Rough fuzzy approximations on two universes of discourse

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various fuzzy generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a covering and three relations are induced to a single universe of discourse. Based on the induced notions, four pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized, and some interesting properties are investigated. Finally, the connections of these approximation operators are made, and conditions under which some of these approximation operators are equivalent are obtained.     

Fuzzy Reasoning as a Control Problem

Different from the dominant view of treating fuzzy reasoning as generalization of classical logical inference, in this paper fuzzy reasoning is treated as a control problem. A new fuzzy reasoning method is proposed that employs an explicit feedback mechanism to improve the robustness of fuzzy reasoning. The fuzzy rule base given a priori serves as a controlled object, and the fuzzy reasoning method serves as the corresponding controller. The fuzzy rule base and the fuzzy reasoning method constitute a control system that may be open loop or closed loop, depending on the underlying reasoning goals/constraints. The fuzzy rule base, the fuzzy reasoning method, and the corresponding reasoning goals/constraints define the three distinct ingredients of fuzzy reasoning. While various existing fuzzy reasoning methods are essentially a static mapping from the universe of single fuzzy premises to the universe of single fuzzy consequences, the new fuzzy reasoning method maps sequences of fuzzy premises to sequences of fuzzy consequences and is a function of the underlying reasoning goals/constraints. The Monte Carlo simulation shows that the new fuzzy reasoning method is much more robust than the optimal fuzzy reasoning method proposed in our previous work. The explicit feedback mechanism embedded in the fuzzy reasoning method does significantly improve the robustness of fuzzy reasoning, which is concerned with the effects of perturbations associated with given fuzzy rule bases and/or fuzzy premises on fuzzy consequences. The work presented in this paper sets a new starting point for various principles of feedback control and optimization to be applied in fuzzy reasoning or logical inference and to explore new forms of reasoning including robust reasoning and adaptive reasoning. It can be also expected that the new fuzzy reasoning method presented in this paper can be used for modeling and control of complex systems and for decision-making under complex environments.     

时变论域下的语言动力学轨迹

为了描述在不同时刻内涵发生变化的模糊集合, 本文通过综合模糊集合、语言动力系统(Linguistic dynamic systems,LDS) 及动态规划, 提出了时变论域与动态模糊规则理论, 将时变论域分为离散型与连续型两类, 每一类分为递增型、递减型及波动型三种, 并讨论如何在时变论域上建立动态模糊规则以及进行词计算, 最后给出时变论域下的语言动力学轨迹.     

空天飞空天飞行器的自适应变论域模糊预测控制行器的自适应变论域模糊预测控制

考虑一类非线性不确定系统的变论域模糊预测控制问题.根据跟踪误差在线调整伸缩因子,使变论域模糊系统一致逼近被控对象中的未知干扰和不确定因素.通过引入鲁棒自适应控制器,消除了模糊建模误差,提高了系统的动态性能.基于泰勒展开的非线性预测控制律,避免了繁重的计算负担.基于Lyapunov理论,给出了伸缩因子的σ调整律,并证明了闭环系统一致最终有界.最后,将该算法用于空天飞行器(ASV)姿态控制系统的设计,仿真结果表明了该算法的有效性.     

模糊推理的合成规则及其合成运算的选择研究

本文首先提出当知识库中含有多条取自于同论域上的模糊推理规则时，应用通常的六条合成推理规则所推得的结论将随着模糊推理规则数的增多而越来越偏离真实度这一问题。本文针对该问题对通常的六条合成推理规则及其三角模下的六条推广合成推理规则进行了理论比较研究，并对合成运算进行了比较研究。研究表明：合成推理规则（或Ｒ￣４）以合成运算ｍａｘ－Ｔ０是克服上述问题的最佳选择。这一重要结果对于模糊知识库的设计具有指导意义。     

基于子空间划分的模糊系统模型辨识

提出了基于子空间划分的模糊系统模型（SPFS）,并给出一种针对SPSF的白适应模型辨识方法．应用遗传算法进行子空间划分方案的优化。降低了最大子空间的辨识误差,从而得到优化的模型辨识结果．理论分析和仿真计算证明了该模型的有效性．所提出的模型有助于缓解规则数爆炸问题．     

