TRANSPLANT EVOLUTION FOR OPTIMIZATION OF GENERAL CONTROLLERS

This paper describes a new method of evolution that is named Transplant Evolution (TE). None of the individuals of the transplant evolution contains genotype as in Grammatical Evolution (GE). Each individual of the transplant evolution contains the phenotype in the tree structure. Reproduction methods as crossover and mutation work with parts of phenotypes (sub-trees). The hierarchical structure of grammar-differential evolution that is used for finding optimal structures and parameters of general controllers is described.


INTRODUCTION
The aim of this paper is to describe a new optimization method that can create control equations of general regulators. For this type of optimization a new method was created and we call it Two-Level Transplant Evolution (TLTE). This method allowed us to apply advanced methods of optimization, for example direct tree reducing of tree structure of control equation. The reduction method was named Arithmetic Tree Reducing (ART). For optimization of control equations of general controllers is suitable combine two evolutionary algorithms. Main goal in the first level of TLTE is the optimization of structure of general controllers. In the second level of TLTE the concrete parameters are optimized and the unknown abstract parameters in structure of equations are set. The method TLTE was created by combination of Transplant Evolution method (TE) [1,2,3,8,9,10] and Differential Evolution method (DE) [7]. The Transplant Evolution (TE) optimizes structure of solution with unknown abstract parameters and the DE optimizes the parameters in this structure. The parameters are real numbers. The real numbers are not easy find directly in TE without DE. For evaluation of quality of found control equation are described new methods, which allow us evaluate their quality. It can be used in the case when the simulation of control process cannot be finished. In results are shown some practical application. In all results we received the control equation that reached better quality of control process, than classical PSD controllers and Takahashi`s modification of PSD controller.

THE PRESENTATION OF OBJECT TREE STRUCTURES
The phenotype representation of the individual is stored in the object tree structure. Each of nodes in the tree structure, including the sub-nodes, is an object that is specified by a terminal symbol and the type of terminal symbols. All nodes are independent and correctly defined mathematical functions that can be calculated, e.g. the function x-3, shown on Fig. 1, is a tree structure containing a functional block (sub-tree).
Creating the object tree is a key part of GEOS, which this method differs from other evolutionary algorithms. When the object tree is generated, similar methods to a traditional grammatical evolution are used. But the GEOS does not store the genotype, because the production rules are selected by randomly generated genes that are not saved in chromosomes of individuals. The final GEOS's individual contains only phenotype expressed in an object tree structure.
The algorithm of GEOS uses a generative grammar [4,5,6] whose translation process starts from the initial symbol S and continues randomly with using the rules of defined grammar [2]. The basic procedure of the translation algorithm is shown on Fig. 2 where is explain to why is unnecessary to store the genotype.

CROSSOVER
The crossover is a distinctive tool for genetic algorithms and is one of the methods in evolutionary algorithms that are able to acquire a new population of individuals. For crossover of object trees can be used following methods:

Crossover the parts of object trees (sub-trees)
The method of crossover object trees is based on the selection of two parents from the population and changing each other part of their sub-trees. For each of the parents cross points are randomly selected and their nodes and sub-trees are exchanged. This is the principle of creating new individuals into subsequent population as is shown on Fig. 3.

Crossover by linking trees or sub-trees
This method, as well as the previous one, is based on the crossover of two parents who are selected from the previous population. But the difference is in the way how the object trees are crossed. This method, unlike the previous one, does not exchange two randomly selected parts of the parents but parts of individuals are linked together with new and randomly generated node. This node will represent a new root of the tree structure of the individual. This principle is shown on Fig. 4.

Non-structural Mutation (NM)
Non-structural mutations do not affect the structure of already generated individual. In the individual who is selected for mutation, chosen nodes of object sub-tree are further subjected to mutation. The mutation will randomly change chosen nodes, whereas used grammar is respected. For example it means that mutated node, which is a function of two variables (i.e. + -× ÷) cannot be changed by node representing function of one variable or only a variable, etc. see

Structural Mutation (SM)
Structural mutations, unlike non-structural mutations, affect the tree structure of individuals. Changes of the sub-tree by extending or shortening its parts depend on the method of structural mutations. Structural mutation can be divided into two types: Structural mutation which is extending an object tree structure (ESM) and structural mutation which is shortening a tree structure (SSM). This type of mutation operator can be subdivided into two types:

Extending Structural Mutation (ESM)
In the case of the extending mutation, a randomly selected node is replaced by a part of the newly created sub-tree that respects the rules of defined grammar (see fig. 3). This method obviously does not always lead to the extension of the sub-tree but generally this form of the mutation leads to extension of sub-tree. (see Fig. ).

