In this paper, we propose a general framework to combine
mathematical model and heuristics into nonlinear adaptive controller
design, where the heuristics are in the form of fuzzy IF-THEN rules. We
consider the case where the following three pieces of information are
available: 1) an approximate model of the nonlinear system under
control; 2) a collection of fuzzy IF-THEN rules describing the error
between the approximate model and the real system; and 3) fuzzy control
rules describing recommended control actions under various conditions.
Our approach combines these three pieces of information into an adaptive
fuzzy controller. Specifically, the first two pieces of information are
used to construct an estimated model of the system and a controller is
designed based on this estimated model. The third piece of information
is used to construct another controller. The final controller is a
weighted average of these two controllers. We develop an adaptive law
for adjusting the free parameters in the controller such that the
closed-loop system follows a desired trajectory
This paper describes the results of inference and hint-effect experiments, and analyses the human decision-making process from fuzzy-theoretic and concept-formational viewpoints. To begin with, we conduct a fuzzy-theoretic experiment on inference and show an interrelation among memory, newly-known information, and inference. Next, we discuss the problem of the decision-making process subject to a specific and newly given information. In other words, the fuzzy-theoretic experiment concerning with some hints in the estimation and decision processes of the card arrangement is conducted. Then, the concept-formational process is analysed clearly by means of the introduction of both the cusp-curved surface in catastrophe theory and fuzziness concept in fuzzy theory. Finally, we show the hint effect in the estimation and decision processes.
The proposed architecture, ASAFES2, is a function approximator which combines the functional reasoning or Sugeno's fuzzy reasoning method with stochastic reinforcement learning-a class of quite powerful neural network training algorithms. It is a simple and versatile mathematical tool for fuzzy computing, featuring smooth and quick convergence and ease of use. The main ideas are the fuzzy partitioning of the input space into fuzzy subspaces (each corresponding to a possible fuzzy rule), and the use of a separate, stochastic reinforcement learning neural unit (ANASA II) for every fuzzy subspace, in order to calculate the optimum consequence parameters. Some preliminary results are presented, proving ASAFES2 superior over backpropagation. A new, and "flexible" membership function is also proposed.
In engineering applications of fuzzy logic, the main goal is not to simulate the way the experts really think, but to come up with a good engineering solution that would (ideally) be better than the expert's control. In such applications, it makes perfect sense to restrict ourselves to simplified approximate expressions for membership functions. If we need to perform arithmetic operations with the resulting fuzzy numbers, then we can use simple and fast algorithms that are known for operations with simple membership functions. In other applications, especially the ones that are related to humanities, simulating experts is one of the main goals. In such applications, we must use membership functions that capture every nuance of the expert's opinion; these functions an therefore complicated, and fuzzy arithmetic operations with the corresponding fuzzy numbers become a computational problem. In this paper, we design a new algorithm for performing such operations. This algorithm uses fast Fourier transforms (FFTs) to reduce computation time from O(n<sup>2</sup>) to O(nlog(n)) (when n is the number of points x at which we know the membership functions μ(x)). To compute FFT even faster, we propose to use special hardware
A rule-base self-extraction and simplification method is proposed
to establish interpretable fuzzy models from numerical data. A fuzzy
clustering technique is used to extract the initial fuzzy rule-base. The
number of fuzzy rules is determined by the proposed fuzzy partition
validity index. To reduce the complexity of fuzzy models without
decreasing the model accuracy significantly, some approximate similarity
measures are presented and a parameter fine-tuning mechanism is
introduced to improve the accuracy of the simplified model. The
simplified fuzzy model has good balance between accuracy and
In robot learning control, the learning space for executing the
general motions of multijoint robot manipulators is very complicated.
