A Differential Beta Quantum-behaved Particle Swarm Optimization
for Circular Antenna Array Design
Leandro dos Santos Coelho
1,5
, Emerson Hochsteiner de Vasconcelos Segundo
2
,
Fabio Alessandro Guerra
3
and Viviana Cocco Mariani
4,6
1
Pontifical Catholic University of Parana, Industrial and Systems Engineering Graduate Program (PPGEPS), Curitiba, Brazil
2
Pontifical Catholic University of Parana, Mechanical Engineering Graduate Program (PPGEM), Curitiba, PR, Brazil
3
Electricity Department, DPEL/DVSE/LACTEC, Institute of Technology for Development, Curitiba, PR, Brazil
4
Pontifical Catholic University of Parana, Mechanical Engineering Graduate Program (PPGEM), Curitiba, Brazil
5
Federal University of Parana (UFPR), Department of Electrical Engineering (DEE/PPGEE), Curitiba, PR, Brazil
6
Federal University of Parana (UFPR), Department of Electrical Engineering (DEE), Curitiba, PR, Brazil
Keywords: Metaheuristics, Particle Swarm Optimization, Quantum Mechanics, Circular Antenna Array Design.
Abstract: The classical particle swarm optimization (PSO) algorithm is inspired on biological behaviors such as the
social behavior of bird flocking and fish schooling. In this context, many significant improvements related
the updating formulas and new operators have been proposed to improve the performance of the PSO
algorithm in the literature. On the other hand, recently, as an alternative to the classical PSO, a quantum-
behaved particle swarm optimization (QPSO) algorithm was proposed. The contribution of this paper is
linked with a modified QPSO based on beta probability distribution and mutation differential operator. The
effectiveness of the proposed modified QPSO algorithm is demonstrated by solving three kinds of
optimization problems including two benchmark functions and a circular antenna design problem.
1 INTRODUCTION
Particle swarm optimization (PSO) is a population-
based algorithm of the swarm intelligence field
proposed in Kennedy and Eberhart (1995) and
Eberhart and Kennedy (1995). Its basic idea was
based on simulation of simplified animal social
behaviors. Over the years, PSO has gained
significant popularity due to its simple structure and
high performance. Furthermore, PSO has been
shown to be efficient in a plethora of applications
(Aote et al., 2013; Rini et al., 2011). However, many
studies and several variants of the PSO algorithm
(Sedighizadeh and Masehian, 2009; Eslami et al.,
2012) have been done to improve the performance
of PSO in continuous optimization.
Recently, novel optimization methods have been
motivated from the concepts of quantum mechanics
and computation (Han and Kim, 2002). One of the
recent developments in PSO proposed by Sun et al.
(2004a, 2004b) called quantum-behaved particle
swarm optimization (QPSO). It is based on the
perspective of quantum mechanics view rather than
the Newtonian rules assumed in previous versions of
PSO. QPSO is characterized by good search ability
and fast convergence. Although QPSO is an efficient
algorithm for solving continuous optimization
problems, it is still necessary to pay enough attention
to the inherent problem of possible premature.
To enhance the searching ability of PSO and
accelerate its convergence, several studies (Coelho
and Mariani, 2008; Fang et al., 2010; Sun et al.,
2012; Mariani et al., 2012; Kamberaj, 2014) propose
modifications in the QPSO.
Based on the mentioned considerations, we
proposed in this paper a modified QPSO (MQPSO)
based on beta probability distribution and mutation
operator inspired by differential evolution paradigm.
The differential evolution (DE) algorithm (Storn and
Price, 1997; Das and Suganthan, 2011) is an
evolutionary algorithm that uses a rather greedy and
less stochastic approach to problem solving than do
some evolutionary algorithms. The advantages of
DE are simple structure, efficiency and robustness.
To judge the performance of the proposed
algorithm, a set of two benchmark functions and a
circular antenna design problem are solved. The
results of simulations and convergence performance
are compared with the classical PSO.
192
dos Santos Coelho L., Hochsteiner de Vasconcelos Segundo E., Alessandro Guerra F. and Cocco Mariani V..
A Differential Beta Quantum-behaved Particle Swarm Optimization for Circular Antenna Array Design.
DOI: 10.5220/0005070201920197
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2014), pages 192-197
ISBN: 978-989-758-052-9
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

