Project Assignment for Fuzzy Logic

8/11/2019 Project Assignment for Fuzzy Logic

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Project 1 Assignment

Developing Fuzzy Inference Systems

September 30, 2012


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Contents

Introduction 2

1 Project Objective 3

2 Theoretical Background 4

2.1 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Types of Membership Functions . . . . . . . . . . . . . . 5

2.2 Fuzzy Logic Operations . . . . . . . . . . . . . . . . . . . . . . . 72.3 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Fuzzy and Linguistic Variables . . . . . . . . . . . . . . . . . . . 92.5 Fuzzy Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1 Inference from Fuzzy Rules . . . . . . . . . . . . . . . . . 112.6 Fuzzy Inference Systems . . . . . . . . . . . . . . . . . . . . . . . 14

2.6.1 Stages of Developing Fuzzy Inference Systems . . . . . . . 19

3 Project Report 20

Appendix. Typical Fuzzy Inference Systems 21

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Introduction

The goal of this project is to develop and verify by testing a Fuzzy InferenceSystem (FIS) for any purpose, e. g., control, classification, or decision making.

AFuzzy Inference Systemis [1] any system whose variables range over statesthat are fuzzy sets. For each variable, the fuzzy sets are defined on some relevantuniversal set, which in this project is taken as an interval of real numbers. Inthis case, the fuzzy sets are fuzzy numbers, and the associated variables arelinguistic variables.

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Chapter 1

Project Objective

In order to complete this project, each team has to carry out the following steps:

1. Choose any control, classification, or recognition problem to be solved us-ing fuzzy logic (typicalexamples are given in Appendix 1). The problemshould be complex enough to be difficult to solve using conventional math-ematical tools (calculus, probability theory, or differential equations).

2. Choose input and output linguistic variables (Sect. 2.4). Define theirfuzzy values by assigning appropriate membership functions (Sect. 2.1).The suggested number of input variables lies between 3 and 5.

3. Build up a system of fuzzy rules connecting input and output fuzzy values(Sect. 2.6.1).

4. Develop a fully-functional code for the project (either using MATLABFuzzy Logic Toolbox, or from scratch). There are no specific requirementsas to the I/O interface, however, it should be clear enough to evaluate thesystem performance.

5. Prepare a Project Report (Chap. 3) in a form of a presentation and defendit in front of other teams.

Steps 13 must be discussed with the lecturer and approved beforedefending the project.

It is requiredto analyze system performance depending on its parameters.Particular ways of modifying system parameters must also be discussed withthe lecturer and approvedbefore defending the project. Possible suggestionsare:

choosing different membership functions for the same variable;

reducing/increasing the number of fuzzy rules;

varying the number of fuzzy values of linguistic variables.

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Chapter 2

Theoretical Background

2.1 Fuzzy Sets

A fuzzy set[2] is a set without a crisp, clearly defined boundary. It can containelements with only a partial degree of membership.

Let denote a (universal) set of objects. Then, a fuzzy set A in ischaracterized [3] by a membership function fA(x) which associates with eachpointx a real number fA(x) [0, 1] representing the membership grade ofx in A (the grade of possibility that x A). A typical membership function isshown in Fig. 2.1.

Figure 2.1: The membership function of a fuzzy set

Stated formally, a fuzzy set is a set of ordered pairs:

A= {(x, A(x)); x , A(X) [0, 1]}. (2.1)

Often, it is convenient to use list notationto write down members of a fuzzyset. For a discrete universe , the fuzzy set can be represented as

A=xi

A(xi)

xi.

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For a continuous universe, this notation changes to

A=x

A(x)x

,

where the integral sign is a mere notational mark, and does not denote anycalculus operation.

Example 1.

Consider the continuous universe of discourse =. Then, the membershipfunctionA(x) =

11+(x5)10

defines a fuzzy setA which can be interpreted as a

set of numbers close to 5.

