component, represents the personal thinking of each particle. Cognitive component encourages the particles to move toward their own best positions found so far. The third part is known as the “social” component, which represents the collaborative effect of the particles, in finding the optimal global solution. The social component always pulls the particles toward the global best particle found so far [5]. Once the velocity has been determined, it is simple to move the particle to its next location.

The new coordinate is computed for each of the dimensions according to the equation (2). Start Specify the parameters for the PSO Generate initial population Evaluate the fitness of each particle Compare each particle’s fitness evaluation with the current particle’s to obtain Pbest Compare fitness evaluation with the population’s overall previous best to obtain Gbest Updating particle’s velocity and position by eq (1) and (2) No

Condition of termination Yes End

Fig.1.

Flowchart of PSO algorithm

III. RELATED WORKS

Several algorithms have been proposed in literatures that have addressed the issue of using fuzzy logic in particle swarm optimization. Shi and Eberhart after introducing a linearly decreasing inertia weight to the PSO over the course of PSO, designed fuzzy systems to nonlinearly changing the inertia weight [12]. This fuzzy system has some measurements of the PSO performance as the input and the new inertia weight as the output of the fuzzy systems. Kang, Wang, and Wu proposed a novel fuzzy adaptive optimization strategy for the PSO [13].

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Noroozi and Meybodi combined fuzzy logics with PSO. In this method C1, C2 are controlled by four fuzzy rules. The result of this algorithm is affected directly by the way of choosing the fuzzy rules [14]. Liu and Abraham proposed a hybrid metaheuristic fuzzy scheme, called as variable neighborhood fuzzy particle swarm algorithm, based on fuzzy particle swarm optimization and variable neighborhood search to solve the QAP [15]. IV. THE PROPOSED METHOD The learning factors C1, C2 represent the weighting of the stochastic acceleration terms that pull each particle. Low values allow particles to roam far from target regions before being tugged back, while high values result in abrupt movement toward target regions. Early experience with PSO led us to set the learning factors C1, C2 equal to 2.0, but it is not a usual rule [16]. In the search process, social behavior sometimes may be more important than cognitive behavior and vice versa. Regarding some statistical parameters, which are calculated in each iteration, it is possible to tune the cognitive parameter and social parameter adaptively to steer the whole swarm and each particle toward the proper trajectory.

Based on this kind of knowledge, a fuzzy system is developed to adjust the inertia weight, and learning factors with the best fitness (BF) and number of generations for unchanged best fitness (NG) as the input variables, and the inertia weight (ω) and learning factors (C1 and C2) as output variables. Where the range of BF and NG are [0, 1].The value for ω is bounded between 0.2 and 1.2 and the values of C1 and C2 are bounded between 1.0 and 2.0, these membership functions are shown in Fig.4, Fig.5 and Fig.6. The flowchart of the fuzzy particle swarm optimization algorithm (FPSO) is detailed in Fig.2. Start Specify the parameter for the PSO Generate initial population Evaluate the fitness of each particle Compare each particle’s fitness evaluation with the current particles’s to obtain Pbest

A fuzzy particle swarm optimization (FPSO) is intended to improve the performance of PSO; a fuzzy system will be employed to adjust the parameters of PSO, the inertia weight ω and learning factors C1 and C2 during the evolution process. From experience, it is known that:  When the best fitness is low at the end of the run in the optimization of a minimum function, low inertia weight and high learning factors are often preferred.  When the best fitness is stuck at one value for a long time, number of generations for unchanged best fitness is large. The system is often stuck at a local minimum, so the system should probably concentrate on exploiting rather than exploring. That is, the inertia weight should be increased and learning factors should be decreased.

Compare fitness evaluation with the population’s overall previous best obtain Gbest Updating particle’s velocity and position by eqs (1) and (2)

No

Fuzzy logic controller adjusting  , C1 and C2

Condition of termination

Yes End Fig.2.

Flowchart of fuzzy PSO algorithm

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INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY,



NG

Defuzzification

Fuzzification

BF Rule base

Inference

C2

Membership function

PM

PB

0

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0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Input variable ((Number of generation for unchanged best fitness)

1

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1

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0.3 0.4 0.5 0.6 0.7 Input variable (Best fitness)

PM

PB

0.8

1.2

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0

1.4 1.5 1.6 Output variable (C1))

PM

PR

1.7

PB

1.8

1.9

2

1.9

2

PR

0.5

1

Fig.5.

