Extremal Fuzzy Dynamic Systems Research Articles

Fuzzy identification problem for continuous extremal fuzzy dynamic system

This work deals with the problems of the Continuous Extremal Fuzzy Dynamic System (CEFDS) optimization and briefly discusses the results developed by Sirbiladze (Int J Gen Syst 34(2):107---138, 2005a; 34(2):139---167, 2005b; 34(2):169---198, 2005c; 35(4):435---459, 2006a; 35(5):529---554, 2006b; 36(1): 19---58, 2007; New Math Nat Comput 4(1):41---60, 2008a; Mat Zametki, 83(3):439---460, 2008b). The basic properties of extended extremal fuzzy measures and Sugeno's type integrals are considered and several variants of their representation are given. Values of extended extremal conditional fuzzy measures are defined as a levels of expert knowledge reflections of CEFDS states in the fuzzy time intervals. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed. A new approach in modeling of CEFDS is developed. Applying the results of Sirbiladze (Int J Gen Syst 34(2) 107---138, 2005a; 34(2):139---167, 2005b), fuzzy processes with possibilistic uncertainty, the source of which are expert knowledge reflections on the states on CEFDS in extremal fuzzy time intervals, are constructed (Sirbiladze in Int J Gen Syst 34(2):169---198, 2005c). The dynamics of CEFDS's is described. Questions of the ergodicity of CEFDS are considered. A fuzzy-integral representation of a continuous extremal fuzzy process is given. Based on the fuzzy-integral model, a method and an algorithm are developed for identifying the transition operator of CEFDS. The CEFDS transition operator is restored by means of expert data with possibilistic uncertainty, the source of which is expert knowledge reflections on the states of CEFDS in the extremal fuzzy time intervals. The regularization condition for obtaining quasi-optimal estimator of the transition operator is represented by the theorems. The corresponding calculating algorithm is provided. The results obtained are illustrated by an example in the case of a finite set of CEFDS states.

ON FUZZY OPTIMAL CONTROLS IN THE WEAKLY STRUCTURABLE CONTINUOUS DYNAMIC SYSTEMS (WSCDS)

We pose problems of the WSCDS optimal control and discuss the results developed by G. Sirbiladze in Ref. 7. Sufficient and necessary conditions are presented for the existence of an extremal fuzzy optimal control process, for which we use R. Bellman's optimality principle and take into consideration the gain-loss fuzzy process. An example of constructing the continuous extremal fuzzy dynamic systems optimal control is presented.

Modeling of extremal fuzzy dynamic systems. Part VI. Problems of states estimation (filtration) of extremal fuzzy process

A fuzzy-integral model of an extremal fuzzy process (EFP) is considered. The model describes the evolution of one class of weakly structurable fuzzy dynamic systems (Klir 1985), i.e. of the so-called extremal fuzzy dynamic systems, which have been constructed in Part III of this work. In the present paper, problems of an optimal filtering of continuous as well as of discrete EFP are solved by means of “past” evaluating information. Sufficient conditions are established for the existence of an optimal estimating fuzzy process. Using only one “past” evaluating fuzzy state of the considered system, variants of fuzzy observers (representations of an optimal estimating EFP) are constructed in terms of approximation of piecewise-constant and extremally measurable filtration functions for continuous and discrete extremal fuzzy dynamic systems. The results obtained are illustrated by a numerical example.

Modelling of extremal fuzzy dynamic systems: Part V. Optimization of continuous controllable extremal fuzzy processes and the choice of decisions

This work deals with the problems of optimization of continuous controllable extremal fuzzy processes that have been presented in parts I, II and III of the current paper. A separate consideration is given to the case where an extremal fuzzy control process acting on an extremal fuzzy dynamic system (EFDS) (i) depends and (ii) does not depend on an EFDS state. Sufficient and necessary conditions for the existence of an optimal control are proved. Applying Bellman's optimality principle and assuming that the gain–loss process exists for an EFDS, a variant of the fuzzy integral representation of an optimal control is given for continuous EFDSs. This variant employs the instrument of extended extremal fuzzy composition measures constructed in parts I and II of this work. Finally, the questions of defining a fuzzy gain relation for an EFDS are considered, taking into account the available expert knowledge on the EFDS subject matter.

Modeling of extremal fuzzy dynamic systems

Based on the fuzzy-integral model, methods and algorithms are developed for identifying the “input–output” operator of continuous and stationary discrete extremal fuzzy dynamic systems (EFDS). The EFDS “input–output” operator is restored by means of experimental data with possibilistic uncertainty, the source of which is extremal fuzzy time intervals. The regularization conditions for obtaining quasi-optimal estimates are substantiated by the proved theorems. The corresponding algorithms are provided. The results obtained are illustrated by examples in the case of a finite set of EFDS states.

Modeling of extremal fuzzy dynamic systems. Part III. Modeling of extremal and controllable extremal fuzzy processes

New approaches in modeling the so-called extremal fuzzy dynamic systems (EFDSs) are developed. Applying the results of Parts I and II, fuzzy processes with possibilistic uncertainty, the source of which is extremal fuzzy time intervals, are constructed. Fuzzy-integral representations of controllable extremal fuzzy processes (CEFPs) are given. The dynamics of EFDSs is described. Questions as to the ergodicity of EFDSs are considered, using the properties of both extended composition extremal fuzzy measures and -compositions over them (Part II). Sufficient conditions are proved for stationary CEFPs to be ergodic.

Modeling of extremal fuzzy dynamic systems. Part I. Extended extremal fuzzy measures

The basic properties of extended extremal fuzzy measure are considered and several variants of their representation are given. In considering extremal fuzzy measures, several transformation theorems are proved for extended lower and upper Sugeno integrals. Extended extremal conditional fuzzy measures are defined. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed.

Modeling of extremal fuzzy dynamic systems. Part II. Extended extremal fuzzy measures on composition products of measurable spaces

This paper continues the investigation of extremal fuzzy measures and their extensions started in Part I. Here, we consider the basic properties of composition extremal fuzzy measures. Several variants of their representations are given. The notion of H-composition over extremal fuzzy measures is introduced, and some algebraic properties of the corresponding structures, which forms the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems, are discussed.