Thus far we have considered fuzzy rules of the form:
IF x is Ai AND IF y is BiTHEN z is Ci (2.61) where Ai, Bi, and Ciare fuzzy states governing the i-th rule of the rule base.
In fact this is the Mamdani approach (Mamdani system or Mamdani model) named after the person who pioneered the application of this approach.
Here, the knowledge base is represented as fuzzy protocols of the form (2.61) and represented by membership functions for Ai, Bi, and Ci, and the inference is obtained by applying the compositional rule of inference. The result is a fuzzy membership function, which typically has to be defuzzified for use in practical tasks.
Several variations to this conventional method are available. One such version is the Sugeno model (or Takagi–Sugeno–Kang model or TSK model).
Here, the knowledge base has fuzzy rules with crisp functions as the con-sequent, of the form
IF x is AiAND IF y is BiTHEN ci= fi(x, y) (2.62) for Rule i, where fiis a crisp function of the condition variables (antecedent) x and y. Note that the condition part of this rule is the same as for the Mamdani model (2.61), where Aiand Biare fuzzy sets whose membership functions are functions of x and y, respectively. The action part is a crisp function of the condition variables, however. The inference L(x, y) of the fuzzy knowledge-based system is obtained directly as a crisp function of the condition variables x and y, as follows.
For Rule i, a weighting parameter wi(x, y) is obtained corresponding to the condition membership functions, as for the Mamdani approach, by using either the “min” operation or the “product” operation. For example, using the “min” operation we form
wi(x, y) = min[µAi(x),µBi(y)] (2.63)
The crisp inference L(x, y) is determined as a weighted average of the indi-vidual rule inferences (crisp) ci= fi(x, y) according to
L(x, y) = (2.64)
2 Fu n dament als of fuzzy logic sy s tems
128 where r is the total number of rules. For any data x and y, the knowledge-based action L(x, y) can be computed from (2.64), without requiring any defuzzification. The Sugeno model is particularly useful when the actions are described analytically through crisp functions, as in conventional crisp control, rather than linguistically. The TSK approach is commonly used in applications of direct control and in simplified fuzzy models. The Mamdani approach, even though popular in low-level direct control, is particularly appropriate for knowledge representation and processing in expert systems and in high-level (hierarchical) control systems.
2.12 Summary
Fuzzy logic was first developed by L.A. Zadeh in the mid-1960s for repres-enting some types of “approximate” knowledge that cannot be represented by conventional, crisp methods. In the crisp Boolean logic, truth is repres-ented by the state 1 and falsity by the state 0. Boolean algebra and crisp logic have no provision for approximate reasoning. Fuzzy logic is an extension of crisp bivalent (two-state) logic in the sense that it provides a platform for handling approximate knowledge. Fuzzy logic is based on fuzzy set theory in a similar manner to how crisp bivalent logic is based on crisp set theory. A fuzzy set is represented by a membership function. A particular “element”
value in the range of definition of the fuzzy set will have a grade of member-ship, which gives the degree to which the particular element belongs to the set. In this manner, it is possible for an element to belong to the set at some degree and simultaneously not belong to the set at a complementary degree, thereby allowing a non-crisp (or fuzzy) membership. This chapter presented the basic theory of fuzzy logic, which uses the concept of fuzzy sets. Applica-tions of soft computing and particularly fuzzy knowledge-based systems rely on the representation and processing of knowledge using fuzzy logic. In particular, the compositional rule of inference is what is applied in decision-making with fuzzy logic. Underlying principles were systematically presented in this chapter. Geometrical illustrations and examples were presented in order to facilitate the interpretation. The application of these concepts in knowledge-based “intelligent” systems and particularly in intelligent control will be treated in future chapters.
Problems
1. Suppose that A denotes a set, X denotes the universal set and f denotes the null set.
The law of contradiction:
A ∧ A = f
and the law of excluded middle:
A ∨ A = X
P roble m s
129are satisfied by crisp sets and hence agreeable to bivalent logic. Show that these two laws are not satisfied by fuzzy sets and hence violated in fuzzy logic. Give an example to illustrate each of these two cases.
2. Show that for crisp sets, the Bold Intersection (Bounded Max) operation is ident-ical to the conventional intersection (min), and the Bold Union (Bounded Min) operation is identical to the conventional union (max). Also show that this is not the case in general for fuzzy sets. You may use a graphical proof.
