[studies in fuzziness and soft computing] fuzzy modeling and control volume 69 || mathematics of...

4. Mathematics of Fuzzy Sets

The main elements of fuzzy sets are logical rules of the type (4.1).

IF (Xl is medium) AND (X2 is smalI) THEN (y is large) (4.1)

To process information in such models many operations, mostly of a logical character, must be performed. A set of these operations and the ideas related to them can be labeled mathematics offuzzy sets (Zimmermann 1994a). Its principles are presented below.

4.1 Basic operations on fuzzy sets

A fuzzy model of a real system contains Iogical rules reflecting the functioning of the system. In a system having two inputs (Xl, X2) and one output y the rule can have the form (4.2):

IF [(Xl is smalI) AND (X2 is medium)] OR [(Xl is medium) AND (X2 is smalI)] THEN (y is medium), (4.2)

where: "smalI" , "medium" are fuzzy sets (fuzzy evaluations of the states of the variables in the system), IF - THEN, AND, OR - Iogical connectives (aggregation operators of fuzzy sets).

If the fuzzy sets (smalI, medium) are used for the evaluation of the input and output states of a system, then the logical connectives determine the qualitative relationship between these states by joining fragments of a rule to get a whole. The accuracy of a fuzzy model depends on both the manner of defining the fuzzy sets used (their number, shape and parameters of membership function) and the kind of logical connectives used.

The main kinds of connectives, called logical operators, are:

• AND, n, 1\ - the intersection (logical product) operator of sets, • OR, U, V - the union (logical sum) operator of sets, • NOT, -, ..., - the negation (logical complement) operator of sets.

A. Piegat, Fuzzy Modeling and Control© Springer-Verlag Berlin Heidelberg 2001

112 4. Mathematics of fuzzy sets

AnB

-----------y----l I I I A ............. I I

I I I I I I I I I~B I

o 1 2 3 4 5 6 7 8 9 10 x

A = {l, 2,3,4, 5,6}, B = {3,4, 5,6, 7,8}, An B = {3,4, 5, 6}

Fig. 4.1. Example of the logical product of the non-fuzzy sets An B

Many various mathematical forms can represent the logical operators and selecting them properly is a problem. The condition for proper selection is, at minimum, knowledge of the basic forms of these operators.

4.1.1 Interseetion operation (logical product) of fuzzy sets

Fuzzy logic has developed on the basis of classical, non-fuzzy, bivalued logic. Its initiator, Lofti Zadeh, noticed imperfeetions of classicallogic in the modeling of reality. By introducing the idea of fuzzy sets (Zadeh 1965) he created possibilities for the improvement of models containing logical connectives. He defined the intersection operation of fuzzy sets to be the expansion of that operation on non-fuzzy sets. This means that operations on non-fuzzy sets should be a particular case of operations on fuzzy sets. This postulate can often be found in the literature, e.g. in (Yager 1994,1995), although practical reasons (attempting to improve the accuracy of fuzzy models) mean that one also intro duces operators not satisfying it (Yager 1994,1995).

In non-fuzzy logic the logical product of sets A and B is defined without the use of membership functions (Driankov 1993,1996; Poradnik 1971), (4.3).

An B = {x: xE A and x E B} (4.3)

Example 4.1.1.1 An example of the calculation of the logical product of the non-fuzzy sets An Bis shown in Fig. 4.1. 000

The main properties of the intersection operation of the non-fuzzy sets A n B defined in the universe of discourse X, to be expected also in the case of fuzzy sets, are the features (4.4) - (4.9) given below.

Commutativity: AnB=BnA. (4.4)

This property means that the ordering of the sets involved in the operation is unimportant for the final result.

4.1 Basic operations on fuzzy sets 113

Associativity: (A n B) n C = An (B n C) . (4.5)

This property means that when creating the logical product of many sets we can calculate it stepwise, using the products of pairs of the sets. The order in which the sets are paired is unimportant for the final result.

Idempotency: AnA = A.

Absorption by the empty set 0:

An0=0.

Identity: AnX=A,

where: X - the universe of discourse.

Exclusive contradiction: AnA = 0.

(4.6)

(4.7)

(4.8)

(4.9)

Below it will be shown that not all these properties can be transferred to the intersection operation of fuzzy sets, e.g. the property (4.9) is such a one. ln the environment of fuzzy sets this operation can be performed with the use of various methods and therefore its sense is non-unique. The non-uniqueness will be presented in Example 4.1.1.2.

Example 4.1.1.2 Two fuzzy sets A and B defined by (4.10) and (4.11) are given. A - the set of cheap cars, Xi - the notation of a car:

(4.10)

B - the set of luxury cars:

(4.11)

Simultaneously there is A = iJ, and this is represented in Fig. 4.2. A set C = A n B containing cheap and at the same time luxury cars has

to be determined. Since A = iJ, then in the case of non-fuzzy sets we would obtain the empty set, according to the property (4.9) (A n A = 0). What will the result be in the case of the fuzzy sets to be considered?

A car X4 is cheap to the degree J.tA(X4) = 0.4 and luxury to the degree J.tB(X4) = 0.6. To which degree J.tAnB(X4) is it cheap and luxury at the same time? How can this degree be determined using the membership grades J.tA(X4) and J.tB(X4) in the respective sets?


o X2 Xs Xi

Fig. 4.2. Discrete membership functions of cars Xi in the set of cheap cars (A) and luxury cars (B)

In (Zadeh 1965) L. Zadeh proposed to calculate the membership function ofthe product ofthe sets by means ofthe MIN operator, the formula (4.12).

JLAnB(X) = MIN(JLA(x), JLB(X)), 't/x EX. (4.12)

This operator was the first operator extending the intersection operation of non-fuzzy sets n to fuzzy sets. By applying the formula (4.12) to calculate the set A n B of the cars that are both cheap and luxurious, we obtain the formula (4.13). The fuzzy set resulting from this operation is illustrated in Fig.4.3.

(4.13)

The MIN operator can be represented in algebraic form:

MIN( ) = Xl + X2 -lXI - X21 = Xl + X2 - (Xl - X2) . sgn(XI - X2) XI,X2 2 2'

(4.14) where:

{-I for Xl - X2 < 0 ,

sgn(xi - X2) = 0 for Xl - X2 = 0, 1 for Xl - X2 > 0 .

The MIN operator in the form (4.14) is called "hardMIN", because the reversal of sign of the difference (Xl - X2) causes a drastic change in the value sgn(xi - X2) and the output of the operator, Fig. 4.4.

To make the operation ofthe MIN(XI -X2) operator softer, one can apply a special form (4.15) ofthe two-input sgn operator that can be found in the literature on this subject.

(4.15)


Xs

o Xs

Fig. 4.3. The product of the fuzzy sets An B of cheap (A) and luxury (B) cars obtained by means of the MIN operator

1

o

-------1-1

Fig. 4.4. "Hard" (a) and "soft" (b) sgn(xl - X2)

where: 8 - a small number, e.g. 0.05. Increasing the value 8 makes the operation of sgn(xi - X2) "softer", Fig. 4.4b.

By using the "soft" sgn, we obtain as a result the "soft" MIN ö (Xl - X2) operator determined by the formula (4.16).

(4.16)


0.990

1.000

1.010

0.990

1.000

1.010

M1N6

M1N6

0.988

0.998

. 0.987

MIN6 .

. 0.988

M1N6 .

Fig. 4.5. Illustration of the fault of the "soft" MIN operator (inaccuracy of ca1culations and dependence on the order of calculations)

The "soft" sgn means that the operation of controllers and fuzzy models becomes smoother. It smoothes out sharp edges of the surface of the input -+ output mapping. However, it also has a disadvantage. When the soft sgn is calculated from a greater (than 2) number of signals Xi, the result of the calculation is dependent on the sequence of the signals and is not exactly equal to any one value Xi, Fig. 4.5.

