A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

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A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

Randy L. Haupt

The Pennsylvania State University Applied Research Laboratory

P. O. Box 30

State College, PA 16804-0030 [email protected]

Abstract: This paper describes a genetic algorithm (GA) that optimizes chromosomes containing a mix of continuous and binary encoded variables. Uniform crossover from the binary GA and a mutation rate akin to that of the continuous GA are fundamental parts of the algorithm.

Parameter selection as well as an application to antenna design are presented.

Keywords: Genetic algorithm, optimization, microstrip antenna

1. Introduction

Most optimization problems work with continuous values. If the variables have integer values, then special algorithms must be used. If continuous values and integers are part of the same optimization problem, then it is known as mixed integer optimization, and popular approaches, such as branch and bound, are used. The genetic algorithm and particle swarm optimization have also been used for mixed integer optimization.

This paper presents a GA formulation that simultaneously works with continuous, integer, and binary values in one chromosome. Operators for his new algorithm is described in the next section and applications to a circularly polarized patch antenna designs are shown in the following section. The beauty of this algorithm is that it can optimize on any type of variable value without a change in the algorithm.

2. Mixed Integer/Binary GA The initial population matrix of the GA is given by

11 12 1

21 22

1

C

R RC

a a a

a a

P

a a

ª º

« »

¬ ¼

"

# % #

"

(1)

(2)

where 0da_rcd . Each row is a chromosome and the values are created by a uniform random 1 number generator. A cost function evaluates each chromosome and returns a cost.

> @

^m¹^, ^m²^, ^, ^mC

cost f a a " a (2)

Inside the cost function, the variables may be converted to a new continuous range by

^max ^min

^min

n mn

x x x a x (3)

or converted to an integer by

^

^max ^min

`

^min

n mn

x roundup x x a x (4)

where roundup rounds to the next highest integer and x values are integers. In some cases, the value is converted to binary. One possibility is to round the value.

^ `

n mn

x round a (5)

Another possibility is to quantize the value.

^ `

n mn

x quantize a (6)

The cost function does all the scaling, quantizing, and rounding, so the GA can operate independent of the type of variable. Uniform crossover works well for the binary GA, so it is implemented here. Two parents are selected and a random binary mask is created. If the mask has a one in the column, then the offspring receives the variable value in parent#1. If it has a zero, then the offspring receives the variable value in parent#2.

1 2 3 4 5 6 7 8 9

parent #1 parent # 2

1 0 1 1 0 1 0 0 1

mask offspring

m m m m m m m m m

n n n n n n n n n

m n m m n m n n m

a a a a a a a a a

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This type of crossover results in a diversity of values if the values are binary, but only interchanges values between chromosomes if the values are integer or continuous. Consequently, the mutation must be responsible for creating diversity within the population for continuous and integer values.

One possible approach to mutation is to randomly select variables in the population and replace them with uniform random values. The mutated chromosome (chromc) is created from the selected chromosome (chrom) by

>

^r¹ ^r² ^r³ ^r⁴ ^r⁵ ^r⁶ ^r⁷ ^r⁸ ^r⁹

@

chromc a a ac a a a a a ac (8)

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where the primed values are uniform random numbers. Another approach is to add a random correction factor. The correction factor may be created by multiplying each element within a chromosome by a random number ( 1 dE_rm d ) and multiplying the entire chromosome by a 1 mutation factor ( 0dD_r d ). 1

>

¹ ¹ ² ² ³ ³ ⁴ ⁴ ⁵ ⁵ ⁶ ⁶ ⁷ ⁷ ⁸ ⁸ ⁹ ⁹

@

c

r r r r r r r r r r r r r r r r r r r

chrom D E a E a E a E a E a E a E a E a E a ⁽⁹⁾

Now, the mutated chromosome is given by

^

^c

`

chromc rem chrom chrom (10)

where rem is the remainder function (digits to the left of the decimal point are dropped).

This algorithm was tested on two cost functions to try to determine an appropriate population size and D_r. In both cases, the GA quit after 400 function evaluations and reported the best results. The first test functions is

⁶ ²

1

1 n n

f ^x

¦

x ⁽¹¹⁾

with a minimum of zero at x_n . The results shown in Figure 1 were averaged over 100 runs 0 for population sizes between 8 and 96 and mutation rates between 0.01 and 0.3. The best results occurred when the population size was 8 and the mutation rate was 0.1.

Figure 1. GA results for f averaged over 100 runs for various population sizes and ₁ mutation rates.

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The second test functions is

⁶ ²

2

1

60 _n 10 cos 2 _n

n

f ^x

¦

ª¬x Sx º¼⁽¹²⁾

with a minimum of zero at x_n . The results shown in Figure 2 were averaged over 500 runs 0 for population sizes between 8 and 96 and mutation rates between 0.005 and 0.3. The best results occurred when the population size was 40 and the mutation rate was 0.01.

Figure 2. GA results for f averaged over 500 runs for various population sizes and ₂ mutation rates.

3. Applications to Antenna Design

The goal is to design a rectangular patch for circular polarization at 10 GHz using FEKO.

Input variables for the cost function are

^{x y}^, position of probe feed

x, y

L L patch length in x and y directions

h substrate thickness (either 1.575mm or 3.15mm)

Hr relative dielectric constant of the substrate (either 2.2 or 2.33) The cost function returns the following value

(5)

max , , 11

2

E E

cost E E s

E E

T M

S ½

° °

® ¾

° °

¯ ¿

(13)

The first two quantities in (13) equal zero for circular polarization, and s₁₁ is zero for a perfect match to 50:. When the patch is circularly polarized and perfectly matched, then cost = 0.

The optimization was performed using the best population size and mutation rate for both test functions in the previous section. Results were averaged over 5 runs and are summarized in Table 1. Figure 3 shows the best of five independent results for a population size of 8 and

r 0.10

D and for a population size of 40 and D_r 0.01. Since the optimizations were terminated after 400 function calls, the number of generations to convergence is different for different population sizes.

Table 1. Cost statistics after 5 independent runs.

maximum minimum average population size = 8

mutation rate = 0.10 0.48995 0.055435 0.22332 population size = 40

mutation rate = 0.01 0.51665 0.30441 0.40838 Nelder-Mead downhill

simplex algorithm 1.0921 0.27444 0.53891

Figure 3. Convergence results for the two best runs with a population size of 40 and

r 0.01

D , and a population size of 8 and D_r 0.10. The top two dashed curves are the population average and the bottom two curves are the best population cost. The GA was terminated after 400 function evaluations.

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The best chromosome translates to the optimum design values of

^{x y}^, ^1.262^mm^,3.451^mm

, 8.618 ,10.138

x y

L L mm mm

1.575

h mm

2.2H_r

The resulting patch is right-hand elliptically polarized with an axial ratio of 1.09 and s₁₁ 0.055.

4. Conclusions

Many antenna designs have variables with integer values and with continuous values.

This paper presented a version of a GA that works with values between zero and one and uses binary uniform crossover and continuous mutation. The patch antenna design is a multiobjective optimization using continuous and binary variables.

References

[1] Y. Li and M. Gen, "Nonlinear mixed integer programming problems using genetic algorithm and penalty function,"

[2] Z. Gaing, "Constrained optimal power flow by mixed-integer particle swarm optimization,"

[3] R.L. Haupt and Sue Ellen Haupt, Practical Genetic Algorithms, 2^nd edition, New York: John Wiley & Sons, 2004.