Automated Antenna Design Using Self-adaptive Differential Evolution

Algorithm

Kun Qin 1*, Sanyou Zeng 1, Zhengjun Li 2, Hongyong Jing 2

1China University of Geosciences (Wuhan) ,Wuhan, 430074, China 2China Academy of Space Technology (Xi'an) ,Xi'an, Shanxi, 710000,China

* qinkun1910@yeah.net Abstract— As we knew, an evolved X-band antenna for NASA’s Space Technology 5 (ST5) spacecraft had been deployed on schedule in 2006. This paper uses Self-adaptive Differential Evolution Algorithm (SaDE) to solve ST5 antenna design problem. The performance of SaDE is reported on the set of 24 benchmark functions provided by CEC2006 special session on real parameter optimization. The experimental result shows SaDE can solve ST5 antenna design problem effectively. The simulation result indicates our evolved antenna’s voltage standing wave ratio performance is better than NASA evolved antenna and gain performance is competitive. Moreover, our evolved antenna was evolved on finite ground plane, the specified size for the ST5 antenna, while NASA’s on infinite ground.

Keywords- Differential Evolution Algorithm; Automated Antenna Design; Self-adaptive

I INTRODUCTION

Evolutionary antenna design and optimization has been widely investigated since early 1990s [1, 2], including wire antennas [3], antenna arrays[4], helical antennas [5]. Some of them like X-band [6], S-band [7], and fish bone [8] antennas are so irregular and weird looking that seems unlikely be thought out by human.

In 2003, a group of NASA had designed an evolved X-band antenna for NASA’s Space Technology 5 (ST5) spacecraft [9], using branching GA. It had been adopted in the final flight on March 22-June 30 2006 and became the first evolved hardware in space.

Nowadays, evolutionary antenna design is also a hot research. NASA also pays attention on the research of evolutionary antenna design [10].

Antenna design can be regarded as a kind of constrained optimization problems with real parameters, while DE [11] is an efficient algorithm for solving real parameter optimization problems, and has been used to solve real world problems, e.g. [12]. In this paper we create a real parameter model for antenna design problems and use SaDE[13] to solve such antenna problems. The performance of the SaDE is reported on the set of 24 benchmark functions provided by CEC2006 special session on real parameter optimization [13]. We take lower earth orbit ST5 antenna designed problem [6] as a real world test problem to verify the effectiveness of our work. Table 1 presents the key requirements of ST5 antenna.

Remaining sections of this paper will be organized as below: section 2 describes the details of new antenna model, section 3 describes the details of antenna design using SaDE, section 4 includes the performance of our evolved antennas, and section 5 presents the conclusion.

Table 1 Key requirements of ST5

Property Requirement

Transmit Frequency 8470MHz

Receive Frequency 7209.125MHz

VSWR <1.2:1 at Transmit Frequency

<1.5:1 at Receive Frequency

Gain Pattern >=0 dBic, where

Finite ground 15 cm diameter

Input Impedance 50�

II ANTENNA MODEL

A. Representation of antenna topological structure

Antenna design is created by starting with an initial feeder and adding four identical arms. Antenna geometric structure is symmetrical about the z-axis, each arm

2011 Fourth International Conference on Intelligent Computation Technology and Automation

978-0-7695-4353-6/11 $26.00 © 2011 IEEE

DOI 10.1109/ICICTA.2011.527

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2011 Fourth International Conference on Intelligent Computation Technology and Automation

978-0-7695-4353-6/11 $26.00 © 2011 IEEE

DOI 10.1109/ICICTA.2011.527

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rotated 90 from its neighbors. We use linear real-values code to represent branch in antenna arms. As shown in Figure 1.

The initial feed wire is a thumbnail lead, starting by origin with a specified length along the positive Z-axis. Its corresponding code can be described as follow: terminal coordinates: (0, 0, L0), where L0 indicates the feed wire length.

The other 4 conductors start from end of the feed wire. Their code can be described as: terminal coordinates: (x, y, z ).Both the feed wire and other wires’ radius is the same: r.

Figure 1. Antenna Structure

Table 2 describes the geometric structure of antenna in figure 1. The corresponding chromosome is (r, L0, x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4).

Table 2 Code of Antenna in Figure 1

Start Point End Point Radius

Feeder (0,0,0) (0,0,L0) r

Wire 1 (0,0,L0) (x1,y1,z1) r

Wire 2 (x1,y1,z1) (x2,y2,z2) r

Wire 3 (x2,y2,z2) (x3,y3,z3) r

Wire 4 (x3,y3,z3) (x4,y4,z4) r

In our experiments, we set the wire number to 5 and the wires with the same radius, because a large radius change between connected antenna wires may decrease accuracy in NEC2 [14].

B. Antenna Design Optimization Problem

We construct the following optimization problem based on the representation of antenna structure and the antenna specifications.

