In this article a design methodology for Mamdani interval type-3 fuzzy systems with center-of-sets type reduction is outlined. The methodology utilizes statistical measures, fuzzy c-means clustering and granular computing, to establish the justifiable footprint of uncertainty (JFOU) of the fuzzy granules, as explainable semantic abstractions that form the fuzzy model. The design methodology is presented in three general steps, first we use the principle of justifiable granularity to build a diagram of the justifiable information granule that contains a data structure with the descriptive measures of the experimental evidence of the data set. These measures are obtained from the partition matrix of the utilized clustering process, and these measures are used to evaluate the parameters of membership functions and characterize their JFOU. Second, we use the data structure of the justifiable information granule to characterize and parameterize the asymmetric interval type-3 membership functions. Lastly, the main procedure to obtain all the justifiable information fuzzy granules that define the knowledge base and the inference process of the fuzzy model is presented. Experiments were made with synthetic and real benchmark data from automated learning repositories, computing R adj 2 ${R}_{\mathrm{adj}}^{2}$ and root-mean-squared error to measure the reliability of the methodology, while keeping the justifiable uncertainty of the model.

Traditional mathematical models work with type-0, which means using precise numbers in the models, but since the seminal work of Prof. Zadeh in 1965, type-1 fuzzy models emerged as a powerful way to represent human knowledge and natural phenomena. Later type-2 fuzzy models were also proposed by Prof. Zadeh in 1975 and more recently have been studied and applied in real world problems by many researchers. In addition, as another extension of the original type-1 fuzzy logic, Prof. Atanassov proposed Intuitionistic Fuzzy Logic, which is a very powerful theory in its own right. Previous works of the author and other researchers have shown that certain problems can be appropriately solved by using type-1, and others by interval type-2, while others by using intuitionistic fuzzy logic. Bio-inspired and meta-heuristic optimization algorithms have been commonly used to find the optimal design of type-1, type-2 or intuitionistic fuzzy models for applications in control, robotics, pattern recognition, time series prediction, just to mention a few. However, the question still remains about if even more complex problems may require even higher types, orders or extensions of type-1 fuzzy models to obtain better solutions to real world problems. In this paper a framework for solving this problem of finding the optimal fuzzy model for a particular problem is presented. To the knowledge of the author, this is the first work to propose a systematic approach to solve this problem, and we envision that in the future this approach will serve as a basis for more efficient algorithms for the same task of finding the optimal fuzzy system.

Nowadays Takagi Sugeno Fuzzy Inference Systems provide an interesting alternative to solve different kind of problems, and this is because this approach offers versatility and easy design. One of the applications of this kind of Fuzzy Inference Systems is in the classification area, specifically in diagnosis problems, in order to be used as computer aided systems, and this provides interesting results. This paper aims at evaluating the performance of this kind of systems in diagnosis problems, but modifying the order of the Sugeno polynomial. This polynomial is originally a first-order polynomial that relates the inputs with the firing force of the rules. However, with the emergence of high-order Sugeno polynomials is interesting to evaluate how this approach can improve the performance of the Takagi Sugeno Fuzzy Inference Systems. On the other hand, we evaluate the performance by changing the order of the Sugeno polynomial for Type-1, Interval Type-2 and General Type-2 Fuzzy Inference Systems, in order to obtain a tendency of the performance with respect to the order of the Sugeno polynomial.

This paper proposes a new General Type-2 Radial Basis Function Neural Network (GT2-RBFNN) that is functionally equivalent to a GT2 Fuzzy Logic System (FLS) of either Takagi-Sugeno-Kang (TSK) or Mamdani type. The neural structure of the GT2-RBFNN is based on the α-planes representation, in which the antecedent and consequent part of each fuzzy rule uses GT2 Fuzzy Sets (FSs). To reduce the iterative nature of the Karnik-Mendel algorithm, the Enhaned-Karnik-Mendel (EKM) type-reduction and three popular direct-defuzzification methods, namely the 1) Nie-Tan approach (NT), the 2) Wu-Mendel uncertain bounds method (WU) and the 3) Biglarbegian-Melek-Mendel algorithm (BMM) are used. Hence, this paper provides four different architectures of the GT2-RBFNN and their parametric optimisation. Such optimisation is a two-stage methodology that first implements an Iterative Information Granulation (IIG) approach to estimate the antecedent parameters of each fuzzy rule. Secondly, each consequent part and the fuzzy rule base of the GT2-RBFNN is optimised using an Adaptive Gradient Descent method (AGD) respectively. A number of popular benchmark data sets, the identification of a nonlinear system and the prediction of chaotic time series are considered. The reported comparative analysis of experimental results is used to evaluate the performance of the suggested GT2 RBFNN with respect to other popular methodologies.

In this work we propose a variant for the adjustment of parameters in galactic swarm optimization (GSO) using fuzzy logic. GSO is a newly created metaheuristic that uses the movement and distribution of stars and galaxies in the universe as inspiration. In galactic swarm optimization, multiple cycles of exploration and exploitation are used to obtain a better balance between the exploration and the exploitation phases, thus trying to improve the search for the best solutions. In this paper, it is proposed to perform an optimization of the parameters of the membership functions used in an initial fuzzy system called fuzzy galactic swarm optimization 1 (FGSO1) with the aim of improving the results obtained with the FGSO1 using a parameter setting based on experimentation. The proposed method and the original FGSO1 fuzzy system are tested with the CEC-2015 functions with different number of dimensions to be able to appreciate its behavior characteristics from a low number of dimensions to a high number of dimensions.

In this paper we present a new variant of the Fuzzy K-Nearest Neighbor algorithm. We propose to use Interval Type-2 Fuzzy Logic to improve the performance of the Fuzzy K-Nearest Neighbor algorithm (Fuzzy KNN algorithm). We have used different measures to calculate the distance between the neighbors and the vector to be classified, such as Euclidean, Hamming, cosine similarity and city block distances. These distances represent the inputs for the Interval Type-2 Fuzzy Inference System. Simulation results show the potential of the proposed approach.

Themain aim of this paper is to use fuzzy inference systems for controlling the relevant parameters within the equations of the FWA algorithm. In other words, parameters that are considered constant in the traditional FWA and are now made dynamic by using fuzzy logic. It is worth mentioning that we also made a small modification to the algorithm with the goal of having a better performance and the modification was to change the stopping criteria of the algorithm. In the conventional fireworks algorithm (FWA) the stopping criteria is based on the function evaluations, including, some variations or modifications of the FWA but they also manage this in the same way. We propose a stopping criteria based on the iterations with the goal of having a more precise control and consequently, have a better way to control the output variables in a Fuzzy Inference System (FIS) of Mamdani type, since the number of iterations is used as the input variable. To demonstrate the validity of this modification we tested the algorithm with 12 benchmark functions with good results, and we call the proposed algorithm as Iterative Fuzzy Fireworks Algorithm and we denoted as IFFWA.

