Comparative analysis of three approaches of antecedent part generation for an IT2 TSK FLS

Since extreme learning machine is a non-iterative estimation procedure, it is faster than gradient-based algorithms which are iterative. Moreover, the extreme learning machine does not have any design parameters such as learning rate, covariance matrix, etc. The rigorous proof of universal approximation of extreme learning machine with much milder conditions makes it a preferable choice in many different approaches. Although this algorithm is optimal for the parameters which appear linearly in the consequent part of interval type-2 fuzzy logic systems, it is not optimal for the parameters of the antecedent part as it uses random parameters. In this paper, heuristic optimization approaches such as genetic algorithm and artificial bee colony are used to optimize the parameters of the antecedent part of interval type-2 fuzzy logic systems. As these methods are global optimizers, there is less possibility that they will fall in a local minima and are suitable for the selection of the parameters of the antecedent part. A comparative analysis of the optimal parameters with the randomly and manually generated parameters is presented here using noise-free and noisy Mackey-Glass time series data sets and a real world data set. Simulation results support this idea over randomly and manually generated parameters.     

Hybrid learning for interval type-2 fuzzy logic systems based on orthogonal least-squares and back-propagation methods

This paper presents a novel learning methodology based on a hybrid algorithm for interval type-2 fuzzy logic systems. Since only the back-propagation method has been proposed in the literature for the tuning of both the antecedent and the consequent parameters of type-2 fuzzy logic systems, a hybrid learning algorithm has been developed. The hybrid method uses a recursive orthogonal least-squares method for tuning the consequent parameters and the back-propagation method for tuning the antecedent parameters. Systems were tested for three types of inputs: (a) interval singleton, (b) interval type-1 non-singleton, and (c) interval type-2 non-singleton. Experiments were carried out on the application of hybrid interval type-2 fuzzy logic systems for prediction of the scale breaker entry temperature in a real hot strip mill for three different types of coil. The results proved the feasibility of the systems developed here for scale breaker entry temperature prediction. Comparison with type-1 fuzzy logic systems shows that hybrid learning interval type-2 fuzzy logic systems provide improved performance under the conditions tested.     

Experimental study of intelligent controllers under uncertainty using type-1 and type-2 fuzzy logic

Uncertainty is an inherent part in control 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 control systems. Type-2 fuzzy sets that are used in type-2 fuzzy systems can handle such uncertainties in a better way because they provide us with more parameters and more design degrees of freedom. This paper deals with the design of control systems using type-2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements, environmental noise, etc. The experimental results are divided in two classes, in the first class, simulations of a feedback control system for a non-linear plant using type-1 and type-2 fuzzy logic controllers are presented; a comparative analysis of the systems’ response in both cases was performed, with and without the presence of uncertainty. For the second class, a non-linear identification problem for time-series prediction is presented. Based on the experimental results the conclusion is that the best results are obtained using type-2 fuzzy systems.     

Interval type-2 fuzzy inverse controller design in nonlinear IMC structure

In the recent years it has been demonstrated that type-2 fuzzy logic systems are more effective in modeling and control of complex nonlinear systems compared to type-1 fuzzy logic systems. An inverse controller based on type-2 fuzzy model can be proposed since inverse model controllers provide an efficient way to control nonlinear processes. Even though various fuzzy inversion methods have been devised for type-1 fuzzy logic systems up to now, there does not exist any method for type-2 fuzzy logic systems. In this study, a systematic method has been proposed to form the inverse of the interval type-2 Takagi-Sugeno fuzzy model based on a pure analytical method. The calculation of inverse model is done based on simple manipulations of the antecedent and consequence parts of the fuzzy model. Moreover, the type-2 fuzzy model and its inverse as the primary controller are embedded into a nonlinear internal model control structure to provide an effective and robust control performance. Finally, the proposed control scheme has been implemented on an experimental pH neutralization process where the beneficial sides are shown clearly.     

