PREFACE
This book is devoted to the study of Fuzzy Reasoning as applied to decision making and control processes. It co...
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PREFACE
This book is devoted to the study of Fuzzy Reasoning as applied to decision making and control processes. It contains a collection of important contributions covering a wide weUselected range of topics within the field. The first pioneering work on fuzzy sets and fuzzy logic was published in 1965 by Zadeh. This work, which has broken down the classical twovalued logic, was not very much appreciated until Zadeh has published his subsequent papers in 1971, 1972 and 1974. These works are now considered as the foundation of fuzzy reasoning and have inspired numerous researchers to work in the field. The first noticeable practical application of fuzzy logic was done at Queen Mary College of London by Mamdani. This work concerns the control of a pilot scale steam engine, and has shown for the first time that fuzzy logic offers the right tool for the construction of a linguistic model of a given process on the basis of the operators' experience. From that point on, fuzzy logic theory has found a tremendous attention with a plythora of theoretical and practical accomplishments. Today one can distinguish five main areas of research on fuzzy reasoning, namely: analysis of relational systems (Ostergaad, Rutherford, Perdrycz), fuzzy optimization / decision making (Gupta, Willaeys), design of fuzzy controllers (Kickert, Mamdani), and mixed fuzzy logic - neural network techniques (Kosko, Keller, Yager). Most of the current work on fuzzy sets and applications is currently conducted in Europe and Asia (China, India, Japan). TTie present book contains twenty one chapters, written by thirty four distinguished contributors and divided in the following five parts: Part 1: General Issues Part 2: Neuro-Fuzzy Systems Part 3: Fuzzy Controllers Part 4: Fuzzy Reasoning and Estimation Part 5: Applications Part 1 involves four chapters providing background material together with useful techniques for the validation of fuzzy knowledge bases and the software representation of fuzzy sets and fuzzy logic. Part 2 presents an overview of neuro-fuzzy expert systems along with an important case study, and a neural network model which is suitable for fuzzy reasoning. Part 3 presents the state of art of fuzzy controllers, including design and implementation aspects. This part starts with a chapter on the demystification of fuzzy control and includes critical evaluations of fuzzy controllers, along with new types of fuzzy controllers such as the sliding mode fuzzy controller. Part 4 involves a chapter on fuzzy parameter and state estimation, which is of fundamental importance in a variety of applications, a chapter on fuzzy reasoning as used in rule-based systems, and a chapter on computing the multivariable shape of an n-D pattern class. Finally, Part 5 presents six important applications dealing with industrial robotic systems, mechanical systems,
manipulators with artificial rubber muscles, Petri nets, biomedical engineering, and nondestructive fruit collection. Although the literature on fuzzy sets and fuzzy logic theory and applications is now very rich, the editors feel that the present book provides an important addition, since it presents new angles of attack and includes many new topics and results not available in other books. Each chapter is self-contained and in many cases involves fresh results and how-to-do issues. The book would not have been possible without the enthusiastic support of the contributors. The editors are indebted to all of them for their up-to-date contributions, and to Kluwer's (Dordrecht) editorial staff members for their care throughout the editorial and printing process. The editors also express their gratitude to the Natural Sciences and Engineering Research Council of Canada for support through an international scientific exchange award which made possible the preparation of their own contributions. The book is suitable for the researcher and practitioner, as well as for the teacher and student in related Master and Doctoral courses. Spyros Tzafestas Anastasios Venetsanopoulos
CONTRIBUTORS BARTOLINl G.
Univ. of Genova, Geneva, Italy
BINAGHI E.
IFCTR-CNR, Milano, Italy
BRUIJN P.
Delft Univ. of Technology, Delft, The Netherlands
CHEN S.-M.
Natl. Chiao Tung Univ., Taiwan
DE MEYER M.
Univ. of Louvain, Louvain-la-Neuve, Belgium
DUBOIS D.
IRIT, Univ. Paul Sabatier, Toulouse, France
FERRARA A.
Univ. of Genova, Genova, Italy
GILLK.
Univ. of Leeds, Leeds, England
GOREZ R.
Univ. of Louvain, Louvain-la-Neuve, Belgium
HATZIVASSILIOU F.
Natl. Tech. Univ. of Athens, Athens, Greece
HENDERSON M.
Univ. of Leeds, Leeds, England
HIROTA K.
Hosei Univ., Tokyo, Japan
JAGER S.
Delft Univ. of Technology, Delft, The Netherlands
JIN S.
Saga Univ., Saga, Japan
KALTSOUNIS S.
Natl. Tech. Univ. of Athens, Athens, Greece
KAWAMURA H.
Kobe Univ., Nada, Kobe, Japan
KUNG C.-C.
Tatung Inst, of Technology, Taipei, Taiwan
LIN S.-C.
Tatung Inst, of Technology, Taipei, Taiwan
LOONEY C.
Univ. of Nevada, Reno, USA.
MANDAL D.
Indian Statist. Inst., Calcutta, India
MITRA S.
Indian Statist. Inst., Calcutta, India
MURTHY C.A.
Indian Statist. Inst., Calcutta, India
PALS.
Indian Statist. Inst., Calcutta, India
PEDRYCZW.
Univ. of Manitoba, Manitoba, Canada
PRADE H.
IRIT, Univ. Paul Sabatier, Toulouse, France
RAYK.
Indian Statist. Inst., Calcutta, India
STAMOU G.
Natl. Tech. Univ. of Athens, Athens, Greece
TANI A.
Kobe Univ., Nada, Kobe, Japan
TERZAKIS S.
Nad. Tech. Univ. of Athens, Athens, Greece
TZAFESTAS S.
Natl. Tech. Univ. of Athens, Athens, Greece
VENETSANOPOULOS A. Univ. of Toronto, Toronto, Canada VERBRUGGEN H.
Delft Univ. of Technology, Delft, The Netherlands
YAMAMOTO S.
Matsushita Electric Ind. Co., Osaka, Japan
WATANABE K.
Saga Univ., Saga, Japan
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PART 1
GENERAL ISSUES
CHAPTER 1
FUZZY SETS AND FUZZY REASONING: AN INTRODUCTION S.G. (*)
(**)
Tzafestas(*), A. N. Venetsanopoulos^**) and S. T e r z a k i s O Intelligent Robotics and Control Unit National Technical University of Athens Zografou Campus, Zografou 15773, Athens, Greece Department of Electrical Engineering University of Toronto, Toronto, Ontario, Canada M1551A4
1. I N T R O D U C T I O N Fuzzy set theory was initiated by Zadeh in 1965 [1] and permits the treatment of vague, uncertain, imprecise and ill defined knowledge and concepts in an exact mathematical way. Throughout the years this theory was fully studied and used for the analysis, modelling and control of technological and nontechnological systems [2-21]. Actually, our life and world obey the principle of compatibility of Zadeh, according to which "the closer one looks at a 'real' world problem, the fuzzier becomes its solution". Stated informally, the essence of this principle is that, as the complexity of a system increases, our ability to make precise and yet significant statements about its behaviour diminishes until a threshold beyond which precision and significance (relevance) become almost exclusive characteristics. Fuzzy controUers and fuzzy reasoning have found particular applications in industrial systems which are very complex and cannot be modelled precisely even under various assumptions and approximations. The control of such systems by experienced human operators was proved to be in many cases more successful and efficient than by classical automatic controllers. The human controllers employ experiential rules which can cast into the fuzzy logic framework. These observations inspired many investigators to work in this area with result being the development of the so called fuzzy logic and fuzzy rulebased control [3, 8, 9, 14, 18, 20]. The purpose of this chapter is to provide a short account of fuzzy set and fuzzy reasoning theory in order to help the unfamiliar reader to study and understand easier the rest of the book. The reader who is familiar with the fuzzy sets can probably find here a ready-to-use material for his(her) applications. Section 2 presents the basic concepts and definitions of fuzzy sets. Section 3 reviews the three fundamental fuzzy logic operations of Zadeh and section 4 provides a set of other fuzzy operations and relations. Section 5 presents a generalization of the three fundamental operations, and section 6 introduces the concept of hypercube as used in fuzzy theory. Section 7 presents the representation theorem, discusses the fuzzy functions (domain of definition and domain of values) and states the fuzzy extension principle that helps in the fuzzification of mathematical concepts and laws. Section 8 provides a brief discussion of categories and lattices in the framework of fuzzy sets, and section 9 examines the theory of fuzzy reasoning (linguistic variables, 3 S.G. Tzafestas andA,N. Venelsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 3—29. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.
linguistic modifiers, generalized modus ponens rule, max-min composition rule, and estimation of the membership function). Finally, section 10 gives a quick look at fuzzy or linguistic control.
2. F U Z Z Y SETS: BASIC CONCEPTS The concept of "set" plays a fundamental role in mathematics. Actually one cannot define in a rigorous unique way what is a set, what is number or what is a straight line [10]. These concepts can be understood better not via definitions but via examples [11]. Let X be a classical set and x an element. Then one of the following holds (how much this holds will be discussed soon): The element x belongs to X (symbolically xeX) or X does not belong to X (xtX). This is the so called principle of dichotomy. By disputing dichotomy, the classical (crisp) set theory breaks down and the fuzzy set theory naturally emerges [12]. 2.1. Definition of Fuzzy Sets Let us start by the following paradoxes of logic. i) Pythagoras' School (400 A.C.): Question: Does the Cretan philosopher tell the truth when it asserts that "All Cretans lie" ? The classical reply is that this assertion includes a contradiction. The fuzzy response is that the philosopher does and does not tell the truth !! ii) Russel's Paradox 1: Question: Does or doesn't exist the set of all sets that are not members of themselves ? Classical reply: The assertion involves a contradiction. Fuzzy reply: The set does and doesn't exist !! iii) Russel's Paradox 2: The barber of a village gives a haircut only to those who do not cut their hair off themselves. Question: Who cuts the barber's hair off? Classical reply: The assertion involves a contradiction. Fuzzy reply: The barber cuts and doesn't cut his hair off !! The above paradoxes are due to the fact that the human word (reason) exceeds very much the capabilities of the mathematical notation and formalism [13]. Thus in classical set theory it is safer to describe a set by its elements, e.g. A = {1,2,3,4,5} than to describe the set through a common property of its elements. Let X= {Xj , Jf2 , Xj , x^ , Xj } be a classical (crisp) set which is called reference super set (universe of discourse). Now let A = {Xj, Xj, Xj } be a crisp subset of X. The set A can be equivalently described by A = {(xj,l), (xj.O), (x^A), (x^fi), (^5,1)}, i-e. as a set of pairs (x,|U^(x)), where x is the element of interest and /U^(x) is the membership function of X in the subset A, where f 1 if xeA The question now arises whether fi^{x) can take only the values 0 and 1 or any value between 0 and 1. In other words, why ju^:A-»{0,l} and not ^^-•[0,1] ? This question was firstly examined by L. Zadeh and is the starting point for the development of fuzzy sets [1].
Definition 2.1 Let X be the reference superset and A a subset of X. Then A is said to be a fuzzy subset of X if and only if A = {ix,fi^{x))
I xeX,Ai^U):X-[0.1]}
It is obvious that in the special case where we have {0,1} instead of [0,1] the fuzzy subset A degenerates to the crisp subset A. Pictorially a fuzzy set A has the representation of Fig.l. fuzzy subset A
crisp subset A
Fig. 1. (a) Representation of crisp and fuzzy subset of X (b) A possible shape of the membership function li^Cx).
If A is finite then it is also denoted by ^ =
I^A(^)/^I
+ •• -^ M/iU„) An = Ei"=i
A = \uj^(x)/x
/^A(^/)/^-
and if it is not finite by
where the symbols" + ", "Z", "J" denote union and the symbol"/" does
not denote division. Definition 2.2 Let ^ be a fuzzy subset of the reference superset X. Then (a) The crisp subset Supp(/1) of X is called support of A if and only if Supp(-4) = { x e X : / i ^ ( x ) ^ 0 } with Supp(/4)cX. (b) The crisp subset L^
of X is called a-cut of A if and only if
L^A = {xeX :
n^(x)ia}
with L ^ c X . Definition 2.3 The fuzzy subset A of X is called (i) convex if and only if (see Fig.2): (ii) norma] if and only if there exists at least one element xeX for which n^(x) = \. \if.W ,,
fVW i
(a)
(b)
Fig. 2. (a) convex fuzzy set (b) nonconvex fuzzy set
Definition 2.4 Let A he a fuzzy subset of the reference superset X. Then (i) The quantity
| A | = Y,^gx I^Ai'')
(ii) The quantity
|A|=|A|/|X1
^^ '^^^^^ ^^^ cardinality of A
is called the relative cardinality of A.
The cardinality measures in a certain way the "size" of a fuzzy set. Using the cardinality one can compare fuzzy sets that have the same reference superset. Example 2.1 (a) Let X=N and "A = integers near to 10". Then one possible A is the following: A = {(7,0.1) , (8,0.5) , (9,0.8) , (10,1) , (11,0.8) , (12,0.5) , (13,0.1)} = 0.1/7 + 0.5/8 + 0.8/9 + 1/10 + 0.8/11 + 0.5/12 + 0.1/13 and has the pictorial representation of Fig.3. (b) Let X=R and "A = real positive numbers near 10". Then one possible A is
^2^-ll
and has the representation of fig.4. ^(x)
7
8
9
10
11
12
13
X
Fig. 3. Pictorial representation of the example's fuzzy set A.
liXx)
i
Fig. 4. Pictorial representation of A
(c)
For A in (a) we have Supp(>i) = {7,8,9,10,11,12,13} L(jA = {7,8,9,10,11,12,13} ^0 5^ = {8,9,10,11,12} Lo8A = {9,10,ll} L^A = {\Q}
For A in (b) we have (d)
Supp(A) = R"' Lo5/4 = {A-eR+ : l/[\ + {x-\Qif]i0.5} For A in (a) we have |A I =0.1 + 0.5 + 0.8+1 + 0.8 + 0.5 + 0.1 =3.8 |X|=7 H i = 3.8/7
3. T H E T H R E E F U N D A M E N T A L OPERATIONS O F Z A D E H The three fundamental operations of fuzzy sets are based on the use of the membership function [1]. Definition 3.1 Consider the fuzzy subsets A and B of the reference superset X. Then the section and union of A, B are defined as Section:
Union:
C = ADB
= I.(X,IJQ(X))
C = ADB = l(x,/j(--(x))
and the complement
| xeX
\ xeX
, ^c(^)
, n^^x)
A'^ oiA as A' = Ux,Hj^,{x))
= '^^'^{V-Ai^)
= miix{\x^{x)
> ^^BC-*^)}}
,
\ xeX , A'^c(Jt) =
\ig{x)}\ l-\x^{x)\
Example 3.1. Let X = {apartments with 1,2,..., 10 rooms}, A = {apartments suitable for 4-member families}, B= {apartments with large space}. The sets A and B are fuzzy subsets of X. Suppose that A = 0.2/1 + 0.5/2 + 0.8/3 +1/4 + 0.7/5 + 0.3/6 and B=0.2/3 + 0.4/4 + 0.6/5 + 0 . 8 / 6 + l / 7 + l / 8 . Then C=Ar\B= {apartments suitable for 4-member families and with large space} = 0.2/3 + 0.4/4 + 0.6/5 + 0.3/6 D = /\u_B= {apartments suitable for 4-member families or with large space} = 0.2/1-1-0.5/2 + 0.8/3+1/4-1-0.7/5 + 0.8/6+1/7-1-1/8 A'^= {apartments not suitable for 4-member families} = 0.8/1 + 0.5/2+0.2/3 + 0.3/5 + 0.7/6 B*-= {apartments with smaU area} = 1/1+1/2+0.8/3 + 0.6/4 + 0,4/5 + 0.2/6 Remark: The fundamental axioms of the Aristotelian logic do not hold in fuzzy logic, i.e. in general AnA'^^0 and AuA'^^X . This can be easily seen from Fig.5.
4. G E N E R A L OPERATIONS OF F U Z Z Y SETS A N D F U Z Z Y RELATIONS 4.1. Fuzzy Operations Here we define some important operations of fuzzy sets.
Definition 4.1. Let A and B fuzzy subsets of X. Then we define the following: (i) Algebraic sum A + B A+B = [Cx,iJ^^g(x))
I xeX , I^^.BM
= fxjx)
+ n^U) -
n^(x)\ig{x)]
Fig. 5. Membership functions of A and A*^.
(ii) Algebraic
product AB=
(iii) Bounded A®B (iv) Bounded
AB \(x,tij^gix))
I xsX , /i^g(x) = n^ix)-Hg{x)
\
sum A«B = { (x,fj^Qg(x))
I xeX , fi^^g(x)
= m i n { l , p ^ U ) + fig(x)]
I x€X , MA.BM
= vaax{0,n^(x)-Hg(x)}
]
product A °B
AoB = { (x,fi^,gM)
}
(v) Left square ^A ^A = { U . / i . ^ ( x ) ) I x^:x€X (vi) Convex combination AcB
= { {x,^iJ^^g^x))
(vii) Ordering
, A«=^(x^) = /i^(*) }
AcB I xeX
, UACB(X) = ^I^AM
+ H'Z/'BW , ^,+^2=1]
of A and B >4cfl - / i ^ ( x ) i / i U )
VxeX
Example 4.1 Let X= {1,2,3,4,5,6,7} and A = 0.8/3 + 1 / 5 + 0.6/6 , B= 0.7/3 +1/4+0.5/6. Then >iuB=0.8/3 + 1 / 4 + 1 / 5 + 0.6/6 / i n B = 0.7/3+ 0.5/6 A''= 1/1+1/2+0.2/3+1/4 + 0.4/6 AB=0.56/3+ 0.3/6 >i2 = ^ s ^ = 0.64/3+ 1/5 + 0.36/6 A«J3=l/3+l/4+l/5+l/6 v4oB=0.1/3+1/5 + 0.1/6 2/1 = 0.8/9+1/25+1/36
10 4.2. Fuzzy Relations and Fuzzy Cartesian Product Definition 4.2 Let Xj , X2 , ... , Xj, be reference supersets and A^, Aj , - , A^^ respective fuzzy subsets of them. Then as a fuzzy relation f{A^ , ... , A^) we define the fuzzy set: / M l . A ) = { i(x^,...,xj
, fifi4„.A.)(^t, ,xj ) I (x^,...,xJeX^x
xX^ ]
where
A crisp and a fuzzy relation are shown in Fig.6.
^L
li^X,.X^
crup relacion R f(x.x^=0
3
Fig. 6. Geometric representation of crisp and fuzzy relations.