2.1.1 Types of Membership Functions

The only condition a membership function must satisfy [2] is that it must varybetween 0 and 1. The function itself can be an arbitrary curve whose shape isdefined to suit the problem from the viewpoint of simplicity, convenience, speed,and efficiency.

The simplest membership functions are formed using straight lines. Themost commonly used among them are triangular and trapezoidalmembershipfunctions (Fig. 2.2):

A(x; a,b,c) =

0, x axaba

, a x bcxcb

, b x c

0, c x

, (2.2)

A(x; a,b,c,d) =

0, x axaba

, a x b

1, b x cdxdc

, c x d

0, d x

. (2.3)

When it is more convenient to use smooth functions rather than piecewise,it is a good practice to use Gaussian or generalized bell membership functions(Fig. 2.3):

A(x; , c) = e(xc)2

22 , (2.4)

A(x; a,b,c) = 1

1 +xc

a2b . (2.5)

Although the Gaussian membership functions and bell membership functionsachieve smoothness, they are unable to specify asymmetric membership func-tions, which are important in certain applications. The sigmoidalmembershipfunction (Fig. 2.4) can be either open left or right:

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(a)

(b)

Figure 2.2: The triangular (a) and trapezoidal (b) membership functions

(a)

(b)

Figure 2.3: The Gaussian (a) and generalized bell (b) membership functions

A(x; a, c) = 1

1 + ea(xc) . (2.6)

Asymmetric and closed (i.e. not open to the left or right) membership

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Figure 2.4: The sigmoid membership function

functions can be synthesized using two sigmoidal functions.

2.2 Fuzzy Logic Operations

It is well known [4] that the complement, union, and intersection of crispsets correspond to the logical operations NOT, OR, and AND, respectively, inthe corresponding crisp, bivalent logic. Also, in the crisp, bivalent logic, theunion of a set with the complement of a second set represents an implicationof the first set by the second set. Set inclusion (subset) is a special case ofimplication in which the two sets belong to the same universe.

These operations (connectives) may be extended to fuzzy sets for correspond-ing use in fuzzy logic and approximate reasoning. For fuzzy sets, the applicableconnectives must be expressed in terms of the membership functions of the setswhich are operated on. However, unlike crisp logic, in general it is possible todefine an infinite number of various kinds of fuzzy complements, unions, in-tersections, and implications. Further on, we will discuss only standard fuzzy

operations.The standard fuzzy complement is given by the following formula:

A(x) = 1 A(x). (2.7)

The complement in fuzzy sets corresponds [4] to the negation (NOT) oper-ation in fuzzy logic.

Example 2. Consider the fuzzy set of hot temperatures. This may berepresented by the membership function from Fig. 2.5 (solid line). The com-plement of this set is represented by the dotted line in Fig. 2.5, and representstemperatures that are not hot.

The standard union of two fuzzy sets A and B is given by the formula:

AB(x) = max[A(x), B(x)]. (2.8)

The union corresponds [4] to a logical OR operation (A B).

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Figure 2.5: Fuzzy complement

Example 3. Consider a universe representing the driving speeds, in km/h.Suppose that the fuzzy set Fast is given by the discrete membership function

F = 0.6/80 + 0.8/90 + 1.0/100 + 1.0/110 + 1.0/120,

and the fuzzy set Medium is given by

M= 0.6/50 + 0.8/60 + 1.0/70 + 1.0/80 + 0.8/90 + 0.4/100.

Then, the fuzzy set Fast OR Medium is given be the membership function

FM= 0.6/50+0.8/60+1.0/70+1.0/80+0.8/90+1.0/100+1.0/110+1.0/120.

The standard intersection of two fuzzy sets A and B is given by the formula:

AB(x) = min[A(x), B(x)]. (2.9)

The intersection corresponds [4] to a logical AND operation (A B).

2.3 Fuzzy Relations

A fuzzy relation is [1] is a fuzzy set defined on the Cartesian product of crispsets X1, X2, . . . , X n. A fuzzy relation can be conveniently represented by ann-dimensional membership function.

Example 4.

