Industrial applications of type-2 fuzzy sets and systems: A concise review

Data, as being the vital input of system modelling, contain dissimilar level of imprecision that necessitates different modelling approaches for proper analysis of the systems. Numbers, words and perceptions are the forms of data that has varying levels of imprecision. Existing approaches in the literature indicate that, computation of different data forms are closely linked with the level of imprecision, which the data already have. Traditional mathematical modelling techniques have been used to compute the numbers that have the least imprecision. Type-1 fuzzy sets have been used for words and type-2 fuzzy sets have been employed for perceptions where the level of imprecision is relatively high. However, in many cases it has not been easy to decide whether a solution requires a traditional approach, i.e., type-1 fuzzy approach or type-2 fuzzy approach. It has been a difficult matter to decide what types of problems really require modelling and solution either with type-1 or type-2 fuzzy approach. It is certain that, without properly distinguishing differences between the two approaches, application of type-1 and type-2 fuzzy sets and systems would probably fail to develop robust and reliable solutions for the problems of industry. In this respect, a review of the industrial applications of type-2 fuzzy sets, which are relatively novel to model imprecision has been considered in this work. The fundamental focus of the work has been based on the basic reasons of the need for type-2 fuzzy sets for the existing studies. With this purpose in mind, type-2 fuzzy sets articles have been selected from the literature using the online databases of ISI-Web of Science, ScienceDirect, SpringerLink, Informaworld, Engineering Village, Emerald and IEEE Xplore. Both the terms “type-2 fuzzy” and “application” have been searched as the main keywords in the topics of the studies to retrieve the relevant works. The analysis on the industrial applications of type-2 fuzzy sets/systems (FSs) in different topics allowed us to summarize the existing research areas and therefore it is expected be useful to prioritize future research topics. This review shows that there are still many opportunities for application of type-2 FSs for several different problem domains. Shortcomings of type-1 FSs can also be considered as an opportunity for the application of type-2 FSs in order to provide a better solution approach for industrial problems.     

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.     

Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets

In this paper, we present a new method to deal with fuzzy multiple attributes group decision-making problems based on ranking interval type-2 fuzzy sets. First, we propose a new method for ranking interval type-2 fuzzy sets. Then, we propose a new method for fuzzy multiple attributes group decision-making based on the proposed ranking method of interval type-2 fuzzy sets. We also use some examples to illustrate the fuzzy multiple attributes group decision-making process of the proposed method. The proposed method is simpler than the methods presented in [Chen and Lee, 2010a] and [Lee and Chen, 2010b] for fuzzy multiple attributes group decision-making based on interval type-2 fuzzy sets. It provides us with a useful way for dealing with fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets.     

基于包含度的区间二型模糊粗糙集

基于区间二型模糊包含度的公理化定义,给出了新的区间二型模糊包含度计算公式.进一步,通过包含度定义了区间二型模糊粗糙集,并讨论了它的一些基本性质.最后,利用区间二型模糊粗糙集研究了连续域决策信息系统的属性约简,给出了新的约简方法.实例说明了该约简方法的具体计算步骤,并且通过实验验证了该算法的有效性和可行性.     

二型直觉模糊集

二型模糊集和直觉模糊集都具有很强的实际应用背景.二型模糊集增强了系统处理不确定性的能力,直觉模糊集为解决人们判断问题所出现的犹豫信息提供了理论依据.本文在二型模糊集和直觉模糊集的基础上,给出了二型直觉模糊集的概念,证明了二型直觉模糊集是一型模糊集、直觉模糊集、区间值模糊集、区间值直觉模糊集的广义形式,讨论了二型直觉模糊集的基本运算和二型直觉模糊关系.最后,研究了基于二型直觉模糊理论的近似推理,并实例说明了二型直觉模糊集的实际应用背景.     

Ⅱ-型模糊集截集的定义及特征研究

经典模糊集的截集概念是经典模糊集合与经典集合联系的桥梁,对于II-型模糊集,该文在分析II-型模糊集、区间值II-型模糊集、I-型模糊集以及经典集合之间关系的基础上,定义了II-型模糊集的截集概念,分析了II-型模糊集截集的特征,仿真证明了II-型模糊集截集的有效性,为基于II-型模糊集的决策、聚类等实际应用提供了新的方法。     

Computing with words and its relationships with fuzzistics

Words mean different things to different people, and so are uncertain. We, therefore, need a fuzzy set model for a word that has the potential to capture their uncertainties. In this paper I propose that an interval type-2 fuzzy set (IT2 FS) be used as a FS model of a word, because it is characterized by its footprint of uncertainty (FOU), and therefore has the potential to capture word uncertainties. Two approaches are presented for collecting data about a word from a group of subjects and then mapping that data into a FOU for that word. The person MF approach, in which each person provides their FOU for a word, is limited to fuzzy set experts because it requires the subject to be knowledgeable about fuzzy sets. The interval end-points approach, in which each person provides the end-points for an interval that they associate with a word on a prescribed scale is not limited to fuzzy set experts. Both approaches map data collected from subjects into a parsimonious parametric model of a FOU, and illustrate the combining of fuzzy sets and statistics—type-2 fuzzistics.     