1.1

1.2

1.3

1.4 1.5 1.6 Output variable (C2))

1.7

1.8

Membership functions for learning factors

PS

PM

PB

PR

0.8

0.6

0.4

0.2

0 0.2

Fig.6.

0.3

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0.5

0.6 0.7 0.8 Output variable (W))

0.9

1

1.1

1.2

Membership functions for inertia weight

PR

The Mamdani-type fuzzy rule is used to formulate the conditional statements that comprise fuzzy logic. The fuzzy rules in Table 1, 2 and 3 are used to adjust the inertia weight (ω) and learning factors (C1 and C2), respectively. Each rule represents a mapping from the input space to the output space.

0.5

0

PB

B. Fuzzy rules

PR

0.5

0

Membership function

PS

1.1

PS

1

Fuzzification is used to associate each of the real inputs, through the membership functions, a degree of membership for each fuzzy subsets defined on the entire speech. The purpose of the fuzzification is to transform the input variables to variables "Linguistic" or fuzzy variables. Thus, in this example, they will be qualified for Little (P), Medium (M) and Large (L).

1

1

1

Structure of fuzzy controller

A. Fuzzification

PM

0.5

0

Membership function

Fig.3.

C1

PS

1

0

Membership function

The membership functions of inputs and outputs of FPSO model are shown in Fig. 4 and Fig. 5. The fuzzy system consists of four principal components [17]: fuzzification, fuzzy rules, fuzzy reasoning and defuzzification, which are described as below:

Membership function

VOL.12, NO.6, NOVEMBER 2017

Fig.4. Membership functions of best fitness and number of generation for unchanged best fitness

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INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY, VOL.12, NO.6, NOVEMBER 2017 Table 1: Fuzzy rules for inertia weight (ω) ω NUMBER_GEN PM PB PM PB PM PB PB PB PB PR

BF PS PM PB PR

PS PS PM PB PB

n



i 1

PR PB PR PR PR

Table 2: Fuzzy rules for learning factor (C1) BF PS PM PB PR

PS PR PB PM PM

C1 NUMBER_GEN PM PB PM PM PM

PB PB PM PS PS

PR PM PS PS PS

Table 3: Fuzzy rules for learning factor (C2) BF PS PM PB PR

PS PR PB PM PM

C2 NUMBER_GEN PM PB PB PM PM PS PM PS PS PS

PR PM PS PS PS

C. Fuzzy reasoning

y

(3)

n

  y dy i 1 Bi



Defuzzified value is directly acceptable values of ω, C1 and C2 parameters, where the input for the defuzzification process is a fuzzy set : µ Bi(y) (the aggregate output fuzzy set) and the output is a single number y. V. SYNTHESIS OF ARRAY PATTERNS A. Linear array The array pattern can approach the desired template by adjusting the exciting current amplitude of each element in a linear uniform array with N isotropic elements. From the antenna theories, the far-field pattern of a linear uniform array is: F ( ,  ) 

The fuzzy control strategy is used to map from the given inputs to the outputs [18]. Mamdani’s fuzzy inference method is used in this study [19]. The AND operator is typically used to combine the membership values for each fired rule to generate the membership values for the fuzzy sets of output variables in the consequent part of the rule. Since there may be several rules fired in the rule sets, for some fuzzy sets of the output variables there may be different membership values obtained from different fired rules.

y. Bi  y dy

y

f ( ,  ) N  a i exp j k 0 X i sin  cos    i  (4) Fmax i 1

Where f(θ,φ) is the radiation pattern, N is the number of elements, k 0: wave number k0 = 2π/λ, θ: angular direction, ai, ψi: amplitude and phase of the complex excitation power. In this paper, we have implemented the both algorithms, FPSO and SPSO for the synthesis of uniformly spaced linear array constituted with 16 rectangular microstrip antennas; the array is shown in Fig.7.

These output fuzzy sets are then aggregated into a single output fuzzy set by OR operator. That is to take the maximum value as the membership value of that fuzzy set.

y

M

 Radiating antenna

D. Defuzzification

Y2

To obtain a deterministic control action, a defuzzification strategy is required. The method of centroid (center-of-sums) is used as shown below:

x

 -1

Fig.7.

1

2

Linear antennas array

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N

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0 FPSO SPSO Fd

-10 -20 -30 -40 Amplitude

In this case, we have introduced an example of simulation, by optimizing only excitation weights for a desired radiation pattern specified by a symmetrical narrow beam pattern with a beam width of 8 degrees and maximum side lobe level of (-20 dB). In our simulation, we have used the inertia weight equal to 0.7 and learning factors C1 and C2 equal to 1.48 for SPSO [20].

