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Recent years have witnessed a widespread in the use of interval type-2 fuzzy logic systems (IT2 FLSs) in real-world applications. It has been shown recently that interval type-2 fuzzy sets (IT2 FSs) are more general than interval-valued fuzzy sets (IV FSs) [1] . Hence, there is a need to explore the capabilities of the more general forms of IT2 FSs (beyond IV FSs) and the applications areas they will be more suitable for. In addition, there is a need to develop the theory of the general forms of IT2 FLSs (gfIT2 FLSs), which employ IT2 FSs that are not equivalent to IV FSs and can have nonconvex secondary membership functions (MFs). Although these systems could be considered within the scope of general type-2 FLSs (GT2 FLSs), the practical implementation of GT2 FLSs has traditionally required the secondary MFs to be convex and normal type-1 fuzzy sets (T1 FSs). In addition, the type-reduction operation still presents a challenge for GT2 FLSs because of its computational complexity. In this paper, we present a complete framework for a type-2 FLS that uses the most recent perception of IT2 FSs (the so called general forms of interval type-2 fuzzy sets, gfIT2 FSs), whose secondary grades can be nonconvex T1 FSs. This framework includes new equations for the meet and join operations of gfIT2 FSs, as well as a new type reduction procedure for the type-2 FLS involving gfIT2 FSs. In addition, we present the type-2 FLS operation for singleton and nonsingleton fuzzification. We will introduce the various operations employed within a gfIT2 FLSs, from fuzzification (including singleton and nonsingleton) to inference, type-reduction, and defuzzification. We will also present two examples in which these gfIT2 FSs arise naturally when modeling sonar sensors input noise and the antecedents/consequents from a survey including different users.

Singleton interval type-2 fuzzy logic systems (FLSs) have been widely applied in several real-world applications, where it was shown that the singleton interval type-2 FLSs outperform their singleton type-1 counterparts in applications with high uncertainty levels. However, one of the main criticisms of singleton interval type-2 FLSs is the fact that they outperform singleton type-1 FLSs solely based on their use of extra degrees of freedom (extra parameters) and that type-1 FLSs with a sufficiently large number of parameters may provide the same performance as interval type-2 FLSs. In addition, most works on type-2 FLSs only compare their results with singleton type-1 FLSs but fail to consider nonsingleton type-1 systems. In this paper, we aim to directly address and investigate this criticism. In order to do so, we will perform a comparative study between optimized singleton type-1, nonsingleton type-1, and singleton interval type-2 FLSs under the presence of noise. We will also present a multiobjective evolutionary algorithm (MOEA) for the optimization of singleton type-1, nonsingleton type-1, and singleton interval type-2 fuzzy systems for function approximation problems. The MOEA will aim to satisfy two objectives to maximize the accuracy of the FLS and minimize the number of rules in the FLS, thus improving its interpretability. Furthermore, we will present a methodology to obtain “optimal” consequents for the FLSs. Hence, this paper has two main contributions: First, it provides a common methodology to learn the three types of FLSs (i.e., singleton type-1, nonsingleton type-1, and singleton interval type-2 FLSs) from data samples. The second contribution is the creation of a common framework for the comparison of type-1 and type-2 FLSs that allows us to address the aforementioned criticism. We provide details of a series of experiments and include statistical analysis showing that the type-2 FLS is able to handle higher levels of noise than its nonsingleton and singleton type-1 counterparts.

In this study, a novel fuzzy controller, which is able to self-design from scratch, while working online, is proposed. The controller does not use the information regarding the differential equations that govern the plant's behavior or any of their bounds. The algorithm presented is able to determine the most-adequate topology for the fuzzy controller based on the data obtained during the system's normal operation. Therefore, the controller can start operating with an empty set of fuzzy rules and needs no offline training. The proposed methodology comprises two phases: adaptation of the consequents for every selected topology and online addition of new membership functions (MFs). Some of the main advantages of this method are its robustness under changes on the plant's dynamics, good performance in noisy situations, and the ability to perform variable selection among a group of candidate variables. Unlike other online methods, the modification of the topology is based on the analysis of the whole operating region of the plant, thus providing higher robustness. Several simulation examples are used to show these features.

This paper presents a new adaptive procedure for the linear and nonlinear separation of signals with nonuniform, symmetrical probability distributions, based on both simulated annealing and competitive learning methods by means of a neural network, considering the properties of the vectorial spaces of sources and mixtures, and using a multiple linearization in the mixture space. Moreover, the paper proposes the fusion of two important paradigms-genetic algorithms and the blind separation of sources in nonlinear mixtures. In nonlinear mixtures, optimization of the system parameters and, especially, the search for invertible functions is very difficult due to the existence of many local minima. The main characteristics of the methods are their simplicity and the rapid convergence experimentally validated by the separation of many kinds of signals, such as speech or biomedical data.

A novel approach to achieve real-time global learning in fuzzy controllers is proposed. Both the rule consequents and the membership functions defined in the premises of the fuzzy rules are tuned using a one-step algorithm, which is capable of controlling nonlinear plants with no prior offline training. Direct control is achieved by means of two auxiliary systems: The first one is responsible for adapting the consequents of the main controller's rules to minimize the error arising at the plant output, while the second auxiliary system compiles real input-output data obtained from the plant. The system then learns in real time from these data taking into account, not the current state of the plant but rather the global identification performed. Simulation results show that this approach leads to an enhanced control policy thanks to the global learning performed, avoiding overfitting.

