main-content

## Über dieses Buch

In the last ten years, a true explosion of investigations into fuzzy modeling and its applications in control, diagnostics, decision making, optimization, pattern recognition, robotics, etc. has been observed. The attraction of fuzzy modeling results from its intelligibility and the high effectiveness of the models obtained. Owing to this the modeling can be applied for the solution of problems which could not be solved till now with any known conventional methods. The book provides the reader with an advanced introduction to the problems of fuzzy modeling and to one of its most important applications: fuzzy control. It is based on the latest and most significant knowledge of the subject and can be used not only by control specialists but also by specialists working in any field requiring plant modeling, process modeling, and systems modeling, e.g. economics, business, medicine, agriculture,and meteorology.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
Information accepted by methods based on conventional mathematics must be precise, for example, the speed of a car v = 111 (km/h) Such information can be represented graphically by means of the so-called singleton, Fig. 1.1.
Andrzej Piegat

### 2. Basic Notions of Fizzy Set Theory

Abstract
Fuzzy sets are generally applied by people for qualitative evaluation of physical quantities, states of plants and systems and for comparison to each other. Each man can evaluate the height of temperature without using a thermometer on the basis of his own feelings, in the coarse scale, Fig. 2.1.
Andrzej Piegat

### 3. Arithmetic of Fuzzy Sets

Abstract
Fuzzy numbers may be applied, for example, in the modeling of a system of a known input/output mapping given in terms of a conventional mathematical model y = f (X), where the input signals cannot be measured precisely but only approximately, e.g.:
$$\begin{array}{*{20}{c}} {{x_1} = approx.9,}\\ {{x_1} = approx.10,}\\ {y = {x_1} + {x_2}.} \end{array}$$
.
Andrzej Piegat

### 4. Mathematics of Fuzzy Sets

Abstract
The main elements of fuzzy sets are logical rules of the type (4.1).
Andrzej Piegat

### 5. Fuzzy Models

Abstract
The structure of a typical fuzzy model for a 2-inputs/l-output System is depicted in Fig. 5.1.
Andrzej Piegat

### 6. Methods of Fuzzy Modeling

Abstract
Chapter 6 presents three methods of fuzzy modeling, i.e. building up the fuzzy models of real Systems:
a)
fuzzy modeling based on the System expert’s knowledge,

b)
creation of self-tuning fuzzy models based on input/output measurement data of the System,

c)
creation of self-organizing and self-tuning fuzzy models based on input/output measurement data of the System.

Andrzej Piegat

### 7. Fuzzy Control

Abstract
Static plants and some dynamic plants can be controlled by static Controllers, transforming the control error e into the control signal u in accordance with the Controller characteristic u = F(e), Fig. 7.1.
Andrzej Piegat

### 8. The Stability of Fuzzy Control Systems

Abstract
According to obligatory industrial regulations issued by Authorities of many countries, the stability of a control System governed by a Controller of a proposed type has to be proved. That requirement is treated as a necessary condition for use of the control System. There are many applications where “delivery” of proof of control System stability has to be perceived as a task of crucial importance. The control Systems affect the safety of people (stabilizing airplane flight, etc.), govern costly plants and complicated industrial processes which are sensitive to loss of stability. The same regulations should be satisfied regardless of whether fuzzy or non-fuzzy Controllers are considered.
Andrzej Piegat

### Backmatter

Weitere Informationen