The Genetic Algorithm

Transcription

1 The Genetic Algorithm The Genetic Algorithm, (GA) is finding increasing applications in electromagnetics including antenna design. In this lesson we will learn about some 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: Lesson Assignment 8: Verify the author s solution of Example 7.6, page 361, of Stutzman and Thiele, for the Taylor line source design. For reading about the GA please refer to: R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms (New York: John Wiley and Sons, Inc., 1998). J. M. Johnson and Y. Rahmat-Samii, Genetic Algorithms in Engineering Electromagnetics IEEE Antennas and Propagation Magazine 39, no. 4 (Aug. 1997): Also for more applications refer to: E. Alshuler and D. S. Linden, Wire Antenna Design Using Genetic Algorithms IEEE Antennas and Propagation Magazine, 39, no. 2 (April 1997): Click icon for explanation by Dr. Ferraro To stop playing the sound file, press Escape. EE 538: Antenna Engineering--Module 1, Lesson 9 1

2 The Genetic Algorithm As a conceptual application, consider the design of a monopole antenna loaded with parallel RLC circuits for the purpose of increasing the bandwidth of the antenna. The monopole consists of 4 sections of wire: l 1, l, 2, l 3 l 4 l 1 θ C 1, R 1, L 1 l 2 C 2, R 2, L 2 l 3 C 3, R 3, L 3 l 4 Figure There are 3 parallel RLC circuits loading the monopole. The goal of the synthesis is to determine the values of the 13 parameters of lengths and component values to meet a desired performance of gain of the antenna over a certain bandwidth. As you can see, none of the previous synthesis methods can tackle the immense job of selecting the parameters. If we talk in terms of generalities, we will lose track of what we are trying to accomplish with the GA in synthesizing an antenna. I propose a little conceptual application just for the sake of discussion. I show, in the figure, a monopole antenna above an infinite perfectly conducting ground plane. It is basically a monopole except that there are 3 parallel tuned circuits. It is well known that by putting parallel tuned circuits into antenna monopoles, you can increase the useful bandwidth of the antenna. EE 538: Antenna Engineering--Module 1, Lesson 9 2

3 The Genetic Algorithm 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 = 90 o ) to be greater than -5 dbi with a goal of 0 dbi whose performance is to be achieved by loading the monopole with lumped components. 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 Figure EE 538: Antenna Engineering--Module 1, Lesson 9 3

4 Genes and Chromosomes The parameters can be coded as binary values into the genes so that the chromosomes might look like the following: l 1 L 1 l 2 C 2 C 1 R 1 R 2 L Figure The number of bits to represent a parameter depends upon the desired resolution and range of that parameter. For instance, 1111 for R 1 could be picked to represent 10 ohms. Other binary bits would give values to R less than 10 ohms. An antenna having the above chromosome values may or may not meet the stated design goal. It has to be computed using the numerical electromagnetic code which is discussed in Module Three. This is basically an analysis tool. In the meantime, suppose we are able to compute the gain, G, of this monopole at any frequency, 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 of chromosomes 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 a method of measuring fitness since in the search strategy we will discard unfit candidates. The coding I have shown in Figure certainly represents some monopole, but it may not meet our design goal. To determine if it meets our design goal, we have 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 this for any number of frequencies on angle, θ, that we desire. By using a random process, we can probably generate many strings, looking like Figure We can therefore, generate a population of chromosomes, each containing coding representing a particular type of loaded monopole. 1 EE 538: Antenna Engineering--Module 1, Lesson 9 4

5 The Objective Function to Measure Fitness To meet the design goals of this antenna as outlined, one possible fitness function might be the following: F G = N i = 1 2 ( G( fi, θ 0 ) G0) (1.9.1) where G(fi, θ0 ) is the antenna gain at frequency f i and at elevation angle θ 0 G0 is the desired goal. The smaller F G, the more fit is the loaded monopole towards the design goal. The f i are selected from the required band of frequencies and should be dense enough to properly represent the design goal. To measure fitness, we come up with what is called objective function or a fitness function. For example, and this is only an example, we can define a fitness function F G as being the summation of the square of the differences between the gain of the antenna at frequency f i, and angle θ 0 with reference to G 0 which is the desired goal. This is summed up over all the frequencies in the band 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 the required bandwidth and should be dense enough to properly represent the design goal over these frequencies. Because we now have a large population of chromosomes looking like Figure 1.9.3, we should evaluate the fitness over the frequency band for each one of the chromosomes. That would require an analysis be made on a large population of monopoles having the given parameters. EE 538: Antenna Engineering--Module 1, Lesson 9 5

6 The Objective Function to Measure Fitness Usually a simple fitness function as (1.9.1) is not satisfactory and other constraints are added. For instance, to ensure smoothness of the gain versus frequency one could examine the fitness function: F S = N 1 i = 1 ( G ( f, ) G ( f θ )) 2 i + 1 θ 0 i, 0 (1.9.2) The smaller F S is, then the more smoothly the gain varies with frequency. The final fitness function might now be F = F S + F G (1.9.3) The object of the GA is to arrive at coded chromosomes, Figure 1.9.3, that minimize Equation (1.9.3). This is done by the process of selection, reproduction and generation replacements. EE 538: Antenna Engineering--Module 1, Lesson 9 6

7 Selection, Reproduction and Generation Replacements We briefly outline some of the concepts and you should read the assigned 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 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 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. Under the concept of selection, we have a population of hundreds of chromosomes like the chromosome Figure We want to keep the most fit. How do we select the most fit from this population? There are many ways, as you will see by reading the articles. One way is to define a minimum level of fitness, which is set by the designer, and throw away all chromosomes whose fitness is less than this minimum; therefore, we have retained those chromosomes containing those genes which lead to a more fit antenna. A better solution is to use a roulette wheel approach, and this is clearly described in the articles. EE 538: Antenna Engineering--Module 1, Lesson 9 7

8 The Genetic Algorithm In summary we can state the following: The GA is a search procedure that iteratively leads a population of randomly selected design parameters to an optimal solution. The success of the method for antenna design is clearly demonstrated in the two suggested readings. You can search the Web for numerous links to the genetic algorithm such as: What this operation does is search through, in an iterative manner, a population of randomly selected design parameters, and arrives at an optimal solution. This is very much like survival of the fittest, or, in the jungle, the weak die and the strong survive; the most fit survive and the most fit create offspring which carry through their genes the fitness characteristics 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 fit antenna. The success of the method for antenna design is clearly demonstrated in the suggested readings. Several interesting examples are discussed, which I think you will enjoy looking at. You can also search on the Web, for there are numerous links to the GA and you can try the link that I have indicated here. It, too, is a tutorial, which could supplement these notes and the papers I have suggested you read. EE 538: Antenna Engineering--Module 1, Lesson 9 8

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

10 Figure EE 538: Antenna Engineering--Module 1, Lesson 9 10

11 Module 1, Lesson 9--The Genetic Algorithm EE 538: Antenna Engineering--Module 1, Lesson 9 11