Introduzione agli Algoritmi Genetici Problema di Ottimizzazione

Introduzione agli Algoritmi Genetici

Università di Roma Tor Vergata

G. Marrocco L. Mattioni

Problema di Ottimizzazione

{ }K!!! ,...,, 21=!

{ }Mggg ,...,, 21=g

Set dei parametri modificabili (geometrici, elettrici..)

Set delle grandezze osservabili (ex. Zin, G)

maxminkkk !!! "" Vincoli (spazio dei parametri)

Problema: determinare i valori del set di parametri della struttura, nei vincoli specificati, in modo che sia massima ( o minima) una opportuna funzione obiettivo:

)]([ !gF

Tipologie di Ottimizzazione

Metodi Locali: fissato un valore di iniziale dei parametri, tali algoritmi convergono in maniera deterministica ad un massimo (minimo) locale della funzione obiettivo

- sono rapidi - sono fortemente influenzati dalla scelta dei parametri iniziali - richiedono che la funzione obiettivo abbia opportune proprietà matematiche - problemi con un alto numero di parametri diventano intrattabili

Metodi Globali: fissato un valore di iniziale dei parametri, convergono in maniera stocastica ad un valore prossimo al massimo (minimo) assoluto della funzione obiettivo.

- dipendono molto poco dalla soluzione iniziale - non richiedono particolari proprietà della funzione obiettivo - producono un insieme di soluzioni sub-ottimali (oltre quella ottimale) - permettono di studiare problemi con un elevato numero di parametri - sono di lenta convergenza - richiedono la taratura (in base all’esperienza) di parecchi parametri di simulazione

Gradiente coniugato Metodi di Newton ..

Genetic Algorithm Random Walk

Ottimizzazione Genetica (o Evolutiva)

Gli algoritmi genetici sono ottimizzatori iterativi di tipo stocastico basati sui concetti Darwiniani di selezione naturale e di evoluzione.

In un algoritmo genetico, un insieme di potenziali soluzioni viene fatto “evolvere” verso una soluzione globale ottimale.

- riproduzione tra individui - mutazione del patrimonio genetico

L’evoluzione (ottimizzazione) è il risultato di un insieme di processi di selezione che nel tempo tendono a scartare configurazioni meno adatte al problema da risolvere.

Ottimizzazione Genetica (GA) Terminologia

Individuo : costituisce una possibile soluzione al problema di ottimizzazione da risolvere. Ciascun individuo è determinato da una combinazione dei valori dei parametri reali da ottimizzare

Gene : codifica (ad esempio binaria) di un parametro da ottimizzare

Cromosoma : sequenza dei geni del singolo individuo. E’ costituito da una stringa di numeri (1 e 0 nel caso più comune di codifica binaria) che rappresenta quindi la codifica di una possibile soluzione del problema da risolvere

)( kB !

)](),...,(),([ 21 Kn BBB !!!=P { }K!!!! ,...,, 21=)( kB !

{ })(),...,(),( 21 Kn BBB !!!=p

np

Ottimizzazione Genetica (GA) Terminologia - cont.

Popolazione : è costituita da un insieme di individui (alcune possibili configurazioni nello spazio dei parametri) codificati sotto forma di cromosomi.

],...,,[ 21 NpppP = colonne: Nbit righe: N (# individui)

Evoluzione : iterazione del processo di ottimizzazione che permette, tramite una sequenza di operatori, di modificare il patrimonio genetico degli individui della popolazione

)1()( +! ii PP

Nbit

N

Riga =>antenna

Ottimizzazione Genetica (GA) Terminologia - cont.

Fitness : indicatore associato a ciascun individuo (set di parametri) in base alla bontà con cui è soluzione del problema. Dipende dai valori reali dei parametri che costituiscono il singolo individuo E’ la parte più delicata dell’algoritmo. Usualmente la Fitness totale è scritta come una media pesata delle fitness associate alle varie grandezze osservabili

)]([)(1

njj

J

jjnp gFwFF !p !