The mentioned fault of the MIN operator diminishes at smaller values J. However, it has another disadvantage, Le. the need of carrying out calculations step by step, for successive pairs of signals (the operator calculates the value of the minimum for only one pair of signals Xl and X2 in any one step). Therefore, for the calculation of the minimum from a greater number of signals, it is advisable to use the form (4.17) of this operator (Berenji 1992).

n

LXi' e-kxi

softMIN(Xl,'" ,xn ) = _i=--,l_n __ _

Le-kxi

i=l

i = 1, ... , n, k > O. ( 4.17)

The operation of the so ftMIN operator is illustrated in Fig. 4.6. Together with the increase in the value of the factor k (k -+ 00), the operation of the softMIN operator becomes more and more similar to the operation of the "hard" MIN. Practically, for the value k > 100 great accuracy of calculation is already obtained. However, it should be remembered that the operation of the operator becomes "harder" as the value k increases.

Summarizing, the "hard" MIN operator has the following advantages and disadvantages.

0.990

1.000

1.010


0.993 softMINö

Fig. 4.6. Example of the calculation of the value sojtMIN(0.990; 1.000; 1.010) from the formula (4.17)

Advantages:

1. The calculations are carried out quickly and in a simple way, resulting in a decrease of the load on computers and microprocessors. It makes it possible to use cheap microprocessors as fuzzy controllers.

2. The possibility of "softening" the action of the MIN operator. However, this results in the increase of the number of calculations associated with decreasing accuracy.

Disadvantages:

1. In general, the accuracy of a model is worse than when other operators are used.

2. The smoothness of the surface of a model is worse than in the case of using other operators.

3. The occurrence of insensitivity and drastic changes at the output of a model and the output of a fuzzy controller containing the MIN operators.

The analysis of the result of the intersection operation of sets An B with the use of the MIN operator represented in Fig. 4.3 attests to disadvantage no. 3. Only membership in the set of luxury cars decides whether the cars Xl, X2, X3 belong to the set An B. The fact, whether the cars are cheaper or more expensive, has no influence on this membership. In the case of the car X3 we obtain the result:

According to this model, irrespective of how cheap car X3 would be (even when the price of that car would amount to zero), the grade of membership J.LAnB(X3) would be the same (0.4). The discussed fault is illustrated in Fig.4.7.

The example represented in this drawing shows that the use of the MIN operator to realize the intersection operation of sets causes part of the information to be lost. The reason is that this operator takes into account only the fact that one membership grade is less than another, whereas the value of the difference of the membership grades is not taken into account by it.


o

JlA2(Xj),JlB(Xj)

o

X2

X2

X2

Xs X6 Xj

Xs

Xs

Fig. 4.7. Identical product of sets Al nB = A2 nB realized with the use of the MIN operator, in the case of different forms of membership functions of sets Al and A2

For this reason, the use of the MIN operator in models of systems usually causes insensitivity of a model to small variations of its inputs and a drastic switching over of the output state after a certain level of the inputs has been exceeded (Piegat 1995a). This feature is very inconvenient in modeling systems having a smooth surface of the input -+ output mapping.

In some systems, where the method of processing information is similar to the logical one (the majority of dependencies between inputs and output of the system are of a logical character), the use of these operators can have advantages. The above-mentioned disadvantages of the MIN operator mean that the range of its use is decreasing. In 1995 the participants in the 5th Workshop "Fuzzy Control" (October 16-17, 1995, Witten, Germany) were asked for their opinion regarding the use of the MIN operator. It showed that most specialists participating in that conference preferred the PROD operator (logical product) rather than the MIN operator (Pfeiffer 1996),


Xs

o Xs

Fig. 4.8. Membership functions of the logical product of the fuzzy sets A and B obtained with the use of the MIN and PROD operators

while earlier opinions to the contrary were given in the literat ure (Kahlert 1995; Knappe 1994). The calculation of the membership function of the product of fuzzy sets using the PROD operator is carried out according to the formula (4.18).

J.tAnB(X) = J.tA(X) . J.tB(X), Vx EX. (4.18)

The advantage of the PROD operator is that the product value J.tAnB(X) is quantitatively dependent on the actual values of both the components of membership function J.tA(X) and J.tB(X) (apart from when the value of one function is zero). You can see that here the loss of information is not so considerable as in the case of the MIN operator where the value J.tAnB(X) is dependent only on the lesser value (in the given range of x) of the components J.tA(X) or J.tB(X). The comparison of the results of logical product calculation using the MIN and PROD operators is represented in Fig. 4.8.


As can be seen from Fig. 4.8, the values of the membership function J-tAnB(X) obtained by the use of the PROD operator are less than when the MIN operator is used. Therefore, this operator is sometimes called the subMIN operator.

The example of possibilities of the use of both the MIN and PROD operator to realize the intersection operations of fuzzy sets shows that the way it is carried out is not unique. There are many opinions concerning the subject of how this operation should be carried out, and so in practice the operator n is often selected by conjecture, experience, intuition and trials. Many intersection operators of sets have been invented; but the respective use of them gives better or worse results according to the actual application. The most often used intersection operators An B are so-called t-norms representing various forms of the realization of this operation (Driankov 1993,1996; Vager 1994,1995; Knappe 1994). The t-norm operator is a function T modeling the intersection operation AND of two fuzzy sets A, B having the properties (4.19) - (4.24) which are satisfied for each x E X.

Mapping spaces: T: [0,1] x [0,1]-+ [0,1]. (4.19)

Zeroing: T(O, 0) = 0. ( 4.20)

Identity of unity:

T(J-tA(X), 1) = J-tA(X), T(J-tB(X), 1) = J-tB(X). (4.21)

Commutativity:

( 4.22)

Union:

( 4.23)

Monotonicity:

J-tA(X) :::; J-tc(x), J-tB(X) :::; J-tD(X) :::} T(J-tA(X),J-tB(X)) :::; T(J-tc(X),J-tD(X)). (4.24)

The property of commutativity says that the order of sets is unimportant for carrying out the operation. The property of union teIls us that the intersection operation of a greater (than 2) number of sets should be carried out successively, but the order of creating pairs of sets has no effect on the final result. The property of monotonicity means that the result of the operation does not diminish when the values of its arguments become larger.

These operators are divided into parameterized and non-parameterized tnorms. The effect of the action of non-parameterized t-norms is constant. In

4,1 Basic operations on fuzzy sets 121

Table 4.1. Some non-parameterized t-norm operators

I Operator name I Formula

minimum (MIN) I'AnB(X) = MIN(I'A(x),I'B(x))

product (PROD) I'AnB(X) = I'A(X) 'I'B(X)

Hamacher product ( ) I'A(X) 'I'B(X) I'AnB x = I'A(X) + I'B(X) -I'A(X) 'I'B(X)

Einstein product ( ) I'A(X) 'I'B(X) I'AnB X = 2 - (I'A(X) + I'B(X) -I'A(X) 'I'B(X»

drastic product ) { MIN(PA(X),"B(X» ro, MAX(PA,"B) ~ 1

I'AnB(X = o otherwise

bounded difference I'AnB(x) = MAX(O, I'A(X) + I'B(X) -1)

Table 4.2. Some parameterized intersection operators of fuzzy sets

I Operator I Formula name

I'A(X) 'I'B(X) Dubois- I'AnB(X, a) = MAX(j.&A(X),I'B(X), a] , a E [0,1] intersection operator a=O: I'AnB(x,a) = MIN(I'A(x),I'B(x))

a = 1: I'AnB(x,a) = PROD(I'A(X),I'B(X))

( ) I'A(X) 'I'B(X) > 0

Hamacher-I'AnB x,'Y = 'Y + (1- 'Y)(I'A(X) + I'B(X) -I'A(X) 'I'B(X)) , 'Y -

intersection 'Y = 0: I'AnB(X,'Y) = HamacherPROD operator 'Y = 1: I'AnB(X,'Y) = PROD(I'A(X),I'B(X»

'Y = 2: ll.AnB(X,'Y) = EinsteinPROD 'Y -t 00: I'AnB(X,'Y) = drasticPROD

Yager- I'AnB(X,p) = 1- MIN( 1, «1 -I'A(X»P + (1 -I'B(x»p)i) , p ~ 1

intersection operator p= 1: I'AnB(X,p) = bounded difference

p -t 00: I'AnB(X,p) = MIN(I'A(x),I'B(x))

contrast, the effect of parameterized t-norms varies qualitatively and quantitatively when any parameter which is the degree of freedom of the operator has been changed, The most weH known non-parameterized t-norms are listed in Table 4,1. The most often used parameterized t-norms are specified in Table 4,2 where the dependence of the operators upon their parameters is also given,


low medium=A high=B

o 37 38 39

Fig. 4.9. Membership functions of the linguistic value "fever"

Example 4.1.1.3 The example illustrates the operation of fuzzy t-norms. In Fig. 4.9 the membership functions of the linguistic variable "fever" are given.