3. Consider the laws of bivalent (crisp) logic. Which of these laws are violated by fuzzy logic? In particular, show that DeMorgan’s Laws are satisfied.
4. Critically investigate why Japanese industry and society accepted and utilized fuzzy logic much sooner and more profitably than western society.
5. List five consumer appliances that use fuzzy logic. Indicate how fuzzy logic is used in each of these appliances.
6. List five areas of possible application of fuzzy logic (not necessarily fuzzy logic control). In each area, discuss the appropriateness or inappropriateness of using fuzzy logic, compared to other (conventional) approaches. Also, for each area, describe one practical application and indicate how fuzzy logic is utilized in that application.
7. Using common knowledge, experience, judgment, and perception, construct and sketch appropriate membership functions for the following sets:
(i) Tall men (v) Hot outside temperature
(ii) Tall women (vi) Cold outside temperature
(iii) Hot room temperature (vii) Fast car speed on interstate highway (iv) Cold room temperature (viii) Fast car speed in city
In each case, you must give either the functional relation for the membership func-tion, with appropriate numerical values for their parameters, or numerical data to completely represent the membership function.
8. Two fuzzy sets A and B are represented by the following two membership functions:
mA( x)= max 0, for x≤ 10
= max 0, for x> 10
mB( x )= max 0, for x≤ 10
= max 0, for x> 10
(a) Sketch these membership functions.
(b) What do A and B approximately represent?
(c) Which one of the two sets is fuzzier?
9. Consider a fuzzy set A in the universe ℜ (i.e., the real line) whose membership function is given by
2 Fu n dament als of fuzzy logic sy s tems
130 (a) Sketch the membership function.
(b) What is the support set of A?
(c) What is the a-cut of A for a= 0.5?
10. The chronology of artificial intelligence may be summarized as follows:
The first technical paper on the subject appeared in 1950. LISP programming language and the first expert system were developed in the 1960s. The first paper on “Fuzzy Sets” was published by Zadeh in 1964. Systems with natural language capabilities appeared in the 1970s. Many hardware tools, develop-ment systems, and expert systems were commercially available, and prolifera-tion of fuzzy logic applicaprolifera-tions, particularly in consumer appliances, took place in the 1980s. Integration of various AI techniques, extensive availability of PC-based systems, and sophisticated user interfaces are noted in the 1990s. Application in decision support systems and other complex industrial systems is continuing in the 2000s.
In your opinion why did it take about two decades for us to accept the versatility to fuzzy logic in AI applications?
11. The characteristic function cAof a crisp set A is analogous to the membership function of a fuzzy set, and is defined as follows:
cA(x)= 1 if x ∈ A
= 0 otherwise Show that
cA′= 1 − cA
cA∨B= max( cA,cB) cA∧B= min( cA,cB)
cA→B( x, y) = min[1, {1 − cA( x) + cB( y)}]
where A and B are defined in the same universe X, except in the last case (impli-cation) where A and B may be defined in two different universes X and Y.
What are the implications of these results?
12. Show that the commutative and associative properties and boundary conditions are satisfied for “min” and “max” norms. Also show that DeMorgan’s Law is satisfied by these two norms. This verifies that “min” is a T-norm and “max” is an S-norm.
13. For two membership functions mAand mBshow that mA·mB≤ min( mA, mB)
Does this indicate that the “max-dot” inference tends to provide fuzzier inferences (more conservative) than the “max-min” inference? Discuss.
14. If mA< 0.5, show that a set A* that is less fuzzy than A satisfies mA*< mA< 0.5.
Similarly, if mA> 0.5, a set A* that is less fuzzy than A satisfies mA*> mA> 0.5.
15. The degree of containment CA,B of a fuzzy set A in another fuzzy set B, both of which being defined in the same universe, is given by
CA,B= 1 if mA≤ mB
= mB if mA> mB
What does this parameter represent?
P roble m s
13116. Consider the S-norm (or T-conorm) given by x + y − xy with 0 < x < 1 and 0 < y < 1.
Show that
x+ y − xy > max( x, y)
where max( x, y) is another S-norm.
17. In the usual notation, the T-norm of two membership functions x and y is denoted by xTy and its complementary (or conjugate) norm, the S-norm, is denoted by xSy.
