Showing papers on "Membership function published in 1997"

An ART-based fuzzy adaptive learning control network

Abstract: This paper addresses the structure and an associated online learning algorithm of a feedforward multilayer neural net for realizing the basic elements and functions of a fuzzy controller. The proposed fuzzy adaptive learning control network (FALCON) can be contrasted with traditional fuzzy control systems in network structure and learning ability. An online structure/parameter learning algorithm, FALCON-ART, is proposed for constructing FALCON dynamically. It combines backpropagation for parameter learning and fuzzy ART for structure learning. FALCON-ART partitions the input state space and output control space using irregular fuzzy hyperboxes according to the data distribution. In many existing fuzzy or neural fuzzy control systems, the input and output spaces are always partitioned into "grids". As the number of variables increases, the number of partitioned grids grows combinatorially. To avoid this problem in some complex systems, FALCON-ART partitions the I/O spaces flexibly based on data distribution. It can create and train FALCON in a highly autonomous way. In its initial form, there is no membership function, fuzzy partition, and fuzzy logic rule. They are created and begin to grow as the first training pattern arrives. Thus, the users need not give it any a priori knowledge or initial information. FALCON-ART can online partition the I/O spaces, tune membership functions, find proper fuzzy logic rules, and annihilate redundant rules dynamically upon receiving online data.

Combination of Rough and Fuzzy Sets Based on α-Level Sets

Abstract: A fuzzy set can be represented by a family of crisp sets using its α-level sets, whereas a rough set can be represented by three crisp sets. Based on such representations, this paper examines some fundamental issues involved in the combination of rough-set and fuzzy-set models. The rough-fuzzy-set and fuzzy-rough-set models are analyzed, with emphasis on their structures in terms of crisp sets. A rough fuzzy set is a pair of fuzzy sets resulting from the approximation of a fuzzy set in a crisp approximation space, and a fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a crisp set in a fuzzy approximation space. The approximation of a fuzzy set in a fuzzy approximation space leads to a more general framework. The results may be interpreted in three different ways.

Fuzzy Maps: A New Tool for Mobile Robot Perception and Planning

Abstract: An essential component of an autonomous mobile robot is the exteroceptive sensory system. Sensing capabilities should be integrated with a method for extracting a representation of the environment from uncertain sensor data and with an appropriate planning algorithm. In this article, fuzzy logic concepts are used to introduce a tool useful for robot perception as well as for planning collision-free motions. In particular, a map of the environment is defined as the fuzzy set of unsafe points, whose membership function quantifies the possibility for each point to belong to an obstacle. The computation of this set is based on a specific sensor model and makes use of intermediate sets generated from range measures and aggregated by means of fuzzy set operators. This general approach is applied to a robot with ultrasonic rangefinders. The resulting map building algorithm performs well, as confirmed by a comparison with stochastic methods. The planning problem on fuzzy maps can be solved by defining various path cost functions, corresponding to different strategies, and by searching the map for optimal paths. To this end, proper instances of the A* algorithm are devised. Experimental results for a Nomad 200™ robot moving in a real-world environment are presented. © 1997 John Wiley & Sons, Inc.

An algorithmic approach for fuzzy inference

Abstract: To apply fuzzy logic, two major tasks need to be performed: the derivation of production rules and the determination of membership functions. These tasks are often difficult and time consuming. This paper presents an algorithmic method for generating membership functions and fuzzy production rules; the method includes an entropy minimization for screening analog values. Membership functions are derived by partitioning the variables into the desired number of fuzzy terms and production rules are obtained from minimum entropy clustering decisions. In the rule derivation process, rule weights are also calculated. This algorithmic approach alleviates many problems in the application of fuzzy logic to binary classification.

Fuzzy model identification for classification of gait events in paraplegics

Abstract: 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

Fuzzy Systems Design Principles: Building Fuzzy If-Then Rule Bases

Abstract: Foreword. Preface. Introduction. Theory. The Basic Fuzzy Inference Algorithm. Conceptual Design. Fuzzy Variable Design. Membership Function Shape Analysis. Composing Fuzzy Rules. Implication Design. Appendix: The Basic Fuzzy Inference Algorithm. Index.

Planning Reservoir Operations with Imprecise Objectives

Abstract: Imprecise and noncommensurable objectives for reservoir operation are addressed through fuzzy dynamic programming using an implicit stochastic approach. Fuzzy membership functions for evaluating the achievement of a linguistically described operational goal and linguistically described constraints are estimated from surveys of decision makers. Summary statistics of the membership function values for optimal operation provide easily interpreted measures of degree of satisfaction among diverse objectives. An example application to the proposed Grey Mountain Reservoir on the Cache la Poudre River in northern Colorado showed that the expected degree of satisfaction for water supply objectives (constraints) would exceed 70% in six of 12 months, while 90% satisfaction of the flood control objective could be expected throughout the year. Expected degrees of satisfaction for storage, recreational, fish habitat, and hydropower objectives are substantially lower. Relative variability in predicted degrees of satisfa...

Fuzzy approximation via grid point sampling and singular value decomposition

Abstract: This paper introduces a new approach for fuzzy approximation of continuous function on a compact domain. The approach calls for sampling the function over a set of rectangular grid points and applying singular value decomposition to the sample matrix. The resulting quantities are then tailored to become rule consequences and membership functions via the conditions of sum normalization and non-negativeness. The inference paradigm of product-sum-gravity is apparent from the structure of the decomposition equation. All information are extracted directly from the function samples. The present approach yields a class of equivalent fuzzy approximator to a given function. A tight bounding technique to facilitate normal or close-to-normal membership functions is also formulated. The fuzzy output approximates the given function to within an error which is dependent on the sampling intervals and the singular values discarded from the approximation process. Trade-off between the number of membership functions and the desired approximation accuracy is also discussed.