LetR be a fuzzy relation between the sets X= {New York City, Paris}andY ={Beijing, New York City, London} which represents the relational conceptvery far. This relation can be written in list notation as

R(X, Y) = 1.0/(NYC, Beijing) + 0.0/(NYC, NYC) + 0.6/(NYC, London)+0.9/(Paris, Beijing) + 0.7/(Paris, NYC) + 0.3/(Paris, London).

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The standard composition of binary fuzzy relations P(X, Y) and Q(Y, Z)with a common setY is defined by

R(x, z) = [P Q](x, z), orR(x, z) = maxyYmin[P(x, y), Q(y, z)] x X, z Z .

(2.10)

2.4 Fuzzy and Linguistic Variables

A variableis characterized [5] by a triple (X, , R(X, u)), whereXis the nameof the variable, is a universal set, u is generic name for the elements if , andR is a subset of which represents a restrictionon the values ofu imposed byX.

Example 5. Consider a variable named age. In this case, might be takenas the set of integers 0, 1, 2, . . ., and R(X, u) might be the subset 0, 1, . . . , 100.

A fuzzy variableis characterized [5] by a triple (X, , R(X, u)), where X isthe name of the variable, is a universal set, u is generic name for the elementsif , and R is a fuzzy subset of which represents a fuzzy restrictionon thevalues ofu imposed by X. The nonfuzzy variableu is called the base variableforX.

The assignment equation forXhas the form

x= u : R(X),

and represents an assignment of a value u to x Xsubject to the restrictionR(X). The degree to which this equation is satisfied (compatibility ofu withR(X)) is given by R(X)(u), the grade of membership ofu in R(X).

Example 6. Consider a fuzzy variable named budget, with = [0,], and

R(budget) =

10000

1/u +

1000

1 +

u 1000

200

21/u .

Then in the assignment equation budget = 1100 : R(budget), the compati-bility of 1100 with R is R(budget)(1100) = 0.80.

A linguistic variable is characterized [5] by a quintiple (L, T(L), , G , M ),where L is the name of the variable, T(L) is the term-set ofL, or the set ofnames of linguistic valuesbeing fuzzy variables (denoted by X) ranging over

the universe (associated with the base variable), G is a syntactic rule forgenerating the elements ofT(L), andM is a semantic rule for association witheach X its meaning, which is a fuzzy subset of . A particular X is called aterm.

The meaning M(X) is defined to be the restriction R(X, u) on the basevariable u imposed by the fuzzy variable named X.

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Example 7. Consider a linguistic variable named Age (L = Age), with = [0, 100]. A linguistic value ofAgemay be named old. Another value may

be very old. The term-set may be expressed asT(Age) ={old, very old, not old, more or less young, quite young, . . .}.Each term is the name of a fuzzy variable. For example, the restriction

R(old) constitutes the meaning ofold. A possible restriction might be definedas

M(old) =

10050

1 +

u 50

5

21/u .

It is always convenient to use the same symbol to denote both a fuzzy set,and the name of that set.

Put informally, we can say that a linguistic variable is a variable whose valuesare fuzzy sets defined on the same universe of discourse. This is illustrated in

Fig. 2.6.

Figure 2.6: Hierarchical structure of a linguistic variable

2.5 Fuzzy Propositions

The fundamental difference between classical propositions and fuzzy proposi-tions is [1] in the range of their truth values. While each classical propositionis required to be either true or false, the truth or falsity of fuzzy propositions isa matter of degree.

In this project, it will be sufficient to discuss only one type of fuzzy proposi-tions, namely,unqualified propositions. The canonical form of fuzzy propositions

p of this type is expressed by the sentence

p: V is F , (2.11)

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where Vis a variable that takes values v from some universal set V, and F isa fuzzy set on V representing a fuzzy predicate (tall, low, cheap, and so

on). Given a particularv, it belongs toFwith membership gradeF(v), whichis also interpreted as the degree of truth T(p) of propositionp.

Example 8.