The two norms are related through the DeMorgan’s Law xSy= 1 − (1 − x)T(1 − y)
Using this relationship, determine the S-norm corresponding to each of the following two T-norms:
(i) xy
(ii) max[0, x+ y − 1]
Clearly indicate all the important steps of your derivations.
In operations with fuzzy sets, what do T-norm and S-norm generally represent?
18. The transitivity property of conventional (crisp) sets states that If A⊂ B and B ⊂ C then A ⊂ C
Is this property satisfied by fuzzy sets? Explain.
19. The involution property of fuzzy sets states that E= A
(i.e., NOT(NOT A) = A). Is this property satisfied by fuzzy sets? Explain.
20. Consider the fuzzy implication A→ B where A and B are fuzzy sets. Give several mathematical operations (e.g., min) that are used in the literature for representing fuzzy implication. Discuss the suitability of each operation in practical applications.
21. Suppose that the state of “fast speed” of a machine is denoted by the fuzzy set F with membership function mF(n ). Then the state of “very fast speed,” where the linguistic hedge “very” has been incorporated, may be represented by mF(n− n0) with n0> 0. Also, the state “presumably fast speed,” where the linguistic hedge
“presumably” has been incorporated, may be represented by mF2(n ).
(a) Discuss the appropriateness of the use of these membership functions to represent the respective linguistic hedges.
(b) In particular, if
F= , , , , , , , ,
in the discrete universe V= {0, 10, 20, . . . , 190, 200} rev/s and n0= 50 rev/s, determine the membership functions of “very fast speed” and “presumably fast speed.”
(c) Suppose that power p is given by the relation (crisp) p = n2
For the given fast speed (fuzzy) F, determine the corresponding membership function for power. Compare/contrast this with “presumably fast speed.”
#
2 Fu n dament als of fuzzy logic sy s tems
132 22. Sketch the membership function mA( x ) = e−l( x−a)nfor l= 2, n = 2, and a = 3 for the support set S = [0, 6]. On this sketch separately show the shaded areas that represent the following fuzziness measures:
where A is the complement of the fuzzy set A. Evaluate the values of M1, M2
and M3for the given membership function.
(i) Establish relationships between M1, M2and M3.
(ii) Indicate how these measures can be used to represent the degree of fuzziness of a membership function.
(iii) Compare your results with the case l= 1, a = 3, and n = 2 for the same support set, by showing the corresponding fuzziness measures on a sketch of the new membership function.
23. The grade of inclusion of fuzzy set A in fuzzy set B is given by:
g( x)= 1 for mA( x) ≤ mB( x)
= mB( x) for mA( x) > mB( x)
Show that g( x) = sup{c ∈ [0, 1]| mA( x)Tc≤ mB( x)}
where the T-norm, T, may be interpreted as the min operation.
24. For a fuzzy set A, prove the representation theorem:
mA( x) = [a mAa( x)]
where mAais the a-cut of mA( x).
25. Show that max[0, x + y − 1] is a T-norm. Also, determine the corresponding T-conorm (i.e., S-norm). Hint: Show that the nondecreasing, commutative, and associative properties and the boundary conditions are satisfied.
26. (a) Establish the “isomorphism” between conventional set theory and binary propositional logic by comparing the primary operations of sets and logic.
(b) Consider the case of ternary (three-valued) logic where in addition to “True”
and “False” with the respective truth values 1 and 0, there is “indeterminacy”
with the corresponding truth value 1/2. In this logic, give suitable truth tables for the operations NOT, AND, OR and IF-THEN.
27. (a) Consider the membership function mA( x) = e−l| x−a|n, for a fuzzy set A.
Interpret the meaning of the parameters a, l and n. In particular, discuss how (1) fuzziness and (2) a fuzzy adjective or fuzzy modifier such as “very” or
“somewhat” of a fuzzy state may be represented using these parameters.
(b) Using the general membership function expression used in part (a), give an analytical representation for temperature inside a living room that has the
sup
P roble m s
133three fuzzy states “cold,” “comfortable,” and “hot.” You must give appropri-ate numerical values for the parameters of the analytical expression.
28. A fuzzy set A is defined in the universe X = [0, 8] and its membership function is given by
mA( x) =
Determine the fuzzy set B which is obtained through the crisp relation y = x + 1.