Definition 4.3 Let Xp...,X^ be reference supersets and Ap...^„ respective fuzzy subsets of them. Then the Cartesian product of A^,...yA^ is defined to be the fuzzy set g(A^,...,A^) = {
((AC,,...,J:J
. ^tg^A^,,^/x^,...^^)) | (x^,...,x^)eX^x•••xX^ }
where fg(A,....A,)(^i'-'''n) = niin{/i^(x,),.„,/i^(xj) Figure 7 shows a crisp and fuzzy Cartesian product. One can see that the crisp Cartesian product is 2-dimensional but the fuzzy one is 3-dimensional.
11 ^ J (lAlxAl (X1,X2)
'^
fuzzy A 2
cnip
xi
fuzzy Ai
\
''
\ / fuzzy A-xAo
Fig. 7. Geometric representation of crisp and Cartesian product.
Example 4.2 Let Xi = X2=(2,4,6) , Ai = 0.5/2+1/4 + 0.6/6 and ^^2= V2 + 0.6/4. AixA2=0.5/(2,2) + 1/(4,2) + 0.6/(6,2) + 0.5/(2,4) + 0.6/(4,4) + 0.6/(6,4)
Then
5. GENERALIZATION OF THE THREE FUNDAMENTAL OPERATIONS The families of T-operators provide suitable generalizations of Zadeh's three fundamental operations max(»,»), min(«,»), l-(») for the union, section, and complement, respectively. We have three kinds of T-operators, namely T-norms, T-conorms and N-negation functions. The definitions of them are as follows. Definition 5.1 The function T:[0,l]x[0,l]-»[0,1] is called T-norm if and only if Tpossesses the following properties for all x,y^€[0,l]: 1. ^x,y) = ^y^) 2. T{x,y)i'Jlx,z) yy^z 3. T{x,T{y,z)) = TXTix,y)^) 4. T{x,l) = \
12
Definition 5.2 The function T :[O,l]x[0,l]-*[O,l] is called T-conorm if and only if T possesses the following properties for all x,y,2e[0,l]: 1. r{x,y) = r{y,x) 2. T (x,y)iT (x,z) \iy4-[0,l] } be a fuzzy subset of the finite reference
superset X. Then the vector
(/i^(A:j),^^(jCj),...,/i^(x^)):X-[0,l]''
is called the (fuzzy
unit) fit vector of the fuzzy set A. Clearly, the fit vector contains (gives) all the information that is provided by the set theory approach. Also, the pair (X,[0,1]") defines a basic metric space in the fuzzy theory. TTie set [0,l]"=[0,l]x...x[0,l] is called unit/j_>j3erci;be of dimension n, and forn=l, n = 2 and /j = 3 has the geometric representation shown in Fig.8. (1.1.1)
(0.1,1) (0.1)
(1,1)
(0.1,0)
/
(1.1,0)
/
L _
(1.0.1)
kL (0.0)
[0,1]
(1.0)
[0,1]
(Pfl,0)
(1.0,0)
[0,1]
Fig. 8. Hypercubes of dimensions n= 1,2,3.
Using the concept of hypercube one can represent both the crisp sets as their vertices, and the fuzzy sets as internal points. The above can be clarified by the following example. Let X= {x^,X2} be a reference superset, and P(x) the set of all subsets of X, i.e. P(x)= {0,{x^^2}'{^i}>{^2}}- Now consider the following fuzzy subset of X: i4 = {(Xj,l/3), (Ar2,3/4)}. The fit vectors of P{x) are as follows: i) The set e> or {(x^fi), (^2,0)} has fit vector (0,0). ii) The set {^1,^2} or {(Xpl), (^2'0} ^^^ ^^ vector (1,1). iii) The set {Xj} or {(.yj,!), (%0)} has fit vector (1,0), and iv) The set {;r2} or {(x^O), (x^j-^)} ^^^ ^^^ vector (0,1). v) The fit vector of the fuzzy set A is (1/3, 3/4). Let us now represent the above fit vectors in the hypercube [0,1]^ (here n = 2 since X has two elements). This is shown in Fig.9. One observes that the crisp sets are represented at the vertices, and the fuzzy sets at internal points of the hypercube.
14
Xi.Xi=(l,l)
(X2l=(0.1)
fuzzy subset A A=((Xi, 1/3) , (X ,23/4)1
O=(0.0)
ix,i=(l,0)
Fig. 9. Representation of the fit vectors in the hypercube [0,1]^.
6.2. Relations Among A, A^ AnA*^ and AuA"^. With the aid of the hypercube concept the "symmetric in space" relation of A, A'^, ArA'^ and AuA'^ can be easily revealed. To this end, consider the fuzzy set yl = (l/3, 3/4). Then ^ ' ' = (3/4,1/3), >\04"^ = (1/3,1/3) and AuA'' = (2/4,3/4). These fuzzy sets are represented in the hypercube [0,1]^ as shown in Fig.lO.
(0,0)
(0,1)
Fig. 10. Geometric representation of A, A^ ArvV^ and AuA'^. From Fig.lO it is now obvious that the sets A, A^, AnA*^ and AuA"^ occupy symmetric, with respect to the centre M, positions in the hypercube. If A = (0,1), then A'^ = (1,0), AnA'^ = (0,0) and AuA'^ = (l,l), i.e. the sets A, A^, AnA"^ and AuA^ coincide with the vertices of the hypercube. If A = M=(l/2,l/2), then ^"^=(1/2,1/2), AnA'' = (1/2,1/2) and ^uy\*^=(l/2,l/2), i.e. the sets A, A'^, AnA^ and AuA'^ all lie at the centre of the hypercube. Thus for minimum uncertainty (i.e. 0,1) the four sets coincide with the hypercube vertices, and for maximum uncertainty (i.e. 1/2) the four sets coincide with the centre M.
15 The centre M of the hypercube behaves in a singular way since it possesses the paradox property: M=M^=MnM^=MuM^ !!! Thus the centre M is considered as the "black hole" of the fuzzy set theory. In other words, at the midpoint of a hypercube nothing is distinguishable, while at the vertices everything is distinguishable. Actually, one can easily verify that the logic paradoxes discussed in Sec.2.1 which according to the classical logic contain contradictions are solvable with the fuzzy logic and geometrically lie at the centre of the hypercube [16]. Indeed in these cases, if S is a proposition and S^ is its complement we have S='S^ and S^=*S i.e. S'^S^. Now, if t{S) is the truth value of 5 then in the classical logic we have t(S) = 1(8*^ i.e. 0 = 1 (contradiction). But in the fuzzy logic we have t(S) = t(S^) or f(S) = l-f(S^) whence t(S) = t(S^= 1/2. Thus S and S^ are half true and half false and the paradox ceases to exist.
6.3. How Big And How Fuzzy Is A Fuzzy Set ? A basic measure of the size of a fuzzy set is its cardinality (see Definition 2.4):
Ml = E A ' . W which coincides with the Hamming-distance and measures the distance of the fuzzy set A (in the hypercube) from the vertex (0,0) as shown in Fig. 11. (X2)=(0,l)
lx,x^=(l,l)
fuzzy subset A A=((x,,l/3),(x^/4))
O=(0,0) ix,)-,.,-, F*ig. II. Geometric representation of the cardinality.
Definition
6.2
The generalized distance of order p ^ l of two fuzzy sets A and B is defined by
L''{A,B) =
EIM^.)-/'B(*\uA'=|=n and £(A) = 0/n = 0. (1.0)
A<JA'^=X,
i.e.
(1,1)
— b
Fig. 13. Quantities for the computation of the fuzzy entropy
7. SOME BASIC THEOREMS AND PRINCIPLES 7.1. The Representation Theorem The possibility of approximating a fuzzy set by a crisp set is a fundamental topic of fuzzy set theory. Dominating role for this approximation plays the concept of a-cut (see Def. 2.2). This is so because the a-cut is a crisp set that approximates the fuzzy set with "accuracy or". We recall that the a-cut of a fuzzy set A is the set A first idea is to approximate the fuzzy set with a sequence of a-cuts where Osa^l. Then it is obvious that LQA=X and a s ^ •• L ^ ^ ^ . The approximation problem is formulated as follows. Let A, B, C,... fuzzy sets and L^, L^, L^C,... their a-cuts. As a first step in the approximation one works with L„A, L^, L^C,... which are crisp sets, and from them produces a set, say ^ , for some ae[0,l]. This is repeated many times, and so from the given families of the a-cuts L^, L^, L^C, ... a sequence of results ^ is produced for all a, ae[0,l]. The question now arises: Does
there exist a fuzzy set Z which would give the above family of sets ^ for all a, Osasl in the a-cut sense? The answer to this question is positive as stated in the following theorem. Theorem 7.1 Let ^ , Osasl be the family of crisp sets with the following properties: i)oZ=X ii) a s ^ - ^ i ^ CO
iii) Oji...sa^
,
lima^ = a -* n-oo
^Z = fl Z n=l ^
Then there exists a unique fuzzy set Z such that
L^=^.
For the proof the reader is referred to [17]. 7.2. Fuzzy Functions: Domain of Definition and Domain of Values Let us first consider two crisp sets X and Y and the classical (crisp) function f:X-+Y. Then P{X) = {A\A^X) and P(y)={B I BcY} are the sets of all subsets of X and Y, respectively. If AcXi.e. A£P{X) and BcYi.e. BeP{Y), then we define the domain of values of f a s the set f{A) = {yeY \ 3x6>4, y=f{x)) : A-*f{A). TTie domain of definition of f is defined as r\B) = {xeX \ f{x)eB} , f^: f-\ByB i.e. f:PiX)^P(Y) and f-hP(B)^P(A). Now consider the fuzzy case. Let X and Y two crisp sets and f.X-*Yii crisp function. We define F(X) to be the set of all fuzzy sets on X and F( Y) to be the set of all fuzzy sets on Y. We use the crisp function f:X-*Y to define the fuzzy definition domain ( s:F(X) ) and the fuzzy domain of values ( cF( Y) ). Let X = {{x,^ixix)) I x€X , fixix) Y= {(y,My(7)) ]yeY.^ly{y)
: X-[0,1]€F(X)} : Y-[0,l]eF(Y)}
Determination of the domain of values: To find where an arbitrary x is mapped through f we apply the following algorithms. For an arbitrary y: (a) If *•=/" (y) is not defined (i.e. f{x) possesses a singularity at the point of concern or f'^(y) = 0), then we set /iy(y) = 0. (b) If x=f'^(y) exists, then we find all other points /that give the same x. Among these points y we select the one for which the corresponding x maximizes the function /j_j({Ar), and set this quantity as jUy(y). Thus y= {(y.Fy(7)) \y^Y,^lxiy):Y^[Q,\]
}
where sup fjj^(x)
,
/-'(jr)#0
f^rW 0 . r'(jf)=0 Clearly, the above procedure gives the image of X via f. It should be emphasized here that f is used as a crisp function, i.e. it is applied to xeX to give yeY.
19 Determination of the domain of definition: To find from which fu2zy set X, a given fuzzy set y has resulted via /, one uses the following scheme. Let Y€F( Y). Then, obviously, f "^(Y) = XeF(X). For any y=fix) it suffices to find /iy(/(;r)) and then lif-i^r)M = Py(/(*))
- It is clear, that given Y, this scheme provides the fuzzy set X.
Again, / is used as crisp function. 7.3. The Fuzzy Extension Principle The extension principle helps in the fuzzification of mathematical laws, and thus plays a dominant role in the fuzzy set theory. Definition 7.1 (Extension Principle) Let Xp X2, ..., X^ be reference supersets; X=XjxX2x...xX^ their Cartesian product; A^ A^, —rA^ respective fuzzy subsets; and /:XjxX2x...xXj-» Ywith y=t{Xy...^^ a crisp function. Then the extension principle transfers the fuzziness of Ap ..., A^ into a fuzzy set B of Y where B = {(y,^B(y))
I y=f(.X^,-,Xr)
, (Xi,...,X^)€XiX...xX^
}
and sup
minlA*. (x)....,A«^ (x)}
,
f'Hy)*B
f'ECy) =
0
, f-Hy) =
For n= 1 the extension principle coincides with the concept of "image of a fuzzy set" (see sec.7.2) i.e. y= {iy,^iYiy))
I y ^ " ^ . M;f(y):"5^-[o,i] }
where sup Pjf(x) fifiy}
.
f'Hx)*0
=•
0
. /-'(x) = 0
The extension principle allows a crisp function f, with definition domain the crisp set X, to change its definition domain to, say, certain fuzzy subsets of X (instead of X itself). For example if A is a fuzzy subset of the crisp set X, and f is a crisp function defined on X, then the extension principle shows how the function f can be applied upon the fuzzy set A. In particular if y4=ju^(xj)/jfj + ...+ju^(x„)/Af„ , then
It is noted that / i s always applied upon the elements of the crisp set X.
20 8. CATEGORIES A N D LATTICES 8.1. Categories Category is a complex mathematical concept which consists of (i) a class of objects • supv maxj^i n min(p.Aj(u), [iB^(y)) = maxj=i_n
HAJ^")-
Thus, we can never establish that (3) does not hold (i.e. that 3u, supy 7tYix(v>u) < 1)-
3.2 - Coherence ofparallel certainty rules For simplicity, we first consider the case of two parallel rules of the form "the more X is Aj, the more certain Y is Bj", i = 1,2. We have itxCU'V) = min[max(l -
|.IAJ(U), U B I M ) .
max(l - lJi/>^^(u), ^ 6 2 ^ ) ] -
38 Then proj(%;X)(u) = supvmax(min(l -^lAl("). 1 -^A2("))'"''"(' -t^Ai(")'I^B2W). '"•"(' - I^A2(")> t^Bi^)' min(HBj(v), |LIB2(V))) i.e. proj(X ; X)(u) = maxd - H A J ( U ) , 1 - HA2(")' cons(Bi,B2)) (12) provided that B j and B2 are normalized (i.e. supy llg. (v) = 1 = supy [I'Q^iy]) and where cons(B 1,82) = supv min(H3 j(^)' ^lB2(^))
(' ^)
estimates the con.sistency of B | and 82Applying the coherence principle (3) to (12) we get proj(X ; X)(u) = 1, Vu cons(Bi,B2) = 1 or Vu, min(Hy^ (u), H;s^^(u)) = 0 1 - HA(") = ' ^ ^ that hgt(B) > hgt(B n G), VG, it can be checked that the only way to have proj(% ; X)(u) = 1 for u e (Hie I support(Ai)) n (Djg I support(Aj)) is to have hgtdHie I ^i) = 1- Indeed in order to have (15) equal to 1, we only have to consider the terms such that 1 - \i.[\. J A ( " ) = ^ which entails that the index set J is a subset of f, the complement of I in CI ,nl, but these terms are of the form min(l - n y . ^ jAi(u). hgtCflkg J B^)) <min(l - u y , ^
JAJC"). hsKflie
I Bj)) = hgt(ni£ J Bi),
40 using (16). Thus we have the following proposition. Proposition : A necessary and sufficient condition for a base of parallel certainty rules of the form "if X is Aj then Y is Bj", i = l,n to be coherent is that V I ^ , (flie I support(Ai)) n (fljg i support(Aj)) 5* 0 => hgKflie i Bi) = 1. This proposition provides a simple procedure for checking the coherence of a family of parallel certainty rules. More explicit conditions can also be given when n is sufficiently small. For instance, forn = 3, (15) writes proj(^ ; X)(u) = max(l - ltAiuA2uA3(")> •"'"(! - liAjuA2(")< hgt(B3)), min(l - HAiuA3(")' hgt(B2)), min(l - 11A2UA3("). hgt(B,)), min(l - H A I ( " ) . H^2 coreCBj) n core(Bi^) * 0. We recognize the above Proposition applied to three rules.
3.3 - Coherence of Parallel Gradual Rules Coherence condition (3) writes for gradual rules Vu, supv minj^i n lifi,A.a) -^ HgjCv) = 1 with |iA-(u) -* M.Bj(v) = 1 if Thus (17) is equivalent to
VLA^^)
(1T>
^ \^Bi ^Bi(^) = I^Bi(v) otherwise.
Vu, 3v, V i e I*u, \iA-(u) < HB/V)
(18)
with I*y = [i, HA(U) > 0}- T^'S 'S still equivalent to VJcIl,nl,Vaie(O,l].niej(Ai)ai^0=*niej(Bi)aj#0
(19)
where A^, B^^ denote the a-cuts of the fuzzy sets A and 8 (AQ( = {u, ^ A ( " ) - "D- These coherence conditions have been ateady pointed out in (Dubois et al., 1988). Easy-to-establish consequences of (18)-(19) are V i e J. J c Il,nl, hgKriieJ Ai) < hgt(nie j 8;).
(20)
This is a necessary condition for coherence (already suggested in Dubois and Prade (1982)). However it is not sufficient as shown on Figure 2. In Appendix, it is shown in the case of triangular membership functions that (20) becomes a necessary and sufficient condition, when the slopes of the decreasing and increasing parts of the Aj's are proportional to the ones of the corresponding parts of the 8j's. This encompasses the case where all the Ai's are symmetrical and identical up to a translation and the same condition holds for the Bj's. A well-known particular case of coherence is encountered in practice in fuzzy control when triangular membership functions
42
such thai Vi=l,n-1, Vu, ^ A ( " ) "•• l^A- i(") = ^ (which entails hgtCAj n Aj^j) = 1/2) are used on the domain of X and the Bj's make a similar fuzzy partition.
Figure 2 : Condition (18) does not hold : {v I (0.3 (v) > IIAI (")) '^ t'^' l^Bj^"^) - M-A7(")l = 0 The case where the Aj's and the Bj's have monolonically increasing (or decreasing) membership functions is much simpler. If ^y^ , ji^^, jig , and lUg, are all increasing, then the two corresponding gradual rules are coherent (provided that the Bj's are normalized). Let us consider the situation where JJ-AI > M-B > I^Ao ^ ^ increasing while jig^ is decreasing. In this case hgt(Ai n A2) = I if Aj and A2 are normalized. Then due to (20) it is necessary to have hgt(Bi n B2) = 1 in order to have the coherence, this is also sufficient to insure (18) here, as it can be easily checked. Indeed the two rules "the larger X, the larger Y" and "the larger X, the smaller Y" can be coherent only if Y increases and decreases in different subdomains.
