1. INTRODUCTION

What is fuzzy logic FL?

Fuzzy logic is a problem solving control system methodology that was invented by Lofti

Zadeh a professor at the University of California at Berkley. It has proven to be an excellent

choice for many control system applications since it mimics human control logic. It can be built

into anything from small hand-held products to large computerized process control systems. It is

imprecise but very descriptive language to deal with input data more like human operator.

Fuzzy logic (FL) offers several unique features that make it a particularly good choice for

many control problems. These include:

It is inherently robust since it does not require precise, noise free input, and can be programmed

to fail safely if a feedback sensor quits or destroyed. The output control is a smooth control

function despite a wide range of input variations.

FL controller processes user-defined rules governing the target control system, therefore it can be

modified and tweaked easily to improve or drastically alter system performance for the better.

FL has no limited inputs feedback and control outputs. It is also not necessary to measure or

compute rate of change parameters in order for it to be implemented, any sensor data that

provides some indication of a system’s action and reaction is sufficient. This allows the sensor to

be inexpensive and imprecise thus keeping the overall cost and complexity low

FL control nonlinear systems that would be difficult or impossible to model mathematically. This

opens doors for control systems that could normally be deemed unfeasible for automation.

2. FUZZY SETS

FL deals with problems that have fuzziness or vagueness , unlike other methods like classical

theory which is based on Boolean logic where a particular object or variable is either a member

of a given set (logic 1) or not (logic 0). Fuzzy logic problem is an input/output, static, nonlinear

mapping problem through a main box as shown in figure 1

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The input information is defined in the input space, it is processed in the main box and

the solution appears at the output space. The main box is a fuzzy system or any other method e.g.

expert system, neural network or a general mathematical system that can give a desired output.

2.1 Membership function (MF)

The MF is a graphical representation of the magnitude of participation of each input. It associates

a weighting with each of the inputs that are processed, define, functional overlap between inputs

and ultimately determines and output response. The fuzzy variables has values that are expressed

by the natural language for instance English.

A membership curve defines how the values of fuzzy variable in a certain region are mapped to a

membership value μ (or degree of membership) between 0 and1. A stator temperature of a motor

for example, as a fuzzy variable can be defined by the qualifying linguistic variables as cold,

mild or hot, where they are represented by a triangles or straight line segment membership

function. Figure 2 shows the features of membership function.

MF can have different shapes like triangular, trapezoidal, two sided Gaussian, generalised

bell, sigmoid-right, sigmoid-left, difference sigmoid, product sigmoid, polynomial-Z,

polynomial-Pi and polynomial-S. Triangular and trapezoidal types are the mostly used MF as

shown in figure 3, which can be symmetrical or asymmetrical. MF can be represented by

mathematical functions, segmented straight lines (for triangular and trapezoidal) and look-up

tables.

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2.2 operations with sets

The basic properties of Boolean logic are also valid for FL, figure 4 shows the logical

operations of OR, AND, and NOT on the fuzzy set A and B using triangular MFs compares with

the corresponding Boolean operations on the right. Let µA(x), µB(x) denote the degree of

membership of a given element x n the universe of discourse X (denoted bye x ∈ X).

Union: if A and B are two fuzzy sets, defined in the universe of discourse X the union Α∪Bis

also a fuzzy set of x with membership function given as:

µ AUB(x) ≡ Max [µA(x), µB(x)]

≡ µA(x) v µB(x)

Where symbol “v” is a maximum operator equivalent to Boolean OR logic.

Intersection: The intersection of two fuzzy sets A and B in the universe of discourse x denoted

by A∩ B has the membership function given by

µ A∩B(x) ≡ Max [µA(x), µB(x)]

≡ µA(x) ^ µB(x)

Where “^” is the minimum operator equivalent to Boolean AND logic.

Complementary or Negation: The complement of a given set A in the universe of discourse X

denoted bye Ā and has a membership function given by:

µĀ (x) ≡ 1- µA (x)

Equivalent to the NOT operation in Boolean logic.

