Definition of fuzzy
Fuzzy – “not clear, distinct, or precise;
blurred”
Definition of fuzzy
logic
A form of knowledge representation suitable for notions that cannot be
defined precisely, but which depend upon their contexts.
Introduction to fuzzy logic
Uncertainty is inherent in
accessing information from large amount of data; for example words like near
and slow in sentences like
“My house is near to the
office”
“He drives slowly”
If we set slow as speeds
<=20 and fast otherwise, then is 20.1 is fast?
Crisp sets
Crisp
sets: In a crisp set, members belong to the group
identified by the set or not
slow = {s such that 0 <= s <= 40}
fast = {s such that 40 < s <70}
40.1 belongs to set fast, hence 40.1 is not
slow
Drawback of crisp sets: Suppose a physical system has to apply brakes if the
speed of the vehicle is fast and release the brake if the speed is slow. If the
speed is in the interval [39, 41], such a system would continuously keep
jerking which is not desired
The
crisp set is defined in such a way as to divide the individuals in some given
universe of discourse into two groups: members and nonmembers.
However,
many classification concepts do not exhibit this characteristic.
For
example, the set of tall people, expensive cars, or sunny days.
Fuzzy sets
A
fuzzy set can be defined mathematically by assigning to each possible
individual in the universe of discourse a value representing its grade of
membership in the fuzzy set.
For
example: a fuzzy set representing our
concept of sunny might assign a degree of membership of 1 to a cloud cover of
0%, 0.8 to a cloud cover of 20%, 0.4 to a cloud cover of 30%, and 0 to a cloud
cover of 75%.
For
example let us evaluate few dates 12, 13, 14, 15, 16 August 2014
Crisp
set { (12,1), (13, 1), (14, 0), (15, 1), (16,0)}
Here
12, 13, 15 belongs to sunny set.
Fuzzy
set {(12, 0.9), (13, 1), (14, 0.8), (15,1), (16,0.3)}
Here
all belongs to sunny set but with definite grade of membership.
A membership function
A
characteristic function: the values
assigned to the elements of the universal set fall within a specified range and
indicate the membership grade of these elements in the set.
Larger
values denote higher degrees of set membership.
A
set defined by membership functions is a fuzzy set.
The
most commonly used range of values of membership functions is the unit interval
[0,1].
Fuzzy Sets
To
reduce the complexity of comprehension, vagueness is introduced in crisp sets
Fuzzy
set contains elements; each element signifies the degree or grade of membership
to a fuzzy aspect
Membership
values denote the sense of belonging of a member of a crisp set to a fuzzy set
Example
of a fuzzy set
Consider
a crisp set A with elements representing ages of a set of people in years
A =
{ 2, 4, 10, 15, 20, 30, 35, 40, 45, 60, 70}
Classify
the age in terms of six fuzzy variables or names given to fuzzy sets as:
infant, child, adolescent, adult, young and old
Membership
is different from probabilities
Memberships
do not necessarily add up to one
Fuzzy Terminology
Universe
of Discourse (U): The range of all possible values that comprise the
input to the fuzzy system
Fuzzy
set: A set that has members with
membership (real) values in the interval [0,1]
Membership
function: It is the basis of a
fuzzy set. The membership function of the fuzzy set A is given by µA: Uà [0,1]
Fuzzy Relations
Generalizes
classical relation into one that allows partial membership
Describes
a relationship that holds between two or more objects
Example: a fuzzy relation “Friend” describe the degree of
friendship between two person (in contrast to either being friend or not being
friend in classical relation!)
A
fuzzy relation is a mapping from
the Cartesian space X x Y to the interval [0,1], where the strength of the
mapping is expressed by the membership function of the relation m (x,y)
The
“strength” of the relation between ordered pairs of the two universes is
measured with a membership function expressing various “degree” of strength
[0,1]
Fuzzy
If-Then Rules
General
format:
If
x is A then y is B
Examples:
If
pressure is high, then volume is small.
If
the road is slippery, then driving is dangerous.
If
a tomato is red, then it is ripe.