4. Redundancy Redundancy is a drawback which is less important than inconsistency for a knowledge base. However, it may have bad consequences from a computation point of view, even if redundancy can make some consequences of the knowledge base explicit in it. Given a collection ^ of n pieces of knowledge represented by possibility distributions rtj, i=l,n, a piece of knowledge represented by a possibility distribution 7t will be said to be redundant with respect to ^ if and only if j r ^ = mini=,i u7ri l^Bi(v) then HA(U) S
HB(V)
(25) or HB(V) ^ ^Bl(v)•
(26)
•
HA(") - I^B(^)' ^u>v, implies that B = V (the whole domain of v) since A is normalized. This is the trivial rule "if X is A, then Y is V" which is redundant with anything ; • The other condition leads to ^B(V) - I ^ B I ( ^ ) ' ^ ^ ^^'^^ ^^^' >M^Bi(^)' choosing u so that
^iA](u)=lMoreover, (25) implies that V v £ core(Bi), HB(^) - ^"Pu 1^A(")' '•^- I^B(^) = 1 ^'"'^'^ ^ •** normalized. So (25), (26) imply that B 3 Bj. Now, given A, Bj and B 3 B], (25) implies Vu, JXA(U) ^ inf{|iB(v) / ^lBl(v) ^ ^lAl(u)l•
(27)
When B = BJ, (27) leads to Aj 2 A, which is a rather expected result. So, the rule "the more X is A, the more Y is B" is redundant with respect to the rule "the more X is Aj, the more Y is Bj" if and only if B a Bj and A satisfies (27).
46 The case of redundancy with respect to a set of two gradual rules is more tricky and left for further research. 5. Concluding Remarks This paper presents preliminary results on the coherence of knowledge bases made of parallel fuzzy certainty rules or parallel fuzzy gradual rules. The study of more general knowledge bases would require a detailed study of the "transitive chaining" of fuzzy rules, i.e. the generation of new rules through chaining. However in the particular case of so-called possibilistic knowledge bases made of weighted classical logical formulas (which encompasses the case of certainty rules with non-fuzzy conditions and weighted conclusions (i.e. Jig. = max(p.3'., 1 - a) with B'j non-fuzzy)), the coherence problem corresponds to the research of "no goods" in a possibilistic Assumption-based Truth Maintenance Systems, where assumptions are the input literals; see (Dubois, Lang and Prade, 1991a ; Benferhat et al., 1992). This enables us to extend coherence checking procedures available in the case of classical logic knowledge bases (e.g. Beauvieux and Dague, 1990 ; Mcseguer, 1991 ; Loiscau, 1992) to possibilistic knowledge bases.
Appendix : Coherence of gradual rules with triangular membership functions Let us consider a collection of fuzzy gradual rules of the form "the more X is Aj, the more Y is Bj" where the Aj's and Bj's have triangular membership functions. Let a; (resp.: b[) denote the peak and [a'i.a'j] (resp.: [b'i, b'j]) the support of A; (resp. : Bj). Sec Figure 3.
32 "1 U
J3'2bl
91 a2
Figure 3
b2
b'l
47
We further assume that g'i ^a'i+i anda'j ^a'i+i.h'i S h'j+j and b'j < b'j^] (i.e. a fuzzy set only overlaps with its immediate neighbours). Then the necessary and sufficient condition for coherence (18) writes Vi=l,n-1, Vu, 3v, |iiAi(u) ^ ^ ^ 8 ; ^ ^n^ HAi+i(") ^ ^^8;+] W or equivalently Vi, Vu. (Bi)^^,(„) n (Bi+i)^^^^^(„) ^ 0. For convenience we make i = 1, i + 1 = 2 in the following. Introducing the slopes «] = 1 / (ai - 1 ' i ) < 0, a2 = 1/ (32 - a'2) > 0, Pi = 1 / (bi - b'l) < 0, P2 = 1/ (b2 - b'2) > 0we have for u as in Figure 3, HAI(U) = " I ( U - 3'l)> I^AoW = ^2(^ " 32) which leads to the coherence condition ^ u - Fi) + b'l > ^ u - a'2) + b'2 Pi §2 i.e.
(«L_|2U|2^.2-^'l+b-,-b'2>0.
(A)
This condition should hold for any u in the interval [max(aj,a'2). niin(a2^'i)]- Moreover the necessary condition for coherence hgt(Ai n A2) S hgt(B j n B2) writes hgt(Ai n A2) = "^ " ^ ^ ( a ' 2 - a ' l ) < ^^ P^(b'2 - b'l) = hgt(Bi n B2).
a2-«i
(B)
fc-Pi
Then it can be easily checked that hgt(Ai n A2) 5 hgt(Bj n B2) becomes a sufficient condition for coherence if _ tt) _ a 2
since then conditions A and B coincide. This expresses the respective proportionality of the slopes of the increasing and decreasing parts of the Aj's with the ones of the Bj's. A particular case where this latter condition holds is when cTi = - a j and Pi = -p2. ie- when the Aj's have symmetrical and identical membership functions, and a similar condition is satisfied by the Bj's; this situation is in agreement with the idea that Aj models "close to
48 a^", which entails the symmetry and the identity of shapes of the Aj's if we want to have the same modelling of the notion of closeness everywhere. References Ayel M., Rousset M.C. (1990) La Coherence dans les Bases de Connaissances. CepaduesEdilions, Toulouse, France. Beauvieux A., Dague P. (1990) A general consistency (checking and restoring) engine for knowledge bases. Proc. of the 9th Europ. Conf. on Artificial Intelligence (ECAr90) (L.C. Aiello, ed.), Stockholm, Sweden, Aug. 6-10, 77-82. Benferhat S., Dubois D., Lang J., Prade H. (1992) Hypothetical reasoning in possibilistic logic : basic notions, applications and implementation issues. In : Advances in Fuzzy Systems : Applications and Theory Vol. I (P.Z. Wang, K.F. Loe, eds.), to appear. Bourrelly L., Chouraqui E., Portafaix V. (1992) Les topoi pour la validation structurclle d'une base de connaissances. Joum6e Francophone de la Validation et de la Verification des Systfemes h Bases de Connaissances, Dourdan, France, Apr. 16. Dubois D., Lang J., Prade H. (1989) Automated reasoning using possibilistic logic : semantics, belief revision, and variable certainty weights. Preprints of the 5th Workshop on Uncertainty in Artificial Intelligence, Windsor, Ont., 81-87. Revised version in IEEE Trans, on Data and Knowledge Engineering, to appear. Dubois D., Lang J., Prade H. (1991a) A possibilistic assumption-based truth maintenance system with uncertain justifications, and its application to belief revision. Proc. of the ECAI Workshop on Truth-Maintenance Systems (J.P. Martins, M. Reinfranck, eds.), Stockholm, Aug, 6, 1990, Lecture Notes in Computer Sciences, n° 515, Springer Verlag, Berlin, 87-106. Dubois D., Lang J., Prade H. (1991b) Fuzzy sets in approximate reasoning — Part 2 : Logical approaches. Fuzzy Sets and Systems, 40,203-244. Dubois D., Martin-Clouaire R., Prade H. (1988) Practical computing in fuzzy logic. In : Fuzzy Computing (M,M. Gupta, T. Yamakawa, eds.). North-Holland, Amsterdam, 11-34. Dubois D., Prade H. (1982) Towards the analysis and the synthesis of fuzzy mappings. In : Fuzzy Sets and Possibility Theory — Recent Developments (R.R. Yager, ed.), Pergamon Press, New York, 316-326. Dubois D., Prade H. (1991a) Fuzzy sets in approximate reasoning — Part 1 : Inference with possibility distributions. Fuzzy Sets and Systems, 40, 143-202. Dubois D., Prade H. (1991b) A note on the validation of possibilistic knowledge bases. BUSEFAL (IRIT, Univ. P. Sabatier, Toulouse, France), 48, 114-116.
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49 Dubois D., Prade H. (1992) Gradual inference rules in approximate reasoning. Information Sciences, 61(1,2), 103-122. Hall L.O., Friedman M., Kandel A. (1988) On the validation and testing of fuzzy expert systems. IEEE Trans, on Systems, Man and Cybernetics, 18, 1023-1028. Kinki61cl6 D. (1992) Detection des incoherences potentielles dans les bases de connaissances floues : vers un modele conceptuel. Presented at "leres Rencontres Nationales des Jeunes Chercheurs en Intelligence Artificiclle", Rcnnes, France, Sept. 7-9. Larsen H.L., Nonfjall H. (1989) Modeling in the design of a KBS validation system. Proc. of the 3rd Inter. Fuzzy Systems Assoc. (IPSA) Congress, Seattle, Aug. 6-11, 341-344. Larsen H.L., Nonfjall H. (1991) Modeling in the design of a KBS validation system. Int. J. of Intelligent Systems, 6, 759-775 ; Erratum : 7, 1992, p. 391. Loiseau S. (1992) Refinement of knowledge bases based on consistency. Proc. of the 10th Europ. Conf. on Artificial Intelligence (ECAr92) (B. Neumann, ed.), Vienna, Austria, Aug. 3-7, 845-849. Meseguer P. (1991) Verification of multi-level rule-based expert systems. Proc. of the 9th National Conf. on Artificial Intelligence (AAAI'91), July 14-19, 323-328. Nguyen T.A., Perkins W.A., Laffery T.J., Pecora D. (1985) Checking an expert systems knowledge base for consistency and completeness. Proc. of the 9th Inter. Joint Conf. on Aruficial Intelligence (IJCAI'SS), Los Angeles, CA, 375-378. Nguyen T.A., Perkins W.A., Laffery T.J., Pecora D. (1987) Knowledge base verification. AI Magazine, 8(2), 69-75. Turksen I.B., Wang Q. (1992) Consistency of fuzzy expert systems with interval-valued fuzzy sets. Proc. of the Abstracts and Summaries of the 1st Inter. Conf. on Fuzzy Theory and Technology (FT&T'92) (P.P. Wang, ed.), 225-230. Yager R.R., Larsen H.L. (1991) On discovering potential inconsistencies in validating uncertain knowledge bases by reflecting on the input. IEEE Trans, on Systems, Man and Cybernetics, 21,790-801. Zadeh L.A. (1979) A theory of approximate reasoning. In : Machine Intelligence, Vol. 9 (J.E. Hayes, D. Michie. L.I. Mikulich, eds.), Elsevier, New York, 149-194.
CHAPTER 3
SOFTWARE REPRESENTATION OF FUZZY SETS AND LOGIC Sozo Yamamoto Central Research Laboratories Matsushita Electric Industrial Co., Ltd. 3-1-1, Yagomo-Nakatnachi, Moriguchi, Osaka 570, Japan
Abstract Problems and their solutions in representing fuzzy sets and logic in software systems are discussed in this article. Fuzzy set theory is getting to be widely used as a tool for managing uncertainty in complicated systems. Interactions of fuzzy set theory and information processing is called 'fuzzy information processing', where software representation of fuzzy sets and logic is an important subject. Fuzzy information processing is an important area of research but is not fully investigated. This is because it has some problems. For one thing, a fuzzy set can be represented with various kinds of complicated d a t a structures. Another problem is that there are effectively infinite number of operations defined on fuzzy sets. Some fuzzy logic based systems have been proposed, like fuzzifications of Prolog, fuzzy control shells, and specially designed languages for fuzzy set processing. But they are not fully acceptable as a uniform platform of fuzzy information processing. T h e trade-off of flexibility, convenience and performance remains. Object-orientation can be a key to solve these problems. Because objectorientation has the ability of d a t a abstraction and information hiding, it is suitable for fuzzy information processing which needs manipulation on complicated data structures. An object-oriented fuzzy set manipulation system named F O P S was developed on such ideas. Two basic classes for fuzzy sets, ArrayedFuzzySet and PairedFuzzySet, are provided and they can be used interchangeably. W i t h its support for fuzzy logic and development environment, F O P S can serve as a good starting point of fuzzy logic based software. Outline of the system and internal d a t a structures are discussed in this article. 51 S.G. Tzafeslas and A.N. Venelsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 51-68. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.
52
1.
Introduction
Great progresses have been made in the application of fuzzy set theory and fuzzy logic. Most remarkable area of application is 'fuzzy control', where fuzzy logic was first applied to plant control systems and its use is expanding to consumer products. Most of fuzzy control systems uses fuzzy inference with max-min or max-product composition, similar to the algorithm that first used by Mamdani in 1970s. Some algorithms are developed to refine fuzzy controls systems but the main part of algorithm stays the same. Triggered by the success of fuzzy control systems, other ways of applying fuzzy set theory are also investigated. They are usually referred t o as 'fuzzy expert syst e m s ' , and their purpose are to combine the idea of fuzzy theory with AI based approach toward knowledge processing. These approaches can be more generally viewed as 'fuzzy information processing', t h a t is to bring fuzzy idea into information processing systems. These systems have variety of algorithm and each of t h e m are built on some software basis like Lisp or C language. But if a system is based on fuzzy set theory or fuzzy logic, the system must have some part t h a t is implementing fuzzy set representation and fuzzy logic. So there will be a need for a concrete basis for fuzzy information processing, which enables users to treat fuzzy sets or fuzzy concepts in the same manner with other components without implementing software for processing fuzzy set. Notations of floating point numbers like '1.23E4' or '5.6e7' are understood by most of common computer languages. Likewise, if fuzzy sets can be treated conveniently in programming languages, it will help software development and researches related to fuzzy logic. In this article, methods of representing fuzzy sets in software are discussed and a brief survey is made on software systems for fuzzy knowledge processing and fuzzy set representation. After t h a t , F O P S , which is an object-oriented software for fuzzy set manipulation, is introduced to show the effectiveness of object-orientation for fuzzy set processing and fuzzy information processing.
2.
Fuzzy Sets and Logic in Software
Fuzzy sets can be described in various kinds of d a t a structures. A fuzzy set is basically a mapping from some universe of discourse to unit interval [0,1]. A fuzzy set A on an universe of discourse U can be defined as, A^
I fiA{u)/u
(1)
where pL^ : U -^ [0,1] is the membership function of fuzzy set A. So a fuzzy set is expressed with a shape in two-dimentional space with grade and universe axis, which typically is triangular, trapezoidal, or bell-shaped. To describe this information, there are several ways.
53
Fuzzy sets in functions(Fig. l-(a)) One simple way is to describe the fuzzy set in a function which express the membership function fi^. For example, a membership function for bell-shaped fuzzy set can be expressed as 1 + p{u — ay This method is achieved with less memory than other ways of representation, but is less general because the result of operation can not necessarily be represented in the same manner. Pointwise Definition(Fig. l-(b)) Another approach is to describe a fuzzy set with a polygonal line, giving ea-"
expression fuzzy set expression
"*Vi)—
empt'i \
•
grade — ^ v / y ~ ^ elements
-
© AFS A Pf S
(a) Fuzzy Set Operation
(b) Extension Principle
(c) Fuzzy Relation Figure 5: Comparisons of Operation Speed Most evident difference is in the result of compose: operation. It is the composition operation between unary and binary fuzzy relations. Relational operations can be treated as matrix operation, so arrayed representation scores higher speed. In Figure 5-(c), PF5 is 10 times faster than PFS at 32 elements, and this difference is greater at more number of elements. This result may not be fair to PFS, for it is assumed that all elements in universe of discourse have non-zero membership. In some cases, some elements in universe of discourse have zero for grade of membership and can be omitted from PFS. Amount of memory and computation depends on the number of elements with non-zero membership, where it depend on the whole number of elements in universe of discourse for PairedFuzzySet. So PFS gives better performance if large portion of elements have zero membership.
67
7.
Conclusion
Software for fuzzy information processing is a important but not sufficiently investigated area of research. Its need is indicated with the fact that each system that is based on fuzzy set theory is implementing diiferent set of codes for fuzzy set processing. Some systems have been developed, but most of fuzzy set based software are still built on conventional computer languages. It is a natural way to treat fuzzy sets as d a t a types, as with other components in software. Fuzzy set manipulation system F O P S was aimed to utilize the merits of object-oriented paradigm for fuzzy set manipulation. By integrating two different types of fuzzy set representation and treating fuzzy sets as numbers, treatment of fuzzy sets in programming language became much clearer. Two fuzzy set representations, ArrayedFuzzySet and PairedFuzzySet, are implemented and interchangeably used in F O P S . These two have different characteristics in memory usage and computation load, so users can choose appropriate d a t a structure for a given application. It is expected that such fuzzy set representations are standardized into some form and get to be included in many computer languages, to promote researches in fuzzy information processing or fuzzy knowledge processing. This work was performed in part at the Laboratory for International Fuzzy Engineering Research(LIFE). The author wishes to thank Dr. Seiji Yasunobu and Mr. Yoshifumi Inoue for their collaboration at LIFE and Professor Umano of Osaka University for helpful discussions.
References [1] L. A. Zadeh: " P R U F - A Meaning Representation Language for Natural Languages", Intl. J. of Man-Machine Studies, 10, 395-460(1978). [2] J. M. Adamo: "L.P.L. A Fuzzy Programming Language: 1. Syntactic Aspects", Fuzzy Sets and Systems, Vol.3, 151 - 179(1980). [3] J. M. Adamo: "L.P.L. A Fuzzy Programming Language: 2. Semantic Aspects ", Fuzzy Sets and Systems, Vol.3, 261 - 289(1980). [4] T.P. Martin, J . F . Baldwin, B.W. Pilsworth: "The Implementation of F P R O L O G - A Fuzzy Prolog Interpreter", Fuzzy Sets and Systems, Vol. 23, 1 1 9 - 129(1987). [5] M. Umano: 753(1987).
"Fuzzy-Set Prolog", Preprints of 2nd IFSA Congress, 750-
[6] W. Siler: " F L O P S : A Fuzzy Expert System Shell", Preprints of Second IFSA Congress, 848-850(1987).
68
[7] J. J. Buckley, W. Siler: "Managing Uncertainty in a Fuzzy Expert System. Part 1: Combining Uncertainties", Preprints of Second IFSA Congress, 737739(1987). [8] W. Siler, J. J. Buckley: "Managing Uncertainty in a Fuzzy Expert System. Part 2: Truth Maintenance System", Preprints of Second IFSA Congress, 744-746(1987). [9] K. S. Leung, W. Lam: "Fuzzy Concepts in Expert Systems", Computer Magazine, IEEE, Vol. 21-9, 43-58(1988). 10] J. Teichrow, E. Horskotte, M. Togai: "The Fuzzy-C Compiler: A Software Tool for Producing Portable Fuzzy Expert Systems", Proc. 3rd IFSA Congress, Intl. Fuzzy Systems Association, 708-711(1989). 11] M. Umano, M. Mizumoto, K. Tanaka: "FSTDS System: A Fuzzy-Set Manipulation System", Information Sciences, 14, 115-159(1978). 12] M. Umano: "Fuzzy-Set Manipulation System in Lisp", Preprints of 2nd IFSA Congress, 840-843(1987). 13] Z. A. Sosnowski: "FLISP - A Language for Processing Fuzzy Data", Fuzzy Sets and Systems, Vol.37, 23 - 32(1990). 14] Z. A. Sosnowski: "Data Structures for Representing and Processing of Fuzzy Information in Lisp", Computers and Artificial Intelligence, Vol. 10, No. 6, 561-571(1991). 15] A. Goldberg, D. Robson: "Smalltalk-80: The Language", Addison Wesley(1989). 16] S. Yamamoto, Y. Inoue, S. Yasunobu: "Object-Oriented Fuzzy Set Manipulation - Internal Data Structures -", IFSA '91 Brussels, AL218-221(1991).
