!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 4.5) Generalization of modus tollens: from Definition 5.1 by using the union; and Goguen's fuzzy implication follows from Definition 5.4 by using the A+B=inf{t~[O,l],B+t p.A(u))v(pU,(v))/(u,v). B; = A" 0 R,, Goguen's fuzzy implication: RA= AxV+U X B >> p.B(v>)/(u,v) The membership function pBL of the fuzzy set B; is pointwise defined for all v E V by where CLA(U) - 422 S 1 .o 1 .o 0.5 0.5 0 0 Fig. 3. Approximate reasoning: generalized modus ponens with Larsen's product operation rule. r 0 1 (b) Fig. 1. Diagrams for calculation of membership functions. (a) flR versus pAwith the parameter j~~.(b) pR,, versus pBwith parametef 0 PA. Fig. 4. Approximate reasoning: generalized modus ponens with Larsen's product operation rule. sp(lgA) 1-BA(() 0.5 . .?'. ------.'\ 0.3''\.\. Fig. 2. Approximate reasoning: generalized modus ponens with Larsen's product operation rule. 0 AA(() Fig. 5. Approximate reasoning: generalized modus ponens with ship function pBLis obtained by Larsen's product operation rule. CLB$V) = SUP min (PA( U)9 PA( U)PB( V) I U€U by a broken line and dotted line, respectively. The mem- pnL = SUP PA(U)CLB(V) bership function is given by U€U PB;,(V) = SUP min{l.~"(.),pA(u),H(V~l =pn(V), CLA(U) =I- U E ti {AA'=A2): The values of S,,(p:(u)) with a parameter =pn( v). = pB(v), say pB(v) 0.3 and 0.8, are indicated in Fig. 3 by a {A' = not A): The values of S,( - pA(u))with a pa- broken line and dotted line, respectively. The member- rameter ps(~),say ps(~)=0.3 and 0.8, are indicated in ship function psi may be expressed as Fig. 5 by a broken line and dotted line, respectively. The psh PB$V) = SUP min(Cl~(u),CLA(u)pB(v)J membership function is given by U€U = minIl-pA(U),pA(U)/*H(v)l P BP,(v) SUP =PB(V). U E ti {A' = The values of S,,(py(u)) with a parame- PAVI ter pB(v), say pJv) = 0.3 and 0.8, are indicated in Fig. 4 LEE: FUZZY LOGIC' IN C'ONTKOL SYSTEMS: FUZZY LOGIC' CONTROI.I.EK-PAKT II 423 0 1 0 NdV) Fig. 8. Approximate reasoning: generalized modus tollens withlarsen's Fig. 6. Approximate reasoning: generalized modus tollens with product operation rule. Larsen's product operation rule. S A 1.o 1.o 0.6 0.5 0 1 0 NdV) Fig. 9. Approximate reasoning: generalized modus tollens with Fig. 7. Approximate reasoning: generalized modus tollens with Larsen's product operation rule. Larsen's product operation rule. Larsen's Product Rule in GMT: Suppose that B'= not TABLE 111 SUMMARYOF INFERENCE RESULTS FOR GENERALIZEDMODUS PONENS B" (a> 0); then the consequence A; is inferred as fol- lows: A Very A More or Less A Not A PB 0.5 A A; = R, o(not B") PB PB - l+PB 4s-1 1 ,l L 6-1 The membership function pA; of the fuzzy set A; is TVPB 1 pointwise defined for all U E U by fi--1 1 pA;(U) = sup min {pA(u)pB( v), 1- v)} 2VPB V€V G 1 = SUP w-P%V>> V€V where Fig. 7 by a broken line and dotted line, respectively. The s,(1- Prig( V 1) min I PA( U)PB( V) > 1- cL% v) 1 . membership function pA;is given by {E=not B}: The values of S,(1- pB(v))with a pa- rameter pA(u), say pA(u)= 0.3 and 0.8, are indicated in Fig. 6 by a broken line and dotted line, respectively. The membership function pA;is given by ~,g;(u)= S~P~~~{CLA(U)I~B(V),~-~B(V)} V€V - PA( U) ' {B'= not The values of S,(1- &j5(v)) with a {B'= not B2}: The values of S,(1- &(v)) with a parameter pA(u),say pA(u)= 0.3 and 0.8, are indicated in parameter pA(u),say pA(u)= 0.3 and 0.8, are indicated in Fig. 8 by a broken line and dotted line, respectively. The 424 IEEE TRANSA(.TIONS ON SYSTEMS, MAN, ANI) (‘YUERNETIC‘S. VOI.. 20, NO. 2, MAR(II/APRIL. 1990 TABLE IV SUMMARYOF INFERENCE RESULTSFOR GENERALIZEDMoous TOI.I.F.NS Not B Not Very B Not More or Less B B R, - .FA) 1 I 2 TABLE V SATISFACTIONOF VARIOUSFuzzy IMPLICATIONFUNCTIONS UNDER INTUITIVE CRITERIA R, R, R,, RP>? R, Rh RI, Criteria 1 0 0 X X 0 X X Criteria 2-1 X X X X 0 X X Criteria 2-2 0 0 X X X X X Criteria 3-1 X X X X 0 X X Criteria 3-2 0 0 X X X X X Criteria 4-1 X X 0 0 0 0 0 Criteria 4-2 X X X X X X X Criteria 5 X X X X 0 X X Criteria 6 X X X X 0 X X Criteria 7 X X X X 0 X X Criteria 8-1 X X 0 X 0 0 0 Criteria 8-2 0 0 X X X X X membership function pA;,is given by consequent of one rule serving as the antecedent of another. In effect, the FLC functions as a one-level for- PLA;(U) = SUP min{PA(U)PB(v),1-P~5(V)} ward data-driven inference (GMP). Thus the backward V€V goal-driven inference (GMT), chaining inference mecha- - 2PAU)+ 1- nisms (syllogisms), and contraposition do not play a role - 4- in the FLC, since there is no need to infer a fuzzy control 2PA(U) action through the use of these inference mechanisms. {B’= B): The values of S,(pB(v))with a parameter Although R, and R, do not have a well-defined logical pA(u),say pA(u)= 0.3 and 0.8, are indicated in Fig. 9 by a structure, the results tabulated in Table V indicate that broken line and dotted line, respectively. The member- they are well suited for approximate reasoning, especially ship function pA; is given by for the generalized modus ponens. R, has a logical structure which is similar to R,. R, is PA;W = SUP IPA(U)PB(V)?PB(V)l V€V based on the implication rule in Lukasiewicz’s logic LAleph. However, R, and R, are not well suited for approximate =PA(U). reasoning since the inferred consequences do not always The remaining consequences [24] inferred by R,, R,, fit our intuition. Furthermore, for multiple-valued logical R,,R,,R,,R, can be obtained by the same method as systems, R, and RA have significant shortcomings. Over- just described. The results are summarized in Tables I11 all, R, yields reasonable results and thus constitutes an and IV. appropriate choice for use in approximate reasoning. By employing the intuitive criteria in Tables I and I1 in Tables 111 and IV, we can determine how well a fuzzy B. Interpretation Sentence Connectives “and, also ’’ implication function satisfies them. This information is of summarized in Table V. In most of the existing FLC’s, the sentence connective In FLC applications, a control action is determined by “and” is usually implemented as a fuzzy conjunction in a the observed inputs and the control rules, without the Cartesian product space in which the underlying variables take values in different universes of discourse. As an TABLE VI SLJlTAH1.F PAIRS OF A F(JZZYIMP1 I(ATI0N FUNCTION illustration, in “if (A and B) then C,” the antecedent is AND CONNECTIVE “LllSO” interpreted as a fuzzy set in the product space U X V, Implication Rule Connective Also with the membership function given by PA XB( U,.