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Reasoning with Uncertainty

Fuzzy Logic

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Fuzzy Logic

Fuzzy Logic is a multivalued logic Rule-based inference system

Membership values indicate degree of truthof predicates

Fuzzy set operations permit reasoning withmembership values

Fuzzy Logic has been applied verysuccessfully to a number of control

problems

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Fuzzy Logic - Applications

Many everyday applications use FuzzyLogic control

Microwaves

 ABS brakes

Camera image stabilization

Cruise control Air conditioning control

Washing machine control

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Fuzzy Sets

 A Fuzzy set  A is a set of items withmembership values

 A is a subset of the universe (all possible

objects)

There is a membership function

µ  A 

(x) ∈ [0..1]

indicating the degree to which x belongs toset A 

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Fuzzy Set Operations

Union of two Fuzzy Sets µ  A ∪ B (x) = µ  A (x) ⊕ µ B (x) 

Often max (µ  A (x) , µ B (x) )

Intersection of two Fuzzy Sets

µ  A ∩ B (x) = µ  A (x) ⊗ µ B (x) 

Often min (µ  A (x) , µ B (x) ) Inversion of a Fuzzy Set

µ¬  A (x) = 1 - µ  A (x) 

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Fuzzy Inference (Control)

Fuzzy Logic uses logic inference rules anddefuzzification

Inference rules are of the form:

If a in A and b in B then c in C Where A , B , and C are Fuzzy sets, a, b, and c, are elementsfrom the universe of discourse.

Multiple rules for the same set are combined using ⊕

 A value for the variable c in C is extracted bydefuzzification - Often as the center of mass of themembership function

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Fuzzy Inference (Control)

Inference rules derive a membershipfunction for the resulting fuzzy set

Rule 1: If a in A and b in B then c in C

Results in a membership function for c which islimited by the degree of truth of the rule’santecedents and the membership function for

µ Rule 1 (c) = min ( µ  A (a) ⊗µ B (b) , µ C (c) )  

 All inference rules that have the samevariable as a consequent

µ Result (c) = µ Rule 1 (c) ⊕µ Rule 2 (c) 

⊕…

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Fuzzy Inference (Control)

 A value for the variable in the consequent isderived from the resulting membership profile

using defuzzification

Defuzzification often uses the center of mass of the

membership function

c is the point where 

- ∞ c µ Result (x) dx =  c 

∞ µ Result (x) dx 

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Fuzzy Logic

 Advantages Simple inference system

Easy to design

Good for simple control

Problems Problems with strings of inference

Non-symmetric inference

Difficulty interpreting resulting membershipvalues.