Design of interval type-2 fuzzy sliding-mode controller

In this paper, an interval type-2 fuzzy sliding-mode controller (IT2FSMC) is proposed for linear and nonlinear systems. The proposed IT2FSMC is a combination of the interval type-2 fuzzy logic control (IT2FLC) and the sliding-mode control (SMC) which inherits the benefits of these two methods. The objective of the controller is to allow the system to move to the sliding surface and remain in on it so as to ensure the asymptotic stability of the closed-loop system. The Lyapunov stability method is adopted to verify the stability of the interval type-2 fuzzy sliding-mode controller system. The design procedure of the IT2FSMC is explored in detail. A typical second order linear interval system with 50% parameter variations, an inverted pendulum with variation of pole characteristics, and a Duffing forced oscillation with uncertainty and disturbance are adopted to illustrate the validity of the proposed method. The simulation results show that the IT2FSMC achieves the best tracking performance in comparison with the type-1 Fuzzy logic controller (T1FLC), the IT2FLC, and the type-1 fuzzy sliding-mode controller (T1FSMC).     

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.     

A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks

In real life, information about the world is uncertain and imprecise. The cause of this uncertainty is due to: deficiencies on given information, the fuzzy nature of our perception of events and objects, and on the limitations of the models we use to explain the world. The development of new methods for dealing with information with uncertainty is crucial for solving real life problems. In this paper three interval type-2 fuzzy neural network (IT2FNN) architectures are proposed, with hybrid learning algorithm techniques (gradient descent backpropagation and gradient descent with adaptive learning rate backpropagation). At the antecedents layer, a interval type-2 fuzzy neuron (IT2FN) model is used, and in case of the consequents layer an interval type-1 fuzzy neuron model (IT1FN), in order to fuzzify the rule’s antecedents and consequents of an interval type-2 Takagi-Sugeno-Kang fuzzy inference system (IT2-TSK-FIS). IT2-TSK-FIS is integrated in an adaptive neural network, in order to take advantage the best of both models. This provides a high order intuitive mechanism for representing imperfect information by means of use of fuzzy If-Then rules, in addition to handling uncertainty and imprecision. On the other hand, neural networks are highly adaptable, with learning and generalization capabilities. Experimental results are divided in two kinds: in the first one a non-linear identification problem for control systems is simulated, here a comparative analysis of learning architectures IT2FNN and ANFIS is done. For the second kind, a non-linear Mackey-Glass chaotic time series prediction problem with uncertainty sources is studied. Finally, IT2FNN proved to be more efficient mechanism for modeling real-world problems.     

Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions

In this paper, we deal with the problem of classification of interval type-2 fuzzy sets through evaluating their distinguishability. To this end, we exploit a general matching algorithm to compute their similarity measure. The algorithm is based on the aggregation of two core similarity measures applied independently on the upper and lower membership functions of the given pair of interval type-2 fuzzy sets that are to be compared. Based on the proposed matching procedure, we develop an experimental methodology for evaluating the distinguishability of collections of interval type-2 fuzzy sets. Experimental results on evaluating the proposed methodology are carried out in the context of classification by considering interval type-2 fuzzy sets as patterns of suitable classification problem instances. We show that considering only the upper and lower membership functions of interval type-2 fuzzy sets is sufficient to (i) accurately discriminate between them and (ii) judge and quantify their distinguishability.     

二型直觉模糊粗糙集

将二型直觉模糊集和粗糙集理论融合,建立二型直觉模糊粗糙集模型。首先,在二型直觉模糊近似空间中,定义了一对二型直觉模糊上、下近似算子,并讨论了二型直觉模糊关系退化为普通二型模糊关系和一般等价关系时,上、下近似算子的具体变化形式。然后,将普通二型模糊集之间包含关系的定义推广到了二型直觉模糊集,在此基础上研究了二型直觉模糊上、下近似算子的一些性质。最后,定义了自反的、对称的和传递的二型直觉模糊关系,并讨论了这3种特殊的二型直觉模糊关系与近似算子的特征之间的联系。该结论进一步丰富了二型模糊集理论和粗糙集理论,为二型直觉模糊信息系统的应用奠定了良好的理论基础。     

On type-reduction of type-2 fuzzy sets: A review

As an undetachable module of type-2 (T2) fuzzy computations and reasoning, type-reduction methods play an important role in various fuzzy disciplines including fuzzy logic systems and fuzzy clustering. Importance of type-reduction techniques lies in the fact that they are the main tools for collecting the entire inherent vagueness of the data. Therefore, type-reduction methods form the output of type-2 fuzzy sets (T2 FSs) as the representative of the entire uncertainty in a given space. Hence, their accuracy, precision, and performance speed is of much interest. This paper, presents a comprehensive review on various type-reduction and defuzzification strategies for general and interval type-2 fuzzy sets and systems. It is tried to analyze the existing approaches from different point of views accompanied by extensive comparisons on different features of type-reduction methods to facilitate further research studies by the fuzzy community.     