===

Set dei parametri dell’individuo n-mo

j-ma grandezza osservabile dell’individuo n-mo

Pesi 11

=!=

J

jjw

E’ l’unica grandezza che dipende dalla fisica del problema. Nel caso della antenne potrebbe essere un misura o un calcolo elettromagnetico

Operatori Genetici Selezione

Selezione : implementa il “principio di sopravvivenza” dei migliori che parteciperanno alla riproduzione. Viene costruita una nuova popolazione PS costituita da NS <N individui della popolazione P(i) )(iPN

N

i

Ns

N

iS

SS

O )()( PP !=

NNsel <

sO

Vi sono diversi criteri di selezione normalmente basati sul valore della fitness di ciascun individuo. Tra i più usati: il metodo della roulette - detta FT la somma dei fitness dei singoli individui. - si estrae un numero 0<s<1. - partendo dal primo individuo, si sommano le fitness relative Fp/FT finchè s< Fp/FT - l’ultimo individuo considerato è quello selezionato

Operatori Genetici Cross-Over

Cross-Over : E’ l’operatore di base per la creazione di due nuovi inividui a partire da una coppia di “genitori” della popolazione selezionata. Viene quindi creata una nuova popolazione PC

SSS N

iS

NC

N

iC

)()( POP !=

CO

1. Si scelgono a caso coppie di cromosomi genitori e si estrae a caso un numero 0<s<1. Se è minore della probabilita di cross-over (circa 0.8) si copia nei figli il DNA dei genitori. Diversamente:

2. Si scelgono a caso una o più posizioni nel cormosoma a partire dalle quali scambiare i bit dei genitori

Operatori Genetici Mutazione

Mutazione : Introduce aleatoriamente nuovo materiale genetico in alcuni cromosomi, originando individui con caratteristiche del tutto diverse. Permette di migliorare l’esplorazione dello spazio dei parametri

SSS N

iC

NM

N

iM

)()( POP !=

MO

1. Vengono scelti a caso dei cromosomi (in base alla probabilità di mutazione (0 - 0.5) 2. Nei cromosomi prescelti vengono cambiati a caso uno o piu bit

Elevate probabilità di mutazione rallentano la convergenza ma permettono di ottenere individui migliori

Nuova Popolazione

!"#

$%&=

'

+

SSSN

i

NNS

N

iMN

i O )()()1( PPP !"

Viene costruita la popolazione (i+1)-ma aggiungendo alla popolazione “mutata” i migliori elementi della popolazione precedente

Popolazione mutata

Ns<N individui

)(iMP

Popolazione iniziale

N individui

Popolazione evoluta

!

Solo i migliori (N-NS) individui )(iP

Riepilogo

Parametri da scegliere: •  numero di bit per la codifica •  probabilità di cross-over •  probabilità di mutazione •  fitness function: pesi wj •  numero di individui della popolazione (comunemente non inferiore al

numero complessivo di bit del cromosoma)

E’ richiesta una attenta calibrazione dell’ottimizzatore

(Calcolo elettromagnetico)

Criteri di arresto

Nel progredire dell’evoluzione, la fitness dell’individuo migliore tende ad aumentare e gli individui della popolazione tendono a diventare tutti uguali. L’evoluzione viene arrestata quando accade uno dei seguenti eventi:

a) viene raggiunto il massimo numero di iterazioni prefissato b) l’individuo migliore continua a rimanere lo stesso ( o il suo fittness non migliora) per piu’ di Nx iterazioni c) gli individui sono tutti uguali o hanno tutti lo stesso fitness

302 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 2, 2003

Gain-Optimized Self-Resonant Meander LineAntennas for RFID Applications

Gaetano Marrocco

Abstract—New meander line antennas with improved gain areproposed as low-profile self-resonant tags for application in passiveradio frequency identification. Antenna shape and size is optimizedby genetic algorithm taking into account the conductor losses. Ex-amples are presented for application at 869 MHz with antennas ofdifferent materials and sizes.Index Terms—Genetic algorithms, miniaturized antenna, small

antenna, transponders.