The remainder of this paper is organized as follows:
The fundamentals of the PSO, QPSO and MQPSO
are provided in Sections 2 and 3, respectively. The
description of the circular antenna array design
problem is given in Section 4. Experiments on
numerical optimization used to illustrate the
efficiency of the proposed MQPSO are given in
Section 5. Finally, a conclusion and future research
are conducted in Section 6.
2 CLASSICAL PSO ALGORITHM
The classical PSO algorithm maintains a swarm of
particles, where each particle represents a potential
solution to the objective problem. The particles are
initially placed at random positions in the search-
space, moving (flying) in randomly defined
directions in the n-dimensional search space. Their
velocities are changed based on the results of the
populational (gbest, global best) or personal (pbest,
personal best) locations search, and they move
toward the function optimum.
The procedure for implementing the global
version of PSO is given by the following steps:
Step 1: Initialization of swarm: Initialize a
population of particles with random positions and
velocities in the n-dimensional problem space using
uniform probability distribution function.
Step 2: Evaluation of particle’s fitness: Evaluate
each particle’s fitness value.
Step 3: Comparison to pbest: Compare each
particle’s fitness with the particle’s pbest. If the
current value is better than pbest, then set the pbest
value equal to the current value and the pbest
location equal to the current location in n-
dimensional space.
Step 4: Comparison to gbest: Compare the
fitness with the population’s overall previous best. If
the current value is better than gbest, then reset gbest
to the current particle’s array index and value.
Step 5: Updating of each particle’s velocity and
position: Change the velocity, v
i
, and position of the
particle, x
i
, according to equations (1) and (2):
)]()([
2
)]()([
1
)()1(
t
i
xt
g
pUdc
t
i
xt
i
pud ct
i
vwt
i
v


(1)
)1)()1( 

 (t
i
vtt
i
xt
i
x
(2)
where i=1,2,…,N indicates the number of particles
of population; t=1,2,…,t
max
indicates the generations
(iterations);

T
,...,
2
,
1 in
v
i
v
i
v
i
v 
stands for the
velocity of the i-th particle,

T
,...,
2
,
1 in
x
i
x
i
x
i
x 
stands for the position of the i-th particle of
population, and


T
,...,
2
,
1 in
p
i
p
i
p
i
p 
represents the
best previous position of the i-th particle. Positive
constants c
1
and c
2
are referred to as the cognitive
and social parameter, respectively. Index g
represents the index of the best particle among all
the particles in the swarm. The stochastic variables
ud and Ud are random numbers generated uniformly
distributed in [0,1]. Equation (2) represents the
position update, according to its previous position
and its velocity, considering
1t .
Step 6: Repeating the evolutionary cycle: Return
to Step 2 until a stop criterion is met. In this paper
the maximum number of generations is adopted.
In this paper, this is done by bounding a
particle’s velocity to the 20% of the full range of the
search space, so the particle can at most move from
one search space boundary to the other in one step.
PSO has some advantages over other similar
optimization techniques such as genetic algorithm. It
is easier to implement and needs fewer parameters to
adjust. On the other hand, the original PSO is easily
fall into local optima in many optimization
problems.
Recent studies (Rini et al., 2011; Aote et al.,
2013) have also attempted various ways to analyze
and improve PSO. Proper selection of these w, c
1
and c
2
parameters can improve the convergence rate.
However, the design of an effective method to select
PSO’s parameters using the relationship between w,
c
1
and c
2
parameters can be a complex task.
3 QPSO ALGORITHM
The core idea of classical PSO is the exchange of
information among the velocity, global best, local
best, and current particles. In the QPSO, the velocity
equation in the PSO algorithm is neglected.
Experimental results performed on some well-
known benchmark functions show that the QPSO
method has better performance than the PSO method
(see Sun et al. (2004a, 2004b, 2011)).
The probability of the particle’s appearing in
position x
i
from probability density function |  (x,t)|
2
,
the form of which depends on the potential field the
particle lies. In this context, each particle in a
quantum state formulated by wavefunction
 (x, t).
Using the Monte Carlo method, the position of
the particles can be obtained at iteration t+1 as (Sun
et al., 2004a, 2004b):
ADifferentialBetaQuantum-behavedParticleSwarmOptimizationforCircularAntennaArrayDesign
193

