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Fuzzy Logic
Presented by: Mahesh Todkar
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Content
� What is Fuzzy?
� Sets Theory
� What is Fuzzy Logic?
� Why use Fuzzy Logic?
� Theory of Fuzzy Sets
� Vocabulary
� Fuzzy if-then Rules
� Fuzzy Logic Operations
� Fuzzy Inference Systems (FIS)
� Fuzzy Inference Process
� References
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What is Fuzzy?
� Fuzzy means
not clear, distinct or precise;
not crisp (well defined);
blurred (with unclear outline).blurred (with unclear outline).
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Sets Theory
Classical Set: An element either belongs or does not
belong to a sets that have been defined.
Fuzzy Set: An element belongs partially or gradually to
the sets that have been defined.
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What is Fuzzy Logic?
� It has two different meanings as,
In narrow sense: Fuzzy logic is a logical system,
which is an extension of multi-valued logic.
In a wider sense: Fuzzy logic (FL) is almost
synonymous with the theory of fuzzy sets, a theory
which relates to classes of objects with unsharpwhich relates to classes of objects with unsharp
boundaries in which membership is a matter of
degree.
� Fuzzy logic (FL) should be interpreted in its wider
sense
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What is Fuzzy Logic?
� A way to represent variation or imprecision in logic
� A way to make use of natural language in logic
� Approximate reasoning
� Definition of Fuzzy Logic:� Definition of Fuzzy Logic:
A form of knowledge representation suitable for
notions that cannot be defined precisely, but which
depend upon their contexts.
� Superset of conventional (Boolean) logic that has been
extended to handle the concept of partial truth - the truth
values between "completely true & completely false".
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Why use Fuzzy Logic?
� Conceptually easy to understand
� Flexible
� Tolerant of imprecise data
� FL can model nonlinear functions of arbitrary complexity� FL can model nonlinear functions of arbitrary complexity
� FL can be built on top of the experience of experts
� FL can be blended with conventional control techniques
� FL is based on natural language
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Theory of Fuzzy Sets
Classical Set Fuzzy Set
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Theory of Fuzzy Sets
� Theory which relates to classes of objects with unsharpboundaries in which membership is matter of degree
� Thus every problem can be presented in terms of Fuzzy Sets
� A set without crisp
� Fuzzy set describes vague concepts� Fuzzy set describes vague concepts
� Fuzzy set admits the possibility of partial membership in it
� Degree of an object belongs to Fuzzy Set is denoted by membership value between 0 to 1
� Membership Function (MF) associated with a given Fuzzy Set maps an input value to its appropriate membership value
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Vocabulary
� Linguistic Variable: Variable whose values are words
or sentences rather than numbers
� It represent qualities spanning a particular spectrum
� Example: Speed, Service, Tip, Temperature, etc.
� Linguistic Value or Term: Values or Terms used to
describe Linguistic Variable
� Example: For Speed (Slowest, Slow, Fast, Fastest), For
Service (Poor, Good, Excellent), For Temperature
(Freezing, Cool, Warm, Hot), etc.
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Vocabulary
� Universe of Discourse or Universe or Input Space (U): Set of all possible elements that can come into consideration, confer the set U in (1).
� It depends on context.
� Elements of a fuzzy set are taken from a Universe of � Elements of a fuzzy set are taken from a Universe of Discourse.
� An application of the universe is to suppress faulty measurement data.
� Example:
Set of x >> 1 could have as a universe of all real numbers, alternatively all positive integer.
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Vocabulary
� Membership Function (MF) is a curve that defines how
each point in the input space is mapped to a membership
value between 0 and 1.
� It is denoted by µ.
� Membership value is also called as degree of membership
or membership grade or degree of truth of proposal.or membership grade or degree of truth of proposal.
� Types of Membership Functions:
Piece-wise linear functions
Gaussian distribution function
Sigmoid curve
Quadratic and cubic polynomial curves
� Singleton Membership Function
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Membership Functions
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Syntax of Fuzzy Set
� A = {x, µA(x) | x X}
Where,
A – Fuzzy Set
x – Elements of Xx – Elements of X
X – Universe of Discourse
µA(x) – Membership Function of x in A
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Fuzzy if-then Rules
� Statements used to formulate the conditional statements
that comprise fuzzy logic
� Example:
if x is A then y is B
where,
A & B – Linguistic values
x – Element of Fuzzy set X
y – Element of Fuzzy set Y
� In above example,
Antecedent (or Premise)– if part of rule (i.e. x is A)
Consequent (or Conclusion) – then part of rule (i.e. y is B)
� Antecedent is interpretation & Consequent is assignment
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Fuzzy if-then Rules
� Antecedent is combination of proposals by AND, OR, NOT
operators
� Consequent is combination of proposals linked by AND
operators. OR and NOT operators are not used in
consequents as these are cases of uncertainty.
� Example:
If it is early, then John can study.