What is the corresponding universe?
29. (a) The extension principle may be considered as a special case of the applica-tion of the composiapplica-tional rule of inference. Describing the extension principle and the compositional rule of inference, justify this statement.
(b) A process is represented by the crisp relation z2= ( x − a)2+ y2
where information in the X × Y domain is mapped to the Z domain, with x, y, and z defined in the universes X, Y, and Z, respectively.
Suppose that the information S in the X × Y domain is represented by the fuzzy relation
mS( x, y) =
What is the membership function mC( z) of the corresponding “inference” C in the Z domain? Assume that a, b, c, d> 0 and b > d.
Sketch mS( x, y) and mC( z).
30. (a) Define the following terms as applied to fuzzy logic:
(i) Projection
(ii) Cylindrical extension (iii) Join
(iv) Composition.
What is the significance of the “composition” operation in fuzzy decision-making?
(b) Consider two fuzzy relations R and S, defined in X× Y and Y × Z, respectively, as given below.
2 Fu n dament als of fuzzy logic sy s tems
134 (iv) What is the composition Ro S of R and S?
(v) What is the projection of Ro S onto Z?
(vi) What is the projection P(R ) of R onto Y ? (vii) What is the composition of P(R ) and S ?
Compare this result with what you obtained in part (v) above.
Note: You may use the “product” operation instead of the “min” operation to represent the “intersection” of two fuzzy sets.
31. Consider the two fuzzy relations R( x, z) and S( z, y ). Their sup-min composition is denoted by Ro S where:
mRoS(x, y) = {min [mR(x, z), mS(z, y)]}
Their inf-max composition is denoted by R⊗ S where:
mR⊗S(x, y) = {max [mR(x, z), mS(z, y)]}
Carefully check whether the DeMorgan’s Laws:
T= N ⊗ O and
U= N o O
are satisfied for these compositions. Proofs have to be provided for your conclusions.
32. Consider the binary relations (rule bases) R ( x, z ), P ( x, z ) and the context fuzzy set S( z ). Check, giving necessary proofs, whether the following relations are satisfied in this case:
(a) (R∪ P) o S = (R o S) ∪ (P o S) (b) (R∩ P) o S = (R o S) ∩ (P o S)
Indicate practical implications of your results in the context of fuzzy knowledge-based decision-making.
33. Consider the knowledge base represented by the set of fuzzy rules Y→ U with
Y= + + + +
and
U= + +
Determine the membership function matrix R of this knowledge base.
Suppose that three fuzzy observations Y′ have been made whose member-ship functions are:
(a) mY′( yi) = [0.4 1.0 0.4 0 0]
(b) mY′( yi) = [0.4 0.8 0.4 0 0]
(c) mY′( yi) = [0.4 0.5 0.4 0 0]
Determine the corresponding fuzzy inference U′ in each case.
0.3
Re fe renc es
13534. For fuzzy information mX( x ) and a fuzzy rule base mR( x, y ), the compositional rule of inference provides the inference
Xo R = min[mX(x), mR(x, y)]
Now consider the special case where the rule base is actually a crisp relation y = f(x). Specifically,
mR= 1 if y = f(x)
= 0 otherwise
Show that the extension principle is obtained in this case.
References
1. De Silva, C.W., and Lee, T.H. (1994) “Fuzzy logic in process control,”
Measurements and Control, vol. 28, no. 3, pp. 114–24, June.
2. De Silva, C.W. (1997) “Intelligent control of robotic systems,” International Journal of Robotics and Autonomous Systems, vol. 21, pp. 221–37.
3. De Silva, C.W. (1995) Intelligent Control: Fuzzy Logic Applications, CRC Press, Boca Raton, FL.
4. Dubois, D., and Prade, H. (1980) Fuzzy Sets and Systems, Academic Press, Orlando.
5. Jain, L.C., and De Silva, C.W. (1999) Intelligent Adaptive Control: Industrial Applications, CRC Press, Boca Raton, FL.
6. Klir, G.J., and Folger, T.A. (1988) Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, NJ.
7. Zadeh, L.A. (1979) “A theory of approximate reasoning,” Machine Intelligence, J.E. Hayes, D. Michie, and L.I. Mikulich (eds), vol. 9, pp. 149–94, Elsevier, New York.
sup
x