Particle Swarm Optimization for Real-Time Adaptive Array Control

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

PARTICLE SWARM OPTIMIZATION FOR REAL-TIME ADAPTIVE

ARRAY CONTROL

M. Donelli, F. De Natale, S. Piffer, and A. Massa

January 2011

for Real-Time Adaptive Array Control

Massimo Donelli,Fran es oDeNatale,StefanoPier,and AndreaMassa

DepartmentofInformationandCommuni ationTe hnology UniversityofTrento,ViaSommarive14,38050Trento-Italy

Tel. +390461882057,Fax+390461882093,E-mail: andrea.massaing.unitn.it Web-site: http://www.eledia.unitn.it

Abstra t. This paper des ribes the appli ation of the parti le swarm optimizer (PSO) to the real-timeadaptive antenna ontrol. The PSOisan evolutionary pro edure similar togeneti algorithms, but generally it requires only few parameters to be alibrated. Furthermore, the PSO optimizer is mu heasier tobe implemented. To assessthe performan e ofsu h ate hnique as ompared to state-of-the-art methods, a set of sele ted experiments is arried out and the obtained results are deeply analyzed froma omputationalpoint of view as well asin terms of the numeri al performan e.

1. INTRODUCTION

The PSO was developed in 1995 [1℄ by Eberhart and Kennedy and it simulates the behavior and distributed intelligen e of swarms. Su h a numeri al pro edure is simple and it an be applied to a wide range of ele tromagneti s appli ations [2 ℄. Re ently, PSO has been su essfully applied to antenna design [3℄[4 ℄ and to inverse s attering problems [5℄. This paper is aimed at assessing the ee tivenessofsu hanapproa hindealingwitha omplexandtime-varyingproblemastheon-line ontrol ofadaptivearrayantennas. Within thisframework,thePSO isusedto adaptively tunethe arrayweightsinordertoseparatethedesiredsignalfromnoiseandinterferingsour esbymaximizing theSINR at the re eiver. This task is obtained maximizing the Signal-to-Interferen e-plus-Noise-Ratio(SINR).

Thepaper isorganized as follows. InSe tion 2, themathemati al details of theappli ation of the PSO to the real-time adaptive array ontrol are presented. Then, a numeri al assessment of the proposed pro edure is presented and the results ompared with those of referen e methods (Se t. 3). Finally,some on lusionsfollows inSe t. 4.

2. MATHEMATICAL FORMULATION

Let us to onsider a linear array where

M

isotropi elements are equally spa ed with an inter-element distan eequal to

d

=

λ

2

,

λ

being thefreespa ewavelength. Under narrow-band onditions andthe assumption of o- hannel interferen e, thesignal-to-noise-plus-interferen e ratio (SINR) at there eiver anbeoptimizedbymaximizingthefollowing ostfun tionarisingfromtheApplebaum theory[6 ℄

φ( ¯

w) =







[α(θ

d

)]

t

_w







2

[w]

t∗

C

T

w

(1)

α(θ

d

)

beinganarray- olumnwhi hjthelementisgivenby

α

m

(θ

d

) = e

jm

2π

λ

dsen(θ

d

)

,

m

= 0, ..., M − 1

;

θ

d

is the in ident angle indi ating the impinging dire tion of the desired signal (DOA);

w

=



w

m

= c

m

e

jϕ

m

; m = 0, ..., M − 1

,and

C

T

isthemeasurable desired-plus-undesired ovarian e

ma-trix. Byassuming onstantamplitude oe ients

c

m

,theantennaarrayis ontrolledby ontinuously tuningthephase oe ients

ϕ

m

for maximizing(1)and a ording to a PSO-basedpro edure. Morein detail, the PSO is an evolutionarypro edure, whi h operateson symboli representations ( alled parti les) of trial solutions. The algorithm onsiders a set of

S

parti les (or swarm),

D

=

{P

s

; s = 1, ..., S}

,and itoperates following so ial intera tion rules. inorder to a hieve thegoal of minimizing or maximizing a suitable tness fun tion that determines thequality of thesolution of theproblemat hand. Ea h parti le

P

s

islo atedat the position

x

s

=

n

ϕ

(s)

m

; m = 0, ..., M − 1

o

and moves in the solution spa e with a velo ity

v

s

=

n

v

(s)

m

; m = 0, ..., M − 1

o

. Su essively, iteration byiteration(

k

beingthe iteration number),theparti le iesfrom urrent position

x

s

(k)

toanother position

x

s

(k + 1)

in order to ee tively sample the sear hing spa e a ording to the following updating relation:

ϕ

(s)

_m

(k + 1) = ϕ

(s)

_m

(k) + v

(s)

_m

(k + 1)

(2) where

v

_m

(s)

(k + 1) = I

w

v

m

(s)

(k) + C

1

U

1

n

p

(s)

_m

(k) − ϕ

(s)

_m

(k)

o

+ C

2

U

2

n

g

m

(k) − ϕ

(s)

m

(k)

o

(3)

p

_s

(k)

being the lo ation with the highest tness value dis overed by the sth parti le up till now

(

p

s

(k) = arg {max

h=1,...,k

[φ(x

s

(h))]}

)and

g(k)

isthe positioninthesolutionspa eofhighestglobal

tness(

g(k) = arg {max

s=1,...,S

[φ(p

s

(h))]}

);

U

1

and

U

2

arerandomnumberssele tedbetween

0

and

1

;

C

1

and

C

2

arepositive onstants alleda eleration oe ients: they modelthe ognition and so ial weightoftheswarmpushingea hparti le

x

s

(k)

towards

p

s

(k)

and

g(k)

. Finally,theinertial weight

I

w

isa s alingfa tor ofthe velo ity

v

s

(k)

.

