handling uncertainty using fuzzy logicpkalra/siv895/old/fuzzy.pdf · fuzzy logic with engineering...

Handling Uncertainty using

FUZZY LOGIC

Conventional (Boolean) Set Theory:

Fuzzy Set Theory

© INFORM 1990-1998 Slide 3

“Strong Fever”

40.1°C

42°C

41.4°C

39.3°C

38.7°C

37.2°C

38°C

Fuzzy Set Theory:

40.1°C

42°C

41.4°C

39.3°C

38.7°C

37.2°C

38°C

“More-or-Less” Rather Than “Either-Or” !

“Strong Fever”

Fuzzy Variable: Old

1

030 50

Age

75 90

Membership

μ(25)= 0

(33)= 0.15

μ(38)= 0.4

μ(44)= 0.7

μ(72)= 1.0

Crisp Logic vs Fuzzy Logic

Crisp logic needs hard decisions

Fuzzy Logic deals with “membership in group” functions

Probability and Possibility One cupboard A contains a number of bottles, some

containing water, and others containing undesirable products. The label on one of the bottle reads,

“ Probability of water being good is 0.93”.

Another cupboard B contains water from various sources, like mineral water, tap water, rain water, pond water, drain water etc. The label on one of the bottle reads,

“ Possibility of water being good is 0.8”.

Which one would you prefer to open for drinking?

Fuzzy Logic deals with Possibility measures.

Possibility indicates the extent of belief.

First proposed by Lufti Zadeh.

Lots of applications:

Digital cameras

Camcorders

Washing machines

Braking systems (trains)

Process control

Image processing

Control of automatic exposure in video cameras,

humidity in a clean room,

air conditioning systems,

washing machine timing,

microwave ovens,

vacuum cleaners.

Continuous Fuzzy sets

Discrete Fuzzy set

x = { 0/0, 0.2/1, 0.4/2, 1/ 3, 0.9/4, 0.3/5, 0.1/6 }

Membership function Crisp set representation

Characteristic function

if

if

Fuzzy set representation Membership function

if x is totally in A

if x is not in A

If x is partly in A

( ) : 0,1Af x X

1,( )

0Af x

x A

x A

( ) 1

( ) 0

0 ( ) 1

A

A

A

x

x

x

Fuzzy set theory basicsFuzzy set operators:

EqualityA = BA (x) = B (x) for all x X

ComplementA’A’ (x) = 1 - A(x) for all x X

ContainmentA BA (x) B (x) for all x X

Fuzzy set theory basicsFuzzy set operators:

UnionA BA B (x) = max(A (x), B (x)) for all x X

IntersectionA BA B (x) = min(A (x), B (x)) for all x X

T-norms & co-normsIntersection of two fuzzy sets

t-norm properties:

If b1 < b2, then

Basic t-norms:

Standard intersection -

Bounded sum -

Algebraic product -

( , )

( ,1) ( )

( , ) ( , )

z T a b

T a T a

T a b T a b

1 2( , ) ( , )T a b T a b

( , ) min( , )

( , ) max(0, 1)

( , )

m

b

p

T a b a b

T a b a b

T a b ab

Example fuzzy set operations

15

A’

A B A B

A B

A

Well known Membership Functions

A given value could have number of possibilities

X has following possibilities

possibility(low) = 0.8

possibility(medium) = 0.4

all others possibilities (high, V.high) =‏‏‏ 0

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!)

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall

Fuzzy Relations

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 (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 Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill

˜ R

R~

Fuzzy Cartesian Product

Let

be a fuzzy set on universe X, and

be a fuzzy set on universe Y, then

Where the fuzzy relation R has membership function


˜ A ˜ B ˜ R X Y

R (x, y) A x B

(x, y) min( A (x), B

(y))

˜ A ˜ B

Fuzzy Cartesian Product: ExampleLet

defined on a universe of three discrete temperatures, X = {x1,x2,x3}, and

defined on a universe of two discrete pressures, Y = {y1,y2}

Fuzzy set represents the “ambient” temperature and

Fuzzy set the “near optimum” pressure for a certain heat exchanger, and the Cartesian product might represent the conditions (temperature-pressure pairs) of the exchanger that are associated with “efficient”operations. For example, let


˜ A

˜ B ˜ A ˜ B

˜ A 0.2

x10.5

x21

x3and

˜ B 0.3

y10.9

y2

} ˜ A ˜ B ˜ R

x1

x2

x3

0.2 0.2

0.3 0.5

0.3 0.9

y1 y2

Fuzzy CompositionSuppose

is a fuzzy relation on the Cartesian space X x Y,

is a fuzzy relation on the Cartesian space Y x Z, and

is a fuzzy relation on the Cartesian space X x Z; then fuzzy max-min and fuzzy max-product composition are defined as


˜ R ˜ S ˜ T

˜ T ˜ R S

maxmin

T (x,z)

yY( R (x,y) S

(y,z))

max product

T (x,z)

yY(

R (x,y)

S (y, z))

