The clustering problem is cast in the framework of possibility theory. The approach differs from the existing clustering methods in that the resulting partition of the data can be interpreted as a possibilistic partition, and the membership values can be interpreted as degrees of possibility of the points belonging to the classes, i.e., the compatibilities of the points with the class prototypes. An appropriate objective function whose minimum will characterize a good possibilistic partition of the data is constructed, and the membership and prototype update equations are derived from necessary conditions for minimization of the criterion function. The advantages of the resulting family of possibilistic algorithms are illustrated by several examples.

This paper discusses a general approach to qualitative modeling based on fuzzy logic. The method of qualitative modeling is divided into two parts: fuzzy modeling and linguistic approximation. It proposes to use a fuzzy clustering method (fuzzy c-means method) to identify the structure of a fuzzy model. To clarify the advantages of the proposed method, it also shows some examples of modeling, among them a model of a dynamical process and a model of a human operator's control action.

A direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control, is capable of incorporating fuzzy if-then control rules directly into the controllers, and guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded is developed. The specific formula for the bounds is provided, so that controller designers can determine the bounds based on their requirements. The direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the Duffing chaotic system to track a trajectory. The simulation results show that the controller worked without using any fuzzy control rules, and that after fuzzy control rules were incorporated the adaptation speed became much faster. It is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control.

Establishing suitable control of pH, a requirement in a number of mineral and chemical industries, poses a difficult problem because of inherent nonlinearities and frequently changing process dynamics. Researchers at the U.S. Bureau of Mines have developed a technique for producing adaptive fuzzy logic controllers (FLC's) that are capable of effectively managing such systems. In this technique, a genetic algorithm (GA) alters the membership functions employed by a conventional FLC, an approach that is contrary to the tactic generally used to provide FLC's with adaptive capabilities in which the rule set is altered. GA's are search algorithms based on the mechanics of natural genetics that are able to rapidly locate near-optimal solutions to difficult problems. The Bureau-developed technique is used to produce an adaptive GA-FLC for a laboratory acid-base experiment. Nonlinearities in the laboratory system are associated with the logarithmic pH scale (pH is proportional to the logarithm of H sub(3)O super(+) ions) and changing process dynamics are introduced by altering system parameters such as the desired set point and the concentration and buffering capacity of input solutions. Results indicate that FLC's augmented with GA's offer a powerful alternative to conventional process control techniques in the nonlinear, rapidly changing pH systems commonly found in industry.

A family of objective functions called fuzzy c-regression models, which can be used too fit switching regression models to certain types of mixed data, is presented. Minimization of particular objective functions in the family yields simultaneous estimates for the parameters of c regression models, together with a fuzzy c-partitioning of the data. A general optimization approach for the family of objective functions is given and corresponding theoretical convergence results are discussed. The approach is illustrated by two numerical examples that show how it can be used to fit mixed data to coupled linear and nonlinear models.

In an earlier companion paper a supervised learning neural network pattern classifier called the fuzzy min-max classification neural network was described. In this sequel, the unsupervised learning pattern clustering sibling called the fuzzy min-max clustering neural network is presented. Pattern clusters are implemented here as fuzzy sets using a membership function with a hyperbox core that is constructed from a min point and a max point. The min-max points are determined using the fuzzy min-max learning algorithm, an expansion-contraction process that refines the author's earlier Fuzzy Adaptive Resonance Theory neural network. The fuzzy min-max clustering neural network stabilizes into pattern clusters in only a few passes through a data set; it can be reduced to hard cluster boundaries that are easily examined without sacrificing the fuzzy boundaries; it provides the ability to incorporate new data and add new clusters without retraining; and it inherently provides degree of membership information that is extremely useful in higher level decision making and information processing. This paper will provide some background concerning the development of the fuzzy min-max clustering neural network and provide a comparison with similar work that has recently emerged. A brief description of fuzzy sets, pattern clustering, and their synergistic combination is presented. The fuzzy min-max clustering neural network will be explained in detail and examples of its clustering performance will be given. The paper will conclude with a description of problems that need to be addressed and a list of some potential applications.

An architecture for neural networks that can handle fuzzy input vectors is proposed, and learning algorithms that utilize fuzzy if-then rules as well as numerical data in neural network learning for classification problems and for fuzzy control problems are derived. The learning algorithms can be viewed as an extension of the backpropagation algorithm to the case of fuzzy input vectors and fuzzy target outputs. Using the proposed methods, linguistic knowledge from human experts represented by fuzzy if-then rules and numerical data from measuring instruments can be integrated into a single information processing system (classification system or fuzzy control system). It is shown that the scheme works well for simple examples.