If 0.5 k then
)/1ln()(
,
)()()1(
,
ut
ji
xt
j
Mbestt
i
pt
ji
x  
(3)
Else
)/1ln()(
,
)()()1(
,
ut
ji
xt
j
Mbestt
i
pt
ji
x  
(4)
where x
i,j
(t+1) is the position for the j-th dimension
of i-th particle in t-th generation (iteration); Mbest
j
(t)
is the global point called Mainstream Thought or
Mean Best (Mbest) for the j-th dimension;

is a
design parameter called contraction-expansion
coefficient in range [0,1]; u and k are numbers
generated according to a uniform probability
distribution in range [0,1]; and p
i
(t) is a local point.
Trajectory analysis in Clerc and Kennedy (2002)
demonstrated that the convergence of the PSO
algorithm may be achieved if each particle
converges to its local attractor.
The Mainstream Thought or Mean Best (Mbest)
is defined as the mean of the pbest positions of all
particles and it given by



N
j
t
jg
p
N
(t)
j
Mbest
1
)(
,
1
,
(5)
where g represents the index of the best particle
among all the particles’ swarm in j-th dimension. In
this case, it is adopted
21
,2,1
)(
cc
ig
pc
ik
pc
t
i
p



,
(6)
where p
k,i
(pbest) represents the best previous i-th
position of the k-th particle and p
g,i
(gbest)
represents the i-th position of the best particle of the
population. In the same form that the classical PSO,
constants c
1
and c
2
are the cognitive and social
components, respectively. The procedure for
implementing the QPSO is given by the following
steps:
Step 1: Initialization of swarm positions:
Initialize a population of particles with random
positions in the n dimensional problem space using a
uniform probability distribution function.
Step 2: Evaluation of particle’s fitness: Evaluate
the fitness value of each particle.
Step 3: Comparison of each particle’s fitness
with its pbest (personal best): Compare each
particle’s fitness with the particle’s pbest. If the
current value is better than pbest, then set a novel
pbest value equals to the current value and the pbest
location equals to the current location in n-
dimensional space.
Step 4:
Comparison of each particle’s fitness
with its gbest (global best): Compare the fitness with
the population’s overall previous best. If the current
value is better than gbest, then reset gbest to the
current particle’s array index and value.
Step 5: Updating of global point: Calculate the
Mbest using equation (5).
Step 6: Updating of particles’ position: Change
the position of the particles using equations (3) or
(4), and (6).
Step 7: Repeating the evolutionary cycle: Loop
to Step 2 until a stopping criterion is met.
3.1 MQPSO Algorithm
QPSO may be trapped in local minima, because it
easily loses the diversity of swarm during the search.
To overcome the disadvantage of QPSO, a modified
QPSO (MPQSO) combining strategy of QPSO and
DE is introduced in this study.
The main point of the proposed MPQSO is to
improve the global performance of the QPSO by
using DE-inspired mutation operator. In the
MQPSO, exploration behavior was enhanced at the
early stage of searching so that more search space
can be explored by particles. While at the later stage
the MQPSO emphasized the exploitation to find the
accurate optimum using a DE-inspired mutation
operator. Furthermore, the MQPSO uses a linearly
decreasing parameter to choice the global or local
search during the generational cycle.
The Steps (1)-(5) of the QPSO are the same to
the MQPSO. However, the Step (6) in MQPSO is
given by following pseudocode:
21
,2,1
)(
cc
ig
pc
ik
pc
t
i
p



If r
3
> 0.8 - 0.6( t / t
max
)
Modified approach:
)
3
10,
2
10(27.0 rrbetarnd 




Uses Equation (3) or (4) to update
)1(
,

t
ji
x
Else (uses the differential mutation)
F = 0.6 + 0.3( t / t
max
)


