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103

4 Fuzzy Inference Systems

4.1 Fuzzy SySteMS

The concepts o sot computing that we introduced in Chapter 1 orm the guiding principles o

uzzy logic. With uzzy logic, the system is made to give the most probable output to any kind o

input based on the predened rules. Fuzzy systems are a modied orm o rule-based approach, in

which rules are applied to nd the output o any input. In place o strict rules applied over the classes

or sets, however, the rules are much soter in the manner in which they are applied. This not only

gives a scope or the unction to perorm well in the presence o uncertainties and noise, but it also

makes it possible to obtain more realistic systems that imitate natural behavior. Fuzzy systems are

used in numerous real lie industrial applications, including biomedical engineering and roboticcontrol. The success o uzzy systems in such varied domains clearly speaks to the eectiveness o

these systems, which orm an integral part o sot computing. Fuzzy logic is the natural choice when

modeling systems that have predened rules governing their behavior.

Other interesting use o uzzy systems is in classication and pattern recognition problems,

where they are able to easily determine the output class that the input corresponds to. The uzzy

nature o these systems serves as an instrumental tool in nding the output class by a set o rules

that may be ramed by looking at the training data.

Fuzzy systems get their name rom the uncertainty or probability they associate with the vari-

ous stages o unctioning as they calculate the outputs rom the applied inputs. The rules governing

the behavior o the uzzy system are based on the classes o inputs. These rules simply denote that

certain types o inputs have a certain types o output. With uzzy systems, we study the inputs and

outputs in groups or classes. The novel concept behind uzzy systems is that the the input belongs

to a class by a certain degree or a certain probability. This concept is urther used to work over

the rules to come up with a certain answer to the problem. The probability-based association o

the uzzy systems allows them to imitate various systems that could not have been built using

traditional approaches. These systems hence act as a boon in the implementation o the rule-based

approach by sot-computing techniques.

Fuzzy systems are entirely rule driven. Mapping o the inputs and outputs is accomplished by the

rules, which are specied during the design. The dierent rules aect the output in their own way.

In other words, the rules try to nd the output according to their own understanding o the system

or which they are made. The nal result is the output calculated by the combined eect o all therules and is given as output o the system. It is very likely that this output was not the result o any

o the rules; rather it is the result o all the rules being put together.

Just like any system, the uzzy system maps the inputs to the outputs. This mapping is derived

rom various rules that are uzzy in their implementation. The rules are written in the orm o nor-

mal English rules, which can easily be ramed ater a study o the system. As we have explained,

the output is the combined eect o all these rules put together. Note that the dierent rules do not

behave in similar manner to one another. Some rules may result in a high output, while others may

result in a low output. The aggregation o the output predicted by all these rules computes the nal

answer to the unknown input that was given to the system.

In the subsequent sections o this chapter, we discuss the various issues and concepts o the uzzysystem, including its design and its usage in real lie applications.

© 2010 by Taylor and Francis Group, LLC

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104 Real Life Applications of Soft Computing

4.2 hiStoriCal note

The history o uzzy logic goes back to the days o Aristotle and the binary logic representing

true and alse, which began the development o logic in the history o humankind. Multilogic also

evolved about the same time, but not to a very good extent. It was not until 1964 that Loti Zadeh

introduced the concept o uzzy logic, when he introduced a ormal method o dealing with andproblem solving with uzzy sets. The eld attracted the attention o numerous researchers worldwide

and initiated a great deal o work in this eld. Fuzzy logic then joined the application domain, where

it has been used in numerous systems and consumer applications, including washing machines,

camcorders, and microwave ovens, to name just a ew.

4.3 Fuzzy logiC

In this section, we introduce the concept o logic and, hence, uzzy logic. We discussed logic in

Chapter 1, where we saw how logic is used or problem solving. In this section, we give an in-depth

analysis o the same and then move to a discussion o uzzy logic.

4.3.1 logic

Every mapping o the inputs to the outputs is done using a set o guidelines, or unctions, that are

the inherent properties o the system being considered. This mapping orms the basis o logic. We

must gure out the knowledge that is available in the system and then determine how to store it in a

usable manner. We try to represent the system by a set o rules or in a way that can be easily under-

stood and implemented by the machine. Knowledge is o deep interest to system developers, as it

provides a means or the machine to understand, act, and make decisions and inerences based on

the common understanding o the general people. This knowledge removes the gap between human

and machine understanding.

Recall rom Chapter 1 the denition o logic: “What a program knows about the world in general,

the acts o the specic situation in which it must act, and its goals are all represented by sentences

o some mathematical logical language. The program decides what to do by inerring that certain

actions are appropriate or achieving its goals” (McCarthy, 2007). We also dened knowledge as “a

unction that maps a domain o clauses onto a range o clauses. The unction may take algebraic or

relational orm depending on the type o applications” (Konar, 2000).

Logic is used to make machines intelligent and to empower them with the ability to make deci-

sions. Logic makes it possible or machines to take input and act in the desired manner. Machines

are able to do this because they ollow a set rules that denote the knowledge assembled or repre-

sented in the system.The rules are simple i-then clauses. The if part, also called the antecedent , denotes the condi-

tion that must be true or the particular rule to re. The condition expresses the particular case in

which the action would hold true. The then part, known as the consequent , consists o the action

or the conclusion that results rom the rule being red. The entire set o knowledge is mapped

onto this rule set. The systems then use these rules or all operations. Once these rules are ready,

we know that all available inormation has been incorporated into the system in the rules. These

systems also have a memory associated with them called the working memory, which stores all

the inormation regarding the state o the system. Based on the state, the rules are red by a rule

implementer.

Consider the rule:

I ( X marks are more than 80) & ( X attendance is more than 75%), then ( X grade is A).

Here the if part states all the conditions that i true lead to the action.


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Fuzzy Inference Systems 105

4.3.2 proBlemS With nonfuzzY logic

Now rom the concept o logic, we move to the concept o uzzy logic. It is clear rom the example

above that any rules speciy certain conditions or antecedents or the corresponding actions or con-

sequents to be activated. This means that either the action will take place or it will not. I the condi-

tion were the set o conditions joined by logical operators, then the same concept holds true. Once

again, the various conditions are evaluated using the state o the system and are worked using the

logical operators. This decides whether the nal condition will be true or alse. I the nal condition

is true, the corresponding action is activated. Thus, in the above example, both the statements must

be true or the system to perorm the action.

In the real world, however, this might not give a very realistic picture o the entire system.

Consider this example:

I (driver experience is high) & (road is bad), then (accident risk is moderate).

I (driver experience is low) & (road is bad), then (accident risk is high).

I (road is good), then (accident risk is low).

In this problem, we have a system that is trying to nd the risk o accident by taking the inputs o

driver experience and road condition. But driver experience, road, and accident have been dened in

quite abstract terms. Suppose that driver experience o more than or equal to 5 years is high and less

than that is low. Further suppose that road condition is measured by a road index that lies between

0 and 1. A bad road means an index o less than or equal to 0.4, while a good road means an index

larger than 0.4. Further let us suppose that accident is measured as a probability. A high accident

means a probability o 0.7. Moderate means a probability o 0.4. Low means a probability o 0.2.

This is summarized in Figures 4.1a and 4.1b. Figure 4.1a depicts the two levels o experience as low

and high, while Figure 4.1b depicts the two levels o road as bad and good .

0 1 2 3 4 5 6 7 8 9 10

M e

m b e r s h i p V a l u e

Experience

Figure 4.1 The variable experience or the accident risk problem.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p V a l u e

Road

Figure 4.1b The variable road or the accident risk problem.


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Working with these rules, it may easily be seen that or a person with 5 years o experience driv-

ing on a road with road index 0.5, the accident probability is moderate (or 0.4). This sounds reason-

able. But suppose that under the same conditions, we have a driver with an experience o 1 day less

than 5 years. In such a case, the accident probability suddenly becomes high (or 0.7). Thus with a

decrease o just 1 day o experience, we see a sudden increase in the probability o an accident. This

is the unrealistic nature that uzzy systems are good at modeling. The output or the various inputs is

given in Figure 4.2, where the three levels o accident are represented by three dierent symbols.

The unrealistic nature o the above system can be better solved by using uzzy logic, as we shall

see later in this chapter.

Let us consider another model o solving the same problem using a nonuzzy system. This time

we use simple mathematical unctions to map the output to the inputs. This system o problem solv-ing is commonly known as a human logic system. The system discussed here is its very basic version

and is given by Equations 4.1 through 4.3.

accident total = accident experience + accident road (4.1)

where accident total is the total probability o an accident (output), accident experience is the accident

probability due to experience, and accident road is the accident probability due to road conditions.

accident experience = 0, i experience ≥ 10 (4.2)

= (1 – experience /10)/2, otherwise

accident road = (1 – road )/2 (4.3)

Analyzing this system, we can easily see that i the driver does not know how to drive (experi-

ence = 0), the probability o an accident or a very good road (road = 1) is 0.5. This means there

is only a 50 percent chance that there will be an accident. In reality, this would be more than 90

percent, because whenever you give a car to a new driver, that person is always accompanied by an

experienced driver because accidents are very likely. The same is also true in the case that the road

is very bad. The surace o this unction is given by Figure 4.3.

These two problem-solving methods are used together in nonuzzy systems. Even these systemsnd interesting applications and can be adapted well or modeling problems. However we do not

study these nonuzz systems in this text. Simply by looking at these two examples, we can easily

see that nonuzzy systems have problems. In the rest o this chapter, we will see how uzzy systems

can solve these problems with better system design.

