25-05-2012, 03:44 PM
A Short Fuzzy Logic Tutorial
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The purpose of this tutorial is to give a brief information about fuzzy logic
systems. The tutorial is prepared based on the studies [2] and [1]. For further
information on fuzzy logic, the reader is directed to these studies.
A fuzzy logic system (FLS) can be dened as the nonlinear mapping of an
input data set to a scalar output data [2]. A FLS consists of four main parts:
fuzzier, rules, inference engine, and defuzzier. These components and the
general architecture of a FLS
Linguistic Variables
Linguistic variables are the input or output variables of the system whose
values are words or sentences from a natural language, instead of numerical
values. A linguistic variable is generally decomposed into a set of linguistic
terms.
Example: Consider the air conditioner in Figure 2. Let temperature (t) is the
linguistic variable which represents the temperature of a room. To qualify the
temperature, terms such as \hot" and \cold" are used in real life. These are the
linguistic values of the temperature. Then, T(t) = ftoo-cold, cold, warm, hot,
too-hotg can be the set of decompositions for the linguistic variable temperature.
Each member of this decomposition is called a linguistic term and can cover a
portion of the overall values of the temperature.
Membership Functions
Membership functions are used in the fuzzication and defuzzication steps
of a FLS, to map the non-fuzzy input values to fuzzy linguistic terms and vice
versa. A membership function is used to quantify a linguistic term. For instance,
in Figure 3, membership functions for the linguistic terms of temperature vari-
able are plotted. Note that, an important characteristic of fuzzy logic is that a
numerical value does not have to be fuzzied using only one membership func-
tion. In other words, a value can belong to multiple sets at the same time.
For example, according to Figure 3, a temperature value can be considered as
\cold" and \too-cold" at the same time, with dierent degree of memberships.
Fuzzy Rules
In a FLS, a rule base is constructed to control the output variable. A fuzzy
rule is a simple IF-THEN rule with a condition and a conclusion. In Table 1,
sample fuzzy rules for the air conditioner system in Figure 2 are listed. Table 2
shows the matrix representation of the fuzzy rules for the said FLS. Row captions
in the matrix contain the values that current room temperature can take, column
captions contain the values for target temperature, and each cell is the resulting
command when the input variables take the values in that row and column. For
instance, the cell (3, 4) in the matrix can be read as follows: If temperature is
cold and target is warm then command is heat.
Fuzzy Set Operations
The evaluations of the fuzzy rules and the combination of the results of the
individual rules are performed using fuzzy set operations. The operations on
fuzzy sets are dierent than the operations on non-fuzzy sets. Let A and B are
the membership functions for fuzzy sets A and B. Table 3 contains possible fuzzy
operations for OR and AND operators on these sets, comparatively. The mostly-
used operations for OR and AND operators are max and min, respectively. For
complement (NOT) operation, Eq. 1 is used for fuzzy sets.