The main goal of the work presented in the paper is to introduce the use of fuzzy logic in the Grey Wolf Optimizer (GWO) algorithm specifically for dynamic simultaneous adaptation of the key parameters, which are crucial in the performance of the metaheuristic. The proposed approach for this modification of GWO using fuzzy logic is presented. In addition, a brief comparison between the traditional GWO algorithm and the Grey Wolf Optimizer using fuzzy logic for dynamic adaptation of parameters is reported. This research shows the individual dynamic adjustment of two parameters and then a proposal of how to simultaneously adjust both parameters and finally we present the performance of these methods when they are tested with a set of benchmark functions, showing the advantage of using the strategy of simultaneous adaptation of parameters.

We propose in this paper the use fuzzy logic to adjust parameters in the fireworks algorithm (FWA), that is, parameters that usually are considered as constants in the algorithm, we have transformed them to be dynamic parameters in the FWA. First, we realized an exhaustive experimentation of the parameters of the FWA algorithm, with the purpose of selecting the parameters that have more effect on the FWA performance, and we concluded that the main parameters of this algorithm are: numbers of sparks and the explosion amplitude of each firework. The modifications made to these parameters help us provide a better exploration and exploitation abilities to the algorithm. The main goal of this paper is to optimize the performance of the FWA. In this paper, we show the results of the modified algorithm, which we called fuzzy fireworks algorithm and we denoted as FFWA. The results of the experiments were obtained with 6 benchmarks functions.

In this paper two bio-inspired methods are used to optimize the type-2 fuzzy inference system integrator in an ensemble of three neural networks with type-2 fuzzy weights. The genetic algorithm and particle swarm optimization are used to optimize the type-2 fuzzy system integrators that work in response integration of the ensemble neural network for obtaining the final output. In this work an optimized type-2 fuzzy inference system integrator to perform the integration for an ensemble of three neural networks and the results for the two bio-inspired methods are presented. The proposed approach is applied to a case of time series prediction, specifically for the Mackey-Glass time series.

Edge detection is an essential method used in the image processing systems and can be applied to image sets before the training phase in pattern recognition systems. An edge detector simplifies the analysis of the images; because, it reduces the dataset processed. In this paper we present the advantage to use a fuzzy edge detector method in a face recognition system. In the methodology, first the General type-2 fuzzy edge detector was applied over three image databases; secondly the recognition system was performed using monolithic neural network, and after that the mean recognition rate was obtained; finally the recognition rate is compared using different edge detectors, such as the Sobel operator, Type-1 and Interval Type-2 fuzzy edge detectors.

In this paper, a comparison of the Choquet and Sugeno integrals is presented. The proposed methods enable the calculation of the Choquet and Sugeno integrals for combining multiple source of information with a degree of uncertainty. The methods are used to combine the modules output of a modular neural network for face recognition. In this paper, the focus is on aggregation operators that use measures as inputs, in particular the Choquet and Sugeno integrals. Recognition results with the Choquet integral are better or comparable to results produced by Sugeno integral.

In this paper a Modular Neural Network (MNN) with a granular approach optimization is proposed, where a firefly optimization is proposed to design a optimal MNN architecture. The proposed method can perform the optimization of some parameters such as; number of sub modules, percentage of information for the training phase and number of hidden layers (with their respective number of neurons) for each sub module. The proposed method is applied to human recognition based on iris biometrics. A benchmark database is used to prove the efficiency and effectiveness of the proposed method, using as objective function the minimization of the error of recognition.

We describe in this paper the architecture of a modular neural network (MNN) for pattern recognition. More recently, the study of modular neural network techniques theory has been receiving significant attention. The design of a recognition system also requires careful attention. The paper aims to use the Ant Colony paradigm to optimize the architecture of this Modular Neural Network for pattern recognition in order to obtain a good percentage of image identification and in the shortest time possible.

This paper presents the optimization of the antecedent parameters for a system of image edge detection based on the Sobel technique combined with interval type-2 fuzzy logic. The optimal design of fuzzy systems is a difficult task and for this reason the use of meta-heuristic optimization techniques is considered in this paper. For the optimization of the fuzzy inference system the Cuckoo Search (CS) algorithm is applied, and the idea is to find the optimal design parameters of interval type 2 fuzzy systems and achieve better results in applications of edge detection for digital images.

This paper describes the optimization of interval type-2 fuzzy integrators in Ensembles of ANFIS (adaptive neuro-fuzzy inferences systems) models for the prediction of the Dow Jones time series. The Dow Jones time series is used to the test of performance of the proposed ensemble architecture. We used the interval type-2 and type-1 fuzzy systems to integrate the output (forecast) of each Ensemble of ANFIS models. Genetic Algorithms (GAs) were used for the optimization of membership function parameters of each interval type-2 fuzzy integrator. In the experiments we optimized Gaussian, Generalized Bell and Triangular membership functions parameter for each of the fuzzy integrators, thereby increasing the complexity of the training. Simulation results show the effectiveness of the proposed approach.

This paper presents an edge-detection method that is based on the morphological gradient technique and generalized type-2 fuzzy logic. The theory of alpha planes is used to implement generalized type-2 fuzzy logic for edge detection. For the defuzzification process, the heights and approximation methods are used. Simulation results with a type-1 fuzzy inference system, an interval type-2 fuzzy inference system, and with a generalized type-2 fuzzy inference system for edge detection are presented. The proposed generalized type-2 fuzzy edge-detection method was tested with benchmark images and synthetic images. We used the merit of Pratt measure to illustrate the advantages of using generalized type-2 fuzzy logic.

In this paper a new model of a Multi-Objective Hierarchical Genetic Algorithm (MOHGA) based on the Micro Genetic Algorithm (μGA) approach for Modular Neural Networks (MNNs) optimization is proposed. The proposed method can divide the data automatically into granules or sub modules, and chooses which data are for the training and which are for the testing phase. The proposed Multi-Objective Genetic Algorithm is responsible for determining the number of granules or sub modules and the percentage of data for training that can allow to have better results. The proposed method was applied to human recognition and its applicability with good results is shown, although the proposed method can be used in other applications such as time series prediction and classification.

In this paper we propose a new Gravitational Search Algorithm (GSA) using fuzzy logic to change alpha parameter and give a different gravitation and acceleration to each agent in order to improve its performance, we use this new approach for mathematical functions and present a comparison with original approach.

This paper describes an architecture for Ensembles of ANFIS (adaptive network based fuzzy inference system), with integrators of type-1 FLS and interval type-2 FLS (Fuzzy Logic System), with emphasis on its application to the prediction of chaotic time series, where the goal is to minimize the prediction error. The time series that was considered is the Mackey-Glass. The methods used for the integration of the ensembles of ANFIS are: Integration by average, the integration by weighted average, integration by type-1 FLS and integration by interval type-2 FLS. The performance obtained with this architecture overcomes several standard statistical approaches and neural network models reported in the literature by various researchers. In the experiments we changed the type of membership functions and the desired goal error, thereby increasing the complexity of the training.

This paper describes an evolutionary genetic algorithm approach for the optimization of a fuzzy reactive controller applied to the design of a fuzzy controller for a mobile robot. The algorithm will optimize the fuzzy inference system evaluating the performance of each individual with a neuro-fuzzy fitness function that considers the robots covered distance, time used, battery life and the pattern of the trajectory.

