Fuzzy Inference and Reasoning

Proposition

Logic variable

Basic connectives for logic variables (1) Negation (2) Conjunction

Basic connectives for logic variables (3) Disjunction (4) Implication

Logical function

Logic Formula

Tautology

Tautology

Predicate logic

Fuzzy Propositions Assuming that truthand falsity are expressed by values 1 and 0, respectively, the degree of truth of each fuzzy proposition is expressed by a number in the unit interval [0, 1].

Fuzzy Propositions

p : temperature (V) is high (F).

Fuzzy Propositions p : V is F is S V is a variable that takes values v from some universal set V F is a fuzzy set onV that represents a fuzzy predicate S is a fuzzy truth qualifier In general, the degree of truth, T(p), of any truth-qualified proposition p is given for each v e V by the equation T(p) = S(F(v)).

p : Age (V) is very(S) young (F).

Representation of Fuzzy Rule

Representation of Fuzzy Rule

Fuzzy rule as a relation

Fuzzy implications

Example of Fuzzy implications

Example of Fuzzy implications

Example of Fuzzy implications

Representation of Fuzzy Rule Single input and single output Multiple inputs and single output Multiple inputs and Multiple outputs

Representation of Fuzzy Rule Multiple rules

Compositional rule of inference The inference procedure is called as the “compositional rule of inference”. The inference is determined by two factors : “implication operator” and “composition operator”. For the implication, the two operators are often used: For the composition, the two operators are often used:

Representation of Fuzzy Rule Max-min composition operator Mamdani: min operator for the implication Larsen: product operator for the implication

One singleton input and one fuzzy output Mamdani

One singleton input and one fuzzy output Mamdani

One singleton input and one fuzzy output Larsen

One singleton input and one fuzzy output Larsen

One fuzzy input and one fuzzy output Mamdani

One fuzzy input and one fuzzy output Mamdani

Ri consists of R1 and R2

Example

Two singleton inputs and one fuzzy output Mamdani

Two singleton inputs and one fuzzy output Mamdani

Example

Two fuzzy inputs and one fuzzy output Mamdani

Two fuzzy inputs and one fuzzy output Mamdani

Two fuzzy inputs and one fuzzy output Mamdani

Example

Multiple rules

Multiple rules

Multiple rules

Example

Mamdani method

Mamdani method

Mamdani method

Mamdani method

Larsen method

Larsen method

Larsen method

Larsen method

Fuzzy Logic Controller

Inference

Inference

Inference

Inference

Defuzzification Mean of Maximum Method (MOM)

Defuzzification Center of Area Method (COA)

Defuzzification Bisector of Area (BOA)