0

1

2

3

45

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

E x p e

r i e n c e

Road

Accident

Risk

Low

Moderate

High

Figure 4.2 The output o the nonuzzy system or the accident risk problem.


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4.3.3 fuzzY logic

In this section, we introduce the concept o uzzy logic and learn how it is dierent rom normal

logic. This discussion orms the basis o our coverage o uzzy inerence systems, which are entire

systems that use uzziness to map inputs to outputs.

In our earlier example, we saw that every input belongs to one and only one class—or example,

experience could be either high or low. This restriction is the basic reason or the problems encoun-

tered in nonuzzy systems. In uzzy logic, however, this restriction is changed or generalized.

In uzzy logic, every input belongs to every class. The degree o association o the input to the

various classes varies. In other words, the input belongs to the dierent classes by dierent degrees

o associations. This association may be very strong to some class but weak or other classes, or

the association may be moderate or all classes. Hence in uzzy logic, we would never say that the

input i is high, low, moderate, and so on. Rather we would say that the input i is high to some extent,

moderate to some other extent, and so orth. The higher the degree o association o the input to

some class, the more characteristics o that particular class it implements.

In our example o the road, we nd that under uzzy logic, experience can be high and low at the

same time. Thus the driver’s experience may be high to the extent o 80 percent and low to the extent

o 10 percent. This means the driver’s behavior closely ollows the behavior o experienced drivers, but

the 10 percent association indicates that this behavior to some extent ollows the behavior o inexperi-

enced drivers. Hence when we apply the rules using the specialized operators that we study next, the

output is the aggregation o both eects. This gives good results when applied over real lie cases.

4.3.4 When not to uSe fuzzY

A uzzy approach is not the best approach or all types o problems; thereore we need to study the

problem to be solved beore applying uzzy approach. In this section, we discuss some types o

problems or which uzzy logic should not be applied.

Suppose we have identied the system. We know the inputs and outputs, but we do not have a

clear idea o the rules that map the inputs to outputs. In this instance, uzzy logic may not be the

best approach. Fuzzy logic ollows simple English rules that must be known or a system to have an

eective design. In the absence o these rules, the perormance might be poor, or we may have toapply many eorts to study the patterns o inputs and outputs in search o the rules.

Consider the natural systems o physics, in which the bounding equations are well known and

perorm well. In this situation, uzzy systems may not perorm as well as the already established

mathematical equations. Say the situation is that o a car moving at speed v and acceleration a.

1098760

1.0

0.8

0.6

0.4

0.2

00

0.2

0.4

A c c i d e n t

0.6

0.8

1.0

1 2 3 4

E xperience

R o a d

5

Figure 4.3 The surace plot o accident risk in the second example.


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Further suppose that the condition is ideal, which means there is no external orce, riction, and so

orth on the car. I we are to nd the speed at time t , it would be better to apply the standard math-

ematical equation rather than uzzy logic. Although in the same problem, i we introduce additional

constraints that mathematics nds it very dicult to cater, the problem may become uzzy.

4.3.5 fuzzY SetS

We have already discussed the concept o degree o membership and various classes in terms o

uzzy logic. We now ormalize the same concept that orms the basis o uzzy sets. According to

the theory o mathematics, sets are collections. In road example, we may regard experience to be a

collection o all possible experiences (in years) that the driver has. This may be any value greater

than or equal to 0 and may be represented by Equation 4.4.

experience = { z: z ≥ 0} (4.4)

In a uzzy approach, we represent each element o a set with a certain probability. This is shown

as a / b, where a denotes the element o the set and b denotes the degree o membership o a in theset. Consider the set o high experience, which would be given by Equation 4.5:

highexperience = { z / μ( z) : z ≥ 0} (4.5)

where we assume that the degree o membership o z in the set is given by the unction μ( z). Thus

it is natural that the degree o membership will increase as z increases, because as experience

increases, the driver will more closely ollow the characteristics o an experienced driver.

4.4 MeMBerShiP FunCtionS

In the previous section, we talked about the degree o membership, which denotes the belonging-ness o any value to any input. Every element is denoted with a certain degree o association that is

given by a unction known as the membership unction (MF).

The MF takes as a single input the element whose membership needs to be ound and returns the

membership degree o that input. The unction may be denoted by μ( z), where z is the element.

Any input may have one or more membership unctions associated with it. In our road example,

we have the input experience associated with two MFs—high and low. There are no set guidelines

as to how many MFs make an ideal system; the choice usually lies with the designer’s implementa-

tion. Having an idea o the rules or how the system works may play an important role in deciding

the number o MFs.

The member unctions or the two classes o experience—low and high—are given in Figures4.4a and 4.4b, respectively.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Road

Figure 4.4 The membership unction low or road in the accident risk problem.


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It can clearly be seen rom Figure 4.4a that as long as experience is low, membership is high. This

means that the lower experience values have a high similarity with the class low, with the highest

being when experience is 0. This situation is desirable, because i we make any rule or the class o low-experienced drivers, these people would be more likely to exhibit the properties o the rule. As

we keep increasing experience, the membership value keeps decreasing and ultimately reaches 0 at

an experience o 10 years. The converse would be true or Figure 4.4b.

MFs are dened by the system designer according to the problem. Normally designers preer to

use standard membership unctions, which have been used in numerous problems. We now discuss

a ew o these membership unctions.

4.4.1 gauSSian memBerShip functionS

The Gaussian MF depicts the Gaussian curve, given in Figure 4.5a. This widely used membership

unction denotes either a sharp Gaussian decrease or a sharp Gaussian increase in the membership

value. The Gaussian MF is given by Equation 4.6:

f x c e

x c

( , , )( )

σ σ =− − 2

22 (4.6)

where c and σ are parameters that may be adjusted to control the behavior o the unction, x is the

given input, and c is the input or which the membership value is maximum (or 1) or the center o

the curve.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M

e m b e r s h i p

Road

Figure 4.4b The membership unction high or road in the accident risk problem.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Input

Figure 4.5 Standard membership unctions: Gaussian.


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4.4.2 triangular memBerShip function

This unction, which denotes a straight-line decrease or increase in the membership value, is used in

situations where there is a simple linear degradation or upgradation o the membership value. The

curve o this unction is shown in Figure 4.5b and in Equation 4.7.

f x a b c

x a

x a

b aa x b

c x

c bb x c

( , , , ) =

≤−

−≤ ≤

−

−≤ ≤

0

0

i

i

i

ii c x≤

(4.7)

where a, b, and c are parameters such that a ≤ b ≤ c. The membership value is 0 until it reaches

point a. From that point, the membership value starts increasing and touches a maximum o 1 whenit is at point b. It then starts decreasing until it reaches 0 at point c. From c onward the membership

value is 0.

4.4.3 Sigmoidal memBerShip function

The sigmoidal MF, which depicts the sigmoidal unction, is given by Figure 4.5c and by Equation

4.8.

f x a ce a x c

( , , )( )

=+ − −

1

1(4.8)

where a and c are parameters.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M

e m b e r s h i p

Input

Figure 4.5b Standard membership unctions: Triangular.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Input

Figure 4.5c Standard membership unctions: Sigmoidal.


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4.4.4 other memBerShip functionS

Other standard membership unctions are summarized in Table 4.1.

The motivation behind the use o uzzy sets is that we must be able to implement a traditional

rule-based approach. Using MF, we can determine the degree o association o any element to any

o the classes o inputs or outputs. This empowers us to replace all conditions o the orm “i a is i,

taBle 4.1Cmm us Mmbsp Fcs

S. n. nm eq gp

1. Generalized bell-shaped f x a b c x c

a

b( , , , ) =

+−

1

12

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2. Gaussian combination f x c e x c

( , , )( )

σ σ =− − 2

22

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3. Dierence sigmoidal

f x a c a c

e ea x c a x c

( , , , ),

( )

1 1 2 2

1

1

1

11 1 2 2=

+−

+− −( ) − −

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4. Product sigmoidal

f x a c a c

e ea x c a x c

( , , , , )

*( ) ( )

1 1 2 2

1

1

1

11 1 2 2=

+ +− − − −

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5. S-shaped

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(continued )


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then b is j” with their membership values. We concentrate only on the antecedents here. Suppose

the condition reads, “I driver is highly experienced.” We can easily replace this condition with the

membership value given by the membership unction o highexperience. In the later sections o this

chapter, we will replace other operations o the rule-based approach until we have the entire uzzy-

based inerence engine ready.

4.5 Fuzzy logiCal oPeratorS

In this section, we study the uzzy way o dealing with logical operators. Any condition in a rule-

based approach may carry a number o logical operators. These operators must be evaluated to get

the value o the entire condition. Consider the condition given by Equation 4.9. Here the various

conditions are joined using the logical operators AND and OR. Any operator may also be appliedwith a unary operator NOT .

c = ( x1 AND x2) AND NOT ( x3 OR x4) (4.9)

where x1, x2, and x3 represent the various conditions.

The various operators according to the rules o Boolean algebra ollow the precedence order

NOT , AND, OR, with NOT having the highest precedence.

Any condition may ultimately be represented using the generalized orm given in Equation 4.10:

c =

[NOT] x1 op [NOT] x2 op [NOT] x3 op x4 ….. op [NOT] xn (4.10)

where x1, x2, x3, . . ., xn represent the various conditions, each condition is o the orm yi = f j, f j is the

membership unction, yi is the variable, [NOT] means that its presence is optional, and op stands

or AND/OR.

taBle 4.1 (Continued)Cmm us Mmbsp Fcs

S. n. nm eq gp

6. Z-shaped

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

7. Pi-shaped f ( x, a, b, c, d ) = S-shaped( x, a, b)

* Z-shaped( x, c, d )

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


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We saw in the previous section that in uzzy systems, each condition x1, x2, x3, …, xn denotes some

value. This value is the degree o association o the variable to the particular class and is given by

the governing membership unction.

