Elitist Genetic Algorithm Performance on the Uniform Circular Antenna Array Pattern Synthesis Problem

Transcription

1 Fatih YAMA, Asım Egm YILMAZ Akara Uivrsity Elitist Gtic Algorithm Prformac o th Uiform Circular Ata Array Pattr Sythsis Problm Abstract. I this papr, th impacts of litism rat o th Gtic Algorithm (GA) prformac for th Uiform Circular Ata Array Pattr Sythsis Problm ar ivstigatd. For this purpos, a circular ata array with uiformly spacd isotropic lmts havig idtical xcitatio amplituds is usd as a modl. Ulik th classical GA, w us a GA structur havig a litist populatio updat approach for improvd GA prformac. I th litist approach, a crtai ratio of th populatio (i.. th fittst idividuals) is dirctly trasfrrd to th xt gratio. Strszczi. W artykul zaprztowao wykorzystai algorytmu gtyczgo do aalizy jdorodj kołowj matrycy at. W tym clu jako modl użyto aty z rgulari rozmiszczoymi lmtami mającymi tę samą amplitudę pobudzaia. Użyto zmodyfikowago algorytmu gtyczgo z populacją listyczą. (Sytza kołowj aty matrycowj z wykorzystaim algorytmu gtyczgo) Kywords: Circular ata array, litist gtic algorithm, litism rat, pattr sythsis. Słowa kluczow: ata kołowa, algorytm gtyczy, sytza. Itroductio Gtic algorithm (GA), diffrig from covtioal sarch tchiqus, is a probabilistic sarch mthod basd o th pricipls of Darwi s atural slctio ad volutio thory. So far, GA has fficitly b usd for th solutio of combiatorial optimizatio problms [1-4]. I this cotxt, it has b quit succssful i may girig applicatios. Additioally, it has also b applid to crtai problms i th lctromagtic thory. Ata array sythsis problm costituts a major portio of such applicatios. Ata array sythsis problm, which is o of th popular rsarch topics i lctromagtics, rquirs othig but to obtai th dsird pattr (grally which caot b achivd with a sigl ata) with miimum rror. I such a problm, lctrical spcificatios ad gomtrical placmts of th array lmts ar importat. Grally, th mai xpctatios of th dsigr from th ata arrays ar: radiatio pattr cosistig of a arrow mai bam i th dsird dirctio, ad low-lvl sid lobs. May diffrt ata array cofiguratios might b costructd; but du to thir simplicity th most widly applid cofiguratios ar liar, plaar ad circular arrays [5-6]. I th litratur, thr ar studis o ata array dsig, sythsis ad pattr formig usig optimizatio mthods such as gtic algorithm ad particl swarm optimizatio [7-1]. I [7] ad [], th sid lob rductio o circular ata arrays was prformd by mas of th gtic algorithm ad particl swarm optimizatio. I [9], radiatio pattr sythsis was prformd through a hybrid ral codd GA algorithm cosistig of simplifid quadratic itrpolatio. I our prvious rsarch [1], th impacts of th mutatio rat ad th crossovr positio o th GA prformac wr ivstigatd via th uiform circular ata array pattr sythsis problm. This tim, w aim to aalys th litist GA prformac, whr th idal litism rat will b ivstigatd. Thr ar studis o th litist GA i th litratur [11-2].I [11], a w tchiqu calld adaptiv litistpopulatio sarch mthod was itroducd. It is basd o th cocpt of adaptivly adjustig th populatio siz accordig to th idividuals dissimilaritis ad a ovl dirctio dpdt litist gtic oprator. Rf. [] dmostrats th possibilitis of adaptig litism-basd GA modls i ordr to optimiz th high volum fly ash cocrt mixs. Rf. [13] prsts a w prmutatio-basd litist gtic algorithm usig srial schdul gratio schm for solutio of a larg-sizd multipl-rsourc costraid projct schdulig problm. I [], a mthod combiig th w Rakd basd Roultt Whl slctio algorithm with Parto-basd populatio rakig Algorithm is proposd. Two litismbasd compact gtic algorithms, prsistt litist compact gtic algorithm ad oprsistt litist compact gtic algorithm wr dscribd i [15]. Rf. [] itroducs a w tchiqu calld adaptiv