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Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Part I1

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Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Part I1 !Err TRANSACTIONS ON SYSTEMS, MAN. ANI) C’YHT.RNT.TI(‘S, VOI.. 20, NO. 2. MAR(.II/APRII. 1000 419 Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Part I1 Abstract -During the past several years, fuzzy control has emerged as I) Basic Properties of a Fuuy Implication Function: one of the most active and fruitful areas for research in the applications The choice of a fuzzy implication function involves a of fuzzy set theory, especially in the realm of industrial processes, which number of criteria, which are discussed in [31, [241, [2l, do not lend themselves to control by conventional methods because of a lack of quantitative data regarding the input-output relations. Fuzzy [711, [181, [521, [191, [116], [SI, [72], and [96]. In particular, control is based on fuzzy logic-a logical system that is much closer in Baldwin and Pilsworth [3] considered the following basic spirit to human thinking and natural language than traditional logical characteristics of a fuzzy implication function: fundamen- systems. The fuzzy logic controller (FLC) based on fuzzy logic provides a tal property, smoothness property, unrestricted inference, means of converting a linguistic control strategy based on expert knowl- symmetry of generalized modus ponens and generalized edge into an automatic control strategy. A survey of the FLC is pre- sented; a general methodology for constructing an FLC and assessing its modus tollens, and a measure of propagation of fuzziness. performance is described; and problems that need further research are All of these properties are justified on purely intuitive pointed out. In particular, the exposition includes a discussion of grounds. We prefer to say that the inference (conse- fuzzification and defuzzification strategies, the derivation of the database quence) should be as close as possible to the input truth and fuzzy control rules, the definition of fuzzy implication, and an function value, rather than be equal to it. This gives us a analysis of fuzzy reasoning mechanisms. more flexible criterion for choosing a fuzzy implication I. DECISIONMAKINGLOGIC function. Furthermore, in a chain of implications, it is necessary to consider the “fuzzy syllogism” [ 1471 associ- S WAS noted in Part I of this paper [150], an FLC ated with each fuzzy implication function .before we can A may be regarded as a means of emulating a skilled talk about the propagation of fuzziness. human operator. More generally, the use of an FLC may Fukami, Mizumoto, and Tanaka [24] have proposed a be viewed as still another step in the direction of model- set of intuitive criteria for choosing a fuzzy implication ing human decisionmaking within the conceptual frame- function that constrains the relations between the an- work of fuzzy logic and approximate reasoning. In this tecedents and consequents of a conditional proposition, context, the forward data-driven inference (generalized with the latter playing the role of a premise in approxi- modus ponens) plays an especially important role. In what mate reasoning. As is well known, there are two impor- follows, we shall investigate fuzzy implication functions, tant fuzzy implication inference rules in approximate rea- the sentence connectives and and also, compositional soning. They are the generalized modus ponens (GMP) operators, inference mechanisms, and other concepts that and the generalized modus tollens (GMT). Specifically, are closely related to the decisionmaking logic of an FLC. premise 1: x is A’ A. Fuzzy Implication Functions Dremise 2: if x is A then y is B (GMP) In general, a fuzzy control rule is a fuzzy relation which consequence: y is B’ is expressed as a fuzzy implication. In fuzzy logic, there premise 1: y is B’ are many ways in which a fuzzy implication may be defined. The definition of a fuzzy implication may be oremise 2: if x is A then v is B (GMT) expressed as a fuzzy implication function. The choice of a consequence: x is A’ fuzzy implication function reflects not only the intuitive criteria for implication but also the effect of connective also. in which A, A’, B, and B’ are fuzzy predicates. The propositions above the line are the premises; and the proposition below the line is the consequence. The pro- Manuscript received May 27, 198X; revised July 1, 19x9. This work wpported in part by NASA under grant NCC-2-275 and in part by posed criteria are summarized in Tables I and 11. We AFOSR under grant 89-0084. note that if a causal relation