Showing papers on "Fuzzy number published in 1994"

Fuzzy Logic with Engineering Applications

Abstract: About the Author. Preface to the Third Edition. 1 Introduction. The Case for Imprecision. A Historical Perspective. The Utility of Fuzzy Systems. Limitations of Fuzzy Systems. The Illusion: Ignoring Uncertainty and Accuracy. Uncertainty and Information. The Unknown. Fuzzy Sets and Membership. Chance Versus Fuzziness. Sets as Points in Hypercubes. Summary. References. Problems. 2 Classical Sets and Fuzzy Sets. Classical Sets. Operations on Classical Sets. Properties of Classical (Crisp) Sets. Mapping of Classical Sets to Functions. Fuzzy Sets. Fuzzy Set Operations. Properties of Fuzzy Sets. Alternative Fuzzy Set Operations. Summary. References. Problems. 3 Classical Relations and Fuzzy Relations. Cartesian Product. Crisp Relations. Cardinality of Crisp Relations. Operations on Crisp Relations. Properties of Crisp Relations. Composition. Fuzzy Relations. Cardinality of Fuzzy Relations. Operations on Fuzzy Relations. Properties of Fuzzy Relations. Fuzzy Cartesian Product and Composition. Tolerance and Equivalence Relations. Crisp Equivalence Relation. Crisp Tolerance Relation. Fuzzy Tolerance and Equivalence Relations. Value Assignments. Cosine Amplitude. Max Min Method. Other Similarity Methods. Other Forms of the Composition Operation. Summary. References. Problems. 4 Properties of Membership Functions, Fuzzification, and Defuzzification. Features of the Membership Function. Various Forms. Fuzzification. Defuzzification to Crisp Sets. -Cuts for Fuzzy Relations. Defuzzification to Scalars. Summary. References. Problems. 5 Logic and Fuzzy Systems. Part I Logic. Classical Logic. Proof. Fuzzy Logic. Approximate Reasoning. Other Forms of the Implication Operation. Part II Fuzzy Systems. Natural Language. Linguistic Hedges. Fuzzy (Rule-Based) Systems. Graphical Techniques of Inference. Summary. References. Problems. 6 Development of Membership Functions. Membership Value Assignments. Intuition. Inference. Rank Ordering. Neural Networks. Genetic Algorithms. Inductive Reasoning. Summary. References. Problems. 7 Automated Methods for Fuzzy Systems. Definitions. Batch Least Squares Algorithm. Recursive Least Squares Algorithm. Gradient Method. Clustering Method. Learning From Examples. Modified Learning From Examples. Summary. References. Problems. 8 Fuzzy Systems Simulation. Fuzzy Relational Equations. Nonlinear Simulation Using Fuzzy Systems. Fuzzy Associative Memories (FAMS). Summary. References. Problems. 9 Decision Making with Fuzzy Information. Fuzzy Synthetic Evaluation. Fuzzy Ordering. Nontransitive Ranking. Preference and Consensus. Multiobjective Decision Making. Fuzzy Bayesian Decision Method. Decision Making Under Fuzzy States and Fuzzy Actions. Summary. References. Problems. 10 Fuzzy Classification. Classification by Equivalence Relations. Crisp Relations. Fuzzy Relations. Cluster Analysis. Cluster Validity. c-Means Clustering. Hard c-Means (HCM). Fuzzy c-Means (FCM). Fuzzy c-Means Algorithm. Classification Metric. Hardening the Fuzzy c-Partition. Similarity Relations from Clustering. Summary. References. Problems. 11 Fuzzy Pattern Recognition. Feature Analysis. Partitions of the Feature Space. Single-Sample Identification. Multifeature Pattern Recognition. Image Processing. Summary. References. Problems. 12 Fuzzy Arithmetic and the Extension Principle. Extension Principle. Crisp Functions, Mapping, and Relations. Functions of Fuzzy Sets Extension Principle. Fuzzy Transform (Mapping). Practical Considerations. Fuzzy Arithmetic. Interval Analysis in Arithmetic. Approximate Methods of Extension. Vertex Method. DSW Algorithm. Restricted DSW Algorithm. Comparisons. Summary. References. Problems. 13 Fuzzy Control Systems. Control System Design Problem. Control (Decision) Surface. Assumptions in a Fuzzy Control System Design. Simple Fuzzy Logic Controllers. Examples of Fuzzy Control System Design. Aircraft Landing Control Problem. Fuzzy Engineering Process Control. Classical Feedback Control. Fuzzy Control. Fuzzy Statistical Process Control. Measurement Data Traditional SPC. Attribute Data Traditional SPC. Industrial Applications. Summary. References. Problems. 14 Miscellaneous Topics. Fuzzy Optimization. One-Dimensional Optimization. Fuzzy Cognitive Mapping. Concept Variables and Causal Relations. Fuzzy Cognitive Maps. Agent-Based Models. Summary. References. Problems. 15 Monotone Measures: Belief, Plausibility, Probability, and Possibility. Monotone Measures. Belief and Plausibility. Evidence Theory. Probability Measures. Possibility and Necessity Measures. Possibility Distributions as Fuzzy Sets. Possibility Distributions Derived from Empirical Intervals. Deriving Possibility Distributions from Overlapping Intervals. Redistributing Weight from Nonconsonant to Consonant Intervals. Comparison of Possibility Theory and Probability Theory. Summary. References. Problems. Index.