To improve limitations of fuzzy PI controller especially when applied to high order systems, we propose two types of fuzzy logic controllers that take out appropriate amounts of accumulated control input according to fuzzily described situations in addition to the incremental control input calculated by conventional fuzzy PI controllers. The structures of the proposed controller were motivated by the problems of fuzzy PI controllers that they generally give inevitable overshoot when one tries to reduce rise time of response especially when a system of order higher than one is under consideration. Since the undesirable characteristics of the fuzzy PI controller are caused by integrating operation of the controller, even though the integrator itself is introduced to to overcome steady state error in response, we propose two fuzzy controllers that fuzzily clear out integrated quantities according to situation. The first controller determines the fuzzy resetting rate by situations described fuzzily by error and error rate, and the second one by error and control input. The two structures both give reduced rise time as well as small overshoot.

Two fuzzy adaptive filters are developed: one uses a recursive-least-squares (RLS) adaptation algorithm, and the other uses a least-mean-square (LMS) adaptation algorithm. The RLS fuzzy adaptive filter is constructed through the following four steps: (1) define fuzzy sets in the filter input space Rn whose membership functions cover U; (2) construct a set of fuzzy IF-THEN rules which either come from human experts or are determined during the adaptation procedure by matching input-output data pairs; (3) construct a filter based on the set of rules; and (4) update the free parameters of the filter using the RLS algorithm. The design procedure for the LMS fuzzy adaptive filter is similar. The most important advantage of the fuzzy adaptive filters is that linguistic information (in the form of fuzzy IF-THEN rules) and numerical information (in the form of input-output pairs) can be combined in the filters in a uniform fashion. The filters are applied to nonlinear communication channel equalization problems.

We will introduce and study different fuzzy-set oriented computational models of neurons. The generic topologies of the neurons emerging there are significantly influenced by basic logic operators(AND, OR, NOT) encountered in the theory of fuzzy sets. The logical flavor of the proposed constructs is expressed in terms of operators used in their formalization and a way of their superposition in the neurons. The two broad categories of neurons embrace basic aggregation neurons (named AND and OR neurons) and referential processing units (such as matching, dominance, inclusion neurons). The specific features of the neurons are flexibly modeled with the aid of triangular norms. The inhibitory and excitatory characteristics are captured by embodying direct and complemented (negated) input signals. We will propose various topologies of neural networks put together with the use of these neurons and demonstrate straightforward relationships coming off between the problem specificity and the resulting architecture of the network. This limpid way of mapping the domain knowledge onto the structure of the network contributes significantly toward enhancements in learning processes in the network and substantially facilitates its interpretation.

Properties of objects and spatial relations between objects play an important role in rule-based approaches for high-level vision. The partial presence or absence of such properties and relationships can supply both positive and negative evidence for region labeling hypotheses. Similarly, fuzzy labeling of a region can generate new hypotheses pertaining to the properties of the region, its relation to the neighboring regions, and, finally, hypotheses pertaining to the labels of the neighboring regions. A unified methodology that can be used to characterize both properties and spatial relationships of object regions in a digital image is presented. The methods proposed for computing the properties and relations of image regions can be used to arrive at more meaningful decisions about the contents of the scene.

We introduce a parameterized family of defuzzification operators called the Semi LInear DEfuzzification (SLIDE) method. This method is based upon a simple transformation of the fuzzy output set of the controller. We suggest an algorithm for the learning of the parameter from a data set. In an attempt to simplify the parameter learning we suggest a modified version of the SLIDE method which results in a simple learning algorithm. The development of the learning algorithm is based upon the use of the Kalman filter.

The feedforward multilayer perceptron (MLP) with back-propagation of error is described. Since use of this network requires a set of labeled input-output, as such it cannot be used for segmentation of images when only one image is available. (However, if images to be processed are of similar nature, one can use a set of known images for learning and then use the network for processing of other images.) A self-organizing multilayer neural network architecture suitable for image processing is proposed. The proposed architecture is also a feedforward one with back-propagation of errors; but like MLP it does not require any supervised learning. Each neuron is connected to the corresponding neuron in the previous layer and the set of neighbors of that neuron. The output status of neurons in the output layer is described as a fuzzy set. A fuzziness measure of this fuzzy set is used as a measure of error in the system (instability of the network). Learning rates for various measures of fuzziness have been theoretically and experimentally studied. An application of the proposed network in object extraction from noisy scenes is also demonstrated.

