Analytical structure and stability analysis of a fuzzy PID controller

Analytical structure for a fuzzy PID controller is introduced by employing two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. This structure is derived via left and right trapezoidal membership functions for inputs, trapezoidal membership functions for output, algebraic product triangular norm, bounded sum triangular co-norm, Mamdani minimum inference method, and center of sums (COS) defuzzification method. Conditions for bounded-input bounded-output (BIBO) stability are derived using the Small Gain Theorem. Finally, two numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controller.     

The simplest fuzzy PID controllers: mathematical models and stability analysis

This paper reveals mathematical models for the simplest fuzzy PID controllers which employ two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. Mathematical models are derived via left and right trapezoidal membership functions for each input, singleton or triangular membership functions for output, algebraic product triangular norm, different combinations of triangular co-norms and inference methods, and center of sums (COS) defuzzification method. Properties of these structures are studied to examine their suitability for control application. For the structure which is suitable for control, bounded-input bounded-output (BIBO) stability proof is presented. An approach to design fuzzy PID controllers is given. Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controllers.     

A trainable transparent universal approximator for defuzzificationin Mamdani-type neuro-fuzzy controllers

A novel technique of designing application specific defuzzification strategies with neural learning is presented. The proposed neural architecture considered as a universal defuzzification approximator is validated by showing the convergence when approximating several existing defuzzification strategies. The method is successfully tested with fuzzy controlled reverse driving of a model truck. The transparent structure of the universal defuzzification approximator allows us to analyze the generated customized defuzzification method using the existing theories of defuzzification. The integration of universal defuzzification approximator instead of traditional methods in Mamdani-type fuzzy controllers can also be considered as an addition of trainable nonlinear noise to the output of the fuzzy rule inference before calculating the defuzzified crisp output. Therefore, nonlinear noise trained specifically for a given application shows a grade of confidence on the rule base, providing an additional opportunity to measure the quality of the fuzzy rule base. The possibility of modeling a Mamdani-type fuzzy controller as a feedforward neural network with the ability of gradient descent training of the universal defuzzification approximator and antecedent membership functions fulfil the requirement known from multilayer preceptrons in finding solutions to nonlinear separable problems     

基于模糊贝叶斯网的危害性分析方法

针对传统的故障模式、影响与危害性分析(FMECA)方法不足的问题,提出了一个基于模糊贝叶斯网的危害性分析方法。该方法将模糊理论与贝叶斯网推理技术结合起来,用三角模糊数来描述专家的模糊评分值;通过模糊集合映射,将其转化为评级的模糊子集;以置信结构的模糊规则,表示故障模式的属性与危害度之间的关系;利用贝叶斯网络推理算法综合置信结构的模糊规则,通过贝叶斯网推理得到模糊子集形式的危害度,再经过去模糊计算,得到故障危害等级的清晰值,从而确定故障模式的危害程度。实验结果表明,所提方法能够提高传统分析方法的准确性和应用范围。     

Cooperative Evolutionary Learning of Linguistic Fuzzy Rules and Parametric Aggregation Connectors for Mamdani Fuzzy Systems

There are two tasks in the design of linguistic fuzzy models for a concrete application: The derivation of the linguistic rule base and the setup of the inference system and the defuzzification method. Traditionally, the derivation of the linguistic rule base has been considered the most important task, but the use of the appropriate aggregation connectors in the inference system and the defuzzification interface can improve the fuzzy system behavior. In this paper, we take in consideration this idea, we propose an evolutionary learning method to learn a linguistic rule base and the parametric aggregation connectors of the inference and defuzzification in a single step. The aim of this methodology is to make possible a high level of positive synergy between the linguistic rule base and the aggregation connectors, improving the accuracy of the linguistic Mamdani fuzzy systems. Our proposal has shown good results solving three different applications. We introduce a statistical analysis of results for validating the model behavior on the applications used in the experimental study. We must remark that we present an experimental study with a double intention: (a) to compare the behavior of the new approach in comparison with those ones that first learn the rule base and then adapt the connectors, and (b) to analyze the rule bases obtained with fixed aggregation connectors and with the adaptive ones for showing the changes on the consequent rules, changes on labels that produce a better behavior of the linguistic model than the classic ones.     

Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables

This paper considers inventory models for items with imperfect quality and shortage backordering in fuzzy environments by employing two types of fuzzy numbers, which are trapezoidal and triangular. Two fuzzy models are developed. In the first model the input parameters are fuzzified, while the decision variables are treated as crisp variables. In the second model, not only the input parameters but also the decision variables are fuzzified. For each fuzzy model, a method of defuzzification, namely the graded mean integration method, is employed to find the estimate of the profit function in the fuzzy sense, and then the optimal policy for the each model is determined. The optimal policy for the second model is determined by using the Kuhn–Tucker conditions after the defuzzification of the profit function. Numerical examples are provided in order to ascertain the sensitiveness in the decision variables with respect to fuzziness in the components.     

