Fuzzy Control Lect 3 Membership Function and Approximate Reasoning Basil Hamed Electrical...

Fuzzy Control

Lect 3 Membership Function and Approximate ReasoningBasil Hamed

Electrical Engineering

Islamic University of Gaza

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Content Membership Function Features of Membership Function Fuzzy Membership Functions Types of Membership Function

Approximate Reasoning Linguistic Variables IF THEN RULES Example

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Membership Function

Membership Functions characterize the fuzziness of fuzzy sets. There are an infinite # of ways to characterize fuzzy infinite ways to define fuzzy membership functions.

Membership function essentially embodies all fuzziness for a particular fuzzy set, its description is essential to fuzzy property or operation.

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Membership Function

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Features of Membership Function

Core: comprises of elements x of the universe, such that

A(x) = 1

Support: comprises of elements x of universe, such that A(x) > 0

Boundaries: comprise the elements x of the universe 0 < A(x) < 1

A normal fuzzy set has at least one element with membership 1

For fuzzy set, if one and only one element has a membership = 1, this element is called as the prototype of set.

A subnormal fuzzy set has no element with membership=1.

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x

1

0

0.5MF

MF Terminology

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cross points

core

width

-cut

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Graphically,

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The most common forms of membership functions are those that are normal and convex.However, many operations on fuzzy sets, hence operations on membership functions, result in fuzzy sets that are subnormal and nonconvex.Membership functions can be symmetrical or asymmetrical.

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Membership Functions (MF’s)

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Convexity of Fuzzy Sets

A fuzzy set A is convex if for any in [0, 1].

1 2 1 2( (1 ) ) min( ( ), ( ))A A Ax x x x

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Mem

bers

hip

valu

e

height

1

0


A fuzzy set is completely characterized by a membership function.

a subjective measure. not a probability measure.

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“tall” in Asia

“tall” in USA

“tall” in NBA

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The degree of the fuzzy membership function µ (x) can be define as possibility function not probability function.

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What is The Different Between Probability Function and Possibility Function

Example:

When you and your best friend go to visit another friend, in the car your best friend asks you, “Are you sure our friend is at the home?"

you answer, "yes, I am sure but I do not know if he is in

the bed room or on the roof “ "I think 90% he is there"

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Look to the answer. In the first answer, you are sure he is in the house but you do not know where he is in the house exactly.

However, in the second answer you are not sure he may be there and may be not there. That is the different between possibility and the probability

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In the possibility function the element is in the set by certain degree,

Meanwhile the probability function means that the element may be in the set or not .

So if the probability of (x) = 0.7 that means (x) may be in the set by 70%. But in possibility if the possibility of (x) is 0.7 that means (x) is in the set and has degree 0.7.

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Fuzzy Membership FunctionsOne of the key issues in all fuzzy sets is how to determine fuzzy membership functionsThe membership function fully defines the fuzzy setA membership function provides a measure of the degree of similarity of an element to a fuzzy setMembership functions can take any form, but there are some common examples that appear in real applications

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Membership functions can either be chosen by the user arbitrarily, based

on the user’s experience (MF chosen by two users could be different depending upon their experiences, perspectives, etc.)

Or be designed using machine learning methods (e.g., artificial neural networks, genetic algorithms, etc.)

There are different shapes of membership functions; triangular, trapezoidal, piecewise-linear, Gaussian, bell-shaped, etc.

Fuzzy Membership Functions

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Types of MF

In the classical sets there is one type of membership function but in fuzzy sets there are many different types of membership function, now will show some of these types.

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Types of MF

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Different shapes of membership functions (a) s_function, (b) π_function, (c) z_function, (d-f) triangular (g-i) trapezoidal (J) flat π_function, (k) rectangle, (L) singleton

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MF Formulation

Triangular MF

Trapezoidal MF

Gaussian MF

Generalized bell MF

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( ; , , ) max min , ,0x a c x

x a b cb a c b

trimf

( ; , , , ) max min ,1, ,0x a d x

x a b c db a d c

trapmf

2

1( ; , , )

1bx a b cgbellmf

x c

b

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2( ; , , )x c

gau x a b c essmf

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MF Formulation

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Types of MFStandard types of membership functions: Z function; function; S function; trapezoidal function; triangular function; singleton.