Q:

decision variables: 1 2( , , , )nx x x x� , )n, )

min ( ) _ _f x T VSWR R VSWR� �)) + 355 80

( , ) ( , )0 40

( _ _ )T Gain R Gain� � � �� �� �

� �� �80

40�

satisfy to:

_ ( ) a r gT V S W Rg x T T V S W R� )) a r) a r

_ ( ) arg 0;R VSWRg x T RVSWR� )) arg) arg

_ ( , ) ( ) arg 0;T Gaing x T Gain� � � )) arg) argarg

_ ( , ) ( ) arg 0;R Gaing x T Gain� � � )) arg) argarg

where 40 80 ,0 360� � 80 0 360� 80 080 080 080 080 080 0 .

T_VSWR is the actual value of the VSWR and TargTVSWR is the maximum VSWR for transmit frequence, while R_VSWR and TargRVSWR represent those value for receive frequency. T_Gain and R_Gain are actual value of antenna gain. TargGain is the minimum gain value for each designated angle. For ST5 antenna design problem, we design as follows:

TargTVSWR = 1.2; TargRVSWR = 1.5; TargGain = 0dBic, where40 80 ,0 360� � 80 0 360� 80 080 080 080 080 080 0

III ST5 ANTENNA DESIGN VIA SADE

A. Self-adaptive Differential Evolution Algorithm (SaDE)

We use the Self-adaptive Differential Evolution Algorithm (SaDE) [13] to solve antenna problems. The framework of SaDE is as follows: Algorithm 1: Framework of SaDE

Step 1 Randomly create population P(0) with size N. Set evolutionary generation counter t = 0;

Step 2 Breed offspring population Q(t) of size N from P(t) (see Algorithm 2);

Step 3 Unite the parents and offspring populations: R(t) = P(t) Q(t) ;

Step 4 Calculate g( xx ) , f( xx ),where xx �R(t) Step 5 Select N solutions from R(t) as next

population: P(t+1) (see section 3.2); Step 6 t = t + 1; Step 7 If stopping criterion satisfied go to Step8, else

go to Step2; Step 8 Output P(t).

Algorithm 2: Framework of breeding offspring operator Step 1 Empty Q(t);

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FOR i = 1 TO N; Step 2 Randomly pick

1 2 1 2, [1,2, , ]p p N i p p� � �]i]i p] ;

Randomly pick [1,2, , ]jrand n� ]n, , where n is the length of chromosome ixx ;

Step3 mutation operation, here we use the following strategy:

FOR j =1 TO n current to best/1: [11]

1 2( 1) ( ) ( ( ) ( )) ( ( ) ( ))ij ij best ij i ip ip ih t x t x t x t F x t x t F� � � � � �h ( 1) F( ) ( ( ) ( )) ( ( ) ( ))) ( ( ) ( )) ( ( ) (( ) ( ( ) ( )) ( ( )( ) ( ( ) ( )) ( (( ( ) ( ))

END FOR j Step4 crossover operation: FOR j=1 TO n

( 1) 1 )( 1)

( ) 1 )ij ij i

ijij i j i

h t rand CR or j jrandv t

x t rand CR or j jrand � ��� � �

� ���

h ( 1)( 1)v ( 1) jj ( )

x t ra( )) (2)

( 1) 1 )( 1)

( ) 1 )ij ij i

ijij i j i

h t rand CR or j jrandu t

x t rand CR or j jrand � ��� � �

� ���

h ( 1)( 1)u ( 1) jj ( )

x t ra( )) Where

1ijrand is a random decimal number� [0,1]

END FOR j Step5 selection operation (see comparison of

individuals in 3.2) : if ivv is better than ixx , then select ivv ; otherwise, if iuu is better than ixx , then select iuu ; otherwise, return ixx . Step6 Adjust parameters iF , which means Scaling

factor, and iCR , which is crossover probability (see

comparison of individuals in 3.2):

1 2 1, 1

,

3 4 2, 1

,

( ) ( ) and

( ) ( ) and

i ii t

i t

i ii t

i t

rand if t is better than t randF

F otherwise

rand if t is better than t randCR

CR otherwise

�

�

�

�

� � ��

� � ��

v x

v x

t is better than t rand( ) ( ) and) ( �( ))( )

t is better than t rand( ) ( ) and) ( �( ))( )

(3)

Where }4,3,2,1{, �jrand j is a random decimal

number � [0,1], and 1 2,� � are the probability if adjusting parameters iF , iCR respectively.

END FOR i Step 7 Output offspring population Q(t). For each individual xx , using the following formula

to calculate its violations of constraints value.

