chapt 3 - fuzzy logic-2

8/12/2019 Chapt 3 - Fuzzy Logic-2

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INTELLIGENT CONTROL SYSTEM (ICS) Semester2 session 20132014


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Contents

Fuzzy Inference

Fuzzification of the input variables

Rule evaluation

Aggregation of the rule outputs

Defuzzification

Mamdani

Sugeno



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Fuzzy Inference

The most commonly used fuzzy inference technique is the so-

called Mamdanimethod.

In 1975, Professor Ebrahim Mamdani of London University built

one of the first fuzzy systems to control a steam engine and

boiler combination. He applied a set of fuzzy rules supplied by

experienced human operators.



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Mamdani Fuzzy Inference

The Mamdani-style fuzzy inference process is performed in

four steps:

1. Fuzzification of the input variables

2. Rule evaluation (inference)

3. Aggregation of the rule outputs (composition)

4. Defuzzification.



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Mamdani Fuzzy Inference

We examine a simple two-input one-output problem that includes

three rules:

Rule: 1 Rule: 1

IF x is A3 IF project_funding is adequate

OR y is B1 OR project_staffing is small

THEN z is C1 THEN risk is low

Rule: 2 Rule: 2

IF x is A2 IF project_funding is marginal

AND y is B2 AND project_staffing is large

THEN z is C2 THEN risk is normal

Rule: 3 Rule: 3

IF x is A1 IF project_funding is inadequate

THEN z is C3 THEN risk is high



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TABLE 2.1Mathematical Characterization of

Triangular Membership Functions



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TABLE 2.2Mathematical Characterization of

Gaussian Membership Functions



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Step 1: Fuzzification

The first step is to take the crisp inputs, x1 and y1 (project

fundingandproject staffing), and determine the degree to which

these inputs belong to each of the appropriate fuzzy sets.

Crisp Input

y1

0.1

0.7

1

0 y1

B1 B2

Y

Crisp Input

0.2

0.5

1

0

A1 A2 A3

x1

x1 X

(x=A1)= 0.5(x=A2)= 0.2

(y=B1)= 0.1(y=B2)= 0.7



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Step 2: Rule Evaluation

The second step is to take the fuzzified inputs, (x=A1)=0.5, (x=A2)= 0.2, (y=B1)= 0.1 and (y=B2)= 0.7, and apply them to

the antecedents of the fuzzy rules.

If a given fuzzy rule has multiple antecedents, the fuzzy operator

(AND or OR) is used to obtain a single number that representsthe result of the antecedent evaluation.

This number (the truth value) is then applied to the consequent

membership function.



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Step 2: Rule Evaluation (cont)

RECAL:

To evaluate the disjunction of the rule antecedents, we use the OR

fuzzy operation. Typically, fuzzy expert systems make use of the

classical fuzzy operation union:

AB(x) = max [A(x), B(x)]

Similarly, in order to evaluate the conjunction of the rule

antecedents, we apply the AND fuzzy operation intersection:

AB(x) = min [A(x), B(x)]



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A3

1

0 X

1

y10 Y

0.0

x1 0

0.1C1

1

C2

Z

1

0 X

0.2

0

0.2C1

1

C2

Z

A2

x1

Rule3:

A11

0 X 0

1

Zx1

THEN

C1 C2

1

y1

B2

0 Y

0.7

B10.1

C3

C3

C30.5 0.5

OR(max)

AND(min)

OR THENRule1:

AND THENRule2:

IFxisA3 (0.0) isB1 (0.1) z isC1 (0.1)

IFxisA2 (0.2) isB2 (0.7) z isC2 (0.2)

IFxisA1 (0.5) z isC3 (0.5)




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Now the result of the antecedent evaluation can be applied to

the membership function of the consequent.

There are two main methods for doing so:

Clipping Scaling




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The most common method of correlating the rule consequentwith the truth value of the rule antecedent is to cut theconsequent membership function at the level of the antecedenttruth. This method is called clipping(alpha-cut).

Since the top of the membership function is sliced, the clippedfuzzy set loses some information.

However, clipping is still often preferred because it involves lesscomplex and faster mathematics, and generates an aggregatedoutput surface that is easier to defuzzify.




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While clipping is a frequently used method, scalingoffers a

better approach for preserving the original shape of the fuzzy

set.

The original membership function of the rule consequent is

adjusted by multiplying all its membership degrees by the truth

value of the rule antecedent.

