Section 4 Inference Methods

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1506-1521

Fuzzy Logic

52

Chapter 4

Systems

fuzzy control systems

1. Introduction

fuzzy set.

55

Abstract

Poli Venkata Subba Reddy

Some Methods of Fuzzy

Conditional Inference for

Application to Fuzzy Control

Zadeh proposed fuzzy logic with single membership function. Two Zadeh, Mamdani and TSK proposed fuzzy conditional inference. In many applications like fuzzy control systems, the consequent part may be derived from precedent part. Zadeh, Mamdani and TSK proposed different fuzzy conditional inferences for "if … then …" for approximate reasoning. The Zadeh and Mamdani fuzzy conditional inferences are know prior information for both precedent part and consequent part. The TSK fuzzy conditional inferences need not know prior information for consequent part but it is difficult to compute. In this chapter, fuzzy conditional inference is proposed for "if…then…" This fuzzy conditional inference need not know prior information of the consequent part. The fuzzy conditional inference is discussed using the single fuzzy membership function and twofold fuzzy member-

ship functions. The fuzzy control system is given as an application.

Keywords: fuzzy logic, twofold fuzzy logic, fuzzy conditional inference,

When information is incomplete, fuzzy logic is useful [10–26]. Many theories [1, 2] deal with incomplete information based on likelihood (probability), whereas fuzzy logic is based on belief. Zadeh defined fuzzy set with single membership function. Zadeh [3], Mamdani [4], TSK [2] and Reddy [5] are studied fuzzy conditional inferences. The fuzzy conditions are of the form "if <. Zadeh, Mamdani and TSK fuzzy conditional inference requires both precedent-part and consequent-part but 5fuzzy inferences don't require consequent part. Precedent-part > then <consequent-part >." Zadeh [6] studied fuzzy logic with single membership function. The single membership function for the proposition "x is A" contains how much truth in the proposition. The fuzzy set with two membership functions will contain more information in terms of how much truth and false it has in the proposition. The fuzzy certainty factor is studied as difference on two membership functions "true" and "false" to eliminate conflict of evidence, and it becomes single membership function. The FCF is a fuzzy set with single fuzzy membership function of twofold

## Chapter 4

## Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

Poli Venkata Subba Reddy

## Abstract

Zadeh proposed fuzzy logic with single membership function. Two Zadeh, Mamdani and TSK proposed fuzzy conditional inference. In many applications like fuzzy control systems, the consequent part may be derived from precedent part. Zadeh, Mamdani and TSK proposed different fuzzy conditional inferences for "if … then …" for approximate reasoning. The Zadeh and Mamdani fuzzy conditional inferences are know prior information for both precedent part and consequent part. The TSK fuzzy conditional inferences need not know prior information for consequent part but it is difficult to compute. In this chapter, fuzzy conditional inference is proposed for "if…then…" This fuzzy conditional inference need not know prior information of the consequent part. The fuzzy conditional inference is discussed using the single fuzzy membership function and twofold fuzzy membership functions. The fuzzy control system is given as an application.

Keywords: fuzzy logic, twofold fuzzy logic, fuzzy conditional inference, fuzzy control systems

## 1. Introduction

When information is incomplete, fuzzy logic is useful [10–26]. Many theories [1, 2] deal with incomplete information based on likelihood (probability), whereas fuzzy logic is based on belief. Zadeh defined fuzzy set with single membership function. Zadeh [3], Mamdani [4], TSK [2] and Reddy [5] are studied fuzzy conditional inferences. The fuzzy conditions are of the form "if <. Zadeh, Mamdani and TSK fuzzy conditional inference requires both precedent-part and consequent-part but 5fuzzy inferences don't require consequent part. Precedent-part > then <consequent-part >."

Zadeh [6] studied fuzzy logic with single membership function. The single membership function for the proposition "x is A" contains how much truth in the proposition. The fuzzy set with two membership functions will contain more information in terms of how much truth and false it has in the proposition. The fuzzy certainty factor is studied as difference on two membership functions "true" and "false" to eliminate conflict of evidence, and it becomes single membership function. The FCF is a fuzzy set with single fuzzy membership function of twofold fuzzy set.

54

The fuzzy control systems are considered in this chapter as application of single fuzzy membership function and twofold fuzzy set.

## 2. Fuzzy log with single membership function

Zadeh [6] has introduced a fuzzy set as a model to deal with imprecise, inconsistent and inexact information. The fuzzy set is a class of objects with a continuum of grades of membership.

The fuzzy set A of X is characterized as its membership function A = μA(x) and ranging values in the unit interval [0, 1]

μA(x): X ➔[0, 1], x Є X, where X is the universe of discourse. <sup>A</sup> <sup>=</sup> <sup>μ</sup>A(x1)/x1 <sup>þ</sup> <sup>μ</sup>A(x2)/x2 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup>A(xn)/xn, where "+" is the union. For instance, the fuzzy proposition "x is High" High = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 Not High = 0.8/x1 þ 0.4/x2 þ 0.1/x3 þ 0.4/x4 þ 0.8/x5 For instance, the fuzziness of "Temperature is high" is 0.8 The graphical representation of young and not young is shown in Figure 1. The fuzzy logic is defined as a combination of fuzzy sets using logical operators. Some of the logical operations are given below. For example, A, B and C are fuzzy sets. The operations on fuzzy sets are given as: Negation If x is not A A<sup>0</sup> = 1 � μA(x)/x Conjunction x is A and y is B➔ (x, y) is A ΛB

Disjunction

For instance,

Concentration μvery A(x) = μA(x)<sup>2</sup>

μmore or less A(x) = μA(x)0.5

considered for fuzzy control systems.

Zadeh [6] fuzzy inference is given as:

<sup>=</sup> min(1, 1 � (A1, A2,…, An) <sup>þ</sup> B)

Diffusion

Implication

Figure 3. Disjunction.

57

If x = y

Figure 2. Conjunction.

> x is A and y is B➔ (x, y) is A V B AVB = max(μA(x), μB(y)}(x,y)

DOI: http://dx.doi.org/10.5772/intechopen.82700

x is A and y is B➔ (x, y) is A V B AVB = max(μA(x), μB(y)}/x

A = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 B = 0.4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.1/x5 AVB = 0.4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 The graphical representation is shown in Figure 3.

If x1 is A1 and x2 is A2 and … and xn is An, then y is B The presidency part may contain any number of "and/or"

If x1 is A1 and x2 is A2 and … and xn is An, then y is B

The graphical representation of concentration and diffusion is shown in Figure 4.

Zadeh [6], Mamdani [7] and Reddy [5] fuzzy conditional inferences are

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

AΛB = min(μA(x), μB(y)}(x,y) If x = y x is A and y is B➔ (x, y) is A ΛB AΛB = min(μA(x), μB(y)}/x For example A = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 B = 0.4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.1/x5 AΛB = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.1/x5 The graphical representation is shown in Figures 1 and 2.

Figure 1. Fuzzy membership function.

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700

#### Figure 2. Conjunction.

The fuzzy control systems are considered in this chapter as application of single

Zadeh [6] has introduced a fuzzy set as a model to deal with imprecise, inconsistent and inexact information. The fuzzy set is a class of objects with a continuum

The fuzzy set A of X is characterized as its membership function A = μA(x) and

The graphical representation of young and not young is shown in Figure 1. The fuzzy logic is defined as a combination of fuzzy sets using logical operators.

