fuzzy sets and fuzzy logic - krchowdhary.com

Fuzzy Sets and Fuzzy Logic

KR Chowdhary, Professor,Department of Computer Science & Engineering, MBM

Engineering College, JNV University, Jodhpur,

CSE Dept., MBM Engg. Col., JNV Univ.

Outline

Crisp Logic Fuzzy Logic Fuzzy Logic Applications Conclusion

“traditional logic”: {true,false}


Crisp Logic Crisp logic is concerned with absolutestrue or false, there is no in

between.

Example: Rule:

If the temperature is higher than 80F, it is hot; otherwise, it is not hot.Cases:

Temperature = 100F Hot Temperature = 80.1F Hot Temperature = 79.9F Not Hot Temperature = 50F Not Hot


Membership function A fuzzy set is a generalization of an ordinary set by by allowing

a degree (or grade) of membership for each element, which varies from 0 to 1, i.e. [0, 1].

Let the people in an organization be the universe. A subset of this is a crisp set.

Consider a set of “young” people in this organization. The “youngness” is not a step function from 0 to 1, for certain age, say 30.


A degree of youngness is associated to each element, like {Ann/0.8, Bob/0.1, Cathy/1}. Perhaps they are 28,40,23. Each element is represented as <element/degree>.

The membership function of a set maps each element to its degree.


Membership function of crisp logic

80F Temperature

HOT

1

If temperature >= 80F, it is hot (1 or true);

If temperature < 80F, it is not hot (0 or false).

0

True

False


Drawbacks of crisp logic

The membership function of crisp logic fails to distinguish between members of the same set.


Concept of Fuzzy Logic

Many decisionmaking and problemsolving tasks are too complex to be defined precisely

However, people succeed by using imprecise knowledge

Fuzzy logic resembles human reasoning in its use of approximate information and uncertainty to generate decisions.


Natural Language is not crisp

Consider: Joe is tall what is tall? Joe is very tall what does this differ from tall?

Natural language (like most other activities in life and indeed the universe) is not easily translated into the absolute terms of 0 and 1.


What is Fuzzy Logic? An approach to uncertainty that combines

real values [0…1] and logic operations

Fuzzy logic is based on the ideas of fuzzy set theory and fuzzy set membership often found in natural (e.g., spoken) language.


Example: “Young” Example:

Ann is 28, 0.8 in set “Young” Bob is 35, 0.1 in set “Young” Charlie is 23, 1.0 in set “Young”

Unlike statistics and probabilities, the degree is not describing probabilities that the item is in the set, but instead describes to what extent the item is the set.


Membership function for “young”

Membership function m(x) to be young:

x M(x)

25 1.0

30 0.5

40 0.1

50 0.1

m x ={1.0 for 0<x≤251

1 x−255

2for x>25 }

Membership function for youngness

M(x)

0 25 30 35 40 Age->


Membership function of fuzzy logic

Age25 40 55

Young Old1

Middle

0.5

DOM

Degree of Membership

Fuzzy sets

Fuzzy values have associated degrees of membership in the set.

0


Crisp set vs. Fuzzy set

A traditional crisp set A fuzzy set


Crisp set vs. Fuzzy set


Benefits of fuzzy logic You want the value to switch gradually as

Young becomes Middle and Middle becomes Old. This is the idea of fuzzy logic.


Fuzzy Set Operations Fuzzy union (∪): Union of two fuzzy sets is the maximum

(MAX) of each element from two sets.

A∪B={x/max(mA(x),mB(x)| x ∈U}

Let a fuzzy set is A = Comfortable house for six persons. A = {(1, .2), (2, .5), (3, .8), (4, 1), (5, .7), (6, .3)}

Let B = Large type of house = {(3, .2), (4, .4), (5, .6), (6, .8), (7, 1), (8, 1)}A ∪ B = {(1, .2), (2, .5), (3, .8), (4, 1), (5, .7), (6, .8), (7, 1), (8, 1)}


Fuzzy Set Operations.. Fuzzy intersection (∩): Intersection of two fuzzy sets is just the MIN

of each element from the two sets. A∩B={x/min(mA(x),mB(x)| x ∈U} E.g. A = {(1, .2), (2, .5), (3, .8), (4, 1), (5, .7), (6, .3)} Let B = Large type of house B = {(3, .2), (4, .4), (5, .6), (6, .8), (7, 1), (8, 1)} A ∩ B = { (3, .2), (4, .4), (5, .6), (6, .3) }


Fuzzy Set Operations.. The complement of a fuzzy variable with DOM x is (1x). Complement ( A’): The complement of a fuzzy set is

composed of all elements’ complement. A′={x/(1mA(x))|x ∈U} Example.A = {(1, .2), (2, .5), (3, .8), (4, 1), (5, .7), (6, .3)} A′ = {(1, .8), (2, .5), (3, .2), (4, 0), (5, .3), (6, .7)}


Crisp Relations Ordered pairs showing connection between two sets: (a,b): a is related to b (2,3) are related with the relation “<“

Relations are set themselves < = {(1,2), (2, 3), (2, 4), ….}

Relations can be expressed as matrices

< 1 2

1 × √

2 × ×


Fuzzy Relations Triples showing connection between two sets: (a,b,#): a is related to b with degree #