CHAPTER 4
FUZZY DYNAMIC SYSTEMS: ANALYSIS, CONTROL AND IDENTIFICATION
A.N.Venetsanopoulos(*) S.G.Tzafestas x«7, x^z ii) X « (xTy) < y iii) X « (x^y) > y For Zadeh 's T-norm i.e. the operator "min" the corresponing "-operator is Operator «: a«b=l or b if asb or a>b The operator « can be applied both between two fuzzy sets X and Y, and between a fuzzy set y and a fuzzy relation R. In the first case:
In the second case: R'-Y : Hji^ri^)
= min(;i^(x,y)«/iy(y))
We are now ready to formulate the above measurement and synthesis problems. Measurement Problem: The measurement problem of a fuzzy system is stated as follows: Given the system R and its output Ycompute the input such that Y=XoR. This problem is not exactly solvable in the sense that from the pair (Y, R) one can compute only the largest fuzzy set Xsuch that Y=XoR. This does not preclude to exist other fuzzy inputs X' smaller than X, i.e. X's:X that satisfy the relation Y=X'oR. The largest fuzzy input X is computed from the pair {R,Y) with the aid of the operator «. More specifically X is given by X = R^Y
V fixix)
= min(/iR(Ar,7)«jUy(7))
Synthesis Problem: The synthesis problem is: Given the fuzzy sets X (input) and Y (output) compute the fuzzy relation R that describes the fuzzy system. This problem is also not exactly solvable in the sense that from the pair {X,Y) one can derive only the largest relation R that satisfies the equality Y=XoR. This does not preclude the existence of other smaller relations R' i.e. R'^R for which the relation Y=XoR' holds. The largest fuzzy relation R is computed from (X,Y) through the relation: /? = X « y
or tijt(x,y)
= ^^(;(r)«/iy(y)
73 4. FUZZY DYNAMIC SYSTEMS As we have already seen, fuzzy systems are described by fuzzy relational equations. Our purporse here is to introduce the concept of fuzzy state equations on the basis of fuzzy relational equations. We shall describe fuzzy systems of first and higher order, and discuss the relation of relational equations with the classical difference equations. 4.1. State Space Models of Fuzzy Systems TTie concept of state space possesses a central position in modern control theory where the state equations are derived using the classical models of differential and difference equations. Here we shall borrow from the classical state space methodology to develop and apply the concept of fuzzy state space. Let X, U and V be discrete reference supersets for the state, the control and the output variables of the system, respectively. For convenience the corresponding fuzzy state, control and output sets are assumed to be finite, i.e X = {xp X2, ..., Xjj} fuzzy state set U = {i/j, U2, ..., u^} fuzzy control set ^ = {/!« Vi' •••' yq} f"2zy output set Then, the state space description of a dynamic fuzzy system via the use of relational equations is:
^k*p =
^k*p°^
where o: Zadeh 's compositional rule of inference t/j^: fuzzy control variable at time instant k ^k+p' ^k+p-i'"^['"i"(/^A;,(^)'M/?(""^'>'))])j The pth-order fuzzy state space model can be converted to a set of Ist-order equations by setting
R' -
^k.\'^k.2'-'^k.p-l'R
^ k*p = ^A+p
s' = s where R' is the fuzzy relation R':UxXx...X-^[0,l] and S' the fuzzy relation 5':Xxy-*[0,l]. The resulting Ist-order fuzzy model is:
Thus one can work with a fuzzy state-space of 1st order without any loss of generality. 4.3 Relation with Difference Equations Finite difference equations constitute a very useful and extensively applicable class of equtaions for the description of discrete dynamic systems and the solution of differential equations. It can be shown [3] that the relational equations are a direct extension of difference equations, when the input and state variables do not take particular numerical values but they constitute fuzzy sets. To see how this is done consider the difference equation
where a, b, Xj^, Xj^^j and Uj^ take real values. These values can be regarded as fuzzy numbers with the following membership functions: M[/ (") = 1
if U = Uj^;
=0
otherwise
i"x (^) ~ ^
if X = Xj^;
=0
otherwise
if y = ax+bu;
=0
otherwise
Hjf{x,y,u)
= 1
Thus if we consider the quantities t/^, X^, X^ +j and R to be fuzzy ones, one can form the relational equation:
75
which in terms of membership functions is written as: MXk^,(y) = max|min|Ai^(u),;i;^i^(x),^^(Y,y,u))| Introducing the respective substitutions we find that here MA" (y) ^ ^ if y = aXj^ + bU]^ ; =0 otherwise The above veriiy (at least in this example) that the relational equations constitute an extension of the usual concept of difference equations when the variables involved are fuzzy sets. 5. CONTROL OF FUZZY DYNAMIC SYSTEMS Let us now consider a first-order fuzzy system Xi^= Uf^oXfOR which has the block diagram shown in Fig. 1. For this system we shall discuss here three particular fuzzy control problems, namely: i) One step control without or with constraints. ii) Multistep control. iii) Construction of the controller relational equation. 5.1 One-Step Control No constraint case: This is the simplest possible case and is formulated as follows. Given: The fuzzy system Xi^^.i=Uj^oX^^oR and a final state Xf, Find: The control Uf^ which brings the system in one step from the present state X;t to the final Xf. This problem is equivalent to solving the relational equation Xf= U^oX^^oR with respect to Uj^. Writing this equation as Xf= UjpEj^ where E^ = Xi^oR one can solve it with the aid of the « operator. The solution is U\
=
E,^X,^{X,OR)«X,
Fig. 1 Block diagram of a fuzzy system of order 1
76
where t/j^'is the largest from all possible controls (if they exist) that satisfy the equation Xj.= 140X1,0/?. If a solution does not exist, i.e. if the abvoe solution gives to some point of l ^ ' negative membership value, then one can use some distance criterion and try to find a control Uj^ that brings the quantity L/^oXj^o/? as near as possible (with respect to this criterion) to the desired final state X/. In this case the problem is to find a Uj^' such that the fuzzy distance d{Uj^oX^^oR, Xf) is minimized, i.e. U'\
: d{U"j^'>x^cR,Xf)
= min cy( l/j^oX^oR,X/.)
where e.g. d may be Hamming fuzzy distance (See Ch. 1, Sec. 6.3). A measure of "how good" is the control U^.' is provided by the concept which is defined as follows.
e-reachability
Definition 5.1 Let e{Uf^") be the quantity
e{U'\)
=
diU'\'>x^oR,Xf)/supd{U^oX^''R,Xf)
where d is the Hamming fuzzy distance. Then X^is said to be EQ-reachable from Xj^, if e{Ui^")<Eg. Clearly, if e^=0, then the state Xf is exactly reachable from Xj^. Constraint
case: Tliis problem is still open.
5.2. Multi-step Control This problem is formulated as follows (the no-constraint case is assumed). Given: The Ist-order system Xj^.^ 1=140X^^0/?, with initial state X^, and a final state Xp Find: The control sequence U^, U^, ..., C/„ j that brings the system (in p steps) from the initial state X^ to the final state Xf. To solve this problem we apply respectively the system equation starting from X^, namely: Xi =
U^oX^oR
X j = t/joXjoft
=
Ui'>(U^oX^oR)oR
= t;,ot;„=x„oj?2
Xf=Xp= Setting U= U ^oU
2°-°^i°^o
Up,,oUp_2^...oUioU^oX^oRP ' " "^^ '^^^ equation we get
77 Xf=
Uo(XgoRP)
which is solved for U as in the one-step control, i.e. U' = {X„oRP)o.Xf 5.3. Construction of the Controller Equation The system under control: Xj^+j =1^0X^0/? has the block diagram of Fig. 1. If the controller 's equation is L'j,=X|.oG, then it has the block diagram of Fig.2.
X,
U,
Fig. 2 Block diagram of the controller G The closed-loop system has the relational equation (Fig. 3):
=
X^^X^^RoG
The control problem for the closed-loop system is to find the fuz2y relation G of the controller such that one can apply to it (i) single step and (ii) multi-step control.
Fig. 3 Block diagram of fuzzy closed-loop system
78
Clearly, the closed-loop equation can be written as Xj^^^ = CoG which has exactly the same form as the equation Xf= Uj^oE^ of the single step control. Thus, the one step solution is readily applicable here too.
6. IDENTIFICATION OF FUZZY DYNAMIC SYSTEMS The identification problem of fuzzy systems consists in estimating the relation R that describes the system, from fuzzy input- output pairs (X,, V,), / = 1,2,..., N. Here this problem will be treated in the no-noise case. The case where some kind of noise is present needs the introduction of two additional concepts, namely the probabilistic sets and the fuzzy clustering, and will not be considered here. 6.1. Estimation of the Fuzzy Relation R Consider a fuzzy system described by the equation Y=XoR, where X, Y, R are assumed finite, i.e.
with respective membership functions /^x = >[^x(^/)]. A'' y =' [^• r(y,)j. P R• = [f^Ri^vYi)]'' • • ' '
. , ' ""'
j=i,z,...,m
Using these membership functions the system equation is written as Fy(7j) = max|min(/j;y(.v,),^;j(;yy,7;))| Then the problem of estimating R is formally stated as follows: Given: The fuzzy vectors \}i)^x>i\, lJUy(y,)]. Find: The fuzzy matrix [}iji{Xj, y)]. The information contained in the input - output pairs (X,-, Vj), i= \,2,...,N is not sufficient for the unique determination of the relation /?, but only for the determination of a lower and an upper bound {R, R respectively) such that
These bounds can be found by using the operators « and « of Sanchez [4]. Theorem 6.1 _ An upper bound R of R is given by the formula:
f^ni^i'yj)
= Mx(^i)"i"y(y,)
1.
MA-C-*',)
^ t^yiyj)
79 Theorem 6.2 A lower bound 5 of R is given by
^Ri^i^yj)
=
i^xi^d'^i^Yiyj)
• P^Y(yj)^
0,
MxC^i) > M y ( 7 j ) >
^f^x(^k)
otherwise
The property K s R c ^ implies that
The upper bound /? satisfies the relational equation Yj=XjOR, which is not satisfied by the lower bound, i.e. XpR^Yj. An algorithm for the estimation of the fuzzy relation R (i.e. an upper and a lower bound for it) is the following: Initialization: Put UpperR*-\, LowerR*0 Counter fc /z s = l - n 7 i Ai Xi(xi)> i=l i=l ;"Xi(xi):xi-^[0.1]. Xi€Xi(i=l,2,...,n)
(10)
J"R. J" I. A^P. A /R
—I—.—S.HEIGHT 350 ICB)
0
VUWiWlO'
D.n.4 0 i D i l . D
ECONOMY
FUNCTION
U 10'
>»-NECE n
MO^cin) "
• D-COLM 0 (on)
(cm)
(oml
10'
58 60 62
60
16
-^N-ALL "
0 "
5
0
-^PERIOD 0
'(YEAR)
Fig.34 Inputs for Structural Optimization
"
UMiU\Ji'
SAFEH
116 111 /i,
n*
lo'l-
(4) Uai. "WJ-STATE3 10 !
srATE2
r
(7) -"A-IMI STATES
10-
• QR-WALL 0 (21 U
SrATE9
1.1
1.?
(5) -"jn
1.5
STATE5
(31 -"jf-aui STATE!
(6) U:^ CTITCR M l « STATE6
10
0 50
(9) A j m 10^
.OR-com „ (t)
»A-WALL
'^' ^i-tEsiw STATEIQ 10
>ACC 800 (gall
200
I10)/4i-i«.l STATE4
10-
60
" "
0.5
^ - « L L STATEll
*-DESICNo (xlO'cn)
^N-*ALL 7 8
9 10
STATE?
»DAMAGE 0 0.3
0
N-WALL
0
N-WALL 5
10
Fig.35 Calculated Variables in Structural Optimization \**^^ I 0 •ECONOMY
0
, FUNCTION SAFETY
N-WALL
5
»N
13
Fig.36 Decision of Optimal Number of Shear Wall Units
ISAFETYI lECONOMYl IFUNCTIONI iSTATEfill tSTATEQi llSTATEIflJ
...... I
I INPUTS
STATE J
i>s««isi iiinaiii nffiemi
Fig.37 Transfonncd Fuzzy Hierarchy for Structural Evaluation
117
/^
^ 1
^WEIGHT (t)
2200
EiGiir
•com
lO'
lO'
10'
I nl—•—L 22
J—.—>D-COLM 60 (cm)
.N-COLM 0
10
,
.HEIGHT
350
/^PERIOD
•lAlL
ID-
0
LO-
I -^t-WALL (cnl
-*L-lfALL 0 (cm)
^.N-HIALL 0
0
•PERIOD (YEAR)
^.
58 50 62
.N-ALL
^IK-NECE 0 (xiO^cnl
20
Fig.38 Inputs for Structural Evaluation U
0.5
SAFETY -> 0
FUNCTION
*
ECONOMY
0
1.0 0.5 1.0 0.5 Fig.39 Evaluation of Economy, Safety and Function
4. Discussions To identify states (i.e., fuzzy relations) in fuzzy systems, in this chapter, neural network, fuzzy confluence rule and fuzzy identifier are proposed. We have to pay much attention to the problem that each of the models has its own merits and demerits. Neural network is available for any type of functions, but it is a "black box". Fuzzy confluence rule is effective for human judgements, and it is not a "black box". Fuzzy identifier is appropriate for engineering problems in which design equations and functions are given. Anyway, it is desirable for us to have many alternatives for such an identification. In this real world, almost events and judgements may be transformed into fuzzy networks as shown in Fig.21, where some variables and states are given and others are not given. The calculation procedure of states as shown in Fig.31 can be called "learning", and the calculation procedure of variables as shown in Figs.30, 33 and 37 can be called "inference". By means of local identifications of states in fuzzy systems and/or macroscopic leaning of state in fuzzy networks, every state in fuzzy networks can be determined, which enables iis to infer unknown variables. When these learning and
inference are performed automatically, fuzzy networks can be called "intelligent fuzzy networks", or "ncuro-fuzzy networks". 5.C0NCLUSI0NS In this chapter, a paradigm of intelligent fuzzy networks is proposed especially for engineering planning and design. Its main features are described as follows: (1) The proposed fuzzy network is composed of fuzzy variables and fuzzy states (constraints). (2) By learning given knowledges, not only neural network but also fuzzy confluence rule and multi-variable identifier (quasi-neural networks proposed by the authors) become available for the fuzzy states. (3) By employing intelligent object-oriented systems, engineering evaluations and/or optimization can be performed in the same manner. (4) The proposed intelligent fuzzy network is a fusion of fuzzy systems and neural networks. Acknowledgement: The parts of this research are performed under the financial supports by Grant-in-Aid for Scientific Research (C) (The Ministry of Education, Science and Culture, Japanese Government), Kansai Electric Power Co. Ltd., Newjec Co. Ltd., and the Secom Science and Technology Foundation. The authors are grateful to all the graduate students who coworked in the authors' laboratory. 6. References [1] Bcllman,R.E. and Zadeh,L.A., "Decision-Making in a Fuzzy Environment," Management Science, 17, 4, pp.141-164: Dec. 1970. [2] Cui,X., Yamada,M., Kawamura,H., and Tani,A., "Support System for Selecting Structural Planning Data of R/C Multi-Story Frames with Cantilever-Type Shear Walls," Extended Abstracts of the IV-ICCCBE'91 Cconference, Tokyo, Japan, pp.263: July 1991. [3] Kawamura,H., Tani,A., Kawamura,M., Matsumoto,S., and Yamada,M., "A General Formulation of the Confluence Rule of Fuzzy Goal and Constraint and its NonNumerical Maximization," Proc., 3rd Fuzzy System Symposium, Osaka, Japan, pp.71-76: June 1987. (in Japanese) [4] Kawamura,H., Tani,A., Yamamoto,Y., and Yamada,M., "Application of Fuzzy Confluence Rule to Subjective Evaluations in Structural Design," lintemational Workshop on Fuzzy System Applications, lizuka, Japan, pp.185-186: Aug. 1988. [5] Kawamura,H., and Yao,J.T.P., "Application of Fuzzy Systems Based on Conditioned Fuzzy Sets to Structural Engineering," Journal of Structural Engineering, Vol.32B, pp.51-56: March 1990. (in Japanese) [6] Kawamura,H., Tani,A., Yamamoto,K., and Yamada,M., "Constitution of Intelligent Fuzzy Network by Frame Knowledge Representation," Proc, International Conference on Fuzzy Logic & Neural Networks, Vol.1, lizuka, Japan, pp.261-265: July 1990.
This page intentionally blank
119
[7] Kawamura,H., and Tani,A-, "Multi-variable Fuzzy Identifier," Proc, 6th Fuzzy System Symposium, Tokyo, Japan, pp.179-182: Sept.1990. (in Japanese) [8] Kawamura,H., Tani,A., Kambara,H., and Yamada,M., "Intelligent Fuzzy Network for Optimum Structural Planning and Design," Proc., 7th Fuzzy System Symposium, Nagoya, Japan, pp.99-102: June 1991. (in Japanese) [9] Kawamura,H., Tani,A., and Kambara,H., "Aseismic Structural Planning System by Fuzzy Network," Proc. 10th World Conference on Earthquake Engineering, Vol.10, Madrid, Spain, pp.6271-6275: July 1992. [10] Kawamura,H., Tani,A., "A Paradigm of Intelligent Fuzzy Networks," Proc, The IEEE International Conference on Systems Engineering, Kobe, Japan, pp.159-164: Sept. 1992. [11] Papis,C.P. and Sugeno,M., "Fuzzy Relational Equations and the Inverse Problem," Fuzzy Sets and Systems, 15, pp.79-90: 1985. [12] Rosenblatt.F., "The Perccptron:A Probabilistic Model for Information Storage and Organization in the Brain," Psychological Review, Vol.65, No.6, pp.386-408: 1958. [13] Rumelhart.D.E., Hinton,G.E., and Williams,R.J., "Learning Representations by Back-propagation Errors," Nature, 323-9, pp.533-536: Oct. 1986. [14] Sanchez,E., "Resolution of Composite Fuzzy Relational Equations," Information and Control, 30, pp.38-48:1976. [15] Yamada,M., Kawamura,H., Tani,A., and Yamamoto,K., "A Determination Method of Hierarchy Models for Decision Making in Aseismic Structural Design by Fuzzy Confluence Rule," Proc., The 12th Symposium on Computer Technology of Information, Systems and Applications, Kyoto, Japan, pp.223-228: Dec. 1989. (in Japanese) [16] Yamada.M., Kawamura,H., and Tani,A., "Research on Quantitative Formulation and Objective Determination Methods of Multi-Objective Decision Making Process in Aseismic Structural Design (Comparison between Fuzzy Confluence Rule and Neural Network)," Proc. of Annual Meeting, Structural Division., Airchtectural Institute of Japan, Kinki Branch, pp.161-164: May 1991. (in Japanese) [17] Yao,J.T.P, Safety and Reliability of Existing Structures, Pitmann Publ. Inc., Boston, London, Melbourne: 1985. [18] Zadch,L.A.,"Fuzzy Sets," Information and Control, Vol.8, pp.338-353:1965. [19] Zadeh,L.A., "Toward a Theory of Fuzzy Systems," in Aspects of Network and System Theory, (Eds. Kalman,R.E. and DeClaris,N.), Holt, Rinehart and Winstcr, Inc.: 1971.