Let Vbe the water temperature, and let the membership function (Fig.2.7) represent the predicate high. Then, the corresponding fuzzy propositionpis expressed by the sentence

p: temperature (V) is high (F).

Figure 2.7: Components of the fuzzy proposition p

The degree of truth T(p) depends on the actual value of the temperature.For example, ifv = 85, T(p) = 0.75.

2.5.1 Inference from Fuzzy Rules

Each fuzzy rule can be expressed as a conditional propositionin the canonicalform

p: IfX is A, then Y is B , (2.12)

where X and Yare variables whose values are in sets X and Y, respectively,andA, B are fuzzy sets onXandY, respectively. These propositions may alsobe viewed as propositions of the form

X,Y is R , (2.13)

whereR is a fuzzy set on X Ydetermined by the formula

R(x, y) = I[A(x), B(y)], (2.14)

whereIdenotes fuzzy implication.

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There are various kinds of fuzzy implication, but in this project it is recom-mended to use the implication in the form I(a, b) = min[a, b].

Usually, the if clause of a fuzzy rule is called an antecedent, and the thenclause is called a consequent.

Let us assume that A and B are fuzzy sets on X and Y, resprectively.Then, ifR andA are given, we can obtain B by the equation:

B =A R (2.15)

called the compositional rule of inference(CRI) (Fig. 2.8).

Figure 2.8: Compositional Rule of Inference

Using relationR obtained from a proposition p by (2.14), and given anotherproposition qof the formq: X is A, we may conclude that Y isB by the CRI(2.15). This procedure is called a generalized modus ponens(GMP), and can beexpressed by the following schema:

Rule: If X is A, thenY is BFact: X is A

Conclusion: Y is B

Such kind of inference is an example of a so-called approximate reasoning.

Example 9.Let sets of values of variables Xand YbeX= {x1, x2, x3} andY ={y1, y2},

respectively. Assume that a proposition ifX isA, thenY isB is given, whereA = 0.5/x1 + 1.0/x2 + 0.6/x3, and B = 1.0/y1 + 0.4/y2. Then, given a factexpressed by the proposition x is A, where A = 0.6/x1+ 0.9/x2+ 0.7/x3,we can use the GMP to derive a conclusion in the form Y is B .

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Using fuzzy implication, we obtain:

R= 0.5/(x1, y1) + 0.4/(x1, y2) + 1.0/(x2, y1)+0.4/(x2, y2) + 0.6/(x3, y1) + 0.4/(x3, y2)

by (2.14). Then, by (2.15), we obtain:

B(y1) = supxXmin[A(x), R(x, y1)]= max[min(0.6, 0.5), min(0.9, 1.0), min(0.7, 0.6)]= 0.9,

B(y2) = supxXmin[A(x), R(x, y2)]= max[min(0.6, 0.4), min(0.9, 0.4), min(0.7, 0.4)]= 0.9.

Thus, we conclude that Y is B , where B = 0.9/y1+ 0.9/y2.

If a fuzzy rule has a complex antecedent:p: IfX1 is A1 andX2 is A2 and . . . andXn is An, then Y is B ,

then it can be reduced to the previous case by introducing a multidimensionalfuzzy set X with a membership function X(x1, x2, . . . , xn) = max[X1(x1),. . . , Xn(xn)]. Of course, if the antecedent contains OR operation instead ofAND, or any combination of both, the multidimesional set can be constructedusing max and min operations in proper order.

When there are several fuzzy rules, we face what is called multiconditionalapproximate reasoningof the following form:

Rule 1: If X is A1, thenY is B1Rule 2: If X is A2, thenY is B2

. . . . . .Rule n: IfX is An, thenY is BnFact: X is A

Conclusion: Y is B

The most common way to determing B is [1] a method of interpolation. Itconsists of the following two steps:

1. Calculate the degree of consistencyrj(A) between the given fact and theantecedent of each rule j :

rj(A) = sup

xX

min[A(x), Aj (x)] ; (2.16)

2. Calculate the conclusion B by truncating each set Bj by the value of

rj(A

), and take the union of the truncated sets:B(y) = sup

j=1,n

min[rj(A), Bj (y)] y Y . (2.17)

The illustration of the method of interpolation is given in Fig. 2.9. In thisproject, it is required to build a FIS using this method.