I. INTRODUCTION

RADIO frequency identification (RFID) of objects andpeople and remote control of devices has become very

popular in logistics, inventory management and bio-engi-neering applications [1]. Data are contactless transferred toa local querying system (reader) from a remote transponder(tag) including the antenna and a microchip transmitter. Asuitable antenna for the tag must have low cost, low profile,and especially small size, whereas the bandwidth requirement(a few kilohertz) is less critical. In passive tag systems, thequerying signal coming from the reader must have enoughpower to activate the tag microchip, perform data processingand transmit back a modulated string up to the required readingrange (typically 0.3–1 m). Since the maximum effectiveisotropic radiated power (EIRP) of the reader is constrained tolocal regulations, high-gain tags are required to increase thereading range. Additionally, to simplify the matching networkand obtain low-cost tags, the antenna should be self resonant.In the ultra-high frequency (UHF) band, especially below

1 GHz, meander line antennas (MLA) are an attractive choicefor the purpose of reducing the tag sizes. As proposed in [2],folding the elements in a meander produces a wire configura-tion with both capacitive and inductive reactance, which mutu-ally cancel. Resonances are therefore produced at much lowerfrequencies than in the case of straight wire antenna of the sameheight at the expense of narrow bandwidth and low gain, espe-cially when the antenna surface needs to be contained in a fewcentimeter-side square (less than to label small ob-jects).This paper discusses the design of gain-optimized MLA by

means of genetic algorithms (GA) and investigates on the in-fluence of the wire conductivity on the optimized shape of theantenna. Following preliminary results in [3], it will be shownthat, for a fixed height, a standard MLA with regular meanders

Manuscript received September 15, 2003; revised October 20, 2003.G. Marrocco is with the Dipartimento di Informatica, Sistemi e Produzione,

University of Roma “Tor Vergata,” Rome, Italy (e-mail: [email protected]).Digital Object Identifier 10.1109/LAWP.2003.822198

Fig. 1. Scheme for nonuniform with indication of parameters to beoptimized. Only half antenna is visible.

does not exhibit the optimum gain especially when the con-ductor losses can not be neglected and that the GA-optimizedshapes with lossless wires are greatly different from the opti-mized antenna involving real conductors. It will be also provedthat the optimized MLA shape converges to a top-loaded dipoleas the available height increases.

II. EVOLUTIONARY DESIGN OF MLAS

With reference to Fig. 1, the parameters of a symmetric rect-angular are: the number of turns, the length of thehorizontal ( and ) and vertical ( and ) segmentsof the th turn and the length of the central segments and. The commonly used configuration is the uniform MLA

(U-MLA), which is a particular geometry described by onlythree parameters— , , and —hereafterdenoted as U- . The most general symmetricalnonuniform is singled out by param-eters.Design curves for self-resonant U- can be easily ob-

tained by setting the height and then iteratively tuning thewidth to have the antenna at resonance. For example,Fig. 2 shows the tuning curves for copper wire U-MLAs wherethe method of moments [4] has been used for antenna modeling.As expected, higher order antennas (N 1) have a smaller res-onant width since the wire is folded along a greater number ofturns within a small area. Moreover, the gain becomes lower forincreasing but approaches the gain of a straight dipole as theheight increases.Up to the first antenna resonance, the currents on the adjacent

horizontal segments of the MLA have opposite phase. Thesetransmission line currents do not give valuable contribution

1536-1225/03$17.00 © 2003 IEEE

MARROCCO: GAIN-OPTIMIZED SELF-RESONANT MEANDER LINE ANTENNAS FOR RFID APPLICATIONS 303

Fig. 2. (a) Design curve for small copper U-MLA. (b) Maximum gain forresonant U-MLA versus the height H for different meanders.

to the radiated power, but they nevertheless produce losses.In particular, a lot of power is wasted close to the antennacenter, where the current reaches highest values. The radiationresistance is therefore mainly affected by the vertical segmentsand, hence, as shown in [5], by the antenna total vertical heightrelative to the resonant wavelength. On the contrary, the lossresistance is primarily determined by the wire diameter andby the total wire length. For a fixed maximum available area