The task is to determine the membership function of fever T in the fuzzy set C = "medium AND high temperature" (C = A n B) using various nonparameterized operators. The results are shown in Fig. 4.10.

According to binary non-fuzzy logic the fever can not be medium AND high at the same time. Fuzzy logic shows that such a set can exist. The membership function of this set is not strictly defined and is dependent upon which t-norm operator has been used. 000

As can be seen in Fig. 4.10, the MIN operator allows us to obtain the highest values of membership functions. Therefore, other t-norm operators are sometimes called sub-MIN-operators or sub-MIN-norms (Knappe 1994), Fig.4.11.

By using the sub-MIN-operators (t-norms) smaller values of the membership function I'AnB(X) of the product of sets are obtained than when the MIN-operator is applied. This means that the sub-MIN-operators are stricter and require the conditions A and B of the fuzzy product to be satisfied to a higher degree. Therefore, the MIN-operator is considered (Driankov 1993) to be the most optimistic among t-norms.

According to the degree of optimism, t-norms can be ordered in the following sequence:

minimum > Hamacher product > algebraic product > Einstein product > bounded difference > drastic product.

To realize the intersection operation operators which are not t-norms (Le. operators not satisfying the conditions of at-norm) are also used. An


JlAnIJ('1) JlAnIJ(n

MIN bounded difference

0.5

0 0 38 39 T(°C) 38 39 T(°C)

JlAnIJ(n JlAnIJ(n

PROD Einstein PROD

0.25 0.2 -----------------0 0

38 39 TeC) 38 39 T(°C)

JlAnIJ(n

Hamacher PROD

0.33

0 38 39 T(°C)

Fig. 4.10. Membership functions ofthe fuzzy set "medium AND high temperature" calculated by means of various t-norms

Jl{x)

---------------------------------------------------

o x

Fig. 4.11. The relationship of the MIN-operator to other t-norms


example of such an operator is the parameterized mean intersection operator (Driankov 1993), (4.25).

Parameterized mean interseetion operator

J.lAnB(X) = 'Y' MIN(J.lA(x) , J.lB(X)) + 0.5(1 - 'Y)(J.lA(X) + J.lB(X)), Vx EX, (4.25)

where: 'Y E [0, 1]. For 'Y = 1 this operator becomes the MIN -operator, for 'Y = 0 it will be

the arithmetie mean-operator (4.26):

J.lAnB(X) = 0.5(J.lA(X) + J.lB (x)) , Vx EX, (4.26)

Because the inequality (4.27):

is always satisfied, the mean operator is called the super-MIN-operator. It is more optimistic than the most optimistic t-norm, Le. the MIN-operator.

In the case of n fuzzy sets Al, ... An the formula (4.28) is applied.

(4.28)

Harmonie mean-operator For n fuzzy sets Al, ... An the membership in the resulting intersection set is calculated from the formula (4.29) (Yager 1994).

n J.lAln, ... ,nAn (x) = 1 1 ,Vx E X (4.29)

-(-) + ... + () J.lAI X J.lAn X

Geometrie mean-operator For n fuzzy sets Al, ... An the membership in the resulting set is calculated according to the formula (4.30).

Generalized mean-operator In this case for n fuzzy sets Al, ... An the formula (4.31) is used.

I

(x) = (J.l~l (x) + J.l~2(X) + ... + J.l~n (X)) ä Vx E X (4.31) J.lAln, ... ,nAn n '

This operator is the parameterized intersection operator: the parameterized quantity is the parameter a. For:


Jl(T) low medium=A high=B

a)

JlAnB(T) 37 38 9 T(°C)

I I

b)MIN(a~oo) I I I I I I

0.5 I -_ .. --------,----I I I I I I I I

JlAnB(T) 37 38 39 T(°C)

e) harmonie mean (a= -1)

0.5

0.375

JlAnB(T) 38.25 T(°C)

d) geometrie mean (a= 0)

0.5 0.433

JlAnB(T) 38.25 T(°C)

e) arithmetie mean

0.5

37 38 39

Fig. 4.12. The comparison of the results of the realization of the intersection of the fuzzy sets A and B carried out with the use of the MIN -operator and meanoperators


0: -+ -00: the generalized mean-operator becomes, at the limit, the MIN-operator,

0: = -1 : the harmonie mean-operator, 0: = 0 : the geometrie mean-operator, 0: = 1 : the arithmetic mean-operator, 0: -+ +00: the MAX -operator.

Fig. 4.12 illustrates the comparison of the effect of different fuzzy operators in the case of the intersection of two sets A and B from the Example 4.1.1.3.

As can be seen from Fig. 4.12, all mean-operators whieh are derived from the generalized mean-operator are the operators of super-MIN type and so they are more optimistie than the MIN operator iso The degree of optimism increases according to the increase in the coefficient 0: in the formula (4.31). The arithmetie mean-operator (0: = 1) has the property of additivity: the membership function in the resulting set varies proportionally to changes of the components of the membership function.

4.1.2 Union (logical sum) of fuzzy sets

In classieal logic the logieal sum of the sets A and B is defined without the use of the idea of membership function, see the formula (4.32) (Driankov 1993,1996; Poradnik 1971).

Au B = {x: xE A or x E B} (4.32)

An example of logieal summation is represented in Fig. 4.13. The result of the union of non-fuzzy sets is here unique because the union

is always carried out in the same manner. In the case of fuzzy sets the union operation can be made in many ways. Therefore the result is not unique.

Considering the postulate (Chapter 4.1.1) relating to fuzzy logie, whieh says that all its operations carried out on non-fuzzy sets should give identi-

A

o

A B

1---------------I I

2 3 4 5 6

B

7 8 9 10 x

A = {1, 2, 3,4, 5,6}, B = {3, 4, 5,6, 7,8}, AU B = {l, 2, 3,4, 5, 6, 7, 8}

Fig. 4.13. Example of the logical sum of non-fuzzy sets


cal results to the operations of classical logic, it is expected that the union operation offuzzy sets would have the properties (4.33) - (4.38).

Commutativity: AUB=BuA. (4.33)

This property means that the order of sets involved in the union operation is unimportant for the final result.

Associativity:

AU (B u C) = (A u B) U C = AU B U C. (4.34)

When creating the union of many sets we can do it successively for pairs of sets. The order of succession of creating pairs is unimportant.

Idempotency: AUA=A. (4.35)

Union with the empty set 0:

Au0=A. (4.36)

Absorption by the universe of discourse X:

AUX=X, (4.37)

Union with the complementary set Ä:

AuÄ=X. (4.38)

In the following example it will be shown that not all properties of the union operation of non-fuzzy sets can be transferred into fuzzy sets.

Example 4.1.2.1 Assurne there is the set A of cheap cars (4.39) and the set B = Ä of luxury cars (4.40), Xi - a car number.