In this section, we model how the logical operator handles uzzy arithmetic. Each operator takes

two membership values (or just one in the case o NOT) and returns as a result the membership value

according to the operator’s logic. Various conditions joined by logical operators may be handled ina manner similar to how we handled the logical operators in Boolean algebra. Similar to boolean

operators, uzzy operators have the rule o precedence, associative law, commutative law, etc.

4.5.1 and operator

AND is a binary operator that takes two inputs and returns a single output. It may be represented

by Equation 4.11.

c = x AND y (4.11)

In Boolean algebra, the unctioning o AND is given by Table 4.2. The output is true (or 1) only

i both o its inputs are 1; otherwise it is 0. In a logical sense, this means that the operator returns a

high only when the rst and the second inputs are high.

The uzzy AND does not have as its inputs 0 or 1. Instead it has a continuous range o values

rom 0 to 1. In uzzy systems, we usually take the AND operator as the min or product , both o

which have their conventional meaning and are represented by Equations 4.12 and 4.13:

c = min { x, y} (or a min system) (4.12)

c = x * y (or a product system) (4.13)

Observe that in both the cases the inputs and outputs are bounded between 0 and 1 and that the

system ollows the outputs o the Boolean algebra system when given Boolean inputs. We take two

sample graphs or the variables x and y. The resultant graph generated by the uzzy AND opera-

tor using both the min method and the product method are given in Figure 4.6a, while Figure 4.6b

shows their binary equivalents.

4.5.1.1 r f M Pc

It may be interesting to observe the behavior o the AND operators in the inputs given in Figure 4.6.

Looking at Figures 4.6a and 4.6b, we may easily observe a very strong correlation between the

uzzy and the nonuzzy counterparts.Consider the system with which we are supposed to nd the vulnerability o intrusion at some

location. We know that or this intrusion, an intruder must break two security doors, one ater the

other. Ater that, the intruder may exploit the system. A simple uzzy rule might say, “I (door 1

taBle 4.2t tb f and

x c = x and y

0 0 00 1 0

1 0 0

1 1 1


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0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m

b e r s h i p V a l u e

x

0

0.20.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e

r s h i p V a l u e

y

0

0.20.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


x AND y by min

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m


x AND y by product

Figure 4.6 The AND logical operator in uzzy arithmetic.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


x

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


y

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


x AND y

Figure 4.6b The AND logical operator in Boolean (nonuzzy) arithmetic.


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security is poor) AND (door 2 security is poor), then (intrusion is high).” The AND may be resolved

by the use o a product operator. A mathematical counterpart may also suggest that the same prob-

lem may be solved by the use o probability, where the probability (P) o intrusion is given by

Equation 4.14. This sounds very similar to the use o product as the AND operator.

P(intrusion) = P(door 1 is passed) * P(door 2 is passed) (4.14)

Consider another example. Suppose you are traveling on a road and need to measure the comort

o traveling. The comort depends on the road condition and the vehicle condition. The general

rule may be ramed as, “I (vehicle condition is bad) AND (road condition is bad), then (comfort is

poor).” In this system, it may easily be observed that i the road has too many curves and trac, no

matter how good the vehicle is, the drive would not be comortable. In addition, i the vehicle is in

very bad shape, the drive would not be comortable. In such a case, it may be seen that the comort

behaves as the minimum o the two actors. We assume that the comort is measured by asking the

person traveling, who gets dissatised when either o the conditions is bad and thus reports the drive

uncomortable. We urther assume that i the person is traveling on a dirt road, it would not makeany dierence whether he travels by a very expensive car or a normal car, since he would not enjoy

the drive in any case.

4.5.2 or operator

OR is another binary operator that takes two inputs and returns a single output. It may be repre-

sented by Equation 4.15. In Boolean algebra, the unctioning o OR is given by Table 4.3. The output

is true (or 1) i any o its inputs are true (or 1); otherwise it is 0.

c = x OR y (4.15)

In uzzy systems, we take the OR operator as the max or the probabilistic or . Both have their

conventional meaning and are represented by Equations 4.16 and 4.17.

c = max { x, y} (or a max system) (4.16)

c = x + y – x * y (or a probabilistic OR system) (4.17)

Again observe that in both cases, the inputs and outputs are bounded between 0 and 1 and that

the system ollows the outputs o Boolean algebra when given Boolean inputs. The graphs or the

OR operator are given in Figure 4.7a, while Figure 4.7b shows their binary equivalents.

4.5.2.1 r f M

Consider the same vulnerability analysis system in which we measure the intrusion risk. Consider

that the same two doors are not sequential this time, but parallel. In this case, the intruder may break

taBle 4.3t tb f or

x c = x or y

0 0 00 1 0

1 0 0

1 1 1


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0

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0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m


x

0

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0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b

e r s h i p V a l u e

y

00.2

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0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


x OR y by max

0

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0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M

e m b e r s h i p V a l u e

x OR y by probabilistic or

Figure 4.7 The OR logical operator in uzzy arithmetic.

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


x

0

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0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


y

0

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0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


x OR y

Figure 4.7b The OR logical operator in Boolean (nonuzzy) arithmetic.


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any one o the doors to exploit the vulnerability. In such a situation, we may write the uzzy rule as,

“I (door 1 security is poor) OR (door 2 security is poor), then (intrusion is high).” Considering the

system rom the intruder’s point o view, he would rst select the door in which the intrusion is most

likely; this is the door with the least security or the highest chance o intrusion. He would then break

the security or intrusion. This situation behaves in a similar way to the max operator.

4.5.3 not operator

NOT is a unary operator that takes one input and returns a single output. It may be represented by

Equation 4.18. In Boolean algebra, the unctioning o NOT is given by Table 4.4. The output is the

reverse o the input.

c = NOT x (4.18)

In uzzy systems, the NOT operator does exactly the same thing—reversal—as represented by

Equation 4.19. The graphs or the NOT are given in Figure 4.8a. Figure 4.8b shows their binary

equivalents.

c = 1 – x (4.19)

taBle 4.4t tb f not

x c = not x

0 1

0

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0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p V a l u e s

x

0

0.2

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0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p V a l u e s

NOT x

Figure 4.8 The NOT logical operator in uzzy arithmetic.


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4.5.4 implication

As we proceed through the text, we will advance toward the conversion o any general rule-based

approach to a uzzy approach. So ar, we have reduced any rule to the orm “i x, then y1 = c1 and y2

and c1 and y2,” or “ x → y”. We now resolve the THEN operator (→), which is known as the impli-

cation operator and is given by Equation 4.20. We may even consider a much more generalized

manner in which a rule may be written considering all the inputs and outputs. The general way o

representing such a rule is given by Equation 4.21.

x → y (4.20)

i [NOT] x1 = f 1 op [NOT] x2 = f 2 op [NOT] x3 = f 3 op x4 = f 4 . . . op [NOT] xn = f n

then y1 = f 1 AND y2 = f 2 AND y3 = f 3 AND yn = f n (4.21)

where x1, x2, x3, . . ., xn are the input variables, f j is the membership unctions, [NOT] means that its

presence is optional, op stands or and/or, and y1, y2, y3, . . ., yn are the output variables.

The nonuzzy systems have a series o rules in the “i . . ., then . . ., else.” ormat. Whenever the

condition is true, the corresponding statements are executed, and the resulting output may be oper-

ated. This procedure, however, does not work in uzzy systems, where no condition is true or alse.

On the contrary, the truth is always to some degree in the interval 0 to 1. Hence we need ormal

methods to carry out implication and, as we shall see later, to combine the results o various rules.

The AND represented in the let part o the expression in Equation 4.21 is dierent rom the one

given on the right side. The AND on the let side represents the combination o the various actors

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1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


x

0

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1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p V a

l u e

NOT x

Figure 4.8b The NOT logical operator in Boolean (nonuzzy) arithmetic.


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to represent the rule. They are combined using the logical equivalent uzzy operators. On the other

hand, the AND on the right is not combined; it represents the same conditions aecting the various

output variables. They are all independent o each other and are treated separately while working.

Similarly the = sign occurs both on the let and the right o THEN. These two = are also dier-

ent. The ormer is the check or equality, which checks the closeness o the input to any membership

unction. The latter is an assignment, where we try to assign the output to a membership. We will

now see how this assignment is done.

Implication is a binary operation that takes two inputs and returns a single output. Implication in

the case o uzzy systems is normally perormed by the minimum or the product unction. This is

the same operation that we used in the AND operator and is given by Equations 4.22 and 4.23. The

graphs are the same as shown in Figure 4.9.

x → y = min( x, y) (4.22)

x → y = x * y (4.23)

where x represents the nal calculated value by the application o the various logical operators and

y represents the selected membership unction o the output variable. We get a single membership

value or a given input. The same value is used or the purpose o calculation. The output is the

entire membership unction graph. The operation is applied by a single membership value on the

entire membership unction.

Unlike the AND operator, the implication unction is not the combination o conditions according

to the laws o the logical operators. Rather it tries to do an assignment. Due to its way o unctioning,

this operation is sometimes known as chopping i using the min operator. It chops o the regions in

the membership graph o the output variable. Similarly the operator may be called squashing when

working with the prod operator, as this squashes the entire graph into a lower length graph.