This note describes an approach to integrating fuzzy reasoning systems with radial basis function (RBF) networks and shows how the integrated network can be employed as a multivariable self-organizing and self-learning fuzzy controller. In particular, by drawing some equivalence between a simplified fuzzy control algorithm (SFCA) and a RBF network, we conclude that the RBF network can be interpreted in the context of fuzzy systems and can be naturally fuzzified into a class of more general networks, referred to as FBFN, with a variety of basis functions (not necessarily globally radial) synthesized from each dimension by fuzzy logical operators. On the other hand, as a result of natural generalization from RBF to SFCA, we claim that the fuzzy system like RBF is capable of universal approximation. Next, the FBFN is used as a multivariable rule-based controller but with an assumption that no rule-base exists, leading to a challenging problem of how to construct such a rule-base directly from the control environment. We propose a simple and systematic approach to performing this task by using a fuzzified competitive self-organizing scheme and incorporating an iterative learning control algorithm into the system. We have applied the approach to a problem of multivariable blood pressure control with a FBFN-based controller having six inputs and two outputs, representing a complicated control structure.

A fuzzy-inference method in which fuzzy sets are defined by the families of their alpha -level sets, based on the resolution identity theorem, is proposed. It has the following advantages over conventional methods: (1) it studies the characteristics of fuzzy inference, in particular the input-output relations of fuzzy inference; (2) it provides fast inference operations and requires less memory capacity; (3) it easily interfaces with two-valued logic; and (4) it effectively matches with systems that include fuzzy-set operations based on the extension principle. Fuzzy sets defined by the families of their alpha -level sets are compared with those defined by membership functions in terms of processing time and required memory capacity in fuzzy logic operations. The fuzzy inference method is then derived, and important propositions of fuzzy-inference operations are proved. Some examples of inference by the proposed method are presented, and fuzzy-inference characteristics and computational efficiency for alpha -level-set-based fuzzy inference are considered.

An improved synthesis method for the multilayered neural network (NN) as function approximator is proposed. The method offers a translation mechanism that maps the qualitative knowledge into a multilayered NN structure. Qualitative knowledge is expressed in the form of representative points, which can be linguistically described as, 'when x is around x/sub i/, then y/sub i/ is around y'. Synthesis equations for the translation mechanism are provided. After the direct synthesis of the initial NN, the NN is tuned by backpropagation (BP), using the training data. The direct synthesis decreases the burden on BP and contributes to improved learning efficiency, accuracy, and stability. It is demonstrated that the translation mechanism is also useful for incremental modeling, i.e., increasing the number of neurons, or representative points, based on the results of BP.

A new fuzzy reasoning method for fuzzy control recently proposed by R. Yager is investigated. A comparison with the most useful fuzzy control schemes, for a first-order with time delay process, is carried out. The results obtained show that Yager's method is superior from the point of view of both computational burden and control system behavior.

Multistage fuzzy inference, where in the consequence in an inference stage is passed to the next stage as a fact, is studied and formulated as a type of linguistic-truth-value propagation, based on a concept of linguistic similarities between conditional propositions in successive stages. The formulation is useful in studying the characteristics of multistage fuzzy inference and reveals its structural relationship to multilayer perceptrons. The learning algorithm for multistage fuzzy inference is then derived, using backpropagating error information. The algorithm provides a means of automatically training the multistage fuzzy inference network, using input-output exemplar patterns. Intermediate membership functions based on simulation results, which are generated automatically in the intermediate stage, are proposed. The intermediate stage fuzzy-classifies the input space using intermediate membership functions. In this way, intermediate membership functions provide information regarding regional characteristics in exemplar patterns.

Investigates a range of phenomena from dynamical systems or chaos theory which appear in a simple fuzzy logic with the introduction of self-reference. Within that logic, self-referential sentences exhibit properties of fixed point attractors, fixed point repellers, and full chaos on the

The objective of this paper is to provide fuzzy control designers with a generalized design tool for stable fuzzy logic controllers in an optimal sense. Given multiple sets of data disturbed by vagueness uncertainty, we generate the implicative rules that guarantee stability and robustness of closed-loop fuzzy dynamic systems. First, the mathematical basis of fuzzy hypercubes and fuzzy dynamic systems is rigorously studied by considering the membership conditions for perfect recall and the evidential combination for reliable reasoning. Second, the author suggests the cell-state transition method, which utilizes Hsu's cell-to-cell mapping concept. As a result, a generic and implementable design methodology for obtaining a fuzzy feedback gain K, a fuzzy hypercube, is provided and illustrated with simple examples. The designed rules or membership functions in K form the cell-state transitions that lead an initial state to the goal state globally. The cell-state transition approach provides flexibility in choosing different controller rule bases depending on optimal strategies.