For example, A, B and C are fuzzy sets. The operations on fuzzy sets are given as:

fuzzy membership function and twofold fuzzy set.

2. Fuzzy log with single membership function

of grades of membership.

Fuzzy Logic

μA(x): X ➔[0, 1], x Є X,

where "+" is the union.

Negation If x is not A A<sup>0</sup> = 1 � μA(x)/x Conjunction

If x = y

Figure 1.

56

Fuzzy membership function.

For example

ranging values in the unit interval [0, 1]

where X is the universe of discourse.

<sup>A</sup> <sup>=</sup> <sup>μ</sup>A(x1)/x1 <sup>þ</sup> <sup>μ</sup>A(x2)/x2 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup>A(xn)/xn,

Some of the logical operations are given below.

x is A and y is B➔ (x, y) is A ΛB AΛB = min(μA(x), μB(y)}(x,y)

x is A and y is B➔ (x, y) is A ΛB AΛB = min(μA(x), μB(y)}/x

A = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 B = 0.4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.1/x5 AΛB = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.1/x5 The graphical representation is shown in Figures 1 and 2.

For instance, the fuzzy proposition "x is High" High = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 Not High = 0.8/x1 þ 0.4/x2 þ 0.1/x3 þ 0.4/x4 þ 0.8/x5

For instance, the fuzziness of "Temperature is high" is 0.8

### Disjunction

x is A and y is B➔ (x, y) is A V B AVB = max(μA(x), μB(y)}(x,y) If x = y x is A and y is B➔ (x, y) is A V B AVB = max(μA(x), μB(y)}/x For instance, A = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 B = 0.4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.1/x5 AVB = 0.4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 The graphical representation is shown in Figure 3. Concentration μvery A(x) = μA(x)<sup>2</sup> Diffusion μmore or less A(x) = μA(x)0.5

The graphical representation of concentration and diffusion is shown in Figure 4. Implication

Zadeh [6], Mamdani [7] and Reddy [5] fuzzy conditional inferences are considered for fuzzy control systems.

If x1 is A1 and x2 is A2 and … and xn is An, then y is B The presidency part may contain any number of "and/or" Zadeh [6] fuzzy inference is given as: If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min(1, 1 � (A1, A2,…, An) <sup>þ</sup> B)

Figure 3. Disjunction.

Figure 4. Fuzzy quantifiers.

Mamdani [4] fuzzy inference is given as: If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min(A1, A2,…, An, B) Zadeh and Mamdani fuzzy inference has prior information of A and B. The relation between A and B is known. Then, B is derived from A. Reddy [2] inference is given by: If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min(A1, A2,…,An) Consider the fuzzy rule: If x1 is A1 and x2 is A2, then x is B For instance, A1 = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 A2 = 0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.3/x5 B = 0.1/x1 þ 0.4/x2 þ 0.6/x3 þ 0.4/x4 þ 0.1/x5 The graphical representation of A1, A2 and B is shown in Figure 5. The graphical representation of fuzzy inference is shown in Figure 6. Composition If some relation between R and A1 than B1 is to infer from R B1 = A1 o R, where R = A➔B Zadeh fuzzy inference is given by: B1 = A1 o R = min{μA(x), μR(x)} = min{μA(x), min(1,1�μA1(x) þ μB(x))} Mamdani fuzzy inference is given by: = min{μA1(x),μA(x) þ μB(x)}

3. Justification of Reddy and Mamdani fuzzy conditional inference

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

DOI: http://dx.doi.org/10.5772/intechopen.82700

Justification of Reddy fuzzy conditional inference may be derived in the

If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min{A1, A2,…, An}.

If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> f(x1, x2,…, xn). The proposed method of fuzzy conditional inference may be defined by

If x1 is A1 and/or A2 and/or,…, and/or An, then y is B <sup>=</sup> f(A1, A2,…, An) If x1 is A1 or A2 and An, then y is B = f(A1, A2, A3) = A1 V A2 Λ –Λ A3 If x1 is A1 or A2 and A3, then y is B = f(A1, A2, A3) = A1 V A2 Λ A3

The fuzzy conditional inference is given by using Mamdani fuzzy inference

If x1 is A1 or A2 and A3, then x is B = min(max(μA1(x1), μA2(x2)), μA3(x3))

If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min{A1, A2,…,An}. Justification of Mamdani fuzzy conditional inference may be derived in the

Zadeh fuzzy conditional inference for "if … then … else …" is given by:

If some relation R between A and B is known, then Mamdani fuzzy conditional

If x is A, then y is B else y is C = If x is A then y is B v If x is A<sup>0</sup> then y is C = AxB

If x1 is A1 or A2 and A3, then y is B = min(A1 or A2 and A3, B)

Thus, the Reddy fuzzy conditional inference is satisfied.

Thus, the Mamdani fuzzy conditional inference is satisfied.

If x is A, then y is B else y is C = AxBvA<sup>0</sup> x C

Consider Reddy fuzzy conditional inference:

Consider TSK fuzzy conditional inference:

replacing x1, x2,…, xn with A1, A2 and … and An

B = min(max(μA1(x1), μA2(x2)), μA3(x3))

following:

Figure 6.

Fuzzy conditional inference.

following:

v A<sup>0</sup> x C

59

If x is A<sup>0</sup>

inference is given by:

If x is A, then y is B = AxB

It is logically divided into: If x is A, then y is B = AxB

If x is A, then y is B = A x B.

, then y is C = A<sup>0</sup> x C

If there is some relation R between A and B, then Reddy fuzzy inference is given by:

= μA1(x)

Figure 5. Fuzzy sets.

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700

Figure 6. Fuzzy conditional inference.

Mamdani [4] fuzzy inference is given as:

<sup>=</sup> min(A1, A2,…, An, B)

<sup>=</sup> min(A1, A2,…,An) Consider the fuzzy rule:

For instance,

Figure 4. Fuzzy quantifiers.

Fuzzy Logic

Composition

given by: = μA1(x)

Figure 5. Fuzzy sets.

58

Reddy [2] inference is given by:

If x1 is A1 and x2 is A2, then x is B

B1 = A1 o R, where R = A➔B Zadeh fuzzy inference is given by: B1 = A1 o R = min{μA(x), μR(x)}

= min{μA1(x),μA(x) þ μB(x)}

= min{μA(x), min(1,1�μA1(x) þ μB(x))} Mamdani fuzzy inference is given by:

If x1 is A1 and x2 is A2 and … and xn is An, then y is B

If x1 is A1 and x2 is A2 and … and xn is An, then y is B

A1 = 0.2/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 A2 = 0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.3/x5 B = 0.1/x1 þ 0.4/x2 þ 0.6/x3 þ 0.4/x4 þ 0.1/x5

relation between A and B is known. Then, B is derived from A.

Zadeh and Mamdani fuzzy inference has prior information of A and B. The

The graphical representation of A1, A2 and B is shown in Figure 5. The graphical representation of fuzzy inference is shown in Figure 6.

If there is some relation R between A and B, then Reddy fuzzy inference is

If some relation between R and A1 than B1 is to infer from R

## 3. Justification of Reddy and Mamdani fuzzy conditional inference

Justification of Reddy fuzzy conditional inference may be derived in the following:

Consider Reddy fuzzy conditional inference:

If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min{A1, A2,…, An}. Consider TSK fuzzy conditional inference:

If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> f(x1, x2,…, xn).

