lecture 32 fuzzy systems
TRANSCRIPT
FUZZY LOGIC
Fuzzy Set (Value)
Let X be a universe of discourse of a fuzzy variable and x be its elements
One or more fuzzy sets (or values) Ai can be defined over X
Example: Fuzzy variable: AgeUniverse of discourse: 0 – 120 yearsFuzzy values: Child, Young, Old
A fuzzy set A is characterized by a membership function µA(x) that associates each element x with a degree of membership value in A
The value of membership is between 0 and 1 and it represents the degree to which an element x belongs to the fuzzy set A
FUZZY LOGIC
Fuzzy Set Representation
Fuzzy Set A = (a1, a2, … an)
ai = µA(xi)
xi = an element of XX = universe of discourse
For clearer representationA = (a1/x1, a2/x2, …, an/xn)
Example: Tall = (0/5’, 0.25/5.5’, 0.9/5.75’, 1/6’, 1/7’, …)
FUZZY LOGIC
Fuzzy Sets Operations
Intersection (A B)
In classical set theory the intersection of two sets contains those elements that are common to both
In fuzzy set theory, the value of those elements in the intersection:
µA B(x) = min [µA(x), µB(x)]
e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75, 0/6)Tall Short = (0/5, 0.1/5.25, 0.5/5.5, 0.1/5.75, 0/6)
= Medium
FUZZY LOGIC
Fuzzy Sets Operations
Union (A B)
In classical set theory the union of two sets contains those elements that are in any one of the two sets
In fuzzy set theory, the value of those elements in the union: µA B(x) = max [µA(x), µB(x)]
e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75)Tall Short = (1/5, 0.8/5.25, 0.5/5.5, 0.8/5.75, 1/6)
= not Medium
FUZZY LOGIC
Fuzzy Sets Operations
Complement (A)
In fuzzy set theory, the value of complement of A is: µ A(x) = 1 - µA(x)
e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Tall = (1/5, 0.9/5.25, 0.5/5.5, 0.2/5.75, 0/6)
FUZZY LOGIC
Fuzzy Relations
Fuzzy relation between two universes U and V is defined as:
µR (u, v) = µAxB (u, v) = min [µA (u), µB (v)]
i.e. we take the minimum of the memberships of the two elements which are to be related
FUZZY RULES
Approximate Reasoning Example: Let there be a fuzzy associative matrix M for the rule: if A then B
e.g. If Temperature is normal then Speed is medium
Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200]B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]
FUZZY RULES
Approximate Reasoning: Max-Min Inference Let A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200]
B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]then
M = (0, 0) (0, 0.6) . . .(0.5, 0) . . .. . .
= 0 0 0 0 00 0.5 0.5 0.5 0 by taking the minimum0 0.6 1 0.6 0 of each pair0 0.5 0.5 0.5 00 0 0 0 0
FUZZY LOGIC
Composition of Fuzzy Relations
Now we need a operator which allows us to infer something about B, given Acurrent
“Composition” is such an operator
FUZZY LOGIC
Composition of Fuzzy Relations
Let there be three universes U, V and W
Let R be the relation that relates elements from U to V
e.g. R = 0.6 0.8 0.7 0.9
And let S be the relation between V and W
e.g. S = 0.3 0.1 0.2 0.8
FUZZY LOGIC
Composition of Fuzzy Relations
With the help of an operation called “composition” we can find the relation T that maps elements of U to W
By max-min rule T = R S = maxvV { min(R(u, v), S(v, w)) }
0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9 0.2 0.8 0.3 0.8
Where element (1,1) is obtained by max{min(0.6, 0.3), min(0.8, 0.2)} = 0.3
Note that S R = 0.3 0.3 R S 0.7 0.8
FUZZY LOGIC
Composition of Fuzzy Relations
R = R(u, v) v1 v2
u1 0.6 0.8u2 0.7 0.9
U
V
u1 u2
v2
v1
0.6 0.7
0.8
0.9
FUZZY LOGIC
Composition of Fuzzy Relations
S = S(v, w) w1 w2
v1 0.3 0.1v2 0.2 0.8
W
V
w1w2
v2
v1
0.10.3
0.80.2
FUZZY LOGIC
Composition of Fuzzy RelationsT = R S = maxvV { min(R(u, v), S(v, w)) }
0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9 0.2 0.8 0.3 0.8
W
V
w1w2
v2
v1
0.10.3
0.80.2
Uu1 u2
0.6 0.7
0.8
0.9
FUZZY LOGIC
Composition of Fuzzy RelationsT = R S = maxvV { min(R(u, v), S(v, w)) }
0.6 0.8 0.3 0.1 = 0.3 0.8 0.7 0.9 0.2 0.8 0.3 0.8
W
V
w1w2
v2
v1
0.3
0.8
0.3U
u1 u2
0.8
Reading Assignment & References
Engelbrecht Chapter 18 & 19