Shortening Structural Mutation (SSM)
Conversely the shortening mutation replaces a randomly selected node of the tree, including its child nodes, by node which is described by terminal symbol (i.e. a variable or a number). This type of mutation can be regarded as a method of indirectly reducing the complexity of the object tree (see Fig. ).
The complexity of the tree structure can be defined as the total number of objects in the tree of individual.

DIRECT TREE REDUCTION
The minimal length of an object tree is often one of the factors required in the optimal problem solution. This requirement can be achieved in several ways: By penalizing the part of the individual fitness which contains a complex object tree, Method of targeted structural mutation of individual (see SSM), The direct shortening of the tree using algebraic adjustments -algebraic reducing tree (ART).
The last-mentioned method can be realised by the GEOS, where all of individuals does not contain the genotype, and then a change in the phenotype is not affected by treatment with genotype. The realisation of above mentioned problem with individual, which use genotype would be in this case very difficult. This new method is based on the algebraic arrangement of the tree features that are intended to reduce the number of functional blocks in the body of individuals (such as repeating blocks "unary minus", etc.). The method described above is shown on Fig. and   In view of the object tree complexity of the individual and also for subsequent crossover is preferable to have a function in the form than x = a + a + a, or more generally x = n × A. Another example is the shortening of the function x = ─ (─ a), where is preferable to have the form x = a (it is removing redundant marks in the object tree individual). The introduction of algebraic modifications of individual phenotype leads to the shorter result of the optimal solution and consequently to the shorter presentation of the individual, shortening the time of calculation of the function that is represented in object tree and also to find optimal solutions faster because of higher probability of crossover in the suitable points with higher probability to produce meaningful solutions. The essential difference stems from the use of direct contraction of trees, which leads to significantly shorter resulting structure than without using this method.

HIERARCHICAL STRUCTURE OF TE (GDEOS) FOR OPTIMISATION OF THE CONTROLLER
The hierarchical structure of the transplant evolution can be used for optimisation of the structure and parameters of a general controller. This structure contains three layers. First two layers (GE + DE) are contained in TE. Those two layers are used for If an individual of grammatical's evolution has some abstract parameters, differential evolution will be run for solve them, otherwise simulation of regulation will be run directly.
Has the grammatical individual some abstract parameters? At the beginning of GDEOS an initial population is created (see Fig. 2) and then fitness of individuals is calculated. In the case of finding the optimal solution in the first generation, the algorithm is terminated, otherwise creates a new population of individuals by crossover and mutation operators, with the direct use of already created parent's object tree structures (it is analogy as transplantation of already created organs, without necessary know-ledge of DNA -"Transplant Evolution (TE)"). If the result of GDEOS needs some numerical parameters (for example num in

RESULTS
The TE and TE + ART methods for optimization of equation for general controller were compared.
The resulting form of the recurrent equation of general controller without using the direct method shortening of the tree (ART) is following (equation 1):
Bellow is shown result of optimisation parameters of PSD controllers and optimisation of the structure and parameters of general controllers. The parameters of PSD controllers were optimised with using DE and structure and parameters of general controller were optimised with using TE + ART method.
The basic criterion of minimal integral control area was used as criterial function for optimisation of PSD or general controllers, (see equation 3) On the Fig. and We tested the TLTE method for optimization of recurrent equation of general controllers. There is some results of optimization for one following system:.

CONCLUSION
The Two-Level Transplant Evolution (TLTE) was successfully use for automatic generation of control programs of general controllers. We tested this algorithm on many problems, only one example was described in this paper. We hope that this new method of controller design will be use in practice, not only for simulation.
Although we are at early stages of experiments, but it seems that it is possible to use parallel grammatical evolution with backward processing to generate combinatorial logic circuits. The grammatical algorithm can be outperformed with algorithms, which are designed specifically for this purpose.

ACKNOWLEDGMENTS
This work has been supported by Czech Ministry of Education No: MSM 00216305529 Intelligent Systems in Automation and GA ČR No: 102/09/1668.