The proposed method of fuzzy conditional inference may be defined by replacing x1, x2,…, xn with A1, A2 and … and An

If x1 is A1 and/or A2 and/or,…, and/or An, then y is B <sup>=</sup> f(A1, A2,…, An)

If x1 is A1 or A2 and An, then y is B = f(A1, A2, A3) = A1 V A2 Λ –Λ A3

If x1 is A1 or A2 and A3, then y is B = f(A1, A2, A3) = A1 V A2 Λ A3

B = min(max(μA1(x1), μA2(x2)), μA3(x3))

The fuzzy conditional inference is given by using Mamdani fuzzy inference If x1 is A1 or A2 and A3, then y is B = min(A1 or A2 and A3, B)

If x1 is A1 or A2 and A3, then x is B = min(max(μA1(x1), μA2(x2)), μA3(x3)) Thus, the Reddy fuzzy conditional inference is satisfied.

If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min{A1, A2,…,An}. Justification of Mamdani fuzzy conditional inference may be derived in the

## following:

If some relation R between A and B is known, then Mamdani fuzzy conditional inference is given by:

If x is A, then y is B = AxB

Zadeh fuzzy conditional inference for "if … then … else …" is given by:

If x is A, then y is B else y is C = AxBvA<sup>0</sup> x C

If x is A, then y is B else y is C = If x is A then y is B v If x is A<sup>0</sup> then y is C = AxB v A<sup>0</sup> x C

It is logically divided into:

If x is A, then y is B = AxB

If x is A<sup>0</sup> , then y is C = A<sup>0</sup> x C

Thus, the Mamdani fuzzy conditional inference is satisfied.

If x is A, then y is B = A x B.

## 4. Fuzzy control systems using single fuzzy membership function

Zadeh introduced fuzzy algorithms. The fuzzy algorithm is a set of fuzzy statements. The fuzzy conditional statement is defined as fuzzy algorithm:

If xi is A1i and xi is A2i and… and xi is An, then y is Bi The consequent part may not be known in control systems The fuzziness may be given for Reddy fuzzy inference as If BZ is low (0.6) and BE is normal (0.7) then reduce fan speed = min (0.6, 0.7) = 0.6

The fuzzy set type-2 is a type of fuzzy set in which some additional degree of information is provided.

Definition: Given some universe of discourse X, a fuzzy set type-2 A of X is defined by its membership function μA(x) taking values on the unit interval [0,1], i.e., μÃ(x)➔[0,1][0.1]

Suppose X is a finite set. The fuzzy set A of X may be represented as

<sup>A</sup> <sup>=</sup> <sup>μ</sup>Ã1(x1)/Ã1 <sup>þ</sup> <sup>μ</sup>Ã2(x2)/Ã2 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup>Ãn(xn)/Ãn

Temperature = {0.4/low, 0.6/medium, 0.9/high}

John has "mild headache" with fuzziness 0.4

The fuzzy control system for boiler consists of a set of fuzzy rules [4].

If a set of conditions is satisfied, then the set of consequences is fired

The fuzzy control system is shown in Figure 7.

The fuzzy control system containing fuzzy variables are represented in decision Table 1.

The fuzzy control system of boiler is given in Table 2.

For instance,

If BZ is low

and BE is normal

then reduce fan speedFor instance, consider the fuzzy control system (Table 3). The computation of proposed method (3.4) is given in Table 4.

## Defuzzification

The centroid technique is used for defuzzification. It finds value representing the centre of gravity (COG) aggregated fuzzy generalized fuzzy set:

COG = Σ Ci μAi(x)/ Σ Ci

Condition Burning zone (BZ) temperature

(0.1 þ 0.3 þ 0.5 þ 0.7 þ 0.9) = 73.6

Speed = {0.1/20 þ 0.3/40 þ 0.5/60 þ 0.7/80 þ 0.9/100} COG = (0.1\*20 þ 0.3\*40 þ 0.5\*60 þ 0.7\*80 þ 0.9\*100)/

A1 A2 … An <sup>B</sup> A11 A12 … A1n B1 A21 A22 … A2n B2 ⁞ ⁞ ⁞ ⁞ Am1 Am2 … Amn Bmn

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

DOI: http://dx.doi.org/10.5772/intechopen.82700

Condition Burning zone (BZ) temperature Back-end (BE) temperature Action

Condition Burning zone (BZ) temperature Back-end (BE) temperature Action

AND Drastically low (0.7) Low (0.6) Reduce Klin speed AND Drastically low (0.7) Low (0.8) Reduce fuel AND Slightly low (.8) Low (.9) Increase fan speed AND Low (0.7) High (0.65) Reduce fuel AND Low (0.6) Normal (0.7) Reduce fan speed

> Back-end (BE) temperature

AND Drastically low (0.7) Low (0.6) Reduce Klin speed

AND Drastically low (0.7) Low (0.8) Reduce fuel (0.7) AND Slightly low (.8) Low (.9) Increase fan speed

AND Low (0.7) High (0.65) Reduce fuel (0.65) AND Low (0.6) Normal (0.7) Reduce fan speed (0.6)

Action

(0.6)

(0.8)

AND Drastically low Low Reduce Klin speed AND Drastically low Low Reduce fuel AND Slightly low Low Increase fan speed AND Low High Reduce fuel AND Low Normal Reduce fan speed

For instance,

Table 1. Fuzzy rules.

Table 2. Boiler controller.

Table 3.

Table 4. Fuzzy inference.

61

Boiler fuzzy controller.

Figure 7. Fuzzy control system.

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700


#### Table 1.

4. Fuzzy control systems using single fuzzy membership function

ments. The fuzzy conditional statement is defined as fuzzy algorithm:

If xi is A1i and xi is A2i and… and xi is An, then y is Bi The consequent part may not be known in control systems The fuzziness may be given for Reddy fuzzy inference as

<sup>A</sup> <sup>=</sup> <sup>μ</sup>Ã1(x1)/Ã1 <sup>þ</sup> <sup>μ</sup>Ã2(x2)/Ã2 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup>Ãn(xn)/Ãn Temperature = {0.4/low, 0.6/medium, 0.9/high} John has "mild headache" with fuzziness 0.4

The fuzzy control system is shown in Figure 7.

The fuzzy control system of boiler is given in Table 2.

The computation of proposed method (3.4) is given in Table 4.

the centre of gravity (COG) aggregated fuzzy generalized fuzzy set:

If BZ is low (0.6) and BE is normal (0.7) then reduce fan speed = min (0.6, 0.7)

information is provided.

i.e., μÃ(x)➔[0,1][0.1]

For instance, If BZ is low and BE is normal

Defuzzification

= 0.6

Fuzzy Logic

Table 1.

Figure 7.

60

Fuzzy control system.

Zadeh introduced fuzzy algorithms. The fuzzy algorithm is a set of fuzzy state-

The fuzzy set type-2 is a type of fuzzy set in which some additional degree of

Definition: Given some universe of discourse X, a fuzzy set type-2 A of X is defined by its membership function μA(x) taking values on the unit interval [0,1],

Suppose X is a finite set. The fuzzy set A of X may be represented as

The fuzzy control system for boiler consists of a set of fuzzy rules [4]. If a set of conditions is satisfied, then the set of consequences is fired

The fuzzy control system containing fuzzy variables are represented in decision

then reduce fan speedFor instance, consider the fuzzy control system (Table 3).