Therefore, in spite of their ability to generalize, the learning
controllers are usually used as subordinates to conventional controllers
or the learning process needs to be repeated each time a new trajectory
is encountered, because the motion variety requires them to consume
excessive amount of memory when they are employed as major roles in
motion governing. To simplify learning space complexity, we propose,
from the standpoint of learning control, that robot motions be
classified according to their similarities. The learning controller can
then be designed to govern groups of robot motions with high degrees of
similarity without consuming excessive memory resources. Motion
classification based on using the PUMA 560 robot manipulator
demonstrates the effectiveness of the proposed approach
Probabilistic set theory is discussed from a mathematical point of view. It is constructed considering ambiguity or variety of objects and subjectivity or evolution of knowledge of observers, and it includes fuzzy set theory. The background of the idea is first explained and some inconveniences of fuzzy set theory are pointed out. A probabilistic set on a total space is defined by a pointwise measurable function from a parameter space to a characteristic space. The main result is that the family of probabilistic sets constitutes a complete pseudo Boolean algebra. Some new concepts are shown such as probabilistic mappings, moment analysis, and expected cardinal numbers. The possibility of moment analysis is an important result and it cannot be seen in other theories.
A sufficient and necessary condition is presented for fuzz functions to be Zadeh-type functions in an arbitrary fuzzy set L. This result is a generalization of the corresponding ones of Y.M. Liu (1987) and G.J. Wang (1984)
This paper presents several methods for functional approximation with variable-knot, variable-degree splines, with variable-knot first order splines, which are relatively easy to find, as the intermediate input. By means of a fuzzy characteristic function determining how "sharp" the angle at each knot is (the higher the polynomial degree, the "sharper" its first order spline approximation bends), we can decide whether we can group certain adjacent segments together to be approximated by a single higher order polynomial segment. Some simulation experiments have also been done.
Fuzzy intervals are classified as regular or irregular based on the defuzzification method. A fuzzy interval is regular if its membership function is maximized at its defuzzified value. If the center-of-area method or the fuzzy-mean method is used, the regularity of a fuzzy interval is preserved after applying an arithmetic operation with nonzero real number. Further, the regularities based on the center-of-area method are preserved after addition and subtraction operations on two or more regular fuzzy intervals
In this paper we define and explore the properties of lower and upper inverses of fuzzy relations which extend multi-valued mappings. With the notion of degree of inclusion of non fuzzy sets, we then relate the preceeding notions to possibility distributions in natural languages and to problems of medical diagnosis.
In this paper, we show how a neural net can be used to solve
A¯X¯=C¯, for X¯, even though for some values of
A¯ and C¯ there is no fuzzy arithmetic solution for X¯.
The neural net solution is identified with our new solution (Buckley and
Qu, Fuzzy Sets & Systems, vol. 39, pp. 291-301, 1991) to fuzzy
In this paper, a parameterized fuzzy processor (PFP) is presented based on a designed fuzzy instruction set of the parameterized fuzzy number (PFN). This PFN is presented by some parameters of the universe of the discourse instead of a series of the membership grades. The algorithms of these instructions are based on the parameter operations to implement the fuzzy arithmetics and the fuzzy reasoning methods. The instruction set supports the versatility of the fuzzy information, e.g., fuzzy arithmetic operations, inference operations, fuzzy logic operations, data translations and data exchangements. Based upon these, the characteristics of the PFN are discussed, and the architecture of the PFP outlined is constructed in this paper and its processing time is saved due to the adaptation of the parameterized fuzzy number (PFN). To evaluate the performance of PFN instructions and PFP structure, two experiments are presented. One is the speed performance evaluated for the PFN to compare with the discrete fuzzy number (DFN). The other is the confirmation of the feasibility and the capability of the instructions, which is demonstrated by three examples related to human inference, intuitive control and decision-making.
The problem of deriving the structure of a non-deterministic system from its behavior is a difficult one even when that behavior is itself well-defined. When the behavior can be described only in fuzzy terms structural inference may appear virtually impossible. However, a rigorous formulation and solution of the problem for stochastic automata has recently been given  and, in this paper, the results are extended to fuzzy stochastic automata and grammars. The results obtained are of interest on a number of counts, (1) They are a further step towards an integrated 'theory of uncertainty'; (2) They give new insights into problems of inductive reasoning and processes of 'precisiation'; (3) They are algorithmic and have been embodied in a computer program that can be applied to the modelling of sequential fuzzy data.
We prove a version of the classical Peano's theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the Hausdorff distances between level sets.