Product of two fuzzy sets: The product of two fuzzy logic sets A and B defined in the same

universe of discourse X is a new fuzzy set A.B with MF that equals the algebraic product of MFs

A and B

µA.B(x) ≡ µA(x). µB(x)

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Multiplying fuzzy sets by a crisp number: The MF of fuzzy set A can be multiplied by a crisp

number k to obtain a new fuzzy set called product k.A its MFs is

µkA(x) ≡ k.µA(x).

Power of a fuzzy set: Fuzzy set A can be raised to a power m(positive real number) by raising

its MFs to m. the m power of A is a new fuzzy set Am with MF denoted by

µAm(x) ≡ [ µ A (x)]m

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3. FUZZY SYSTEM.

Fuzzy system consists of a formulation of the mapping from a given input set to and

output using fuzzy logic, which consists of the following five steps.

Step 1: Fuzzification of input variables, defining the control objectives and criteria.

Step 2: application of fuzzy operators (AND, OR, NOT) in the IF (antecedent) part of the rule.

Determine the output and input relationships and choose a minimum number of variables

for input to the fuzzy logic engine.

Step 3: implication from antecedent to the consequent (THEN part of the rule) for the desired

system output response for a given system input conditions.

Step 4: aggregation of the consequents across the rules by creating fuzzy logic membership

functions that define the meaning (values) of input/output terms used in the rule.

Step 5: defuzzification to obtain a crisp result.

The following example shows how the above five steps can be implemented in a non technical

environment for a restaurant tipping where food and service are the inputs fuzzy variable (0 -10

range) and tip is the output variable (0-25% range). The input variable service is represented by

three fuzzy sets poor, good, and excellent which corresponds to curved MFs. While variable food

is represented by two fuzzy sets bad and delicious this corresponds to straight-line MFs. The

output variable tip is represented by three sets cheap, average, and generous which correspond to

triangular MFs. Three rules are developed as shown in figure 5.

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If the quality of service is 3 which implies MF poor gives the output µ=0.3 which is a

result of fuzzification (step 1) and if the score for food is 8 which is referred to a MF bad, the

result of fuzzification is µ=0. After all the inputs have be fuzzified and each degree of the

antecedent if a rule has been satisfied, the OR or max operator is specified and therefore between

the two values 0.3 and 0, the result of the operator is 0.3 which is selected in (step 2) this defined

as the degree of fulfillment (DOF). The implication stage helps to evaluate the consequent part of

a rule. For this rule the output MF cheap is truncated at the value µ=0.3 to give a fuzzy output

(step 3). All the rules are evaluated the same manner and their contributions are shown in figure

6 the outputs are combined or aggregated in a cumulative manner to result a final fuzzy

output (step 4). Finally, the fuzzy output (area) is converted into crisp which is defined as

defuzzification (step 5)

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3.1 Implication methods

There are three major methods of implication which are; mamdani, lusing lerson and

sugeno types but the frequently use is the mamdani because of its simplicity. Given the following

rules of a fuzzy system;

Rule 1: If X is negative small (NS) and Y is zero ZE THEN Z is positive small (PS)

Rule 2: If X is zero (ZE) and Y is zero (ZE) THEN Z is zero (ZE)

Rule 3: If X is zero (ZE) and Y is positive small (PS) THEN Z is negative small (NS)

Where X and Y are the input variables while Z is the output and NS, ZE, and PS are the fuzzy

sets. Figure 7 explains the fuzzy inference system with mamdani method for inputs x=3 and Y= 1.5

The DOF of the rules are;DOF1 = µNS(x) ^ µZE(Y) = 0.8 ^ 0.6 = 0.6

DOF2=µZE(X) ^ µZE(Y) = 0.4 ^ 0.6 = 0.4

DOF3= µZE(X) ^ µPS(Y) = 0.4 ^ 1.0=0.4

The total output is the union (OR) of all the components MFs µout (Z) =µPS’ (Z) v µZE’ (Z) v µNS’ (Z).