Optimization of interval type-2 fuzzy systems for image edge detection

This paper presents the optimization of a fuzzy edge detector based on the traditional Sobel technique combined with interval type-2 fuzzy logic. The goal of using interval type-2 fuzzy logic in edge detection methods is to provide them with the ability to handle uncertainty in processing real world images. However, the optimal design of fuzzy systems is a difficult task and for this reason the use of meta-heuristic optimization techniques is also considered in this paper. For the optimization of the fuzzy inference systems, the Cuckoo Search (CS) and Genetic Algorithms (GAs) are applied. Simulation results show that using an optimal interval type-2 fuzzy system in conjunction with the Sobel technique provides a powerful edge detection method that outperforms its type-1 counterparts and the pure original Sobel technique.     

Optimization of interval type-2 fuzzy logic controllers using evolutionary algorithms

A method for designing optimal interval type-2 fuzzy logic controllers using evolutionary algorithms is presented in this paper. Interval type-2 fuzzy controllers can outperform conventional type-1 fuzzy controllers when the problem has a high degree of uncertainty. However, designing interval type-2 fuzzy controllers is more difficult because there are more parameters involved. In this paper, interval type-2 fuzzy systems are approximated with the average of two type-1 fuzzy systems, which has been shown to give good results in control if the type-1 fuzzy systems can be obtained appropriately. An evolutionary algorithm is applied to find the optimal interval type-2 fuzzy system as mentioned above. The human evolutionary model is applied for optimizing the interval type-2 fuzzy controller for a particular non-linear plant and results are compared against an optimal type-1 fuzzy controller. A comparative study of simulation results of the type-2 and type-1 fuzzy controllers, under different noise levels, is also presented. Simulation results show that interval type-2 fuzzy controllers obtained with the evolutionary algorithm outperform type-1 fuzzy controllers.     

Type-1 and type-2 fuzzy inference systems as integration methods in modular neural networks for multimodal biometry and its optimization with genetic algorithms

We describe in this paper a comparative study between fuzzy inference systems as methods of integration in modular neural networks for multimodal biometry. These methods of integration are based on techniques of type-1 fuzzy logic and type-2 fuzzy logic. Also, the fuzzy systems are optimized with simple genetic algorithms with the goal of having optimized versions of both types of fuzzy systems. 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 fuzzy systems automatically. Then the response integration of the modular neural network was tested with the optimized fuzzy systems of integration. The comparative study of the type-1 and type-2 fuzzy inference systems was made to observe the behavior of the two different integration methods for modular neural networks for multimodal biometry.     

Intuit before tuning: Type-1 and type-2 fuzzy logic controllers

Although a considerable amount of effort has been put in to show that fuzzy logic controllers have exceptional capabilities of dealing with uncertainty, there are still noteworthy concerns, e.g., the design of fuzzy logic controllers is an arduous task due to the lack of closed-form input–output relationships which is a limitation to interpretability of these controllers. The role of design parameters in fuzzy logic controllers, such as position, shape, and height of membership functions, is not straightforward. Motivated by the fact that the availability of an interpretable relationship from input to output will simplify the design procedure of fuzzy logic controllers, the main aims in this work are derive fuzzy mappings for both type-1 and interval type-2 fuzzy logic controllers, analyse them, and eventually benefit from such a nonlinear mapping to design fuzzy logic controllers. Thereafter, simulation and real-time experimental results support the presented theoretical findings.     

Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2 fuzzy sets

In this study, a new approach for the formation of type-2 membership functions is introduced. The footprint of uncertainty is formed by using rectangular type-2 fuzzy granules and the resulting membership function is named as granular type-2 membership function. This new approach provides more degrees of freedom and design flexibility in type-2 fuzzy logic systems. Uncertainties on the grades of membership functions can be represented independently for any region in the universe of discourse and free of any functional form. So, the designer could produce nonlinear, discontinuous or hybrid membership functions in granular formation and therefore could model any desired discontinuity and nonlinearity. The effectiveness of the proposed granular type-2 membership functions is firstly demonstrated by simulations done on noise corrupted Mackey–Glass time series prediction. Secondly, flexible design feature of granular type-2 membership functions is illustrated by modeling a nonlinear system having dead zone with uncertain system parameters. The simulation results show that type-2 fuzzy logic systems formed by granular type-2 membership functions have more modeling capabilities than the systems using conventional type-2 membership functions and they are more robust to system parameter changes and noisy inputs.     

A review on the design and optimization of interval type-2 fuzzy controllers

A review of the methods used in the design of interval type-2 fuzzy controllers has been considered in this work. The fundamental focus of the work is based on the basic reasons for optimizing type-2 fuzzy controllers for different areas of application. Recently, bio-inspired methods have emerged as powerful optimization algorithms for solving complex problems. In the case of designing type-2 fuzzy controllers for particular applications, the use of bio-inspired optimization methods have helped in the complex task of finding the appropriate parameter values and structure of the fuzzy systems. In this review, we consider the application of genetic algorithms, particle swarm optimization and ant colony optimization as three different paradigms that help in the design of optimal type-2 fuzzy controllers. We also mention alternative approaches to designing type-2 fuzzy controllers without optimization techniques. We also provide a comparison of the different optimization methods for the case of designing type-2 fuzzy controllers.     

Modeling the permeability of carbonate reservoir using type-2 fuzzy logic systems

In this work, the use of type-2 fuzzy logic systems as a novel approach for predicting permeability from well logs has been investigated and implemented. Type-2 fuzzy logic system is good in handling uncertainties, including uncertainties in measurements and data used to calibrate the parameters. In the formulation used, the value of a membership function corresponding to a particular permeability value is no longer a crisp value; rather, it is associated with a range of values that can be characterized by a function that reflects the level of uncertainty. In this way, the model will be able to adequately account for all forms of uncertainties associated with predicting permeability from well log data, where uncertainties are very high and the need for stable results are highly desirable. Comparative studies have been carried out to compare the performance of the proposed type-2 fuzzy logic system framework with those earlier used methods, using five different industrial reservoir data. Empirical results from simulation show that type-2 fuzzy logic approach outperformed others in general and particularly in the area of stability and ability to handle data in uncertain situations, which are common characteristics of well logs data. Another unique advantage of the newly proposed model is its ability to generate, in addition to the normal target forecast, prediction intervals as its by-products without extra computational cost.     

Interval type-2 neuro-fuzzy system with implication-based inference mechanism

Neuro-fuzzy systems have been proved to be an efficient tool for modelling real life systems. They are precise and have ability to generalise knowledge from presented data. Neuro-fuzzy systems use fuzzy sets – most commonly type-1 fuzzy sets. Type-2 fuzzy sets model uncertainties better than type-1 fuzzy sets because of their fuzzy membership function. Unfortunately computational complexity of type reduction in general type-2 systems is high enough to hinder their practical application. This burden can be alleviated by application of interval type-2 fuzzy sets. The paper presents an interval type-2 neuro-fuzzy system with interval type-2 fuzzy sets both in premises (Gaussian interval type-2 fuzzy sets with uncertain fuzziness) and consequences (trapezoid interval type-2 fuzzy set). The inference mechanism is based on the interval type-2 fuzzy Łukasiewicz, Reichenbach, Kleene-Dienes, or Brouwer–Gödel implications. The paper is accompanied by numerical examples. The system can elaborate models with lower error rate than type-1 neuro-fuzzy system with implication-based inference mechanism. The system outperforms some known type-2 neuro-fuzzy systems.     