The centroid technique is used for defuzzification. It finds value representing

Fuzzy rules.


### Table 2.

Boiler controller.


#### Table 3.

Boiler fuzzy controller.


#### Table 4.

Fuzzy inference.

COG = Σ Ci μAi(x)/ Σ Ci For instance, Speed = {0.1/20 þ 0.3/40 þ 0.5/60 þ 0.7/80 þ 0.9/100} COG = (0.1\*20 þ 0.3\*40 þ 0.5\*60 þ 0.7\*80 þ 0.9\*100)/ (0.1 þ 0.3 þ 0.5 þ 0.7 þ 0.9) = 73.6


Table 5.

Twofold fuzziness.

## 5. Fuzzy logic with twofold fuzzy sets

Generalized fuzzy logic is studied for incomplete information [8, 9].

Given some universe of discourse X, the proposition "x is A" is defined as its twofold fuzzy set with membership function as

μvery A(x) = {μ<sup>A</sup>

Fuzzy membership function.

"x is more or less A" μmore or less A(x) = (μ<sup>A</sup>

Diffusion

Figure 8.

Implication

min (1, 1 � min(μA1

TrueFalse(x),…, <sup>μ</sup>An

= {min(μA1

= {min(μA1

False(x))}(x,y)

μA2

μAn

63

= {min (1, 1 � min(μA1

True(x)2

DOI: http://dx.doi.org/10.5772/intechopen.82700

, μ<sup>A</sup>

A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} B = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5}

0.9/x1 þ 0.8/x2 þ 0.7/x3 þ 0.8/x4 þ 0.9/x5}

0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5}

0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5}

0.4/x1 þ 0.3/x2 þ 0.4/x3 þ 0.5/x4 þ 0.6/x5}

Zadeh fuzzy conditional inference given as

False(x), μA2

Mamdani fuzzy conditional inference given as

False(x),μ<sup>B</sup>

Reddy [5] fuzzy conditional inference given by

1/x1 þ 1/x2 þ 1/x3 þ 0.8/x4 þ 1/x5}

True(x), μA2

True(x), μA2

True(x)1/2, μ<sup>A</sup>

A<sup>0</sup> = not A = {0.5/x1 þ 0.3/x2 þ 0.1/x3 þ 0.3/x4 þ 0.5/x5,

AVB = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5,

A Λ B = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5,

0.01/x1 þ 0.04/x2 þ 0.09/x3 þ 0.04/x4 þ 0.01/x5}

0.31/x1 þ 0.44/x2 þ 0.54/x3 þ 0.44/x4 þ 0.31/x5} A➔ B = {1/x1 þ 0.8/x2 + /x3 þ 0.9/x4 þ 1/x5,

AoB = {0.8/x1 þ 0.7/x2 þ 0.7/x3 þ 0.5/x4 þ 0.5/x5,

Very A = {0.25/x1 þ 0.49/x2 þ 0.81/x3 þ 0.49/x4 þ 0.25/x5,

The presidency part may contain any number of "and"/"or."

True(x), μA2

True(x),…, <sup>μ</sup>An

True(x),…, <sup>μ</sup>An

Consider the fuzzy condition "if x is A1 and x is A2, then x is B" The presidency part may contain any number of "and"/"or."

More or less A = {0.70/x1 þ 0.83/x2 þ 0.94/x3 þ 0.83/x4 þ 0.70/x5,

Consider the fuzzy condition "if x is A1 and x is A2 and .. and x is An, then y is B."

TrueFalse(x),…, <sup>μ</sup>An

False(y))}(x,y)

True(x),…, <sup>μ</sup>An

True(x), μ<sup>B</sup>

True(x)), min(μA1

True(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup>

True(y)), min(μA1

False(x), μA2

False(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup>

True(y)),

False(y))}(x,y)

False(x),

TrueFalse(x),…,

False(x)μA(x)<sup>2</sup>

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

}

False(x)μA(x)0.5

μA(x) = {μ<sup>A</sup> True(x), μ<sup>A</sup> False(x)} or A = {μ<sup>A</sup> True(x), μ<sup>A</sup> False(x)} where A is the seneralized fuzzy set and x Є X, 0 < = μ<sup>A</sup> True(x) < = 1 and, 0 < = μ<sup>A</sup> False(x) < = 1 A = {μ<sup>A</sup> True(x1)/x1 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup><sup>A</sup> True(xn)/xn, μA False(x1)/x1 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup><sup>A</sup> True(xn)/xn, xi Є X, μA True(x) <sup>þ</sup> <sup>μ</sup><sup>A</sup> False(x) < 1, μA True(x) <sup>þ</sup> <sup>μ</sup><sup>A</sup> False(x) >1 and μA True(x) <sup>þ</sup> <sup>μ</sup><sup>A</sup> False(x) = 1

The conditions are interpreted as redundant, insufficient and sufficient, respectively.

For instance,

A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5}

The graphical representation is shown in Figure 8.

The fuzzy logic is defined as a combination of fuzzy sets using logical operators. Some of the logical operations are given below.

Let A, B and C be the fuzzy sets. The operations on fuzzy sets are given below for twofold fuzzy sets.

Negation A<sup>0</sup> = {1�μ<sup>A</sup> True(x), 1�μ<sup>A</sup> False(x)}/x Disjunction AVB = {max(μ<sup>A</sup> True(x), μ<sup>A</sup> True(y)), max(μ<sup>B</sup> False(x), μ<sup>B</sup> False(y))}(x,y) Conjunction AΛB = {min(μ<sup>A</sup> True(x), μ<sup>A</sup> True(y)), min(μ<sup>B</sup> False(x), μ<sup>B</sup> False(y))}/(x,y) Composition AoR = {minx (μ<sup>A</sup> True(x), μ<sup>A</sup> True(x)), minx(μ<sup>R</sup> False(x), μ<sup>R</sup> False(x))}/y The fuzzy propositions may contain quantifiers like "very", "more or less".

These fuzzy quantifiers may be eliminated as follows:

## Concentration

"x is very A"

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700

#### Figure 8.

5. Fuzzy logic with twofold fuzzy sets

twofold fuzzy set with membership function as

False(x)} where A is the seneralized fuzzy set and x Є X,

True(x) < = 1 and, 0 < = μ<sup>A</sup>

False(x)}

True(x), μ<sup>A</sup>

True(x1)/x1 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup><sup>A</sup>

False(x) < 1,

False(x) = 1

Some of the logical operations are given below.

True(x), μ<sup>A</sup>

True(x), μ<sup>A</sup>

True(x), μ<sup>A</sup>

These fuzzy quantifiers may be eliminated as follows:

True(x), 1�μ<sup>A</sup>

False(x) >1 and

A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} The graphical representation is shown in Figure 8.

False(x)}/x

True(y)), max(μ<sup>B</sup>

True(y)), min(μ<sup>B</sup>

True(x)), minx(μ<sup>R</sup>

The fuzzy propositions may contain quantifiers like "very", "more or less".

