the genetic algorithm - penn state college of engineering

EE 538: Antenna Engineering--Module 1, Lesson 9 1

Module 1, Lesson 9--The Genetic Algorithm

The Genetic Algorithm

For reading about the GA please refer to:

Practical Genetic Algorithms R. L. Haupt and S. E. Haupt, (New York: John Wiley and Sons, Inc., 1998).

“Genetic Algorithms

in Engineering ElectromagneticsJ. M. Johnson and Y. Rahmat-Samii,

IEEE Antennas and PropagationMagazine 39, no. 4 (Aug. 1997): 7-25.

Also for more applications refer to:“Wire Antenna Design Using Genetic Algorithms” E. Alshuler and D. S. Linden,

IEEE Antennas and Propagation Magazine , 39, no. 2 (April 1997): 33-43.

Lesson Assignment 8:Verify the author’s solution of Example 7.6, page 361, ofStutzman and Thiele, for the Taylor line source design.

To stop playing the sound file, press Escape.

The Genetic Algorithm, (GA) is finding increasing applications in electromagnetics including antenna design. In this lesson we will learn aboutsome of these techniques so you are aware of this emerging area. There is a lot of literature available and in this lesson you will be required to read 2 of these important tutorials. The mathematics associated with the GA is relatively simple compared to our previous lessons on synthesis. Before going any further, I am assigning the last of the synthesis assignments:

”

Click icon for explanation by Dr. Ferraro

Kathie Merrill



As a conceptual application, consider the design of a monopole antenna loadedwith parallel RLC circuits for the purpose of increasing the bandwidth of theantenna.

The monopole consists of 4

sections of wire:l1

l2

l3

l4

θC1, R1, L1

C2, R2, L2

C3, R3, L3

There are 3 parallel RLC circuits loading the monopole. The goal of thesynthesis is to determine the values of the 13 parameters of lengths and componentvalues to meet a desired performance of gain of the antenna over a certainbandwidth.

As you can see, none of the previous synthesis methods can tackle the immensejob of selecting the parameters.

432 llll ,,,1

Figure 1.9.1

If we talk in terms of generalities, we will lose track of what we are tryingto accomplish with the GA in synthesizing an antenna. I propose a littleconceptual application just for the sake of discussion. I show, in thefigure, a monopole antenna above an infinite perfectly conducting groundplane. It is basically a monopole except that there are 3 parallel tunedcircuits. It is well known that by putting parallel tuned circuits intoantenna monopoles, you can increase the useful bandwidth of theantenna.




The specific goal might be to design a monopole

• whose total length is 2 meters

• which operates from 50 to 500 MHz

• which has a gain at the horizon (θ = 90o) to be greater than -5dBi with a goal of 0 dBi

• whose performance is to be achieved by loading the monopole withlumped components.

Figure 1.9.2


The concept of the genetic algorithm applied to this problem uses a search strategy to find the unknown parameters that pattern the natural selection and evolution of Darwin. We therefore code the unknown parameters into genes and create a chromosome by stringing the genes together as shown in Figure 1.9.2.

Kathie Merrill


Genes and Chromosomes

The parameters can be coded as binary values into the genes so that thechromosomes might look like the following:

The number of bits to represent a parameter depends upon the desiredresolution and range of that parameter. For instance, 1111 for R

1 could be picked

1

An antenna having the above chromosome values may or may not meet the stateddesign goal. It has to be computed using the numerical electromagnetic codewhich is discussed in Module Three. This is basically an analysis tool. In themeantime, suppose we are able to compute the gain, G, of this monopole at anyfrequency, f, and angle, θ, given the parameters of length and component values.

Using a random process of “flipping” a coin, one could generate a population ofchromosomes and each would contain the coding of a particular loaded monopole.Some might be fit or unfit towards meeting the desired goal. Thus we need amethod of measuring fitness since in the search strategy we will discard unfitcandidates.

The coding I have shown in Figure 1.9.3 certainly represents some monopole,but it may not meet our design goal. To determine if it meets our design goal, wehave to do some analysis on this antenna. As we will later see, deeper into Module Three of this course, we are going to study the numerical electromagnetic code. This is an analysis tool that would allow us to compute for this monopole, having the parameters outlined in the chromosome, the gain, the impedance, the standing wave ration, and many other parameters. We would be able to do thisfor any number of frequencies on angle, , that we desire. By using a random process, we can probably generate many strings, looking like Figure 1.9.3.We can therefore, generate a population of chromosomes, each containing codingrepresenting a particular type of loaded monopole.