litist populatio sarch mthod, which allows xtsio of uimodal fuctio optimizatio mthods to locat all optima of multimodal problms i a fficit mar. I [17], a rplacmt stratgy for stady-stat gtic algorithms cosidrig two faturs of th cadidat chromosom to b icludd ito th populatio: a masur of th cotributio of divrsity to th populatio ad th fitss fuctio. I [], th rlatioship btw th litist gtic algorithm prformac ad algorithm trmiatio critria was xamid. Rf. [19] proposs a hybrid algorithm cosistig of hyprmutatio ad litist stratgis for computd aidd aalog circuit dsig sythsis problms. Rf [2] prsts a w cross-layr rsourc allocatio modl for multiusr packt-basd Orthogoal Frqucy Divisio Multiplxig (OFDM) systms proposig a litist slctio adaptiv gtic algorithm. Ulik th othr studis i th litratur, w ivstigatd mthods for prformac improvmt i this study i ordr to obtai a dsird pattr of circular ata array with litist GA. For this purpos, diffrt litism approachs ar trid to solv Uiform Circular Ata Array Pattr Sythsis Problm. Th ffcts of litism hav so far b studid may tims. I our opiio, th origiality of this study coms from th followig raso: Ev though i som prvious publicatios (such as [11-]) it has b mtiod that litist GA provid bttr prformac, to our kowldg, thr xists o publicatio xplicitly ivstigatig how to chag or maipulat th litism rat idally. I additio, to th bst of our kowldg, thr is o study about litist GA prformac o ata array sythsis problm. PRZEGLĄD ELEKTROTECHICZY (Elctrical Rviw), ISS , R. R 1b/2 273

2 I this papr, i Sctio 2, basic dfiitios of GAs ar rvisitd brifly. I Sctio 3, formulatios rgardig th circular ata arrays ad th rlvat sythsis problm ar prstd. I Sctio 4, matrial ad mthod ar prstd togthr with th obtaid rsults ad rlvat discussios. Sctio 5 cocluds th papr with som rmarks. Gtic Algorithm (GA) Gtic Algorithm (GA) was first itroducd i 1975 by Hollad [1]. Ds usag of this algorithm has b ralizd spcially aftr Goldbrg s studis [2]. Gtic Algorithm provids th cssary solutio yildig th global miimum or maximum valus of multidimsioal ad complicatd fuctios [2-3]. Thy ar usd i widly for th solutio of th problms, which ar cosidrd vry difficult for covtioal optimizatio mthods. GA simulats th survival of th fittst amog idividuals ovr coscutiv gratios throughout th solutio of a problm. Each gratio cosists of a populatio of charactr (usually biary) strigs that ar aalogous to th chromosoms. Each idividual rprsts a poit i th sarch spac ad a solutio cadidat. Th idividuals i th populatio ar th xposd to th procss of volutio. Gs from good idividuals propagat throughout th populatio. Thus, ach succssiv gratio will litrally bcom mor suitd to thir viromt. I th optimizatio trmiology, this corrspods to th situatio that wr gratios hav bttr fitss valus [1-4]. Fig.1 prsts th gral structur of th covtioal/classical Gtic Algorithm. vry wid applicatio spctrum for th ata arrays. Arrays may b locatd diffrt gomtric shaps. If lmts of array ar locatd o a circl, this typ of array is calld th Circular Ata Array as s i Fig.2. Fig.2. Gomtry of a uiform circular ata array with isotropic lmts. Radiatio pattr of th circular ata array cosistig of isotropic lmts ca b xprssd by mas of th array factor. Th array factor of circular ata array formulatd as follows []: Grat th iitial populatio radomly (1) AF cos( ) cos( ) (, I jka, dm ) I 1 Calculat th palty, fitss ad probability valus of idividuals i th currt populatio Slct good idividuals from populatio through roultt whl Obtai a w gratio usig crossovr ad mutatio oprators Rpat util th itratio umbr Hr, AF is th array factor, is th umbr of lmts i th array, I is th xcitatio amplitud of th th lmt, a is th radius of th circl, k=2π/λ is th wavumbr, is agl of dirctio of th mai