Adaptive Fuzzy Systems and Control: Design and Stability Analysis

Abstract: Description and analysis of fuzzy logic systems training of fuzzy logic systems using back-propagation training of fuzzy logic systems using orthogonal least squares training of fuzzy logic systems using a table-lookup scheme training of fuzzy logic systems using nearest neighbourhood clustering comparison of adaptive fuzzy systems with artificial neural networks stable indirect adaptive fuzzy control of nonlinear systems stable direct adaptive fuzzy control of nonlinear systems design of adaptive fuzzy controllers using input-output linearization concepts design and stability analysis of fuzzy identifiers of nonlinear dynamic systems fuzzy adaptive filters.

Fuzzy systems as universal approximators

Abstract: An additive fuzzy system can uniformly approximate any real continuous function on a compact domain to any degree of accuracy. An additive fuzzy system approximates the function by covering its graph with fuzzy patches in the input-output state space and averaging patches that overlap. The fuzzy system computes a conditional expectation E|Y|X| if we view the fuzzy sets as random sets. Each fuzzy rule defines a fuzzy patch and connects commonsense knowledge with state-space geometry. Neural or statistical clustering systems can approximate the unknown fuzzy patches from training data. These adaptive fuzzy systems approximate a function at two levels. At the local level the neural system approximates and tunes the fuzzy rules. At the global level the rules or patches approximate the function. >

A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer

Abstract: A robust stabilization problem for fuzzy systems is discussed in accordance with the definition of stability in the sense of Lyapunov. We consider two design problems: nonrobust controller design and robust controller design. The former is a design problem for fuzzy systems with no premise parameter uncertainty. The latter is a design problem for fuzzy systems with premise parameter uncertainty. To realize two design problems, we derive four stability conditions from a basic stability condition proposed by Tanaka and Sugeno: nonrobust condition, weak nonrobust condition, robust condition, and weak robust condition. We introduce concept of robust stability for fuzzy control systems with premise parameter uncertainty from the weak robust condition. To introduce robust stability, admissible region and variation region, which correspond to stability margin in the ordinary control theory, are defined. Furthermore, we develop a control system for backing up a computer simulated truck-trailer which is nonlinear and unstable. By approximating the truck-trailer by a fuzzy system with premise parameter uncertainty and by using concept of robust stability, we design a fuzzy controller which guarantees stability of the control system under a condition. The simulation results show that the designed fuzzy controller smoothly achieves backing up control of the truck-trailer from all initial positions. >

Metric Spaces of Fuzzy Sets: Theory and Applications

Abstract: Fuzzy Sets Spaces of Subsets of Rn Compact Convex Subsets of Rn Set Valued Mappings Crisp Generalizations The Space En Metrics on En Compactness Criteria Generalizations Fuzzy Set Valued Mappings of Real Variables Fuzzy Random Variables Computational Methods Fuzzy Differential Equations Optimization Under Uncertainty Fuzzy Iterations and Image Processing.