Universe: U = {4,8,12,16,20,24}; time of day
Input Fuzzy set: early = {(4,0),(8,1),(12,0.9),(16,0.7),(20,0.5),(24,0.2)}
Output Fuzzy set: can study=singleton Fuzzy set (assume) so µstudy =1
i.e. at 20 (8 pm), µearly (20) = 0.5
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Fuzzy if-then Rules
� Interpreting if-then rule is a three–part process
1) Fuzzify Input: Resolve all fuzzy statements in the
antecedent to a degree of membership between 0 and 1.
2) Apply fuzzy operator to multiple part antecedents:
If there are multiple parts to the antecedent, apply fuzzy
logic operators and resolve the antecedent to a single logic operators and resolve the antecedent to a single
number between 0 and 1.
3) Apply implication method: The output fuzzy sets
for each rule are aggregated into a single output fuzzy
set. Then the resulting output fuzzy set is defuzzified, or
resolved to a single number.
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Fuzzy if-then Rules
Interpreting if-then rule is a three–part process:
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Fuzzy Logic Operations
� Fuzzy Logic Operators are used to write logic combinations between fuzzy notions (i.e. to perform computations on degree of membership)
� Zadeh operators
1) Intersection: The logic operator corresponding to the intersection of sets is AND.
µ = MIN(µ , µ )µ(A AND B) = MIN(µ(A), µ(B))
2) Union: The logic operator corresponding to the union of sets is OR.
µ(A OR B) = MAX(µ(A), µ(B))
3) Negation: The logic operator corresponding to the complement of a set is the negation.
µ(NOT A) = 1 - µ(A)
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Fuzzy Logic Operations
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Fuzzy Inference Systems (FIS)
� Fuzzy Inference is the process of formulating the mapping
from a given input to an output using fuzzy logic.
� Process of fuzzy inference involves Membership Functions
(MF), Logical Operations and If-Then Rules.
� FIS having multidisciplinary nature, so cab called as
fuzzy-rule-based systems, fuzzy expert systems, fuzzy fuzzy-rule-based systems, fuzzy expert systems, fuzzy
modeling, fuzzy associative memory, fuzzy logic
controllers, and simply (and ambiguously) fuzzy systems.
� Types of FIS:
1) Mamdani-type: Most commonly used. Expects the output MF’s to
be fuzzy sets.
2) Sugeno-type: Output MF’s are either linear or constant.
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Fuzzy Inference ProcessTo describe the fuzzy inference process, lets consider the
example of two-input, one-output, two-rule valve control
problem.
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Fuzzy Inference ProcessStep 1: Fuzzify Input (Fuzzification)
� Take the inputs and determine the degree to which they belong to each of the appropriate fuzzy sets via membership functions.
� Input is always a crisp numerical value limited to the universe of discourse of the input variable.universe of discourse of the input variable.
� Output is a fuzzy degree of membership in the qualifying linguistic set.
� Each input is fuzzified over all the qualifying membership functions required by the rules.
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Fuzzy Inference Process
Step 1: Fuzzify Input (Fuzzification)
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Fuzzy Inference Process
Step 2 : Apply Fuzzy Operator
� If the antecedent of a given rule has more than one
part, the fuzzy operator is applied to obtain one
number that represents the result of the antecedent
for that rule.
� The input to the fuzzy operator is two or more
membership values from fuzzified input variables.
� The output is a single truth value.
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Fuzzy Inference Process
Step 2 : Apply Fuzzy Operator
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Fuzzy Inference Process
Step 3: Apply Implication Method
� First must determine the rule’s weight.
� Operation in which the result of fuzzy operator is used to
determine the conclusion of the rule is called as
implication.implication.
� The input for the implication process is a single number
given by the antecedent.
� The output of the implication process is a fuzzy set.
� Implication is implemented for each rule.
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Fuzzy Inference Process
Step 3: Apply Implication Method
Antecedent Consequent
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Fuzzy Inference Process
Step 4 : Aggregate All Outputs
� Aggregation is the process by which the fuzzy sets that
represent the outputs of each rule are combined into a
single fuzzy set.
� Aggregation only occurs once for each output variable.� Aggregation only occurs once for each output variable.
� The input of the aggregation process is the list of
truncated output functions returned by the implication
process for each rule.
� The output of the aggregation process is one fuzzy set
for each output variable.
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Fuzzy Inference ProcessStep 4 : Aggregate All Outputs
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Fuzzy Inference Process
Step 5: Defuzzify
� Move from the “fuzzy world” to the “real world” is
known as defuzzification.
� The input for the defuzzification process is a fuzzy set.
� The output is a single number.
The most popular defuzzification method is the � The most popular defuzzification method is the
centroid calculation, which returns the center of area
under the curve
� Other methods are bisector, middle of maximum (the
average of the maximum value of the output set),
largest of maximum, and smallest of maximum.
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Fuzzy Inference Process
Step 5: Defuzzify
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References
� Fuzzy Logic Toolbox™ 2 User’s Guide
� Tutorial On Fuzzy Logic by Jan Jantzen
� Fuzzy Logic by Cahier Technique Schneider
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Thank You…Thank You…
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Any Questions? Any Questions?