We know that i the condition is x → y, it means that we are trying to associate the output o the

variable x by that represented by the MF y. I we assume that the condition given to the let o THEN

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


y

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Implication by min with x

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Implication by prod with x

Figure 4.9 The min and product implication operators ( x = 0.6) in uzzy arithmetic.


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was completely true—that is, it has a membership o 1—then the output would exactly ollow the

membership graph o the selected MF o the output variable. Suppose the condition is, “I road is

bad, then accident is high.” We can say that i there is only this condition in the entire system and

the road is given to be bad (bad with high membership), the output would be a high accident (high

with a large membership degree). This needs to be true, because the output has to ollow the rules.

In case o bad road, it should give a high accident.I we reduce the membership degree, however, the eect o this rule also reduces, which means

we have an idea o the output, but we are not that sure o the output. For this reason, we minimize

the output’s membership degree. The lower membership degree signies that the condence, or

belongingness, o the output to that class is low. In the same example, i there is lower membership

o the rules, the accident would remain high, but its membership value would reduce. This means we

are not that sure o the accident being high. This is exactly what the implication operator does.

4.6 More oPerationS

We have converted a signicant amount o the rule-based approach to model it on the lines o uzzylogic. In this section, we proceed with our discussion o other operations, including aggregation and

deuzzication. Ater discussing these operators, we put everything together to produce the nal

model that will be the uzzy implementation o the rule-based approach, also known as the uzzy

inerence engine.

4.6.1 aggregation

In the previous section, we saw how an i-then clause can be used to nd the output or any class. We

saw that the output class was identied and its member unction was operated according to the value

o the condition. This gave us a membership unction that was the output o the class or that rule.

In any uzzy system, numerous rules exist. This means by using knowledge o inputs and systems

so ar, we are able to obtain a set o unctions or each and every rule or every output class. To

complete the system, we need a means or deriving the nal output rom these individual unctions.

This will enable us to use the various rules to generate the nal output. The nal output is aected

by each rule and by the decided membership unctions.

The work o aggregating all the rules together to orm a single output is done by the aggregation

operator, which may be visualized as a summation o the various rules to get the nal output. This

summation represents an MF that is the combination, or the aggregation, o the constituent MFs.

We mainly use three kinds o unctions or the aggregation: sum, maximum, and probabilistic or .

In whichever o these methods we ollow, the nal outcome must always be between 0 and 1. This is

with regard to the property o the membership unction. The graphs or all three o the aggregationor three dierent rules x, y, and z are given in Figure 4.10.

4.6.1.1 r f Sm M

We use sum, max, and probabilistic or or the purpose o aggregation. In this section, we present the

novelty behind the use o these unctions and how they are catering to the needs o the unctions or

the purpose o eective uzzy system design.

The rst unction used is sum. The motivation behind this operation is simply the way in which

one would normally handle multiple rules. The eect o the dierent rules is simply added. Any

value greater than 1 is taken to be 1 itsel. Suppose you have to decide between ast driving or slow

driving. You leave the decision to dierent people who will decide it or you. Normally you would

add the number o people suggesting ast driving and the number o people suggesting slow driving.

This is the voting mechanism, which works on the principles o addition.

Similarly assume that you asked the dierent people the same question. This time it was a het-

erogeneous group o people with dierent levels o understanding. All try to answer the question


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according to their own understanding. In this case, you might trust the person who is supposed to be

the best or who has the greatest condence in her responce. You might ollow her decision blindly.

This is the motivation behind the use o the max operator.

4.6.2 defuzzification

So ar we have the aggregated output as a result o applying the various rules. Now we need to

return the crisp output, or the numeral output, that the system is expected to give. This is done by the

deuzzication operator. This process converts the calculated membership to a single numeric out-

put or each output variable. In concept, this is the opposite o the uzzication unction, in which

we converted the numeric inputs into membership degrees by the use o membership unctions. The

deuzzication process is the last step that gives the nal output o the system.

Deuzzication is applied to the obtained membership degrees to generate the crisp output. The

process o deuzzication involves analysis o the entire membership unction to nd the most opti-mal value according to the logical or problem requirements.

Various methods are used to deuzziy the outputs. The most prominent methods are centroid,

bisector, largest o maximum (LOM), mean o maximum (MOM), and smallest o maximum

(SOM). Here all unctions have their usual meaning.

0

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M a m b e r s h i p V a l u e

x

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M a m b e r s h i p V a l u e

y

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M e m


z

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Aggregation of x, y, and z with sum

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M e

m b e r s h i p V a l u e

Aggregation of z , y, and z with max

0

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Aggregation of x, y, and z with probabilistic or

Figure 4.10 The sum, max, and probabalistic or aggregation operators in uzzy arithmetic.


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Centroid: The centroid nds the centroid o the total area represented by the membership

curve. The centroid is a concept similar to that o the center o mass or a body. The cen-

troid may be calculated by Equation 4.24:

o x m dx

m dx

i i

i

=∫

∫ * *

*(4.24)

where o is the nal deuzzied output, xi is the range o values o the output variable, and

mi is the corresponding membership unction. Figure 4.11a shows a sample membership

curve and the corresponding deuzzied output calculated by centroid method.

Bisector: The bisector nds the bisector o the total area represented by the membershipcurve. The area bisector divides the whole membership unction area into two equal halves,

as given by Equation 4.25:

m dx m dxo

x

x

o

* *= ∫ ∫ 2

1

(4.25)

where o is the nal deuzzied output, x1 and x2 are the ranges o output, and m is the cor-

responding membership unction. Figure 4.11b shows a sample membership curve and the

corresponding deuzzied output calculated by bisector method.

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x

Centroid

Figure 4.11 Deuzzication by centroid.

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x

Bisector

Figure 4.11b Deuzzication by bisector.


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MOM: The mean o maximum is the average maximizing at which the membership unctionis the maximum. This method works similar to the max operation in implication in that it

tries to obtain the solution with the maximum membership degree. The general equation

is given by Equation 4.26:

o

x dx

dx=

∫

∫

*(4.26)

where o is the nal deuzzied output and x covers all values o the output range where

membership is maximum. Figure 4.11c shows a sample membership curve and the corre-

sponding deuzzied output calculated by MOM.SOM: The smallest o maximum is the smallest value o the output variable at which the

membership unction is the maximum. This method explores all the values o the output

variable where the maximum membership degree is ound and then gives the smallest o

those values. The general equation is given in Equation 4.27:

o = min( x) (4.27)


membership is maximum. Figure 4.11d shows a sample membership curve and the corre-

sponding deuzzied output calculated by SOM.

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M e m b e r s

h i p V a l u e

x

MOM

Figure 4.11c Deuzzication by mean o minimum.

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x

SOM

Figure 4.11 Deuzzication by smallest o maximum.


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LOM: This is same as SOM, except that we select the maximum value. The equation is givenin Equation 4.28 and the graph is in Figure 4.11e.

o = max( x) (4.28)


membership is maximum.

4.7 Fuzzy inFerenCe SySteMS

In the previous section, we learned about the means and methods with which we can convert any

general rule-based system into a uzzy logic–based system. The motivation behind this task was

to make use o rules that might be generally known in a system to model the system that is driven

by these rules. Because the rules were known beorehand, it is natural or the system to ollow the

desired outputs. This gives rise to a complete system that can be used to model the complexities in

real lie problems.

In this section, we present a step-by-step approach to how the system nds the correct output to

any problem. We cover all the concepts that were presented earlier to engineer a complete system.

We also learn how the various parts o the system perorm, one ater the other, to give the correct

output rom the inputs.

The uzzy inerence system (FIS) is an intelligent system that is built to give the correct outputs

to the known and unknown inputs. The outputs are mapped to the inputs by a set o rules that arecautiously ramed ater a study o the system’s input and output behavior. FIS has a great ability to

change the common-language description consisting o rules into a complete system. FISs are hence

good at modeling real lie problems once we know the common characteristics or the general rules

o the system.

We start by discussing the general methodology and characteristics o FIS. We then provide a

step-by-step guide to working with these systems.

4.7.1 fuzzY inference SYStem deSign

This section covers the general design principles o the uzzy systems that so ar we have beendiscussing in general. A good uzzy system design needs to correctly map the inputs to the outputs,

which is done by designing the correct rules and the correct adjustments o those rules.

The major task, as with any sot-computing system, is identiying the inputs and outputs. The

inputs are decided based on the system, rather than on the approach. Hence, uzzy logic does not

0

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x

LOM

Figure 4.11 Deuzzication by largest o maximum.


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play that important a role as ar as selection o inputs and outputs is concerned. The important

aspect regarding selection o the inputs and outputs is that they must lie within a nite range and

their variation must be known. Knowing the variation is helpul at the time o setting up the MFs o

the inputs and outputs. In addition, we must know the manner in which the various inputs relate to

the outputs. We cannot take any randomly distributed data and try the uzzy approach. Instead we

must know the general guiding rules that relate the inputs to the outputs. This is how uzzy systemsdier rom articial neural networks. In articial neural networks, we simply give the inputs and

the outputs to the network so it can make the rules on its own.

Along with inputs and outputs, the design consists o selecting the correct rules that relate the

inputs to the outputs. These rules should be known in general, which means we must have an idea

o how the inputs map to the outputs by the application o the various rules. In real lie systems, we

generally need to know the manner in which the output behaves upon the increase or decrease o

any particular input, as this helps when raming the rules. Thus we can say that when a particular

input combination is low, the output is low, and vice versa. The rules may be specically studied

by looking at the system’s behavior. In most real lie applications, we try to correlate the change in

the values o input variables to that o the output variables in the presence o multiple inputs andoutputs.