4. DEFUZZIFICATION7

This the final stage in fuzzy system where the fuzzy output needs to be converted to a crisp (non

fuzzy) form. A fuzzy rule-based controller for instance, uses such step to generate a crisp control

command.

There three major types of defuzzification techniques

(i) The Mean Of Maxima (MOM) method.

The MOM defuzzification calculates the average of all variables with maximum

membership degree. The crisp output is given by the equation;

Zo=∑m=1

m Zm

M

Where Zm = mth element in the universe of discourse, when the output MF

is at the maximum value, and M = number of such element.

(ii) Sugeno method

From the sugeno implication method where K1,K2,K3,……Kn represent the

consequent part of each rule, the output MF in each rule is a singleton spike which, is

multiplied by the respective DOF to contribute the fuzzy output of each rule hence the

output will be given by the formula;

Zo=K1 DOF1+ K2 DOF 2+K3 DOF3+… K n DOFn

DOF1+DOF2+DOF3+… DOFn

(iii) Center of area (COA) or centroid method

This method calculates the weighted average a fuzzy output set. The crisp output of

Zo is taken to the geometric center of the output fuzzy value µout (Z) area, where µout

(Z) is formed by taking the union of all the contributions of values whose DOF>0

with a discrete universe of discourse as shown by the expression;

Zo=∑i=1

n

Z i µout (Z i)

∑i=1

n

µout(Z i)

For example: given a two rule fuzzy system output shown in figure 8, the crisp

output using COA method will be as shown;

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Therefore:

Zo =1. 1

3+2. 2

3+3. 2

3+4. 2

3+5. 1

3+6. 1

3+7. 1

313+ 2

3+ 2

3+ 2

3+ 1

3+ 1

3+ 1

3+¿¿

= 3.7

FUZZY CONTROL

A fuzzy control system embeds the experience an intuition of human plant operator and

those of a designer and/or researcher of the plant. It gives robust performance for linear or

nonlinear plant with parameters variations and can be applied in a complex process such as

cement plant, nuclear reactors etc.

5. APPLICATIONS

For the output of a fully control thyristor power converter, the output voltage is determine by the

following equation, Vmean¿2V max

πcos∝.

Where Vmean = the average dc output of the inverter

∝= delay angle in degrees

Using a single phase voltage source, a simple fuzzy rule-based system using five simple rules

can be used to approximate the output voltage. The universe of discourse for the input variable ∝

will be interval [ 0, 90] in degrees, the input voltage can vary [80-260] line voltage, and the

universe of discourse for the output Vmean is the interval [0, 216] volts. The input variable is

partition into five membership functions as follows: Very Small VS, Small S, Medium M, Large

L, Very Large VL and three for input voltage; Low L, Medium M, and High H while the out is

partition into three membership functions; Small S, Medium M, High H. using the fuzzy logic

tool box under MATLAB environment the following were obtained.

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Figure 9: FIS editor for power converter

Figure 10: Rule editor for a power converter

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Figure 11: Rule Viewer for a power converter

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6. REFERENCE:

1. John Yen, Reza Langari, Fuzzy logic: intelligence, control, and information. Pearson

Education: patparganji India 2005.

2. Guanrong Chen, Trung Lat Pham: Introduction to fuzzy sets, fuzzy logic and fuzzy

control systems: CRC Press LLC Florida 2001.

3. Marco Russo, Lakhmi C Jain, Fuzzy Learning and application: New York 2001.

4. http://www.prairiedigital.com/PDI_Website/PDI_Model40.htm  

5. Fuzzy Logic Toolbox, The MathWorks http://www.mathworks.com/products/fuzzylogic/.

6. Ajith Abraham, Rule-based Expert Systems, pp 909-919, Oklahoma State University,

Stillwater, OK, USA.

7. Nikola K. Kasabov, Foundations of Neural Networks, Fuzzy Systems, and

Knowledge Engineering, The MIT Press Cambridge. 1996.

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