 Open access peer-reviewed chapter

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

Written By

Poli Venkata Subba Reddy

Submitted: September 27th, 2018 Reviewed: November 24th, 2018 Published: October 23rd, 2019

DOI: 10.5772/intechopen.82700

From the Edited Volume

## Fuzzy Logic

Edited by Constantin Volosencu

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## 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, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 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 , Mamdani , TSK  and Reddy  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  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.

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  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.

A = μA(x1)/x1 + μA(x2)/x2 + … + μA(xn)/xn,

where “+” is the union.

For instance, the fuzzy proposition “x is High”

High = 0.2/x1 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.2/x5

Not High = 0.8/x1 + 0.4/x2 + 0.1/x3 + 0.4/x4 + 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′ = 1 − μA(x)/x

Conjunction

x is A and y is B➔ (x, y) is A ΛB

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 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.2/x5

B = 0.4/x1 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.1/x5

AΛB = 0.2/x1 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.1/x5

The graphical representation is shown in Figures 1 and 2.

Disjunction

x is A and y is B➔ (x, y) is A V B

A V B = 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 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.2/x5

B = 0.4/x1 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.1/x5

AVB = 0.4/x1 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.2/x5

The graphical representation is shown in Figure 3.

Concentration

μvery A(x) = μA(x)2

Diffusion

μmore or less A(x) = μA(x)0.5

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

Implication

Zadeh , Mamdani  and Reddy  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  fuzzy inference is given as:

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

= min(1, 1 − (A1, A2,…, An) + B)

Mamdani  fuzzy inference is given as:

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

= 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  inference is given by:

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

= 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 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.2/x5

A2 = 0.5/x1 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.3/x5

B = 0.1/x1 + 0.4/x2 + 0.6/x3 + 0.4/x4 + 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) + μB(x))}

Mamdani fuzzy inference is given by:

= min{μA1(x),μA(x) + μB(x)}

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

= μA1(x)

## 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 = 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 = 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 = 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 = 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 = A x B

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

If x is A, then y is B else y is C = A x B v A′ 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′ then y is C = A x B v A′ x C

It is logically divided into:

If x is A, then y is B = A x B

If x is A′, then y is C = A′ 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

A = μÃ1(x1)/Ã1 + μÃ2(x2)/Ã2 + … + μÃn(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 .

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.

A1A2AnB
A11A12A1nB1
A21A22A2nB2
Am1Am2AmnBmn

### Table 1.

Fuzzy rules.

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

ConditionBurning zone (BZ) temperatureBack-end (BE) temperatureAction
ANDDrastically lowLowReduce Klin speed
ANDDrastically lowLowReduce fuel
ANDSlightly lowLowIncrease fan speed
ANDLowHighReduce fuel
ANDLowNormalReduce fan speed

### Table 2.

Boiler controller.

For instance,

If BZ is low

and BE is normal

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

ConditionBurning zone (BZ) temperatureBack-end (BE) temperatureAction
ANDDrastically low (0.7)Low (0.6)Reduce Klin speed
ANDDrastically low (0.7)Low (0.8)Reduce fuel
ANDSlightly low (.8)Low (.9)Increase fan speed
ANDLow (0.7)High (0.65)Reduce fuel
ANDLow (0.6)Normal (0.7)Reduce fan speed

### Table 3.

Boiler fuzzy controller.

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

ConditionBurning zone (BZ) temperatureBack-end (BE) temperatureAction
ANDDrastically low (0.7)Low (0.6)Reduce Klin speed
(0.6)
ANDDrastically low (0.7)Low (0.8)Reduce fuel (0.7)
ANDSlightly low (.8)Low (.9)Increase fan speed (0.8)
ANDLow (0.7)High (0.65)Reduce fuel (0.65)
ANDLow (0.6)Normal (0.7)Reduce fan speed (0.6)

### Table 4.

Fuzzy inference.

ConditionBurning zone (BZ) temperatureBack-end (BE) temperatureAction
AND/ORDrastically low (0.7,0.1)Low (0.8,0.1)Reduce Klin speed (0.6,0.2)
AND/ORDrastically low (0.8,0.1)Low (0.9,0.1)Reduce fuel (0.7,0.2)
AND/ORSlightly low (1.0,0.2)Low (1.0,0.1)Increase fan speed (0.9,0.2)
AND/ORLow (0.8,0.1)High (0.9,0.2)Reduce fuel (0.6,0.1)
AND/ORLow (0.7,0.1)Normal (0.8,0.2)Reduce fan speed (0.5,0.1)

### Table 5.

Twofold fuzziness.