The rules are the driving actors o uzzy systems. A system with deective rules will not be able

to perorm very well, especially in the presence o a high amount o data. Even a system with cor-

rectly designed rules may be urther optimized by the adjustment o the dierent parameters.

The uzzy systems also depend on the selection and the correct parameterization o the MFs.

Fuzzy systems may be ne-tuned by adjusting the parameters o the various MFs o inputs and

outputs. Because application o dierent types o MFs may oten have a deep impact on the uzzy

system’s perormance, we must clearly identiy the input as well as the type o data that the system

would encounter. Based on this, the MF may be selected. Doing so is more o an art and experience

rather than a deep knowledge o uzzy systems.

4.7.2 the fuzzY proceSS

In this section, we study the step-by-step process that maps an input to the output. We have already

studied the various steps involved. Here we present a complete picture o the system. Let us return

to our road example. Consider the ollowing three rules:

R0: I (driver experience is high) & (road is bad), then (accident risk is moderate).

R1: I (driver experience is low) & (road is bad), then (accident risk is high).

R2: I (road is good), then (accident risk is low).

This system takes in two inputs—driver experience and road. Let each input be rated on a scale

o 0 to 1. The system gives one output—the risk o the accident. Let the risk o the accident be mea-

sured on a scale o 0 to 1. The higher the value o the input, the higher the risk o accident. Suppose

we apply any arbitrary inputs x and y. We explain the process or getting the nal output rom this

system. It is assumed that the system has already been designed.

Fuzzifcation: The process starts with the uzzication o the inputs. In this step, we calcu-

late the value o the degree o membership or each input to each o the needed classes by

using the associated membership unctions. This step is done or each input and or each

MF per the requirement o the rules. In our road example, while solving or R0, we wouldrst have to nd the degree o membership o x to high and o y to bad . Suppose the mem-

bership unctions o a high experience and bad road are μ A and μ B, respectively. From the

uzzication, we get two membership degrees. The rst denotes the membership o x to the

class o high experience, or μ A( x), and the second is the membership degree o y to the class


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o bad road , or μ B( y). We can clearly see that each input is associated with some class, and

the corresponding membership degree is thus ound out.

Logical operations: We must combine the various logical operations in each and every rule.

This is done using the uzzy logical operators that we studied in the earlier sections o this

chapter. Each logical operator takes one or two operands, and each operand is a number.

The answer is another single number that is the result o the logical operation over theoperands. In the road example, the min method may be applied over μ A( x) and μ B( y) to give

a single number that may replace the let part o the then expression. The number is actu-

ally the membership degree o the inputs to the particular rule that is being considered.

Implication: In the implication step, we assign the output variable some value or member-

ship degree. This membership degree is in the orm o a graph or a set o values or every

output, as we saw in the previous section. The implication is applied to each output vari-

able, resulting in a membership graph per output variable. In our example, implication on

condition R0 would result in a graph being made or the output variable accident .

Aggregation: In the aggregation step, we combine the dierent rules to study their combined

eects by using the operators we discussed in the previous section. Aggregation combinesall rules into one. This results in the combination o the dierent membership graphs

to generate a common membership graph. In our example, the three rules—R0, R1, and

R2—are combined to produce a common graph that will then be urther processed.

Deuzzifcation: This step completes the uzzy system by giving back the output as desired

by the system—that is, a crisp, or numeric, output. Deuzzication is carried out using any

o the operators we discussed in the previous sections. It converts the graph we obtained in

the aggregation step into a numeral that is given as the output. This step is done or each

output variable in the system.

4.7.3 illuStrative example

Consider once again the road example. Suppose the membership unctions o the two input vari-

ables experience and road and one output variable accident are as given in Figures 4.12a, 4.12b,

and 4.12c, respectively. The input variables road and experience have two membership unctions.

The only output variable accident has three membership unctions. This matches the rules we have

considered.

Suppose we apply an input o 2.5 to experience and 0.4 to road . Now we want to study the sys-

tem’s output. First we must to uzziy the inputs. I we are solving or rule R0, we must try to nd

the membership degree o the 2.5 input to high experience. This comes out to be 0.1605, as shown

in Figure 4.13a. Similarly or an input o 0.4 or bad road , the degree o membership o MF bad

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5

Experience

0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Low

High

Figure 4.12 The membership unctions or experience or the accident risk problem.


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m

b e r s h i p

Road

Bad

Good

Figure 4.12b The membership unctions or road or the accident risk problem.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Accident

Low

Moderate

High

Figure 4.12c The membership unctions or accident or the accident risk problem.

0

0.2

0.4

0.6

0.8

1.0

0 1.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

M e m b e r s h i p

Experience

0.25

Figure 4.13 Fuzzication or rule 0 or experience in the accident risk problem.


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comes out to be 0.1443. This is given in Figure 4.13b. The values or the rest o the rules may be

calculated in a similar manner.

We now apply the AND operator between the two values o 0.1605 and 0.1443. Suppose that the

min operator is used or AND. This would give the result given in Equation 4.29. The values or the

rest o the rules may be calculated in a similar manner.

y = min(0.1605, 0.1443) = 0.1443 (4.29)

The next step is implication, which we need to carry out along the point y = 0.1443. For rule R0,

the result is given in Figure 4.14. The other rules may be handled in a similar manner.Then we use aggregation or the three rules R0, R1, and R2. Suppose we are using max as the

method o aggregation; the resultant graph is shown in Figure 4.15.

At the end, deuzzication o the resultant graph is done to yield the net result or the output

accident . In this example, we used centroid as a deuzzication operator. The nal answer comes

out to be 0.385, as shown in Figure 4.16.

In this way, we can get the output rom any set o inputs. The complete FIS system is given in

Figure 4.17. It denotes the manner in which the output is mapped to the inputs.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m

b e r s h i p

Road

Figure 4.13b Fuzzication or rule 0 or road in the accident risk problem.

0

0.400

0.600

0.800

1.000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Accident

0.144

Figure 4.14 The implication operator.


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4.7.4 Surface diagramS

To get a better understanding o the system and its behavior, we oten use surace diagrams. A sur-

ace diagram is a multidimensional representation o the entire system. It tries to show the entire

input space. However, because the input space is highly dimensional, it cannot be represented on

the screen because we cannot see more than three dimensions. One o these dimensions is xed to

show the output. The output is hence plotted against any two inputs taken on the other two axes. The

surace diagram shows the eect o changing these inputs on the output. The other inputs must be

kept constant. Figure 4.18 plots the surace diagram o the road problem.

0.400

0.600

0.800

1.000

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1.0

M

e m b e r s h i p

Accident

0.144

0.400

0.600

0.800

1.000

0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0

M e m b e r s h i p

Accident

0.144

0.200

0.600

0.800

1.000

0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0

M e m b e

r s h i p

Accident

0.405

0.2

0.4

0.6

0.8

1.0

0 0.1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0

M e m b e r s h i p

Accident

Figure 4.15 The aggregation o the three rules in the accident risk problem.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Accident

0.385

Figure 4.16 The deuzzication o the output in the accident risk problem.


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4.8 tyPe-2 Fuzzy SySteMS

Our entire discussion so ar has ocued on the initial model o uzzy logic. Proessor Zadeh proposed

another model known as type-2 (T2) uzzy logic. The initial model that we discussed earlier was

henceorth called type-1 (T1) uzzy logic. The T2 uzzy logic system (T2 FLS) uses the T2 uzzy

sets (T2 FS) or operations. T2 FS is a more generalized orm o sets that can model the uzziness to

an even greater extent. In this section, we briefy discuss the T2 uzzy sets and T2 uzzy systems.

4.8.1 t2 fuzzY SetS

From our discussion so ar, we know that the T1 FS denotes the uzziness or impreciseness present

with any input. We use MFs to measure the belongingness o any input to any o the membership

classes. We even plotted these MFs on a graph to see the membership values or dierent inputs.

The T2 FSs are a higher level o abstraction that denote the uncertainties associated with MFs.

Hence they may be reerred to as the uzzy uzzy models, because they denote the uzziness in the

T1 uzzy model. The graph o any trivial membership unction is given in Figure 4.19a. Now sup-

pose the let corner o the triangle depicted in the graph is not well known. Say it can lie anywhere

in the small region, as shown in Figure 4.19b. Now suppose that the same uncertainty exists in

1

2

3

0 1

Experience = 0.5 Accident = 0.32

0 1

0 1

Road = 0.5

Figure 4.17 The complete uzzy inerence system or the accident risk problem.

1.0

0.8

0.6

R o a

d

E x pe r ie nc e

0.4

0.2

010987654321

0

0.2

0.4

0.6

A c c i d e n t 0.8

1.0

Figure 4.18 The surace diagram or the accident risk problem.


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the entire curve, as is represented in Figure 4.19b. Here the points required by the T1 FLS can lie

anywhere in the associated region. We cannot be sure o where these points will lie.

Suppose that you are to nd the membership o any input x to any class. In T1 FLS, the naturalapproach would be to use the associated membership unction and calculate the value—say, μ’( x).

The same may be seen rom the membership unction graph, where the membership values or the

dierent inputs were plotted. However, the same is not true or a T2 FLS, in which you would rst

take the input x and consult the graph only to nd the range o values within which the membership

degree can lie (see Figure 4.19b). The membership degree μ( x) associated with the input is uzzy.