In this paper a neural network learning method with lower and upper type-2 fuzzy weight adjustment is proposed. The general mathematical analysis of the proposed learning method architecture and the adaptation of the interval type-2 fuzzy weights are presented. The proposed method is based on research of recent methods that manage weight adaptation and especially type-2 fuzzy weights. In this paper the neural network architecture managing lower and upper type-2 fuzzy weights and the obtained lower and upper final results are presented. The proposed approach is applied to a case of Mackey-Glass time series prediction.

This paper describes an architecture for Ensembles of Neuro-Fuzzy models with interval type-2 and type-1 fuzzy integrators, with emphasis on its application to the prediction of time series, where the objective is obtained the goal is to minimize the prediction error. The time series that was considered is the Mackey-Glass. The methods used for the integration of the ensembles of neuro-fuzzy (we used the ANFIS models "adaptive network based fuzzy inference system") are: integration by average, the integration by weighted average, interval type-2 and type-1 fuzzy inference systems (FIS) integrators. The performance obtained with this architecture overcomes several standard statistical approaches and neural network models reported in the literature by various researchers.

In this paper we describe a new methodology to optimize fuzzy logic controllers using Ant Colony Optimization (ACO); in particular, the fuzzy logic controller for the water tank benchmark problem. The proposed methodology is applied in the optimization of membership function parameters and type of membership functions, using a set of constraints for the construction of the solution matrix of an ACO algorithm.

This issue marks the beginning of the IEEE Transactions on Neural Networks and Learning Systems (TNNLS). By adding "Learning Systems" to the title, we now state explicitly the scope of the Transactions to include neural networks as well as related learning systems. This issue marks a new era in the history of our Transactions. The Transactions is now ready to face the challenges of the next 10-20 years. With the evolution of the fields of neural networks in particular and computational intelligence in general, the IEEE Transactions on Neural Networks and Learning Systems will continue to grow and to succeed in this ever-changing world. Also included are a few comments about the review process of TNN manuscripts and the introduction of 14 new TNNLS Associate Editors. Short biographies are included for the new Associate Editors.

This paper describes an optimization method for ensemble neural network models with fuzzy aggregation of responses for forecasting complex time series using genetic algorithms. The time series under consideration for testing the hybrid approach is the Mackey-Glass data, and results for the optimization of type-1 fuzzy response aggregation in the ensemble neural network are presented. Simulation results show the effectiveness of the proposed approach.

In this paper we describe the application of a Modular Neural Network (MNN) for iris, ear and voice recognition for a benchmark database. The proposed MNN architecture consists of three modules; iris, ear and voice. Each module is divided into other three sub modules. Each sub module contains different information, this means one third of the database for each sub module. We considered the integration of each biometric measure separately. Later, we proceed to integrate these modules with a fuzzy integrator. Also, we performed optimization of the modular neural networks and the fuzzy integrators using genetic algorithms, and comparisons were made between optimized results and the results without optimization.

We described in this paper the use of Modular Neural Networks (MNN) for pattern recognition in parallel using a cluster of computers with a master-slave topology. In this paper, we are proposing the use of MNN to face recognition with large databases to validate this approach. Also, a parallel genetic algorithm to optimization architecture was used.

This paper presents a new validation index for the Fuzzy C-Means algorithm, which is composed of two metrics, the modified partition entropy index and the sum of the distances between the means of the fuzzy partitions. The modified partition entropy represents the variation of the data in clusters of the dataset, and the sum of the distances between the means of the fuzzy partition represents the separation between clusters in the data set. The proposed index was tested with synthetic and benchmark datasets with good results.

In this paper, a modification of the Sugeno integral with interval type-2 fuzzy logic is proposed. The modification includes changing the original equations of the Sugeno Measures and the Sugeno integral that were initially proposed for type-1 fuzzy logic. The proposed modification enables calculation of the interval type-2 Sugeno integral for combining multiple source of information with a higher degree of uncertainty than with the traditional type-1 Sugeno integral. The advantages of the interval type-2 Sugeno integral are illustrated by reporting improved recognition rates in benchmark face databases. This new concept could also be a useful tool in other areas of applications. Also, the improvement provided by the type-2 integral is verified to be statistically significant in the recognition results for complex face databases (like the FERET database) when compared with the type-1 Sugeno integral. The proposed Sugeno integral is used to combine the modules' outputs of a modular neural network for face recognition. Simulation results show that the interval type-2 Sugeno integral is able to improve the recognition rate for the benchmark face databases. Recognition results are better or comparable to results produced by alternative approaches present in the literature reported for the same benchmark problems.

In this paper we describe an evolutionary method for the optimization of type-2 fuzzy systems based on the level of uncertainty. The proposed evolutionary method produces the best fuzzy inference systems (based on the memberships functions) for particular applications. The optimization of membership functions of the type-2 fuzzy systems is based on the level of uncertainty considering three different cases to reduce the complexity problem of searching the solution space.

We describe in this paper a new hybrid approach for mathematical function optimization combining Particle Swarm Optimization (PSO) and Genetic Algorithms (GAs) using Fuzzy Logic for parameter adaptation and integrate the results. The new evolutionary method combines the advantages of the fuzzy logic to give us an improved FPSO+FGA hybrid method. Fuzzy Logic is used to combine the results of the PSO and GA in the best way possible. The new hybrid FPSO+FGA approach is compared with the PSO and GA methods with a set of benchmark mathematical functions. The new hybrid FPSO+FGA method is shown to be superior to the individual evolutionary methods on the set of benchmark functions.

This paper presents the modular neural network architecture as a system for recognizing persons based on the iris biometric measurement of humans. In this system, the human iris database is enhanced with image processing methods, and the coordinates of the center and radius of the iris are obtained to make a cut of the area of interest by removing the noise around the iris. The inputs to the modular neural network are the processed iris images and the output is the number of the person identified. The integration of the modules was done with a gating network method.

We propose a modular neural network architecture in order to make easy and fast the recognition process of the ear as a biometric. Comparing with other biometrics, ear recognition has one of the best performances, even when it has not received much attention. To improve the performance for ear recognition and make a comparison with other existing methods, we used the 2D wavelet analysis with Global Thresholding method, and Sugeno Measure and Winner-Takes-All (WTA) as modular neural network integrator. Recognition results achieved was up to 97%.

In this paper we describe an evolutionary method for the optimization of a modular neural network for multimodal biometry. The proposed evolutionary method produces the best architecture of the modular neural network (number of modules, layers and neurons) and fuzzy inference systems (memberships functions) as fuzzy integration methods. The integration of responses in the modular neural network is performed by using optimal interval type-2 fuzzy inference systems. The optimization of membership functions of the type-2 fuzzy systems is based on the level of uncertainty with application to fuzzy response integration.

In this paper we propose a feature extraction method based on brightness control of the pixels of an image using a fuzzy logic inference system. The proposed method was used in a hybrid pattern recognition system of the fingerprints, to improve the identification rate. The fuzzy extraction method preprocesses the fingerprint images and adjusts the brightness. The obtained image is input to the ensemble neural network (ENN), the ENN output is based on response integration with type-1 and type-2 fuzzy logic to obtain the fingerprint identification.