-50 -60 -70 -80 -90 -100 -100

-20 0 20 theta [degree]

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0.1

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Fig.9. Fitness evolution of FPSO and SPSO algorithms

(5) FPSO 1

Where S is the space spanned by the angle θ excluding the main lobe and ρ represents the unknown parameter vector, such as element positions and phases. This objective function minimizes all the side lobe level and maximizes the power in the main lobe located at θ=θ0. From Fig. 9, the approaching speed of the global optimal of FPSO is much quickly than that of SPSO, and the fitness values of the best individuals of FPSO are almost lower than that of SPSO in every population.

Amplitude

AF  0 

-40

0.12

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8 10 Sources SPSO

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1

Amplitude



f   Max S

-60

0.14

We have chosen a suitable fitness function that can guide the FPSO optimization toward a solution that meets the desired radiation pattern. The fitness function to be minimized is selected from the work of Chuan Lin [21] which is described by the equation below:

AF  

-80

Fig.8. Radiation pattern of a linear array with 16 elements optimized by both FPSO and SPSO algorithms

Fitness

For FPSO, the fuzzy controller is used to adjust the inertia weight and learning coefficients of PSO algorithm, with a population size equal to 30 individuals. Fig. 8 illustrates an example of linear antenna array synthesis by the optimization of amplitude excitation coefficients using the both of FPSO and SPSO. The two algorithms are used for the determination of elements amplitude excitations, which are shown in Fig.10. It is clearly seen that the radiation pattern obtained by FPSO meet better the desired pattern than the obtained by the SPSO. The maximum side lobe level obtained by FPSO optimization (-48.41 dB) is much better than in the case of SPSO (-33.29 dB).

0.5

0

Fig.10. Optimized sources amplitude excitations

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1.55 FPSO SPSO

1.54

Learning Factor LC1

1.53 1.52 1.51 1.5 1.49 1.48 1.47

0

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300 Iterations

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600

Fig.11. Adjusting C1 during PSOs run

1.58 FPSO SPSO

B. Planar array For a linear array, the synthesis is reduced to check the feed law on an axis, of number of elements fixed in advance, while in a planar array, the research synthesis are consisted of the complex weighting of the sources power supply in a plan [22]. This generalization of the planar array given in Fig.14 is considered by replacing the direction θ by the pair of directions (θ, φ). Let us consider a planar antenna array constituted of MxN equally spaced rectangular antennas arranged in a regular rectangular array in the x-y plane, with an inter-element spacing of d=dx=dy=λ/2 and whose outputs are added together to provide a single output. Mathematically, the normalized array far-field pattern is given by:

1.56

Learning Factor LC2

F(,)  1.54

f  , Fmax

M

I

mn

exp j  m 1 k0 sin cos dx  jmn 

m1

N

.exp  j  n 1 k0 sin cos dy  jmn 

1.52

n1

1.5

1.48 0

100

200

300 Iterations

400

500

600

Where f (θ,φ) : Represents the radiation pattern of an element. Imn : Amplitude coefficient at element (m, n). ψmn: Phase coefficient at element (m, n). k0: Wave number.

Fig.12. Adjusting C2 during PSOs run

y



d

0.8 FPSO SPSO

M

d

0.75 Inertia Weight W

(6)

0.7

2 1 0.65

0

100

200

300 Iterations

400

Fig.13. Adjusting ω during PSOs run

500

600

o

1

2

Fig.14. Planar antennas array

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x

460


We have adopted a desired radiation pattern specified by a narrow beam with a side lobe level of (-20 dB). Fig. 15 shows the synthesis result planar array constituted by 4x4 half wavelength spaced rectangular microstrip antennas with 0.906 cm width and 1.186 cm long working at the frequency of 10 GHz. In our simulation, we have used the inertia weight equal to 0.7 and learning factors C1 and C2 equal to 1.48 for SPSO [23]. For FPSO, it sets about adapting inertia weight and learning coefficients using a fuzzy controller depending on the best individual fitness, the population size equal to 30 individuals. In Fig. 15, we have presented the result of planar array optimization by amplitude only excitation coefficients using both FPSO and SPSO.

From Fig. 16, it is clear that a convergence characteristic of FPSO is better than the SPSO. The programming has been written in MATLAB language using MATLAB R2010a on Intel(R) Core (TM) i3-2348M, 2.30 GHz with 4 GB RAM. The optimized amplitudes excitations coefficients obtained by both optimizing algorithms are represented in Fig. 17. 0.14 FPSO SPSO

0.12

0.1

Fitness

The FPSO algorithm is able to model and to optimize the antennas arrays, by acting on radioelectric parameters of the feed law (amplitude) of the radiating sources. In that, the synthesis of uniformly spaced planar array of 4x4 rectangular patch antennas is presented.

