Intelligent Control Methods Lecture 12: Genetic Algorithms Slovak University of Technology Faculty of Material Science and Technology in Trnava

2 Genetics algorithms Optimising method Template: Nature Generation Quality

3 GA imitate natural evolution: Natural selection:  Individuals input to reproduction process with probability, which depends on their adaptation to natural conditions. Random genetic drift:  Accidental events occur in the process (mutation of genetic material, random death,...). Reproductive process:  Descendants inherent properties (genetic information) of their parents.

4 Main idea: A solution of a task is an individual in population. So as the individual develops and adapts to nature in more generations, so that the solution of the task approaches to the optimum. Applicable on different kinds of problems, primarily on optimization. It is not necessary to know the mathematical model of solved problem. The only condition is the possibility of funded solution assessment. The method is competent to escape from the local extreme and to proceed to global one. GA works parallel. GA master tens as far as hundreds of parameters.

5 Basic terms: String (chromosome, individual)  A sequence of bits, numbers or symbols, which represent properties Genes:  Elements of string, i.e. properties Generation:  A group of strings of defined number, which responds to 1 iteration of evolutional (= solving) process. Base operations:  cross-over  mutation

6 Cross-over operation: Operation, where two parent strings split up in the same random position and reciprocate genes behind the splitting point. Parents: Descendants: a1 a2 a3 a4 a5 a6a1 a2 a3 a4 b5 b6 b1 b2 b3 b4 b5 b6b1 b2 b3 b4 a5 a6 => More-points cross-over: b1 b2 a3 a4 b5 b6 a1 a2 b3 b4 a5 a6

7 Mutation: Operation, where one or more genes accidental change their values. The new values are random, but from allowed area. a1 a2 a3 a4 a5 a6  a1 b a3 a4 a5 a6 (a2 min < b < a2 max )

8 Genetic algorithms: Initialization of population Objective function (fitness) End ? Solution Cross-over Mutation SelectionNew generation Y n N a b p c n-a a pmpm

9 GA example: 5 digits should be nicked (12345) String consists of 5 genes with values Objective function: The number of nicked digits Population: 8 individuals Selection:  2 best individuals go over direct without changes to positions 1 and 2 into new generation.  3 best and 3 accidentally chosen individuals go over to positions 3 – 8. The neighboring couples cross-over with probability p c = 1 and after that the descendants mutate with probability p m = 0.1. To derive 2 next generations.

10 GA example: 5 digits should be nicked (12345) Iteration: Selektion Cross-over Mutation Generation N+1 Generation N

11 GA application example: Static optimization of technological process (1) 40<T 1 <4674<T 2 <7620<T 3 <25 T 0 = 20T1T1 T2T2 T3T3 u 1 = ? u 2 = ? u 3 = ? Chemical technological process  3 consecutive operations  for each operation should be ensured by heating or by cooling (actuating signal u i [kW]) specified temperature T 1 = -1,5 u ,6 u 1 + T 0 T 2 = -0,25 u ,0 u 2 + T 1 T 3 = -0,31 u ,56 u 3 + T 2 N 1 = 1,1 u ,68 u 1 N 2 = 4,33 u ,75 u 2 N 3 = -0,88 u ,8 u 3 N = N 1 + N 2 + N 3 —> min [Sk/m 3 ]

12 GA application example: Static optimization of technological process (2) Chromozome:[u 1, u 2, u 3 ] After 500 iterations:  u 1 = 1,47 kW u 2 = 2,63 kW u 3 = -3,64 kW  T 1 = 45,7 o C T 2 = 75,0 o C T 3 = 24,7 o C

13 GA application example: Static optimization of technological process (3) Costs according to iterations:

14 GA application example: Control loop synthesis (1): PI controller: PIprocess w + e u y - The goal of GA: To design P and I so, that the required control quality (given by the criterion of absolute regulation area) will be obtained.

15 GA application example: Control loop synthesis (2):  Strings in a population: [P,I]  Allowed intervals for P and I must be specified (e.g. from stability conditions) before the first population generation.  The criterion evaluates for closed control loop.

16 GA application example: Control loop synthesis (3): 1010 g = 1 g = 10 g = 50

17 Remarks to selection: It is necessary to ensure, that the better individuals survive with greater probability Elitism  If the best individual goes over to next generation (without changes), the best solution in the new generation can not be worse Base selection types:  roulette: probability p(h) of the survival of individual h with value F(h) is: p(h) = F(h)/  F(h i )  tournament: random pairs are chosen, the probability of survival of the better individual is p, the survival probability of the worse one is 1-p  rank: Individuals are ordered by objective function values. The probability of survival depends on the rank in the sequence.

18 Effectivity of GA (own trials with the best solution by different input parameters): Best results: p mut = 0.1, n = 80, direct: 40%, best strings to genetic operations: %. In literature: p mut = , n = , direct: ? %, best strings to genetic operations: ? %.