Edge detection is a previous step for image recognition systems that helps to extract the most important shapes in an image, ignoring the homogeneous regions and remarking the real object to classify or recognize. Traditional and fuzzy edge detectors can be used, but it's very difficult to demonstrate which one is better before the recognition results are obtained. In this work we present an experiment where several edge detectors were used to preprocess the same image sets. Each resultant image set was used as training data for a neural network recognition system, and the recognition rates were compared. The goal of this experiment is to find the better edge detector that can be used for the training data on a neural network to improve image recognition.

In this work we describe a fuzzy inference system to determine the relevance of each module in modular neural networks for images recognition. The tests were made with Type 1 and Interval type-2 fuzzy inference system, to compare the performance. In both cases the fusion operator for the modules is the Sugeno integral, and the parameters to estimate are the fuzzy densities.

We describe in this paper a new hybrid approach for mathematical function optimization combining Particle Swarm Optimization (PSO) and Genetic Algorithms (GAs) using Fuzzy Logic to integrate the results. The new evolutionary method combines the advantages of PSO and GA to give us an improved PSO+GA hybrid method. Fuzzy Logic is used to combine the results of the PSO and GA in the best way possible. The new hybrid PSO+GA approach is compared with the PSO and GA methods with a set of benchmark mathematical functions. The new hybrid PSO+GA method is shown to be superior that the individual evolutionary methods. The mathematical functions were evaluated with 2, 4, 8 and 32 variables to validate this approach.

We describe in this paper the application of evolutionary methods for the optimization of type-2 fuzzy logic controllers. These optimal type-2 fuzzy logic controllers are used for the trajectory tracking control of autonomous mobile robots. The evolutionary method to consider is the genetic algorithm, presenting simulations results in Simulink©of Matlab.

An interval type-2 fuzzy logic system (IT2-FLS) is designed to evaluate the parameters of a population growth model based on the environment resources stochasticity, in time and space. Interval type-2 fuzzy sets are used to measure the uncertainties of the environment resources. The main goal of this work is to demonstrate how the IT2-FLS integrated into a population growth model can make a suitable evaluation of the parameters required to make that the population size reach a stable equilibrium level on which it fluctuates into a time interval and after that, population size goes down as consequence of insufficient resources. The simulation results are compared against the results of others mathematical models made through the years in Ecology to study the population dynamics.

Soft computing can be used to build hybrid intelligent systems for achieving different goals in real-world applications.Soft computing techniques include, at the moment, fuzzy logic, neural networks, genetic algorithms, chaos theory methods, and similar techniques that have been proposed in recent years.Each of these techniques has advantages and disadvantages, and several real-world problems have been solved, by using one of these techniques.However, many real-world complex problems require the integration of several of these techniques to really achieve the efficiency and accuracy needed in practice.In particular, evolutionary computing can be used to optimize the topology of a fuzzy or a neural system.Also, there are neuro-fuzzy approaches or even neuro-fuzzy-genetic approaches for designing the best intelligent system for a particular application.This special issue consists of five papers that consider the use and integration of different soft computing techniques for the development of hybrid intelligent systems for modeling, simulation, and control of nonlinear dynamical systems.The five papers, of this special issue, describe different applications of soft computing techniques to real-world problems and can be considered a significant contribution to the field of hybrid intelligent systems.The first paper, "An Artificial Bee Hive for Continuous Optimization" by Mario A. Muñoz et al., deals with an artificial bee-hive algorithm for optimization in continuous search spaces based on a model aimed at individual bee behavior.The algorithm defines a set of behavioral rules for each agent to determine what kind of actions must be carried out.In addition, the proposed algorithm includes some adaptations not considered in the biological model to increase the performance in the search for better solutions.To compare the performance of the algorithm to other swarm-based algorithms a statistical analysis was performed.The second paper, "A Levenberg-Marquardt Learning Applied for Recurrent Neural Identification and Control for a Wastewater Treatment Bioprocess" by Ieroham Baruch and Carlos R. Mariaca-Gaspar, describes a new recurrent neural network (RNN) model for systems identification and states estimation of nonlinear plants.The proposed RNN identifier is implemented in direct and indirect adaptive

Edges detection in a digital image is the first step in an image recognition system. In this paper, we show an efficient edges detector using an interval type-2 fuzzy inference system (FIS-2). The FIS-2 uses as input the original images after applying Sobel filters and attenuation filters, then the fuzzy rules infer normalized values for the edges images, especially useful to enhance the performance of neural networks. To illustrate the results, we built frequency histograms of some images and compare the results of the FIS-2 edge's detector with the gradient magnitude method and a type-1 fuzzy inference system (FIS-1). The FIS-2 results are better than the gradient magnitude and FIS-1, because the edges preserve more detail of the original images, and the backgrounds are more homogeneous than with FIS-1 and the gradient's magnitude method. © 2009 Wiley Periodicals, Inc.

We describe in this paper a new hybrid approach for mathematical function optimization combining particle swarm optimization (PSO) and genetic algorithms (GAs) using fuzzy logic to integrate the results. The new evolutionary method combines the advantages of PSO and GA to give us an improved PSO+GA hybrid method. Fuzzy logic is used to combine the results of the PSO and GA in the best way possible. The new hybrid PSO+GA approach is compared with the PSO and GA methods with a set of benchmark mathematical functions. The new hybrid PSO+GA method is shown to be superior than the individual evolutionary methods.

In this paper, a class of Interval Type-2 Fuzzy Neural Networks (IT2FNN) is proposed, which is functionally equivalent to interval type-2 fuzzy inference systems. The computational process envisioned for fuzzy neural systems is as follows: it starts with the development of an ”Interval Type-2 Fuzzy Neuron”, which is based on biological neural morphologies, followed by the learning mechanisms. We describe how to decompose the parameter set such that the hybrid learning rule of adaptive networks can be applied to the IT2FNN architecture for the Takagi-Sugeno-Kang (TSK) type of reasoning.

The paper proposed to use recurrent fuzzy-neural multi-model (FNMM) identifier for decentralized identification of a distributed parameter anaerobic wastewater treatment digestion bioprocess, carried out in a fixed bed and a recirculation tank. The distributed parameter analytical model of the digestion bioprocess is reduced to a lumped system using the orthogonal collocation method, applied in three collocation points (plus the recirculation tank), which are used as centers of the membership functions of the fuzzyfied plant output variables with respect to the space variable. The local and global weight parameters and states of the proposed FNMM identifier are implemented by a hierarchical fuzzy-neural multi-model sliding mode controller (HFNMM-SMC). The comparative graphical simulation results of the digestion wastewater treatment system identification and control, obtained via learning, exhibited a good convergence, and precise reference tracking out performing the optimal control.

This paper presents the development and design of two software tools for computational intelligence. The software tools include a graphical user interface for construction, edition and observation of the intelligent systems. The software tool are for interval type-2 fuzzy logic and modular neural networks. The interval type-2 fuzzy logic system toolbox (IT2FLS), is an environment for interval type-2 fuzzy logic inference system development. Tools that cover the different phases of the fuzzy system design process, from the initial description phase, to the final implementation phase, build the toolbox. The toolboxpsilas best qualities are the capacity to develop complex systems and the flexibility that permits the user to extend the availability of functions for working with the use of type-2 fuzzy operators, linguistic variables, interval type-2 membership functions, defuzzification methods and the evaluation of interval type-2 fuzzy inference systems. The toolbox for modular neural networks has similar advantages.