Max-Min Composition The max-min composition of two fuzzy relations R1

(defined on X and Y) and R2 (defined on Y and Z) is

Properties:

Associativity:

Distributive over union:

)],(),([),(2121

zyyxzx RRy

RR

R S T R S R T ( ) ( ) ( )

R S T R S T ( ) ( )

Fuzzy Composition: Example (max-min)


X {x1, x2},

T (x1,z1)

yY( R (x1,y) S

(y,z1))

max[min( 0.7,0.9),min(0.5, 0.1)]

0.7

Y {y1, y2},and Z {z1,z2, z3}

Consider the following fuzzy relations:

˜ R x1

x2

0.7 0.5

0.8 0.4

y1 y2

and ˜ S y1

y2

0.9 0.6 0.5

0.1 0.7 0.5

z1 z2 z3

Using max-min composition,

} ˜ T x1

x2

0.7 0.6 0.5

0.8 0.6 0.4

z1 z2 z3

Max-Product Composition The max-product composition of two fuzzy relations R1

(defined on X and Y) and R2 (defined on Y and Z) is

Properties:

Associativity:

Distributive over union:

)],(),([),(2121

zyyxzx RRy

RR

R S T R S R T ( ) ( ) ( )

R S T R S T ( ) ( )

Fuzzy Composition: Example (max-Prod)


X {x1, x2},

T (x2, z2 )

yY( R (x2 , y) S

(y, z2))

max[( 0.8,0.6),(0.4, 0.7)]

0.48

Y {y1, y2},and Z {z1,z2, z3}

Consider the following fuzzy relations:

˜ R x1

x2

0.7 0.5

0.8 0.4

y1 y2

and ˜ S y1

y2

0.9 0.6 0.5

0.1 0.7 0.5

z1 z2 z3

Using max-product composition,

} ˜ T x1

x2

.63 .42 .25

.72 .48 .20

z1 z2 z3

Application: Fuzzy Relation Petite


Fuzzy Relation Petite defines the degree by which a person with

a specific height and weight is considered petite. Suppose the

range of the height and the weight of interest to us are {5’, 5’1”,

5’2”, 5’3”, 5’4”,5’5”,5’6”}, and {90, 95,100, 105, 110, 115, 120,

125} (in lb). We can express the fuzzy relation in a matrix form as

shown below:

˜ P

5'

5'1"

5' 2"

5' 3"

5' 4"

5' 5"

5' 6"

1 1 1 1 1 1 .5 .2

1 1 1 1 1 .9 .3 .1

1 1 1 1 1 .7 .1 0

1 1 1 1 .5 .3 0 0

.8 .6 .4 .2 0 0 0 0

.6 .4 .2 0 0 0 0 0

0 0 0 0 0 0 0 0

90 95 100 105 110 115 120 125


˜ P

5'

5'1"

5' 2"

5' 3"

5' 4"

5' 5"

5' 6"

1 1 1 1 1 1 .5 .2

1 1 1 1 1 .9 .3 .1

1 1 1 1 1 .7 .1 0

1 1 1 1 .5 .3 0 0

.8 .6 .4 .2 0 0 0 0

.6 .4 .2 0 0 0 0 0

0 0 0 0 0 0 0 0

90 95 100 105 110 115 120 125

Once we define the petite fuzzy relation, we can answer two

kinds of questions:

• What is the degree that a female with a specific height and a specific

weight is considered to be petite?

• What is the possibility that a petite person has a specific pair of height

and weight measures? (fuzzy relation becomes a possibility

distribution)



Given a two-dimensional fuzzy relation and the possible values

of one variable, infer the possible values of the other variable

using similar fuzzy composition as described earlier.

Definition: Let X and Y be the universes of discourse for

variables x and y, respectively, and xi and yj be elements of X

and Y. Let R be a fuzzy relation that maps X x Y to [0,1] and

the possibility distribution of X is known to be Px(xi). The

compositional rule of inference infers the possibility distribution

of Y as follows:

max-min composition:

max-product composition:

PY(yj ) maxx i

(min(PX (xi),PR (xi , yj)))

PY(yj ) maxx i

(PX (xi) PR(xi , yj ))



Problem: We may wish to know the possible weight of a petite

female who is about 5’4”.

Assume About 5’4” is defined as

About-5’4” = {0/5’, 0/5’1”, 0.4/5’2”, 0.8/5’3”, 1/5’4”, 0.8/5’5”, 0.4/5’6”}

Using max-min compositional, we can find the weight possibility

distribution of a petite person about 5’4” tall:

Pweight

(90) (0 1) (0 1) (.41) (.81) (1 .8) (.8 .6) (.4 0)

0.8

˜ P

5'

5'1"

5' 2"

5' 3"

5' 4"

5' 5"

5' 6"

1 1 1 1 1 1 .5 .2

1 1 1 1 1 .9 .3 .1

1 1 1 1 1 .7 .1 0

1 1 1 1 .5 .3 0 0

.8 .6 .4 .2 0 0 0 0

.6 .4 .2 0 0 0 0 0

0 0 0 0 0 0 0 0

90 95 100 105 110 115 120 125

Similarly, we can compute the possibility

degree for other weights. The final result is

Pweight {0.8 /90,0.8 /95,0.8 /100,0.8/105,0.5 /110,0.4 /115, 0.1/120,0 /125}

Fuzzy Inference Systems Fuzzy logical operations

Fuzzy rules

Fuzzification

Implication

Aggregation

Defuzzification

Fuzzifier Inference Engine De-fuzzification

FuzzyKnowledge base

input output

Fuzzifier

Converts the crisp input to a linguistic variable using the membership functions stored in the fuzzy knowledge base.

