**11. References**

Flux 2D and 3D Tutorial, Cedrat Co. 2002.

**15** 

*Poland* 

**Fuzzy Modelling Stochastic Processes** 

Wiener process, as a special mathematical model of Brownian motions, has been investigated and modelling in many probabilistic examples. In the topic literature it is easy to find many procedures of numeric probabilistic simulations of the Wiener process. Fuzzy modelling does not give us more accurate models than probabilistic modelling. Fuzzy knowledge-based modelling allows to determine linguistic description of non-precise relationships between variables and to derive the reasoning procedure from non-crisp facts. More over, using the notions of probabilities of fuzzy events, it is possible to determine a

Wiener process and a random walk are very often used for modelling phenomena in physics, engineering and economy. In the area of robot control theory these processes can represent some time-varying parameters of the environments where the object of control operates. Fuzzy models of these processes can constitute a part of a fuzzy model of a tested complex system. In paragraph 2. of this chapter, the mathematical descriptions of Brownian motions has been reminded, according to the theory of probability and stochastic processes. Some basics of fuzzy modelling has been presented in paragraph 3., to show the method of creating the knowledge base and rules of reasoning. Attention is focused on identification techniques for building empirical probabilities of fuzzy events from input-output data. Exemplary calculations of knowledge bases for real stochastic processes, as well as, some remarks on

The Brownian motion it is well known in physics, a random movement of a particle suspended in a liquid or a gas. The name of the movement is given after the botanist Robert Brown (1827), who was studying the movement of pollen grains suspended in water. There are many similar phenomena, where the time evolutions of the object depend on stochastic, microscopic contacts (collisions) with elements of the surrounded system. In mathematics, many models describing Brownian motion are well known and applied, e.g. the random walk stochastic process, Wiener stochastic process, Langevin stochastic differential equation,

Observations of the microscopic particle behavior show, that at any time step, the particle is changing its position in the space, according to collisions with liquid particles. Crashes of

frequency of a conclusion as well as its expected value.

future works have been presented in paragraphs 4 and 5.

**2. Mathematical models of Brownian motion** 

general diffusion equations and others.

**1. Introduction** 

**Describing Brownian Motions** 

Anna Walaszek-Babiszewska *Opole University of Technology* 


Motor-CAD v3.1 software manual, April 2006.

