The document presents information about fuzzy logic and its applications. It begins with general definitions of fuzzy logic and how it is used to represent expert knowledge that uses vague terms. It then discusses the history and development of fuzzy logic. Applications mentioned include ABS brakes, expert systems, control units, and consumer products. The document also provides formal definitions of fuzzy sets and fuzzy logic operations. It concludes with an example of a fuzzy logic controller for an air conditioner.
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1. Presented by:
Bhanu Fix Poudyal
066BEL305
Department Of Electrical Engineering
Pulchowk Campus
Institute Of Engineering, Pulchowk Campus2/20/2012
2. Agenda
General Definition
Applications
Formal Definitions
Operations
Rules
Fuzzy Air Conditioner
Controller Structure
Institute Of Engineering, Pulchowk Campus2/20/2012
3. Definition
Experts rely on common sense when they solve problems.
How can we represent expert knowledge that uses vague and
ambiguous terms in a computer?
Fuzzy logic is not logic that is fuzzy, but logic that is used to describe
fuzziness. Fuzzy logic is the theory of fuzzy sets, sets that calibrate
vagueness.
Fuzzy logic is based on the idea that all things admit of degrees.
Temperature, height, speed, distance, beauty – all come on a sliding scale.
The motor is running really hot.
Tom is a very tall guy.
Fuzzy Logic
Institute Of Engineering, Pulchowk Campus2/20/2012
4. Many decision-making and problem-solving tasks are too complex to be
understood quantitatively, however, people succeed by using
knowledge that is imprecise rather than precise.
Fuzzy set theory resembles human reasoning in its use of approximate
information and uncertainty to generate decisions.
It was specifically designed to mathematically represent uncertainty and
vagueness and provide formalized tools for dealing with the imprecision
intrinsic to many engineering and decision problems in a more natural
way.
Boolean logic uses sharp distinctions. It forces us to draw lines between
members of a class and non-members. For instance, we may say, Tom is tall
because his height is 181 cm. If we drew a line at 180 cm, we would find
that David, who is 179 cm, is small.
Is David really a small man or we have just drawn an arbitrary line in the
sand?
Definition
Fuzzy Logic
Institute Of Engineering, Pulchowk Campus2/20/2012
5. Bit of History
Fuzzy, or multi-valued logic, was introduced in the 1930s by Jan
Lukasiewicz, a Polish philosopher. While classical logic operates with
only two values 1 (true) and 0 (false), Lukasiewicz introduced logic that
extended the range of truth values to all real numbers in the interval
between 0 and 1.
For example, the possibility that a man 181 cm tall is really tall might be
set to a value of 0.86. It is likely that the man is tall. This work led to
an inexact reasoning technique often called possibility theory.
In 1965 Lotfi Zadeh, published his famous paper “Fuzzy sets”. Zadeh
extended the work on possibility theory into a formal system of
mathematical logic, and introduced a new concept for applying natural
language terms. This new logic for representing and manipulating
fuzzy terms was called fuzzy logic.
Fuzzy Logic
Institute Of Engineering, Pulchowk Campus2/20/2012
6. Why Fuzzy Logic?
Why fuzzy?
As Zadeh said, the term is concrete, immediate and descriptive; we all
know what it means. However, many people in the West were repelled by
the word fuzzy, because it is usually used in a negative sense.
Why logic?
Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of
that theory.
The term fuzzy logic is used in two senses:
Narrow sense: Fuzzy logic is a branch of fuzzy set theory, which deals
(as logical systems do) with the representation and inference from
knowledge. Fuzzy logic, unlike other logical systems, deals with
imprecise or uncertain knowledge. In this narrow, and perhaps correct
sense, fuzzy logic is just one of the branches of fuzzy set theory.
Broad Sense: fuzzy logic synonymously with fuzzy set theory
Fuzzy Logic
Institute Of Engineering, Pulchowk Campus2/20/2012
7. Applications
ABS Brakes
Expert Systems
Control Units
Bullet train between Tokyo and Osaka
Video Cameras
Automatic Transmissions
Washing Machines
Institute Of Engineering, Pulchowk Campus2/20/2012
8. Formal Definitions
Definition 1: Let X be some set of objects, with elements noted as x.
X = {x}.
Definition 2: A fuzzy set A in X is characterized by a membership function
mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x)
approaches 1.0, the "grade of membership" of x in A increases.
Definition 3: A is EMPTY iff for all x, mA(x) = 0.0.
Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB].
Definition 5: mA' = 1 - mA.
Definition 6: A is CONTAINED in B iff mA mB.
Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)).
Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x),
mB(x)).
Institute Of Engineering, Pulchowk Campus2/20/2012
9. Fuzzy Logic Operators
Fuzzy Logic:
NOT (A) = 1 - A
A AND B = min( A, B)
A OR B = max( A, B)
Institute Of Engineering, Pulchowk Campus2/20/2012
10. Operations
A B
A B A B A
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14. Fuzzy Controllers
Used to control a physical system
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15. Controller Structure
Fuzzification
Scales and maps input variables to fuzzy sets
Inference Mechanism
Approximate reasoning
Deduces the control action
Defuzzification
Convert fuzzy output values to control signals
Institute Of Engineering, Pulchowk Campus2/20/2012
16. Structure of a Fuzzy Controller
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17. Fuzzification
Conversion of real input to fuzzy set values
e.g. Medium ( x ) = {
0 if x >= 1.90 or x < 1.70,
(1.90 - x)/0.1 if x >= 1.80 and x < 1.90,
(x- 1.70)/0.1 if x >= 1.70 and x < 1.80 }
Institute Of Engineering, Pulchowk Campus2/20/2012
18. Inference Engine
Fuzzy rules
based on fuzzy premises and fuzzy consequences
e.g.
If height is Short and weight is Light then feet are Small
Short( height) AND Light(weight) => Small(feet)
Institute Of Engineering, Pulchowk Campus2/20/2012
19. Fuzzification & Inference Example
If height is 1.7m and weight is 55kg
what is the value of Size(feet)
Institute Of Engineering, Pulchowk Campus2/20/2012
20. Defuzzification
Rule base has many rules
so some of the output fuzzy sets will have membership
value > 0
Defuzzify to get a real value from the fuzzy outputs
One approach is to use a centre of gravity method
Institute Of Engineering, Pulchowk Campus2/20/2012
21. Defuzzification Example
Imagine we have output fuzzy set values
Small membership value = 0.5
Medium membership value = 0.25
Large membership value = 0.0
What is the deffuzzified value
Institute Of Engineering, Pulchowk Campus2/20/2012
23. Rule Base
Air Temperature
Set cold {50, 0, 0}
Set cool {65, 55, 45}
Set just right {70, 65, 60}
Set warm {85, 75, 65}
Set hot {, 90, 80}
Fan Speed
• Set stop {0, 0, 0}
• Set slow {50, 30, 10}
• Set medium {60, 50, 40}
• Set fast {90, 70, 50}
• Set blast {, 100, 80}
Institute Of Engineering, Pulchowk Campus2/20/2012
24. Rules
Air Conditioning Controller Example:
IF Cold then Stop
If Cool then Slow
If OK then Medium
If Warm then Fast
IF Hot then Blast
default:
The truth of any
statement is a
matter of degree
Membership function is
a curve of the degree
of truth of a given
input value
Institute Of Engineering, Pulchowk Campus2/20/2012