CHAPTER 6 N E U R O - F U Z Z Y E X P E R T SYSTEMS: O V E R V I E W WITH A C A S E S T U D Y
Sushniita Mitra and Sankar K. Pal Electronics and Communication Sciences Unit Indian Statistical Institute, Calcutta - 700 035, India
I.
Introduction
Artificial neural networks or connectionist models [1,2,3] are massively parallel interconnections of simple neurons that function as a collective system. They are designed perhaps as an attempt to emulate human performance and function intelligently. An advantage of neural nets lies in their high computation rate provided by massive parallelism, so that real-time processing of huge data sets becomes feasible with proper hardware. Information is encoded among the various connection weights in a distributed manner! .The multilayer perceptron (MLP) [2] is a feed-forward neural network model consisting of multiple layers of simple, sigmoid processing elements (nodes) or neurons. Alter a lowermost input layer there are usually any number of intermediate or hidden layers followed by an output layer at the top. The learning procedure has to determine the internal parameters of the hidden units based on its knowledge of the inputs and desired outputs. An expert system [4,5] is a computer program that functions in a narrow domain dealing with specialized knowledge generally possessed by human experts. Such programs are very useful due to the usual shortage of qualified human experts in real life. The primary characteristics of an expert system are a knowledge base designed with the help of a human expert, a narrow problem domain, and a performance on par with a human expert. The knowledge base is a problem-specific module containing information that controls inferencing. Traditional rule-based expert systems encode this information as If-Thtn rules while the connectionist expert system [6] uses the set of connection weights of a trained neural net model for this purpose. The inference engine is problem independent while the user interface links the external environment to the system. Connectionist expert systems are usually suitable in data-rich environment. They help in minimizing human interaction and associated inherent bias during the phase of knowledge base formation (which is time-consuming in case of traditional modeb) and also reduce the possibility of generating contradictory rules. The rule generation phase of such connectionist models are usually completely automated. An expert system is expected to be able to draw conclusions without seeing all possible external inforrfiation. It should be capable of directing the acquisition of new information in an efficient manner and also be able to justify a conclusion reached. The knowledge base is generally a major source of uncertain information. Besides, 121 S.G. Tzafestas and A.N. Venelsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 121-143. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.
122
inferencing based on partial information also involves some amount of uncertainty. The utility of fuzzy sets [7,8,9,10,11] lies in their capability in modelling uncertain or ambiguous data so often encountered in real life. Fuzzy expert systems [12], incorporating the concept of fuzzy sets at various stages, are a step towards handling such situations. There have been several attempts recently [13,14,15,16,17] i^ making ^ fusion of fuzzy logic and neural networks for better performance in decision making systenas. The uncertainties involved in the input description and output decision are taken care of by the concept of fuzzy sets while the neural net theory helps in generating the required decision regions. Fuzzy neural networks are used to design neuro-fuzzy expert systems. Such models help in combining the advantages of fuzzy neural nets with those of connectionist expert systems. The present article initially provides a brief account of the various types of uncertainties associated with traditional expert systems. The relevance of fuzzy expert systems is then considered in this context. Next, the utility of making a fusion of fuzzy logic with neural networks is discussed and its application in designing neuro-fuzzy expert systems reviewed. Finally, a fuzzy version of the MLP using a modified version of the conventional backpropagation algorithm to incorporate the logical operators And and Or (in place of the weighted sum and sigmoid functions^ is presented. The input vector (which can be in quantitative/linguistic/set form) is represented in terms of the linguistic properties low, medium and high while the output decision is in terms of class membership values [18]. Fig. 1 shows a three-layered network with multiple outputs. The conjugate pair of t-norm T and t-conorm S [19], viz. the product-probabilistic sum, are utilised to model the And and Or operations. The hidden layer consists of And nodes while the output layer is made up of Or nodes.
Output
OR
AND
Input Figure 1 :, A three-layered neural network implementation And and Or logic functions at successive layers.
123
This fuzzy logical model broadly performs two main tasks. First we construct the three-layered fuzzy logical network for classifying multi-class patterns. Then, the trained network is used to generate rules. The connection weights in this stage constitute the knowledge base (in embedded form) for the classification problem under consideration. The model is now capable of inferring the output decision for complete and/or partial inputs along with a certainty measure and querying the user for the more essential missing input information. If asked by the user, the model is capable of justifying its decision in If-Then rule form (in terms of the salient features) with the antecedent and consequent parts produced in linguistic and natural terms. The effectiveness of the model is demonstrated on a set of speech data.
II.
Traditional Expert Systems : Some Problems and Relevance of Fuzzy Sets
The major components of an expert system are the knowledge base, inference engine and user interface. The knowledge base contains expert-level information necessary to solve problems in a specific domain. This information is generally represented in the form of a set of rules, although frames, semantic nets [20] and belief networks [21] are also in vogue. We shall consider rule-based systems in this discussion. Knowledge bases, being domain-specific, are nontransferable. The inference engine interacts both with the knowledge base and a working memory (that records facts about the current problem and is updated with the availability of new information). Pattern matching occurs between the rules in the knowledge base and the facts in the working memory to select the relevant rules applicable. Note that when no matching occurs, no rule is selected, whereas when multiple rules apply, conflict resolution strategies are used to select the most specific one. The same inference engine can be used with different knowledge bases. The expert system should be able to justify its reaLSoning since in some domains, like medicine, a doctor must accept the -ultimate responsibility for a diagnosis even if it was arrived at with considerable help from a program.
A.
Involvement of uncertainties
However the knowledge base itself b a major source of uncertain information [5] in expert systems, the causes being : • unreliable information t imprecise descriptive languages • inferencing with incomplete information • poor combination of knowledge from different experts Firstly, ill-defined domain concepts or inaccurate data result in unreliable information leading to weak correlation between a rule's premise and its conclusion. Mycin (22] uses numeric certainty factors to quantify the degree of this correlation. Secondly, the numerous ambiguities in natural language often result in imprecise
124
expression of rules in formal language. Therefore the meaning of the facts have to be approximately matched with those of the premises. Thirdly, when the available information is incomplete the system accepts the value unknown while evaluating the premise's degree of certainty during the approximate pattern matching. Finally, generation of a consensus knowledge base by combining the views of multiple experts is often difficult, especially when the experts have contradictory viewpoints. Moreover, problems occur as human beings generally do not have uniform levels of expertise throughout a domain. Some conflict resolution strategies need to be followed, like separately weighting the knowledge of each expert, to calculate a composite conclusion. Good expert systems are expected to be capable of handling uncertainty, as in most cases the data are inherently inexact, incomplete or unmeasurable. Some of the existing methods in this direction include the subjective probability theory, Dempster-Shafer theory [23], possibility theory [24] and certainty factors [22]. Using the subjective Bayesian method, Prospector [25] has been designed to provide advice on mineral exploration. Mycin is a rule-based expert system (using certainty factors) which attempts to reconmiend appropriate therapies for patients with bacterial infections. It interacts with the physician to acquire the clinical data it needs. The associated Teiresias system enables the doctors to interact with Mycin to cisk it questions about its recisoning and to modify and augment its knowledge base. The knowledge base of an expert system contains human knowledge, most of which is imprecise and qualitative. To describe situations where the boundary between competing hypotheses is vaguely defined, human experts use terms such as very likely, likely, more or less likely, low, medium, high, etc. Encoding this sort of expertise by probabilities causes this fuzziness or imprecision to be lost. Fuzzy logic may be used to express these vague terms with precision and accuracy. Besides, facts about the world are rarely known with certainty. Conventional rulebased systems, with two-valued logic, usually evade this issue of partial matching.
B.
The role of fuzzy logic
Generally linguistic variables are used in the representation of the experts' knowledge or rules. Such representation schemes enable a knowledge engineer to capture the essence of the experts' experience and judgement without attempting to overquantify intuition. The importance of fuzzy logic to the management of uncertainty in expert systems lies in its abilitj' for dealing with fuzzy quantifiers and modifiers. Fuzzy logical systems allow a proposition or conclusion to range over fuzzy subsets (like very true, more or less true, likely true, etc.) of truth-value sets'characterised by their possibility distributions. Besides, the predicates may be crisp or fuzzy. Fuzzy modifiers like not, very, more or less, extremely, slightly, much, a little, etc. can also be represented. A fuzzy certainty factor is associated with the conclusion to analyse the transmission and cumulation of uncertainty from the premises to the conclusion. Deduction of conclusions from observations and rules in the knowledge base are made using either truth value restriction or compositional rule of inference. Hence, partial match can occur between the antecedent of a rule and a fcict supplied by the user. In short, fuzzy logic or approximate reasoning [12] provides a natural conceptual framework for knowledge representation and inferencing from knowledge bases that are imprecise, incomplete or not totally reliable. The advantage of using fuzzy reasoning is that it can yield an approximate answer even when mathematical
125
theories are not applicable, as the latter often require idealized assumptions such as the independence of evidence and the mutual exclusiveness and exhaustiveness of hypotheses. Some approaches in fuzzy inferencing for traditional expert systems include the approximate analogical reasoning based on similarity measures by Turksen and Zhong [26], the problem-reduction method of Ishizuka et al [27], modelling of physician decision processes by Esogbue and Elder [28], inferencing in the framework of inflammatory protein variations by Sanchez and Bairtolin [29] fusing weighting) and the extension to Clips for manipulating fuzzy facts and rules by Sosnowski [30].
C.
A few other problems
However, the various uncertainty management schemes of traditional expert models share some common problems. A willing human expert able to accurately quantify expertise is needed. The transfer of the knowledge takes place gradually through many interviews between the expert and the system and is therefore very time consuming. Usually humans are prone to be easily biased and thus the quality of knowledge extracted from the experts depends greatly on the methods used for assessment. Moreover, large knowledge bases need to be searched quickly and it is also very important to check that this knowledge base remains consistent as more information is accummulated. It would be welcome if knowledge assessment could be automated by freeing it from human intervention, thereby avoiding human bias and subjectivity.
III.
Connectionist Expert Systems
Expert systems having connectionist networks as their knowledge bases are called connectionist expert systems [6]. Powerful learning techniques exist for generating connectionist networks from training samples. This enables us to automate the construction of knowledge bases for classification-type expert systems. It is worth mentioning that the most difficult, time-consuming and expensive task in building an expert system is constructing and debugging its knowledge base. In practice the knowledge base construction can be said to be the only real task in building an expert system, given the proliferating presence of expert shells. Several approaches have been explored for easing this knowledge-acquisition bottleneck. Connectionist expert systems offer an alternative approach both to the knowledge base construction as well as the inferencing phase, providing interaction with the user accompanied by justification(s) of the conclusion(s) reached. Rules are not required to be supplied by humans. Instead, the connection weights encode among themselves, in a distributed fashion, the information conveyed by the input-output combinations of the training set. Such models are especially suitable in data-rich environments and enable human intervention to be minimized. Thereby the problems faced by traditional expert systems regarding the difficulties in normalizing across different experts' scales, conversion from human expressions to numerical terms, bias of the expert(s), generation of contradictory rules by the experts, etc., may be overcome. The use of the learning technique of connectionist networks enables the model to extract the information inherent in the data (that is not utilised
126
in the traditional models) and allows dynamical adjustments to changes in the environment. Besides, the various characteristics of neural nets, viz., generalization, tolerance to noise, graceful degradation at the border of the domain of expertise, ability to discover new relations between vziriables, etc., are in-built and hence can be utilised by the connectionist expert system model. In this article we restrict our discussion to classification-type expert systems. A set of environmental stimuli (features or symptoms) is clcissified into one or more of many possible categories (output classes or diseases). The output is expressed as continuous membership values to the various classes. This is suitable for handling data in fuzzy input feature space where a pattern may have finite belongingness to more than one class. In the medical domain this is analogous fo diagnosing a particular symptom set as affliction by more than one disease. Using fuzzy neural nets for this purpose, helps one to incorporate the advantages of approximate reasoning into the connectionist expert system. In this section we provide a discussion on the various existing neurofuzzy approaches to inferencing and expert system design. Sections IV and V present respectively a fuzzy logical version of the MLP and its application to automatic rule generation.
A.
Inferencing in neurofuzzy systems
There have been several attempts recently [13,14,15,16,17,31] in making a fusion of fuzzy logic and neural networks for better performance in decision making systems. The uncertainties involved in the input description and output decision are taken care of by the concept of fuzzy sets while the neural net theory helps in generating the required (linearly nonseparable) decision regions. Takagi [32j provides a review on related work in this direction. Fuzzy logic expresses the qualitative reasoning capability of humans and thereby enables easier formulation of substantially smaller number of rules (for the problem under consideration). The membership function that maps the relationship between quantitative data and fuzzy sets may be learned and adapted by training a neural network with real-life data suitable for the environment. The MLP-based approach by Keller'and Tahani [33] receives' the possibility distributions of the antecedent clauses at the input, uses a hidden layer to generate an internal representation of the relationship and finally produces the possibility distribution of the consequent at the output. The model is expected to function as an inference engine with each small sub-network learning the functional input-output relationship of a rule. Trapezoidal possibility distributions, sampled at discrete points, are used to represent fuzzy linguistic terms and modifiers. The network is supposed to be able to extrapolate to other inputs (for a rule) following modus ponens. Conjunctive antecedent clauses are also modelled using separate groups of hidden nodes for each clause. In [34], Keller et al explicitly encode the knowledge of each rule among the connection weights of the neural net. A measure of disagreement between the input possibility distribution and the antecedent clause distribution is used at the clause-checking and combination layers to determine the uncertainty in the consequent part of the fired rule. Theoretical properties of the variotis combination schemes are also investigated. Ishibuchi et al [35], on the other hand, use interval vectors to represent fuzzy input and output in a MLP. A modified version of the backpropagation algorithm is used to incorporate interval airithmetic. Different fuzzy If-Then rules are interpolated from a few sample rules (used during training).
127
Fuzzy inferencing has been used by Nakanishi and Takagi [36] for the recognition of noisy patterns (English alphabets distorted by ink marks). An alphabet is divided into n blocks X = {Xi,X^,...,X„), with each block being processed by a separate neural network. Each block is divided into m smaller subblocks such that the neural network NNj for block Xj has m inputs ^],^], • • •, ^T- Each NNj has the number of outputs equal to the number of patterns to be classified and each output is mapped into membership values to the three fuzzy sets zero, positive medium, and positive big. The fuzzy outputs from the n neural nets are then used to formulate the antecedents of fuzzy inference rules to recognise a noisy pattern when its typical feature is still visible in a small block of the irik-blofched pattern. An approach to fuzzy inferencing by Zhuang et al [37], for controlling a mathematically intractable system, uses a multilayer network with each node standing for a fuzzy subset of a linguistic variable. The truth values flow from the input nodes to the output nodes according to the rule of propagation being modelled. The interconnection weights are associated with ordered pairs of real numbers lying in the range [-1,1] and are indicative of the certainty factors used to represent the reliability of the fuzzy control rules. The basic operations in fuzzy logic, viz., union, intersection and negation, are implemented at the neuron levels. The linguistic variables used are positive large, positive medium, positive small, zero, negative small, negative medium and negative large. The role of fuzzy logic in the control of the activation, training, reliability and performance of neural networks has been investigated by Yen [38] using a hybrid architecture. Fuzzy rules are used to detect situations under which certain actions need to be invoked for neural network modules based on their performance measures. The networks process data obtained either from external sensor devices or from the knowledge base of the symbolic system. The global knowledge base consists of a fuzzy database and a neural network taxonomy (of neural net classes) that describes meta-level knowledge about the neural nets themselves (using crisp or fuzzy attribute values). The fuzzy database stores data and hypotheses that can be uncertain, imprecise or vague. A production system that takes into account the degree of partial matching of the fuzzy action rules thus enables the system to respond in a robust way even in the face of incomplete or noisy data. Masuoka et al [39] use knowledge in the form of membership functions and fuzzy rules (in And-Or form), extracted from experts, to build and preweight the structured neural network which is then tuned using selected learning data. This neural model consists of the input variable membership net, the rule net, and the output variable net. Modified fuzzy rules, extracted from the trained neural network using pruning, can then be evaluated and unsuitable rules corrected using relearning. Okada et al [40] use a similar approach to examine the bond rating of investors. Changes in the weights and threshold levels of the neurons (after training) are interpreted as adaptations in the membership functions and fuzzy rules. Yager [41] employs membership neural modules for the antecedents, inverse membership modules for the consequents and a rule neural module with a combiner (using min or product functions) for modelling the rules of fuzzy logic controllers. The various weights 2Lre learned and the importance of the antecedent clauses simulated. The neural network based fuzzy reasoning scheme by Takagi and Hayashi [42] is capable of learning the membership function of the If part and determining the amoimt of control in the Then part of the inference rules. The input data is clustered to find the best number of partitions corresponding to the number of inference rules applicable to the reasoning problem, a single neural net block
128
modelling one rule. The optimum number of cycles required is determined to avoid overlearmng and the minimal number of input variables selected for inferring the control values. A few of the other existing neurofuzzy approaches include the rule-based phoneme recognition method by Amano and Aritsuka [43] for Japanese words, Takahashi and Minami's technique [44] of modelling the subjective evaluation of humans, the linguistic fuzzy modelling scheme of Horikawa et al [45], the knowledge-based image interpretation system by Zahzah et al [46], and the self-learning rule-based controller by Lee [47]. Among the applications in the medical domain are the MLP Eind radial basis function network models of Bounds et al [48j that detect low back disorders using crisp input-output values and considering networks of multiple single-class nets in the process. Some image processing techniques are combined by Silverman and Noetzel [49] to allow the MLP to recognize tumors (at various scales, orientations and locations) from ultrasonic images. Pixel information is used as input along with crisp output values, and the model acts as a classifier.
B.