This method, which generally loses less information, can be

very useful in fuzzy expert systems.




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Degree ofembership

1.0

0.0

0.2

Z

Degree ofembership

Z

C2

1.0

0.0

0.2

C2



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Step 3: Aggregation of the rule outputs

Aggregation is the process of unification of the outputs of all

rules.

We take the membership functions of all rule consequents

previously clipped or scaled and combine them into a single

fuzzy set.

The input of the aggregation process is the list of clipped or

scaled consequent membership functions, and the output is one

fuzzy set for each output variable.



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0

0.1

1C1

Cis 1 (0.1)

C2

0

0.2

1

Cis 2 (0.2)0

0.5

1

Cis 3 (0.5)ZZZ

0.2

Z0

C30.5

0.1

Step 3: Aggregation of the rule outputs (cont)



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Step 4: Defuzzification

The last step in the fuzzy inference process is defuzzification.

Fuzziness helps us to evaluate the rules, but the final output of a

fuzzy system has to be a crisp number.

The input for the defuzzification process is the aggregate output

fuzzy set and the output is a single number.



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There are several defuzzification methods, but probably themost popular one is the centroid technique. It finds the point

where a vertical line would slice the aggregate set into two equal

masses. Mathematically this centre of gravity(COG) can be

expressed as:

Step 4: Defuzzification (cont)

b

a

A

b

a

A

dxx

dxxx

COG



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Centroid defuzzification method finds a point representing thecentre of gravity of the fuzzy set,A, on the interval, ab.

A reasonable estimate can be obtained by calculating it over a

sample of points.

(x)1.0

0.0

0.2

0.4

0.6

0.8

160 170 180 190 200

a b

210

A

150

X



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1.0

0.0

0.2

0.4

0.6

0.8

0 20 30 40 5010 70 80 90 10060

Z

Degree ofembership

67.4

4.675.05.05.05.02.02.02.02.01.01.01.0

5.0)100908070(2.0)60504030(1.0)20100(

COG



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Sugeno Fuzzy Inference

Mamdani-style inference, as we have just seen, requires us tofind the centroid of a two-dimensional shape by integrating

across a continuously varying function. In general, this process

is not computationally efficient.

Michio Sugeno suggested to use a single spike, a singleton, as

the membership function of the rule consequent.

A singleton, or more precisely a fuzzy singleton, is a fuzzy set

with a membership function that is unity at a single particularpoint on the universe of discourse and zero everywhere else.



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Sugeno-style fuzzy inference is very similar to the Mamdanimethod. Sugeno changed only a rule consequent. Instead of afuzzy set, he used a mathematical function of the input variable.The format of the Sugeno-style fuzzy ruleis

IF xisAAND yis B

THEN zis f(x, y)

wherex, yand zare linguistic variables;Aand Bare fuzzy sets

on universe of discoursesXand Y, respectively; and f(x, y) is amathematical function.

Sugeno Fuzzy Inference (cont)



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The most commonly used zero-order Sugeno fuzzy modelapplies fuzzy rules in the following form:

IF xisA

AND yis B

THEN zis k

where kis a constant.

In this case, the output of each fuzzy rule is constant. All

consequent membership functions are represented by singletonspikes.

Sugeno Fuzzy Inference (cont)



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Sugeno Rule Evaluation

A3

1

0 X

1

10 Y

0.0

x1 0

0.1

1

Z

1

0 X

0.2

0

0.2

1

Z

A2

x1

IFxisA1 (0.5) zis k3 (0.5)Rule3:

A11

0 X 0

1

Zx1

THEN

1

y1

B2

0 Y

0.7

B1

0.1

0.5 0.5

OR(max)

AND(min)

OR isB1 (0.1) THEN zis k1 (0.1)ule1:

IFxisA2 (0.2) AND isB2 (0.7) THEN zis k2 (0.2)Rule2:

k1

k2

k3

IFxisA3 (0.0)

A3

1

0 X

1

10 Y

0.0

x1 0

0.1

1

Z

1

0 X

0.2

0

0.2

1

Z

A2

x1

IFxisA1 (0.5) zis k3 (0.5)Rule3:

A11

0 X 0

1

Zx1

THEN

1

y1

B2

0 Y

0.7

B1

0.1

0.5 0.5

OR(max)

AND(min)

OR isB1 (0.1) THEN zis k1 (0.1)ule1:

