A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design


 Damon Townsend
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1 A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA Abstract: This paper describes a geetic algorithm (GA) that optimizes chromosomes cotaiig a mix of cotiuous ad biary ecoded variables. Uiform crossover from the biary GA ad a mutatio rate aki to that of the cotiuous GA are fudametal parts of the algorithm. Parameter selectio as well as a applicatio to atea desig are preseted. Keywords: Geetic algorithm, optimizatio, microstrip atea 1. Itroductio Most optimizatio problems work with cotiuous values. If the variables have iteger values, the special algorithms must be used. If cotiuous values ad itegers are part of the same optimizatio problem, the it is kow as mixed iteger optimizatio, ad popular approaches, such as brach ad boud, are used. The geetic algorithm ad particle swarm optimizatio have also bee used for mixed iteger optimizatio. This paper presets a GA formulatio that simultaeously works with cotiuous, iteger, ad biary values i oe chromosome. Operators for his ew algorithm is described i the ext sectio ad applicatios to a circularly polarized patch atea desigs are show i the followig sectio. The beauty of this algorithm is that it ca optimize o ay type of variable value without a chage i the algorithm.. Mixed Iteger/Biary GA The iitial populatio matrix of the GA is give by a a a a1 a P ar1 a C RC (1) 591
2 where 0 a rc 1. Each row is a chromosome ad the values are created by a uiform radom umber geerator. A cost fuctio evaluates each chromosome ad returs a cost. m1 m mc cost f a, a,, a () Iside the cost fuctio, the variables may be coverted to a ew cotiuous rage by or coverted to a iteger by x x x a x (3) max mi m mi x roudup x x a x (4) max mi m mi where roudup rouds to the ext highest iteger ad x values are itegers. I some cases, the value is coverted to biary. Oe possibility is to roud the value. x m roud a (5) Aother possibility is to quatize the value. x m quatize a (6) The cost fuctio does all the scalig, quatizig, ad roudig, so the GA ca operate idepedet of the type of variable. Uiform crossover works well for the biary GA, so it is implemeted here. Two parets are selected ad a radom biary mask is created. If the mask has a oe i the colum, the the offsprig receives the variable value i paret#1. If it has a zero, the the offsprig receives the variable value i paret#. paret #1 a a a a a a a a a m1 m m3 m4 m5 m6 m7 m8 m9 paret # a a a a a a a a a mask offsprig a a a a a a a a a m1 m3 m4 5 m6 7 8 m9 (7) This type of crossover results i a diversity of values if the values are biary, but oly iterchages values betwee chromosomes if the values are iteger or cotiuous. Cosequetly, the mutatio must be resposible for creatig diversity withi the populatio for cotiuous ad iteger values. Oe possible approach to mutatio is to radomly select variables i the populatio ad replace them with uiform radom values. The mutated chromosome ( chrom ) is created from the selected chromosome (chrom ) by chrom a a a a a a a a a (8) r1 r r3 r4 r5 r6 r7 r8 r9 59
3 where the primed values are uiform radom umbers. Aother approach is to add a radom correctio factor. The correctio factor may be created by multiplyig each elemet withi a chromosome by a radom umber ( 1 rm 1) ad multiplyig the etire chromosome by a mutatio factor ( 0 r 1). c chrom r r1ar1 rar r3ar3 r4ar4 r5ar5 r6ar6 r7ar7 r8ar8 r9ar9 Now, the mutated chromosome is give by (9) c (10) chrom rem chrom chrom where rem is the remaider fuctio (digits to the left of the decimal poit are dropped). This algorithm was tested o two cost fuctios to try to determie a appropriate populatio size ad r. I both cases, the GA quit after 400 fuctio evaluatios ad reported the best results. The first test fuctios is f 1 6 x x (11) with a miimum of zero at x 0. The results show i Figure 1 were averaged over 100 rus for populatio sizes betwee 8 ad 96 ad mutatio rates betwee 0.01 ad 0.3. The best results occurred whe the populatio size was 8 ad the mutatio rate was Figure 1. GA results for f 1 averaged over 100 rus for various populatio sizes ad mutatio rates. 593
4 The secod test fuctios is cos f x x x (1) with a miimum of zero at x 0. The results show i Figure were averaged over 500 rus for populatio sizes betwee 8 ad 96 ad mutatio rates betwee ad 0.3. The best results occurred whe the populatio size was 40 ad the mutatio rate was Figure. GA results for f averaged over 500 rus for various populatio sizes ad mutatio rates. 3. Applicatios to Atea Desig The goal is to desig a rectagular patch for circular polarizatio at 10 GHz usig FEKO. Iput variables for the cost fuctio are x, y positio of probe feed L, L patch legth i x ad y directios x y h substrate thickess (either 1.575mm or 3.15mm) relative dielectric costat of the substrate (either. or.33) r The cost fuctio returs the followig value 594
5 E E cost max, E E, s E E 11 (13) The first two quatities i (13) equal zero for circular polarizatio, ad s 11 is zero for a perfect match to 50. Whe the patch is circularly polarized ad perfectly matched, the cost = 0. The optimizatio was performed usig the best populatio size ad mutatio rate for both test fuctios i the previous sectio. Results were averaged over 5 rus ad are summarized i Table 1. Figure 3 shows the best of five idepedet results for a populatio size of 8 ad r 0.10 ad for a populatio size of 40 ad r Sice the optimizatios were termiated after 400 fuctio calls, the umber of geeratios to covergece is differet for differet populatio sizes. Table 1. Cost statistics after 5 idepedet rus. maximum miimum average populatio size = 8 mutatio rate = populatio size = 40 mutatio rate = NelderMead dowhill simplex algorithm Figure 3. Covergece results for the two best rus with a populatio size of 40 ad r 0.01, ad a populatio size of 8 ad r The top two dashed curves are the populatio average ad the bottom two curves are the best populatio cost. The GA was termiated after 400 fuctio evaluatios. 595
6 The best chromosome traslates to the optimum desig values of x, y 1.6 mm,3.451mm Lx, Ly mm,10.138mm h 1.575mm. r The resultig patch is righthad elliptically polarized with a axial ratio of 1.09 ad s Coclusios May atea desigs have variables with iteger values ad with cotiuous values. This paper preseted a versio of a GA that works with values betwee zero ad oe ad uses biary uiform crossover ad cotiuous mutatio. The patch atea desig is a multiobjective optimizatio usig cotiuous ad biary variables. Refereces [1] Y. Li ad M. Ge, "Noliear mixed iteger programmig problems usig geetic algorithm ad pealty fuctio," [] Z. Gaig, "Costraied optimal power flow by mixediteger particle swarm optimizatio," [3] R.L. Haupt ad Sue Elle Haupt, Practical Geetic Algorithms, d editio, New York: Joh Wiley & Sos,
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