Indirect adaptive fuzzy observer and controller design based on interval type-2 T-S fuzzy model

This paper presents the design scheme of the indirect adaptive fuzzy observer and controller based on the interval type-2 (IT2) T-S fuzzy model. The nonlinear systems can be well approximated by IT2 T-S fuzzy model, in which the fuzzy rules’ antecedents are interval type-2 fuzzy sets and consequents are linear state equations. The proposed IT2 T-S fuzzy model is a combination of IT2 fuzzy system and T-S fuzzy model, and also inherits the benefits of type-2 fuzzy logic systems, which is able to directly handle uncertainties and can minimize the effects of uncertainties in rule-based fuzzy system. These characteristics can improve the accuracy of the system modeling and reduce the number of system rules. The proposed method using feedback control, adaptive laws, and on-line object parameters are adjusted to ensure observation error bounded. In addition, using Lyapunov synthesis approach and Lipschitz condition, the stability analysis is conducted. The simulation results show that the proposed method can handle unpredicted disturbance and data uncertainties very well in advantage of the effectiveness of observation and control.     

基于Nie-Tan算法的区间二型模糊集质心计算:改变主变量采样个数

计算区间二型模糊集的质心(也称降型)是区间二型模糊逻辑系统中的一个重要模块。Karnik-Mendel(KM)迭代算法通常被认为是计算区间二型模糊集质心的标准算法。尽管如此,KM算法涉及复杂的计算过程,不利于实时应用。在各种改进类算法中,非迭代的Nie-Tan(NT)算法可节省计算消耗。此外,连续版本NT(CNT,continuous version of NT)算法被证明是计算质心的准确算法。本文比较了离散版本NT算法中求和运算和连续版本NT算法中求积分运算,通过四个计算机仿真例子证实了当适度增加区间二型模糊集主变量采样个数时,NT算法的计算结果可以精确地逼近CNT算法。     

Explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents

This paper is devoted to the inversion of fuzzy systems expressed by fuzzy rules with singleton consequents if input variables are described using strong triangular partitions. As pointed out in recent works, such fuzzy systems can be decomposed into collections of multi-linear subsystems. In this paper, an analytical formulation of the system output is explicitly developed and directly used in order to determine solutions to the inversion problem. Based on this analytical methodology, an algorithm is proposed for computing inverse solutions. As the inversion is handled analytically, the exactness of the obtained solutions is guaranteed. Furthermore, according to the decomposability of the studied fuzzy systems, all inverse solutions are found. Finally, whatever the fuzzy system under consideration, there is no need to study its invertibility beforehand since the algorithm is able to handle all possible situations (no solution, one unique solution, multiple solutions, an infinity of solutions).The proposed approach can be easily extended to other types of fuzzy systems provided that decomposability is preserved. In other words, with regard to exact inversion which often plays a key role in engineering applications such as control or diagnosis, decomposability is probably the first criterion that should be considered when choosing a specific fuzzy system structure.     

Exact analytical inversion of TSK fuzzy systems with singleton and linear consequents

In literature, exact inversion methods for TSK fuzzy systems exist only for the systems with singleton consequents. These methods have binding limitations such as strong triangular partitioning, monotonic rule bases and/or invertibility check. These extra limitations lessen the modeling capabilities of the TSK fuzzy systems. In this study, an exact analytical inversion method for TSK fuzzy systems with singleton and linear consequents is presented. The only limitation of the proposed method is that the inversion variable should be represented by piecewise linear membership functions (PWL-MFs). In this case, the universe of discourse of the inversion variable is divided into specific regions in which only one linear piece exists for each PWL-MF at most. In the proposed method, the analytical formulation of TSK fuzzy system is expressed in terms of the inversion variable by using linear equations of PWL-MFs. Thus, the fuzzy system output in any region can be obtained by using the appropriate parameters of the linear equations of PWL-MFs defined within the related region. This expression provides a way to obtain linear and quadratic equations in terms of the inversion variable for TSK fuzzy systems with singleton and linear consequents, respectively. So, it becomes very easy to find exact inverse solutions for each region by using explicit analytical solutions for linear or quadratic equations. The proposed inversion method has been illustrated through simulation examples.     