between “x is A” and “y is The author is with the Electronics Research Laboratory, Department B” is not strong in a fuzzy implication, the satisfaction of of Electrical Engineering and Computer Sciences, University of Califor- nia, Berkeley, CA 94720. criterion 2-2 and criterion 3-2 is allowed. Criterion 4-2 is IEEE Log Number X032014. interpreted as: if x is A then y is B, else y is not B. 0018-9472/90/0300-0419$01.00 01990 IEEE 420 IEEE TRANSACTIONS ON SYSTTMS. MAN, ANI) ('YBERNETI('S. Vol.. 20. NO. 2, MAlZ('II/Al'KII. 1990 TABLE I TABLE 11 INTUITIVECRITEKIA RTLATING PREI AND CONS INTUITIVE CRITERIA REIATINGPRF.1 ANI) CONS FOR GIVEN PRE2 IN GMP FOR GIVENPRE~ IN GMT x is A'(Pre1) y is BYCons) v is B'(Pre1) x is AYCons) Criterion 1 x is A y is B Criterion 5 y is not B x is not A Criterion 6 y is not very B x is not very A Criterion 2-1 x is very A y is very B Criterion 7 y is not more or less B x is not more or less A Criterion 2-2 x is very A y is B Criterion 8-1 y is B x is unknown Criterion 3-1 x is more or less A y is more or less B Criterion 8-2 y is B x is A Criterion 3-2 x is more or less A y is B Criterion 4-1 x is not A y is unknown Criterion 4-2 x is not A y is not B defined for all x,y E [O, 11: union x v y = max{x, y) Although this relation is not valid in formal logic, we often make such an interpretation in everyday reasoning. algebraic sum x*y=x+y-xy The same applies to criterion 8. bounded sum xey = min{l,x + y) 2) Families of Fuzzy Implication Functions: Following Zadeh's [146] introduction of the compositional rule of x y=o inference in approximate reasoning, a number of re- drastic sum xuy= y x=o searchers have proposed various implication functions in i1 x,y>o which the antecedents and consequents contain fuzzy variables. Indeed, nearly 40 distinct fuzzy implication disjoint sum xAy = max(min(x, 1 - y), functions have been described in the literature. In gen- min(1- x,y)}. eral, they can be classified into three main categories: the fuzzy conjunction, the fuzzy disjunction, and the fuzzy The triangular norms are employed for defining conjunc- implication. The former two bear a close relation to a tions in approximate reasoning, while the triangular co- fuzzy Cartesian product. The latter is a generalization of norms serve the same role for disjunctions. A fuzzy con- implication in multiple-valued logic and relates to the trol rule, "if x is A then y is B," is represented by a extension of material implication, implication in proposi- fuzzy implication function and is denoted by A + B, tional calculus, modus ponens, and modus tollens [18]. In where A and B are fuzzy sets in universes U and V with what follows, after a short review of triangular norms and membership functions pA and pB, respectively. triangular co-norms, we shall give the definitions of fuzzy Definition 3: Fuuy Conjunction: The fuzzy conjunction conjunction, fuzzy disjunction, and fuzzy implication. is defined for all U E U and v E V by Some fuzzy implication functions, which are often em- A+B=AXB ployed in an FLC and are commonly found in the litera- ture, will be derived. =/I,x~pa(u)*pB(v)/(u,v) Definition 1: Triangular Noms: The triangular norm * where * is an operator representing a triangular norm. is a two-place function from [O, 11 x [O, 11 to [O, 11, i.e., *: Definition 4: Fuuy Disjunction: The fuzzy disjunction is [O, 11 X [0, 13 + [O, 11, which includes intersection, algebraic defined Cor all U E U and v E V by product, bounded product, and drastic product. The A+B=AXB greatest triangular norm is the intersection and the least one is the drastic product. The operations associated with triangular norms are defined for all x, y E [0,1]: where + is an operator representing a triangular co-norm. Definition 5: Fuzzy Implication: The fuzzy implication is intersection x A y = min (x,y) associated with five families of fuzzy implication functions algebraic product x.y = xy in use. As before, * denotes a triangular norm and + is a triangular co-norm. bounded product xOy = max(0,x + y - 1) 4.1) Material implication: x y=l drastic product xny= y x=l A + B =(not A)+ B i0 x,y<l. 4.2) Propositional calculus: A --+ B = (not A)+(A * B) Definition 2: Triangular CO-Noms: The triangular co- 4.3) Extended propositional calculus: norms is a two-place function from [O, l] x [O, l] to [O, l], + A -+ B = (not A x not B)+ B i.e. + : [O, 11 X [O, 11 to [O, 11, which includes union, alge- braic sum, bounded sum, drastic sum, and disjoint sum. 4.4) Generalization of modus ponens: The operations associated with triangular co-norms are A 4 B =sup{cE [OJ], A *c< B} LEE: FUZZY LOGIC' IN ('ONTROI.
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