The determination of solutions in group decision making is considered. The point of departure is a collection of individual fuzzy preference relations. A solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation. As opposed to conventional approaches in which a crisp (threshold type) majority rule is used, we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier, e.g., ‘most’, ‘much more than 50%’, etc. A calculus of linguistically quantified propositions is applied. Using the fuzzy majority, various solution concepts are derived, mainly of the type of core, minimax (opposition) set and consensus winner.
The paper presents an introduction to the theory of fuzzy binary relations and some related concepts. First, proximity and similarity relations, their classes and fuzzy partitions are introduced and their properties are investigated. Then, major classes of fuzzy orderings are defined and classified with respect to the duality relation. Transitivity properties of fuzzy orderings and the Ferrers property of induced fuzzy orderings are established.
We concentrate on a shortest path problem on a network in which a fuzzy number, instead of a real number, is assigned to each arc length. Introducing an order relation between fuzzy numbers based on “fuzzy min”, a nondominated path or Pareto Optimal path from the specified node to every other node is defined. An algorithm for solving the problem is developed on the basis of the multiple labeling method for a multicriteria shortest path. As a result, a number of nondominated paths can be obtained and is offered to a decision maker. However, a number of nondominated paths derived from large scale network may be too numerous for him to choose a preferable path. Due to this situation, we propose a method to reduce the number of paths according to a possibility level. The proposed algorithm is numerically evaluated on large scale random networks.
Management of uncertainty is an intrinsically important issue in the design of expert systems because much of the information in the knowledge base of a typical expert system is imprecise, incomplete or not totally reliable.In the existing expert systems, uncertainty is dealt with through a combination of predicate logic and probability-based methods. A serious shortcoming of these methods is that they are not capable of coming to grips with the pervasive fuzziness of information in the knowledge base, and, as a result, are mostly ad hoc in nature. An alternative approach to the management of uncertainty which is suggested in this paper is based on the use of fuzzy logic, which is the logic underlying approximate or, equivalently, fuzzy reasoning. A feature of fuzzy logic which is of particular importance to the management of uncertainty in expert systems is that it provides a systematic framework for dealing with fuzzy quantifiers, e.g., most, many, few, not very many, almost all, infrequently, about 0.8, etc. In this way, fuzzy logic subsumes both predicate logic and probability theory, and makes it possible to deal with different types of uncertainty within a single conceptual framework.In fuzzy logic, the deduction of a conclusion from a set of premises is reduced, in general, to the solution of a nonlinear program through the application of projection and extension principles. This approach to deduction leads to various basic syllogisms which may be used as rules of combination of evidence in expert systems. Among syllogisms of this type which are discussed in this paper are the intersection/product syllogism, the generalized modus ponens, the consequent conjunction syllogism, and the major-premise reversibility rule.
As a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches. To do so we briefly introduce fuzzy relations on which the third class of approaches is based. Then we recall some transitivity-related concepts and an ordering procedure based on a acyclic fuzzy relation. Acyclicity is a very weak restriction on a fuzzy relation. We prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity. So we give a widely applicable formulation to derive a total ranking order from a fuzzy relation. With our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part I.
Modeling, storing and retrieving geographical information has become an important part of our information society. Geographical information is typically specified in terms of collections of entities and phenomena that are structured aggregations of spatial entities. GIS features tend to form natural class hierarchies. Another characteristic of geographical information is that often it may be inexact or vague. With respect to these characteristics, the confluence of the two technologies fuzzy set theory and object-oriented databases could provide a powerful tool for knowledge representation underlying geographical information systems. The fuzzy object data model is currently being developed and prototype implementations have been undertaken using an integrative approach with existing software including an expert system shell and a commercial object-oriented database system. In this paper, the benefits of a fuzzy object data model for geographical information systems are examined, an overview of the model is presented, and the current prototype implementations are described.
Up to now, these are five methods of ranking n fuzzy numbers in order, but these methods contain some confusions and occasionally conflict with intuition. This paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers. In addition, we give a method for calculating the ordering value of each fuzzy number with triangular, trapezoidal, and two-sided drum-like shaped membership functions.