bam, is th agular positio of th th lmt with rspct to. ka ad ar giv by: (2) 2 a ka dm i (3) i1 i1 2 dmi i1 dm i Dtrmi th bst idividual as solutio ad trmiat th algorithm Fig.1. Gral structur of th classical gtic algorithm. Circular Ata Arrays Ata arrays ar ata systms, which ar cratd by combiig diffrt or similar atas i diffrt forms. Ata arrays ar usd i ordr to provid th dsird spcificatios such as low-lvl sid lob, arrow mai lob ad high dirctivity [5-6]. Furthrmor, thaks to addig togthr powr of th lmts i array, high-powr radiatio pattr is achivd ad without rquirig mchaical movmt, mai bam ca b strd to a dsird dirctio [5-6]. This yilds a Matrial ad Mthod I our study, a circular ata array with uiformly spacd isotropic lmts havig idtical xcitatio amplituds is usd as a modl. For this purpos, radiatio pattr of a circular ata array, for which: - th agl of dirctio for th mai bam is qual to zro dgr; - th agular positios of th lmts ( ) ar.5, 67.5, 1.5, 157.5,.5, 7.5, ad ; ad - th radius of th circl is 2 cm, is dtrmid as rfrc. Th Half Powr Bam Width of th rfrc pattr is chos to b 56, ad th opratig frqucy is chos to b 75 MHz. To obtai th dsird pattr through gtic algorithm, a iitial populatio cosistig of 6 idividuals is cratd. Th, agular positios of lmts ar covrtd to biary cods. Aftrwards, th cratd populatio is 274 PRZEGLĄD ELEKTROTECHICZY (Elctrical Rviw), ISS , R. R 1b/2

3 subjctd to th gtic algorithm oprators: slctio, crossovr ad mutatio. Throughout th study; all comparisos i all xprimts wr prformd by cosidrig idpdt xcutios of ach GA schm i ordr to limiat th chac factor ad to b abl to prform a fair compariso. Tabl 1 lists th paramtr stup usd throughout th xprimts i this study. Tabl 1. Paramtr valus usd i th gtic algorithm ad th xampl problm. Paramtr Valus usd i GA umbr of idividuals i th populatio 6 umbr of lmts i th array umbr of itratios (gratios) 1 Crossovr rat.2 Mutatio rat.1 umbr of idpdt xcutios 1 I th implmtatio, th radiatio diagram is dividd ito 36 qual parts i (-π, π) radias. Thus, a ssitivity of o-dgr is achivd. GA is implmtd i MATLAB 7.3., but o third party tool or aothr stadard product or library (such as MATLAB Gtic Algorithm Toolbox) is usd. Th mai raso for this is to prsrv th flxibility to modify th GA architctur i our ogoig/futur studis i ordr to ivstigat furthr prformac improvmt possibilitis. I classical or covtioal GA, GA oprators such as slctio, crossovr ad mutatio ar applid to all idividuals i th populatio. I this study, w trid to improv gtic algorithm prformac usig th litist populatio structur. I litist populatio mthod, th bst idividuals of currt populatio i a crtai ratio ar dirctly trasfrrd to th xt gratio. Hc, throughout this procss, th trasfr of good idividuals to th xt gratio is guaratd. I ordr to dtrmi th idal litism rat, th prformacs of diffrt litism rat chagig schms ar xamid. For this purpos, primarily, fixd litism rat is tstd. xt, icrasig ad dcrasig litism rats ar tstd. Rsults show that icrasig litism rat mthods caot improv th GA prformac. For this raso, oly o icrasig litism rat is prstd. I th followig quatios, r rprsts th litism rat, i rprsts th umbr of itratio, ad f av rprsts th avrag fitss valu of populatio. W us fiv diffrt litism rat schms. Ths ar: 1 st Schm: Costat litism rat I this schm, th litism rat is kpt fixd for all itratios. (4) r. 2 2 d Schm: Icrasig litism rat throughout th itratios I this schm, th litism rat icrass dpdig o th umbr of itratio as i th followig quatio: (5) 4 r.1 ( 2 1 ) I Figur 3, variatio of th litism rat alog th 3 rd Schm: Dcrasig litism rat throughout th itratios I this schm, th litism rat dcrass dpdig o th itratio umbr as i th followig quatio. i (6).4 r.25 (.1 ) i I Figur 4, variatio of th litism rat alog th Fig.3. Icrasig litism rat throughout th itratios Fig.4. Dcrasig litism rat throughout th itratios. 