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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.
Department of Computer Science and Engineering, College of Engineering, Sri Venkateswara University, Tirupati, India
*Address all correspondence to: pvsreddy@hotmail.co.in
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 [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.
The fuzzy control systems are considered in this chapter as application of single fuzzy membership function and twofold fuzzy set.
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.
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.
Figure 1.
Fuzzy membership function.
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.
Figure 2.
Conjunction.
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.
Figure 3.
Disjunction.
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.
Figure 4.
Fuzzy quantifiers.
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”
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 [4].
If a set of conditions is satisfied, then the set of consequences is fired
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
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.
Figure 17.
GFCF for Table 5.
The graphical representation of fuzzy inference for condition part containing “AND” is shown in Figure 18.
Figure 18.
Fuzzy conditional inference for “AND.”
Graphical representation of fuzzy inference for condition part containing “OR” is shown in Figure 19.
Figure 19.
Fuzzy conditional inference for “OR.”
Defuzzification
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 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.
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.
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Written By
Poli Venkata Subba Reddy
Submitted: 27 September 2018Reviewed: 24 November 2018Published: 23 October 2019
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