Approximation theory of fuzzy systems/spl minus/SISO case

Abstract: In this paper, the approximation properties of MIMO fuzzy systems generated by the product inference are discussed. We first give an analysis of fuzzy basic functions (FBF's) and present several properties of FBF's. Based on these properties of FBF's, we obtain several basic approximation properties of fuzzy systems: 1) basic approximation property which reveals the basic approximation mechanism of fuzzy systems; 2) uniform approximation bounds which give the uniform approximation bounds between the desired (control or decision) functions and fuzzy systems; 3) uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions; and 4) universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect. The similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds. >

Adaptive control of a class of nonlinear systems with fuzzy logic

Abstract: An adaptive tracking control architecture is proposed for a class of continuous-time nonlinear dynamic systems, for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture employs fuzzy systems, which are expressed as a series expansion of basis functions, to adaptively compensate for the plant nonlinearities. Global asymptotic stability of the algorithm is established in the Lyapunov sense, with tracking errors converging to a neighborhood of zero. Simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance. >

Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis

Abstract: A bipolar fuzzy set theory is presented for cognitive modeling and multiagent decision analysis. Firstly, notions of bipolar fuzziness are introduced. Secondly an interval-based bipolar fuzzy logic is defined which generalizes a real-valued bipolar fuzzy logic by allowing interval-based linguistic variables x and y to be substituted into (S, =, V, /spl otimes/) or (S, =, /spl cup/, /spl odot/) where S={/spl forall/(x,y)|(x,y)/spl isin/([-1,0]/spl times/[0,+1])}. Thirdly, a fuzzy number-based bipolar logic is presented which. Further generalizes the interval-based model by allowing /spl alpha/-level fuzzy number-based linguistic variables x and y to be substituted into (S, =, V, /spl otimes/) or (S, =, /spl cup/, /spl odot/), S={/spl forall/(x,y)|(x,y) maps ([-1,0]/spl times/[0,+1]) to [0,1]}. Bipolar fuzzy set operations of disjunction composition (V-/spl otimes/), union-composition (/spl cup/, /spl odot/), are proved commutative and associative; V respect to /spl otimes/ and /spl cup/ respect to /spl odot/ are proved distributive. It is shown that a interval-based bipolar variable is a nesting of a real-valued bipolar variable; a trapezoidal-fuzzy number-based bipolar variable is an 2-level nesting of an interval-based bipolar variable; and an /spl alpha/-level (/spl alpha/ is an integer) fuzzy number-based bipolar variable is an /spl alpha/+1 level nesting of an interval-based bipolar variable. Based on the nesting features, it is proved that /spl alpha/-level fuzzy number-based bipolar operations can be converted to interval-based and then real-valued bipolar operations. The conversions lead to significant computational simplification on bipolar fuzzy relations. Major advantages of the bipolar fuzzy set theory include: (1) it formalizes a unified approach to polarity and fuzziness; (2) it captures die bipolar or double-sided (negative and positive, or effect and side effect) nature of human perception and cognition; and (3) it provides a basis for bipolar cognitive modeling and multiagent decision analysis. >

Reinforcement structure/parameter learning for neural-network-based fuzzy logic control systems

Abstract: This paper proposes a reinforcement neural-network-based fuzzy logic control system (RNN-FLCS) for solving various reinforcement learning problems. The proposed RNN-FLCS is constructed by integrating two neural-network-based fuzzy logic controllers (NN-FLC's), each of which is a connectionist model with a feedforward multilayered network developed for the realization of a fuzzy logic controller. One NN-FLC performs as a fuzzy predictor, and the other as a fuzzy controller. Using the temporal difference prediction method, the fuzzy predictor can predict the external reinforcement signal and provide a more informative internal reinforcement signal to the fuzzy controller. The fuzzy controller performs a stochastic exploratory algorithm to adapt itself according to the internal reinforcement signal. During the learning process, both structure learning and parameter learning are performed simultaneously in the two NN-FLC's using the fuzzy similarity measure. The proposed RNN-FLCS can construct a fuzzy logic control and decision-making system automatically and dynamically through a reward/penalty signal or through very simple fuzzy information feedback such as "high," "too high," "low," and "too low." The proposed RNN-FLCS is best applied to the learning environment, where obtaining exact training data is expensive. It also preserves the advantages of the original NN-FLC, such as the ability to find proper network structure and parameters simultaneously and dynamically and to avoid the rule-matching time of the inference engine. Computer simulations were conducted to illustrate its performance and applicability. >

Measuring fuzzy uncertainty

Abstract: First, this paper reviews several well known measures of fuzziness for discrete fuzzy sets. Then new multiplicative and additive classes are defined. We show that each class satisfies five well-known axioms for fuzziness measures, and demonstrate that several existing measures are relatives of these classes. The multiplicative class is based on nonnegative, monotone increasing concave functions. The additive class requires only nonnegative concave functions. Some relationships between several existing and the new measures are established, and some new properties are derived. The relative merits and drawbacks of different measures for applications are discussed. A weighted fuzzy entropy which is flexible enough to incorporate subjectiveness in the measure of fuzziness is also introduced. Finally, we comment on the construction of measures that may assess all of the uncertainties associated with a physical system. >