> = min {PA(U)?PA 41 or PA xn( U 2 v) = PA( U 1 .PAv 1 “It depends on the shape of reproduced curve which forms the set of fuzzy control rules. where U and V are the universes of discourse associated with A and B, respectively. though they yield reasonably good results in approximate When a fuzzy system is characterized by a set of fuzzy reasoning. control rules, the ordering of the rules is immaterial. This From a practical point of view, the computational as- necessitates that the sentence connective “also” should pects of an FLC require a simplification of the fuzzy have the properties of commutativity and associativity control algorithm. In this perspective, Mamdani’s R, and (see sections 111-A and 111-C in Part I and Part D in this Larsen’s R, with the connective “also” as the union section). In this connection, it should be noted that the operator appear to be better suited for constructing fuzzy operators in triangular norms and co-norms possess these models than the other methods in FLC applications. We properties and thus qualify as the candidates for the will have more to say about these methods at a later interpretation of the connective “also.” In general, we use point. the triangular co-norms in association with fuzzy conjunc- tion and disjunction, and the triangular norms in associa- C. Compositional Operators tion with fuzzy implication. The experimental results [52]-[54], [96], [731 and the theoretical studies [181, [851, In a general form, a compositional operator may be [116], [191 relate to this issue. expressed as the sup-star composition, where “star” de- Kiszka et al. [52] described a preliminary investigation notes an operator-e.g., min, product, etc.-which is of the fuzzy implication functions and the sentence con- chosen to fit a specific application. In the literature, four nective “also” in the context of the fuzzy model of a dc kinds of compositional operators can be used in the series motor. In later work, they presented additional compositional rule of inference, namely: results for fuzzy implication functions and the connective sup-min operation [Zadeh, 19731, “also” in terms of the union and intersection operators sup-product operation [Kaufmann, 19751, [SI, [MI. sup-bounded-product operation [Mizumoto, 19811, Our investigation leads to some preliminary conclu- sup-drastic-product operation [Mizumoto, 19811. sions. First, the connective “also” has a substantial influ- ence on the quality of a fuzzy model, as we might expect. In FLC applications, the sup-min and sup-product Fuzzy implication functions such as R,, RA,and R, with compositional operators are the most frequently used. the connective “also” defined as the union operator, and The reason is obvious, when the computational aspects of Rc, R,, R,,, and R,,, defined as the intersection, yield an FLC are considered. However, interesting results can satisfactory results. These fuzzy implication functions dif- be obtained if we apply the sup-product, sup-bounded- fer in the number of mathematical operations which are product, and sup-drastic-product operations with differ- needed for computer implementation. ent fuzzy implication functions in approximate reasoning Recently, Stachowicz and Kochanska [96] studied the [70], [72]. The inferred results employing these composi- characteristics of 38 types of fuzzy implication along with tional operators are better than those employing the nine different interpretations (in terms of triangular norms sup-min operator. Further investigation of these issues in and co-norms) of the connective “also,” based on various the context of the accuracy of fuzzy models may provide forms of the operational curve of a series motor. Based interesting results. on their results, we tabulate in Table VI a summary of the D. Inference Mechanisms most appropriate pairs for the FLC of the fuzzy implica- tion function and the connective “also.” The inference mechanisms employed in an FLC are Additional results relating to the interpretation of the generally much simpler than those used in a typical expert connective “also” as the union and the intersection are system, since in an FLC the consequent of a rule is not reported in [73]. The investigation in question is based on applied to the antecedent of another. In other words, in a plant model with first-order delay. It is established that FLC we do not employ the chaining inference mecha- the fuzzy implication functions Rc,R,, Rhp,R,, with the nism, since the control actions are based on one-level connective “also” as the union operator yield the best forward data-driven inference (GMP). control results. Furthermore, the fuzzy implications R, The rule base of an FLC is usually derived from expert and R, are not well suited for control applications even knowledge. Typically, the rule base has the form of a 426 IEEE TKANSAC‘TIONS ON SYSTEMS, MAN, AND (‘YBEKNETIC‘S, VOL.. 20, NO. 2, MAK(.H/APKIL 1990 MIMO system process state variables and the control variable, respec- tively; A,, B,, and C, are linguistic values of the linguistic R={RLIMO, RhIMO,’ ‘-9 R“,MO} variables x, y, and z in the universes of discourse U, V, where R‘,,,, represents the rule: if (x is A; and..., and W, respectively, with i = 1,2; a, n. and y is B;) then (z, is C;; . ., zq is 0;).The antecedent The fuzzy control rule “if (x is A, and y is B,) then (z of R,,,, forms a fuzzy set A; x - x E; in the product is C,))’is implemented as a fuzzy implication (relation) R, space U X . . . X V. The consequent is the union of q and is defined as independent control actions. Thus the ith rule R‘,,,, may be represented as a fuzzy implication PR, ~P-(A,dndB,~C,)(U,V,W) =[PA,(U) +PC,(W) R‘,,,,: (AiX . . . X B;) -+ ( Z, + . . . + zq) and PB,(V)] from which it follows that the rule base R may be repre- where “A, and B,” is a fuzzy set A, X B, in U X K sented as the union R, (A, and B,)-+ C, is a fuzzy implication (relation) in U x V x W7and -+ denotes a fuzzy implication function. The consequence C’ is deduced from the sup-star compositional rule of inference employing the definitions of a fuzzy implication function and the connectives “and” and “also.” ={ ij [(A;x ... XB;)-+(z,+ *** +zq,]} i= 1 In what follows, we shall consider some useful proper- ties of the FLC inference mechanism. First, we would like ={ ij [(A;x... XB;)-+z,)], to show that the sup-min operator denoted by 0 and the i=l connective “also” as the union operator are commutative. n Thus the fuzzy control action inferred from the complete . U [(A;x ... XB;) +z2)];-, set of fuzzy control rules is equivalent to the aggregated i= I result derived from individual control rules. Furthermore, as will be shown later, the same properties are possessed ij [(A;x ... xB;)-+zq)]} by the sup-product operator. However, the conclusion in i=l question does not apply when the fuzzy implication is used in its traditional logical sense [lS], [19]. More specifi- ={ U U [(A;X *** xB;)+zx)]) cally, we have k=l i=l Lemma I: (A‘,B’) 0 U := ,RI = U := ,(A‘,B’) 0 RI. = {~LIISO7 RG41SO ,. * . , RB&ISOJ. Proofi In effect, the rule base R of an FLC is composed of a set of sub-rule-bases RBL,,,, with each sub-rule-base n C’=(A’,B’)oU R, RBL,,, consisting of n fuzzy control rules with multiple r=l process state variables and a single control variable. The n general rule structure of a MIMO fuzzy system can there- = (A‘,B’) 0 U (A, and B, -+ C,). fore be represented as a collection of MISO fuzzy sys- r=l tems: The membership function pc. of the fuzzy set C’ is R = I~~I,,~w41so 9 * * , Wl1so) pointwise defined for all w E W by where RB;,,, represents the rule: if (x is A; and.. . , Pc,( W)= (PA’W> PB4 VI) 0 max (PRf4v, w) 1 and y is Bi) then (2, is D;),i = 1,2; . *, n. u,v,w Let us consider the following general form of MISO .PRJU,v,w),. . . ,PR,,(U,V,W)) fuzzy control rules in the case of two-input/single-output fuzzy systems: = SUP min ((PA4 U)9 Pus-( v)17 max (PR f U ,v ,w ) 7 u,v u,v,w input: x is A’ and y is B’ if x is A, and y is B, then z is C, RI: .PRJ U ,v7 w)9. . . 