Ernest Czogala

FUZZY RELATION EQUATIONS ON A FINITE SET

Abstract The family of all the solutions of a fuzzy relation equation on a finite set is considered. It is characterized by the set of all the lower solutions, which can be obtained by a combinatorial algorithm.

Associative monotonic operations in fuzzy set theory

Abstract The properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets, on fuzzy numbers and on fuzzy probabilistic sets.

Control problems in fuzzy systems

Abstract This paper deals with some problems of the control of fuzzy systems described by means of a relational equation of the type: X κ + 1 = U κ ∘ X κ ∘ R . The basic aspects such as mutual identification and control, stabilization control, control with constraints, generation of control rules of a fuzzy logic controller, important from the theoretical and applicational point of view are discussed and illustrated by means of numerical examples.

Some problems concerning the construction of algorithms of decision-making in fuzzy systems

The fuzzy set theory established by L. A. Zadeh has started anew period in formalizing a decision-making processes in ill-defined systems where a human being (operator) is an important element in the control loop. The control algorithm used here is based on a generalized fuzzy version of modus ponens (compositional rule of inference) where a set of decision-making rules forming a control algorithm is given. The aim of this paper is to present some problems which may appear in the initial stage off design of a decisionmaking algorithm and discuss a method of their formulation. We introduce notions like the completeness of algorithm, interactivity and the competitivity of control rules and consider indices illustrating the presented design aspects.

On distribution function description of probabilistic sets and its application in decision making

The paper deals with some problems of decision making described and solved by the use of the concept of a probabilistic set. The distribution function description of probabilistic sets which is important from a theoretical point of view is discussed, and its application in decision making is illustrated by means of numerical examples.

Decision making in uncertain environments

Abstract In the paper some problems of decision making in fuzzy and probabilistic fuzzy environments are formulated. The goals and constraints in such decision situations may be simultaneously fuzzy and random by their nature. Looking for a decision set which should be any reasonable function of goals and constraints, it leads to operations on probabilistic sets which are also of interest in that work. Some numerical examples of decision making are illustrating the theoretical considerations presented in this paper.

Contribution to application of energy measure of fuzzy sets

Abstract The works of De Luca & Termini continued by, for example, Knopfmacher, Loo and Gottwald, are the most important on the topic of determination of measures of fuzzy sets. The matter is to evaluate how fuzzy a fuzzy set is. There are two general concepts of measures of fuzzy set, i.e. entropy and energy measures. We show that the special kind of energy measure is better suited than the entropy kind of measure in many practical situations. Applications of the use of energy measure discussed in detail include decision making, fuzzy process control and prediction in fuzzy systems.

The aggregation operations for decision making in probabilistic fuzzy environment

This paper deals with some important classes of aggregation operations on various kinds of sets applied to decision making problems. These operations are mainly based on general concepts such as triangular norms (t- and s-norms). In this paper we focus particularly on operations on probabilistic sets and their distribution function representation. The considerations are illustrated by means of numerical examples.

On the concept of fuzzy probabilistic controllers

The paper deals with a new approach to the construction of a so-called fuzzy probabilistic controller where input-output data are treated as probabilistic sets. It is shown that the notion of a fuzzy controller is embedded in the concept discussed here. First of all the distribution characterization of probabilistic sets is discussed, some operations on probabilistic sets are presented and next the construction of the controller is performed. Numerical examples form an illustration of theoretical investigations.

FUZZY RULES GENERATION FOR FUZZY CONTROL

The paper deals with the design problem of control algorithms in fuzzy systems described by means of fuzzy relational equations, which can be implemented in the framework of fuzzy controllers applied to control of ill-defined systems. Some ways leading to assigning varying grades of importance to goals and constraints imposed in control process and their influence on the relation of the fuzzy controller is discussed in detail.