We describe in this paper a comparative study of fuzzy inference systems as methods of integration in modular neural networks (MNNpsilas) for multimodal biometry. These methods of integration are based on type-1 and type-2 fuzzy logic. Also, the fuzzy systems are optimized with simple genetic algorithms. First, we considered the use of type-1 fuzzy logic and later the approach with type-2 fuzzy logic. The fuzzy systems were developed using genetic algorithms to handle fuzzy inference systems with different membership functions, like the triangular, trapezoidal and gaussian; since these algorithms can generate the fuzzy systems automatically. Then the response integration of the modular neural network was tested with the optimized fuzzy integration systems. The comparative study of type-1 and type-2 fuzzy inference systems was made to observe the behavior of the two different integration methods of modular neural networks for multimodal biometry.

This paper describes a new approach for response integration in ensemble neural networks using interval type-2 fuzzy logic. When using ensemble neural networks it is important to choose a good method of response integration to obtain a better identification in pattern recognition. In this paper a comparative analysis between interval type-2 fuzzy logic, type-1 fuzzy logic and the Sugeno integral, as response integration methods, in ensemble neural networks is presented. Based on simulation results interval type-2 fuzzy logic is shown to be a superior method for response integration.

In this paper a fuzzy logic approach to determine the relevance of each module in modular neural networks for images recognition is presented. The tests were made with Type-1 and Interval Type-2 Fuzzy Inference Systems, to compare the performance of the proposed approach. In both cases the fusion operator for the modules is the Sugeno Integral, and the estimated parameters are the fuzzy densities.

We describe in this paper a new hybrid approach for optimization combining particle swarm optimization (PSO) and genetic algorithms (GAs) using fuzzy logic to integrate the results. The new evolutionary method combines the advantages of PSO and GA to give us an improved PSO+GA hybrid method fuzzy logic is used to combine the results of the PSO and GA in the best way possible. The new hybrid PSO+GA approach is compared with the PSO and GA methods with a set of benchmark mathematical functions. The proposed hybrid method is also tested with the problem of neural network optimization. The new hybrid PSO+GA method is shown to be superior with respect to both the individual evolutionary methods.

This paper presents the development and design of a graphical user interface and a command line programming toolbox for construction, edition and observation of interval type-2 fuzzy inference systems. The interval type-2 fuzzy logic system toolbox (IT2FLS), is an environment for interval type-2 fuzzy logic inference system development. Tools that cover the different phases of the fuzzy system design process, from the initial description phase, to the final implementation phase, build the toolbox. The toolbox's best qualities are the capacity to develop complex systems and the flexibility that permits the user to extend the availability of functions for working with the use of type-2 fuzzy operators, linguistic variables, interval type-2 membership functions, defuzzification methods and the evaluation of interval type-2 fuzzy inference systems.

Uncertainty is an inherent part in controllers used for real-world applications. The use of new methods for handling incomplete information is of fundamental importance in engineering applications. We simulated the effects of uncertainty produced by the instrumentation elements in type-1 and type-2 fuzzy logic controllers to perform a comparative analysis of the systems' response, in the presence of uncertainty. We are presenting an innovative idea to optimize interval type-2 membership functions using an average of two type-1 systems with the Human Evolutionary Model, and we show comparative results of the optimized proposed method. We found that the optimized membership functions for the inputs of a type-2 system increases the performance of the system for high noise levels.

We describe in this paper a new method for response integration in modular neural networks using type-2 fuzzy logic. The modular neural networks were used in human person recognition. Biometric authentication is used to achieve person recognition. Three biometric characteristics of the person are used: face, fingerprint, and voice. A modular neural network of three modules is used. Each module is a local expert on person recognition based on each of the biometric measures. The response integration method of the modular neural network has the goal of combining the responses of the modules to improve the recognition rate of the individual modules. We show in this paper the results of a type-2 fuzzy approach for response integration that improves performance over type-1 fuzzy logic approaches.

Uncertainty is an inherent part of intelligent systems used in real-world applications. The use of new methods for handling incomplete information is of fundamental importance. Type-1 fuzzy sets used in conventional fuzzy systems cannot fully handle the uncertainties present in intelligent systems. Type-2 fuzzy sets can handle such uncertainties in a better way because they provide us with a more complete model of real-world uncertainty. Experimental results are also presented for forecasting chaotic time series in which interval type-2 fuzzy logic outperforms some hybrid intelligent approaches. Neural networks provide a comparable result with type-2 fuzzy systems.

We describe in this paper the evolution of modular neural networks using hierarchical genetic algorithms for pattern recognition. Modular neural networks (MNN) have shown significant learning improvement over single neural networks (NN). For this reason, the use of MNN for pattern recognition is well justified. However, network topology design of MNN is at least an order of magnitude more difficult than for classical NNs. We describe in this paper the use of a hierarchical genetic algorithm (HGA) for optimizing the topology of each of the neural network modules of the MNN. The HGA is clearly needed due to the fact that topology optimization requires that we are able to manage both the layer and node information for each of the MNN modules. Simulation results prove the feasibility and advantages of the proposed approach.

In this paper we consider a Modular Neural Network combined with two Interval Type-2 Fuzzy Inference Systems (FIS 2) for image recognition. The first FIS 2 is used for edges detection in training data, and the second one to find the best parameters for the Sugeno Integral as decision operator. Once again Fuzzy Logic is shown to be a tool that can help improve the results of a neural system facilitating the representation of the human perception.

We describe in this paper a new method for response integration in modular neural networks using type-2 fuzzy logic. The modular neural networks were applied to human person recognition. Biometric authentication is used to achieve person recognition. Three biometric characteristics of the person are used: face, fingerprint, and voice. A modular neural network of three modules is used. Each module is a local expert on person recognition based on each of the biometric measures. The response integration method of the modular neural network has the goal of combining the responses of the modules to improve the recognition rate of the individual modules. We show in this paper the results of a type-2 fuzzy logic approach for response integration that improves performance over type-1 fuzzy logic approaches.

Stability is one of the more important aspects in the traditional knowledge of automatic control. Type-2 fuzzy logic is an emerging and promising area for intelligent control (in this case, fuzzy control) applications. We present a design methodology based on the work by Margaliot (2000) for the design of stable Mamdani type-2 fuzzy logic controllers.

On this research work we present a software tool to experiment with new neural multi-net structures, incuding ensemble and modular approaches. This tool allow us to draw models, set parameters, save as project and generate files with results, always in a user friendly graphic enviroment. Other feature is the implementation of the Sugeno Integral formulas, this program was developed to allow the combination of any number of elements.

We describe in this paper the application of several neural network architectures to the problem of simulating and predicting the dynamic behavior of complex economic time series. We use several neural network models and training algorithms to compare the results and decide at the end, which one is best for this application. We also compare the simulation results with the traditional approach of using a statistical model. In this case, we use real time series of prices of consumer goods to test our models. Real prices of tomato and green onion in the U.S. show complex fluctuations in time and are very complicated to predict with traditional approaches. For this reason, we have chosen a neural network approach to simulate and predict the evolution of these prices in the U.S. market.