7 PR,( U ,v, w))} also R,: if x is A, and y is B, then z is C, ... = SUP max (min [(PA~(”),PB~(v)),c~~U,v,w)]7 ... u,v [(,V,W . . .,min [(PA, = max { [ (PA’(4,PB’( 4)”PRfU9 Vd]7 z is C’ K,V,W where x, y, and z are linguistic variables representing the . . . 3 [ (PA’(4PB’(V))O PR,,(U,v,w)]}. LEE: FUZZY L.OGl(~ IN C'ONTROI. SYSTEMS: FUZ7.Y I O(il(' C'ONTK0I.I ER-PART II 427 Therefore C'= [(A',B'). RI] U [(A',B')oR,] .U ... U [ ( A',B') 0 R,,] n = U (A',B') 0 R, ,=I n = U (A',B')o(A,and B,-+C,) /=I n U c/f. ,=I Lemma 2: For the fuzzy conjunctions R,, R,, R,,, and R,,, we have (A',B') 0 ( A, and B, -+ C,) As will be seen in following section, the last lemma not =[A'~(A,-+C,)]~[B'O(~,-+C,)]only simplifies the process of computation but also pro- if PA,XB, = PA, A PB, vides a graphic interpretation of the fuzzy inference mechanism in the FLC. Turning to the sup-product oper- (A',B') 0 (A,and B, -+ C,) ator, which is denoted as -,we have the following. n n = [A'.( A, -+ C,)][ B'.(B, -+ C,)] Lemma 2': (A',B')*U Rj= U (A',B')*Rj. if PA,XB, PA,'PB,. i-1 i=l Lemma 2': For the fuzzy conjunctions R,, R,, R,,, Proofi and R,,, we have C;=(A',B')o(A,and B,-+C,) (A',B') -(A,and B, -+ C,) c~c;=(l~A,,~B,)o(llA,xB,-+Pc,) = [A'.( A, -+ c,)]n [ B' .( B, -+ c,)] if ~A,xe,= PA, A PB, = (PA',PB') (min (PA,7 PB,) -+ PC,) (A',B') .(A, and B, -+ C,) = ( PA',PB') min [ ( PA, PC, ) ? ( PB, -+ PC, I] = [A'*(A, + C,)] . [ B'*(B, -+ C,)] = sup min {[(PA,, PB.) PA,~B,=PA,*PB; u,v if 'min[(PA,-+PC,)~(PB,-+)lC,)]} Lemma 3: If the inputs are fuzzy singletons, namely, A'= U(), B'= vO, then the results derived by employing = supmin {min [ P~,~(PA, -+ clC,>], Mamdani's minimum operation rule R, and Larsen's u,v product operation rule R,, respectively, may be expressed . min [ PB,?( PB, -+ Pc,I] 1 simply as =min{[ PA"(PA, -+ PC,)]?[ I*B"(PB, -3 PC,)]}. R,: ffr/\APC,(W) R,: ff;A Pc,(W) 1) 2) Hence we obtain R,: ff,"~PCLc,(W) R,: ff,.*PC,(W) C; = [A'.(A, -+ C,)] n [ B'o(B, -+ C,)]. Q.E.D. where a," = pA,(~())ApS,(vo) and a;= EI.A,(U()).PB,(VO). Let us consider two special cases that follow from the Therefore we can assert that preceding lemma and that play an important role in FLC n applications. R,: Pes= U a,A Pc, ,=I Lemma 3: If the inputs are fuzzy singletons, namely, n A'= U(), B'= vo, then the results dervied by employing R,: Pea= Uff,.PC, Mamdani's minimum operation rule R, and Larsen's ,=I product operation rule R,, respectively, may be expressed where the weighting factor (firing strength) a; is a mea- simply as sure of the contribution of the ith rule to the fuzzy control action. The weighting factor in question may be R,: %*A PC,(W) determined by two methods. The first uses the minimum operation in the Cartesian product, which is widely used 428 IEEC TRANSACTIONS ON SYSTTMS, MAN. AND (‘YUERNFTIC‘S, VOL. 20, NO. 2, MAR(.H/APRII 1990 hs.. (...... -----.-- .....,... * ...... ys !, *[;>,,,p,,ll[~ <, , I, 0 0 0 U V Fig. 10. Graphical interpretation of Lemma 2 under a A and Rc. Fig. 11. Graphical interpretation of Lemma 2 under a’and R,,. in FLC applications. The second employs the algebraic and second rules may be expressed as product in the Cartesian product, thus preserving the = PAI(X,))A PB,(YO) contribution of each input variable rather than the domi- nant one only. In this respect, it appears to be a reason- a2 = PA2(XO) A PBZ(Y0) able choice in many FLC applications. where kA,(xO)and pBfyo)play the role of the degrees of For simplicity, assume that we have two fuzzy control partial match between the user-supplied data and the rules, as follows: data in the rule base. These relations play a central role in the four types of fuzzy reasoning currently employed in FLC applications, and are described in the following. RI: if x is A, and y is B, then z is C,, I) Fuzzy Reasoning of the First Type -Mamdani’s Min- R,: if x is A, and y is B, then z is C,. imum Operation Rule as a Fuzzy Implication Function: Fuzzy reasoning of the first type is associated with the use of Mamdani’s minimum operation rule R, as a fuzzy Fig. 10 illustrates a graphic interpretation of Lemma 2 implication function. In this mode of reasoning, the ith under R, and a:. Fig. 11 shows a graphic interpretation rule leads to the control decision of Lemma 2 under R, and a:. In on-line processes, the states of a control system play PC,W = “r A Pc,(W) an essential role in control actions. The inputs are usually which implies that the membership function (U, of the measured by sensors and are crisp. In some cases it may inferred consequence C is pointwise given by be expedient to convert the input data into fuzzy sets. In CLAW) = llc; v Pc; general, however, a crisp value may be treated as a fuzzy singleton. Then the firing strengths a,and a, of the first =[“I APCI(W)]V[LYZAPCZ(W)]. 1.F.F: FUZZY I-OGIC' IN C'ONTROL. SYSTEMS: FUZZY 1.OGIC C'ONTROI.I.F.K--PART !I 429 4 j ti 0 XO YO Fig. 12. Diagrammatic representation of fuzzy reasoning 1 W XO YO Fig. 13. Diagrammatic representation of fuzzy reasoning 2. To obtain a deterministic control action, a defuzzifica- process is illustrated in Fig. 13, which shows a graphic tion strategy is required, as will be discussed at a later interpretation of Lemma 3 in terms of Larsen's meth- point. The fuzzy reasoning process is illustrated in Fig. 12, od R,. which shows a graphic interpretation of Lemma 3 in 3) Fuzzy Reasoning of the Third Type -Tsukamoto's terms of Mamdani's method R,. Method with Linguistic Terms as Monotonic Membership 2) Fuzzy Reasoning of the Second Type- Larsen's Functions: This method was proposed by Tsukamoto [1171. Product Operation Rule as a Fuzzy Implication Function: It is a simplified method based on the fuzzy reasoning of Fuzzy reasoning of the second type is based on the use of the first type in which the membership functions of fuzzy Larsen's product operation rule R, as a fuzzy implication sets A,, B,, and C, are monotonic. However, in our function. In this case, the ith rule leads to the control derivation, A, and B, are not required to be monotonic decision but C, is. In Tsukamoto's method, the result inferred from the P~:(w)= ai*Pc,(w)- first rule is a, such that aI= C,(y,).The result inferred a2= Consequently, the membership function pc of the in- from the second rule is a2 such that C,(y2).Corre- ferred consequence C is pointwise given by spondingly, a crisp control action may be expressed as the weighted combination (Fig. 14) PC(W) = Pc; v Pc; aIYl+ %Y2 20 = + a2 = [ a,*Pc,(W)] v [ %.PC,(W)]. 