Connectionist expert models
Neurofuzzy techniques have recently been used by Endo and'Isshiki [50], Rocha [51,52] and Machado [53] to design expert systems. Hayashi et al [54] have shown that any fuzzy expert system may be approximated by a neural network and vice versa. Here we discuss a few of the existing layered connectionist expert system models (mostly in the medical domain)) The model by Gallant [6] dealing with sacrophagal problems uses crisp inputs/ outputs and a linear discriminant network (with no hidden nodes) that is trained by the simple Pocket Algorithm. The absence of the hidden nodes and nonlLnearity limits the utility of the system in modelling complex decision surfaces [l]. Dependency information regarding the variables, in the form of an adjacency matrix, are provided by the expert. Each variable (symptom, disease or treatment) corresponds to some node of the network. The model incorporates inferencing/ forward chaining, confidence estimation, question generation/backward chaining and explanation of conclusions by If-Then rules. Yin and Liang |55] employ a gradually-augmented-node learning algorithm, with binary inputs ana outputs, to incrementally build a dynamic knowledge base capable of both acquiring new knowledge as well as relearning existing information. The rules are explicitly represented among the condition nodes, rule nodes and action nodes and the algorithm gradually builds the multilayer feedforward network. This connectionist incremental expert model is used as an animal identification system whose network structure is changed dynamically according to the new environment or through human intervention. A Distributed single-layer perceptron-based model using the Pocket-Algorithm has been used as an expert system for diagnosing hepatobiliary disorders by Yoshida et al [56]. Here real-life fuzzy data are defuzzified using the Level Set representation to produce the crisp inputs {-hi, - 1 , 0} required by the algorithm. Besides, all contradictory training data are excluded, as these cannot be tackled by the model. Extraction of fuzzy If-Then production rules is possible. Hayashi [57,58] has extended this to include linguistic relative importance terms like very important and moderately important in each proposition; linguistic truth values like completely true, true, possibly true, unknown, possibly false, false and completely false can
129
also be assigned to the extracted rules. The approach by Hudson et al [59] is for detecting carcinoma of the lung. They use a feed-forward neural network model to extract information directly from the accummulated data and then combine it with a rule-based exp'ert system incorporating approximate reasoning techniques. The input nodes represent data values for signs, symptoms and test results (may be continuous or discrete). The interactive nodes account for the interactions which may occur between these parameters. The learning method is an adaptation of the potential function approach to pattern recognition and is used to determine the weighting factors as well as the relative strengths of rules for the two-class problem. Further extension of this work has been made in [60,61,62]. Sanchez [63] associates primary linguistic weights and secondary numerical weights to generate the knowledge base for a biomedical application (inflammatory protein variations) using a feedforward network. Triangular membership functions like negative large, negative medium, negative small, approximately zero, positive small, positive medium and positive large; or, decreased, normal and increased account for the linguistic weights while the quantitative weights lie in the range [0,1]. The linguistic weights are tuned according to the information provided from the input-output examples while the numeric weights and the netv.'ork topology are determined by solving fuzzy relation equations. Saito and Nakano [64] use a multi-layer network to design a medical diagnostic expert system for detecting headache. A patient responds to a questionnaire regarding her perceived symptoms and these constitute the input to the network. The doctor is supplied with information regarding possible diagnoses based on the output nbde values. Relation factors, estimating the strength of the relationship between symptom(s) and disea5e(s), are extracted from the network and used to help doctors. Rules are generated from the changes in levels of input and output units; the connection weights are not involved in the process. These rules are then used to allow the patient to confirm the symptoms initially provided by her to the system, in order to eliminate noise from the answers. A cell recruitment learning algorithm, capable of forgetting previously learned facts by learning new information, is employed by Romaniuk and Hall [65] to build a fuzzy connectionist expert system for determining the creditworthiness of credit applicants. Fuzzy functions maximum, minimum and negation are applied at the neuronal levels depending upon the corresponding bias values. This incremental learning algorithm can be used in conjunction with an existing rule knowledge base or alone. Learned knowledge can be extracted from the network in the form of rules (during an explanation phase) by a top-down traversal involving analysis of the cell activations, their bias and the associated link weights. The network consists of positive and negative collector cells along with unknown and intermediate cells and can handle fuzzy or uncertain data.
IV.
Fuzzy M L P using Logical Operators
The model discussed here consists of logical neurons employing conjugate pairs of t-norms T and t-conorms S in place of the weighted sum and sigmoid functions of the conventional MLP. The hack propagation algorithm is modified to incorporate the logical operations in the error derivative term. The components of the input
130
vector consist of the membership values to the overlapping partitions of linguistic properties low, medium and high corresponding to each input feature. During training, supervised learning is used to assign output membership values lying in the range [0,1] to the training vectors. A heuristic for gradually decreasing the learning rate and the momentum is used to help avoid spurious local minima and usually prevents oscillations of the mean square error in the weight space, in the process of convergence to a minimum error solution.
A.
The conventional MLP
Consider the layered network given in Fig. 1. The conventional MLP [2] is made up of simple neurons implementing the weighted sum and sigmoid functions (in place of the And and OT functions in Fig. 1). The output of a neuron in any layer other than the input layer is a monotonic non-linear function of its total input and is given as
where yf is the state of the t"* neuron in the preceding /i"* layer and w^- is the weight of the connection from the i"" neuron in layer h to the j " * neuron in layer ha. For nodes in the input layer, y° corresponds to the j " " component of the input vector. The Least Mean Square (LMS) error in output vectors, for a given network weight vector w, is defined ELS
EH = \Uyf.cH
- d,,r
(2)
where yyf(.(w) is the state obtained for output node j in layer H in input-output case c and djc is its desired state specified by the teacher. The error E[w) is minimized by the back propagation algorithm using gradient-descent. We start with any set of weights and repeatedly update each weight by an ajnount dE ^""^'^^^ " ''d^i
^ aA Ein] can be represented as a 3n-dimensional vector
iF 0, let us define if;i = i otherwise where the T{S) operation at layer h is performed over all / S[T) operation outputs from the neurons in the preceding layer h — 1, provided / ^ i, for h = 1(2) respectively; also let
smf={
S{yt-\w^n
if/» = l
y \ ' V ,{ otherwise
T{y^-\w^rA
(15)
133 Using equations (9), (10), (14) and (15), we have
r(5m*,fim^)j dwii
1
«
ifA = l
^fsm^jSmfj
otherwise
(16)
where the t-norm T and t-conorm S aire given by eqn. (8) in order to model the logical operators And cind Or. Expanding, we have dwji
1 (1 - sm'')y^~^
otherwise
(17)
Analogously, we compute the sensitivity mecisure as
^
= (1 - . m ' - X -
(18)
Substituting the values of ^ ^ and j ^ from eqns. (17-18) into eqns. (11,13) enables one tp evaluate the error derivative j ~ _ of eqn. (3) and,thereby update the connection weights during training. This constitutes the back propagation algorithm for a network incorporating logical nodes. Note that the e of eqn. (3) is gradually decreased in discrete steps, taking values from the chosen set {2, 1, 0.5, 0.3, 0.1, 0.05, 0.01, 0.005, 0.001}, while the momentum factor a is also decreased. Let the various values of e be indicated by Eo = 2,£i — l , . . . , f 5 — 0.001 such that EJ indicates the (t-f- l)"" value of e. Let ao = 0.9 and ai - a2 = ... = a^ = 0.5. We use
{:^^
if mse[nt — kn) — m&t[nC) < 6
f^^s
otherwise
^ '
where i = 0 initially, |e| == g -I- 1 and Q < 5 < 0.0001. Note that nise{nt) is the mean square error at the end of the nt*'^ sweep through the training set and kn is a positive integer such that mse is sampled at intervals of kn sweeps. The process is terminated when i > q.
V.
Rule Generation From The Trained Net
After the design and training of the network is complete, it is expected to be able to infer the correct claissification for the test data. Handling of imprecise inputs is possible and natural decision is obtained associated with a certainty measure denoting the confidence in the decision. Rules can also be generated from the trained network. Fig. 2 gives an overall view of the various stages involved in the process of inferencing and rule generation.
A.
Input form
The input for a test pattern can be in quantitative, linguistic or set forms or a combination of these. It is represented as memberships to the three primary
134 If o o r t r
* weights •
Trained NM
Tnput patlern
known or unknown
(Connection weights )
components
I
>
known or unobVo noble
Output decision
+
then part
Rule generation
Certainty measure When output neuron (s) unknown
Figure 2 : Block diagram of the inferencing and rule generation phase of the proposed fuzzy logical neural network.
linguistic properties low, medium and high as in eqn. (4), modelled as 7r-functions. When the information is in exact numerical form like Fj is rj, say, we use eqn. (5). However, when the input is given as Fj is prop (say), where prop stands for any of the primary linguistic properties low, medium or high, eqn. (6) is used. The model can also handle the linguistic hedges [67] very, more or less and not using the Concentration (Con) and Dilation (Dil) operators [8]. In case of set form usage, the input is a mixture of linguistic hedges and quantitative terms. Since the linguistic term increases the impreciseness in the information, the membership value of a quantitative term should be lower when modified by a hedge [67]. The modifiers used are about, less than, greater than and between. If any input feature Fj is not available or missing, we clamp the three corresponding neurons x° = ij_i = i j ^ j = 0.5, such that k = (j — 1) * 3 + 1. We use rO.5 0.5 0.51 (20) no information as 0.5 represents the most ambiguous value in the fuzzy membership concept. We also tag these input neurons with noinf^ — noinf°^^ — noinf°j^^ = 1. Note that in all other cases the variable noinf^ is tagged with 0 for the corresponding input neuron k, indicating absence of ambiguity in its input information.
B.
Forward pass
Associated with each neuron j in layer h + 1 are its confidence estimation factor conf^*^, a variable unknown'^'^^ providing a measure of the weighted information from the preceding ambiguous neurons t in layer h (having noinf^ = l) and a variable inoiyny"*"' giving a measure of the weighted information from the remaining non-ambiguous preceding neurons (with noinf^ = 0). For neuron j in layer h > 0
135
we define for A = 1
unknown^ —
otherwbe
(21)
and (22) I
for all » having noinf^"^ — 1, and S[yt\w''-'
known j =
T{yi
h-l
u
fOT
h=l
otherwise
(23)
for all i with noinf-^~^ = 0, where T and 5 stand for the conjugate pair of t-norm Bind t-Conorm defined in eqn. (8). Here the T[S) operation at layer h is performed over all t S{T) operation outputs from the neurons in the preceding layer /i — 1 for /i = 1(2) respectively. Note that for /i > 0 we have noinfj
. 1 if known^ > unknown^ = < 1 if knowrVj < unknown^ 0
for h = 1 for /i = 2 other^vise
(24)
Using eqns. (9-10,21-24), we define conf^ =
li. X unden'
if noinfj
y'
otherwise
= 1 and h > 0
(25)
If there is no neuron j with noinfj = 1, then the system finalises the decision inferred irrespective of whether the input information is complete or partial. In case of partial inputs, this implies presence of all the necessary features required for taking the decision. A certainty measure (for each output neuron) is defined as cert
(26)
where 0 < cert^ < 1. The higher the value of certf, the lower is the difficulty in deciding an output class j and hence the greater is the degree of certainty of the output decision. Depending on the value of certf, the final inferred output may be given in natural form. C.
Querying
If there is any neuron j in the output layer H with noinff = 1 by eqn. (24), we begin the querying phase. We select the unknown output neuron j'l from among the neurons with noinfj — 1 such that confj^^ by eqn. (25) (among them) is maximum.
136
Then we pursue the path from neuron j i in layer H, in a top-down manner, to find the ambigupus neuron t'l in the preceding layer [h = H — l) with the greatest absolute influence on neuron ji. This process is repeated down to the input layer (A = 0). We select : = I'l such that with noinf^ = 1, for 0 < A < ff — 1, we have ^Ku'V')
= ^lin[5(u;^,y^)]
for A = 0
^K-VJ/')
= max[r(u;j',,,yf)]
otherwise
For node t'l in the input layer (/i = 0), the model queries the user for the value of the corresponding input feature ui. Note that if a missing input variable by eqn. (20) is queried and found to be missing once again, we now tag it as unobtainable. The inferencing mechanism treats such variables as known with values i j ^ = a;°,+i = arj^^j = 0.5 but with noinf^^ = noinf°^^y^ = noinf°^^2 ~ 0) s^,....,b„) Therefore, for input A - = Wi,--->ai,,}, n = m^,...,m„
for / = l,...,n
one must get the output B' = {bi,...,b'„} b . In case the sets Aj are crisp sets the network must produce results that agree with the rules of classical logic. In the followmg, it is proved that every neural network of the type shown in Fig.4 with describing equations
c/fr
=u{l{a'ij,w,j)] )
S = C(u( Nu-
method cog fm wfm
*,/ Ng + l Nu + 1
+ ,2 K - 1)
2(^„ - 1) SAT,+ 1 2(iV„ - 1)
Comparing the defuzzification methods, it appears that the choice of the defuzzification m e t h o d determines to a W g e extend the "quality" of control as well as t h e computational cost of the controller. Comparing the results based on control and computational performance, one can choose the appropriate defuzzification m e t h o d for a specific application. Figure 10 shows experimental results of applying several defuzzification methods a.s well as some tests for comparison. As can be seen in figure 10a using the mean-ofm a x i m a defuzzification method often results in a steady state error. This is due t o the fact t h a t the only dominant fuzzy rule is: if e is AZ and Ae is AZ then u is AZ T h e m a x i m u m value depends on the "width" of t h e membership functions AZ, of both the error e and error change A e . Solving this problem can b e done by splitting u p the membership function AZ of the error into two, for example negative zero (NZ) and positive zero (PZ). This however, leads to a limit cycle in the control signal and thus a possible limit cycle in the process output (Jager et o/., 1992). Applying the centre-of-gravity (see figure 10b) does not result in a steady state error. Knowing t h e linear relation between the number of quantisations Ng used for the centre-of-gravity related defuzzification methods (see table 1) and the neglectable difference in the resulting control performance (Jager et al., 1992), one can conclude that t h e (weighted-)fuzzy-mean method is more preferable t h a n the centre-of-gravity m e t h o d , which uses more quantisations in its calculation. Using this (weighted)fuzzy-mean defuzzification method in fact transforms a fuzzy controller according to Mamdani (Mamdani & Assilian, 1975) into a Sugeno-type (Takagi & Sugeno, 1983) of fuzzy controller (see section 3.2 for description of Mamdani- and Sugenotype of fuzzy controllers), some examples can b e found in B a t u r and Kasparian (1991), Harris and Moore (1989) and Matsuoka (1991).
186
5
Fuzzy versus linear control
In this section we will discuss the relation between fuzzy and linear control. It is shown that any linear controller can be described as a fuzzy controller: fuzzy control can be seen as a superset of linear control or linear control as a subset of fuzzy control. The most widely used controller is the PID-controller £ind most fuzzy controllers described in literature imitate a conventional PID-controller. The second subsection will address these fuzzy PID-controllers. Finally, fuzzy control of non-linear processes is discussed.
5.1
Fuzzy control as superset of linear control
When designing a fuzzy controller and applying specific choices for membership functions, logical operators and scaling of in- and outputs, the fuzzy controller can emulate a linear controller. From this point of view linear control can be seen as a subset of fuzzy control. First let us start with looking at a fuzzy controller as a controller which represents a mapping / ( ) from the N^ inputs x to the output u: n = f{x} = f{x„...,XN^)
(10)
In case of a linear controller this mapping is a linear algebraJc equation:
u — a^x + 6 = y^ ajXi + b
(11)
i=l
The fuzzy controller function (10) can emulate the linear controller (H) when meeting the following criteria: 1. the membership functions on the universe of discourse of the inputs are triangular shaped; 2. there are no more than two overlapping membership functions on a universe of discourse; 3. the membership functions form a complete set: the sum of the membership functions on a universe of discourse equals 1; 4. the fuzzy rule base is complete.
187 5. the intersection operator for combining the premises of the fuzzy rules is the product operator (probabilistic intersection); 6. the defuzzified conclusions of t h e individual fuzzy rules Eire chosen according to equation (11); 7. the union operator Lukasiewicz);
is
the
summation
operator
(union
according
to
8. t h e fuzzy-mean defuzzification m e t h o d is used. In a lot of fuzzy controllers found in literature criteria 1 ^ axe met. Although not explicitly stated they normally also meet criterium 8. This leaves criteria 5-7 to be the main differences between "standard" fuzzy controllers and linear controllers. Using t h e product a n d summation operators instead of the m a x and m i n operators respectively, is necessary because t h e emulation of linear controllers requires linear operators. T h e most important criterium to be m e t is 6, because due to criteria 1-5 and 7-8 the o u t p u t of a fuzzy controller, with N^ inputs results in a linear interpolation between 2 ' points in a iVj.-diniensional space and the numerical conclusions of the at most 2 ' fuzzy rules, which contribute to the output, determine wether or not a linear relation, by connecting these points, (hyperplane) exists. When this hyperplane exists it equation (11) holds. We can write this more formally by stating that the controller o u t p u t u is described by applying the fuzzy-mean defuzzification, as defined in (8c)) (subsection 4.1), and criterium 2, as listed above, is met:
^='-%
(12a) k=\
2Nx
(12b)
where Uk is the defuzzification of membership function Uk and u^ is the numerical consequent of fuzzy rule rj (defuzzification of fuzzy consequent [/•'). W h e n we
assume criteria 2-5 and 7 we can write:
(13a) t=l
3=1
with, as shown in figure 11:
Mil + Mi2 = 1
(13b)
F i g u r e 1 1 : Relation between controller input, centres of membership functions and membership values.
This is also the reason why in (12a) we use 2^" as the number of active rules. T h e number of active rules is in fact greater t h a n 0 and equal or less than 2^"^. Assuming some non-active rules to be active but not contributing to the output, (12a) is correct when criteria 2-5 and 7 are met. When (11) describes the control hyperplane: the controller o u t p u t as linear function of the controller inputs and the numerical result of each individual fuzzy rules lies on this hyperplcine, the controller output from (12a) can be described by: (14a) j=i
(14b) J=l
189
(14c) Because the fuzzy controller should behave like a linear controller:
u = a x[t) + b
(15)
which equals (14a) when:
£(*) = Y.
(16a)
Ij^j
j=i
N^ ( M I I ^ I I + IJ-12X12) Ylifikl
+ fJ-k2)
k=2
{HilXii + Hi2Xi2) Ylinkl fc=l
+ IJ'k2) n
(^'^l + '^'•2)
(16b)
*=i+l JVi-1
[flN^lXN^l + fJ.N^2XN^2) J J (MW +/^fc2) fc=l
Mil a^ii + Mi2a'i2 t^N^l^Nzl
+
(16c)
f^N^2XNz2
which is t r u e , because of (13) and thus it is proven t h a t t h e controller output is a linear combination of the controller inputs in ca^e t h e criteria as stated in the beginning of this section Eire met. In case of Sugeno-type of rules there is a very simple proof: all rules should have (11) as consequent and the fuzzy-mean defuzzification methods should be used. This result in a controller o u t p u t , which equals (11): Nr
(17a)
E7. 3=1
190 A',
=
'^^^.