(4.39)

B _ {~ 0.2 0.4 0.6 0.8 -.!..} - , , , , , Xl X2 X3 X4 X5 X6

(4.40)

The universe of discourse:

x = {:1 ' :2 ' :3 ' :4 ' :5 ' :6 } . A set C = A u B = A u Ä of cheap or luxury cars has to be determined.


o Xs

o Xs

Fig. 4.14. The membership functions of the sets A and B and their logical sum

H the operator of the type of algebraic sum (4.41) were used to realize the union of the sets Au B, then we would obtain the set (4.42), as a result.

J.l.c(X) = J.l.A(X) + J.l.B(X) - J.l.A(X) . J.l.B(X) (4.41)

C - A u B _ {~ 0.84 0.76 0.76 0.84 ~} - - , , , , , Xl X2 X3 X4 Xs X6

(4.42)

The membership functions of the sets A, B, C are shown in Fig. 4.14. 000

As can be seen from the above represented example, if the operator of algebraic sum has been used to carry out the union of fuzzy sets A U A, then the result is not conformable to the sixth property (A U A = X) of this operation that is always true in the case of non-fuzzy sets.

The first operators which were proposed to realize the union of fuzzy sets were (Zadeh 1965) the MAX -operator and algebraic sumo As fuzzy logic has been developing, the number of those operators has been increasing. At present, the most popular union operators of sets are the t-conorms also called the s-norms.

The s-norm or t-conorm operator is the function S realizing the union operation OR of two fuzzy sets A and B having the properties (4.43) - (4.48) for each X E X.


Table 4.3. Non-parameterized s-norrns

I Operator name I Formula

maximum (MAX) I'AUB(X) = MAX(I'A(X),I'B(X»

algebraic sum I'AUB(X) = I'A(X) + I'B(X) -I'A(X) . I'B(X)

Hamacher sum () I'A(X) + I'B(X) - 2 'I'A(X) 'I'B(X) I'AuB X =

1 - I'A(X) 'I'B(X)

Einstein sum () I'A(X) + I'B(X) I'AUB X = 1 + I'A(X) 'I'B(X)

drastic sum {MAX("A(X),"B(X)) f", MIN(PA,"B) ~ 0

I'AUB(X) = 1 otherwise

bounded sum I'AUB(X) = MIN(l, I'A(X) + I'B(X»

Input/output mapping space:

8: [0,1] x [0, 1] ~ [0,1]. ( 4.43)

The property of zeroing:

8(0,0) = 0. (4.44)

The action when a pair contains a natural element J.LB(X) = 0:

8(I'A(X), 0) = 8(O,J.LA(X» = I'A(X), ( 4.45)

The property of commutativity:

(4.46)

The property of associativity (association of any pairs):

8(J.LA(X), 8(I'B(X), J.Lc(x))) = 8(S(J.LA(X),J.LB(X»,J.Lc(x». (4.47)

The feature of monotonicity:

J.LA(X) ~ J.Lc(x), J.LB(X) ~ I'D(X) => 8(J.LA(X),I'B(X» ~ 8(J.LC(X),J.LD(X». (4.48)

The s-norm operators are divided into the parameterized and non-parameterized ones. The non-parameterized operators act in a constant way. The most often used operators of this type are listed in Table 4.3. The most often used parameterized s-norms are specified in Table 4.4.

The particular s-norms differ one from another in the degree of optimism. The biggest result of calculations is given by the operator drastic sum, the


Table 4.4. Parameterized s-norms

I Operator I '" 1 rormu a name

( ) _ I'A(X) + I'B(X) + (-y -1). I'A(X)· I'B(X) > 1 I'AuB X, 'Y - () () ,'Y-1 + 'Y . I'A X . I'B X -Hamacher-

union 'Y = -1: I'AUB(X,'Y) = Hamacher sum operator 'Y = 0: I'AUB(X,'Y) = algebraic sum

'Y = 1: I'AUB(X,'Y) = Einstein sum 'Y -+ 00: I'AUB(X,'Y) = drastic sum

Yager- I'AUB(X,p) = MIN {1, [(I'A(X))P + (I'B(X))P];} , p;::: 1 union p= 1: I'AUB(X,p) = bounded sum operator p -+ 00: I'AUB(X,p) = MAX(I'A(X),I'B(X))

p,(x)

- - - - ---_.

o x

Fig. 4.15. The relationship between the MAX -operator and other s-norm operators

smallest one by the MAX-operator. The sequence of the s-norms according to the degree of optimism is as follows:

drastic sum > bounded sum > Einstein sum > algebraic sum > Hamacher sum > MAX

Because the calculation of the membership in the set AUB with the use of the MAX -operator gives the smallest result, all the s-norms operators except for the MAX-operator are called super-MAX-operators, Fig. 4.15.

The t-norm and s-norm operators create complementary pairs satisfying the condition (4.49).

( 4.49)

If the t-norm is given, the complementary (in relation to it) s-norm can be calculated. In Table 4.5 the complementary pairs of the t-norms and s-norms are given.


Table 4.5. The complementary pairs of the t-norms and s-norms

I t-norm I complementary s-norm

MIN MAX

algebraic product algebraic sum

Hamacher product Hamacher sum

Einstein product Einstein sum

drastic product drastic sum

bounded product bounded sum

parameterized Hamacher- parameterized Hamacher-intersection operator union operator

parameterized Yager- parameterized Yager-intersection operator union operator

For the realization of the union operation of sets, the OR-operators which are not s-norms are also applied (they do not satisfy the conditions of a snorm). An example of such an operator is the parameterized union meanoperator of sets (4.50) (Driankov 1993).

J.LAUB(X) = 7· MAX[J.LA(X),J.LB(X)] + 0.5· (1 - 7) . [J.LA(X) + J.LB(X)] ,

7 E [0,1], 'Vx EX. (4.50)

For 7 = 1 the operator (4.50) becomes the MAX-operator, for 7 = 0 it becomes the arithmetic mean-operator.

For the union operation offuzzy sets the algebraic sum operator (4.51) can also be used.

J.LAIU, ... ,uAn (x) = J.LA1 (x)+, . .. ,+J.LAn (x), 'Vx EX. (4.51)

This operator is the most optimistic of aH union operators of sets and has the property of additivity. The resulting membership function increases proportionally to the increase in the component functions in the formula (4.51). Therefore, this operator as weH as the arithmetic mean-operator can be called a linear operator. The use of them in fuzzy models is conducive to obtaining linear surfaces of the sectors in input/output mapping of these models. The linear operators transform fuzzy models involved in the operation in the so-caHed fuzzy bags (Yager 1994,1995). They can be used for performing operations on fuzzy bags described in Chapter 2.

Example 4.1.2.3 Let A be the set of fast cars and B the set of comfortable cars, Xi - the designation of a car,


A= {~ ~ ~ ~} , " , Xl X2 X3 X4

B _ {0.6 0.7 0.9 ~} - , , , . Xl X2 X3 X4

We want to purchase a fast OR comfortable car. By applying the MAX s-norm we obtain the result (4.52).

C = AUB = {~,~,~,~} u {0.6, 0.7, 0.9,~} = {~,~,~,~} Xl X2 X3 X4 Xl X2 X3 X4 Xl X2 X3 X4

(4.52) As a result of the use of the MAX -operator we obtain the information

that each car can be purchased because their membership in the set "fast OR comfortable" is the same and equal to 1. By using the arithmetic sumoperator the sets A and B are transformed in a fuzzy bag (4.53) in the first step and then in a fuzzy set (4.54) in the second step.