)(
,
2
)(
,
1
)(
,
)1(
,
t
js
xt
js
xFt
ji
xt
ji
x 



End
where r
1
, r
2
and r
3
are uniformly distributed random
numbers bound within the range [0,1];
s
1
,s
2
{1,2,...,N}, and betarnd is a number
generated by a beta probability distribution, where
2
10 r

and
3
10 r
generate the shape parameters of
the beta distribution. In this case, the Matlab's script
called betarnd.m was employed.
The beta distribution is very flexible for
ECTA2014-InternationalConferenceonEvolutionaryComputationTheoryandApplications
194

modeling data that are measured in a continuous
scale on the open interval (0,1)
since its density has
quite different shapes depending on the values of the
two parameters that index the distribution.
4 CIRCULAR ANTENNA ARRAY
DESIGN PROBLEM
Consider a circular array of N antenna elements
spaced on a circle of radius r in the x-y plane. This is
shown in Fig. 1 and the antenna elements are said to
constitute a circular antenna array. The array factor
for the circular array is written as follows (Das and
Suganthan, 2010):
]
0
cos
cos[
1
exp)(
i
i
ang
jar
i
ang
jar
Ne
i
i
IAF





















(6)
where Ne is the number of antennas,
12




is the
angular position of the ith element on the x-y plane,
ar = Nd where a is the wave-number, d is the
angular spacing between elements and r is the radius
of the circle defined by the antenna array,
0

is the
direction of maximum radiation,

is the angle of
incidence of the plane wave, I
i
represents the
amplitude excitation of the i-th element of the array
and β
i
is the phase excitation of the i-th element.
Figure 1: Geometry of circular antenna array (source: Das
and Suganthan, 2010).
The purpose of the adopted optimization task is
modify the current and phase excitations of the
antenna elements and try to suppress side-lobes,
minimize beamwidth and achieve null control at
desired directions. The objective function f is taken
as:




 




num
k
kdes
sll
IARIDIR
IARIARf
1
000
0max0
,,,,,1
,,,/,,,










(7)
where the first component attempts to suppress the
sidelobes.
sll

is the angle at which maximum
sidelobe level is attained. The second component
attempts to maximize directivity (DIR) of the array
pattern. Nowadays directivity has become a very
useful figure of merit for comparing array patterns.
The third component strives to drive the maxima of
the array pattern close to the desired maxima
.
des

The fourth component penalizes the objective
function if sufficient null control is not achieved.
num is the number of null control directions and
k

specifies the kth null control direction.
We adopt the number of antenna elements in
circular array as Ne=12 for a uniform separation
distance of d = 0.5
  , where  is the wavelength.
The desired maxima
des

is set to 180
o
and null =
[50,120] in radians (no null control). In this case
study, 6 excitation amplitudes (I
i
) and 6 phase
perturbations (β
i
) are optimized. It is considered the
following search space:
12.0 
i
I
and
6,...,1 ,180180  i
o
i
o

. The source code in
Matlab of the circular array design benchmark
problem is provided by IEEE-CEC (2011).
5 OPTIMIZATION RESULTS
In this section, two benchmark functions and the
circular antenna array design are carried out to test
the validity of the proposed MQPSO, and the results
are compared with those of PSO and QPSO.
In the tests, the value of function is defined as
the fitness function of algorithm. Due to the
stochastic nature of the proposed approach, these
four systems were repeatedly solved 25 times by the
PSO and QPSO approaches.
The settings adopted in the tested PSO
approaches for the benchmarks functions is the
swarm size (population size) equal to 25 particles,
25 runs, and the stopping criterion is 10,000
generations. In the antenna case, 70 particles, 25
runs and the stopping criterion is 800 generations
was adopted.
In terms of PSO setting, c
1
= c
2
= 2.05 and the
inertia factor linear decreasing of 0.9 to 0.4 during
the iterations is adopted. QPSO and MQPSO use a
linearly decreasing contraction-expansion coefficient
ADifferentialBetaQuantum-behavedParticleSwarmOptimizationforCircularAntennaArrayDesign
195

(

) which starts at 1 and ends at 0.5.
5.1 Benchmark Functions
The Griewank function, first introduced in
(Griewank, 1981), has been employed as a test
function for global optimization algorithms in many
papers. The function is defined as follows:















n
i
i
i
x
n
i
i
xxf
1
1cos
1
2
4000
1
)(
1
(8)
within search space given by [-600,600]
n
. The global
minimum value is 0 and the global minimum is
located in the origin, but the function also has a very
large number of local minima, which regularly
distributed, exponentially increasing with n. It is
similar to the Rastrigin function, but the number of
local optima is larger in this case. Rosenbrock
function (Rosenbrock, 1960) is non-convex, non-
separable and quadratic function defined by