True(x), μ<sup>A</sup>

Condition Burning zone (BZ) temperature

False(x1)/x1 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup><sup>A</sup>

True(x) <sup>þ</sup> <sup>μ</sup><sup>A</sup>

True(x) <sup>þ</sup> <sup>μ</sup><sup>A</sup>

True(x) <sup>þ</sup> <sup>μ</sup><sup>A</sup>

for twofold fuzzy sets. Negation A<sup>0</sup> = {1�μ<sup>A</sup>

Disjunction AVB = {max(μ<sup>A</sup>

Conjunction AΛB = {min(μ<sup>A</sup>

Composition AoR = {minx (μ<sup>A</sup>

Concentration "x is very A"

62

μA(x) = {μ<sup>A</sup>

0 < = μ<sup>A</sup>

A = {μ<sup>A</sup>

or A = {μ<sup>A</sup>

Table 5. Twofold fuzziness.

Fuzzy Logic

μA

μA

μA

μA

respectively. For instance,

Generalized fuzzy logic is studied for incomplete information [8, 9].

True(xn)/xn,

The conditions are interpreted as redundant, insufficient and sufficient,

The fuzzy logic is defined as a combination of fuzzy sets using logical operators.

Let A, B and C be the fuzzy sets. The operations on fuzzy sets are given below

False(x), μ<sup>B</sup>

False(x), μ<sup>B</sup>

False(x), μ<sup>R</sup>

False(y))}(x,y)

False(y))}/(x,y)

False(x))}/y

True(xn)/xn, xi Є X,

Given some universe of discourse X, the proposition "x is A" is defined as its

Back-end (BE) temperature

AND/OR Drastically low (0.7,0.1) Low (0.8,0.1) Reduce Klin speed

AND/OR Drastically low (0.8,0.1) Low (0.9,0.1) Reduce fuel (0.7,0.2) AND/OR Slightly low (1.0,0.2) Low (1.0,0.1) Increase fan speed

AND/OR Low (0.8,0.1) High (0.9,0.2) Reduce fuel (0.6,0.1) AND/OR Low (0.7,0.1) Normal (0.8,0.2) Reduce fan speed

Action

(0.6,0.2)

(0.9,0.2)

(0.5,0.1)

False(x) < = 1

Fuzzy membership function.

μvery A(x) = {μ<sup>A</sup> True(x)2 , μ<sup>A</sup> False(x)μA(x)<sup>2</sup> } Diffusion "x is more or less A" μmore or less A(x) = (μ<sup>A</sup> True(x)1/2, μ<sup>A</sup> False(x)μA(x)0.5 A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} B = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} A<sup>0</sup> = not A = {0.5/x1 þ 0.3/x2 þ 0.1/x3 þ 0.3/x4 þ 0.5/x5, 0.9/x1 þ 0.8/x2 þ 0.7/x3 þ 0.8/x4 þ 0.9/x5} AVB = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} A Λ B = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} Very A = {0.25/x1 þ 0.49/x2 þ 0.81/x3 þ 0.49/x4 þ 0.25/x5, 0.01/x1 þ 0.04/x2 þ 0.09/x3 þ 0.04/x4 þ 0.01/x5} More or less A = {0.70/x1 þ 0.83/x2 þ 0.94/x3 þ 0.83/x4 þ 0.70/x5, 0.31/x1 þ 0.44/x2 þ 0.54/x3 þ 0.44/x4 þ 0.31/x5} A➔ B = {1/x1 þ 0.8/x2 + /x3 þ 0.9/x4 þ 1/x5, 1/x1 þ 1/x2 þ 1/x3 þ 0.8/x4 þ 1/x5} AoB = {0.8/x1 þ 0.7/x2 þ 0.7/x3 þ 0.5/x4 þ 0.5/x5, 0.4/x1 þ 0.3/x2 þ 0.4/x3 þ 0.5/x4 þ 0.6/x5} Implication Consider the fuzzy condition "if x is A1 and x is A2 and .. and x is An, then y is B." The presidency part may contain any number of "and"/"or." Zadeh fuzzy conditional inference given as = {min (1, 1 � min(μA1 True(x), μA2 True(x),…, <sup>μ</sup>An True(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup> True(y)), min (1, 1 � min(μA1 False(x), μA2 TrueFalse(x),…, <sup>μ</sup>An False(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup> False(y))}(x,y) Mamdani fuzzy conditional inference given as = {min(μA1 True(x), μA2 True(x),…, <sup>μ</sup>An True(x), μ<sup>B</sup> True(y)), min(μA1 False(x), μA2 TrueFalse(x),…, <sup>μ</sup>An False(x),μ<sup>B</sup> False(y))}(x,y) Reddy [5] fuzzy conditional inference given by = {min(μA1 True(x), μA2 True(x),…, <sup>μ</sup>An True(x)), min(μA1 False(x), μA2 TrueFalse(x),…, μAn False(x))}(x,y) Consider the fuzzy condition "if x is A1 and x is A2, then x is B"

The presidency part may contain any number of "and"/"or."

For instance, A1 = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} A2 = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} B = {0.5/x1 þ 0.7/x2 þ 1/x3 þ 0.7/x4 þ 0.5/x5, 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5} Zadeh fuzzy conditional inference given as ={min (1, 1�min(μA1 True(x), μA2 True(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup> True(x)), min (1, 1�min (μA1 False(x), μA2 TrueFalse(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup> False(x))} = {1/x1 þ 0.1/x2 þ 1/x3 þ 1/x4 þ 1/x5, 1/x1 þ 1/x2 þ 1/x3 þ 1/x4 þ 1/x5} Mamdani fuzzy conditional inference given as = {min(μA1 True(x), μA2 True(x),…, <sup>μ</sup>An True(x), μ<sup>B</sup> True(x)), min(μA1 False(x), μA2 TrueFalse(x),…, <sup>μ</sup>An False(x), μ<sup>B</sup> False(x))} = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} Reddy fuzzy conditional inference given as ={min(μA1 True(x), μA2 True(x)), min(μA1 False(x), μA2 TrueFalse(x))} = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} Composition If some relation R between A and B is known and some value A1 than B1 is inferred from R, B1 = A1 o R, where R = A➔B Zadeh fuzzy inference is given by B1 = A1 o R == A1o{min (1, 1 � μ<sup>A</sup> True(x) <sup>þ</sup> <sup>μ</sup><sup>B</sup> True(x)), min (1, 1 � μA False(x) <sup>þ</sup> <sup>μ</sup><sup>B</sup> False(x))} = min{μA(x), min(1,1- μA1(x) þ μB(x)} Mamdani fuzzy inference is given by = A1o{min (μ<sup>A</sup> True(x), μ<sup>B</sup> True(x)), min (μ<sup>A</sup> TrueFalse(x), μ<sup>B</sup> False(x)} If some relation R between A and B is not known, according to Reddy fuzzy inference, = {min (μA1 True(x), μ<sup>A</sup> True(x)), min (μA1 TrueFalse(x), μ<sup>A</sup> False(x)} The fuzzy set A of X is characterized as its membership function A = μA(x) and ranging values in the unit interval [0, 1] μA(x): X ➔[0, 1], x Є X, where X is universe of discourse. <sup>A</sup> <sup>=</sup> <sup>μ</sup>A(x) <sup>=</sup> <sup>μ</sup>A(x1)/x1 <sup>þ</sup> <sup>μ</sup>A(x2)/x2 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup>A(xn)/xn, "+" is union The generalized fuzzy certainty factor (GFCF) is defined as μA GFCF(x) = μ<sup>A</sup> True(x) � <sup>μ</sup><sup>A</sup> False(x) The generalized fuzzy certainty factor becomes single fuzzy membership function. μA GFCF(x): X➔[0, 1], x Є X, where X is universe of discourse. The generalized fuzzy certainty factor (GFCF) will compute the conflict of evidence in the uncertain information. For example, A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} μA GFCF(x) <sup>=</sup> {0.5/x1 <sup>þ</sup> 0.7/x2 <sup>þ</sup> 0.9/x3 <sup>þ</sup> 0.7/x4 <sup>þ</sup> 0.5/x5 � 0.1/x1 <sup>þ</sup> 0.2/x2 <sup>þ</sup> 0.3/ x3 þ 0.2/x4 þ 0.1/x5} = 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5