This uzziness o the uzzy system orms the concept o T2 FIS.

Because the membership unction μ’( x) in case o T2 FS is uzzy, the uzziness must have some

value. This uzziness denotes the degree o certainty in the T1 FS equivalent MF. Hence we may

represent the MF in the case o T2 FLS as μ( x,u), where u is the point over which the member-

ship value must be calculated. It may easily be visualized that this unction represents a three-dimensional graph showing the nal membership value x and u. The membership value may even

be denoted by the grayness o the curve—that is, higher membership value curves may be darker

than their counterparts.

Figure 4.20 shows a T2 FS membership unction that we represented by μ( x,u). Let there be a

total o N membership unctions. This means that u can take N values corresponding to each o

these N membership unctions or the corresponding x. These values may be denoted as u1 = MF 1( x),

u2= MF 2( x), u3= MF 3( x), . . ., u N = MF N ( x).

The graph given in Figure 4.19b presents a uniormly colored curve. It may hence be interpreted

that the probability or membership value is equal or all points in the curve. Let this value be 1.

Hence i a point lies inside the region, its membership value would be 1; otherwise 0. A three-

dimensional equivalent o this graph would be a graph having a discrete vertical axis where points

1

x

0

µ ( x

)

Figure 4.19 The type-1 membership unction.

µ ( x

)

1

x = x

0 x

Figure 4.19b The type-2 (T2) membership unction.


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may only lie at a value o 0 or 1. These orm a very special type o systems known as the interval

type-2 uzzy sets (IT2 FS). For these sets, all μ( x,u)=

1.The IT2 FS is commonly used. In act, most T2 FLS applications use IT2 FS. The latter’s under-

lying assumption results in lesser computation and complexity, making it possible to use these

systems or many applications. Their counterparts, unortunately, require a great deal o time due

to the large computation involved.

For better understanding, we usually represent the membership curves in two-dimensional maps

only. This depiction is similar to the one shown in Figure 4.19b, where the nonzero membership

areas are colored. The curve so obtained is called the ootprint o uncertainty (FOU), which depicts

the areas and their associated membership degrees in a two-dimensional graph.

4.8.2 repreSentationS of t2 fSThe T2 FS is commonly represented using vertical slice representation or wavy slice representa-

tion. The vertical style is a simpler representation and is more commonly used or computational

purposes. The wavy slide representation, however, is more commonly used or theoretical purposes.

We study each on its own.

I we slice the three-dimensional membership plot at any value o x, we would get a two-dimen-

sional gure with axes o μ( x,u) and u, where x is a constant across which the plot was cut. This plot

is called the vertical slice at any particular x.

The vertical slice representation uses these properties to represent the uzzy system. Here we

monitor two membership unctions—the upper membership unction (UMF) and the lower mem-

bership unction (LMF). The UMF contains the maximum u or any xed x, while the LMF con-tains the least (see Figure 4.21).

The wavy slice representation uses embedded MF in its representation. This representation

is also known as the Mendel-John Representation Theorem. The embedded uzzy system is a

general curve that is within the least and the maximum values. It may easily be seen that the

union o all such curves gives the FOU. The embedded membership unction is depicted in

Figure 4.22.

4.8.3 Solving a t2 fS

The basic approach or solving a T2 FS is quite similar to the T1 FS we discussed earlier. The sys-

tem takes in crisp inputs. The process o uzzication is carried out. The uzzied inputs re the

rules. The results o various rules are aggregated to orm the uzzied output. In a T1 system, the

uzzied output had to be deuzzied. However, in a T2 system, we need to apply the additional

operation o type reduction beore the output can be deuzzied.

µ ( x

)

1

0 xu

i = MF

i( x), i = 1, …, N

Figure 4.20 The membership unctions in T2 uzzy sets.


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One o the major dierences between the T1 and T2 systems is how they deal with the rules. In T2

uzzy sets, every input and every output MF is in turn an MF. Hence the rules need to be worked out

accordingly. The added uncertaininty increases the complexity to a very large extent. Every combina-

tion o MF or every rule to be red must be handled separately, and the outputs are later combined.

For simplicity, consider the simplest rule, having one antecedent and one consequent. Let the rule

be, “I x is P1, then y is Q1,” where P1 and Q1 are in turn made up o MFs. Let us say that P1 is made

up o membership unctions P11, P1

2, P13, . . ., P1

np, and Q1 is made up o membership unctions Q11,

Q12, Q1

3, . . ., Q1nq. Now every combination o P1

i and Q1i must be worked out separately. This way

we would be able to deal with the rules in a similar way that we dealt with them in T1 systems. The

results o all the combinations are then combined. There will be nP x nQ combinations possible,where nP and nQ represent the number o MFs o the MF P1 and the MF Q1, respectively. The ring

o rules is diagrammatically shown in Figure 4.23.

The implication in a T1 FS consists o taking the minimum o the antecedents and slicing the

output MF at the same level. The T2 FS is, to a reasonable extent, the same, except that this opera-

tion must be done or both the UMF and the LMF (see Figure 4.24).

The type reduction (TR) step converts a T2 FS into a T1 FS. This step may be perormed by

the center o sets (COS) mechanism, which makes use o the Karnik-Mendel (KM) algorithm.

Ater the TR, the deuzzication operation may be carried out, or we may alternatively use COS

deuzzication.

µ ( x )

1

x = x

0 x

LMF

UMF

Figure 4.21 The lower membership unction (LMF) and upper membership unction (UMF).

µ ( x

)

1

x = x

0

x

LMF

UMF

Figure 4.22 The embedded uzzy sets.


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4.9 other SetS

So ar we have studied uzzy sets. But two other sets, based on similar ideas, are commonly used in

real lie applications—vague sets and rough sets. Readers may want to recollect the set undamen-

tals, as well as the basics o uzzy sets, beore proceeding with the text.

4.9.1 rough SetS

We have only considered the sets in which some uzziness o an element belongs to the set. The

rough sets have evolved as tools to better analyze experimental data. These data suer rom noise

as well as another major problem—sometimes some values in these data may be totally absent.

Thus the rough set is a generalized concept in which the existence o any element in a set is vague.

This means that the existence o that element cannot be determined. The rough set theory denes a

P 11

Q1

1

P 1

np

P 1

i

Q1

nq

Q1

i

Q1

1

Q1

nq

Q1

i

x

y

O11

O1i

y

O21

O2i

O2np

O1np

O

Figure 4.23 The rule-ring methodology.

µ ( x

)

1

0 x

µ (

x )

1

0 x

M i

M i

x1 x2

µ ( y

)

1

0 y

Figure 4.24 Implication in T2 uzzy systems.


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boundary that consists o elements whose existence in a set is vague or not precisely known. I the

boundary is o zero width or does not contain any element, the rough set is called a crisp set and

becomes a traditional mathematical set with no impreciseness.

4.9.2 vague SetS

Vague sets (VS) are sets in which each element has both a degree o trueness and a degree o alse-

ness associated with it. This means there is some degree to which the existence o an element is true

and some degree to which it is alse. The sum o the two degrees is not necessarily 1, as was the

case with uzzy logic. As a matter o act, the sum is always less than or equal to 1. Both degrees

lie between 0 and 1. The membership o any element x in a VS may hence be represented as <α( x),

1 – β ( x)>, where α( x) denotes the degree o trueness, and β ( x) denotes the degree o alseness. It is

evident that α( x) + β ( x) ≤ 1. The membership unction in the case o VS may be plotted as shown inFigure 4.25.

4.9.3 intuitioniStic fuzzY SetS

Intuitionistic uzzy sets (IFS) are a similar concept to that o the VS. In IFS, two degrees o mem-

bership are associated with any element o the set. The rst degree measures the membership o

the element, and the second measures the nonmembership. It is denoted by < μ( x), V ( x)>, where μ( x)

measures the degree o membership and V ( x) the degree o nonmembership. The plot or this mem-

bership unction is given in Figure 4.26.

1

0

α( x)

1 – β( x)

D e g r e e o f M e m b e r s h i p

x

Figure 4.25 The vague set membership unction.

1

0

µ( x)

V ( x)

x

D e g r e e o f M e m b e r s h i p

Figure 4.26 The intuitionistic uzzy sets membership unction.


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4.10 Sugeno Fuzzy SySteMS

The T1 FIS that we have discussed is known as Mamdani FIS and is widely used in real lie applica-

tions. There is another FIS known as Sugeno FIS or Takagi-Sugeno-Kang FIS. This model is sameas that o the Mamdani FIS; however the output membership unctions in Sugeno can only be linear

or constant. The rule in such a system is o the orm, “I x is P, then y is Q,” where Q is a constant

or crisp number. This rule represents the zero-order Sugeno FIS. In these systems, the implication

method is simply multiplication or minimum, and the aggregation operator includes outputs o the

various rules. Figure 4.27 shows the implication operation in such systems. The aggregation is given

in Figure 4.28. Deuzzication in these systems is simply the weighted mean, as shown by dotted

line in Figure 4.28.

First-order Sugeno FIS is a more generalized FIS. In this system, rules may be o the orm “I

x is P, then y = a * x + b,” where a and b are constants. This is a similar concept to the zero-order

system, except that the output can move about in a linear ashion. The higher-order Sugeno systemsare computationally very expensive and hence not used in real lie applications.

4.11 exaMPle: Fuzzy Controller

Among the numerous applications o uzzy logic, we have controllers. Fuzzy logic has ound

immense applications in such systems, where we try to control the output o a machine to attain

some predened output catering to the machine’s constraints. To ully understand uzzy logic, we

take the example o a uzzy controller used in robotic control.