4) Fuzzy Reasoning of the Fourth Type-The Conse- From C, a crisp control action can be deduced through quence of a Rule is a Function of Input Linguistic Variables: the use of a defuzzification operator. The fuzzy reasoning Fuzzy reasoning of the fourth type employs a modified 430 lEFK TRANSACTIONS ON SYSTEMS. MAN, ANU ('YBEKNETI('S, VOI.. 20, NO. 2, MARC"/APRIL 1990 it. At present, the commonly used strategies may be described as the max criterion, the mean of maximum, and the center of area. A. The mar criterion method The max criterion produces the point at which the possibility distribution of the control action reaches a maximum value. l&l...... 1 l&-r 'v. B. The Mean of Maximum Method (MOM) The MOM strategy generates a control action which 0 0 0 represents the mean value of all local control actions U Y min W whose membership functions reach the maximum. More X YO y2 specifically, in the case of a discrete universe, the control Fig. 14. Diagrammatic representation of fuzzy reasoning 3. action may be expressed as Iw version of state evaluation function. In this mode of Z() = c 2 ]=I 1 reasoning, the ith fuzzy control rule is of the form where wI is the support value at which the membership RI: if(xis A,;.. andyis B,)thenz=f,(x;..,y) function reaches the maximum value pz(wl),and 1 is the where x, . . . , y, and z are linguistic variables represent- number of such support values. ing process state variables and the control variable, re- spectively; A,,. . ., B, are linguistic values of the linguistic C. The Center of Area Method (COA) variables x, * * e, y in the universes of discourse U,.. . , V, The widely used COA strategy generates the center of respectively, with i = 1,2,. . -,n; and f, is a function of the gravity of the possibility distribution of a control action. process state variables x,. . . ,y defined in the input sub- In the case of a discrete universe, this method yields spaces. n For simplicity, assume that we have two fuzzy control c cLZ(WI)*Wl rules as follows: /=I 20 = n RI: if x is A, and y is B, then z = fl(x,y) c PAW]) R,: if x is A, and y is B, then z = f2( x,y). ]=I The inferred value of the control action from the first rule where n is the number of quantization levels of the is alfl(xo,yo). The inferred value of the control action output. from the second rule is a2f, Angle NL NM NS ZR PS PM PL NL 0 NM Change NS NS ZR / 1 \” of ZR NM ZR PM MAX COA MOM Angle PS ZR PS PM Fig. 15. Diagrammatic representation of various defuzzification strate - PL gies. has emerged as one of the most active and fruitful areas for research in the application of fuzzy set theory. In a:,; . ., and a:, identified by training. The training process many applications, the FLC-based systems have proved to involves a skilled operator who guides the fuzzy model car be superior in performance to conventional systems. under different conditions. In this way, Sugeno’s car has Notable applications of FLC include the heat exchange the capability of learning from examples. [80], warm water process [47], activated sludge process 2) FLC Hardware Systems: A higher-speed FLC hard- [113], [35], traffic junction [82], cement kiln [59], [118], ware system employing fuzzy reasoning of the first type aircraft flight control [SI, turning process [92], robot has been proposed by Yamakawa [130], 11311. It is com- control [1191, [941, [1061, [SI, [34], model-car parking and posed of 15 control rule boards and an action interface turning [971-[991, automobile speed control [74], [751, wa- (i.e., a defuzzifier based on the COA). It can handle fuzzy ter purification process [127], elevator control [23], auto- linguistic rules labeled as NL, NM, NS, ZR, PS, PM, PL. mobile transmission control [40], power systems and nu- The operational speed is approximately 10 mega fuzzy clear reactor control [4], [511, fuzzy memory devices [107], logical inferences per second (FLIPS). [108], [120], [1281, [1291, [133], and the fuzzy computer The FLC hardware system has been tested by an appli- [132]. In this connection, it should be noted that the first cation to the stabilization of inverted pendulums mounted successful industrial application of the FLC was the ce- on a vehicle. Two pendulums with different parameters ment kiln control system developed by the Danish cement were controlled by the same set of fuzzy control rules plant manufacturer F. L. Smidth in 1979. An ingenious (Table VII). It is worthy of note that only seven fuzzy application is Sugeno’s fuzzy car, which has the capability control rules achieve this result. Each control rule board of learning from examples. More recently, predictive fuzzy and action interface has been integrated to a 40-pin chip. control systems have been proposed and successfully ap- 3) Fuzzy Automatic Train Operation (ATO) Systems: plied to automatic train operation systems and automatic Hitachi Ltd. has developed a fuzzy automatic train opera- container crane operation systems [135]-[139]. In parallel tion system (ATO) which has been in use in the Sendai- with these developments, a great deal of progress has City subway system in Japan since July 1987. In this been made in the design of fuzzy hardware and its use in system, an object evaluation fuzzy controller predicts the so-called fuzzy computers [132]. performance of each candidate control command and selects the most likely control command based on a skilled B. Recent Developments human operator’s experience. More specifically, fuzzy AT0 comprises two rule bases Sugeno’s Fuzzy Car: One of the most interesting 1) which evaluate two major functions of a skilled operator applications of the FLC is the fuzzy car designed by based on the criteria of safety, riding comfort, stop-gap Sugeno. Sugeno’s car has successfully followed a crank- accuracy, traceability of target velocity, energy consump- shaped track and parked itself in a garage [98]-[991. tion, and running time. One is constant-speed control The control policy incorporated in Sugeno’s car is rep- (CSC), which starts a train and maintains a prescribed resented by a set of fuzzy control rules which have the speed. The other is the train automatic stop control form: (TASC), which regulates a train speed in order to stop at RI: if x is A,, . . . and y is B, then the target position at a station. Each rule base consists of twelve object-evaluation fuzzy control rules. The an- z = a;, + aix . . . + aiy + tecedent of every control rule performs the evaluation of where x; . ., and y are linguistic variables representing train operation based on safety, riding comfort, stop-gap the distances and orientation in relation to the bound- accuracy, etc. The consequent determines the control aries of the track; A,; . ., and B, are linguistic values of action to be taken based on the degree of satisfaction of x; . ., and y; z is the value of the control variable of the each criterion. The control action is the value of the train ith control rule; and a;,; *, and a; are the parameters control notch, which is evaluated every 100 ms from the entering in the identification algorithm [ 1031, [99]. maximal evaluation of each candidate control action, and The inference mechanism of Sugeno’s fuzzy car is based it takes as a value a discrete number; positive value means on fuzzy reasoning of the fourth type, with the parameters “power notch,” negative value means “break notch.” 432 IELE TRANSACTIONS ON SYSTIIMS, MAN, ANI) <‘YHF.KNF.TI(‘S. VOI . 