The GFCF is 0.6

Negation A<sup>0</sup> = 1 � μ<sup>A</sup>

Conjunction

Disjunction

Concentration

μvey A

Figure 9.

Figure 10. Negation.

65

Generalized fuzzy certainty factor.

GFCF(x)/x

DOI: http://dx.doi.org/10.5772/intechopen.82700

AΛB = min(μA(x), μB(x)}/x

AVB = max(μA(x), μB(y)}/x

GFCF(x) = μ<sup>A</sup>

= 0.6/x1 þ 0.5/x2 þ 0.4/x3 þ 0.5x4 þ 0.6/x5

The graphical representation is shown in Figure 10.

AΛB = 0.3/x1 þ 0.4/x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5 The graphical representation is shown in Figure 11.

AVB = .4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 The graphical representation is shown in Figure 12.

GFCF(x)2

The graphical representation of GFCF is shown in Figure 9.

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

For example, A and B are generalized fuzzy sets. A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5 � 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5 B = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5 = 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.3/x1 þ 0.4x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5 The operations on GFCF are given as follows:

For instance, "x is high temperature" with fuzziness {0.8,0.2}

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700

The GFCF is 0.6 The graphical representation of GFCF is shown in Figure 9. For example, A and B are generalized fuzzy sets. A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5 � 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5 B = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5 = 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.3/x1 þ 0.4x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5 The operations on GFCF are given as follows: Negation A<sup>0</sup> = 1 � μ<sup>A</sup> GFCF(x)/x = 0.6/x1 þ 0.5/x2 þ 0.4/x3 þ 0.5x4 þ 0.6/x5 The graphical representation is shown in Figure 10. Conjunction AΛB = min(μA(x), μB(x)}/x AΛB = 0.3/x1 þ 0.4/x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5 The graphical representation is shown in Figure 11. Disjunction AVB = max(μA(x), μB(y)}/x AVB = .4/x1 þ 0.6/x2 þ 0.9/x3 þ 0.6/x4 þ 0.2/x5 The graphical representation is shown in Figure 12. Concentration μvey A GFCF(x) = μ<sup>A</sup> GFCF(x)2

#### Figure 9.

For instance,

Fuzzy Logic

={min (1, 1�min(μA1

False(x), μA2

= {min(μA1

={min(μA1

Composition

inferred from R, B1 = A1 o R, where R = A➔B

False(x) <sup>þ</sup> <sup>μ</sup><sup>B</sup>

= A1o{min (μ<sup>A</sup>

= {min (μA1

GFCF(x) = μ<sup>A</sup>

For example,

x3 þ 0.2/x4 þ 0.1/x5}

μA

function. μA

μA

64

TrueFalse(x),…, <sup>μ</sup>An

(μA1

μA2

μA

A1 = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} A2 = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} B = {0.5/x1 þ 0.7/x2 þ 1/x3 þ 0.7/x4 þ 0.5/x5, 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5} Zadeh fuzzy conditional inference given as

True(x), μA2

True(x),…, <sup>μ</sup>An

True(x)), min(μA1

TrueFalse(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup>

Mamdani fuzzy conditional inference given as

False(x), μ<sup>B</sup>

= {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} Reddy fuzzy conditional inference given as

= {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5}

= {1/x1 þ 0.1/x2 þ 1/x3 þ 1/x4 þ 1/x5, 1/x1 þ 1/x2 þ 1/x3 þ 1/x4 þ 1/x5}

True(x), μA2

True(x), μA2

Zadeh fuzzy inference is given by B1 = A1 o R == A1o{min (1, 1 � μ<sup>A</sup>

False(x))}

= min{μA(x), min(1,1- μA1(x) þ μB(x)} Mamdani fuzzy inference is given by

True(x), μ<sup>B</sup>

True(x) � <sup>μ</sup><sup>A</sup>

A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5, 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5}

= 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5

according to Reddy fuzzy inference,

True(x), μ<sup>A</sup>

ranging values in the unit interval [0, 1]

evidence in the uncertain information.

If some relation R between A and B is not known,

True(x)) <sup>þ</sup> <sup>μ</sup><sup>B</sup>

True(x), μ<sup>B</sup>

False(x), μA2

False(x))}

False(x))}

If some relation R between A and B is known and some value A1 than B1 is

True(x)), min (μ<sup>A</sup>

True(x)), min (μA1

<sup>A</sup> <sup>=</sup> <sup>μ</sup>A(x) <sup>=</sup> <sup>μ</sup>A(x1)/x1 <sup>þ</sup> <sup>μ</sup>A(x2)/x2 <sup>þ</sup> … <sup>þ</sup> <sup>μ</sup>A(xn)/xn, "+" is union The generalized fuzzy certainty factor (GFCF) is defined as

The generalized fuzzy certainty factor becomes single fuzzy membership

The generalized fuzzy certainty factor (GFCF) will compute the conflict of

GFCF(x) <sup>=</sup> {0.5/x1 <sup>þ</sup> 0.7/x2 <sup>þ</sup> 0.9/x3 <sup>þ</sup> 0.7/x4 <sup>þ</sup> 0.5/x5 � 0.1/x1 <sup>þ</sup> 0.2/x2 <sup>þ</sup> 0.3/

False(x)

GFCF(x): X➔[0, 1], x Є X, where X is universe of discourse.

For instance, "x is high temperature" with fuzziness {0.8,0.2}

μA(x): X ➔[0, 1], x Є X, where X is universe of discourse.

True(x) <sup>þ</sup> <sup>μ</sup><sup>B</sup>

The fuzzy set A of X is characterized as its membership function A = μA(x) and

True(x)), min (1, 1�min

True(x)), min(μA1

TrueFalse(x))}

True(x)), min (1, 1 �

False(x)}

False(x)}

TrueFalse(x), μ<sup>B</sup>

TrueFalse(x), μ<sup>A</sup>

False(x),

Generalized fuzzy certainty factor.

Figure 10. Negation.

If x1 is A1 and x2 is A2 and … and xn is An, then y is B The presidency part may contain any number of "and"/"or."

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

If x1 is A1 and x2 is A2 and … and xn is An, then y is B

If x1 is A1 and x2 is A2 and … and xn is An, then y is B

If x1 is A1 and x2 is A2 and … and xn is An, then y is B

A1 = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5 �

The graphical representation of A1, A2 and B is shown in Figure 14.