Triangular membership functiona, b and c represent the x coordinates of the three vertices of µA(x) in a fuzzy set A (a: lower boundary and c: upper boundary where membership degree is zero, b: the centre where membership degree is 1)

cxif

cxbifbc

xc

bxaifab

ax

axif

xA

0

0

)(

a b c x

µA(x)1

0

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Gaussian membership function

c: centre s: width m: fuzzification factor (e.g., m=2)

µA(x)

m

A s

cxmscx

2

1exp),,,(

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

c=5

s=2

m=2

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Manipulating Parameter of theGeneralized Bell Function

2

1( ; , , )

1bx a b cgbellmf

x c

b

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Membership Functions

Degree of Truth or "Membership"Temp: {Freezing, Cool, Warm, Hot}

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50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

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Membership FunctionsHow cool is 36 F° ?

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Temp. (F°)


0

1

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Membership Functions

How cool is 36 F° ?It is 30% Cool and 70% Freezing

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Temp. (F°)


0

1

0.7

0.3

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A convex fuzzy set has a membership whose value is:

1. strictly monotonically increasing, or2. strictly monotonically decreasing, or3. strictly monotonically increasing, then

strictly monotonically decreasing

Or another way to describe:(y) ≥ min[(x), (z)], if x < y < z

If A and B are convex sets, then A B is also a convex set

Crossover points have membership 0.5

Height of a Fuzzy set is the maximum value of the membership: max{A(x)} 39Basil Hamed


If height < 1, the fuzzy set is subnormal.

Fuzzy number: like a number is close to 5. It has to have the properties:

1. A must be a normal fuzzy set.2. A must be closed for all (0,1].3. The support, 0A must be bounded.

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Fuzzy PartitionFuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:

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Approximate Reasoning

Linguistic VariablesWhen fuzzy sets are used to solve the problem without analyzing the system; but the expression of the concepts and the knowledge of it in human communication are needed. Human usually do not use mathematical expression but use the linguistic expression.

For example, if you see heavy box and you want to move it, you will say, "I want strong motor to move this box" we see that, we use strong expression to describe the force that we need to move the box. In fuzzy sets we do the same thing we use linguistic variables to describe the fuzzy sets.

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Linguistic VariablesLinguistic variable is “a variable whose values are words or sentences in a natural or artificial language”.

Each linguistic variable may be assigned one or more linguistic values, which are in turn connected to a numeric value through the mechanism of membership functions.

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Natural LanguageConsider: 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.

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Motivation

Conventional techniques for system analysis are intrinsically unsuited for dealing with systems based on human judgment, perception & emotion.

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Linguistic VariablesIn 1973, Professor Zadeh proposed the concept of linguistic or "fuzzy" variables. Think of them as linguistic objects or words, rather than numbers. The sensor input is a noun, e.g. "temperature", "displacement", "velocity", "flow", "pressure", etc. Since error is just the difference, it can be thought of the same way. The fuzzy variables themselves are adjectives that modify the variable (e.g. "large positive" error, "small positive" error ,"zero" error, "small negative" error, and "large negative" error). As a minimum, one could simply have "positive", "zero", and "negative" variables for each of the parameters. Additional ranges such as "very large" and "very small" could also be added

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Fuzzy Linguistic Variables

Fuzzy Linguistic Variables are used to represent qualities spanning a particular spectrumTemp: {Freezing, Cool, Warm, Hot}Membership FunctionQuestion: What is the temperature?Answer: It is warm.Question: How warm is it?

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Example

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if temperature is cold and oil is cheap

then heating is high

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Example

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if temperature is cold and oil is cheap

then heating is high

LinguisticVariable

LinguisticVariable

LinguisticVariable

LinguisticValue

LinguisticValue

LinguisticValue

cold cheaphigh

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Definition [Zadeh 1973]

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A linguistic variable is characterized by a quintuple

, ( ), , ,x T x U G MName

Term Set

Universe

Syntactic Rule

Semantic Rule Basil Hamed

Example

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A linguistic variable is characterized by a quintuple

, ( ), , ,x T x U G Mage

old, very old, not so old,

(age) more or less young,

quite young, very young

G

[0, 100]

old(old) , ( ) [0,100]M u u u

12old

0 [0,50]

( ) 501 [50,100]

5

u

u uu

Example semantic rule:

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Example

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Linguistic Variable : temperature

Linguistics Terms (Fuzzy Sets) : {cold, warm, hot}

(x)

cold warm hot

20 60

1

x

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Fuzzy If-Than Rules

IF (input1 is MFa) AND (input2 is MFb) AND…

AND (input n is MFn) THEN (output is MFc)

Where MFa, MFb, MFn, and MFc are the linguistic values of

the fuzzy membership functions that are in input 1,input 2…

input n, and output.

