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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.
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.
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.: $$\begin{array}{*{20}{c}} {{x_1} = approx.9,}\\ {{x_1} = approx.10,}\\ {y = {x_1} + {x_2}.} \end{array}$$ .
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.
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.
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...
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