4 th Schm: Partial fixd litism rat I this schm, th litism rat is fixd (2%) i th first half of itratio umbr ad litism rat qual to zro i th scod half of itratio umbr. I Figur 5, variatio of th litism rat alog th 5 th Schm: Dcrasig litism rat dpdig o th avrag fitss valu of populatio (adaptiv litism) I this schm, th litism rat dcrass dpdig o th avrag fitss valu of populatio as i th followig quatio: (7) r.25 (5 f ) av I Figur 6, variatio of th litism rat alog th 2 1 Fig.5. Partial fixd litism rat For rfrc throughout th prformac comparisos, th avrag fitss valus of populatio obtaid by classical GA without litism is prstd i Figur 7. PRZEGLĄD ELEKTROTECHICZY (Elctrical Rviw), ISS , R. R 1b/2 275

4 Avrag fitss valus blogig to all proposd ad tstd litism schms ar prstd i Figur. Ths rsults show that partial fixd litism rat (4th schm) outprforms to th othr schms. Th dsird radiatio pattr togthr with th obtaid radiatio pattr by mas of 4th schm ca b s i Fig Fig.6. Dcrasig litism rat (dpdig o th avrag fitss valu of populatio). Avrag Fitss Valu gtic algorithm via th uiform circular ata array pattr sythsis problm. For this purpos, a circular ata array with uiformly spacd isotropic lmts havig idtical xcitatio amplituds is usd as a modl. Thaks to litist populatio approach, th bst idividuals of currt populatio i a crtai ratio ar dirctly trasfrrd to th xt gratio. Rgardig th litism rat, th followig obsrvatios ca b mad: Th slctio, crossovr ad mutatio oprators may caus corruptio of th gs of idividuals. This ffct may rduc th fitss valus of idividuals. By usig th litist GA, this gativ ffct ca b limiatd. Howvr, to obtai th bst GA prformac, th litism rat should b maipulatd i a itlligt mar. Our study shows that for this ata array sythsis problm, Partial Fixd Elitism Rat (i which th litism rat is fixd (2%) i th first phas of th algorithm; ad qual to zro i th scod phas of th algorithm) provids th bst prformac compard to othrs. This phomo ca b xplaid as follows: i th arly gratios, th ovrall populatio is of rlativly low quality; ad hc, litism is a prformac-improvig factor, which icrass th rat of good idividuals i xt populatio. But wh th populatio gts qualifid, litism starts to caus saturatio, v distractio i th populatio. Thrfor, aftr a crtai itratio umbr, litism rat should b rducd umbr of Fitss Valu Evaluatio Fig.7. Avrag fitss valus of populatio obtaid by classical GA. Powr (db) Avrag Fitss Valu Schm-1 Schm-2 Schm-3 Schm-4 Schm Fig.. Avrag fitss valus of populatio (diffrt litisim rat schms). Th most importat cotributio of litist GA is rductio i th umbr of fitss valu valuatios throughout th solutio. This cotributio provids timsavig i th solutio of problms. I our study, thaks to th Partial Fixd Elitism Rat (4th schm), 1% rductio is achivd cocrig th umbr of fitss valu valuatios. I additio to this succss, 23% prformac improvmt (solutio accuracy) is achivd compard to th classical GA. Th avrag fitss valus of th populatio (obtaid by Partial Fixd Elitism Rat Mthod ad Classical GA) vrsus th umbr of fitss valu valuatios ar s i Figur 1. It ca b s that about 23% mor avrag fitss valus ar achivd aftr 1% lss fitss fuctio valuatios. Coclusios This papr discussd how to maipulat th litism rat i a litist gtic algorithm for prformac improv Ø Agl (dgr) Dsird Pattr GA Pattr Fig.9. Th compariso of dsird ad obtaid radiatio pattrs. Thaks to Partial Fixd Elitism Rat, 1% fitss valu valuatio rductio ad 23% solutio accuracy prformac improvmt is obtaid compard to th classical GA. This advatag of th Partial Fixd Elitism Rat approach would b much mor apprciatd for th problms i which th computatioal cost of fitss fuctio valuatios ar quit high. Avrag Fitss Valu Classical GA Partial Fixd Elitism umbr of Fitss Valu Evaluatio Fig.1. Avrag fitss valus of populatio (obtaid by Partial Fixd Elitism Rat ad classical GA) vrsus th umbr of fitss fuctio/valu valuatios. 276 PRZEGLĄD ELEKTROTECHICZY (Elctrical Rviw), ISS , R. R 1b/2