If
the speed is high, then apply the brake a little.
LINGUISTIC
VARIABLES
A
linguistic variable is a fuzzy variable.
The
linguistic variable speed ranges between 0 and 300 km/h and includes the fuzzy
sets slow, very slow, fast, …
Fuzzy
sets define the linguistic values.
Hedges
are qualifiers of a linguistic variable.
All
purpose: very, quite, extremely
Probability:
likely, unlikely
Quantifiers:
most, several, few
Possibilities:
almost impossible, quite possible
TRUTH TABLES
Truth
tables define logic functions of two propositions. Let X and Y be two propositions, either of which
can be true or false.
The
operations over the propositions are:
Conjunction
(Ù): X AND Y.
Disjunction
(Ú): X OR Y.
Implication
or conditional (Þ): IF X THEN Y.
Bidirectional
or equivalence (Û): X IF AND ONLY IF Y.
FUZZY RULES
A
fuzzy rule is defined as the conditional statement of the form
If
x is A
THEN
y is B
where
x and y are linguistic variables and A and B are linguistic values determined
by fuzzy sets on the universes of discourse X and Y.
The
decision-making process is based on rules with
sentence conjunctives AND, OR and ALSO.
Each
rule corresponds to a fuzzy relation.
Rules
belong to a rule base.
Example: If (Distance x to second car is SMALL) OR (Distance
y to obstacle is CLOSE) AND (speed v is HIGH) THEN (perform
LARGE correction to steering angle q) ALSO (make MEDIUM reduction in speed
v).
Three
antecedents (or premises) in this example give rise to two outputs
(consequences).
FUZZY INFERENCE SYSTEMS (FIS)
Fuzzy
rule based systems, fuzzy models, and fuzzy expert systems are also known as
fuzzy inference systems.
The
key unit of a fuzzy logic system is FIS.
The
primary work of this system is decision-making.
FIS
uses “IF...THEN” rules along with connectors “OR” or “AND” for making necessary
decision rules.
The
input to FIS may be fuzzy or crisp, but the output from FIS is always a fuzzy
set.
When
FIS is used as a controller, it is necessary to have crisp output.
Hence,
there should be a defuzzification unit for converting fuzzy variables into
crisp variables along FIS.
There
are two types of Fuzzy Inference Systems:
Mamdani
FIS(1975)
Sugeno
FIS(1985)
MAMDANI
FUZZY INFERENCE SYSTEMS (FIS)
Fuzzify
input variables:
Determine
membership values.
Evaluate
rules:
Based
on membership values of (composite) antecedents.
Aggregate
rule outputs:
Unify
all membership values for the output from all
rules.
Defuzzify
the output:
COG:
Center of gravity (approx. by summation).
SUGENO FUZZY INFERENCE SYSTEMS
(FIS)
The
main steps of the fuzzy inference process namely,
fuzzifying
the inputs and
applying
the fuzzy operator are exactly the same as in MAMDANI FIS.
The
main difference between Mamdani’s and Sugeno’s methods is that Sugeno output
membership functions are either linear or constant.
FUZZY EXPERT SYSTEMS
An
expert system contains three major blocks:
Knowledge
base that contains the knowledge specific to the domain of application.
Inference
engine that uses the knowledge in the knowledge base for performing suitable
reasoning for user’s queries.
User
interface that provides a smooth communication between the user and the system.
Fuzzy
Inference Processing
1.There
are three models for Fuzzy processing based on the expressions of consequent
parts in fuzzy rules
Suppose
xi are inputs and y is the consequents in fuzzy rules
Mamdani
Model: y = A
where
A is a fuzzy number to reflect fuzziness
Though
it can be used in all types of systems, the model is more suitable for
knowledge processing systems than control systems
2. TSK (Takagi-Sugano-Kang) model:
y =
a0 + Ʃ ai xi where ai are constants
The
output is the weighted linear combination of input variables (it can be expanded to nonlinear combination
of input variables)
Used
in fuzzy control applications
3.
Simplified fuzzy model: y = c
where
c is a constant
Thus
consequents are expressed by constant values
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