20. NO. 2, MAR<~ll/APRIl 1990 The Sendai-City subway system has been demonstrated employs MAX and MIN operations, which are imple- to be superior in performance to the conventional PID mented by the emitter coupled fuzzy logic gates (ECFL AT0 in riding comfort, stop gap accuracy, energy con- gates) in voltage-mode circuit systems. The linguistic in- sumption, running time, and robustness [1351, [1361, [1391. puts, which are represented by analog voltages distributed 4) Fuzzy Automatic Container Crane Operation (ACO) on data buses, are fed into each inference engine in Systems: In the application of FLC to the automatic oper- parallel. The results inferred from the rules are aggre- ation of container-ship loading cranes, the principal per- gated by a MAX block, which implements the function of formance criteria are safety, stop-gap accuracy, container the connective “also” as a union operation, yielding a sway, and carrying time. consequence which is a set of analog voltages distributed Fuzzy ACO involves two major operations: the trolley on output lines. In the FLC applications, a crisp control operation and the wire rope operation. Each operation command necessitates an auxiliary defuzzifier. In this comprises two function levels: a decision level and an implementation, a fuzzy computer is capable of process- activation level. Field tests of fuzzy ACO systems with ing fuzzy information at the very high speed of approxi- real container cranes have been performed at the port of mately 10 mega-FLIPS. It is indeed an important step not Kitakyusyu in Japan. The experimental results show that only in industrial applications but also in common-sense cargo handling ability of Fuzzy ACO by an unskilled knowledge processing. operator is more than 30 containers per hour, which is IV. FUTURESTUDIES AND PROBLEMS comparable to the performance of a veteran operator. The tests have established that the fuzzy ACO controller In many of its applications, FLC is either designed by has the capability of operating a crane as safely, accu- domain experts or in close collaboration with domain rately, and skillfully as a highly experienced human opera- experts. Knowledge acquisition in FLC applications plays tor [ 1371-[ 1391. an important role in determining the level of performance 5) Fuzzy Logic Chips and Fuzzy Computers: The first of a fuzzy control system. However, domain experts and fuzzy logic chip was designed by Togai and Watanabe at skilled operators do not structure their decisionmaking in AT&T Bell Laboratories in 1985 [107]. The fuzzy infer- any formal way. As a result, the process of transferring ence chip, which can process 16 rules in parallel, consists expert knowledge into a usable knowledge base of an of four major parts: a rule-set memory, an inference- FLC is time-consuming and nontrivial. Although fuzzy processing unit, a controller, and an input-output cir- logic provides an effective tool for linguistic knowledge cuitry. Recently, the rule-set memory has been imple- representation and Zadeh’s compositional rule of infer- mented by a static random access memory (SRAM) to ence serves as a useful guideline, we are still in need of realize a capability for dynamic changes in the rule set. more efficient and more systematic methods for knowl- The inference-processing unit is based on the sup-min edge acquisition. compositional rule of inference. Preliminary timing tests An FLC based on the fuzzy model of a process is indicate that the chip can perform approximately 250000 needed when higher accuracy and reliability are required. FLIPS at 16-MHz clock. A fuzzy logic accelerator (FLA) However, the fuzzy modeling of a process is still not well based on this chip is currently under development [l08], understood due to difficulties in modeling the linguistic [ 1201. Furthermore, in March 1989 the Microelectronics structure of a process and obtaining operating data in Center of North Carolina successfully completed the fab- industrial process control [131, [841, [llll, [1251, 11041, rication of the world’s fastest fuzzy logic chip, designed by [loll. Watanabe. The full-custom chip comprises 688 000 tran- Classical control theory has been well developed and sistors and is capable of making 580000 FLIPS. provides an effective tool for mathematical system analy- In Japan, Yamakawa and Miki realized nine basic fuzzy sis and design when a precise model of a system is logic functions by the standard CMOS process in available. In a complementary way, FLC has found many current-mode circuit systems [ 1281. Later, a rudimentary practical applications as a means of replacing a skilled concept of a fuzzy computer was proposed by Yamakawa human operator. For further advances, what is needed at and built by OMRON Tateishi Electric Co. Ltd [132]. The this juncture are well-founded procedures for system de- Yamakawa-OMRON computer comprises a fuzzy mem- sign. In response to this need, many researchers are ory, a set of inference engines, a MAX block, a defuzzi- engaged in the development of a theory of fuzzy dynamic fier, and a control unit. The fuzzy memory stores lin- systems which extends the fundamental notions of state guistic fuzzy information in the form of membership 161, controllability [311, and stability [771, [441, 1891, [SI. functions. It has a binary RAM, a register, and a member- Another direction of recent exploration is the concep- ship function generator [128]. A membership function tion and design of fuzzy systems that have the capability generator (MFG) consists of a PROM, a pass transistor to learn from experience. In this area, a combination of array, and a decoder. Every term in a term set is repre- techniques drawn from both fuzzy logic and neural net- sented by a binary code and stored in a binary RAM. The work theory may provide a powerful tool for the design of corresponding membership functions are generated by systems which can emulate the remarkable human ability the MFG via these binary codes. The inference engine to learn and adapt to changes in environment. LEE: FUZZY LOGIC IN CONTROL SYSTEMS: FUZZY LOGIC CONTROI.L.ER-PART II 433 [241 S. Fukami. M. Mizumoto, and K. Tanaka. “Some considerations ACKNOWLEDGMENT of fuzzy conditional inference,” Fuzzy Sets Syst., vol. 4, DD.I, 243-273. 1980. 1 am greatly indebted to Professor btfi A. &de-, of [25] B. R. Gaines and L. J. Kohout, “The fuzzy decade: A bibliogra- phy of fuzzy systems and closely related topics,” Int. J. Man. the University of California-Berkeley for his encourage- Mach. Studies, vol. 9, pp. 1-68, 1977. ment of this research. The assistance of Professor Zadeh [26] s. Gottwald and W. Pedrycz, “Problems of the design of fuzzy is gratefully acknowledged. The author would like to controllers,” in Approximate Reasoning in Expert Systems, M. M. Gupta, A. Kandel, W. Bandler, and J. B. Kiszka, Ed. Amster- thank Professor M. Tomizuka of the University of Califor- dam: North-Holland, 1985, pp. 393-405. nia-Berkeley and the reviewers for their helpful corn- [27] B. P. Graham and R. B. Newell, “Fuzzy identification and ments and suggestions. control of a liquid level rig,” Fuzzy Sets Syst., vol. 26, pp. 255-273, 1988. -~“Fuzzy adaptive control of a first-order process,” Fuzzy Sets Syst., vol. 31, pp. 47-65, 1989. 