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

For instance,

Speed = {0.1/20 + 0.3/40 + 0.5/60 + 0.7/80 + 0.9/100}

COG = (0.1*20 + 0.3*40 + 0.5*60 + 0.7*80 + 0.9*100)/

(0.1 + 0.3 + 0.5 + 0.7 + 0.9) = 73.6

## 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

μA(x) = {μATrue(x), μAFalse(x)}

or

A = {μATrue(x), μAFalse(x)}

where A is the seneralized fuzzy set and x Є X,

0 < = μATrue(x) < = 1 and, 0 < = μAFalse(x) < = 1

A = {μATrue(x1)/x1 + … + μATrue(xn)/xn,

μAFalse(x1)/x1 + … + μATrue(xn)/xn, xi Є X,

μATrue(x) + μAFalse(x) < 1,

μATrue(x) + μAFalse(x) >1 and

μATrue(x) + μAFalse(x) = 1

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

For instance,

A = {0.5/x1 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5,

0.1/x1 + 0.2/x2 + 0.3/x3 + 0.2/x4 + 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′ = {1−μATrue(x), 1−μAFalse(x)}/x

Disjunction

AVB = {max(μATrue(x), μATrue(y)), max(μBFalse(x), μBFalse(y))}(x,y)

Conjunction

AΛB = {min(μATrue(x), μATrue(y)), min(μBFalse(x), μBFalse(y))}/(x,y)

Composition

A o R = {minxATrue(x), μATrue(x)), minxRFalse(x), μRFalse(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”

μvery A(x) = {μATrue(x)2, μAFalse(x)μA(x)2}

Diffusion

“x is more or less A”

μmore or less A(x) = (μATrue(x)1/2, μAFalse(x)μA(x)0.5

A = {0.5/x1 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5,

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

B = {0.4/x1 + 0.6/x2 + 0.8/x3 + 0.6/x4 + 0.4/x5,

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

A′ = not A = {0.5/x1 + 0.3/x2 + 0.1/x3 + 0.3/x4 + 0.5/x5,

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

A V B = {0.5/x1 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5,

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

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

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

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

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

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

0.31/x1 + 0.44/x2 + 0.54/x3 + 0.44/x4 + 0.31/x5}

A➔ B = {1/x1 + 0.8/x2 + /x3 + 0.9/x4 + 1/x5,

1/x1 + 1/x2 + 1/x3 + 0.8/x4 + 1/x5}

A o B = {0.8/x1 + 0.7/x2 + 0.7/x3 + 0.5/x4 + 0.5/x5,

0.4/x1 + 0.3/x2 + 0.4/x3 + 0.5/x4 + 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(μA1True(x), μA2True(x),…, μAnTrue(x)) + μBTrue(y)), min (1, 1 − min(μA1False(x), μA2TrueFalse(x),…, μAnFalse(x)) + μBFalse(y))}(x,y)

Mamdani fuzzy conditional inference given as

= {min(μA1True(x), μA2True(x),…, μAnTrue(x), μBTrue(y)), min(μA1False(x), μA2TrueFalse(x),…, μAnFalse(x),μBFalse(y))}(x,y)

Reddy  fuzzy conditional inference given by

= {min(μA1True(x), μA2True(x),…, μAnTrue(x)), min(μA1False(x), μA2TrueFalse(x),…, μAnFalse(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 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5,

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

A2 = {0.4/x1 + 0.6/x2 + 0.8/x3 + 0.6/x4 + 0.4/x5,

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

B = {0.5/x1 + 0.7/x2 + 1/x3 + 0.7/x4 + 0.5/x5,

0.4/x1 + 0.5/x2 + 0.6/x3 + 0.5/x4 + 0.4/x5}

Zadeh fuzzy conditional inference given as

={min (1, 1−min(μA1True(x), μA2True(x)) + μBTrue(x)), min (1, 1−min(μA1False(x), μA2TrueFalse(x)) + μBFalse(x))}

= {1/x1 + 0.1/x2 + 1/x3 + 1/x4 + 1/x5,

1/x1 + 1/x2 + 1/x3 + 1/x4 + 1/x5}

Mamdani fuzzy conditional inference given as

= {min(μA1True(x), μA2True(x),…, μAnTrue(x), μBTrue(x)), min(μA1False(x), μA2TrueFalse(x),…, μAnFalse(x), μBFalse(x))}

= {0.4/x1 + 0.6/x2 + 0.8/x3 + 0.6/x4 + 0.4/x5,

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

Reddy fuzzy conditional inference given as

={min(μA1True(x), μA2True(x)), min(μA1False(x), μA2TrueFalse(x))}

= {0.4/x1 + 0.6/x2 + 0.8/x3 + 0.6/x4 + 0.4/x5,

0.1/x1 + 0.2/x2 + 0.3/x3 + 0.2/x4 + 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 − μATrue(x) + μBTrue(x)), min (1, 1 − μAFalse(x) + μBFalse(x))}

= min{μA(x), min(1,1- μA1(x) + μB(x)}

Mamdani fuzzy inference is given by

= A1o{min (μATrue(x), μBTrue(x)), min (μATrueFalse(x), μBFalse(x)}

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

according to Reddy fuzzy inference,

= {min (μA1True(x), μATrue(x)), min (μA1TrueFalse(x), μAFalse(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.