Uncertainty is an inherent part in controllers used for real-world applications. The use of new methods for handling incomplete information is of fundamental importance in engineering applications. This paper deals with the design of controllers using type-2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements. We simulated type-1 and type-2 fuzzy logic controllers to perform a comparative analysis of the systems' response, in the presence of uncertainty

We describe in this paper the evolution of modular neural networks using hierarchical genetic algorithms. Modular neural networks (MNN) have shown significant learning improvement over single neural networks (NN). For this reason, the use of MNN for pattern recognition is well justified. However, network topology design of MNN is at least an order of magnitude more difficult than for classical NNs. We describe in this paper the use of a hierarchical genetic algorithm (HGA) for optimizing the topology of each of the neural network modules of the MNN. The HGA is clearly needed due to the fact that topology optimization requires that we are able to manage both the layer and node information for each of the MNN modules. Simulation results shown in this paper prove the feasibility and advantages of the proposed approach. The method of integration of response is based on fuzzy integral and Sugeno measures, where parameter /spl lambda/ also is optimized by means of the hierarchical genetic algorithms.

We describe in this paper a new approach for pattern recognition using modular neural networks with a fuzzy logic method for response integration. We proposed a new architecture for modular neural networks for achieving pattern recognition in the particular case of human fingerprints. Also, the method for achieving response integration is based on the fuzzy Sugeno integral. Response integration is required to combine the outputs of all the modules in the modular network. We have applied the new approach for fingerprint recognition with a real database of fingerprints from students of our institution.

We describe in this paper a new approach for pattern recognition using modular neural networks with a fuzzy logic method for response integration. We proposed a new architecture for modular neural networks for achieving pattern recognition in the particular case of human faces. Also, the method for achieving response integration is based on the fuzzy Sugeno integral. Response integration is required to combine the outputs of all the modules in the modular network. We have applied the new approach for face recognition with a real database of faces from students of our institution.

We describe a new approach for face recognition using modular neural networks with a fuzzy logic method for response integration. We proposed a new architecture for modular neural networks for achieving pattern recognition in the particular case of human faces. Also, the method for achieving response integration is based on the fuzzy Sugeno integral. Response integration is required to combine the outputs of all the modules in the modular network. We have applied the new approach for face recognition to a real database of faces of students at our institution. Recognition rates with the modular approach were compared against the monolithic single neural network approach to measure the improvement. The results of the new modular neural network approach were excellent overall and also in comparison to the monolithic approach. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 275–291, 2005.

In this article we analyzed the explorative behavior of the Breeder Genetic Algorithm. We tested this algorithm using fuzzy recombination and the classical discrete mutation operator. A general review of the most common linear model structures is given for recommending where to use this type of evolutionary algorithm. We selected for convenience a finite impulse response structure model because it has global optima; we considered this characteristic advisable for analyzing the evolution of the best and poorest individuals in the population through the generations. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 293–311, 2005.

We describe A new approach for plant monitoring and diagnostics using type-2 fuzzy logic and fractal theory. The concept of the fractal dimension is used to measure the complexity of the time series of relevant variables for the process. A set of type-2 fuzzy rules is used to represent the knowledge for monitoring the process. In the type-2 fuzzy rules, the fractal dimension is used as a linguistic variable to help in recognizing specific patterns in the measured data. The fuzzy fractal approach has been applied before in problems of financial time series prediction and for other types of problems, but now it is proposed to the monitoring of plants using type-2 fuzzy logic. We also compare the results of the type-2 fuzzy logic approach with the results of using only a traditional type-1 approach. Experimental results show a significant improvement in the monitoring ability with the type-2 fuzzy logic approach.

We describe in this paper the use of type-2 fuzzy logic for achieving adaptive noise cancellation. The objective of adaptive noise cancellation is to filter out an interference component by identifying a model between a measurable noise source and the corresponding un-measurable interference. We propose the use of type-2 fuzzy logic to find this model. The use of type-2 fuzzy logic is justified due to the high level of uncertainty of the process, which makes difficult to find appropriate parameter values for the membership functions.

We describe in this paper a new approach for plant monitoring and diagnostics using type-2 fuzzy logic and fractal theory. The concept of the fractal dimension is used to measure the complexity of the time series of relevant variables for the process. A set of type-2 fuzzy rules is used to represent the knowledge for monitoring the process. In the type-2 fuzzy rules, the fractal dimension is used as a linguistic variable to help in recognizing specific patterns in the measured data. The fuzzy-fractal approach has been applied before in problems of financial time series prediction and for other types of problems, but now it is proposed to the monitoring of plants using type-2 fuzzy logic. We also compare the results of the type-2 fuzzy logic approach with the results of using only a traditional type-1 approach. Experimental results show a significant improvement in the monitoring ability with the type-2 fuzzy logic approach.

We describe in this paper adaptive model-based control of non-linear plants using type-2 fuzzy logic and neural networks. First, the general concept of adaptive model-based control is described. Second, the use of type-2 fuzzy logic for adaptive control is described. Third, a neuro-fuzzy approach is proposed to learn the parameters of the fuzzy system for control. A specific non-linear plant is used to test the hybrid approach for adaptive control. A specific plant was used as test bed in the experiments. The non-linear plant that was considered is the "Pendubot", which is a non-linear plant similar to the two-link robot arm. The results of the type-2 fuzzy logic approach for control were good, both in accuracy and efficiency.

There is a need for shortening the time of formation of the batteries at the production line, and this is due the great demand of this product. In this paper we are proposing a novel technique in this area to optimize the time of manufacturing. This technique uses a fuzzy logic approach to increase the ampere rate in order to minimize the formation time. This paper shows a fuzzy logic system for the formation of batteries using a high scale integration device, such as the FLASH PSD813F1, which is a micro-controller. It includes a CPLD (COMPLEX PLD) embedded, which allows implementing logical functions defined by the user in its 73 inputs 19 outputs AND-OR array. The fuzzy logic algorithm is summarized in a look-up table, described by Boolean equations implemented in the PSD813F1 CPLD. The fuzzy controller, for the formation of batteries for the CPLD of the PSD813F1, basically controls the current intensity applied to the battery depending on its temperature and the change of temperature.

We describe in this paper adaptive model-based control of non-linear plants using type-2 fuzzy logic and neural networks. First, the general concept of adaptive model-based control is described. Second, the use of type-2 fuzzy logic for adaptive control is described. Third, a neuro-fuzzy approach is proposed to learn the parameters of the fuzzy system for control. A specific non-linear plant is used to test the hybrid approach for adaptive control. A specific plant was used as test bed in the experiments. The non-linear plant that was considered is the "Pendubot", which is a non-linear plant similar to the two-link robot arm. The results of the type-2 fuzzy logic approach for control were good, both in accuracy and efficiency.