= a^x + 6
(17b)
(17c)
As (17) shows, there are no restrictions on t h e membership functions or the T-norm used in the rule of inference for t h e and connective. T h e only restriction is t h a t there should at least be one active fuzzy rule.
5.2
Fuzzy PID-control
Although PID-controUers are known to be linear controllers, they are not in practical implementations. First of all t h e control signal is limited. Secondly anti-windup mechanisms axe used in practical set-ups. T h e fuzzy equivalent of a PD-controUer is normally build u p by rules relating the error and t h e error change to t h e controller output, for example (see figure 3 for abbreviations): if e is P M and Ae is NS then u is P B In table 2 several families of fuzzy rules and their corresponding linear controller type are given. Using derivatives of the error instead of previous values resembles more closely the way humans use error signals to clcissify the deviation between the process state and the reference signal. Applications of fuzzy controllers which use the error and its derivatives as inputs emd satisfy conditions 1-4, 6 and 8 as stated in the previous subsection (5.1), are in fact unnecessary approximations of linear controllers. T h e only difference between such an application and the pure emulation of the linear equivalent is t h a t the moas-min inference rule is applied. Due to the use of these non-linear operators t h e control hypersurface will be a non-linear interpolation between the numerical results of t h e individual fuzzy rules, defining the characteristics of the control hypersurface. This non-linecir interpolation, however, is a non-trivial and it is more trivial to perform the most objective interpolation between these chju-acteristic points on the control hypersurface: a linear interpolation. Figure 12 shows the comparison of a linear fuzzy PD-controUer and a "full" fuzzy version of this controller. As can be seen in figure 12b the use of linear operators will lead to a linear control hypersurface (not taking in account the limiting of signals). Figure 12a shows the case where the non-linear operators were used. This will
191 T a b l e 2: Type of fuzzy rale and corresponding
PID controller
type.
fuzzy rule type
P
if e is . . . then « is . . . if e is . . . then A« is . . , if e[fe] is . . . and Ae[fe] is . . . then n[k] is . . . if e[k] is , . . and e[k— 1] is . . . then u[k] is . . . if e\k\ is . . . and Ae[fel is , . . then i-ktt[fe] is . . . if e\k] is . . . and e[k—l] is . . . then Ati[&] is . . . if e[fe] is . . . and Ae[fc] is . . . and A'e[fc] is . . . then A«,[&] is . . . if e[ife] is .. .and e[A—1] is .. .and e[fc —2] is .. .then Ati.[ls] is , . .
X
X
X
X
X
X
X
X
X
X
X
X
X
X
(b) Linear operators (product and summ,atioii).
F i g u r e 12: Control surface of fuzzy
PD-controllers.
D
X X
f
\
(a) Non-linear opermtors (miuimum and maximum,).
I
192 result in "waved" version of the control hypersurf£ice defined by linear case. Figure 12 shows also another interesting detail of the control surface. Because t h e limiting of the controller o u t p u t , to prevent wind-up, is explicitly embedded in the rule base, there exist non-linear characteristics due to t h e transition from the "linear" paxt of the rule base to the "limiting" part. These non-linear characteristics can be recognized in figure 12 by the b u m p s marked by arrows. Using fuzzy controllers looks quit sophisticated, to use a fuzzy controller t o mimic an operator. However, applying fuzzy controllers in this way ignores the possible strength of fuzzy controllers: being able to define a non-linear controller using a fairly stretightforward and easy to use concept of design. One should note t h a t a fuzzy P ID-controller is more understandable to non-control engineers than a conventional PID-controller. This is probably also a reason of the popularity of fuzzy control: you do not have to be a control engineer to understand how it works or how to build
However, a fuzzy controller is superior to a corresponding FID-controller and therefore can be used to obtain better control (for example in terms of speed) of a linear process t h a n a PID-controller is capable of. Because of more degrees of freedom a fuzzy controller caji be used to obtain at least the same performance t h a t would be obtained with the equivalent linear controller. Fuzzy PID-controUers can be used to implement non-linear PID-controllers. Using an "extension" of the Sugeno-type of fuzzy rules it is, for example, very easy to implement a PID-controller where the I-action is only activated when t h e "error is small": r i : if e is big then u = kp\e r2 : if e is small then w = u -f- ^^26 -|- /cp^Ae T h e "extension" of the Sugeno-type of fuzzy rules is t h a t we allow the consequents of the fuzzy rules to have additional signals besides the ones used in the condition part of the fuzzy rules. In the example above the previous controller output is used in the consequent of fuzzy rule 7*2. A set of rules, which meets the basic definition of the Sugeno-rules, can easily be obtained by extending the condition part of the fuzzy rules. This will result in more fuzzy rules, resulting in the same control hypersurface as obtained by using ri and ^2.
5.3
Fuzzy control of non-linear processes
In the previous subsection it was stated t h a t with a fuzzy controller a more "optimal" control can be obtained. This holds cdso in case of non-linear processes. Often fuzzy
193
controllers axe said to be superior to their corresponding linear controller to control non-linear processes. For fuzzy PID-controUers this is only t r u e for a small set of problems, nsimely when the non-linearity of t h e process can be written as a function of the error and its derivatives, being the input signals of the controller. This is normally not the case, because the error and error change are not solely determined by the process, but also by the externally defined reference signal. For controlling non-linear processes, fuzzy controllers axe superior to their lineeir corresponding controller in case: • the non-linearity of the process is a function of the process o u t p u t y; • this function is symmetrical around y — 0; • set-point changes A r are equal in magnitude; • t h e set-point is changed when t h e process is more or less in steady state: e ft; 0 as well as its derivatives. T h e fuzzy controller is not an "optimal" average for the whole input space like a PID-controller is, b u t an interpolation between more local "optimal" averages. This, however, has nothing t o do with t h e non-linearity of the process: a fuzzy controller can control a non-linear process as least as good as its corresponding linear controller, just because a fuzzy controller cem control a linear process as least as good as its corresponding linear controller. If it is desired to have a non-linear controller which is able to obtain better control of non-linear processes, the controller should not be beised on the error and its derivatives: one should use, for example, the reference signal or the process o u t p u t (and their derivatives) as additioncd inputs of the controller. In practise already non-linear control is applied for non-linear processes. These kinds of non-linear controllers can be easily implemented using Sugeno-type of fuzzy controllers. An example of this is the control of a bi-linear process using a fuzzy controller with only two fuzzy rules of the Sugeno-type (using t h e "extension" as explained in the previous subsection): Tp : if Ay is positive then u — /p(e, Ae) r „ : if Ay is negative then u = fnie, Ae) where fp and / „ represent PI(D)-algorithms with different peirameters, appropriate for t h e situation as stated in the condition part of t h e fuzzy rules. Using Mamdanitype of rules one could obtain a similar control hypersurface, but it would require
194 more rules. W h e n we extend the example, as show above, to more "local" PI(D)controllers for specific situations a^s described in the condition p a r t of t h e fuzzyrules, one obtains the same effect as is obtained when using gain-scheduling. T h e difference between the classic2d gain-scheduling and these type of fuzzy controllers is t h a t the fuzzy aspect of the fuzzy controllers causes smooth transitions between the control hyperplanes, each determined by a parameter set (and thus a fuzzy rule).
6
Conclusions and discussion
In the introduction it was stated t h a t the popularity of fuzzy control could be the need for a control concept to build controllers t h a t cire able to deal with changes in production rate, product mix and production procedures and recipes. Looking at applications of fuzzy controllers nowadays, it seems t h a t very few applications for solving these kind of problems have been reported in literature. A major problem in the application of fuzzy control is the fact t h a t using fuzzy control to implement operator knowledge and experience will bring u p the knowledgeengineering-bottleneck as major obstacle to the use of knowledge based systems (Feigenbaum & McCorduck, 1983), and thus to the use of fuzzy expert systems. It is indeed very difficult to model h u m a n knowledge, reasoning and experience in a way t h a t can easily handled by computers. On the other hand, control problems which need non-linear controllers can be solved by using fuzzy controllers, but the main problem in this case is t h a t to design t h e proper fuzzy controller (read: non-linear controller) requires a non-linear control theory or a good description of the process to control. A good applicable non-linear control theory has not yet been proposed and in case the last d e m a n d could be fulfilled, there are also other ways to achieve similar results. In our opinion fuzzy control is very useful for prototyping and designing non-linear controllers and offers a great tool for the translation from high-level descriptions to low-level algorithms and the other way around. Fuzzy control can be used to provide a high-level design concept for the design of non-linear controllers. T h e membership functions on universes of discourse in combination with fuzzy rules can be "trcinslated" into a straight-forward non-linear controller definition, which can be used for real-time control. This non-linear controller definition exists of a simple look-up table, which contains the characteristic points of the control hypersurface. Interpolation between these characteristic points can be done using standard interpolation techniques.
195
Applications of fuzzy control in consumer electronics quite often axe implemented using look-up tables. Whether it was recognized or not, implementing fuzzy controllers with look-up tables result in similar behaviour as implementing fuzzy controllers according to theory. The main difference between the two is the interpolation between the numerical results of the individual fuzzy rules. In case if look-up tables the interpolation ccui be done explicitly using a lineaj interpolation (if there is a need for interpolation with respect to the quantisation of the controller inputs and output). Using fuzzy controllers according to the theory, this interpolation is done implicitly by the logical operators in combination with the membership functions. Currently work is done to specify the "translation" of high-level fuzzy controller descriptions to simple non-lineaj controller representations. Implementation of this "translator" in the form of a pre-compiler is under consideration.
Symbols and abbreviations
n u A V 7j
Ae e Mfl, MR
Aty l^u, f^Ui
N,
K Nu
N, r To
R. R
intersection operator union operator
X
T-norm operator T-conorm (S-norm) operator support value for rule TJ derivative of e error signal {r — y) fuzzy set of Rj fuzzy set of fuzzy controller fuzzy set of controller o u t p u t fuzzy set of Uj fuzzy set of U^ number of quantisations number of rules number of fuzzy sets for w number of inputs reference signal j rule of fuzzy controller fuzzy relation of rule Vj fuzzy relation of controller
X,
Xi
Xi u Uj U-'
U Uj
W y coa cog ctrd
fm wfm mom
idfz hgt
inputs of controller i*'' input of fuzzy controller fuzzification of Xi fuzzy set for Xi in rule Vj numerical controller o u t p u t numerical o u t p u t due to rule TJ numerical consequent of rule rj fuzzy controller output fuzzy output due to rule rj fuzzy consequent of rule rj process output centre-of-area centre-of-gravity centroid fuzzy-mean weighted-fuzzy-mean mean-of-maxima indexed- defuzzification height
196
References AOKI, S., S. K A W A C H I A N D M . S U G E N O ( 1 9 9 0 ) . Application of fuzzy control logic for dead-time processes in a glass melting furnace. Fuzzy Sets and Systems, vol. 38, pp. 251-265. ASSILIAN, S. (1974). AHificial intelligence techniques in the control of real dynamic systems. P h D - t h e s i s , Queen Mary College, University of London. B A T U R , C . A N D V . K A S P A R I A N (1991). Predictive fuzzy expert controllers. puters in Industrial Engineering, vol. 20, no. 2. B U C K L E Y , J . A N D W . S I L L E R (1987). Fuzzy operators for possibility sets. Fuzzy Sets cind Systems, vol. 22, p p . 215-227.
Cominterval
D U B O I S , D . A N D H . P R A D E (1980). Fuzzy sets and systems: theory and applications, vol. 144 in Mathematics in science and engineering. Academic Press. F E I G E N B A U M , E . A N D P . M C C O R D U C K (1983). The fifth generation: artificial intelligence and Japan's computer challenge to the world. Addison-Wesley. F L I N T H A M , T . J . M . (1991). Expert systems in control, why so few? l E E Control 91. Edinburgh, Scotland, U.K., March 1991.
Proceedings
H A R R I S , C . J . A N D C . G . M O O R E (1989). Intelligent identification and control for autonomous guided vehicles using adaptive fuzzy-based algorithms. Engineering Applications of Artificial Intelligence, vol. 2, p p . 267-285. December 1989. H A S P E L , D . (1991). A new approach on the cement making process through Linkman. Proceedings of 3^ International N C B Seminar on Cement and Building Materials. Dehli, India. H E L L E N D O O R N , H . (1993). Design and development of fuzzy systems at Siemens R&D. Proceedings of 2 I E E E International Conference on Fuzzy Systems (FUZZ-IEEE 1993), p p . 1365-1370. San Fransisco ( C a ) , U.S.A., March 1993. H E L L E N D O O R N , J . (1990). Reasoning with fuzzy logic. P h D dissertation. Delft University of Technology. Delft, T h e Netherlcinds. J A G E R , R . , H . B . V E R B R U G G E N A N D P . M . B R U I J N (1992). The role of
defuzzi-
fication methods in the application of fuzzy control. Proceedings IFAC Symposium on Intelligent Components and Instruments for Control Applications 1992, A. Ollero and E . F . Camacho (editors), p p . 111-116. Malaga, Spain, May 1992. K L I R , G . J . A N D T . A . F O L G E R (1988). Uncertainty Hall.
and information.
Prentice
L A R K I N , L . I . (1985). A fuzzy logic controller for aircraft flight control. Industrial Applications of Fuzzy Control, M. Sugeno (editor), p p . 87-103.
This page intentionally blank
197 L E E , C . C . (1990a). Fuzzy logic in control systems: fuzzy logic controller - part I. I E E E Transactions on Systems, Man and Cybernetics, vol. 20, no. 2, pp. 404^18. L E E , C . C . (1990b). Fuzzy logic in control systems: fuzzy logic controller - part II. I E E E Transactions on Systems, Man and Cybernetics, vol. 20, no. 2, pp. 419^35. M A M D A N I , E . H . (1974). Applications of fuzzy algorithms for control of simple dynamic plant. Proceedings l E E , , no. 121, p p . 1585-1588. M A M D A N I , E . H . A N D S . A S S I L I A N (1975). An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies, vol. 7, p p . 1-13. M A T S U O K A , H . (1991). A simple fuzzy simulation model for nuclear reactor tem dynamics. Nuclear Technology, vol. 94, pp. 228-241. May 1991. O S T E R G A A R D , J . J . (1990). FUZZY II - The new generation control. Zement-Kalk-Gips, no. 11. P E D R Y C Z , W . (1989). Fuzzy control and fuzzy systems. Wiley & Sons.
sys-
of high level kiln
Research Studies P r e s s / J .
SCHWEIZER, B . AND A. S K L A R (1963). Associative functions and abstract groups. Publicationes Mathematicae Debrecen, vol. 10, pp. 69-81.
semi-
SUGENO, M . AND K. MURAKAMI (1985). An experimental study on fuzzy parking control using a model car. Industrial applications of fuzzy control, M. Sugeno (editor), pp. 125-138. Elsevier Science Publishers B.V. (North-Holland). SUGENO, M . AND M . NiSHIDA (1985). Fuzzy control of model car. Fuzzy Sets and Systems, vol, 16, p p . 103-113. T A K A G I , T . AND M . SUGENO (1983). Derivation of fuzzy control rules from human operator's control actions. Proceedings of IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, p p . 55-60. Marseilles, France, July 1983. TURKSEN, I . E . (1986). Interval valued fuzzy sets based on normal forms. Sets and Systems, vol. 20, pp. 191-210.
Fuzzy
Y A G E R , R . R . A N D D . P . F I L E V (1993). SLIDE: A simple adaptive defuzification method. I E E E Transactions on Fuzzy Systems, vol. 1, no. 1, pp. 69-78. Y u , C , Z. C A O AND A. K A N D E L (1990). Application of fuzzy reasoning to the control of an activated sludge plant. Fuzzy Sets and Systems, vol. 38, pp. 1-14.
CHAPTER 9
ANALYSIS, DESIGN, IMPLEMENTATION AND CRITICAL APPRECIATION OF FUZZY LOGIC CONTROLLER Kumar S. Ray Electronics & Communication Science Unit Indian Statistical Institute 203 B.T. Road Calcutta -700 035, India
1.
Introduction
Immediate after World War II people were very keen to develop sophisticated tool for communication and control.
Despite the landmark achievement of
the classical control theory through the launching of the first sputnik in 1957 and the subsequent developments of the clcissical control theory to modern control theory which has been tested through a number of important hightechnology projects (viz the U.S. Apollo project), there are still serious problems in the control of complex system.
In manufacturing technology such as in
chemical processes or the steel industry, in power generations industry etc. the conventional control algorithms are unable to manage the huge uncertainties involved in the entire process and thus often require human interventions for readjustments of the designed scheme.
Zadeh first realized that people can base decisions on imprecise, nonnumerical information. In 1965, he was implicitly advancing a thesis which indicates that 199 S.G. Tzafestas and A.N. Venetsanopoulos (eds.) Fuzzy Reasoning in Information, Decision and Control Systems, 199-275. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.
200
under uncertain complex situations people are better at control than Machine. In this connection Zadeh's significant achievements are the seminal papers on the linguistic approach and system analysis based on the theory of fuzzy sets [17, 18, 19, 20, 21, 22).
Being motivated by the above said contributions of zadeh, in mid 70's Mamdani and his colleagues first demonstrated the successful applications of the fuzzy logic controller (FLC). About the same time the first significant industrial application of the FLC came up in Denmark at F.L. Smidth corp's cement kiln.
During the past several years, fuzzy control has emerged as one of the most potential areas for research in the application of fuzzy set theory. The concepts of FLC is now an important adjunct to conventional control theory. The tremendous applications of FLC indicate its effective utilization in the context of complex ill-defined systems that can be controlled by a skilled human operator without the quantitative knowledge (in terms of deterministic algebra and differential equations) of their dynamics.
The essential component of FLC is a set of linguistic control rules which are generated by an experienced operator and which can be related by the dual concepts of fuzzy implication and the compositional rule of inference.
Experience shows that the FLC sometimes yields results superior to those obtained by conventional control algorithms. In fact the methodology of FLC appears very useful because it can easily combine heterogeneous control laws over a wide range of operating conditions within which almost every real sys-
201 tern becomes nonlinear in n a t u r e . Further FLCs are easier to design and so cheaper t o p r o d u c e . T h e design of PLC is now a t e x t b o o k example of how a theory becomes a practice.
2.
M e t h o d of approximate Reasoning
Since the first formal description of fuzzy reasoning by Zadeh in [22j many researchers have discussed different forms of fuzzy reasoning. Mizumoto presented a possible extension of fuzzy reasoning in [23].