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Figure 2.9: Illustration of the method if interpolation

2.6 Fuzzy Inference Systems

Fuzzy inference is [2] a process of formulating the mapping from a given inputto an output using fuzzy logic. There are two major types of fuzzy inferencesystems, namely, Mamdani-type and Sugeno-type. In this project, it is requiredto develop the Mamdani-type FIS initially described by Ebrahim Mamdani in [6]

as an attempt to control a steam engine and boiler combination by synthesizinga set of linguistic control rules obtained from experienced human operators.

A general scheme of a typical FIS is presented in Fig. 2.10. Thefuzzy rulebaseconsists of fuzzy rules in the form (2.12). Antecedents of the rules in thebase represent input linguistic variablesof the FIS, whereas consequents stand

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for output linguistic variables.

Figure 2.10: A general scheme of a Fuzzy Inference System

A typical fuzzy inference system operates by repeating a cycle of the followingsteps [1]:

1. Inputs from the environment are taken in the crisp form.

2. These inputs are fuzzified in the fuzzification module by means of a fuzzi-fication function fe, whose purpose is to interpet measurements of inputvariables more realistically. In this project, it is highly recommended totakefe(x) = x, i.e., we will not fuzzify input measurements.

3. The resultant values are used by the inference engine to evaluate each rulefrom the rule base by CRI. Fuzzy sets obtained for each rule are aggregatedinto one by (2.8).

4. The aggregated fuzzy set is converted into a single crisp value that is insome sense the best representative of the set. This conversion is called adefuzzification.

There are several ways of defuzzifying resultant fuzzy sets. One of the mostpopular methods is [2] the centroid calculation:

d(C) = c

cC(z)zdz

cc

C(z)dz , (2.18)

whereCis the fuzzy set to be defuzzified, with the support [c, c]. For a discretecase, when Cis defined on a finite universal set {z1, z2, . . . , zn}, (2.18) changesto

d(C) =

nk=1 C(zk)zknk=1 C(zk)

. (2.19)

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Example 10.

Consider the so-called tipping problem: what is the right amount to tip

your waitperson? Let us state this problem in a following manner:

Given a number between 0 and 10 that represents the quality ofservice at a restaurant (10 being excellent), and another numberbetween 0 and 10 that represents the quality of food (10 being deli-cious), what should the tip be, given that the average tip is usually15%?

Consider the following Fuzzy Inference System for solving this task. The in-put variables are Service Quality and Food Quality, and the output variableis Tip. The values of each variable are defined using the following membershipfunctions (Fig. 2.11):

poor

(x) = ex2

21.52 ;

good(x) = e(x5)2

21.52 ;

excellent(x) = e(x10)2

21.52 ;

rancid(x) =

0, x < 01, 0 x 13x

3 , 1 x 3

0, 3 x

;

delicious(x) =

0, x 7x7

2 , 7 x 9

1, 9 x 10

0, 10< x

;

cheap(x) =

0, x 0x5

, 0 x 510x

5 , 5 x 100, 10 x

;

average(x) =

0, x 10x10

5 , 10 x 1520x

5 , 15 x 20

0, 20 x

;

generous(x) =

0, x 20x20

5 , 20 x 25

30x

5 , 25 x 300, 30 x

.

The fuzzy rule base contains the following fuzzy rules:

1. If Service Quality is poor, or Food Quality is rancid, then Tipis cheap.

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(a)

(b)

(c)

Figure 2.11: The membership functions for the tipping problem FIS

2. If Service Quality is good, then Tip is average.

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3. If Service Quality is excellent, or Food Quality is delicious, thenTip is generous.

The basic structure of the FIS for the tipping problem is shown in Fig. 2.12.

