Slide 1: FUZZY LOGIC Presented by:
A.Gupta
J.Jain
K.Kedia
N.Gupta
R.Bafna 1
Index : Index Brief History
What is fuzzy logic?
Fuzzy Vs Crisp Set
Membership Functions
Fuzzy Logic Vs Probability
Why use Fuzzy Logic?
Fuzzy Linguistic Variables
Operations on Fuzzy Set
Fuzzy Applications
Case Study
Drawbacks
Conclusion
Bibliography 2
Brief History : Brief History Classical logic of Aristotle: Law of Bivalence “Every proposition is either True or False(no middle)”
Jan Lukasiewicz proposed three-valued logic : True, False and Possible
Finally Lofti Zadeh published his paper on fuzzy logic-a part of set theory that operated over the range [0.0-1.0] 3
What is Fuzzy Logic? : What is Fuzzy Logic? Fuzzy logic is a superset of Boolean (conventional) logic that handles the concept of partial truth, which is truth values between "completely true" and "completely false”.
Fuzzy logic is multivalued. It deals with degrees of membership and degrees of truth.
Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true). Boolean
(crisp) Fuzzy 4
: For example, let a 100 ml glass contain 30 ml of water. Then we may consider two concepts: Empty and Full.
In boolean logic there are two options for answer i.e. either the glass is half full or glass is half empty. 100 ml 30 ml In fuzzy concept one might define the glass as being 0.7 empty and 0.3 full. 5
Crisp Set and Fuzzy Set : Crisp Set and Fuzzy Set 6 μ a(x)={ 1 if element x belongs to the set A
0 otherwise
} Classical set theory enumerates all element using A={a1,a2,a3,a4…,an} Set A can be represented by Characteristic function A fuzzy set can be represented by:
A={{ x, A(x) }}
where, A(x) is the membership grade of a element x in fuzzy set
SMALL={{1,1},{2,1},{3,0.9},{4,0.6},{5,0.4},{6,0.3},{7,0.2},{8,0.1},{9,0},{10,0},{11,0},{12,0}} In fuzzy set theory elements have varying degrees of membership Example: Consider space X consisting of natural number<=12
Prime={x contained in X | x is prime number={2,3,5,7,11}
Fuzzy Vs. Crisp Set : Fuzzy Vs. Crisp Set A A’ a a b b c Fuzzy set Crisp set a: member of crisp set A
b: not a member of set A a: full member of fuzzy set A’
b: not a member of set A’
c:partial member of set A’ 7
Fuzzy Vs. Crisp Set : Crisp set Fuzzy Vs. Crisp Set Fuzzy set 8
: Features of a membership function core support boundary 1 0 μ (x) x Core: region characterized by full membership in set A’ i.e. μ (x)=1.
Support: region characterized by nonzero membership in set A’ i.e. μ(x) >0.
Boundary: region characterized by partial membership in set A’ i.e. 0< μ (x) <1 9 A membership function is a mathematical function which defines the degree of an element's membership in a fuzzy set.
Membership Functions : Membership Functions adult(x)= { 0, if age(x) < 16years
(age(x)-16years)/4, if 16years < = age(x)< = 20years,
1, if age(x) > 20years
} 10
Fuzzy Logic Vs Probability : Fuzzy Logic Vs Probability Both operate over the same numeric range and at first glance both have similar values:0.0 representing false(or non-membership) and 1.0 representing true.
In terms of probability, the natural language statement would be ”there is an 80% chance that Jane is old.”
While the fuzzy terminology corresponds to “Jane’s degree of membership within the set of old people is 0.80.’
Fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance. 11
Why use Fuzzy Logic? : Why use Fuzzy Logic? Fuzzy logic is flexible.
Fuzzy logic is conceptually easy to understand.
Fuzzy logic is tolerant of imprecise data.
Fuzzy logic is based on natural language. 12
Fuzzy Linguistic Variables : Fuzzy Linguistic Variables Fuzzy Linguistic Variables are used to represent qualities spanning a particular spectrum
Temp: {Freezing, Cool, Warm, Hot} 13
Operations on Fuzzy Set : Operations on Fuzzy Set A B μA μB A= {1/2 + .5/3 + .3/4 + .2/5} B= {.5/2 + .7/3 + .2/4 + .4/5} Consider: >Fuzzy set (A) >Fuzzy set (B) >Resulting operation of fuzzy sets 14
Slide 15: INTERSECTION
(A ^ B) UNION
(A v B) COMPLEMENT
(¬A) μA ∩ B μA U μA ‘ μA∩ B = min (μA(x), μB(x))
{.5/2 + .5/3 + .2/4 + .2/5} μAUB = max (μA(x), μB(x))
{1/2 + .7/3 + .3/4 + .4/5} μA’ = 1-μA(x)
{1/1 + 0/2 + .5/3 + .7/4 + .8/5} 15
: Example Speed Calculation How fast will I go if it is
65 F°
25 % Cloud Cover ? 16
Slide 17: Input: Temp: {Freezing, Cool, Warm, Hot} Cover: {Sunny, Partly cloudy, Overcast} Output: Speed: {Slow, Fast} 17
Rules : If it's Sunny and Warm, drive Fast
Sunny(Cover)Warm(Temp) Fast(Speed)
If it's Cloudy and Cool, drive Slow
Cloudy(Cover)Cool(Temp) Slow(Speed)
Driving Speed is the combination of output of these rules... Rules 18
Fuzzification:Calculate Input Membership Levels : 65 F° Cool = 0.4, Warm= 0.7
25% Cover Sunny = 0.8, Cloudy = 0.2 Fuzzification:Calculate Input Membership Levels 19
Calculating: : Calculating: If it's Sunny and Warm, drive Fast
Sunny(Cover)Warm(Temp)Fast(Speed)
0.8 0.7 = 0.7
Fast = 0.7
If it's Cloudy and Cool, drive Slow
Cloudy(Cover)Cool(Temp)Slow(Speed)
0.2 0.4 = 0.2
Slow = 0.2 20
Defuzzification:Constructing the Output : Speed is 20% Slow and 70% Fast
Find centroids: Location where membership is 100%
Speed = weighted mean
= (2*25+7*75)/(9)
= 63.8 mph Defuzzification:Constructing the Output 21
Fuzzy Applications : Fuzzy Applications Automobile and other vehicle subsystems : used to control
the speed of vehicles, in Anti Braking System.
Temperature controllers : Air conditioners, Refrigerators
Cameras : for auto-focus
Home appliances: Rice cookers , Dishwashers , Washing
machines and others 22
Drawbacks : Fuzzy logic is not always accurate. The results are perceived as
a guess, so it may not be as widely trusted .
Requires tuning of membership functions which is difficult to
estimate.
Fuzzy Logic control may not scale well to large or complex
problems
Fuzzy logic can be easily confused with probability theory, and
the terms used interchangeably. While they are similar concepts,
they do not say the same things. Drawbacks 23
Conclusion : Fuzzy Logic provides way to calculate with imprecision and
vagueness.
Fuzzy Logic can be used to represent some kinds of human
expertise .
The control stability, reliability, efficiency, and durability of fuzzy
logic makes it popular.
The speed and complexity of application production would not be
possible without systems like fuzzy logic. Conclusion 24
Bibliography : Bibliography BOOK :
Artificial Intelligence by Elaine Rich, Kelvin Knight and Shivashankar B Nair
Internet 25