5 Ths rsults ar valid for this particular problm oly. Gralizatio ad dirct applicatio of ths rsults to all problms would b icorrct. As a futur work, similar aalyss i lctromagtics will b prformd usig othr volutioary algorithms such as th Diffrtial Evolutio Algorithm. REFERECES [1] Hollad, J., Adaptatio i atural ad Artificial Systms, Uivrsity of Michiga Prss, 1975 [2] Goldbrg, D.E., Gtic Algorithms i Sarch Optimizatio ad Machi Larig, Addiso Wsly Logma, 199. [3] M ichalwi cz, Z., Gtic Algorithms + Data Structurs = Evolutio Programs, Sprigr-Vrlag, 1992 [4] W h itly, L. D., Foudatios of Gtic Algorithms-2, Morga Kaufma Publishrs Ic., 1993 [5] B a l ais, C.A., Ata Thory Aalysis ad Dsig, Joh Wily & Idrscic Publicatios, 25 [6] V issr, H.J., Array ad Phasd Array Ata Basics, Joh Wily & Sos Publicatios, 25 [7] Paduro, M.A., Mdz, A.L., Domiguz, R., R o m r o, G, Dsig of o-uiform Circular Ata Arrays for Sid Lob Rductio Usig th Mthod of Gtic Algorithms, AEÜ - Itratioal Joural of Elctroics ad Commuicatios, 6 (26), [] Shihab, M., ajjar, Y., Dib,., Khodir, M., Dsig of o Uiform Circular Ata Arrays Usig Particl Swarm Optimizatio, Joural of Elctrical Egirig, 59 (2), o. 4, 2- [9] Xu, Z., Li, H., Liu, Q.-Z., Pattr Sythsis of Coformal Ata Array by th Hybrid Gtic Algorithm, Progrss i Elctromagtics Rsarch (PIER), 79 (2), 75 9 [1] Yama, F. ad Yilmaz, A.E., Impacts of Gtic Algorithm Paramtrs o th Solutio Prformac for th Uiform Circular Ata Array Pattr Sythsis Problm, Joural of Applid Rsarch ad Tchology, Vol. (21), o.3, [11] Liag, Y. ad Lug, K.-S., Gtic Algorithm with Adaptiv Elitist-Populatio Stratgis for Multimodal Fuctio Optimizatio, Applid Soft Computig, 11 (211), [] Jayaram, M.A., ataraja, M.C. ad Ravikumar, C.., Elitist Gtic Algorithm Modls:Optimizatio of High Prformac Cocrt Mixs, Matrials ad Maufacturig Procsss, (29), 5-9 [13] K i m, J.-L., Prmutatio-Basd Elitist Gtic Algorithm Usig Srial Schm for Larg-Sizd Rsourc-Costraid Projct Schdulig, Procdigs of th 27 Witr Simulatio Cofrc, 27, [] Al Jadaa, O., Rajamai, L., Rao, C.R., o- Domiatd Rakd Gtic Algorithm for Solvig Multi-Objctiv Optimizatio Problms: RGA, Joural of Thortical ad Applid Iformatio Tchology, 2, 6-67 [15] Ah, C.W. ad Ramakrisha, R.S., Elitism-Basd Compact Gtic Algorithms, IEEE Trasactios o Evolutioary Computatio, Vol. 7, o. 4, (23), [] Lug, K.-S. ad Liag, Y., Adaptiv Elitist-Populatio Basd Gtic Algorithm for Multimodal Fuctio Optimizatio, Sprigr-Vrlag Brli Hidlbrg, GECCO (23), [17] Lozao, M., Hrrra, F., Cao, J.R., Rplacmt Stratgis to Prsrv Usful Divrsity i Stady-Stat Gtic Algorithms, Iformatio Scics, 17 (2), [] K i m, J.-L., Examiig th Rlatioship Btw Algorithm Stoppig Critria ad Prformac Usig Elitist Gtic Algorithm, Procdigs of th 21 Witr Simulatio Cofrc, (21), 3-37 [19] Liu, M. ad H, J., Automatd Aalog Circuit Dsig Sythsis Usig a Hybrid Gtic Algorithm with Hypr-Mutatio ad Elitist Stratgis, Itratioal Joural of Iformatio Tchology ad Computr Scic, 1 (29), [2] Tag, Z., Zhu, Y., Wi, G., Zhu, J., A Elitist Slctio Adaptiv Gtic Algorithm for Rsourc Allocatio i Multiusr Packt-basd OFDM Systms, Joural of Commuicatios, Vol. 3, o. 3 (2), Authors: PhD studt Fatih Yama, Elctroics Egirig Dpartmt, Akara Uivrsity, 61 Tadoga, Akara, Turky, fyama@g.akara.du.tr; assist. prof. dr. Asım Egm Yılmaz, Elctroics Egirig Dpartmt, Akara Uivrsity, 61 Tadoga, Akara, Turky, ayilmaz@g.akara.du.tr. PRZEGLĄD ELEKTROTECHICZY (Elctrical Rviw), ISS , R. R 1b/2 277