Adaptive Rule Weights in Neuro-Fuzzy Systems

Neuro-fuzzy systems have recently gained a lot of interest in research and application. They are approaches that use learning techniques derived from neural networks to learn fuzzy systems from data. A very simple ad hoc approach to apply a learning algorithm to a fuzzy system is to use adaptive rule weights. In this paper, we argue that rule weights have a negative effect on the linguistic interpretation of a fuzzy system, and thus remove one of the key advantages for applying fuzzy systems. We show how rule weights can be equivalently replaced by modifying the fuzzy sets of a fuzzy system. If this is done, the actual effects that rule weights have on a fuzzy rule base become visible. We demonstrate at a simple example the problems of using rule weights. We suggest that neuro-fuzzy learning should be better implemented by algorithms that modify the fuzzy sets directly without using rule weights.     

Multiple Fuzzy Relation and Its Application to Coupled Fuzzy Control

Using semi‐tensor product (STP) of matrix, this paper investigates the fuzzy relation of multiple fuzzy and uses this to design coupled fuzzy control is designed. First of all, under the assumption that the universe of discourse is finite, a fuzzy logical variable can be expressed as a vector, which unifies the expression of elements, subsets, and fuzzy subsets of a universe of discourse. Then, the matrix expression of set mappings is naturally extended to fuzzy sets. Second, based on STP, logic‐based matrix addition and product are proposed. These are particulary useful for the calculation of compounded fuzzy relations. Third, a dual fuzzy structure is introduced, which assures the finiteness of the universe of discourse, and is used for fuzzification and defuzzification. Finally, using the results obtained, a new technique is developed to design a coupled fuzzy controller for multi‐input multi‐output (MIMO) systems with coupled multiple fuzzy relations.     

An adaptive ordered fuzzy time series with application to FOREX

An adaptive ordered fuzzy time series is proposed that employs an adaptive order selection algorithm for composing the rule structure and partitions the universe of discourse into unequal intervals based on a fast self-organising strategy. The automatic order selection of FTS as well as the adaptive partitioning of each interval in the universe of discourse is shown to greatly affect forecasting accuracy. This strategy is then applied to prediction of FOREX market. Financial markets, such as FOREX, are generally attractive applications of FTS due to their poorly understood model as well as their great deal of uncertainty in terms of quote fluctuations and the behaviours of the humans in the loop. Specifically, since the FOREX market can exhibit different behaviours at different times, the adaptive order selection is executed online to find the best order of the FTS for current prediction. The order selection module uses voting, statistical analytic and emotional decision making agents. Comparison of the proposed method with earlier studies demonstrates improved prediction accuracy at similar computation cost.     

规则不完备的模糊系统新规则在线生成

由专家经验和输入输出样本数据得到的模糊规则常常是不完备的,规则不完备的模糊系统因为缺少一些规则,对于某些可能的输入值产生的输出会很不合理。文章提出了用插值法在必要时在线生成新规则,并设计了一个模糊系统来插值填充规则库空格。实验结果表明,这种方法大大增强了规则不完备的模糊系统在整个输入域上输出的连续性和稳定性。     

Hierarchical TS fuzzy system and its universal approximation

An efficient tool to deal with the ‘rule explosion’ problem is the hierarchical system by which a fuzzy system can be decomposed into a number of hierarchically connected low-dimensional systems. In this paper a generalized hierarchical Tagaki–Sugeno (TS) system is built. It is shown that the input–output (I/O) relationship of this generalized hierarchical system can be represented as one of a standard TS fuzzy system. And the system approximation capability is analyzed by taking piecewise linear functions as a bridge. By constructive method it is proven that the hierarchical fuzzy systems (HFS’s) can be universal approximators. For the given approximation accuracy, an estimation formula about the number of the rules needed in the HFS is established. Finally some simulation examples confirm that the HFS’s with smaller size rule base can approximate the given functions with high accuracy. The results obtained here provide us with the theoretical basis for various applications of HFS’s.     

时变论域下红绿灯配时的语言动力学分析

城市道路不同时刻的车流量变化很大,建立与车流量变化相适应的红绿灯动态配时模型有利于缓解交通拥堵,减少出行者的等待时间.本文通过综合时变论域、平行控制理论、语言动力系统（Linguistic dynamic system,LDS）,提出了一种新的红绿灯控制方法.该方法以红绿灯不同时刻周期时长所形成的序列为时变论域,由各相位的排队长度确定对应的通行序列与时长,得到时变论域下红绿灯配时方案.该方案形成一个由实时车流数据驱动的动态模糊规则库来对红绿灯配时周期及相位通行序列与时长进行动态调整,进而形成红绿灯配时演化过程的语言动力学轨迹,最后通过实例验证该方案的有效性.     

