Type-2 Fuzzy System

Type-2 Fuzzy System

Alpha-plane

Alpha-plane

Fuzzy inference

Rule: IF is AND is then is

The t-norm of PRODUCT operations:

Type Reduce

For a type-1 fuzzy set

For a interval type-2 fuzzy set

Type Reduce

Let [ ] are the type-1 fuzzy set, and each is different.

[ ] [ ]

Karnik-Mendel algorithm is a way to find and .

Defuzzifier

Result

Example 1:T1m=2T1v=0.3T2v=0.1X={0,0.1,0.2,0.3,…,4.0}alpha=[0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0];

Result

Example 2:T1m=2T1v=0.3T2v=0.1T2b=0.1X={0,0.1,0.2,0.3,…,4.0}alpha=[0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0];

Result

Example 3:T1m=2T1v=0.3T2v=0.1T2b=0.15X={0,0.1,0.2,0.3,…,4.0}alpha=[0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0];

Result

Example 4:T1m=2T1v=0.3T2v=0.1T2b=0.2X={0,0.1,0.2,0.3,…,4.0}alpha=[0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0];