Step 1

(4.53)

Step 2

(4.54)

As a result of the use of the arithmetic Bum-operator we obtain the information that the car which should be purchased (car X4) is the one whose membership in the bag "fast or comfortable" is the highest (the membership in (4.54) can be normalized to the interval [0,1]). It seems that in many situations people making decisions apply such an operator because a man is still inclined to take into account many circumstances when he makes a decision. 000

4.1.3 Compensatoryoperators

Both the t-norms and s-norms are the so-called supposed operators, i.e. they are based on suppositions concerning the way intersection and union operations of sets are performed in the human mind. Up to now the problem has not been completely explained, perhaps because various people use miscellaneous ways to realize the above-mentioned operations, depending on their character, mood and the actual situation.


high (H) Il(V) 1 -----------------------.,..-! ---

Il(d) 1

0.75

0.5 i

! o 25 ---- ------------------1 " , , , , , ,

60 70 100 v (km/h)

speed

short (S)

o 25 50 100 d(m)

distance

Fig. 4.16. The membership functions of the fuzzy sets "high" speed and "short" distance

Research concerning the operators applied by a man, which was carried out by Zimmermann (Altrock 1993; Zimmermann 1979,1987), led to the formulation of the idea of the compensatory operators. The relevance of the compensation will be explained in the example of the reasoning of a driver who approaches an obstacle in the street at a fast speed - the rule(4.55).

IF (the speed is high) AND (an obstacle is near) THEN (brake very strongly) (4.55)

By denoting the speed of the car by v (km/h) and the distance to the obstacle by d (m), the premise A can be given in the form (4.56).

A = Al AND A2 = (v = H) AND (d = S) (4.56)

The greater the truth-value of the premise is, the stronger the braking must be.

Let us assume that the membership functions of the linguistic values "high" speed and "short" distance have, respectively, the forms represented in Fig. 4.16.

Now, let us investigate 3 possible situations to which the car can be exposed and how the driver using the PROD-operator evaluates them.

Situation 1 - the car approaches the obstacle

v = 70 (km/h) J.tH(V) = 0.25

d = 50 (m) J.ts(d) = 0.5

The degree of satisfying the premise A and in turn the degree of the braking force (the degree of the activation of the conclusion resulting from the rule (4.55)) is equal to:

J.tA(V, d) = J.tH(V) . J.Ls(d) = 0.25·0.5 = 0.125.


AND I I o

Ir I OR i I I I I I

1 r

Fig. 4.17. The dependence ofthe character ofthe operator h upon the degree of compensation -y

Situation 2 - the car is very close to the obstacle

v = 70 (kmjh) J1.H(V) = 0.25

d = 25 (m) J1.s(d) = 0.75

The degree of satisfying the premise:

J1.A(V, d) = J1.H(V) . J1.s(d) = 0.25·0.75 = 0.1875.

Situation 3 - the car crashes into the obstacle

v = 70 (kmjh) J1.H(V) = 0.25

d = 0 (m) J1.s(d) = 1

The degree of satisfying the premise A:

J1.A(V, d) = J1.H(V) . J1.s(d) = 0.25·1 = 0.25.

The analysis of situations 1 - 3 shows that when the PROD-operator is used to realize the operation AND in the premise, the degree to which it is satisfied J1.A(V, d), and thus also the degree of the braking force, does not vary appropriately fast in spite of the fast approaching danger. The driver keeping the rule strictly (4.55) with the PROD-operator would crash into the obstacle. Considering the above, is the rule (4.55) false? No, it is not.

The analysis of people's behavior shows that people apply the so-called compensatory principle modifying the AND-operation by combining it to a certain degree with the OR-operation. The measure of compensation is the degree of compensation -y, Fig. 4.17.

On the basis of the experimental results of investigations of people's decisions Zimmermann proposed the I -y-intersection operator in the form of (4.57).

( m )(l-'Y) [m ]'Y J1.A = gJ1.Ai 1- g(1- J1.Ai) , (4.57)


where: "I - the degree of compensation, 0 ~ "I ~ 1, J.tA - the degree of satisfying of the whole premise A = Al n ... nAn ,

J.tAi - the degrees of satisfying of the component premises. If "I = 0, then the whole premise is evaluated on the basis of the AND

intersection operation only - by means of the PROD operator, the formula (4.58).

m

J.tA = IIJ.tAi. (4.58) i=l

If "I = 1, then the whole premise is evaluated on the basis of the formula (4.59) realizing only the aR-operation:

m

J.tA = 1 - II (1 - J.tAi) . (4.59) i=l

The functioning of the operator (4.59) is approximate to that of the MAX operator, although, as is easily noticeable, its functioning is better because the I "I-operator takes into account all component premises and not only the premise satisfied to the highest degree.

A driver most likely changes the value "I depending on the situation: for a distant obstacle he applies a small value and dose to an obstade a large value "I. By using the 1'Y-operator, when "I = 1, in situation 3, we obtain a completely different result from that when the PROD-operator is used. Situation 3 - the car crashes into the obstacle

v = 70 (kmjh)

d = O(m)

J.tH(V) = 0.25

J.ts(d) = 1

The degree of satisfying of the premise A calculated with the use of the operator 1"1, "I = 1, is equal to:

J.tA(v,d) = 1- (1- 0.25)(1-1) = 1.

The fuH degree of satisfying the premise imposes, according to the rule (4.55), to press the brake as strongly as possible, which is, of course, the most natural reaction in such a situation. Naturally, the rule (4.55) with the h -operator would suggest stronger and stronger braking earlier, as the distance to the obstade is decreasing.

Because the compensation coeflicient "I can vary in the range ofO ~ 'Y ~ 1, there exists a problem connected with the selection of its optimal value. In (Altrock 1993] it is recommended to select 'Y for technical applications from the range:

0.1 ~ "I ~ 0.4.

A pragmatic method is the choice of the mean value from this range, 'Y = 0.25, at the beginning, and then the examination of the accuracy of a


AI -------------------------~----_<r---

o I I I I I

o I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

A2 1 ------------------~----~----------

0 I 0 I I I I I

Ci> I I I I I I

I I I I I I I I I I I I I I I I I I I I I I I I I I I

SI S2 S3 S4 Ss Si

Fig. 4.18. The membership function in the subsets Al (good students) and A2

(highly proficient students)

fuzzy model based on this value. If the accuracy is not satisfactory, it is recommended to correct the coefficient stepwise, assuming a step of Ll'Y = 0.01, and to examine the model accuracy.

4.2 Fuzzy relations

In Section 4.1 operations on fuzzy sets defined over one-dimensional universe of discourse X have been considered. It is illustrated in Example 4.2.1.

Example 4.2.1 In a set of students two subsets have been separated: the subset Al of

good students and the subset A2 of highly proficient students. The set of good and highly proficient students: Al 1\ A2 should be determined. X - the set of students:

Al - the subset of good students:

A2 - the subset of highly proficient students:

By using the MIN -operator for the realization of the intersection operation of the subsets 1\ we obtain:

Al 1\ A2 = {(Si, MIN(JLAl(Si),JLA2(Si))} = {(SI, 0), (S2, 0.3), (S3, 0.7), (S4, 1), (S5, 0.7)} .

4.2 Fuzzy relations 137

-------------------------{j>-----------

Ci' , !

Q , , , , , , , , , , , , :

, : , , , ! , , , 1 , :

Q , , ! , , ! , , :

Si

Fig. 4.19. The membership function in the set A I l\A2 (good and highly proficient students)

The set Al /\ A 2 is represented in Fig. 4.19. Owing to the fact that both the subsets have been defined over the same

one-dimensional universe of discourse X, the result of the intersection operation can be represented on the surface, in two-dimensional space. Both the subset Al and the subset A2 can be considered to be simple sets because the grades of membership /-LAi(Si) are assigned to the single elements Si of the universe of discourse X. 000

Apart from one-dimensional universes of discourse there are multidimensional domains being the Cartesian product X of the component uni verses of discourse X I, ... , X n of various quantities. This is illustrated by Example 4.2.2.

Example 4.2.2 Xl - the set of citizens,

X 2 - the set of banks, X 2 = {bI, b2 , ... , b5 } .

