2.2 Type-1 Fuzzy Sets and Systems
2.2.3 Type-1 fuzzy logic systems
A type-1 fuzzy system is a rule based system which can be viewed as a process that maps crisp inputs to outputs by using the theory of fuzzy sets (Negnevitsky, 2002, p. 106). The well-known Mamdani fuzzy model (Mamdani, 1974) contains four com- ponents: fuzzifier, rules, inference engine and output processor (defuzzifier) (Mendel, 2001, p. 6). The Takagi-Sugeno-Kang fuzzy model (TSK) model (Takagi and Sugeno, 1985) is different from the Mamdani model in the inference engine and the output processor. Figure 2.2shows these components which are (Mendel, 2001, p. 6):
TYPE-1 FUZZY LOGIC SYSTEM
Inference Engine Defuzzification Fuzzification
Fuzzy sets Rule base
T1
SETS SETST1
CRISP
INPUT OUTPUTCRISP
Figure 2.2: Type-1 fuzzy logic system
2.2.3.1 Fuzzifier
Fuzzifier maps crisp inputs to fuzzy sets by evaluating the crisp inputs x = (x1, x2, . . . , xn)
based on the antecedents part of the rules and assigns each crisp input a degree of mem- bership µAi(xi) in its input fuzzy set. There are two types of fuzzifiers, singleton and
non-singleton. The singleton fuzzifier maps crisp inputs to fuzzy sets by evaluating the crisp inputs x = (x1, x2, . . . , xn) based on the antecedents part of the rules and assigns
each crisp input to its fuzzy set A(x) in X with its degree of membership in each fuzzy set. The non-singleton fuzzifier maps each given input xi into a fuzzy set (known as
variability set) with a unity membership grade for xi while their neighbours values are
given lesser membership values as they move away from xi (Mouzouris and Mendel,
1994). In this case the fuzzifier considers xi as the most likely correct value among its
neighbours values and normally is the centre of the fuzzy set (Mendel, 2001, p. 188). Non-singleton fuzzification allows better modelling of input uncertainties (using the variability set) and linguistic uncertainties (using antecedent fuzzy sets) in two stages (Wagner and Hagras, 2010b). Figure 2.3 shows how these two fuzzifiers fuzzify crisp inputs.
M e m b e r s h ip G r a d e 0 0.2 0.4 0.6 0.8 1 Domain 0 5 x 10 15 20 Fuzzy set A Variability set Fuzzification result M e m b e r s h ip G r a d e 0 0.2 0.4 0.6 0.8 1 Domain 0 5 x 10 15 20 Fuzzy set A Fuzzification result
Figure 2.3: The fuzzification of a crisp input x using singleton (left) and non- singleton (right) fuzzification
2.2.3.2 Rules
A fuzzy rule is a conditional statements in the form of IF-THEN where it contains two parts, the IF part called the antecedent part and the THEN part called the consequent part. For example :
IF job risk is high THEN salary is high.
These rules are written in fuzzy forms using words such as high, short and slow. To acquire these rules, many methods can be used such as getting them from experts or using data driven methods.
2.2.3.3 Inference engine
The inference engine in the Mamdani model maps the input fuzzy sets into the output fuzzy sets then the defuzzifier converts them to a crisp output. The rules in the Mam- dani model have fuzzy sets in both the antecedent part and the consequent part. For
example, the i th rule in the Mamdani model can be described as follows:
Ri: IF x1 is Ai1 and x2 is Ai2... and xp is Aip
THEN y is Bi
Where x1, x2, ..., xp are the input variables to the fuzzy systems and Ai1, Ai2, .., Aip are
input fuzzy sets in the antecedent part. The consequent part consists of output variable y and its output fuzzy set Bi. The operation “and” is normally modelled by a t-norm operator.
2.2.3.4 Defuzzifier
Defuzzifier in Mamdani model converts the output fuzzy sets that have been produced by inference engine into crisp values. Defuzzification can be done by many methods that have been proposed in the literature such as centroid defuzzifier, centre of sets defuzzifier, height defuzzifier and centre of sums defuzzifier. The inference and defuzzi- fication processes in Takagi and Sugeno model (Takagi and Sugeno, 1985) are different from the Mamdani model as the rules in TSK model are based on input fuzzy sets in the antecedent part and a mathematical linear function in the consequent part. The i th rule in the first-order TSK model is described as follows:
Ri: IF x1 is Ai1 and x2 is Ai2... and xp is Aip
THEN yi(x) = ci0+ ci1∗ x1+ ci2∗ x2... + cip∗ xp
Where i represents the rule number and ci0, ci1, ci2 and cip are the the coefficients of the consequent part of the fuzzy system rules. The final control value Y is computed as follows: Y = Pn i=1αi∗ yi Pn i=1αi
Where αi is known as the firing level for the i th rule which is derived by using a t-
norm operator such as minimum or product. It is important to mention that while the Mamdani model can handle uncertainty in both antecedent and consequent parts, TSK can handle uncertainty only in the antecedent part where fuzzy sets are used which limits its applicability in situations either where there is no uncertainty or the uncertainty exists only in the antecedent part (Mendel, 2001, p. 188).