EVOLUTIONARY DESIGN OF GEOMETRIC-BASED FUZZY SYSTEMS Carlos Kavka To cite this version: Carlos Kavka. EVOLUTIONARY DESIGN OF GEOMETRIC-BASED FUZZY SYSTEMS. Other [cs.OH]. Université Paris Sud - Paris XI, 2006. English. tel-00118883 HAL Id: tel-00118883 https://tel.archives-ouvertes.fr/tel-00118883 Submitted on 6 Dec 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. These` de Doctorat de l’Universite´ Paris-Sud Specialite:´ Informatique Present´ ee´ per Carlos Kavka pour obtenir le grade de Docteur de l’Universite´ Paris-Sud EVOLUTIONARY DESIGN OF GEOMETRIC-BASED FUZZY SYSTEMS Soutenue le 6 Juillet 2006 devant le jury compose´ de Cyril Fonlupt Professeur a` l’Universite´ du Littoral, examinateur Laurent Foulloy Professeur a` l’Universite´ de Savoie, rapporteur Jean-Sylvain Lienard Directeur de Recherche CNRS, examinateur Zbigniew Michalewicz Professeur a` l’Universite´ d’Adelaide, rapporteur Marc Schoenauer Directeur de Recherche INRIA, directeur de these` Contents Introduction 7 1 Evolutionary Computation 11 1.1 Overview . 11 1.2 History . 12 1.2.1 Genetic algorithms . 12 1.2.2 Evolution strategies . 13 1.2.3 Evolutionary programming . 13 1.2.4 Genetic programming . 13 1.3 Representations . 14 1.3.1 Bit strings . 14 1.3.2 Vectors of real numbers . 14 1.3.3 Permutations . 15 1.3.4 Parse trees . 15 1.3.5 Production rules . 15 1.3.6 Specific representations . 16 1.4 Selection . 18 1.4.1 Deterministic selection . 18 1.4.2 Proportional selection . 18 1.4.3 Rank selection . 19 1.4.4 Tournament selection . 19 1.5 Variation operators . 20 1.5.1 Recombination . 20 1.5.2 Mutation . 22 1.5.3 Parameters setting . 25 1.6 Conclusions . 26 2 Fuzzy Systems 29 2.1 Fuzzy Sets . 29 2.2 Linguistic variables . 31 2.3 Fuzzy inference . 32 2.3.1 An example of fuzzy inference . 33 2.4 Fuzzy Systems . 36 2.4.1 The Mamdani Fuzzy System . 37 3 Contents 2.4.2 The Takagi-Sugeno Fuzzy System . 38 2.4.3 MIMO vs. MISO fuzzy systems . 40 2.5 Partition of the Input Space . 40 2.6 Fuzzy Controllers . 41 2.6.1 The ANFIS model . 43 2.6.2 The GARIC model . 44 2.6.3 The genetic learning model of Hoffmann . 45 2.6.4 The SEFC model . 46 2.7 Recurrent fuzzy controllers . 46 2.7.1 Fuzzy temporal rules . 47 2.7.2 The RFNN model . 48 2.7.3 The RSONFIN model . 49 2.7.4 The DFNN model . 50 2.7.5 The TRFN model . 51 2.8 Conclusions . 52 3 The Voronoi Approximation 53 3.1 Basic Computational Geometry Concepts . 53 3.1.1 Voronoi Diagrams . 53 3.1.2 Delaunay Triangulations . 54 3.1.3 Barycentric Coordinates . 56 3.2 The Voronoi approximation . 57 3.2.1 Examples . 61 3.3 Evolution of Voronoi approximators . 66 3.3.1 Representation . 66 3.3.2 Recombination . 69 3.3.3 Mutation . 71 3.3.4 Experimental Study of the Variation Operators . 73 3.4 Function approximation . 78 3.4.1 First experiment . 78 3.4.2 Second experiment . 80 3.5 Conclusions . 83 4 Voronoi-based Fuzzy Systems 85 4.1 The Voronoi-based Fuzzy System . 85 4.2 Properties . 88 4.2.1 Fuzzy finite partition . 88 4.2.2 ǫ-completeness property . 90 4.3 Evolution of Voronoi-based fuzzy systems . 90 4.3.1 Representation . 90 4.3.2 Properties . 91 4.4 Control experiments . 94 4.4.1 Cart-pole system . 94 4.4.2 Evolutionary robotics . 107 4.5 Conclusions . 116 4 Contents 5 Recurrent Voronoi-based Fuzzy Systems 117 5.1 Recurrent Voronoi-based fuzzy systems . 117 5.2 Properties . 119 5.2.1 Recurrent rules representation . 119 5.2.2 Recurrent units with semantic interpretation . 120 5.3 Evolution of recurrent Voronoi-based fuzzy systems . 120 5.4 Experiments . 121 5.4.1 System identification . 121 5.4.2 Evolutionary robotics . 124 5.5 Conclusions . 132 Conclusions 135 Bibliography 150 Index 151 5 Introduction Fuzzy systems are a technique that allows the definition of input-output mappings based on linguistic terms of every day common use. This approach has been particularly success- ful in the area of control problems. In these problems, the objective is to define a mapping (controller) that produces the correct outputs (control signals) given a set of inputs (measure- ments). Traditional approaches in control, of which the proportional integral derivative (PID) control method is maybe the best representative, define the controller as a parameterized mathemat- ical function and then determine the optimum values for its set of parameters in order to approximate the desired mapping. Fuzzy systems use a different approach. The first step in the definition of a fuzzy system is the partition of the input and output domains in a set of linguistic terms (or linguistic labels) defined in natural language. These terms, which are imprecise by their own linguistic nature, are however defined very precisely by using fuzzy theory concepts. As a second step, a set of rules is defined in order to associate inputs with outputs. As an example, let us consider a typical vehicle control problem where the objective is to define a controller that determines the pressure to be applied to the break pedal, based on the vehicle speed and the distance to an obstacle. The two inputs are the speed of the vehicle and the distance to the obstacle, and the single output is the break pressure. The input variable speed can be partitioned in the three linguistic terms low, medium and high, which refer to the concepts of the vehicle moving at low speed, medium speed and high speed respectively. In the same way, the input variable distance is partitioned in the three linguistic terms near, medium and far. The output variable pressure can be partitioned in low, medium and strong. Based on these linguistic terms, a set of rules that establish the correspondence between inputs and outputs can be defined. For example, one of these rules can establish that the pressure on the break pedal should be strong if the speed is high and the distance to the obstacle is near. In principle, a set of nine rules has to be defined in order to determine the action that must be performed by considering all combinations of speed and distance linguistic terms. The fuzzy system is evaluated by determining the degree of application of all rules to the actual situation, and then computing the output of the system based on the values determined by the consequents of the rules, weighted by their corresponding degree of applicability. In this way, the rules that are more adequate to the actual situation have a larger weight (or higher responsibility) in the computation of the output value. As it can be appreciated, the method is simple and has an interesting intuitive appeal. However, the method presents two main disadvantages: 7 Contents The number of rules grows exponentially with the number of input variables and the • number of linguistic terms used in their definition. The selection of adequate parameters for the definition of linguistic terms is a complex • process and becomes more complicated when the number of input variables grows. To overcome these difficulties, several techniques have been proposed. Since from a strictly mathematical point of view, a fuzzy system in its base form is a parameterized system which can be analitically represented, optimization methods that perform parameter learning, like gradient descent or evolutionary algorithms, have been widely used. Gradient descent based methods, like.