IFxisA2 (0.2) AND isB2 (0.7) THEN zis k2 (0.2)Rule2:

k1

k2

k3

IFxisA3 (0.0)



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Sugeno Aggregation of the Rule Outputs

zis k1 (0.1) zis k2 (0.2) zis k3 (0.5)

0

1

0.1

Z 0

0.5

1

Z0

0.2

1

Zk1 k2 k3 0

1

0.1

Zk1 k2 k3

0.20.5



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Sugeno Defuzzification

Weighted Average (WA)

655.02.01.0

805.0502.0201.0

)3()2()1(

3)3(2)2(1)1(

kkk

kkkkkkWA

0 Z

Crisp Output

z1

z1



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Mamdani or Sugeno?

Mamdani method is widely accepted for capturing expertknowledge. It allows us to describe the expertise in more

intuitive, more human-like manner. However, Mamdani-type

fuzzy inference entails a substantial computational burden.

On the other hand, Sugeno method is computationally effective

and works well with optimisation and adaptive techniques, which

makes it very attractive in control problems, particularly for

dynamic nonlinear systems.



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Takagi-Sugeno Fuzzy Systems

The fuzzy system defined in the previous sections will be referred to as a

standard fuzzy system.

In this section we will define a functional fuzzy system, of which the Takagi-

Sugeno fuzzy system is a special case.

For the functional fuzzy system, that uses singleton fuzzification, and the ith

MISO rule has the form

where simply represents the argument of the function giand the bi are

not output membership function centers. The premise of this rule is defined

the same as it is for the MISO rule for the standard fuzzy system



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chooseomay want tinstance,forpossible,areothersmanybutfunctionsaffineandlineardiscusswillweBelow,

.consideredbeingnapplicatioon thedependsfunctiontheofchoiceThe

used.bealsomayvariables

otherbut21,termsthecontainsofargumentoften thethatNotice

function.membership

associatedanhavenotdoesthatsystem)fuzzylfunctionanamethe(hence

()functionauseweconsequentin thefunction,membershipassociatedanwith

termlinguisticaofInsteadhowever.different,arerulestheofsconsequentThe

,, . . ., n,iug

gb

ii

ii



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For the functional fuzzy system, we can use an appropriate operation for

representing the premise (e.g., minimum or product), and defuzzification

may be obtained using

whereiis defined in Equation bellow:

It is assumed that the functional fuzzy system is defined so that no matter

what its inputs are, we have

One way to view the functional fuzzy system is as a nonlinear interpolatorbetween the mappings that are defined by the functions in the

consequents of the rules.



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An Interpolator Between Linear Mappings

In the case where

mappings.linearbetweenioninterpolatnonlineara

performsyessentiallsystemfuzzySugeno-Takagithat theseeweOverall,

e.conveniencformapping

linearaasmappingaffinetherefer towillwestandard,isashowever,Often,

affine.calledismapping

then the,0ifandmappinglinearaismapping()then the,0If

system.fuzzySugeno-Takagia

astoreferredissystemfuzzyfunctionalthenumbersrealarethewhere

00 i,ii,

i,j

aga

a



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As an example, suppose that n = 1, R = 2, and that we have rules

havewe,~

representsand~

representsthatso2.5Figureingivenfordiscourseofuniversewith the

2

12

1

1

11

AA

u

For u1> 1,1= 0, so y = 1+u1, which is a line. If u1< 1,2= 0, so y = 2+u1,

which is a different line.

Between 1 u1 1, the output y is an interpolation between the two lines.



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Figure 2.5Membership functions for Takagi-Sugeno fuzzy system example.

Finally, it is interesting to note that if we pick

(i.e., ai,j = 0 forj > 0), then the Takagi-Sugeno fuzzy system is equivalent to

a standard fuzzy system that uses center-average defuzzification with

singleton output membership functions at ai,0.

It is in this sense that the Takagi-Sugeno fuzzy systemor, more generally,

the functional fuzzy systemis sometimes referred to as a general fuzzy

system.



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An Interpolator Between Linear Systems

It is important to note that a Takagi-Sugeno fuzzy system may have any linear

mapping (affine mapping) as its output function, which also contributes to its

generality.