0.08

0.06

0.04

0.02

0

0

100

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300 Iterations

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Fig.16. Fitness evolution of FPSO and SPSO algorithms FPSO 1

FPSO SPSO Fd

-10 -20

Amplitude

0

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-30 0

0

2

4

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8 10 Sources SPSO

12

14

16

0

2

4

6

8 10 Sources

12

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16

-50 1 -60 Amplitude

Amplitude

-40

-70 -80

0.5

-90 0 -100 -100

-80

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-40


40

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100

Fig.15. Radiation pattern of a planar array with 16 elements optimized by both FPSO and SPSO

It is noticed that the radiation pattern is contained within the limits imposed by the template and the maximum side lobe level is lower than (-20 dB) in such way that the FPSO is better than SPSO and reaches them respectively (-43.23 dB) and (33.44 dB).

Fig.17. Optimized sources amplitude excitations

Fig. 18, Fig. 19 and Fig. 20 show the adjusting of the inertia weight and learning coefficients during the evolution process.

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VI. COMPARATIVE STUDY

1.56 FPSO SPSO

Learning Factor LC1

1.54

1.52

1.5

1.48

1.46

1.44 0

100

200

300 Iterations

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600

Fig.18. Adjusting C1 during PSOs run 1.55 FPSO SPSO

1.54

Learning Factor LC2

1.53 1.52 1.51 1.5 1.49 1.48 1.47

0

100

200

300 Iterations

400

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600

Fig.19. Adjusting C2 during PSOs run 0.8 FPSO SPSO

Simulation results of this comparative study are given to show the difference between the proposed method and the published methods by the hybridization between the PSO algorithm and fuzzy logic. The synthesis of antennas array by using the improved algorithm (FPSO) was proven more powerful that using the standard algorithms (PSO and GA), in term of the accuracy, computing time, maximum side lobe level reduction and main lobe steering. Moreover, it can be noticed that FPSO is robuster than PSO and GA, since it has presented to avoid entrapment in local optima and improve the convergence speed and the precision in the array synthesis.

0.75 Inertia Weight W

In this section, two examples of simulation are performed to illustrate the effectiveness of the proposed method. In the first case, the proposed method FPSO is compared with GA algorithm [24] and PSO algorithm [25] to synthesize 16 elements with half wavelength spacing. Fig. 21 shows the radiation pattern obtained by the proposed FPSO compared to GA and PSO. The maximum side lobe level obtained by FPSO is (39.03 dB), which is better than the value obtained by PSO (-30.07 dB) and GA (-21.78 dB). In the second example, 36 elements (6×6) are synthesized by FPSO compared also with GA [26] and PSO [27]. From results given by Fig. 22, it is clear that the proposed algorithm pattern achieve a maximum side lobe level of (-40.61 dB), as for the both algorithm GA and PSO set a maximum side lobe level of (-26.64 dB) and (21.78 dB) respectively.

0.7

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Fig.20. Adjusting ω during PSOs run

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from the premature convergence. In other words, when the number of iterations increases the quality of the solution by the PSO cannot improve and it converges to the local optima which may not be the global optima.

0 FPSO GA in [24]

-10

PSO in [25]

-20 -30

Amplitude

-40 -50

To overcome the premature convergence of standard PSO algorithms (SPSO), we proposed to dynamically adjust the value of the parameters during the evolution process by the insertion of a fuzzy controller.

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Fig.21. Radiation pattern of a linear array with 16 elements optimized by FPSO, GA [24] and PSO [25] algorithms

The aim of this optimization technique is thus to seek for the optimal combination of these different parameters so that the array complies with the requirements of the user and according to precise specifications. REFERENCES

0 FPSO GA in [26] PSO in [27]

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[1]

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[2]

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Fig.22. Radiation pattern of a planar array with 16 elements optimized by FPSO, GA [26] and PSO [27] algorithms

[3]

[4]

VII. CONCLUSION In this paper, we studied the synthesis problem of patch antenna arrays. Generally, the synthesis is expressed as a minimization problem of a fitness function evaluating the error between a radiation pattern and a template. Various optimization techniques such as PSO for the aim to obtain the global minimum and to avoid remaining to trap in a local minimum like in the case of the deterministic methods. Although PSO is one of the good techniques to find a good solution much faster than the other algorithms, but it suffers

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