1. Graham and P. L. Jones, Expert Sy,stems Knowledge, Uncertainty REFERENCES and Decision. New York: Chaman and Hall, 1988. M. M. Gupta and Y. Tsukamoto, “Fuzzy logic controllers- N. Baaklini and E. H. Mamdani, “Prescriptive methods for A perspective,” in Proc. Joint Automatic Control Conf., San deriving control policy in a fuzzy-logic controller,” Electron. Lett., Francisco, Aug. 1980. pp. FAIO-C. vol. 11, pp. 625-626, 1975. 1311 M. M. Gupta, G. M. Trojan, and J. B. Kiszka, “Controllability of J. Baldwin and N. Guild, “Modeling controllers using fuzzy fuzzy control systems,” IEEE Trans. Syst. Man Cybern., vol. relations,” Kybernetes, vol. 9, pp. 223-229, 1980. SMC-16, no 4, pp. 576-582, 1986. J. F. Baldwin and B. W. Pilsworth, “Axiomatic approach to K. Hirota, Y. Arai, and S. Hachisu, ”Moving mark recognition implication for approximate reasoning with fuzzy logic,” Fuzzy and moving object manipulation in fuzzy controlled robot,” Cont. Sets Syst., vol 3, pp. 193-219, 1980. Theory Adv. Techno[.,vol. 2, no. 3, pp. 399-418, 1986. J. A. Bernard, “Use of rule-based system for process control,” K. Hirota and W. Pedrycz, “Analysis and synthesis of fuzzy IEEE Contr. Syst. Mag., vol. 8, no. 5, pp. 3-13, 1988. systems by the use of fuzzy sets,” Fuzzy Sets Syst., vol. 10, no 1, M. Braae and D. A. Rutherford, “Fuzzy relations in a control pp. 1-14, 1983. setting,” Kybernetes, vol. 7, no. 3, pp. 185-188, 1978. C. Isik,“Identification and fuzzy rule-based control of a mobile -, “Theoretical and linguistic aspects of the fuzzy logic robot motion,” in Proc. IEEE Int. Symposium Intelligent Control, controller,” Automat., vol. 15, no 5, pp. 553-577, 1979. Philadelphia, PA, 1987. -, “Section of parameters for a fuzzy logic controller,” Fuzzy 0. Itoh, K. Gotoh, T. Nakayama, and S. Takamizawa, “Applica- Sets Syst., vol. 2, no. 3, pp. 185-199, 1979. tion of fuzzy control to activated sludge process,” in Proc. 2nd K. Ciliz, J. Fei, K. Usluel, and C. Isik, “Pratical aspects of the IFSA Congress, Tokyo, Japan, July, 1987, pp. 282-285. knowledge-based control of a mobile robot motion,” in Proc. A. Jones, A Kaufmann, and H. J. Zimmermann, Fuzzy Sets 30th Midwest Symp. on Circuits and Systems, Syracuse, NY, 1987. Theory and Applications. Boston: D. Reidel Publishing Com- E. Czogala, Probabilistic Sets in Decision Making and Control. pany, 1986. Cologne: Verlag TUV Rheinland, 1984. J. Kacprzyk and S. A. Orlovski, Optimization Models using Fuzzy E. Czogala, “Probabilistic fuzzy controller as a generalization of Sets and Possibiliry Theory. Boston: D. Reidel Publishing Com- the concept of fuzzy controller,” Fuzzy Sets Syst., vol. 26, pp... pany, 1987. 215-223, i988. . J. Kacprzyk and M. Fedrizzi, Combining Fuzzy Imprecision with E. Czoeala and K. Hirota. Probabilistic Sets: Fuzzv and Stochastic Probabilistic Uncertainty in Decision Making. New York: Y Approach to Decision, Control and Recognition Processes. Springer-Verlag, 1988. Cologne: Verlag TUV Rheinland, 1986. A. Kandel, Fuzzy Techniques in Pattern Recognition. New York: E. Czogala and W. Pedrycz, “Some problems concerning the Wiley-Interscience, 1982. construction of algorithms of decision-making in fuzzy systems,” Y. Kasai and Y. Morimoto, “Electronically controlled continu- Inr. J. Man. Mach. Studies, vol. 15, pp. 201-221, 1981. ously variable transmission,” in Proc. Int. Congress on Transporta- -, “On identification in fuzzy systems and its applications in tion Electronics, Dearborn, MI, 1988. control problems,” Fuzzy Sets Syst., vol. 6, no. 1, pp. 73-83, 1981. A. Kaufmann, Introduction to Theory of Fuzzy Subsets. New -, “Control problems in fuzzy systems,” Fuzzy Sets Syst., vol. York: Academic, 1975. 7, no. 3, pp. 257-274, 1982. A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic. -, , “Fuzzy rule generation for fuzzy control,” Cybem. Syst., New York: Van Nostrand, 1985. vol. 13, no. 3, pp. 275-293, 1982. -, Fuzzy Mathematical Models in Engineering and Manage- E. Czogala and T. Rawlik, “Modelling of a fuzzy controller with ment Science. Amsterdam: North-Holland, 1988. application to the control of biological processes,” Fuzzy Sets M. Kelemen, Fuzziness and structural stability,” Kybernetes, vol. Syst., vol. 31, pp. 13-22, 1989. IO, pp. 67-69, 1981. B. Bharathi Devi and V. V. S. Sarma, “Estimation of fuzzy W. J. M. Kickert, “Towards an analysis of linguistic modeling,” memberships from histograms,” Inform. Sci., vol. 35, pp. 43-59, Fuzzy Sets Syst., vol. 2, pp. 293-307, 1979. 1985. -, “Further analysis and application of fuzzy logic control.” D. Dubois and H. Prade, “Fuzzy logics and the generalized Internal Rep. F/WK2/75, Queen Mary College, London, 1975. modus ponens revisited,” Cybern. Syst., vol. 15, pp. 3-4, 1984. W. J. M. Kickert and H. R. Van Nauta Lemke, “Application of a -, “The generalized modus ponens under sup-min composi- fuzzy controller in a warm water plant,” Automat., vol. 12, no. 4, tion-A theoretical study.” in Approximate Reasoning in fipert pp. 301-308, 1976. Systems, M. M. Gupta, A. Kandel, W. Bandler, and J. B. Kiszka, W. J. M. Kickert and E. H. Mamdani, “Analysis of a fuzzy logic Eds. Amsterdam: North-Holland, 1985, pp. 21 7-232. controller,” Fuzzy Sets Syst.. vol. 1, no. 1, pp. 29-44, 1978. -, “Unfair coins and necessity measures: Toward a possi- P. J. King and E. H. Mamdani, “The application of fuzzy control bilistic interpretation of histograms,” Fuzzy Sets Syst., vol. 10, no. systems to industrial processes,” in IFAC World Congress, MIT, 1, pp. 15-20, 1985. Boston, 1975. [211 __, Possibility Theory: An Approach to Computerized Processing -, “The application of fuzzy control systems to industrial of Uncertainty. New York: Plenum Press, 1988. processes,” Automat., vol. 13, no. 3, pp. 235-242, 1977. [221 J. Efstathiou, “Rule-based process control using fuzzy logic,” in M. Kinoshita, T. Fukuzaki, T. Satoh, and M. Miyake, “An Approximate Reasoning in Intelligent Systems, Decision and Con- automatic operation method for control rods in BWR plants,” in trol, E. Sanchez and L. A. Zadeh, Eds. New York: Pergamon, Proc. Specialists’ Meeting on In-Core Instrumentation and Reactor 1987, pp. 145-158. Core Assessment, Cadarache, France, 1988. F. Fujitec, “FLEX-8800 series elevator group control system,” J. B. Kiszka, M. E. Kochanska, and D. S. Sliwinska, “The Fujitec Co., Ltd., Osaka, Japan, 1988. influence of some parameters on the accuracy of fuzzy model,” in 434 irrr TRANSA~~TIONSON sYs,rms, MAN. ANI) (‘YHERNETICS, VOL. 20, NO. 2, MARCH/APKII. 1990 Industrial Applications of F11zzy Cvntrol. M. Sugeno, Ed. Amster- J. J. Ostergdad. “Fuzzy logic control of a heat exchange process,” dam: North-Holland, 1985, pp. 187-230. in Fuzzy Automata and Decision Processes, M. M. Gupta, G. N. [53] -, “The influence of some fuzzy implication operators on the Saridis, and B. R. Gaines, Eds. Amsterdam: North-Holland, accuracy of a fuzzy model-Part 1.” Fuzzy Sets Syst., vol. 15, pp. 1977, pp. 285-320. lll-l28,1985. S. K. Pal and D. K. Dutta Majumder, Fuzzy Mathematical Ap- [54] -, “The influence of some fuzzy implication operators on the proach to Pattern Recognition. New Delhi: Wiley Eastern Lim- accuracy of a fuzzy model-Part 11:’’ Fkrzy Sets Sist., vol. 15, pp. ited, 1986. 