Adaptive fuzzy regulation of the DC-bus capacitor voltage in a wind energy conversion system (WECS)

This paper proposes a new voltage regulator of the DC-bus capacitor of a variable speed wind power generation system based on adaptive fuzzy system. The change in the fuzzy rule base is done using a variable-structure direct adaptive control algorithm to achieve the pre-defined control objectives. This algorithm has two merits. First, it has a good performance in the training phase as it makes use of the initial rule base defined for the fuzzy logic controller. Second, it has a robust estimator since it depends on variable structure technique. The adaptive nature of the new controller significantly reduces the rule base size and improves its performance.     

A fuzzy neural network with fuzzy impact grades

Fuzzy rule derivation is often difficult and time-consuming, and requires expert knowledge. This creates a common bottleneck in fuzzy system design. In order to solve this problem, many fuzzy systems that automatically generate fuzzy rules from numerical data have been proposed. In this paper, we propose a fuzzy neural network based on mutual subsethood (MSBFNN) and its fuzzy rule identification algorithms. In our approach, fuzzy rules are described by different fuzzy sets. For each fuzzy set representing a fuzzy rule, the universe of discourse is defined as the summation of weighted membership grades of input linguistic terms that associate with the given fuzzy rule. In this manner, MSBFNN fully considers the contribution of input variables to the joint firing strength of fuzzy rules. Afterwards, the proposed fuzzy neural network quantifies the impacts of fuzzy rules on the consequent parts by fuzzy connections based on mutual subsethood. Furthermore, to enhance the knowledge representation and interpretation of the rules, a linear transformation from consequent parts to output is incorporated into MSBFNN so that higher accuracy can be achieved. In the parameter identification phase, the backpropagation algorithm is employed, and proper linear transformation is also determined dynamically. To demonstrate the capability of the MSBFNN, simulations in different areas including classification, regression and time series prediction are conducted. The proposed MSBFNN shows encouraging performance when benchmarked against other models.     

A novel and fast MIMO fuzzy inference system based on a class of fuzzy clustering algorithms with interpretability and complexity analysis

A data driven Fuzzy Inference System (FIS) employs Membership Functions (MFs) with adjustable parameters in its IF part to fuzzify the input data. The input space is partitioned simply by dividing universe of discourse of each input variable into some fuzzy subspaces. The MFs are then defined on the fuzzy subspaces of the input variables. Parameters of the MFs are tuned for maximum accuracy of the system (which demands high runtime) without considering the data structure which impairs interpretability of the FIS and degenerates the system into a black-box tool. Such a FIS does not represent actual structure of the data and its MFs are not necessarily in accord with the data distribution in the input space. In addition, the FIS suffers from exponential complexity of order O(T^r) where T is number of linguistic terms (number of subspaces on the universe of discourse of input variables) and r is number of input variables. This article presents a novel Multiple-Input and Multiple-Output Clustering based Fuzzy Inference System (MIMO CFIS) which is made directly from a class of fuzzy clustering algorithms to overcome these shortcomings. CFIS identifies dense regions of the input data using fuzzy clustering and then places a cluster on each of these regions. These fuzzy clusters represent actual structure of the data and serve as fuzzy rules in the rule base of CFIS and provide MFs that exactly fit the dense regions of the data that makes the system more interpretable and avoids redundant rules. These MFs are normal, convex, and continuous and have no parameter to be tuned (which makes CFIS much faster than other FISs) and fuzzify the input data according to their membership in the clusters. THEN part of CFIS is generalized form of THEN part of Takagi-Sugeno (TS) fuzzy system which accommodates any function of input variables. Despite less number of adjustable parameters, testing error of CFIS is less than that of TS system and its modified versions. Moreover, number of fuzzy rules in CFIS rule base is the same as the number of linguistic terms (or fuzzy clusters) and consequently its complexity is of orderO(T). Also, CFIS is a MIMO system and avoids inconsistent (contradictory) rules by generating well-separated fuzzy clusters whereas TS system is MISO and never guarantees generation of consistent rules. In addition, CFIS satisfies most of the interpretability criteria of FISs.     