This paper presents a multiobjective evolutionary algorithm to optimize radial basis function neural networks (RBFNNs) in order to approach target functions from a set of input-output pairs. The procedure allows the application of heuristics to improve the solution of the problem at hand by including some new genetic operators in the evolutionary process. These new operators are based on two well-known matrix transformations: singular value decomposition (SVD) and orthogonal least squares (OLS), which have been used to define new mutation operators that produce local or global modifications in the radial basis functions (RBFs) of the networks (the individuals in the population in the evolutionary procedure). After analyzing the efficiency of the different operators, we have shown that the global mutation operators yield an improved procedure to adjust the parameters of the RBFNNs.

Interest in hybrid methods that combine artificial neural networks and evolutionary algorithms has grown in the last few years, due to their robustness and ability to design networks by setting initial weight values, by searching the architecture and the learning rule and parameters. This paper presents an exhaustive analysis of the G-Prop method, and the different parameters the method requires (population size, selection rate, initial weight range, number of training epochs, etc.) are determined. The paper also the discusses the influence of the application of genetic operators on the precision (classification ability or error) and network size in classification problems. The significance and relative importance of the parameters with respect to the results obtained, as well as suitable values for each, were obtained using the ANOVA (analysis of the variance). Experiments show the significance of parameters concerning the neural network and learning in the hybrid methods. The parameters found using this method were used to compare the G-Prop method both to itself with other parameter settings, and to other published methods.

The Web newspaper pagination problem consists of optimizing the layout of a set of articles extracted from several Web newspapers and sending it to the user as the result of a previous query. This layout should be organized in columns, as in real newspapers, and should be adapted to the client Web browser configuration in real time. This paper presents an approach to the problem based on simulated annealing (SA) that solves the problem on-line, adapts itself to the client's computer configuration, and supports articles with different widths.

The identification of a model is one of the key issues in the field of fuzzy system modeling and function approximation theory. There are numerous approaches to the issue of parameter optimization within a fixed fuzzy system structure but no reliable method to obtain the optimal topology of the fuzzy system from a set of input-output data. This paper presents a reliable method to obtain the structure of a complete rule-based fuzzy system for a specific approximation accuracy of the training data, i.e., it can decide which input variables must be taken into account in the fuzzy system and how many membership functions (MFs) are needed in every selected input variable in order to reach the approximation target with the minimum number of parameters.

Most genetic algorithm (GA) users adjust the main parameters of the design of a GA (crossover and mutation probability, population size, number of generations, crossover, mutation, and selection operators) manually. Nevertheless, when GA applications are being developed it is very important to know which parameters have the greatest influence on the behavior and performance of a GA. The purpose of this study was to analyze the dynamics of GAs when confronted with modifications to the principal parameters that define them, taking into account the two main characteristics of GAs; their capacity for exploration and exploitation. Therefore, the dynamics of GAs have been analyzed from two viewpoints. The first is to study the best solution found by the system, i.e., to observe its capacity to obtain a local or global optimum. The second viewpoint is the diversity within the population of GAs; to examine this, the average fitness was calculated. The relevancy and relative importance of the parameters involved in GA design are investigated by using a powerful statistical tool, the analysis of the variance (ANOVA).

To date, clustering techniques have always been oriented to solve classification and pattern recognition problems. However, some authors have applied them unchanged to construct initial models for function approximators. Nevertheless, classification and function approximation problems present quite different objectives. Therefore it is necessary to design new clustering algorithms specialized in the problem of function approximation. This paper presents a new clustering technique, specially designed for function. approximation problems, which improves the performance of the approximator system obtained, compared with other models derived from traditional classification oriented clustering algorithms and input-output clustering techniques.

In this paper, a systematic design is proposed to determine fuzzy system structure and learning its parameters, from a set of given training examples. In particular, two fundamental problems concerning fuzzy system modeling are addressed: 1) fuzzy rule parameter optimization and 2) the identification of system structure (i.e., the number of membership functions and fuzzy rules). A four-step approach to build a fuzzy system automatically is presented: Step 1 directly obtains the optimum fuzzy rules for a given membership function configuration. Step 2 optimizes the allocation of the membership functions and the conclusion of the rules, in order to achieve a better approximation. Step 3 determines a new and more suitable topology with the information derived from the approximation error distribution; it decides which variables should increase the number of membership functions. Finally, Step 4 determines which structure should be selected to approximate the function, from the possible configurations provided by the algorithm in the three previous steps. The results of applying this method to the problem of function approximation are presented and then compared with other methodologies proposed in the bibliography.

The four papers in this special section address distinct subjects focusing on new, significant novel lines of development on genetic fuzzy systems (GFSs).

Implementations of fixed-template Cellular Neural Networks (CNN's) with reduced circuit complexity are presented. Considerable improvements in area without performance degradation have been obtained by: (1) using single-polarity signals that reduce the number of transistors required for signal replication and to generate the pseudo-linear output function; (2) using simple current-mode circuits to implement the output pseudo-linear function; and (3) searching for network parameter configurations that solve a particular application using the proposed circuit implementation with less hardware complexity. Experimental results for a CCD-CNN chip prototype with a density of 230 cells per millimetersquared (mm/sup 2/) are also reported.