On methods of defuzzification of parametrically represented fuzzy numbers

A comparative analysis of such methods of defuzzification of fuzzy numbers as WABL (Weighted Averaging Based on Levels), centroid, and mean of maxima (MOM) is presented in the study. Analytic formulas are presented for calculating the defuzzification values for parametrically represented fuzzy numbers of triangular and trapezoidal form.     

Fuzzy arithmetic based reliability allocation approach during early design and development

During early design and development stages, every engineering system has to meet its specific reliability goals. The target reliability of the system is achieved through application of an effective reliability apportionment technique to its subsystems. There are various traditional methods exist to perform the reliability allocation based on engineering factors that are assessed in a subjective manner. The conventional reliability allocation approach requires the assessment of factors like complexity, cost, and maintenance. This may not be realistic in real applications if they are assessed in a crisp manner by the domain experts of their varied expertise and background.In this paper, we treat allocation factors as fuzzy numbers, which are evaluated in fuzzy linguistic terms. As a result, fuzzy proportionality factor scales are proposed for the subsystems. In order to accomplish fuzzy division to evaluate the fuzzy proportionality factor, an approximation method based on linear programming for trapezoidal fuzzy numbers is also proposed in this paper. For the evaluation of weighting factors from fuzzy proportionality factors, centroid method of defuzzification is being employed. The allocated reliability of each subsystem is computed with the help of weighting factor thereafter. An example is provided to illustrate the potential application of the proposed fuzzy based reliability allocation approach.     

A fuzzy ID3 induction for linguistic data sets

In real life, humans communicate by means of words. Computing with words enables flexibility via fuzzy logic to reach more informative results for the classification and decision‐making. Fuzzy logic handles the imprecise information. In our paper, we propose a novel fuzzy ID3 algorithm for the classification on linguistic data set, where data can be given as linguistic variables. Linguistic variables are defined by using triangular fuzzy numbers given as LR (left‐right) fuzzy numbers. And weighted averaging based on levels (WABL) method is used as the defuzzification method for each data. Then, fuzzy c‐means algorithm is performed to handle the membership degrees for each variable given in each data set used in an experimental study. At last, the fuzzy ID3 algorithm is applied. The rules are generated, and the reasoning is done by different T‐operators. Our study is encouraged by (using) statistical analysis. In conclusion, it is seen that our algorithm proposed for linguistic data is as good as the proposed approach for numeric data. Also, it is shown that the proposed linguistic approach by using different T‐operators on linguistic data gives better results than numerical approach on some data sets.     

Multiple attribute group decision-making methods based on trapezoidal fuzzy two-dimension linguistic power generalized aggregation operators

On the basis of two-dimension uncertain linguistic variables, in this paper, we further presented a trapezoidal fuzzy two-dimension linguistic variable in which the first dimensional linguistic uncertain information is extended to trapezoidal fuzzy number. First, the definition, operational laws, characteristics, expectation, comparative method and distance of trapezoidal fuzzy two-dimension linguistic information are proposed. Then, the trapezoidal fuzzy two-dimension linguistic power generalized aggregation operator and the trapezoidal fuzzy two-dimension linguistic power generalized weighted aggregation (TF2DLPGWA) operator are developed, and some properties and special cases of these operators are analyzed. Furthermore, based on the TF2DLPGWA operator and the comparative formula of the trapezoidal fuzzy two-dimension linguistic variables, an approach to group decision making with trapezoidal fuzzy two-dimension linguistic variables is established. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

自调整模糊控制器的软件仿真系统设计

该文提出一种具有修正因子的自调整模糊控制器软件仿真系统RMDFCSS。它包含有编辑器、调试器、模糊推理机和代码生成器功能,用编辑器定义模糊规则、隶属函数、特定的推理方法和反模糊化方法;用调试器能检查整个推理过程的每一个步骤,完成模糊控制算法模型的确立、论证和优化操作。     

Fuzzy Inference Neural Network for Fuzzy Model Tuning

In fuzzy modeling, it is relatively easy to manually define rough fuzzy rules for a target system by intuition. It is, however, time-consuming and difficult to fine-tune them to improve their behavior. This paper describes a tuning method for fuzzy models which is applicable regardless of the form of fuzzy rules and the used defuzzification method. For this purpose, this paper proposes a fuzzy neural network model which can embody fuzzy models. The proposed model provides the functions to perform fuzzy inference and to tune the parameters for the shape of antecedent linguistic terms, the relative importance degrees of rules, and the relative importance degrees of antecedent linguistic terms in rules. In addition, to show its applicability, we perform some experiments and present the results     

Handling forecasting problems based on fuzzy time series model and model error learning

Classical fuzzy time series forecasts are comprised of three steps: fuzzification, identification of fuzzy relation, and defuzzification. In this paper, we propose a new approach and add an error learning step to improve forecasts. In the fuzzification step, a hybrid method, based on the fuzzy c-means clustering and the fuzzy Silhouette criterion, is employed to determine the optimal number of intervals, which avoids time-consuming iterations of the whole algorithm. In the defuzzification step, an optimization model is set up to explain the rule of defuzzification. In the model structure, an error term is assembled into the traditional model to express model error, which is predicted by linear fitting and abnormal errors processing. Learning of model errors and considering of data characteristics guarantee good interpretability and accuracy. The numerical results show that the proposed approach has superior forecast performance to existing methods.     

