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Dvouúrovňová evoluční optimalizace regulátorů |Bio-inspirované metody a jejich využití Two level Evolutionary Optimization of |Controllers Basic Principles of Transplant evolution Pave l Ošmera Institute of Automation and Computer Science Brno University of Technology, Technická 2, Brno, Czech Republic

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Introduction We are trying to piece together the knowledge of evolution with the help of biology, informatics and physics to create a complex evolutionary structure. It can speed up the creation of optimization algorithms with high quality features. The emergent properties of complex adaptive systems are presented. One of the most highly developed skills in contemporary Western civilization is the capability to split up problems into their smallest possible components. But we often forget to put the pieces together again. In this way we can ignore the complex interaction between our problems and the rest of the universe.

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Basic strategy of the evolutionary optimization

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Classical and evolutionary design Example: antenna designs NASA Ames (http://ic.arc.nasa.gov/projects/esg/research/antenna.htm)http://ic.arc.nasa.gov/projects/esg/research/antenna.htm Classical method Evolutionary approach all solutions

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Grammatical evolution The PGE is based on the grammatical evolution GE [1], where BNF grammars consist of terminals and non-terminals. Terminals are items, which can appear in the language. Non-terminals can be expanded into one or more terminals and non-terminals. Grammar is represented by the tuple {N,T,P,S}, where N is the set of non-terminals, T the set of terminals, P a set of production rules which map the elements of N to T, and S is a start symbol which is a member of N. For example, below is the BNF used for our problem: N = {expr, fnc} T = {sin, cos, +, -, /, *, X, 1, 2, 3, 4, 5, 6, 7, 8, 9} S = and P can be represented as 4 production rules: 1. := 2. := sin cos + * - U- 3. := X 4. := 0,1,2,3,4,5,6,7,8,9

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Forward processing (classical approach) 0 I N 0 Z D 0 N (NON) 1 Z ZZ 1 N R 0 D 0 2 N Z 1 D 1 0 R x 2 D 2 1 R Z 3 D 3 2 R Z.Z 4 D 4 3 R N^N 5 D 5 0 O + 6 D 6 1 O - 7 D 7 2 O * 8 D 8 3 O / 9 D 9 phenotype tree: translation table DNA sequence (genotype): The symbol "I" is always root. The rules are processed from the left to the right. The rules are applied on the leftmost non-terminal symbol. The rule chosen is determined by the result of the modulo operation: 13 mod 1 = 06 mod 3 = 02 mod 3 = 25 mod 2 = 18 mod 2 = 017 mod 10 = 7 I N NON ZON ZZON DZON 7 7DON N +R 74+x 2 mod 2 = 04 mod 10 = 412 mod 4 = 04 mod 3 = 14 mod 4 = 0

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Crossover When using grammatical evolution the resulting phenotype coded by one gene depends on the value of the gene and on its context. If a chromosome is crossed at random point, it is very probable that the context of the genes in second part will change. This way crossover causes destruction of the phenotype, because the newly added parts code different phenotype than in the original individual. This behavior can be eliminated using a block marking system. Crossover is then performed as an exchange of blocks. The crossover is made always in an even number of genes, where the odd gene must be BB gene and even must be EB gene. Starting BB gene is presently chosen randomly; the first gene is excluded because it encapsulates (together with the last used gene) the whole individual. The operation takes two parent chromosomes and the result is always two child chromosomes. It is also possible to combine the same individuals, while the resulting child chromosomes can be entirely different. Given the parents: 1) cos( x + 2 ) + sin( x * 3 ) 2) cos( x + 2 ) + sin( x * 3 ) The operation can produce children: 3) cos( sin( x * 3 ) + 2 ) + sin( x * 3 ) 4) cos( x + 2 ) + x This crossover method works similar to direct combining of phenotype trees, however this method works purely on the chromosome. Therefore phenotype and genotype are still separated. The result is a chromosome, which will generate an individual with a structure combined from its parents. This way we receive the encoding of an individual without backward analysis of his phenotype. To perform a crossover the phenotype has to be evaluated (to mark the genes), but it is neither used nor know in the crossover operation (also it doesnt have to exist).