223-240, 1985. (821 C. P. Pappis and E. H. Mamdani, “A fuzzy logic controller for a J. B. Kiszka, M. M. Gupta, and P. N. Nikiforuk, “Energetistic traffic junction,” IEEE Trans. Syst. Man Cybern., vol. SMC-7, no. stability of fuzzy dynamic systems,’’ IEEE Trans. Syst. Man 10, pp. 707-717, 1977. Cybern., vol. SMC-15. no. 5, pp. 783-792, 1985. W. Pedrycz, “An approach to the analysis of fuzzy systems,” Int. G. J. Klir and T. A. Folger, Fuzzy Sets, Uncertainry, and Informa- J. Control, vol. 34, no. 3, pp. 403-421, 1981. tion. Englewood Cliffs, NJ: Prentice Hall, 1988. ~, “An identification of fuzzy relational systems,” Fwzy Sets S. V. Komolov, S. P. Makeev, and I. F. Shaknov, “Optimal Syst., vol. 13, pp. 153-167, 1984. control of a finite automation with fuzzy constraints and a fuzzy -, “Applications of fuzzy relational equations for methods of target,” Cybern., vol. 16, no. 6, pp. 805-810. 1979. reasoning in presence of fuzzy data,” Fuzzy Sets Syst., vol. 16, pp. I581 L. 1. Larkin, “A fuzzy logic controller for aircraft flight control,” 163-175, 1985. in Industrial Applications of Fuzzy Control, M. Sugeno, Ed. Ams- H. Prade and C. V. Negoita, Fuzzy Logic in Knowledge Engineer- terdam: North-Holland, 1985, pp. 87-104. ing. Gesamtherstellung: Verlag TUV Rheinland, 1986. P. M. Larsen, “Industrial applications of fuzzy logic control,” Int. T. J. Procyk and E. H. Mamdani, “A linguistic self-organizing J. Man Mach. Studies, vol. 12, no. I, pp. 3-10, 1980. process controller,” Automat., vol. 15, no. 1, pp. 15-30, 1979. E. Lembessis, “Dynamical learning behavior of a rule-based G. P. Rao and D. A. Rutherford, “Approximate reconstruction self-organising controller,” Ph.D. thesis, Queen Mary College, of mapping functions from linguistic description in problems of Univ. of London, 1984. fuzzy logic applied to system control,” Cybernet., vol. 12, no. 3, I611 Y. F. Li and C. C. Lau, “Development of fuzzy algorithms for pp. 225-236, 1981. servo systems,’’ IEEE Contr. Syst. Mag., vol. 9, no. 3, pp. 65-72, 1891 K. S. Ray and D. D. Majumder, “Application of circle criteria for 1989. stability analysis of linear SISO and MIMO systems associated J. Maiers and Y. S. Sherif, “Applications of fuzzy set theory,” with fuzzy logic controller,” IEEE Trans. Syst. Man Cybern., vol. IEEE Trans. Syst. Man Cybern., vol. SMC-15, no. I, pp. 175-189, SMC-14, no. 2, pp. 345-349, 1984. 1985. I901 D. A. Rutherford and G. C. Bloore, “The implementation of I631 E. H. Mamdani, “Applications of fuzzy algorithms for simple fuzzy algorithms for control,” in Proc. IEEE, vol. 64, no. 4, pp. dynamic plant,” Proc. IEE, vol. 121, no. 12, pp. 1585-1588, 1974. 572-573, 1976. I641 E. H. Mamdani and S. Assilian, “An experiment in linguistic ~911 D. A. Rutherford and G. A. Carter, “A Heuristic adaptive synthesis with a fuzzy logic controller,” Int. J. Man Mach. Stud- controller for a sinter plant,” in Proc. 2nd IFAC Symp. on ies, vol. 7, no 1, pp. 1-13, 1975. Automation in Mining, Mineral and Metal Processing, Hohannes- E. H. Mamdani, “Advances in the linguistic synthesis of fuzzy burg, RSA, Sept. 1976. controllers,” Int. J. Man Mach. Studies, vol. 8, no. 6, pp. 669-678, Y. Sakai, “A fuzzy controller in turning process automation,” in 1976. Industrial Applications of Fuzzy Control, M. Sugeno, Ed. Amster- I661 -, “Application of fuzzy logic to approximate reasoning dam: North-Holland, 1985, pp. 139-152. using linguistic synthesis,” IEEE Trans. Computer. vol. C-26, no. E. Sanchez and L. A. Zadeh, Approximate Reasoning in Intelligent 12, pp. 1182-1191, 1977. Systems, Decision and Control. New York: Pergamnon, 1988. 1671 ~, “Applications of fuzzy set theory to control systems: a E. M. Scharf and N. J. Mandic, “The application of a fuzzy survey,’’ in Fuzzy Automata and Decision Processes, M. M. Gupta, controller to the control of a multi-degree-freedom robot arm,” G. N. Sardis, and B. R. Gaines, Eds. Amsterdam: North- in Industrial Applications of Fuzzy Control, M. Sugeno, Ed. Ams- Holland, 1977, pp. 77-88. terdam: North-Holland, 1985, pp. 41-62. E. H. Mamdani and B. R. Gaines, Fuzzy Reasoning and its [951 S. Shao, “Fuzzy self-organizing controller and its application for Applications. London: Academic, 1981. dynamic processes,” Fuzzy Sets Syst., vol. 26, pp. 151-164, 1988. K. Matsushima and H. Sugiyama, “Human operator’s fuzzy model 1961 M. S. Stachowicz and M. E. Kochanska, “Fuzzy modeling of the in man-machine system with a nonlinear control object,” in First process,” in Proc. 2nd IFSA Congress, Tokyo, Japan, July 1987, Int. Conf. on Fuzzy Information Processing, 1984. pp. 86-89. I701 M. Mizumoto, “Note on the arithmetic rule by Zadeh for fuzzy M. Sugeno and K. Murakami, “Fuzzy parking control of model conditional inference,” Cybern. Syst., vol. 12, pp. 247-306, 1981. car,” in 23rd IEEE Conf. on Decision and Control, Las Vegas, 1711 M. Mizumoto and H. Zimmermann, “Comparison of fuzzy rea- 1984. soning methods,” Fuzzy Sets and Systems, vol. 8, p~.253-283, M. Sugeno and M. Nishida, “Fuzzy control of model car,” Fuzzy 1982. Sets Syst., vol. 16, pp. 103-113, 1985. 1721 M. Mizumoto,“Comparison of various fuzzy reasoning methods,” M. Sugeno and K. Murakami, “An experimental study on fuzzy in Proc. 2nd IFSA Congress, Tokyo, Japan, July 1987, pp. 2-7. parking control using a model car,” in Industrial Applications of I731 -, “Fuzzy controls under various approximate reasoning Fuzzy Control, M. Sugeno, Ed. Amsterdam: North-Holland, methods,” in Proc. 2nd IFSA Congress, Tokyo, Japan, July, 1987, 1985, pp. 125-138. pp. 143-146. M. Sugeno, “An introductory survey of fuzzy control,” Inform. I741 S. Murakami, “Application of fuzzy controller to automobile Sei., vol. 36, pp. 59-83, 1985. speed control system,” in Proc. of the IFAC Symp. on Fuzzy M. Sugeno and G. T. Kang, “Structure identification of fuzzy Information, Knowledge Representation and Decision Analysis, model,” Fuzzy Sets Syst., vol. 28, no. I, pp. 15-33, 1988. Marseille, France, 1983, pp. 43-48. K. Sugiyama, “Analysis and synthesis of the rule based self- S. Murakami and M. Maeda, “Application of fuzzy controller to organising controller,” Ph.D thesis, Queen Mary College, Univ. automobile speed control system,” in Industrial Applications of of London, 1986. Fuzzy Control, M. Sugeno, Ed. Amsterdam: North-Holland, T. Takagi and M. Sugeno, “Derivation of fuzzy control rules 1985, pp. 105-124. from human operator’s control actions,” in Prvc. of the IFAC Y. Murayama and T. Terano, “Optimizing control of diesel Symp. on Fuzzy Information, Knowledge Representation and Deci- engine,” in Industrial Applications of Fuzzy Control, M. Sugeno, sion Analysis, Marseilles, France, July 1983, pp. 55-60. Ed. Amsterdam: North-Holland, 1985, pp. 63-72. 11041 -, “Fuzzy identification of systems and its applications to C. V. Negoita, “On the stability of fuzzy systems,” in Proc. IEEE modeling and control,” IEEE Trans. Syst. Man Cybern., vol. Int. Conf. on Cybernetics and Svciery, 1978, pp. 936-937. SMC-15, no. I, pp. 116-132, 1985. -, Expert Systems and Fuzzy Systems. Menlo Park: Ben- K. L. Tang and R. J. Mulholland, “Comparing fuzzy logic with jamin/Cummings Publishing Company, 1985. classical controller designs,” IEEE Trans. Syst. Man Cyhern., vol. A. Di Nola and A. G. S. Ventre, The Mathematics of Fuzzy SMC-17, no. 6, pp. 1085-1087, 1987. Systems. Gesamtherstellung: Verlag TUV Rheinland, 1986. R. Tanscheit and E. M. Scharf, “Experiments with the use of a LEE: FUZZY LOGIC IN CONTROI. SYSTEMS: FUZZY 1.0cilC CONTR0I.I FR-PART II 435 rule-based self-organising controller for robotics applications,” [ 1291 T. Yamakawa and K. Sasaki, “Fuzzy memory device,” in Prvc. Fuzzy Sets Syst., vol. 26, pp. 195-214, 1988. 