One mapping that has proven to be particularly useful is to have a linear

dynamic system as the output function so that the ithrule has the form

system.fuzzytheinput toldimensiona

-theis)](.,..),(),([)(anddimension,eappropriatofmatricesinputandstatetheare21,andinput,modelldimensiona

-theis)](.,..),(),([)(inputs,ofnumberthey)necessaril

notis(nowstateldimensiona-theis)](.,..),(),([)(Here,

T

21

T

21

T

21

ptztztztz, . . ., R,iBA

mtutututu

nntxtxtxtx

p

ii

m

n



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This fuzzy system can be thought of as a nonlinear interpolator between

R linear systems, it takes the input z(t) and has an output

If R = 1, we get a standard linear system.



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ruleswith2and,1),()(thatsupposeexample,anAs

)].(),([)(sometimesor),()(chooseinstance,for),(forpossiblearechoicesMany

output.thetocontributeand

onturnwillrulescertainonly),(ofegiven valuaand1forGenerally,

Rmnptxtz

tutxtztxtztz

tzR

).with

2.5Figureofaxishorizontaltherelabelwe(i.e.,lyrespective,~and~for

functionsmembershiptheas2.5Figurefromandusethat weSuppose

1

2

1

1

1

21

x

AA



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Takagi-Sugeno fuzzy system interpolates between the two linear systems.

We see that for changing values ofx1(t), the two linear systems that are in the

consequents of the rules contribute different amounts.

We think of one linear system being valid on a region of the state space that

is quantified via1and another on the region quantified by2(with a fuzzy

boundary in between).

For the higher-dimensional case, we have premise membership functions for

each rule quantify whether the linear system in the consequent is valid for aspecific region on the state space.

As the state evolves, different rules turn on, indicating that other combinations

of linear models should be used.

Overall, we find that the Takagi-Sugeno fuzzy system provides a very intuitiverepresentation of a nonlinear system as a nonlinear interpolation between R

linear models.



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Figure 2.6: Commonly used fuzzy if-then rules and fuzzy mechanism.


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Lets consider the nonlinear system below:

The goal is to derive a T-S fuzzy model from the above given nonlinear system

equations by the sector nonlinearity approach as if the response of the T-S

fuzzy model in the specified domain exactly match with the response of the

original system with the same input u.

The following steps should be taken to derive the T-S fuzzy model of aboveequation. For simplicity, we assume thatx1 [0.5, 3.5] and x2 [-1, 4]. Here x1

and x2 are nonlinear terms in the equations in the last equations so we make

them as our fuzzy variables.

Generally they are denoted as z1, z2 and are known as premise variables that

may be functions of state variables, input variables, external disturbances

and/or time. Therefore z1 = x1 and z2 = x2. Equation above can be written as


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wherex(t) = [x1(t) x2(t)]T. The first step for any kind of fuzzy modeling is to

determine the fuzzy variables and fuzzy sets or so-called membership

functions. Although there is no general procedure for this step and it can be

done by various methods predominantly trial and error, in exact fuzzy modeling

using sector nonlinearity, it is quite routine. It is assumed that the premise

variables are just functions of the state variables for the sake of simplicity. Thisassumption is needed to avoid a complicated defuzzification process of the

fuzzy controllers [9].

To acquire membership functions, we should calculate the minimum and

maximum values of z1(t) and z2(t) which under x1 [0.5, 3.5] and x2 [-1. 4],

they are obviously obtained as follows:

max z1(t) = 3.5; min z1(t) = 0.5;max z2(t) = 4; min z2(t) = -1:


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Figure 2.7: Membership functions M1(z1(t)), M2(z2(t)), N1(z2(t)) and N2(z2(t)).

We name the membership functions "Positive", "Negative," "Big," and "Small,"

respectively. Figure 2.7 shows these membership functions.


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Here, we can generalize that the ithrule of the continuous T-S fuzzy models

are of the following forms:

Model Rule i:

IF z1(t) is Mi1 and and zp(t) is Mip,

Here, Mij is the fuzzy set and r is the number of model rules; x(t) is the state

vector, u(t) is the input vector, y(t) is the output vector, Ai is the square matrix

with real elements and z1(t),..,zp(t) are known premise variables asmentioned before. Each linear consequent equation represented byAix(t) +

Biu(t) is called a subsystem.


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Therefore, the nonlinear system equation previous is modeled by the

following fuzzy rules (we don't consider input u(t) in this stage):


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Therefore, if z1 = x1 = 2.75 and z2 = x2 = 0.25, the T-S fuzzy modeling

implication can be derived as:

Figure 2.8: Given the value of z1 = 2.75 and z2 = 0.25 to the membership

functions


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chapt 3 - fuzzy logic-2

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