Jean J. Saade

Ordering fuzzy sets over the real line: an approach based on decision making under uncertainty

Abstract The ordering of fuzzy sets over the real line is approached from the point of view of ordering intervals rather than ordering numbers. First, the maximax and maximin criteria which are commonly used for ordering intervals are expressed in terms of characteristic functions. These criteria and the Hurwicz criterion for decision making under complete ignorance are then reformulated in a manner that allows for their generalization to the fuzzy case. Transitivity is established for these ordering rules. A criterion based on the principle of diminishing marginal utility is also presented.

A unifying approach to defuzzification and comparison of the outputs of fuzzy controllers

This paper addresses the defuzzification of the fuzzy set outputs of fuzzy controllers from a comparison or ranking perspective. This is done by emphasizing the fuzzy controller as a decision-making system. Based on the extensive study and justification of the fuzzy-set comparison criteria that were developed and published elsewhere by the author, a ranking and, thus, a defuzzification index is introduced. This index is shown to overcome the disadvantages of the commonly used defuzzification methods whose attempted justifications based on probabilistic arguments have not been successful. In addition, the proposed index is based on the generalization of the Hurwicz criterion that is usually adopted in decision making under nonprobabilistic uncertainty and it encompasses the pessimistic maximin and the optimistic maximax criteria as special cases.

Fuzzy hypothesis testing with hybrid data

Abstract The formulation of binary hypothesis testing, where the available data under one hypothesis is a superposition of a random and a fuzzy component, is addressed. In this formulation the likelihood ratio, which is normally utilized in such types of decision making problems, is fuzzified and compared to a threshold. This involves the application of principles of ordering fuzzy sets which have been developed elsewhere by the authors. Decision rules are described which are independent of the shape of the membership function representing the fuzzy data. These decision rules are illustrated for the case where the random component of the data is normally distributed. Probability of error performance curves, using different ordering criteria, are obtained and compared to the performance of non-fuzzy hypothesis testing.

Extension of fuzzy hypothesis testing with hybrid data

Abstract The problem of binary hypothesis testing, where the available data under both hypotheses is a superposition of a random and a fuzzy component, is addressed. A rule that compares two fuzzy likelihood functions is used in the decision making process. This rule is proved to result from the application of a fuzzified version of the Bayes criterion. In order that a crisp decision is ultimately obtained, use is made of a criterion for ordering fuzzy sets over the real line which was developed elsewhere by the author. A specific case which assumes normally distributed random component of the data is considered. Probability of error performance curves for fuzzy hypothesis testing are plotted and compared with the curves that correspond to the minimax probability of error criterion and the generalized likelihood ratio test when non-fuzzy hypothesis testing is concerned.

Defuzzification techniques for fuzzy controllers

Based on the features and disadvantages of the commonly used defuzzification techniques and on the elements involved in the structure of a fuzzy controller, a new and advantageous defuzzification technique is introduced and justified. This is done by integrating the defuzzification problem into the global structure of fuzzy controllers. Another related defuzzification strategy, which has been introduced elsewhere by the principle author of this study, is also given and commented upon. Further, case studies are considered and comparative conclusions are drawn.

An efficient data-driven fuzzy approach to the motion planning problem of a mobile robot

A data-driven fuzzy approach is developed for solving the motion planning problem of a mobile robot in the presence of moving obstacles. The approach consists of devising a general method for the derivation of input-output data to construct a fuzzy logic controller (FLC) off-line. The FLC is constructed based on the use of a recently developed data-driven and efficient fuzzy controller modeling algorithm, and it can then be used on-line by the robot to navigate among moving obstacles. The novelty in the presented approach, as compared to the most recent fuzzy ones, stems from its generality. That is, the devised data-derivation method enables the construction of a single FLC to accommodate a wide range of scenarios. Also, care has been taken to find optimal or near optimal FLC solution in the sense of leading to a sufficiently small robot travel time and collision-free path between the start and target points. Furthermore, since the algorithm has been shown efficient in the representation of non-linear control functions, in terms of combating noise and possessing a good generalization capability, these aspects are also tested in this practical control problem. Comparison of the results with those obtained by fuzzy-genetic and another hybrid and data-driven design approach is also done.

Application of fuzzy hypothesis testing to signal detection under uncertainty

Abstract Fuzzy hypothesis testing with hybrid data as developed elsewhere by the authors is applied to signal detection when the uncertainty associated with a received signal parameter cannot be modelled objectively using probability. The presence of additive Gaussian noise is assumed, and a fuzzy amplitude as well as a fuzzy signal shape are considered. A numerical example is given and the resulting probability of error performance curves are plotted and compared with the curves obtained using the minimax criterion.

Maximization of a function over a fuzzy domain

Abstract The problem of maximizing a real-valued function over a fuzzy domain, when a crisp maximum is of concern, is examined. Some previously introduced maximizing formulas, which rely on the use of a maximizing set, are reviewed. Their advantages, disadvantages and relationships are emphasized. The reasonability of one of the maximizing set definitions given by Zadeh is established. As a result, it is argued that if this maximizing set definition is used properly, in the context of function maximization, it leads to a sound maximizing formula. In addition, other maximizing sets and formulas are provided.

Mapping convex and normal fuzzy sets

Abstract This study stresses the issue of mapping convex and normal fuzzy sets by a function. It is proved that a function mapping a space into another induces from a normal fuzzy set in the domain space a normal fuzzy set. It is also proved that if the function is real-valued of a real variable and possesses the continuity property, then it maps a convex fuzzy set, defined on its domain, into a convex fuzzy set on its range. This, however, holds under some restrictions on the support and the membership function of the fuzzy set defined over the domain of the function. In addition, an algorithm for fast approximate plot of the fuzzy set induced from a convex one defined over the real line by a continuous real-valued function is described.

Performance testing of refrigerators using fuzzy inference methodology under Labview/sup (R)/

The purpose of this paper is to present the use of fuzzy inference methodologies in the performance testing of refrigerators, as an efficient alternative to the classical time-consuming and relatively complex techniques. The basic elements that are used to optimize the performance of a refrigerator are fuzzified and then used in fuzzy inference models to represent the intelligent behavior of a human tester. These models are simulated and the results are compared with the ones obtained by classical algorithms. Further, conclusive comments are provided regarding the effectiveness of the introduced models and the avenues that need to be explored in order to make them increasingly efficient and fully automated.