Fuzzy model identification for classification of gait events inparaplegics

Fuzzy system identification was applied to a biomedical system for classification purposes. Gait achieved through functional electrical stimulation (FES) of paraplegics was divided using sensor measurements of kinematic variables as inputs to five discrete events. Two identification algorithms were used to estimate the system model. Both max-min and max-product composition were used. Membership functions were either trapezoidal or triangular and all membership functions in a particular universe of discourse had the same shape and size. The universe of discourse was varied by altering the overlap between membership functions. The classification performance was assessed quantitatively, by measuring the percentage of time steps in which the correct event was found, and qualitatively, by observing types of errors. The identification algorithm affected system performance. No difference in classification was found between max-min and max-product composition. The performance was dependent on membership function overlap. A comparison of the classification found using the fuzzy rule base versus that found using a traditional look-up table demonstrated that the fuzzy approach was superior. It is speculated that the use of fuzzy logic decreased errors stemming from sensor noise and/or small variations in the input signals. The performance of this approach was compared to that of a feedforward neural network and the fuzzy system is found superior     

Robust adaptive fuzzy controller for non‐affine nonlinear systems with dynamic rule activation

This paper describes the design of a robust adaptive fuzzy controller for an uncertain single‐input single‐output nonlinear dynamical systems. While most recent results on fuzzy controllers considers affine systems with fixed rule‐base fuzzy systems, we propose a control scheme for non‐affine nonlinear systems and a dynamic fuzzy rule activation scheme in which an appropriate number of the fuzzy rules are chosen on‐line. By using the proposed scheme, we can reduce the computation time, storage space, and dynamic order of the adaptive fuzzy system without significant performance degradation. The Lyapunov synthesis approach is used to guarantee a uniform ultimate boundedness property for the tracking error, as well as for all other signals in the closed loop. No a priori knowledge of an upper bounds on the uncertainties is required. The theoretical results are illustrated through a simulation example. Copyright © 2002 John Wiley & Sons, Ltd.     

Fuzzy rough set on probabilistic approximation space over two universes and its application to emergency decision‐making

Probabilistic approaches to rough sets are still an important issue in rough set theory. Although many studies have been written on this topic, they focus on approximating a crisp concept in the universe of discourse, with less effort on approximating a fuzzy concept in the universe of discourse. This article investigates the rough approximation of a fuzzy concept on a probabilistic approximation space over two universes. We first present the definition of a lower and upper approximation of a fuzzy set with respect to a probabilistic approximation space over two universes by defining the conditional probability of a fuzzy event. That is, we define the rough fuzzy set on a probabilistic approximation space over two universes. We then define the fuzzy probabilistic approximation over two universes by introducing a probability measure to the approximation space over two universes. Then, we establish the fuzzy rough set model on the probabilistic approximation space over two universes. Meanwhile, we study some properties of both rough fuzzy sets and fuzzy rough sets on the probabilistic approximation space over two universes. Also, we compare the proposed model with the existing models to show the superiority of the model given in this paper. Furthermore, we apply the fuzzy rough set on the probabilistic approximation over two universes to emergency decision‐making in unconventional emergency management. We establish an approach to online emergency decision‐making by using the fuzzy rough set model on the probabilistic approximation over two universes. Finally, we apply our approach to a numerical example of emergency decision‐making in order to illustrate the validity of the proposed method.     

Fuzzy inference to risk assessment on nuclear engineering systems

This paper presents a nuclear case study, in which a fuzzy inference system (FIS) is used as alternative approach in risk analysis. The main objective of this study is to obtain an understanding of the aging process of an important nuclear power system and how it affects the overall plant safety. This approach uses the concept of a pure fuzzy logic system where the fuzzy rule base consists of a collection of fuzzy IF–THEN rules. The fuzzy inference engine uses these fuzzy IF–THEN rules to determine a mapping from fuzzy sets in the input universe of discourse to fuzzy sets in the output universe of discourse based on fuzzy logic principles. The risk priority number (RPN), a traditional analysis parameter, was calculated and compared to fuzzy risk priority number (FRPN) using scores from expert opinion to probabilities of occurrence, severity and not detection. A standard four-loop pressurized water reactor (PWR) containment cooling system (CCS) was used as example case. The results demonstrated the potential of the inference system for subsiding the failure modes and effects analysis (FMEA) in aging studies.