Bearing condition prediction considering uncertainty: An interval type-2 fuzzy neural network approach

Rolling-element bearings are critical components of rotating machinery. It is important to accurately predict in real-time the health condition of bearings so that maintenance practices can be scheduled to avoid malfunctions or even catastrophic failures. In this paper, an Interval Type-2 Fuzzy Neural Network (IT2FNN) is proposed to perform multi-step-ahead condition prediction of faulty bearings. Since the IT2FNN defines an interval type-2 fuzzy logic system in the form of a multi-layer neural network, it can integrate the merits of each, such as fuzzy reasoning to handle uncertainties and neural networks to learn from data. The interval type-2 fuzzy linguistic process in the IT2FNN enables the system to handle prediction uncertainties, since the type-2 fuzzy sets are such sets whose membership grades are type-1 fuzzy sets that can be used in failure prediction due to the difficult determination of an exact membership function for a fuzzy set. Noisy data of faulty bearings are used to validate the proposed predictor, whose performance is compared with that of a prevalent type-1 condition predictor called Adaptive Neuro-Fuzzy Inference System (ANFIS). The results show that better prediction accuracy can be achieved via the IT2FNN.     

New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule

The centroid of an interval type-2 fuzzy set (IT2 FS) provides a measure of the uncertainty of such a FS. Its calculation is very widely used in interval type-2 fuzzy logic systems. In this paper, we present properties about the centroid of an IT2 FS. We also illustrate many of the general results for a T2 fuzzy granule (FG) in order to develop some understanding about the uncertainty of the FG in terms of its vertical and horizontal dimensions. At present, the T2 FG is the only IT2 FS for which it is possible to obtain closed-form formulas for the centroid, and those formulas are in this paper.     

II型模糊集合与系统研究进展

Ⅱ型模糊集合是传统Ⅰ型模糊集合的扩展,其本质是模糊集合中隶属度值的再次模糊化表示.Ⅱ型模糊集合可以直接处理模糊规则的不确定性,是解决现实环境高不确定性问题的有效手段.本文首先简要给出了Ⅱ型模糊集合与系统的基本概念,然后分别回顾了广义和区间Ⅱ型模糊理论的发展历史.接着分别讨论了广义和区间Ⅱ型模糊系统的计算复杂性问题研究进展,并进一步介绍了基于区间Ⅱ型模糊集合的词计算理论发展状况.最后给出了本文的结论和进一步研究问题的展望.     

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.     

Type-2 fuzzy sets applied to multivariable self-organizing fuzzy logic controllers for regulating anesthesia

In this paper, novel interval and general type-2 self-organizing fuzzy logic controllers (SOFLCs) are proposed for the automatic control of anesthesia during surgical procedures. The type-2 SOFLC is a hierarchical adaptive fuzzy controller able to generate and modify its rule-base in response to the controller's performance. The type-2 SOFLC uses type-2 fuzzy sets derived from real surgical data capturing patient variability in monitored physiological parameters during anesthetic sedation, which are used to define the footprint of uncertainty (FOU) of the type-2 fuzzy sets. Experimental simulations were carried out to evaluate the performance of the type-2 SOFLCs in their ability to control anesthetic delivery rates for maintaining desired physiological set points for anesthesia (muscle relaxation and blood pressure) under signal and patient noise. Results show that the type-2 SOFLCs can perform well and outperform previous type-1 SOFLC and comparative approaches for anesthesia control producing lower performance errors while using better defined rules in regulating anesthesia set points while handling the control uncertainties. The results are further supported by statistical analysis which also show that zSlices general type-2 SOFLCs are able to outperform interval type-2 SOFLC in terms of their steady state performance.     

Ⅱ型模糊控制综述

Ⅱ型模糊集合是传统Ⅰ型模糊集合的扩展,其特征是隶属度值本身为模糊集合.基于Ⅱ型模糊集合的Ⅱ型模糊控制器可以同时有效地处理语言和数据不确定性,在高小确定场合具有明显超过相应Ⅰ型控制器的性能表现.本文首先对Ⅱ型模糊集合及系统理论进行了概述,然后对Ⅱ型非自适应模糊控制器Ⅱ型自适应模糊控制器和Ⅱ型自组织模糊控制器的研究进展分别...     

二型模糊粗糙集

基于二型模糊关系,研究二型模糊粗糙集.首先,在二型模糊近似空间中定义了二型模糊集的上近似和下近似;然后,研究二型模糊粗糙上下近似算子的基本性质,讨论二型模糊关系与二型模糊粗糙近似算子的特征联系;最后,给出二型模糊粗糙近似算子的公理化描述.