1
1
2
)1(
2
)
1
2
(100)(
2
n
i
i
x
i
x
i
xxf
(9)
with search space given by [-30,30]
n
and the global
minimum value is 0.
Simulation results presented in Tables 1 and 2
(best results in boldface) showed that the MQPSO
outperform the adopted PSO and QPSO on the basis
of mean and standard deviation of the best objective
function value of the total runs for the two
benchmark functions with n = 10.
Table 1: Results of f
1
(Griewank function).
Index PSO QPSO MQPSO
Maximum
(Worst)
4.48×10
-2
2.20×10
-1
8.35×10
-2
Mean 2.74×10
-2
9.36×10
-2
5.07×10
-2
Minimum
(Best)
1.11×10
-2
3.23×10
-2
2.46×10
-2
Standard
Deviation
7.82×10
-2
4.68×10
-2
1.75×10
-2
Table 2: Results of f
2
(Rosenbrock function).
Index PSO QPSO MQPSO
Maximum
(Worst)
19.7794 12.6433 16.2680
Mean 10.3934 6.3008
6.0639
Minimum
(Best)
5.22×10
0
4.28×10
-1
5.90×10
-2
Standard
Deviation
3.5496 3.1631 3.7946
5.2 Circular Antenna Array Design
A comparison with PSO, QPSO and MQPSO of
results presented in IEEE-CEC (2011) shows that
the MQPSO approach provides quite encouraging
results. As it is clear from Table 3 (best results in
boldface), the MQPSO is able to find the global
minimum and mean f values that outperform other
10 algorithms mentioned in IEEE-CEC (2011). The
best result (minimum) using MQPSO presented f=-
21.7586 is presented in Table 4.
Table 3: Results in terms of the best f values (25 runs).
Optimization
method
f
minimum
*B
f
mean
#M
PSO -20.7274 - -16.9347 -
QPSO -21.2292 - -19.1884 -
MQPSO -21.7586 - -21.4360 -
GA-MPC -21.8425 2
-21.7022
1
WI_DE -21.8000 6 -21.7000 2
SAMODE -21.8216 4 -21.6589 3
OXcoDE
-21.8650
1 -21.5910 4
ED-DE -21.8320 3 -21.4210 5
EA-DE-MA -21.7956 8 -21.2554 6
Mod_DE_LS -21.7691 9 -21.0897 7
AdapDE171 -21.8084 5 -20.9583 8
mSBX-GA -21.2545 11 -20.8860 9
DE-RHC -20.5000 13 -18.3000 10
RGA -21.0188 12 -17.2908 11
DE-Acr -21.6010 10 -16.7560 12
ENSML_DE -21.8000 6 -15.6000 13
CDASA -19.0310 14 -13.5400 14
*B: ranking based on the best results in terms of f in CEC-2011
#M: ranking based on the
mean results in terms of f in CEC-2011
Table 4: Best solution found using MQPSO.
Element i
Amplitude
excitation (I
i
)
Phase
perturbation (β
i
)
1 0.9993 -31.1885
2 0.3916 31.3933
3 0.4194 -89.5185
4 0.2026 -56.7929
5 0.4232 83.9948
6 0.6186 -16.7231
SSL (dB) -21.8191
DIR (dB) 10.0084
6 CONCLUSIONS
QPSO is a complex nonlinear system, and accords to
states superposition principle. In this paper, a
MQPSO based on beta probability distribution and
mutation differential operator is proposed and
validated.
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196

Simulation results illustrates that the incorporation
of the beta probability distribution and mutation
differential operation scheme enhances the search
moves of a MQPSO by generating the more
promising exemplars as the guidance particles.
Furthermore, it provides the necessary trade-off
between exploration and exploitation to global
optimization. In this context, the simulation results
show the effectiveness of our approach.
In future research, statistical significance tests to
compare different optimization approaches with
MQPSO will be carried out to monobjective and
multiobjective cases.
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