A = μA(x) = μA(x1)/x1 + μA(x2)/x2 + … + μA(xn)/xn, “+” is union

The generalized fuzzy certainty factor (GFCF) is defined as

μAGFCF(x) = μATrue(x) − μAFalse(x)

The generalized fuzzy certainty factor becomes single fuzzy membership function.

μAGFCF(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 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5,

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

μAGFCF(x) = {0.5/x1 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5 − 0.1/x1 + 0.2/x2 + 0.3/x3 + 0.2/x4 + 0.1/x5}

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

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

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 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5

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

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

B = {0.4/x1 + 0.6/x2 + 0.8/x3 + 0.6/x4 + 0.4/x5 =

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

= 0.3/x1 + 0.4x2 + 0.5/x3 + 0.4/x4 + 0.3/x5

The operations on GFCF are given as follows:

Negation

A′ = 1 − μAGFCF(x)/x

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

The graphical representation is shown in Figure 10.

Conjunction

AΛB = min(μA(x), μB(x)}/x

AΛB = 0.3/x1 + 0.4/x2 + 0.5/x3 + 0.4/x4 + 0.3/x5

The graphical representation is shown in Figure 11.

Disjunction

AVB = max(μA(x), μB(y)}/x

AVB = .4/x1 + 0.6/x2 + 0.9/x3 + 0.6/x4 + 0.2/x5

The graphical representation is shown in Figure 12.

Concentration

μvey AGFCF(x) = μAGFCF(x)2

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

Diffusion

μmore or less AGFCF(x) = μAGFCF(x)0.5

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

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

Implication

Zadeh , Mamdani  and Reddy  fuzzy conditional inferences are considered keeping in view of 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 fuzzy inference is given as follows:

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

= min(1, 1 − (A1, A2,…,An) + 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

= 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

= 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 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5

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

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

A2 = {0.4/x1 + 0.6/x2 + 0.8/x3 + 0.6/x4 + 0.4/x5 =

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

= 0.3/x1 + 0.4x2 + 0.5/x3 + 0.4/x4 + 0.3/x5

B = {0.5/x1 + 0.7/x2 + 1/x3 + 0.7/x4 + 0.5/x5,

0.4/x1 + 0.5/x2 + 0.6/x3 + 0.5/x4 + 0.4/x5}

= 0.1/x1 + 0.2x2 + 0.4/x3 + 0.2/x4 + 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) + B)

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

Mamdani fuzzy inference is given as

min(A1, A2,…, An, B)

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

Reddy fuzzy inference is given as

min(A1, A2,…, An)

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

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

Composition

The GFCF is a single fuzzy membership function

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

B1 = A1 o R = min{μA1GFCF(x), μRGFCF(x)}/x

Zadeh fuzzy inference is given by

B1 = A1 o R = min{μA1GFCF(x), μRGFCF(x)}

= min{μA1GFCF(x),min(1,1- μA1GFCF(x) + μBGFCF(x))}

Mamdani fuzzy inference is given by

= min{μA1GFCF(x), μA1GFCF(x), μBGFCF(x)}

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

= μA1GFCF(x)

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

A = {0.5/x1 + 0.7/x2 + 0.9/x3 + 0.7/x4 + 0.5/x5

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

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

B = {0.5/x1 + 0.7/x2 + 1/x3 + 0.7/x4 + 0.5/x5,

0.4/x1 + 0.5/x2 + 0.6/x3 + 0.5/x4 + 0.4/x5}

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

A1 = more or less A

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

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

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

Zadeh  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

μAGFCF(x) = {μATrue(x) − μAFalse(x)}

For instance, “x has fever”

The GFCF for fever given as

μLowGFCF(x) = {μLowTrue(x) − μLowFalse(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.

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 μAiGFCF(x)/ Σ Ci

For instance,

Speed = {0.1/20 + 0.3/40 + 0.5/60 + 0.7/80 + 0.9/100}

COG = (0.1*20 + 0.3*40 + 0.5*60 + 0.7*80 + 0.9*100)/

(0.1 + 0.3 + 0.5 + 0.7 + 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 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.