Adaptive type-2 fuzzy backstepping control of uncertain fractional-order nonlinear systems with unknown dead-zone

A new problem of adaptive type-2 fuzzy fractional control with pseudo-state observer for commensurate fractional order dynamic systems with dead-zone input nonlinearity is considered in presence of unmatched disturbances and model uncertainties; the control scheme is constructed by using the backstepping and adaptive technique. To avoid the complexity of backstepping design process, the dynamic surface control is used. Also, Interval type-2 Fuzzy logic systems (IT2FLS) are used to approximate the unknown nonlinear functions. By using the fractional adaptive backstepping, fractional control laws are constructed; this method is applied to a class of uncertain fractional-order nonlinear systems. In order to better control performance in reducing tracking error, the PSO algorithm is utilized for tuning the controller parameters. Stability of the system is proven by the Mittag–Leffler method. It is shown that the proposed controller guarantees the boundedness property for the system and also the tracking error can converge to a small neighborhood of the origin. The efficiency of the proposed method is illustrated with simulation examples.     

Approximation of centroid end-points and switch points for replacing type reduction algorithms

Despite several years of research, type reduction (TR) operation in interval type-2 fuzzy logic system (IT2FLS) cannot perform as fast as a type-1 defuzzifier. In particular, widely used Karnik–Mendel (KM) TR algorithm is computationally much more demanding than alternative TR approaches. In this work, a data driven framework is proposed to quickly, yet accurately, estimate the output of the KM TR algorithm using simple regression models. Comprehensive simulation performed in this study shows that the centroid end-points of KM algorithm can be approximated with a mean absolute percentage error as low as 0.4%. Also, switch point prediction accuracy can be as high as 100%. In conjunction with the fact that simple regression model can be trained with data generated using exhaustive defuzzification method, this work shows the potential of proposed method to provide highly accurate, yet extremely fast, TR approximation method. Speed of the proposed method should theoretically outperform all available TR methods while keeping the uncertainty information intact in the process.     

A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets”

Zhang and Zhang (2013) proposed the arithmetic operations of trapezoidal interval type-2 fuzzy numbers having different left and right heights and hence the arithmetic operations of trapezoidal interval type-2 fuzzy soft sets having different left and right heights. In this paper, it is pointed out that the complement operation of a trapezoidal interval type-2 fuzzy number, proposed by Zhang and Zhang, is not valid and hence, the complement operation of trapezoidal interval type-2 fuzzy soft set as well as all the results, proposed by Zhang and Zhang in which complement operation is used, are not valid. The results, proposed by Zhang and Zhang, are valid only for such trapezoidal interval type-2 fuzzy numbers and trapezoidal interval type-2 fuzzy soft sets in which left and right heights are equal.     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

A cloud computing based Big-Bang Big-Crunch fuzzy logic multi classifier system for Soccer video scenes classification

Soccer video summarization and classification is becoming a very important topic due to the world wide importance and popularity of soccer games which drives the need to automatically classify video scenes thus enabling better sport analysis, refereeing, training, advertisement, etc. Machine learning has been applied to the task of sports video classification. However, for some specific image and video problems (like sports video scenes classification), the learning task becomes convoluted and difficult due to the dynamic nature of the video sequence and the associated uncertainties relating to changes in light conditions, background, camera angle, occlusions and indistinguishable scene features, etc. The majority of previous techniques (such as SVM, neural network, decision tree, etc.) applied to sports video classifications did not provide a consummate solution, and such models could not be easily understood by human users; meanwhile, they increased the complexity and time of computation and the associated costs of the involved standalone machines. Hence, there is a need to develop a system which is able to address these drawbacks and handle the high levels of uncertainty in video scenes classification and undertake the heavy video processing securely and efficiently on a cloud computing based instance. Hence, in this paper we present a cloud computing based multi classifier systems which aggregates three classifiers based on neural networks and two fuzzy logic classifiers based on type-1 fuzzy logic and type-2 fuzzy logic classification systems which were optimized by a Big-Bang Big crunch optimization to maximize the system performance. We will present several real world experiments which shows the proposed classification system operating in real-time to produce high classification accuracies for soccer videos which outperforms the standalone classification systems based on neural networks, type-1 and type-2 fuzzy logic systems.     