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Example

In a system where the inputs of the system are Serves and Food

and the output is the Tip.

Food may be (good, ok, bad), and serves can be (good, ok,

bad), the output tip can be (generous, average, cheap), where

good, ok, bad, generous, average, and cheap are the linguistic

variables of fuzzy membership function of the inputs (food, and

serves) and the output (tip).

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Example

We can write the rules such as.IF (Food is bad) and (Serves is bad) THEN (Tip is cheap)IF (Food is good) and (Serves is good) THEN (Tip is generous)

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Fuzzy If-Than Rules

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If x is A then y is B. antecedent

orpremise

consequenceor

conclusion

A B

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Fuzzy If-Than Rules

Premise 1 (fact): x is A,Premise 2 (rule): IF x is A THEN y is B, Consequence (conclusion): y is B.

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Examples

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If x is A then y is B. A B

If pressure is high, then volume is small.

If the road is slippery, then driving is dangerous.

If a tomato is red, then it is ripe.

If the speed is high, then apply the brake a little.Basil Hamed

Fuzzy Rules as Relations

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If x is A then y is B. A B

, ,R A Bx y x y

R

A fuzzy rule can be defined as a binary relation with MF

Depends on how to interpret A B

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Generally, there are two ways to interpret the fuzzy rule A → B. One way to interpret the implication A → B is that A is coupled with B, and in this case,

The other interpretation of implication A → B is that A entails B, and in this case it can be written as

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Interpretations of A B

A

B

A entails B

xx

y

A coupled with B

A

B

xx

y

, , ?R A Bx y x y

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EXAMPLE

Inputs: Temperature

Temp: {Freezing, Cool, Warm, Hot}

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50 70 90 1103010

Temp. (F°)


0

1

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Inputs: Temperature, Cloud Cover

Temp: {Freezing, Cool, Warm, Hot}

Cover: {Sunny, Partly, Overcast}

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Temp. (F°)


0

1

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1

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Output: Speed

Speed: {Slow, Fast}

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50 75 100250

Speed (mph)

Slow Fast

0

1

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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...

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Example Speed Calculation

How fast will I go if it is 65 F° 25 % Cloud Cover ?

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Calculate Input Membership Levels

65 F° Cool = 0.4, Warm= 0.7

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50 70 90 1103010

Temp. (F°)


0

1

0.4

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Calculate Input Membership Levels

25% Cover Sunny = 0.8, Cloudy = 0.2

0.8

0.2

70

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1

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25%

...Calculating...

If it's Sunny and Warm, drive FastSunny(Cover)Warm(Temp)Fast(Speed)

0.8 0.7 = 0.7 Fast = 0.7

If it's Cloudy and Cool, drive SlowCloudy(Cover)Cool(Temp)Slow(Speed)

0.2 0.4 = 0.2 Slow = 0.2

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Constructing the Output

Speed is 20% Slow and 70% Fast

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50 75 100250

Speed (mph)

Slow Fast

0

1

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Constructing the OutputSpeed is 20% Slow and 70% Fast

Find centroids: Location where membership is 100%

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50 75 100250

Speed (mph)

Slow Fast

0

1

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Constructing the OutputSpeed is 20% Slow and 70% Fast

Speed = weighted mean = (2*25+...

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50 75 100250

Speed (mph)

Slow Fast

0

1

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Constructing the Output

Speed is 20% Slow and 70% Fast

Speed = weighted mean = (2*25+7*75)/(9)= 63.8 mph

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50 75 100250

Speed (mph)

Slow Fast

0

1

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HW 2

Choose a project to present in this course Due 9/10/2011

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Fuzzy Control Lect 3 Membership Function and Approximate Reasoning Basil Hamed Electrical...

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