In ordinary fuzzy reasoning as suggested by Zadeh in [22], we make inferences of t h e form
P: if X is A then Y \s B q: X is A'
(Concl.)r Ci also
Rule 2 •.A2andB2 => C^ also
Rule n : A„andBn
^
C„
Fact : A' a n d B'
Consequence : C = C[UC'^U
... U C'^
where C, = {A'andB')o{AiandBi
(1)
=> Q)V,
Ai, A' are fuzzy sets in U Bi,B'
are fuzzy sets in V
Ci, C are fuzzy sets in W
3.3
Methods of defuzzification [11]:
T h e m e t h o d of obtaining a singleton w^ which is a representative point for the resulting fuzzy set C of (l) is called a defuzzifier m e t h o d . In the following we shall list several defuzzifier m e t h o d s . (i) Center of Gravity M e t h o d : This m e t h o d is widely used in the fuzzy controls.
T h e center of
219 gravity of C is adopted as the desired singleton Woi t h a t is,
(ii) Average of M a x i m a Metho: T h e representative point Wo is obtained as an average of the elements which give the mciximal grade in C, t h a t is, ^0 —
m
(iii) Midpoint of M a x i m a M e t h o d : This m e t h o d is a simplified version of Average of M a x i m a M e t h o d . Instead of taking all elements Wj which give the maximal grade, the smallest element w' and the largest element w" among t h e m are picked up a n d the midpoint of w' and w" is given eis the representative point Wo, t h a t is. 2
T h e above defuzzifier m e t h o d s of obtaining a representative point Wo are all derived from the calculation of the fuzzy set C which was aggregated from C i , . . . , C ^ as in ( l ) . (iv) Height M e t h o d : This m e t h o d obtains w^ as an weighted average of t h e representative points Wi of C, by the heights hi of C\. Namely, _ "'o
ml-M+m2h2+....+wnhn hl+fc2 + ....+An
(v) Maximal Height M e t h o d : A representative of point wj of Cj which corresponds to the maximal height hj among hi[i ~ l , . . . , n ) is a d o p t e d as Wo Wd = wj (hj is the maximal height) (vi) Area M e t h o d : Wo is obtained as the weighted average of the representative points wi by the areas Si of Ci'.
220 _ ^°
ml-51+iD2-52+...+wnSn Sl+S2+...+Sn
(vii) Maximal Area Method: A representative point Wj of Cj which corresponds to the maximal area Sj among Si[i — 1, ...,n) is selected as Wo, that is Wo = Wj {Sj is the maximal area).
4.
Some useful properties of FLC [23] [60].
In this section we discuss some useful properties of the FLC inference mechanism under sup-min operator denoted by 0 and sup-product operator denoted by •. The connective "also" is treated as union operator.
Proposition 1
Proof: n
C = {A',B') o\jRi 1=1
n
= {A',B') o [J{AiandB, -^ C,). 1=1
The membership function ^lc' of the fuzzy set C is point wise defined for a.\\weW by
221 l^c'{W)
=
(/^^'(«),MB'(f)) ° max(/i/j,(u, tj,u;),/jyij(u, u, ui),....,/i/j„(u, ti.uj))
=
snpmin{{nA'(u),
l^B'iv)), m!ix(fiR,lu,v,w),
u.v
HK^(U,V,W)
HR
{u,v,w))}
u.u.w
=
supmax{min[(M^.(u),/iij'(f)),;i/j,(u, u, u^)|,..-,min|(/i^/(u),AJB'(i;))),/ijj„(u, u, tu))} „,„ u.v,w
=
m a x { [ ( / i ^ . ( « ) , /^B' i^)) ° M R , ( " , " , "')],••••, ((/^A'(«), /^B' («)) ° M J ? „ ( " , «. " j ) | } -
Therefore
C
=
\{A',B')OR^]U\{A',B')OR^]LI...\J[(A',B')OR„]
=
\J(A',B')oR, t= l n
=
U (^', B') o (AtandBi -^ C.)
1=1
Preposition 2
{A\B')*\}URi^\^U{A',B')*IU. Proposition S For the fuzzy translation Rc,Rp,Ri,p, and R^^, we have (.4', B') o [Ai and Bi -* C.) = \A' o (^, - . Ci)\ n [ 5 ' o ( 5 , - C,)l if MA.XB, = /^/l,
A/iB,
= [A' o (^, -> C,)l • \B' o ( 5 . -» C.)l i f / ^ A . x B , = /^>l, • A«fl,
222
Proof:
Mc;
=
( M A S M B ' ) ° (MA.XB, - • Mc.)
=
( M A ' , M B ' ) ° (min(MA,,MB.) - ^ Mc,)
=
(M/1',MB') ° min[(/X4. -> fJ,C;)Af^Bi -* Mc,)l
=
supmin{[(/XA',Mfl')>min[(MA. -* M C , ) , ( M B . -* Mc,)]}
=
supinin{ininl/i^', (MA, " • Mc,))>min|MB', ( ^ s . -^ Mc.))} U,tJ
=
inin{[^A' o ifJ'Ai -» Mc,)], [MB' ° (MB, -• fJ'C,)]}-
Hence we obtain C' = \A' o (/I. ^ C,)I n [B' o {B, Proposition
C,)]. Q.E.D.
4
For t h e fuzzy translation Rc,Rp,Rhp,
(A',B')*(A,andBi
and ii^p, we have
-^ C,) = \A' *[A, - C.)| n | B ' . (B, ^ C,)|t//i^,xB, = MA, A ^ B ,
{^'.BOM-^.andB. ^ C.) = 1 ^ ' • (^. --C.)] . jB' ( f l ; - . C,))i7/i^,^B, = / i ^ , /XB,
223
5.
Design aspects of FLC [60] [62]
In the previous sections we have described the basic components and properties of FLC. Now we focus our attention on two fundamental aspects of FLC design; namely static analysis and dynamic analysis.
5.1
Static analysis of FLC
In this section we consider the completeness of FLC, number of fuzzy control rules, their interactions and consistency.
5.1.1
C o m p l e t e n e s s of FLC
By completeness, we mean that a FLC should always be able to infer an appropriate control action for any input fuzzy state. If we consider the baisic equation of a FLC with sup-min composition, then an empty fuzzy set of control corresponding to a specified nonempty fuzzy set of input cannot be accepted. That means, a given set of control rules, if X is A{ then j / is fij, i = 1, ...,n. is complete in this condition ; V u eU ^ (Vu)u
3R, I Ai(u) > e 1 G) 1 < i' < n
224 where G (0, l]
In other words, the union of fuzzy relations, At be greater than zero for all u e U; (Vu)u(sup^/XA,(w)) >G
(2)
the above relation holds because we deal with fuzzy labels which usually overlap. The inequality condition (2) will be violated if, at the time of construction of FLC, we miss some label or forget to describe the relevant "if-then" control rule. Under such circumstances an additional rule should be added to satisfy the inequality condition. Thus from the above discussion it is obvious that the completeness of a FLC relates to its database, rule base or both.
5.1.2
N u m b e r of fuzzy control rules
So far, there is no specific procedure to determine the optimal number of fuzzy control rules. Many important factors namely; performance of the FLC, computational efficiency, human behaviour, choice of linguistic variables, etc. are involved in this decision process.
5.1.3
Interaction of Fuzzy control rules
If we have a set of control rules, Ri : ifxisAithanyisBi,i
=
l,...,n
and if the input is A, then we would expect that the control action is B,. But, in fact the derived control action may be a sub set or super set of the
225
original control depending upon the definition of fuzzy implication and the rule of composition[52]. This phenomenon occurs as a consequence of interaction between the rules. Now we stipulate the following results; Proposition 5 If the fuzzy relational matrix R of the FLC is computed as a union of the cartesian products of A, and B, and if the fuzzy relations Ai are normal for V,, the resulting fuzzy set of control satisfies ; Vi < i sup min[yli(u), Bi(t;)| ueu
which, by virtue of normality of Af, yields {AioR)(v)^
Bi{v),\/(v)ev
Q.E.D.
The following result specifies the conditions for noninteractions between the control rules. Proposition 6 If for the fuzzy relations Ai, V^ are pairwise disjoint, i.e., Ai n Aj = ,fori ~ j,i,j
= l,...,n which means, min( Ai{u), A j{u)) = 0
and if they are normal, then Vj < j < „ Ai oR = Bi. Proof: Let supp {Aj) be a support of the fuzzy relation Aj and assume the sujvmin composition of Aj and the relational matrix R.
226 {AjoR){v)
=
sup{min[A.j{u),
=
inax{
sup
R(u,v\} {min(A,(u),iJ(u, u)|},
u 6 supp{Ai)
=
sup « € fiuppiA,
5.1.4
sup ug
{min|j4j(u),it(u, u)]}}
tikpp{Aj)
{min{Ay{u), max[min[j4j(u), Bj(u)l|}} )
C o n s i s t e n c y of f u z z y c o n t r o l r u l e s
If the derivation of fuzzy control rules is based upon the experience of an o p e r a t o r , the rules may be subjected to different performance criteria. In practice, we check the consistency of fuzzy control rules in order to minimize the possibility of contradiction [57],[58). We conclude this section by drawing the nonlinear relay characteristics of FLC [4][14] [17]. If we assume t h a t the fuzzy controller is specified by R = U"_i(Ai X Bi] a n d by sup-min composition, t h e n we can formulate the following sufficient conditions for multilevel relay; if (i) Bi are normal fuzzy set symmetric around their m a x i m a (ii) a m a x - m e t h o d is selected to defuzzify t h e control action, then t h e characteristics y = f(x)
( i - i n p u t of the controller and u— is the
o u t p u t of the controller) have a multilevel relay character. T h e relay characteristics of the F L C are analysed in [4][l4][l7] in relation to the closed loop stability of the system.
227
5.2
Dynamic analysis of FLC
Generally FLC can be successfully applied to ill-defined process. But, when FLC is based on the decision of the operator then it reaches the set point quickly without having significant overshoot. But it oscillates around the set point. Thus steady state precision is not as good as with conventional PID controllers where integral action eliminates the steady state error. The above phenomenon suggests to combine the FLC and the PID controller. For distant control, the fuzzy controller is preferable while for close by control (where the system is close to the setpoint) the PID controller is preferable. Such ideas are discussed in [59][l6][l3] and some encouraging simulation results are provided. But we need to determine the appropriate switch criterion for such combined control or we can use them sinaultaneously. In case of multivariable system (linear or nonlinear) use of FLC alone may produce poor system response due to high interaction between inputs and outputs. Under such circumstances simultaneous use of FLC and conventional algorithm for decoupling the multivariable system is recommended in[l6][13].
6.
Implementations of FLC
The FLC has emerged as one of the most useful areas of research in the application of fuzzy set theory. In many complex situations use of FLCbased systems have proved to be superior in performance to conventional systems.
228
Remarkable applications of FLC include the heat exchanger [62], warm water process [3], activated sludge process [64] [65] traffic junction [66] cement kiln [67] [68] aircraft flight control [29] turning process [72], robot control [70] [33] model-car parking and turning [72], automobile speed control [73], water purification process [74], elevator control [75], automobile transmission control [76] power systems and nuclear reactor control [77][13][16], fuzzy memory devices [78] and the fuzzy memory devices [78] [79], Recently, predictive fuzzy control systems have been proposed and successfully applied to automatic train operation systems and automatic container crane operation systems [80]. Now, in the following we demonstrate a successful application of FLC to 210 MW steam generating unit. The detail descriptions of the steam generating unit, schematic representation of the closed-loop control are given in the appendix. We test the asymptotic responses of the closedloop plant under different operating conditions. To perform the tests we first state the control rules of the closed-loop plant as shown in Fig.A3 in Table VI and VIII. The corresponding distribution of the membership functions are given in Tables X, XI, XII, XIII.
6,1
Response to perturbation in state
We achieve the perturbed condition by reducing the load demand. Fig.5 shows the response of the closed-loop system to the perturbation in the initial states. The set-points in throttle pressure, drum water level and steam flow to H.P. turbine are 154.15(A:g./cm.^),64(cm) and 188.8(A;5f./5ec) respectively. Fig.5 shows that the closed-loop system returns rapidly to the desired state (i.e. the set-point), from the perturbed conditions of
229
throttle pressure = 185(kg./crri^), drum water level = 60(cm). It is seen from the figure mentioned above that well damped responses in throttle pressure, drum water level and steam flow to H.P. turbine are obtained.
[kg/cm'] 1850 103 U ,
t
I [sec]
t
2.5
50
35
Throtlle pressure
[kg/tec] 188.8 I /f\ji'^^''-mm ISO'-I-S
I
3
I 50
Sleom flow to HP turbine
[cm]
60
K
tisec) Drum level
50
Fig 5 • Response to perfurbotion in m* iTMtiol slate (lorge perturbohon ).
230 C o n t r o l u n d e r change of set-points: In this cEise, the set-points of the plant are chosen as follows : Throttle pressure
=
140.0 (Kg/cm*)
Drum water level
=
60 (cm)
Steam flow to H.P. turbine
=
160.0 (kg./sec)
under perturbed condition, which is obtained by reducing the load demand and increasing the feed water flow, we obtain well damped responses as shown in fig.6. The steady state responses of the systems are almost equal to zero.
T'
^"K
SOlsec] ThroHl* prrssurv
[kg/s»c] 160 1*0
• T» IT • - J ^ ^ ^ S ^
SO (see 1
'V2,b
SUam flow lo H P turbm*
[cm] 68 60 ^—^-
t 15
'-«>r»r»
so [s*c] Orum (9V«t
Fig.6 Response under chonge of set points.
231
P a r a m e t e r sensitivity: We assume that the systems parameters always vary due to uncertainty in coal specification, variation in spray water flow, boiler blow down etc. Hence variation of ±10(%) in different parameters is introduced. Under such conditions, the responses of the closed-loop systems are tested with perturbation in initial states of throttle pressure, steam flow to H.P. turbine and drum water level. Well damped responses are obtained as shown in fig.7.
0.5[s«?c] 15A.)5 US
I
•ir^=@ Fig 9
Drum
woter
/fjx >^Ztiiiy=@ Wje\
control.
235
Both the control loops mentioned above are tested for 20% load change at 60% MCR and the plant responses are shown in fig.10.
urve-fitted
controller
Lkg/cm J 154 IS
U2.0 C kg/secj 200,0
fuzzy logic
controller
188.d
curve-tilted controller FiglO' Plant responses under 20C%21 positive change in load
236
Table I: INTUITIVE CRITERIA IN GMP [81]
X is A' (Pre q)
v is B' (concl r)
Criterion 1
X is A
v is B
Criterion 2-1
X is very A
y is very B
Criterion 2-2
X is very A
v is B
Criterion 3-1
X is more or less A
v is more or less B
Criterion 3-2
X is more or less A
v is B
Criterion 4-1
X is not A
v is unknown
Criterion 4-2
X is not A
y is not B
237
Table II: PERFORMANCE OF FUZZY IMPLICATION FUNCTIONS UNDER INTUITIVE CRITERIA OF TABLE I [60] [811
hline
key:
M«.
MRp
M«a
fJ'Rn.
M«.
M«A
M«t
Criteria 1
0
o
X
X
0
X
X
Criteria 2-1
X
X
X
X
0
X
X
Criteria 2-2
o
o
X
X
X
X
X
Criteria 3-1
X
X
X
X
o
X
X
Criteria 3-2
o
o
X
X
X
X
X
Criteria 4-1
X
X
0
o
0
0
o
Criteria 4-2
X
X
X
X
X
X
X
o =
Satisfaction
X=
Failure
238 Table III: Q U A N T I Z A T I O N AND P R I M A R Y FUZZY S E T S USING A NUMERICAL DEFINITION Range
NB
NM
NS
ZE
PS
PM
PB
Co < -6.4
1.0
0.2
0.0
0.0
0.0
0.0
0.0
-6.4< Co < -3.2 -3.2< £„ < -1.6 -1.6< Co < -0.8
0.8 0.4 0.2
0.6 0.8 1.0
0.0 0.2 0.6
0.0 0.0 0.2
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0 0.0
-o.8< e„ < -0.4
0.0
0.8
0.8
0.4
0.0
0.0
0.0
-0.4< e„ < -0.2
0.0
0.6
1.0
0.8
0.0
0.0
0.0
-0.2< Co < 0.2
0.0
0.2
0.8
1.0
0.4
0.0
0.0
0.2< e„ < 0.4
0.0
0.0
0.6
0.8
0.8
0.0
0.0
0.4< e„ < 0.8
0.0
0.0
0.2
0.4
1.0
0.4
0.2
0.8< e„ < 1.6
0.0
0.0
0.0
0.2
0.8
0.8
0.4
1.6< e„ < 3.2
0.0
0.0
0.0
0.0
0.4
1.0
0.6
3.2< e„ < 6.4
0.0
0.0
0.0
0.0
0.0
0.8
0.8
6.4 < e„
0.0
0.0
0.0
0.0
0.0
0.4
1.0
Key : NB = Negative big, NM = Negative m e d i u m , NS = Negative small, ZE = Zero, PS = Positive small, P M = Positive m e d i u m , P B = Positive big.
239
Table IV: NORMALIZATION AND PRIMARY FUZZY SETS USING A FUNCTIONAL DEFINITION [60] Normalized
Normalized
universe
segments
(-1.0,1.0)
Range
u/
oj
Primary fuzzy sets
(-1.0, -0.6)
(-13.8,-8.2)
-1.0
0.5
NB
(-0.6, -0.3)
(-8.2, -4.4)
-0.6
0.3
NM
(-0.3, -0.0)
(-4.4, -0.0)
-0.3
0.3
NS
(-0.0,0.2)
(-0.0, 2.0)
0.0
0.3
ZE
( 0.2, 0.6)
( 2.0, 5.0)
0.3
0.3
PS
( 0.6, 1.0)
( 5.0, 9.0)
0.6
0.3
PM
1.0
0.5
PB
240
Table V: MANUAL CONTROL RULES FOR A LIME KILN [30l Case
Condition
Action to be taken
1
BZ low
When BZ drastically low:
OX low
(a) reduce kiln speed
BE low
(b) reduce fuel When BZ is slightly low: (a) increase I.D. speed (b) increase fuel rate
2
3
BZ low
(a) reduce kiln speed
OX low
(b) reduce fuel rate
BE O.K.
(c) reduce I.D. fan speed
BZ low
(a) reduce kiln speed
OX low
(b) reduce fuel rate
BE high
(c) reduce I.D. fan speed
BE ~ back end temperature, BZ = burning zone temperature, OX = percentage of oxygen gas in kiln exit gas.
241
Table VI: FUZZY MODEL If ei
= NULL then uj
= very SMALL
also if ei
= very SMALL then vi
= SMALL
abo if ei
= SMALL then tii
= more or less MEDIUM
also if ei
= more or less MEDIUM then Ui
= MEDIUM
also if ei
= MEDIUM then vi
= very MEDIUM
also if ei
= more or less BIG then Uj
= more or less BIG
also if ei
= BIG then ui
= BIG
also if Ci
= very BIG then Ui
= very BIG
ei