A simple design method for interval type-2 fuzzy pid controllers

In this study, a design method for single Input interval type-2 fuzzy PID controller has been developed. The most important feature of the proposed type-2 fuzzy controller is its simple structure consisting of a single input variable. The presented simple structure gives an opportunity to the designer to form the type-2 fuzzy controller output in closed form formulation for the first time in literature. This formulation cannot be achieved with present type-2 fuzzy PID controller structures which have employed the Karnik-Mendel type reduction. The closed form solution is derived in terms of the tuning parameters which are chosen as the heights of lower membership functions of the antecedent interval type-2 fuzzy sets. Elaborations are done on the derived closed form output and a simple strategy is presented for a single input type-2 fuzzy PID controller design. The presented interval type-2 fuzzy controller structure still keeps the most preferred features of the PID controller such as simplicity and easy design. We will illustrate how the extra degrees of freedom provided by the antecedent interval type-2 fuzzy sets can be used to enhance the control performance on linear and nonlinear benchmark systems by simulations. Moreover, the type-2 fuzzy controller structure has been implemented on experimental pH neutralization. The simulation and experimental results will illustrate that the proposed type-2 fuzzy controller produces superior control performance and can handle nonlinear dynamics, parameter uncertainties, noise and disturbances better in comparison with the standard PID controllers. Hence, the results and analyses of this study will give the control engineers an opportunity to draw a bridge and connect the type-2 fuzzy logic and control theory.     

Genetic learning and performance evaluation of interval type-2 fuzzy logic controllers

Type-2 fuzzy sets, which are characterized by membership functions (MFs) that are themselves fuzzy, have been attracting interest. This paper focuses on advancing the understanding of interval type-2 fuzzy logic controllers (FLCs). First, a type-2 FLC is evolved using Genetic Algorithms (GAs). The type-2 FLC is then compared with another three GA evolved type-1 FLCs that have different design parameters. The objective is to examine the amount by which the extra degrees of freedom provided by antecedent type-2 fuzzy sets is able to improve the control performance. Experimental results show that better control can be achieved using a type-2 FLC with fewer fuzzy sets/rules so one benefit of type-2 FLC is a lower trade-off between modeling accuracy and interpretability.     

Application of interval type-2 fuzzy neural networks in non-linear identification and time series prediction

Neural networks (NNs), type-1 fuzzy logic systems and interval type-2 fuzzy logic systems (IT2FLSs) have been shown to be important methods in real world applications, which range from pattern recognition, time series prediction, to intelligent control. Recent research shows that embedding an IT2FLS on an NN can be very effective for a wide number of non-linear complex systems, especially when handling imperfect or incomplete information. In this paper we are presenting several models of interval type-2 fuzzy neural networks (IT2FNNs) that use a set of rules and interval type-2 membership functions for that purpose. Simulation results of non-linear function identification using the IT2FNN for one and three variables and for the Mackey–Glass chaotic time series prediction are presented to illustrate that the proposed models have potential for real world applications.     

A type-2 fuzzy logic controller design for buck and boost DC–DC converters

Conventional (type-1) fuzzy logic controllers have been commonly used in various power converter applications. Generally, in these controllers, the experience and knowledge of human experts are needed to decide parameters associated with the rule base and membership functions. The rule base and the membership function parameters may often mean different things to different experts. This may cause rule uncertainty problems. Consequently, the performance of the controlled system, which is controlled with type-1 fuzzy logic controller, is undesirably affected. In this study, a type-2 fuzzy logic controller is proposed for the control of buck and boost DC–DC converters. To examine and analysis the effects of the proposed controller on the system performance, both converters are also controlled using the PI controller and conventional fuzzy logic controller. The settling time, the overshoot, the steady state error and the transient response of the converters under the load and input voltage changes are used as the performance criteria for the evaluation of the controller performance. Simulation results show that buck and boost converters controlled by type-2 fuzzy logic controller have better performance than the buck and boost converters controlled by type-1 fuzzy logic controller and PI controller.     

