Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 144

4.3.1 Mamdani Type of Fuzzy Logic Systems

Mamdani (1977) proposed the first fuzzy inference system with the objective to
control a combination of a steam engine and a boiler, based on a set of linguistic
control rules built as the extracted knowledge of a human expert.
When applied to Boyle's law, as described in Section 4.2, the following fuzzy
linguistic rules can be written:

Rule-1: IF the pressure is HIGH and the temperature is CONSTANT, THEN

the Volume is LOW.

Rule-2: IF pressure is MEDIUM and the temperature is CONSTANT, THEN

the volume is MEDIUM.

Rule-3: IF the pressure is LOW and the temperature is CONSTANT, THEN

the volume is HIGH.

These rules are known as Mamdani-type fuzzy rules (first introduced and used by
Mamdani in 1977). The main features of such rules are that both the IF
(antecedents) parts and the THEN (consequents) parts of the rules are fuzzy
(imprecise) in nature. That is, fuzzy sets are used here in order to describe both the
input and the output variables of the system.
As another example of the Mamdani-type fuzzy rules, consider a single input-
single output system that describes the relationship between the heater current and
the temperature trend as follows:

x IF the heater current is HIGH, THEN the temperature rise is FAST
x IF the heater current is MEDIUM, THEN the temperature rise is
MODERATE
x IF the heater current is LOW, THEN the temperature rise is SLOW.

Note that in the above Mamdani-type fuzzy rules the heater current and
temperature rise are the two linguistic variables (input and output of the system
respectively), whereas HIGH, MEDIUM, and LOW are the three fuzzy sets,
represented by suitable (triangular/Gaussian) membership functions and provide
the means to express the states of the linguistic input variables. Similarly, FAST,
MODERATE, and SLOW are the three output fuzzy sets - also represented by
suitable membership functions - representative of the states of linguistic output
variables of the systems.

4.3.2 Takagi-Sugeno Type of Fuzzy Logic Systems

With Takagi-Sugeno (TS) type fuzzy rules the IF (antecedent) part is fuzzy in
nature, whereas the THEN (consequent) part is a crisp function of an antecedent
variable (as a rule, a linear equation) rather than a fuzzy proposition. The example
presented above for Boyle's law could be written correspondingly as:

Rule-1: IF P is LOW and T is CONSTANT, THEN V = a 1 P + b 1 T + c 1

Rule-2: IF P is HIGH and T is CONSTANT, THEN V = a 2 P + b 2 T + c 2

Rule-3: IF P is MEDIUM and T is CONSTANT, THEN V = a 3 P + b 3 T + c 3 .