, the optimum gain is therefore expected to beachieved with that antenna design having the highest radiationresistance and the smallest total wire length. The above opti-mization problem, involving a tradeoff between miniaturizationwith self resonance (long wire length) and loss minimization(short wire length), requires all the vertical and horizontalsegments to be independently designed and can be efficientlyhandled by the GA approach that has been widely used as anelectromagnetic design tool [6]. For the actual problem, thelength of each segment of the MLA is encoded into 7 bits andeach antenna is solved by the method of moment, provided thatthe minimum segment length is , which stillpermits to perform stable electromagnetic analysis. For eachth antenna of the GA population at the th generation, thefollowing fitness function is then evaluated:

(1)

where , , are the th antenna height, maximum gainand input reactance, respectively. Parameters have been chosenas: , (maximum gain of half-wavelength perfect conductor dipole), . The fitnessfunction converges to as the antenna gain equals ,the height equals and the antenna is at resonance.

III. MLA FOR 869-MHz APPLICATIONS

Some numerical experiments are reported at 869 MHz (atypical European frequency for RFID devices) for 0.1 mmwire MLA. A first optimized design set refers to antennaswith maximum allowed size , ,typical of book barcodes. The following wire conductivitieshave been considered: (perfect electric conductor),

(good conductor: copper) and(poor conductor: a metallo-organic conducting ink [7]). A rel-evant parameter for these antennas is the activation range

, which is the distance from thereader where the tag collects enough power to activate the mi-crochip transmitter. has been computed for an

Fig. 3. Optimization results at 869 MHz, for , with sizes (in cm) notexceeding . Copper U-MLA in the first column. Thecircles indicate the source points.

Fig. 4. Optimization results (sizes in cm) at 869 MHz for .

(highest value allowed by European regulations) and chip inputpower [1].Optimization results, with weight , ,

, are shown in Fig. 3 for antennas and inFig. 4 for antennas. The copper U-MLAs obtainedby curves of Fig. 2 are also reported for reference. It can benoted that the optimum usage of the wire current and the spaceoccupation is different and it is strongly related to the wireconductivity as it is visible from the different shapes. Loss-less-wire antennas are the shortest ones and exhibit nearly thesame gain as a half wavelength dipole, with size reduction factor

of about 23%. Lossy conductorMLAs tend to fill all the available area and their horizontal seg-ments originating transmission line stubs are shorter close tothe source in order to minimize the transmission line current,which does not contribute to radiation. In comparison with theU-MLA of the same order, the optimized copper antennas showa gain enhancement of 0.3 dB and 1 dBand consequently the reading range has been improved by about3 cm and 5 cm , respectively. The total wirelength is similar for NU-MLA of the same order and rangesbetween 0.6 and 0.67 . The input impedance isfor the copper-wire antenna and it is close to the load impedanceof commonly used microchip transmitters [1]. Although not anoptimization parameter, the bandwidth has been improved byabout 10% in moving from copper U- to the optimizedNU- since the latter tends to occupy more space.To discuss the performances of optimized self-resonant

copper MLAs of a same size, two new design sets have beenproduced (Fig. 5) having fixed the available size to that of twoU- with and

, respectively. The following optimizationweights in (1) favor gain maximization: , ,

. A top-loaded dipole (“C” antenna) tuned by

304 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 2, 2003

Fig. 5. Optimized antennas with the same size (W,H in cm) of two U- . A top loaded dipole is also shown for comparison. and are radiation andloss resistances, respectively. The gain of the two U- are 0.30 dB and 0.81 dB .