4.11.1 proBlem deScription

Robotic uzzy controller is used to move robots rom a source to a destination, or goal. This problem

has relevance in the eld o robotics, which applies intelligent systems to make a map and decide

the path. Then it becomes the duty o the robotic controller to move the robot ollowing the desired

path.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M

e m b e r s h i p

Input 1

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M

e m b e r s h i p

Input 2

00.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Output

Figure 4.27 Implication in the Sugeno uzzy inerence system.


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The robot basically consists o wheels and motors that drive the wheels. All robotic movements

are governed by wheels. For example, a carlike robot rotates by the rotation o its wheels. Considering

these acts, the robot cannot make every possible move. It is able to turn only by a certain amount

o angle. Furthermore a very sharp turn would make the journey very unsmooth and would urther

require a reduction o speed. This is undesirable.

For the sake o simplicity, we assume that the robot’s speed is constant and cannot change. We

urther assume that no obstacles exist anywhere in the map. Under these constraints and assump-

tions, we must move the robot.

The robot in this example is more o a carlike robot. It can only move orward. O course, in

such a robot, backward motion is possible, but that is seldom used in real lie situations or in experi-

mental purposes. In addition, we can turn the robot in both a clockwise and a counterclockwise

direction by any desired amount. As in a car, the turning o robot is done by turning the wheels at

the required angle.

4.11.2 inputS and outputS

At any time, the robot’s motion depends on the angle and the goal. These orm the inputs to the

system. The angle α is the angle by which the robot must turn in order to ace the goal. This angle

is measured by taking the dierence between the robot’s current angle φ and the angle o the goal

measured by the robot’s current position θ . The result is always between –180 degrees and 180

degrees, as shown in Figure 4.29.

The other input is the goal, or the distance between the robot’s current position and the position o

the goal. The distance is normalized by multiplication o a constant so that it lies between 0 and 1.

The system has a single output—the angle by which the robot may be turned at the next move.

The robot then physically moves and turns by this angle in its next move. This angle may be positiveor negative, depending on whether the desired move is in a clockwise or counterclockwise direction.

As we proceed, this angle usually gets smaller and smaller, because over time the robot orients itsel

in the direction o the goal. Ater this, the robot just needs to move toward the goal or march in a

straight line toward the goal.

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Input 1

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Input 2

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Input 3

0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h i p

Output

Final Output

Figure 4.28 The aggregation and deuzzication in Sugeno uzzy inerence systems.


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4.11.3 memBerShip functionS

Ater deciding the inputs and outputs, we orm the membership unctions or each input and out-

put, as given by Figures 4.30a, 4.30b, and 4.30c or angle, goal, and output turn, respectively. All

the membership unctions used are Gaussian, except or the extremes o the angle, which are trap

membership unction in nature.

It may be seen rom these gures that any MF starts rom the midpoint, or extreme, o the

neighboring MFs. This helps us rame the rules. At the time at which some input corresponds to the

maxima o some MF, it also happens to lie at the minima o other MFs. It may hence be seen that at

these inputs, only one MF is active with a membership value o 1, while all others are inactive with

a membership value o 0. The output corresponding to this input may be the precise output at theconsequent o the rule. When the rule is evaluated, there happens to be a direct mapping o the input

that has a membership value o 1 to the output that may again have a membership value o 1. Hence

using this mechanism, we can create a type o lookup between known inputs and outputs, and we

can perectly match the desired output using rules. As we deviate slowly rom this point, the other

MFs start getting active and start infuencing the output.

Consider the input angle. At the time when α = 0, only one membership unction (called no) is

active. This MF has a value o 1 at α = 0. The corresponding output can be mapped to the membership

Goal

θ φ

α = θ –φ

x

Figure 4.29 The measurement o angle α in the robotic control problem.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

–180 –144 –108 –72 –36 0 36 72 108 144 180

M e m b e r s h i p D e g r e e

Angle

More negative

Less negative

No

Less positive

More positive

Figure 4.30 Membership unctions or angle in the robotic control problem.


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unction no o the output, which also has a value o 0 as the output with only one active MF. Hence on

application o α = 0, this rule gives a perect output o 0, which is desirable.

The angle is composed o ve MFs that cover the range rom –180 degrees to 180 degrees.

These are named (rom least to right, according to Figure 4.30a) more negative (moren), less nega-

tive (lessn), no difference (no), less positive (lessp), and more positive (morep). Except or the two

extreme MFs, these are more or less equally distributed. The inspiration or this comes rom the

ollowing act: Suppose two conditions—α = 150 and α = 100. Even though the dierence in value

o α is very large, the turn or both these cases would be around the maximum possible value that is

comortable. This is because in any other case, the robot would take too long to orient itsel. Hence

it can be seen that there is almost no dierence in the output, even or a large change in inputs.

Again, as discussed above, the MF at α = 0 stands or no turn. This is the region where we require

making no turns or very small turns, as the robot is almost acing the goal. The region between

these is covered by one MF on each side (lessp and lessn), which has been cautiously placed around

45 degrees because it was easy or us to visualize the preerable turn around this angle.

Similarly the input distance has three MFs: near , far , and distant . The distant MF ollows the

same philosophy as the extreme MFs o the input angle. I the distance is large, we would preer not

to make turns so as to avoid sharp turns. It does not make much dierence i the goal is so distant

that the robot cannot see or so near that the robot might just see; the output is not much aected. The

MF near covers the region where the goal is so near that we need to make very sharp turns to reach it.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M e m b e r s h

i p D e g r e e

Goal

Near

Far

Distant

Figure 4.30b Membership unctions or distance in the robotic control problem.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

–10 –8 –6 –4 –2 0 2 4 6 8 10

M e m b e r s h i p D e g r e e

Output Turn

More left

Less left

No

Less right

More right

Figure 4.30c Membership unctions or output turn in the robotic control problem.


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I we do not do so, we might miss the goal. The length o this MF is less that may be attributed to the

philosophy o motion. The MF far covers cases where the goal is within the comortable regions.

The output MF is divided into ve regions: more left (morel), less left (lessl), no turn (no), less

right (lessr ), and more right (morer ). The extreme MFs are larger in coverage, as this avoids too

large o turns being taken, which would be undesirable. The location o each MF matches the desir-

able angle o turn o the MFs o the input angle.

4.11.4 ruleS

Rules orm the basis o the FIS, because the behavior o the system largely depends on the rules.

We use the same understanding and philosophy o inputs that we used to orm the MFs. However,

although we discuss MFs, rules, and results sequentially in this book, the design o any uzzy sys-

tem does not perectly go in that order. In act, it happens in iterations o these steps, in which we

rst orm MFs, then rame rules, then simulate the system, and nally see results. Aterward the

system’s errors and shortcomings are noted, and accordingly the MFs and rules are modied.

We ramed a total o nine rules or the system. These rules, which are given in Figure 4.31, maybe understood rom our understanding o the inputs and the MFs.

4.11.5 reSultS and Simulation

The model we made was validated and tested by a simulation engine. The general approach ol-

lowed was that we rst calculated the input needed by the FIS according to the present conditions.

This was then entered into the FIS to get the angle o the next move. This angle was then imple-

mented in the next move. This procedure was repeated until the goal was reached. This simulation

is shown in Figure 4.32.

Based on this simulation methodology, the path traced by the robot or various runs with dier-

ent initial positions and angles is given in Figures 4.33a through 4.33d. In each gure, the robot is

moving upward. Hence the lower point in the path is the initial position and the upper point is the

nal position. The angle o the robot is the direction in which it initially moves.

Rule1: If (α is morep) then (output is morer)

Rule2: If (α is lessp) then (output is lessr)

Rule3: If (α is no) then (output is no)

Rule4: If (α is lessn) then (output is lessl)

Rule5: If (α is moren) then (output is morel)

Rule6: If (α is not morep) and (goal is distant) then (output is no)

Rule7: If (α is not moren) and (goal is distant) then (output is no) Rule8: If (α is lessp) and (goal is near) then (output is morer)

Rule9: If (α is lessn) and (goal is near) then (output is morel)

Figure 4.31 The uzzy rules or the robotic control problem.

Initial

conditions

Calculate

inputsGoal

reached?

Exit

FIS

Implement

move

Yes

No

Figure 4.32 The simulation process in the robotic control problem.


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100

90

80

70

60

50

40

30

20

20 30 40 50

x

y

60 70 80 90 100

10

100

Figure 4.33 Path traced by robot at rst run.

100

90

80

70

60

50

40

30

20

20 30 40 50

x

y

60 70 80 90 100

10

100

Figure 4.33b Path traced by robot at second run.

100

90

80

70

60

50

40

30

20

20 30 40 50

x

y

60 70 80 90 100

10

100

Figure 4.33c Path traced by robot at third run.


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ChaPter SuMMary

This chapter presented an in-depth analysis o uzzy systems. We started our discussion with uzzy

logic, where we discussed the basics o uzzy logic and its dierence rom normal logic. Fuzzy and

nonuzzy systems were then compared and contrasted. The loopholes o nonuzzy systems ormed

our motivation or using uzzy systems. Fuzzy sets were another major topic that ormed the basic

oundation o the chapter.

To develop and understand a uzzy system, the essential parameters o a rule-based approach and

its uzzy counterparts were presented. This enabled modeling o the uzzy systems along the lines

o a basic rule–based approach that is well understood and commonly used. Fuzzy membership

unctions enabled us to determine the degree o membership or belongingness o an element to a

uzzy set. Numerous commonly used uzzy sets were illustrated.