Figure 1.9.3

l 1 l 2 C2C1 R2R1 L2L1

0011 1010 0111 1000 0101 0001 1001 1100

to represent 10 ohms. Other binary bits would give values to R less than 10 ohms.

θ


The Objective Function to Measure Fitness

∑=

−=N

iiG GfGF

1

200 )),(( θ (1.9.1)

where ),G(f 0i θ is the antenna gain at frequency fi and at elevation angle θ0

0G is the desired goal.

• The smaller FG

, the more fit is the loaded monopole towards the designgoal.

• The fi are selected from the required band of frequencies and should be

dense enough to properly represent the design goal.


To meet the design goals of this antenna as outlined, one possible fitness function might be the following:

To measure fitness, we come up with what is called objective function or afitness function. For example, and this is only an example, we can define afitness function F

G as being the summation of the square of the differences

between the gain of the antenna at frequency fi, and angle θ0 with reference to

G0 which is the desired goal. This is summed up over all the frequencies in theband of interest. The smaller F

G, then the more fit the loaded monopole is

towards meeting our design goal. The frequencies have to be selected from therequired bandwidth and should be dense enough to properly represent thedesign goal over these frequencies. Because we now have a large population ofchromosomes looking like Figure 1.9.3, we should evaluate the fitness over thefrequency band for each one of the chromosomes. That would require ananalysis be made on a large population of monopoles having the givenparameters.

Kathie Merrill

Kathie Merrill


The Objective Function to Measure Fitness

Usually a simple fitness function as (1.9.1) is not satisfactory and other constraintsare added. For instance, to ensure smoothness of the gain versus frequency one couldexamine the fitness function:

( ) ( )( )∑−

=+ −=

1

1

2001 ,,

N

iiiS fGfGF θθ (1.9.2)

The smaller FS

is, then the more smoothly the gain varies with frequency.

The final fitness function might now be

The object of the GA is to arrive at coded chromosomes, Figure 1.9.3, that minimizeEquation (1.9.3). This is done by the process of selection, reproduction andgeneration replacements.

(1.9.3)F = FS

+ FG



Selection, Reproduction and GenerationReplacements

We briefly outline some of the concepts and you should read the assigned articles.

Under the concept of selection, we have a population of hundreds ofchromosomes like the chromosome Figure 1.9.3. We want to keep themost fit. How do we select the most fit from this population? There aremany ways, as you will see by reading the articles. One way is to define aminimum level of fitness, which is set by the designer, and throw away allchromosomes whose fitness is less than this minimum; therefore, we haveretained those chromosomes containing those genes which lead to a morefit antenna. A better solution is to use a roulette wheel approach, and thisis clearly described in the articles.


SELECTION: From a population of chromosomes, like Figure 1.9.3, keep the most fit. The minimum fitness is set by the designer. A better selection process uses the roulette wheel approach described in the assigned papers.

REPRODUCTION:

From this population, pairs of chromosomes (parents) are selected to reproduce and yield 2 new chromosomes (children) by the crossover operation. Here some genes from parents 1 and 2 are copied to children 1 and 2. Figure 1.9.4 shows the crossover operation. On a small percentage of the children, mutation is introduced. A randomly selected element is changed (i.e., 0 to 1 or 1 to 0). Figure 1.9.5 shows the mutation operation for bit a

9.

GENERATION REPLACEMENT:

The new generation (children) replaces the previous generation and the process of selection, reproduction, and generation replacement is repeated. See the flow chart, Figure 1.9.6, for the cycle.

Kathie Merrill



In summary we can state the following:

The success of the method for antenna design is clearly demonstrated in the twosuggested readings.

You can search the Web for numerous links to the genetic algorithm such as:

http://cs.felk.cvut.cz/~xobitko/ga/

The GA is a search procedure that iteratively leads apopulation of randomly selected design

parameters to an optimal solution.

What this operation does is search through, in an iterative manner, apopulation of randomly selected design parameters, and arrives at anoptimal solution. This is very much like survival of the fittest, or, in thejungle, the weak die and the strong survive; the most fit survive and themost fit create offspring which carry through their genes the fitnesscharacteristics of the parents. This method, as shown on the flow chart,Figure 1.9.6, if repeated many, many times, absolutely leads to a more fitantenna. The success of the method for antenna design is clearlydemonstrated in the suggested readings. Several interesting examples arediscussed, which I think you will enjoy looking at. You can also search onthe Web, for there are numerous links to the GA and you can try the linkthat I have indicated here. It, too, is a tutorial, which couldsupplement these notes and the papers I have suggested you read.


http://cs.felk.cvut.cz/~xobitko/ga/



Figure 1.9.4


Figure 1.9.5


the genetic algorithm - penn state college of engineering

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