We describe the application of type-2 fuzzy logic to the problem of automated quality control in sound speaker manufacturing. Traditional quality control has been done by manually checking the quality of sound after production. This manual checking of the speakers is time-consuming and has occasionally been the cause of errors in quality evaluation. For this reason, we developed an intelligent system for automated quality control in sound speaker manufacturing. The intelligent system has a type-2 fuzzy rule base containing the knowledge of human experts in quality control. The parameters of the fuzzy system are tuned by applying neural networks using, as training data, a real time series of measured sounds as given by good sound speakers. We also use the fractal dimension as a measure of the complexity of the sound signal.

We describe in this article a new hybrid fuzzy-fractal approach for plant monitoring. We use the concept of the fractal dimension to measure the complexity of a time series of observed data from the plant. We also use fuzzy logic to represent expert knowledge on monitoring the process in the plant. In the hybrid fuzzy-fractal approach, a set of fuzzy if-then rules are used to classify different conditions of the plant. The fractal dimension is used as an input linguistic variable in the fuzzy system to improve the accuracy in the classification. An implementation of the proposed approach is shown to describe in more detail the method. © 2002 Wiley Periodicals, Inc.

In this paper, we describe a new method for the estimation of the fractal dimension of a geometrical object using fuzzy logic techniques. The fractal dimension is a mathematical concept, which measures the geometrical complexity of an object. The algorithms for estimating the fractal dimension calculate a numerical value using as data a time series for the specific problem. This numerical (crisp) value gives an idea of the complexity of the geometrical object (or time series). However, there is an underlying uncertainty in the estimation of the fractal dimension because we use only a sample of points of the object, and also because the numerical algorithms for the fractal dimension are not completely accurate. For this reason, we have proposed a new definition of the fractal dimension that incorporates the concept of a fuzzy set. This new definition can be considered a weaker definition (but more realistic) of the fractal dimension, and we have named this the "fuzzy fractal dimension." We can apply this new definition of the fractal dimension in conjunction with soft computing techniques for the problem of time series prediction. We have developed hybrid intelligent systems combining neural networks, fuzzy logic, and the fractal dimension, for the problem of time series prediction, and we have achieved very good results.

We describe a new theory of chaos using fuzzy logic techniques. Chaotic behavior in nonlinear dynamical systems is very difficult to detect and control. Part of the problem is that mathematical results for chaos are difficult to use in many cases, and even if one could use them, there is an underlying uncertainty in the accuracy of the numerical simulations of the dynamical systems. For this reason, we can model the uncertainty of detecting the range of values where chaos occurs, using fuzzy set theory. Using fuzzy sets, we can build a new theory of fuzzy chaos, where we can use fuzzy sets to describe the behaviors of a system. We apply this new theory of fuzzy chaos to the case of simulation and control of robotic dynamic systems.

Stepping motors are widely used in robotics and in the numerical control of machine tools, where they have to perform high-precision positioning operations. However, the variations of the mechanical configuration of the drive, which are common to these two applications, can lead to a loss of synchronism for high stepping rates. Moreover, the classical open-loop speed control is weak and a closed-loop control becomes necessary. In this paper, fuzzy logic is applied to control the speed of a stepping motor drive with feedback. A neuro-fuzzy hybrid approach is used to design the fuzzy rule base of the intelligent system for control. In particular, we used the ANFIS (Adaptive Network-based Fuzzy Inference System) methodology to build a Sugeno fuzzy model for controlling the stepping motor drive. An advanced testbed is used in order to evaluate the tracking properties and the robustness capacities of the fuzzy logic controller.

We describe the application of several neural network architectures to the problem of simulating and predicting the dynamic behavior of complex economic time series. We use several neural network models and training algorithms to compare the results and decide which one is best for this application. We also compare the simulation results with fuzzy logic models and the traditional approach of using a statistical model. In this case, we use real time series of prices of consumer goods to test our models. Real prices of tomato and green onion in the US show complex fluctuations in time and are very complicated to predict with traditional statistical approaches. For this reason, we have chosen neural networks and fuzzy logic to simulate and predict the evolution of these prices in the US market.

We describe different hybrid approaches for controlling dynamical systems in electrochemical applications. The hybrid approaches combine soft computing techniques and mathematical models to achieve the goal of controlling the electrochemical process to follow a desired production plan. We develop several hybrid architectures that combine fuzzy logic, neural networks, and genetic algorithms, compare the performance of each of these combinations, and decide on the best one for our purpose. Electrochemical processes, like the ones used in battery charging, are very complex and for this reason very difficult to control. We achieved very good results using the fuzzy logic for control, neural networks for modelling the process, and genetic algorithms for tuning the hybrid intelligent system.

We describe a new method for the automated simulation of non-linear dynamical systems. This method is based on a hybrid fuzzy-genetic approach to achieve, in an efficient way, automated simulation of a particular dynamical system given its mathematical model. The use of genetic algorithms is to achieve automated parameter selection for the mathematical models. Our genetic algorithm uses a Lotka-Volterra approach to change the size of the population in each generation. The use of fuzzy logic simulates the process of expert behavior identification, by implementing the knowledge of identification by a set of fuzzy rules. We also use the concept of the fractal dimension of a time series to help make this identification more accurate. Experimental results for the case of robotic dynamic systems show the efficiency and accuracy of this new method for the simulation of complex non-linear dynamical systems.

We describe the application of a new method for automated simulation of nonlinear dynamical systems, using a fuzzy-genetic approach, to the problem of simulating companies as they compete for the market of their products. A particular company can be viewed as a dynamical system evolving in time and also competing with similar companies for the market of their products. Also, within an international trade agreement there are also competing companies from foreign countries, which complicates the problem even more. We can apply our new method for automated simulation (O. Castillo and P. Melin, 1998) to simulate the evolution of a company or a group of companies as they compete for a fixed market. As a result of these simulations, we can formulate specific mathematical conditions for a specific country to go bankrupt or specific conditions for another company to have success.

We describe a new method for automated simulation of nonlinear dynamical systems. This method is based on a hybrid fuzzy-fractal-genetic approach to achieve, in an efficient way, automated simulation of a particular dynamical system given its mathematical model. The use of genetic algorithms is to achieve automated parameter selection for the mathematical models. The use of fuzzy logic is to simulate the process of expert behavior identification, by implementing the knowledge of identification by a set of fuzzy rules. We also use the concept of the fractal dimension of a time series to help make this identification more accurate. Experimental results for the case of robotic dynamic systems show the efficiency and accuracy of this new method for the simulation of complex nonlinear dynamical systems.

Describes a method for adaptive model-based control of non-linear dynamic plants in the food industry using neural networks, fuzzy logic and fractal theory. The neuro-fuzzy-fractal method combines soft computing (SC) techniques with the concept of the fractal dimension for the domain of non-linear dynamic plant control. The new method for adaptive model-based control has been implemented as a computer program to show that our neuro-fuzzy-fractal approach is a good alternative for controlling non-linear dynamic plants. We illustrate our methodology with the case of controlling biochemical reactors in the food industry. For this case, we use mathematical models for the simulation of bacteria growth for several types of food.