The next topic o discussion was the uzzy logical operators. Here we studied the uzzy coun-

terparts o various logical operations, including AND, OR, NOT, and implication. The other uzzy

operators included aggregation and deuzzication. Then uzzy inerence systems were presented.

We studied the manner in which FIS maps the inputs to the outputs, as well as other design issues

o uzzy systems.

Another topic o discussion was the type-2 uzzy system. These systems have been shown to

model impreciseness or uzziness better than the type-1 uzzy systems. We also moved rom the

uzzy sets to the other sets—namely, rough sets, vague sets, and intuitionistic uzzy sets.

At the end o the chapter, a real lie example o a uzzy controller was built using the uzzy iner-ence system. This system could move a robot rom the known initial position to a nal position by

making a smooth transition in its path.

Solved exaMPleS

1. Discuss the general methodology o problem solving using uzzy inerence systems

(FIS).

Answer: Problem solving in FIS is an iterative process in which we make a model and

keep modiying the model until satisactory results are achieved.First we must study the problem and decide the inputs and outputs. We discussed the

ways and means to do this in Chapter 1. Once the inputs and the outputs are known, the

next step is to decide the membership unctions. Initially it would be preerable to go with a

limited number o MFs, rather than crowding the model with too many. Another important

100

90

80

70

60

50

40

30

20

20 30 40 50

x

60 70 80 90 100

10

100

Figure 4.33 Path traced by robot at ourth run.


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aspect is the placement o the MFs. These are normally placed by developing an insight

into the kind o input the system would give or the kind o output that is expected rom the

system. In the input side, suppose that the lower inputs are very likely and appear most o

the time, while the higher inputs are very rare. In such a case, more MFs may be placed at

the lower regions. Also suppose that the output is expected to drastically change or small

changes in input at the lower inputs. At the higher inputs, however, even a large change ininput produces a small change in output. In such a case as well, it would be advisable to

place more MFs at the lower region.

The MFs are usually placed at points around which the corresponding output is airly

known. This helps map the input to the output according to the rules. In addition, an MF

usually reaches its maxima o 1 when the previous MF reaches its minima o 0. At this

point, there is only one active MF with a membership value o 1. This urther allows us to

easily map the inputs to outputs by rules. Next the rules are ramed using common sense.

Once this model is ready, it is tested by the simulation engine, by known inputs, or by

common sense. The discrepancies and errors are noted. Now we must modiy the model

accordingly. Many times the wrong outputs may result rom the act that we did not con-

sider many cases and hence did not rame rules or these cases. In many other cases, the

errors may be due to the wrong placement o the MFs. These issues may be ne-tuned

according to the requirements. I the errors are not due to either o these reasons, we may

consider adding up the MFs and replacing MFs in regions where the output was wrong.

The modied FIS is again tested; this process continues until we get the desired output

or all inputs. This process is given in Figure 4.34.

Suppose we already know some inputs and outputs. Reiterating in search o the most

optimal solution is a time-consuming step that requires a great deal o patience and energy.

These problems require some algorithm to perorm these tasks and nd the most optimal

structure. We will see in Chapter 6 that the neuro-uzzy systems are an eective way o

doing such things. We will also see the application o genetic optimizations as anothersolution.

2. Compare an artifcal neural network (ANN) with FIS or problem solving.

Answer: Both ANN and FIS are intelligent systems. They are used to give correct outputs

to the inputs presented. The ANN learns rom the historical database itsel, whereas the

FIS must be tuned manually so that it imitates the historical database, i available.

Knowledge exists in every intelligent system. Through this knowledge, the system is

able to map inputs to outputs or, in other words, give the correct output or the inputs.

This knowledge needs some kind o knowledge representation and usage. In the case o

ANNs, this knowledge exists in the orm o weights between neurons. In the case o uzzy

systems, knowledge is in orm o rules.

ANN training usually happens with great ease in that the rules used to map the inputs to

the outputs are simple enough. This training is when the bulk o the data is in agreement by

Select inputsand outputs.

Frame rules.Select MFs.

Modify FIS.

Simulate Final FIS

Errorstolerable

Errors

Figure 4.34 Problem solving in uzzy inerence systems.


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rules, without any great anomalies. I the rules are not simple, however, the data training

requires a great deal o eort and may be ullled with the addition o neurons. Every kind

o data cannot be trained by ANN. We normally try to train the ANN with more neurons

or or more time to get a decent perormance. But using more neurons or more time may

not do well in the testing data due to loss o generalization, as discussed in Chapter 2. In

short, the ANN training is simple or most simple and clearly dened problems.

The same is true or the FIS. I an FIS does not perorm well, we may make necessary

modications. I we are unable to get high perormance, we may add more rules or MFs,

though this would not be required in most o the simple problems. Simpler problems would

give good perormances even with a low number o rules and MFs.

Hence the neurons and layers in ANN are similar to the rules and MFs in FIS.The FIS however needs a air idea o the initial rule and an understanding o the system.

This helps rame the correct rules, which are very necessary or system. This is not the

case with ANN, where rules are automatically extracted.

3. Suppose that the quality o road Q is measured on a scale o 0 to 1. Two actors aect

Q: the average evenness o the road and the average pebble density. Both inputs can

be measured by standard practices and are normalized to lie between 0 and 1. Both

have an equal eect on Q. Design a uzzy system or fndingQ. Is there any other way

to solve the problem?

0.55

0.35

1.0 1.00.8

0.6 0.60.8

0.4P e b b l e D e ns i t y E ve n ne s s

0.40.2 0.2

0

R o a

d Q u a

l i t y

0.45

0.50

0.40

Figure 4.35 The surace or Example 3 in the case o FIS with one membership unction.

1.0

0.8

0.6

0.4

0.2

0

1.01.0

0.8

0.8

0.6 0.60.4

0.4P e b b l e D e ns i t y E v e

n n e s s

0.2 0.2

0 0

R o a

d Q u a

l i t y

Figure 4.35b The surace or Example 3 in the case o FIS with three membership unctions.


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Answer: Let the average evenness o the road be given by R and the average pebble densitybe given by P. It is given that both inputs equally aect the output and are equally distrib-

uted over their input ranges. The uzzy model that is generated has one MF per input and

output. Let the MF o R be r , P be p, and Q be q. All the MFs are triangular in nature, with

maxima at the input corresponding to the input/output o 0 and minima corresponding to

the input/output o 1. The system has one rule: “I ( R is r ) and (P is p), then (Q is q).” The

surace model o the resultant system is shown in Figure 4.35a. When we change the num-

ber o MFs to three and number o rules to three, the resultant surace is given in Figure

4.35b.

This problem can also be solved by a simple mathematical equation, as given by

Equation 4.30:

Q = R * (1 – P) + P * (1 – R) (4.30)

The plot o this surace is given in Figure 4.35c. Although it is natural that a uzzy

approach would ultimately lead to this equation, using FIS can add computational over-

heads to the system.

From Figures 4.35a through 4.35c, we can make two major inerences

Adding rules and MFs makes the plot more complex. The same behavior was exhibited•

by ANNs.

Many simple problems can be solved by very ew rules and MFs.•

exerCiSeS

general queStionS

1. Compare and contrast uzzy sets with (a) vague sets and (b) rough sets.

2. What is the dierence between binary logic and uzzy logic?

3. What are membership unctions?

4. Compare a production rule-based problem-solving approach with uzzy logic.

5. What are rules in an FIS?6. Compare and contrast Sugeno and Mandami type-1 uzzy logic systems.

7. Compare and contrast Type I and Type II uzzy logic systems.

8. Suppose a = 0.05 and b = 0.3. Calculate (with respect to uzzy logic) (a) a AND b, (b) a OR

b, and (c) (NOT a) AND (NOT b). Make suitable assumptions wherever necessary.

1.0

0.5

0

1.00.8

0.60.4

P e b b l e D e n s i t y

E ve n ness

0.2

0.4

0.6

0.8

1.0

0.200

R o a

d Q u a

l i t y

Figure 4.35c The surace or Example 3 in the case o nonuzzy systems.


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9. The ollowing identities hold in Boolean logic. Do they hold in uzzy logic?

(a) a OR b = b OR a

(b) a OR (b OR c) = (a OR b) OR c

(c) NOT (a OR b) = (NOT a) OR (NOT b)

10. Consider the FIS given in Section 4.7.3. Find the output or the input <0.23, 0.87>.

11. Does the order o ring rules aect the output in FIS? Why or why not?12. What is the role o deuzzication in FIS? Name a ew deuzzication methods.

13. Why is there a need or so many deuzzication methods, when any one o them can be

used or the same purpose?

14. Explain type-2 uzzy inerence systems.

15. What do we mean by aggregation in an FIS?

16. What are surace diagrams? Why are they needed?

17. Explain uzzy controllers.

18. What is the dierence between uzzication and deuzzication?

19. What is a crisp number?

20. What is the role o attaching weights to rules in FIS?

practical queStionS

1. Imagine a car moving toward a wall at some speed. You are supposed to stop the car beore

it crashes against the wall. Simulate this problem using a uzzy controller.

2. Suppose the health o a person depends on his age and medical history. Make an FIS o this

system. Try using various AND, OR, implication, and deuzzication methods.

3. In the solution generated in Practical Question 1, study the eect o adding or deleting

membership unctions o each input and output. What is the least number o MFs required

or a desirable behavior o the system?

4. Make an FIS that ranks (assigns scores to) cricket-playing nations. What parameters do

you consider? Make suitable assumptions wherever necessary.

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