The existing data envelopment analysis (DEA) models for measuring the relative efficiencies of a set of decision making units (DMUs) using various inputs to produce various outputs are limited to crisp data. To deal with imprecise data, the notion of fuzziness has been introduced. This paper develops a procedure to measure the efficiencies of DMUs with fuzzy observations. The basic idea is to transform a fuzzy DEA model to a family of conventional crisp DEA models by applying the α-cut approach. A pair of parametric programs is formulated to describe that family of crisp DEA models, via which the membership functions of the efficiency measures are derived. Since the efficiency measures are expressed by membership functions rather than by crisp values, more information is provided for management. By extending to fuzzy environment, the DEA approach is made more powerful for applications.
By relating to the conventional PID control theory, we propose a new fuzzy controller structure, namely PID type fuzzy controller. In order to improve further the performance of the transient state and the steady state of the PID type controller, we develop a method to tune the scaling factors of the PID type fuzzy controller on line. Simulation of the PID type fuzzy controller with the self-tuning scaling factors shows a better performance in the transient and steady state response.
In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.
The traditional event tree analysis uses a single probability to represent each top event. However, it is unrealistic to evaluate the occurrence of each event by using a crisp value without considering the inherent uncertainty and imprecision a state has. The fuzzy set theory is universally applied to deal with this kind of phenomena. The main purpose of this study is to construct an easy method to evaluate human errors and integrate them into event tree analysis by using fuzzy concepts. A systematic fuzzy event tree analysis algorithm is developed to evaluate the risk of a large-scale system. A practical example in a nuclear power plant is used to demonstrate this procedure.
Dubois and Prade introduced the mean value of a fuzzy number as a closed interval bounded by the expectations calculated from its upper and lower distribution functions. In this paper introducing the notations of lower possibilistic and upper possibilistic mean values we define the interval-valued possibilistic mean and investigate its relationship to the interval-valued probabilistic mean. We also introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions, which are consistent with the extension principle. We also show that the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory.
The concept of fuzzy random variable was introduced as an analogous notion to random variables in order to extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets. But in contrary to the classical statistical methods no unique definition has been established yet. In this paper a set-theoretical concept of fuzzy random variable will be presented. This notion provides a useful framework to compare different concepts of fuzzy random variables, using methods of general topology and drawing on results from topological measure theory and the theory of analytic spaces. As the main result, it will be shown that the introduced concept of fuzzy random variable is a unification of the already known ones.
Hájek's BL logic is the fuzzy logic capturing the tautologies of continuous t-norms and their residua. In this paper we investigate a weaker logic, MTL, which is intended to cope with the tautologies of left-continuous t-norms and their residua. The corresponding algebraic structures, MTL-algebras, are defined and completeness of MTL with respect to linearly ordered MTL-algebras is proved. Besides, several schematic extensions of MTL are also considered as well as their corresponding predicate calculi.
Neural networks are currently finding practical applications, ranging from ‘soft’ regulatory control in consumer products to accurate modelling of non-linear systems. This paper presents the development of improved neural networks based short-term electric load forecasting models for the power system of the Greek Island of Crete. Several approaches including radial basis function networks, dynamic neural networks have been considered. In addition, a novel approach, based on neural-fuzzy approach has been proposed and discussed in this paper. Their performances are evaluated through a simulation study, using metered data provided by the Greek Public Power Corporation. The results indicate that the load forecasting models developed provide more accurate forecasts compared to the conventional backpropagation network forecasting models. Finally, the embedding of the new model capability in a modular forecasting system is presented.
We introduce a deductive probabilistic and fuzzy object-oriented model where a class property (i.e., an attribute or a method) can contain fuzzy set values, and uncertain class membership and property applicability are measured by lower and upper bounds on probability. Each uncertainly applicable property is interpreted as a default probabilistic logic rule, which is defeasible, and probabilistic default reasoning on fuzzy events is proposed for uncertain property inheritance and class recognition. This provides a formal basis for the design and implementation of FRIL++, the object-oriented extension of FRIL, a logic programming language dealing with both probability and fuzziness. The basic features of FRIL++ and its application as a programming language for deductive probabilistic and fuzzy object-oriented databases are presented.
The transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed. For the solution of the problem the technique of parametric programming is used. This makes it possible to obtain not only the maximizing solution (according to the Bellman-Zadeh criterion) but also other alternatives close to the optimal solution.