The Cartesian product X = Xl X X 2 is the set of all possible pairs (Ci, bj ),

i = 1, ... ,5, j = 1, ... ,5, Table 4.6. 000

Before the idea of fuzzy relations is explained, it will be advantageous to acquaint oneself with the concept of classical relation (Empacher 1970).

The classical relation (bivariant, binary) - one of the prime ideas of mathematicallogic - is a property of pairs of objects and describes a certain interrelation existing between the objects. The idea of the classical relation is illustrated in Example 4.2.3.


Table 4.6. The discrete Cartesian product X = Xl X X2 represented in the form of a table (two-dimensional universe of discourse )

I~ CI C2 C3 C4 Cs

bj

b l Chbl c2,b l c3,b l c4,b l cs,b l

b2 Chb2 c2,b2 c3,b2 c4,b2 cs,b2



bs ChbS c2,bs c3,bs c4,bs cs,bs

Example 4.2.3 The one-dimensional component sets Xl and X 2 are given. Xl - the set of citizens,

X 2 - the set of banks, X 2 = {bl ,b2 , ••• ,bs}.

An example of a classical relation over the set X = Xl X X 2 is the relation "one having an account in", Table 4.6. The set X is here the universe of discourse. The relation can have the following form:

R = {(Cl, b2 ), (C3, b4), (C4, bl ), (cs, b3 )} •

The relation R consists of the pairs (Ci, bj ) and so it is the binary relation assigning the citizens Ci to the banks bj in which they have opened their accounts. This relation can be described by using the membership function f..t(ci,bj ) which can be represented in three-dimensional space, Fig. 4.20.

The relation R can be also represented in the form of the matrix R.

[000101 10000

R= 00001 00100 00000

Because citizen C2 has not opened an account anywhere, there are only zeros in the second column of the matrix R. 000


R CI C2 c3 C4 Cs

1 bl 0 0 0 1 0

Q b2 1 0 0 0 0 b3 0 0 0 0 1 b4 0 0 1 0 0 bs 0 0 0 0 0

1

Q 1 1 1

1 1 1 Ql 1 1

" 1 : : 1 Q 1 C3 : I 1 lCI C2 C4 Cs 1

" 1

Fig. 4.20. The representation of the relation R in the form of the three-dimensional membership functionJ.'( Ci, bj) and in the form of the relation matrix

The matrix R need not be quadratic. It depends on the number of elements in the component universes of discourse Xi. The relation from Example 4.2.3 has a discrete character. In Example 4.2.4 a continuous relation is represented.

Example 4.2.4 Assume that two sets of real numbers Xl, X 2 are given:

Xl = {Xl: 2:::; Xl :::; 4}, X 2 = {X2: 1:::; X2 :::; 5}.

Let us determine the relation "smaller , equal to" or ":::;" defined over the Cartesian product X = Xl x X 2 :

The relation is of a continuous character. Its membership function is represented in Fig. 4.21. 000

The Definition 4.2.1 determines a c1assical n-variant relation R defined over the universe of discourse X = Xl X •.. X X n .

Definition 4.2.1 The c1assical n-variant relation R determined over the universe of discourse:

X = Xl X •.• X X n ,


1

Fig. 4.21. The membership function I'R(Xl, X2) in the form of a continuous surface spread over the continuous universe of discourse X = Xl X X2

is an ordered set of n-tuples having the form:

where: ( ) _ {I if (Xl,' .. ,Xn ) ER,

J.LR Xl,' .. ,Xn - ° otherwise , is the membership function of the relation R.

As we know, the membership nmction of a classic relation realizes the mapping of the universe of discourse X onto the discrete set {O, I}:

J.LR: Xl X '" X X n -t {O, I}.

The difference between the fuzzy relation and the classical relation is that for the membership nmction the continuous interval [0, 1] has been introduced instead of the discrete set consisting of two elements {O, I}.

Definition 4.2.2 The fuzzy n-variant relation R determined on the universe of discourse X = Xl X ••. X X n is the ordered set of n-tuples having the form of:

where:


1

Fig. 4.22. Example of the continuous membership function of the fuzzy relation

/-LR(Xl, ... ,Xn ): Xl X ... X X n -+ [0, 1],

is the membership function of the relation R realizing the mapping of the universe of discourse X onto the continuous interval [0, I].

In the general case the membership function /-LR of the relation is a hypersurface in (n + 1 )-dimensional space. An example of the membership function for n = 2 is shown in Fig. 4.22.

The membership functions of fuzzy relations for discontinuous universes of discourse can be represented in the form of relation tables giving the grade of membership /-LR(Xl, ... , x n ) for each discrete n-tuple. This is illustrated by Example 4.2.5.

Example 4.2.5 Two discrete component universes of discourse Xl and X 2 are given:

Xl = {1,2,3,4}, X 2 = {1,2,3,4,5}.

Table 4.7 exemplifies grades of membership for the relation "(Xl, X2) about (3,2)" defined over the set X = Xl X X 2•

In Table 4.7 we can see, for example, that the pair (Xl,X2) = (2,3) belongs to the relation R to the degree 0.5 or, otherwise, the pair (2,3) is similar, in the sense of relation, to the pair (3, 2) to the degree 0.5. 000

Fuzzy relations can be defined directly over the n-tuples of a multidimensional universe of discourse Xl X .•. X X n , as in Example 4.2.5. However,


Table 4.1. The fuzzy relation "(Xl, X2) about (3,2)" in tabular form

~ 1 2 3 4

X2

1 0 0 0 0

2 0 0.5 1 0

3 0 0.5 0.66 0

4 0 0.33 0.33 0

5 0 0 0 0

in fuzzy modeling and control we deal most often with relations created by aggregation of fuzzy sets defined over various one-dimensional spaces. An example of this is the rule of the type of IF ... THEN ... , (4.60)

IF (Xl = small) AND (X2 = large) THEN (y = medium) , (4.60)

where the component premises (Xl = small), (X2 = large) are combined by the logical connectives AND, OR. They create the binary fuzzy relation R having the membership function Jl.R(XI, X2) which determines the degree to which the premise is satisfied for the actual numerical values of the arguments Xl, X2. The aggregation ofth~ fuzzy sets "smali" (8) and "Iarge" (L) can be accomplished using operators of the t-norm type in the case of the connective AND, (4.61), or the s-norm type for the connective OR.

(4.61)

Assume that the membership functions Jl.S(XI) and Jl.L(X2) have the form shown in Fig. 4.23.

Both the fuzzy sets 8 and L are defined over various universes of discourse Xl and X 2 which can represent, in general cases, different physical quantities (e.g. current voltage and intensity). Therefore, these sets can not be aggregated directly, as in the case of fuzzy sets defined on one universe of discourse. Firstly, such sets should be transformed in certain special fuzzy relations defined on the Cartesian product Xl x X 2 and called the cylindrical extensions. Only when this has been done can the aggregation be carried out. The cylindrical extension is determined by the Definition 4.2.3.

Definition 4.2.3 If Xl and X 2 are non-fuzzy sets and a fuzzy set A is defined on Xl, then the cylindrical extension A" of the set A onto the universe of discourse Xl x X 2

is the relation determined as the Cartesian product ofthe sets A and X 2 , Le. AXX2:


small (S) large (L)

1

2 3 4

Fig. 4.23. The membership functions of the fuzzy sets "smali" and "large" used in the relation (4.61)

for all the pairs (Xl, X2) E Xl X X 2·

In the case of the cylindrical extension of a set A( Xl) onto an-dimensional universe of discourse Xl x ... X X n the extension operation is accomplished according to the formula:

for all n-tuples (Xl' ... ' Xn ) E Xl X ... X X n . The cylindrical extension is illustrated by Example 4.2.6.

Example 4.2.6 Let us determine the cylindrical extension of the set A(XI) onto the discrete set Xl x X 2 . The universes of discourse are:

the fuzzy set is: A = {I/al, O.5/a2, O/a3}.