The graphical representation of fuzzy inference is shown in Figure 15.

0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5 A2 = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5 =

0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.3/x1 þ 0.4x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5 B = {0.5/x1 þ 0.7/x2 þ 1/x3 þ 0.7/x4 þ 0.5/x5, 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5} = 0.1/x1 þ 0.2x2 þ 0.4/x3 þ 0.2/x4 þ 0.1/x5

= 0.8/x1 þ 0.8/x2 þ 0.9/x3 þ 0.8/x4 þ 0.8/x5

= 0.1/x1 þ 0.2/x2 þ 0.4/x3 þ 0.2/x4 þ 0.1/x5

= 0.2/x1 þ 0.4/x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5

The GFCF is a single fuzzy membership function

Zadeh fuzzy inference is given as follows:

Mamdani fuzzy inference is given as follows:

<sup>=</sup> min(1, 1 � (A1, A2,…,An) <sup>þ</sup> B)

DOI: http://dx.doi.org/10.5772/intechopen.82700

Reddy inference is given as follows:

If x1 is A1 and x2 is A2, then x is B

Zadeh fuzzy inference is given as = min(1, 1 � (A1, A2) þ B)

Mamdani fuzzy inference is given as

Reddy fuzzy inference is given as

min(A1, A2,…, An, B)

min(A1, A2,…, An)

Composition

Figure 14. GFCF for fuzzy rule.

67

<sup>=</sup> min(A1, A2,…, An, B)

<sup>=</sup> min(A1, A2,…, An) Consider the fuzzy rule:

For instance,

Figure 11. Conjunction.

= 0.16/x1 þ 0.25/x2 þ 0.36/x3 þ 0.25/x4 þ 0.16/x5 Diffusion μmore or less A GFCF(x) = μ<sup>A</sup> GFCF(x)0.5 = 0.63/x1 þ 0.71/x2 þ 0.77/x3 þ 0.71/x4 þ 0.63/x5

The graphical representation of concentration and diffusion are shown in Figure 13.

## Implication

Zadeh [9], Mamdani [7] and Reddy [5] fuzzy conditional inferences are considered keeping in view of fuzzy control systems.

#### Figure 13. Implication.

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700

If x1 is A1 and x2 is A2 and … and xn is An, then y is B The presidency part may contain any number of "and"/"or." Zadeh fuzzy inference is given as follows: If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min(1, 1 � (A1, A2,…,An) <sup>þ</sup> B) Mamdani fuzzy inference is given as follows: If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min(A1, A2,…, An, B) Reddy inference is given as follows: If x1 is A1 and x2 is A2 and … and xn is An, then y is B <sup>=</sup> min(A1, A2,…, An) Consider the fuzzy rule: If x1 is A1 and x2 is A2, then x is B For instance, A1 = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5 � 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5 A2 = {0.4/x1 þ 0.6/x2 þ 0.8/x3 þ 0.6/x4 þ 0.4/x5 = 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.3/x1 þ 0.4x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5 B = {0.5/x1 þ 0.7/x2 þ 1/x3 þ 0.7/x4 þ 0.5/x5, 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5} = 0.1/x1 þ 0.2x2 þ 0.4/x3 þ 0.2/x4 þ 0.1/x5 The graphical representation of A1, A2 and B is shown in Figure 14. Zadeh fuzzy inference is given as = min(1, 1 � (A1, A2) þ B) = 0.8/x1 þ 0.8/x2 þ 0.9/x3 þ 0.8/x4 þ 0.8/x5 Mamdani fuzzy inference is given as min(A1, A2,…, An, B) = 0.1/x1 þ 0.2/x2 þ 0.4/x3 þ 0.2/x4 þ 0.1/x5 Reddy fuzzy inference is given as min(A1, A2,…, An) = 0.2/x1 þ 0.4/x2 þ 0.5/x3 þ 0.4/x4 þ 0.3/x5 The graphical representation of fuzzy inference is shown in Figure 15. Composition

The GFCF is a single fuzzy membership function

Figure 14. GFCF for fuzzy rule.

= 0.16/x1 þ 0.25/x2 þ 0.36/x3 þ 0.25/x4 þ 0.16/x5

= 0.63/x1 þ 0.71/x2 þ 0.77/x3 þ 0.71/x4 þ 0.63/x5

considered keeping in view of fuzzy control systems.

GFCF(x)0.5

The graphical representation of concentration and diffusion are shown in

Zadeh [9], Mamdani [7] and Reddy [5] fuzzy conditional inferences are

GFCF(x) = μ<sup>A</sup>

Diffusion μmore or less A

Implication

Figure 13.

Figure 11. Conjunction.

Fuzzy Logic

Figure 12. Disjunction.

Figure 13. Implication.

66

6. Fuzzy control systems using two fuzzy membership functions

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

If a set of conditions is satisfied, then a set of consequences is inferred. The fuzzy set with twofold membership function will give more information

statements. The fuzzy conditional statement is defined as follows: If xi is A1i and xi is A2i and … and xi is An, then yi is Bi

The fuzzy control system consist of a set of fuzzy rules.

The generalized fuzzy certainty factor (GFCF) is given as

True(x) <sup>μ</sup>Low

False(x)}

The precedence part may contain and/or/not.

True(x) <sup>μ</sup><sup>A</sup>

Consider the rule in fuzzy control system

For instance, fuzziness may be given as follows:

Fuzziness of GFCF may be given as follows:

than the single membership function.

DOI: http://dx.doi.org/10.5772/intechopen.82700

For instance, "x has fever" The GFCF for fever given as

GFCF(x) = {μLow

If BZ is low (0.9,0.2) and BE is normal (0.8,0.2) then reduce fan speed (0.6, 0.3)

If BZ is low (0.7) and BE is normal (0.6) then reduce fan speed (0.3)

"AND" is shown in Figure 18.

is shown in Figure 19.

GFCF(x) = {μ<sup>A</sup>

μA

μLow

system.

Figure 17.

Figure 17. GFCF for Table 5.

69

If BZ is low and BE is normal then reduce fan speed

Zadeh [6] introduced fuzzy algorithms. The fuzzy algorithm is a set of fuzzy

False(x)}

For instance, consider the twofold fuzzy relational model of fuzzy control

The graphical representation of twofold fuzzy relational model is shown in

The graphical representation of fuzzy inference for condition part containing

Graphical representation of fuzzy inference for condition part containing "OR"

Figure 15. Implication.

If some relation R between A1, then B1 is to infer from R: B1 = A1 o R = min{μA1 GFCF(x), μ<sup>R</sup> GFCF(x)}/x Zadeh fuzzy inference is given by B1 = A1 o R = min{μA1 GFCF(x), μ<sup>R</sup> GFCF(x)} = min{μA1 GFCF(x),min(1,1- μA1 GFCF(x) <sup>þ</sup> <sup>μ</sup><sup>B</sup> GFCF(x))} Mamdani fuzzy inference is given by = min{μA1 GFCF(x), μA1 GFCF(x), μ<sup>B</sup> GFCF(x)} If there is some relation R between A and B, then Reddy fuzzy inference is given by = μA1 GFCF(x) where A, B, A1, and B1 are the GFCF. A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5 � 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5 B = {0.5/x1 þ 0.7/x2 þ 1/x3 þ 0.7/x4 þ 0.5/x5, 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5} = 0.1/x1 þ 0.2x2 þ 0.4/x3 þ 0.2/x4 þ 0.1/x5 A1 = more or less A = = 0.55/x1 þ 0.63/x2 þ 0.71/x3 þ 0.63/x4 þ 0.55/x5

The composition of Zadeh, Mamdani and Reddy fuzzy inference is shown in Figure 16.