Fuzzy relations are set themselves Fuzzy relations can be expressed as matrices

…


Fuzzy Relations Matrices

Example: ColorRipeness relation for tomatoes

R1(x, y) unripe semi ripe ripe

green 1 0.5 0

yellow 0.3 1 0.4

Red 0 0.2 1


Inference rule The inference rule is rule of composition

[a bc d ] ο [ xy ]= [ax ∨by

cx ∨dy ]

Relation of composition on crisp sets:R°S={(a,c) | (a,b)∈R,(b,c)∈S,a∈A,b∈B,c∈C}

Fuzzy Compositional rule:


Inference rule

u n se ri hi low

gryere [

1 .5 o. 3 1 .40 .2 1 ] °

unseri [

.1 1

.6 .41 .2 ]

hi lo

=gryere [

. 5 1

. 6 . 41 . 2 ]

Consider that there are two relation sets. One related the color oftomato to ripeness, and other relates ripeness to prices(hi,low).Given this find out the relation from color to price.


Example: Fuzzy Inference

Two temperature Inputs (x, y) and one output (z) Membership functions:

low(t) = 1 ( t / 10 ); is o/p as function of i/phigh(t) = t / 10 ; is o/p as a fn. of i/p

Low High

1

0t

0.39 0.61

0.32

0.68

Low(x) = 0.68, High(x) = 0.32, Low(y) = 0.39, High(y) = 0.61Crisp Inputs

yx


Create rule base

Rule 1: If x is low AND y is low Then z is high (i.e., if I/P x,y have membership of low, then the o/p z has membership of high. x and y are ANDed)

Rule 2: If x is low AND y is high Then z is low

Rule 3: If x is high AND y is low Then z is low

Rule 4: If x is high AND y is high Then z is high

X

YX Y Z

0 0 1

0 1 0

1 0 0

1 1 1

Z

Z = X ⊕ Y


Inference step #1(conjunction part of composition)

Rule1: low(x)=0.68, low(y)=0.39 => high(z)=min(0.68,0.39)=0.39

Rule2: low(x)=0.68, high(y)=0.61 => low(z)=min(0.68,0.61)=0.61

Rule3: high(x)=0.32, low(y)=0.39 => low(z)=min(0.32,0.39)=0.32

Rule4: high(x)=0.32, high(y)=0.61 => high(z)=min(0.32,0.61)=0.32

Conjunction is performed of x and y.


Inference step #2 (disjunction part of composition)

Low High1

0t

•Low(z) = max(rule2, rule3) = max(0.61, 0.32) = 0.61

•High(z) = max(rule1, rule4) = max(0.39, 0.32) = 0.39

0.61

0.39

Z low = .61

Z high =.39


Fuzzy Resolution and other

Fuzzy De Morgans: ¬(F∧G) = ¬F∨¬G and ¬(F∨G) = ¬F ∧ ¬G

L∨C1, ¬L∨C2

C1∨C2

Fuzzy Resolution:


Other operations on fuzzy sets

Concentration CON(A)={x/mA(x)2|x∈U}

Dilation DIL(A)={x/sqrt(mA(x) |x∈U}

Normalization NORM(A)={x/(mA(x)/Max) |x∈U}


Operations on Fuzzy System

Crisp Input

Fuzzy Input

Fuzzy Output

Crisp Output

Fuzzification

Rule Evaluation

Defuzzification

Input Membership Functions

Rules / Inferences

Output Membership Functions


In what areas are fuzzy systems effective and why?

Difficult cases where traditional techniques do not work

Used in fuzzy control of physical or chemical characteristics such as temperature, electric current, flow of liquid/gas, motion of machines, etc.

Fuzzy logic can be applied in fuzzy knowledge based systems, which uses fuzzy if then rules; fuzzy software engineering, that may incorporate fuzziness in programs and data; fuzzy databases that store and retrieve fuzzy information; fuzzy pattern recognition that deals with fuzzy visual or audio signals; applications to medicine, economics, and management problems that involve fuzzy information processing.


In what areas are fuzzy systems effective and why?.. Fuzzy systems are useful for approximate reasoning where

mathematical model are hard to derive

It allows for decision making with estimated values under incomplete information

For hard systems, conventional non fuzzy systems are expensive and depend on mathematical approximation (e.g., linearization of nonlinear problems), which may lead to poor performance.

Response of fuzzy systems are smoother.


Control System Based on rules of logic obtained from train

drivers so as to model real human decisions as closely as possible

Task: Controls the speed at which the train takes curves as well as the acceleration and braking systems of the train


Control System…

This system is still not perfect; humans can do better because they can make decisions based on previous experience and anticipate the effects of their decisions

This led to use of fuzzy systems


Decision Support: Predictive Fuzzy Control Can assess the results of a decision and determine if the

action should be taken

Has model of the motor and break to predict the next state of speed, stopping point, and running time input variables

Controller selects the best action based on the predicted states.


Decision Support: Predictive Fuzzy Control…

The results of the fuzzy logic controller for the Sendai subway (Japan) are excellent!!

The train movement is smoother than most other trains

Even the skilled human operators who sometimes run the train cannot beat the automated system in terms of smoothness or accuracy of stopping


Fuzzy Expert Systems Fuzzy expert system is a collection of membership

functions and rules that are used to reason about data. Usually, the rules in a fuzzy expert system have the

following form: “if x is low and y is high then z is medium”Classical: “if x and y then z “

fuzzy sets and fuzzy logic - krchowdhary.com

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