The cylindrical extension of the set A(XI) onto Xl x X 2 is given in Table 4.8. 000

The next example illustrates the continuous cylindrical extension.

Example 4.2.7 Assurne that there are two fuzzy sets "smalI' (8) and "Iarge" (L), Fig. 4.23, defined over the sets Xl and X2. The universes of discourse are:

Xl = [1,4J, X 2 = [1,5J.


Table 4.8. The membership function of the cylindrical extension of the set A(Xl) onto Xl x X2

a\ a2 a3

b\ 1 0.5 0

b2 1 0.5 0

b3 1 0.5 0

Fig. 4.24. The cylindrical extensions S·(Xl,X2) and L·(Xl,X2) of the fuzzy sets S(xI) and L(X2) onto the two-dimensional universe of discourse Xl x X2

The membership nmctions PS(XI) and PL(X2) are given in Fig. 4.23. Fig. 4.24 represents the cylindrical extensions S*(XI,X2) and L*(XI,X2) of the fuzzy sets S(XI) and L(X2) onto Xl x X2' 000

The membership function of the relation R(XI,X2) determined by the formula (4.61) can be calculated using, for example, the MIN-operator as a t-norm.

PR(XI,X2) = MIN(ps·(XI,X2),PL.(XI,X2)), (XI,X2) E Xl X X 2. (4.62)

The result of this operation is represented in Fig. 4.25. If the rule (4.60) had, in the premise, the logical connective of the type

OR: IF (Xl =8) OR (X2 =L) THEN (y=M),

then for calculating the membership function of the premise, one of the snorms should be used, for instance the MAX:


a) b)

Fig. 4.25. The formation of the membership function of the relation JLR(Xl, X2) using the cylindrical extension of the component sets S and L (a) and the result of the operation (b) obtained with the use of the MIN -operator

Fig. 4.26. The membership function of the relation JLR(Xl, X2) according to the formula (4.63) created by means of the MAX-operator

Then we would obtain the membership function J.tR(Xl, X2) shown in Fig.4.26.


In fuzzy modeling, rules containing complex premises are encountered. They contain both the connectives AND and OR.

IF (Xl = small) AND (X2 = large) OR (Xl = large) AND (X2 = small) THEN (y = medium)

(4.64)

In order to evaluate the truth value of the premise in this rule the membership functions ofthe component relations I'Rl(XI,X2) and I'R2(XI,X2) should be determined, where:

I'Rl(XI,X2) = T(I'S(Xt},I'L(X2)), (XI,X2) E Xl X X 2 , I'Rl (Xl> X2) = T(I'L(Xt}, I'S(X2)), (Xl, X2) E Xl X X 2 . (4.65)

T means at-norm operator, e.g. the MIN-operator. Then the resulting relation R, which is the logical sum of the component relations R = RI U R2 ,

should be calculated on the basis of Definition 4.2.4.

Definition 4.2.4 Two binary relations R I and R 2 defined on the same universe of discourse Xl x X 2 are given. Then the sum of these relations R I U R 2 is given by the formula:

I'RIUR2 = S(I'RI(XI,X2),I'R2(XI,X2)),

where S means an s-norm (e.g. the MAX). If two binary relations are combined by the logical connectives of the type

AND, then the Definition 4.2.5 should be used.

Definition 4.2.5 Two binary relationsRI and R 2 defined on the identical universe of dis

course Xl x X 2 are given. The logical product R I n R 2 of these relations is determined by the formula:

I'RlnR2 = T(I'Rl(XI,X2),I'R2(XI,X2)),

where T means at-norm, e.g. the MIN. The above definitions should be respectively extended for the case of n

variant relations. The aggregation of the binary relations is illustrated by Example 4.2.8.

Example 4.2.8 Let us determine the membership function I'R(XI, X2) of the premise composed oftwo component premises (4.66).

IF (Xl = S) AND (X2 = L) OR (Xl = L) AND (X2 = S) (4.66)

The membership functions of individual fuzzy sets are given in Fig. 4.27. The membership function of the first component premise can be calculated

using the MIN -operator. In the case of the second component premise the membership function can be calculated similarly.


f..I.(X2)

s L 1 ------ ---------------------------------

o 2 3 4 5 Xl 0 2 3 4 5 6 X2

Fig. 4.27. The membership functions of the fuzzy sets in the premise of the rule ( 4.66)

,4

f..I.S(X2) /!

Xl ,/

,,/ , / "

X2

, , , , , , / : , ,

1

Fig. 4.28. The membership functions of the component relations Rl and R 2 of the premise of the rule (4.67)

/-LR1(XI,X2) = MIN(/-Ls(xd,/-LL( x2)) /-LR1(XI,X2) = MIN(/-LL(xd,/-Ls(x2)) (4.67)

The membership functions of the component relations are represented in Fig.4.28.

To carry out the OR-operation in the rule (4.66) the MAX-operator can be selected as an s-norm. Then the membership function of the resulting relation R = R I U R 2 is calculated from the formula (4.68).

/-LR(XI,X2) = MAX(/-LR1(XI,X2),/-LR2(XI,X2))

This function is represented graphically in Fig. 4.29. 000

(4.68)

In fuzzy models an opposite operation with respect to the cylindrical extension is also used. It is called projection. The cylindrical extension increases the dimensionality of the universe of discourse Xl of the given fuzzy


1

Fig. 4.29. The resulting membership function of the relation R(Xl, X2) determining the truth value of the complex premise in the rule (4.66)

set A(xt} by creating the relation A*(Xl,X2) over the universe of discourse Xl x X2. The projection of the relation (Xl,X2) defined on the universe of discourse Xl x X2 gives the fuzzy set A*(xd defined on the universe of discourse Xl of lower dimensionality. Thus, projection is the opposite operation with respect to the cylindrical extension.

Definition 4.2.6 If A is a fuzzy relation defined over the universe of discourse Xl x X 2 , then the projection of this relation onto the universe of discourse Xl is the fuzzy set A * determined as folIows:

A*(Xl) = ProjA(xl,X2) = MAX[A(Xl,X2)) Xl X2

The projection is illustrated by Example 4.2.9.

Example 4.2.9 The relation A, Table 4.9, defined on the universe of discourse X = Xl X X 2 ,

is given. Let us determine its projection onto the universe of discourse Xl.

ProjA = MAX[A(Xl,X2)) = A*(xd = (~, 0.5,~) "'2 al a2 a3

The accomplished projection is represented in Fig. 4.30. 000

4.3 Implication 149

Tab1e 4.9. The discrete membership function of the relation A(Xl, X2)

I~ X2 al a2 a3

b1 1 0.5 0

b2 05 0.5 0

b3 0 0 0

1 1

o o

a) b)

Fig. 4.30. Graphical illustration of fuzzy projection of a discrete (a) and continuous (b) ;elation

4.3 Implication

Implication is a kind of relation having the form of a rule used in the process of reasoning. A distinction can be made between classical and fuzzy implication.

Classical implication has the form (4.69) (Poradnik 1971).

IF p THEN q (4.69)

Its abbreviated form is expressed by (4.70).

(4.70)

where: p - the statement called the antecedent (premise), q - the statement called the consequent (conclusion, result).

In classicallogic, statements can be absolutely true (Jl.p = 1, Jl.q = 1) or untrue (Jl.p = 0, Jl.q = 0). The implication can be true or untrue according to the actual value of Jl.p and Jl.q (the truth of the antecedent and consequent).


Table 4.10. The membership function of the classical implication /Jp-+q

I /Jp I /Jq I /Jp-+q I 1 1 1

1 0 0

0 1 1

0 0 1

The truth-value of the implication is determined by its membership function ,",p-+q which can have only two values 0 and 1. The membership nmction of the c1assical implication is uniquely determined in the form of Table 4.10 (Poradnik 1971; Knappe 1994, Kahlert 1994).

As can be easily checked, the membership nmction of the c1assical implication can be calculated from the formula (4.71).