Figure 2.12: The basic structure of the FIS for the tipping problem

Let the input value for the Service Quality variable be Service0

= 3, andfor the Food Quality variable be Food0 = 8. Then, to compute the output tip,we need to carry out the following steps according to the method of interpolationexplained in Sect. 2.5.1. Keep in mind that, since we decided not to fuzzify ourinputs, the fact in our case is represented by a fuzzy set consisting of onlyone element: A = 1/(3, 8):

1. For each rule, calculate the degree of consistencyrj(A):

r1(A) = min[A(3, 8), A1(3, 8)] =A1(3, 8) =max[poor(3), rancid(8)] = 0.135 ;

r2(A) = min[A(3, 8), A2(3, 8)] =A2(3, 8) =good(3) = 0.411 ;

r3(A) = min[A(3, 8), A3(3, 8)] =A3(3, 8) =max[excellent(3), delicious(8)] = 0.500.

2. For each rule, obtain fuzzy sets by truncating rule consequents byrj :

B1(y) = min[r1(A), B1(y)] = min[0.135, cheap(y)] ;

B2(y) = min[r2(A), B2(y)] = min[0.411, average(y)] ;

B3(y) = min[r3(A), B3(y)] = min[0.500, generous(y)].

3. Unite resulting sets into one:

B

(y) = max[B

1(y), B

2(y), B

3(y)].

4. Defuzzify the result to obtain crisp output. In our case, applying centroidmethod yields 16.7%.

The fuzzy inference diagram for this example is shown in Fig. 2.13.

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Figure 2.13: The fuzzy inference diagram for the tipping problem

2.6.1 Stages of Developing Fuzzy Inference Systems

To solve a problem with the help of a Fuzzy Inference System, it is necessary to

follow several steps:

1. Identify relevant input and output linguistic variables of the system, andranges of their values.

2. Select meaningful fuzzy values for each variable and express them by ap-propriate fuzzy sets.

3. Formulate fuzzy rules representing the knowledge needed to solve the prob-lem.

4. Choose appropriate defuzzification method to convert the output fuzzyset to a crisp value.

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Chapter 3

Project Report

The Project Report must cover the following topics:

description of the problem to be solved; it is necessary to stress why itis better (cheaper, faster, more convenient) to use fuzzy logic instead ofconventional tools;

description of the fuzzy inference system, including input and output vari-ables, membership functions defined for their values, and appropriate fuzzyrules;

analysis of the results of applying FIS to solving the problem at hand, andits comparison with results obtained by conventional tools;

analysis of system performance depending on its parameters.

Particular structure of the report, or its esthetical appeal, is irrelevant aslong as the report contains all the parts stated above, and the presentationsatisfies requirements imposed by the lecturer.

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Bibliography

[1] Klir, G. J., Yuan, B.Fuzzy Sets and Fuzzy Logic: Theory and Applications.Prentice Hall, 1995.

[2] Fuzzy Logic Toolbox. For Use with MATLAB. The MathWorkds, Inc., 1999.

[3] Zadeh, L. A. Fuzzy sets,Information and Control, 8, 1965, pp. 338353.

[4] Karray, F. O., de Silva, C.Soft Computing and Intelligent Systems Design.Theory, Tools and Applications. Pearson Education Limited, 2004.

[5] Zadeh, L. A. The concept of a linguistic variable and its application toapproximate reasoning, Information Sciences, 8, 1975, pp. 199249.

[6] Mamdani, E. H., Assilian, S. An experiment in linguistic synthesis witha fuzzy logic controller, International Journal of Man-Machine Studies,Vol. 7, No. 1, 1975, pp. 113.

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Appendix. Typical Fuzzy

Inference Systems

Fuzzy logic based controller for the climate control system of automobiles.

Fuzzy inference system for classification of different species of flowers.

Fuzzy logic controller for robot maze traversal.

Fuzzy traffic light controller.

Fuzzy controller for a washing machine.

Fuzzy inference system for handwritten digit recognition.

Fuzzy system for controlling the liquid level in a tank.

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Project Assignment for Fuzzy Logic