Figure 16. Composition.

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700

## 6. Fuzzy control systems using two fuzzy membership functions

Zadeh [6] introduced fuzzy algorithms. The fuzzy algorithm is a set of fuzzy statements. The fuzzy conditional statement is defined as follows:

If xi is A1i and xi is A2i and … and xi is An, then yi is Bi

The precedence part may contain and/or/not.

The fuzzy control system consist of a set of fuzzy rules.

If a set of conditions is satisfied, then a set of consequences is inferred.

The fuzzy set with twofold membership function will give more information than the single membership function.

The generalized fuzzy certainty factor (GFCF) is given as μA GFCF(x) = {μ<sup>A</sup> True(x) <sup>μ</sup><sup>A</sup> False(x)} For instance, "x has fever" The GFCF for fever given as μLow GFCF(x) = {μLow True(x) <sup>μ</sup>Low False(x)} Consider the rule in fuzzy control system If BZ is low and BE is normal then reduce fan speed For instance, fuzziness may be given as follows: If BZ is low (0.9,0.2) and BE is normal (0.8,0.2) then reduce fan speed (0.6, 0.3) Fuzziness of GFCF may be given as follows: If BZ is low (0.7) and BE is normal (0.6) then reduce fan speed (0.3)

For instance, consider the twofold fuzzy relational model of fuzzy control system.

The graphical representation of twofold fuzzy relational model is shown in Figure 17.

The graphical representation of fuzzy inference for condition part containing "AND" is shown in Figure 18.

Graphical representation of fuzzy inference for condition part containing "OR" is shown in Figure 19.

Figure 17. GFCF for Table 5.

If some relation R between A1, then B1 is to infer from R:

GFCF(x), μ<sup>R</sup>

GFCF(x), μ<sup>R</sup>

GFCF(x), μ<sup>B</sup>

A = {0.5/x1 þ 0.7/x2 þ 0.9/x3 þ 0.7/x4 þ 0.5/x5 � 0.1/x1 þ 0.2/x2 þ 0.3/x3 þ 0.2/x4 þ 0.1/x5} = 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5 B = {0.5/x1 þ 0.7/x2 þ 1/x3 þ 0.7/x4 þ 0.5/x5, 0.4/x1 þ 0.5/x2 þ 0.6/x3 þ 0.5/x4 þ 0.4/x5} = 0.1/x1 þ 0.2x2 þ 0.4/x3 þ 0.2/x4 þ 0.1/x5

= = 0.55/x1 þ 0.63/x2 þ 0.71/x3 þ 0.63/x4 þ 0.55/x5

GFCF(x)}/x

GFCF(x)}

GFCF(x))}

GFCF(x) <sup>þ</sup> <sup>μ</sup><sup>B</sup>

GFCF(x)} If there is some relation R between A and B, then Reddy fuzzy inference is

The composition of Zadeh, Mamdani and Reddy fuzzy inference is shown in

B1 = A1 o R = min{μA1

B1 = A1 o R = min{μA1

GFCF(x)

A1 = more or less A

= min{μA1

= min{μA1

given by = μA1

Figure 15. Implication.

Fuzzy Logic

Figure 16.

Figure 16. Composition.

68

Zadeh fuzzy inference is given by

GFCF(x),min(1,1- μA1

Mamdani fuzzy inference is given by

where A, B, A1, and B1 are the GFCF.

GFCF(x), μA1

fuzzy conditional inference based on single membership function and twofold fuzzy set are studied. The FCF is studied as difference between "True" and "False" membership functions to eliminate conflict of evidence and to make as single fuzzy membership function. FCF = [True-False] will correct truthiness of single membership function. The methods of Zadeh, Mamdani and Reddy fuzzy conditional

Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems

The author states that he has no conflict of interest and that he has permission to

inference studied for fuzzy control systems are given as application.

use parts of his previously published work from the original publisher.

Department of Computer Science and Engineering, College of Engineering,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Conflict of interest

DOI: http://dx.doi.org/10.5772/intechopen.82700

Author details

71

Poli Venkata Subba Reddy

Sri Venkateswara University, Tirupati, India

provided the original work is properly cited.

\*Address all correspondence to: pvsreddy@hotmail.co.in

Figure 18. Fuzzy conditional inference for "AND."

Figure 19.

Fuzzy conditional inference for "OR."

#### Figure 20. Defuzzification.

## Defuzzification

Usually, centroid technique is used for defuzzification. It finds value representing the centre of gravity (COG) aggregated fuzzy generalized fuzzy set.

COG = Σ Ci μAi GFCF(x)/ Σ Ci For instance, Speed = {0.1/20 þ 0.3/40 þ 0.5/60 þ 0.7/80 þ 0.9/100} COG = (0.1\*20 þ 0.3\*40 þ 0.5\*60 þ 0.7\*80 þ 0.9\*100)/ (0.1 þ 0.3 þ 0.5 þ 0.7 þ 0.9) = 73.6 The defuzzification is shown in Figure 20.

## 7. Conclusion

The fuzzy set of two membership function will give more information than single fuzzy membership function for incomplete information. The fuzzy logic and Some Methods of Fuzzy Conditional Inference for Application to Fuzzy Control Systems DOI: http://dx.doi.org/10.5772/intechopen.82700

fuzzy conditional inference based on single membership function and twofold fuzzy set are studied. The FCF is studied as difference between "True" and "False" membership functions to eliminate conflict of evidence and to make as single fuzzy membership function. FCF = [True-False] will correct truthiness of single membership function. The methods of Zadeh, Mamdani and Reddy fuzzy conditional inference studied for fuzzy control systems are given as application.

## Conflict of interest

The author states that he has no conflict of interest and that he has permission to use parts of his previously published work from the original publisher.

## Author details

Defuzzification

Figure 18.

Fuzzy Logic

Figure 19.

Figure 20. Defuzzification.

Fuzzy conditional inference for "AND."

Fuzzy conditional inference for "OR."

COG = Σ Ci μAi

For instance,

7. Conclusion

70

Usually, centroid technique is used for defuzzification. It finds value representing the centre of gravity (COG) aggregated fuzzy generalized fuzzy set.

The fuzzy set of two membership function will give more information than single fuzzy membership function for incomplete information. The fuzzy logic and

GFCF(x)/ Σ Ci

(0.1 þ 0.3 þ 0.5 þ 0.7 þ 0.9) = 73.6 The defuzzification is shown in Figure 20.

Speed = {0.1/20 þ 0.3/40 þ 0.5/60 þ 0.7/80 þ 0.9/100} COG = (0.1\*20 þ 0.3\*40 þ 0.5\*60 þ 0.7\*80 þ 0.9\*100)/ Poli Venkata Subba Reddy Department of Computer Science and Engineering, College of Engineering, Sri Venkateswara University, Tirupati, India

\*Address all correspondence to: pvsreddy@hotmail.co.in

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 5

Expert Systems

75
