International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

In this paper, fuzzy initial value problems for modelling aspects of uncertainty in dynamical systems are introduced and interpreted from a probabilistic point of view. Due to the uncertainty incorporated in the model, the behavior of dynamical systems modelled in this way will generally not be unique. Rather, we obtain a large set of trajectories which are more or less compatible with the description of the system. We propose so-called fuzzy reachable sets for characterizing the (fuzzy) set of solutions to a fuzzy initial value problem. Loosely spoken, a fuzzy reachable set is defined as the (fuzzy) set of possible system states at a certain point of time, with given constraints concerning the initial system state and the system evolution. The main-part of the paper is devoted to the development of the numerical methods for the approximation of such sets. Algorithms for precise as well as outer approximations are presented. It is shown that fuzzy reachable sets can be approximated to any degree of accuracy under certain assumptions. Our method is illustrated by means of an example from the field of economics.

In this paper, a fuzzy sliding mode controller (FSMC), which is synthesized by a collection of linguistic control rules whose membership functions of THEN-part is adapted, is proposed. Both the membership functions of IF-part and THEN-part are arranged symmetrically and distributed equally in the individual universe of discourse. In particular, the membership functions of the THEN-part can be adapted via one parameter adaptation to meet the required system specification. The proposed direct adaptive FSMC can be synthesized through the following stages. First, the control rules are constructed according to the concepts of SMC, and the fuzzy sets whose membership functions are symmetrically covered in state space. Then, the derived adaptive law is used to adjust the membership functions of the THEN-part. The FSMC is employed to approximate the equivalent control of SMC without knowing the mathematical model of the controlled system. Third, a hitting control is developed to guarantee the stability of the control system. Finally, we apply this FSMC to control a nonlinear inverted pendulum system for confirming the validity of the proposed approach.

Defuzzification is an essential problem in fuzzy systems that it is always solved in a heuristic way. The aim of this work is to give a semantic interpretation to this process with the help of indistinguishability operators.

We consider the problem of decision making under uncertainty where the payoffs are measured on an ordinal scale. First the case in which no information is available with respect to the uncertainty is studied. Here decision functions are obtained which reflect the attitude of the decision maker. Next the case where a possibility distribution exists with respect to uncertain variable is considered. Finally we look at the case where both possibilistic and probabilistic uncertainty co-exist.

On real processes many functional states are usually observed. however some of them may represent difference significant levels (or rates) of the same functional mode. They do not represent by themselves a functional mode, they are only sub-modes of the same functional mode (e.g. one of the functional mode of my car is tank empty; some functional states observed may be 10% empty tank, 50% empty tank, 96% empty tank). Thus the interest is not only to diagnose the functional mode (e.g. tank empty, sleeping driver) but also to highlight the gravity level of this functional mode (e.g. the reaction in the middle of desert, confronted to a 4% empty tank is not the same as confronted to a 98% empty tank). The aim of this paper is to present a diagnosis of the current functional mode of a process and its gravity level. Usually in pattern recognition area, a membership function is a monotonic decreasing function of a Euclidean distance between two objects. Those objects represent two states of the process and a distance here is a dissimilarity measure between those states. Such a distance is defined in all the fuzzy subset associated with this membership function. So that function is monotonic decreasing in all subset directions. In this paper directional membership functions are proposed. In this case the distance is defined only by reference to a path describing the evolution from one functional state to another one. Then the obtained membership function is oriented according to this path and do not decrease identically within all directions in the subset. Such membership functions are then suitable in order to diagnose the state associated with data evoluting between known functional modes. An application to the French telephone network illustrates this method.

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to vladik@cs.utep.edu, or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.