Theiterative pro ess isrepeateduntil

g(K) ≤ η

where

η

isa xedthresholdand

K

is theiteration ofthe onvergen e of the optimization pro edure.

3. NUMERICAL ASSESSMENT

Inordertoassestheee tivenessofthePSO-basedreal-time ontrolstrategy,alineararray onsist-ingof

M

= 20

isotropi elements was onsidered. The weight amplitudes

c

m

was hosen a ording the Dolph-Chebys hev riterion. As far as the time-varying environment is on erned, the inter-feren es enario was modeled a ording to the sto hasti modeldes ribed in[7℄. In parti ular, the life-time of theinterfering signals was hosen

L

t

= 5

and thePoissonfrequen y of theinterferen e arrival was assumed to be equal to

1 Hz

. Moreover, the amplitude of the interfering signals

s

i

,

i

= 1, ..., I

(

I

beingthenumberofthe interferingsignals) was assumedto be

30 dB

above the desired

signal

s

d

. Su h a referen e signal was onsidered to impinge on the me hani al bore-sight of the arrayantenna. Finally,aba kground noise

s

n

ofabout

30 dB

belowthelevelof

s

d

wasaddedat the re eived signal.

Asan example, Fig. 1 gives a representative plot of the sto hasti interferen e s enario by show-ing the distribution of the angles of arrival of the interfering signals during the iterative pro ess. Con erningthePSO parameters, thefollowing valueswasheuristi ally determined:

S

= 40

(swarm dimension),

C1 = C2 = 2.0

(a eleration terms),andthe onstant inertialweight equalto

I

w

= 0.4

. For omparison purposed, the same s enario was deal with other state-of-the-art ontrol methods in order to point out the advantages and possible limitations of the proposed approa h. Within su ha framework,the optimal theoreti alstrategy ortheoptimal Applebaum'smethods [6 ℄ aswell as a deterministi pro edure based on the least mean square error riterion (LMS) [8℄ was taken intoa ount. Moreover,for ompleteness,amodiedversionofaGeneti Algorithm, alledLearned Real-Time Geneti Algorithm (LRTGA)and detailedin[9℄,was onsidered, aswell.

0

30

60

90

120

150

180

0

100

200

300

400

500

600

700

800

900

θ

i

Iteration Number (k)

Figure1. Anglesofarrival(

θ

i

,

i

= 1, ..., I

)oftheinterferingsignalsversustheiterationnumber

k

.

ThestrategybasedonthePSOgenerallyoutperformedothermethodologiesintermsof onvergen e rateas well asrobustness to the noise-interferen es. Interms of empiri altuning of meta-heuristi parameters,the alibrationphaserequired averyshorttimeas omparedtothatofothersto hasti pro eduresbeingthe number of ontrol parameters very limited.

-10

0

10

20

30

40

50

60

0

100

200

300

400

500

600

700

800

900

SINR [dB]

Iteration Number (k)

PSO

Applebaum

LMS

LRTGA

Figure2. ComparisonsofthebehavioroftheSINRobtainedbyusingdierentkindsofminimization pro edures.

Asarepresentative example, Figure2 shows the behavior oftheSINR during theiterative pro ess for dierent strategies. It an be observed that on average the performan e of thePSO turns out tobe greaterofabout

4 dB

than thatof the best numeri al method(namely theLRTGA).

An optimization method based on the Parti le Swarm Optimizer has been applied to the real-time ontrol of linear antenna arrays. By means of some preliminary numeri al experiments, the ee tiveness of the proposed approa h has been pointed out and the a hieved results have been ompared with referen e losed-form solutions as well as with other referen e numeri al methods. Futureworkwillaimedatextendingtheproposedpro eduretotheadaptive ontrolofvariousarray geometries and to further assess the ability of the ontrol strategy in dealing with more realisti environments.

Referen es

[1℄ J. Kennedy and R.C. Eberhart, Parti le swarm optimization, Pro . IEEE Int. Conf. Neural Networks IV,Pis ataway, NJ,1995.

[2℄ J. Robinson and Y. Rahmat-Samii, Parti le swarm optimization in Ele tromagneti s, IEEE Trans. Antennas Propagat., vol.52,pp.397-407, 2004.

[3℄ W.BoeringerandH.Werner,Parti le swarmoptimizationversusgeneti algorithmsforphased arraysynthesis, IEEE Trans. Antennas Propagat., vol. 52,pp.771-779, 2004.

[4℄ D. Gies and Y. Rahmat-Samii, Parti le swarm optimization for re ongurable phase-dierentiated array design, Mi rowave and Opti al Te hnology Letters, vol. 38, pp. 168-175, 2003.

[5℄ S.Caorsi,M.Donelli,A.Lommi,andA.Massa,Lo ationandimagingoftwo-dimensional s at-terersbyusingaparti leswarmalgorithm, JournalofEle tromagneti WavesandAppli ations, vol. 18,pp.481-494,2004.

[6℄ S. P. Applebaum, Adaptive arrays, IEEE Trans. Antennas Propagat., vol. 24, pp. 585-598, 1976.

[7℄ M. Donelli, A. Lommi, A. Massa, and C. Sa hi, Assessment of the ga-based adaptive array ontrolstrategy: the aseofsto hasti life-time o- hannelinterferen es, Mi rowaveandOpti al Te hnology Letters, vol. 37,pp.198-238,2003.

[8℄ B.Widrow,B.Mantley,P.Griths, andL.Goode,Adaptiveantennasystem, Pro . IEEE,vol. 55,pp.2143-2159, 1967.

[9℄ S. Caorsi, M. Donelli, A. Lommi, and A. Massa, A real-time approa h to array ontrol based onalearnedgeneti algorithm, Mi rowave andOpti al Te hnology Letters,vol.36,pp.235-238, 2003.