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The PGE algorithm was tested with the group of 6 computers in the computer network. Five computers calculated in the structure of five subsystems MR1, MR2, MR3, MR4, and MR5 and one master MR. The male subpopulation M of MR in the higher level follows the convergence of the subsystem. Every figure presents 10 runs of the PGE- program. The shortest time of computation is only 10 generation. All calculation were finished before 40 generation. This is better to compare with backward processing on one computer. The forward processing on one computer was the slowest.

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Results:(a) forward and (b) backward processing, (c) the PGE with 5 PC using backward processing (average in bold )

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Logical function XOR Input values are two integer numbers a and b; a, b 2. Output number c is the value of logical function XOR. Training data is a set of triples (a, b, c): P = {(0, 0, 0); (0, 1, 1); (1, 0, 1); (1, 1, 0)}. Thus the training set represents the truth table of the XOR function. The function can be expressed using _, ^, ¬ functions: a + b = (a ^ ¬b) _ (¬a ^ b) = (a _ b) ^ (¬a _ ¬b) = (a _ b) ^ ¬(a ^ b) The grammar was simplified so that it does not contain conditional statement and numeric constants, on the other hand three new terminals were added to generate functions _, ^, ¬. Thus the grammar generates representations of the XOR functions using other logical functions. 1a) function xxor($a,$b) { $result = "no_value"; $result = ($result) | ((((~(~(~(~(~($result)))))) | (($a) & (($a) & (~($b))))) & ($a)) | ((~($a)) & ($b))); return $result; } Number of generations: 32 1b) function xxor($a,$b) { $result = "no_value"; $result = ($result) | (((~$result | ($a & ($a & ~$b))) & $a) | (~$a & $b)); return $result; }

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Logical function XOR 2a) function xxor($a,$b) { $result = "no_value"; $result = ($result) | ((((~(~(~(~(~($b)))))) & (($a) & (($a) & (~($b))))) & ($a)) | ((~($a)) & ($b))); return $result; } Number of generations: 53 2b) function xxor($a,$b) { $result = "no_value"; $result = ($result) | (((~$b & ($a & ($a & ~$b))) & $a) | (~$a & $b)); return $result;

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This paper describes the application of Two-Level Transplant Evolution (TE) that can evolve control programs using a variable length linear genome to govern the mapping of a Backus Naur Form grammar definition. TE combines Grammatical Evolution (on the genotype level) with Genetic Programming (tree structures on the phenotype level). To increase the efficiency of Transplant Evolution (TE) the parallel Diferential Evolution was added. The adaptive significance of Parallel Transplant Evolution (PTE) with male and female populations has been studied.

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.. Fitness criterions of control process Flow chart

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. Step response for integration system with time delay

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Each of these systems is a network of many agents acting in parallel. There is Complex adaptive systems share certain crucial properties (a nonlinearity, a complex mixture of positive and negative feedback, nonlinear dynamics, emergence, collective behavior, spontaneous organization, and so on). There is not the master agent. A complex adaptive system has many levels of organization (the hierarchical structure).

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COMPLEX ADAPTIVE SYSTEMS the traditional science in the Age of the Machine tended to emphasize stability, order, uniformity, and equilibrium. todays science: disorder, instability, diversity, disequilibrium, nonlinear relationships (in which small inputs can trigger massive consequences), and temporality ( a sensitivity to the flow of time).

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Conclusions The Parallel Transplant Evolution can be used for the automatic generation of programs. We are far from supposing that all difficulties are removed but first results with TPEs are very promising.

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MENDEL 2012 Call for Papers 18th International Conference on Soft Computing Evolutionary Computation, Genetic Programming, Bayesian Method, Fuzzy Logic, Neural Networks, Rough Sets, Fractals June , 2012 Brno, CZECH REPUBLIC

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