Parallel Interval Type-2 Subsethood Neural Fuzzy Inference System

Neuro-fuzzy models are being increasingly employed in the domains like weather forecasting, stock market prediction, computational finance, control, planning, physics, economics and management, to name a few. These models enable one to predict system behavior in a more human-like manner than their crisp counterparts. In the present work, an interval type-2 neuro-fuzzy evolutionary subsethood based model has been proposed for its use in finding solutions to some well-known problems reported in the literature such as regression analysis, data mining and research problems relevant to expert and intelligent systems. A novel subsethood based interval type-2 fuzzy inference system, named as Interval Type-2 Subsethood Neural Fuzzy Inference System (IT2SuNFIS) is proposed in the present work. Mathematical modeling and empirical studies clearly bring out the efficacy of this model in a wide variety of practical problems such as Truck backer-upper control, Mackey–Glass time-series prediction, Narazaki–Ralescu and bell function approximation. The simulation results demonstrate intelligent decision making capability of the proposed system based on the available data. The major contribution of this work lies in identifying subsethood as an efficient measure for finding correlation in interval type-2 fuzzy sets and applying this concept to a wide variety of problems pertaining to expert and intelligent systems. Subsethood between two type-2 fuzzy sets is different from the commonly used sup-star methods. In the proposed model, this measure assists in providing better contrast between dissimilar objects. This method, coupled with the uncertainty handling capacity of type-2 fuzzy logic system, results in better trainability and improved performance of the system. The integration of subsethood with type-2 fuzzy logic system is a novel idea with several advantages, which is reported for the first time in this paper.     

Interval type-2 fuzzy expert system for prediction of carbon monoxide concentration in mega-cities

This paper presents an indirect approach to interval type-2 fuzzy logic system modeling to forecaste the level of air pollutants. The type-2 fuzzy logic system permits us to model the uncertainties among rules and the parameters related to data analysis. In this paper, we propose an indirect method to create an interval type-2 fuzzy logic system from a historical data, where Footprint of Uncertainties of fuzzy sets are extracted by implementation of an interval type-2 FCM algorithm and based on an upper and lower value for the level of fuzziness m in FCM. Finally, the proposed model is applied for prediction of carbon monoxide concentration in Tehran air pollution. It is shown that the proposed type-2 fuzzy logic system is superior in comparison to type-1 fuzzy logic systems in terms of two performance indices.     

Ant colony optimization with dynamic parameter adaptation based on interval type-2 fuzzy logic systems

A dynamic parameter adaptation methodology for Ant Colony Optimization (ACO) based on interval type-2 fuzzy systems is presented in this paper. The idea is to be able to apply this new ACO method with parameter adaptation to a wide variety of problems without the need of finding the best parameters for each particular problem. We developed several fuzzy systems for parameter adaptation and a comparison was made among them to decide on the best design. The use of fuzzy logic is to control the diversity of the solutions, and in this way controlling the exploration and exploitation abilities of ACO. The travelling salesman problem (TSP) and the design of a fuzzy controller for an autonomous mobile robot are the benchmark problems used to test the proposed methodology.     

Optimal design of interval type 2 fuzzy controllers based on a simple tuning algorithm

Evolutionary algorithms are one of the most common choices reported in the literature for the tuning of fuzzy logic controllers based on either type-1 or type-2 fuzzy systems. An alternative to evolutionary algorithms is the simple tuning algorithm (STA-FLC), which is a methodology designed to improve the response of type-1 fuzzy logic controllers in a practical, intuitive and simple ways. This paper presents an extension of the simple tuning algorithm for fuzzy logic controllers based on the theory of type-2 fuzzy systems by using a parallel model implementation, it also includes a mechanism to calculate the feedback gain, new integral criteria parameters, and the effect of the AND/OR operator combinations on the fuzzy rules to improve the algorithm applicability and performance. All these improvements are demonstrated with experiments applied to different types of plants.