Inference Engine Using If-Then type fuzzy rules converts the fuzzy input to

the fuzzy output.

COMPOSITIONAL RULE OFINFERENCE

In order to draw conclusions from a set of rules (rule base) one needs a mechanism that can produce an output from a collection of rules. This is done using the compositional rule of inference.

B’= A’ o R

“o“ is the composition operator. The inference procedure is called “compositional rule of inference”.

The inference mechanism is determined by two factors:

1. Implication operators: Mamdani: min

Larsen: algebraic product

2. Composition operators: Mamdani: max-min

Larsen: max-product

35

Rule Evaluation

distance

small

o.55

Clipping approach (others are possible):

Clip the fuzzy set for “slow” (the consequent) at the height given by our belief in the premises (0.55)

We will then consider the clipped AREA (orange) when making our final decision

Rationale: if belief in premises is low, clipped area will be very smallBut if belief is high it will be close to the whole unclipped area

acceleration

slow

36

Rule Evaluation

Distance is not growing, then keep present acceleration

delta

=

0.75

acceleration

slow present fast fastestbrake

37

Rule Evaluation

Distance is not growing, then keep present acceleration

delta

=

0.75

acceleration

present

38

Rule Aggregation

acceleration

present

acceleration

slow

From distanceFrom delta (distance change)

To make a final decision: From each rule we haveObtained a clipped area. But in the end we want a singleNumber output: our desired acceleration

Washing Machine

INPUT:

Load (Quantity)

small, medium, large

Fabric Softness:

Hard, Not so Hard, Soft, Not so soft

OUTPUT: (Wash Cycle)

Light, Normal, Strong

If

Laundry quantity is large (Fuzzy)

then

wash cycle is strong (Fuzzy)

Washing Machine1

0.6

0.3

1

0

Small Medium Large

Laundry Quantity

Light Normal Strong

.

Wash Cycle

If Laundry quantity is large (Fuzzy) then wash cycle is strong (Fuzzy)

Washing machine needs a NON-fuzzy information.

Rule 1: If Laundry quantity is LARGE and

Laundry softness is HARD then wash cycle is

strong.

Rule 2: If Laundry quantity is MEDIUM and

Laundry softness is NOT SO HARD then wash

cycle is normal.

All rules in rule base get fired

Washing Machine

0.6

0.3

1

0

Small Medium Large

Laundry Quantity

Light Normal Strong

1

0

Hard N.H N.S Soft

Laundry Softness

0.2

.75

Wash Cycle

Rule 1: If Laundry quantity is LARGE and Laundry softness is HARD then wash cycle is strong.

Rule 2: If Laundry quantity is MEDIUM and Laundry softness is NOT SO HARD then wash cycle is normal.

44

Summary: If-Then rules1. Fuzzify inputs:

Determine the degree of membership for all terms in the

premise.

If there is one term then this is the degree of support for

the consequence.

2. Apply fuzzy operator:

If there are multiple parts, apply logical operators to

determine the degree of support for the rule.

45

Summary: If-Then rules3. Apply implication method:

Use degree of support for rule to shape output fuzzy set of

the consequence.

How do we then combine several rules?

46

Multiple rules We aggregate the outputs into a single fuzzy set which

combines their decisions.

The input to aggregation is the list of truncated fuzzy sets and the output is a single fuzzy set for each variable.

Aggregation rules: max, sum, etc.

As long as it is commutative then the order of rule exec is irrelevant.

47

Defuzzify the output Take a fuzzy set and produce a single crisp number that

represents the set.

Practical when making a decision, taking an action etc.

Center of gravity

Defuzzification

Center of Gravity

Low High

1

0

0.61

0.39

tCrisp output

Max

Min

Max

Min

dttf

dtttf

C

)(

)(

Center of Gravity

Sugeno Fuzzy Models

Also known as TSK fuzzy model

Takagi, Sugeno & Kang, 1985

Goal: Generation of fuzzy rules from a given input-output data

set.

Fuzzy Rules of TSK Model:

If x is A and y is B then z = f(x, y)

Fuzzy Sets Crisp Function

Examples

R1: if X is small and Y is small then z = x +y +1

R2: if X is small and Y is large then z = y +3

R3: if X is large and Y is small then z = x +3

R4: if X is large and Y is large then z = x + y + 2

51

Issues with Fuzzy logic How to determine the membership functions? Usually

requires fine-tuning of parameters

How to generate Fuzzy rules?

3 input variables, each can take 4 possible linguistic values

GA can help

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