Aerospace and Electronic Systems - July 2019 - 24

Design of a TTC Antenna Using Simulation and Multiobjective Evolutionary Algorithms

Figure 14.
3-D radiation pattern of the antenna.

In general, there are multiple valid solutions that are
defined using the concept of Pareto-optimal Front.
The Pareto-optimal Front is the set of the best possible
solutions for the problem. In Section "PROBLEM
DEFINITION," we defined the solution objectives so as a
vector of n objective values, i.e., so ¼ ðso1 ; . . . ; son Þ.
To obtain the Pareto-optimal Front of a problem, it is necessary to sort all solutions according to their relationship
of dominance. We say that a solution s dominates a solution v, denoted as s " v if the objective values of s are
partially less (at least one less and equal the rest) than the
objectives values of v, i.e., 8i 2 ð1; . . . ; nÞ; soi voi ^
9i 2 ð1; . . . ; nÞ : soi < voi . This definition considers that
we are minimizing all objective values. To maximize, just
change the less than operator by greater than operator.
Note that the relationship of order " is partial and therefore there may be solutions that do not dominate each
other. A set of solutions that do not dominate each other is
said to belong to the same front. Those solutions that are
not dominated by any other, belong to the first front, called

Pareto-optimal Front. The solutions dominated by those
belonging to the first front, but which do not dominate
each other, form the second front. And so, successively,
all the solutions are grouped in different fronts. To illustrate the previous concepts, let us provide an example
with the problem that concerns us. In our case, we have
four objectives to optimize, i.e., cross-polar polarization
level (dB) and gain RHCP (dBi) for 1.81 and 2.55-GHz
frequencies. Table 1 shows the objective values of six solutions obtained from the experimentation. Solutions 1 and
2 correspond to solutions SPEA-2 5 and SPEA-2 6 shown
in Table 2. Last column of the table shows the front to
which each solution belongs. All solutions in the first front
belong to the Pareto-optimal Front.
Note that solutions in the same front do not dominate
each other, but they do dominate solutions in lower fronts.
Solution 1, for example, dominates solution 2 for the
1.81-GHz values, but is dominated for the 2.55-GHz values by solution 2. Figure 15 shows a 4-D chart (the fourth
axis is the color range) with values obtained from the
experimentation (some of them are shown in Table 1).
Solutions in the Pareto-optimal Front (front 1) correspond
to the most top-left plane.

METAHEURISTICS AND EVOLUTIONARY ALGORITHMS
Metaheuristics are a family of approximate optimization
techniques for solving the computational problem. There
are multiple metaheuristic techniques available for solving
MOOPs.
Evolutionary algorithms (EAs) are a set of algorithms
inspired in the biologic evolution. Algorithm 1 shows the
pseudocode of a standard EA. At each generation (loop
iteration), an auxiliary population (with the same size as
the original one) is generated by iteratively applying the
genetic operators (crossover and mutation), then, both
the current and the auxiliary populations are merged into
one single new population. Worst individuals of the new

Table 1.

Objective Values of Six Solutions
Cross Polar Level (dB)
Solution

Gain RHCP (dBi)

1.81 GHz

2.55 GHz

1.81 GHz

2.55 GHz

Front

À19.13

À13.82

5.81

3.82

À12.87

À19.2

3.33

4.02

À9.2

À8.83

À1.7

À3.1

À5.9

À9.01

À5.2

À5.0

À1.3

À5.2

À8.2

À5.1

À3.9

À0.4

À5.3

À9.75

IEEE A&E SYSTEMS MAGAZINE

JULY 2019

Aerospace and Electronic Systems - July 2019

Table of Contents for the Digital Edition of Aerospace and Electronic Systems - July 2019