(4.71)

The operator of the c1assical implication has some properties which make it difficult to use in fuzzy modeling and control.

Example 4.3.1 Let us consider the implication (4.72).

IF (the age of a car x = new) THEN (fuel consumption y = small) (4.72)

The universe of discourse X of the variable "the age of a car" will be represented in binary form (new: x = 1, old: x = 0). 8imilarly, the universe of discourse Y of the variable "fuel consumption" (smali: y = 1, large: y = 0).

The statement: (the age of a car = new) = p = the antecedent, The statement: (fuel consumption = low) = q = consequent.

The question can be put: when is the implication (4.72) true (J.tp-+q = 1) and when is it untrue (J.tp-+q = O)?

By substituting possible linguistic values (new, old) for x and (smali, large) for y we obtain four states Si of the implication.

81: IF (the age of a car = new) THEN (fuel consumption = small),

J.tp = 1, J.tq = 1, J.tp-+q = 1.

For x = new, y = small, the implication is true.

82: IF (the age of a car = new) THEN (fuel consumption = large),

J.tp = 1, J.tq = 0, J.tp-+q = O.

4.3 Implication 151

/l /lp-+q(x,y)

1 1 /lp

/lq n 0 n 0

x x (age of acar)

• y y (fuel consumption)

a) b)

Fig. 4.31. The discrete membership functions ofthe premise /Jp and conclusion /Jq

(a) and the membership function of the implication /Jp-+q defined on the Cartesian product X x Y (n - new, 0 - old, s - small, l - large) (b)

For X = new and y = large the implication (4.72) is untrue. It is intelligible because the premise (the age of a car = new) has not changed. And so, the changed conclusion (fuel consumption = large) can not be true.

S3: IF (the age of a car = old) THEN (fuel consumption = smaIl),

J.tp = 0, J.tq = 1, J.tp--+q = 1.

For x = old and y = small the implication (4.72) is true. This results from the fact that the considered implication (4.72) concerns only the state: (the age of a car = new), whereas it says nothing ab out the opposite state (the age of a car = old). According to classicallogic, both conclusions (fuel consumption = small) and (fuel consumption = large) - given in S4 - can be true here.

S4: IF (the age of a car = old) THEN (fuel consumption = large),

J.tp = 0, J.tq = 0, J.tp--+q = 1 .

The implication (4.72) is true for these x,y values. Explanation as in S3. The membership function of the discrete implication J.tp--+q considered in the example is shown in Fig. 4.31.

The fault of the operator of the classical implication is that if its premise pis not satisfied even to the lowest degree (J.tp = 0), then the implication is true and this activates the mutually exclusive conclusions (fuel consumption is small) and (fuel consumption is large), Fig. 4.32. 000


1

• I

I n

o

J)p-+q( S ,y)

• I I I I I I I I I I I I o

x

Fig. 4.32. The membership functions of the implication JJp-+q(x, y) for x = 0 (old)

In the case of fuzzy control systems, when many fuzzy rules are activated at the same time, the use of the operator of c1assical logic would disadvantageously affect the control process (Kahlert 1994). In this case implication operators of more unique operation are needed. Therefore, in fuzzy control and in many problems of fuzzy modeling another implication, namely the Mamdani implication, is most often used. The Mamdani implication will be described in the following.

Fuzzy implication The fuzzy implication is the rule R whose simplest form is expressed by (4.73),

IF (x = A) THEN (y = B) , (4.73)

where: (x = A) is apremise (antecedent) and (y = B) is a conc1usion (consequent).

A and B are the fuzzy sets defined by their membership functions JLA(X) and JLB(Y) and the universes of discourse X and Y. The fuzzy implication is determined by the symbol (4.74).

A~B (4.74)

The difference between the c1assical and fuzzy implication is that in the case of the c1assical implication the premise and conc1usion is either absalutely true or absolutely untrue, whereas in the case of the fuzzy implication their partial truth, contained in the continuous interval [0,1], is allowed. Such an approach is very advantageous because in practice we rarely deal with situations where the premises of rules are completely satisfied. For this reason it can not be assumed that the conc1usion is absolutely true.

4.3 Implication 153

Table 4.11. The operators of fuzzy implication

Lukasiewicz implication MIN(1, 1 - J.'A(X) + J.'B(Y))

Kleene-Dienes implication MAX(1 - J.'A(X), J.'B(Y)) Kleene-Dienes-Lukasiewicz

1 - J.'A(X) + J.'A(X) . J.'B(Y) implication

Gödel implication { 1 1o, "A(X)" "B(Y) J.'B (y) otherwise

Vager implication (J.'A(XWB(II)

Zadeh implication MAX(1 - J.'A(X), MIN(J.'A(x),J.'B(Y)))

The fuzzy implication is determined, as every other fuzzy relation, by the membership function J.tA-tB(X, y) defined in the universe of discourse which is the Cartesian product X x Y of the domains of the premise and conclusion.

The membership function of the implication J.tA-tB(X,y) is the base for the so-called fuzzy reasoning (Chapter 5.1.2) which makes it possible to calculate the output of a fuzzy model (controller) for a given state of its inputs. To determine this function on the basis of the membership function of the premise J.tA(X) and conclusion J.tB(Y) an appropriate operator ofthe implication should be used. The Mamdani implication operator is based on the assumption that the truth-value of the conclusion J.tB(Y) can not be higher than the degree to which the premise J.tA(X) is satisfied, formula (4.75).

(4.75)

Such an assumption is intuitively intelligible. For example, in the case of the rule (4.76):

IF (the age of a car = new) THEN (fuel consumption = small), (4.76)

it is intelligible that if a car is not completely new, then its fuel consumption can not be so small as in the case of a completely new car. Apart from the Mamdani operator in fuzzy control the algebraic product operator PROD (4.77) is also applied:

(4.77)

Besides the operators of fuzzy implication represented so far, many other operators have also been elaborated. Their use can have various results according to the actual problem. The operators are represented in Table 4.11.

According to the research results published in (Knappe 1994) the operator which has the best properties, considering a certain set of criteria assumed in this publication, is the Lukasiewicz operator. The remaining operators given


Jl{x) Jl{y)

1 1

o 1 2 3 x o 1 2 3 4 5 Y age of a car (years) fuel consumption

Fig. 4.33. The membership functions of the fuzzy sets "new" and "smali" used in the premise and conclusion

J.I.

1

,/,/,/,,/ X (age of a car)

y (fuel consumption)

Fig. 4.34. The membership function IJA~B(X, y) ofthe implication (4.78) obtained with the use of the Mamdani implication operator

in Table 4.11 are ordered according to the decreasing degree to which these criteria are satisfied. Example 4.3.2 shows the method of creating the membership functions ofthe implication J.l.A~B(X, y) with the use ofthe Mamdani operator.

Example 4.3.2 Let us consider the fuzzy implication (4.78),

IF (the age of a car x = new) THEN (fuel consumption y = small) (4.78)

4.3 Implication 155

J.L J.L J.LB~) 1 -----------------.,..J

" " J.LA .... B(1.5,y) 1

I ,

o 4 5 6 7 8 Y

a) b)

Fig. 4.35. The membership function of the implication /-LA .... B(X, y) for the given value of the variable x = Xo = 1.5 (a) and its projection onto the plane {/-L, y}

where the fuzzy sets "new" and "small" are defined by the membership functions f.-Ln(x) and f.-Ls(Y) given in Fig. 4.33.

The membership function of the implication (4.78) is represented in Fig. 4.34. By using an appropriate method of inference and having the given value Xo of the variable x in the premise the membership function of the conclusion f.-LA-tB(Xo,Y) can be determined, which can then be used for calculating the crisp value of the output Yo of a fuzzy model, Fig. 4.35. This problem is described in Sections 5.1.2. and 5.1.3. 000

[studies in fuzziness and soft computing] fuzzy modeling and control volume 69 || mathematics of...

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