We describe in this paper a new method for automated simulation of non-linear dynamical systems. This method is based on a hybrid fuzzy-fractal-genetic approach to achieve, in an efficient way, automated simulation of a particular dynamical system given its mathematical model. The use of genetic algorithms is to achieve automated parameter selection for the mathematical models. The use of fuzzy logic is to simulate the process of expert behavior identification, by implementing the knowledge of identification by a set of fuzzy rules. We also use the concept of the fractal dimension of a time series to help make this identification more accurate. Experimental results for the case of robotic dynamic systems show the efficiency and accuracy of this new method for the simulation of complex non-linear dynamical systems.

We describe the application of our new method for adaptive model based control (using a neuro-fuzzy-genetic approach) to the problem of controlling international trade dynamics. The problem of international trade between two or more countries is a very complex one because of the nonlinearities involved in the mathematical models (O. Castillo and P. Melin, 1998). We describe the methodology to develop an intelligent system for controlling international trade that can be used by the government of a specific country to maximize the profit from its international trade with other countries. Our method for adaptive model based control of nonlinear dynamical systems consists of using a fuzzy rule base for model selection, a genetic algorithm for identification and a neural network for control (O. Melin and P. Castillo, 1998). Accordingly, an intelligent control system based on our methodology has an architecture with three main modules: model selection, identification and control. For the case of international trade, we have developed the fuzzy rule base for selecting the appropriate mathematical models for the problem, the genetic algorithm for parameter identification, and the neural network for control.

We describe a new method for adaptive model-based control of robotic dynamic systems using a new hybrid neuro-fuzzy-fractal approach. Intelligent control of robotic systems is a difficult problem because the dynamics of these systems is highly nonlinear. Optimal control of many robotic systems also requires methods which make use of predictions of future behavior. We describe an intelligent system for controlling robot manipulators to illustrate our neuro-fuzzy-fractal hybrid approach for adaptive control. We use a new fuzzy inference system for reasoning with multiple differential equations for model selection based on the relevant selection parameters for the problem. We use neural networks for identification and control of robotic dynamic systems.

We describe a general method for adaptive model-based control of nonlinear dynamic systems using neural networks, fuzzy logic and fractal theory. The new neuro-fuzzy-fractal method combines soft computing techniques with the concept of the fractal dimension for the domain of nonlinear dynamic system control. The new method for adaptive model-based control has been implemented as a computer program to show that our neuro-fuzzy-fractal approach is a good alternative for controlling nonlinear dynamic systems. We illustrate our new methodology with the case of controlling aircraft dynamic systems. For this case, we use mathematical models for the simulation of aircraft dynamics during flight. The goal of constructing these models is to capture the dynamics of the aircraft, so as to have a way of controlling this dynamics to avoid dangerous behavior of the system.

The authors describe non-linear mathematical models that can be used to study the dynamics of international trade. Mathematical models of international trade (IT), between three or more countries, can show very complicated dynamics in time (with the possible occurrence of chaotic behavior). The simulation of these models is critical in understanding the behavior of the relevant financial and economical variables for the problem of IT. Also, performing the simulations for different parameter values of the models will enable the forecasting of future IT. The problem of simulation and forecasting of IT has been solved by using soft computing (SC) techniques. An intelligent system for automated simulation of IT, combining fuzzy logic techniques and genetic algorithms, has been developed for the simulation and behavior identification of the mathematical models. The intelligent system uses a specific genetic algorithm to generate the best set of parameter values for performing numerical simulation of the dynamical system (of IT) and a fuzzy rule base for behavior identification. The importance of simulation and forecasting of IT can be measured if one considers that one of the goals for a specific country is to find the optimum benefit from its international trade with other countries.

We describe in this paper a computer program for automated mathematical modelling and simulation of robotic dynamic systems using fuzzy logic techniques, genetic algorithms and fractal theory. The computer program combines soft computing (SC) techniques with mathematical methods and can be considered an intelligent system for the domain of modelling and simulation of robotic systems. This domain is quite complex because robotic systems can be viewed as nonlinear dynamical systems, and it is a well known fact that even very simple nonlinear dynamical systems can exhibit "chaotic" behavior. The computer program simulates the reasoning of a human expert in the process of developing mathematical models of robotic dynamic systems (RDS). The program contains the knowledge of the human experts expressed as fuzzy rules (in the knowledge base) for mathematical modelling and simulation (MMS) of robotic systems. The computer program uses efficiently SC techniques and fractal theory for MMS of RDS. Mathematical modelling and simulation of robotic systems is very important because it can help in the control of an actual system or in the design of a new system using the results of the simulations.

The authors describe paper non-linear mathematical models, from dynamical systems theory, that can be used to study the dynamics of international trade. Mathematical models of international trade (IT), between three or more countries, can show very complicated dynamics in time (with the possible occurrence of behavior known as chaos). The simulation of these models is critical in understanding the behavior of the relevant financial and economical variables for the problem of IT. Also, performing the simulations for different parameter values of the models will enable the prediction of future IT. The problem of modelling and simulation of IT has been solved in this paper by using artificial intelligence (AI) techniques. An intelligent system for automated modelling and simulation of IT has been developed to obtain the "best" mathematical model for a particular situation and then to obtain the "best" simulations for the model. The use of fuzzy logic techniques enables automated modelling of IT using as input real data (time series) for this problem and the use of expert systems technology enables the selection of the appropriate parameter values for the simulation, to obtain all the dynamic behaviors of the corresponding mathematical model. The importance of modelling, simulation and forecasting of IT can be measured if one considers that one of the goals for a specific country is to find the optimum benefit from its international trade with other countries.

We describe a new method for performing automated mathematical modelling for financial time series prediction using fuzzy logic techniques, dynamical systems and fractal theory. The main idea is that using fuzzy logic techniques we can simulate and automate the reasoning process of human experts in mathematical modelling for financial time series prediction. Our new method for automated modelling consists of three main parts: time series analysis, developing a set of admissible models, and selecting the "best" model. Our method for time series analysis consists of using the fractal dimension of a set of points as a measure of the geometrical complexity of the time series. Our method for developing a set of admissible dynamical systems models is based on the use of fuzzy logic techniques to simulate the decision process of the human experts in modelling financial problems. The selection of the "best" model for financial time series prediction (FTSP) is done using heuristics from the experts and statistical calculations. This new method can be implemented as a computer program and can be considered an intelligent system for automated mathematical modelling for FTSP.

Describes a computer program that can be considered as an intelligent system for the domain of financial time series prediction. The computer program is an implementation of a new algorithm for discovering mathematical models for financial time series prediction, combining artificial intelligence methodology with dynamical systems theory, fractal theory and statistical methods. Given a financial time series for an specific problem, the intelligent system develops mathematical models for the problem based on the geometry of the data, using three different approaches. First, the computer program develops regression models for the time series using traditional statistical methods, then it develops nonlinear mathematical models based on dynamical systems theory and chaos theory, and finally it develops fractal mathematical models based on the theory of fractal geometry. The intelligent system then analyzes all of the mathematical models obtained before making a selection of the model that gives the "best" prediction for the financial time series. This selection is done by the intelligent system using a combination of heuristics and calculations that are contained in the knowledge base. An intelligent system that can learn models from financial data would be very useful in practice in making the task of prediction easier and less time-consuming.