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Unit-04/Lecture-01 |
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Fuzzy
sets vs. crisp sets;(Jun2012,13,14) Crisp sets are the sets that we have used most of
our life. In a crisp set, an element is either a member of the set or not.
For example, a jelly bean belongs in the class of food known as candy. Mashed
potatoes do not. Fuzzy sets, on the other hand, allow elements to
be partially in a set. Each element is given a degree of membership in
a set. This membership value can range from 0 (not an element of the set) to
1 (a member of the set). It is clear that if one only allowed the extreme
membership values of 0 and 1, that this would actually be equivalant
to crisp sets.
A membership function is the relationship between
the values of an element and its degree of membership in a set. An example of
membership functions are shown in Figure. In this example, the sets (or classes) are numbers that are negative
large, negative medium, negative small, near zero, positive small, positive
medium, and positive large. The value, µ, is the amount of membership in the
set. Figure : Membership Functions for the Set of All Numbers (N = Negative, P =
Positive, L = Large, M = Medium, S = Small) Union A ∪ B = {x | x ∈ A or x ∈ B} The union
between the two sets, denoted A ∪ B, represents all those elements in the universe that
reside in (or belong to) the set A, the set B, or both sets A and B. This
operation is also called the logical
or
Union of sets A and B (logical or)
in terms of Venn diagrams Intersection A ∩ B = {x | x ∈ A and x ∈ B} The intersection
of the two sets, denoted A ∩ B, represents all those elements in the
universe X that simultaneously reside in (or belong to) both sets A and B.
This operation is also called the logical
and
Intersection of sets
A and B. The complement of a
set A,is defined as the
collection of all elements in the universe that do not reside in the set A.
Complement of set A Difference A | B = {x | x ∈ A and x is not ∈ B} The difference of a
set A with respect to B, denoted A | B, is defined as the collection of all
elements in the universe that reside in A and that do not reside in B
simultaneously
Difference operation
A | B Properties of Classical (Crisp) Sets · Commutativity A ∪ B = B ∪ A A ∩ B = B ∩ A · Associativity A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B
∩ C) = (A ∩ B) ∩ C · Distributivity A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (2.7) · Idempotency A ∪ A = A A ∩ A = A · Identity A ∪ ∅ = A A ∩ X = A A ∩ ∅ = ∅ A ∪ X = X · Transitivity If A ⊆ B and B ⊆ C, then A ⊆ C Fuzzy
Operations
Union The union is the maximum degree of membership
of sets A and B.
Intersection The intersection is the minimum degree of membership
of sets A and B.
Complement The complement of the membership of set A is
Fuzzy Set Operations
Applications Fuzzy sets are appropriate for pattern
classification because a given gesture or pattern may in fact have partial
membership in many different classes. Several companies already have products
based on fuzzy pattern recognition: 1. Hand Writing Recognition: CSK, Hitachi 2. Hand Printed Character Recognition: Sony 3. Voice Recognition: Ricoh, Hitachi
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Unit-04/Lecture-02 |
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Crisp and Fuzzy Relations Crisp Relation A crisp relation is used to represents the
presence or absence of interaction, association, or interconnectedness
between the elements of more than a set. This crisp relational concept can be
generalized to allow for various degrees or strengths of relation or
interaction between elements. Operations on Crisp Relations Let A and B be two relations defined on X x Y and are represented by relational matrices. The
following operations can be performed on these relations A and B Union A ∪ B (x,y) = max [ A (x,y) , B
(x,y) ] Intersection Fuzzy relation Degrees of association can be represented by
grades of the membership in a fuzzy relation in the same way as degrees of
set membership are represented in the fuzzy set. In fact, just as the crisp
set can be viewed as a restricted case of the more general fuzzy set concept,
the crisp relation can be considered to be a restricted case of the fuzzy
relations. Cartesian product The Cartesian product of two crisp sets X and Y, denoted by
Cartesian is the crisp set of all ordered
pairs such that the first element in each pair is a member of X and the
second element is a member of Y. Formally,
Cartesian Relation among sets A relation among crisp sets
so
Crisp and Fuzzy Relations Thus, a fuzzy relation is a fuzzy set defined on the Cartesian product
of crisp sets
Cartesian product of crisp sets, may have varying
degrees of membership within the relation. The membership grade is usually
represented by a real number in the closed interval, A fuzzy relation can also conveniently be
represented by an n-dimensional membership array whose entries correspond to
n-tuples in the universal set. These entries take values representing the
membership grades of the corresponding n-tuples. Examples Let Q be a crisp relation among the two sets A={dollar, pound, franc, mark} and Y={United States,
France, Canada, Britain, Germany}, which associates a country with a currency
as follows: R(A,B)={(dollar,UnitedStates),(franc,France),(dollar,Canada),(pound,Britain),
(mark,Germany)} This relation can also be represented by the
following two dimensional membership array:
Let R be a fuzzy relation among the two sets the
distance to the target X={far, close, very close}
and the speed of the car Y={very slow, slow, normal, quick, very quick},
which represents the relational concept “the break must be pressed very
strong”. This relation can be written in list notation as R(X,Y) = {0/(far, very
slow) + .3/(close, very slow) + .8/(very close, very slow) + 0/(far, slow) + .4/(close, slow) + .9/(very
close, slow) + 0/(far, normal) + .5/(close, normal) + 1/(very close, normal)
+ .1/(far, quick) + .6/(close, quick) + 1/(very close, quick) + .2/(far,very quick)+ .7/(close,very
quick)+ 1/(very close,very quick)} This relation can also be represented by the
following two dimensional membership array:
Binary Relation Example
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Unit-04/Lecture-03 |
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MEMBERSHIP FUNCTIONS;(Jun-2013) The membership function is a
graphical representation of the magnitude of participation of each input. It
associates a weighting with each of the inputs that are processed, define
functional overlap between inputs, and ultimately determines an output
response. The rules use the input membership values as weighting factors to
determine their influence on the fuzzy output sets of the final output
conclusion. Once the functions are inferred, scaled, and combined, they are defuzzified into a crisp output which drives the system.