A new approach for ranking fuzzy numbers based on a distance measure is introduced. A new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first, and then it is used to formulate the distance measure for fuzzy numbers. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches.
The classes of the Lp,∞- and Lp-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each Lp,∞-metric there exists some Lp-metric which induces the same topology. This paper deals with the completion of the Lp,∞- and Lp-metrics. We can also show that the relationship between the classes of Lp,∞- and Lp-metrics still holds for the obtained respective classes of their completions.
Let be two convex fuzzy random variables on . Using a suitable metric we prove that the conditional expectation is the best approximation of by measurable functions of . This generalizes the analogous and well known property for real random variables. A further topic is the approximation of by a linear function of . In special cases and by use of Hukuhara's difference between fuzzy sets, we obtain formulas which are analogous to the classical structure. Contrary to the classical fact, however, the conditional expectation of Gaussian fuzzy random variables in general does not coincide with the linear regression function.
An interactive system is introduced which supports a decision maker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals. One part of the system is the determination of membership functions representing goals. To this purpose fuzzy extreme solutions are computed and are presented to the decision maker. These and each of the proposed compromise solutions are fuzzy-efficient.
A concept of fuzzy objective based on the Fuzzification Principle is presented. In accordance with this concept, the Fuzzy Linear Mathematical Programming problem is easily solved. A relationship of duality among fuzzy constraints and fuzzy objectives is given. The dual problem of a Fuzzy Linear Programming problem is also defined.
The generation of membership functions for fuzzy systems is a challenging problem. We show that for Mamdani-type fuzzy systems with correlation-product inference, centroid defuzzification, and triangular membership functions, optimizing the membership functions can be viewed as an identification problem for a nonlinear dynamic system. This identification problem can be solved with an extended Kalman filter. We describe the algorithm and compare it with gradient descent and with adaptive neuro-fuzzy inference system (ANFIS) based optimization of fuzzy membership functions. The methods discussed in this paper are illustrated on a fuzzy filter for motor winding current estimation, and are compared with Butterworth filtering. We demonstrate that the Kalman filter can be an effective tool for improving the performance of a fuzzy system.
Since Zadeh introduced fuzzy sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, others try to handle imprecision and uncertainty in a different (better?) way. Kerre (Computational Intelligence in Theory and Practice, Physica-Verlag, Heidelberg, 2001, pp. 55–72) has given a summary of the links that exist between fuzzy sets and other mathematical models such as flou sets (Gentilhomme), two-fold fuzzy sets (Dubois and Prade) and L-fuzzy sets (Goguen). In this paper, we establish the relationships between intuitionistic fuzzy sets (Atanassov, VII ITKR's Session, Sofia, June 1983 (Deposed in Central Sci.—Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)), L-fuzzy sets (J. Math. Anal. Appl. 18 (1967) 145), interval-valued fuzzy sets (Sambuc, Ph.D. Thesis, University of Marseille, France, 1975), interval-valued intuitionistic fuzzy sets (Intuitionistic fuzzy set, Physica-Verlag, Heidelberg, New York, 1999).
Multiple criteria decision making (MCDM) shows signs of becoming a maturing field. There are four quite distinct families of methods: (i) the outranking, (ii) the value and utility theory based, (iii) the multiple objective programming, and (iv) group decision and negotiation theory based methods. Fuzzy MCDM has basically been developed along the same lines, although with the help of fuzzy set theory a number of innovations have been made possible; the most important methods are reviewed and a novel approach — interdependence in MCDM — is introduced.
In the paper a fuzzy adaptive control algorithm is presented. It belongs to the class of direct model reference adaptive techniques based on a fuzzy (Takagi–Sugeno) model of the plant. The plant to be controlled is assumed to be nonlinear and predominantly of the first order. Consequently, the resulting adaptive and control laws are very simple and thus interesting for use in practical applications. The system remains stable in the presence of unmodelled dynamics (disturbances, parasitic high-order dynamics and reconstruction errors are treated explicitly). The global stability of the overall system is proven in the paper, i.e. it is shown that all signals remain bounded while the tracking error and estimated parameters converge to some residual set that depends on the size of disturbance and high-order parasitic dynamics. The proposed algorithm is tested on a simulated three-tank system. Its performance is compared to the performance of a classical MRAC.