1( ) max{( _ 1.2),0};G x T VSWR� �) {) max{ (4)

2 ( ) max{( _ 1.5),0};G x R VSWR� �) {) max{ (5)

( , )( ) max{ arg _ ,0};iG x T Gain T Gain � �� �) {) max{) max{ (6)

( , )( ) max{ arg _ ,0},jG x T Gain R Gain � �� �) {) max{) max{ (7)

( 3, ,2 ; 3 , ,2 2 ).i nm j nm nm� � � � �,2 ; 3 , ,2 2 ).j,2 ; 3 , ,2 2; 3 , ,2 2 ),2 ; 3 , ,2; 3; 3 , ,2

Where TargGain is 0 if 40 80 ,0 360� � 80 0 360� 80 080 080 080 080 080 0 ; and n is

the number of � , m is the number of � .

1( )G x) , 2 ( )G x) , ( )iG x) , ( )jG x) are original violations

for antenna problems.

B. Comparison of individuals via SaDE

While breeding offspring in step 2 and selecting solutions in step 5, SaDE needs to compare two individuals. Suppose there are two individuals:

1xx and

2xx . We divide all of the violations to three classes: VSWR violations

1( )V x) ; gain violations at transmit frequency:

2 ( )V x) ; and gain violations at receive frequency:

3( )V x) .

1 1 2( ) ( ) ( );V x G x G x� �) ( ) ( );) ( ) () ( ) (( )( (8)

3

23

( ) ( );nm

ii

V x G x�

�

� �3

) ( );nm

(()�

� (9)

2

33

( ) ( );nm

inm

V x G x�

�

� �2

) ( );nm

(()�

� (10)

Then the total violation of xx is :

1 2 3( ) ( ) ( ) ( );V x V x V x V x� � �) ( ) ( ) ( );) ( ) ( ) () ( ) ( ) (( ) ( )( ) (( ) (11) With this violations, 1xx and 2xx can be described

as:

1 1 1( ( ), ( ));x f x V x� ( ( ) ( ));x ( ( ) (( ) (

2 2 2( ( ), ( ));x f x V x� ( ( ) ( ));x ( ( ) (( ) ( The comparison between 1xx and 2xx follows such

two criteria: (1) Both the individuals, the one with small violation

wins; (2) If the violations of them are equal, then the one

with better objective value wins.

C. SaDE Setup

The parameters in the SaDE during solving the ST5 antenna problem are set as follow:

(1) Population size, N=50. (2) Number of Parents, M =5. (3) Crossover probability, CR = 0.3. (4) Scaling factor, F=0.5.

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(5) Mutation probability, MP = 0.1

(6)Max function evaluations: 500000, which means the max evolutionary generations: 50000 / 50 = 1000.

IV RESULTS OF EXPERIMENT

We use Numerical Electromagnetic Codes, Version 2 (NEC2) [14] to evaluate all antenna designs. A finite ground plane with 14cm diameter was used in the evaluation.

The performance of the SaDE is reported on the set of 25 benchmark functions provided by CEC2006 special session on real parameter optimization [13].

We use SaDE to solve low earth orbit ST5 antenna design problem, and produced an antenna (Called SaDE-03-01). Its performances satisfy the key requirements of ST5 antenna (as shown from Figure 2 to Figure 6). Its structure is presented in figure 2, 3. Figure 4 is the comparison of VSWR performance between SaDE-03-01 and ST5 antenna. Figure 5, 6 are the comparison of gain performance between SaDE_03_01 and ST5 antenna.

SaDE_03_01 antenna achieves high gain across a wide range of elevation angles and its VSWR for transmit and receive frequencies are sufficient small. Table3 describes the comparison of SaDE_03_01 and ST5.

Table 3 Comparison of SaDE_03_01 and ST5 Property SaDE_03_01 ST5

gain similar

VSWR(Transmit) 1.2 2.4

VSWR(Receive) 1.5 2.6

Ground plane Finite Infinite

V CONCLUSION

We use a new mathematical model: Real Parameter Optimization Model, while NASA using Combination Optimization Model. So we use SaDE which is effective in solving real parameter optimization problems to solve this antenna design problem. A large number of experiments show our method can find good solutions for this problem.

Evolved antenna SaDE_03_01 was shown to be compliant with respect to the ST5 antenna performance requirements. Its performance of VSWR is better than ST5 and its performance of gain pattern is competitive

with ST5. Moreover, it was evolved on finite ground plane, while ST5 on infinite ground plane.

Our future work will focus on automated antenna design through SaDE and other evolutionary algorithms.

VI ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (No.s: 60871021, 60473037).

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Figure 2. Structure of SaDE-03-01

(0.0003,0.003,0.002,0.015555,0.021268, 0.017212,0.002,0.020375,0.009918,0.008933,0.004621,0.011084,0.012647,0.019024)

Figure 3. Genotype of SaDE-03-01

(a) SaDE-03_01

(b) NASA’s ST5_3_10

Figure 4. VSWR

(a) SaDE_03_01

(b) NASA’s ST5_3_10

Figure 5. Maximum and minimum gain at 8.47GHZ for antennas

(a) SaDE_03_01

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(b) NASA’s ST5_3_10

Figure 6. Maximum and minimum gain at 7.2GHZ for antennas

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