Powerset operator foundations for point-set lattice-theoretic (poslat) fuzzy set theories and topologies

Abstract: This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology. In particular, those aspects of fuzzy sets, hinging around (crisp) powersets of fuzzy subsets and around powerset operators between such powersets lifted from ordinary functions between the underlying base sets, are examined and characterized using point-set and lattice-theoretic methods. The basic goal is to uniquely derive the powerset operators and not simply stipulate them, and in doing this we explicitly distinguish between the “fixed-basis” case (where the underlying lattice of membership values is fixed for the sets in question) and the “variable-basis” case (where the underlying lattice of membership values is allowed to change). Applications to fuzzy sets/logic include: development a...

Fuzzy logic applications to multisensor-multitarget correlation

Abstract: A consistent tactical picture requires data fusion technology to combine and propagate information received from diverse objects and usually vague situations. The information may be contained in two types of data; numerical data received from sensor measurements, and linguistic data obtained from human operators and domain experts. In real world situations, the numerical data may be noisy, inconsistent, and incomplete, and the linguistic information is imprecise and vague. To deal with these two types of data simultaneously, fuzzy sets and fuzzy logic provide a methodology to obtain an approximate but consistent tactical picture in a timely manner for very complex or ill-defined engineering problems. A functional paradigm for fuzzy data fusion is presented. It consists of four basic elements: (1) fuzzification of crisp elements, (2) fuzzy knowledge base derived from numerical input/output relations and humans, (3) fuzzy inference mechanism based on a class of fuzzy logic, (4) defuzzification of fuzzy outputs into crisp outputs for use by a plant. For real-time practical systems, the on-line determination of a fuzzy membership function from a given set of crisp inputs is vital. To this end, a methodology for estimating an optimal membership function from crisp input data has been implemented. This is based on the possibility/probability consistency principle as proposed by L.A. Zadeh. A relationship between the fuzzy membership function and the confidence level of statistical input data has been developed and it serves as a design parameter for fuzzification. This technique has been applied to a two-dimensional multisensor-multitarget tracking system. Fuzzy system performance evaluations have been presented. With simulated data in the laboratory environment, the simulation has been performed to evaluate the Mission Avionics Sensor Synergism (MASS) Systems. These results show better performance for the data correlation function using the fuzzy logic techniques.

A methodology for constructing fuzzy algorithms for learning vector quantization

Abstract: This paper presents a general methodology for the development of fuzzy algorithms for learning vector quantization (FALVQ). The design of specific FALVQ algorithms according to existing approaches reduces to the selection of the membership function assigned to the weight vectors of an LVQ competitive neural network, which represent the prototypes. The development of a broad variety of FALVQ algorithms can be accomplished by selecting the form of the interference function that determines the effect of the nonwinning prototypes on the attraction between the winning prototype and the input of the network. The proposed methodology provides the basis for extending the existing FALVQ 1, FALVQ 2, and FALVQ 3 families of algorithms. This paper also introduces two quantitative measures which establish a relationship between the formulation that led to FALVQ algorithms and the competition between the prototypes during the learning process. The proposed algorithms and competition measures are tested and evaluated using the IRIS data set. The significance of the proposed competition measure is illustrated using FALVQ algorithms to perform segmentation of magnetic resonance images of the brain.

Finding the Brain Cortex Using Fuzzy Segmentation, Isosurfaces, and Deformable Surface Models

Abstract: A method for finding the cortical surface of the brain from magnetic resonance images using a combination of fuzzy segmentation, isosurface extraction, and a deformable surface is presented After MR images are acquired and preprocessed to remove extracranial tissue, fuzzy membership functions for gray matter, white matter, and cerebrospinal fluid are computed An iterative procedure using isosurfaces of filtered white matter membership functions is then used to obtain a topologically correct estimate of the cortical surface This estimate forms the initialization of a deformable surface, which is then allowed to converge to peaks of the gray matter membership function We demonstrate the results of each step and show the final parameterized map of the medial layer of the cortex

An algorithm for fuzzy multi-criteria decision making

Abstract: The paper presents a new algorithm for solving the general fuzzy multi-criteria decision making (MCDM) problem involving fully data expressed by means of linguistic terms. A fuzzy performance matrix representing the overall assessment of alternatives with respect to each criterion is obtained by using interval arithmetic. To avoid the complex and unreliable ranking process of fuzzy numbers, the /spl alpha/-cut concept is used to transform the fuzzy performance matrix into an interval performance matrix. Incorporated with the decision maker's attitude towards risk, an overall preference index is obtained by applying the concept of the degree of similarity to the ideal solution using a vector matching function. The algorithm developed is simple and comprehensible with easy computation which facilitates its use in a decision support system for the general fuzzy MCDM problem. An example is presented to demonstrate its applicability.

Adaptive fuzzy controller that modifies membership functions

Abstract: A direct adaptive fuzzy controller having an adaptation mechanism that modifies the locations of output membership functions to improve performance of the fuzzy controller. The controller has a fuzzy rule base that has a different or unique output membership function for each fuzzy rule. The adaptation mechanism modifies the location of the output membership functions in response to the performance of the controller system, to continuously improve its performance. The controller is a feedback mechanism that functions not only using measured outputs of the process controlled, but has feedforward compensation that causes the controller to anticipate feedback due to measured disturbances or other parameters of the process. The adaptive fuzzy controller can be used for various processes, including combi-boiler domestic hot water temperature control, automobile cruise control, a residential thermostat, duct air temperature or static pressure control for the air handling unit in a heating ventilation and air conditioning system, and furnace temperature control for curing parts.