Asymmetric subsethood-product fuzzy neural inference system (ASuPFuNIS)

This work presents an asymmetric subsethood-product fuzzy neural inference system (ASuPFuNIS) that directly extends the SuPFuNIS model by permitting signal and weight fuzzy sets to be modeled by asymmetric Gaussian membership functions. The asymmetric subsethood-product network admits both numeric as well as linguistic inputs. Input nodes, which act as tunable feature fuzzifiers, fuzzify numeric inputs with asymmetric Gaussian fuzzy sets; and linguistic inputs are presented as is. The antecedent and consequent labels of standard fuzzy if-then rules are represented as asymmetric Gaussian fuzzy connection weights of the network. The model uses mutual subsethood based activation spread and a product aggregation operator that works in conjunction with volume defuzzification in a gradient descent learning framework. Despite the increase in the number of free parameters, the proposed model performs better than SuPFuNIS, on various benchmarking problems, both in terms of the performance accuracy and architectural economy and compares excellently with other various existing models with a performance better than most of them.     

A fuzzy inference and categorization approach for supplier selection using compensatory and non-compensatory decision rules

Fuzzy set theory has been used as an approach to deal with uncertainty in the supplier selection decision process. However, most studies limit applications of fuzzy set theory to outranking potential suppliers, not including a qualification stage in the decision process, in which non-compensatory types of decision rules can be used to reduce the set of potential suppliers. This paper presents a supplier selection decision method based on fuzzy inference that integrates both types of approaches: a non-compensatory rule for sorting in qualification stages and a compensatory rule for ranking in the final selection. Fuzzy inference rules model human reasoning and are embedded in the system, which is an advantage when compared to approaches that combine fuzzy set theory with multicriteria decision making methods. Fuzzy inference combined with a fuzzy rule-based classification method is used to categorize suppliers in qualification stages. Classes of supplier performance can be represented by linguistic terms, which allow decision makers to deal with subjectivity and to express qualification requirements in linguistic formats. Implementation of the proposed method and techniques were analyzed and discussed using an illustrative case. Three defuzzification operators were used in the final selection, yielding the same ranking. Factorial design was applied to test consistency and sensitivity of the inference rules. The findings reinforce the argument that including stages of qualification based on fuzzy inference and categorization makes this method especially useful for selecting from a large set of potential suppliers and also for first time purchase.     

基于前景理论和VIKOR的风险型模糊多准则决策方法

针对准则值均为模糊数的风险型多准则决策问题,提出一种基于前景理论和VIKOR的多准则决策方法。首先,进行区间数、三角模糊数、梯形模糊数、直觉模糊数和语言值的无量纲化处理;然后,基于各个准则各种状态下各个方案的准则值排序,确定中位数参考点以及各个方案在各个准则下的综合前景值;接着,基于前景价值矩阵,给出基于VIKOR的扩展方法;最后,通过具体实例验证了所提出方法的有效性和可行性。     

模糊控制器输出值不变的两个充分条件

模糊控制器通常由模糊化、模糊推理以及清晰化三部分构成, 而模糊推理决定了一个由输入论域到输出论域的模糊映射. 当模糊映射为常值映射时, 任意选择模糊化和去模糊化方式, 模糊控制器的输出值不因输入信号变化而改变. 本文给出了模糊映射为常值映射的两个充分条件, 并将结论从单入单出模糊系统推广到多入单出模糊系统.     

A new trapezoidal Pythagorean fuzzy linguistic entropic combined ordered weighted averaging operator and its application for enterprise location

Recently some new models based on Pythagorean fuzzy sets (PFSs) have been proposed to deal with the uncertainty in multiple attribute group decision making (MAGDM) problems. In this paper, considering linguistic variables and entropic, we propose a new trapezoidal Pythagorean fuzzy linguistic entropic combined ordered weighted averaging operator to solve MAGDM problems. Next, we study some main properties by utilizing some operational laws of the trapezoidal Pythagorean fuzzy linguistic variables. Finally, a numerical example concerning the enterprise location is given to illustrate the practicality and effectiveness of the proposed operator.     

泛蕴涵推理机及其应用

本文提出模糊系统中基于泛逻辑的泛蕴涵推理机，给出其在描绘函数图形时的应用，同时比较了它与Mamdani型和Lasen型两种模糊系统在描绘函数图形时的误差。分析和比较表明，在相同规则下含有泛蕴涵推理机的模糊系统产生的误差最低。     

一种输入数据为模糊数的模糊支持向量机

支持向量机所处理的数据绝大多数是精确值,但当训练样本中含有模糊信息时,支持向量机将无能为力。基于此,针对输入数据是模糊数的分类问题,提出一种带有去模糊函数的模糊支持向量机（FSVM*）。该算法采用模糊数间的距离作为模糊数去模糊的度量,从而构造去模糊函数将模糊值转化为精确值,同时将去模糊函数与模糊支持向量机相结合完成模糊数据的分类。数值结果表明：相比Forghani提出的FSVDD*算法,该算法更有效。