基于模糊测度与累积前景理论的区间二型模糊多准则决策方法

针对准则值为区间二型模糊数且准则间存在关联关系的风险型多准则决策问题, 本文提出一种基于模糊测度理论与累积前景理论的区间二型模糊多准则决策方法。首先, 为全面反映准则间的关联关系, 本文提出Shapley区间二型模糊Choquet积分算子, 并证明该算子的一些性质。其次, 为反映专家行为偏好, 本文定义区间二型模糊前景效应与前景价值函数, 并提出累积前景Shapley区间二型模糊Choquet积分算子。然后, 为确定准则集的模糊测度, 本文建立基于区间二型模糊双向投影与Shapley函数的权重优化模型。在此基础上, 本文给出一种用于解决准则值为区间二型模糊数, 准则间存在关联关系, 专家存在风险偏好以及准则权重部分未知的多准则决策方法。最后, 通过风险投资实例佐证所提出的方法的适用性与科学性。     

Fuzzy events and fuzzy logics in classical information systems

Classical information systems are introduced in the framework of measure and integration theory. The measurable characteristic functions are identified with the exact events while the fuzzy events are the real measurable functions whose range is contained in the unit interval. Two orthogonality relations are introduced on fuzzy events, the first linked to the fuzzy logic and the second to the fuzzy structure of partial a Baer¹-ring. The fuzzy logic is then compared with the “empirical” fuzzy logic induced by the classical information system. In this context, quantum logics could be considered as those empirical fuzzy logics in which it is not possible to have preparation procedures which provide physical systems whose “microstate” is always exactly defined.     

Comment on “Based on interval type-2 adaptive fuzzy H∞ tracking controller for SISO time-delay nonlinear systems”

In this comment, we point out the inappropriateness of Theorem 1 in the article [Tsung-Chih Lin, Mehdi Roopaei. Based on interval type-2 adaptive fuzzy H^∞ tracking controller for SISO time-delay nonlinear systems. Commun Nonlinear Sci Numer Simulat 2010;15:4065–75]. For solving this problem, some formular mistakes are corrected and novel parameter adaptive laws of interval type-2 fuzzy neural network system are given.     

The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making

QUALIFLEX, a generalization of Jacquet-Lagreze’s permutation method, is a useful outranking method in decision analysis because of its flexibility with respect to cardinal and ordinal information. This paper develops an extended QUALIFLEX method for handling multiple criteria decision-making problems in the context of interval type-2 fuzzy sets. Interval type-2 fuzzy sets contain membership values that are crisp intervals, which are the most widely used of the higher order fuzzy sets because of their relative simplicity. Using the linguistic rating system converted into interval type-2 trapezoidal fuzzy numbers, the extended QUALIFLEX method investigates all possible permutations of the alternatives with respect to the level of concordance of the complete preference order. Based on a signed distance-based approach, this paper proposes the concordance/discordance index, the weighted concordance/discordance index, and the comprehensive concordance/discordance index as evaluative criteria of the chosen hypothesis for ranking the alternatives. The feasibility and applicability of the proposed methods are illustrated by a medical decision-making problem concerning acute inflammatory demyelinating disease, and a comparative analysis with another outranking approach is conducted to validate the effectiveness of the proposed methodology.     

一种基于新的区间二型梯形模糊相似测度的多属性群决策方法

本文首先基于区间二型梯形模糊数的周长、面积、负指数距离提出了一种新的区间二型梯形模糊相似测度,讨论了其性质。其次,基于该相似测度公式分别构建了区间二型梯形模糊专家权重和属性权重确定模型,然后通过集结区间二型梯形模糊决策信息与权重信息,给出了一种基于该相似测度的群决策方法。最后,通过投资方案选择实例说明了该方法的合理性和有效性。     

Discovering the dynamic behavior of unknown systems using fuzzy logic

To know the dynamic behavior of a system it is convenient to have a good dynamic model of it. However, in many cases it is not possible either because of its complexity or because of the lack of knowledge of the laws involved in its operation. In these cases, obtaining models from input–output data is shown as a highly effective technique. Specifically, intelligent modeling techniques have become important in recent years in this field. Among these techniques, fuzzy logic is especially interesting because it allows to incorporate to the model the knowledge that is possessed of the system, besides offering a more interpretable model than other techniques. A fuzzy model is, formally speaking, a mathematical model. Therefore, this model can be used to analyze the original system using known systems analysis techniques. In this paper a methodology for extract information from unknown systems using fuzzy logic is presented. More precisely, it is presented the exact linearization of a Takagi–Sugeno fuzzy model with no restrictions in use or distribution of its membership functions, as well as obtaining its equilibrium states, the study of its local behavior and the search for periodic orbits by the application of Poincaré.     