There are different membership functions associated with each input and
output response. For any set Membership functions on
Membership function of a fuzzy set Sometimes, a more general definition
is used, where membership functions take values in an arbitrary fixed algebra
or structure
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![[0,1]](CS-801_SC_UNIT-4_files/image031.gif){kind=small}
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Unit-04/Lecture-04 |
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Fuzzy rule base
system Fuzzy systems are built to replace
the human expert with a machine using the logic a human would use to perform
the tasks. Suppose we ask someone how hot it is today. He may tell us that it
is hot, moderately hot or cold. He cannot tell us the exact temperature.
Unlike classical logic which can only interpret the crisp set such as hot or
cold, fuzzy logic has the capability to interpret the natural language. Thus,
fuzzy logic can make human-like interpretations and is a very useful tool in
artificial intelligence, machine learning and automation. Fuzzy logic
operates on the basis of rules which are expressed in the form of If-Then
constructs, also known as horn clauses. The concept of linguistic variable
was introduced to process the natural language. The linguistic variable is
temperature. The linguistic variable can take the verbal values such as hot,
moderately hot or cold. The terms temperature
is hot and temperature is cold andtemperature
is moderate are
known as fuzzy propositions. 5.2 Fuzzy Proposition
A fuzzy proposition can be an atomic
or compound sentence. For example "Temperature is hot" is an atomic fuzzy proposition. "Temperature is hot and humidity
is low" is a compound fuzzy proposition. Compound fuzzy relations are
expressed with fuzzy connectives such as and, or and complement. 5.3 Syntax for IF and THEN rules
The fuzzy rules are written as If <fuzzy
proposition> then <fuzzy proposition> The fuzzy proposition can be atomic
or compound. Aggregation of fuzzy
rules „ 1. Conjunctive
system of rules. In the case of a system of rules that must be jointly
satisfied, the rules are connected by ‘‘and’’ connectives. In this case the
aggregated output(consequent), y, is found by the fuzzy intersection of all
individual rule consequents, yi , where i = 1, 2, . . . r as
2. Disjunctive
system of rules. The aggregated output is found by the fuzzy union of all
individual rule contributions, as
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Unit-04/Lecture-05 |
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Fuzzy Rule-Base 1)
It is the collection
of fuzzy IF-THEN rules in which the preconditions and consequences are
linguistic terms 2)
Fuzzy rules relate
inputs to outputs (control logic) 3)
Rules are formed
using linguistic variables, so it is not precise 4)
Output is also a
linguistic value representing a fuzzy set 5)
Determine degree of
match of fuzzy input with rule antecedent and assign this to the rule
conclusion 6)
Antecedent is the
intersection or union of fuzzy inputs 7)
It is also called
the degree of truth Example Assume two fuzzy linguistic variables
Speed and Position
v If
Speed = Fast and Position = Centered then Change in
speed = Faster v If
Speed = Fast and Position = Right then Change in speed = Zero v If
Speed = Medium and Position = Centered then Change
in speed = Faster v If
Speed = Fast (0.65) and Position = Centered (0.4)
then Change in speed = Faster (0.4) v If
Speed = Fast (0.65) and Position = Right (0.6) then Change in speed = Zero
(0.6) v If
Speed = Medium (0.35) and Position = Centered (0.4)then Change in speed = Faster (0.35) Apply “Union” or “Intersection”
according to “And or “Or” |
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Unit-04/Lecture-06 |
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FUZZY INFERENCE SYSTEM; (Jun-2012) A fuzzy inference
system (FIS) essentially defines a nonlinear mapping of the input data vector into a scalar
output, using fuzzy rules. The mapping process involves input/output
membership functions, FL operators, fuzzy if–then rules, aggregation of
output sets, and defuzzification. An FIS with multiple
outputs can be considered as a collection of independent multiinput, single-output
systems. A general model of a fuzzy inference system (FIS) is shown in below Figure. The FLS maps crisp inputs into crisp
outputs. It can be seen from the figure that the FIS contains four components:
the fuzzifier, inference engine, rule base, and defuzzifier.