Fig. 6. Copper-wire GA-optimized NU-MLA antennas for different maximum available areas (dashed shapes). The gain of the correspondingU-MLA antennas are also reported.

some foldings, which is a simple way to miniaturize a dipole,has been also considered for reference. While no sensibleimprovement over the U- has been appreciated inoptimized NU- , better performances have been obtainedwith NU- . It is possible to verify that the antenna lossincreases with the wire length while the radiation resistance ishigher in those designs with prevalent vertical shape near thevoltage source, where the current peaks (e.g. the NU-and the top-loaded dipole). Therefore, having fixed the antennaheight, the radiation resistance depends on “how effective”is the current usage in the wire path. The highest radiationresistance is that of the top-loaded dipole whose central verticalsegment has the longest length which is permitted in theavailable space. However, the wire length of this antenna and,hence, the loss resistance are the highest. Accordingly, the gainis lower than that of the U- . The best antennas, in thesense of (1), are the NU- which show the best tradeoffbetween wire length minimization and best current utilization.The gain improvement over the two U- has been of about0.2 dB in both the considered sizes. It is interesting to note thatabove results partially disagree with what stated in [5], e.g., thatthe radiation resistances of shaped dipoles of fixed height aresimilar, independent of antenna geometry and total wire length.

Nevertheless, no information about the antenna horizontal sizeis given in that paper.Finally, to further investigate on the ability of the GA opti-

mization to enhance MLA gain, some more designs have beenperformed for different maximum size rangingfrom to . As depicted in Fig. 6, the GA-op-timization is as much effective over the uniform MLA as themaximum available area decreases. For sizes smaller that

a antenna has been considered to obtain selfresonance. As the available space increases, the antenna tuningrequires shorter horizontal segments, mainly localized at thewire’s ends to minimize losses. The MLAs therefore convergeto a single-fold top-loaded dipole which, as discussed above,has the best current utilization.

IV. CONCLUSION

Miniaturized self-resonant meander-line antennas have beendesigned by GA optimization. It has been numerically experi-enced that: 1) the optimum usage of the wire current is stronglyrelated to the wire loss strength; 2) for a fixed maximumavailable space, optimized NU-MLAs perform better thansame-order U-MLAs (up to 1-dB gain improvement for copper

MARROCCO: GAIN-OPTIMIZED SELF-RESONANT MEANDER LINE ANTENNAS FOR RFID APPLICATIONS 305

wire); 3) same order MLAs (uniform and optimized) of samesize exhibit similar performances, while gain enhancement ispossible by adding further degrees of freedom, e.g. by higherorder NU-MLAs; and 4) in all the considered cases, optimizedMLAs perform better than top-loaded dipoles having the samesize. This class of antennas may be useful as an extremely smalltag for radiofrequency identification applications involving areading range of about 0.5 m and with a bandwidth of a fewmegahertz.

ACKNOWLEDGMENT

The author would like to thank the reviewers for suggestionsand A. Fonte, Prof. F. Bardati, and M. Sabbadini, for supportand helpful discussions.

REFERENCES[1] K. Finkenzeller, RFID Handbook. London, U.K.: Wiley, 1999.[2] T. J. Warnagiris and T. J. Minardo, “Performance of a meandered line

as electrically small transmitting antenna,” IEEE Trans. Antennas Prop-agat., vol. 46, pp. 1797–1876, Dec. 1998.

[3] G. Marrocco, A. Fonte, and F. Bardati, “Evolutionary design of minia-turized meander-line antennas for RFID applications,” in Proc. IEEEAP-S Int. Symp., vol. 2, San Antonio, TX, 2002, pp. 362–365.

[4] G. J. Burke and A. J. Poggio, “Numerical Electromagnetic Code (NEC)Method of Moments,” Naval Ocean Syst. Ctr., CA, Tech. Doc. 116, pt.I and II, 1980.

[5] S. Best, “On the resonant properties of the Koch fractal and other wiremonopole antennas,” IEEE Trans. Antennas Wireless Propagat. Lett.,vol. 1, pp. 74–76, 2002.

[6] D. S. Weile and E. Michielsen, “Genetic algorithm optimization appliedto electromagnetics: a review,” IEEE Trans. Antennas Propagat., vol.45, pp. 343–353, Mar. 1997.

[7] Inks for RFID Antennae. Parmod VLT Tech. Rep. [Online]. Available:www.parelecusa.com/parelec3/product/images/RFID.pfd