An interactive method for multiple criteria group decision analysis based on interval type-2 fuzzy sets and its application to medical decision making

The theory of interval type-2 fuzzy sets provides an intuitive and computationally feasible way of addressing uncertain and ambiguous information in decision-making fields. The aim of this paper is to develop an interactive method for handling multiple criteria group decision-making problems, in which information about criterion weights is incompletely (imprecisely or partially) known and the criterion values are expressed as interval type-2 trapezoidal fuzzy numbers. With respect to the relative importance of multiple decision-makers and group consensus of fuzzy opinions, a hybrid averaging approach combining weighted averages and ordered weighted averages was employed to construct the collective decision matrix. An integrated programming model was then established based on the concept of signed distance-based closeness coefficients to determine the importance weights of criteria and the priority ranking of alternatives. Subsequently, an interactive procedure was proposed to modify the model according to the decision-makers’ feedback on the degree of satisfaction toward undesirable solution results for the sake of gradually improving the integrated model. The feasibility and applicability of the proposed methods are illustrated with a medical decision-making problem of patient-centered medicine concerning basilar artery occlusion. A comparative analysis with other approaches was performed to validate the effectiveness of the proposed methodology.     

Enhanced adaptive fuzzy sliding mode control for uncertain nonlinear systems

In this article, a novel Adaptive Fuzzy Sliding Mode Control (AFSMC) methodology is proposed based on the integration of Sliding Mode Control (SMC) and Adaptive Fuzzy Control (AFC). Making use of the SMC design framework, we propose two fuzzy systems to be used as reaching and equivalent parts of the SMC. In this way, we make use of the fuzzy logic to handle uncertainty/disturbance in the design of the equivalent part and provide a chattering free control for the design of the reaching part. To construct the equivalent control law, an adaptive fuzzy inference engine is used to approximate the unknown parts of the system. To get rid of the chattering, a fuzzy logic model is assigned for reaching control law, which acting like the saturation function technique. The main advantage of our proposed methodology is that the structure of the system is unknown and no knowledge of the bounds of parameters, uncertainties and external disturbance are required in advance. Using Lyapunov stability theory and Barbalat’s lemma, the closed-loop system is proved to be stable and convergence properties of the system is assured. Simulation examples are presented to verify the effectiveness of the method. Results are compared with some other methods proposed in the past research.     

Similarity,inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral

Similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets. Inclusion measures for type-2 fuzzy sets are the degrees to which a type-2 fuzzy set is a subset of another type-2 fuzzy set. The entropy of type-2 fuzzy sets is the measure of fuzziness between type-2 fuzzy sets. Although several similarity, inclusion and entropy measures for type-2 fuzzy sets have been proposed in the literatures, no one has considered the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, new similarity, inclusion and entropy measure formulas between type-2 fuzzy sets based on the Sugeno integral are proposed. Several examples are used to present the calculation and to compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are more reasonable than existing measures. On the other hand, measuring the similarity between type-2 fuzzy sets is important in clustering for type-2 fuzzy data. We finally use the proposed similarity measure with a robust clustering method for clustering the patterns of type-2 fuzzy sets.     

生物系统冗余结构的Type-2型数学模型

生态系统本身就是一个自组织的进化系统,生物与环境的交流和相互作用中,生物利用自身的冗余结构和冗余补充,使整个系统的进化始终朝着有利于生物发展的方向进化。本文利用Type-2模糊系统建立了生物群落冗余结构的数学模型,提出了生物群落稳定性的度量方法,研究了生物群落冗余结构与生物进化可靠性和稳定性的关系。