Genetic-based new fuzzy reasoning models with application to fuzzy control

Abstract: The successful application of fuzzy reasoning models to fuzzy control systems depends on a number of parameters, such as fuzzy membership functions, that are usually decided upon subjectively. It is shown in this paper that the performance of fuzzy control systems may be improved if the fuzzy reasoning model is supplemented by a genetic-based learning mechanism. The genetic algorithm enables us to generate an optimal set of parameters for the fuzzy reasoning model based either on their initial subjective selection or on a random selection. It is shown that if knowledge of the domain is available, it is exploited by the genetic algorithm leading to an even better performance of the fuzzy controller. >

Classification by fuzzy integral: performance and tests

Abstract: This paper presents an attempt to characterize the performance of methods of classification based on fuzzy integral. After an introductory explanation about the approach, a lower bound of the minimal number of training samples is found, and it is shown that a minimum squared error criterion leads to the best approximate for the optimal Bayes classifier. Some tests on simulated and real data are provided.

Fuzzy decision trees by fuzzy ID3 algorithm and its application to diagnosis systems

Abstract: A popular and particularly efficient method for making a decision tree for classification from symbolic data is ID3 algorithm. Revised algorithms for numerical data have been proposed, some of which divide a numerical range into several intervals or fuzzy intervals. Their decision trees, however, are not easy to understand. We propose a new version of ID3 algorithm to generate an understandable fuzzy decision tree using fuzzy sets defined by a user. We apply it to diagnosis for potential transformers by analyzing gas in oil. >

Anomaly detection

Abstract: Computer are finite discrete machines, the set of real numbers is an infinite continuum. So real numbers in computers are approximation. Rough set theory is the underlying mathematics. A 'computer' version of Weistrass theorem states that every sequence, within the radius of error, repeats certain terms infinitely many times. In terms of applications, the theorem guarantees that the audit trail has repeating patterns. Examining further, based on fuzzy-rough set theory, hidden fuzzy relationships (rules) in audit data are uncovered. The information about the repeating data and fuzzy relationships reflect "unconscious patterns" of users' habits. They are some deeper "signatures" of computer users, which provide a foundation to detect abuses and misuses of computer systems, A "sliding window information system" is used to illustrate the detection of a 'simple' virus attack. The complexity problem is believed to be controllable via rough set representation of data.

Common fixed points of maps on fuzzy metric spaces

Abstract: Following Grabiec's approach to fuzzy contraction principle, the purpose of this note is to obtain common fixed point theorems for asymptotically commuting maps on fuzzy

Fuzzy cash flow analysis using present worth criterion

Abstract: In practice, engineering economic analysis involves uncertainty about future cash flows To deal quantitatively with imprecision or uncertainty, fuzzy set theory is primarily concerned with vagueness in human thoughts and perceptions As an alternative to conventional cash flow models where cash flows are defined as either crisp numbers or risky probability distributions, we propose an engineering economic decision model in which the uncertain cash flows and discount rates are specified as triangular fuzzy numbers The present worth formulation of this fuzzy cash flow model is derived The result of the present worth is also a fuzzy number with nonlinear membership function The present worth can be approximated by a triangular fuzzy number Deviation between exact present worth and its approximate form is examined Finally, the fuzzy project selection is performed by applying different dominance rules To demonstrate the application of the fuzzy present worth function, a comprehensive numerical

Construction of fuzzy classification systems with rectangular fuzzy rules using genetic algorithms

Abstract: This paper proposes a genetic-algorithm-based approach to the construction of fuzzy classification systems with rectangular fuzzy rules. In the proposed approach, compact fuzzy classification systems are automatically constructed from numerical data by selecting a small number of significant fuzzy rules using genetic algorithms. Since significant fuzzy rules are selected and unnecessary fuzzy rules are removed, the proposed approach can be viewed as a knowledge acquisition tool for classification problems. In this paper, we first describe a generation method of rectangular fuzzy rules from numerical data for classification problems. We next formulate a rule selection problem for constructing a compact fuzzy classification system as a combinatorial optimization problem with two objectives: to minimize the number of selected fuzzy rules and to maximize the number of correctly classified patterns. We then show how genetic algorithms are applied to the rule selection problem. Last, we illustrate the proposed approach by computer simulations on numerical examples and the iris data of Fisher.