This paper deals with a shortest path problem on a network in which a fuzzy number, instead of a real number, is assigned to each arc length. Such a problem is “ill-posed” because each arc cannot be identified as being either on the shortest path or not. Therefore, based on the possibility theory, we introduce the concept of “degree of possibility” that an arc is on the shortest path. Every pair of distinct paths from the source node to any other node is implicitly assumed to be noninteractive in the conventional approaches. This assumption is unrealistic and involve inconsistencies. To overcome this drawback, we define a new comparison index between the sum of fuzzy numbers by considering interactivity among fuzzy numbers. An algorithm is presented to determine the degree of possibility for each arc on a network. Finally, this algorithm is evaluated by means of large-scale numerical examples. Consequently, we can find this approach is efficient even for real world practical networks.
Possibilistic logic is a weighted logic introduced and developed since the mid-1980s, in the setting of artificial intelligence, with a view to develop a simple and rigorous approach to automated reasoning from uncertain or prioritized incomplete information. Standard possibilistic logic expressions are classical logic formulas associated with weights, interpreted in the framework of possibility theory as lower bounds of necessity degrees. Possibilistic logic handles partial inconsistency since an inconsistency level can be computed for each possibilistic logic base. Logical formulas with a weight strictly greater than this level are immune to inconsistency and can be safely used in deductive reasoning. This paper first recalls the basic features of possibilistic logic, including information fusion operations. Then, several extensions that mainly deal with the nature and the handling of the weights attached to formulas, are suggested or surveyed: the leximin-based comparison of proofs, the use of partially ordered scales for the weights, or the management of fuzzily restricted variables. Inference principles that are more powerful than the basic possibilistic inference in case of inconsistency are also briefly considered. The interest of a companion logic, based on the notion of guaranteed possibility functions, and working in a way opposite to the one of usual logic, is also emphasized. Its joint use with standard possibilistic logic is briefly discussed. This position paper stresses the main ideas only and refers to previous published literature for technical details.
We propose an interpretation of fuzzy set theory (both from a semantic and a syntactic point of view) in terms of conditional events and coherent conditional probabilities. During past years, a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty, or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory. In this paper we emphasize the role of conditioning (in a proper framework, i.e. de Finetti's coherence) to get rid of many controversial aspects. Moreover, we introduce suitable operations between fuzzy subsets, looked on as corresponding operations between conditional events endowed with the relevant conditional probability. Finally, we show how the concept of possibility function naturally arises as a coherent conditional probability.
The aim of this paper is to introduce a fuzzy control model with well-founded semantics in order to explain the concepts applied in fuzzy control. Assuming that the domains of the input and output variables for the process are endowed with equality relations, that reflect the indistinguishability of values lying closely together, the use of triangular and trapezoidal membership functions can be justified and max-⨅ inference where ⨅ is a t-norm turns out to be a consequence of our model. Distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like MOM or COA are appropriate or lead to undesired results.
Dynamic data mining is increasingly attracting attention from the respective research community. On the other hand, users of installed data mining systems are also interested in the related techniques and will be even more since most of these installations will need to be updated in the future. For each data mining technique used, we need different methodologies for dynamic data mining. In this paper, we present a methodology for dynamic data mining based on fuzzy clustering. Using the implementation of the proposed system we show its benefits in two application areas: customer segmentation and traffic management.
This work mainly develops a discrete singular fuzzy model and discusses its stability. The standard discrete T–S model is generalized into the so-called discrete singular Takagi–Sugeno (DST-S) model, which can represent a larger class of nonlinear systems, and the stability criteria of this system are then investigated. These criteria are described by non-strict linear matrix inequalities, and we thus involve the projection method for ensuring stability. Based on some derived projection operators, a projection algorithm is proposed for solving a feasible solution. An illustrating example demonstrates the validity and effectiveness of the proposed method.
In this paper, we study the class of fuzzy differential equations where the dynamics is given by a continuous fuzzy mapping which is obtained via Zadeh's extension principle. We get a fuzzy solution for this class of fuzzy differential equations and several illustrative examples are presented. We also give some properties and we show the relationships with others interpretation. Finally, we propose a numerical procedure for obtaining the fuzzy solution.