The rule base contains linguistic rules that are provided by experts. It is
also possible to extract rules from numeric data. Once the rules have been
established, the FIS can be viewed as a system that maps an input vector to
an output vector. The fuzzifier maps input numbers into corresponding fuzzy
memberships. This is required in order to activate rules that are in terms of
linguistic variables. The fuzzifier takes
input values and determines the degree to which they belong to each of the
fuzzy sets via membership functions. The inference engine defines mapping
from input fuzzy sets into output fuzzy sets. It determines the degree to
which the antecedent is satisfied for each rule. If the antecedent of a given
rule has more than one clause, fuzzy operators are applied to obtain one
number that represents the result of the antecedent for that rule. It is
possible that one or more rules may
fire at the same time. Outputs for all rules are then aggregated. During
aggregation, fuzzy sets that represent the output of each rule are combined
into a single fuzzy set. Fuzzy rules are
fired in parallel, which is one of the important aspects of an FIS. In an
FIS, the order in which rules are fired does not affect the output. The defuzzifier maps output fuzzy sets into a crisp number.
Given a fuzzy set that encompasses a range of output values, the defuzzifier
Figure: Block diagram of a fuzzy inference system. returns one number,
thereby moving from a fuzzy set to a crisp number. Several methods for
defuzzification are used in practice, including the centroid, maximum, mean
of maxima, height, and modified height defuzzifier.
The most popular defuzzification method is the centroid, which calculates and
returns the center of gravity of the aggregated
fuzzy set. FISs employ rules. However, unlike rules in conventional expert
systems, a fuzzy rule localizes a region of space along the function surface
instead of isolating a point on the surface. For a given input, more than one
rule may fire. Also, in an FIS, multiple regions are combined in the output
space to produce a composite region.
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Unit-04/Lecture-07 fuzzy decision making; (June-2012) Most decisions that people make are logical
decisions, they look at the situation and make a decision based on the
situation. The generalized form of such a decision is called a generalized
modus ponens, which is in the form: If P, then Q. This form of logical reasoning is fairly
strict, Q can only be if P. Fuzzy logic loosens this strictness by saying
that Q can mostly be if P is mostly or: If P, then Q. Where P and Q are now fuzzy numbers. The
reasoning above requires a set of rules to be defined. These rules are
linguistic rules to relate different fuzzy sets and numbers. The general form
of these rules are: "if x is A then y is B," where x
and y are fuzzy numbers in the fuzzy sets A and B respectivly.
These fuzzy sets are defined by membership functions. There can be any number
of input and output membership functions for the same input as well,
depending on the number of rules in the system. For example, a system could
have membership functions that represent slow, medium, and fast as inputs. The linguistic rules are
used to define the relation between the input and the output, but how exactly
are the output fuzzy values determined? There are several ways to determine
the answer based on the inputs, mainly the Mamdani, Larsen, Takagi-Sugeno-Kang, and Tsukamoto inference and aggregation
methods. Firstly, we must describe the basic general set of rules, they will
bet a set of rules that have one input in a fuzzy set and one output in a
fuzzy set: If x is Ai then y is Bi,
i=1,2,...n Fuzzy Logic Applications Almost any control
system can be replaced with a fuzzy logic based control system. This may be
overkill in many places however it simplifies the design of many more
complicated cases. So fuzzy logic is not the answer to everything, it must be
used when appropriate to provide better control. If a simple closed loop or
PID controller works fine then there is no need for a
fuzzy controller. There are many cases when tuning a PID controller or
designing a control system for a complicated system is overwhelming, this is
where fuzzy logic gets its chance to shine. 1)
One of the most famous applications
of fuzzy logic is that of the Sendai Subway system in Sendai, Japan. This
control of the Nanboku line, developed by Hitachi,
used a fuzzy controller to run the train all day long. This made the line one
of the smoothest running subway systems in the world and increased efficiency
as well as stopping time. This is also an example of the earlier acceptance
of fuzzy logic in the east since the subway went into operation in 1988. 2)
The most tangible applications of
fuzzy logic control have appeared commercial appliances. Specifically, but
not limited to heating ventillation and air
conditioning (HVAC) systems. These systems use fuzzy logic thermostats to
control the heating and cooling, this saves energy by making the system more
efficient. It also keeps the temperature more steady than a traditional
thermostat 3)
Another significant area of
application of fuzzy control is in industrial automation. Fuzzy logic based
PLCs have been developed by companies like Moeller.
These PLCs, as well as other implementations of fuzzy logic, can be used to
control any number of industrial processes. 4)
Fuzzy logic also finds applications
in many other systems. For example, the MASSIVE
3D animation system for generating crowds uses fuzzy logic for artificial
intelligence. This program was used extensively in the making of the Lord of
the Rings trilogy as well as The Lion, The Witch and the Wardrobe films. 5)
As a final example of fuzzy logic, it
can be used in areas other than simply control. Fuzzy logic can be used in
any decision making process such as signal processing or data analysis. An example
of this is a fuzzy logic system that analyzes a power system and diagnoses
any harmonic disturbance issues. The system analyzes the fundamental voltage,
as well as third, fifth and seventh harmonics as well as the temperature to
determine if there is cause for concern in the operation of the system. |
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