2nd IFSA Congrevs, Tokyo. Japan. July 1987, pp. 551-555. [I071 M. Togai and H. Watanabe. “Expert system on a chip: An engine [I301 T. Yamakawa. “High speed fuzzy controller hardware system.” in for real-time approximate reasoning,” IEEE Expert Syst. Mag., Prvc. 2nd Fuzzy Sywm Symp.. Japan, 1986. pp. 122-130. vol. 1, pp. 55-62, 1986. [I311 -, “Fuzzy controller hardware system,” in Prvc. 2nd IFSA [lo81 M. Togai and S. Chiu, “A fuzzy accelerator and a programming Congress, Tokyo. Japan, July 1987. environment for real-time fuzzy control,” in Prvc. 2nd IFSA [I321 ~, “A simple fuzzy computer hardware system employing Congress, Tokyo, Japan, July 1987, pp. 147-151. min and max operations-A challenge to 6th generation com- [lo91 R. M. Tong, “Analysis of fuzzy control algorithms using the puter,” in Proc. 2nd IFSA Congress, Tokyo, Japan, July 1987. relation matrix,’’ Int. J. Man Mach. Studies, vol. 8, no. 6, pp. [133] T. Yamakawa, “Fuzzy microprocessors-Rule chip and defuzzi- 679-686, 1976. fier chip,” in Int. Workshop on Fuzzy System Applications, Iizuka, [110] -, “A control engineering review of fuzzy systems,” Automa., Japan, Aug. 1988, pp. 51-52. vol. 13, no. 6, pp. 559-569, 1977. 1341 T. Yamazaki and M. Sugeno, “A microprocessor based fuzzy [111] -, “Synthesis of fuzzy models for industrial processes,” Int. controller for industrial purposes,” in Industrial Applications of Gen. Syst., vol. 4, pp. 143-162, 1978. Fuzzy Control, M. Sugeno, Ed. Amsterdam: North-Holland, [112] -, “The evaluation of fuzzy models derived from experimen- 1985, pp. 231-240. tal data,” Fuzzy Sets Syst., vol. 4, pp. 1-12, 1980. 1351 S. Yasunobu, S. Miyamoto, and H. Ihara, “Fuzzy control for [113] R. M. Tong, M. B. Beck, and A. Latten, “Fuzzy control of the automatic train operation system,” in Prvc. 4th IFAC/IFIP/ activated sludge wastewater treatment process,” Automat., vol. IFORS Int. Congress on Control in Transportation Sy.stems, Baden- 16, no. 6, pp. 695-701, 1980. Baden, April 1983. [114] R. M. Tong, “A retrospective view of fuzzy control systems,” 1361 S. Yasunobu and S. Miyamoto, “Automatic train operation by Fuzzy Sets Syst., vol. 14, pp. 199-210, 1984. predictive fuzzy control,” in Industrial Application of Fuzzy Con- [115] -, “An annotated bibliography of fuzzy control,” in Indus- trol, M. Sugeno, Ed. Amsterdam: North-Holland, 1985, pp. trial Applications of Fuzzy Corkol, M. Sugeno, Ed. Amsterdam: 1-18. North-Holland, 1985, pp. 249-269. S. Yasunobu and T. Hasegawa, “Evaluation of an automatic E. Trillas and L. Valverde, “On mode and implication in approxi- container crane operation system based on predictive fuzzy con- mate reasoning,” in Approximate Reasoning in Expert Systems, trol,” Control Thevty Adv. Technol., vol. 2, no. 3, pp. 419-432, M. M. Gupta, A. Kandel, W. Bandler, and J. B. Kiszka. Eds. 1986. Amsterdam: North-Holland, 1985, pp. 157-166. ~, “Predictive fuzzy control and its application for automatic Y. Tsukamoto, “An approach to fuzzy reasoning method,” in container crane operation system,” in Proc. 2nd IFSA Congress, Advances in Fuzzy Set Theory and Applications, M. M. Gupta, Tokyo, Japan, July 1987, pp. 349-352. R. K. Ragade, and R. R. Yager, Eds. Amsterdam: North- [139] S. Yasunobu, S. Sekino, and T. Hasegawa, “Automatic train Holland, 1979. operation and automatic crane operation systems based on pre- I. G. Umbers and P. J. King, “An analysis of human-decision dictive fuzzy control,” in Proc. 2nd IFSA Congress, Tokyo, Japan, making in cement kiln control and the implications for automa- July 1987, pp. 835-838. tion,” Int. J. Man Mach. Studies, vol. 12, no. 1, pp. 11-23, 1980. M. Yonekara, “The application of fuzzy sets theory to the tem- M. Uragami, M. Mizumoto, and K. Tananka, “Fuzzy robot perature of box annealing furnaces using simulation techniques,” controls,” Cybem., vol. 6, pp. 39-64, 1976. in Proc. 8th IFAC World Congress, Tokyo, 1981. H. Watanabe and W. Dettloff, “Reconfigurable fuzzy logic pro- L. A. Zadeh, “Fuzzy sets,” Informat. Control, vol. 8, pp. 338-353, cessor: A full custom digital VLSI,” in Int. Workshop on Fuzzy 1965. System Applications, Iizuka, Japan, Aug. 1988, pp. 49-50. -, “Fuzzy algorithm,” Informal. Control, vol. 12, pp. 94-102, 11211 D. Willaeys, “Optimal control of fuzzy systems,” in Proc. Inr. 1968. Congress on Applied Systems Research and Cybernetics, Acapulco, ~, “Toward a theory of fuzzy systems,” in Aspects of Network Dec. 1980. and System Theory, R. E. Kalman and N. DeClaris, Ed. New [ 1221 -, “Decision making with possibilistic dynamic systems,” in York: Holt, Rinehart and Winston, 1971, pp. 469-490. Proc. 20th IEEE Conf. on Decision and Control, San Diego, Dec. -, “Similarity relations and fuzzy orderings,” Informat. Sci., 1981. vol. 3. ..pp. 177-200, 1971. D. Willaeys and N. Malvache, “Use of fuzzy model for process ~, “A rationale for fuzzy control,” Trans. ASME, J. Llynam. control,” in Proc. IEEE Int. Conf. on Cybernetics and Society, Syst. Measur. Control, vol. 94, pp. 3-4, 1972. 1978, pp. 942-946. -, “Outline of a new approach to the analysis complex -, “Contribution of the fuzzy sets theory to man-machine systems and decision processes,” IEEE Trans. Syst. Man Cybern., system,” in Advances in Fuzzy Set Theory and Applications, M. M. vol. SMC-3, pp. 28-44, 1973. Gupta, R. K. Ragade, and R. R. Yager, Eds. Amsterdam: -, “The concept of a linguistic variable and its application to North-Holland, 1979, pp. 481-500. approximate reasoning I, 11, 111,” Informat. Sci., vol. 8, pp. C. W. Xu and Y. Zailu, “Fuzzy model identification and self- 199-251, pp. 301-357; vol. 9, pp. 43-80, 1975. learning for dynamic systems,” IEEE Trans. Syst. Man Cybem., H. J. Zimmermann, Fuzzy Set Theory and Its Applications. vol. SMC-17, no. 4, pp. 683-689, 1987. Boston: Kluwer Academic Publishers, 1985. R. R. Yager, S. Ovchinnikov, R. M. Tong, and H. T. Nguyen, H. J. Zimmermann, Fuzzy Sets, Decision Making, and Grpert Fuzzy Sets and Applications: Selected Papers by L. A. Zadeh. Systems. Boston: Kluwer Academic Publishers, 1987. New York: Wiley-Interscience, 1987. C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller 0. Yagishita, 0. Itoh, and M. Sugeno, “Application of fuzzy -Part I,” IEEE Trans. Syst. Man Cybem., vol. 20, no. 2, pp. reasoning to the water purification process,” in Industrial Appli- 404-418, Mar./Apr. 1990. cations of Fuzzy Control, M. Sugeno, Ed. Amsterdam: North- Holland, 1985, pp. 19-40. [I281 T. Yamakawa ‘and T. Miki, “The current mode fuzzy logic integrated circuits fabricated by the standard CMOS process,” Chuen Chien Lee (S’88) for a biography and photograph, please see IEEE Trans. Computer, vol. C-35, no. 2, pp. 161-167, 1986. page 418 of this TRANSACTIONS.